1

Department of Engineering

CFD models of transitional flows

ByDavide Di Pasquale

Department of Engineering,University of Leicester,

Leicester, LE1 7RH, UK.

Thesis submitted to the University of Leicester in accordance with the

requirements for the degree of Master of Philosophy

Supervisor: Dr. Aldo RonaCo-supervisor: Dr. Shian Gao

June 2012

2

Abstract

CFD models for transitional flows

Davide Di Pasquale

Department of Engineering

University of Leicester

In favour of the more widely investigated laminar and turbulent regimes, the study ofthe transitional flow regime has received a lower research effort. The important effect oflaminar-turbulent transition is not included in the majority of todays engineering CFDsimulations. This thesis deals with the problem of modelling a low Reynolds numberzero pressure gradient boundary layers in an incompressible flow without any heat-transfer effects. The zero-pressure gradient transitional boundary layer is one of thecanonical shear flows important in many applications and of large theoretical interest.An overview of the more widely used approaches to model transition in ComputationalFluid Dynamics (CFD), has shown the challenge of modelling transition by CFD. Theapproaches are compared to one another, highlighting their respective advantages anddrawbacks. This work then progress to document some of the precautions that arerequired to interpret and use the European Research Community on Flow, Turbulenceand Combustion (ERCOFTAC) dataset to calibrate CFD codes.Finally, this work implements and tests the transition model of Suzen and Huang andthe laminar kinetic energy transition method by Walter and Leylek in the in-housecomputational fluid dynamics scheme Cosmic. Cosmic is a finite-volume in-house CFDcode written in FORTRAN90. It is a multi-block Navier-Stokes solver, which usesMPI(Message Passing Interface) and is capable of handling complex geometries. Thetest cases are simple flat plate experiments. The test cases are used to test the predictingcapabilities of two different transitions model, under various stream turbulenceintensity, Reynolds number variations.After having investigated both models it is possible to state that the laminar kineticenergy method is more reliable with respect to the intermittency transport method whenthe flow-field is subjected to a lower free-stream turbulence intensity case, but bothmodels have shown similar behaviour in case of higher free-stream turbulence intensity.

3

Acknowledgements

I would like to express my profound gratitude to my supervisor Dr. A. Rona for his

support and suggestions over the period of my studies. I would also like to especially

thank Prof. Paul Gostelow for his invaluable help and for many interesting discussion

throughout the years.

I cannot forget my colleagues and friends at the University of Leicester, above all

Marco Grottadaurea, Mohammed Fekry Farah El-Dosoky, Ivan Spisso, Manuele Monti,

and Pietro Ghillani. Useful discussions took place in front of a lovely Italian coffee or

just over my desk.

I wish to thank Gian Franco Marras, Cristiano Calonacci and Nicola Varini, Cineca,

Casalecchio di Reno, Italy, who provided me with useful suggestions on how porting

the code in the Cineca new machine that is based on CPUs having a completely

different architecture (IBM Power6) with respect to the clusters previously used (based

on standard x86-64 processors) and with useful advices that made possible to improve

the parallelization algorithm. This work was supported by the HPC-Europa2

Transnational Access programme, financed by the European Community - Research

Infrastructure Action Structuring the European Research Area of the Seventh

Framework Programme. This gave me access to the SP6 High Performance Computing

facilities at Cineca.

Finally, I would like to thank my family for the never ending support they rendered to

me throughout my life. They gave me the opportunity to travel as well as study a subject

that I was passionate about, without regrets and without questioning my choice. I would

also like to thank all my friends for their cooperation during my stay here at University

of Leicester. I also wish to thanks my girlfriend Daniela for her support and patient.

This research project has been supported by a Marie Curie Early Stage Research

Training Fellowship of the European Communitys Sixth Framework Programme under

contract number MEST CT 2005 020301. The grant offered me the opportunity to

network at international conferences and find new friends and colleagues among the

AeroTraNet Marie Curie Early Stage Training fellows.

4

Contents1. Introduction......................................................................................................................... 10

1.1. Motivation of research ................................................................................................ 10

1.2. Background ................................................................................................................. 11

1.2.1. Transition Overview............................................................................................ 11

1.2.2. Different mechanisms of transition ..................................................................... 13

Natural transition............................................................................................................. 13

Bypass transition ............................................................................................................. 15

Separated flow transition................................................................................................. 17

Wake-induced transition ................................................................................................. 18

Reverse transition............................................................................................................ 19

1.2.3. Parameters affecting transition............................................................................ 19

Free stream turbulence .................................................................................................... 19

Effect of pressure gradient .............................................................................................. 20

Effect of surface roughness............................................................................................. 21

Effect of compressibility ................................................................................................. 21

Effect of curvature........................................................................................................... 22

Effect of heat transfer...................................................................................................... 23

1.3. Summary remarks on transition flow physics ............................................................. 24

2. Review of transition models for CFD ................................................................................. 25

2.1. Theoretical framework ................................................................................................ 25

2.2. Transitional methods for CFD .................................................................................... 26

2.2.1. Stability theory approach .................................................................................... 26

2.2.2. Low Reynolds number turbulent closure approach............................................. 29

2.2.3. The intermittency transport method with empirical correlations ........................ 30

2.2.4. The laminar fluctuation energy method .............................................................. 32

2.2.5. DNS for transition ............................................................................................... 35

2.2.6. LES for transition................................................................................................ 36

2.2.7. The ...............................................................................................model 36

2.2.8. The intermittency and vorticity Reynolds number approach .............................. 37

2.3. Summary remarks on transition modelling approaches .............................................. 38

3. Review of experimental transition data for benchmarking CFD schemes .......................... 41

3.1. Introduction................................................................................................................. 41

3.2. Database integrity........................................................................................................ 41

3.3. Summary remarks on the benchmark data for transition methods.............................. 47

5

4. Numerical scheme............................................................................................................... 48

4.1. Introduction................................................................................................................. 48

4.2. Governing equations ................................................................................................... 48

4.3. Short-time Reynolds averaged Navier-Stokes equations ............................................ 50

4.4. Space discretization..................................................................................................... 52

4.5. Calculation of inviscid fluxes...................................................................................... 55

4.6. MUSCL data reconstruction ....................................................................................... 57

4.7. TVD scheme ............................................................................................................... 59

4.8. Entropy correction for the Roe scheme....................................................................... 62

4.9. Calculation of viscous fluxes ...................................................................................... 63

4.10. Time integration...................................................................................................... 65

4.10.1. Runge-Kutta scheme ........................................................................................... 65

4.11. Boundary Conditions .............................................................................................. 67

4.11.1. Inviscid wall ........................................................................................................ 68

4.11.2. Non-slip wall....................................................................................................... 68

4.11.3. Far-field boundary condition............................................................................... 69

4.11.4. Symmetry plane .................................................................................................. 70

4.11.5. Subsonic outflow................................................................................................. 70

4.11.6. Subsonic inflow................................................................................................... 71

5. Implementing and testing two transition models ................................................................ 73

5.1. Turbulence closure without any transition model ....................................................... 73

5.1.1. turbulence model ..................................................................................... 73

5.2. Source term ................................................................................................................. 81

5.3. The Menter Shear Stress Transport (SST) model ....................................................... 81

5.4. Turbulence closure with transition model................................................................... 84

5.4.1. The classical correlation method of Suzen and Huang ....................................... 84

5.4.2. Model implementation in the 2D version of the in-house CFD code Cosmic .... 87

5.4.3. Test cases ............................................................................................................ 91

5.4.4. Zero pressure gradient transitional boundary layer results ................................. 93

5.5. The Laminar Kinetic Energy Approach...................................................................... 98

5.5.1. Transport equation for the turbulent kinetic energy............................................ 98

5.5.2. Transport equation for the laminar kinetic energy ............................................ 101

5.5.3. Transport equation for the specific turbulent dissipation rate ....................... 102

5.5.4. Model implementation in the 3D version of the in-house CFD code Cosmic .. 104

5.5.5. Zero pressure gradient transitional boundary layer results ............................... 105

5.6. Comparison between the two models ....................................................................... 109

6

6. Code parallelization using MPI......................................................................................... 113

