Review and assessment of turbulence models for hypersonic ﬂows

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Progress in Aerospace Sciences 42 (2006) 469–530

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Review and assessment of turbulence models forhypersonic flows

Christopher J. Roya,�, Frederick G. Blottnerb

aAerospace Engineering Department, Auburn University, 211 Aerospace Engineering Building, Auburn, AL 36849-5338, USAbConsultant, Sandia National Laboratories, Albuquerque, NM, USA

Abstract

Accurate aerodynamic prediction is critical for the design and optimization of hypersonic vehicles. Turbulence modeling

remains a major source of uncertainty in the computational prediction of aerodynamic forces and heating for these

systems. The first goal of this article is to update the previous comprehensive review of hypersonic shock/turbulent

boundary-layer interaction experiments published in 1991 by Settles and Dodson (Hypersonic shock/boundary-layer

interaction database. NASA CR 177577, 1991). In their review, Settles and Dodson developed a methodology for assessing

experiments appropriate for turbulence model validation and critically surveyed the existing hypersonic experiments. We

limit the scope of our current effort by considering only two-dimensional (2D)/axisymmetric flows in the hypersonic flow

regime where calorically perfect gas models are appropriate. We extend the prior database of recommended hypersonic

experiments (on four 2D and two 3D shock–interaction geometries) by adding three new geometries. The first two

geometries, the flat plate/cylinder and the sharp cone, are canonical, zero-pressure gradient flows which are amenable to

theory-based correlations, and these correlations are discussed in detail. The third geometry added is the 2D shock

impinging on a turbulent flat plate boundary layer. The current 2D hypersonic database for shock–interaction flows thus

consists of nine experiments on five different geometries. The second goal of this study is to review and assess the validation

usage of various turbulence models on the existing experimental database. Here we limit the scope to one- and two-

equation turbulence models where integration to the wall is used (i.e., we omit studies involving wall functions).

A methodology for validating turbulence models is given, followed by an extensive evaluation of the turbulence models on

the current hypersonic experimental database. A total of 18 one- and two-equation turbulence models are reviewed, and

results of turbulence model assessments for the six models that have been extensively applied to the hypersonic validation

database are compiled and presented in graphical form. While some of the turbulence models do provide reasonable

predictions for the surface pressure, the predictions for surface heat flux are generally poor, and often in error by a factor

of four or more. In the vast majority of the turbulence model validation studies we review, the authors fail to adequately

address the numerical accuracy of the simulations (i.e., discretization and iterative error) and the sensitivities of the model

predictions to freestream turbulence quantities or near-wall yþ mesh spacing. We recommend new hypersonic experiments

be conducted which (1) measure not only surface quantities but also mean and fluctuating quantities in the interaction

region and (2) provide careful estimates of both random experimental uncertainties and correlated bias errors for the

measured quantities and freestream conditions. For the turbulence models, we recommend that a wide-range of turbulence

models (including newer models) be re-examined on the current hypersonic experimental database, including the more

recent experiments. Any future turbulence model validation efforts should carefully assess the numerical accuracy and

model sensitivities. In addition, model corrections (e.g., compressibility corrections) should be carefully examined for their

e front matter r 2007 Elsevier Ltd. All rights reserved.

erosci.2006.12.002

ing author. Tel.: +1334 844 5187; fax: +1 334 844 6803.

ess: [email protected] (C.J. Roy).

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ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469–530470

effects on a standard, low-speed validation database. Finally, as new experiments or direct numerical simulation data

become available with information on mean and fluctuating quantities, they should be used to improve the turbulence

models and thus increase their predictive capability.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Computational Fluid Dynamics (CFD); Turbulence model; Hypersonic flow; Boundary layer; Shock wave; Compressible flow

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

1.2. Scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

1.3. Molecular transport for hypersonic flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

1.4. Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

1.4.1. Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

1.4.2. Compressibility effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

2. Turbulence model validation methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

2.1. Cases examined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

2.2. Turbulence models examined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

2.3. Model implementation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

2.4. Efforts to establish numerical accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

2.4.1. Grid convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

2.4.2. Iterative convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

2.5. Turbulence model sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

2.6. Turbulence model validation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

3. Turbulence validation database for 2D/axisymmetric hypersonic flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

3.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

3.2. Previous experimental databases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

3.2.1. AGARD experimental review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

3.2.2. Experimental reviews by Settles and Dodson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

3.2.3. ERCOFTAC database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

3.2.4. Holden database. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

3.2.5. Other limited reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

3.3. Theory-based correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

3.3.1. Correlations for the flat plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

3.3.2. Correlations for the sharp cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

3.4. Direct numerical simulation database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

3.5. Updated hypersonic turbulence model validation database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

3.5.1. Previous flow geometries with adverse pressure gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

3.5.2. New flow geometries with and without pressure gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

3.6. Conclusion and recommendation of adequacy of experimental database . . . . . . . . . . . . . . . . . . . . . . 490

4. Usage of the hypersonic validation database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

4.1. Validation of theory-based correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

4.1.1. Flat plate/cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

4.1.2. Sharp circular cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

4.2. Validation of turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

4.2.1. One-equation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

4.2.2. Two-equation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

4.2.3. Physical freestream turbulence quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

4.3. Turbulence model application to the hypersonic validation database . . . . . . . . . . . . . . . . . . . . . . . . . 498

4.4. Previous flow geometries with adverse pressure gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

4.4.1. Case 1: 2D compression corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

4.4.2. Case 2: cylinder with conical flare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

4.4.3. Case 3: cone with conical flare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469–530 471

4.4.4. Case 4: axisymmetric impinging shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

4.5. New flow geometries with and without pressure gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

4.5.1. Case 5: 2D impinging shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

4.5.2. Case 6: flat plate/cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

4.5.3. Case 7: sharp circular cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

4.6. Conclusion and recommendation on turbulence model validation usage. . . . . . . . . . . . . . . . . . . . . . . 510

5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

6. Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

Appendix A. Compressibility corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

A.1. Two-equation k2� turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

Appendix B. Turbulent flat plate correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

B.1. Correlation of skin-friction data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

B.2. Correlation of heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

B.3. Mean temperature profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

B.4. Mean velocity profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

Appendix C. Turbulent sharp cone to flat plate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Appendix D. Perfect gas air model and molecular transport properties for hypersonic flows . . . . . . . . . . . . 524

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

1. Introduction

1.1. Background

Turbulence plays a key role in determining theaerodynamic forces and heating for hypersonicvehicles. However, experimental data for turbulencemodel validation are difficult to obtain. Thereare very few flight tests in the open literature, andthese tests generally provide only small amountsof data, usually with large experimental uncertain-ties. There are many more ground-based windtunnel tests on simplified geometries in hypersonicflow. These ground tests generally provide muchmore data than the flight tests, and usually withsmaller experimental uncertainties. However, due tothe extremely high velocities found in hypersonicflow, the hypersonic ground tests generally do notmatch the same high total enthalpy and lowfreestream turbulence levels typical of hypersonicflight. The validation of turbulence models withwind tunnel data thus generally involves significantextrapolation to flight enthalpies. Because of thesedifficulties in obtaining validation data for turbu-lent, hypersonic flows, designers are forced torely heavily on computational fluid dynamics andthe associated models for turbulence, chemistry,ablation, etc.

The current effort builds on the reviews bySettles and Dodson [1–4] conducted in the early1990s. Differences between the Settles and Dodson

reviews and the current work are that the currenteffort:

1.
has a different scope since only hypersonic flowsare considered,
2.
includes new experimental data since 1994, 3. addresses the steps required for validating
turbulence models, and
4. takes the additional step of reviewing and
assessing turbulence models as applied to theexisting hypersonic experimental database.

The current article can be considered both as anupdate to the Settles and Dodson work, as well asan extension which includes the steps of validatingthe turbulence models. Finally, we soften the Settlesand Dodson requirement that the upstream bound-ary layer be fully characterized by the experiment incases where the predictive capabilities of theturbulence model are judged to be sufficiently good,i.e., flat plates or cylinders with natural transition.

1.2. Scope

The validation of turbulence models shouldnecessarily include a wide range of flows. However,the extremely wide range of turbulent flows andavailable experimental data are enormous, so we areforced to limit the scope of this article. Prediction ofthe more basic turbulent flows needs to be improved

ARTICLE IN PRESS

Nomenclature

a speed of sound, m/sA incompressible transformation functionB incompressible transformation functioncp specific heat at constant pressure, J/

(kgK)Cf skin-friction coefficientCm turbulence modeling constantf general solution variableF compressible to incompressible transfor-

mation functionF s safety factor in the GCIG Mangler transformation functionhc heat transfer coefficient, W=ðm2 KÞH total enthalpy, J/kgk turbulent kinetic energy, m2=s2

l turbulent length scale, mL reference length, mm incompressible transformation function,

flat plate to sharp cone scaling exponentM Mach numberp order of accuracy of the numerical

method, pressure, N=m2

Pr Prandtl numberq square root of turbulent kinetic energy,

m/s, heat flux, W=m2

qw wall heat flux, W=m2

r grid refinement factor, recovery factorR specific gas constant, J/(kgK)Raf Reynolds analogy factorRn nose radius, mRe Reynolds numberSf flat plate to sharp cone scaling factorSt Stanton numbert time, sT temperature, KT� dimensionless total temperatureTu turbulence intensityu x-component of velocity, m/suþ wall-tangent component of velocity in

turbulence coordinatesut turbulence friction velocity ðut ¼

ðtw=rwÞ1=2Þ, m/s

v y-component of velocity, m/sV total fluid velocity, m/sx spatial coordinate (usually main flow

direction), mX axial distance from the leading edge, my spatial coordinate (usually wall-normal

direction), m

yþ wall-normal mesh spacing in turbulencecoordinates ðyþ ¼ uty=vÞ

Greek letters

a incompressible transformation functionb turbulence modeling constant, incom-

pressible transformation functiong ratio of specific heatsd boundary-layer thickness, mDy height of first cell off the wall, m� specific dissipation rate, m2=s3

k Karman constant (typically k ¼ 0:41)r density, kg=m3

m absolute molecular viscosity, kg/(m s)mT turbulent eddy viscosity, kg/(m s)n kinematic molecular viscosity, m2=sy momentum thickness, mt shear stress, N=m2

o turbulence frequency, 1/sz enstrophy, 1=s2

Subscripts

1 freestream quantityaw adiabatic wall valuee boundary-layer edge propertyinc incompressible valuek grid level ð1 ¼ finest gridÞ0 total (stagnation) conditionsRE Richardson extrapolated quantityT turbulence quantityw wall value

Superscript

� incompressible valuef denotes Reynolds-average of f~f denotes Favre-average of f

Acronyms

DNS direct numerical simulationERCOFTAC European Research Community

on Flow, Turbulence and CombustionGCI Roache’s grid convergence indexNASA National Aeronautics and Space Admin-

istrationRANS Reynolds-averaged Navier–StokesRMS root mean square

C.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469–530472


and validated before the more complex flows areinvestigated. Furthermore, while we have endea-vored to include all appropriate experimental andcomputational studies, it is inevitable that somequalified studies will be overlooked. We apologize inadvance for such omissions.

Herein we consider only two-dimensional (2D)/axisymmetric hypersonic flows, where the free-stream Mach number is limited to values greaterthan or equal to approximately five. In addition,only wall-bounded flows are considered, thuseliminating flows such as hypersonic mixing layersand jets. While there are ongoing research efforts inadvanced turbulence models such as Reynolds stressmodels and large Eddy simulation (LES), the mostcomplex models currently employed in designstudies (where a large number of parametric casesmust be considered) are one- and two-equationturbulence models. We therefore limit the currentstudy to these models. We also limit this study tomodels where integration of the governing equa-tions to the wall is performed, thereby eliminatingthe use of wall functions. This choice was primarilydriven by the fact that a majority of the cases ofinterest for hypersonic flows include shock-bound-ary-layer interactions, where the assumptions in-herent in the use of wall functions are difficult tojustify. We further limit our scope to cases where thetransition from laminar to turbulent flow occursnaturally, and where this transition location isspecified in the experimental description. The focushere is not on the prediction of transition, whichitself is a difficult challenge for hypersonic flows.Finally, the effects of surface roughness, ablation,chemical reactions, real gases, and body rotation areall neglected as the existing experimental databasedoes not yet adequately address these phenomena.

In most cases, turbulence models are expected tobe valid for a wide range of problems and not‘‘tuned’’ for a very limited class of turbulent flows(this latter approach more closely resembles modelcalibration or parameter fitting than a true predic-tion). Therefore, the testing of a turbulence modelfor high speed flows should include the evaluationof the model for all speeds and various flowgeometries to determine its limitations. Here welimit our study to include only those models whichhave a well-established validation history over awide range of flow conditions including low-speedflows. We therefore will not discuss efforts wherethe researchers propose model improvements, butdo not address the effects of these model improve-

ments on the prior model validation heritage. Westrongly recommend that future high-speed turbu-lence modelers test their compressible flow modelimprovements on a standard set of incompressibleflows as well, or at least give arguments as to whytheir corrections will not impact low-speed flows.See Marvin and Huang [5] for a recommended set ofexternal aerodynamics test cases in the subsonicthrough supersonic regime.

1.3. Molecular transport for hypersonic flows

Due to the difficulties of reproducing highenthalpy environments in ground-based facilities,experimental freestream static temperatures areoften quite low, sometimes on the order of 50Kor below. In addition, the most common test gasesare air, nitrogen, and helium. For these reasons,appropriate molecular models for viscosity andthermal conductivity should be used. See AppendixD for recommended low temperature moleculartransport models for both air and nitrogen.

1.4. Turbulence

1.4.1. Physics

The Navier–Stokes equations contain all of thephysics necessary to simulate turbulent flows.However, due to the wide range of length and timescales associated with simulating turbulence atReynolds numbers typical of flight vehicles, thisdirect simulation approach for turbulence is wellbeyond the capabilities even of today’s fastestcomputers. Engineers are thus forced to rely onturbulence models, which account for the effects ofthe turbulence rather than simulate it directly. Thesimplest turbulence modeling approach is Reynolds-Averaged Navier–Stokes (RANS), where all of theturbulent length and time scales are modeled viatemporal filtering of the Navier–Stokes equations.For compressible flows, density-weighted (or Favre)averaging is used. See Wilcox [6] for details of theReynolds and Favre averaging procedures.

1.4.2. Compressibility effects

Typically turbulence models have been developedfor incompressible flows and then extended withoutmuch change to compressible flows. This approach inmany cases is not adequate. For complex turbulentflows, Coakley et al. [7] have recommended correc-tions to apply to the two-equation k2� and k2oturbulent eddy viscosity models. In addition, Aupoix


and Viala [8] have proposed corrections to the k2�model for compressible flows. The authors have usedflat plate flows and mixing layers to assess thecompressible corrections introduced. Significant ef-forts to assess turbulence models for compressibleflows have occurred at NASA Ames ResearchCenter. The results of these investigations have beenpublished by Horstman [9], Horstman [10], Coakleyand Huang [11], Huang and Coakley [12], Coakley etal. [7], Bardina et al. [13], and Bardina et al. [14]. SeeA.1 for additional discussion of the compressibilityeffects for hypersonic flows.

2. Turbulence model validation methodology

The turbulence model validation methodologypresented herein is influenced heavily by the work ofMarvin [15] and Marvin and Huang [5]. Theproposed validation framework [16] includes guide-lines for documentation, model sensitivities, andmodel validation. In addition, it is recommendedthat a significant effort be made to estimate thenumerical accuracy of the simulations as part of thevalidation procedure. Listed below are six criteriafor assessing the models. The first three criteria(2.1–2.3) focus on the thorough documentation ofthe model evaluation efforts. Details of the flow caseand the models used must be given in enough detailso that the results are reproducible by otherresearchers. The last three criteria (2.4–2.6) list thespecific standards for evaluating the models. Theturbulence models should be evaluated by firstestablishing the numerical accuracy of the simula-tions, then by examining model sensitivities, andthen finally by validation comparisons to experi-mental data.

2.1. Cases examined

Details of (or references to) the specific flowproblem examined should be given including flow-field geometry and relevant physics (ideal gas versusequilibrium thermochemistry, transport properties,etc.). All required boundary conditions should belisted including inflow and outflow conditions, wallboundary conditions for temperature, incomingboundary-layer thickness, freestream turbulenceintensities, a measure of the freestream turbulencedissipation rate, etc. One of the difficulties encoun-tered in the specification of computational bound-ary conditions is that the level of informationrequired may not be fully characterized in the

experiment. For example, a large number ofotherwise excellent hypersonic validation data setsfail to report the thickness of the turbulentboundary-layer upstream of the interaction region;this information is especially important when theboundary layer is tripped to force transition toturbulence. It should be clearly stated whether theflow is fully turbulent or transitional. Finally, thedata available for model validation should be given(feature location, surface quantities, turbulent fieldprofiles, etc.).

2.2. Turbulence models examined

It should be clearly stated which form of theturbulence model is employed. It is stronglyrecommended that the standard model constantsbe used so as to build on prior turbulence modelvalidation efforts. Where applicable, the form of thelow Reynolds number wall damping functions usedshould be stated. The treatment of the near-wallregions should also be listed (i.e., integration to thewall versus wall functions).

2.3. Model implementation issues

The form of the governing equations should begiven. For example, different results may be foundwhen employing the full Navier–Stokes, thin-layerNavier–Stokes, parabolized Navier–Stokes, viscousshock layer equations, or boundary-layer equations.The boundary conditions employed in the simula-tion, including both flow properties and turbulencequantities, should be specified. Finally, any limitingof the turbulence quantities should be discussed.For example, limiting of the ratio of production todissipation of turbulent kinetic energy to some ratio(e.g., P=r�p5) is often used. In addition, realize-ability constraints on the turbulence variables and/or normal turbulent stresses [17] should also bediscussed.

2.4. Efforts to establish numerical accuracy

The numerical accuracy of the simulations is animportant factor to consider when comparing toexperimental data; for example, if the numericalaccuracy of pressure distributions is estimated to be�20%, then agreement with experimental datawithin 5% does not mean the model is accuratewithin 5%. The first step towards determining theaccuracy of the simulations is code verification, i.e.,


building confidence that the code is solving thegoverning equations correctly. Code verification canbe performed by comparison of the code results toexact solutions to the governing equations, highlyaccurate numerical benchmark solutions, or by themethod of manufactured solutions [18–21]. Onceone has confidence that the code is verified, then theaccuracy of the individual solutions must beverified. Solution accuracy includes assessing theerrors due to incomplete iterative convergence [16],temporal convergence for unsteady problems, andgrid convergence [18,21]. Methods for estimatingthe grid convergence errors based on systematic gridrefinement [22] tend to be the most reliable and areapplicable to any type of discretization includingfinite-difference, finite-volume, and finite-element.Grid convergence error estimates for hypersonicflows are complicated by the presence of shockwaves, which tend to reduce the spatial order ofaccuracy to first order on sufficiently refined meshes[23,24], regardless of the nominal order of thespatial discretization scheme.

2.4.1. Grid convergence

Discretization error is defined as the differencebetween the exact solution to the discrete equationsand the exact solution to the original partialdifferential equations. Discretization error can beestimated by performing computations on two ormore meshes. The Richardson extrapolation proce-dure [18] can be used to obtain an estimate of theexact solution from the relation

f RE ¼ f 1 þf 1 � f 2

3, (1)

where 1 denotes the fine mesh and 2 the coarsemesh. This relation assumes that the numericalscheme is second-order, that both mesh levels are inthe asymptotic grid convergence range, and that amesh refinement factor of two (i.e., grid doubling) isused. A more general expression for the Richardsonextrapolated value is given by

f RE ¼ f 1 þf 1 � f 2

rp � 1, (2)

where r is the grid refinement factor and p is theorder of accuracy (either formal or observed). Theformal order of accuracy can be found from atruncation error analysis of the discretizationmethod. If solutions are available on three meshes,

then the observed order of accuracy can becalculated from

p ¼ln½ðf 3 � f 2Þ=ðf 2 � f 1Þ�

lnðrÞ, (3)

where 2 now denotes the medium mesh and 3 thecoarse mesh. Here it is assumed that the refinementfactor between the coarse and medium mesh is equalto that between the medium and fine mesh.

The accuracy of the solutions can be estimatedusing the exact solution approximated by f RE whichgives the discretization error as

% Error of f k ¼ 100%�f k � f RE

f RE

, (4)

where k ¼ 1; 2, etc. is the mesh level. Since it isequally possible that the true exact solution is aboveor below this estimate, it is generally recommendedthat some factor of safety be included in the errorestimate. Roache [22] combines the concept of afactor of safety along with absolute values to producean error band rather than an error estimate. Theresulting error (or numerical uncertainty) estimate isreferred to as the Grid Convergence Index, or GCI.The GCI thus produces an error (or uncertainty)band around the fine mesh solution and is given by

GCI ¼Fs

rp � 1

f 2 � f 1

f 1

��. (5)

When solutions from only two meshes are available,Roache recommends a factor of safety of three. Forthree meshes where the observed order of accuracyagrees with the formal order of accuracy, a much lessconservative value of Fs ¼ 1:25 is suggested byRoache.

2.4.2. Iterative convergence

When iterative or relaxation methods are em-ployed, an additional error source arises due toincomplete iterative convergence. The numericalerror due to incomplete iterative convergence isusually assessed by evaluating norms of theresiduals, where the residual is defined by substitut-ing the current numerical solution into the dis-cretized governing equations. For steady-stateflows, the residual is calculated with the steady-state terms only, even if the temporal terms areincluded to speed up the convergence process. Theresiduals will approach zero as the steady-statesolution is reached and the current solution satisfiesthe discretized form of the steady equations. These


residuals can generally be driven to zero withinmachine round-off tolerance; however, this extremelevel of iterative convergence is generally notnecessary. Many studies (e.g., [16,20]) suggest thatfor computational fluid dynamics simulations, theresidual reduction levels correlate quite well with theactual iterative error in the flow properties.

2.5. Turbulence model sensitivities

Model sensitivity studies should be performedto determine practical guidelines for model use.A systematic study of the effects of the freestreamturbulence levels on the numerical predictionsshould be performed. The normal spacing at thewall ðyþÞ should also be varied in order to testmodel robustness and accuracy for both integrationto the wall and wall functions. In addition toestablishing the solution accuracy, a mesh refine-ment study can also be used to determine a giventurbulence model’s sensitivity to the mesh density.

The sensitivity to the freestream turbulence levelscan manifest in two forms: changes in the locationof transition from laminar to turbulent flow andchanges in the eddy viscosity levels in the turbulentregion. The former may actually be a desirablecharacteristic when bypass transition is beingmodeled, while the latter is generally undesirable.Experimental evidence [25,26] suggests that surfaceproperties (e.g., shear stress) in the fully-developedturbulent region are generally not affected byfreestream turbulence intensity, at least in the caseof low-speed flows.

2.6. Turbulence model validation results

Model validation results should be presented in aquantitative manner rather than qualitatively. Forexample, the percent difference between the predictions(with demonstrated numerical accuracy) and experi-ment should be plotted or explicitly stated. Wheneverpossible, experimental error bounds should be givenfor all measurements used for validation. These errorbounds should include contributions from instrumentuncertainty, experimental run-to-run uncertainty, phy-sical model alignment uncertainty, flowfield non-uniformities, etc. Bias errors are generally difficult toquantify, so if possible, multiple measurement techni-ques should be employed and, furthermore, tests inmultiple facilities should be performed. Techniques areavailable for converting some experimental bias errorsinto random uncertainties [27].

3. Turbulence validation database for 2D/

axisymmetric hypersonic flows

3.1. Overview

The validation of turbulence models must rely onreal-world observations, i.e., experimental data, toestablish model accuracy. The experimental datahave been mainly obtained from wind tunnels, wheredetailed measurements can be performed, rather thanin flight. There is a long history of high speedturbulent wind tunnel flow experiments. Compilationof experimental data for compressible turbulentboundary layers up to approximately 1980 is givenin AGARD reports by Fernholz and Finley [28–30].For high speed compressible turbulence, an experi-mental database has been developed by Settles andDodson [1–4] for two- and three-dimensional shock-wave boundary-layer interaction flows, attachedboundary layers, and free shear flows. The hyperso-nic portions of these databases are described belowalong with other more limited reviews. In addition, adiscussion is provided on the role of both correla-tions and direct numerical simulation (DNS) data inturbulence model validation.

3.2. Previous experimental databases

3.2.1. AGARD experimental review

There has been a significant effort by Fernholzand Finley [28–30] to document available experi-mental data for compressible turbulent flow up toabout 1980. A total of 77 experiments are reviewedwith 59 given in Ref. [28] and 18 given in Ref. [29].A further compilation of compressible boundary-layer data is given in Ref. [30]. The number ofhypersonic experiments is limited. In addition, thesereports do not provide a clear recommendation fora limited list of experiments that should be used forvalidation of turbulence models.

3.2.2. Experimental reviews by Settles and Dodson

A very careful assessment of validation experimentsfor compressible turbulent flow was performed bySettles and Dodson in the early 1990s [1–4]. Theydeveloped a list of eight necessary criteria for validationexperiments which is given below (in abbreviated form)in the order in which they were applied.

1.
Baseline applicability: Supersonic or hypersonicturbulent flow with shock wave/boundary layerinteraction.


2.
Simplicity: Experimental geometries sufficientlysimple that they may be readily modeled by CFDmethods.
3.
Specific applicability: Must provide useful experi-mental data for testing turbulence models.
4.
Well-defined experimental boundary conditions:Sufficient boundary condition data must besupplied to allow CFD solutions to be performedwithout any assumptions.
5.
Well-defined experimental error bounds: Mustprovide an analysis of the accuracy and repeat-ability of the data.
6.
Consistency criterion: All data must be self-consistent (i.e., different measurements cannotbe contradictory).
7.
Adequate documentation of data: Data must beavailable in tabulated form and capable of beingput into machine-readable form.
8.
Adequate spatial resolution of data: Sufficientdata must be presented such that key features ofthe flow are clearly resolved.
In addition to the above necessary criteria, thefollowing desirable criteria were also used in theevaluation of the experiments:

1.
Turbulent data: Turbulent properties (Reynoldsstress, etc.) of the flow field are given.
2.
Realistic test conditions: Flow conditions andboundary conditions typical of actual hypersonicflight.
3.
Non-intrusive instrumentation: Preference is givento non-intrusive experimental data.
4.
Redundant measurements: Preference is given toexperiments in which redundant data are taken.
5.
Flow structure and physics: Preference is given tothose experiments that reveal flow structure andphysical mechanisms.
The initial study [1] examined 105 experimentalstudies of shock wave interactions with turbulentboundary layers at Mach 3 or higher. There are fiveexperiments at hypersonic conditions that wereconsidered as acceptable while seven experimentsat supersonic conditions that were considered asacceptable. The second study [2] examined 39experiments of attached boundary layers withpressure gradients and 45 supersonic turbulentmixing layer experiments. The authors recom-mended nine experiments as acceptable for attachedboundary layers with pressure gradients and threeexperiments as acceptable for supersonic turbulent

mixing layers. The last report [3] has reviewed sevenadditional experiments and has corrections to threeof the previously reviewed experiments. A summaryof the supersonic and hypersonic shock/boundary-layer interaction experiments has been published inRef. [4].

