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GROUP FOR AERONAUTICAL RESEARCH AND TECHNOLOGY IN EUROPE

FRANCE . GERMANY . ITALY . THE NETHERLANDS . SPAIN . SWEDEN . UNITED

KINDOM

ORIGINAL: ENGLISHJanuary 7th, 2003

GARTEUR/TP-137

GARTEUR Open

Application of Transition Criteria in

Navier Stokes Computations

by

GARTEUR AD(AG35)This report has been published under auspices of

the Aerodynamics Group of Responsablesof the Group for Aeronautical Research and

Technology in EURope (GARTEUR)

Group of Resp. : AD/GoRRepport Resp. : R. HoudevilleProject Manager : R. HoudevilleMonitoring Resp. : J.J. Thibert

Action Group : AD/(AG35)Version :Completed :©GARTEUR [1999] :

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List of Authors

M.T. Arthur (QinetiQ)H. S. Dol (NLR)A. Krumbein (DLR)R. Houdeville (ONERA)J. Ponsin (INTA)

Summary

Introduction of transition criteria in Navier-Stokes (NS) codes needs to overcome thelack of precision obtained on the boundary layer velocity pro�les with respect to bound-ary layer solvers. This is con�rmed in the �rst step of the study by considering theAEROSPATIALE �B-airfoil� with imposed transition location. All the NS solvers usedby the partners give poor quality results as concerns the boundary layer parameters. Fac-ing this di�culty, two strategies of implementation of transition criteria have been chosenby the partners in the second step: i) coupling Navier-Stokes and boundary layer solvers,ii) adapting transition criteria to have them less sensitive to the precision problem. Withthe �rst strategy, high quality results regarding the transition prediction are expectedthanks to the use of BL solvers well suited to the incorporation of elaborate transitioncriteria. This is obtained at the cost of a coupling process between two di�erent solvers.The second strategy is more straightforward, and right now useful in 3D �ows providingintroduction of cross�ow instability criterion, but at the cost of a more or less limitedprecision level depending on the �ow con�guration. The two strategies have been suc-cessfully implemented and validated for 2D �ows by considering the AS �B-airfoil� at lowMach number and, in the third step, the transonic CAST10 airfoil in a very di�cult con-�guration relatively to transition prediction. The case of a multi-element airfoil at highincidence has also been considered by NLR.

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Contents

1 The AG35 group 11.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Initial framework of Action Group 35 . . . . . . . . . . . . . . . . . . . . . 2

2 First step: imposed transition location 42.1 Experimental con�guration . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 RANS computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 DLR contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 INTA contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 NLR contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.4 ONERA contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.5 QinetiQ contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Synthesis and �rst conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Second step: implementation of the transition computation strategies 293.1 DLR contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Coupling Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2 Transition Prediction Algorithm . . . . . . . . . . . . . . . . . . . . 303.1.3 First validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 INTA contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.1 Implementation of the transition criteria . . . . . . . . . . . . . . . 373.2.2 First validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 NLR contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.2 Transition prediction techniques . . . . . . . . . . . . . . . . . . . . 443.3.3 Transition mechanisms and prediction criteria . . . . . . . . . . . . 463.3.4 Application to a single-element airfoil . . . . . . . . . . . . . . . . . 49

3.4 ONERA contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.4.1 Implementation of the transition criteria . . . . . . . . . . . . . . . 583.4.2 First validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5 QinetiQ contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5.1 Description of the computational procedure . . . . . . . . . . . . . 613.5.2 Transition criteria employed . . . . . . . . . . . . . . . . . . . . . . 613.5.3 Calculation performed for the AS-B aerofoil . . . . . . . . . . . . . 63

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4 Third step: Validation 684.1 CAST10 airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1.1 Flow conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.2 Navier-Stokes computations . . . . . . . . . . . . . . . . . . . . . . 694.1.3 Boundary layer computations . . . . . . . . . . . . . . . . . . . . . 69

4.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2.1 DLR contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2.2 INTA contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.3 NLR contribution - Multi-element airfoil . . . . . . . . . . . . . . . 914.2.4 ONERA contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2.5 QinetiQ contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Conclusion and perspectives 107References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

List of Figures

2.1 Cp-distributions for the AS-B airfoil test case . . . . . . . . . . . . . . . . . 72.2 Cf1-distributions for the AS-B airfoil test case . . . . . . . . . . . . . . . . 72.3 Viscous lengths for Airfoil B (displacement and momentum thicknesses) . . 102.4 Shape parameter for Airfoil B . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Momentum thickness comparison between BL and NS code . . . . . . . . . 112.6 Shape parameter comparison between BL and NS code . . . . . . . . . . . 112.7 Comparison between BL and NS code: Velocity pro�les B-Airfoil upper side 122.8 �B-Pro�le�, pressure distribution . . . . . . . . . . . . . . . . . . . . . . . . 172.9 �B-Pro�le�, �xed transition. Skin friction coe�cient . . . . . . . . . . . . . 172.10 �B-Pro�le�, �xed transition. Incompressible shape factor . . . . . . . . . . 172.11 �B-Pro�le�, �xed transition. Reynolds number based on momentum thickness 172.12 Comparison of pro�les of tangential velocity and its normal gradient. AS-B

aerofoil, M0 = 0:15, � = 7Æ, Re = 2:0 106 . . . . . . . . . . . . . . . . . . . 212.13 Comparison of pro�les of tangential velocity and its normal gradient. AS-B

aerofoil, M0 = 0:15, � = 7Æ, Re = 2:0 106 . . . . . . . . . . . . . . . . . . . 222.14 Cp-distributions for the AS-B airfoil test case: algebraic turbulence models 242.15 Cp-distributions for the AS-B airfoil test case: transport quation turbulence

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.16 Cp-distributions for the AS-B airfoil test case: algebraic turbulence models,

BLOW UP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.17 Cp-distributions for the AS-B airfoil test case: transport equation turbu-

lence models, BLOW UP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.18 Cf1-distributions for the AS-B airfoil test case: algebraic turbulence models 262.19 Cf1-distributions for the AS-B airfoil test case: transport equation turbu-

lence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.20 Cf1-distributions for the AS-B airfoil test case: algebraic turbulence mod-

els, BLOW UP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.21 Cf1-distributions for the AS-B airfoil test case: transport equation turbu-

lence models, BLOW UP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Coupling structure of the RANS solver and the transition prediction module 303.2 Natural laminar airfoil of Somers . . . . . . . . . . . . . . . . . . . . . . . 323.3 Convergence history of the �-residual, the lift coe�cient Cl and the drag

coe�cient Cd and the iteration process of the transition locations . . . . . 33

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3.4 Cl = Cl(Cd)-polars of the Somers-airfoil and experimental and computedtransition locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Cp-distributions and the Cf -distributions of the results with prescribedand predicted transition locations . . . . . . . . . . . . . . . . . . . . . . . 36

3.6 Skin friction distribution with several turbulence models . . . . . . . . . . 393.7 Initial and �nal skin friction �B Airfoil� . . . . . . . . . . . . . . . . . . . . 413.8 Convergence evolution, forces and transition location (Granville) . . . . . 413.9 Density residual (Granville) . . . . . . . . . . . . . . . . . . . . . . . . . . 413.10 Convergence evolution, forces and transition location (Drela) . . . . . . . . 413.11 Density residual (Drela) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.12 Convergence evolution, forces and transition location (Drela) . . . . . . . . 423.13 Skin friction, �B-Airfoil� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.14 Detail of the computational grid for the B-airfoil at the leading edge. . . . 503.15 Detail of the computational grid for the B-airfoil at the trailing edge. . . . 503.16 Initial transition-location update for the B-airfoil's upper surface. . . . . . 513.17 Re� distributions for the initial transition prediction (at cycle 4001). . . . . 513.18 Re� distributions for the second transition prediction (at cycle 5001). . . . 523.19 Re� distributions for the third transition prediction (at cycle 6001). . . . . 523.20 Re� distributions for the fourth transition prediction (at cycle 7001). . . . 533.21 Computed transition-location history for the B-airfoil �ow calculation; up-

dates activated at cycles 4001, 5001, 6001 and 7001. . . . . . . . . . . . . . 533.22 Convergence history for the B-airfoil computations involving four transition-

location modi�cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.23 E�ect of the transition location on the surface-pressure coe�cient distrib-

ution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.24 E�ect of the transition location on the skin-friction coe�cient distribution. 553.25 E�ect of the transition location on the momentum-loss thickness distribu-

tion (unphysical decay near trailing edge due to hybrid grid de�nition). . . 553.26 E�ect of the transition location on the y+ value of the �rst grid points. . . 563.27 E�ect of the transition location on the k+r value of the �rst grid points. . . 563.28 �B-Pro�le�, computed transition. Skin friction coe�cient. . . . . . . . . . . 623.29 �B-Pro�le�, computed transition. Skin friction coe�cient. . . . . . . . . . . 623.30 Pressure distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.31 Predicted upper surface transition location. AS-B aerofoil, M0 = 0:15,

� = 7Æ, Re = 2:0 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.32 Comparison of Cp distributions with predicted and imposed transition on

the upper surface. AS-B aerofoil, M0 = 0:15, � = 7Æ, Re = 2:0 106 . . . . . 653.33 N factors for a range of disturbance modes in the Navier-Stokes solution.

AS-B, M0 = 0:15, � = 7Æ, Re = 2:0 106 . . . . . . . . . . . . . . . . . . . . 663.34 Comparison ofCf distributions with predicted and imposed transition on

the upper surface. AS-B aerofoil, M0 = 0:15, � = 7Æ, Re = 2:0 106 . . . . . 67

4.1 CAST10 airfoil, natural transition - Wall isentropic Mach number. . . . . . 694.2 CAST10 airfoil, natural transition - Pressure coe�cient. . . . . . . . . . . . 69

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4.3 CAST10 airfoil, �xed transition - Isentropic Mach number using the pro-posed corrected conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4 CAST10 airfoil, application of the simpli�ed eN method (also called �paraboles�) 714.5 CAST10 airfoil, application of the simpli�ed eN method. Comparison with

experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.6 CAST10 airfoil - Boundary layer computation at Reynolds number equal

to RC = 3:9 106 - Suction side . . . . . . . . . . . . . . . . . . . . . . . . . 734.7 CAST10 airfoil - Boundary layer computation at Reynolds number equal

to RC = 3:9 106 - Pressure side . . . . . . . . . . . . . . . . . . . . . . . . . 744.8 CAST10 airfoil - Boundary layer computation at Reynolds number equal

to RC = 13:5 106 - Suction side . . . . . . . . . . . . . . . . . . . . . . . . 754.9 CAST10 airfoil - Boundary layer computation at Reynolds number equal

to RC = 13:5 106 - Pressure side . . . . . . . . . . . . . . . . . . . . . . . . 764.10 DLR C-H grid for the CAST10 airfoil . . . . . . . . . . . . . . . . . . . . . 784.11 Cp-distributions for the CAST10 airfoil, run 154: up, run158: down . . . . 804.12 Cp-distributions for the CAST10 airfoil, run 154 with new settings . . . . . 824.13 convergence history of the RANS computation, up, and the convergence

history of the transition location iteration, down . . . . . . . . . . . . . . . 834.14 �nal Cf -distributions of the CAST10 airfoil for run 154, up, and run 158,

down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.15 Density residual CAST10 Re=3:9 106 . . . . . . . . . . . . . . . . . . . . . 854.16 Convergence evolution, forces and transition location . . . . . . . . . . . . 864.17 Skin friction, CAST10, Re=3:9 106 . . . . . . . . . . . . . . . . . . . . . . 864.18 Cp distribution, CAST10, Re=3:9 106 . . . . . . . . . . . . . . . . . . . . . 874.19 Density Residual CAST10, Re=13:5 106 . . . . . . . . . . . . . . . . . . . . 874.20 Convergence evolution, forces and transition location . . . . . . . . . . . . 884.21 Skin friction, CAST10, Re=13:5 106 . . . . . . . . . . . . . . . . . . . . . . 884.22 Cp distribution, CAST10, Re=13:5 106 . . . . . . . . . . . . . . . . . . . . 894.23 Skin friction with two di�erent transition locations . . . . . . . . . . . . . 894.24 Detail of the computational grid for the multi-element airfoil at the leading

edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.25 Detail of the computational grid for the multi-element airfoil at the trailing

edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.26 Convergence history for the multi-element airfoil computations involving

transition-location modi�cations. . . . . . . . . . . . . . . . . . . . . . . . 944.27 History of the stagnation and transition locations on the upper surface of

the slat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.28 Geometrical de�nition of the airfoil elements, complemented with stream-

lines, stagnation points and transition locations. The inset displays a mag-ni�ed view of the slat's stagnation zone. . . . . . . . . . . . . . . . . . . . 95

4.29 History of the transition locations on the upper surface of the slat. . . . . . 964.30 History of the transition locations on the upper surface of the main-wing

element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.31 History of the transition locations on the upper surface of the �ap. . . . . . 97

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4.32 Comparison of the Cp distributions from the initial and �nal �ow solutionsfor the three airfoil elements. . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.33 Comparison of the cf distributions from the initial and �nal �ow solutionsfor the three airfoil elements. . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.34 Comparison of the Cp distributions from the transitional and fully-turbulent�ow solutions for the three airfoil elements. The inset displays a horizontalzoom of the peak on its left. . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.35 CAST10 airfoil - Isentropic wall Mach number - RC = 3:9 106 . . . . . . . . 1004.36 CAST10 airfoil - Convergence history - RC = 3:9 106 . . . . . . . . . . . . 1014.37 CAST10 airfoil - Momentum Reynolds number - RC = 3:9 106 . . . . . . . 1024.38 CAST10 airfoil - Skin friction - RC = 3:9 106 . . . . . . . . . . . . . . . . . 1024.39 CAST10 airfoil - Incompressible shape parameter - RC = 3:9 106 . . . . . . 1024.40 RAE2822 airfoil - Skin friction - RC = 6:5 106 . . . . . . . . . . . . . . . . 1034.41 RAE2822 airfoil - Incompressible shape parameter - RC = 6:5 106 . . . . . 1034.42 Comparison of pressure distributions from computations using three dif-

ferent transition prediction strategies. CAST10 aerofoil, M0 = 0:725,� = �0:35Æ, Re = 3:9 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.43 Predicted transition locations (after under-relaxation) for three di�erentstrategies. CAST10 aerofoil, M0 = 0:725, � = �0:35Æ, Re = 3:9:0 106 . . . . 105

4.44 Predicted distribution of Cf from the �nal calculation. CAST10 aerofoil,M0 = 0:725, � = �0:35Æ, Re = 3:9:0 106 . . . . . . . . . . . . . . . . . . . . 106

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Notations

Cf0, Cf1 : wall friction coe�cient �w1

2�1U2

1

Cl, Cd : lift and drag coe�cientsCp, Kp : pressure coe�cient p�p1

1

2�1U2

1

f , F : physical and reduced frequency of the instability wavesHi, H12 : incompressible shape parameterk2, k4 : arti�cial dissipation coe�cient (2nd and 4th di�erences)L�s : reference length in normal direction to the wall sqrtRs

M : Mach NumberN : total ampli�cation coe�cient of the instability wavesPi, pi : stagnation pressureRet : turbulence Reynolds number �t

�

Rx : Reynolds number based on x reference lengths : arc lengthTi : stagnation temperatureTu : external turbulence level

q2k3U2

e

u� : friction velocityq

�w�

U : velocityy+ : wall distance in wall units yu�

�

� : incidence� : ampli�cation coe�cient of the instability waves�r, �i : real and imaginary parts of �Æ : conventional boundary layer thicknessÆ�, Æ1 : displacement thickness�, �11 : momentum thickness�2, �4 : arti�cial dissipation coe�cient (2nd and 4th di�erences)�, �, �2 : pressure gradient parameter �2

�dueds

�, �, �2 : mean values of �, � and �2

� : molecular viscosity� : molecular viscosity �

�

�t : eddy viscosity

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Chapter 1

The AG35 group

Since the EG37 group in 1999 followed by the AG35, the following institutes have con-tributed to the present work:

BAe : A. GouldDERA/ARA then QinetiQ : C. Atkin and Moira Maina, then M.T. ArthurDLR : H.W. Stock, then A. KrumbeinFOI/FFA : D. Henningson, then A. Hani�INTA : J. PonsinNLR : K.M.J. De Cock, then H.S. DolONERA : R. Houdeville and D. ArnalAEROSPATIALE : Ph. Colin, then L. TourretteDassault Aviation : J.J. Vallée

1.1 Context

Navier Stokes codes (NS) are intensively used for CFD applications and aircraft design.For a long time, simple turbulence models based on algebraic formulations have been useddue to their supposed robustness. Nowadays, more and more sophisticated turbulencemodels are available in Navier Stokes codes and more and more realistic con�gurations canbe simulated. Similarly, great progress has been made in the understanding of transitionmechanisms and very e�cient transition criteria, such as data base methods, are available.In contrast with these improvements, transition prediction remains very crude in manyNS codes. Very often, the boundary layer is assumed �turbulent� from its origin, thenumerical transition process being left under the turbulence model responsibility whichrarely, if ever, includes the physics of the phenomenon.

In the best cases, the transition location is imposed. Moreover, the predictedtransition length, corresponding to the intermittency region, is more dependent on theturbulence model itself than on the physics of the phenomenon.

This situation is not satisfactory because various important problems are stronglydependent on transition mechanisms. One can cite: laminar �ow control, high lift separa-

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tion and shock - boundary layer interaction. On one hand, there is clearly a strong needto include transition prediction in Navier-Stokes computation for industrial applicationsand, on the other hand, the corresponding tools seem available but they remain dispersedand mainly used in conjunction with the boundary layer approach.

Some preliminary attempts to include transition predictions in NS codes havebeen published, in the �eld of turbomachinery applications [33, 24] and also for externalaerodynamics [22, 18, 32, 53]. A small subtask was devoted to this problem within theEuropean EUROTRANS project, but the results, although encouraging, were limited to afew test cases. One can cite also the ERCOFTAC activity for industrial turbomachineryapplications [43].

Finally it was decided to undertake a more ambitious and more systematic studyof this problem in the framework of a GARTEUR project. The Exploratory Group (EG37)established at the beginning of 1998 became the Action Group (AG35) the next year.

1.2 Initial framework of Action Group 35

All the transition prediction techniques are based on the accurate knowledge of the laminarboundary layer subjected to the transition process. The transition location may be greatlyin�uenced by lack of precision on the shape of the velocity pro�le and integral boundarylayer parameters such as the shape parameter. These considerations have suggested threedi�erent steps for the Action Group:

� �rst step: this step is devoted to the validation of Navier-Stokes calculations withrespect to the prediction of the laminar boundary layer parameters. To this end, Navier-Stokes calculations for the AEROSPATIALE �B pro�le� with an imposed location of theonset of the transition are done in order to compare the laminar boundary layer velocitypro�les (integral thicknesses and shape parameter) with a boundary layer calculationdone using the Cp distribution from the NS codes. This will give an estimate of theaccuracy which can be expected regarding the transition location itself.

� second step: The objective of this step is to perform actual transition prediction fromNS computations. To this end, a transition criterion and a calculation strategy mustbe de�ned and introduced in the NS codes. As the second point is the most critical,each partner may follow his own way: i) using a coupling technique with boundarylayer calculations, ii) remaining completely within the �Navier-Stokes world�.

The �rst way is probably safer than the second one as long as 2D �ows are concernedbut extension to 3D �ows can prove to be more di�cult for complex geometries. Thesecond way is really very ambitious because various intricate problems must be solved:accuracy (need of mesh re�nement), calculation strategy and numerical stability.

As concerns the transition prediction itself, the choice is reduced to two possibilities:either an analytical criterion (Granville, Michel: : :) or the eN method (exact compu-tations or data base methods). The former technique is easier to implement into aNS code. On the other hand, the latter technique o�ers a larger domain of applicationand can be extended to various �ow conditions more easily.

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For this step, the �B pro�le� is used as �rst validation test case.

� third step: This step is devoted to the application of the methods developed in thesecond step to the CAST10 transonic airfoil at small incidence to check the validity ofthe developments. The CAST10 con�guration is a di�cult case for transition criteriabecause there is no signi�cant overshoot on the pressure coe�cient and compressibilitye�ects on the development of the instability waves cannot be neglected. Only limitedexperimental results exist and comparisons are mainly done with boundary layer com-putations. In this step each partner may also use his own validation test case, e.g.high-lift airfoil con�guration.

Chapter 2

First step: imposed transition location

This step is devoted to the validation of Navier-Stokes calculations with respect to theprediction of the laminar boundary layer parameters. To this end, Navier-Stokes compu-tations for the AEROSPATIALE �B pro�le� are made with an imposed location of theonset of the transition in order to compare the laminar boundary layer velocity pro�les(integral thicknesses and shape parameter) with a boundary layer calculation done usingthe pressure distribution from the NS codes.