6.1. Single domain decomposition ................................................................................... 113

6.2. Recursive domain decomposition ............................................................................. 115

6.3. Input/output............................................................................................................... 119

6.3.1. General strategy ................................................................................................ 119

6.3.2. Input of parameters from external text file........................................................ 120

6.3.3. Use of external cgns file.................................................................................... 120

6.4. Parallelization performance ...................................................................................... 121

7. Conclusion and Recommendations for future Work......................................................... 124

8. References......................................................................................................................... 125

7

List of Figures

1.1 Simplified turbulent spot structure. 14

1.2 Schematic of transition process. 15

1.3 Schematic of disturbance level affecting transition, adapted from Morkovin et al.(1994). 17

2.1 Illustration of wall-limiting concept leading to splat mechanism" for production ofkl. 33

3.1 Cross-stream velocity measurements for T3A (a) and T3AM (b). 42

3.2 Pressure vs. distance in x1 for T3A case. 41

3.3 Pressure vs. distance in x1 for T3AM case. 44

3.4 Free-stream velocity along the flat plate for the T3A test case. 45

3.5 Free-stream velocity along the flat plate for the T3AM test case. 45

3.6 Streamwise pressure gradient evaluated by the Thwaites method for the T3A test

cases in the laminar part of the boundary layer. 46

3.7 Streamwise pressure gradient evaluated by the Thwaites method for the T3AM test

cases in the laminar part of the boundary layer. 47

4.1 The interface between two adjacent cells. 56

4.2 The four-cell stencil used to build the MUSCL scheme. 57

4.3 The TVD second order scheme region. 60

4.4 The constructed control volume for the diffusive fluxes calculation. 62

4.5 The ghost cells (dashed line) around the computational domain (the solid thick line).

66

5.1 Flow chart of the in-house code Cosmic without intermittency transport model. 88

5.2 Transition model low chart. 89

5.3 Computational mesh: For clarity, one point every 12 in both the x and y directions is

shown, and the x-axis to y-axis ratio is 0.4. 90

8

5.4 Inflow velocity profile for the T3A test case. 91

5.5 Inflow velocity profile for the T3AM test case. 91

5.6 Free stream turbulence intensity decay for test case T3A. 92

5.7 Comparison of the experimental shape factor coefficient against computational

results for the T3A test case. 95

5.8 Comparison of the experimental shape factor coefficient against computational

results for the T3AM test case. 95

5.9 Comparison of the experimental and skin friction coefficient against computational

results for the T3AM test case. 96

5.10 Computational mesh. The total length of the test section is 1850mm; 150mm

before the leading edge. 105

5.11 Comparison of the experimental skin friction coefficient against computational

results for the T3A test case. 106

5.12 Comparison of the experimental skin friction coefficient against computational

results for the T3AM test case. 107

5.13 Comparison of the predicted skin friction coefficient for the T3A case. 108

5.14 Comparison of the predicted skin friction coefficient for the T3AM case. 108

5.15 Variation of Reynolds number based on momentum thickness along the flat

plate for T3A case. 109

5.16 Variation of Reynolds number based on momentum thickness along the flat

plate for T3AM case. 110

6.1 Example of the decomposition of one block model via SDD MPI Cartesian

communicators. 112

6.2 Exchanged cells between processor 0 and processor 1 in green. 113

9

6.3 Example of the decomposition of a two blocks model via MPI Cartesian

communicators. 114

List of Tables

3.1 Fluctuating velocity and their ratio at two different boundary layer thicknesses forthe two test cases. 43

4.1 The values of MUSCL parameters. 58

5.1 Table of stretching factor. 104

6.1 RRD code variables. 115

6.2 SSD performance on the CINECA cluster. 121

List of abbreviations

CFD Computational Fluid Dynamics

CGNS CFD General Notation System

DES Detached Eddy Simulations

DNS Direct Numerical Simulation

ERCOFTAC European Research Community on Flow, Turbulence and Combustion

FST Free-Stream Turbulence

HPC High Performance Computing

LES Large Eddy Simulation

LPT Low Pressure Turbine

MPI Message Passing Interface

OpenMP Open Multi-Processing

PDE Partial Differential Equations

RANS Reynolds-Averaged Navier-Stokes

RDD Recursive domain decomposition

SDD Single domain decomposition

SIG Special Interest Group

SMC Second-Moment-Closure

T-S Tollmien-Schlichting

10

1. Introduction

1.1. Motivation of research

Understanding, predicting and controlling laminar-to-turbulent flow transition is of

great interest to engineers because of the wide range of practical applications in which

transition is significant. Despite the enormous amount of research effort devoted to it,

the current understanding of the problem is still far from complete. Most of the

difficulty in the understanding lies in the large number of inter-linked factors that affect

transition. In order to better understand this area, research in this topic is essential.

While the importance of transition phenomena for aerodynamic and heat transfer

simulations is widely accepted, it is difficult to include all of these effects in a single

model. If the whole process can be understood much better, namely if it is possible to

evaluate the influence of each parameter on the transition, it may be possible in the

future to design specific system that control transition.

A turbulent boundary layer flow has typically one order of magnitude higher skin

friction and heat transfer than an equivalent laminar boundary layer due to increased

mixing between the boundary layer and free-stream flow. The increased wall shear

stresses due to turbulence result in significantly larger viscous drag on aircraft wings. It

is estimated that fuel savings of up to 25% would be possible for a large commercial

aircraft if laminar flow could be maintained over the wings (Thomas 1985, Saric 1994).

On the other hand, separation and stall of low Reynolds number aerofoils and turbine

blades can be significantly improved if the boundary layer is turbulent. Turbomachinery

designers have often tended to induce transition to a turbulent boundary layer at a fixed

location in order to avoid working with intermittent flow.

An accurate prediction of flow transition is imperative in designing more efficient

turbomachines. Losses have been observed on turbine blades working in a low static

pressure environment, particularly affecting aircraft at cruise altitude or the later stages

of steam turbines. Some of these reductions in performance are attributed to separation

of laminar boundary layers (Suzen and Huang, 2004).

The better mixing properties of a turbulent flow are an asset for designing efficient

combustion systems and the increased heat transfer coefficient of a turbulent flow is

desirable in some heat exchanger designs. In gas turbine engines, an accurate estimation

of the thermal loads on turbine blades strongly depends on the prediction of the onset of

turbulence. Wind turbines for renewable energy generation stand to benefit from this

11

research, as it is may be possible to improve the relative fraction of time in which a

wind turbine blades work in the laminar flow regime. Specifically, off-shore

aerogenerators can have a blade span of 70m. The large tangential velocity gradient

along the span enables the blade to work over a large range of chord based 'local'

Reynolds numbers. This implies that different transitional flows may co-exist along the

blade span, with competing transitional modes changing the flow during one revolution.

The opportunity for increasing the performance of these aerogenerators, in terms of lift

to drag ratio, is therefore attractive, given the relative maturity of conventionally

designed blades by the established RANS CFD methods. The heat-shielding

requirement of hypersonic re-entry vehicles and the performance and detection of

submarines and torpedoes are other applications for which predicting and controlling

transition is essential. Therefore, a better understanding of transition will lead to better

prediction of a wide range of wall-bounded flows with an overall improvement of the

design.

1.2. Background

1.2.1. Transition Overview

Whenever a fluid flows over a solid body, such as the hull of a ship or an aircraft,

frictional forces retard the motion of the fluid in a thin layer close to the solid body. A

boundary layer is formed by the retardation of particles near to a surface, generating a

cross-stream velocity gradient. The development of this layer is a major contributor to

flow resistance and is of great importance in many engineering problems. The concept

of a boundary layer is due to Prandtl (1904) who showed that the effects of friction

within the fluid (viscosity) are significant only in a very thin layer close to the surface.

Fluid flows are generally referred to as being either laminar or turbulent. Transition is a

complex phenomenon, defined as the whole process of change from laminar to turbulent

flow. Laminar flow is ordered and layered while turbulent flow is chaotic and not

predictable. Almost all laminar flows eventually develop into turbulent flows. If the

flow velocity is high enough, the flow in this layer will eventually become unordered,

swirling and chaotic or simply described as being turbulent. One simple example is

water streaming out from the water tap; at low velocities, the flow is laminar but at

higher velocities, the flow becomes turbulent.