None of the references to the acceptable super-sonic experiments are included in the list ofreferences of this review, but are available in Ref.[4]. For hypersonic flow conditions, seven experi-ments on six flow geometries are classified asacceptable for validation of turbulence models.The data for all of the acceptable experiments aretabulated in the Settles and Dodson reports andavailable in electronic format. The Settles andDodson flow geometries have been numbered withthe first four being 2D or axisymmetric and the nextfour being 3D experiments. Additional flow geome-tries or flow problems will be discussed in thispaper, with a total of seven geometries beingrecommended for turbulence model validation.The Settles and Dodson flow geometries numbersare as follows:

(1)
Two-dimensional compression corner. For a free-stream Mach number of 9, the wall pressure andheat flux have been determined in the experi-ment of Coleman and Stollery [31].
(2)
Cylinder with conical flare. For a freestreamMach number of 7, the wall pressure and heatflux have been measured and flow field surveyshave been made in the experiment of Kussoyand Horstman [32].
(3)
Cone with conical flare. For a freestream Machnumber of 11 and 13, the wall pressure and heatflux have been measured in the experiments ofHolden et al. [33] and Holden [34].
(4)
Axisymmetric impinging shock. For a freestreamMach number of 7, the wall pressure, skinfriction and heat flux have been measured andflow field surveys have been made in theexperiment of Kussoy and Horstman [35].
(5)
Flat plate with two fins (crossing shocks). For afreestream Mach number of 8.3, the wallpressure and heat flux have been measured,and surveys in the flow field have been made inthe experiment of Kussoy and Horstman [36].
(6)
Flat plate with 3D fin. For a freestream Machnumber of 6, the wall pressure and heat fluxhave been measured in the experiment of Law[37]. For a freestream Mach number of 8.2, thewall pressure and heat flux have been measured


and surveys in the flow field have been made inthe experiment of Kussoy and Horstman [38,39].For hypersonic flow at Mach 4.9, the Rodi andDolling experiments [40] are also acceptable.

(7)
Cylinder with skewed flare. For hypersonic flow,Settles and Dodson have no available experi-ments.
(8)
Three-dimensional compression corner with
sweep. For hypersonic flow, Settles and Dodsonhave no available experiments.

3.2.3. ERCOFTAC database

A comprehensive database of European workis being developed on the web [41] and hasthe name European Research Community onFlow, Turbulence and Combustion (ERCOFTAC).A complete review of this database has not yet beenperformed, but most of the databases are presentlyvery limited as this effort is in the early stages ofdevelopment.

3.2.4. Holden database

A review of the hypersonic experiments that havebeen performed at Calspan has been made byHolden and Moselle and reported in Ref. [42].There are numerous experiments that have beenperformed that are of interest:

(1)
Sharp and blunted cones at Mach 11 withlaminar, transitional and turbulent flow havebeen investigated. This work is documented byHolden [43].
(2)
Flow over a cone/flare model at Mach 11–16 hasbeen investigated. Earlier experiments on thisflow geometry is one of the Settles and Dodsonacceptable experiments. This work is documen-ted by Holden [44].
(3)
Two-dimensional compression corner [42]. (4) Flat plate with 3D fin [34,42]. (5) Flat plate [42,45,46]. (6) Two-dimensional impinging shock [42,47,48].
3.2.5. Other limited reviews

It has been nearly 15 years since a comprehensivereview has been performed on the new experimentsin hypersonic flow. There have been several limitedreviews of hypersonic experiments at the CaliforniaInstitute of Technology by Hornung [49] andhypersonic flow research in Europe by Groenigand Olivier [50].

3.3. Theory-based correlations

Theory-based correlations exist for two of thesimpler geometries discussed herein: the flat plateand the sharp cone. In some respects, theory-basedcorrelations can be considered superior to any singleexperimental data set since they mitigate theexperimental bias errors that vary from facility tofacility as well as bias errors associated with a givenmeasurement technique.

3.3.1. Correlations for the flat plate

The turbulence properties of interest are thewall skin friction, heat transfer, and profiles ofvelocity and temperature across the boundary layer.While a summary of the flat plate correlations ispresented here, a detailed discussion can be found inAppendix B.

Correlation of skin-friction data: The basic ap-proach which transform the experimental compres-sible local skin friction and momentum thicknessReynolds number to incompressible values for a flatplate is the Van Driest II theory [51]. Squire [52]estimates that the accuracy of the Van Driest IIcorrelation is within �3% for the flat plate. Basedon the sometimes erratic agreement between experi-ments and the correlation (e.g., see [53,45]), we feelthat this error estimate is somewhat optimistic andshould be increased to at least �5% for hypersonicflows.

Correlation of heat transfer: Reynolds’ analogy isused to predict the wall heat flux, which is expressedas the compressible Stanton number St ¼ qw=½reueðHw �HawÞ�. Reynolds’ analogy is written interms of the compressible local skin friction2St=Cf ¼ Raf , where Raf is the Reynolds analogyfactor. Experiments indicate that 0:9oRafo1:3.While there are insufficient reliable experimentaldata to establish the Reynolds analogy factor, forhypersonic flows a reasonable choice at presentappears to be Raf ffi 1. Additional work is needed toestablish the appropriate value for the Reynoldsanalogy factor for hypersonic flow.

Mean velocity profiles: In the log-law region,similarity of the Favre-averaged velocity ~u can beobtained with the Van Driest velocity transforma-tion (see Appendix B). Huang et al. [54] have alsoobtained the transformed velocity from the wall tothe edge of the boundary by taking into account theviscous sublayer and by including a wake function.This procedure gives the skin friction, velocity, andtemperature profiles as a function of the Reynolds


number. It has been developed as a seven stepprocedure with iteration of the solution untilconverged (see Appendix B for details).

Mean temperature profiles: The general form ofthe mean temperature across the zero-pressuregradient turbulent compressible boundary layer asa function of the mean turbulent velocity andturbulent kinetic energy is

T ¼ Tw � a ~u� b ~u2 � gTk. (6)

Huang, Bradshaw, and Coakley (HBC) [54] havedeveloped the temperature equation by neglectingthe convective terms in the momentum and energyequations. Their analysis (see Appendix B) yieldsthe following relations for the variables in Eq. (6):

a ¼ ðPrT=cpÞðqw=twÞ; b ¼ PrT=2cp,

gT ¼ PrT=cp. (7)

3.3.2. Correlations for the sharp cone

One of the problems with the sharp cone is thelack of a theoretical correlation of the experi-mental data to use as a benchmark solution. Forlaminar flow, the skin friction and heat transferfor a flat plate are multiplied by

ffiffiffi3p

to obtainthe cone values. There does not appear to be awell established approach to transform the turbu-lent flat plate results to the cone. Van Driest [55]has suggested an approximate approach that hasbeen developed further in the book by White [56]using the von Karman momentum integral relation(see Appendix C for more details). The flatplate skin friction and wall heat flux are multipliedby a scale factor G that gives the Cone Rule asfollows:

ðCf Þcone ¼ GðCf Þplate; ðqwÞcone ¼ GðqwÞplate, (8)

where G ¼ffiffiffi3p¼ 1:732 for laminar flow and G �

1:13 for turbulent flow.A correlation of the heat transfer on axisym-

metric flight vehicles with flat plate relations hasbeen investigated by Zoby and Sullivan [57] and anadditional correlation including ground data hasbeen investigated by Zoby and Graves [58]. Theformer includes six references for experimental dataon sharp cones where the Mach number varies from2.0 to 4.2. An assessment of the theoreticalcorrelations for sharp cones was given in Roy andBlottner [59].

3.4. Direct numerical simulation database

The numerical solution of the unsteady Navier–Stokes equations with refined grids as formulated inthe DNS and LES methods has the potential ofproviding an accurate numerical simulation data-base with limited or no turbulence modelingassumptions. However, the increased fidelity ofthese approaches requires additional temporal andspatial information for the specification of the initialand boundary conditions. As computer speed andmemory size have increased, the accuracy andcapabilities of these computational fluid dynamicapproaches have increased and will increase in thefuture. Next a brief indication is given of theturbulent flat plate and sharp cone flow problemsthat are being solved with the DNS and LESmethods.

Martin [60]: Martin [60] has started to develop aDNS database of hypersonic turbulent boundary-layer flows over a flat plate. She provides a review ofprevious DNS solutions that have been obtained forhigh speed compressible flows. The list includes areview of other work as well as her previous paperswith coworkers. Martin has obtained DNS solu-tions for perfect gas and reacting air flows over a flatplate. Martin presented DNS solutions for perfectgas flow with the gas viscosity modeled with apower law dependence on temperature. The simula-tions use freestream conditions corresponding to analtitude of 20 km and the Mach number varies from3 to 8. The wall temperature is specified to be nearlythe adiabatic temperature. At Mach 8, the walltemperature is 2713K and would result in signifi-cant dissociation of the oxygen in air. The perfectgas model is not adequate to simulate these physicalflow conditions. From the simulation solutionsobtained, the mean flow velocity across the bound-ary layer has been determined, then transformedwith the Van Driest transformation to incompres-sible form, and presented in figures for the casessimulated. No information is presented on the wallskin friction and heat transfer. This work isimportant as it is starting to provide useful DNSsolutions at hypersonic flow conditions. However,there is a need to extend this work by obtainingsimulations with a gas model that are moreappropriate for flight conditions or modify the flowto wind tunnel conditions.

Yan et al. [61]: Yan et al. [61] have obtained LESsolutions with the Monotonically Integrated LES(MILES) technique for flat plate flow at Mach


number 2.88 and 4. Both adiabatic and isothermalwall boundary conditions are used. The authorshave provided a list of researchers that have studiedcompressible LES with no work performed athypersonic flow conditions. The authors’ velocityprofile predictions are compared to the law of thewall analysis and experimental data of Zheltovodovat Mach 2.9 and 3.74, although the computationsare at a lower Reynolds number than the experi-ment. LES temperature profiles are also comparedto experimental data. The comparisons for velocityand temperature at Mach 2.9 are good but thecomparison at Mach 4 is poor. The interestingresults of this investigation are concerned with heattransfer and Reynolds analogy. The authors in-dicate that Reynolds analogy factor is Raf ¼

2St=Cf ¼ 1=Prtm ¼ 1:124 where the mean Prandtlnumber Prtm ¼ 0:89. At Mach 4, the LES solutiongives Raf ¼ 1:23 while the experimental value isRaf ¼ 1:12. The Reynolds analogy factors differ by10%. The simulations obtained in this articleindicate the potential of the LES technique to helpvalidate turbulence models; however, the subgridscale model required in LES can affect the resultsand therefore is a limitation of this approach.

Pruett and Chang [62]: The Pruett and Chang [62]investigations are concerned with DNS of hyperso-nic boundary-layer flows on sharp cones and cone-flare models. The initial work in 1995 is anapproximate simulation of the geometry and flowconditions in the wind tunnel experiment of Stetsonet al. [63]. A 7 half-angle cone in a Mach 8 flow issimulated. The inviscid flow at the edge of theboundary layer is specified, and the wall tempera-ture is specified as the laminar adiabatic walltemperature, which is given as 611K. The free-stream properties are estimated from the Sims tables[142], which give the total temperature as 733K,static temperature as 53K, and the unit Reynoldsnumber as 3:407� 106=m. Pruett and Chang [64] in1998 published an investigation of DNS of hyper-sonic boundary-layer flow on a flared cone. TheDNS solution is for the quiet (low freestreamturbulence) wind tunnel experiment of Lachowiczet al. [65], where the freestream turbulence has beenreduced significantly. The axial length of the cone-flare model is 0.51m and the sharp cone axial lengthis 0.254m. The freestream flow conditions for thesimulation are specified as Mach number 8, statictemperature 55K, total temperature 450K, and unitReynolds number 8:85� 106=m. In both of theabove DNS the air viscosity is determined with

Sutherland viscosity law, and a perfect gas model isused. Although the Pruett and Chang DNScomputations do not provide useful informationon fully developed turbulent flow on the conicalpart of the models, the numerical simulationsindicate the future potential for providing valuabledata for validation of compressible turbulencemodels.

Comments on DNS database: For flows withoutchemical reactions and for typical flight conditions,the wall temperature needs to be sufficiently low.The maximum gas temperature occurs in theboundary layer due to viscous dissipation and canbe sufficiently high to produce vibrational excita-tion. Complete simulation without chemical reac-tions requires a vibrational non-equilibrium model.The solutions from the complete model can bebounded by using perfect gas and thermally perfectgas models, which makes DNS solutions with thesemodels valuable. The gas models need a moreappropriate viscosity model than Sutherland visc-osity law. For hypersonic wind tunnel conditions,the stagnation temperature is sufficiently high tohave vibrational excitation while the freestreamtemperature in the test section is low. The modelrequirements are the same as for flight conditions asvibrational non-equilibrium effects can be impor-tant. Keyes viscosity model should be used due tothe low gas temperatures. The desired databaseshould include a matrix of accurate solutions whichdepend on Mach number, boundary-layer momen-tum thickness Reynolds number, and wall tempera-ture. A series of solutions should be obtained withonly one of the variables varying and with the othertwo variables held constant. These solutions wouldprovide a database that can be used to validate theVan Driest transformation approach to correlatecompressible turbulent skin friction and heattransfer (Stanton number) and to determine theReynolds factor in the Reynolds analogy. Inaddition, the DNS method should be extended toflow over sharp cones. The database would helpto determine the Mangler transformation requiredto transform compressible turbulent flow for theaxisymmetric case to the 2D case.

3.5. Updated hypersonic turbulence model validation

database

In this section we present our recommendationsfor the current 2D/axisymmetric experimental data-base for validating turbulence models. We adhere to

ARTICLE IN PRESS

Table 1

Turbulence validation database for 2D/axisymmetric hypersonic

flows

Case

no.

Flow geometry Experiments

1 2D compression

corner

Coleman and Stollery/Elfstrom

[31,66,67]

Holden [48]

2 Cylinder with

conical flare

Kussoy and Horstman [32]

Babinsky and Edwards [68,69]

3 Cone with conical

flare

Holden [44]

4 Axisymmetric

impinging shock

Kussoy and Horstman

[35,70,71]

Hillier et al. [72–75]

5 2D impinging shock Kussoy and Horstman [38]

Schulein et al. [76–78]

6 Flat plate/cylinder Van Driest (VDII)a [51]

Huang et al. (HBC)a [54]

Aupoix et al. (AVC) [8]

Hopkins and Keener [79–81]

Horstman and Owen [82–84]

Coleman et al. [85]

Kussoy and Horstman [36,38]

Hopkins et al. [86,87]

Holden et al. [42,45,46,48,88]

7 Sharp circular cone Van Driest (VDII)a[51] and

Whitea [56]

Kimmel [89,90]

Rumsey et al. [91,92]

Chien [93]

Hillier et al. [94–97]

Holden et al. [42–44,46,48,98]

aDenotes a data correlation.

C.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469–530 481

Settles and Dodson’s necessary criteria for accep-tance of experiments into the validation database[1–4] with one exception. Here we relax criteria #4(well-defined experimental boundary conditions)and allow cases where the boundary layer is notcharacterized upstream of the interaction region aslong as this upstream boundary layer can beadequately predicted by turbulence models orcorrelations (i.e., for flat plates, cylinders, or sharpcones where an equilibrium boundary layer hasbeen established). Note that in the original Settlesand Dodson validation database, they also appearto have relaxed this requirement for the Colemanand Stollery/Elfstrom [31,66,67] compression cornerexperiment.

3.5.1. Previous flow geometries with adverse pressure

gradient

This section describes the four 2D/axisymmetrichypersonic flow geometries that were assessed in theSettles and Dodson database [1–4]. For each ofthese flow geometries, we present both the experi-ments that were included in the Settles and Dodsondatabase, as well as new experiments conductedsince 1994 which we recommend for inclusion in a2D/axisymmetric hypersonic validation database.The experiments discussed in this section areincluded as Cases 1–4 in Table 1.

3.5.1.1. Case 1: 2D compression corner. There aretwo hypersonic experiments for the 2D compressioncorner which are deemed acceptable with somecaveats. The first experiment was conducted in theMach 9 nitrogen gun tunnel and includes heattransfer measurements (see [31,66]) and surfacepressures from a separate experiment (see [67]).The second experiment was conducted at theCalspan 48 in and 96 in shock tunnels at Mach 8by Holden [48].

Coleman and Stollery/Elfstrom experiment

[31,66,67]: Elfstrom [67] reported surface pressuremeasurements and Coleman and Stollery [31] andColeman [66] reported surface heat transfer mea-surements for a Mach 9.22 flow over a 2D wedge/compression corner. The wedge angle was variedbetween 15 and 38, and includes flows which arenominally attached, at incipient separation, andfully separated. It is not fully clear whether or notthe pressure [67] and heat transfer [31] measure-ments had the same upstream length for the flatplate. While this experiment is considered accepta-ble by Settles and Dodson, there are no numerical

uncertainties on the surface quantities given inRefs. [31,67], nor are any uncertainties presented inthe Settles and Dodson reviews [1,4]. Holden [48]points out that the interaction region in the Cole-man and Stollery/Elfstrom experiments may be tooclose to the transition zone, resulting in a differenttrend of separation zone size versus Reynoldsnumber than seen in equilibrium turbulent bound-ary layers, which require a distance of approxi-mately 50–100 boundary-layer thicknesses betweenthe transition point and the interaction region. Oncethe oncoming turbulent boundary layer has reachedan equilibrium state, the trend should be a decreaseof separation zone size (and incipient separationpoint) with increasing Reynolds number, while theseexperiments showed the opposite trend.


Holden experiment [48]: Due to the concernsregarding the equilibrium nature of the upstreamboundary layer, the compression corner experi-ments conducted by Holden [48] are also includedhere, although they too fail to report uncertaintieson the surface measurements. While no upstreamboundary-layer profile is measured in this experi-ment, the surface quantities compared well to theVan Driest II correlations [99]; furthermore, thetransition location can be easily determined fromthe surface skin friction and heat transfer measure-ments made on a flat plate and reported in the samereference. The freestream Mach number for thiscase is approximately 8, and the ratio of walltemperature to the freestream stagnation tempera-ture was 0.3. The Reynolds number based on theboundary-layer thickness at the interaction locationwas varied between 100,000 and 10 million.Measurements are reported for surface pressure,skin friction, and heat transfer in the interactionregion for wedge angles of 27, 30, 33, and 36 andalong the flat plate in a configuration without thewedge. Span effects were also investigated andshown to be negligible. This was the first experimentto show the reversal of the separation zone size andincipient separation with Reynolds number withinthe same experiment. To our knowledge, thisexperiment has not been employed for validatingturbulence models.

3.5.1.2. Case 2: cylinder with conical flare. Thereare two experiments which meet the Settles andDodson criteria for the axisymmetric cylinder-flaregeometry. The first was included in the Settles andDodson review and was performed by Kussoy andHorstman [32] at NASA-Ames Research Center.The second is a more recent experiment performedin the HSST supersonic blow-down wind tunnel andis detailed by Babinsky [69] and Babinsky andEdwards [68].

Kussoy and Horstman Experiment [32]: Kussoyand Horstman [32] studied the flow over axisym-metric ogive-cylinder-flares at a freestream Machnumber of 7 for flare angles between 20 and 35.Data include surface pressure and surface heattransfer both upstream of the shock/boundary-layerinteraction (see Case 6: flat plate/cylinder flow) andin the interaction region. Pitot-probe surveysthrough the boundary layer are presented at variousaxial locations for the 20 flare case only, with theboundary-layer surveys upstream of the interactionconfirming a fully-developed turbulent boundary

layer. Information is given on freestream RMSvalues for stagnation temperature and mass flux aswell as temperature for the water-cooled modelsurface. The data set includes experimental uncer-tainty estimates for each measured quantity. De-rived boundary-layer quantities (displacementthickness, momentum thickness, etc.) are alsoreported upstream of the interaction. Surface datawere taken at 90 locations to confirm that the flowwas axisymmetric, and multiple runs were con-ducted to reduce run-to-run uncertainty. A rela-tively long model was employed to allow for natural(non-tripped) transition to occur at approximately0.4 to 0.8m from the tip, which is at least 0.6mupstream of the interaction region.

Babinsky and Edwards Experiment [68,69]: Ba-binsky and Edwards [68] conducted careful experi-mental studies of cylinder-flare flows at Mach 5.1for flare angles between 3 and 20, with additionaldetails presented by Babinsky [69] for flare angles of15 and 20. The experiments were conducted in thesupersonic blow-down wind tunnel (HSST) at DRAFort Halstead, Great Britain using a Mach 5 nozzlethat included a cylindrical centerbody which ex-tended upstream of the test section to the nozzlethroat. This centerbody was deemed necessary toallow for the formation of a fully-developedturbulent boundary layer without the use of flow-intrusive boundary-layer tripping mechanisms.However, the use of the centerbody led to thepresence of non-negligible axial gradients of pitotpressure (10% variation) and Mach number (3%variation) in the test section. Data were presentedfor surface pressure, surface heat transfer (via high-resolution liquid crystal thermography), and pitotpressure through the boundary layer at variousaxial stations. Detailed experimental uncertaintieswere also provided for each of the measuredquantities. Derived quantities presented includevelocity profiles, skin friction, boundary-layer thick-ness, and displacement thickness. Conventionaltheory suggests that for flare angles of 20 andbelow, no separation will occur. However, investi-gations using shear stress sensitive liquid crystalsshowed a small separated region (possibly in thelaminar sublayer) for both the 15 and 20 flarecases. The authors suggest that the presence ofthis small separation zone destroys the similaritybetween pressure and heat transfer.

3.5.1.3. Case 3: cone with conical flare. There isonly one experiment for the cone/conical flare case


that is appropriate for turbulence model validation.Holden [44] performed experiments in Calspan’s 96in shock tunnel at Mach numbers of 11 and 13. Thisis one of Settles and Dodson’s accepted hypersonicexperiments; however, there is some discrepancyregarding the references. Settles and Dodson [1,4]reference a 1984 AIAA Paper [34], a 1986 CUBRCinternal report [33], and a 1988 AFOSR technicalreport [100], all by Holden and coworkers. How-ever, the initial reporting of these cone/conical flareexperiments was not until 1991 in Ref. [44], and thisis confirmed by examining Refs. [42,101] which arereviews of the experimental hypersonic programconducted at Calspan. In any case, the datapresented by Settles and Dodson [1] does appearto be the same data given in Refs. [44,101]. It shouldbe noted that there is also some question regardingthe flare angle for this case. Ref. [44] does not makeit clear whether the flare angle is measured from thesymmetry axis or from the initial cone angle of 6,while Ref. [101] clearly shows that the flare angleshould be measured from the symmetry axis.However, Settles and Dodson [1] state that the flareangles should be measured from the 6 forecone,and crude angles measured from Schlieren photo-graphs in Refs. [44,101] seem to support thisconclusion. Subsequent communications with theauthor of Ref. [44] confirmed that the flare angle

should be measured from the 6 cone, not thesymmetry axis [102].

Holden experiment [44]: Holden [44] studied theflow over 6 (half angle) cones with conical flares atfreestream Mach numbers of 11 and 13 for flareangles of 30 and 36 as measured from the forecone(36 and 42 from the symmetry axis). The smallerflare angle represents an incipient separated flowcase, while the larger angle a fully separated flow.Data include surface pressure and heat transfer aswell as pitot pressure and total temperature withinthe interaction region. Experimental uncertaintiesare given for the freestream conditions as well asheat transfer coefficient ð�5%Þ and pressure coeffi-cient ð�3%Þ.

3.5.1.4. Case 4: axisymmetric impinging shock. Thereare two different axisymmetric impinging shockexperiments which are deemed acceptable for turbu-lence model validation. The first is a series ofexperiments conducted by Kussoy et al. at a Machnumber of 7 on a cone-ogive-cylinder model[35,70,71]. The second is a more recent experimental

investigation by Hillier et al. at Mach 9 on a hollowcylinder model [72–75].

Kussoy and Horstman Experiment [35,70,71]:Kussoy and Horstman studied the flow over anaxisymmetric cone-ogive-cylinder model at a free-stream Mach number of 6.9. Axisymmetric cowls of7:5 and 15 were used to impinge axisymmetricshock waves onto the turbulent boundary layer onthe cylinder. Surface data include surface pressure,heat transfer, and skin friction [35,70]. Pitotpressure, static pressure, and total temperature weresurveyed throughout the interaction region [35,70].Ref. [71] also presents turbulence intensity andReynolds stress profiles for these two cases. Thecowl length is relatively short, thus the leadingshock wave and subsequent expansion fan mergebefore the shock impinges on the surface. It istherefore strongly recommended that future com-putations of this experiment also include the viscousboundary layer on the outer cowl itself. The lengthof the model cylinder is 3.3m, thus suggesting afully-developed turbulent boundary layer is formedwell ahead of the interaction region. The modelsurface is water-cooled to maintain a temperature of300K. The data set includes experimental uncer-tainty estimates for each measured quantity, withthe exception of the turbulence measurements ofRef. [71]. Derived boundary-layer quantities (dis-placement thickness, momentum thickness, etc.) arealso reported upstream of the interaction. Settlesand Dodson [1] give the experimental data for the15 shock generator case in tabular form.

