AIAA 2002-3346

THE CURRENT STATUS OF UNSTEADY CFD APPROACHES FOR AERODYNAMIC

FLOW CONTROL

Mark H. Carpenter � Bart A. Singer � Nail Yamaleev y Veer N. Vatsa � Sally A. Viken z Harold L. Atkins �

1 Abstract

An overview of the current status of time depen-dent algorithms is presented. Special attention isgiven to algorithms used to predict uid actuator ows, as well as other active and passive ow controldevices. Capabilities for the next decade are pre-dicted, and principal impediments to the progress oftime-dependent algorithms are identi�ed.

2 Introduction

Continuously expanding computer capabilities al-low more attention to be devoted to the simulationof unsteady ows. At the turn of the millennium,practitioners routinely compute complex 3-D steady ows involving 106 � 107 grid points, and 2-D un-steady ows involving 105� 106 points. These featsare performed while carrying �ve or more variablesper node! If Moore's law persists (a �xed cost dou-bling of computer resources every 1.5 years) the nextdecade will provide practitioners with the resourcesto routinely simulate 3-D unsteady ows on 106 gridpoints. This computer capability will enable theburgeoning �eld of aerodynamic ow control (AFC),which is often time-dependent.Active ow control o�ers the aerospace commu-

nity the opportunity to expand the ight envelopethrough the use of steady suction/blowing, zero netmass synthetic jet actuators, or pulsed jets. These ow control devices exhibit promising ow controlcapabilities including separation control, thrust vec-toring, mixing enhancement, noise control, and vir-tual shape change. Bene�ts of ow control in-clude reduction in part-card count, empty weight,

�Senior Research Scientists, Computational Methods and

Simulation Branch, NASA Langley Research Center, Hamp-

ton, VA 23681-0001yNRC Resident Research Associate, NASA Langley Re-

search Center, Hampton, VA 23681-0001zSenior Research Scientist, Flow Physics and Control

Branch, NASA Langley Research Center, Hampton, VA

23681-0001

Copyright c 2002 by the American Institute of Aeronau-

tics and Astronautics, Inc. No copyright is asserted in the

United States under Title 17, U.S. Code. The U.S. Govern-

ment has a royalty-free license to exercise all rights under the

copyright claimed herein for government purposes. All other

rights are reserved by the copyright owner.

manufacturing costs, operating cost, fuel burn, andnoise. A number of active ow control concepts havebeen tested in the laboratory and ight. Examplesinclude leading-edge suction for transition delay,93

zero net mass separation control 123; 124; 125; 126; 127

and thrust vectoring uidic injection.111 Computa-tional studies have demonstrated that Reynolds av-eraged Navier-Stokes (RANS) methodologies pro-vide qualitative insight into active ow control ap-plications. However, quantitative agreement is lack-ing between the computational and experimental re-sults. To get from the bench-top to real applicationsof ow control, reliable computational uid dynam-ics (CFD) design tools must be developed and val-idated with the experimental and ight databases.An extensive amount of research is still needed todevelop a production-type tool for active ow con-trol applications for the design engineer.

A critical assessment of the current capabilities oftime-dependent CFD, and identi�cation of impedi-ments that still exist is timely. We focus on identi-fying the critical areas (algorithmic and modeling)that possess notable leverage to the success of 3-DAFC computations.

The review of this material will be presented withthe following strategy. Each section will begin witha broad overview of current state of the art in that�eld, followed by a description of general bottle-necks, and speci�c impediments for time-dependentAFC computations. Finally, each section will con-clude with a brief summary of NASA Langley Re-search Center's (LaRC) present research aimed atalleviating the bottlenecks, recognizing that someimpediments are not being addressed due to limitedresources. The �elds of CFD and turbulence model-ing are nearly boundless! To limit the scope of thereview, only those methodologies which have shownpromise in AFC simulations will be addressed.

The paper focuses on the general areas of algo-rithmic issues and turbulence models, and on thespeci�c area of uid actuators. The paper is orga-nized as follows. Section 3 describes discretizationsin space and time. The section begins with a broaddiscussion of the advantages of high-order schemes,followed by speci�c discussions of temporal and spa-

1

tial discretizations. Section 4 describes algorithmicconsiderations related to convergence acceleration.Section 5 describes the current state of turbulencemodeling for time-dependent ows. Section 6 de-scribes speci�c considerations for e�ective actuatorboundary conditions. Section 7 presents conclusions.

3 Discretizations: Time and Space

3.1 Why High-Order?

For reasons of e�ciency, high-order schemes havelong been advocated for use in time-dependent prob-lems. In 1902, Kutta recognized the virtues of inte-grating ordinary di�erential equations (ODEs) withhigh-order schemes. The following simple exampleillustrates this point. Local error ei committed dur-ing one step of a temporal integration is describedby the formula

keik � (�t)p+1i (1)

where p is the temporal order of the integration for-mula. Global error at time Tf is estimated by sum-ming all local errors after transporting each to the�nal time Tf . Estimates of global error, though notsharp, are generally expressed in the form 55

kEk � (�t)pmaxC�exp[L(Tf � To)]� 1

�(2)

where To is the initial time, and C and L are problemdependent constants related to solution smoothness,etc. In equation (2) the time-step satis�es �t < 1,and a given error tolerance can be achieved by in-creasing the order p while increasing the time-step.The resulting algorithm is more e�cient if any addi-tional work accrued at each large time-step, is morethan compensated by a reduced number of steps.Work, however, increases with the order p. Near op-timal values of p in the range 3 � p � 5 existfor countless sti� and non-sti� model problems. (seexIV.10 in Hairer and Wanner 56).While it is generally recognized that high-order

temporal schemes result in greater e�ciency fortime-dependent problems, it is less well appreciatedthat higher order spatial schemes additionally con-tribute to time-dependent e�ciency! The virtues ofhigh-order spatial schemes were �rst recognized andquanti�ed by Kreiss and Oliger.83 To illustrate thisadvantage we provide an overview of the original ar-gument presented in Kreiss and Oliger.83

Consider the wave equation and initial data

Ut + aUx = 0; U (x; 0) = ei k x (3)

on the space and time intervals 0 � x � 2� and0 � t � Tf , with the exact solution:

U (x; t) = ei k (x� a t) (4)

Assume a uniform grid xj = j�x with �x = 2�=N .A general nl + nr + 1 point spatial discretization is

Uxjj =1

�x

nrXl=�nl

�lU (xj+l); j = 0; N � 1 (5)

Substituting equation (5) into equation (3) and solv-

ing the system of ODEs (~Ut +M~U = 0) in Fourierspace yields the modal solution

U (x; t) = ei k (x� a(k) t) (6)

where a(k) is the wave speed of the semi-discreteproblem and is related to the Fourier image of thespatial operator. For example, the second- andfourth-order central di�erence waves speeds are

a(k)2 = 2asin(k�x)

2 k�x(7)

a(k)4 = 2a8 sin(k�x) � sin(2k�x)

12 k�x(8)

For real a(k), the di�erence (error) between the ex-act and numerical solutions (eqs. 4 and 6) obtainedusing trigonometric relations is �(k) = 2sin(k[a �a(k)]t=2). Expanding the error in small phase anglesyields the simple expression for the phase error:

�(k) = j k [a � a(k)] t j (9)

Taylor series arguments produce the leading orderterm for the di�erence in wave speeds

a � a(k) ' a�p(k�x)p

(10)

with �p a scheme and order dependent constant.Substituting equation (10) into (9), de�ning thepoints per wavelength as P = N=k, and substi-tuting t = Tf yields

�(k) ' [�pk a Tf (2�

P)p

] (11)

The semi-discrete solution error accumulates lin-early with time Tf , and is a strong function of thespatial truncation error. Rearranging equation (11)in terms of a maximumacceptable target error �T (k)yields the expression:

Pp � 2 � p

s�pkaTf�T (k)

(12)

The grid points per wavelength necessary to achievethe speci�ed target error �T (k) increases for prob-lem size (2�). The other dependencies rapidly de-crease as the order of the spatial approximation is in-creased, motivating high-order spatial formulations

2

in time-dependent problems. The cost of the com-putations increase with increasing order of accuracy,and a global minimum is reached for a �nite valueof p. Kreiss and Oliger 83 suggest 4 � p � 6 forproblems of practical interest. Note that the optimalorder for spatial and temporal operators is similar.

In summary, error accumulates linearly in time.The global error at Tf is the sum of local errorsthat accumulate from each time-step in the integra-tion. The local error at each time-step is the sumof three components: the temporal truncation error,the spatial truncation error, and the algebraic er-ror. A simulation requiring many time-steps to reachTf requires extremely small local errors. High-ordermethods are the most e�cient means of achievingthese small local error tolerances.

An example will help to clarify this point. Con-sider a steady-state problem requiring lift to an engi-neering accuracy of three signi�cant digits. A secondorder method could easily achieve this accuracy re-quirement. Now consider a similar time-dependentproblem requiring lift (at the speci�ed Tf ) to an en-gineering accuracy of three signi�cant digits. Fur-ther assume that 100 time-steps are required to inte-grate from T0 to Tf . The local error (temporal, spa-tial, algebraic) at each time-step must be less than10�5 to achieve the desired error tolerance of threesigni�cant digits. The constraint on spatial error(10�5) in the time-dependent problem is much moresevere than that required for the steady-state prob-lem (10�3). This example demonstrates the com-pelling need for high-order spatial operators espe-cially for time-dependent simulations.

3.2 Temporal Algorithms

3.2.1 Overview

The application of method of lines (MOL) to time-dependent partial di�erential equations (PDEs) re-sults in an initial value problem (IVP) for a systemof ODEs. Dozens of excellent texts with detailed de-scriptions of multi step, multi stage, and linear multistep methods have been written on the numericalintegration of ODEs. 26; 40; 55; 56; 86; 128 After morethan 100 years of theoretical development, the math-ematical framework for solving ODEs is relativelymature. In a general context, it is doubtful thatdramatic (factors of 10) e�ciency improvements cancome from new methods.

The potential for dramatic e�ciency improve-ments is greater in the �eld of time-dependent CFD,where current methodologies are surprisingly primi-tive. This schism between tidy mathematical theory,and rough CFD practices is not without good reason.Fluids practitioners are preoccupied with more ur-

gent issues such as algebraic solvers, dimensionalityissues, discontinuities, nonlinear instability, turbu-lence models, grid generation, etc. Nevertheless, thecurrent objective is to identify mature technologiesin the ODE literature that could have an immediateimpact in CFD.The hallmark of current ODE software is the abil-

ity to perform automated integration for sti� ODEs.The �rst widely available multi step integration li-brary was that developed by Gear,51 later modi-�ed and improved by Hindmarsh,60 resulting in theLSODE family of codes. Other variants have prolif-erated over the past two decades to account for thede�ciencies of the original approaches (see VODE61).Automated integration begins with the user spec-

ifying 1) the system of ODEs, 2) the system Ja-cobian, and 3) the desired solution error tolerance.The software then automatically integrates the equa-tions, using the most e�cient numerical method cho-sen from a variety of candidate methods (second-order backward di�erentiation formulae (BDF2) isfrequently used). A reliable solution error estima-tor allows variable time-stepping. The time-step isadjusted to match the desired error tolerance. Theresultant nonlinear system of algebraic equations issolved at each time-step using a Newton or modi�edNewton method. Direct matrix inversions are usedwithin the Newton methods whenever possible. Thealgebraic error is reduced to a predetermined level,a constant multiple below the speci�ed error toler-ance. The Jacobian used in the nonlinear iterationis periodically reevaluated and stored based on theconvergence rate of the iteration.In contrast, the second-order accurate multi step

BDF2 method is extensively used in the CFD com-munity. System dimensionality prohibits the use ofdirect inverse methods useful for Newton or modi-�ed Newton methods. Iterative techniques such asNewton-Krylov methods are usually not as e�cientas other more highly tuned methods (multigrid orcombinations of methods). Error estimation or vari-able time-stepping mode is not perceived as neces-sary (correct or otherwise).Three technologies presently used in the ODE

community could have an immediate impact onCFD:

� high-order integrators : p � 3

� error estimation/variable time-stepping

� iteration termination strategies

To support this assertion, a brief summary of eacharea is presented.