2.1 Experimental con�guration

The �ow parameters correspond to experiments from F2 wind tunnel at ON-ERA/FAUGA [25]. They are:

M1=0.15 incidence=7Æ Rc=2 106 (based on airfoil chord).

For this case, the transition takes place at x=c = 0:42 on the upper side through a smallseparation bubble. On the lower side, it is arti�cially triggered at x=c = 0:3.

2.2 RANS computations

A structured C-type grid around the B-airfoil has been generated by DLR and given toall partners. It is made of a single block, 337�73. To generate the grid, the trailingedge geometry has been changed to close the airfoil because the actual trailing edge hasa relative thickness equal to 0.5%. The stretching in the normal direction is such thatsmall y+ values of order 0:15 are obtained at the wall.

For the RANS computations the experimental �ow conditions are used withoutany correction. The transition location is imposed at x=c = 0:42 on the upper side andat 0:3 on the lower side.

4

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2.2.1 DLR contribution

2.2.1.1 RANS computations

For the computation of the AEROSPATIALE �B airfoil� test case and all validations theDLR �ow solver FLOWer was used.

FLOWer solves the Reynolds-averaged Navier-Stokes equations (RANS) for 3-dimensional compressible, steady and unsteady �ows on body-�tted (curvilinear), struc-tured multiblock-meshes of any topology in Cartesian co-ordinates. The space discretiza-tion is based upon the �nite volume method and uses a cell-vertex approach and centraldi�erences to approximate the convective �uxes. The viscous �uxes are approximated bya node-centered-scheme using a compact cell and central di�erences. On smooth meshesthe �uxes are 2nd order accurate in space. To maintain full solver accuracy two layers ofdummy cells are used around each block. Both `thin-layer' and full Navier-Stokes optionsare available.

To damp high frequency oscillations and to allow su�ciently sharp resolution ofshock waves, scalar dissipation (Jameson) is added to the discretized �uxes using a blendof 2nd and 4th di�erences of the �ow variables and the Martinelli scaling factors for thedi�erent co-ordinate directions for cells with high aspect ratio in viscous calculations. Thescaling factors include the spectral radii of the �ux Jacobians in the di�erent co-ordinatedirections.

The discrete time integration of the spatially discretized RANS equations is doneby explicit hybrid multi-stage Runge-Kutta schemes (usually: (5,2)-Runge-Kutta schemefor Euler and (5,3)-Runge-Kutta-scheme for NS with weighted evaluation of the dissipativeterms) using coe�cients optimized with respect to numerical stability and smoothingproperties. In steady computations the following acceleration techniques can be used:local time stepping, implicit residual smoothing, enthalpy-damping (Euler) and di�erentmulti-grid strategies. In order to more e�ciently solve low Mach number �ows a pre-conditioning option is available.

Several boundary condition types are available: slip and no-slip wall; far-�eldin�ow and out�ow and free stream; symmetry; engine inlet, fan and engine core outlets;internal �ow inlet (pressure-extrapolation, velocity extrapolation or characteristics) andoutlet (constant pressure) conditions; cuts to the same or to other blocks. Automaticrecognition and treatment of singularities is to a limited extent available.

For turbulent calculations the algebraic Baldwin-Lomax model, a number of one-equation transport models of Spalart-Allmaras type (SA, SA with Edwards modi�cation,SALSA) and a number two equation k � ! transport models (standard Wilcox k � !,k � ! LLR, k � ! SST, LEA k � !) can be selected. The convective �uxes of theturbulence equations have been discretized using a �rst-order upwind approach. To reducethe sti�ness of the source terms, a �rst version of a point implicit formulation for theturbulent terms has been added to the Runge-Kutta time-marching scheme.

Transition onset is imposed by prescribing transition points on the upper andlower side of an airfoil or by two sets of transition points forming poly-lines on the upperand lower side of a wing. The complete transition lines are given then by linear interpo-lation between the prescribed points. Grid points with a transition-relevant co-ordinate

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(x-co-ordinate in FLOWer, assumed to be `approximately parallel' to the main �ow di-rection) lower than the transition-relevant co-ordinate of a related transition point of thetransition line are treated as `fully laminar' for algebraic turbulence models by setting theeddy viscosity �t to zero, �t = 0. For transport equation models, in the fully laminar �owregions production sources are limited to be equal or less than the destruction sources,such that 1) �t does not grow in prescribed laminar regions, and 2) turbulent e�ects fromupstream components are included, see [36, 34, 35].

FLOWer is able to deal with arbitrarily prescribed transition lines on wing-likeand fuselage-like parts of aircrafts and helicopters with both convex and concave sur-face elements in a fully topology independent manner on a arbitrary number of aircraftcomponents.

The code, written in FORTRAN 77, is fully portable for either scalar and vectorarchitectures on sequential or parallel computers. Parallelisation is realised through theuse of a high level communication library CLIC-3D, which is based on the message passinginterface, MPI.

For more than four years now, the FLOWer code which is regularly extended andimproved in terms of new functionalities, accuracy, e�ciency and robustness has becomea standard CFD tool for design work in German aircraft industry.

2.2.1.2 Results

The computations of the test case were performed using the algebraic Baldwin-Lomaxturbulence model and the standard Wilcox k � ! turbulence model. The results for thepressure coe�cients, Cp, and the friction coe�cients, Cf1 = Cf , are shown in �g. 2.1and 2.2.

The results of both computations show a good prediction of the suction peak nearthe nose of the airfoil, �g. 2.1. The overall characteristics of the Cp-distribution are caughtvery well. However, some details of the �ow were not simulated accurately. The separationat the trailing edge has nearly dissapeared due to the change of the geometry from a thickto a closed trailing edge. The separation bubble on the upper side directly downstreamof the upper transition point was not reproduced in the computations. Di�erent possiblereasons may have caused this latter e�ect: The computations were done applying `pointtransition' to mimic the transition process, i.e. that from one surface point to anotherthe �ow changes from laminar to turbulent without a physically modeled transitional�ow region. The point of transition onset was �xed at x/c = 0.42 which probably is thebeginning of the separation region and normally is a good approximation for the transitionlocation caused by a separation bubble. In this case, this approximation together with thefact that the computational grid was not made to resolve a separation bubble - thereforethe number of grid points along the airfoil contour should be doubled at least - mayprevent the forming of the separation bubble in the simulation.

All these features may be reasons why there are deviations between the exper-imental and the computed Cp-distributions which are clearly visible along the completeupper side of the airfoil.

In the Cp-distribution resulting from the computation using the Baldwin-Lomax

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x/c

c p

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

experimentDLR k-ω WilcoxDLR BL

transition locations:upper side: xtr,u/c = 0.42lower side: xtr,l/c = 0.3

Ma = 0.15α = 7 degRe = 2.0 E+06

Aerospatiale B-airfoil

Figure 2.1: Cp-distributions for the AS-B airfoil test case

x/c

c f,inf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

experimentDLR k-ω WilcoxDLR BL


Ma = 0.15α = 7 degRe = 2.0 E+06


Figure 2.2: Cf1-distributions for the AS-B airfoil test case

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model the disturbance around the upper transition point is clearly visible. This dis-turbance is caused by the steep increase of the friction coe�cient Cf in the arti�ciallysimulated transitional �ow region due to the application of point transition, �g. 2.2.

The extent of this arti�cial transition region is about 13% of chord for the com-putation using the Baldwin-Lomax model and about 17% of chord for the computationusing the Wilcox k � ! model.

In both computations, one can see a strong upstream in�uence in theCf -distribution on upper and lower side. The extent of the upstream in�uence is about7% of chord in both cases.

As shown in [52], the computation of highly accurate boundary layer data deter-mined directly from a computational grid which has been made for a RANS computationis possible only when the laminar and the turbulent boundary layers are resolved in thecomputational grid in a special way. Here, the computational grid was generated to re-solve a turbulent boundary layer such that one can not expect the velocity pro�les in thelaminar boundary layer to be of su�cient quality to yield boundary layer data that areaccurate enough for any stability analysis or a transition criterion. As the proof of thisfact has been already done, [52], it is not repeated here.

2.2.2 INTA contribution

2.2.2.1 Navier-Stokes computations with �xed transition

The selected Navier-Stokes code to be used in AG35 is called STNS2D [23, 21]. Thiscode solves the compressible RANS equations in a structured curvilinear mesh using a�nite di�erences discretisation method. An implicit approximate factorisation schemeadapted from Beam and Warming is used for the time discretisation. For the spacediscretisation, the convective and viscous terms central di�erences are used. In order toensure the numerical stability of the method, a blend of explicit and implicit second orderand explicit fourth order scalar dissipation (of Jameson type) are added to the numericalscheme [40]. Martinelli coe�cients are used for taking into account the anisotropy ofthe dissipation scaling when dealing with high aspect ratio cells. The method is secondorder accurate in space and �rst order in time. All the computations which will be shownin this report have been obtained using local varying time step as the only convergenceacceleration technique.

The turbulence models selected for the �rst step task of AG35 are the Baldwin-Lomax, Spalart-Allmaras and k � ! BSL. These three models are compared in order toexamine the behaviour of the turbulence models in the prediction of the transitional zone.The performance in the transitional zone could a�ect the convergence of the transitionprediction coupling algorithm. In the NS code used for this project transition is imposedby setting turbulent viscosity to zero in the laminar regions. The way to activate theturbulence is di�erent in each of the tested turbulence models. For the Spalart-Allmarasmodel, the original model [49] with the transition trip term is used. As for the k�! BSL,the production and dissipation terms are set to zero in the laminar areas for both transportequations.

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2.2.2.2 Results

The mesh used for the �B-airfoil� computations has been provided by DLR. The mesh isa 337�73 structured single block C-mesh that has been generated in order to have y+

below one in the �rst node o� the wall and approximately thirty points inside the laminarboundary layer. This mesh is representative of a type of mesh used for 2D/3D turbulent�ows computation in terms of resolution and point distribution. The parameters of thearti�cial dissipation terms used in the computations carried out by INTA are k2 = 0:25and k4 = 0:01. A systematic decrease of k4 has not been carried out in the course ofthis project. Although the coe�cient of the fourth order scalar dissipation term couldbe decreased in order to diminish the numerical dissipation inside the boundary layer,several authors [2, 44] have shown that while decreasing this coe�cient does improve theresults, the overall e�ect is not su�cient to obtain the accuracy levels required by thetransition prediction criteria. Indeed, the excessive amount of dissipation stems from thescalar nature of the numerical dissipation formulation which produces an over-scaling ofthe dissipation in the normal direction. Therefore a more elaborate dissipation model asfor example the matrix dissipation or the upwind schemes must be used to avoid the lackof accuracy in NS solvers. Moreover, additional techniques, as the boundary layer meshadaptation, are required in order to obtain good results. The results obtained with theINTA NS code, with special emphasis on pressure distribution and skin friction coe�cient,are shown in section 2.3 and compared with those obtained by other partners. In thepresent section, the attention is focused on the quality of the boundary layer integralparameters obtained from the NS results. To assess the NS accuracy with respect tothese parameters, a comparison has been carried out using a boundary layer code whichsolves the incompressible laminar 2D boundary layer equations using a �nite di�erencesscheme. In order to perform the comparison of the boundary layer integral parameters,a procedure to evaluate the boundary layer edge must be implemented in the NS code.Among the possible available procedures to determine the viscous lengths, two criteriahave been tested. The �rst one is the so-called diagnostic function [52] while the secondone is based on the computation of the isentropic velocity [41]. The viscous length scalesobtained with both methods are shown in Fig 2.3. Fig 2.4 displays the shape parameter,which is a key ingredient parameter for transition prediction. The diagnostic functionseems to improve slightly the shape parameter prediction. However, due to the relativecoarse grid used for this computations, the viscous lengths obtained with the diagnosticfunction have wiggles which are not desirable when applying transition criteria. Thesewiggles can be eliminated, if for example, the diagnostic function is interpolated withnew points prior to determine its maximum. The comparison between the boundary layerparameters obtained with the NS code (with the diagnostic function) and the boundarylayer (BL) code is shown in �g 2.5 and 2.6 for the airfoil upper side. As can be seen, theincompressible shape parameter is under-estimated by approximately a 15%, while themomentum thickness is overestimated by approximately 15-20%. Such large di�erencesare clearly inadmissible for accurate transition prediction. If a transition criteria basedon the shape parameter is used, no transition would be detected using the NS results,whereas if a momentum thickness based criteria is used, the predicted transition would

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0 0.1 0.2 0.3 0.4x/c

4.0E-04

8.0E-04

δ*,θ

Diagnostic FunctionDiagnostic Function (interpolated)Isentropic velocity

Airfoil B, upper side

Figure 2.3: Viscous lengths for Airfoil B (displacement and momentum thicknesses)

0 0.1 0.2 0.3 0.4x/c

1

1.5

2

2.5

3

3.5

4

H12

Diagnostic FunctionDiagnostic Function (interpolated)Isentropic velocity

Airfoil B, upper side

Figure 2.4: Shape parameter for Airfoil B

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0 0.25 0.5X/C

0.0x10+00

1.0x10-04

2.0x10-04

3.0x10-04

4.0x10-04

5.0x10-04

θ

BL codeNS code

B Airfoil, upper side

Figure 2.5: Momentum thickness comparison between BL and NS code

0 0.25 0.5X/C

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

H1

2

BL codeNS code

B Airfoil, upper side

Figure 2.6: Shape parameter comparison between BL and NS code

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Figure 2.7: Comparison between BL and NS code: Velocity pro�les B-Airfoil upper side

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probably be far upstream of the `correct' one, due to the overestimation of this parameter.These results are in consonance with those obtained by other partners, and with thosepublished by other authors [44, 52]. It is expected that these results can be improvedusing a laminar grid adaptation [44] in combination with a less dissipative numericalmethod. Nevertheless, the required grid adaptation could be very costly for a complex3D con�guration. Finally, the comparison among the velocity pro�les at di�erent stationsobtained from the NS solution (with the diagnostic edge function) and those obtainedfrom the BL solution are shown in Fig 2.7. As could be expected, the accuracy of the NSsolution degrades as the local Reynolds number, which scales the arti�cial dissipation inthe normal direction, increases. Therefore, the more downstream the transition point islocated the more discrepancy can be expected in the shape parameter computed by a NSsolver. This situation is typical for transonic laminar airfoils.

2.2.3 NLR contribution

The present study reports on progress made in the prediction of the laminar-to-turbulenttransition location for the 2D high-lift �ow around a single-element and multi-elementairfoil in the framework of Reynolds-Averaged Navier-Stokes (RANS) computations. Cor-rect prediction of the transition location is crucial for obtaining realistic surface-pressureand skin-friction distributions, especially for incidence angles near maximum lift and forwind-tunnel �ow conditions characterised by a moderate Reynolds number. Transition isnot predicted by the RANS equations and the onset location has to be modelled sepa-rately. An algorithm has been developed and tested, which utilises a compound criterionfor four transition mechanisms � linear instability, laminar separation, free-stream tur-bulence and surface roughness � to determine and adapt the transition locations on theindividual airfoil elements during the RANS computations.

The CFD system in which the transition modelling algorithm has been integratedsolves the steady 2D compressible Navier-Stokes equations, discretised on an unstructuredgrid using the �nite-volume method. A FAS multi-grid method is applied to speed upconvergence. The turbulence is modelled with a slightly modi�ed version of the standardk-! model. With this high-Reynolds-number turbulence model, the laminar-to-turbulenttransition has to be delayed by enforcing laminar �ow upstream of the explicit transitiontrigger location sex. The transition modelling algorithm, developed in this study, is basedon the following streamwise locations in both boundary layers of each airfoil element:

1. The �rst location sls satisfying the Granville linear-stability criterion.

2. The �rst location ssep where laminar or turbulent separation occurs.

3. The �rst location sfst where su�cient free-stream turbulence is entrained to inducetransition.

4. The �rst location ssr where a surface roughness `strip' will cause transition.

Furthermore, the computed transition location sct, i.e. the location where the turbulencemodel equations e�ectively become active, is de�ned by the �rst point in the boundary

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layer where the turbulent eddy viscosity exceeds the molecular viscosity. The locationssls, ssep, sfst and sct are computed at the beginning of each new multi-grid cycle, whereaslocation sex is updated at a user-speci�ed interval as follows:

sex = min[ssep; ssr]if sct < sex then

sex = min[sls; ssep; sfst; ssr]endif

An initial algorithm validation exercise is performed for two well-documentedwind-tunnel �ow cases: the B-airfoil of Aerospatiale and the 59% span wing section ofthe Airbus A310 aircraft. The �rst case mainly validates the implemented linear insta-bility mechanism (Granville criterion) since that is the only active mechanism for thesingle-element airfoil under the considered �ow conditions. The complete �ow algorithmconverges within �ve updates of the explicit transition location. The second case, athree-element airfoil, is a realistic 2D high-lift �ow problem for which all four transitionmechanisms come into play. Unfortunately, the complete (steady) �ow algorithm doesnot converge due to unsteady laminar �ow separation on the upper surface of the slat,resulting in an oscillating transition location. Nevertheless, the transition location on theupper surface of the main-wing element converges within �ve to eight updates. The activetransition mechanism is linear instability (Granville criterion). On all remaining surfaces,transition is triggered at the surface roughness strip. Interestingly, in the �nal stage ofthe iteration process the computed transition location on the main wing's upper surfaceseems to converge to the location predicted by linear stability analysis independent fromthe explicit transition location, which could imply that the turbulent-�uctuation ampli-�cation by the k-! model complies with the non-turbulent �uctuation growth given bylinear stability theory.

Concluding, a transition modelling method for 2D high-lift �ows is successfullycoupled with a 2D RANS �ow solver. The foreknowledge and e�ort required from the userto perform an accurate transitional computation is substantially decreased. This gives theproposed algorithm the potential of being a useful engineering tool. Future investigationswill focus on the laminar/turbulent transition modelling capabilities of low-Reynolds-number single-point turbulence closure models.

The NLR results for the �B-Pro�le� test case with �xed transition are plotted in�gures 2.15, 2.17, 2.19 and 2.21 in section 2.3.

2.2.4 ONERA contribution

2.2.4.1 Navier-Stokes computations

To compute the �B-Pro�le� test case, the CANARI code [12, 37, 55] was �rst used. It solvesthe Navier-Stokes equations in structured mesh using a cell-centered �nite-volume dis-cretization technique. The time integration is done by a four steps Runge-Kutta method,with a local time step and an implicit residual smoothing. For the space discretization,

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the Jameson centered scheme is used. It is stabilized by a blend of 2nd and 4th di�er-ence arti�cial dissipation with the Martinelli correction. The corresponding �2 and �4

coe�cients are generally taken equal to 1 and 0.016.The �rst computations with transport equation models experience convergence

problems and only the use of a mixing length model gave satisfactory results. Nevertheless,it has benn prefered to re-mesh the airfoil with biggest cells at the wall. Two blocks havealso used to mesh exactly the thick trailing edge. With the new �C+H� mesh containing257x91 and 81x201 nodes respectively, the y+ value of the �rst cells at the wall is oforder 1, instead of 0.15 for the original mesh.

During the course of the AG35 group, ONERA developed the new elsA CFDplatform for structured meshes [9]. All the ONERA results which will be presentedin this report have been obtained with this new technology software based on ObjectOriented programming technique. elsA retains the numerical approach presented abovefor CANARI but new capabilities exist. In the present application we use the low velocitypreconditioning technique to accelerate convergence.

The computation of the boundary layer integral thicknesses is of primary impor-tance for the application of transition criteria. A �rst di�culty comes from the fact thatpressure is not constant in the direction normal to a curved wall. This induces velocityvariations at the boundary layer edge and the integral thicknesses become dependent onthe choice of the conventional boundary layer thickness Æ. The usual de�nition corre-sponding to the location where the velocity reaches a prescribed fraction, close to 1, ofthe external velocity is no longer relevant because the external velocity is not preciselyde�ned. It is preferable to use characteristic quantities related to the shear stress, eitherthe vorticity or the total stress:

Æ ! jjjjmax

= 1� " (2.1)

Æ� ! j� jj� jmax

= 1� "� (2.2)

jjmax and j� jmax are the maximum values of the vorticity and the total shear stress inthe boundary layer at the considered location. The above two de�nitions are equivalentfor laminar boundary layers. Typically, one chooses " ' 10�3 and "� ' 10�2.

A second di�culty is related to the direction of the integration. From the de�ni-tion of the integral thicknesses, the normal direction to the wall must be used. It is muchsimpler to use mesh lines. For inclined lines, the error is reduced by taking the actualdistance to the wall (by projecting in the normal direction). In that case, the remainingerror is due to the velocity variation in the �ow direction which is small with respect tothe variation in the normal direction.