Hillier et al. Experiment [72–75]: Hillier et al. [72]studied the flow over an axisymmetric cone-ogive-cylinder model at a freestream Mach number of 8.9and Reynolds number of 52� 106=m. An axisym-metric cowl with a quadratic expression for theshock-generating surface is used to impinge anaxisymmetric shock wave onto the turbulentboundary layer formed on the cylinder. Surfacedata include surface pressure and heat transfer.Transition of the boundary layer on the cylinderbegins at 80mm and ends at 170mm, with the shockinteraction occurring at roughly 520mm. Due to thecurved nature of the shock-generating cowl, it isrecommended that computations of this experimentalso include the viscous boundary layer on the outercowl itself. Although not reported in the experi-mental description, the model surface temperaturewas 300K [103]. The data set includes experimentaluncertainty estimates for the surface pressure andheat transfer. An additional discussion of the


experimental uncertainties is given in Ref. [73]which quotes uncertainties of �4% and �7% forsurface pressure and heat transfer, respectively.More recent data for shock generator angles of4:7 (attached flow) and 10 (separated flow) aregiven by Murray and Hillier [74,75] for a hollowcylinder forebody. Although no experimental un-certainties were quoted in these last two references,the same experimental techniques that wereused in Refs. [72,73] were employed. Personalcommunication with one of the authors confirmedthe uncertainty levels given above [103]. Forthe 10 shock generator case, the length of theshock-generating outer cowl had to be reduced toprevent choking of the shock system. This recentexperiment has not yet been used in the validationof one- or two-equation turbulence models but isrecommended.

3.5.2. New flow geometries with and without pressure

gradient

In this section we discuss the three new flowgeometries which should be added to the 2D/axisymmetric hypersonic turbulence database. Thefirst is the 2D impinging shock problem and isreferred to as Case 5. Cases 6 and 7 are zero-pressure gradient flows (flat plate/cylinder and coneflow) where the simplicity of the flow also allows thedevelopment of theoretical correlations.

3.5.2.1. Case 5: 2D impinging shock. A 2D imping-ing shock occurs when an externally generatedoblique shock impinges on a flat plate boundarylayer. There are two experimental data sets thatsatisfy the Settles and Dodson criteria: Kussoy andHorstman [38] conducted a careful experimentalstudy of the 2D impinging shock case in the Ames3.5 ft Hypersonic Wind Tunnel at Mach 8.2, andSchulein et al. [76–78] studied a similar geometry atMach 5.

Kussoy and Horstman Experiment [38]: Kussoyand Horstman [38] studied a 2D oblique shockimpinging on a turbulent flat plate boundary layerat a freestream Mach number of 8.2 for effectivewedge angles of 5, 8, 9, 10, and 11. Datainclude surface pressure and heat transfer bothupstream of the shock/boundary-layer interaction(see Case 6: flat plate/cylinder flow) and in theinteraction region. Mean flow surveys through theboundary layer are given for the undisturbedboundary layer (i.e., without the shock generator)in the vicinity of the shock interaction. Surveys in

the interaction are alluded to in the report butare not presented (these may be included on acomputer disk which is mentioned in the report).The model is water-cooled to maintain a surfacetemperature of 300� 5K. The data set includesextensive experimental uncertainty estimates foreach measured quantity, but not for the freestreamconditions. A relatively long 2.2m model wasemployed to allow for natural (non-tripped)transition to occur approximately 0.5 to 1mfrom the leading edge, which is at least 0.5mupstream of the interaction region. The modellength results in a fully developed, equilibriumturbulent boundary layer (confirmed by the flow-field surveys) that is nearly 4 cm thick near theinteraction region.

Schulein et al. Experiment [76–78]: Schulein andcoworkers [76–78] have also studied a 2D obliqueshock impinging on a turbulent flat plate boundarylayer. The flow is at Mach 5 and wedge angles of 6

to 14 were studied. The flat plate was 0.5m longand 0.4m wide, with natural transition occurring0.1m from the leading edge, as judged by peak skin-friction. For all shock generator angles, the inter-action occurs approximately 0.35m from the lead-ing edge. The spanwise extent was chosen to ensure2D flow, which was further confirmed by surfacepressure and pitot survey data taken at variousspanwise locations. The Reynolds number based ondistance from the leading edge at the interactionregion was 13� 106, and the boundary-layer thick-ness at this location was approximately 5mm.The wall temperature is 300� 5K. The originalreport [76] gives pitot surveys in the interactionregion, as well as wall static pressures with anestimated uncertainty of �5% [78]. Surface skinfriction (obtained via optical means) and heattransfer data are also available in Ref. [77] anduncertainties in the interaction region for both aregiven as �10% [78].

3.5.2.2. Case 6: flat plate/cylinder. The uniformviscous flow over a flat plate or cylinder isconsidered, where a laminar to turbulent boundarylayer develops along the surface. The 2D zero-pressure gradient turbulent boundary-layer flowproblem is unique. Theoretical analyses of this casefor perfect gas flows have been performed whichresult in correlations of the experimental results.The turbulent properties of interest are the wall skinfriction, heat transfer, and profiles of velocity andtemperature across the boundary layer.


A survey has been performed of authors that haveused hypersonic turbulent boundary-layer experi-mental data with zero-pressure gradient to validate

Table 2

Model geometries and freestream conditions for hypersonic flat plate/c

Investigator/geometry Date Me C

Sommer–Short [104] (hollow

cylinder, flight range)

1953 5.63, 6.9, 7.0 Y

Korkegi [106] 1956 5.79 Y

Hill [107] (conical nozzle, nitrogen

gas)

1956 8.3–9.1 Y

Hill [108] (conical nozzle, nitrogen

gas)

1959 8.27, 9.07,

10.04

Y

Tendeland [109] (cone-cylinder) 1958 5.04 N

Brevoort–Arabian [111]

(downstream inside cylinder)

1958 5.05 N

Winkler–Cha [112] 1959 5.14, 5.22,

5.25

Y

Winkler–Cha [112] 1961 5.14, 5.22,

5.25

N

Matting et al. [113] (flat wind tunnel

wall, helium gas)

1961 6.7, 9.9 Y

Moore [114] 1962 5 N

Young [115] 1965 5 Y

Wallace–McLaughlin 1966 7.4, 8.1, 10.7 Y

Wallace [117] 1967 7.4, 8.1, 10.7 N

Heronimus [118] 1966 4.6 to 11.7 Y

Neal[120] 1966 6.8 Y

Cary–Morrisette [121] (wedge

a ¼ �5; 0; 5)1968 6.8, 6.0, 5.3 N

Hopkins et al. [86] 1969 6.5 Y

Hopkins et al. [87] 1970 6.5 Y

Hopkins et al. [122] 1972 6.5 Y

Cary [123] 1970 6 N

Weinstein [124] (10 deg wedge) 1970 8, 6.8, 6, 5.2 N

Hopkins–Keener [80] (tunnel wall) 1972 7.4 Y

Keener–Hopkins [81] (thermally

perfect gas)

1972 6.2–6.5 Y

Owen–Horstman [82] (cone-ogive-

cylinder)

1972 7.2 Y

Horstman–Owen [83] (cone-ogive-

cylinder)

1972 7.2 Y

Owen et al. [84] (cone-ogive-

cylinder)

1975 7.0 Y

Holden [45] 1972 6.8–13 Y

Holden electronic database [46] 2003

Coleman et al. [85] (flat plate

negative angle)

1972 7.15–9.22 N

Watson et al. [125] (wedge, helium) 1973 9–10 Y

Watson [126] (wedge, helium) 1978 10–11.6 Y

Laderman–Demetriades[127] (wind

tunnel wall)

1974 9.4 Y

Kussoy–Horstman [38] 1991 8.18 Y

Kussoy–Horstman [36] 1992 8.28 Y

AdW, adiabatic wall; F&F x, Fernholz and Finley case x.

correlations of skin friction and Stanton numberand correlations of velocity and temperature pro-files. Tabulation of the survey is given in Table 2

ylinder database

f St Profiles Tabulated data reference

es No No Peterson [105]

es AdW No Peterson [105]

es Yes Yes F&F 5901

es Yes No Hill [108], Peterson [105]

o Yes No Spalding–Chi [110], Author

o Yes No Spalding–Chi [110], Author

es Yes Yes F&F 5902

o Yes Authors [112], Peterson [105]

es AdW Yes Peterson [105]

o No Yes F&F 6201

es Yes Yes F&F 6506

es Yes No Cary–Bertram [53]

o Yes No No tabulation

es Yes,

AdW

No Cary [119]

es Yes No No tabulation

o Yes No Cary–Bertram [53]

es AdW Yes Authors [86]

es Yes Yes Authors [87]




es No Yes F&F 7203

es No Yes F&F 7204

es Yes Yes F&F 7205, Authors [82]


es Yes Yes No tabulation

es Yes No Author [45]

Author [46]

o Yes Yes Cary–Bertram [53]

es Yes Yes F&F 7305

es Yes Yes F&F 7804

es No Yes F&F 7403




where the experiments are for flat plate flows in airunless indicated otherwise. Since the correlationshave been developed for perfect gas flows and fullydeveloped zero-pressure gradient flow, the experi-mental database is limited to flat plates, wedges,wind tunnel walls and cylindrical body flows. Alsothe flow properties should be measured sufficientlyfar downstream where the turbulent flow is fullydeveloped. Upstream history effects can signifi-cantly influence the database as the non-equilibriumturbulent effects sometimes persisting 1000s ofboundary-layer thicknesses downstream [48].

For validation of correlation theory and turbu-lence models, direct measurement of skin friction,heat transfer, velocity profiles, and temperatureprofiles should be made. In addition, the hypersonicexperimental database should be limited to the sameflow geometry (flat plate) with zero-pressure gra-dient and same freestream perfect gas model. Thesharp wedge and flat plate produce the sameturbulent boundary layer as the flow Reynoldsnumber approaches infinity. Flow along hollowcylindrical geometries with decreasing transversecurvature effects approaches flat plate flow. Whilecylindrical geometries with conical or ogive nosegeometries produce initial disturbances that influ-ence the boundary layer, far downstream a flatboundary layer is obtained. Wind tunnel wallboundary layers also have upstream history effectsthat impact the attainment of a flat plate flow.

Many of the earlier experiments did not measureboth the skin friction and the wall heat transfer andare not as useful. While most experiments areperformed in air, two experiments use helium gasand one uses nitrogen gas. The ideal hypersonicexperimental database for validation of correlationtheories becomes very small if one applies all of therestrictions mentioned above. However, many ofthe experiments in Table 2 have been useful inestablishing the accuracy of the correlation theoriesand determining the validity of turbulence models(see Section 4.5.2). Many of the experiments areuseful in direct comparison between turbulentmodel predictions and experimental results toestablish turbulent model validation. This approachis a significantly larger computational effort, butwould also help in the evaluation of the importanceof the differences in the experiments.

An extensive database of compressible turbulentflows in a standard form has been compiled byFernholz and Finley [28–30] as discussed in 3.2.1.This database is limited to 2D flows where flow

profile data are available in tabular form. Assess-ment of the data quality or significance of the data isnot complete. Tabulated data is given for the edgeand wall flow properties and survey propertiesacross the boundary layer. The hypersonic databasegiven in the Fernholz and Finley reports include anumber of experiments where either real gas effectswere an issue, a fully turbulent boundary layer wasnot established after transition process, or the windtunnel side wall was used to generate the boundarylayer (thus bringing the equilibrium nature of theboundary layer into question). As a result, theFernholz and Finley reports are of limited value.

Two experiments which will be included from theFernholz and Finley database are those of Hop-kins–Keener and Horstman–Owen (discussed indetail below). Additional hypersonic experimentsthat have not been included in earlier databasereviews and are considered useful are discussedbelow.

Hopkins– Keener [79–81] (Fernholz & Finley Cases

6601, 7203 and 7204): The initial work by Hopkinsand Keener [79] was concerned with measuring theproperties of the turbulent boundary layer on theside wall of the NASA Ames 8� 7 ft supersonicwind tunnel at Mach 2.4–3.4. The next investiga-tions were performed in the NASA Ames 3.5 fthypersonic wind tunnel. Hopkins and Keener [80]measured the local skin friction, total-temperatureprofiles, and pitot-pressure profiles on the hyperso-nic wind tunnel wall. Although the pressure gradientis small near the measurement location, thereappears to be significant upstream history effectsin this experiment. Keener and Hopkins [81]investigated the wind tunnel air flow over a flat

plate at Mach 6.2–6.5. The total temperature ofthe freestream was 764–1028K and the temperatureat the edge of the boundary layer was 73K orgreater. The analysis of the air flow propertiesincluded corrections for calorically imperfect gaseffects. The boundary-layer properties were investi-gated with forced and natural transition. Surfaceproperties measured: pressure, temperature, andwall shear stress. Properties measured acrossboundary layer: static pressure, pitot pressure, andtotal temperature.

Horstman– Owen [82–84] (Fernholz & Finley Case

7205): This investigation was performed for air flowover an axisymmetric cone-ogive-cylinder at Mach7.2 in the NASA Ames 3.5 ft hypersonic windtunnel. The total temperature of the freestream airwas 667K and the temperature at the edge of the


boundary layer was 59K. Natural transitionoccurred along the body and the boundary layerbecame an equilibrium, constant pressure flowdownstream on the body. The transverse curvatureeffects are considered to be negligible for thisgeometry. Fluctuating properties of the flow werealso measured and hu0i=ut ¼ 0:17 at y=d ¼ 1:1,which gives an indication of the turbulent intensityin the freestream flow. Surface properties measured:pressure, temperature, wall shear stress. Propertiesmeasured across boundary layer: pitot pressure,total temperature.

Additional experiments that have not beenincluded in the Fernholz and Finley database[28,30] follow below.

Coleman– Elfstrom– Stollery [85] (see also Refs.

[31,66,67]): The compressible turbulent boundarylayer on a flat plate was studied at a freestreamMach number of 9 in the Imperial College no. 2 guntunnel. The total temperature of the freestream airwas 1070K. The local boundary-layer edge Machnumber was varied (Mach 3, 5, and 9) by changingthe incidence of the plate from 0 to 26:5. Bothnatural and tripped boundary layer flows wereinvestigated. Theory based on Spalding–Chi skin-friction and Reynolds analogy was used to predictthe Stanton number for the three Mach numbers.There was an increased discrepancy between mea-surements of heat transfer and the prediction of thetheory as the Mach number was increased. Surfaceproperties measured: static pressure and heattransfer. Properties measured across boundarylayer: pitot pressure.

Kussoy– Horstman [36,38]: The experiments wereconducted with flow over a water-cooled flat plate inthe NASA Ames 3.5 ft hypersonic wind tunnel atMach 8.2. The flat plate without a sharp fin is thedatabase that is being considered. The plate surfacewas maintained at a constant surface temperature of300� 5K. In the first experiment [38], the proper-ties of the boundary layer 1.87m from the leadingedge were determined. In the second experiment[36], the properties of the boundary layer 1.62mfrom the leading edge were determined. Naturaltransition occurred between 0.5 and 1.0m fromthe leading edge. A fully developed, equilibriumboundary layer was established at the measurementlocation. Tabulated results are presented forboundary-layer surface and edge properties.Tabulated results for the velocity, density andtemperature profiles are also given for the measure-ment location. Surface properties measured: static

pressure and heat transfer. Properties measuredacross boundary layer: pitot pressure, static pres-sure, and total temperature.

Hopkins et al. [86,87,122]: The initial experimentswere performed by Hopkins et al. [86] on simpleshapes for turbulent boundary layers with nearlyzero-pressure gradient in the NASA Ames 3.5 fthypersonic wind tunnel. Local skin friction and heattransfer data were measured on flat plates (Mach6.5 and 7.4), cones (edge Mach 4.9, 5.0, and 6.6),and the wind tunnel wall (Mach 7.4). Skin-frictiondata are given in tabulated form. The next experi-ments were performed by Hopkins et al. [87] on asharp leading edge flat plate in the same Amesfacility. Flat plate skin-friction was measureddirectly with an edge Mach number of 6.5. Theskin-friction experimental database at various mo-mentum thickness Reynolds numbers and adiabaticwall temperature ratios are given in tabulated form.Hopkins et al. [122] conducted further flat plateexperiments in the same Ames facility. This studyprovides additional results to those previouslyreported by Keener and Hopkins [81], but at ahigher Reynolds number. The model was injectedinto the airstream at various angles of attack, whichresulted in local Mach numbers at the measuringstation of 5.9, 6.4, 6.9, 7.4, and 7.8. No boundary-layer trips were used. The model surface tempera-ture was nearly isothermal. Direct measurements ofskin friction and velocity profiles were made for thevarious Mach numbers and for Tw=Taw ¼ 0:3 and0.5. Real gas corrections as given in Ref. [128] wereused in the analysis of the data. Tabulated results ofthe database are given in the article. Surfaceproperties measured: wall shear stress with a skin-friction balance. Properties measured across bound-ary layer: pitot pressure.

Holden [42,45,46,48,88]: Experiments [48,45] wereconducted on a flat plate in the Calspan 48 in and 96in shock tunnels at Mach numbers 6.8–13. Steadyflow was established in these facilities in 1 or 2ms.The investigation measured the wall shear stress andthe heat flux. The wall skin friction and heattransfer results were transformed with the VanDriest, Eckert, and Spalding–Chi methods andcompared in figures to the incompressible results.The experimental data is approximately within 30%of the incompressible results, which is more scatterthan expected from experimental results. The paper[45] provides tabulated test conditions, heat trans-fer, and skin-friction. Another experiment [88] wasconducted in the Calspan tunnels on a flat plate


with a constant curvature surface downstream witha freestream Mach number of 8. The upstream partof the database on the flat plate could be useful, butneeds further evaluation. A brief summary ofexperiments performed on flat plates is given byHolden and Moselle [42]. An electronic database[46] of results from the many experiments per-formed by Holden is now available on the internetto qualified users. A further evaluation of theusefulness of the Holden flat plate database needsto be performed.

3.5.2.3. Case 7: sharp circular cone. The sharp conemodel is defined by the cone half-angle (anglebetween cone axis and cone surface) and the lengthL along the cone axis. The axial distance along thecenter of the cone from the tip is defined as X anddistance along the cone surface from the tip isdefined as x to be consistent with flat plate notation.The basic flow properties in the freestream aredefined by specifying the Mach number, totaltemperature, and freestream unit Reynolds numberReu1, which are tabulated in Table 3 for the currentsharp cone experimental database being evaluated.The cone surface temperature is obtained from thespecification of the wall to total temperature ratio.The other properties of the freestream flow areobtained from the following perfect gas relations:

T1 ¼ T0 1þg� 1

2

� �M21

� ��1,

Tw ¼ ðTw=T0ÞT0; g ¼ 1:4,

a1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffigRT1

p; u1 ¼ a1M1,

r1 ¼ m1Reu1=u1,

Table 3

Model geometries and freestream conditions for cone database

Investigator Cone half-

angle

Cone

length (m)Rn � 105 (m

Rumsey et al. [91,92] 7:5 0.7874 —

Stainback et al. [129] 10 � 0:5 —

Chien [93] 5 0.656 —

Kimmel [89,90] 7 1.016 5.0

Hillier et al. [94–96,130] 7 0.5783 —

Holden: cone

[42,43,46,48,98]

6 0.7073 —

Holden: cone-flare [44] 6 2.667 —

Pruett (DNS) [62] 7 1.427 0

Pruett (DNS) [64] 7 0.254 0

p1 ¼ r1RT1 ¼ p0 1þg� 1

2

� �M21

� ��g=ðg�1Þ.

The nose radius Rn of the sharp cones is onlyavailable for the Kimmel experiment.

Rumsey et al. [91,92]: Rumsey and coworkers[91,92] performed a number of flight tests withvehicles with a sharp-cone nose and at varioussupersonic/hypersonic Mach numbers. The Rumseyand Lee [91] report has data at Mach 5.15 and hasbeen included as a potential part of the presentsharp cone database. The authors present much ofthe needed database information in figures. Thefreestream unit Reynolds number is not consistentwith the value determined from the 1976 StandardAtmosphere conditions with the altitude given (10%difference). The accuracy of this database has notbeen estimated, but it is valuable as limitedhypersonic flight data is available.

Stainback et al. [129]: The experimental results[129] were obtained in the NASA Langley 20 inhypersonic wind tunnel in air. The authors obtainedthe Stanton number along the surface of a 10 sharpcone at Mach 6 and 8. The Mach 6 results are givenin their Figs. 3 and 13. The study was alsoconcerned with boundary-layer transition to turbu-lent flow and the authors measured the unsteadywall pressure. Tabulated test conditions and bound-ary-layer edge properties are given. This databasehas been used for validation of transition models.

Chien [93]: Chien [93] performed a wind tunnelinvestigation on the skin friction and heat transferon a 5 half-angle sharp cone of length 0.656m at afreestream Mach number of 7.9. The experimentalinvestigation was conducted in the Naval OrdnanceLaboratory Hypersonic Wind Tunnel in air. Chienhas tabulated the test conditions for 11 runs. The

) M1 T0 (K) Tw=T0 Reu1=m� 10�6

5.15 1265 0.591 31.0

6 500 0.6 32.9

7.90 816 0.351 35.2

7.93 722 0.420 6.60

9.16 1063 0.273 55.0

13.04 1739 0.173 15.60

10.96 1509 0.199 12.07

8.0 733 0.834 3.407

6.0 450 0.865 8.950


Stanton number as a function of the boundary-layeredge properties and surface distance is tabulated forthe 11 runs with different freestream conditions. Inaddition, surface skin-friction measurements wereobtained in four of the runs and these values arealso tabulated. The measured Stanton numbers arecompared to four analytical turbulence models.

Kimmel [89,90]: The experiment by Kimmel[89,90] was conducted in the Hypersonic WindTunnel B at Arnold Engineering DevelopmentCenter where six test conditions were used. Theinvestigation is concerned with boundary-layertransition on a 7 sharp cone model of length 1mat Mach 7.9. In addition, the aft part of the modelcould be flared or an ogive. Results of thisexperimental investigation were initially publishedin the proceedings of an ASME meeting [89] andlater published in Ref. [90] with limited changes.The flow conditions are specified with the Machnumber and total temperature held constant, whilethe unit freestream Reynolds number is varied bychanging the total pressure. The surface pressure,surface temperature, and wall heat transfer weremeasured along the model. The heat transfermeasurements are given in figures as the Stantonnumber as a function of x=L or boundary-layeredge Reynolds number with length scale x. Thesharp cone results for a freestream unit Reynoldsnumber 6.6 million per meter can be more readilyobtained from this article. Boundary-layer edgeconditions are not specified.

Hillier et al. [94–96,130]: At the Antibes Work-shop on Hypersonic Re-entry Flows, which isdocumented in the books by Desideri et al. [94],the sharp cone is the first hypersonic turbulent flowproblems to be solved by participants. Denmann etal. [95] obtained the experimental database in theImperial College Number 2 Gun Tunnel, whereMach 9.2 nitrogen flow over a 7 cone of length0.58m is investigated. The flow in the nozzle is notuniform, but is like a spherical source flow, whichgives a Mach number gradient along the nozzle.Mallinson et al. [96] have performed furthercalibration of the gun tunnel flow to determineimproved input conditions for hypersonic flowcomputations. Measurements obtained are pressureand heat transfer (Stanton number) along the conesurface. The pitot pressure is measured across theboundary layer at two locations along the cone.With the assumption that the static pressure isconstant across the boundary layer, the Machnumber across the boundary layer is obtained from

the Rayleigh pitot formula. The wall pressure andStanton number along the cone surface are given.Lawrence [130] presented at the workshop resultsthat compare his prediction of total pitot pressureacross the boundary layer with the experimentaldata. Hillier et al. [131] have obtained further datafor a new cone test model. The authors present theStanton number as a function of a Reynoldsnumber (distance along the surface and freestreamproperties). In 1993, Abgrall et al. [132] present anupdate of the European Hypersonic Database andthe number one problem in the database is the sharpcone problem.

Holden [42–44,46,48,98]: Over more than 30years, Holden has investigated many hypersonicflow problems experimentally and recently created adatabase of the measured results. Two experimentshave been performed that can contribute to thesharp cone database. Holden performed the tests atthe experimental facilities at Calspan in the 96 inshock tunnel in air. The testing time in this shocktunnel is approximately 25ms, which makes thechange of the model wall temperature very smallduring a run. The models used in the two tests are asfollows:

6 sharp cone: An initial investigation wasperformed by Holden [48] in 1977 and has limitedinformation provided on the model description andon the freestream flow properties in the tunnel.Documentation of the next experimental resultswith the same model is presented in Ref. [43]. Thisstudy is mainly concerned with boundary-layertransition on 6 sharp and blunt cones at angle ofattack in Mach 11 and 13 flows. However, the heatflux as a function of distance along the sharp cone atzero angle of attack is given and the boundary layerhas transitioned from laminar to turbulent flow. Itappears the distance is along the surface of the coneand measured from the nose-tip junction point,which is not specified. It is estimated that thejunction point is 0.15m from the cone tip and thecone length is 0.71m. In 1992, Holden [42] reviewedhis experiments concerned with hypersonic flow andcreated a database of work performed from 1965 to1991. The database includes the sharp cone workreported in the 1985 paper; however, little newinformation is presented on the turbulent boundaryflow properties. Holden performed further experi-ments on this model in 1995 and Ref. [98] is mainlyconcerned with the transition issue. Tabulated dataof freestream flow conditions for a list of tunnelruns is given and includes the conditions for Run 2.


In addition, tabulated data on the model config-uration, angle of attack, freestream Mach number,and unit Reynolds number for the list of runs arealso given. New measured wall pressure and heatflux along the cone surface for the sharp cone atzero angle of attack are given. For these tunnel testruns, the turbulent boundary layer has not becomefully turbulent. Described in Ref. [46] is the furtherdevelopment of Holden’s hypersonic database. Thesharp cone database is available on a CD ROM andon the CUBRC website (http://www.cubrc.org/aerospace/index.html) to qualified applicants. Itdoes not appear that a complete database forexperiments performed on the 6 sharp cone modelis available in the open literature and the informa-tion available at Holden’s website has not yet beeninvestigated. Tabulation is needed of the skinfriction and heat flux along the cone for the varioustest conditions. The tabulated properties at the edgeof the boundary layer are required for the correla-tion of the skin friction and Stanton number.

6 cone with 30 flare: The experimental study byHolden was conducted at Mach 11, 13 and 15 andthe results documented in a 1991 AIAA paper [44].Tabulation of the stagnation and freestreamtest conditions for the three shock tunnel runs isgiven. For these flow conditions, the boundarylayer is fully turbulent well upstream of the cone/flare junction. The measured wall pressure andheat transfer along the cone/flare model for threeruns are given. Holden measured the pitot pressureand total temperature across the boundarylayer, and the velocity profiles across the boundarylayer have been determined from the measurements.Tabulation of the measured turbulent boundary-layer properties are not available in the referencesreviewed here. Location of the cone/flare junctionpoint is not specified in this paper but is notnecessary for the solution on the 6 cone withthe reasonable assumption that the upstreameffect of the flare can be neglected. However,for the profile data, distance is measured from thecone/flare junction point. In the 1992 and2003 database papers, Holden et al. do notprovide additional information on the cone/flareexperiment.