3

3.2.2 High-Order IntegratorsAll general purpose solvers must integrate equa-

tions of considerable sti�ness. We begin with abroad overview of sti�ness, and identify the mathe-matical properties that enable a temporal integratorto e�ciently integrate sti� equations.Consider the integration of the system of ordinary

di�erential equations represented by the equation

dU

dt= S(U(t))

In the present case, the vector S results from thesemi-discretization (spatial and source terms) of theequations of uid mechanics plus a suitable turbu-lence model. The integrator must integrate any Swith which it is provided. Numerical di�culties of-ten arise when the Jacobian of S, J = @S=@U, haslarge eigenvalues. A useful de�nition for sti�nessstates that a problem is sti� when the largest eigen-value of the Jacobian J (scaled by the time-step)jjz = �(�t)jj contained in the complex left-half-plane (LHP) becomes much greater than unity. Theresulting sti�ness is then governed by both the Jaco-bian and the chosen time-step. Ideally, the time-stepis selected solely based on error considerations anda good method simply executes this step-size in astable and robust fashion. Time integration meth-ods that do not amplify any LHP scaled eigenvaluesare called A-stable. While A-stability is generallynecessary, it is often not su�cient. We further de-mand that all eigenvalues jjz !�1jj be completelydamped. The combination of these two properties,A-stability and damping of �1 eigenvalues, is de-�ned as L-stability. General purpose solvers invari-ably rely on L-stable methods (and the partially sta-ble L(�) methods with suitable error controllers) tosuppress temporal numerical instability and facili-tate convergence of the nonlinear equation solver.Popular implicit ODE integration methods are

generally either distinctly multi step or multistagemethods. Each has di�erent strengths and weak-nesses. Implicit multi step BDF methods computeeach U-vector update to design order of accuracyusing one nonlinear equation solve per step. Unfor-tunately, they are not A-stable above second-order.Additionally, they are not self-starting and have di-minished stability properties when used in a vari-able step-size context. (Stability proofs are formu-lated assuming constant time-steps. Variable time-step cases may not be stable.) Practical experienceindicates that large-scale engineering computationsare seldom stable if run with BDF4.102 The BDF3scheme, with its smaller regions of instability, isoften stable but diverges for certain problems and

some spatial operators. Thus, a conservative prac-titioner uses the BDF2 scheme exclusively for largescale computations due to its L-stability rather thanL(�)-stability.Practical Runge-Kutta (RK) methods such as ex-

plicit, singly diagonal implicit, Runge-Kutta (ES-DIRK) methods can be made arbitrarily high-orderwhile retaining L-stability but possess intermedi-ate U-vectors with a reduced order of accuracy andlesser stability. This reduced stage order may giverise to order reduction phenomena in the presenceof substantial sti�ness. ESDIRK schemes with sstages require (s-1) nonlinear equation solves perstep. Achieving progressively higher stage-ordermethods is possible with fully implicit methods suchas the Radau IIA family. The cost of fully implicitmethods greatly exceeds that of ESDIRK methodsin the current context. Much less experience existswith implicit RK methods than BDF methods in thecomputation of large-scale engineering ows.The general formula for a k-step, order-k, BDF

scheme can be written as

U(n+k) = �k�1Xi=0

�iU(n+i) + (�t)�kS

(n+k)(13)

where n is the time-step index. At each time-stepthe BDF involve the storage of k+1 levels of the so-lution vector U, and the implicit solution of one setof nonlinear equations. Stability diagrams for thesemethods may be found in Hairer and Wanner.56 Atorder k > 2 an unstable zone for scaled eigenvaluesin the complex LHP exists. At orders f1; 2; 3; 4;5;6gthe methods are L(�)-stable where � is given byf90o; 90o; 86:03o; 73:35o; 51:84o; 17:84og. (The dif-fusion terms yield negative real eigenvalues. Theconvective terms discretized with nondissipative op-erators yield eigenvalues clustered on the imaginaryaxis � = 90o. Numerical dissipation and boundaryconditions displace both sets of eigenvalues into theLHP. Spatial operators with high levels of dissipa-tion are more likely to be stable with BDF3.)ESDIRK methods 78; 85 are implemented as

Uk = Un + (�t)kX

j=1

akjS (Uj) ; k = 1; s

Un+1 = Un + (�t)sX

j=1

bjS (Uj) (14)

Un+1 = Un + (�t)sX

j=1

bjS (Uj)

where s is the number of stages, akj are the stage

weights, bi and bj are the main and embedded

4

scheme weights. The vectors U and U are thepth-order and (p� 1)th-order solutions at time leveln + 1. The vector U is used solely for estimat-ing error and is calculated at little extra cost. ES-DIRK schemes di�er from traditional SDIRK meth-ods (see xIV.6 in Hairer and Wanner 56) by thechoice a11 = 0, which permits stage-order two meth-ods. The sti�y accurate assumption (asj = bj)makes the new solution Un+1 independent of anyexplicit process within the integration step.

Many methods that combine multi step and multistage schemes, generally referred to as linear multistep (LMS) methods, have been proposed in attemptto overcome the problems of each. Surprisingly, theresults obtained with LMS methods (with a few ex-ceptions) have been disappointing compared with ei-ther multi step or multi stage schemes. Notable LMSmethods include the works of Cash.34; 35 To increasethe stability of the BDFmethods, Cash proposed theextended backward di�erentiation formulae (EBDF)and the modi�ed EBDF (MEBDF) schemes. TheMEBDF schemes involve three stages to advance thesolution one time-step. The �rst two stages are builtfrom existing (p � 1)th-order BDF formulas, whilethe last stage combines the two previous BDF resultsinto a pth-order solution. Note that the second BDFstage predicts a \super-future" point one time-stepbeyond the target time level, and substantially con-tributes to the A-stability of the method. At ordersf1; 2; 3; 4; 5; 6g the methods are L(�)-stable where �is given by f90o; 90o; 90o; 90o; 88:36o; 83:07og. Themachinery involved with implementing the MEBDFalgorithm is nearly identical to that involved in theBDF formulations. MEBDF schemes have the ad-ditional advantage that very accurate solution dataare available on the �rst and third stages, based onprevious information. This information can be usedto provide the starting guess for the nonlinear it-eration, and to establish time-step error estimates.The second stage typically uses the trivial guess asthe starting point for the nonlinear iteration and noerror estimate is made.

Butcher 28 proposed a class of LMS methods forsti� di�erential equations. The new methods com-bine the properties of A- and L-stability, and are rea-sonably simple to implement. They have a stabilityregion that is identical to that of a RK method, buthave high stage order. Uniformly high stage ordereliminates the possibility of order reduction. Thenew methods were identi�ed by focusing speci�callyon a diagonally implicit subclass of schemes referredto as DIMSIM.27

3.2.3 Error EstimationTemporal error management in the CFD com-

munity is presently accomplished by systematicallyhalving the time-step until the solution is indepen-dent of further reduction. This strategy, while ac-complishing the desired goal, can be streamlined byusing an error estimator at each time-step and ad-justing each time-step to attain the desired error.Error estimation is accomplished by comparing

two solutions of di�erent orders (Un+1 and Un+1)at the same time-step. For reasons of e�ciency, theauxiliary solution Un+1 should be available at littleadditional cost. For example, in ESDIRK schemes(see eqn. 15), as well as MEBDF 35 schemes, bothUn+1 and Un+1 are constructed from available data.The di�erence kUn+1�Un+1k is proportional to thetruncation error of the lower order formula Un+1.The estimate predicts the magnitude of the error inthe solution, and gives insight into its overall qual-ity. Frequently, linear and nonlinear instability canbe predicted by the estimator well before the simu-lation diverges.Figure (1) shows the error estimate (MEBDF4)

for various �t. The test problem is for periodicshedding from the turbulent circular cylinder. Theestimates are accurate to the correct order basedon grid-converged data. The error estimate predictsthat certain portions of the shedding cycle are moredi�cult to resolve in time. Variable time-steppingcould easily increase the e�ciency of the calculationby adjusting the time-step so that the same amountof error is produced at each time-step.Variable time-stepping can introduce instability

into some temporal integrators. The stability func-tion of multi step schemes (BDF, MEBDF, LMS)is derived assuming constant time-steps. Large de-partures from constant step-size can lead to solutioninstability (although a good error estimator shouldforewarn this possibility). Conversely, the stabilityof multi stage schemes (ESDIRK) is independent ofvariable time-steps, because they are self-starting.The time-steps in a variable time-step formula-

tion, are chosen by a controller. A simple explicitcontroller is (see xIV.2 in Hairer and Wanner 56)

(�t)n+1 = (�t)n� Tol

kUn+1 � Un+1k

�p(15)

Similar yet more elaborate controllers exist for im-plicit formulations. The stability characteristics ofa controller can be tuned/optimized in conjunc-tion with the integration technique it is control-ling. Together, they should meet the design ob-jective and not introduce instability into the inte-gration. Note that in �gure (1) the predicted error

5

at �ner tolerances has a high frequency componentthat the controller must suppress. (See Kennedyand Carpenter78 for details on the feedback errorcontrollers used with the ESDIRK scheme.)

3.2.4 Termination Strategy

An accurate error estimate can also be used toautomate the termination strategy of the nonlineariteration. Two competing components of temporalerror are the truncation and algebraic errors. Trun-cation error is related to �t and the order of ac-curacy p, while algebraic error is the residual errorgenerated each time-step by approximately solvingthe algebraic system. The local temporal error isthe sum of the two components. To see full designorder from the temporal scheme, the algebraic er-ror must be driven below the truncation error ateach time-step. This requires an accurate measureof truncation error, and must be provided by theerror estimator.

The iteration termination strategy is complicated.Our experience indicates that design-order temporalconvergence is achieved by maintaining a toleranceratio of 10�2 � T � 10�1. Here T is de�ned as theratio of nonlinear algebraic error to temporal inte-gration error at each time-step (or stage). Algebraicerror for the nonlinear iteration is based on the L1norm of the density residual. Choosing the time-stepbased on accuracy considerations alone may not bethe most e�cient strategy for a temporal calcula-tion. Decreasing the time-step can possibly greatlyincrease the convergence rate of the nonlinear alge-braic system, thus increase e�ciency. Gustafssonand S�oderlind 54 devised optimal criteria for adjust-ing �t. They assumed that either �xed point itera-tions, or modi�ed Newton iterations is used for solv-ing the algebraic system. The time-step is adjustedso that the iteration convergence rate approximatelyequals the optimal value. Because typical CFD alge-braic solvers fall somewhere between �xed point andmodi�ed Newton iterations, additional work to re-�ne these estimates is needed in the context of CFDtime-dependent solvers.