To summarize, the integral thicknesses are obtained as follow :

� computation of jjmax and j� jmax along a mesh line, starting from the wall,

� computation of the conventional boundary layer Æ (minimum value of Æ and Æ� ),

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� computation of the velocity pro�les in the boundary layer reference frame (aligned withthe edge velocity UÆ and perpendicular to the wall),

� integration of the velocity pro�les (stream-wise and crosswise directions).

2.2.4.2 Boundary layer computations

The validation of the boundary layer parameters is done from comparison to bound-ary layer computations. To this end, the general boundary layer solver 3C3D has beenused [31, 20]. The velocity distribution needed for these computations is deduced fromthe wall pressure given by the RANS computations.

2.2.4.3 Results

The computed pressure coe�cient is compared in �gure 2.8 to the experimental results.A small under-prediction of the Kp-level is observed on the suction side, exactly as forall the other computations. Lowering the arti�cial viscosity from �4 = 0:016 to 0:008has no e�ect on the results. The skin friction coe�cient is compared in �gure 2.9 to theresults of the boundary layer solver. Over the upper side, the transition location is �xedat x=c = 0:42. In the Navier-Stokes computation, the skin friction starts to increase 2cells upstream of that point due to arti�cial viscosity e�ects and probably also becauseof the use of central derivatives in all directions. In the boundary layer computation, theincrease of the skin friction level is steeper because a mixing length model is used withoutintermittency damping function whereas in elsA the two-equation k� l turbulence modelis used and turbulence needs more time to develop. This phenomenon is more pronouncedon the pressure side because transition occurs in a stabilizing pressure gradient. Despitethese di�erences in turbulence models, the overall agreement is good. The discrepancyobserved on the lower side between x=c = 0:15 and the transition point is more di�cult toexplain. It may be due to a mesh e�ect or to a lack of convergence. It has been observedthat the skin friction coe�cient converges more slowly than the other parameters in thelaminar region.

The evolution of the incompressible shape factor is given in �gure 2.10 for twovalues of the arti�cial viscosity coe�cient �4. The comparison with the boundary layercomputation clearly demonstrates the poor prediction of Hi with the Navier-Stokes code.On the upper side, the shape factor given by the boundary layer computation remainsclose to 2.6 in agreement with the fact that the pressure is nearly constant from x=c =0:05 to 0:3. The Navier-Stokes computation predicts a decrease of Hi which reaches 2:3at x=c = 0:3 for �4 = 0:016, instead of 2:6. Decreasing �4 by a factor 2 improves theresult but only a small part of the error is recovered. At x=c = 0:3, the error reaches 8%for �4 = 0:008. On the lower side, the shape factor is also underestimated: 2:2 insteadof 2:35, always at x=c = 0:3.

It is well known that the development of instability waves is very sensitive to theshape of the velocity pro�les, closely related to the shape parameter. Either transitioncriteria or stability computations need very accurate velocity pro�les. The typical required

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x/c

Kp

0 0.25 0.5 0.75 1

-1

-0.5

0

0.5

1

1.5

2

0 0.25 0.5 0.75 1

-1

-0.5

0

0.5

1

1.5

2

Figure 2.8: �B-Pro�le�, pressure distrib-ution

x/c

Cf 0

0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

3C3D suc. side3C3D pres. side

0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Figure 2.9: �B-Pro�le�, �xed transition.Skin friction coe�cient

x/c

Hi

0 0.25 0.5 0.75 11

1.25

1.5

1.75

2

2.25

2.5

2.75

3


0 0.25 0.5 0.75 11

1.25

1.5

1.75

2

2.25

2.5

2.75

3

χ4 = 0.016χ4 = 0.008

Figure 2.10: �B-Pro�le�, �xed transition.Incompressible shape factor

x/c

Rθ

0 0.25 0.5 0.75 10

250

500

750

1000

1250

1500

1750

2000


0 0.25 0.5 0.75 10

250

500

750

1000

1250

1500

1750

2000

χ4 = 0.016χ4 = 0.008

Figure 2.11: �B-Pro�le�, �xed transition.Reynolds number based on momentumthickness

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accuracy on Hi is a few 10�3 around 2:6. This is one order of magnitude smaller thanwhich can be currently expected from Navier-Stokes solvers.

Another important parameter for transition criteria is the Reynolds number basedon the momentum thickness �. This parameter is compared in �gure 2.11, always for two�4-values, 0:016 and 0:008, to the boundary layer computation results. The in�uenceof the arti�cial viscosity is clearly less important on this quantity than on the shapeparameter. Nevertheless, the over-prediction of order 15% at x=c = 0:35 on the upperside will cause a too early transition prediction by any criteria using only this parameter.The error will be all the more important that d�=dx will be small (accelerated �ows). Inthe present case, this error would correspond to a shift of the order of 7% in x=c.

2.2.5 QinetiQ contribution

2.2.5.1 Navier-Stokes computations

The solution of the Navier-Stokes equations for the �ow around the Aerospatiale B-pro�leaerofoil was obtained with the code RANSMB using the common grid generated by DLR.RANSMB was developed by BAE SYSTEMS and is used extensively by Airbus UK.It is similar to the ONERA CANARI code in that it is a Jameson-type, cell-centred,�nite-volume formulation designed for application on block-structured meshes. It usesfour-stage Runge- Kutta local time stepping to march the solution towards the steadystate. It employs multigrid convergence acceleration. The scheme is stabilised throughthe addition of second and fourth order dissipative terms. The code incorporates severalturbulence models but all the results presented here were obtained with the k � g model(g = 1=

p!) in a form implemented by Hutton et al. (QinetiQ). No attempt has been

made to implement a transition model directly in the RANSMB code and no attempt hasbeen made to extract boundary layer integral thicknesses from the calculated solutions.Prediction of the location of the onset of transition for the second and third steps of thisexercise has been through application of the boundary layer code BL2D with empiricaltransition criteria (Granville) and stability analysis.

2.2.5.2 Boundary layer computations

The BL2D code solves the compressible laminar boundary layer equations on a swept-tapered or in�nite-swept wing using the method described by Atkin [8]. Swept-taperedor in�nite-swept similarity is imposed by assuming the edge conditions to be invariantalong the wing generators. This is a valid approximation if the isobars coincide withgenerators and is therefore appropriate for most transport wing applications. The laminarboundary layer equations are linearised and then transformed to a system of �rst-orderequations. The latter system is discretised using a compact scheme and solved usingNewton's method. The solution develops downstream from the attachment line until thepoint of laminar separation is reached. Solutions to the equations are very sensitive tothe edge velocity gradient, the position of the attachment line and the spanwise velocityat the attachment line. The latter is, of course, not an issue for two-dimensional �owcalculations. The edge velocity gradient is not provided as input and three numerical

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di�erentiation schemes are o�ered by BL2D to aid the accuracy and robustness of themethod. It remains for the user to examine and assess the calculated derivatives. Thelocation of the attachment point (in two-dimensional �ow) in the RANSMB solution isestimated from the extrapolated surface pressures using a robust procedure which assumesa quadratic variation in pressure in the neighbourhood of the maximum predicted pressure.This is equivalent to a linear variation in the �uid speed in the equivalent inviscid �ow andis taken to extend to the edge centres of the cells adjacent to that in which the predictedpressure is a maximum.

Considerable care has been taken to ensure that the solutions from the boundarylayer calculations are directly comparable with those from the Navier-Stokes equations.Potential problems arise from the fact that the RANSMB scheme is cell-centred, so thatthe pressure is not calculated directly at the wall, and that the polygonal surface ofthe aerofoil de�ned by the computational grid cells di�ers markedly from the true aerofoilsurface (by as much as two or three cell heights in some places). In an attempt to minimiseerrors arising from these factors, the following steps have been taken. The static pressurecalculated at cell centres by RANSMB is extrapolated linearly to the centre of the cell edgede�ning the (polygonal) aerofoil surface. The �ow speed in the equivalent inviscid �ow isthen obtained from the wall pressure and interpolated linearly along the cell edges to thenodes. If required, the pressure coe�cient at each node is obtained from the �ow speedusing the same isentropic �ow relations as in the previous step. BL2D accepts either �owvelocities or pressure coe�cients to compute the boundary layers. Pro�les of tangentialvelocity and its derivative with respect to wall-normal distance are extracted from theRANSMB solution along normals to the curved aerofoil surface through the surface gridnodes. The points on the normals to which velocity components are interpolated arede�ned by the interpolation procedure as follows. Let the family of grid lines runningaround the aerofoil be labelled j=constant and the family approximately normal to theaerofoil be i=constant. The aerofoil contour is j=1. The cell labelled i,j has the node (i,j)at its bottom left-hand corner. For each value of j, the cells i,j and i+1,j with centreseither side of the normal are identi�ed. The velocity components at the cell centres arethen interpolated linearly with distance along the line joining the mid points of the edgesi=constant of the cells, i.e. along lines which could be labelled j+1/2. The tangentialcomponent of velocity is then determined and its derivative is calculated by second orderaccurate �nite di�erences.

In order for the boundary layers to be computed, the surface distance from theattachment point to the nodes at which pressure coe�cients are given must be provided.The surface distance is measured along the (straight) cell edges rather than the curvedsurface of the aerofoil to reproduce the geometry on which RANSMB operates.

2.2.5.3 Results

The results were obtained using the regular, �C� grid provided by DLR and described insection 2.2.2.2. The computed pressure coe�cient distribution is compared in �gures 2.15and 2.17 with that of other participants and the experimental results. It shows a suddenincrease in the pressure gradient just ahead of the onset of transition. This can also be

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seen in the experimental pressure distribution but is less evident in some other computedsolutions. Elsewhere, the agreement with other computed solutions is good and the de-viation from the measured pressure coe�cients is comparable with that exhibited by theother solutions. This is discussed further in the next section, and comments about theadequacy of the grid for predicting the extent of the separation bubble are given in sec-tion 2.2.1.2. The computed lift and drag coe�cients are 1.2802 and 0.011196 respectively.The variation of the skin friction coe�cient is shown in �gures 2.19 and 2.21. As thepressure increases rapidly through transition, so the skin friction rises sharply, reachinga peak value of about 0.0098. The rise is signi�cantly more rapid, and the coe�cientreaches a higher value than in the other calculated solutions. This is consistent with thevariation in pressure through transition. It is clear that the predicted maximum skinfriction coe�cient exceeds the experimental value, though it is not possible to determinethe rate of rise of skin friction through transition from the experimental data.

Pro�les of tangential velocity and its normal gradient computed by RANSMBand BL2D are compared in �gures 2.12 and 2.13 for a sequence of stations close tothose selected by INTA for �gure 2.7. BL2D predicts laminar separation at or ahead ofx/c=0.406, the station closest to INTA's station 5, and so only four pro�les are presented.Pro�le number 1 is at the attachment point and successive pro�les are at the surface nodesof the computational grid. Thus pro�le 30 corresponds to x=c = 0:0465, pro�le 40 tox=c = 0:1064, pro�le 51 to x=c = 0:2063 and pro�le 59 to x=c = 0:2999. The agreementbetween Navier- Stokes and boundary layer predictions of velocity pro�le is very goodclose to the wall but the boundary layer pro�les are more full, as in the results of INTA.The agreement in predicted velocity gradients is less good but typical of results for thistype of aerofoil. It has not been possible within this programme to explore the e�ect onthe predictions of adjusting the turbulence model or the added dissipation, or of usingalternative turbulence models.

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Figure 2.12: Comparison of pro�les of tangential velocity and its normal gradient. AS-Baerofoil, M0 = 0:15, � = 7Æ, Re = 2:0 106

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Figure 2.13: Comparison of pro�les of tangential velocity and its normal gradient. AS-Baerofoil, M0 = 0:15, � = 7Æ, Re = 2:0 106

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2.3 Synthesis and �rst conclusions

Presentation of the comparison of the Cp-coe�cients for all codes

Fig. 2.14 and 2.15 show a comparison of the Cp-distributions provided by the partners,the results from computations using algebraic turbulence models in �g. 2.14, the resultsfrom computations using transport equation turbulence models in �g. 2.15.

The global view shows a good matching of the results from all the di�erent codes,for algebraic, �g. 2.14, as well as for transport equation turbulence models, �g. 2.15. Allcodes yield the same deviation from the experimental values on the upper side of theairfoil.

The di�erences between the codes become obvious in a blow up of the regionbetween the nose and the separation bubble on the upper side, �g. 2.16 and 2.17, theregion where the computational results deviate most from the experimental values. Thecomputational results di�er most in the area of the suction peak at the nose and in theregion of the pressure disturbance due to the abrupt transition process.

Presentation of the comparison of the Cf-coe�cients for all codes

In �g. 2.18 and 2.19 the distributions of the friction coe�cient Cf from all partners aredepicted. Here, the di�erences in the solutions are clearly visible even in the global view.The solutions di�er most signi�cantly in the arti�cal transitional region. The di�erencesare: the extent of the arti�cial transitional region, the Cf -rise from fully laminar to fullyturbulent �ow, the slope of the Cf -curve in the arti�cial transitional region and the extentof the upstream in�uence.

The blow ups of the arti�cal transitional regions show a dramatic scatter es-pecially for the solutions of computations using transport turbulence models. Possiblereasons for this scatter are e.g. di�erent spatial discretization schemes in the �ow solvers,di�erent properties of the arti�cial, numerical dissipation in the di�erent schemes, thedi�erent turbulence models, di�erent implementations of the same turbulence model, dif-ferent treatment of fully laminar grid points and �nally di�erences in the calculation ofthe value of the friction coe�cient.

As concerns the boundary layer parameter predictions, all the Navier-Stokes codesused by the partners give poor results: large under-estimation of the shape factor andover-prediction of the momentum thickness. These results could probably be improvedusing mesh re�nement techniques.

It has been demonstrated by DLR, [52], that the required precision of the bound-ary layer parameters can be obtained directly from the RANS grid when a special meshadaptation procedure is applied and when enough grid points are put between airfoilsurface and boundary layer edge.

It could be shown that about 70 grid points in the wall normal direction should belocated inside the laminar boundary layer. In this case, the quality of the velocity pro�lesis high enough such that the 1st and 2nd derivates of the velocity which are necessaryto compute the boundary layer parameters are of su�cient accuracy. Additionally, thespacing of the 70 grid points should be more or less equidistant in the laminar part of the

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x/c

c p

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

experimentINTA BLDERA BLDLR BL


Ma = 0.15α = 7 degRe = 2.0 E+06


Figure 2.14: Cp-distributions for the AS-B airfoil test case: algebraic turbulence models

x/c

c p

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

experimentINTA SADERA k-ωDLR k-ωBAe k-gNLR k-ωQinetiQ k-gONERA k-l


Ma = 0.15α = 7 degRe = 2.0 E+06


Figure 2.15: Cp-distributions for the AS-B airfoil test case: transport quation turbulencemodels

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x/c

c p

0 0.1 0.2 0.3 0.4 0.5 0.6

-1.5

-1.25

-1

experimentINTA BLDERA BLDLR BL


Ma = 0.15α = 7 degRe = 2.0 E+06


Figure 2.16: Cp-distributions for the AS-B airfoil test case: algebraic turbulence models,BLOW UP

x/c

c p

0 0.1 0.2 0.3 0.4 0.5 0.6

-1.5

-1.25

-1

experimentINTA SADERA k-ωDLR k-ωBAe k-gNLR k-ωQinetiQ k-gONERA k-l


Ma = 0.15α = 7 degRe = 2.0 E+06


Figure 2.17: Cp-distributions for the AS-B airfoil test case: transport equation turbulencemodels, BLOW UP

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x/c

c f,in

f(u

pper

side

)

c f,inf

(low

ersi

de)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045experimentINTA BLDERA BLDLR BL


Ma = 0.15α = 7 degRe = 2.0 E+06


upper side

lower side

Figure 2.18: Cf1-distributions for the AS-B airfoil test case: algebraic turbulence models

x/c

c f,in

f(u

pper

side

)

c f,inf

(low

ersi

de)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045experimentINTA SADERA k-ωDLR k-ωBAe k-gNLR k-ωQinetiQ k-gONERA k-l


Ma = 0.15α = 7 degRe = 2.0 E+06


upper side

lower side

Figure 2.19: Cf1-distributions for the AS-B airfoil test case: transport equation turbu-lence models

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x/c (upper side)

x/c (lower side)

c f,in

f(u

pper

side

)

c f,inf

(low

ersi

de)

0.4 0.5 0.6

0.2 0.3 0.4

-0.005

0

0.005

0.01

0.015

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04experimentINTA BLDERA BLDLR BL


Ma = 0.15α = 7 degRe = 2.0 E+06


upper side

lower side

Figure 2.20: Cf1-distributions for the AS-B airfoil test case: algebraic turbulence models,BLOW UP

x/c (upper side)

x/c (lower side)

c f,in

f(u

pper

side

)

c f,inf

(low

ersi

de)

0.4 0.5 0.6

0.2 0.3 0.4

-0.005

0

0.005

0.01

0.015

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04experimentINTA SADERA k-ωDLR k-ωBAe k-gNLR k-ωQinetiQ k-gONERA k-l


Ma = 0.15α = 7 degRe = 2.0 E+06


upper side

lower side

Figure 2.21: Cf1-distributions for the AS-B airfoil test case: transport equation turbu-lence models, BLOW UP

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boundary layer and the distribution of the complete laminar boundary layer edge must beknown. Thus, the laminar boundary layer edge must be determined every now and thenduring the computation. Downstream of the transition point the grid properties mustbe changed again such that in the turbulent part of �ow the turbulent boundary layer isresolved in an adequate way, i.e. the grid lines must be clustered near the solid surfacesas is normally done in RANS grids which are usually generated for fully turbulent �owsand the point spacing follows a certain law, e.g. an exponential law.

As the transition prediction process is an iterative process for the locations ofthe transition points the mesh adaptation procedure must be performed after every iter-ation step of the transition prediction process. This leads to a very expensive amount ofadditional grid generation work during the RANS computation.

In view of these di�culties, di�erent strategies have been chosen by the partners:

� DLR, INTA, QinetiQ decide to couple their Navier-Stokes solvers with a boundarylayer code. In 2D �ows, the pressure distribution given by the Navier-Stokes solvercan easily be used as input for the boundary layer code which computes very preciselythe transition onset location. This information is then given to the NS solver and theprocess iterates until convergence is reached. This technique can be extended to simple3D geometries as in�nite swept wings or even actual 3D wings.

� NLR uses a Navier-Stokes solver which includes transition criteria adapted to 2Dindustrial high lift con�gurations. ONERA chooses to introduce transition criteria inelsA, well aware of the precision problems. In order to try to minimize these problems,minor changes could be introduced in the criteria when necessary to make them lesssensitive to the error in the shape factor estimation.

Chapter 3

Second step: implementation of the

transition computation strategies

This chapter presents the various implementation strategies which have been chosen byeach partner to take into account the transition computation in the Navier-Stokes solvers.

3.1 DLR contribution

The computation of �ows with transition prediction in industrial applications using aRANS solver must be performed automatically and autonomously, i.e. without externalinterference by the user and without speci�c knowledge about the transition process itself(black box). For wings and tails of transport aircraft the transition locations can be de-termined during the run of the RANS solver using a transition prediction module coupledto the solver. The coupling of the solver to an eN -method [52, 41], provides a solution,that promises su�ciently high accuracy of the predicted transition locations. The use ofan eN -database method [51], results in a program system that is able to automaticallyhandle transition prediction.

3.1.1 Coupling Structure

In order to compute transition on wings of transport aircraft FLOWer was coupled to atransition prediction module which is called after a certain number of iteration cycles,kcyc:, of the RANS solution process. With the call of the module the solution process isinterrupted and the module analyses the laminar boundary layers of previously speci�edcomponents of the con�guration, e.g. of an 2-dimensional airfoil or a wing section. Thedetermined transition locations, xTj (cycle = kcyc:) with j = 1; : : : ; nloc:, are communi-cated to the RANS solver, which performs transition prescription applying the transitionsetting algorithm outlined above and continues the solution process of the RANS equa-tions. In so doing, the determination of the transition locations becomes an iterationprocess itself. The structure of the approach is outlined graphically in �g. 3.1.

The transition prediction module consists of a compressible boundary layer

29

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boundary layer method

prediction method

transition predictionmodule

RANS solver

iteration locationiteration

transition

TXj (cycle=kcyc.)