The transition from laminar to turbulent flow state was first investigated by Reynolds

(1883) who made experiments on the flow of water in glass tubes, visualizing the flow

12

state using ink as a passive marker. He found that the flow state was determined solely

by a non-dimensional parameter that is since then called the Reynolds number,

=

(1.1)

where

V is the mean velocity of the object relatively to the fluid (m/s)

L is a characteristic linear dimension (m)

is the dynamic viscosity of the fluid (kg/ms)

is the density of the fluid (kg/m3)

The Reynolds number is a measure of the ratio between the inertial and the viscous

forces in the flow, so that a high Reynolds number flow is dominated by inertial forces.

Inertial forces associated with fluid mass try to amplify disturbances in the flow.

Viscous forces try to damp disturbances. Depending on the flow characteristics and the

fluid properties, like density, viscosity, velocity gradient, proximity to the boundaries,

etc., disturbances can be damped or amplified. If these disturbances are damped the

result is a laminar flow, otherwise the flow is turbulent.

The origin of turbulence and the accompanying transition from a laminar to a turbulent

regime is of fundamental importance for the whole science of fluid mechanics.

Flow transition is an important phenomenon in many aspects of fluid flow, compared to

the more widely modelled laminar and turbulent regimes; the study of this topic

received comparatively less attention. Transition may be directly relevant to those

physical situations where a flow is observed to change from laminar to turbulent, as it

often happens, for example, on aircraft wings or past turbine blades. Transition affects

strongly the evolution of losses and other factors of practical significance, such as the

distributions of wall shear stress and surface heat transfer (Gostelow, J. P et al., 1989).

It is accompanied by many changes in flow characteristics. Specifically, an abrupt

change in the law of resistance occurs with transition and both the skin friction and heat

transfer increase considerably in the turbulent flow. The most important feature of the

phenomenon of transition is the increased diffusivity in the flow. Transition is a

complex phenomenon. The main reason of its complexity is that, besides the

simultaneous presence of turbulent and laminar flow, there is also the interaction

between the two phases. To put it simply, during transition, the two phases of a laminar

and turbulent flow exist together and alternate as a function of time. Furthermore,

transition in boundary layers is affected by several parameters, such as the stream-wise

13

pressure gradient, noise, the Reynolds number, the surface roughness, the wake

unsteadiness and turbulence intensity, making its prediction very difficult. Moreover,

transition occurs through different mechanisms in different applications.

1.2.2. Different mechanisms of transition

There are a number of different transition mechanisms documented in the literature.

These depend on the turbulence level of the external flow, the pressure gradient along

the laminar boundary layer, the geometrical details, and the surface roughness.

Transition to turbulence in boundary layer flows may follow different routes depending

on the flow situation. In all cases, where disturbances that enter the boundary layer

grow in amplitude, transition may occur.

The different modes of transition are natural transition, bypass transition, separated-

flow transition, periodic-unsteady transition, and reverse transition. Bypass transition

and periodic-unsteady transition are caused by the disturbances in the external flow,

such as free stream turbulence and pressure gradient, and they are observed in majority

of the turbomachinery applications. Separated-flow transition occurs in the free shear

layer and may or may not involve Tollmien-Schlichting (T-S) waves. This mode of

transition is observed in compressors. Reverse transition is observed in nozzles with

highly accelerating flows.

Natural transition

The early research on transition was based on inviscid stability theory, which suggested

that all boundary layer flows are only unstable if there is an inflexion point in the

velocity profile. It was later predicted physically by Prandtl and then proven

mathematically by Tollmien that a laminar boundary layer can be destabilized by the

presence of viscous instability waves, often referred to as Tollmien-Schlichting (T-S)

waves.

In aerodynamic flows, transition is typically the result of flow instabilities, such as T-S

waves or cross-flow instability. This type of transition begins with a weak instability in

the laminar boundary layer at a critical value of the momentum thickness Reynolds

number. These instabilities are one-dimensional. These weak instabilities proceed

through various stages of amplifications and amplify into two-dimensional instabilities

and finally into three-dimensional instabilities. These instabilities further grow and form

loop vortices and then, during the growth of the waves, spanwise distortions and three-

14

dimensional non-linear interactions become relevant. Finally, areas of turbulence,

denoted as turbulent spots, start to develop in the streamwise direction. Turbulent spots,

discovered by Emmons in 1951, initiate with an irregular shape and grow in the

streamwise direction during which the initial shape is preserved. The turbulent spot are

the the key point of transition. The amount and size of the spots can be affected by

changing the velocity of the flow or by placing disturbances in the flow. These spots

grow in the streamwise direction, while the leading edge velocity of the spot is larger

than the trailing edge velocity. The spots also grow in the lateral direction as they travel

downstream. Due to their growth, the spots start to overlap each other and thus coalesce

until a complete turbulent boundary layer is obtained and thus transition is completed.

This instability becomes three-dimensional (3D) and non-linear by the formation of a

vortex loop and eventually results in a non-linear breakdown to turbulence. Therefore,

spot appears in a streamwise very narrow region in the boundary layer (BL). For

simplification, as illustrated in figure 1.1, the turbulent spot is supposed to have a

triangular shape.

Fig. 1.1: Simplified turbulent spot structure.

This process is often referred to as natural transition. It occurs at low free-stream-

turbulence (FST) level of less than 0.5%. This process of transition gradually appears

after a critical value of Re is exceeded.

The instability that initiates natural transition is via a subtle mechanism, whereby

viscosity destabilizes the waves and they begin to grow very slowly. In a low-

disturbance environment, such as the one found in free flight and in some marine

applications, quasi-two-dimensional T-S waves precede transition in a flat-plate

boundary layer and similar flow fields. Physically, T-S waves are streamwise-travelling

structures of spanwise-oriented vorticity. Various receptivity processes are responsible

for generating these instabilities, which are probably initially three-dimensional and

randomly distributed. However, quasi-two-dimensional T-S waves are preferentially

15

amplified in the boundary layer. Further downstream, the amplitude of the most

amplified waves becomes sufficiently large for non-linear effects to become important.

This process is depicted in figure 1.2

Fig 1.2: Schematic of transition process.

Bypass transition

Although the T-S transition scenario is now well understood, transition to turbulence

does not follow this path when the initial disturbances are large. Morkovin (1969) first

coined the term bypass transition for cases in which known instability mechanisms (at

the time, T-S waves only) are bypassed. Transition triggered by large-amplitude surface

roughness and free-stream turbulence is the prototypical bypass scenario.

In turbomachinery applications, the main transition mechanism is bypass transition

imposed on the boundary layer by high levels of turbulence in the free-stream (due to

the combustion process). In this case, low-frequency oscillations in the streamwise

velocity appear in the boundary layer. These oscillations are due to a streamwise streak

of alternating high and low velocity and flow visualization studies have shown that the

streaks meander slowly sideways and thereby give rise to the observed low-frequency

variations. If the energy of the streamwise velocity fluctuations is measured in the

boundary layer, it is found to have its maximum in the centre of the boundary layer and

to exhibit an initial amplification that is linear with downstream distance, in contrast to

amplified T-S waves that grow exponentially. The idea behind bypass transition is that

the disturbances in the flow cause laminar fluctuations in the boundary layer that initiate

spots, or that disturbances are strong enough to enter the boundary layer and initiate

turbulent spots immediately. For transition at high free-stream turbulence levels of

above 1%, the first and possibly the second and third stages of the natural transition

16

process are bypassed, such that turbulent spots are directly produced within the

boundary layer by the influence of the free-stream disturbances. As the spot is born, it

grows in the streamwise direction, merging with neighbouring spots, until a complete

turbulent boundary layer is obtained. The occurrence of Tollmien-Schlichting waves,

spanwise vorticity, and three-dimensional breakdowns is bypassed, which explains

the name bypass transition. Linear stability theory is irrelevant, as the turbulent spots

are generated more towards the leading edge of a plate (or turbine blade) compared to

natural transition. As a means of categorizing transition scenarios, Morkovin (1969)

introduced a transition roadmap. This roadmap was updated by Morkovin et al.