3.2.5 Bottlenecks

Algebraic solvers that exhibit poor convergencebehavior are an impediment for high-order schemes.Huge time-steps are needed to utilize the favor-able aspects of high-order formulations. Algebraicsolvers are needed that exhibits time-step indepen-dent convergence characteristics. If the convergencerate varies considerably with the time-step, then itmay be more e�cient to use a low-order schemewith small time-steps. Thus, high-order temporalschemes need fast and robust algebraic solvers. Tur-

Time

Log

10

(Pre

dic

ted

Err

or)

10 20 30-9

-8

-7

-6

-5

Dt = 0.1

Dt=0.2

Dt=0.4

Dt=0.75

Figure 1. Time-dependent variation of predictedL2 density error as calculated with the MEBDF4scheme. The test case is turbulent ow around acircular cylinder, at Re = 104, and Ma = 0.25.

bulent cases that have little or no convergence (oneorder of magnitude) present a second obstacle. Asmall number of cases are extremely di�cult to con-verge, yielding dubious solutions at best. Neverthe-less, solutions are still sought. It may be di�cultto keep high-order formulations stable under thesecircumstances.

A perceptual impediment is the implementationof error estimation technology. A change in atti-tude about the nature of temporal error and theimportance of its control is necessary. In spite ofthe perceived adequacy of existing temporal errorpractices, the CFD community should immediatelyadopt the practice of reporting a time-step error es-timate as a necessary requirement of a high �delitytime-dependent simulation. Ideally, the estimateshould include the component of primary interestin the simulation. For example, if lift and drag arethe object of the study, then the estimate should in-clude stepwise error estimates of these quantities, aswell as information on which formulas were used toobtain the estimate.

Another common yet dangerous practice in theCFD community is to use a �xed number of itera-tions for each time-step. This approach eliminatesthe need for an iteration termination strategy andin most circumstances is satisfactory. Global tem-poral error [see equation (2)] strongly depends ontime-steps with large local error. The error from just

6

one nonconvergent time-step potentially could domi-nate the error from all other time-steps combined! Ifan intrinsic feature of the ow signi�cantly changesthe convergence rate of the algebraic solver, then a�xed number of iterations is not a good strategy.The periodic blowing from zero mass uid actuatorsis a prime example. Di�erent phases of the cycleconverge at di�erent rates because convergence rateis sensitive to boundary conditions. A terminationstrategy that ensures a uniformly bounded algebraicerror at each time-step is needed.

3.2.6 Langley e�ortAn ongoing e�ort focuses on the e�cacy and e�-

ciencies of several time integration schemes for theunsteady compressible Navier-Stokes equations. Ex-isting and newly developed multi step and multistage schemes are being studied, with particular at-tention to high-order (p � 3) schemes. Past workincludes comparisons of the high-order (ESDIRK4)78 Runge-Kutta scheme with �rst- and second-orderBDF on laminar problems. Bijl, et al.20 showed thatthe e�ciency of the ESDIRK4 scheme exceeds thatof the BDF2 by a factor of 2:5 at engineering er-ror tolerance levels (10�1-10�2). E�ciency gains aremore dramatic at smaller tolerances. No problems ofnonlinear instability were noted with the high-orderESDIRK4 scheme on the problems tested.Carpenter et al.32 has shown that stage order two

Runge-Kutta schemes are susceptible to order re-duction for sti� systems, although none is experi-enced for laminar problems with sti�ness levels ofO(103). However, turbulence models exhibit con-siderable sti�ness at Reynolds numbers in the rangeof 105 � 107. Signi�cant order reduction is expe-rienced with ESDIRK4 for cases experiencing sti�-ness from strong turbulence �elds. Ongoing stud-ies include investigating the e�ciency of ESDIRKschemes on other one- and two-equation turbulencemodels.Figure 2 shows the convergence behavior of the

ESDIRK4 scheme, the BDF2 and BDF3 schemes,and the MEBDF4 scheme. The test problem is thecircular cylinder at Reynolds number 104, with aMach number of 0.25. The calculations are runwith the unstructured Fun2D code.2 Design orderslopes are obtained for each scheme: 2, 3, 4, and3 for BDF2, BDF3, MEBDF4, and ESDIRK4, re-spectively (note that we have accounted for the the-oretical order of ESDIRK4 in accordance with orderreduction). The MEBDF4 scheme was added to thecomparison because it is a stage order three methodand is not as susceptible to order reduction as theESDIRK4 scheme. Research continues on establish-ing reliable error estimators and iteration termina-

2

22

22

2

22

2 2 22 2

3

33

33

33

3 3 33 33 3 3

M

M

MM

MM

MM

MMMMMMM

R

RR

RR

R

RR

RR

RR

RR

Log 10 (Dt)

Log

10

(L2

Err

or)

-2 -1.5 -1 -0.5 0 0.5 1-10

-9

-8

-7

-6

-5

-4

-3

BDF2BDF3MEBDF4ESDIRK4

23MR

Figure 2. A comparison of temporal density errorobtained with BDF2, BDF3, MEBDF4, and ES-DIRK4 schemes on a circular cylinder, with Re= 104, and Ma = 0.25. The turbulence model isSpalart-Allmaras.

tion strategies. A comparison of e�ciency will bemade between all integration schemes once each isautomated.

A �nal observation is relevant to help focus futurework. The BDF2 scheme yields engineering accuracyif each temporal mode in a time periodic ow is re-solved with approximately 50� 100 time-steps. Thefourth-order ESDIRK formulation attains a similaraccuracy (in spite of order reduction) using 5 � 10time steps per period, with �ve stages per step,yielding approximately 25�50 time-samples/period.State-of-the-art LMS methods could lower this esti-mate to 15 � 30 time-samples/period, an improve-ment of approximately O(101=2) over existing tem-poral e�ciency. The theoretical lower bound fortemporal schemes, based on in�nite order Chebyshevoperators, is � samples/period. High-order schemesare presently asymptotically close to this theoreti-cal lower bound. Another factor of three reductionin samples/period is perhaps all that remains and isbecoming increasingly more di�cult to attain. Al-gorithmic work focusing on other aspects of solvertechnology will have a greater chance of producingmeaningful improvements during the next decade.The next section describes high-order spatial algo-rithms and their potential to increase the temporale�ciency, and section (4) describes the current andfuture status of algebraic solvers.

7

3.3 Spatial Algorithms

3.3.1 OverviewThe spatial algorithms used currently in general

purpose aerodynamics solvers have not changed ap-preciably during the past decade. Most current pro-duction codes (structured or unstructured) rely onsome form of second-order upwind formulation with ux limiting to provide necessary robustness in thevicinity of unresolved features in the ow. Excellenttexts describing these methodologies can be foundelsewhere. For �nite-di�erence methodologies seeHirsch 62; 63, and LeVeque.88 For basic �nite-elementmethodologies see Hughes 70, Zienkiewicz and Tay-lor 149; 150 and Baker and Pepper.10

In section 3:1, we established that spatial algo-rithms play a important role in determining tem-poral e�ciency. High-order methodologies will sig-ni�cantly contribute to the ultimate goal of e�-cient, general-purpose, time-dependent aerodynamicsolvers. A broad overview of the spatial discretiza-tion landscape is now presented. Enabling technolo-gies that allow extension of high-order methods intothe general purpose aerodynamic solver arena areidenti�ed. A wealth of scienti�c literature supportsthe assertion that high-order general purpose algo-rithms will likely mature within the unstructured�nite element framework within the next decade.Spatial operators are categorized by the kind of

grids on which they are formulated: structuredor unstructured. A structured grid has large re-gions of the interior vertices that are topologicallyalike, which results in well-established connectiv-ity patterns. Accommodation of complex geome-tries requires an arbitrary subdivision of the struc-tured grid into what is referred to as a hybrid ormulti block formulation. Three examples of struc-tured codes currently used at Langley as general-purpose aerodynamics solvers include the block-structured TLNS3D,144 and CFL3D,142 and theoverset-structured OVERFLOW 72. An unstruc-tured mesh is one in which vertices may have ar-bitrarily varying local neighbors. Three examplesof unstructured codes used at Langley (ICASE) areUSM3D49, FUN3D2, and NSU3D.94 The distinctionbetween structured and unstructured meshes usu-ally (although not necessarily) extends to the shapeof the elements: 2-D structured meshes typically usequadrilaterals, while unstructured meshes use trian-gles, with similar analogous element shapes in 3-D(hexahedra vs. tetrahedra).Structured solvers o�er simplicity, easy data ac-

cess, and thus e�ciency. The data structure andalgorithmic simplicity of structured solvers leads tomore e�ciency and lower memory requirements for

a given accuracy tolerance. A discrete derivativerequires simple increments/decrements in array in-dices, in stark contrast to an unstructured formu-lation. The structured advantage in CPU time andmemory can be as much as a factor of three on prob-lems not requiring signi�cant grid adaptation. How-ever, on a complicated geometric domain a struc-tured mesh may require many more elements thanan unstructured mesh, because elements in a struc-tured mesh cannot vary in size as rapidly. Struc-tured grid generation approaches are far from beingfully automated, and require user guidance in thedecomposition step. A complicated 3-D structuredmesh can take a month to generate. The current andfuture role of structured formulations is for repeti-tive computations late in the design cycle where gridtemplates might exist and grid generation and adap-tation are not important components in the solutionprocess.

Unstructured meshes o�er exibility in �ttingcomplicated domains, rapid variation from smallto large elements, and relative ease in re�nementand de-re�nement. Unlike structured mesh gener-ation, unstructured mesh generation has been au-tomated in mainstream computational geometry forsome years. The major approaches for generating,re�ning, and improving unstructured meshes relyon unconstrained and constrained Delaunay trian-gulation, quad trees algorithms, or combinations ofthe above.19 A highly e�ective combination of tech-niques for high Reynolds number ows is an advanc-ing layers method (ALM) 114 in the near-wall re-gion, and an advancing front method (AFM) 91 inthe far-�eld. Highly stretched viscous grids can begenerated in a reasonably automated fashion withthis approach. Automation begins to break down asaspect ratio increases on complex geometries.

Element shape has a profound impact on the accu-racy and e�ciency (direct and indirect) of a formu-lation. Meshes with unintended large aspect ratiocells lead to both poorly conditioned matrices andpoor solution accuracy. Poor solution accuracy re-quires more grid points for a given accuracy. Theadditional cost of �xing a bad mesh can usuallybe mitigated by the faster convergence of the iter-ative solver.17 Babu�ska and Aziz7 showed that con-vergence on triangular elements is achieved only forangles bounded away from 180o. This rather weakcondition becomes an issue for strongly anisotropicmeshes used in high Reynolds number turbulentNavier-Stokes simulations. Near-wall aspect ratioson these grids can be in the range 104 � 105. For-mulations typically try to limit the maximum an-gle in a grid (for example 179o) even though cur-

8

rent evidence is divided on the necessity of this con-dition. Quadrilateral and hexahedral meshes havean advantage in accuracy over triangular and tetra-hedral meshes for these problems. The faces ofhexahedral elements in the boundary layer are ei-ther almost parallel or almost orthogonal to the sur-face. Shock fronts and shear layers, which are alsostrongly anisotropic, require high aspect ratio cellsfor which the direction and location cannot be pre-dicted in advance. Generation of these meshes canbe di�cult.