Cp(cycle=kcyc.)

ouput

input

Figure 3.1: Coupling structure of the RANS solver and the transition prediction module

method for swept, tapered wings [30], and an eN -database method for Tollmien-Schlichtingwaves, [51]. At every call of the module the surface pressure, Cp(cycle = kcyc:), along anairfoil computed by the RANS solver is used as input to the boundary layer calcula-tion. The viscous data calculated by the boundary layer method are then subsequentlyanalysed by the database method at every surface point. The application of a boundarylayer method for the computation of all viscous data necessary for the transition predic-tion method ensures the high accuracy of the viscous data required by the eN -method forthe analysis of the laminar boundary layers. Thus, as shown in [52], the large number ofgrid points near the wall for a high resolution of the boundary layers, the adaptation ofthe grid in the laminar and turbulent boundary layer regions and the generation of a newadapted grid after every step of the transition location iteration are avoided.

3.1.2 Transition Prediction Algorithm

1. The RANS solver is started as if a computation with prescribed transition locationsshould be performed with transition locations set far downstream on upper and lowersides of the airfoil, e.g. at the trailing edge. The RANS solver now computes a fullylaminar �ow over the airfoil.

2. During the solution process of the RANS equations the laminar �ow is checked forlaminar separation. If laminar separation is detected, the separation point is used asapproximation of the transition location.

3. The RANS equations are iterated until the lift, or the lift coe�cient Cl respectively,which can be represented as a function of the iteration cycles, Cl = Cl(cycles), hasbecome constant with respect to the iteration cycles,

Cl(cycles) = const: in cycles: (3.1)

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4. The transition prediction module is called.

5. The determined transition locations xTj (cycle = kcyc:) are underrelaxed, i.e. as newtransition locations ~xTj (cycle = kcyc:) coordinates located downstream of the coordi-nates xTj (cycle = kcyc:) are used,

~xTj (kcyc:) = CTj (kcyc:)x

Tj (kcyc:) with j = 1; : : : ; nloc: ; (3.2)

with CTj (kcyc:) > 1. Only after the last step of the transition location iteration

CTj (kcyc:) = 1 is applied. The underrelaxation of the determined transition locations

prevents the case that at an unconverged stage during the transition location iterationtransition coordinates are determined too far upstream which might not be shifteddownstream again.

6. As convergence criterion�~xT;lj < " (3.3)

is applied with �~xT;lj = j~xTj (klcyc:)� ~xTj (kl�1cyc:)j, l being the current iteration step. In the

case that the criterion is satis�ed, the iteration for xTj is �nished, else the algorithmloops back to station 2.

3.1.2.1 eN-Database Method

The eN -method used, [51], applies

N jxT = �Z xT

x0

�i dx (3.4)

as transition criterion, with N jxT , the experimentally determined limiting N -factor atthe transition location xT , which represents the total ampli�cation of a perturbation ofthe mean �ow of frequency f at the transition location, x0, the x-coordinate of the pointwhere this perturbation enters the unstable zone, and �i, the local spatial ampli�cationrate. The perturbation of the mean �ow V of frequency f � V is assumed to be steadyand well parallel � is described by a harmonic Tollmien-Schlichting wave,

v0 = v00(y) exp [i(�x� !t)] ; (3.5)

with the circular frequency ! = 2�fÆ1=Ue and the displacement thickness Æ1, Æ1 =R Æ0 (1� u(y)=Ue)dy , and � = �r + i�i. �i can be expressed as a function of the shape pa-rameter H12, ReÆ1 , the Reynolds number with respect to the displacement thickness ReÆ1 ,and the reduced frequency F ,

�i = �i(H12; ReÆ1 ; F ) (3.6)

with

H12 =Æ1Æ2

; Æ2 =Z Æ

0

u(y)

Ue

1� u(y)

Ue

!dy (3.7)

ReÆ1 =Ue�eÆ1�e

(3.8)

F =2�f�e�eU2

e

=!

ReÆ1(3.9)

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x

y

0 0.25 0.5 0.75 1

-0.1

0

0.1

0.2

Figure 3.2: Natural laminar airfoil of Somers

Æ2 being the momentum loss thickness. As described in [51] and [53], for 13 shape pa-rameters H12, which cover the range from highly accelerated up to separating �ows, theboundary layers of the similarity solution using the approach of Falkner and Skan weregenerated. The growth of the boundary layer was simulated by varying ReÆ1 . For eachboundary layer, stability computations were completed for a su�cient large range of ex-cited frequencies. The results for the ampli�cation rates �i were stored in a database.

The stability computation for a real boundary layer, using the database method,is executed for a given frequency f in Hz in the following way: At each grid point onthe airfoil the properties f , ReÆ1 , H12, Ue, �e and �e are known. Evaluating F fromequation (3.9), �i is obtained from equation (3.6) via interpolation in the database. In [51]and [53] it was shown that the use of the database is a high quality approximation for thevalues resulting from a local linear stability code.

3.1.3 First validation

The transition prediction algorithm has been applied to the natural laminar airfoil ofSomers, [48], using a 1 block mesh with about 24.000 grid points, 332 in i-direction and72 in j-direction (i: tangential direction, j: wall normal direction), as shown in �g. 3.2.

The aerodynamic parameters are Mach number Ma = 0:1 and Reynolds numberRe = 4 106. In the computations the algebraic Baldwin-Lomax turbulence model andthe k � ! transport equation model were used. First, the application of the transitionprediction algorithm is illustrated for a single angle of attack, � = �6:2Æ, in a computationwith the Baldwin-Lomax model. The prediction of the transition locations was performedusing the database method applying N jxT = 11. This value was determined by Stock [50],by comparison of the experimental data in [48] with results of linear stability calculations.Fig. 3.3 shows the convergence history of the �-residual, the lift coe�cient Cl and the dragcoe�cient Cd on the left side and the iteration process of the transition locations on theright side. The initial transition locations were set at 75% of chord on upper and lower

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cycle

ρ-re

sid

ual

c l

c d

500 1000

10-6

10-5

10-4

10-3

10-2

10-1

100

-0.25

-0.24

-0.23

-0.22

0.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

0.0085

0.009

0.0095

0.01ρ-residualcl

cd

cycle

xT upp,

xT uppun

derr

elax

ed,x

T low

c l

c d

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.25

-0.2

-0.15

-0.1

-0.05

0

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

cd

cl

xTlow

xTupp

underrelaxed

xTupp

Figure 3.3: Convergence history of the �-residual, the lift coe�cient Cl and the dragcoe�cient Cd and the iteration process of the transition locations

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side of the airfoil at the beginning of the computation. The left picture illustrates thecoupling of the RANS solver with the transition prediction module and shows how theRANS solution is interrupted by the single iteration steps of the transition predictioniteration, marked by peaks in the curve of the �-residual. It shows the constant levels ofthe lift coe�cient, which were reached before a determination of the transition locationsby the database was performed, and the impact of the changed transition locations on the�-residual and the force coe�cients during the solution process of the RANS equations.The database was activated only when the RANS solution had been converged withtemporarily �xed transition locations which were changed afterwards by the next callof the transition prediction module. The convergence of the transition location iterationitself is illustrated in the right picture. It shows the convergence of the transition locationsand the values of the force coe�cients in the moments when the database was activated.The two curves of the transition locations start with the preset values from the startof the RANS solver. After 100 iteration cycles, for the �rst time the �ow, which iscomputed as a laminar �ow from the leading edge up to 75% of chord on upper andlower side, was checked for laminar separation. Laminar separation was found on thelower side at 2% of chord and this location is now used as approximation of the transitionlocation on the lower side. After 250 cycles the database was activated for the �rst time.During the ongoing RANS solution only on the upper side excited frequencies reachingN jxT = 11 were found. All the transition locations determined directly by the databasewere set underrelaxed before the RANS solution continued. After its last call the valuexTupp = 0:4983 was set without underrelaxation for a converged solution of the RANSiterations.

Fig. 3.4 shows the Cl = Cl(Cd)-polars of the experiment with free transition [48],of fully turbulent computations and of computations with transition using the transitionprediction algorithm and the database on the left side. For both turbulence models astrong improvement of the computational results is achieved. For all points, there is aclear tendency towards the experimental data. The drag deviations at high values of �(� � 6Æ) still have to be investigated.

The right side of �g. 3.4 compares the experimentally determined transition loca-tions, as given in [48], with the transition locations determined by the transition predic-tion algorithm using the database. Indicated by �lled, black symbols are the transitionlocations determined by the database. The hollow, black symbols mark the transitionlocations approximated by the x-coordinates of laminar separation, whose locations weredetected by the laminar boundary layer method [30]. The depicted curves are the resultsfrom the computations using the k � ! model, the results for the Baldwin-Lomax modelare of the same high quality [36].

3.1.3.1 Aerospatiale B airfoil test case

For the prediction of the transition locations for the Aerospatiale B airfoil test case thesame procedure as described in the chapter before was applied.

The aerodynamic parameters are M0 = 0:15, Re = 2 106, � = 7:0Æ, the transitionlocation on the lower side was precribed and �xed xlowtr = 30% of chord, the transition

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cd

c l

0.01 0.02 0.03

0

0.5

1

1.5

2

exp.: free transitionBL fully turb.k-ω fully turb.BL predictedk-ω predicted

α

xT uppe

r,

xT low

er

-5 0 5 10 15

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

upper sideexp.databaselam. sep.

lower sideexp.databaselam. sep.

upper side

lower side

k-ω

Figure 3.4: Cl = Cl(Cd)-polars of the Somers-airfoil and experimental and computedtransition locations

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x/c

c p

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

experimentDLR k-ω prescribedDLR BL prescribedDLR k-ω predictedDLR BL predicted


0 0.25

-1.7

-1.6

-1.5

-1.4

x/c

c f,inf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

experimentDLR k-ω prescribedDLR BL prescribedDLR k-ω predictedDLR BL predicted


0.2 0.3 0.4 0.5 0.6 0.70

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Figure 3.5: Cp-distributions and the Cf -distributions of the results with prescribed andpredicted transition locations

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location on the upper side xupptr should be predicted and its initial value was set to xupptr =95% of chord. The simulation was done with both the Baldwin-Lomax and the Wilcoxk � ! turbulence model. The limiting F -Factor was set to the high value of N jxT = 12as transition is known to be caused by laminar separation.

For both turbulence models the results are the same: The transition point on theupper side was determined at xupptr = 39:27% of chord as a laminar separation point in thetransient phase of the computation at RANS cycle = 150 by the boundary layer code.

Between the results of the computations with prescribed and predicted transitionlocations exist the following di�erences in terms of the global force coe�cients:

k � ! : Cpresl � Cpred

l = 0:0025 cdpres � cdpred = �1:73 countsBL : Cpres

l � Cpredl = 0:003 cdpres � cdpred = �1:8 counts

In Fig. 3.5 a comparison of the Cp-distributions, left, and the Cf -distributions, right, ofthe results with prescribed and predicted transition locations is depicted. As one can see,the Cp-distribution is almost not a�ected at all and the Cf -distribution is only slightlya�ected by the predicted transition location whose position was determined a little bittoo far upstream.

The di�erence between the experimental and computed value of the transitionlocation is 2.7% of chord.

3.2 INTA contribution

3.2.1 Implementation of the transition criteria

As mentioned in section 2.3, INTA decided to use the strategy of implementing a Navier-Stokes Boundary layer coupling algorithm. This strategy seems to give good results atlow computational cost. In addition, it is not necessary to worry about mesh resolutionand point distribution in the laminar areas, i.e. the same original turbulent mesh canbe used for transition prediction. This implies that the algorithm is mesh independent,which is a very valuable feature in the CFD practice.

In the process of the algorithm implementation, special emphasis has been putin obtaining a full automation of the algorithm. The goal is to compute the con�gurationfrom scratch and to obtain the solution with the �nal transition position determined bythe algorithm without using any �man on the loop�. In order to achieve this goal, specialcare has to be taken concerning the robustness of each component of the algorithm,specially the boundary layer code. Moreover, the sensitivity of the coupling method tothe numerical control parameters must be studied in order to get a robust and stabletool with good convergence behaviour. (The calibration of the method depends on thefeatures of the components of the algorithm, specially on the Navier-Stokes solver).

The implementation of the INTA coupling algorithm has been carried out in avery straightforward manner. A description of the implemented coupling algorithm, interms of pseudocode language, would be the following:

Set XtrIni

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Do Nact = 1; Nmax

Call NS�

solver(Cl; Cf) �!NS Solver Step

if (mod(Nact; Ntr) = 0) then �!Every Ntr iterations

compute �Cl=Cl or �Cf=Cf

if (�Cl < e�

cl) then �!Activate BL+Transition Module

call bltrans (XtrNew)

XtrNew = XtrOld+ ! � (XtrNew �XtrOld) �! Relax new Transition location

call Initia�

Trans(XtrNew)

end if

End if

End do

The numerical parameters that control the coupling algorithm and consequently, theconvergence of the process, are XtrIni, Ntr, "tr or "cl and !. XtrIni denotes the initialprescribed transition position, which is normally located far downstream and close to thetrailing edge. Ntr is the iteration number at which the rate of change of, for example, thelift coe�cient, will be tested in order to activate the transition module. If the lift coe�cienthas remained nearly constant (i.e. the relative variation is less than "cl) for the previousNtr iterations, the transition module is activated. Finally, ! � 1 is an under-relaxationparameter which makes the update of the transition location farther downstream than theone actually determined by the transition module. The under-relaxation is used mainlyto avoid situations in which too early transition prescription could prevent the methodto converge to the `correct' transition location. Typical values used in the computationsshown below are Ntr = 100 to 500, "cl = 1: 10�3 and ! = 0:5 to 0:9.

Next, the components of the coupling algorithm are described. The NS solverhas been described in section 2.2.2.1. The transition module is composed by a BL codeand the transition criteria itself. The BL code solves the 2D incompressible boundarylayer by a �nite di�erences method which uses the modi�ed Keller box scheme [10]. The�nite di�erences method is more time consuming than an integral method but by usingthis kind of method it is possible to carry out a more sophisticated transition prediction(such as LST, PSE). The implemented transition criterion are the classical Granvillecriteria [46] and the Drela's approximate envelope method [15]. The Granville criteriontakes into account the history of the evolution of the boundary layer and the e�ect ofthe pressure gradients in the transition. For that purpose, it combines classical linearstability theory with experimental correlations. In order to apply the criteria, three keyparameters need to be computed locally: The Reynolds momentum thickness number,the pressure gradient parameter, � = (�2=�)(du=ds), and the average gradient parameter,which is de�ned by:

� =

Z Strans

Sinst

�ds

Strans � Sinst(3.10)

In a �rst step, the Granville method determines the instability point. To thisend, the local pressure gradient parameter is compared to that obtained from neutralcurve correlations which are function of Re�. In a second step, an average pressuregradient parameter must be computed locally from the instability point. This parameteris compared for each station with that obtained from the experimental correlations in

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1X

-0.005

0

0.005

0.01

0.015

Cf

Baldwin LomaxSpalart-AllmarasK-w BSL

Influence of the turbulence model on the transitional zoneAirfoil B, Xtru = 0.42, Xtrl = 0.5

Figure 3.6: Skin friction distribution with several turbulence models

function of the Reynolds number based on the momentum thickness. When the averagepressure gradient parameter calculated from the experimental correlation is greater thanthe value determined from the boundary layer development, transition is predicted.

The Drela's envelope method uses an eN method to predict transition. The tran-sition is predicted when the Tollmien-Schlichting waves have been ampli�ed (relative tothe initial amplitude in the instability point) eN times. To predict the N -factor a sim-pli�ed database of the stability characteristics of the Falkner-Skan pro�le family is used,which has been previously computed by solving the Orr-Sommerfeld equation. The di�er-ence between the approximate envelope method and the database method is that in theformer the ampli�cation rate is computed from the envelope of the ampli�ed frequencies,approximated by a straight line. In this way, the frequency need not be taken into con-sideration when computing the stability characteristics. The two key parameters whichhave to be computed locally are the Reynolds number based on the momentum thicknessand the incompressible shape factor. These parameters can be obtained with su�cientaccuracy from the BL method.

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The k � ! BSL model has been selected for all INTA computations in this project. Thisturbulence model (or INTA's implementation of this model) has a good behaviour inthe transitional zone and low upstream in�uence from the transition point onset. Thesefeatures are desirable for helping the convergence of the coupling method. Fig 3.6 displaysthe skin friction distribution obtained with some of the turbulence models implementedin INTA's NS code for the �B Airfoil� with upper �xed transition at x=c = 0:42. As canbe seen from that �gure, the Baldwin-Lomax model solution is characterized by a sharprising of the skin friction coe�cient in the transitional zone and also by a remarkabledecrease in the skin friction and displacement thickness just before the rising of the skinfriction. This kind of algebraic models needs an intermittency function in order to smoothand scale properly the transitional zone. In the case of the Spalart-Allmaras model withthe original trip function, although the transitional zone is smooth, there is neverthelessa remarkable upstream in�uence. Moreover, it has been found that, in some situations,the model produces an `spontaneous' early transition far from the zone where the tripterm is active. Therefore, this model with the original trip term may not be a robustmodel for using in an automated algorithm. Finally, from the �gure it can be seen thatthe k� ! BSL produces a smooth transitional region with almost no upstream in�uence.Obviously the above description is a simple sketch of the behaviour of the turbulencemodels in the transition region.

As a �rst validation test, the �B-Airfoil� transition locations at �ow conditionsspeci�ed in section 2.1 have been predicted using both criteria, the Granville and theDrela's approximate envelope method. The computations have been carried out startingfrom an initial converged solution obtained with the transition positions located at x/c= 0.6 on the upper side and 0.5 on the lower side. The numerical parameters used in thecomputations are Niter = 100, ! = 0:7 and "cl = 10�3. With these parameters the codehas been run 7000 additional iterations, with the transition module active, until someconvergence criterion is met, i.e. stabilized force coe�cients and transition position.

The �nal transition position on the upper side, obtained with the Granville cri-terion, is x=c = 0:42. Fig 3.7 shows the initial starting skin friction distribution with thetransition prescribed downstream and the �nal one obtained with the transition moduleafter 7000 iterations. Fig 3.8 shows the convergence history of the method in terms ofglobal force coe�cients (Cl and Cd scaled by 10) and transition position. Fig 3.9 showsthe evolution of the density residual during the computation. It is worthwhile to noticethat the �nal predicted transition location is very close to the laminar separation pointpredicted by the NS solver and also by the boundary layer method (separation point closeto x=c = 0:42). This fact means that, when using the Granville criterion, a slight changein the predicted pressure distribution, arising from using a di�erent turbulence model,produces the same �nal transition location but induced by laminar separation instead ofTS instability. This situation occurs when the Baldwin-Lomax turbulence model is usedinstead of the k � !.

With Drela's approximate database method and the critical N -factor, Ncrit = 8,the �nal transition position on the upper side is obtained at x=c = 0:42. Fig 3.10 and 3.11

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1X

-0.005

0

0.005

0.01

0.015

0.02

Cf

Initial solutionFinal solution

Figure 3.7: Initial and �nal skin friction�B Airfoil�

1 2501 5001 7501 10001 12501Iterations

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Cl,

Cd

*10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xtrU

p,xt

rLow

clxtr-upxtr-lowcd*10

Aerospatiale BAlpha=7 M=0.15 Re=2MillK-W BSL + Granville

Figure 3.8: Convergence evolu-tion, forces and transition location(Granville)

1 2501 5001 7501 10001 12501Iterations

10-13

10-12

10-11

10-10

10-9

10-8

10-7

L2-d

ensi

tyre

sid

ual

Figure 3.9: Density residual (Granville)

1 5001 10001Iterations

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Cl,

Cd

*10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xtrU

p,xt

rLow

clcd*10xtr-upxtr-low

Ncrit = 8

K-W BSL + Drela eN simplified method

Figure 3.10: Convergence evolution,forces and transition location (Drela)

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1 2501 5001 7501 10001 12501Iterations

10-13

10-12

10-11

10-10

10-9

10-8

10-7L2

-den

sity

resi

dua

l

K-W BSL + Drela eN simplified method

Ncrit = 8

Figure 3.11: Density residual (Drela)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45X/C

0

1

2

3

4

5

6

7

8

9

10

N-f

acto

r

Figure 3.12: Convergence evolution,forces and transition location (Drela)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1X

0

0.01

0.02

Cf

Initial (Xtru = 0.6)Final (Ncrit = 8)MSES (Ncrit = 8)

Figure 3.13: Skin friction, �B-Airfoil�

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show the forces, transition location and evolution of the density residual for this case.Fig 3.12 shows the �nal computedN -factor and Fig 3.13 the �nal skin friction distribution.Also plotted in this �gure is the skin friction distribution corresponding to the initialstarting solution and that obtained with the MSES code [15], which uses a viscous-inviscidinteraction method with a compressible integral boundary layer method combined with theDrela's database method. The agreement between the results obtained with both methodsis very good. As additional tests, transition has been predicted for two additional criticalN -factors. For Ncrit = 7:5, the transition is predicted slightly upstream at x=c = 0:4. ForNcrit = 9 the transition is predicted at x=c = 0:42 but as a result of laminar separationdetected by the boundary layer method instead of the transition criteria module.