(1994) and this updated roadmap is shown in figure 1.3. The transition process can be

categorized into three main stages: receptivity, disturbance growth and breakdown.

Receptivity is the means by which disturbances enter the boundary layer and provide

the initial conditions for disturbance growth. Disturbances like freestream turbulence,

surface roughness and sound enter the boundary layer as steady and/or unsteady

perturbations. This stage is not yet well understood in all situations but is critical

because it provides the initial amplitudes, phases and frequencies of the disturbances.

Once disturbances enter the boundary layer, they proceed along one of the paths A-E on

figure 1.3 depending not only on their initial amplitudes (as first put by Morkovin) but

also, as recent thinking suggests, on other characteristics such as spatial wave numbers

and temporal frequencies. They grow or decay depending on the stability characteristics

of the flow and multiple types of disturbances may coexist and possibly interact once

they reach large amplitudes. The final stage is breakdown to turbulence. In the case of

TS waves (primary modes, path A) once the disturbances reach an amplitude near 1% of

the freestream velocity, laminar flow can no longer be sustained and a more complex

flow arises. This instability of the large-amplitude TS waves is referred to as a

secondary instability because the primary disturbance is so large that it supports a

second instability growth at frequencies that are not unstable in the original,

unperturbed basic state. The secondary instability rapidly breaks down, turbulent bursts

appear and these coalesce into fully developed turbulent flow. In the case of bypass

transition, the disturbances break down without going through secondary instability and

form turbulent bursts directly.

17

Fig 1.3: Schematic of disturbance level affecting transition, adapted from Morkovin et al. (1994).

Separated flow transition

Another important transition mechanism is separation-induced transition (Mayle, 1996),

by which a laminar boundary layer separates under the influence of a pressure gradient

and transition develops within the separated shear layer (which may or may not

reattach) as a result of an inviscid instability mechanism. Where the flow does reattach,

this reattachment forms a laminar-separation/turbulent-reattachment bubble on the

surface. The bubble length depends on the transition process within the shear layer and

may involve all of the stages listed in Figure 1.2 for natural transition. Because of this, it

is generally accepted that the free-stream turbulence level plays a large role in

determining the length of the separation bubble. Traditionally, separation bubbles have

been classified as long or short, based on their effect on the pressure distribution around

an aerofoil (Mayle, 1991).

Short bubbles reattach shortly after separation and only have a local effect on the

pressure distribution. Long bubbles can completely alter the pressure distribution

around an aerofoil. Since long bubbles produce large losses and large deviations in exit

flow angles, they should be avoided (Mayle, 1991). Short bubbles, on the other hand,

can be used to trip the boundary layer and thus allow larger adverse pressure gradients

downstream of the reattachment point. One of the major challenges lies in determining

whether or not a separation bubble will be long or short. This is aggravated by the fact

18

that small changes in either the Reynolds number or the angle of attack of an aerofoil

can cause a bubble to change from short to long (Mayle and Schulz, 1997). The sudden

change in bubble length is often referred to as bursting. It can result in a significant loss

of lift and could even cause the aerofoil to stall if the bubble fails to reattach. Separation

induced transition can also occur around the leading edge of an aerofoil if the leading

edge radius is small enough. The size of the leading edge separation bubble is a strong

function of the free-stream turbulence intensity, the leading edge geometry, the angle of

attack, and, to a lesser extent, of the Reynolds number (Walraevens and Cumpsty,

1995). Tain and Cumpsty (2000) found that the size of the leading edge bubble has a

profound effect on the downstream boundary layer. They concluded that the larger the

leading edge separation, the thicker the downstream boundary layer and thus the more

likely it is to separate under an adverse pressure gradient. Consequently, compressor

blade losses at off-design conditions are thought to be strongly influenced by

separation-induced transition near the leading edge.

Wake-induced transition

Another type of transition process is called wake-induced transition or periodic

unsteady transition (Kyriakides et al., 1999). This process is caused by the periodic

passing of wakes from upstream aerofoils or obstructions. Transition induced by wakes

or shocks, compared to the stages of transition listed in figure 1.2, appear to bypass the

first stage of natural transition. The turbulent spots are formed and convect downstream.

Turbulent spots immediately coalesce after their formation and immediately grow and

propagate downstream to form a fully turbulent boundary layer. The wake influences

the boundary layer and thus the transition process. This flow phenomenon is unsteady

as, during a certain part of the blade-passing period, the flow along turbomachinery

blades is the flow with the background turbulence, and during the remaining part of the

period, the flow is a flow with the wakes. Therefore, transition of this type is denoted as

unsteady transition. For bypass transition, the high background turbulence level drives

the transition process, while for unsteady transition the turbulence in the wake has even

higher intensity levels compared to the background turbulence.

In turbomachines, viscous wakes from the proceeding stator or rotor blade row pass

through the next rotor or stator blade row, generating unsteady pressure, surface heat

transfer, and affecting the boundary layers. This is called unsteady wake/blade

interaction (Fan and Lakshminarayana, 1996).

19

Reverse transition

Transition from turbulent to laminar flow also exists and is called reverse transition or

re-laminarisation. Reverse transition happens in cases of very strong acceleration as

occurring near the leading edge or along the rear part of the pressure side of a Low

Pressure Turbine (LPT) blade. It can also happen in flows through nozzles with strong

acceleration (Mayle, 1991). This is because the acceleration on the pressure side of most

aerofoils near the trailing edge, in the exit ducts of combustors, and on the suction side

of turbine aerofoils near the leading edge is generally higher than that for which reverse

transition occurs. Few details are known about the reverse transition process. It is

assumed that streamwise vortex lines associated with the turbulence in the boundary

layer become highly stretched as a result of the great acceleration and that vorticity

dissipates through viscous effects. Re-laminarisation involves a balance between

convection, production, and dissipation of the turbulent kinetic energy within the

boundary layer.

1.2.3. Parameters affecting transition

Transition is influenced by various factors, such as the free stream turbulence level, the

pressure gradient, surface roughness, curvature, compressibility, heat transfer, film

cooling, and acoustic disturbances. Although transition is affected by various other

secondary factors, the above-mentioned factors often determine the transition process.

The effect of above mentioned parameters on transition is detailed in the following

section.

Free stream turbulence

In a low pressure gas turbine, the key parameter affecting the onset of transition and the

spot production rate is the free-stream turbulence. The Reynolds number does not affect

the transition process itself and has no effect on the spot production rate. It corresponds

rather to a sensitization of the boundary layer to perturbations, such that the onset

location can alter.

Free stream turbulence increases the heat transfer rate in the turbulent boundary layer

and the rate of heat transfer seems to be more sensitive to free-stream turbulence.

Increasing the free-stream turbulence reduces the Reynolds number at which transition

onset occurs. By increasing the free-stream turbulence, the production of turbulent spots

increases and thus the transition length decreases. At higher turbulence levels, transition

20

occurs in a bypass mode and is completely independent of the Tollmien-Schlichting

instability.

Effect of pressure gradient

The pressure gradient has a large effect on both the transition onset and the transition

length. The general trend is that a favourable pressure gradient has a stabilizing effect

on the flow, and therefore the transition onset is more downstream. The trailing edge

velocity of the spots is found to be larger compared to the velocity in a zero pressure

gradient flow. In combination with the observation that the leading edge velocity

remains nearly unchanged, the result is that the transition length becomes larger. An

adverse pressure gradient has a destabilizing effect on the flow. Therefore, the transition

start is situated more upstream while the transition length decreases. Very strong

adverse pressure gradients may result in boundary layer separation.