For Holden’s experiments, the total temperatureis sufficiently high that freestream conditions mayinclude real gas effects due to vibrational excitation.Holden has assumed the gas is in thermodynamicequilibrium in the shock tunnel so vibrationalnonequilibrium effects are neglected. This review

assumes that the wall temperature of the model is atroom temperature of 300K.

3.6. Conclusion and recommendation of adequacy of

experimental database

The current recommended database for 2D/axisymmetric hypersonic experiments is composedof seven different geometries. The two zero-pressuregradient geometries, the flat plate/cylinder (Case 6)and the sharp cone (Case 7), have seen the mostextensive experimental study. As a result, theory-based correlations exist for these two cases. Thesecorrelations are expected to be more accuratethan any single experimental data set since theyhave been shown to match a wide range ofexperimental data and tend to mitigate the effectsof experimental bias errors due to the choice ofmeasurement technique and facility (e.g., flowfieldnon-uniformity).

For the five geometries involving shock/turbulentboundary-layer interactions there are a total of ninerecommended experiments. In general, these experi-ments include surface pressure and surface heat fluxmeasurements, and a few also have skin-frictionmeasurements. Most of these experiments includeflow-intrusive pitot/static surveys in the interactionregion. Given the elliptic mathematical character ofthe separated flow region, these pitot surveys shouldbe used with caution.

Since the designer of hypersonic vehicles isprimarily interested in the prediction of surfacepressures for vehicle aerodynamics and surface heatflux for thermal protection systems, the current 2D/axisymmetric database for shock/turbulent bound-ary-layer interaction appears to be sufficient. How-ever, if this database is to be used to improve theturbulence models (see Section 4), then additionalexperiments are required. In addition to surfacequantities, future experiments should measureprofiles of both mean properties and turbulencestatistics (rms velocities, Reynolds stresses, turbu-lent kinetic energy, etc.) in the interaction region,preferably using non-intrusive measurement techni-ques. More detailed turbulence information fromexperiments or DNS might aid in the determinationof where these turbulence models break down,ideally on a term-by-term basis. Significant effortsshould be made to quantify and reduce theexperimental uncertainties in the measurement andfreestream quantities. Approaches for convertingexperimental bias errors into random uncertainties

http://www.cubrc.org/aerospace/index.html

http://www.cubrc.org/aerospace/index.html


(e.g., Ref. [27]) should also be employed. Finally,preference should be given for axisymmetric geo-metries instead of 2D ones due to the possibility of3D end wall effects.

4. Usage of the hypersonic validation database

4.1. Validation of theory-based correlations

4.1.1. Flat plate/cylinder

The first step in using the flat plate/cylindricalexperimental or DNS database is concerned withthe validation of the accuracy of the theoreticalcorrelations for skin friction, heat transfer, velocityprofiles, and temperature profiles. Theories havebeen developed to correlate the skin friction orStanton number of compressible turbulent bound-ary-layer flows with zero-pressure gradient into asingle correlation curve. In addition, the velocityand temperature profiles in the inner or outerregions of the turbulent boundary layer can alsobe correlated into a single profile (see Section 3.3and Appendix B). The correlation curves or profilesare based on analysis and validation with a largedatabase to establish the accuracy of the varioustheories. The following investigators have used thehypersonic experimental data given in Table 2 toassess the accuracy of theoretical correlationsfor turbulent boundary layers with zero-pressuregradient.

Van Driest [133,134]: In the initial article [133](Van Driest I) a theory is developed for predictingproperties of compressible turbulent boundary-layerflows with the Prandtl mixing-length model, but nocomparison of theory to experimental data is given.In the second article [134] (Van Driest II) the theoryis modified to incorporate the von Karman mixing-length turbulence model. The turbulent Prandtlnumber is still assumed to be one, but in thetemperature relation the recovery factor is intro-duced. The usual approach of plotting normalizedskin-friction as a function of Mach number wasused by Van Driest and he compared the twotheories to experimental data. The experimentaldatabase included supersonic data of Coles [135]and Chapman–Kester [136] and hypersonic data ofSommer–Short [104] and Korkegi [106]. With thissupersonic/hypersonic database, no conclusioncould be made on which of the two Van Driesttheories provided the best prediction of skinfriction.

Peterson [105]: Peterson [105] compares seventheories for predicting the skin friction with anexperimental database for turbulent boundary-layer flows with zero-pressure gradient. Thetheories transform the experimental skin frictionand Reynolds number to incompressible (trans-formed) values. The theoretical prediction of theincompressible skin friction as a function ofincompressible Reynolds number is obtained fromthe Karman–Schoenherr formula, which is consid-ered the most accurate fit of the incompressibleexperimental database. Peterson uses an experimen-tal compressible skin-friction database obtainedfrom 21 references and the data from the referencesare tabulated. For hypersonic flow, there are tworeferences for an adiabatic wall [106,113] and thereare four references for a non-adiabatic wall[112,104,107,108]. The significance of this paper isthat the author uses the transformations of thevarious theories to correlate all of the experimentaldata at different Mach numbers, wall temperatureratios, and Reynolds numbers into a single curve.Also, Peterson recognizes the work of Wilson [137]where he developed a skin-friction transformationfor zero heat transfer with the von Karman mixing-length law. Van Driest [133] developed a skin-friction transformation with the Prandtl mixing-length (now referred to as Van Driest I). Van Driest[134] extended the theory of Wilson to the casewith heat transfer (Van Driest II, or more appro-priately, Peterson refers to this theory as Wilson–Van Driest).

Spalding– Chi [110,138]: Spalding–Chi [110] de-veloped an analytical prediction theory for theskin friction on a smooth surface with zero-pressure gradient at various momentum thicknessReynolds numbers ð400oReyo7500Þ, Mach num-bers ðMo10Þ, and surface temperatures to freestream temperature ½0:1oðTw=T1ÞoðTaw=T1Þ�.A significant number of the experiments are athypersonic Mach numbers but the wall temperaturerange is limited. The hypersonic database includesinvestigations by Sommer–Short [104], Korkegi[106], Hill [107,108], Brevoort and Arabian [111],Matting et al. [113], and Winkler [112]. The RMSvalue of ðCf ;exp � Cf ;thÞ=Cf ;th for the total databaseused in this paper is 11% for van Driest II and 9.9%for the Spalding–Chi theory.

Chi and Spalding [138] developed a theoreticalanalysis to correlate Stanton number (heat transfer)as a function of the Reynolds number. Thecompressible Stanton and Reynolds numbers for


experimental data points are transformed intoincompressible values and should reduce into asingle curve. The authors use a database of 11experiments on isothermal surfaces in air flow withzero-pressure gradient to establish the accuracy ofthe theoretical correlation. The Chi–Spalding theoryis shown to be reasonably accurate. Three of theexperiments are for 5:04oMo5:25 with the datafrom Brevoort and Arabian [111], Tendeland [109],and Winkler [112]. The experiment of Hill [108]provides data at Mach 8.27, 9.07, and 10.04 whichwas obtained in a conical nozzle with nitrogen gasflow. The Brevoort–Arabian geometry is an axiallysymmetric annular nozzle which consists of an innershaped center body and an outer cylindrical sleeve.Boundary-layer measurements were made on theinside of the sleeve which gives essentially flat plateresults. The Tendeland experiment uses the turbu-lent boundary layer along a cone-cylinder geometry.In the determination of the Stanton number, aReynolds analogy factor of 1.16 is used for thecomplete database. Further details of the Spal-ding–Chi theory is given in Appendix B.

Hopkins et al. [81,86,87,139]: Hopkins et al. [86]investigated the accuracy of the correlation theoriesof Sommer and Short, Spalding and Chi, Van DriestII, and Coles for the local skin friction. Theprediction of heat transfer for these theories wasalso investigated. Hopkins et al. [87] investigatedeight local skin-friction transformation theories ofVan Driest II, Spalding–Chi, Sommer–Short, Eck-ert, Moore, Harkness, Coles, and Baronti–Libby.These theories were assessed against the Mach 6.5experimental database of the authors. It wasconcluded that the methods of Van Driest II andColes predict the skin friction within about 5%. Theother six theories underpredicted the skin frictionfrom 10% to 25% for this experimental data set.The survey article by Hopkins and Inouye [139] isbased on a NASA Technical Note [86] and includesadditional skin friction and heat transfer data. Fourtheories are investigated further and these theoriesare described in the survey article. The incompres-sible skin-friction formula of Karman–Schoenherris used to determine the skin friction as a function ofthe momentum thickness Reynolds number. Thehypersonic database for the adiabatic flat plate isonly the experiment of Korkegi [106] while thedatabase for the non-adiabatic flat plate isSommer and Short [104] (hollow cylinder), Hopkinset al. [86], Hopkins–Keener–Louie [87], Wallace–McLaughlin [116], Young [115], and Neal [120].

The authors suggested for hypersonic flat plateflows that Van Driest II theory be used to predictturbulent skin friction, and that heat transfer beobtained with a Reynolds analogy factor equal to1.0 and a recovery factor equal to 0.9. In the ensuingarticle by Hopkins et al. [122], the four theories forcorrelating experimental skin-friction data werefurther investigated. Their experimental data wascompared to numerical turbulent boundary-layersolutions with an algebraic eddy viscosity model ofCebeci. In addition the Baronti–Libby and VanDriest methods for correlating mean velocityprofiles were investigated. The authors determinedthat the Van Driest II, Coles, and numericalturbulent boundary-layer solutions give the bestpredictions of skin friction and are within �10%.The authors state ‘‘The Van Driest theory gave themost satisfactory transformations of the velocity–profile data onto the incompressible law of the walland velocity–defect curves.’’ Keener and Hopkins[81] have investigated five velocity profile correla-tion methods for their Mach 6.5 database: wallreference temperature, T 0 method of Sommer andShort, Coles, Baronti–Libby, and Van Driesttransformations. It is stated that the use of eitherthe Prandtl mixing length (Van Driest I) or the vonKarman mixing length (Van Driest II) result inidentical transformation functions. The Van Driestmethod gives the best correlation for both the law ofthe wall and velocity–defect law when compared toColes’ incompressible velocity profile data. How-ever, the correlations deteriorate with decreasingmomentum thickness Reynolds number.

Cary and Bertram [53]: Cary and Bertram [53]made an investigation of the Reynolds analogy andprediction methods for skin friction and heat transferon flat plates and cones for high speed flows. Theincompressible local skin friction in this paper isobtained from the Spalding–Chi relation. TheSpalding–Chi relation is 2.0% lower than theKarman–Schoenherr equation at Rey ¼ 103 and2.4% high at Rey ¼ 106. The database used in thiswork has a Mach number range from 4 to 13 andratio of wall to total temperature from 0.1 to 0.7. TheReynolds analogy factor for Mach numbers less thanapproximately 5 is adequately approximated withRaf ¼ 1:16 for the available wind-tunnel database.For turbulent flow with significant wall cooling andMach numbers greater than 5 at any ratio of wall tototal temperature Raf is ill defined. The von Karmanexpression for Raf is approximately 10% higher thanexperimental data of Keener and Polek [140] and


Holden [141]. In the authors’ summary, they statethat the Spalding and Chi transformation methodincorporating virtual-origin concepts was found tobe the best prediction method for Mach numbers lessthan 10. The Spalding and Chi transformationmethod using Karman’s Reynolds analogy wasshown to give the best predictions based on eitherthe length or momentum thickness Reynolds numberwhen the proper virtual origin was specified. Thesmall amount of experimental data for Machnumbers greater than 10 were not correlated wellby any of the transformation approaches. Thehypersonic database used in this investigation ofthe accuracy of correlation theories was obtainedfrom the data of Heronimus [118], Wallace [117],Cary and Morrisette [121], Hopkins et al. [86], Cary[123], Weinstein [124], Hopkins et al. [122], Holden[45], and Coleman et al. [66,85].

Owen et al. [84]: Boundary-layer measurementswere made downstream on a cone-ogive-cylindermodel at a freestream Mach number of 7.0 [84].Fluctuating and mean flow measurements wereobtained at one location sufficiently far downstreamwhere the pressure gradient is zero. Mean velocityand total temperature boundary-layer profiles aregiven. The relation between T� (the total tempera-ture in non-dimensional form, see Appendix B) andvelocity is linear except in the region close to thewall. The velocity profile was transformed to theincompressible form with Van Driest theory andcompared to the incompressible velocity correlationcurve of Coles. The data are in good agreementwith the incompressible law of the wall correlation.In the outer region of the boundary layer, the dataare in good agreement with the incompressiblevelocity–defect correlation.

Fernholz and Finley [29]: The authors investigatedthe accuracy of some of the cases in the compres-sible experimental database for the turbulent meantemperature and velocity profiles with comparisonto theoretical correlations [29]. The hypersonic casesinvestigated are as follows:

Keener and Hopkins [81] (Fernholz & Finley Case7204): The sharp flat plate experiments have zero-pressure gradient with no upstream historyeffects. The static temperature is in good agree-ment with the theoretical correlation in the outerpart of the boundary layer. Five methods areevaluated for correlating the measured velocityprofiles with the incompressible form of the lawof the wall and the velocity defect law. The Van

Driest method gives the best correlation of thevelocity profiles. The experimental velocity pro-file database is concluded to be in good agree-ment with the law of the wall and the velocitydefect law.

Horstman and Owen [83] (Fernholz & Finley Case7205): This experiment investigated the turbulentboundary-layer flow over an axisymmetric cone-

ogive-cylinder body where downstream the Machnumber is 7.2 at the edge of the boundary layer.The nose of the body might have introduced aslight favorable pressure gradient. However, it ishighly probable that the boundary layer hasreached equilibrium at the three measurementstations. Agreement between the velocity profilemeasurements and the law of the wall is verygood. There is good agreement with the outervelocity law. Agreement between measured andtheoretical temperature profiles is satisfactory.

Kussoy and Horstman [36,38]: The first flat plateexperiment of Kussoy and Horstman [38] withoutfins or wedges provides data at Mach 8.2 and at onelocation, 1.87m from the sharp leading edge. Theexperimental mean velocity profile was transformedinto incompressible coordinates with Van Driest IItheory and compared to Coles’ universal law of thewall. Since the data and the theory were inreasonable agreement, the authors concluded thatthe turbulent boundary layer is fully developed. Thesecond experiment [36] is for a flat plate at Mach 8.3with the data obtained at 1.62m from the leadingedge. Again the transformed mean velocity profile iscompared to Coles’ law of the wall profile in innervariables. The authors conclude that the turbulentboundary layer is fully developed. Further analysisof the outer region correlation of the boundary layeralso needs evaluation.

Huang et al. [54]: In Ref. [54] a self-consistenttransformation method (denoted as HBC) wasdeveloped to predict skin friction and velocityprofiles of compressible boundary-layer flows withzero-pressure gradient. The paper is also concernedwith the assessment of the authors’ HBC transfor-mation by comparing the predictions to the wellaccepted Van Driest transformation and experi-mental data. For an adiabatic and non-adiabaticwall, the prediction of skin friction with the twotheories was compared to the experimental databaseused by Hopkins and Inouye [139]. The databasewas not sufficiently accurate to determine the bettertheory. The database of Watson [126] was also used


to assess the accuracy of the two theories for skinfriction. The HBC theory was more accurate thanVan Driest II for this case. In addition, thepredictions obtained with the HBC transformationmethod are in good agreement with the experi-mental velocity and temperature profiles of Kussoyand Horstman [38]. The investigation of Fernholzand Finley [29] has shown that the Van Driest IItransformation does indeed transform the compres-sible velocity profile data into a profile that matchesthe incompressible law of the wall.

4.1.2. Sharp circular cone

One of the problems with the sharp cone is thelack of a sufficiently accurate theoretical correlationof the experimental data to use as a benchmarksolution. For laminar flow, the skin friction andheat transfer for a flat plate are multiplied by

ffiffiffi3p

toobtain the cone values. There does not appear to bea well-defined approach to transform the turbulentflat plate results to the cone. However, the flat platecorrelation approach for skin friction and heattransfer has been extended to the sharp cone withsome approximations. The sharp cone geometry iswell suited to wind tunnel testing and avoids the2D/3D issues involved with flat plate flows.

In the correlation of the surface skin friction andheat transfer, the flow properties at the edge of theboundary layer are required. For high Reynoldsnumber flows, the boundary layer is thin and theedge properties can be obtained from the inviscidconical flow solutions, where the edge properties areapproximated with the wall properties. Perfect gastables of the inviscid surface properties as a functionof Mach number and cone half-angle have beendeveloped by Sims [142] (Sims discusses earliertables developed by Taylor–Maccoll and Kopal.).As interpolation is required with the use of thetables, the wall properties obtained from thenumerical solution of the governing ordinarydifferential equations is a better approach. Theconical inviscid perfect gas flow is determined withthe cone half-angle yc and the freestream Machnumber M1 specified. The following cone surfaceproperties are obtained from the tables with linearinterpolation: Me, Te=T1, and pe=p1. Then theedge properties are obtained from the relations

Te ¼ T1ðT e=T1Þ; pe ¼ p1ðpe=p1Þ,

ae ¼ffiffiffiffiffiffiffiffiffiffiffigRT e

p; ue ¼Meae,

re=r1 ¼ ðpe=p1Þ=ðT e=T1Þ; re ¼ pe=RT e,

me ¼ mðTeÞ; Sutherland or Keyes viscosity,

Reue ¼ reue=me; Unit Reynolds number.

The shear stress tw and heat flux qw at the surface ofthe cone are two of the quantities desired from theexperiments, turbulence modeling, and the numer-ical solutions. The wall shear stress is usuallywritten as the non-dimensional skin-friction para-meter, which is defined in two forms

Cf1 ¼ 2tw=r1u21; Cfe ¼ 2tw=reu

2e .

The wall heat flux is usually written as the non-dimensional Stanton number, which is defined intwo forms

St1 ¼ qw=r1u1cpðT01 � TwÞ; cp ¼ gR=ðg� 1Þ,

Ste ¼ qw=reuecpðTaw � TwÞ ¼ qw=reueðHaw �HwÞ,

H ¼ cpT þu2

2. (9)

The second form of the Stanton number becomesindeterminate when the heat flux is zero. The heattransfer coefficient hc is also sometimes used and isdefined as qw ¼ hcðTaw � TwÞ. The flow propertiesacross the turbulent boundary layer are also usefulin the evaluation and validation of turbulencemodeling.

Hopkins– Inouye [139]: Hopkins and Inouye [139]have assessed four transformation theories for flatplate hypersonic flows which already have beendiscussed. However, in the database, the heattransfer on sharp circular cones obtained by Mateer[86,143–145] is included. In a NASA TechnicalNote, Hopkins et al. [86] have shown that theexperimental data for the Stanton number as afunction of energy thickness Reynolds number forthe cone and a flat plate are essentially the samewhen the edge Mach numbers and wall temperatureratio ðTw=TawÞ are nearly the same. A Reynoldsanalogy factor of 1.16 was used for this case. Noindication of the use of a geometry transformationfactor is mentioned.

Chien [93]: Chien [93] measured skin friction andheat transfer (Stanton number) along the conesurface and compared the data to four theories forpredicting skin friction and heat transfer for zero-pressure gradient boundary-layer flows. The Kar-man formula for the Reynolds analogy factor withthe Bertram–Neal modification is used to determine


heat transfer. The paper does not indicate that anyMangler-type transformation is used to modify theflat plate predictions to the cone case. The methodsof Spalding–Chi, Van Driest II, Sommer–Short, andClark–Creel are evaluated. The Van Driest II andClark–Creel skin-friction predictions are withinabout 10% of the experimental data. The VanDriest II method gives reasonable prediction of heattransfer for Tw=T040:2. For Tw=T0 ¼ 0:11, Spal-ding–Chi method results are within 10% for theheat transfer.

Holden [44,48]: The experimental database ob-tained by Holden [48] in 1977 is used to correlate theheat transfer expressed as Stanton number as afunction of Reynolds numbers Rex and Rey. Thetransformation models that transform the compres-sible Stanton number and Reynolds number toincompressible values are Spalding–Chi, Van DriestII, and Eckert. It is concluded that Van Driest IImethod gives the best overall agreement with theexperimental database. Holden [44] has presentedthe cone correlation again in a 1991 AIAA paperwhere a better plot is given for the Van Driest IIcorrelation of Stanton number as a function of Rex.There is approximately a 30% scatter of theexperimental data about the theoretical incompres-sible correlation curve. For further discussion seeAppendix C.

Dinavahi [97]: A boundary-layer computer codewith a Baldwin–Lomax algebraic turbulence modelis used by Dinavahi [97] to predict the laminar toturbulent flow on a sharp cone at Mach 6. Theexperiment of Stainback et al. [129] for a 10 halfangle cone is used to evaluate the transition andturbulent models. The prediction and experimentalresults for the Stanton number along the conesurface are in reasonable agreement; however, thelength of the measured turbulent region is short andthe turbulent flow might not be fully developed.

Pironneau [94]: Pironneau (‘‘A Synthesis ofResults for Test Case 1 and 2: HypersonicBoundary layer and Base Flow,’’ pages 92–94 inDesideri et al. [94]) reviewed the work on the firsttest case in the modeling and computational Work-shop on Hypersonic Flows for Reentry Problems.The first test case is the perfect gas turbulentboundary-layer flow on a cone. The cone modelexperiment was performed by Denman et al. [95] ina contoured Mach 9 axisymmetric nozzle whichproduced a weak source-like (spherical) flow thatshould be modeled in the computation. Lawrence[130] solved the flow field with a Parabolized

Navier–Stokes code and the algebraic turbulencemodel of Baldwin–Lomax was used. The conesurface static pressure and Stanton number weremeasured and predicted. In addition, boundary-layer profile data for pitot pressure were measuredand predicted. Pironneau concluded that the in-vestigations were extremely well done by allinvestigators with the influence of all parameterscarefully studied, yet there was a 10% differencebetween computational and experimental results.Their results were inconclusive since it is unclearwhether the discrepancies arise due to modelingdeficiencies or uncertainty in the experimentaldata.

McKeel et al. [146]: McKeel et al. [146] areconcerned with modeling the transition problemwith the Baldwin–Lomax algebraic model, Wilcox1988 k–o two-equation model, and k–� Lam–Brem-horst model. One of the problems investigated istransition on a sharp cone in hypersonic flow usingthe database of Stainback et al. [129], where the freestream Mach number is 6 (see Section 4.5.3). Thisdatabase gives the Stanton number variation alongthe surface of the cone. The authors compare thetransition/turbulent model predictions with theexperimental heat transfer measurements. Theturbulent predictions with the three models are inreasonable agreement with the data, and the k–omodel is a little more accurate.

4.2. Validation of turbulence models

There are 18 different turbulence models that areassessed in the current work. These models are listedin Table 4 along with the notation used to referencethe model. Recall that we focus only on one- andtwo-equation turbulence models where integrationto the wall is employed (i.e., no wall functions) andwhich have also been previously validated for a widerange of non-hypersonic flows. We thus omitcompressibility corrections which have not yet beenapplied to a comprehensive low-speed validationdatabase, or which do not vanish at lower speeds.

The turbulence models that have been evaluatedfor 2D/axisymmetric hypersonic flows are listedbelow, along with the shorthand notation for themodels used throughout this review. Note that thediscussion of the turbulence models given here isbrief. The interested reader is encouraged to see theoriginal references for specific details of the models.Of these 18 turbulence models, only six have seenextensive validation for the current 2D/axisymmetric

ARTICLE IN PRESS

Table 4

Turbulence model notation

Turbulence model Notation

One-equation models

Spalart–Allmaras [147,148] SA

Goldberg [149,150] UG

Menter [151] MTR

Two-equation models

k–� Jones–Launder [152] k�JLk–� Launder–Sharma [153] k�LSk–� Chien [154] k�CHk–� Nagano and Hishida [155] k�NH

k–� Rodi [156] k�Rk–� So [157,158] k�SOk–� Huang–Coakley [11] k�HC

k–o Wilcox (1988) [159] ko88k–o Wilcox (1988) low Reynolds number [159] ko88LRk–o Wilcox (1998) [6] ko98k–o Menter with SST [160] SST

k–o Menter with BSL [160] BSL

k–l Smith [161,162] kl

k–z Robinson–Hassan [163,164] kzq–o Coakley [165] qo


hypersonic validation database: SA, k�JL, k�LS,k�R, ko88, and qo.

4.2.1. One-equation models

4.2.1.1. Spalart– Allmaras (SA). A transport equa-tion for determining the eddy viscosity with near-wall effects included has been developed by Spalartand Allmaras [147,148]. The accuracy of thepredictions with the Spalart–Allmaras (SA) modelis fairly insensitive to the yþ spacing at the wallrelative to the two-equation models, at least forhigh-speed flows [166]. Our experience with thismodel suggests that it has a good combination ofaccuracy and robustness for attached flows. Whilestable for large yþ values, the maximum foraccurate solutions should be roughly yþp1.

4.2.1.2. Goldberg (UG). Goldberg has developedthe one-equation Rt turbulence model [149,150].This model has been shown to provide goodpredictions for the hypersonic compression ramp(Case 1), but has not yet seen widespread applica-tion to the experimental hypersonic database. Thismodel does not require a wall distance to becalculated.

4.2.1.3. Menter one-equation model (MTR). Men-ter has developed a one-equation eddy viscosity

transport model [151]. This model is derived fromthe standard k–� model, and this relationship isexplored in detail in Ref. [151].

4.2.2. Two-equation models

4.2.2.1. Jones and Launder high Reynolds number

k– � ðk�JLÞ. The basic k–� model was developed byJones and Launder [152] in 1972. The modelconstants were later refined by Launder and Sharma[153] (see below).

4.2.2.2. Launder and Sharma (standard) k– �ðk�LSÞ. The Jones and Launder k–� model [152]was revised by Launder and Sharma [153] in 1974. Itis this 1974 revised model that is generally referredto as the ‘‘standard’’ k–� model. This model isgenerally good for free shear flows, but will not beas accurate for wall-bounded flows as the k–omodels, especially in the presence of adversepressure gradients. Marvin and Huang recommendthat the yþ values at the wall be kept below 0.3 forthis model [5].