General purpose aerodynamic solvers have pro-gressively shifted from hybrid/structured methodsto unstructured formulations over the past decade.The principal motivations driving this change aregrid generation on complex con�gurations and gridadaptation. These compelling reasons are likely tobecome more important during the next decade, par-ticularly as the grid adaptation �eld matures fortime-dependent simulations.

A large amount of inertia persists in the struc-tured grid world, which is not entirely counterpro-ductive. A time-dependent niche exists for computa-tionally e�cient formulations over the next decade.Unlike 3-D steady-state computations, realistic 3-D time-dependent computations are presently con-strained by processor speed rather than memory re-quirements. The increased e�ciency of structuredmethods is a notable advantage when run times canbe decreased by a factor of two to three. The addi-tional hybrid/structured grid generation time can beamortized if a calculation is likely to run for months.

3.3.2 High-Order Spatial Operators

High-order spatial operators need fewer pointsthan second-order operators, to resolve the same in-formation. The exact reduction strongly dependson the desired accuracy. Steady-state problems re-quiring an accuracy of three signi�cant digits can beachieved with fourth-order schemes in half as manypoints in each spatial dimension. The total reduc-tion in the number of points is approximatelyO(101)in 3-D. Time dependent simulations that require so-lution accuracy to four or even �ve signi�cant digitsat each time-step, will favor high-order formulationsto a larger degree. High-order spatial methods canincrease the e�ciency the time-dependent simula-tions by O(101 � 102).

The constraints necessary to expedite grid gener-ation and grid adaption will guide the next genera-tion of high-order solvers. High-order methods mustmove beyond proof of concept and into the realm ofbeing tools used to increase the e�ciency of aerody-namic solvers.

The implementation of high-order methods isstrongly dependent on whether the grid is structuredor unstructured. High-order �nite-elements (FE) arenatural candidates for structured or unstructuredmeshes, while high-order �nite-di�erence (FD) tech-niques are usually implemented on block-structuredor overset grids. Finite-volume (FV) techniques ex-ist in both forms; a close similarity between linearelement FE methods and FV methods exists. Allthree approaches solve di�erent forms of the govern-ing integral equation. FV directly solves the integralequation by approximating the numerical uxes. FDsolves the divergence form of the integral equationby approximating the derivatives. FE take the di-vergence of the integral equations, multiply by anarbitrary test function, and integrate by parts. Thesolution itself is the resulting approximation.

Not all current general purpose spatial discretiza-tion algorithms are natural candidates for high-order extensions. For example, based on 2-D re-sults, Casper and Atkins 36 noted that a 3-D hy-brid/structured essentially nonoscillatory (ENO)-FV formulation would be extremely expensive toimplement relative to comparable FD techniques.Barth and Frederickson12, and Barth13 extendedtheir unstructured FV solver to account for k-exactreconstruction. They note that comparing quadraticwith linear reconstruction on a triangle requiresroughly quadruple the number of solution unknowns.All high-order formulations require more work thansecond-order formulations, but some are more e�-cient than others.

Many di�erent approaches to high-order FE havebeen adopted in developing numerical schemes tosolve the compressible Euler equations. Two majorclasses have emerged as candidate schemes: 1) sta-bilized methods (continuous across interfaces), and2) discontinuous methods (discontinuous across in-terfaces).

Standard Galerkin FE dis-cretizations of convection-dominated Navier-Stokesequations produce wildly oscillating solutions un-less dissipation terms are added to the formulation.Since the early 1980s stabilized FE methods havebecome increasingly popular in CFD. Early devel-opment motivated by the success of upwind FD/FVschemes included the streamline di�usion �nite el-ement Method (SDFEM),68; 75 which later evolvedinto the streamline upwind Petrov-Galerkin (SUPG)scheme of Brooks and Hughes.25 A stabilizing termis added into the weak statement motivated by in-viscid terms and results in a perturbed standardGalerkin test function. The stabilization createsan upwind e�ect by weighting more heavily the up-

9

stream nodes within each element. Hughes and Tez-duyar 69 generalized the SUPG method to �rst-orderhyperbolic systems and included work on the Eulerequation. The original SUPG formulation su�eredfrom oscillation in steep gradient regions (shocks)which led to the introduction of entropy variablesand ultimately to stabilization terms including thee�ects of both the inviscid and viscous terms in theirGalerkin least-squares (GLS) formulation.71 A vari-ety of methodologies have been proposed to provideadditional stability to the convection terms, mono-tone discrete solutions and ease of implementation.

Another approach that has been gaining in pop-ularity in recent years is the discontinuous Galerkin(DG) method. The DG method originally intro-duced by Reed and Hill 117 exhibits several distinctadvantages when applied to complex unstructuredgrids. Local polynomials are used to represent thedata to arbitrary order, with the data on elementinterfaces treated as discontinuities. The approachis advantageous because the solution accuracy is rel-atively insensitive to mesh smoothness and can beextended to arbitrarily shaped elements. In 1986,Johnson and Pitkarata76 proved that the conver-

gence rate of the method is (�x)k+1=2 for general tri-angulations. The method generates a local mass ma-trix that can easily be inverted, making the methode�cient for explicit time integration. An entropyinequality for any scalar nonlinear equation 73 ex-ists, implying discrete nonlinear L2-stability for dis-continuous solutions. (This assumes wellposednessand boundedness of the continuous nonlinear prob-lem.) Several researchers have demonstrated super-convergence with DG.37 Lowrie et al.92 obtainedconvergence rates of 2p + 1, and Hu and Atkins 67

showed that the dispersion of the DG method is gov-erned by 2p+ 1 for polynomials of order p. The DGformulation produces a matrix with many dense butsmall sub-matrices weakly coupled to their neigh-bors. Algorithmically, this matrix structure excelsin a parallel environment and has high cache e�-ciency. Atkins and Shu6 developed a quadrature-free approach that allows the precomputation andstorage of much of the algorithm, thereby increasingthe e�ciency.

In the early 1980s, the p- and the hp-FEM meth-ods were introduced by Babu�ska, and Szab�o8. Theyshowed that for elliptic problems exponential conver-gence could be achieved with the hp-FEM method.The degree of the approximating polynomial canvary by elements so both grid re�nement and orderre�nement are used simultaneously to attack solu-tion error. The methods show design order for p�xed in the limit h ! 0, and convergence for h

�xed p !1. The behavior for high Reynolds num-ber Navier-Stokes equations is less clear; neverthe-less the hp-FEM methods have great potential in thecontext of complex geometries and grid adaptation.

High-order FD excel in their simplicity and e�-ciency, and in the richness of linear and nonlinearalgorithmic permutations that can be formulated.The Achilles heels of FD are \boundaries" and \sta-bility".

Achieving numerical stability near boundarieswith high-order FD stencils is di�cult. This insta-bility is closely related to the classical Runge oscil-lations exhibited by high-order polynomials on uni-form grids near the boundaries. The solution in bothcases is to lower the polynomial order, compress thegrid, or increase the stencil width. Gustafsson 53

showed that to maintain spatial design order accu-racy, the boundary stencil order must not deviateby more than one from the interior order of accu-racy: fourth-order interior stencils require boundaryclosures of at least third-order accuracy! Note thatin uid dynamics applications, near-wall regions areprecisely the regions were high-order accuracy is de-sirable. In general, low order treatments for bound-ary closures are not an acceptable alternative unlessspecial near-wall grid re�nement is used to compen-sate for reduced accuracy.

Strand 137 following the work of Kreiss andScherer,84 partially resolved the boundary closuredilemma by presenting constructive procedures fordeveloping stable and accurate boundary schemes.Stability is ensured in an L2 norm using a dis-crete summation-by-parts (SBP) procedure. Car-penter et al.30 and Olsson 109; 110 showed how toimpose the physical boundary conditions to pre-serve the SBP energy estimate. To date, bound-ary closures have been formulated for central andupwind FD schemes and Hermitian compact FD.FD schemes require smooth structured meshes thatare often di�cult to generate on complex geome-tries. Multi block/overset grids relax the griddingconstraints, allowing piecewise smooth grids aroundcomplex geometries, but create a new set of compli-cations. Conservation is a major concern on multiblock grids and is extremely di�cult to achieve onoverset grids. Shocks and other discontinuities nearinterfaces must be treated carefully to ensure correctshock speeds and locations. In addition, interfaces,like boundaries, can cause linear and nonlinear in-stability and lead to decreased levels of solver ro-bustness.

Multiple attempts have been made to overcomethe di�culties of complex geometries for high-orderFD schemes. By far, the most common solution to

10

ensure conservation and stability has been to reduceboundary or interface accuracy. As a general rule,this approach works well if little structure exists nearthe boundary or interface. Solution accuracy in com-plicated ow scenarios, however, is di�cult to pre-dict. Carpenter, Nordstrom, and Gottlieb 31; 106; 107

have developed L2-stable interface conditions basedon SBP energy estimates. The interface points aretreated discontinuously through a penalty term andprovide conservative, high-order solutions on multiblock grids. The only grid requirement is C0 inter-face continuity between blocks, a mild restriction.

3.3.3 BottlenecksThe major obstacle facing all high-order spatial

discretization methods FE or FD, structured or un-structured, is nonlinear instability in the presenceof unresolved features. Shock and sliplines are no-table examples. The classical approaches to dealwith discontinuities in FD and FV are the additionof local arti�cial viscosity and/or �ltering, and totalvariation diminishing (TVD) or limiting approaches.Equivalent approaches exist in FE, though they aretermed \stabilization." The amount of added dissi-pation depends on the simulation objectives. Mono-tone solutions can be obtained with any formulationat the expense of reduced accuracy. In principle, theminimum amount of dissipation necessary for non-linear stability is advisable. Unfortunately, precisemathematical theory does not exist to determine theoptimal dissipation. Thus, most approaches reducethe approximation/ ux/solution near the disconti-nuity to �rst order to achieve monotonicity and ro-bustness. Reduction to �rst-order accuracy locallyresults in the undesirable second-order 53 global ac-curacy if a uniform h-re�nement is then performed.Arti�cial dissipation/�ltering approaches are

quite e�cient and simple to implement in FD formu-lations. Unfortunately, they are often problem de-pendent, vary considerably on shock strengths, andare often user dependent. Although TVD and uxlimiting approaches maintainmonotonicity near dis-continuities, they unfortunately degenerate to �rst-order accuracy near smooth extrema. (See LeVeque88 for an overview of TVD techniques.)Essentially non oscillatory (ENO) 57 and later

Weighted ENO (WENO) 90; 74 schemes were de-veloped to circumvent nonlinear instability. ENOschemes choose the smoothest stencil from all avail-able design order stencils, thereby avoiding as muchas possible interpolation/di�erentiation across dis-continuities. ENO schemes have been extremely suc-cessful algorithms over the past 10 years for prob-lems where both discontinuities and features requir-ing high-order spatial accuracy are required. See Shu

129 for a detailed account of ENO/WENO schemesand their applications. The principal di�culty withstructured grid ENO schemes is their extension tocomplex geometries. The mathematical foundationsfor ENO/WENO schemes (bounded total variationproofs, etc.) are predominantly based on periodicor in�nite domains, and are outside the context ofboundaries. ENO/WENO schemes can not be im-plemented at several points next to boundaries be-cause they do not have smooth data outside theboundary to build high-order non-oscillatory sten-cils. The extension of ENO/WENO schemes to mul-tiple domains is complicated by numerous bound-ary interfaces throughout the domain. Another dif-�culty with ENO schemes is that stencil searchingalgorithms have stencils that \switch" sometimes ar-bitrarily, which makes convergence to steady-statedi�cult. Atkins 5 proposed a smoothly varyingstencil biasing technique to eliminate this problem.Integer stencil shifts were not allowed in the ap-proach. In addition, WENO schemes that are asmooth weighted sum of stencils in principle shouldnot su�er from this di�culty.