The acceleration parameter is an equivalent measurement of the pressure gradient for

favourable pressure gradient flows. With an increase in the acceleration parameter, the

transition Reynolds number increases, indicating a delay in transition. At low free-

stream turbulence levels, the effect of acceleration is significant, while for flows in gas

turbines where the turbulence level is high, the effect of acceleration is negligible as the

onset of transition is controlled by the free-stream turbulence level. For adverse pressure

gradient cases, an increase in negative acceleration decreases the transition Reynolds

number, bringing an early transition onset. The effect of the turbulence level on

transition is less significant for adverse pressure gradient flows than for favourable

pressure gradient flows (Mayle, 1991). It is found in literature (e.g. Abu-Ghannam and

Shaw [1980]) that at high free-stream turbulence levels the effect of the pressure

gradient on boundary layer transition diminishes. This also supports the flat plate

approach as proposed in the abstract of this thesis. The pressure gradient can be

characterised in terms of the pressure gradient parameter at the onset of transition,

which is defined as:

=

(1.2)

where t is the boundary layer momentum thickness, is the flow kinematic viscosity,

and U is the free-stream velocity that changes with the tangential distance x along the

solid boundary.

21

Effect of surface roughness

In general, surface roughness eases the transition from laminar to turbulent flow.

Surface roughness plays a dominant role on the transition process over an aerofoil.

Increasing the surface roughness decreases the transition Reynolds number and thus the

transition occurs more upstream on a rough surface when compared to a smooth surface

aerofoil. At a high free stream turbulence level, a very rough aerofoil surface decreases

the transition length by 60% when compared to a hydraulically smooth aerofoil surface.

For a smaller roughness height, a smaller effect can be observed. The change in the

transition Reynolds number affects the heat transfer rate at the surface. Guo et al. (1998)

calculated that, for a roughness height of 25 m, the heat transfer rate increases by up to

30%. For roughness heights from 0.8 m to 2.3 m, which are representative values for

turbine blades, Abuaf et al. (1998) showed that surface polishing improves the

aerodynamic performance and reduces the heat transfer load. However, they concluded

that, on the basis of the heat load alone, the addition of the polishing steps that represent

additional time and cost provide only a limited benefit. Surface roughness promotes

transition by producing additional large-amplitude disturbances that require less

amplification to break down into a turbulent spot. However, if the roughness elements

are very small, the resulting perturbations are below the characteristic level of those

generated by free-stream turbulence and the roughness will have no great impact on

transition. From the works of Feindt (1972) and Mick (1987), Mayle (1991) concluded

that transition usually takes place due to external disturbances and so the effect of

surface roughness can be neglected for most turbomachine engine sizes. Nevertheless,

the work of Bons (2002) suggests that, when turbulence and roughness are both present,

synergies are generated that create larger effects on the skin friction factor, Cf, and on

the Stanton number, St, than by simply adding their individual effects.

Effect of compressibility

The majority of the flows in gas turbines are compressible. Measurements that study the

effect of compressibility on bypass transition are still scarce. From the experiments

available in the literature, it follows that the bypass transition mechanism remains

unchanged compared to low Mach numbers. Compressibility has only a very slight

influence on stability and transition at subsonic speeds. The Mach number has an effect

on the onset, the production rate, and spreading angle of the turbulent spot. Aside from

where transition forced by a shock wave interacting with a boundary layer, an increase

22

of the Mach number delays the onset of transition and decreases the spot production

rate. Two models can be found in literature to account for the compressibility effects on

the spot production rate: Chen & Tyson (1971) proposed a spot production rate

evolution proportional to

(+ . .) (1.3)

while Narasimha (1999) gave the spot production rate as proportional to

(+ . .). (1.4)

Clark et al. (1993, 1994, 1996) showed from heat transfer measurement by means of

thin film that, although the characteristics spot boundary velocities were unaffected by

the Mach number, the spreading angle was varying over the Mach number range 0.24 to

1.86. They noticed an important reduction of the spreading angle, by up to 30%, when

the Mach number increases.

However, at higher supersonic speeds, the compressibility has a more complex effect on

transition. Thus, the effect of Mach number on the transition onset and on the spot

production rate has to be taken into account along with the effect of shock waves. As

the Mach number increases, the onset of transition is delayed and the spot production

rate is decreased, thus increasing the transition length roughly by between 8% and 30%.

A passing shock wave from an upstream aerofoil induces a small concentrated vortex on

the pressure side of the aerofoil near the leading edge and this can cause transition as

the shock moves along the surface (Mayle, 1991).

Effect of curvature

It is generally accepted that even a relatively small longitudinal surface or streamline

curvature has a significant effect on the transport of momentum and heat in turbulent

boundary layers. By a linear perturbation analysis of the conservation of momentum

equation in which the viscous terms are neglected, it is possible to explain the tendency

for a boundary layer to be stabilized when it develops along a convex wall and the

opposite effect along concave surface (Gortler 1940). The effects of curvature on the

onset of transition under low disturbances conditions are clear; a concave curvature

leads to an early and more rapid transition and the opposite is true for a convex

curvature. This was previously known but little documentation of the transport process

in the flow was available. The effects of curvature surface on transition were studied by

Grtler (1940) and Liepmann (1943). The effect of surface curvature on the stability of

23

a laminar boundary layer against two-dimensional disturbances is very small for the

range of surface curvature that is likely to occur in practice. The Tollmien-Schlichting

waves are thus expected to behave almost the same on a concave or convex surface as

on a flat plate. However, the flow on a concave surface exhibits a different instability

due to centrifugal pressure gradient, producing a three-dimensional system of

alternating vortices with axes in the streamwise direction within the layer, as studied

theoretically by Grtler. Liepman showed that transition might happen substantially

earlier on a concave surface while it is only slightly delayed on a convex surface.

Effect of heat transfer

Heating or cooling the flow affects the transition at low levels of free stream turbulence

intensity. The wall heat transfer influences the stability and transition because viscosity

depends on temperature. A heated or cooled wall also heats/cools the fluid in its vicinity

and thus changes the fluid viscosity. The stabilizing or destabilizing effect of the wall

heat transfer is essentially due to the dependency of the dynamic viscosity on

temperature. As shown by Schlichting (1979) in a gas flow, the heat transfer from the

boundary layer to the wall stabilizes the boundary layer and leads to an increase in the

transition Reynolds number. The reduced near-wall viscosity stabilizes the flow owing

to the increased velocity gradient and decreased shape factor that affect the flow

transition. Taking the temperature dependence of the viscosity into account, the

contribution of the viscosity gradient to the curvature of the velocity at the wall of a flat

plate is:

= 1

(1.5)

If the wall is warmer than the gas outside the boundary layer, Tw> T , the temperature

gradient at the wall is negative, ( ) < 0 , and as the viscosity grows with

increasing temperature for gases, then also ) ) < 0. Since the velocity gradient

at the wall is positive, it therefore follows that ( ) > 0. Consequently, there is

a point of inflection of the velocity profiles where the curvature vanishes, leading to an

unstable boundary layer according to the point of inflection criterion.

At high free stream turbulence levels, it was observed that heat transfer has a negligible

effect on spot production rate and thus on the transition length (Mayle, 1991).

24

1.3. Summary remarks on transition flow physics

Despite the amount of research effort devoted to it, the current understanding of the

transition flow physics is still far from complete. Most of the difficulty in the

understanding lies in the large number of inter-linked factors that affect transition. In

order to better understand this area, further research in this topic is essential.

While the importance of transition phenomena for aerodynamic and heat transfer

simulations is widely accepted, it is difficult to include all of these effects in a single

model. If the whole process can be understood much better, namely if it is possible to

evaluate the influence of each parameter on the transition process, it may be possible in

the future to design specific systems that control transition. Delaying transition, hence

maintaining as much as possible laminar flow, it would lead to a significant reduction of

drag and of other losses and an improvement in engineering component performance.

On the other hand, inducing an early transition can suppress flow separation and reduce

shape drag in selected engineering applications.