4.2.2.3. Chien k– � ðk�CHÞ. Chien has developed alow Reynolds number k–� model [154]. This modelwas shown to provide better predictions of peakturbulent kinetic energy than the Jones–Laundermodel when applied to fully-developed channelsand turbulent flat plates.

4.2.2.4. Nagano and Hishida k– � ðk�NHÞ. Naganoand Hishida have developed a low Reynoldsnumber k–� model [155]. While this model has beeninvestigated for zero-pressure gradient cases athypersonic speeds, it has seen little validation usagefor shock interaction flows.

4.2.2.5. Rodi k– � ðk�RÞ. Rodi has developed a two-layer low Reynolds number k–� model [156]. In theouter layer, the standard dissipation rate � equationis used, while in the inner layer, an analyticexpression for � is used. This model has seenextensive validation usage for the hypersonicexperimental database.

4.2.2.6. So k– � ðk�SOÞ. So et al. have developed alow Reynolds number k–� model [157] by modifyingthe near-wall behavior of � based on DNS andexperimental data.


4.2.2.7. Huang and Coakley k– � ðk�HCÞ. Coakleyand Huang have developed a low Reynolds numberk–� model [11] which is based on DNS data.

4.2.2.8. Wilcox 1988 k–o ðko88Þ. The Wilcox 1988k–o model [159] is generally better than the k–�model for wall-bounded flows, especially in thepresence of adverse pressure gradients. It is recom-mended that the yþ values at the wall be kept wellbelow one. One problem with this original Wilcoxk–o model is the sensitivity of the results to thefreestream o levels.

4.2.2.9. Wilcox 1988 k–o low Reynolds number

ðko88LRÞ. Wilcox has also developed a low Rey-nolds number version of his 1988 k–o model (seeChapters 4.9.2 and 4.9.3 of Ref. [6] for details).

4.2.2.10. Wilcox 1998 k2o ðko98Þ. In 1998, Wil-cox updated his original k–o turbulence model tomore accurately predict free shear flows [6]. Thisupdated version will be referred to as the Wilcox1998 k–o model, or ko98, herein. While thesensitivity of the results to the freestream o levelsis indeed reduced in the 1998 version of the model[6], some sensitivity effects remain, at least for high-speed flows [16].

4.2.2.11. Menter baseline k–o (BSL). The baseline(BSL) Menter k–o model is a blending of theWilcox 1988 k–o model [6] near walls and atransformed k–� model in shear layers and thefreestream [160]. The BSL model utilizes theblending to reduce the sensitivity to freestreamturbulence levels that afflicts the Wilcox k–o model.This model has obtained good results for a widerange of flows.

4.2.2.12. Menter shear stress transport k–o(SST). The Menter Shear Stress Transport (SST)k–o model utilizes the same blending between thek–o and k–�models as the BSL model; however, theSST model also employs a modified form of theeddy viscosity definition which accounts for thetransport of the Reynolds stress [160]. This mod-ification improves the SST model’s predictiveaccuracy for flows with adverse pressure gradients.

4.2.2.13. Smith k2l ðklÞ. Smith has developed atwo-equation k2l model [161,162] as an improve-ment to an earlier k2kl model [167]. This model hasbeen shown to provide accurate velocity profiles on

compressible flat plate flows where typical k–�models fail.

4.2.2.14. Robinson and Hassan k2z ðkzÞ. It isgenerally acknowledged that the failure of thestandard k–� model to accurately predict a widevariety of flows is due to inadequate modeling of thedissipation equation. Robinson and Hassan[163,164], have developed a new two-equationturbulence model based on the vorticity variance(enstrophy) equation which has demonstratedgood predictive capability for a wide-variety offlows. A number of modeled terms in the enstrophyequation are included with the goal of incorporatingadditional physics into the equation governing thedissipation of turbulent kinetic energy. The k2zmodel does not employ damping or wall functions.

4.2.2.15. Coakley q2o ðqoÞ. A two-equation q–omodel ðq ¼ k1=2

Þ was developed by Coakley [165] topredict low-Reynolds number transition and in-crease numerical robustness over other two-equa-tion models. This model has been demonstrated tohave more favorable numerically stability behaviorthan standard k2o models when integrated to thewall.

4.2.3. Physical freestream turbulence quantities

One method for determining the freestreamturbulence properties is as follows. For the two-equation models, the specification of a freestreamturbulence intensity ðTuÞ can be used to determinethe turbulent kinetic energy in the freestream from

k ¼1:2

2ðTuV1Þ

2, (10)

where, for example, Tu ¼ 0:1 corresponds to afreestream turbulence intensity of 10%. However,the experimental measurement of � (or o; z etc.) isextremely difficult. As a result, the dissipationvariable is often determined by specifying the ratioof turbulent to laminar viscosity, mT=m, i.e.,

� ¼Cmrk2=mmT=m

(11)

or

o ¼rk=mmT=m

. (12)

For one-equation eddy viscosity turbulence models,the transported variable is simply found from themT=m ratio.


4.2.3.1. Effects on transition. High freestream tur-bulence intensity levels can lead to early transitionfrom laminar to turbulent flow. This phenomenon isoften referred to as bypass transition (since thenatural transition mechanisms are bypassed), ormore recently as transition due to a high distur-bance environment [168]. While some turbulencemodels also provide a transition prediction cap-ability, the transition process is complex, especiallyfor high-speed flows, and its modeling is beyond thescope of the current work.

4.2.3.2. Effects on turbulence. Experimental evi-dence [25,169] suggests that surface properties(e.g., shear stress) in the fully-developed turbulentregion are generally not affected by freestreamturbulence intensity, at least in the case of low-speedflows. Thus, it is expected that there should be littleor no effect of the freestream turbulence levels onthe mean flow predictions. Note that this is not thecase in free shear layers, where the freestreamturbulence levels can have a significant effect on theflow.

4.3. Turbulence model application to the hypersonic

validation database

A listing of hypersonic validation experiments ispresented in Table 5 along with the turbulencemodels from Table 4 that have been used with eachexperiment for validation purposes. The flowgeometries in Table 5 include the accepted 2D/axisymmetric experiments of Settles and Dodson[1–4] as Cases 1–4. Case 5 should also become astandard benchmark case for hypersonic shock/turbulent boundary-layer interaction flows. Cases 6and 7 are zero-pressure gradient flows and havereceived extensive validation usage. For the sharpcircular cone (Case 7), an accurate correlation of thesharp cone database similar to Van Driest II for theflat plate is needed.

A summary of the turbulence model validationusage is presented in Table 6. It is clear fromthe table that of the 18 turbulence models thathave been applied to this 2D/axisymmetric hyper-sonic validation database, only a limited numberhave seen extensive validation. In fact, only fiveturbulence models have been applied to the majorityof these seven geometries (SA, k�JL, k�LS, k�R,and ko88). A sixth model, the qomodel of Coakley,has been applied to three of the five shockinteraction geometries (Cases 1–5). Our review will

therefore emphasize the assessments of thesesix turbulence models, which are summarized inTable 7.

4.4. Previous flow geometries with adverse pressure

gradient

4.4.1. Case 1: 2D compression corner

There are two hypersonic experiments for the 2Dcompression corner which are deemed acceptablewith some caveats: the experiments of Coleman andStollery [31] and Coleman [66] with surface pres-sures by Elfstrom [67], and the experiment ofHolden [48]. An overview of the model validationusing the Coleman/Stollery/Elfstrom experiment issummarized graphically in Figs. 1 and 2 for the sixturbulence models given in Table 7.

Coleman and Stollery/Elfstrom experiment

[31,66,67]: Horstman [170] has used the Colemanand Stollery/Eflstrom experiment (among others) toperform validation computations for two-equationk–� models. The two turbulence models examinedare the high-Reynolds number Jones–Launder k–�model ðk�JLÞ and the low Reynolds number k–�model of Rodi ðk�RÞ. A third k–� model employingvarious compressibility corrections was also exam-ined. However, this model was calibrated usingsome of the hypersonic validation experiments, thusblurring the line between model prediction andcalibration. The third k–� model is not included inthis review. A yþ study was performed on a different(unspecified) geometry for this case using surfaceheat flux, and showed that the k�JL model wassensitive to yþ values above 0.15, while the k�Rmodel showed some mild sensitivity above 0.5.A grid refinement study was also performed on afew of the test cases (again, which cases were notspecified) with no change in the predicted values ongrids of 40� 100 and 60� 150. No sensitivities tothe freestream turbulence quantities were discussed.For the 2D compression ramp, the k�JL model wasfound to match the pressure well everywhere exceptfor the constant pressure plateau on the ramp whereit is overpredicted by 20% (see Fig. 1). The heattransfer predicted by this model is given in Fig. 2and greatly overpredicts the heating both in theinteraction region and in the plateau region, in somelocations overpredicting by an order of magnitudeor more. The k�R model performed much better,matching the experimental pressure data within10% and accurately predicting the heat transfereverywhere except within the interaction region,

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Table 5

Turbulence models assessed using the hypersonic validation database

Case no. Flow geometry Experiments Validation usage Turbulence models assessed

1 2D compression corner [31,66,67] [7,11,12] qo, ko88, k�LS, k�CH,

k�SO, k�HC

[170] k�JL, k�R[149,150] UG, SA, MTR

[171] ko88LR, SA, SST

[172,173] kz[48] None

2 Cylinder with conical Flare [32] [10,170] k�JL, k�R[7,11,12] qo, ko88, k�LS, k�CH,

k�SO, k�HC

[174] ko88[175] SA, SST

[68,69] None

3 Cone with conical flare [44] [170] k�JL,k�R

4 Axisymmetric impinging shock [35,70,71] [7,12] ko88, k�LS[170] k�JL, k�R[176] qo

[72–75] None

5 2D impinging shock [38] [170] k�JL, k�R[161] kl

[76,78] [172,173] kz[177] ko88

6 Flat plate/cylinder VDII [51] [178] Various

[13] ko88, k�LS, SST, SA[171] ko88, SST, SA[16] SA, ko98, k�NH, BSL

[175] SA, ko88, SSTHBC [54] [179] ko88, k�LS

[13] ko88, k�LS, SST, SAAVC [8] [8] ko88, k�LS

[180,181] ko88, k�CH, kl, SA

[79–81] None

[82–84] None

[85] None

[36,38] None

[86,87] None

[42,46,48,88] None

7 Sharp circular cone VDII [51]& [16] SA, ko98, k�NH, BSL

White [56]

[89,90] [182,183] kz[16] SA, ko98, k�NH, BSL

[91,92] None

[93] None

[94–97] None

[42–44,46,48,98] None


where the model overpredicts the heating by a factorof two.

Coakley, Huang, and coworkers [7,11,12] alsoused the Coleman and Stollery/Eflstrom experimentfor turbulence model validation purposes. A num-

ber of different two-equation turbulence modelswere examined including: the q–o model of CoakleyðqoÞ, the 1988 k–o model of Wilcox ðko88Þ, the k–�model of Launder and Sharma ðk�LSÞ, the k–�model of Chien ðk�CHÞ, the k2� model of So

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Table 6

Summary of turbulence models assessed using the hypersonic validation database

Turbulence model Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

SA X X X X

UG X

MTR X

k�JL X X X X X

k�LS X X X X

k�CH X X X

k�NH X X

k�R X X X X X

k�SO X X

k�HC X X

ko88 X X X X X

ko88LR X

ko98 X X

SST X X X

BSL X X

kl X X

kz X X X

qo X X X

Table 7

Summary of turbulence model assessment using the hypersonic validation database for selected models with a significant hypersonic

validation history

Turbulence model Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

SA X X X X

k�JL X X X X X

k�LS X X X X

k�R X X X X X

ko88 X X X X X

qo X X X

Fig. 1. Surface pressure turbulence model comparisons for Case

1: 2D compression corner at 34 (experiment by Elfstrom [67]).

Fig. 2. Surface heat flux turbulence model comparisons for Case

1: 2D compression corner at 34 (experiment by Coleman and

Stollery [31]).



ðk�SOÞ which includes compressibility extensionsgiven by Zhang et al. [158], and the k–� model ofHuang and Coakley ðk�HCÞ. A number of modelingcorrections designed specifically for high-speedseparated flows were also investigated by theseauthors, but it is unclear how these correctionsimpact the previous validation efforts for the model,especially at low speeds. These corrected models willtherefore not be included in the current review. Gridrefinement studies were discussed, but no resultswere presented and no estimates of the discretiza-tion error were given. A wall-spacing study showedthat yþ values greater than one gave errors in theskin friction ð42%Þ and also gave stability pro-blems with some models; however, varying the yþ

values from 0.1 to 1.0 showed no changes.Sensitivities to the freestream turbulence quantitieswere not addressed. Only the k�LS and ko88models were examined for the 15 ramp case, andboth models gave good predictions of surfacepressure as shown in Fig. 1, with a slight under-prediction in the interaction region. The heattransfer was not predicted as well (see Fig. 2), withthe k�LS model yielding heat transfer levels 25%higher than the data in the interaction region andthe ko88 model overpredicting the heating by asmuch as 50% in the interaction region and in theplateau region on the ramp. All five models wereapplied to the 34 ramp case, with the k�SO andk�HC models giving accurate predictions for thesurface pressure, while the other three models tendto overpredict the pressure in the interaction regionand significantly underpredicting the size of theseparated region as judged by the initial upstreampressure rise. All five models greatly overpredict theheat transfer in the interaction region by at least afactor of three, and the ko88 model also over-predicts the plateau heating downstream on theramp by 50%.

Goldberg and coworkers [149,150] have com-puted the 38 ramp case of Coleman and Stollery/Elfstrom with three one-equation turbulencemodels: Goldberg (UG), SA, and Menter (MTR).A mesh refinement study was performed for the UGmodel only using 200� 150 and 250� 200 cellmeshes with some minor effects on the results.While the effects of changing the yþ values arenot discussed, the yþ values in all cases are keptnear 0.1. No effects of the freestream turbulencelevels are examined. The SA and MTR models areshown to underpredict the size of the separationzone, thereby predicting an earlier peak in the

pressure. The UG model accurately predicts thepressure and provides fairly good estimates of thewall heating. The SA model also gives goodpredictions for the surface heating, while theMTR model greatly overpredicts the peak heatinglevels in the interaction region by as much as afactor of four.

Coratekin et al. [171] have computed the 38

ramp case of Coleman and Stollery/Elfstrom withthree turbulence models: a low Reynolds numberversion of the Wilcox 1988 k–o model ðko88LRÞ,the SA model, and the hybrid k–o=k–� model ofMenter with the SST option. They also examinedvarious compressibility corrections to the ko88LRmodel, but these corrections have not been eval-uated over a wide variety of flowfields and thus willnot be included here. A single grid of 128� 64 cellsis used for this case, with a grid refinement studyusing three grid levels being performed on aMach 3, 24 compression corner and assumed toextend to the hypersonic case. The yþ valuesemployed are not discussed, and no sensitivity isperformed for the freestream turbulence values. Theko88LR and STT models match the surfacepressure levels reasonably well, but underpredictthe extent of flow separation as judged by the initialrise in surface pressure. The SA model gives goodestimates of both the surface pressure and separa-tion extent. The peak surface heat flux levels areoverpredicted by a factor of two for all the models,and local values of heat flux are as much as fivetimes the experimental measurements in the inter-action region.

Nance and Hassan [172] and Xiao et al. [173]have used the k2z turbulence model ðkzÞ to examinethe Coleman and Stollery/Elfstrom experiment. Inboth papers, a grid sensitivity study is mentioned,but no results are presented. In addition, sensitiv-ities to wall yþ values and freestream turbulencelevels are not discussed. Fairly good agreement withis shown for the 15 ramp [173], but predictions forthe 34 and 38 ramps [172] overpredict themagnitude of the initial pressure rise and greatlyoverpredict (by up to a factor of five) the heating inthe interaction region. In all three cases, therecovery pressure and heating appear to be accu-rately predicted. The later study [173] also examinesvariable turbulent Prandtl number effects, but withlittle improvement for this case.

Holden experiment [48]: To our knowledge, thisexperiment has not been employed for validatingturbulence models.

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2: cylinder with conical flare at 35 (experiment by Kussoy and

Horstman [32]).


4.4.2. Case 2: cylinder with conical flare

There are two experiments which meet the Settlesand Dodson criteria for the axisymmetric cylinder-flare geometry. The first was included in the Settlesand Dodson review and was performed by Kussoyand Horstman [32] at NASA-Ames ResearchCenter. The second is a more recent experimentperformed in the supersonic blow-down wind tunnel(HSST) at DRA Fort Halstead, Great Britain and isdetailed by Babinsky [69] and Babinsky andEdwards [68]. The former experiment has seenextensive validation usage, while to the authors’knowledge, the latter experiment has not yet beencomputed in the literature. An overview of theturbulence model validation for this case asdiscussed below is shown graphically in Figs. 3and 4 for the five of the six turbulence models fromTable 7.

Kussoy and Horstman experiment [32]: TheKussoy and Horstman cylinder-flare experiments[32] have been used for turbulence model validationpurposes by Horstman [10,170]. A brief synopsis ofthe results was presented in Ref. [10], while a moredetailed discussion is given in Ref. [170]. Horstmanexamined two different two-equation turbulencemodels: the low Reynolds number k–� model ofJones and Launder ðk�JLÞ and the low Reynoldsnumber k–� model of Rodi ðk�RÞ (a third ‘‘com-pressible’’ k–� model is omitted from the presentdiscussion as mentioned in Section 4.4.1). Mesh


2: cylinder with conical flare at 35 (experiment by Kussoy and

Horstman [32]).

resolution studies were discussed; however, noresults were shown, and no estimates of thediscretization error were reported. The sensitivityto wall-normal mesh spacing was examined and isreported in Section 4.4.1 above for the 2Dcompression corner. For the 20 flare case, whichwas nominally attached flow, both k–� models gaveaccurate predictions of the surface pressure. Thek�JL model gave reasonable predictions of the heattransfer (within approximately 30%), while the k�Rmodel predicted heat transfer within the experi-mental uncertainty bounds everywhere except pos-sibly in the recovery region where the heat transferis underpredicted by as much as 25%. For the 35

flare case with flow separation, the surface pressurewas reasonably well predicted by both models asshown in Fig. 3. The heat transfer was overpredictedby an order of magnitude or more by the k�JLmodel and underpredicted by almost a factor of twoby the k�R model (see Fig. 4). The size of theseparation zone for this case is underpredicted byover 50% as judged by the initial pressure rise andthe peak pressure.

Coakley, Huang, and co-workers [7,11,12] alsoused the Kussoy and Horstman cylinder-flareexperiments [32] for turbulence model validationpurposes. A number of different two-equationturbulence models were examined including: theq–o model of Coakley ðqoÞ, the 1988 k–o model ofWilcox ðko88Þ, the k–� model of Launder and

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3: cone with conical flare at 36 (experiment by Holden [44]).


3: cone with conical flare at 36 (experiment by Holden [44]).


Sharma ðk�LSÞ, the k–� model of Chien ðk�CH), thek–� model of So ðk�SOÞ which includes compressi-bility extensions given by Zhang et al. [158], and thek–� model of Huang and Coakley ðk�HCÞ. For amore detailed discussion of this study, see Section4.4.1 above for the 2D compression corner. Only thek�LS and ko88 models were applied to the 20 flarecase [11], and both models provided good predic-tions of the surface pressure. The heat transferpredictions were as much as twice the experimentalvalues. All six models were applied to the 35 flarecase [12]. The k�LS, ko88, and qo models gave anadequate prediction of the surface pressure levelsbut underpredicted the size of the separation zoneby 60% (see Fig. 3). These models predicted anearly peak heating location (Fig. 4), with maximumerrors of a factor of 6.5, 3, and 4, respectively. Thek�HC and k�SO models gave reasonable pressurepredictions and only underpredicted the separationzone size by approximately 20%. The peak heating,however, was still overpredicted by a factor of 2.5.

Bedarev et al. [174] used the Wilcox 1988 k–omodel ðko88Þ to study the Kussoy and Horstmancylinder flare experiments. They discussed a gridsensitivity study, but did not report the results. Inaddition, they did not discuss the sensitivities tofreestream turbulence levels or yþ wall spacing.They examined 20, 30, and 35 flares. Their resultsdo not appear to be as good as those of Huang andCoakley [12] with the same turbulence model. Thereasons for these discrepancies are not known.

Olsen et al. [175] used the SA and Menter k–oSST model (SST) to study the Kussoy and Horst-man cylinder flare experiments. Their domainincluded the entire ogive-cylinder-flare, and anextensive grid study was performed for a modifiedk–�model known as the Lag model. The sensitivitiesto wall yþ spacing and freestream turbulence levelswere not addressed. Both models perform well forall of the flare angles except 35, where the upstreampressure and heating rise is not accurately predicted.It is notable that both models give reasonablesurface heat flux predictions, even in the interactionregion.

Babinsky and Edwards experiment [68,69]: Thisexperiment has not yet been used for turbulencemodel validation but is recommended.

4.4.3. Case 3: cone with conical flare

There is only one experiment for the cone/conicalflare case that is appropriate for turbulence modelvalidation. Holden [44] performed experiments in

Calspan’s 96 in shock tunnel at Mach numbers of 11and 13. As noted earlier, personal communicationswith Holden (the author of Ref. [44]) confirmed thatthe flare angle should be measured from the 6 cone,not the symmetry axis [102]. An overview of theturbulence model validation for the two modelsapplied to this case (as discussed below) is showngraphically in Figs. 5 and 6.

Holden experiment: The Holden cone/conical flareexperiment [44] at Mach 11 with a flare angle of 36

(as measured from the forecone) has been used forturbulence model validation by Horstman [170],

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4: axisymmetric impinging shock at 15 (experiment by Kussoy

and Horstman [35,70]).


who examined two different two-equation turbu-lence models: the low Reynolds number k–� modelof Jones and Launder ðk�JLÞ and the low Reynoldsnumber k–� model of Rodi ðk�RÞ (a third ‘‘com-pressible’’ k–� model is omitted from the presentdiscussion as mentioned in Section 4.4.1). Meshresolution studies were discussed; however, noresults were shown and no estimates of thediscretization error were reported. The sensitivityto wall-normal mesh spacing was examined and isreported in Section 4.4.1 above for the 2Dcompression corner. As shown in Fig. 5, the surfacepressure was reasonably well predicted by the k�Rmodel in the interaction region, with the onset ofseparation (judged by the initial rise in pressure)occurring slightly downstream of the experimentallocation. The k�JL model greatly underpredicts thesize of the separated zone, and both models fail tocapture a secondary peak in the pressure in thevicinity of the downstream plateau region. The k�JLmodel also overpredicts the peak heating level bynearly a factor of two, but accurately matches theheating in the recovery region downstream of theinteraction (see Fig. 6). The k�R model under-predicts both the peak heating and the heating levelsin the recovery region by 50%.

4.4.4. Case 4: axisymmetric impinging shock

There are two different axisymmetric impingingshock experiments which are deemed acceptable forturbulence model validation. The first is a series of


4: axisymmetric impinging shock at 15 (experiment by Kussoy

and Horstman [35,70]).

experiments conducted by Kussoy et al. at a Machnumber of 7 on a cone-ogive-cylinder model[35,70,71]. The second is a more recent experimentalinvestigation by Hillier et al. at Mach 9 on a hollowcylinder model [72–75]. An overview of the turbu-lence model validation for this case as discussedbelow is shown graphically in Figs. 7 and 8.

Kussoy and Horstman experiment [35,70,71]:Marvin and Coakley [176] used the Kussoy andHorstman axisymmetric impinging shock experi-ment [35,70] with a shock generator angle of 15 forthe validation of the q–o model of Coakley ðqo).The authors fail to report the effects of meshrefinement, variations of the yþ values, or the effectsof varying the freestream turbulence values. Predic-tions with the qo model are given in Figs. 7 and 8for surface pressure and heat transfer, respectively.Although a mild amount of flow separation isshown by the experimental data and the qo model,the model significantly underpredicts the size of theinteraction region. As a result, the model greatlyoverpredicts the peak levels of pressure, skinfriction, and heat flux (although the scale chosenfor the original figures does not include the peakvalues from the model).

The Kussoy and Horstman axisymmetric imping-ing shock experiment [35,70] with a shock generatorangle of 15 has been used for turbulence modelvalidation by Horstman [170] who examined twodifferent two-equation turbulence models: the lowReynolds number k–� model of Jones and Launder

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5: 2D impinging shock at 10 (experiment by Kussoy and

Horstman [38]).


ðk�JLÞ and the low Reynolds number k2� model ofRodi ðk�RÞ (a third ‘‘compressible’’ k–� model isomitted from the present discussion as mentioned inSection 4.4.1). Mesh resolution studies were dis-cussed; however, no results were shown, and noestimates of the discretization error were reported.The sensitivity to wall-normal mesh spacing wasexamined and is reported in Section 4.4.1 above forthe 2D compression corner. Neither model is able topredict the mild amount of flow separation indi-cated by the experimental data, and both alsounderpredict the size of the interaction region. As aresult, the peak pressures are overpredicted by 25%and the upstream pressure rise is not captured at all(Fig. 7). The peak skin-friction levels are over-predicted by a factor of two with the k�R model anda factor of three with the k�JL model. Both modelsmatch the surface pressure and skin friction in therecovery region well within the experimental un-certainty bounds. The surface heating levels, shownin Fig. 8, are overpredicted by 50% with the k�Rmodel and by at least a factor of two with the k�JLmodel, while the recovery heat flux is underpre-dicted by 30% with k�JL and 50% by k�R.

Huang and Coakley [12] and Coakley et al. [7]used both shock generator angles of the Kussoy andHorstman experiment [35,70] for turbulence modelvalidation. Two two-equation turbulence modelswere examined: the 1988 k–o model of Wilcoxðko88Þ and the k–� model of Launder and Sharmaðk�LSÞ. A mesh refinement study was discussed, andthe authors state that changing the mesh had noeffect of the predictions. While a yþ sensitivity studywas not conducted explicitly, the yþ values werekept below 0.5, which had been shown to besufficient for these models in a related study [11].The effects of changing the freestream turbulencequantities were not assessed in these studies. For the7:5 shock generator case, both models accuratelypredict the extent of the interaction region and thesurface pressure; however, both models also over-predict the peak heating and skin-friction levels by35% and 70%, respectively. As shown in Fig. 7 forthe 15 shock generator case, the width of theinteraction region is underpredicted, and the initialpressure rise in the vicinity of the separated flowregion is not captured. The peak levels of pressureand skin friction are overpredicted by approxi-mately 30% and 100%, respectively. The k�LSmodel overpredicts the heating by more than afactor of two, while the ko88 model is 60% higherthan the data (Fig. 8). The ko88 model overpredicts

all three surface quantities by at least a factor of twoin the recovery region. The k�LS model accuratelycaptures the wall pressure and skin-friction in therecovery region, but underpredicts the heating in therecovery region by 50%.