Durlofsky, et al.47 and Abgrall 1 attempted toovercome the geometric complexity problems bybuilding fully unstructured stencils using stencilsearching algorithms. Ollivier-Gooch 108 suggestedusing a least-squares reconstruction approach for un-structured mesh ENO. Encouraging results were ob-tained for AGARD test case 1 with second- throughfourth-order ENO, although convergence problemswere experienced in the fourth-order case.

The use of overset meshes is another approachused to overcome geometric di�culties. Wang andHuang146 developed a compact ENO scheme and ap-plied it to a multi domain overset code. Boundaryissues with the ENO formulation are still present,and additional complications of non conservation atthe interfaces exist. Wang et al. 145 partially addressthe issue of interface conservation, but signi�cant is-sues still remain for time-dependent discontinuous ows.

Most high-order formulations are only guaranteedstable on linear problems, and some cannot evenclaim linear stability. The high-order DG FE formu-lations can claim a stronger form of stability. Barth14 and Barth and Chirrier 16 have designed numeri-cal uxes that satisfy a nonlinear energy condition.They assume a convex entropy extension of the Eu-ler equations and bound the nonlinear \energy" ofthe system for all time in terms of the initial data.Simpli�ed interface ux functions are derived to al-low this result. These results and others 66 providean encouraging step towards nonlinearly stable for-

11

mulations that maintain high resolution.Although theoretically advantageous, high-order

spatial discretizations in their present form still haveseveral obstacles to overcome. Considerable workhas been devoted to these methods over the pastdecade, yet few studies demonstrate the increasede�ciency of high-order spatial discretizations forgeneral-geometry aerodynamic simulations, includ-ing turbulence models. De Rango and Zingg 43; 44

address this speci�c question and achieve encourag-ing results for which high-order methods give moreaccurate solutions on a given grid. Additional workneeds to be done to demonstrate increased e�ciencyfor a given accuracy.Geuzaine et al.52 and Delanaye et al.41 apply

the quadratic reconstruction FV scheme of Barthand Frederickson12 to high Reynolds number ows.They achieve suitable convergence rates with gener-alized minimal residual (GMRES) and bi-conjugategradient stabilized (Bi-CGSTAB) algorithms, andshow second- and third-order convergence on irreg-ular and smooth meshes, respectively. They do notcompare the e�ciency of second- and third-order for-mulations, but note that the quadratic method con-verges more slowly than the comparable linear re-construction. Delanaye and Liu 42 report signi�cantimprovements in e�ciency and accuracy in inviscid2-D calculation over a multi-element airfoil, compar-ing the quadratic and linear formulations. Resultsin three dimensions are not as dramatic.

3.3.4 Langley E�ortLangley has had a strong presence over the last

decade in the following high-order spatial disciplines:1) structured grid FD, 2) structured grid ENO-FDand ENO-FV, 3) unstructured grid DG-FEM, and4) unstructured grid SUPG-FEM. The following in-formation is presented to summarize our experiencesand provide guidance when comparing the di�erenthigh-order methods.Table (1) can be used to compare the important

attributes of current high-order formulations. Cat-egories are rated on a scale from one to �ve, with�ve being the best currently available, and one be-ing a capability representative of 1980. The cate-gories are 1) complex geometry, 2) grid adaptation,3) robustness (nonlinear), and 4) cost for a givenaccuracy requirement. At some level, all categoriesare closely related; however, we assume that eachis independent from all others when assigning a nu-merical value. Speci�cally, the complex geometrycategory rates the capability of each method to ac-commodate complex 3-D con�gurations. This cat-egory is closely related to the locality of the dis-crete scheme. Grid adaptation is used to attack so-

lution error. The second category, grid adaptation,describes each method's success on adapted grids,including sensitivity to grid smoothness and ease ofgrid generation. The robustness category describesthe robustness of the method for under-resolved fea-tures and discontinuities. In simple terms, this cat-egory rate whether the code \runs" (converges forsteady-state cases, and does not diverge in time-dependent cases) with minimal user support. Thecost category describes the cost of achieving a givenaccuracy. It is assumed that the necessary grid hasbeen generated by whatever means are necessary ineach case.

The candidate schemes include two broad classes:1) unstructured database schemes, and 2) semi-structured databases. We include the high-orderFEM methods as unstructured methods because thestructure within each element does not present a sig-ni�cant burden on the exibility of the method. Theunstructured candidate schemes are DG, SUPG, k-exact �nite-volume, and k-exact ENO FV. The can-didate structured grid schemes are Upwind FD, Up-wind FV, and WENO-FD. The numbers are sub-jective, and should only be used as a relative guidefor the purpose of comparing strengths and weak-nesses. In general, researchers hold strongly varyingopinions about the relative merits of each scheme.

Table 1: Comparison of high-order schemes. Cat-egories are 1) complex geometry, 2) grid adaptation,3) nonlinear robustness, and 4) cost for a given ac-curacy.

Method Geometry Adapt Robust Cost

UnstructuredDG FE 5 5 3 2SUPG FE 5 5 2 3k-exact FV 5 4 2 2LS-ENO FV 4 4 4 1

Multi-blockUpwind FD 3 2 2 5Upwind FV 3 2 3 3WENO-FD 2 1 5 3

4 Convergence Acceleration

A second algorithmic issue that contributes signif-icantly to the e�ciency of the temporal algorithm isthe convergence rate of the algebraic solution algo-rithm. At some point in the solution process thealgebraic system

A ~x = ~b (16)

12

is solved for the vector of data ~x, with

A =I

�t+@F

@U(17)

~x = ~Uk+1 � ~Uk ; ~b = ~R( ~Uk) (18)

The matrixA is sparse. Equation (16) can be solvedeither directly, or iteratively. The dimensionality of~x is O(107) for 3-D problems, which makes directmethods uncompetitive with iterative solvers. Shur-complement methods can be used to attack equation(16) by subdividing A into numerous subproblemsand interface conditions. Each subproblem is thensolved directly, but the interface conditions are dif-�cult (dense). (See Saad 120 for details.)Iterative methods fall into two broad categories:

stationary and nonstationary.11 Stationary methodscan be expressed in the simple form

~xk+1 = B~xk + ~c (19)

with the matrix B independent of the iteration k.These methods are older, simple to implement, andusually not very e�ective when used alone. Sim-ple methods used in uid mechanics are Jacobi,Gauss-Seidel (GS), symmetric GS, successive-over-relaxation (SOR), and symmetric SOR (SSOR). Nu-merous permutations of these methods exist includ-ing matrix reordering. (For details see Barrett etal.11). More powerful stationary methods include in-complete factorization ILU(k), and block factoriza-tion. An ILU(k) approximately factors the originalmatrix A using an LU decomposition with the levelof �ll governed by the parameter k. Block factoriza-tion is motivated by tensor product grids (line datadependencies) where implicit solves along directionsare e�cient. Both are somewhat more expensivethan basic stationary methods, but have consider-ably faster rates of convergence. Implementation ofsweeping algorithms is complicated in a parallel en-vironment.Nonstationary methods involve information that

changes at every iteration. They are more recentlydeveloped, more di�cult to understand, and morepowerful. Commonly used examples in uid me-chanics include conjugate gradient (CG), general-ized minimal residual (GMRES), bi-conjugate gra-dient (BiCG), quasi-minimal residual (QMR), andbi-conjugate gradient stabilized (Bi-CGStab). Thesemethods update the solution in certain \directions"by considering inner products of current residualsand other Krylov space vectors arising during thecourse of the iteration. (See Saad 120 for details).On di�erent problem classes, the convergence rate ofnonstationary methods varies considerably. Nachti-gal et al.104 showed that a class of problems exists

for which each of the aforementioned nonstationarymethods is a clear winner (in terms of e�ciency),and a clear loser (in terms of e�ciency).A preconditioner is a matrix used to rotate a lin-

ear system into a new system that has the samesolution but is easier to solve in some sense. Aleft-preconditioner acts on equation (16) yielding thenew system,

M�1 A ~x = M�1~b (20)

where the new matrix M�1A is easier to solve.The preconditioner changes the eigenstructure of theoriginal system into a more compact set of eigenval-ues that an iterative method can attack more e�ec-tively. Ideally, a preconditioner should change theeigenstructure dramatically but at a minimal ad-ditional cost. Simple preconditioners used in uiddynamics include the block Jacobi, GS, and SSORmethods. More powerful preconditioners include in-complete factorization ILU(k) and block factoriza-tion.Multigrid, at least in terms of elliptic problems,

is a mechanism for rapid communication of mul-tiscale information.23 Multigrid methods are usu-ally de�ned as a strategy to accelerate any sta-tionary or nonstationary iterative procedure. Thesolution is obtained on a sequence of grids, rang-ing from coarse to �ne. Each grid smoothes thehigh-frequency components of the residual on thatgrid. Restrictions and prolongations communicatethe data between grids. The resulting algorithmrapidly communicates long wavelength data via thegrids, while damping short wavelength data by usinge�cient local smoothing operators. The exact choiceof grid structure, restriction and prolongation oper-ators, and smoothers greatly in uences the overallperformance of the procedure.Multigrid methods are e�ective techniques for

accelerating convergence of elliptic and hyperbolicproblems. Convergence rates easily approach 0.1per cycle on elliptic problems such as the Poissonequation. The theoretical lower bound (if we relyon coarse grid corrections) on the convergence forhyperbolic equations is 0.75 per cycle for a second-order spatial discretization, and is routinely achievedby general-purpose Euler solvers. (See Molder103

for details.) The convergence rate su�ers consider-ably for high-Reynolds number (turbulent) viscous ow solutions. The primary cause for this slow-down is the highly stretched wall normal grids usedto resolve turbulent boundary layers. Wall nor-mal sti�ness can introduce sti�nesses of the orderof 104 � 105. A secondary cause of slowdown is theexistence of signi�cant regions of low Mach num-

13

ber ow around stagnation points and in recircula-tion regions. A third cause of slowdown comes fromthe imperfections of realistic grids. Generating gridsthat have 107 points without having grid anoma-lies is di�cult. All general-purpose CFD solvers ad-dress each of these problems ultimately in the sameway. For wall anisotropies, semi-coarsening and/ordirectional-implicit techniques precondition the wallnormal boundary sti�ness. Low Mach number pre-conditioning is added in those regions of the owbelow a critical Mach number. Grid anomalies areaddressed with grid smoothing and movement algo-rithms in problem regions.

Solution techniques within the aerodynamics com-munity are far from being \black-box" algorithms.Two decades of experience have shown that noneof these algorithms is well suited for solving broadclasses of high Reynolds number turbulent ows.Practitioners rely on combinations of a wide vari-ety of methods including 1) modi�ed Newton-Krylovmethods, 2) algebraic multigrid methods and 3) ge-ometric multigrid methods, 4) defect-correction it-eration techniques, and 5) sparse matrix methods.