3.3 NLR contribution

3.3.1 Introduction

The objective of the present study is to improve the accuracy and automation level ofCFD systems for the computation of 2D high-lift �ows around multi-element airfoilsin the framework of Reynolds-Averaged Navier-Stokes (RANS) simulations. The CFDsystem under consideration � FANS � simulates the viscous �ow by solving the steady2D compressible RANS equations, discretised on an unstructured grid using the �nite-volume method. A FAS multi-grid method is applied to speed up convergence. Theturbulence is modelled with a slightly modi�ed version [13] of the standard k-! model[56, 57]. The current improvements are achieved by coupling the CFD system with aseparate laminar-to-turbulent transition modelling algorithm to determine and adapt thetransition onset locations on the individual airfoil elements iteratively during the RANScomputations. Correct prediction of the transition location is crucial for obtaining realisticsurface-pressure and skin-friction distributions, especially for incidence angles near maxi-mum lift characterised by boundary-layer separation and for wind-tunnel �ow conditionscharacterised by a moderate Reynolds number compared to full-scale �ight conditions.

As transition is not predicted by the RANS equations themselves, transition oflaminar �ow to turbulent �ow is imposed explicitly at a prescribed location during the�ow calculations and is updated at a user-speci�ed interval. This approach is straight-forward for well-de�ned �ows in which the transition mechanism is unique. For example,triggering of boundary-layer instabilities by means of tripping with surface-roughnessstrips for single-element airfoils. For �ows around multi-element airfoils, however, multi-ple transition mechanisms may be involved that alter the transition location. Typically,these transition mechanisms are interference phenomena like the con�uence of a turbulentwake and a laminar boundary layer. For these con�gurations, the traditional transitiontriggering is unsatisfactory since the transition location depends on the active transitionmechanism which is in turn a function of the incidence angle. Therefore, the transitionalgorithm developed and tested in this study utilises a compound criterion for four tran-sition mechanisms commonly encountered in 2D high-lift �ows: linear instability, laminarseparation, free-stream turbulence and surface roughness.

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The next sections provide an overview of existing transition prediction tech-niques and the identi�cation of relevant transition mechanisms for 2D high-lift �ows.The implementation of the transition criteria for the individual mechanisms is discussedin conjunction with the established logic to determine the active transition mechanism.The characteristics of the developed algorithm are investigated by performing turbulent�ow computations for both a single- and multi-element airfoil con�guration. The single-element airfoil �ow computations show that for this con�guration the active mechanism islinear instability and that the transition location based on the Granville correlation con-verges (within �ve updates). The prediction of the transition location is observed to be atrue iterative procedure since transition location modi�cations alter the stagnation pointlocation. In turn, the linear instability mechanism predicts a new transition location sincethe properties of the laminar boundary layer have been changed. Computations for themulti-element airfoil con�guration show that the active transition mechanism is di�erentfor the individual elements and that the �ow �eld over the slat is quite intricate. Theunsteady nature of (laminar) �ow separation on a smooth surface hampers convergenceof the transition location on the slat. The fact that the CFD method does not model thetransition process itself may be more important on the slat than on the other elements.

3.3.2 Transition prediction techniques

This section will set the transition prediction mechanisms, used by the FANS CFD sys-tem, in a broader context by providing some background information on the varioustechniques that have been reported in the literature. It will brie�y address simple cor-relation methods, the eN method, the PSE method, linear-combination models and low-Reynolds-number turbulence models as CFD tools for predicting the laminar-to-turbulenttransition. The PSE method can be used for validation of simpler prediction methods forstandard �ows (calibration purposes) in the same way as measurements are traditionallybeing used. Another possible source of validation data is Direct Numerical Simulation(DNS). Validation exercises are performed in a systematic way in the framework of theERCOFTAC Special Interest Group activities on transition modelling [42].

3.3.2.1 Correlation methods

This class of transition prediction methods provides explicit correlations, derived fromlinear stability theory, between the transition location and the �ow conditions. In a CFDcode applying a RANS turbulence model, the transition will then be triggered at thepredicted transition location, which (hopefully) converges along with the velocity �eld.One such linear-theory based method is devised by Granville [26, 39] and this method hasbeen applied in the present work. Although better than assuming a �xed user-speci�edtransition location, the application of these correlation methods is limited to the narrowrange of �ows for which linear instability is the actual transition mechanism. Furthermore,the method of Granville has been designed for 2D incompressible laminar boundary layers.

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3.3.2.2 eN method

A more elaborate linear-theory based transition prediction method is the eN method,suggested independently by both Smith and Gamberoni [47] and Van Ingen [54]. It useslinear stability analysis to determine the envelope around the ampli�cation curves for theTollmien-Schlichting waves in the laminar boundary layer. Transition is then predicted atthe streamwise location where the envelope curve has a certain speci�ed value (around 9),called the N factor. At that point, the amplitude of the most ampli�ed wave is eN timesas large as its amplitude at the beginning of the unstable region. Despite its increasedre�nement compared to linear-theory based correlation methods, its usage is limited toparallel �ows with weak free-stream turbulence (no receptivity e�ects). Fortunately, thisincludes the initial development of the boundary-layer �ow along airfoils. The methodhas been extended to 3D incompressible �ows, see Ref. [5] for a review.

3.3.2.3 PSE method

When a general expression for the disturbance of the mean �ow is substituted into theNavier-Stokes equations which are subsequently parabolised, the Parabolised StabilityEquations (PSE) are obtained [29]. The evolution of the disturbances predicted by theseequations is not restricted to the initial linear development as nonlinear e�ects are takeninto account. The PSE method predicts the transition trajectory more reliably due tothe added �ow physics. In fact, even the linearised version is an improvement as 3Dand non-parallel �ows are supported automatically. Furthermore, adjoint PSE methodsare used for optimisation problems (laminar-�ow control). Disadvantages of PSE are therequired computational e�ort and the sensitivity to the initial and boundary conditions.

3.3.2.4 Linear-combination models

A di�erent class of methods is formed by linear-combination RANS models. These modeltypes specify an empirical function for the intermittency factor which varies from zeroin the laminar regime to one in the fully turbulent regime. Some models then multiplythe expression for the eddy viscosity with this function, assuming that the eddy-viscositymodel provides a turbulent solution in the transition regime otherwise. A well-knownapplication of this model type is the algebraic model of Cebeci and Smith [11]. Otherauthors use the intermittency factor to compute a weighted average of the laminar andturbulent solutions in the transition regime [38] or to create a `uni�ed' model by combiningequations for non-turbulent and turbulent �uctuation growth [17].

3.3.2.5 Low-Ret turbulence models

By introducing appropriate low-Reynolds-number modi�cations into high-Ret RANS tur-bulence models, both laminar and turbulent solutions can be obtained without usingintermittency functions. An explanation of �at-plate transition prediction, based on non-turbulent ampli�cation of low levels of k and ! by the low-Ret k-! model, is given byWilcox [58] and is summarised below. Examples of successful applications of a low-Ret

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turbulence model to a transitional �ow are provided by Hanjali¢ and Hadºi¢ [28, 27]. Ofcourse, low-Ret models are not universal and need tuning for di�erent classes of problems(but this also applies to e.g. the N-factor of the eN method). Furthermore, they oftenneed a minimum level of free-stream turbulence to assure transition from the laminar tothe turbulent branch in the correct area. On the other hand, they promise to o�er �more�exibility and better prospects for predicting real complex �ows with (laminar/turbulent)transitions than any classical linear stability theory� [27].

Wilcox [58] explains the non-turbulent ampli�cation mechanism in the single-point turbulence closure model (k-!) that allows one to integrate the RANS equationsfrom laminar to turbulent �ow. To understand why and how, the k-! model predictionof transition on a �at plate is considered. The �rst observation is that the k-! model hasa non-trivial laminar �ow solution (with �t = 0) for !. The second observation is thatthe sign of the net production terms (i.e. including dissipation) in the k and ! equationsdetermines whether k and ! are ampli�ed or reduced in magnitude. It is found that, aslong as the eddy viscosity �t remains small compared to the molecular viscosity (Ret � 1),the precise locations where the net production of k and ! change sign can be speci�ed interms of the closure coe�cients, a�ecting the beginning and end of transition. In this way,the minimum critical Reynolds number at which linear-stability theory predicts growthof Tollmien-Schlichting waves in a �at plate boundary layer can be matched with thelocation at which non-turbulent ampli�cation of the turbulent kinetic energy k starts.The width of the transition region is controlled by the coe�cient of the net productionterm in the ! equation.

3.3.3 Transition mechanisms and prediction criteria

3.3.3.1 Identi�cation of transition mechanisms in high-lift �ows

Transition mechanisms encountered in 2D high-lift �ows were identi�ed in GARTEURAG-25 [14] and are summarised below:

1. transition initiated by linear instability in attached laminar �ow (natural transition);

2. transition initiated by linear instability in laminar part of laminar/turbulent separationbubble at upper slat surface (separation-induced transition; separation can be causedby shock);

3. transition initiated by free-stream turbulence or con�uence of turbulent wake and lam-inar boundary layer (bypass transition; bypassing linear stability mechanism);

4. transition initiated by user-created surface-roughness strips on geometrical discontinu-ities (bypass transition);

5. transition initiated by distributed surface roughness (bypass transition).

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3.3.3.2 Global transition prediction algorithm

With the high-Reynolds-number turbulence model applied in this study (standard k-!model), the laminar-to-turbulent transition has to be delayed by enforcing laminar �ow(production of k put to zero) upstream of the explicit transition trigger location sex (s isarc length, s = 0 in stagnation zone). The transition modelling algorithm, developed inthis study, is based on the following streamwise locations in both boundary layers of eachairfoil element (corresponding to the above mechanisms with the same number assignedto):

1. The �rst location sls satisfying the Granville criterion [26, 39]. This is the transitionlocation predicted by (simpli�ed) linear stability analysis.

2. The �rst location ssep where laminar or turbulent separation occurs. At the laminarseparation location, transition is assumed to take place immediately, i.e. the laminarpart of the transition bubble is assumed to be negligibly small.

3. The �rst location sfst where su�cient free-stream turbulence is entrained by the bound-ary layer to induce transition. The criterion is that the turbulence Reynolds numberat the boundary-layer edge exceeds a certain threshold value.

4. The �rst location ssr where a surface roughness `strip' on the wing surface will causetransition. This location is set by the user (near the trailing edge in absence of geo-metrical trip locations) and is �xed during the computations.

Furthermore, the computed transition location sct, i.e. the location where the turbulencemodel equations e�ectively become active, is de�ned by the �rst point in the boundarylayer where the turbulent (eddy) viscosity exceeds the molecular viscosity (or equivalently,where the turbulence Reynolds number Ret = �t=� exceeds 10 times the free-stream tur-bulence Reynolds number Ret;1 = 0:1). The locations sls, ssep, sfst and sct are computedat the beginning of each new multi-grid cycle (highest level, passed through to lowerlevels), whereas location sex is updated at a user-speci�ed interval as follows:

sex = min[ssep; ssr]if sct < sex then

sex = min[sls; ssep; sfst; ssr]endif

Notice the key role assigned to laminar separation. The other criteria are only appliedwhen the separation appears to be a turbulent separation. Some aspects of the transitionalgorithm are discussed in more detail below.

Determination of stagnation point and laminar branches The above-mentionedstreamwise locations are searched for on the two laminar-�ow streamlines, called laminarbranches (one upper-surface and one lower-surface branch), connecting the theoreticalstagnation point s = 0 with the user-de�ned trip locations ssr. The theoretical stagnationpoint on a closed surface is de�ned as the point:

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� where the skin friction changes sign;

� which is lying in the input-speci�ed laminar-�ow zone between the geometrical triplocations on the closed surface;

� where the total pressure loss is minimal.

In general, it is unlikely that the theoretical stagnation point will coincide with a gridpoint. For this reason, two stagnation grid points for further use are selected, being thegrid points belonging to the edge on which the theoretical stagnation point is found. Fromeach stagnation grid point a laminar branch starts.

Laminar-to-turbulent transition prediction based on Granville criterion Inorder to apply the Granville criterion, the following data need to be approximated in eachgrid point of all branches:

� the arc length s;

� the boundary-layer thickness Æ;

� the wall-parallel velocity U at the wall-normal location y = Æ, being Ue;

� the momentum thickness � =R Æ0 [U(Ue � U)=U2

e ]dy;

� the kinematic viscosity � at the wall-normal location y = Æ, being �e;

� the velocity gradient dU=ds at the wall-normal location y = Æ, being (dU=ds)e.

The computation of the `pressure-gradient' parameter � = (�2=�e)(dU=ds)e isthen straightforward. This parameter � is a function of the arc length s, as well as of theReynolds number based on the momentum thickness Re� = Ue�=�e and of the instabilityReynolds number Re�;i which is de�ned by the following relations:

Re�;i = exp[2:821 + 34:26 exp(74�)]; � � �0:060Re�;i = exp[5:025 + 30�]; �0:060 < � � �0:024Re�;i = exp[5:257 + 50�+ 430�2]; �0:024 < � � 0:024Re�;i = exp[9:009� 4:7 exp(�29:7�)]; 0:024 < �

Next, the intersection point si between the curve Re�(s) and Re�;i(s) is computed (ifan intersection point exists). This point is the arc-length location where the laminar�ow is supposed to become unstable. The procedure is continued by the computation ofa transition Reynolds number Re�;tr based on the momentum thickness, being a curveobtained from the curve Re�;i(s) by a shift in Re� direction:

Re�;tr = Re�;i +�Re�

with the increment momentum-thickness Reynolds number given by:

�Re� = 642 exp[12:5�m]; �m � �0:020�Re� =

P4n=0Cn�

nm; �0:020 < �m � 0:024

�Re� = 3000� 4714 exp[�64�m]; 0:024 < �m

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andC0 = 0:820571� 103

C1 = 0:282738� 105

C2 = 0:707219� 106

C3 = 0:516769� 107

C4 = 0:223023� 108

In order to evaluate the above equations, the mean pressure-gradient parameter �m =(s � si)

�1R ssi�(s)ds is computed numerically. Finally, the intersection point sls between

the curve Re�(s) and Re�;tr(s) is computed (if an intersection point exists). This point isthe arc-length location where the laminar �ow is supposed to become turbulent accordingto (simpli�ed) linear stability theory (Granville criterion).

Determination of laminar separation and bypass transition The theoretical lam-inar separation point on a closed surface is de�ned as the �rst point found (when runningfrom the stagnation point towards the geometrical trip location) where the skin frictionchanges sign. This location is characterised by the arc length sse. The location may oscil-late between two or more grid points due to numerical instability. This is characterisedby one or more grid points where the skin friction changes sign every multi-grid cycle.

The point on the edge of the boundary-layer grid (ranging from stagnation zone togeometrical trip location) is searched where turbulent �ow `hits' the edge of the boundarylayer, de�ned by Ret = 10�5Re (approximately 40 for the multi-element airfoil). Thislocation is characterised by the arc length sfst. As a �rst approximation, bypass transitionis assumed to happen immediately at this location, i.e. the physical receptivity process ofthe boundary layer is not modelled.

3.3.4 Application to a single-element airfoil

The single-element B-airfoil con�guration of Aerospatiale is selected to study the char-acteristics of the coupling procedure in which the Granville transition prediction methodand a Navier-Stokes method are employed. In fact, the full transition modelling algorithmof the previous section is applied, but the linear-instability mechanism is the only activemechanism for this con�guration.

The viscous �ow around the B-airfoil is computed for the incidence angle � =7 degrees, Mach number M = 0:15 and Reynolds number Re = 2 � 106 (free-streamconditions). An unstructured grid has been generated around the airfoil containing 92,333points. Details of the grid are presented in Figures 3.14 and 3.15. The grid characteristicssatisfy the requirements for Navier-Stokes simulations in terms of high-aspect-ratio cellswith the �rst grid-point spacing corresponding to y+ = 1. The boundary layer is resolvedby at least 32 grid points. The utilised computational strategy consists of obtainingan initial �ow solution for a �xed transition location. This solution is required to havean accurate representation of the laminar boundary layer in terms of the momentum-loss thickness and chordwise velocity gradients. Subsequently, the transition location isupdated corresponding to the Granville transition criterion at a user-de�ned iterationinterval. In the present case, the initial solution is obtained from free-stream values and

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Figure 3.14: Detail of the computational grid for the B-airfoil at the leading edge.

Figure 3.15: Detail of the computational grid for the B-airfoil at the trailing edge.

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0.0 0.2 0.4 0.6 0.8 1.0x/c

−0.2

0.0

0.2

0.4

0.6

0.8

y/c

geometry initial estimate first update

Figure 3.16: Initial transition-location update for the B-airfoil's upper surface.

0.0 0.1 0.2 0.3 0.4 0.5arc length

−1000

0

1000

2000

3000

4000

Rey

nold

s nu

mbe

r

Reynolds (theta) Reynolds (instability) Reynolds (transition)

Figure 3.17: Re� distributions for the initial transition prediction (at cycle 4001).

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0.0 0.1 0.2 0.3 0.4 0.5arc length

−1000

0

1000

2000

3000

4000

Rey

nold

s nu

mbe

r


Figure 3.18: Re� distributions for the second transition prediction (at cycle 5001).

0.0 0.1 0.2 0.3 0.4 0.5arc length

−1000

0

1000

2000

3000

4000

Rey

nold

s nu

mbe

r


Figure 3.19: Re� distributions for the third transition prediction (at cycle 6001).

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0.0 0.1 0.2 0.3 0.4 0.5arc length

−1000

0

1000

2000

3000

4000

Rey

nold

s nu

mbe

r


Figure 3.20: Re� distributions for the fourth transition prediction (at cycle 7001).

0 2000 4000 6000 8000multi−grid cycles

0.1

0.2

0.3

0.4

0.5

x/c

Figure 3.21: Computed transition-location history for the B-airfoil �ow calculation; up-dates activated at cycles 4001, 5001, 6001 and 7001.

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−8

−6

−4

−2

log(

dens

ity r

esid

ual)

Figure 3.22: Convergence history for the B-airfoil computations involving four transition-location modi�cations.

0.0 0.2 0.4 0.6 0.8 1.0x/c

−2

−1

0

1

2

pres

sure

coe

ffici

ent

initial solution final solution

Figure 3.23: E�ect of the transition location on the surface-pressure coe�cient distribu-tion.

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0.0 0.2 0.4 0.6 0.8 1.0x/c

−0.01

0.00

0.01

0.02

skin

−fr

ictio

n co

effic

ient


Figure 3.24: E�ect of the transition location on the skin-friction coe�cient distribution.

0.0 0.2 0.4 0.6 0.8 1.0x/c

0.000

0.001

0.002

0.003

0.004

0.005

mom

entu

m−

loss

thic

knes

s


Figure 3.25: E�ect of the transition location on the momentum-loss thickness distribution(unphysical decay near trailing edge due to hybrid grid de�nition).

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0.0 0.2 0.4 0.6 0.8 1.0x/c

0.0

0.5

1.0

1.5

2.0

y+


Figure 3.26: E�ect of the transition location on the y+ value of the �rst grid points.

0.0 0.2 0.4 0.6 0.8 1.0x/c

0

2

4

6

8

10

kr+


Figure 3.27: E�ect of the transition location on the k+r value of the �rst grid points.

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cycle count transition location

1 0:4210

4001 0:2436

5001 0:2370

6001 0:2396

7001 0:2409

Table 3.1: Upper-surface transition locations x=c for the B-airfoil as function of the multi-grid cycles.

advanced in time by performing 4000 multi-grid cycles. The transition location is �rstupdated after cycle count 4000 and after every additional 1000 multi-grid cycles. Theinitial transition location (user-speci�ed) was set at 42% and 30% of the chord for theupper and lower wing surface, respectively.

The �rst transition-location update results in an upstream correction where thetransition location is �xed at x=c = 24:36% (c is the chord length), see Table 3.1 and Fig-ure 3.16. The distribution of the Re� functions utilised in the Granville method is shownin Figure 3.17. The Re� distributions corresponding to the subsequent updates are doc-umented in Figures 3.18�3.20. In view of the coupling procedure, it is relevant to discussthe properties of the depicted functions. Due to the upstream speci�cation of transition,the upper boundary layer becomes thicker along the airfoil, which is characterised byincreasing values of Re�. Similarly, the values for the transition Reynolds number Re�;trdecrease downstream the turbulent boundary layer. The transition location, de�ned asthe intersection point of Re� and Re�;tr, can initially only move upstream for a givenlaminar boundary layer. The �rst update by the Granville transition prediction methodis therefore restricted to upstream corrections.