Hillier et al. experiment [72–75]: To our knowl-edge, this experiment has not been used in thevalidation of one- or two-equation turbulencemodels.

4.5. New flow geometries with and without pressure

gradient

4.5.1. Case 5: 2D impinging shock

A 2D impinging shock occurs when an externallygenerated oblique shock impinges on a flat plateboundary-layer. There are two experimental datasets that satisfies the Settles and Dodson criteria.Kussoy and Horstman [38] conducted a carefulexperimental study of the 2D impinging shock casein the Ames 3.5 ft Hypersonic Wind Tunnel at Mach8.2. An overview of the turbulence model validationfor this case as discussed below is shown graphicallyin Figs. 9 and 10. Recently, Schulein and coworkers[76,77] also studied the 2D impinging shock case.

Kussoy and Horstman experiment [38]: TheKussoy and Horstman 2D impinging shock experi-ment [38] with an effective wedge angle of 10 hasbeen used for turbulence model validation byHorstman [170], who examined two different two-equation turbulence models: the low Reynolds

ARTICLE IN PRESS

Fig. 10. Surface heat flux turbulence model comparisons for

Case 5: 2D impinging shock at 10 (experiment by Kussoy and

Horstman [38]).


number k–�model of Jones and Launder (k�JL) andthe low Reynolds number k–� model of Rodi ðk�RÞ(a third ‘‘compressible’’ k–� model is omitted fromthe present discussion as mentioned in Section4.4.1). Mesh resolution studies were discussed;however, no results were shown, and no estimatesof the discretization error were reported. Thesensitivity to wall-normal mesh spacing was exam-ined and is reported in Section 4.4.1 above for the2D compression corner. As shown in Fig. 9, thesurface pressure was accurately predicted by bothmodels, with the predictions falling just outside theexperimental uncertainty bars over the entire inter-action region. The models do appear to slightlyunderpredict the upstream separation point asjudged by the initial rise in the surface pressure.The k�JL model overpredicts the heating by 60% inthe interaction region but accurately matches theheating in the recovery region downstream of theinteraction (see Fig. 10). The k�R model accuratelypredicts the heating in the interaction region butunderpredicts the heating levels in the recoveryregion by up to 20%.

The Kussoy and Horstman experiment [38] for 5

and 10 wedge angles was also used by Smith [161]in the validation of a two-equation k2l model ðklÞ.A compressibility correction designed specificallyfor high-speed separated flows [7] was also investi-gated by the author, but it is unclear how thiscorrection impacts the previous validation effortsfor the model, especially at low speeds. Results for

the corrected model will therefore not be includedhere. The transition onset and extent were set to 50and 100 cm, respectively, (as suggested by theexperiment) to achieve the best fit to the undis-turbed boundary-layer profile. A grid refinementstudy was performed for the model with thecompressibility correction for the 10 wedge casewith minor effects on the surface properties. The yþ

values in each case are discussed, but no yþ

sensitivity was performed and the effects of varyingthe freestream turbulence levels were not examined.For the 5 wedge case, the surface pressure isaccurately predicted by the model, with a slightunderprediction of the upstream extent of thepressure rise. The heat transfer was overpredictedby as much as 30% within the interaction region forthis case. The results for the 10 wedge were similar.

Schulein et al. experiment [76,77]: Nance andHassan [172] and Xiao et al. [173] have used the k–zturbulence model ðkzÞ to examine the Schuleinexperiment. In both papers, a grid sensitivity studyis mentioned, but no results are presented. Inaddition, sensitivities to wall yþ values and free-stream turbulence levels are not discussed. The 10

and 14 shock generators were studied, and goodagreement was found for both wall pressure andskin friction. The heat flux in the interaction andrecovery regions was overpredicted, with the peakheating being too high by a factor of two.

Fedorova et al. [177] used the Wilcox 1988 k2omodel ðko88Þ to study the Schulein impinging shockexperiment for shock generator angles of 6, 10,and 14. No grid study was discussed, and thesensitivities of the results to freestream turbulencelevels and wall spacing were not addressed. Theyfound good agreement for the surface pressure at allshock generator angles. The skin friction wasaccurately predicted for the smaller generatorangles, but the recovery skin-friction levels wereunderpredicted for the 14 case. The peak heatingwas overpredicted for all cases in the interactionregion by roughly a factor of two, but the heatingrate in the recovery region appears to be accurate.

4.5.2. Case 6: flat plate/cylinder

For the numerical solution of the hypersonicturbulent flow on a sharp flat plate with a Navier–Stokes code, there are many choices for the grid. Itis recommended that a parabolic grid be used asdescribed in Roy and Blottner [166]. The boundaryconditions for a parabolic grid are well defined andare continuous without a singularity at the leading


edge. The assessment of accuracy of turbulencemodels can be performed with numerical predic-tions for a group of experiments compared directlywith the experimental data. However, a betterapproach is to transform numerical predictionswith the correlation theory into a curve whichshould match the theoretical correlation curvewithin a small error. The following investigatorshave used the hypersonic experimental data given inTable 2 to assess the accuracy of one- and two-equation turbulence models for boundary layerswith zero-pressure gradient.

Huang et al. [179]: An assessment of two-equationturbulence models has been performed in Ref. [179],and it has been determined that the k–� Launder–Sharma ðk�LSÞ and Wilcox 1988 k–� ðko88Þ modelsdo not give the expected law of the wall behavior.The k2o model is much less sensitive to densityeffects than the k2� model. A density correction tothe closure coefficients was developed that im-proved the accuracy of the two-equation models inthe logarithmic part of the turbulent velocity profile.

Aupoix et al. [8,180,181]: In the paper by Aupoixand Viala [8] the standard turbulence models areassessed with supersonic and hypersonic boundary-layer flow on an adiabatic flat plate. The authors usethe following adiabatic wall experimental databaseto evaluate the local skin-friction correlations andto obtain reference test cases to evaluate turbulentmodel predictions (note: these cases are referred toas AVC experiments in Table 1):

Supersonic: Coles [135], Kistler [184], Hasting–Sawyer [185], Mabey et al. [186,187], Richmond[188].

Hypersonic: Winkler–Cha [112], Moore [114],Watson et al. [125,126], Laderman and Deme-triades [127].

The Van Driest II and HBC approaches correlatethe experimental database for adiabatic flat plateflows within a scatter of �10%. The data is notsufficiently accurate to determine which correlationis more accurate. Also the authors investigate theinfluence of the non-dimensional form of yþ in thewall damping functions. For high Mach numberflows, density gradients influence the logarithmicbehavior of the velocity profile and the turbulencemodels require additional modeling to retain thelogarithmic region. A density gradient correction tothe turbulence models is investigated using theapproach of Huang et al. [179]. The database given

above is used to assess the accuracy of densitygradient corrections added to standard turbulencemodels. The Wilcox 1988 k–o model ðko88Þ is lesssensitive to density gradient effects than the k–�models.

The initial work of Catris and Aupoix was givenin an AIAA paper [180] and was later published asRef. [181] with additional work included. Theauthors have proposed modifications of the diffu-sion term in the compressible turbulent transportequations for the various turbulence models. Themodels investigated are the Chien k–� modelðk�CHÞ, Wilcox k2o 1988 model ðko88Þ, the Smithk2l model ðklÞ, and the SA model. The modifiedturbulence models are assessed for supersonic andhypersonic zero-pressure gradient boundary-layerswhere the accuracy of the velocity profile and skinfriction are determined with the reference test cases.Velocity profile predictions with the various turbu-lence models are compared to the following flatplate experimental databases: Mabey et al. [186](Mach 4) and Winkler and Cha [112] (Mach 5.3).Skin-friction predictions with the various turbulencemodels are compared to the following experimentaldatabases: Mabey et al. [187], Winkler and Cha[112], Watson [126] (Mach 10–11.6), and Owen et al.[84] (Mach 7.2). The modified turbulence modelsolutions generally improve the prediction accuracy.

Bradshaw et al. [178]: As a part of the StanfordCollaborative Testing of Turbulence Models, one ofthe entry cases is the compressible flow over a flatplate at a Reynolds number of 104 based onmomentum thickness. The final problem definitionrequested turbulence modelers to obtain the follow-ing solutions: (Case A) Mach 2, 3, 5, and 8 with anadiabatic wall and (Case B) Mach 5 flow withTw=Taw equal 0.2, 0.4, 0.6, 0.8, and 1.0. The VanDriest II values of compressible local skin frictionand Stanton number with Reynolds analogy factorof 1.16 were determined by Bradshaw as thereference solution for comparison. For Case A,the average of the modeler predictions for skinfriction were 1% below reference values at Mach 2and 4% above reference values at Mach 8. For CaseB, the average of the modeler predictions for skinfriction were 5.2% high for Tw=Taw ¼ 0:2 and 2.5%high at Tw=Taw ¼ 0:8. Some further refinement ofspecification of the viscosity law and equation ofstate for a perfect gas is needed.

Bardina et al. [13]: The Mach 5 boundary-layerflow over an adiabatic flat plate was investigated atmomentum thickness Reynolds numbers of 5000,

ARTICLE IN PRESS

Fig. 11. Skin friction turbulence model comparisons for Case 6:

flat plate/cylinder (correlation is Van Driest II [51]).


10,000, 20,000, 50,000, and 100,000. The authorsperformed careful numerical solutions to ensurethat the solutions for four turbulence models(Launder–Sharma k2�, Wilcox 1998 k2o, MenterSST k2o, and Spalart–Allmaras) had small numer-ical errors. The Van Driest II transformationtheory was used with the von Karman–Schoenherrincompressible skin-friction relation to obtain thecompressible skin friction for comparison withthe numerical predictions from the four turbulencemodels. For this case, the Van Driest approachprovides a good approximation to flat plate skinfriction experimental data. The k�LS model showeda significant underprediction of the skin friction(as much as 20%). In addition, the transfor-mation of HBC gives larger skin friction (5–10%)than the Van Driest II transformation. The com-pressible velocity profiles for the four turbulencemodels were transformed to incompressible formand compared to the log law of HBC. Thek�LS turbulence model again gave poor results.The SA turbulence model gave the best overallpredictions for the skin friction and the velocityprofile.

Roy and Blottner [16]: Roy and Blottner [16]examined Mach 8, calorically perfect gas flow over aflat plate using four different turbulence models:SA, Nagano and Hishida k2� ðk�NHÞ, Wilcox 1998k2o ðko98Þ, and Menter’s BSL k2o (BSL). Theconditions correspond to 15 km altitude and a walltemperature of 1000K was used. The plate was 1mlong, and transition was specified at 0.12m to allowa significant amount of both laminar and fully-developed turbulent flow. The simulation resultswere compared to the accurate laminar andturbulent results obtained for this case by VanDriest [99,133]. The validation methodology dis-cussed in Section 2 was used. Multiple grids wererun in order to estimate the discretization error. Inthe fully-developed turbulent region, the discretiza-tion error for the SA model was approximately0.5%, while the error for the two-equation modelswas near 1%. These estimates increase to 0.6% and1.25% when a safety factor of 1.25 is included. Theeffects of varying the wall yþ values between 0.01and 1.0 were studied, and the models were found tobe relatively insensitive to yþ variations below 0.25.Skin friction as a function of Reynolds number inthe turbulent region are shown in Fig. 11 for eachmodel. The results appear to reach an approxi-mately constant error relative to the Van Driestcorrelation by the end of the plate. At this location,

the Wilcox (1998) k2o model underpredicts theVan Driest II curve by 6.7%, while the SA, Menterk2o, and low Reynolds number k2� overpredictthe skin-friction by 1.4%, 3.1%, and 6.3%, respec-tively. Accounting for the grid convergence errors,the skin-friction predictions from the SA andMenter k2� models are within the error tolerances,while the low Reynolds number k2� and Wilcox(1998) k2o models are not. In addition, the surfaceshear stress values for the Wilcox (1998) k2omodelshowed a sensitivity of up to 4% to the freestream ovalues.

Coratekin et al. [171]: The authors have developeda compressible Navier–Stokes code and are con-cerned with the performance of the numericalscheme and accuracy of three linear turbulencemodels for hypersonic perfect gas flows [171]. Theturbulence models in the code are the following:Wilcox k2o model ðko88Þ (with two compressiblecorrections—Coakley et al. [7] have developed alength scale correction for reattachment boundary-layer flows while Coakley and Huang [7,11] haveintroduced a correction in flow regions with strongcompression effects), SA one-equation model, andMenter k2o (SST) model. The turbulent boundarylayer on an isothermal flat plate at Mach 5 is solvedwith the code and compared with the Van Driest II[51] correlation of the skin friction. At a givenmomentum thickness Reynolds number, the turbu-lent model predictions for skin-friction are lowerthan the values obtained with the Van Driest IItheory.

ARTICLE IN PRESS

Fig. 12. Skin friction turbulence model comparisons for Case 7:

sharp circular cone (correlations are Van Driesta [55] and White’s

cone ruleb [56]).

Fig. 13. Surface heat flux turbulence model comparisons for

Case 7: sharp circular cone (correlations are Van Driesta [55] and

White’s cone ruleb [56] and experiment by Kimmel [89,90]).


4.5.3. Case 7: sharp circular cone

The supersonic/hypersonic flow over a sharp coneat zero angle of attack is of interest as the flowproperties at the edge of the boundary layer areapproximately constant along the cone. The sharpcone is an extension of the flat plate geometry and isbasic to the understanding of turbulent boundary-layer flows. From a computational point of view,this geometry is not ideal because the singularity atthe sharp tip can make it difficult to obtain accuratenumerical solutions. With the appropriate extensionof the flat plate type of grid [166], the tip singularityproblem can be handled.

McDaniel et al. [183]: McDaniel et al. [183] areconcerned with the modeling and prediction ofboundary-layer transition in high speed flows. Thek2z ðkzÞ Robinson–Hassan turbulence model isused for the fully turbulent flow. The experimentaldatabase of Kimmel (see Kimmel [89, 4.5.3]) is usedto evaluate the validity of the transition model andalso shows the accuracy of the k2z model forhypersonic turbulent flow. The turbulent heat fluxprediction has an accuracy of approximately 10%and the prediction is smaller than the experimentalvalue.

Roy and Blottner [16]: Flow over a sharp conewith a half angle of 7 was examined by Roy andBlottner [16] using four different turbulence models:SA, Nagano and Hishida k2� ðk�NHÞ, Wilcox 1998k2o ðko98Þ, and Menter’s BSL k2o (BSL). Theflow conditions correspond to a wind tunnel testperformed by Kimmel [89,90], where transitionoccurs at approximately 0.5m downstream of thecone tip. The gas is air, and the temperatures aresuch that the perfect gas assumption with g ¼ 1:4 isappropriate. The discretization error in surfaceheating for the SA model was estimated to be0.25% in the turbulent region. The two-equationmodels had numerical error estimates of less than1.5% in the turbulent region. The effects of varyingthe wall yþ values between 0.01 and 1.0 werestudied, and the models were found to be relativelyinsensitive to yþ variations below 0.25. In this case,the surface heating values for the Wilcox (1998)k2o model showed a sensitivity of up to 4% to thefreestream o values.

Skin-friction predictions versus surface distanceReynolds number are presented in Fig. 12 for thefour turbulence models as well as the correlations ofVan Driest [55] and White [56]. Taking the averageof the two correlations as the true experimentalvalue, all of the models are within the estimated

uncertainty of �5% except for the Wilcox 1998k2o model, which underpredicts the skin frictionby roughly 10%. Surface heating results versussurface distance Reynolds number are presented inFig. 13 for the four turbulence models along withlaminar boundary-layer code results and the turbu-lent Van Driest cone theory. Note that (a) refers tothe transformed Van Driest [55], while (b) denotesWhite’s cone rule [56]. In addition, experimentaldata is taken from Kimmel [90] and includes the

ARTICLE IN PRESS

Fig. 14. Surface heat flux turbulence model comparisons

(enlarged view) for Case 7: sharp circular cone (correlations are

Van Driesta [55] and White’s cone ruleb [56] and experiment by

Kimmel [89,90]).


conservative 10% error bounds suggested by theauthor. Although the surface heating predictions inthe transitional region do not match the experi-mental data, the predictions in both the laminar andturbulent regions are generally within the experi-mental error bounds. An enlarged view of theturbulent heating region is presented in Fig. 14. Atthe end of the cone, the two theoretical correlationsagree to within 4%. This difference is well within theaccuracy of the correlations, which is estimated tobe approximately �5–10%. Taking the theoreticalvalue to be the average of these two curves, theWilcox (1998) k2o model is roughly 5.7% belowthe theory at the end of the cone. Both the Menterk2o model and the low Reynolds number k2�model predict heating values approximately 2.5%high, while the SA model is 4.3% high. Accountingfor the discretization errors, all of the turbulencemodels are well within the estimated error bounds.

Summary of turbulence model validation for the

sharp cone: The use of the sharp cone for validationof zero-pressure gradient turbulent boundary-layerflows has been limited. Hypersonic cone flow isconsidered a simple problem that is not computa-tionally expensive as the flow is axisymmetric. Theexperimental results have been mainly measure-ments of wall heat transfer, which has been used forvalidation of turbulence models. The theoreticalprediction of surface heat transfer is less accuratethan skin-friction prediction as the Reynoldsanalogy factor and a Mangler transformation are

required, which are of limited accuracy (seeAppendix C). There is a need to further predictconical and flat plate flows with various turbulencemodels and compare the results with experimentalmeasurements. This will help establish turbulencemodeling accuracies and the relation betweenplanar 2D and axisymmetric turbulent boundary-layer flows.

4.6. Conclusion and recommendation on turbulence

model validation usage

Of the 18 turbulence models examined in thisreview, only six of them have seen extensivevalidation usage on the current 2D/axisymmetrichypersonic experimental database. In many cases,the effects of grid refinement on the predictions werenot demonstrated. Furthermore, in none of theshock interaction cases were the numerical errorsestimated with regard to grid refinement, nor werethe sensitivities to the freestream turbulence quan-tities assessed. We recommend that future modelvalidation efforts include a comprehensive gridrefinement study, along with estimates of thediscretization error and iterative error.

The ability of the models to predict surfacepressure was mixed, with Rodi’s k–� model per-forming the best; it was accurate for a majority ofthe cases with the exception of the region immedi-ately upstream of the interaction. The ability of themodels to predict skin friction cannot yet bedetermined since only a few of the experimentsinclude detailed skin-friction data. The heat fluxpredictions were generally poor, with the best model(again Rodi’s k–� model) still off by a factor of twofor most of the shock interaction cases. While thesemodel validation results should be used withcaution due to the failure of most authors toadequately address numerical errors and modelsensitivities, they do suggest that these turbulencemodels are not yet capable of accurately predictinghypersonic shock/boundary-layer interactions, evenfor the simple 2D/axisymmetric geometries. Thisshortfall needs to be addressed before attempting topredict complex, three dimensional (3D) flowsinvolving shock wave/turbulent boundary-layerinteraction.

For the two zero-pressure gradient cases (the flatplate and the sharp cone), the available correlationsfor skin friction are widely accepted as being moreaccurate than any single experimental data set. Forexample, the skin-friction correlations for the flat


plate are within �5%. The correlations for heat fluxhave additional uncertainties related to the choicefor Reynolds analogy factor and require furtherstudy, as does the transformation to convert skinfriction and heat transfer from the flat plate to thesharp cone for turbulent flows (the current recom-mend value for this transformation is 1.13 forturbulent flows, see Section 3.3.2).

There is a need for new model validation studieswith a focus on the newer experiments in thecurrent 2D/axisymmetric hypersonic database (e.g.,[68,69,72–78]). We find it surprising that theMenter’s SST [160] model and, to a lesser extent,the SA [147,148] model, have seen only limitedassessment with this hypersonic database. Thesetwo turbulence models are arguably the mostcommonly used turbulence models for externalflows in the subsonic, transonic, and supersonicspeed range.

Compressibility corrections should be implemen-ted in the baseline turbulence models. Some of thesecorrections (e.g., [180]) vanish as the mean densityvariations are reduced, thus ensuring prior modelvalidation efforts at low speeds are still valid. Forother corrections (e.g., [7]), the low-speed modelvalidation test cases should be revisited (thesubsonic through supersonic test cases of Marvinand Huang [5] are recommended). Future modelingefforts should also investigate the effects of shockunsteadiness (e.g., [189,190]) and the effects ofvariations in the turbulent Prandtl number (e.g.,[173,191]). In general, the use of ad hoc modelcorrections applied to a limited class of flowsresembles calibration or parameter fitting andshould be avoided for turbulence models that willbe applied to general hypersonic flows; thus theeffects of any turbulence model modificationsshould be assessed for a wide range of flows.

5. Conclusions

The current recommended database for 2D/axisymmetric hypersonic experiments is composedof seven different geometries. For the cases invol-ving shock/boundary-layer interaction, we haveadded one additional geometry, the 2D impingingshock, to the previous hypersonic validation data-base. There are two new validation experimentsdiscussed on this geometry. For the original fourgeometries in the Settles and Dodson review [1–4],three new experiments have been added. Thesenew experiments generally provide higher spatial

resolution data and use newer measurement techni-ques, and thus are highly recommended for tur-bulence model validation. The current 2D/axisym-metric hypersonic experimental database for shockinteracting flows appears to be sufficient forvalidating turbulence models for predicting surfacepressure and heat flux; however, there are notsufficient data for validating skin-friction predictions.Furthermore, the current database is not sufficient forimproving the turbulence models since there arevery few measurements of mean and fluctuatingturbulence quantities in the interaction region.

The two zero-pressure gradient cases, the flatplate/cylinder and the sharp cone, have been thesubject of extensive experimental investigation. As aresult, the available correlations for skin friction onthe flat plate are estimated to be accurate to within�5% and are widely accepted as being moreaccurate than any single experimental data set.The flat plate correlations for heat flux haveadditional uncertainties related to the choice forReynolds analogy factor and require further study.For the flat plate, theoretical results for the meanprofiles of velocity and temperature are alsoavailable [54]. The correlations for both skin frictionand heat transfer for the sharp cone have largeruncertainties due to the difficulties in determiningthe proper flat plate/cone transformation.

Of the 18 turbulence models examined in thisreview, only six of them have seen extensivevalidation usage on the 2D/axisymmetric hyperso-nic experimental database. For the models that havebeen assessed on the database, most providereasonable predictions of surface pressure (and skinfriction when available), but not for surface heating.The heating rates are generally overpredicted by themodels by as little as a factor of two or as much asan order of magnitude in the interaction region and,to a lesser extent, in the recovery region. In only aminority of cases have these turbulence modelassessments included an adequate assessment ofnumerical errors. In addition, the model assess-ments rarely included sensitivities to wall yþ spacingor freestream turbulence quantities.

6. Recommendations

There is an urgent need for new hypersonic flowexperiments be conducted. In addition to surfacequantities (pressure, skin friction, and heat flux),these experiments should measure profiles of bothmean properties and turbulence statistics (rms


velocities, Reynolds stresses, turbulent kinetic en-ergy, etc.) in the interaction region. Despite thedifficult challenges (short flow residence times,particle seeding, etc.), these turbulence profilesshould be measured with non-intrusive opticaltechniques if possible. The more detailed turbulenceinformation from non-intrusive experiments orDNS might aid in the determination of where theturbulence models break down, ideally on a term-by-term basis. For any new experiments, significantefforts should be made to quantify and reduce theexperimental uncertainties in the measured andfreestream quantities.

We recommend that a comprehensive study beundertaken to assess a wide range of turbulencemodels (including current popular models) on thecurrent 2D hypersonic experimental database. Thisstudy should follow the turbulence model validationmethodology discussed in Section 2, which includescareful documentation of the test cases and turbu-lence models employed, estimates of the numericalerror, model sensitivities to wall yþ spacing andfreestream turbulence values, and quantitativecomparisons with experimental data. The effectsof model corrections should also be examined (e.g.,compressibility, shock unsteadiness, variations inturbulent Prandtl number), but only in a mannerwhich does not destroy the prior validation historyof the model (including low-speed incompressibleflows). As detailed experimental measurements andDNS data become available for mean and fluctuat-ing turbulence quantities, the modeling of specificphysics in the turbulence models can be examined,and the predictive capability of turbulence modelsfor hypersonic flows can be improved.

Acknowledgments

The authors would like to thank Joe Marvin ofELORET Corp., Tom Coakley of NASA-Ames,and Matt Barone and Ryan Bond of SandiaNational Laboratories for their extremely helpfulreviews of this work. We would also like to thankthe first author’s graduate students Ravi Duggiralaand Pavan Veluri for their aid in preparing thefigures presented in this review.

Appendix A. Compressibility corrections

With the two-equation turbulent turbulencemodels, the standard form of these equation is notadequate for obtaining the logarithmic region of the

velocity profile for compressible flows (e.g., see [54]).For the k2� turbulence model, the standard andmodified equations developed by Catris and Aupoix[180] are presented. In addition, the modifiedequation for the eddy viscosity for compressibleflow developed by Catris and Aupoix [180] is alsopresented.

A.1. Two-equation k2� turbulence models

The standard or classical form of the turbulentkinetic energy equation is

rDk

Dt¼

qqtðrkÞ þ

qqxj

ðrujkÞ ¼ Dk þ SPk � SDk,

(A.1)

where the diffusion, production, and destructionterms are

Dk ¼qqxj

mþmT

sk

� �qk

qxj

� �; SPk ¼ Pk; SDk ¼ r�.

In the above Pk is the turbulent kinetic energyproduction. Huang et al. [179] have shown for two-equation turbulence models, that density correc-tions to the incompressible closure coefficients arerequired to obtain a logarithmic region of thevelocity profile for compressible flows. Catris andAupoix [180] have modified the turbulent transportequation for two-equation turbulence models. Acompressibility correction to the turbulent kineticenergy equation has been developed by Catris andAupoix, which gives the same form as Eq. (A.1)except the diffusion term is modified as follows:

Dk ¼qqxj

mrþ

mT

rsk

� �qðrkÞ

qxj

� �.