Anderson et al.3 compared the e�ciencies ofseveral iterative strategies in the context of anunstructured, 3-D incompressible, Navier-Stokessolver. Multi element airfoils and high-lift sys-tems were used as test problems in 2-D and 3-D.The turbulence model used was that of Spalart andAllmaras.132 GMRES was the Krylov method usedin the study, with Gauss-Seidel or incomplete LU-decomposition used as a preconditioner. Newton-type solvers were shown to converge in the fewest it-erations. In terms of work and storage the multigridalgorithms are the most e�ective means of reducingthe residual on the problems studied.

In spite of all the powerful iterative techniquesbrought to bear on aerodynamic problems, the con-vergence rates for high-Reynolds problems can ap-proach 0.98 per cycle. Mavriplis demonstrated thecapabilities of his unstructured code NSU3D 94 oncomplex 3-D con�gurations.95; 96; 97; 98; 99; 100 Thefeatures in NSU3D are among the most advancedpresently used in the CFD community. A wide va-riety of 3-D test problems was run including butnot con�ned to 1) a realistic high-lift con�gurationincluding a wing, pylon, and nacelle, 2) the trape-zoidal wing 99, and 3) an ONERA M6. Grids rangedfrom 1-10 million vertices. Reynolds numbers werein the 1-10 million range with wall-normal spacing of10�5 � 10�6 based on chord length. Published con-vergence rates for these cases ranged from 0.96 - 0.98per cycle. Approximately 500-1000 multigrid cycleswere required to achieve residual levels converged to

engineering tolerances.

4.1 Bottlenecks

The convergence rates quoted in the previous sub-section are based on steady-state solvers. The e�-ciency of time-dependent and steady computationsare closely related, as the underlying nonlinear ma-trices are nearly identical. (The time-dependent for-mulation converges slightly faster for time-steps gov-erned solely by accuracy considerations). Thus, thecost of the unsteady calculations can be related tothat of the steady. Simple back-of-the-envelope cal-culations reveal that at a minimum the unsteadycomputation will be equivalent to the solution of 100steady-state problems each having the same compu-tational complexity . Dropping an equation residualthree orders of magnitude at a rate of 0.98 requires ofapproximately 400 iterations. Thus, modifying theconvergence rate of the algebraic solution algorithmhas a profound e�ect on the e�ciency of the tem-poral algorithm. An order of magnitude increase incomputational e�ciency could be achieved by suc-cessful convergence acceleration e�orts.The convergence characteristics of high-order spa-

tial formulations has not been extensively studied.The dissipation level of the spatial operator fre-quently a�ects the convergence rate of the alge-braic system, with more dissipation producing fasterconvergence. High-order methods inherently haveless dissipation, and could be more di�cult to con-verge. An additional concern is the e�ciency ofmultigrid methods as spatial order increases. Thetheoretical lower bound of multigrid methods (as-suming coarse grid corrections) on linear advectionis � 1 � 1=2p where is the convergence rate,and p is the order of spatial approximation. Fourth-order methods should converge no faster than 0.9375per cycle. Ollivier-Gooch 108 used multigrid onan unstructured high-order FV scheme and expe-rienced increasing di�culty with convergence as theorder of the approximation increased. Delanaye etal.41 notes that quadratic elements converge moreslowly than equivalent linear elements, when a di�er-ence GMRES algorithm is used. Interestingly, Bon-haus 21 using SUPG with a Newton-Krylov GMRESmethod and diagonal preconditioning, experiencedno changes in convergence rate with increasing or-der.

4.2 Langley E�ort

Current iterative nonlinear solvers require O(103)iterations to converge, based on high Reynolds num-ber (� 106 � 107), turbulent, separated, 3-D, com-plex geometry ows. Newton solvers converge theseproblems in O(101) iterations if a reasonable ini-

14

tial guess is given. Currently, a group at Lang-ley is studying methods that potentially have text-book multigrid e�ciency (TME).22; 23 Methods thatachieve TME converge at rates that are independentof the number of degrees of freedom (grid), and con-verge to the truncation error of the discretizationin approximately 101 work units. A work unit isde�ned as the work equivalent to the evaluation ofthe residual. Thus, TME methods have a poten-tial increase in e�ciency of O(102) compared withexisting state of the art solvers. The potential e�-ciency of TME methods relies the \factorizability"property of the Navier-Stokes (NS) equations. Split-ting the NS equations into factors decouples the sys-tem somewhat, and allows optimal (fast and inex-pensive) operators to be constructed for each por-tion. This \divide and conquer" approach yieldsnearly optimal e�ciency for the entire problem. TheLangley e�ort was showcased at the 2001 AIAACFD conference.24; 45; 119; 130; 139 Work continues onthis revolutionary method to implement TME algo-rithms on general purpose aerodynamic solvers.

5 Turbulence Modeling

5.1 Overview: Current practices

An equally important aspect of temporal algo-rithms is the underlying turbulence models beingsolved. Numerous turbulence models exist for at-tached steady turbulent ows. During the last twodecades great strides have been made in tuning theseturbulence models to increase their robustness andgenerality. Practitioners routinely utilize these mod-els to predict the ow behavior of a surprisinglybroad class of complex problems with acceptablecon�dence levels in their solutions. Unfortunately,the same level of maturity does not exist for nonsta-tionary ows, where time-averaged turbulence quan-tities inadequately describe the important dynamicsof the ow. Oftentimes, these ows are dominatedby large-scale ow features that are not properlymodeled by conventional turbulence models. Flowswith massive separations such as blu�-body wakes,cavity ows, shock-induced separations, and recircu-lation zones almost always fall into this category.

5.1.1 LES

In 1970 Deardor�39 published the �rst results ofa large-eddy simulation (LES). The objective of anLES is to simulate, or directly compute, the largeenergy-containing uid motions and to model onlythe small scales that are unresolved by the grid (thesubgrid scales). A subgrid-scale (SGS) model acts toremove the energy associated with the small scalesand therefore facilitates the global energy transfer

from the large scales to the small scales. When prop-erly implemented, LES can be used to simulate theturbulence in a ow at low to moderate Reynoldsnumbers. For more complete discussions of the con-cepts and applications of LES see Piomelli115; 116

and the references therein.In an LES calculation, the smallest resolved scales

are determined by the grid-cell size. On �ner grids,more of the ow is simulated and less is modeled.The SGS model therefore has an explicit dependenceon the local grid-cell size. This feature of LES hasled some to believe that the grid-cell size is unre-stricted and simply corresponds to the break in re-solved and subgrid scales. This sort of thinking usu-ally leads to poor calculations. An important as-sumption in an LES calculation is that the energy-containing scales are actually simulated. To do this,the peak in the energy spectrum must be in the re-solved range of scales and the cuto� between theresolved and subgrid scales should be in the inertialwavenumber range.The development of dynamic subgrid-scale mod-

els for LES has motivated the use of LES on non-stationary turbulent ows over extremely complexcon�gurations. However, estimates of the grid re-quirements for LES computations over realistic ge-ometries at realistic Reynolds numbers indicate thatLES is not likely to be a viable option for most owsfor several decades to come.134

5.1.2 URANSDespite the abovementioned shortcomings of ex-

isting turbulence models, most time-dependentcodes use unsteady extensions of popular steady-state algorithms. These calculations are typicallyreferred to as unsteady Reynolds-averaged NavierStokes (URANS). The turbulence models used inURANS span the spectrum of available models.H�old et al.65 used a Baldwin-Lomax model to

compute the unsteady ow about a rounded half-cylinder embedded in a at plate. Their time-averaged pressure coe�cients agreed reasonably wellwith experimental measurements. Surface pressurespectra, however, showed far more disagreementwith the measured spectra. H�old et al.65 suggestedthat the disagreements were primarily temporal andspatial resolution issues rather than a turbulencemodeling problem. Their paper illustrates a signif-icant problem associated with assessing turbulencemodels for unsteady ows; unless care is taken to�rst assess grid and time-step issues, the in uence ofthe turbulence model is almost impossible to evalu-ate. Because of the computational expense of doingthorough resolution checks for unsteady problems,these numerical issues are often left unresolved.

15

Khorrami, Berkman, and Choudhari79 and laterKhorrami, Singer, and Berkman80 used the two-equation SST (k-!) model of Menter101 for 2-Dunsteady calculations of ow in the vicinity of aleading-edge slat. In their work the grid resolutionand time-step were su�ciently �ne for them to sus-pect that the turbulence model was responsible forexcessive damping of the large eddies in the slatcove. Subsequent work by Khorrami, Singer, andLockard81 locally eliminated the turbulence produc-tion term in much of the slat cove and observed un-steady coherent vortex motions that closely resem-bled the PIV measurements reported in the work ofothers.112; 140; 141

Scotti and Piomelli121 assessed four low Reynoldsnumber turbulence models in pulsating channel owsgenerated by oscillating the imposed pressure gra-dient. The apparent simplicity of these ows wasdeceptive. In fact the pulsating ows representedsigni�cant challenges to the URANS models. Theturbulence was out of equilibrium, in some casesso much so that partial relaminarization occurred,followed by a re-transition. Somewhat surpris-ingly, the phase-averaged streamwise velocity pro-�les matched corresponding LES results reasonablywell when plotted in wall coordinates. Howeverthe time-averaged velocities of the URANS mod-els were always less than those determined from theLES. Signi�cant di�erences in the Reynolds shearstresses and the turbulent kinetic energies were alsoobserved.

5.1.3 Composite LES/RANS SchemesRecently, a class of composite models has been de-

veloped for unsteady ows. These composite modelsattempt to blend the unsteady capabilities of LESwith a method having grid requirements that aremore like those conventionally used in Reynolds-averaged Navier-Stokes (RANS) calculations. Acomposite model will typically involve a RANS tur-bulence model and RANS-type grid in regions nearsolid surfaces, where the resolution of turbulent ed-dies would require exceptionally �ne grid resolution.In these regions typical RANS models are usuallyadequate for modeling the e�ects of turbulent ow.Away from wall regions, where large unsteady owstructures dominate the ow, the composite modelshifts from a RANS-type turbulence model to a less-dissipative LES-type subgrid-scale model. Similarly,away from wall regions, the gridding strategy willshift from a RANS-type grid to a grid amenable toresolving at least some of the unsteady turbulencescales of motions.A variety of di�erent composite models have been

proposed in the last few years. The composite

models go by di�erent names and involve di�erentturbulence models for the RANS and LES regionsand di�erent switching functions to shift betweenthe RANS and LES regions. Batten, Goldberg,and Chakravarthy18 proposed limited Navier Stokes;Zhang, Bachman, and Fasel148 developed their owsimulation methodology; Arunajatesan, Shipman,and Sinha4 developed the hybrid RANS-LES. How-ever, the �rst and perhaps the most thoroughlytested of these composite models has been the de-tached eddy simulation (DES) model of Spalart.135

In its most common form, the DES model is imple-mented in combination with the Spalart-Allmaras(SA) turbulence model,133 although recently DESresults with Menter's SST model101 have also beenpublished.138 In either case, the switch from theRANS region to the LES region is governed by a ra-tio of grid-cell size with distance from the wall. Vis-cous grid cells (high-aspect ratio with lengths longcompared to the distance from the wall) characterizethe RANS regions. Small, approximately uniformlysized grid cells far from solid surfaces characterizethe LES region. In theory, the RANS portion cor-rectly models the rather benign attached turbulent ow regions, while the LES portion has su�cientgrid resolution and su�ciently reduced eddy viscosi-ties to simulate the large-scale turbulent ow struc-tures.