25

2. Review of transition models for CFD

2.1. Theoretical framework

The theoretical framework for understanding transition is based on the stability of small

perturbations in a base flow governed by the Navier-Stokes equations. Rayleigh (1880)

derived simplified governing equations for the evolution of a small disturbance in a

parallel flow neglecting viscous and non-linear terms. Rayleigh assumed a wave-like

solution for the perturbation in both space and time of the form

(,,,) = (() ) (2.1)

By using a Fourier transform, Rayleigh reduced the simplified governing equations to

an eigenvalue problem for exponentially growing or decaying disturbances. He showed

that a necessary but not sufficient condition for inviscid instability is that the basic

velocity profile has an inflection point. However, flows without inflection points, like

boundary layers with a favourable pressure gradient, are observed to be unstable at

finite Reynolds numbers. Following a procedure similar to Rayleighs (1880), Orr

(1907) and Sommerfeld (1908) included the viscous terms that resulted in the Orr-

Sommerfeld (OS) equation that describes the evolution of two-dimensional velocity

disturbances. Squire (1933) derived a similar equation for the wall-normal disturbance

vorticity to describe three-dimensional disturbances and proved that two-dimensional

(2D) disturbances grow faster than and become unstable upstream of three-dimensional

(3D) disturbances. This result is known as Squires Theorem. This result led researchers

to focus on 2D disturbances for detecting the onset of transition until recently. However,

3D disturbances are also found to cause transition through the transient growth

mechanism that will be explained in detail later in this chapter. The first solutions of the

OS equation for a Blasius boundary layer were obtained by Tollmien (1929) and these

were later refined by Schlichting (1933). Tollmien and Schlichting analytically

predicted the existence of two-dimensional unstable waves that are the eigen-mode

solutions that grow or decay exponentially. For a zero-pressure-gradient flat plate

boundary layer, these waves are referred to as Tollmien-Schlichting (T-S) waves. The

existence of T-S waves was experimentally verified by Schubauer and Skramstad

(1947) once wind tunnels with a sufficiently low free-stream turbulence level were

developed that allowed the detection of these waves. Although the T-S transition

scenario is now well understood, transition to turbulence does not follow this path when

the initial disturbances are large.

26

2.2. Transitional methods for CFD

The modelling efforts by different research groups have resulted in a range of

turbulence models that can be used in different applications, while balancing the

accuracy requirements and the computational resources available to the CFD user.

However, the important effect of laminar-turbulent transition is not included in the

majority of todays engineering CFD simulations. The reason for this is that transition

modelling does not offer the same wide spectrum of CFD-compatible model

formulations that is currently available for turbulent flows, even though a large body of

publications is available on the subject. There are several reasons for this unsatisfactory

situation. Firstly, the transition process involves a wide range of scales, with energy and

momentum transfer predominately affected by non-linear (inertial) processes between

eddies of different scales and it is very sensitive to physical flow features such as

pressure gradients and the free-stream turbulence level. Secondly, transition occurs

through different mechanisms in different applications, such as natural transition,

bypass transition, and separation-induced transition. The third complication arises from

the fact that conventional Reynolds Averaged Navier-Stokes (RANS) procedures do not

lend themselves easily to the description of transitional flows, where both linear and

non-linear effects are relevant. RANS averaging eliminates the effects of linear

disturbance growth and it is therefore difficult to apply to the transition process.

2.2.1. Stability theory approach

An established transition modelling approach is based on stability theory that avoids the

aforementioned RANS limitation. Stability theory is based on the study of the behaviour

of small flow disturbances to see whether they grow or not. An input disturbance is

assumed of the form

(,,) = ((() (2.1)

and any arbitrary two-dimensional disturbance is assumed to be representable in the

form of equation (2.1) by expanding it in a Fourier series. If the disturbance amplitude

grows, the flow is unstable and transition to turbulent flow is expected. The advantage

of this approach is that the equations can be linearised, which makes this problem

amenable to an analytical approach. Making use of the continuity and momentum

equations for two-dimensional, incompressible, unsteady flow and neglecting quadratic

terms in the disturbance velocity components results in the Orr-Sommerfeld equation

27

( )( ( =

( +2 ( (2.2)

where the primes denote differentiation with respect to the dimensionless coordinate

y/, is the wavelength of the disturbance, c is the wave speed, is the eigenvector

and R is the Reynolds number of the mean flow. The term on the left hand side are

inertia terms (2nd order), those on the right hand side are viscous terms (4th order). The

problem of stability thus reduces to an eigenvalue problem: for a given Reynolds

number and wavenumber pair, equation (2.2) has eigenvalues k with corresponding

eigenfunctions ek. The stability of each eigenmode is given by the imaginary part of k.

If this is positive, the amplitude of the corresponding disturbance grows exponentially

and if it is negative, the amplitude decays. It could be argued that it is sufficient to study

the stability of each individual mode in order to determine whether the flow stays

laminar or not. This is however not the case and the reason is that the eigenfunctions are

non-orthogonal. Due to this fact, a sum of eigenmodes might show an initial growth

even if linear stability predicts that all eigenmodes decay exponentially. This is known

as the transient growth mechanism. Based on recent advances in hydrodynamic

instability theory, it has been recognised that the dynamics of many wall-bounded shear

flows is better described by a superimposition of normal modes rather than by a single

(the least stable) mode. Even though the Laplace transform resides entirely in the stable

half-plane, transient effects can cause energy amplification that may subsequently

trigger non-linear saturation followed by secondary instabilities. The same viewpoint

should hold for global modes, that is, the superimposition of mutually non-orthogonal

global modes may result in a substantially different perturbation dynamics that the one

that is predicted by the global spectrum. In addition, by means of the stability equation,

a theoretical critical Reynolds number is obtained that indicates the point on the wall at

which amplification of some individual disturbances begins and proceeds downstream

of it. The transformation of such amplified disturbances into turbulence takes place over

some finite streamwise distance. It must therefore be expected that the observed

position of the point of transition will be downstream of the calculated one, in other

words, the experimental critical Reynolds number exceeds its theoretical value. Because

the growth is so slow, transition to turbulence might not be complete until a streamwise

distance that can be as large as 20 times farther downstream from the leading edge than

the initial starting position of linear instability (Durbin and Jacobs, 2002)Moreover,

since the transition model is based on the linear stability theory, it cannot predict the

28

transition due to non-linear effects, such as high free-stream turbulence or surface

roughness. Where the exponential growth of two-dimensional waves results in a finite-

amplitude wave, the linear theory ceases to be valid. In fact, during the growth of the

waves, spanwise distortion and three-dimensional non-linear interactions become

relevant. The more widely used method based on the stability theory is the so-called en

method of Smith & Gamberoni (1956) and Van Ingen (1956). They proposed to

correlate the onset of transition with the amplification rate of the most unstable wave at

each position to determine the amplitude disturbance ratio

=

= ] ] (2.4)

where A0 is the initial disturbance amplitude at the first neutral stability point. en

methods are not compatible with general-purpose CFD methods as typically applied to

complex geometries. The reason is that these methods require a priori knowledge of the

geometry and of the grid topology. In addition, they involve numerous non-local

operations, such as tracking the disturbance growth along each streamline, that are

difficult to implement into todays CFD methods (Stock and Haase, 2000) and the

typical industrial Navier Stokes solutions are not accurate enough to evaluate the

stability equation. However, even the en method is not free from empiricism. This is

because the transition n-factor is not universal and depends on the wind tunnel and on

the free-stream environment. This means that it works well for flows that are not too

different from the ones used for its calibration. The main obstacle to the use of the en

model is that the required infrastructure needed to apply the model is complex. The

stability analysis is typically based on velocity profiles obtained from highly resolved

boundary layer codes that must be coupled to the pressure distribution of a RANS CFD

code. The output of the boundary layer method is then transferred to a stability method,

which then provides information back to the turbulence model in the RANS solver. The

complexity of this set-up is mainly justified for special applications where the flow is

designed to remain close to the stability limit for drag reduction, such as in laminar

wing design. The problem is considerably more complex in 3D flows, in which stream-

wise and cross-flow disturbances can coexist. Moreover, when bypass transition occurs,

this method does not work at all.

29

2.2.2. Low Reynolds number turbulent closure approach

The second way to predict transition is by using low Reynolds number turbulence

models. However, the ability of these turbulent models to predict transition is

questionable. They typically suffer from a close interaction between the transition

capability and the viscous sub-layer modelling and this can prevent an independent

calibration of both phenomena (Savill, 1993). At best, low Reynolds number models

can only be expected to simulate bypass transition, which is dominated by diffusion

effects from the free-stream. This is because standard low Reynolds number models rely

exclusively on the ability of the wall damping terms to capture the effects of transition.