The standard or classical form of the dissipationequation is

rDjDt¼

qqtðrjÞ þ

qqxj

ðrujjÞ

¼ D� þ SP� � SD�; j ¼ �, ðA:2Þ

where the diffusion, production, and destructionterms are

D� ¼qqxj

ðmþ mT=s�Þqjqxj

� �; SP� ¼ c�1f 1Pk

jk,

SD� ¼ c�2f 2rj2

k.

A compressibility correction to the dissipationequation has been developed by Catris and Aupoix


[180], which gives the same transport equation asEq. (A.2) but with j ¼ r�. The diffusion termbecomes

D� ¼qqxj

mffiffiffirp þ

mT

s�ffiffiffirp

� �qffiffiffirp

jqxj

� �,

SP� ¼ c�1f 1Pk

jk,

SD� ¼ c�2f 2

j2

k.

The production, and destruction terms are the sameform as above but j ¼ r�.

Spalart– Allmaras model: The transport equationfor the eddy viscosity was originally developed bySA. Spalart [192] states, ‘‘Note that the S-A paperwas silent on large density variations, and there-fore. . .’’. The transport equation is written with thedependent (working) variable j ¼ n in the followingform, which is appropriate for SA model for

incompressible flows:

DjDt¼

qjqtþ ~uj

qjqxj

¼ D� þ S�p � S�D. (A.3)

The turbulent eddy viscosity is related to thedependent variable by the relation mT ¼ rjf n1.The diffusion term D� is

D� ¼qqxj

m�qjqxj

� �þ

cb2

sqjqxj

qjqxj

,

m� ¼ ½ðm=rÞ þ j�=s.

The production term is S�P ¼ cb1 ~Sj and the destruc-tion term is S�D ¼ cw1f wðj=dÞ2. The various termsintroduced are defined in the original paper ofSpalart and Allmaras. The gas density r does notappear in the transport equation for f except in theviscous sublayer. However, the density does appearin the relation for determining the eddy viscosity.

In fluid dynamics codes, the governing equationsare generally written in conservation form. Theabove kinematic eddy viscosity transport Eq. (A.3)is rewritten in conservation form by multiplying theequation by the density and using the conservationof mass equation to obtain the following conserva-

tion form for the SA turbulence model:

rDjDt¼

qqtðrjÞ þ

qqxj

ðrujjÞ ¼ Dþ Sp � SD. (A.4)

The diffusion term with a variable density includedis

D ¼qqxj

mefqjqxj

� �þ

cb2

sqjqxj

qjqxj

� m�qjqxj

qrqxj

,

mef ¼ r m� ¼ ½mþ rj�=s.

The production term is

SP ¼ rS�P ¼ cb1 ~Srj. (A.5)

The destruction term is

SD ¼ rS�D ¼ cw1f wrðj=dÞ2. (A.6)

The last term in the diffusion term above, whichinvolves the density gradient, is zero for incompres-sible flows and is usually neglected for compressibleflows (see for example: [13,166], and the FLUENTcode [193]).

The SA model has been modified by Catris andAupoix [180] to account for compressibility effects.The resulting Catris and Aupoix equation for the SA

turbulence model for compressible flow is

DðrjÞDt¼

qqtðrjÞ þ uj

qqxj

ðrjÞ

¼ Dþ SP � SD,

j ¼ ðmT=rÞf n1. ðA:7Þ

The diffusion term is

D ¼qqxj

msqð

ffiffiffirp

jÞqxj

þðffiffiffirp

jÞs

qðffiffiffirp

jÞqxj

� �

þcb2

sqqxj

ðffiffiffir

pjÞ

qqxj

ðffiffiffir

pjÞ.

The form of Eq. (A.7) in conservation form on theleft-hand side becomes

DðrjÞDt¼

qqtðrjÞ þ

qqxj

ðrujjÞ � rjquj

qxj

¼ Dþ Sp � SD. ðA:8Þ

The production term is defined in Eq. (A.5) and thedestruction term is defined in Eq. (A.6).

Appendix B. Turbulent flat plate correlations

B.1. Correlation of skin-friction data

The standard approach for correlation of com-pressible skin friction on a flat plate is the van DriestII transformation theory [51]. This approach trans-forms the compressible skin friction at a givenReynolds number (Rex or Rey) into the incompres-sible skin friction at an incompressible Reynoldsnumber. A more complete analysis for the correla-tion of the skin friction has been developed byHung, Bradshaw, and Coakley (HBC) [54]. Thecorrelation theories transform the experimental


compressible skin friction and Reynolds numbersinto incompressible values as follows:

C�f ¼ F cCf ,

F c skin-friction transformation function.

Re�x ¼ FxRex; F x surface distance

Reynolds no. transformation function.

Re�y ¼ FyRey; F y momentum thickness

Reynolds no. transformation function. ðB:1Þ

Therefore, if the theory is accurate, the transformedskin-friction C�f and Reynolds numbers Re�x and Re�yshould be the same as the incompressible values.

C�f � Cf ;i; Re�x � Rex;i; Re�y � Rey;i.

The compressible skin friction, Reynolds numbers,and transformation functions for the van Driest IItheory are

Cf ¼ 2tw=reu2e ; Rex ¼ reuex=me,

Rey ¼ reuey=me, (B.2)

F c ¼ rm=ðsin�1 aþ sin�1 bÞ2,

Fx ¼ F y=F c; F y ¼ me=mw, (B.3)

where

m ¼M2eðg� 1Þ=2; F ¼ Tw=Te,

A ¼ffiffiffiffiffiffiffiffiffiffiffiffirm=F

p; B ¼ ð1þ rm� F Þ=F ,

r ¼ Recovery factor ¼ 0:9,

a ¼ ð2A2 � BÞ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4A2 þ B2p

; b ¼ B=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4A2=B2

q.

The local incompressible skin friction is evaluatedfrom the Karman–Schoenherr relation, which isconsidered one of the most accurate fits to theincompressible experimental data:

Cf ;i ¼1

logð2ReyjincÞ½17:075 logð2ReyjincÞ þ 14:832�.

(B.4)

The accuracy of the theories relative to theexperimental Cf data is illustrated by plotting thetransformed skin friction F cCf as a function ofthe transformed Reynolds number F yRey. Thetransformed data should be in close agreement withthe Karman–Schoenherr curve plotted on thisfigure. A more sensitive illustration of the accuracyis to use percent error E of experimental skin

friction relative to theoretical value Cf ;i, which isobtained from

E ¼ ½ðF cCf=Cf ;iÞ � 1� � 100.

Then the percent error E is plotted versus F yRey.Compressible turbulent model predictions can betreated in the same manner. Correlation of theexperimental measured skin friction of compressibleflows ðMacho5Þ has proven to be reasonablyeffective with the Van Driest II approach. Thecorrelation approach of Huang et al. [54] givesresults of similar accuracy. Squire [52] estimates thatthe accuracy of the Van Driest II correlation iswithin �3% for the flat plate. Based on thesometimes erratic agreement between experimentsand the correlation, we feel that this error estimateis somewhat optimistic and should be increased to�5%.

B.2. Correlation of heat transfer

Reynolds analogy [194] is used to predict the wallheat flux and was developed for incompressibleflow. The compressible Reynolds analogy is writtenin the same form which gives

2St=Cf ¼ Raf ¼ Reynolds analogy factor, (B.5)

where St is the compressible Stanton number, Cf isthe compressible skin friction, and Raf is theReynolds analogy factor. The above is multipliedby F c which is defined in Eq. (B.3) and gives theincompressible Stanton number and the trans-formed Reynolds numbers

St� ¼ F cSt ¼ RafC�f =2,

Re�x ¼ F xRex; Re�y ¼ FyRey. (B.6)

The transformed skin friction C�f can be approxi-mated with the Karman–Schoenherr relation givenin Eq. (B.4). Assume the Reynolds analogy factor isknown ðRaf � 1Þ. Therefore, Eq. (B.6) can be usedto determine the compressible Stanton number andReynolds number or experimental compressibleStanton and Reynolds numbers can be transformedinto incompressible Stanton and Reynolds numbers.The transformed compressible experimental dataplotted as F cSt versus FyRey should be near to thecorrelation curve given by Eq. (B.6) with C�fdetermined from Eq. (B.4) evaluated with Re�y.Turbulent model predictions can be correlated inthe same manner.


Two forms of the Stanton number are used withdifferent locations for the evaluation of the en-thalpy. The edge and adiabatic Stanton numbers forreal and perfect gas models are

Ste ¼ qw=reueðHe � hwÞ ¼ qw=reuecpðT e � TwÞ,

Staw ¼ qw=reueðhaw � hwÞ ¼ qw=reuecpðTaw � TwÞ,

cp ¼ gR=ðg� 1Þ. ðB:7Þ

The adiabatic wall enthalpy for a real gas isobtained from the following first equation and thesecond equation gives the adiabatic wall tempera-ture for a perfect gas:

haw ¼ he þru2e2; Taw ¼ T eð1þ rmÞ,

m ¼u2e

2cpTe¼

g� 1

2

� �M2

e .

The first form of the Stanton number Ste becomesinfinite when Tw ¼ T e. The second form of theStanton number Staw becomes indeterminate whenthe heat flux is zero and Tw ¼ Taw. The Stantonnumbers can use the free stream conditions for thedensity and velocity rather than boundary-layeredge conditions. Fernholz and Finley [29] use thesecond form in Eq. (B.7), but they recommend thatthe heat flux be written in the form suggested byGreen (see in Fernholz and Finley [29, p. 50]), whichis CQ ¼ Nqw ¼ qw=reueHe. This relation avoids thedifficulties with having a indeterminate equation.Also the heat transfer coefficient hc is used and isdefined as qw ¼ hcðTaw � TwÞ.

The Reynolds analog factor is used to predict theheat transfer and is defined in two forms

Rafe ¼ SteðCfe=2Þ ¼ueqw

twðHe � hwÞ

¼ueqw

twcpðTe � Tw þmÞ,

Raf ¼ StawðCfe=2Þ ¼ueqw

twðhaw � hwÞ

¼ueqw

twcpðTaw � TwÞ. ðB:8Þ

The ratio of the Reynolds factors becomes

Rafe=Raf ¼ ðTaw � TwÞ=ðTe � Tw þmÞ.

As the Mach number approaches zero, m! 0,Taw! T e, and the ratio of the Reynoldsfactors Rafe=Raf ! 1. For low-speed flows, authorsusually use Rafe, while for compressible flow authorsuse Raf .

Many of the models for the Reynolds analogyfactor for incompressible flow are of the form

Raf ¼ ½PrTð1þ gffiffiffiffiffiffiffiffiffiffiffiCf=2

pÞ��1; g ¼ Cslð1� PrÞ.

(B.9)

The value of the constant Csl depends on the modelused for the velocity in the viscous sublayer. Thereare many models for the Reynolds analogy factorRaf , some are given below (note: unless otherwisestated, it is assumed that Pr ¼ 0:71 in these models).The various models described below are evaluatedwith PrT ¼ 0:90 and Rey ¼ 104 which givesCf ;i ¼ 2:6345� 10�3. The foregoing values of theparameters are used unless indicated otherwise.Some of the often referred to models or importantmodels are briefly described next. In addition,results are included from an experimental investiga-tion of the hypersonic Reynolds analogy factor.

Reynolds [194] assumed the laminar and turbulentPrandtl numbers are one, the flow is incompressible,and neglected the viscous sublayer, which givesRaf ¼ 1.

Prandtl [195] and Taylor [196] were concernedwith low-speed flows and used the Stanton numberSte and the Reynolds analogy factor as given in Eq.(B.9) with PrT ¼ 1

Prandtl: Csl ¼ 8:7; g ¼ 2:523; Raf ¼ 1:101,

Taylor: Csl ¼ 11:5; g ¼ 3:335; Raf ¼ 1:138.

This model is described by von Karman [197] andby Schlichting [198] (p. 709). The authors assumethe turbulent Prandtl number is one and the flow isincompressible. The velocity profile model includesthe viscous sublayer which is neglected in theReynolds model. Rubesin [199] gives Csl ¼ 11:5for the Prandtl–Taylor model. von Karman indi-cates that Prandtl uses Csl ¼ 8:7 while Schlichtingindicates Prandtl uses Csl ¼ 5:0.

Colburn [200] developed an empirical model forincompressible flow with PrT ¼ 1 which givesRaf ¼ Pr�2=3 ¼ 1:256.

von Karman [197] considers low-speed flows anduses the Stanton number Ste and the Reynoldsanalogy factor is given in Eq. (B.9) with g defined as

g ¼ 5fð1� PrÞ � ln½1� 5ð1� PrÞ=6�g ¼ 2:833,

PrT ¼ 1; Raf ¼ 1:115.

The turbulent Prandtl numbers is assumed to have avalue of one and the flow is incompressible. A threelayer turbulent boundary-layer model is used with aviscous sublayer, a log layer, and a buffer layer,


which is approximated with a linear variation. The g

function is more complex due to the three layermodel.

van Driest [51,201] models compressible flow anduses the Stanton number Staw and the Reynoldsanalogy factor is given in Eq. (B.9) with

g ¼ Csl B� ln 1�5

6B

� �� 0:5A

p2

6þ 1:5A

� � ¼ 1:574; Csl ¼ 5,

A ¼ 1� PrT ¼ 0:1; B ¼ 1� Pr=PrT ¼ 0:21111,

Raf ¼ 1:061.

The model assumes arbitrary constant turbulentPrandtl number. When PrT ¼ 1, the above equationis the same as the von Karman model given above.The turbulent Prandtl number PrT ¼ 0:86 and Pr ¼

0:72 in the paper of van Driest.Chi and Spalding [138] use the transform relations

in Eqs. (B.6) and (B.1). The van Driest transforma-tion functions given in Eq. (B.3) are evaluated withthe Spalding–Chi transformation theory [110]. Chiand Spalding developed the following skin frictionrelation which was then used in the compressibleReynolds analogy given in Eq. (B.6):

Cf ;i=2 ¼ ½3:8þ 5:77 logðReyiÞ��2. (B.10)

The Stanton number correlation curve becomes

Ste;i ¼ Raf ½3:8þ 5:77 logðRey;iÞ��2,

Cf=2 ¼ 1:60ðlnRexÞ0:772.

When Rex;i is specified, the incompressible skinfriction is obtained from the above Schultz–Grunowrelation. The authors use Cf ;i and Ste;i from theexperimental data of Reynolds, Kays, and Kline[202] to determine the Reynolds analogy factor thatmatches the Stanton number database with theStanton number correlation curve. The minimumerror is obtained when Raf ¼ 1:16. The relation usedto correlate the compressible Stanton number databecomes

St�e � F cSte ¼ 1:16½3:8þ 5:77 logðFyRey;iÞ��2,

Re�y � FyRey,

Fy ¼ ðTw=TeÞ�0:702ðTaw=TwÞ

0:772.

The transformation functions F x and F c are thesame as van Driest II relations given Eq. (B.3) and F y

is given above. The incompressible skin friction termCfi=2 can also be obtained from Eq. (B.4). The

Chi–Spalding incompressible skin friction Eq. (B.10)gives values that are 5.5–4.5% higher as Reyi goesfrom 103 to 106 than obtained with the Kar-man–Schoenherr skin-friction relation. Therefore, ifthe Karman–Schoenherr relation is used for Cfi=2,then the Reynolds factor must be changed to Raf ¼

1:22 in order to obtain agreement with the databaseof Reynolds et al. [202]. The database used in testingthe accuracy of the correlation of compressibleStanton numbers are from 11 experiments. With thisdatabase Chi and Spalding indicate that the Stantonnumber correlation function has been validated forRey from 500 to 104, Mach numbers up to 10,Taw=Tw from 0.5 to 2.7, and Rex from 105 to 108.

Keener and Polek [140] made direct measurementsof skin-friction and heat transfer on a smooth flatplate with a hypersonic turbulent boundary layer.The edge Mach number varied from 5.9 to 7.8 andTw=Taw was 0.32 and 0.50. The measured skinfriction and heat transfer are estimated to beaccurate within 5%. The authors are concernedwith compressible flow and use the Stanton numberStaw. For these experimental conditions, the authorsrecommend that Raf ¼ 1:0 with the maximum datascatter of 9% and with most of the data within 4%.This paper is further evaluation of the work of Cary[119] as seven of the data points in his database werepreliminary measurements reported in Hopkinset al. [86].

Cebeci and Bradshaw [203] indicate that a numberof investigators have developed the Reynoldsanalogy factor of the form of Eq. (B.9) with variousvalues of the coefficients. Cebeci and Bradshawsuggest the following values:

PrT ¼ 1=1:11 ¼ 0:901; g ¼ 1:20; Raf ¼ 1:160.

Summary of Reynolds analogy: Experimentsindicate that 0:9oRafo1:3, but may be close tounity for hypersonic flows. There is insufficientreliable experimental data to establish the Reynoldsanalogy factor for a wide range of flow conditions.Free stream turbulence has a significant impact onincreasing the Reynolds analogy factor while sur-face roughness decreases the value. Also there isconfusion on which Stanton number definition isbeing used when the Reynolds analogy factor Raf isbeing determined and compared with other results.

B.3. Mean temperature profiles

A review of the analysis used to obtain analyticalsolutions to the boundary-layer energy equation are


given in the report of Fernholz and Finley [29]. Theinitial relation developed by Crocco and Busemannassumes that the laminar and turbulent Prandtlnumbers are one ðPr ¼ PrT ¼ 1Þ. A solution to thetotal enthalpy energy equation is that the totalenthalpy is constant across the boundary layer.Since ~u ¼ 0 and k ¼ 0 at the wall and h ¼ cpT , thefollowing relations are obtained for the totalenthalpy and the temperature:

H ¼ hþ ~u2=2þ k ¼ Hw ¼ hw,

T ¼ Tw � a ~u� b ~u2 � gTk,

a ¼ 0; b ¼ 1=2cp; gT ¼ 1=cp. (B.11)

The second relation developed by Crocco andBusemann assumes a zero-pressure gradient with anisothermal wall and that the laminar and turbulentPrandtl numbers are one. The momentum andenergy equations are similar, which givesH ¼ C1 þ C2 ~u. With the wall and edge boundaryconditions applied, the energy equation in terms oftemperature becomes Eq. (B.11) with the coefficients

a ¼ ½Tw � Te � b ~u2e � gTke�= ~ue,

b ¼ 1=2cp; gT ¼ 1=cp. (B.12)

Van Driest extended the Crocco analysis forcompressible laminar boundary flows to turbulentflows with a variable Prandtl number. The develop-ment of this Van Driest [133] temperature relationbecomes very complex and not very useful. Themixed Prandtl number was initially introduced in thisarticle by Van Driest and is defined as

Prm ¼ cpmþ mTk þ kT

� �¼

mþ mTðm=PrÞ þ ðmT=PrTÞ

,

Pr ¼cpmk; PrT ¼

cpmTkT

.

Fernholz and Finley [29] have presented the workof Walz where the temperature equation is developedfor a constant Prandtl number which is restricted to0:7oPro1.

Huang et al. [54] (HBC) have developed thetemperature equation by neglecting the convectiveterms in the momentum and energy equations. Thereduced boundary-layer momentum equation withthe pressure gradient neglected can be integratedonce to obtain

ðmþ mTÞd ~u

dy¼ tw. (B.13)

The total enthalpy form of the reduced energyequation can be integrated once to obtain

�qþ ~utw þ ðmþ mTÞdk

dy¼ constant.

Since at the wall ~u ¼ 0 and k ¼ cky2 þ , theconstant in the above equation is qw and the energyequation becomes

q ¼ qw þ ~utw þ ðmþ mTÞdk

dy. (B.14)

The heat flux normal and near to the wall becomeswith the use of the momentum Eq. (B.13)

q ¼ �mþ mT

Prm

� �qh

qy¼ �

cp

Prmðmþ mTÞ

qT

qy

¼ �cptwPrm

qT

qu. ðB:15Þ

From Eq. (B.15) and the temperature relation in Eq.(B.11), the wall heat flux is qw ¼ acptw=Prm andwhen solved for qw=tw gives

qw=tw ¼ ½Tw � T e � b ~u2e � gTke�ðcp= ~uePrmÞ,

b ~u2e ¼ ~u2e=2cp ¼ mT e; m ¼ ~u2e=2cpT e ¼

g� 1

2

� �M2

e .

Using Eqs. (B.14) and (B.15), the differential formof the temperature equation becomes

dT ¼ �Prm

cp

qw

twþ ~u

� �d ~uþ dk

� �. (B.16)

Integration of this equation with Prm constant givesthe temperature Eq. (B.11) with the coefficients

a ¼ ðPrm=cpÞðqw=twÞ,

b ¼ Prm=2cp; gT ¼ Prm=cp. (B.17)

Eqs. (B.13) and (B.14) have been used by HBCwith m and k neglected to obtain the energyEq. (B.11) with coefficients given in Eq. (B.17)where Prm ¼ PrT. The energy equation developedby Fernholz and Finley [29] is essentially the sameas given above, except the turbulent Prandtl numberin the coefficients a and b is replaced with therecovery factor r. The above coefficient a can alsobe written as

a ¼ ½Tw � T e � b ~u2e � gTke�= ~ue.

For an adiabatic wall a ¼ 0 and the aboveequation with Eq. (B.17) gives the adiabatic walltemperature Taw ¼ T eð1þ PrTmþ gTkeÞ wherePrm ¼ PrT. Since the adiabatic wall temperature is


defined as Taw ¼ Teð1þ rmÞ, the recovery factorr ¼ PrT for this analysis. The total temperatureT t ¼ T þ ~u2=2cp is written in non-dimensional formas T� ¼ ðT t � TwÞ=ðT te � TwÞ and is plotted asfunction of ~u= ~ue. At the wall T� ¼ 0 and at theedge of the boundary layer T� ¼ 1. When PrT ¼ 1,the non-dimensional total temperature has alinear variation, T� ¼ ~u= ~ue. When the wall isadiabatic Tw ¼ Taw and the recovery factorequals the turbulent Prandtl number, the non-dimensional total temperature has quadratic varia-tion, T� ¼ ð ~u= ~ueÞ

2.The Van Driest form of the temperature or

density equation with the turbulent kinetic energyneglected is

T=Tw ¼ rw=r ¼ 1þ Bðu=ueÞ � A2ðu=ueÞ2, (B.18)

where

A2 ¼ bu2e=Tw ¼ PrTu2

e=2cpTw,

B ¼ �aue

Tw¼

Te

Tw

� �� 1þ A2 ¼ �

PrTueqw

cpTwtw.

In the 1951 paper of Van Driest [133], he assumedthat PrT ¼ r ¼ 1. In a 1955 paper, Van Driest [134]considered a variable Prandtl number, and theanalysis becomes more complex with the evaluationof a number of integral relations required.

Another form of the temperature or densityequation with the turbulent kinetic energy neglectedis

T=Tw ¼ rw=r ¼ 1� auþ � buþ2

, (B.19)

where

a ¼ aut=Tw ¼ PrTqwut=cpTwtw ¼ PrTBq

¼ ½ð1� T e=TwÞ=uþe � � buþe ¼ 2R2H,

b ¼ bu2t=Tw ¼ PrTu2

t=2cpTw

¼ PrTM2tðg� 1Þ=2 ¼ R2,

Bq ¼ qw=rwcpTwut,

Mt ¼ ut=aw; H ¼ ðqw=twÞ=ut.

B.4. Mean velocity profiles

In the inner region of the turbulent boundary layer,the total shear stress is approximately constant asgiven by Eq. (B.13). The Reynolds stress is writtenin terms of the eddy viscosity which is approximated

with the Prandtl mixing-length approach. The totalshear stress equation, eddy viscosity mT, and mixinglength lm become

rl2md ~u

dy

� �2

þ md ~u

du� tw ¼ 0,

mT ¼ rl2md ~u

dy

��; lm ¼ kyDf , (B.20)

where Van Driest damping function is used in theviscous sublayer and is

Df ¼ 1� e�yþ=Aþ ; Aþ ¼ 25:53.

Introducing inner variables

uþ ¼ ~u=ut; yþ ¼ rwyut=mw,

ut ¼ffiffiffiffiffiffiffiffiffiffiffiffiffitw=rw

p; lþm ¼ kyþDf ,

the total shear stress Eq. (B.20) becomes

aduþ

dyþ

� �2

þ bduþ

dyþ� 1 ¼ 0,

a ¼ lþmffiffiffiffiffiffiffiffiffiffiffir=rw

p; b ¼ m=mw. (B.21)

This equation is solved for the first derivative andthen can be integrated numerically to obtain thecompressible velocity across the inner layer

duþ

dyþ¼

2=b

1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ð2a=bÞ2

q ,

uþ ¼

Z yþ

0

fð2=bÞ=½1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ð2a=bÞ2

q�gdyþ. (B.22)

Also Eq. (B.21) can be solved with a velocitytransformation by introducing the Van Driesttransformed velocity uþc , which is defined as

duþcdyþ¼

ffiffiffiffiffiffiffiffiffiffiffir=rw

p duþ

dyþ,

Zduþc ¼

Z ffiffiffiffiffiffiffiffiffiffiffir=rw

pduþ ¼

Zduþ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiT=Tw

p,

(B.23)

Eq. (B.21) becomes

aduþcdyþ

� �þ b

duþcdyþ� 1 ¼ 0,

a ¼ lþm; b ¼ ðm=mwÞ=ffiffiffiffiffiffiffiffiffiffiffir=rw

p. (B.24)


This equation is solved for the first derivative andthe transformed velocity becomes

duþcdyþ¼

2=b

1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ð2a=bÞ2

q ,

uþc ¼

Z yþ

0

fð2=bÞ=½1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ð2a=bÞ2

q�gdyþ. (B.25)

The solution of this equation for uþc as a function ofyþ requires a numerical solution since b variesacross the inner region of the boundary layer.

In the logarithmic region b! 0, a ¼ kyþ, andsolving Eq. (B.21) for the first derivative gives

duþ

dyþ¼

ffiffiffiffiffiffiffiffiffiffiffirw=r

p=lþm; lþm ¼ kyþ. (B.26)

For incompressible flow (constant density caserw=r ¼ 1), the above becomesZ uþ

uþ0

du ¼

Z yþ

yþ0

dyþ

kyþ; uþ � uþ0 ¼

1

kln yþ �

1

kln yþ0 .