The history of the transition location as computed by the transition predictionmethod for each cycle count is presented in Figure 3.21. The numerical values, provided inTable 3.1, indicate that the transition location is corrected in downstream direction afterthe �rst updates. These downstream corrections are due to changed laminar boundary-layer characteristics. The e�ective wing camber is reduced due to the chordwise increase ofthe turbulent boundary-layer thickness. The associated reduction of the adverse pressuregradients leads to a reduced growth of the laminar boundary layer, characterised bysmaller values of Re�. The intersection of Re� and Re�;tr is thus more downstream.The convergence history for the complete simulation, including four transition locationupdates, is illustrated by Figure 3.22.

The e�ect of the transition location on the �ow solution is demonstrated in a com-parison of the initial and �nal solutions (obtained after resp. 4000 and 8000 cycles). Thesurface-pressure distributions presented in Figure 3.23 indicate that the trailing-edge �owis modi�ed. The upstream speci�cation of the transition location on the upper wing sur-face is evident in the skin-friction distribution given by Figure 3.24. The momentum-lossthickness distributions are illustrated in Figure 3.25. The e�ect of the transition loca-

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tions on y+ and k+r (dimensionless sublayer-scaled wall-roughness height) are illustratedby Figures 3.26 and 3.27, respectively.

3.4 ONERA contribution

3.4.1 Implementation of the transition criteria

First transition criteria have been obtained from experimental correlations [26, 16, 45, 1].They are generally of the form R� equal to a function of a pressure gradient parameterand of the free-stream turbulence level. More recently, stability computation based onself-similarity solutions have been used to produce criteria with a stronger theoreticalsupport [7]. The data base methods directly rely on such computations. As long asself-similarity boundary layer are considered, it is equivalent to choose �2 = �2=� due=dxor the incompressible shape parameter Hi to characterize the pressure gradient e�ect.The choice of Hi results in the desire to include some history e�ects for practical �owsin which �2 does not remain constant. In practice, Hi < 2:6 means that the boundarylayer is more stable than for a �at plate and Hi > 2:6 corresponds to a destabilizationof the boundary layer, all the more important that Hi is far from 2:6 (�2 = 0). It hasbeen seen that in NS computations, Hi is generally greatly underestimated although thepressure coe�cient is closely obtained. This implies that �2 is obtained with the correctsign. Choosing this parameter instead of the shape factor will give the correct stabilizingor destabilizing tendency which is probably far more important to respect than includingthe history e�ect. From the Falkner-Skan self-similarity solutions, we propose to correlateHi to �2 using:

Hi = 4:02923�q�8838:4�4

2 + 1105:1�32 � 67:962�2

2 + 17:574�2 + 2:0593 (3.11)

with �2 in the range [�0:068253; 0:1]. The lower value corresponds to separation (Hi = 4).This estimation of Hi can replace the value directly obtained from the integration of thevelocity pro�les in the NS computations every time it is needed.

To discuss more precisely the precision which can be expected on Hi from �2, onecan consider a case with a small pressure gradient (�2 equal to a few 10�3). Assumingthe error on dp=dx small with respect to the error on the momentum thickness, one have��2=�2 ' 2��=� and �Hi ' 12�2��=� for small values of �2. With ��=� ' 10% thisleads to �Hi ' 1; 2�2. This shows that the precision required for the criteria is obtainedprovided �2 is smaller than a few 10�3. Within these conditions, the error on R� becomesthe most important contribution to the overall precision of the criteria.

3.4.1.1 Abu-Ghannam & Shaw criterion

This criterion [1] is very popular for turbomachinery computations. It expresses themomentum Reynolds number at the onset of the transition region as a function of thelocal Pohlhausen parameter �2 and the turbulence level, expressed in % :

R�T = 163 + exp

"F� (�2T )� F� (�2T )Tu

6; 91

#(3.12)

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F� (�2) = 6; 91 + 12; 75�2 + 63; 64 (�2)2 for �2 < 0 (3.13)

F� (�2) = 6; 91 + 2; 48�2 � 12; 27 (�2)2 for �2 > 0 (3.14)

This criterion has been calibrated for Tu values up to 9%. It was obtained by a correlationof a large number of experimental results in which �2 was in the range �0; 1. For applica-tion in NS codes, �2 must be bounded in this range because the parabola correspondingto �2 < 0 reaches a minimum for �2 = �0:1 and the branch for �2 < �0:1 has no physicalmeaning and must not be used.

3.4.1.2 Arnal-Habiballah-Delcourt criterion (AHD)

This criterion was developed by Arnal et al. [6, 7]. It is based on the linear stabilitytheory and more particularly devoted to the fast estimation of the onset of the transitionlocation for �ows with small external turbulence levels. It is limited to incompressible�ows (M < 0:6) over adiabatic walls.

The basic idea is to use the Falkner-Skan self similarity solutions to represent thelaminar boundary layer pro�les which are characterized by the local Pohlhausen parame-ter �2. For each solution, the total ampli�cation coe�cient can be expressed by :

n = n (R� �R�cr;�2) (3.15)

in which R�cr is the critical Reynolds number based on the momentum thickness � at thepoint where some frequencies become unstable. The transition location is characterizedby n = nT , R� = R�T . Combining (3.15) written at the transition location and the Mackrelationship (nT = �2; 4 lnTu� 8; 43), nT can be eliminated to obtain

R�T �R�cr = f (�2; Tu) (3.16)

The above relationship is strictly valid for self similarity boundary layers. It is extendedto actual �ows by replacing �2 by its mean value along the streamline corresponding tothe region where the boundary layer is unstable :

�2 =1

s� scr

Z s

scr

�2

�

dueds

ds =1

s� scr

Z s

scr

�2ds (3.17)

Arnal has proposed the following analytical form of the criterion :

R�T �R�cr = �206 exp�25; 7�2T

� hln (16; 8Tu)� 2; 77�2T

i(3.18)

The two parameters R�T �R�cr and �2T where previously introduced by Granville [26] forhis criterion corresponding to an analytical correlation of experimental results for smallturbulence levels, of the order of 1%. The relationship (3.18) relies on theoretical basisand includes the Tu e�ect.

For practical applications, R�cr must be known, as well as the scr abscissae. Thisis done by comparing the actual R� value to a critical value given by :

R�cr f = exp�52

Hi

� 14; 8�

(3.19)

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For R� < R�cr f , the boundary layer remains stable. scr is obtained for R� = R�cr f . Theexpression (3.19) is very sensitive to the value of the shape parameter Hi. To overcomethe precision problem on that parameter in NS computations, Hi is obtained from �2 byequation(3.11). Moreover, Hi is imposed to remain in the range (2.2 to 4).

To avoid numerical di�culties in computing �2 which needs to know due=dx atthe boundary layer edge, this later quantity is replaced by dp=dx at the wall (�rst orderboundary layer theory).


A �rst validation of the implementation has been done using the �B-Pro�le� con�guration.The transition is imposed at x=c = 0:3 on the lower side and transition criteria areapplied on the upper side. The computations have been done with the arti�cial viscositycoe�cient �4 equal to 0:016 and the Martinelli correction. The transition criteria arecomputed every 15 iterations. This choice is not at all critical. A small value (less than5 iterations) is time consuming because the algorithm for the transition criteria is notfully vectorized. A large value (100 iterations or more) may lead to a divergence of thecomputation, mainly because of the development of separated laminar regions of largeextent.

Abu Ghannam & Shaw criterion

For the �rst calculations, we consider the Abu Ghannam & Shaw criterion with threedi�erent conditions:

� Tu = 1%, initial condition from the converged solution of the imposed transition case,

� Tu = 0:1%, same initial condition as case Tu = 1%,

� Tu = 0:1%, initial condition from results of case Tu = 1%.

For all cases the preconditioning acceleration technique is used with a two-level multi-gridprocess. The results relative to the skin friction are given in �gure 3.28. As expected,transition occurs earlier for Tu = 1% than for Tu = 0:1%. More interesting, it can be seenthat the transition location can move downstream without any problem and the samesolution is obtained whatever the direction of displacement of the transition point duringthe convergence process. However, the convergence process takes more time when thetransition point moves downstream. In that case, turbulence must disappear and this isobtained by imposing very small �t values in the laminar region. This leads to suppressthe production of k whereas its dissipation remains active. In cases 2 and 3 the samenumber of iterations is done and the remaining small di�erence (see �gure 3.28) comesfrom a lack of convergence for case 2 (transition moving upstream).

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Arnal-Habiballah-Delcourt criterion

This criterion is used with the same numerical conditions as previously. The externalturbulence level is set to 0:1% and the converged solution of the imposed transition case isused as initial condition. Figure 3.29 shows the skin friction evolution on the upper side ofthe airfoil. The �AHD� criteria gives a more downstream transition location than the AbuGhannam & Shaw criteria not really suited for small Tu values. The 3C3D boundary layercomputation results are those of �step 1�, �xed transition and no intermittency function.The too early transition location obtained with the criterion is mainly explained by theoverestimation of the momentum thickness in the NS computation.

The pressure distribution is given in �gure 3.30 for the three cases :

� �xed transition at x=c = 0:42,

� computed transition with Tu = 0:1% and �AHD� criterion,

� computed transition with Tu = 1%.

The result obtained with the Abu Ghannm & Shaw criterion with Tu = 0:1% gives exactlythe same result as the �AHD� criterion, as concerns the pressure. The small di�erenceregarding the transition location between the the �AHD� and the �xed cases has nearlyno e�ect on the pressure distribution, except near x=c = 0:35. However, setting thetransition location too early, at x=c = 0:12 with Tu = 1% and the Abu Ghannm & Shawcriterion, has a large in�uence. The Kp plateau is greatly decreased and the Kp valueat the trailing edge is increased, in better agreement with the experiment. This suggestthat it would be necessary to take into account the thickening of the boundary layer inthe transition bubble to improve the results.

3.5 QinetiQ contribution

3.5.1 Description of the computational procedure

The procedure adopted at QinetiQ for calculating the results for steps 2 and 3 of thisGARTEUR exercise is a two-level iterative procedure in which the RANSMB Navier-Stokes solver is used to calculate the �ow around the element at speci�ed conditions (whichinclude the location of transition), and BL2D is used to calculate the boundary layersfrom the RANSMB solution, perform boundary layer analysis and predict the transitionlocations. These new transition locations are then provided as input to RANSMB forfurther cycles of this solver. The process is repeated in an automated fashion until boththe RANSMB solution and the predicted transition locations have converged.

3.5.2 Transition criteria employed

BL2D incorporates a range of empirical transition criteria including three simple methodsfor two-dimensional, incompressible �ow, two methods for two-dimensional, compressible

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x/c

Cf 0

0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01


0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

123

1 : Tu = 1%,2 : Tu = 0:1%, transition from up-

stream during convergence,3 : Tu = 0:1%, transition from down-

stream during convergence.

Figure 3.28: �B-Pro�le�, computed tran-sition. Skin friction coe�cient.

x/c

Cf 0

0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01


0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

123

1 : Abu Ghannam, Tu = 1%,2 : Abu Ghannam, Tu = 0:1%,3 : �AHD� criterion Tu = 0:1%.

Figure 3.29: �B-Pro�le�, computed tran-sition. Skin friction coe�cient.

x/c

Kp

0 0.25 0.5 0.75 1

-1

-0.5

0

0.5

1

1.5

2

x0 0.25 0.5 0.75 1

-1

-0.5

0

0.5

1

1.5

2

123

1 : �xed transition (x=c = 0:42)2 : �AHD� or Abu Ghannam & Shaw criterion, Tu = 0:1%3 : Abu Ghannam & Shaw criterion, Tu = 1%

Figure 3.30: Pressure distribution

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�ow and two methods for three-dimensional instability. In the present work, the Granvillecriterion has been used. This relates the stability of the boundary layer to the di�erencein Reynolds numbers based on momentum thickness from the neutral stability point totransition:

R� � R�0 < g3 + g4�� g5

�3(3.20)

where � is an averaged Pohlhausen parameter � = �2=� dUe=dx , given by

� =1

x� x0

Z x

x0

�2

�

dUe

dxdx: (3.21)

Granville's neutral stability criterion gives the momentum thickness Reynolds number atthe neutral stability point as R�0 = 10G where G = g2�

2 + g1� + g0. The values of thevariables g0 to g5 are given in the tables below.

g3 g4 g5 �470.0 5781250.0 0.04 < 0:011192.7 42877.0 -0.01 � 0:01

g0 g1 g2 �2.3818 18.3125 90.625 � 0:012.36 21.43 0.0 < 0:042.143 33.85 -175.9 � 0:04

These values have been obtained by calibrating the model through comparisonwith eN predictions for an N factor of 9. Using the eN method with N=9 to predictthe transition location, rather than the present implementation of the Granville criterion,should therefore yield essentially the same results. Similarly, using the eN method with adi�erent N factor will yield di�erent results.

3.5.3 Calculation performed for the AS-B aerofoil

In the calculations performed for the AS-B aerofoil for step 2, the transition location wascalculated for the suction surface and speci�ed as 30% chord for the pressure surface,as in the experiment. The calculations represent the �rst validation of the numericalscheme and no attempt has been made to optimise performance, either in the degreeof convergence obtained with RANSMB between calculations of the transition locations,or in the use of relaxation factors when recalculating transition locations. Furthermore,e�ects of varying parameters in the turbulence model, such as the free-stream value of g,have not been investigated.

Initial cycles of RANSMB were performed with transition set at 90% chord on thesuction surface. When the calculation had converged, in the sense that the variation inpredicted lift coe�cient was small, the boundary layer edge velocities were extracted andpassed to BL2D for calculation and analysis of the boundary layers. The suction surfacetransition location was calculated according to Granville's criterion and passed back to

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Iteration

XT up

p

1 2 3 4 5 6 7 8 9 10 11 12 130

0.2

0.4

0.6

0.8

1

With under-relaxationWithout relaxation

Figure 3.31: Predicted upper surface transition location. AS-B aerofoil, M0 = 0:15,� = 7Æ, Re = 2:0 106

RANSMB after under-relaxation for further cycles of the Navier-Stokes solver. Whenthese cycles had converged, the process was repeated. Two calculations were performed.In the �rst, the relaxation factor for updating the transition location was set to 1 (i.e. nounder- relaxation). In the second, the relaxation factor was set to 0.5 until the predictedlocation of transition onset had remained constant for �ve successive predictions, afterwhich it was set to 1. The variation of transition location (after under-relaxation) withiteration number of the outer loop is shown in �gure 3.31. The relaxation scheme takesthe form

xnewtr (i) = �ixptr(i) + (1� �i)x

newtr (i� 1); (3.22)

where xtr is x=c at transition onset, superscripts p and new indicate values predicted byBL2D and used by RANSMB respectively, i is the transition prediction iteration numberand �i is the relaxation factor, typically 0:5 � �i � 1:0.

Transition occurs through laminar separation associated with a rapid rise in pres-sure, as seen in the results for step 1 (2.2.5.3). It seems likely that once the rapid rise inpressure is established, it will be very di�cult for the predicted location of transition on-set to move downstream since the pressure gradient will provoke separation. The resultsobtained with the use of under-relaxation are therefore likely to provide a more accuraterepresentation of the �ow. It is these results that are presented in the remainder of thesection.

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x/c

Cp

0 0.25 0.5 0.75 1

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Predicted transitionImposed transition

Figure 3.32: Comparison of Cp distributions with predicted and imposed transition onthe upper surface. AS-B aerofoil, M0 = 0:15, � = 7Æ, Re = 2:0 106

3.5.3.1 Smoothing of the computational data

In extracting boundary layer edge velocities from the Navier-Stokes solution, it was ob-served that the surface pressure coe�cient invariably �uctuated in the vicinity of theattachment point. The magnitude of the �uctuations was such that, on occasions, pres-sures in excess of stagnation were predicted. This led to di�culties in determining thelocation of the stagnation point and provided unsatisfactory boundary conditions for thesubsequent calculation of the boundary layers by BL2D. The problem was overcome byapplying a one-dimensional Schumann �lter to the cell-centred values of pressure predictedby RANSMB. The �lter was applied in the direction around the aerofoil from trailing edgelower surface to leading edge to trailing edge upper surface in each of the �rst two rowsadjacent to the aerofoil surface.

3.5.3.2 Results

The results were obtained using the grid provided by DLR and described previously.The pressure distribution calculated using under-relaxation in the transition predictionis shown in �gure 3.32 and is compared with the result obtained with imposed transitionlocations for step 1 of this exercise (�gures 2.15 and 2.17). Laminar separation fromthe suction surface is predicted by the boundary layer code at 37.9% chord, somewhatupstream of the laminar separation observed experimentally at 42% chord. There is an

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0

2

4

6

8

10

12

0 0.1 0.2 0.3 0.4 0.5 0.6

Loc

al N

-fac

tor

x/c

CoDS v4.0 incompr. N-factors for various f, β modes (kHz, k/m).E876544333222

Figure 3.33: N factors for a range of disturbance modes in the Navier-Stokes solution.AS-B, M0 = 0:15, � = 7Æ, Re = 2:0 106

associated reduction in lift coe�cient, from 1.2802 when transition is imposed on bothsurfaces to 1.2726, since the peak suction has been reduced and the sudden increase inpressure associated with transition seen previously (section 2.2.5.3) has, of course, movedforward. There is a corresponding increase in drag coe�cient from 0.01120 to 0.01155.The loss in lift is greater than found by DLR (section 3.1.3.1), for example, but in that casetransition was predicted at 39.27% chord. Although the present result was obtained usinga particular implementation of the Granville criterion calibrated to yield results similarto those from the eN method with an N factor of 9, it may be noted that INTA predictedtransition at 42% chord using both the Granville criterion and the Drela envelope methodwhich uses the eN method with, for this case, a critical N factor of 8 (section 3.2.2). Fur-thermore, NLR predicted transition at 24.09% using the Granville criterion (section 3.3.4).and ONERA predicted a wide range of locations depending strongly on the free streamturbulence assumed as well as the transition criterion used (section 3.4.2). Nonetheless,none of the solutions of other participants exhibits the rapid recompression at transitionseen in the present solution. Stability analysis of the Navier-Stokes solution con�rms thelaminar separation at 37.9% chord if a critical N factor of 9 is assumed. The criticalN factor would need to be less than 6.66 for transition to occur through ampli�cation ofdisturbances. N factors for a range of modes are shown in �gure 3.33. The distributionof the coe�cient of skin friction for the Navier-Stokes solution is shown in �gure 3.34.The behaviour is similar to that observed for the case with transition imposed on bothsurfaces, discussed in section 2.2.5.3. The �gure shows that the upstream in�uence oftransition extends over more than 8% chord.

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x/c

Cf

0 0.25 0.5 0.75 10

0.002

0.004

0.006

0.008

0.01

0.012

Predicted transitionImposed transition

Lowersurface

Uppersurface

Figure 3.34: Comparison ofCf distributions with predicted and imposed transition on theupper surface. AS-B aerofoil, M0 = 0:15, � = 7Æ, Re = 2:0 106

Chapter 4

Third step: Validation

4.1 CAST10 airfoil

4.1.1 Flow conditions

The proposed test case corresponds to experiments conducted in the cryogenic T2 ONERAwind tunnel, in Toulouse. Although adaptative walls are used to reproduce uniformconditions at in�nity, small correction remains necessary because of the lateral wall e�ects.The chord of the airfoil is equal to 0:18m and its trailing edge has a relative thicknessequal to 0:005. Imposed and natural transition conditions have been studied as well asReynolds number e�ect. The external turbulence level Tu in the wind tunnel is equalto 0.001.

In the experiment, the transition location is obtained from oil �ow visualizationand pressure �uctuations at the wall. The oil �ow visualizations do not give the transitiononset but the location of the maximum of the skin friction.

The chosen validation cases correspond to natural transition conditions at smallincidence. These conditions are:M0 = 0:73, � = �0:25Æ, Pi = 1:67 106, Ti = 293K, RC = 3:9 106

M0 = 0:73, � = �0:25Æ, Pi = 1:65 106, Ti = 120K, RC = 13:5 106

Figure 4.1 shows the isentropic Mach number at the wall for the two Reynoldsnumbers. Mwall is obtained from the measured pressure distribution using the isentropicrelationship:

p

pi=�1 +

� 1

2M2

� �

�1

(4.1)

The classical pressure coe�cient is given in �gure 4.2. For M0 = 0:73, one has:

Kp =p=pi � 0:7016

0:2617(4.2)

68

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x/c

Mw

all

0 0.25 0.5 0.75 10.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Rc = 3.9 106

Rc = 13.5 106

Figure 4.1: CAST10 airfoil, naturaltransition - Wall isentropic Mach num-ber.

x/c

-Kp

0 0.25 0.5 0.75 1

-1

-0.5

0

0.5

1

Rc = 3.9 106

Rc = 13.5 106

Figure 4.2: CAST10 airfoil, naturaltransition - Pressure coe�cient.