Realistically, it would be very surprising if these models that were calibrated for viscous

sub-layer damping could faithfully reproduce the physics of transitional flows. It should

be noted that there are several low Reynolds number models where transition prediction

was considered specifically during the model calibration (Wilcox 1994, Langtry &

Sjolander 2002, Walters & Leylek 2004). However, these model formulations still

exhibit a close connection between the sub-layer behaviour and the transition

calibration. The re-calibration of one functionality also changes the performance of the

other. It is therefore not possible to introduce additional experimental information

without a substantial re-formulation of the entire model. Models like the Launder-

Sharma model (1978), where the near-wall behaviour is described by the turbulence

Reynolds number (Ret = k2/), perform better than those that use the local wall

distance. However, no model gives a reliable result for any arbitrary combination of

Reynolds number, Free-Stream Turbulence (FST) level, and pressure gradient.

Moreover, the model predictions are sensitive to the initial conditions, boundary

conditions, and to numerical aspects such as the grid resolution and the computational

domain extent. Westin and Hankes (1997) and Craft et al. (1997) improved the low

Reynolds number closure approach for bypass and separation-induced transition by

means of a non-linear eddy viscosity model that in general produces better results than a

linear eddy viscosity model, but the non-linear eddy viscosity model is still sensitive to

boundary conditions and numerical aspects. Some improvements have been obtained

with the low Reynolds number Second-Moment-Closure (SMC) as used by Hanjalic et

al. (1998). The benefits are located in the provision to account for the anisotropy of the

free-stream and of the near-wall Reynolds stress field, particularly in the ability to

reproduce the normal-to-the-wall velocity fluctuations. Another merit of this model is

its exact treatment of the turbulent production and of the effects of streamline curvature.

30

These characteristics help also in handling other forms of non-equilibrium phenomena,

such as separation and re-attachment that are frequently encountered with different

forms of transition. With this model, Hanjalic (1998) was able to predict the onset of

transition for several known test cases without having to use any empirical triggering, as

recommended by the Special Interest Group on transition (SIG10) of ERCOFTAC. This

technique has been successfully applied at high levels of FST intensity but not to flows

with a FST intensity level lower than 3%. In addition, it is more complex to implement

and also more computationally expensive than more empirical models (Wilcox 1994,

Langtry & Sjolander 2002, Walters & Leylek 2004).

2.2.3. The intermittency transport method with empirical correlations

The third approach to predict transition, which is favoured by the turbomachinery

community, is to use the concept of intermittency, as introduced by Dhawan and

Narasimha (1959). This approach consists in blending together laminar and turbulent

flow regimes as done by Abu-Ghannam and Shaw (1980), Mayle (1991), and Suzen &

Huang (2000), based on empirical values of the critical Reynolds number. Around this

critical value of Reynolds number, the flow becomes intermittent, which means that it

alternates in time between being laminar and turbulent. The physical nature of this flow

can be properly described with the aid of the intermittency factor , which is defined as

the fraction of time during which the flow at a given position remains turbulent, or in

other words, it is the fraction of time that the flow is turbulent during the transition

phase. By letting the intermittency grow from zero to unity, the start and the evolution

of transition can be imposed. This is commonly done by multiplying the eddy viscosity

in a two-equation turbulence model by the intermittency factor. In other words, once

is determined, it is multiplied by the eddy viscosity in the mean-flow equations. In the

pre-transitional regime, is set to zero and assumes a positive value only where the

model is required to initiate transition. Suzen & Huang (2000) developed an

intermittency transport model that can produce both the experimentally observed

streamwise variation of intermittency and a realistic profile in the cross-stream

direction. The model combines the transport equation models of Steelant & Dick (1996)

and Cho & Chung (1992). Specifically, the transport of intermittency, , is given by

31

+ = (1 (1( ()(

+

()

+

+ (1 +( (1 (

(2.3)

where the modelling constant are t = 1.0, C0 = 1.0; C1 = 1.6; C2 = 0.16; and C3 =

0.15.

This approach neglects the interaction between the turbulent and non-turbulent parts of

the flow during transition. In order to capture this interaction, a conditional averaging

technique leading to a set of turbulent and a set of non-turbulent equations for mass,

momentum, and energy is necessary, as used by Steelant & Dick (1996). The

conditional averaging is usually seen as too computationally expensive for engineering

applications, as the number of equations doubles. Therefore, the intermittency concept

is typically used in combination with globally averaged Navier-Stokes equations and the

loss of some physical information is accepted. Despite its inability to capture the

essence of the actual transition mechanism, single-point RANS turbulence closures

offer more flexibility and better prospects for predicting a real flow with transition than

the classical linear stability theory. Although much more limited in capturing the real

physics than DNS or LES, statistical modelling is still the only viable method to

compute complex flows with transition phenomena. It is worth noting that natural

transition is much rarer in industrial flows than bypass and separation induced

transition. The RANS intermittency statistical models typically correlate the transition

momentum thickness Reynolds number to local free-stream conditions, such as the

turbulence intensity and the pressure gradient. These models are relatively easy to

calibrate and are often sufficiently accurate to capture the major effects of transition. In

addition, correlations can be developed for the different transition mechanisms, ranging

from bypass to natural transition as well as cross-flow instability or surface roughness.

The main shortcoming of these models lies in their inherently non-local formulation.

They typically require information on the integral thickness of the boundary layer and

the state of the flow outside the boundary layer. While these models have been used

successfully in special-purpose turbomachinery codes, the non-local operations

involved with evaluating the boundary layer momentum thickness and with determining

the free-stream conditions have precluded their implementation into general-purpose

32

CFD codes. Still, statistical RANS models can adequately capture the effects of

transition in situations where most of the natural transition development stages are

bypassed by some strong external disturbance.

2.2.4. The laminar fluctuation energy method

A new and interesting class of transition models is based on the description of the

laminar fluctuation energy in the pre-transitional region of a boundary layer. The pre-

transitional region of boundary layers subject to free-stream turbulence resembles a

laminar boundary layer in terms of the mean velocity profile. As the FST level is

increased, the profile becomes noticeably distorted from the typical Blasius profile, with

an increase in momentum in the inner region and a decrease in the outer region, even for

a FST level as low as about 1%. This shift in mean velocity profile is accompanied by

the development of relatively high-amplitude streamwise fluctuations, which can reach

intensities several times that of the free-stream turbulence. This process results in an

increase in skin friction and heat transfer in the pre-transitional region and eventually

leads to bypass transition through the breakdown of the streamwise fluctuations (Jacobs

and Durbin, 2001). It is important to note that these streamwise fluctuations are not

turbulence in the usual sense of that word. This distinction was made for modelling

purposes by Mayle and Schulz (1997), who proposed a laminar kinetic energy kl

equation to describe the development of such fluctuations upstream of transition.

Structurally, these fluctuations are very different from turbulent fluctuations, since the

energy is almost entirely contained in the streamwise component of the fluctuating

velocity. Their dynamics is also considerably different. The familiar cascade of energy

from larger to smaller scales is not present. Instead, fluctuations are amplified at certain

scales determined by the boundary layer itself and remain at a relatively low

wavenumber. Dissipation is therefore also expected to be relatively low, except very

near the wall due to the no-slip condition. All of these considerations have led to adopt a

second kinetic energy equation by Mayle and Schultz (1997) to describe these

fluctuations. The growth of kl has been shown experimentally (Volino and Simon, 1997)

and analytically (Leib et al., 1999) to correlate with low-frequency normal velocity

fluctuations (v) in the free-stream. The scale selectivity of the boundary layer was

clearly demonstrated by Johnson and Ercan (1999), who plotted the amplification of six

frequency bands in a pre-transitional boundary layer. The reasons for this selectivity and

amplification are not yet completely understood. Volino (1998) considered the

33

possibility that the growth of kl is due to a splat mechanism, similar to the one

discussed by Bradshaw (1994). It is thought that the wall redirects the normal velocity

fluctuation into a streamwise component, at the same time as creating local pressure

gradients in the boundary layer, leading to disturbance amplification. This mechanism is

decidedly different from typical turbulence production and is adopted herein as a

reasonable explanation of the development and amplification of kl. Splats are likely to

occur only for eddies with a large length-scale relative to the wall distance. Therefore,

the turbulent energy spectrum can be divided into wall-limited (large scales) and non-

wall-limited (small scales) sections in the near-wall region (see Fig. 2.1), where the cut-

off eddy size is designated by eff , which is the effective turbulent length scale threshold

for the small scale turbulence.