The ease with which DES can be implemented ina code is a mixed blessing. Upgrading an existingSA model to accommodate the DES model requiresminimal additional logic. The triviality of the modi-�cations required for this transformation has led nu-merous well-intentioned users to perform and pub-lish DES-like calculations. Unfortunately, it is easierto obtain results than to verify and critically inter-pret them.

Proper grid resolution is always important inCFD. However, with DES, the grid takes on newsigni�cance, as the grid determines the switch be-tween the RANS region and the LES region. For-tunately, Spalart136 provides some guidance for thedevelopment of appropriate grids. Importantly, oncethe RANS region of the ow is adequately resolved,improved grid resolution should be undertaken lo-cal to the LES portion of the simulation, not glob-ally. Over-resolving the RANS regions can displacethe RANS/LES switch-over surface into the bound-ary layer, where the grid is likely to be insu�cientfor a reliable LES calculation. Of course, continuedgrid re�nement will eventually lead to adequate LESgrids over the entire ow, although for aerodynamic ows of commercial interest, solving the ow on sucha grid will likely be a decades-long endeavor.

16

With a DES grid well designed for a particular ap-plication, global grid re�nement is likely to adverselya�ects the results. Nikitin et al105 discusses thesecircumstances in the context of a series of channel- ow calculations. Although modifying the switchingfunction, as is done in other composite LES/RANSmethods, can alleviate some of the deleterious ef-fects, a key to successful implementation of thesemethods is going to involve careful attention to grid-ding.

Another concern with the implementation of com-posite methods is the numerical di�usion, especiallyin the LES region where di�usion associated with themodel is far less than in the RANS region. In simula-tions of the unsteady ow about a sphere, Constan-tinescu and Squires38 used both second-order and�fth-order accurate upwind di�erences for the con-vective terms. All other operators were discretizedusing three-point, second-order accurate central dif-ferences. On the same grid, they found that theresults of calculations with the �fth-order accuratescheme agreed better with a separate LES calcula-tion than the results obtained with the second-orderupwind scheme.

In later work, Strelets138 was concerned about thedi�usion of upwind schemes in the LES region andthe lack of stability of central-di�erence schemes inthe RANS regions. To weaken the adverse e�ectsof upwinding in the LES regions, he used a hybridcentral/upwind approximation of the inviscid uxes.The hybrid is designed so that in the RANS regionthe scheme is \almost upwind" and in the LES re-gions the scheme is \almost centered."

Recent work at NASA Langley suggests thatsecond-order central-di�erence schemes can be usedfor both the RANS and LES regions, although thegrid resolution must be chosen appropriately. Thekey is to ensure that the turbulence model, and notthe numerics, controls the di�usion.

DES has been used in unstructured grid environ-ments, �rst by Forsythe, Ho�mann, and Dietiker.48

Their research suggested that CDES, the one addi-tional free parameter in DES (as compared to SA)needed to be adjusted when DES is used in conjunc-tion with tetrahedrons. Because CDES weights ameasure of grid-cell size in the RANS/LES switch-ing function, the need to adjust CDES when di�er-ently shaped grid elements are used is not surprising.Pelaez, Mavriplis, and Kandil 113 used the standardvalue of CDES for their unstructured grid calcula-tions and ultimately concluded that the value shouldbe determined by performing a decaying homoge-neous turbulence test case.

In spite of the many unanswered questions associ-ated with DES, in the right hands it can be a usefultool for simulating unsteady complex ows. Travinet al.143 used DES for ows around a circular cylin-der. They explored the e�ects of grid, Reynoldsnumber, 2-D versus 3-D, DES versus URANS, andlaminar versus turbulent separation. An importantconclusion, �rst suggested by Spalart et al.134 is that2-D DES calculations are not fruitful. To achieve theadvantages of DES, a true LES should be done inthe LES regions which implies that the full 3-D owneeds to be simulated. Without the third dimension,important turbulence dynamics are inoperative. An-other important conclusion is that the bene�ts ofDES vary with the ow and the information desired.In the case of a circular cylinder with laminar sepa-ration, DES proved itself superior to URANS, evenfor obtaining time-averaged quantities. However, forthe case of turbulent separation, the time-averagedquantities were the same, regardless of whether DESor URANS was used. Hedges, Travin, and Spalart59

found that URANS performed almost as well as DESfor time-averaged quantities on a four-wheel landing-gear model. The unsteady ow around the landinggear appeared much more turbulent-like in the caseof DES compared to URANS, but unfortunately un-steady ow data was not available for comparison.

5.2 Bottlenecks

In this section we will review some of the bottle-necks that are holding back further progress in re-solving turbulence modeling issues. In particular, wewill discuss experimental validation problems, con-sistency concerns for the composite methods, andthe high cost of the calculations.

5.2.1 Validation by ExperimentsOne of the great di�culties with new turbulence

models for unsteady ows is the lack of relevant ex-perimental data with which to validate the mod-els. As discussed earlier, di�erent types of turbu-lence models with very di�erent unsteady character-istics can provide very similar time-averaged results.Hence, if the time-dependent behavior of the ow isimportant in an application, then experiments thatprovide time-dependent data are indispensable.Although ow past circular cylinders and spheres

has provided useful validation tests in the past, forapplications that involve ow actuators, particu-larly those that involve unsteady mass addition, newtypes of experiments are required.A typical unsteady ow actuator comprises a noz-

zle or ori�ce that communicates with a cavity. Thecavity includes either a exible membrane or a mov-ing wall. Unsteady deformation of the exible mem-

17

ber inside the cavity results in a local pressurechange that drives uid through the nozzle and intoan external ow. In practical applications, the ex-ternal ow is turbulent without the presence of theactuator; the ow inside the actuator is often a mixof laminar and turbulent regions.A good place to start with the experiments is

where a number of calculations of ow actuatorshave begun, i.e., with actuators in no- ow and lam-inar ow environments. Such experiments are cur-rently being performed at Langley. These experi-ments are time-consuming and require attention todetails not typically recognized as important. For in-stance, the deformation of the exible member mayneed to be simulated or modeled with greater �delitythan expected, time lags between the electrical ac-tuation of the exible member and its actual me-chanical actuation need to addressed, and ow mea-surements in largely inaccessible locations, like theactuator cavity, are desired.The problems prove much more di�cult when the

external ow is turbulent. In that case, character-istics of the external turbulent boundary layer mustbe measured. In addition, the interaction of the owstructures in the turbulent boundary layer with the ow in the cavity must be characterized. Validatedcriteria for choosing a particular turbulence model-ing approach for a speci�c type of actuator ow havenot been established. However, we speculate that ifa signi�cant amount of the external ow is ingestedinto the actuator cavity, then a full LES calculationmay be the only reasonable approach. On the otherhand, if the actuator is biased such that it does notingest the external ow, a DES calculation might beable to calculate the relevant ow features. One con-cern with DES in this sort of application is that therelevant unsteady ow regions are close to the wall;hence the model may revert to its RANS character-istics in these regions and be too di�usive. Othercomposite LES/RANS models may be more appro-priate for such ows. Although explorations of theseproblems are ongoing at Langley, no de�nitive guid-ance can be provided to the CFD community yet.

5.2.2 Consistency ConcernsGatski50 raises questions as to the formal con-

sistency of the composite methods. The compos-ite methods assume that the ow variables from theRANS regions match smoothly with the correspond-ing variables from the LES regions. Gatski50 showsthat such a match cannot be taken for granted, evenwithout considering the consistency issues associ-ated with the coupling of the temporal averages usedto derive RANS models and the spatial �ltering usedto derive SGS models for the LES regions. To date,

these consistency issues have not been addressed bythe modeling community.

5.2.3 Calculation Cost

One of the problems with using a compositeLES/RANS method is that for the LES region tobe reasonably simulated, it needs to be not onlytime-dependent, but also three-dimensional. With-out three-dimensionality, the vortex stretching andtilting mechanisms, which are so important in real ows, cannot be appropriately simulated. The ne-cessity to do 3-D calculations dramatically raises thecosts of the calculations, both in terms of computermemory and in terms of run time. Turnaround timefor circular cylinder calculations on eight processorsof an SGI Origin 2000 can easily be multiple weeks.Grid and time-step studies therefore turn the activ-ity into a very long process, even for such a relativelysimple ow. The time can be signi�cantly reducedthrough the use of more extensive parallel process-ing, but the parallel e�ciency must be maintainedover approximately 100 processors for the calcula-tion turnaround times to become su�ciently shortfor good systematic studies to be performed. Thoselike Travin et al.143who have performed such studieshave added greatly to our understanding of the per-formance of these methods. More work along thoselines needs to follow.

5.3 Langley E�ort

The e�orts at Langley in this area have concen-trated on the development of appropriate validationexperiments and also in doing some of the carefulstudies needed to calibrate the performance of DESwith di�erent codes. Some additional developmen-tal work on new families of composite LES/RANSmodels has been funded by NASA Langley ResearchCenter. Although the DES model has by far thegreatest following, and therefore the most thoroughtesting, some aspects of the model raise questionsas to its suitability for actuator-type computations.In particular, the fact that the boundary betweenthe RANS region and the LES region is entirely dic-tated by grid size and distance from the wall makesthe simulation of near-wall recirculation regions dif-�cult. Other versions of composite models may workbetter in ows where near-wall recirculation regionsare anticipated.

Another important role played by Langley is toact as a clearinghouse for ideas related to unsteadyturbulence models. Langley researchers can provideappropriate suggestions and criticism of emergingconcepts so that the developers can address theseconcerns early in the models' development. As anexample of this interaction, the consistency concerns

18

discussed earlier are now being addressed in at leastone new model under development.

At NASA Langley Research Center a concertede�ort is underway to measure and compute the owin active ow control devices. This e�ort brings to-gether experimentalists, turbulence modelers, andcomputationalists who all have been working to-gether to develop appropriate techniques for the sim-ulation of actuator ows.

6 Actuator Boundary Conditions

6.1 Overview: Current Practices

Interest in active ow control for drag or noisereduction, ow vectoring, mixing enhancement, andseparation control has stimulated the recent develop-ment of innovative synthetic jet actuators that cre-ate localized disturbances in a ow�eld. Syntheticjets are generated by a dynamic uid actuator con-sisting of a cavity enclosed by one (or more) movingdiaphragm(s) driven into transverse oscillations attheir resonance frequency. The distinctive featureof these actuators is that they have minimal powerrequirements and have jet-like characteristics with-out the need for mass injection. Although net massinjection into the overall system occurs over eachcycle of operation, the momentum transfer into theembedding ow is nonzero. These features enablesynthetic jets to e�ect signi�cant global modi�ca-tions in the embedding ow on scales that are oneto two orders of magnitude larger than the charac-teristic length or force scale of the jets themselves.

In recentyears, because of considerable improvements in com-puter resources, more attention is being devoted tonumerical simulation and optimization of syntheticjet actuators.82; 46; 58; 33; 147; 29; 64; 89; 118; 77; 87 Thepresence of several temporal and spatial scales andmoving boundaries in the problem makes simula-tion of such unsteady ows computationally very ex-pensive. To reduce the computational cost, severalapproaches have been developed. All the methodscan be divided into two classes: 1) simpli�ed modelswithout simulation of the ow inside the actuator,and 2) numerical simulation of both the exterior andcavity ows.