4.1.2 Navier-Stokes computations

For the computations, a two block mesh has been done by ONERA to take into accountthe thick trailing edge of the airfoil. The �rst block is 257�107 C-type around the airfoil,ending at the trailing edge. The second block starts at the trailing edge. It is made of2�107+19 nodes in the normal direction (19 points in the wake) and 51 nodes in the �owdirection.

To estimate the correction of the �ow parameters, various computations havebeen done with the k-l turbulence model using a coarse mesh ( every 2 points). The bestresult corresponds to the following conditions:M0 = 0:725, � = �0:35Æ, Pi = 1:67 106, Ti = 293K, RC = 3:9 106

Figure 4.3 shows the isentropic wall Mach number computed with these conditions andwith the �ne mesh. The agreement is far to be perfect, probably for various reasons:the large sensitivity of the airfoil to �ow conditions, the complex side wall e�ects whichcannot be simply corrected by a change on the incidence and Mach number and alsoby the underestimation of the boundary layer downstream the separation bubble in theNavier-Stokes computation because transition is imposed just before this bubble.

4.1.3 Boundary layer computations

Due to the limited number of results relative to transition in the experiment, boundarylayer computations have been done using the experimental pressure distribution. Thesecomputations provide more information to validate the implementation of the transitioncriteria in the Navier-Stokes solvers.

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x/c

Mw

all

0 0.25 0.5 0.75 10.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

x0 0.25 0.5 0.75 1

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Figure 4.3: CAST10 airfoil, �xed transition - Isentropic Mach number using the proposedcorrected conditions

In the boundary layer computations, the �AHD� criterion as well as the �en� database method of Arnal [3] have been used. In the �en� method, the Mack relationshipn(Tu) is used to detect the transition onset. From that relation, for Tu = 0:001, n is equalto 8.15.

In the ONERA �gures, the eN data base methode is labeled �paraboles�.

Suction side, Rc = 3:9 106

The results corresponding to the suction side are given in �gure 4.6. The transitionlocation given by the two criteria corresponds to:

R� T x=cTAHD 864 0.3�en� 914 0.414

With the �en� criterion, the �n� evolution corresponds to:

x=cT 0.2 0.3 0.4 0.414n 2.18 5.73 8.0 8.15

The shift between the transition onset location and the increase of the skin friction coe�-cient (�gure 4.6) is due to the intermittency function and the di�erence between the twocriteria is due to compressibility e�ects, not present in the �AHD� criterion. To explainthis, one can consider �gure 4.4 from Arnal [4] which shows the evolution of the N -factor

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Figure 4.4: CAST10 airfoil, application of the simpli�ed eN method (also called�paraboles�)

Figure 4.5: CAST10 airfoil, application of the simpli�ed eN method. Comparison withexperiment.

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along the airfoil with and without compressible e�ects in the stability computations. Thislatter case corresponds to Me = 0. Figure 4.5 gives the transition location at three inci-dences. The �AHD� criterion is in good agreement with the N -factor method neglectingcompressibility e�ects. This result is encouraging because it seems to indicate that theindroduction of compressibility e�ects in the simpel �AHD� criterion will leed to goodresults.

The location of the maximum of the friction coe�cient given by the �en� criterionis in good agreement with the �ow visualizations.

Pressure side, Rc = 3:9 106

On the pressure side none of the two criteria detect transition before the laminar boundarylayer separation point. This con�rms that transition occurs through a laminar separationbubble.

Suction side, Rc = 13:5 106

At this Reynolds number none of the criteria detect transition which occurs through alaminar separation bubble at x=c = 0:18 (�gure 4.8). The boundary layer computationcan be continued after that point only if transition is imposed at x=c = 0; 16.

Pressure side, Rc = 13:5 106

Figure 4.8 shows that a transition bubble must exist at x=c = 0:52. The boundary layercomputation can be continued after that point only if transition is imposed at x=c = 0; 5.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

1000

2000

3000

4000R

3C3D - AHD3C3D - Paraboles

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1

2

3

Hi


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x/L

-2.0

0.0

2.0

4.0

6.0

*10-3

Cf 0


Figure 4.6: CAST10 airfoil - Boundary layer computation at Reynolds number equal toRC = 3:9 106 - Suction side

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

1000

2000

3000

4000R

3C3D - AHD3C3D - Paraboles3C3D - Transi. imposee

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1

2

3

4

Hi


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x/L

-2.0

0.0

2.0

4.0

6.0

*10-3

Cf 0


Figure 4.7: CAST10 airfoil - Boundary layer computation at Reynolds number equal toRC = 3:9 106 - Pressure side

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

1000

2000

3000

4000R


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1

2

3

Hi


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x/L

-2.0

0.0

2.0

4.0

6.0

*10-3

Cf 0


Figure 4.8: CAST10 airfoil - Boundary layer computation at Reynolds number equal toRC = 13:5 106 - Suction side

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

1000

2000

3000

4000R


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1

2

3

4

Hi


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x/L

-2.0

0.0

2.0

4.0

6.0

*10-3

Cf 0


Figure 4.9: CAST10 airfoil - Boundary layer computation at Reynolds number equal toRC = 13:5 106 - Pressure side

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4.2 Validation

The CAST10 airfoil was proposed as a common test case but partners had the possibilityto use their own test case. NLR preferred to concentrate on a multi-element airfoil andONERA used the RAE2828 airfoil in addition to the common test case. For the test casevalidations, the two main strategies developed during step 2 have been used: couplingwith a boundary layer solvers for DLR, INTA and QinetiQ, direct application of transitioncriteria in Navier-Stokes solvers for NLR and ONERA.

4.2.1 DLR contribution

For the CAST10 airfoil test case DLR had to generate a new grid based on the originalgrid provided by ONERA. It turned out that the original grid contained discontinuities inthe metrics which were too strong to be handled by the cell-vertex discretization schemein FLOWer which was used by DLR throughout the work in this AG. Using the originalgrid none of the computations converged.

The new grid consists of a C-block around the contour with 356 x 106 cells, 256on the airfoil surface, 35 in the boundary layer, and an H-block for the wake resolutionhaving 50 x 60 cells, 50 cells in longitudinal direction of the wake from the trailing edgeto the far�eld edge of the grid, 60 cells along the thick trailing edge, �g. 4.10.

The predictions were performed using the standard Wilcox k�! model and werestarted with initial transition location xupptr = xlowtr = 90% of chord.

The computations for both, the low (run 154) and the high (run 158) Reynoldsnumber case were done for original and for modi�ed aerodynamic parameters (M0 and �)as they were given in section 4.1.1. The computations were done with prescribed transitionlocations and with the prediction strategy running.

The prescribed transition locations were the following:

run 154 : xupptr = 47% of chord xlowtr = 61% of chordrun 158 : xupptr = 18% of chord xlowtr = 48% of chord

For the predictions the limiting N -Factor N jxT was set to the value of N jxT =8:15 according to the Mack relationship, see above in section 4.1.3.

The results of the predictions are:

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x

y

0 0.25 0.5 0.75 1

-0.1

0

0.1

0.2

x1 1.1 1.2-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

x1 1.01 1.02-0.018

-0.017

-0.016

-0.015

-0.014

-0.013

-0.012

-0.011

-0.01

-0.009

-0.008

Figure 4.10: DLR C-H grid for the CAST10 airfoil

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run 154, original: xupptr = 25.34% of chordlaminar separation point at cycle = 1530determined by boundary layer code

xlowtr = 52.2% of chordlaminar separation point at cycle = 1050determined by boundary layer code

run 154, modi�ed: xupptr = 26.22% of chordlaminar separation point at cycle = 2250determined by boundary layer code


run 158, original: xupptr = 24.48% of chordlaminar separation point at cycle = 1200determined by boundary layer code


run 158, modi�ed: xxupptr = 20.52% of chordlaminar separation point at cycle = 1200determined by boundary layer code


The comparison of the Cp-distributions, �g. 4.11, shows good results for thepredicted transition locations of run 158 and a reasonable distribution of the pressure forthe modi�ed parameter settings.

For run 154, the Cp-distribution for the modi�ed parameter settings seems to bereasonable too. The transition location on the lower side is predicted about 10% of chordupstream of the clearly visible location of the separation bubble on the lower side. Thetransition location on the upper side however, does not seem to be correct at all. Also onthe upper side, the position of the transition point is clearly visible as a Cp-disturbance.

The reason for this may be a value for the limiting N -factor N jxT = 8:15, whichis too high, as up to the point where the laminar boundary layer code detects laminarseparation on the upper side no ampli�ed disturbances reaching N jxT = 8:15 were found.

To get a more accurate approximation for the limiting N -factor the experimentalCp-distribution of the upper side of run 154 was analysed using the laminar boundarylayer code and the eN database method. The enveloppe of the N -curves at x/c = 47% ofchord yields a limiting N -factor N jxT = 7:84.

A computation with modi�ed aerodynamic parameters again leads to a transitionlocation at about 25% of chord due to a laminar separation on the upper side detected

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x/c

c p

0 0.25 0.5 0.75 1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

exp.fixed transitionfixed transition, mod.predicted transitionpredicted transition

CAST 10, GARTEUR AG35, run 154Wilcox k-ω

fixed

predicted

x/c

c p

0 0.25 0.5 0.75 1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

exp.fixed transitionfixed transition, mod.predicted transitionpredicted transition, mod.


predictedfixed

no MOD

MOD

Figure 4.11: Cp-distributions for the CAST10 airfoil, run 154: up, run158: down

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by the boundary layer method.

This behaviour leads to the assumption that this aerodynamic case is very sen-sitive to the values of the aerodynamic parameters with respect to the gradients of theCp-distribution, which �nally are responsible for the success or the failure of the transitionprediction.

Thus, the settings of the aerodynamic parameters were modi�ed once more, suchthat the pressure gradient of the computed Cp-distribution becomes as similar as possiblecompared to the experimental Cp-distribution downstream of the suction peak on theupper side. The corresponding values are:

M0 = 0:71, � = �0:3Æ.With these settings and with N jxT = 7:84 one gets the following results:

run 154, modi�ed new: xupptr = 46.4% of chordTS instability at cycle = 17000determined by eN database


A comparison of the Cp-distribution shows a much better agreement of exper-imental and computed pressure with respect to the Cp-gradients, �g. 4.12. The gapbetween the experimental and computed pressure curve downstream of the suction peakhas decreased visibly.

The transition location on the upper side was predicted with excellent accuracy.

In contrast to the computations done before, in the latter case the intervals be-tween the calls of the transition module are much larger. Thus, the convergence rate ofthe transition location iteration is extremely low, as the transition module was called forthe 1st time after 5000 RANS cycles and then only every 2000 RANS cycles. This wasdone to ensure that, in any case, the Cp-distribution is smooth enough before the laminarboundary layer method was applied. To improve the convergence rate, a sensitivity inves-tigation with respect to the number of RANS cycles after which the transition module canbe called has to be undertaken to �nd out the minimum number of RANS cycles possiblebefore the transition module is called. For all the other cases this has already been done,for the latter case this has still to be done. From this point of view, this solution is ofintermediate character.

Fig. 4.13 shows the convergence history of the RANS computation, left, andthe convergence history of the transition location iteration, right. It is obvious that theoverall number of RANS cycles can be reduced signi�cantly, as the transition module canbe called when the lift coe�cient has reached a stable region.

The convergence history of the transition location iteration shows clearly the ne-cessity to underrelax the transition locations that are directly determined by the database.

Fig. 4.14 shows the �nal Cf -distributions for run 154, left, and run 158, right,which are documenting the high quality of the predicted transition locations.

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x/c

c p

0 0.25 0.5 0.75 1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

exp.fixed transitionpredicted transitionfixed transition, mod. newpredicted transition, mod. new


Figure 4.12: Cp-distributions for the CAST10 airfoil, run 154 with new settings

4.2.2 INTA contribution

In order to assess the transition coupling method for transonic �ows the ONERA CAST10pro�le has been computed. The Navier-Stokes solver used by INTA in this project can notdeal with structured multiblock meshes. Therefore the computations have not been carriedout on the mesh delivered by ONERA (with two blocks). Two additional C-type meshes,with 328x81 points have been generated by INTA. These two meshes have a typical meshresolution corresponding to a 2D/3D turbulent computation, i.e. y+ ' 1 and around 30points inside the boundary layer. The �ow conditions used for transition prediction arethose recommended by ONERA, � = �0:35Æ, M = 0:725 and Re = 3:9 and 13:5 Millionsbased on the chord. The critical N factor is set to Ncrit = 8:15, which corresponds to aturbulence level of Tu = 0:001 when using Mack's relation and gives, as has been describedin section 4.1.3, a transition position close to the experimental one when using the ONERA3C3D code together with the �AHD� criterion. In order to test the fully automation of thealgorithm, the computations start from scratch, assuming an initial transition position atx=c = 0:6 on both sides (upper a lower) of the pro�le. Although one can prescribe theinitial transition farther downstream, i.e. at the trailing edge, it seems to make more senseon physical grounds to prescribe the initial transition upstream which additionally helpsthe convergence of the algorithm. The values used for the numerical control parametersare Ntr = 250, != 0.5 and "cl = 1: 10�3. A low under-relaxation parameter has been

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cycle

ρ-re

sid

ual

lift

dra

g

1 5001 10001 15001 20001

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0.2

0.3

0.4

0.5

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

residualliftdrag(total)

cycles

x tr,u

pper

side

x tr,l

ower

side

0 5000 10000 150000.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

-0.100.10.20.30.40.50.60.70.80.91

lam. sep. FLOWerlam. sep. BL codextr underrelaxedxtr by database

upper side

lower side

Figure 4.13: convergence history of the RANS computation, up, and the convergencehistory of the transition location iteration, down

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x/c

c f,inf

0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006fixed transitionfixed transition, mod.predicted transitionpredicted transition, mod.


x/c

c f,inf

0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006fixed transitionfixed transition, mod.predicted transitionpredicted transition, mod.


Figure 4.14: �nal Cf -distributions of the CAST10 airfoil for run 154, up, and run 158,down

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1 5001 10001Iterations

10-12

10-11

10-10

10-9

10-8

10-7

L2-

den

sity

resi

dual

Figure 4.15: Density residual CAST10 Re=3:9 106

used in the computations. This transonic case is extremely sensitive to the intermediatetransition locations, i.e. the transition position itself strongly in�uences the pressuredistribution. Therefore, a more conservative set of numerical control parameters needs tobe prescribed in order to obtain a reliable convergence of the algorithm.

Fig 4.15 and 4.16 display the density residual and the convergence history of theforce coe�cients as well as the upper and lower transition position after 15000 iterationsfor the lowest Reynolds case, Re = 3:9 106. The convergence is acceptable and the �naltransition position on the upper side obtained with the coupling method using Drela'scriterion is around x=c = 0:38, which is quite close to the experimental one obtained from�ow visualisation. In �g 4.17 is shown the obtained �nal skin friction distribution on thesurface pro�le. Additionally, the skin friction obtained with the MSES code and thatobtained by ONERA from the analysis of the experimental pressure distribution with the3C3D code in combination with the �AHD� criteria are plotted. The agreement among thethree methods is fairly good. The di�erences are bigger at the transitional region for the�AHD� criteria but this discrepancy could be explained by the intermittency function usedin the ONERA computations. The transition obtained by the coupling method on thelower side comes from a laminar separation. A comparison among the obtained pressurecoe�cient distribution, the experimental one and that obtained from the MSES code isplotted in �g 4.18. It can be seen in this �gure that the pressure suction on the upperside is over-predicted by the computations. Furthermore, there is a strong in�uence of the

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1 2001 4001 6001 8001 10001 12001 14001Iterations

0

0.25

0.5

Cl,

Cd

*10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xtrU

p,xt

rLow


CAST10Alpha=-0.35 M=0.725 Re=3.9 MillK-W BSL + Drela DBM with N=8.15

Figure 4.16: Convergence evolution, forces and transition location

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1X/C

-0.005

0

0.005

0.01

Cf,

mcf

NS+BL+Drela DBMMSES VIIExp pressures + 3C3D

Ncrit = 8.15

Figure 4.17: Skin friction, CAST10, Re=3:9 106

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0 0.2 0.4 0.6 0.8 1X

-0.8

-0.4

0

0.4

0.8

-Cp

NS+BL+Drela DBMMSES VIIExperiment

Ncrit = 8.15

Figure 4.18: Cp distribution, CAST10, Re=3:9 106

1 2501 5001 7501 10001 12501Iterations

10-13

10-12

10-11

10-10

10-9

10-8

10-7

Rsm

den

sity

resi

dual

CAST 10

M=0.725, Re = 13.5 Mill, Alpha = -0.25

Figure 4.19: Density Residual CAST10, Re=13:5 106

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1 5001 10001 15001Iterations

0

0.25

0.5

Cl,

Cd

*10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xtrU

p,xt

rLow


CAST10Alpha=-0.35 M=0.725 Re=13.5 MillK-W BSL + Drela DBM with N=8.15

Figure 4.20: Convergence evolution, forces and transition location

0 0.2 0.4 0.6 0.8 1x/c

-0.006

-0.003

0

0.003

0.006

cf

NS+BL+Drela s DBMVII MSESExp prssures+ 3C3D

Figure 4.21: Skin friction, CAST10, Re=13:5 106

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0 0.2 0.4 0.6 0.8 1x/c

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

-Cp

NS+BL+DrelaMSES VII codeExperimental

Figure 4.22: Cp distribution, CAST10, Re=13:5 106

0 0.2 0.4 0.6 0.8 1X/C

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

Cf

XtrUp = 0.55XtrUp = 0.65

CAST10 Re = 3.9 Mill, Mach = 0.725, alpha = -0.35Kw-BSL

Figure 4.23: Skin friction with two di�erent transition locations

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transition point on the pressure distribution obtained with both computational methodswhich is not detected by the experiments.

The results for CAST10 at Re = 13.5 millions are shown in �gures 4.19, 4.20,4.21 and 4.22. For this case, the transition on the upper side is predicted at x=c = 0:23by natural transition. Nevertheless, in the course of the computations the position ofthe laminar separation remains very close to the transition position detected by the TSlinear ampli�cation. Therefore, the situation is similar to the B-airfoil case in which slightchanges on the predicted (by the NS solver) pressure distribution may produce a di�erentmechanism for the transition prediction (laminar separation or TS waves ampli�cation).The transition location obtained with INTA's method and the MSES code is locatedfurther downstream than the corresponding one obtained by the ONERA computationsfrom the experimental pressure distribution. These di�erences, see �g 4.22, may comefrom the di�erences of the pressure distribution on the upper side at the region locatedbetween 15 and 25% of the chord. At this region the experimental pressures have a steeperadverse pressure gradient than the one obtained in the numerical computations.

There are two additional points about the implemented transition predictionmethod which are worth to remark. The �rst one is related to compressibility e�ectsfor the transonic CAST10 case. As has been discussed above, the transition module ispresently based on an incompressible boundary layer code with Drela's approximated en-velope method. In addition, this method is based on the incompressible linear stabilitycharacteristics of the falkner-Skan similarity pro�les. Therefore, the implemented transi-tion prediction module does not take into account explicitly the compressibility e�ects.For the CAST10 case, cross checks with the MSES code, which solves the compressibleboundary layer via an integral method, have been carried out. The comparison of theboundary layer results in both cases indicates that the use of an incompressible BL codemay not be critical for this test case. Nevertheless, the compressibility e�ects on thetransition criteria have not been checked, and these e�ects a�ect the prediction of the�nal transition position as seen in section 4.1.3.

The second question is related to the sensitivity of the �nal solution to the numer-ical control parameters for this transonic test case. A parametric study has been carriedout on the CAST10 at Re = 3:9 106 using di�erent sets of numerical parameters ( Ntr,! and "cl). The conclusion of this study is that di�erent �nal transition locations canbe obtained with di�erent sets of control parameters. Only when the under-relaxationparameter of the transition update is set to a low value, a �good� �nal result can be ob-tained using the INTA method. As has been mentioned above, there is a strong couplingbetween the transition position and the obtained �ow �eld in the CAST10 case. In thiscase the resulting pressure distribution and skin friction coe�cient strongly depend on theupper transition position. Therefore, a laminar separation could be detected by the NSsolver near x=c = 0:25 depending on the transition position. To illustrate this situation,�g 4.23 displays the �nal skin friction coe�cient obtained by setting a �xed transitionin two di�erent positions. It can be observed that for transition location at x=c = 0:55there is no laminar separation detected by the NS solver, whereas if transition is locatedat x=c = 0:65, there is a laminar separation which �xes the transition for the remaining ofthe computations (i.e. if the transition module is activated from this solution). Therefore,

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in the present implementation of the method, a low under-relaxation transition updatingparameter needs to be used in order to avoid an early �xing of the transition (in patho-logical situations as the CAST10). This is a clear weakness of the method which will besolved in the near future.