Fig. 2.1: Illustration of wall-limiting concept leading to splat mechanism" for production of kl.

Scales smaller than eff interact with the mean flow as typical turbulence does and larger

scales contribute to the production mechanism for kl. The laminar kinetic energy

represents the magnitude of the non-turbulent streamwise fluctuations in the pre-

transition boundary layer.

Another region of interest is the transition zone itself. Jacobs and Durbin (2001) showed

that bypass transition is initiated by an instability of the upstream fluctuations, which

leads to turbulent spot development and progression to full turbulence. It is not clear

what initiates the instability. In the kl model, a local transition parameter is implemented

that depends on the specific turbulent kinetic energy, the effective length scale, and on

the fluid viscosity, based in part on measurements by Andersson et al. (1999). Once this

parameter reaches a certain value, transition is assumed to begin, which results in a

34

transfer of energy from the streamwise fluctuations kl to the turbulent fluctuations kt.

This is accompanied by a change in the length scale of the turbulence, as it would occur

in an actual spot breakdown process. Downstream of transition, the model predicts a

fully turbulent boundary layer. Almost all of the fluctuation energy is turbulent but a

small amount of kl is still present within the viscous sub-layer. This agrees qualitatively

with experimental observations indicating the presence of streamwise-oriented streaky

structures in the viscous sub-layer and in the buffer region that bear a resemblance to

those in the pre-transitional region. Although bypass transition is recognised to occur

for a FST level greater than 1%, the downstream location of transition is in fact

shortened by turbulence intensities greater than about 0.1% (Schlichting, 1979).This

suggests that there is a mixed transition regime involving elements of both natural and

bypass transition. In order to include natural transition and mixed mode transition into

the kl model, modifications must be made to both the kl production terms and the

transition production term that governs the transfer of energy between the streamwise

fluctuations and the kinetic energy of turbulence. These modifications do not depend

directly on turbulence quantities but depend instead on the local mean flow and the

laminar kinetic energy kl. Recent examples of such models were formulated by Walters

& Leylek (2004) and Laurdeau et al. (2004). A one-equation system is used to describe

the non-turbulent fluctuations prior to transition,

+ = + )

) (2.4)

Where is the production of laminar kinetic energy by large-scale turbulent

fluctuations, is near-wall dissipation that arises from the no-slip condition on laminar

fluctuations, R represents the averaged effect of the breakdown of streamwise

fluctuations into turbulence during bypass transition. The breakdown to turbulence due

to instabilities is included as a separate natural transition production term .

This equation lacks the usual shear-stress/strain related generation term, but it contains

a source term that is argued to arise from the pressure-diffusion correlation. Thus,

equation (2.4) returns, on calibration, the requisite rise in the fluctuation energy level in

the laminar regime, despite the absence of a shear stress kl production term, which is

presumed to be zero. Information from this system is used to start and let grow the

turbulent kinetic energy in a conventional two-equation k- RANS model. The model is

based on an eddy viscosity coefficient, determined by using three transport equations

for the turbulent kinetic energy k, the laminar kinetic energy kl, and the specific

35

turbulent kinetic energy dissipation rate . The model automatically predicts the onset

of transition without any intervention from the user and is based strictly on local

variables; therefore it does not require the evaluation of any integral parameter. The

principle is physically sound, but the technique is still too new to allow a judgement on

its quality. It has not been extensively validated except for a few flat plate test cases and

a turbine blade case (Walters & Leylek 2004, 2005, Walters & Cokljat 2008). However,

the initial results from this model are promising and indicate that the model appears to

have the correct sensitivity to the free-stream turbulence level. It remains to be seen

how accurately this model can predict the effects of pressure gradient and separation on

transition, particularly at a low FST level, below < 1%. A recent Large Eddy Simulation

(LES) performed by Lardeau and Leschziner (2007) shed some light on the validity of

the assumption underlying the RANS closure for the kinetic energy fluctuation level

observed upstream of the transition onset. This simulation has shown that, from a

statistical point of view, shear-stress/strain-induced production is mainly responsible for

the elevation of the pre-transitional laminar fluctuation energy, a process that is akin to

that observed in the turbulent state, although here it is mainly confined to the upper part

of the boundary layer. Indeed the ratio /k is quite high over a significant portion of

the pre-transitional boundary layer, that contradicts the base assumption of the model,

that is the shear stress term in the production of kl is zero. This indicates that further

studies should be done on this model to improve its representation of pre-transitional

flow.

2.2.5. DNS for transition

Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) are suitable

tools to predict transition (e.g. Durbin and Jacobs, 2002), although the proper

specification of the external disturbance level and structure at the computational domain

boundaries poses substantial challenges. In principle, laminar flow breakdown, the

development of turbulent spots, and transition to fully turbulent flow can be simulated

very accurately using DNS. A DNS computation is performed by solving the full time-

dependent Navier-Stokes equations. Since there is no Reynolds averaging, then there is

no requirement for turbulence closure by a turbulence model. In order to capture the

small scales of turbulence, a DNS computation requires a very fine computational mesh.

Unfortunately, these methods are far too costly for typical engineering applications. For

instance, a DNS simulation by Zheng et al. (1998) of a flat plate transitional boundary

36

layer used approximately 50 million grid points and was performed in about four weeks

on a parallel computer with 64 processors. Due to its large computational requirements,

DNS is clearly not yet at the stage where it can serve as a practical tool for engineering

design. This will be the case for a long time, simulations of industrial flows usually

involve larger and more complex geometries that require even higher computing

resources. DNS simulations are currently used mainly as research tools and as a

substitute for controlled experiments.

2.2.6. LES for transition

Because of the significant computational costs associated with DNS, a number of

researchers have applied the concept of Large Eddy Simulation to transitional flow. In

LES computations, only the large scale eddies are resolved, while the small scale eddies

are modelled using an eddy viscosity approach such as that proposed by Smagorinsky

(1963). One of the main problems with LES is that the predicted transition location is

very sensitive to the choice of the Smagorinsky constant that is used to calibrate the

sub-grid eddy viscosity (Germano et al., 1991). Germano et al. (1991) have since

proposed the dynamic sub-grid-scale model to compute the Smagorinsky constant

locally. The dynamic model has the advantage that, in laminar boundary layers, the sub-

grid eddy viscosity is automatically reduced to zero. Consequently, it is believed that

this model should be more appropriate for predicting transitional flows. Nevertheless,

the dynamic LES model is not a complete solution to the issues associated with

applying LES to predict transitional flows. LES computations performed by Michelassi

et al. (2003) on a low-pressure turbine blade with periodically impinging wakes have

indicated that, while the dynamic LES model was in good qualitative agreement with

DNS results, noticeable differences were observed in the quantitative comparison of

results from the two simulations.

2.2.7. The model

A Large Eddy Simulation (LES) for bypass transitional flow (Yang et al., 1994)

suggested that v, the turbulence fluctuation in the wall-normal direction, plays an

important role in the transition process. A wall-normal velocity disturbance slowly

increases from close to the wall to regions of higher velocity. This motivated the use of

the model (Durbin, 1995), without the inclusion of , for transitional flows. The

model consists of three transport equations for the turbulent kinetic energy k, the

37

specific turbulent kinetic energy dissipation rate , and the flow-normal component of

the kinetic energy along the streamlines. In addition, the model includes a Helmholtz

type equation for a quantity f which models the pressure-strain term. The turbulent

velocity and time scale are calculated from the standard k- equations. The transport

equation is

+2 =2 2

+ +

2 (2.7)

where

= +

(2.8)