In the �rst class of methods, a synthetic jet gener-ated by harmonic motion of the actuator diaphragmis modeled by using simpli�ed boundary conditionsimposed at the ori�ce exit. In the work of Donovan,Kral, and Cary,82 and Kral et al.46, the ow withinthe cavity is not calculated and the perturbation tothe ow�eld is introduced through the wall-normal

component of velocity at the ori�ce exit

v(x; y = 0; t) = VA(x)sin(!t)u(x; y = 0; t) = 0

(21)

where x and y are the ow streamwise and cross-stream directions, respectively. Di�erent spatialvariations of VA(x) over the ori�ce have been con-sidered. Numerical experiments 46; 82 indicate thata \top hat" distribution most closely matches theexperimental data. A modi�ed boundary conditionon the pressure at the ori�ce is introduced througha consideration of the normal momentum equation.Taking into account time-harmonic velocity pertur-bations, this condition, obtained under the assump-tion that the ow is incompressible, becomes

@p

@y= ��VA(x)!cos(!t) (22)

The numerical simulations based on the boundarycondition (eqs. 21 and 22) show good qualita-tive agreement with the experiments.123; 122; 131 Asimilar approach modeling the actuator as a blow-ing/suction boundary condition, which can be fullyspeci�ed in advance of the calculation, is used byothers.29; 58; 64; 89

An alternative technique for modeling syntheticjet actuators has been proposed by Carpenter et al.33

This theoretical model for the actuator is based onclassic thin plate theory for the diaphragm dynam-ics. The ow through the ori�ce is modeled usingunsteady pipe- ow theory. This approach is basedon the assumption that streamlines in the ori�ce exitare parallel to its axis, which is an adequate approxi-mation if the length-to-diameter ratio is much largerthan unity. The governing equation for the ori�ce ow is given by

�@v

@t+�vjvj

2l= �

@p

@y+ �r

@

@r

�1

r

@v

@r

�(23)

where y and r are the axial and radial coordinates,respectively, v is the axial velocity, l is the ori�celength, and density � is taken to be the instanta-neous mean of the cavity and external densities. Theinertial term �vvy is modeled approximately by thesecond term on the left-hand side of equation (23).The dynamics of the air in the cavity is ignored andthe pressure there is calculated by means of the per-fect gas law. The cavity and the external boundarylayer ow�elds are linked by requiring continuity ofvelocity and pressure at the ori�ce exit. In Carpen-ter, Lockerby and Davies,33 only the blowing phaseof the actuator dynamics has been studied.

19

The second class of methods is based on a directnumerical simulation of the entire problem includ-ing the ow inside the actuator. Rizzetta, et al. 118

solve the unsteady compressible Navier-Stokes equa-tions in the external region, the cavity itself, andthe throat of the actuator on separate grids that arelinked with each other through a Chimera method-ology. The membrane motion is simulated by vary-ing the position of appropriate boundary points. Asfollows from the numerical calculations presented inreference [118], the internal cavity ow becomes peri-odic after several cycles. Therefore the velocity pro-�le across the jet exit at each time-step was recordedand was used as a boundary condition in subsequentruns involving the external domain only. For com-putations that consider only the upper exterior do-main, the transverse and span-wise velocity compo-nents (orthogonal to jet axis) are set to zero and theinviscid normal momentum equation, expressed as

@p

@y= ��

�@v

@t+ v

@v

@y

�(24)

is used to establish the pressure. The ori�ce exitdensity is extrapolated from the interior solution.This approach provides more accurate descriptionof the ow details at the ori�ce than the simpli�edboundary condition of equations (21) and (22). Sim-ilar direct numerical simulation of the external andcavity ows has been performed by Joslin et al.77

and shown good agreement with experimental re-sults.To avoid the integration of the Navier-Stokes

equations on a moving grid, an alternative techniqueis used by Lee and Goldstein.87 The main idea of themethod is to impose a localized body force along de-sired points in the computational mesh to bring the uid there to a speci�ed velocity so that the force hasthe same e�ect as a solid boundary. The desired ve-locity is incorporated in an iterative feedback loopto determine the appropriate force. For a movingboundary with velocity VA(x; t), an expression forthe body force is

F (x; t) = �

tZ0

(v � VA)dt0 + �(v � VA) (25)

where v is the uid velocity, and � and � areuser-de�ned constants that are negative and can betreated as the gain and damping of the force �eldwith dimensions of M=(L3T 2) and M=L3T , respec-tively. This approach allows a moving diaphragmwithout using a time-dependent coordinate trans-formation.

6.2 Bottlenecks

In spite of the fact that the methods mentionedabove have successfully been used for modeling syn-thetic jet actuators, several issues persist. One ofthe main drawbacks of the �rst class of methods isthat the simpli�ed boundary conditions do not pro-vide conservation of mass, momentum, and energythrough the actuator ori�ce. Because these methodsuse the normal momentum equation to calculate thepressure, whereas the other quantities are extrapo-lated or prescribed analytically, this boundary con-dition does not satisfy the governing equations at theboundary and, therefore, does not provide the con-servation properties. Another disadvantage of theboundary condition [eqs. (21) and (22)] is its in-ability to account for changes in the pressure �eldcaused by the interaction of the external boundarylayer and actuator. Furthermore, as has been shownin Lee and Goldstein,87 the real streamwise velocitypro�le and the velocity component in the cross- owdirection are far from the analytical expressions ofequation (21).

The main problem associated with the secondclass of methods is complexity/cost. The numericalcalculation of the cavity ow requires a large numberof grid points. For geometries �tted with multipleactuators, grid requirements for the actuators couldexceed those of the exterior ow, and would con-tribute extensively to the computational cost. An-other consideration is the actuator Mach number,which varies from 0.001 (near the diaphragm) to 0.1(at the ori�ce exit). This variation of the ow pa-rameters from fully incompressible in the actuatorto fully compressible in the exterior region imposesvery severe requirements on a numerical method andincreases the algorithm complexity.

6.3 Langley E�ort

As follows from the literature overview presentedabove, the research in the area of active ow controlis of empirical nature, mainly due to the computa-tional cost involved and lack of con�dence in compu-tational methods for such complex time-dependent ows. A strong e�ort 147 is currently underwayat Langley toward constructing a new methodologythat combines the accuracy and conservation prop-erties of the simulation methods with the e�ciencyof the techniques based on simpli�ed boundary con-ditions. In contrast to the methods found in theliterature, the new approach uses a reduced-ordermodel of 2-D or 3-D actuators. In other words,the multidimensional actuator is simulated by solv-ing the time-dependent 1-D Euler equations similarto those used for the quasi-one-dimensional nozzle

20

Time

Ma

ssR

ate

Err

or

0 200 400 600 800

0

0.05

0.1

0.15

0.2blowing/suction boundary conditionreduced order model

Figure 3. Comparison of the mass rate errorsfor the 2-D actuator problem obtained with thequasi-1-D actuator model and the nonconservativeboundary condition (21,22).

problem. The simpli�ed actuator model has severaladvantages. First, this approach provides conser-vation of not only mass, but also momentum andenergy. Second, the new method is much more e�-cient in terms of computational time compared withthe 2-D or 3-D numerical simulation of the ow�eldin the actuator. Third, this reduced-order modelretains some multidimensional features of the realis-tic actuator that are governed by the area variationof the quasi-one-dimensional nozzle, and, therefore,can be used for qualitative study of the resonancecharacteristics of the actuator. As follows from thecomparison presented in �gure (3), the conventionalboundary conditions based on the normal momen-tum equation do not provide mass conservation. Themaximum mass rate error, which occurs during thesuction stage, is of the order of 15 percent if thenormal momentum equation is used as a boundarycondition for pressure. In contrast to the conven-tional approach, the new method provides conserva-tion of mass, momentum, and energy. As a result,the mass rate error is reduced by one to two ordersof magnitude compared with that obtained with theblowing/suction boundary condition [eqs. (21) and(22)]. These preliminary results are very encourag-ing. In our future work, we will focus on calibrationof the new methodology with readily available ex-perimental data and numerical simulation of time-dependent ows encountered in active ow control

environment.

7 Conclusions

The current status of time-dependent algorithmsis presented. Special attention is given to algo-rithms used to predict uid actuator ow�elds. Theoverview begins by considering algorithmic issuesthat could greatly improve the temporal e�ciencyof actuator simulations. The general state of time-dependent turbulent models for nonstationary owsis then assessed. Finally, an e�cient new uid actu-ator boundary condition is presented. Each sectionbegins by describing the current state of the art in-cluding notable impediments in the �eld, and con-cludes with a summary of current Langley e�orts.Profound improvements in the e�ciency of tempo-

ral algorithms could be achieved in the next decade.Notable leverage in time-dependent methods existsin the following algorithmic areas: 1) implementa-tion of high-order (p � 3) temporal schemes, 2)implementation of high-order (p � 3) spatial algo-rithms, and 3) convergence acceleration techniquesfor complex high-Reynolds number ows. Signi�-cant impediments exist in each of these three cate-gories.High-order schemes need huge time steps to uti-

lize their full potential. Algebraic solvers with rapid,time-step independent, convergence rates are neces-sary for these schemes. The principal impedimentfacing the implementation of high-order temporalschemes is the need for robust and rapid algebraicequation solvers. Error estimation, variable time-stepping and automated iteration termination willimmediately follow. Current fourth-order schemesare asymptotically close to being optimal, and fur-ther increases in e�ciency are di�cult to obtain.High-order spatial operators must be exible

enough to accommodate complex geometries, gridadaptation and nonlinear instability. Methods uti-lizing compact locality (unstructured methods) areadvantageous when addressing complex geometriesand grid adaptation. Unstructured high-order for-mulations include Finite-Elements (FE), k-exact Up-wind Finite-Volume (FV), or k-exact ENO FV. Theprincipal impediment facing the implementation ofhigh-order spatial operators is their lack of nonlinearstability resulting from unresolved features and dis-continuities. Extensive research is currently under-way within the FE community. Some formulationspossess a nonlinear L2 stability property. It is stillunknown whether this stability is a strong enoughfor general purpose solvers, or whether stronger sta-bility conditions (TVD, TVB, ENO) must be pur-sued.

21

A wide variety of di�erent iterative methods arecurrently used. Nevertheless, the convergence rateof current state of the art iterative solvers is poor.Complex high Reynolds number 3-D simulationsconverge at per cycle rates in the range 0:96� 0:98.The principal impediments for rapid convergence arehigh aspect ratio cells in the turbulent boundary lay-ers, and low Mach number regions in the ow.

Conventional URANS turbulence models are notvery accurate for nonstationary turbulent ows. Anew class of composite LES/RANS schemes has beendeveloped to address these inadequacies. Amongthese is the detach eddy simulation (DES) approach,and is currently being veri�ed and validated. TheDES scheme is very expensive because is requiresa full 3-D time-dependent simulation of the ow inquestion. The principal impediment facing all com-posite approaches is validation against high qualityexperimental data. The accuracy and consistency ofthe blended region between the outer LES and innerRANS regions is a concern with composite methods.The accuracy of this region is extremely importantin uid actuator simulations.

An optimistic estimate indicates that successfulresearch could improve overall e�ciency by O(101=2)for temporal algorithms, by O(103=2) for spatial al-gorithms, and by O(103=2) for convergence accelera-tion. Improvements in the modeling of uid actua-tors could account for a factor of two. Moore's lawpredicts that increases in computer hardware willyield O(101) improvements. All these e�ects can becombined in a multiplicative sense to yield poten-tial improvements of O(104). Improvements of thismagnitude would allow us to do 3-D optimizationstudies based on DES turbulence models, including uid actuator design and resonance, and actuatorplacement and coupling studies.

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