The incompressible solution becomes

uþinc ¼1

klnðyþÞ þ C; C ¼ uþ0 �

1

klnðyþ0 Þ. (B.27)

With the velocity uþ0 ¼ 0, the constant C ¼

�ð1=kÞ lnðyþ0 Þ, and the value of the coordinateyþ0 ¼ expð�kCÞ. Bradshaw suggest that the vonKarman constant k ¼ 0:41, and the constantC ¼ 5:20, which gives yþ0 ¼ 0:1186. The value ofthe constants are based on a database of incom-pressible zero-pressure gradient boundary-layerexperiments. The appropriate values of these con-stants are still being debated. Eq. (B.27) is only validwhen yþ is approximately 40 or larger and uþinc is 14or larger.

For compressible flow in the logarithmic region

b! 0, a ¼ kyþ, and the governing Eq. (B.25) andsolution become

duþcdyþ¼

1

kyþ; uþc ¼

1

kln yþ þ C,

C ¼ uþc0 �1

klnðyþ0 Þ; yþ0 ¼ exp½kðuþc0 � CÞ�. (B.28)

With uþc0 ¼ 0, the coordinate yþ0 ¼ expð�kCÞ, andC ¼ �ð1=kÞ lnðyþ0 Þ, which are the same as theincompressible values. In the logarithmic region,the transformed compressible velocity uþc becomesthe same as the incompressible velocity uþinc as givenin Eq. (B.27).

Also for the compressible flow in the logarithmic

region, the density ratio is obtained from thetemperature relation given by Eq. (B.19) with theturbulent kinetic energy neglected, then Eq. (B.26) issolved for the velocity uþ as a function of yþ. Thegoverning equation becomes

Z uþ

uþ0

duþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� auþ � buþ

2p ¼

Z yþ

yþ0

dyþ

kyþ.

The evaluation of the integrals gives

1ffiffiffib

p asin2buþ þ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ 4b

p !

�1ffiffiffib

p asin2buþ0 þ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ 4b

p !

¼1

klnðyþÞ �

1

klnðyþ0 Þ. ðB:29Þ

Eq. (B.29) can be written in the notation ofBradshaw with new variables R and H wherea ¼ 2R2H, b ¼ R2, and with the use of Eq. (B.28)the resulting equation is

1

Rasin

Ruþ þ RH

D

� �þ C1 ¼

1

klnðyþÞ þ C ¼ uþc ,

C1 ¼ uþc0 �1

Rasin

Ruþ0 þ RH

D

� �,

R ¼ ut

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrT=2cpTw

q,

H ¼ qw=twut; D ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðRHÞ2

q. (B.30)

From Eqs. (B.27) and (B.28) at yþ0 ¼ expð�kCÞ,uþ0 ¼ uþc0 ¼ 0. The compressible velocity transfor-mation of Van Driest is obtained from Eq. (B.30) as

uþc ¼1

Rasin

Ruþ þ RH

D

� �� asin

RH

D

� �� . (B.31)

The inverse of this equation has been given byBradshaw as

uþ ¼1

RsinðRuþc Þ �H½1� cosðRuþc Þ�. (B.32)

In the original notation, the Van Driest velocity

transformation is obtained from Eq. (B.29) wherethe temperature is given by Eq. (B.18),


a ¼ �But= ~ue, and b ¼ ðAut= ~ueÞ2, which gives

uþc ¼~uc

ut¼ð ~ue=utÞ

A

� asin2A2ð ~u= ~ueÞ � Bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B2 þ 4A2p

" #(

�asin�Bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B2 þ 4A2p

!)

¼1

klnðyþÞ þ C, ðB:33Þ

where A and B are defined after the Van Driesttemperature Eq. (B.18). The velocity ratio is writtenas follows in the Van Driest article

uþe ¼ ~ue=ut ¼ 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðCf=2ÞðTw=TeÞ

p.

Also in the Van Driest transformation the turbulentPrandtl number is set equal to one. In the Bradshawnotation A ¼ R ~ue=ut, B ¼ �2ARH, and Eq. (B.33)is the same as Eq. (B.31) when the turbulent Prandtl

number is one. A plot of uþc ¼ ~uc=ut versus yþ forexperimental data or numerical solutions shouldmatch Eq. (B.27), which is the incompressible log-law. Also the above Eq. (B.33) should approach Eq.(B.27) as the Mach number at the edge of theboundary layer becomes very small and thetemperature becomes uniform across the boundarylayer.

The Fernholz velocity transformation uses thePrandtl mixing-length concept with a recoveryfactor of r ¼ 0:896. The transformation evaluatesthe integration constant at the lower boundarywhere ð ~u= ~ueÞ � 0:5, uþc0 � 14:5, and yþ0 ¼

exp½kðuþc0 � CÞ� � 43. The von Karman constantk ¼ 0:40 and the constant C ¼ 5:10 in the Fernholzanalysis. With Fernholz notation the velocitytransformation is

~uc ¼~ue

basin

2b2ð ~u= ~ueÞ � a

d

� �, (B.34)

where the coefficients are

a ¼ ð1þ rmÞðT e=TwÞ � 1 ¼ �PrTueqw=cpTwtw,

b2¼ rmðTe=TwÞ ¼ r ~u2

e=2cpTw; d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ 4b2

p.

Eq. (B.34) can be obtained in the Bradshawnotation by starting with Eq. (B.30) which is

uþc ¼1

Rasin

Ruþ þ RH

D

� �þ C1 ¼

1

klnðyþÞ þ C,

(B.35)

where C1 is defined in Eq. (B.30) and C is defined inEq. (B.28). The coefficients in the Bradshaw form ofthe Fernholz velocity transformation are

R ¼ utb= ~ue ¼ ut

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir=2cpTw

q; H ¼ qw=twut,

RH ¼ �a=2b; D ¼ d=2b ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðRHÞ2

q.

Fernholz has shown for an adiabatic wall ðH ¼ 0Þ,that C1 is small and can be neglected in Eq. (B.35).The Van Driest transformation given in Eq. (B.31)is the same as the Fernholz Eq. (B.35) if r ¼ PrT andasin ðRH=DÞ is neglected in the Van Driesttransformation.

In the outer region of the turbulent boundary layer,the similarity of the velocity profiles is obtained withthe use of the velocity defect ð ~uce � ~ucÞ=ut. Thevelocity defect outside the viscous sublayer isapproximated as

ð ~uce � ~ucÞ=ut ¼ �1

klnðy=dÞ þ

Pk

wðy=dÞ. (B.36)

Fernholz and Finley [29] has shown the above canbe approximated as

UD ¼ ð ~uce � ~ucÞ=ut ¼ �4:70 lnðy=D�Þ � 6:74,

D� ¼ dZ 1

0

UD dðy=dÞ. ðB:37Þ

Fernholz and Finley [29] use this relation to assessthe accuracy of flat plate turbulent boundary layersin the outer region.

Huang et al. [54] have obtain the transformedvelocity from the wall to the edge of the boundaryby taking into account the viscous sublayer andby including a wake function. This procedure givesthe skin friction, velocity, and temperatureprofiles as a function of the Reynolds number. Ithas been developed as a 7 step procedure withiteration of the solution until converged. Theprocedure is described for the case when themomentum thickness Reynolds number is usedand the momentum thickness is specified. Thefollowing properties are specified:

y;re; cp;PrT; p;Tw; ~ue; mw.

Viscosity at the wall is determined from Sutherlandor Keyes viscosity law with the specified walltemperature. From the above specified properties,


the following parameters are calculated:

mw ¼ mðTwÞ; rw ¼ p=RTw; T e ¼ p=Rre,

me ¼ nðT eÞ.

The solution procedure is as follows:

1.
Guess the thickness ratio y=d and the wall friction
velocity ut ¼ffiffiffiffiffiffiffiffiffiffiffiffiffitw=rw

p; then determine the bound-

ary-layer thickness d ¼ y=ðy=dÞ. The thicknessratio for incompressible flow is estimated asy=d � 7=72, while for compressible flow therelation developed by Smits and Dussauge [204](see p. 194) can be used.

2.
Calculate the momentum thickness Reynoldsnumbers Rey ¼ re ~uey=me and Reyw ¼ re ~uey=mw ¼ ðme=mwÞRey. Then determine the wall func-tion PðReyÞ from Fig. 1a in the Huang et al.paper or use Cebeci–Smith correlation.
3.
Calculate the non-dimensional boundary-layerthickness dþ ¼ yþe ¼ rwutd=mw and wall densityrw ¼ p=RTw. Then determine the law of the wallprofile from the wall ðyþ ¼ 0Þ to the edge of theboundary layer ðyþ ¼ dþÞ by numerical evalua-tion of the following relation:
uþcb ¼

Z yþ

0

2dyþ

1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4ðlþÞ2

q ,

lþ ¼ kyþð1� e�yþ=AþÞ,

Aþ ¼ 25:53; k ¼ 0:41.

4.
Obtain compressible velocity at the edge of theboundary layer, Z ¼ yþ=dþ ¼ 1.
uþe ¼ ð1=RÞ sinðRuþceÞ �H½1� cosðRuþceÞ�

uþce ¼ uþcbðZÞ þ ðP=kÞwðZÞ; Z ¼ 1 wð1Þ ¼ 2.

5.
Update shear velocity ut ¼ ~ue= ~uþe and local skin
friction cf ¼ 2ðT e=TwÞðut= ~ueÞ2.

6.
Tabulate the transformed velocity, the compres-sible velocity, and the temperature across theboundary layer using the following relations:
uþc ¼ uþcbðZÞ þ ðP=kÞwðZÞ,

uþ ¼ ð1=RÞ sinðRuþc Þ �H½1� cosðRuþc Þ�,

T ¼ Twð1� auþ � R2uþ2Þ; a ¼ 2R2H,

R ¼ ut

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrT=2cpTw

q,

H ¼ ½ð1� Te=TwÞ=ðR2uþe Þ � uþ�=2.

Update the thickness ratio
7.
y=d ¼Z 1

0

ruþ

reuþe1�

uþ

uþe

� �dZ; r=re ¼ T e=T .

Steps 1–7 are repeated until the solution converges.

Appendix C. Turbulent sharp cone to flat plate

transformations

Theories for cone to flat plate Mangler transfor-mation are usually of the form

ðCf ÞCone=ðCf ÞFlatPlate ¼ ðStÞCone=ðStÞFlatPlate

¼ GðRe;Me;Tw=T eÞ.

The Mangler transformation parameter G forcompressible flow could be a function of theboundary-layer edge Reynolds number based on x

or y, Me, and Tw=Te. Below is a brief indication ofsome of the contributions to this issue.

Van Driest [55] has developed a simple rule fortransforming local flat plate skin friction and heattransfer to cones at zero angle of attack for fullyturbulent boundary layers (no transition fromlaminar flow) in supersonic/hypersonic flows. Hismethod is different than the standard approach. Theflat plate compressible skin friction is determinedfrom Van Driest II theory, which givesðCf ÞFlatPlate ¼ F ðRex;Me;Tw=T eÞ. Van Driest deter-mined that the cone compressible skin friction maybe calculated from ðCf ÞCone ¼ F ðRex=2;Me;Tw=T eÞ

where the flat plate skin friction relation isevaluated at one half the edge Reynolds number.The transformation parameter G ¼ F ðRex=2;Me;Tw=TeÞ=ðF ðRex;Me;Tw=TeÞÞ with G ¼ 1:14 atRex ¼ 105 and G ¼ 1:08 at Rex ¼ 108.

Seiff [205] has taken into account that theturbulent boundary layer begins down stream onthe flat plate and has determined the effective orvirtual origin of the turbulent boundary layer. It isassumed that the boundary layer is initially laminarand instantaneously transitions to turbulent flow atxtr, which must be specified. The Blasius relation forincompressible skin friction is transformed to acompressible skin-friction relation which is usedwith the Karman momentum–integral relation foraxisymmetric boundary-layer flow. The combinedrelation is a differential equation for the compres-sible skin friction which is integrated downstreamfrom the transition location to obtain the local coneskin friction ðCf ÞFlatPlate ¼ FConeðxtr;Me;Tw=T eÞ.


For the case of fully turbulent flow on the cone andflat plate, transformation parameter G ¼ 1:17 wherethe edge Reynolds number, edge Mach number, andwall temperature ration are the same for the coneand flat plate.

Reshotko and Tucker [206] use the compressibleturbulent boundary-layer integral equations formomentum thickness y and form factor H ¼ d�=y,which are transformed into incompressible formwith the Dorodnitsyn transformation. The com-pressible shear stress (skin friction) is required inthese equations and is determined from the Lud-wieg–Tillmann incompressible skin-friction relationwhich is transformed with the Eckert referenceenthalpy method (variables with subscript r). Thecompressible skin friction is of the form Cf ¼

2tw=reu2e ¼ K0ðrruey=mrÞ

m where m ¼ 0:268 and K0

is function of T e=T r and the form factor H. For theflat plate, the investigation gives the momentumthickness and skin friction as

yFlatPlate ¼ 0:0259KTx0:823,

ðCf ÞFlatPlate ¼ 0:086K fx�0:220.

For the cone, the investigation gives the momentumthickness and skin friction as

yCone ¼ 0:0135KTx0:823; ðCf ÞCone ¼ 0:102K fx�0:220,

where KT and K f are functions of Te=T r, T0=T e,and Mer0a0=m0. The authors obtain for themomentum thickness, skin friction, and heat trans-fer ratios

yCone=yFlatPlate ¼ 0:521,

G ¼ ðCf ÞCone=ðCf ÞFlatPlate ¼ StCone=StFlatPlate ¼ 1:192,

where the boundary-layer edge properties andstagnation conditions on the cone and flat plateare the same.

Bertram and Neal [207] investigated the influenceof the location of the virtual origin of the turbulentboundary layer and the relationship of the resultsobtained on cones to those obtained on flat plates.The theories usually assume the origin of theturbulent boundary layer is at the tip of the conewhile most experiments have laminar flow near thetip with transition occurring at xtr from the tip. Theauthors use the Mangler transformation to trans-form the cone boundary-layer equations into theflat plate boundary-layer equations. The details ofthe development of the theory are not presented;only the final results are given in an Appendix.

The authors assume the virtual origin is at thelocation where the peak shear stress or peak heatingoccurs. One transformation presented is applied todata obtained on cones to change the results to thevalues that would be obtained with the flowturbulent from the cone tip. The following Reynoldsnumbers are defined with the distance on the conefrom the virtual origin xv ¼ x� xtr, the distance onthe flat plate from the virtual origin xv ¼ x� xtr,and the parameter Reue ¼ reue=me.

Rexv ¼ Reuexv; Rexv ¼ Reuexv,

Rex ¼ Reuex; R�x ¼ Rextr=Rexv.

The ratio of the local skin friction on a truncatedcone (TC) to that on a pointed cone is

G� ¼ ðCfRexvÞTC=ðCfRexÞCone

¼ f1þ R�x � R�x½R�x=ð1þ R�xÞ�

n=ðn�1Þg�1=n.

The ratio of the local skin friction on a TC to thaton a flat plate is

ðCfRexvÞTC=ðCfRexvÞFlatPlate

¼ ½ð2n� 1Þ=ðn� 1Þ�1=nG�.

The authors suggest n ¼ 4 for turbulent cone flow.Tetervin [208] has extended the Mangler trans-

formation to compressible boundary-layer flows.The flat plate incompressible skin friction isobtained from the Ludwieg–Tillmann relationwhich is modified to compressible flow with theEckert reference enthalpy method. The compressi-ble wall shear stress becomes

2tw=rru2e ¼ kðReyÞ

�m; Reyr ¼ rruey=mr,

m ¼ 0:268,

k ¼ 0:246e�1:561Hi ðrr=reÞ1�mðmr=meÞ

m.

The transformed flat plate compressible skin fric-tion becomes

Cf ¼ kðReyeÞ�m; Reye ¼ reuey=me.

The ratio of the axisymmetric to flat plate skinfriction as given by Tetervin is

G ¼ ðCf ÞCone=ðCf ÞFlatPlate

¼ ½ðx=xÞðrw=LÞmþ1ðRex=RexÞ�m=ðmþ1Þ,

Rex ¼ reuex=me; Rex ¼ reuex=me.

The Mangler transform of the turbulent boundary-layer equations gives the distance x along theaxisymmetric body as a function of the distance x


along the flat plate as

x ¼Z x

0

ðrw=LÞmþ1 dx L ¼ Reference length.

For a cone ðrw=LÞ ¼ aðx=LÞ where a ¼ sin yc and

x=L ¼ ½amþ1=ðmþ 2Þ�ðx=LÞmþ2.

The above equation for a cone with x ¼ x becomes

G ¼ ðCf ÞCone=ðCf ÞFlatPlate ¼ ð2þmÞm=ðmþ1Þ ¼ 1:189.

If ðCf ÞCone ¼ ðCf ÞFlatPlate, then the above equationbecomes

Rex=Rex ¼ 1=ðmþ 2Þ ¼ 0:441.

White [56,209] has developed the Cone Rule withthe Karman momentum–integral equation foraxisymmetric, compressible flow ðj ¼ 1Þ which is

dydxþ j

yrw¼

Cf

2.

The compressible skin friction is approximated asCf ¼ KðReyÞ

�m¼ Ay�m where A ¼ Kðreue=meÞ

�m.For a flat plate ðj ¼ 0Þ the solution of the integralequation gives the momentum thickness and skinfriction as

yFlatPlate ¼ ½ð1þmÞAx=2�1=ð1þmÞ,

ðCf ÞFlatPlate ¼ A½ð1þmÞAx=2��m=ð1þmÞ.

For a cone ðj ¼ 1Þ, the solution of the integralequation gives the momentum thickness and skinfriction as

yCone ¼ð1þmÞAx2ð2þmÞ

� �1=ð1þmÞ

,

ðCf ÞCone ¼ Að1þmÞAx2ð2þmÞ

� ��m=ð1þmÞ

.

The G transformation becomes with the cone andflat plate locations the same (x ¼ x)

G ¼ ðCf ÞCone=ðCf ÞFlatPlate ¼ ð2þmÞm=ð1þmÞ.

The skin-friction relation becomes, with m ¼ 14,

ðCf ÞCone=ðCf ÞFlatPlate ¼ 1:176 and ðRexÞCone ¼

ðRexÞFlatPlate. If ðCf ÞCone ¼ ðCf ÞFlatPlate, then x ¼ ð2þmÞx and ðRexÞCone ¼ ð2þmÞðRexÞFlatPlate. The con-stant K in the skin-friction relation has beendetermined by Young [210] for incompressible flowand is given on page 158 in his book. In thedevelopment of the above relation, it is assumedthat the turbulent boundary layer begins at the tipof the cone and the leading edge of the flat plate.

Zoby et al. [211] have determined the power lawvelocity profile exponent as a function of themomentum thickness Reynolds number.

u=ue ¼ ðy=dÞ1=n; n ¼ 12:67� 6:6 logðReyÞ

þ 1:21½logðReyÞ�2.

The skin-friction relation parameter is obtainedfrom m ¼ 2=ðnþ 1Þ.

Seiler et al. [212] have used the Hantzsche andWendt transformations to first transform thecompressible boundary-layer equations on a cone(in spherical coordinates) to a new set of conetransformed governing equations. The compressibleboundary-layer equations on a flat plate aretransformed to a new set of flat plate transformedgoverning equations. The transformed governingequations for the cone and the flat plate are of thesame form. This approach needs further develop-ment to determine the Mangler transformationparameter G.

Zoby et al. [57,58,211]: In a NASA TechnicalNote Zoby and Sullivan predicted the heating rate(Stanton number) on axisymmetric sharp cones atzero angle of attack and compared the results to sixsupersonic flight experiments. The flat plate heatrate is obtained from the Colburn form of Reynoldsanalogy, which is expressed as

Ste ¼12CfePr�2=3; Pr ¼ 0:71.

The incompressible skin-friction for a flat plate isobtained from the Blasius or Schultz–Grunowrelations, which are of the form ðCfeÞFlatPlate ¼

function of the surface distance Reynolds numberðRexÞFlatPlate. The incompressible Reynolds numberis modified for compressible flow with the Eckertreference-enthalpy method. The cone inviscid flowconditions at the edge of the boundary layer areobtained from the Sims tables. The cone Reynoldsnumber ðRexÞCone is related to the flat plateReynolds number by the Van Driest relationðRexÞCone ¼ 2ðRexÞFlatPlate. The calculated heat ratesdiffer from the experimental heat rates by approxi-mately 20% or less. In a synoptic journal article,Zoby and Graves [58] compared a larger experi-mental turbulent heating database including windtunnel and flight experiments (no references fordatabase) with prediction using the transformationto an incompressible plane approach as investigatedby Peterson [105].

The compressible cone skin friction is trans-formed to compressible skin friction on a flat plate


with the relations

ðCfCÞCone ¼ GðCfCÞFlatPlate; ðReÞCone ¼ ðReÞFlatPlate,

where the Reynolds number is held constant. Thegeometric parameter G has been given by White’scone rule (p. 561) to have a value between 1.087 and1.176. Zoby et al. [211] have shown that G is afunction of the momentum thickness Reynoldsnumber, G ¼ 1:201 at Rey ¼ 104 and G ¼ 1:123 atRey ¼ 105.

From the Colburn Reynolds analogy given abovefor the compressible flow, the geometric transfor-mation for the Stanton number is ðStecÞCone ¼

GðStecÞFlatPlate. Since no value of G is specified inthis article, it is assumed that G ¼ 1. The compres-sible flat plate Stanton number and Reynoldsnumber are transformed to incompressible flat platevalues with the relations

Stei ¼ F cStec; ReLi ¼ FLReLc; L ¼ x or y.

The length scale in the Reynolds number is thedistance along the surface x in this article. Theprediction of the incompressible Stanton number asa function of incompressible Reynolds number isobtained from the Colburn Reynolds analogy givenabove where the incompressible skin friction isobtained from one of three relation investigated.With the Van Driest II transformation, Van Driestskin-friction relation, and with all of the experi-mental database used, the rms error of thetransformed experimental data relative to theincompressible prediction is between 17.7% and23%, depending on surface distance used in theReynolds number.

In the paper by Zoby et al. [211], the skin frictionis evaluated from Cf=2 ¼ C1ðReyÞ

�m.Hopkins et al. [86,139]: The investigation of

Hopkins and coworkers on the correlation of skinfriction and heat transfer for zero-pressure gradientflows at hypersonic Mach numbers uses mainly flatplate data but includes cones and hollow cylinderflows. This work has already been discussed in theflat plate case. The initial work was documented in aNASA technical note [86] and the completeinvestigation in a journal article [139]. The conedatabase is from the experiments of Mateer (seecone experimental database). The correlation ofheat transfer as a function of wall temperature ratiofor Mach 4.9–7.4 includes all three geometriesand shows no influence of geometries on thecorrelation. There is no indication that the conedata has been transformed to flat plate data

(geometry transformation is discussed above in theZoby et al. section). This investigation does notresolve the appropriate geometry transformationfor cones and the Reynolds analogy factor for conesand flat plates.

Holden [48] has correlated his experimental heattransfer data into incompressible form where theexperimental Stanton number St� is plotted as afunction of the transformed Reynolds number Re�x.The best documentation of this work is given inHolden [44]. Holden uses the Bertram and Nealcone to flat plate transformation to transform theexperimental data to incompressible flat plateStanton number. It appears that the Reynoldsanalogy factor has been set to one. In the Bertramand Neal transformation theory the virtual origin ofthe turbulent boundary layer must be specified andno information is given on this issue. The experi-mental data is correlated into reasonable agreementwith the incompressible curve, but there is signifi-cant scatter of the data about the curve.

Appendix D. Perfect gas air model and molecular

transport properties for hypersonic flows

Air is a multi-component gas mixture of nitrogen,oxygen and other components. The US StandardAtmosphere [213] has the following properties at sealevel:

p0 ¼ 101325:0N=m2; T0 ¼ 288:15K,

r0 ¼ 1:2250 kg=m3. ðD:1Þ

The properties of a gas mixture can be written interms of the mass fraction cs ¼ rs=r of the variousspecies. The molecular weight of the mixture is thenobtained from

Mw ¼ 1XNs

s¼1

cs=Ms

!,¼ 28:9644 kg=ðkgmolÞ.

(D.2)

The gas constant is determined from the relation

R ¼ Ru=Mw ¼ 287:0583J

kgK,

Ru ¼ 8314:472J

kgmolK. ðD:3Þ

The molecular weight and gas constant of air at sealevel are given in Eqs. (D.2) and (D.3)

In NACA Report 1135 [214], several terms areused to define types of perfect gases where there isno chemically activity. For a thermally perfect gas,


the equation of state is given as

p ¼ rRT ¼ ðg� 1Þre; g ¼ cp=cv. (D.4)

At sufficiently low gas temperatures, where there isno significant vibrational excitation, the internalenergy of a mixture of diatomic molecules, whichis air without the trace species included, becomes

e ¼5

2RT ¼ cvT ; cv ¼

5

2R. (D.5)

The specific heat at constant pressure cp and thespecific heat at constant volume cv are constant andbecome for air without the trace species

cv ¼R

ðg� 1Þ¼

5

2R ¼ 717:646

J

kgK,

cp ¼gR

ðg� 1Þ¼

7

2R ¼ 1004:704

J

kgK, ðD:6Þ

where

g ¼7

5. (D.7)

A calorically perfect gas is defined as a gas withconstant specific heats. For a perfect gas model thespecific heats of the gas are constant and theequation of state is given by Eq. (D.4). A perfect

gas model is a thermally and calorically perfect gas.At standard temperatures, Sutherland’s law can

be used for the absolute molecular viscosity of air,and is given by

m ¼ 1:458� 10�6T3=2=ðT þ 110:4Þ

units are kg=m=s, ðD:8Þ

where T is given in Kelvin. For air at lowertemperatures (say below 100K) and for nitrogen,Keyes model [215] for viscosity should be used

m ¼ a0 � 10�6ffiffiffiffiTp

=ð1þ a1T1=TÞ; T1 ¼ 10�a2=T .

Air: a0 ¼ 1:488; a1 ¼ 122:1; a2 ¼ 5:0.

Nitrogen: a0 ¼ 1:418; a1 ¼ 116:4; a2 ¼ 5:0: ðD:9Þ

For a perfect gas, the thermal conductivity can then bedetermined from the Prandtl number and the specificheat at constant pressure from cp ¼ gR=ðg� 1Þ wherefor air

R ¼ 287:0583J

kgK(D.10)

and g ¼ 1:4. For diatomic nitrogen with molecularweight 28.01344 and g ¼ 1:4, the specific gas constantR and the specific heats can be determined from thepreceding equations.

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Documents

Review and assessment of turbulence models for hypersonic ﬂows