4.2.3 NLR contribution - Multi-element airfoil

4.2.3.1 Multi-element airfoil

The second test case concerns a 2D three-element (slat, main wing and �ap) airfoil con-�guration constructed from the Airbus A-310 wing section at 59% wing span. The free-stream �ow conditions for the multi-element airfoil con�guration are given by Machnumber M = 0:218915, incidence angle � = 22:3647 degrees and Reynolds numberRe = 4; 094; 910 (take-o� conditions, �ow case from GARTEUR AG-25). The com-putational domain is discretised by 214,802 nodes. Examples of the grid are presented inFigures 4.24 and 4.25. The extent of the laminar portions of the boundary layer are setat x=c = �0:0516, x=c = 0:1946 and x=c = 0:9857 for the upper surface of the slat, wingand �ap, respectively.

The computational strategy for the multi-element airfoil �ow simulation consistsof performing 3000 multi-grid cycles in order to obtain the laminar boundary-layer char-acteristics for each element. Subsequently, a new location for transition is establishedbased on the compound criterion of transition mechanisms, as given in section 3.3.1. Theexplicit transition location is updated after the �rst 3000 multi-grid cycles and for everyadditional 1000 multi-grid cycles, see Table 4.1. The convergence history of the compu-tation, illustrated by Figure 4.26, indicates that convergence problems occur, which areattributed to �ow separations on the slat. The history of the stagnation-point location onthe slat, presented in Figure 4.27, indicates that the stagnation point moves downstreamduring the course of the simulation. As a result, the laminar boundary layer continues toseparate. Figure 4.27 also presents the history of the trigger location sex and the locationwhere the boundary layer is (fully) turbulent sct. Observe that a transition region existsfor the �ow to become turbulent. The extent of the transition region is determined by thedynamics of the employed turbulence model and by the applied criterion for sct (Ret = 1).Figure 4.28 displays the �nal locations of the stagnation points on the airfoil elements.

The transition-location history for the slat as predicted by the individual transi-tion mechanisms, shown in Figure 4.29, indicates that the criteria for linear instabilities(Granville) and �ow separation are active during the last 5000 multi-grid cycles. The�nal update is due to the laminar separation criterion though. The unsteady nature of�ow separations on a smooth surface is re�ected in a periodical behaviour of the explicittransition location as well as of the computed transition location, as can be observed fromFigure 4.27.

The transition-location history for the main-wing element, shown in Figure 4.30,indicates that the Granville criterion is active. After 10,000 multi-grid cycles, the explicittrigger location is converged to the value provided by linear stability analysis. Noticethat the transition location is corrected in downstream direction. The transition onset

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Figure 4.24: Detail of the computational grid for the multi-element airfoil at the leadingedge.

Figure 4.25: Detail of the computational grid for the multi-element airfoil at the trailingedge.

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cycle count slat wing �ap

1 �0:0516 0:1946 0:9857

3001 �0:0884 0:0786 0:9688

4001 �0:0860 0:0679 0:9792

5001 �0:0836 0:0858 0:9857

6001 �0:0827 0:1003 0:9857

7001 �0:0851 0:1085 0:9857

8001 �0:0836 0:1112 0:9857

9001 �0:0851 0:1092 0:9857

10001 �0:0836 0:1099 0:9857

Table 4.1: Upper-surface transition locations x=c for the multi-element airfoil as functionof the multi-grid cycles.

is pushed beyond its current location due to the calculation of laminar properties in theturbulent boundary layer. Bypass transition due to turbulence contamination from theslat wake (free-stream turbulence criterion) is identi�ed but only becomes active moredownstream, near the roughness strip. Laminar separation does not play a role in thetransition process at the upper surface of the main-wing element.

The transition-location history for the �ap, presented in Figure 4.31, indicatesthat the explicit transition location is initially modi�ed according to the free-stream-turbulence criterion due to turbulence contamination of the main-wing wake. The bypasscriterion probably is not suitable for the �ap element (Ret � 40, possibly too high).After 5000 multi-grid cycles, the considered portion of the boundary layer is su�cientlylaminar to �x the transition onset at the geometrical limit (roughness strip), which isthe most downstream location of the boundary layer region for which laminar boundary-layer quantities are calculated. Notice that after 10,000 multi-grid cycles the computedtransition criterion (Ret = 1, possibly too low) is still satis�ed upstream of the geometricallimit.

Figure 4.28 shows that apart from the �ap's upper surface, transition is alsotriggered by the roughness strip at the geometrical trip location for all lower surfaces.This does also hold for the lower surface of the B-airfoil and in general of most high-liftairfoil con�gurations. Overall, the computed transition location sct lags a certain distancebehind the explicit transition location sex (the transition region mentioned earlier). Inthat light, it is interesting to notice from Figure 4.30 that sct approaches sls independentfrom sex during the last 2000 multi-grid cycles. This might suggest that the turbulent-�uctuation ampli�cation by the k-! model complies with the non-turbulent �uctuationgrowth given by linear stability theory.

Figures 4.32 and 4.33 display the distributions of the surface-pressure coe�cientCp and the skin-friction coe�cient cf for both upper and lower surfaces of all three ele-ments. The results for the initial and �nal distributions (obtained after resp. 3000 and10,000 cycles) are compared, showing the in�uence of the transition modelling algorithm

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−6

−5

−4

−3

−2

−1

log(

dens

ity r

esid

ual)

Figure 4.26: Convergence history for the multi-element airfoil computations involvingtransition-location modi�cations.

3000 4000 5000 6000 7000 8000 9000 10000multi−grid cycles

−0.10

−0.08

−0.06

−0.04

−0.02

x/c

stagnation point explicit transition (ex) computed transition (ct)

Figure 4.27: History of the stagnation and transition locations on the upper surface ofthe slat.

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−0.25 0.00 0.25 0.50 0.75 1.00 1.25x/c

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

y/c

geometry streamlines stagnation points trip locations transition locations−0.0295 −0.0270

−0.1140

−0.1115

slat

main wingflap

slat

Figure 4.28: Geometrical de�nition of the airfoil elements, complemented with stream-lines, stagnation points and transition locations. The inset displays a magni�ed view ofthe slat's stagnation zone.

on these �ow properties. The erratic behaviour of the (not fully converged) initial solutionon the slat is due to the instability of the (separating) laminar boundary layer. Finally,Figure 4.34 displays again the distribution of the pressure coe�cient Cp for both upperand lower surfaces of all three elements, but now together with measurements performedin GARTEUR AG-08. The results for transitional and fully turbulent (laminar �ow notenforced) computations are compared with the experimental data. It is clear that thesurface-pressure distribution from the transitional computations agrees better with theexperiments, especially regarding the prediction of a transition `bubble' on the slat (seeinset in Figure 4.28).

4.2.3.2 NLR conclusions and recommendation

The present study addresses the coupling of a transition modelling algorithm with a(Reynolds-averaged) Navier-Stokes �ow computation algorithm to allow for airfoil �owsimulations including relevant transition locations. The transition prediction methodconsiders four transition mechanisms that are typical for high-lift �ows. Transition cri-teria are established for linear stability analysis of the laminar boundary layer, laminar

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−0.10

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

x/c

linear instab. (ls) free−stream (fst) roughness (sr) separation (sep)

Figure 4.29: History of the transition locations on the upper surface of the slat.


0.05

0.10

0.15

0.20

x/c

1. linear instab. (ls) 3. free−stream (fst) 4. roughness (sr) computed trans. (ct) explicit trans. (ex)

1

3

4

Figure 4.30: History of the transition locations on the upper surface of the main-wingelement.

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0.85

0.90

0.95

1.00

x/c

1. linear instab. (ls) 3. free−stream (fst) 4. roughness (sr) computed trans. (ct) explicit trans. (ex)

1,4

3

Figure 4.31: History of the transition locations on the upper surface of the �ap.

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2x/c

−15

−10

−5

0

pres

sure

coe

ffici

ent


slat

main wing

flap

Figure 4.32: Comparison of the Cp distributions from the initial and �nal �ow solutionsfor the three airfoil elements.

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−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2x/c

−0.03

0.00

0.03

0.06

skin

−fr

ictio

n co

effic

ient

initial solution final solutionslat

main wing flap

Figure 4.33: Comparison of the cf distributions from the initial and �nal �ow solutionsfor the three airfoil elements.

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2x/c

−15

−10

−5

0

pres

sure

coe

ffici

ent

trigger locations transitional comp. fully turbulent comp. measurements

slat

flapmain wing

Figure 4.34: Comparison of the Cp distributions from the transitional and fully-turbulent�ow solutions for the three airfoil elements. The inset displays a horizontal zoom of thepeak on its left.

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�ow separation, bypass transition due to the con�uence of wakes and laminar boundarylayers, and user-de�ned transition trips. The characteristics of the coupling method areinvestigated for a single- and multi-element airfoil. Convergence and consistency of themethod is obtained within �ve iterations in the case of the single-element airfoil calcula-tions. Consistency of the method could not be demonstrated for the multi-element airfoilcase because convergence of the transition onset location on the slat was hampered dueto unsteady �ow separation. Prediction of the onset location for the main-wing elementand the �ap converged though.

Fluid dynamics mechanisms that impose requirements for the iterative procedureare observed during the calculations. The camber reducing e�ect of a chordwise increasedturbulent boundary layer modi�es the stagnation point location and thereby the propertiesof the laminar boundary layer. Signi�cant initial upstream corrections for transition onsetare followed by smaller downstream corrections and therefore the linear stability criterionshould be applied with some under-relaxation. Transition to turbulent �ow is achievedby the �ow model in a short transition region and �ow separations take place almostinstantly. Transition onset modi�cations based on laminar separation may result in asituation where separation occurs downstream of the onset location but upstream of thelocation where the boundary layer is fully turbulent. The prediction of the transition onsetlocation in case of laminar �ow separation should account for the transition region by asmall o�set distance, as demonstrated by Hadºi¢ [27] for a laminar/turbulent transitionalseparation bubble. Hadºi¢ remarks that single-point turbulence closure models, like low-Reynolds-number k-" or low-Reynolds-number k-! models [58], o�er more �exibility andbetter prospects for predicting real complex �ows with laminar/turbulent transitions thanany classical linear stability theory. It is therefore recommended to continue studyingthe laminar/turbulent transition modelling capabilities o�ered by low-Ret single-pointturbulence closure models.

4.2.4 ONERA contribution

4.2.4.1 CAST10 airfoil

The computation are done using the corrected conditions given in paragraph 4.1.2 andthe �ne grid. The k � l two-transport equation model from Smith is used.

As concerns the numerical parameters, arti�cial viscosity is �xed to a mediumvalue (�4 = 0:016) and the Martinelli correction is active. The RK4-IRS method is usedwith a CFL number equal to 4 and a two-level multi-grid.

Figure 4.35 shows the pressure distribution for 2 di�erent computations. In the�rst case, transition is imposed (x=c = 0:42 on the suction side and 0:58 on the pressureside). In the second case, the �AHD� non local criterion is used with Tu = 0:001%. At �rstlook, the pressure distribution seems better when using the transition criterion but thismay be due to error compensations. The �AHD� criterion detects transition at x=c = 0:42instead of 0:58 with a separation bubble. This underestimation has been explained bythe compressibility e�ects in section 4.1.1. By imposing transition at this later point, thetransition bubble is missed and the thickening of the boundary layer is underestimated,

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0 0.25 0.5 0.75 10.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

imposed transition

AHD criterion (Tu=0,1%)

x

Mpa

roi

0 0.25 0.5 0.75 10.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.25 0.5 0.75 10.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

Figure 4.35: CAST10 airfoil - Isentropic wall Mach number - RC = 3:9 106

leading to a too thin boundary layer at the trailing edge, on the lower side. By imposingtransition earlier, the �AHD� criterion gives a thicker boundary layer in the rear part.This improves the pressure values on the lower side and modi�es the pressure level at thetrailing edge leading to a lower Kp value near the leading on the suction side.

Figure 4.36 shows the evolution of the residuals during the convergence process.Using the transition criterion, the boundary layer is laminar everywhere except over thethick trailing edge where it is imposed turbulent. This explains the delay on the turbulencedevelopment with the l-residual which remains very small during the �rst 200 iterations.The history of the k and l-residuals is very di�erent when the transition is computedwith respect to an imposed transition because of the slow displacement of the transitionlocation in the �rst case (large peaks on k-residuals from 200 to 1200 iterations). After1200 iterations the �nal transition location is reached. The remaining hight residual levelsis not due to the application of the transition criterion but to instabilities localized at thecorner cells of the thick trailing edge, in the second block.

The R� evolution is compared in �gure 4.37 to boundary layer computation resultsobtained with �3C3D� code and the �AHD� criterion. The large overestimation of R�

on the lower side probably comes from a lack of mesh points in the boundary layer,at least for a large part. The R� oscillation at transition is caused by the interactionbetween the steep turbulence development at transition and the centered discretizationscheme. Without the Martinelli correction which increases the arti�cial dissipation in thelongitudinal direction, the amplitude of oscillation would be greater. In the worst casesthis may lead to divergence.

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iteration0 1000 2000 3000 4000 5000 6000 7000

10-4

10-3

10-2

10-1

100

101

102

103

ρρuρwρEρkρl

Transition computed every 15 iterations

iteration0 1000 2000 3000 4000 5000 6000 7000

10-4

10-3

10-2

10-1

100

101

102

103

ρρuρwρEρkρl

Fixed transition

Figure 4.36: CAST10 airfoil - Convergence history - RC = 3:9 106

Figure 4.38 compares the skin friction coe�cient from the NS computation to theboundary layer computation which includes the intermittency function. This function,proposed by Arnal, increases the di�erence on the location of the point where Cf0 leavesits laminar value, as well as the location of the maximum value of the skin friction. Onthe upper side, transition is detected at x=c = 0:2 in the NS computation and 0:3 inthe boundary layer computation. On the lower side NS computation gives the transitionlocation at x=c = 0:4 instead of 0:55.

The shape factor evolution is plotted in �gure 4.39. The comparison with theboundary layer results con�rms the under-estimation of this parameter in the NS results.

4.2.4.2 RAE2822 airfoil

We now consider the RAE2822 airfoil with the �ow conditions of �case number 9�M0 = 0:73, � = 2:79Æ, RC = 6:5 106

The only di�erence is that transition is imposed in the experiment whereas it is free in thecomputation. The �AHD� criterion is still used with Tu = 0:001. Results are comparedto boundary layer computations using the same wall pressure distribution. In the Navier-Stokes computations, the Spalart-Allmaras turbulence model is used with the transitiontriggering term (ft1 function of the model). The numerical scheme is the following:

� spatial centered di�erence for the mean �ow, �4 = 0:008 with Martinelli correction,upwind Roe scheme for the transport equation,

� time integration with a 4 step RGK scheme with implicit residual smoothing,

� CFL equal to 5 with a 2-level multi-grid acceleration convergence,

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x/c

Rθ

0 0.25 0.5 0.75 10

250

500

750

1000

1250

1500

1750

2000

elsA "AHD"

0 0.25 0.5 0.75 10

250

500

750

1000

1250

1500

1750

2000

3c3d suction side3c3d pressure side

Figure 4.37: CAST10 airfoil - Momen-tum Reynolds number - RC = 3:9 106

x/c

Cf 0

0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01


0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

elsA "AHD"

Figure 4.38: CAST10 airfoil - Skin fric-tion - RC = 3:9 106

x/c

Hi

0 0.25 0.5 0.75 11

1.5

2

2.5

3

3.5

4

elsA "AHD"

0 0.25 0.5 0.75 11

1.5

2

2.5

3

3.5

4


Figure 4.39: CAST10 airfoil - Incom-pressible shape parameter - RC =

3:9 106

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x/c

Cf 0

0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008


0 0.25 0.5 0.75 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Figure 4.40: RAE2822 airfoil - Skin fric-tion - RC = 6:5 106

x/c

Hi

0 0.25 0.5 0.75 11

1.25

1.5

1.75

2

2.25

2.5

2.75

3


0 0.25 0.5 0.75 11

1.25

1.5

1.75

2

2.25

2.5

2.75

3

Figure 4.41: RAE2822 airfoil - In-compressible shape parameter - RC =

6:5 106

� initialization from a uniform �ow,

� transition criteria computed every 15 iterations.

Figures 4.40 and 4.41 show a comparison with boundary layer results of the skin frictioncoe�cient and of the shape parameter. On the upper side, transition is detected in NScomputation at x=c = 0:11 and at 0:13 in the boundary layer code (3C3D). The di�er-ence between the two computations is ampli�ed by the intermittency function which isnot used in elsA. The overshoot on Cf0 with 3C3D is due to the Arnal intermittencyfunction which reaches 1.5 at its maximum value in the transition region. The goodagreement between the two computations comes from the fact that transition takesplace closely after the location of the minimum of the pressure coe�cient in a regionwhere R� increases rapidly. On the lower side, the di�erence in the transition onsetlocation is more important: x=c = 0:35 in elsA, 0:45 in 3C3D, just before separation.

The comparison of the shape factor always shows the underestimation of this parameterin the NS computation (�gure 4.41). This is mainly due to the lack of mesh points inthe outer part of the boundary layer. Near the wall the grid is �ne enough to give thecorrect value of the skin friction (�gure 4.40).

4.2.5 QinetiQ contribution

Calculations have been performed for the CAST10 aerofoil using the multiblock meshprovided by ONERA and described in section 4.1.2. It was recast as a four-block mesh,the original block of �H� mesh downstream of the aerofoil being divided into three blocks,

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x/c

Cp

0 0.25 0.5 0.75 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Relaxation scheme 1Relaxation scheme 2No relaxation

Cp*

Figure 4.42: Comparison of pressure distributions from computations using three di�erenttransition prediction strategies. CAST10 aerofoil, M0 = 0:725, � = �0:35Æ, Re = 3:9 106

one downstream of the thick trailing edge (20�50 cells) and one each above and belowthe latter block (106�50 cells) and abutting the `C' mesh around the aerofoil. It wasnot found necessary to re�ne the block behind the thick trailing edge in order to obtainconverged solutions with RANSMB. An inviscid �ow boundary condition was applied onthe base of the aerofoil, commensurate with the mesh interval normal to the base.

The procedure used to calculate the �ow around the aerofoil with predicted tran-sition locations was the same as that described above for the AS-B aerofoil ( 3.5.1). TheGranville criterion as described in section 3.5.2 was again used to predict the location oftransition.

Only the low Reynolds number case has been attempted. The �ow conditionswere those recommended by ONERA, namely M1 = 0:725, � = �0:35Æ, T1 = 293K andRe = 3:9 106 based on the aerofoil chord. Other authors of this report have commentedon the sensitivity of the computed solution to the initial conditions and to the procedureby which the transition location is predicted. The computed pressure distribution is verysensitive to the transition location (especially on the suction surface), while the transitionmechanism is sensitive to the pressure distribution. Thus the �nal computed solutiondepends upon the initial speci�cation of transition location and the under-relaxation factor

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Cp*

Iteration

XT up

p

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

Relaxation scheme 1 - upperRelaxation scheme 1 - lowerRelaxation scheme 2 - upperRelaxation scheme 2 - lowerNo relaxation - upperNo relaxation - lower

Figure 4.43: Predicted transition locations (after under-relaxation) for three di�erentstrategies. CAST10 aerofoil, M0 = 0:725, � = �0:35Æ, Re = 3:9:0 106

used. This is illustrated in �gures 4.42 and 4.43. Three calculations have been performed.The �rst had the initial transition location set at x/c=0.9 on both surfaces and useda relaxation factor of 1. The second had the initial transition location set at x/c=0.24on the upper surface and x/c=0.496 on the lower surface; the relaxation factor was setat 0.5 initially but increased to 1 after the predicted location of transition remainedconstant for three successive iterations. The �nal calculation had the initial transitionlocation set at x/c=0.55 on both surfaces and a relaxation factor set at 0.5 initially butincreased to 1 when the predicted transition location moved downstream. Figure 4.42shows the �nal pressure coe�cient distributions from the three cases. All the solutionsare well converged. The values of transition location after under-relaxation are shownin �gure 4.43. The initial transition locations of x/c=0.24 and 0.496 for the secondcalculation were chosen for the following reasons. x/c=0.24 is downstream of the strongrecompression and recovery associated with the transition location predicted in the �rstcalculation. x/c=0.496 was the transition location predicted for the lower surface in the�rst calculation but was not associated with severe changes in pressure gradient and soit was believed that it would not be prevented from moving downstream if necessary.The initial locations for the �nal calculation were chosen to be downstream of transition

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Cp*

x/c

Cf

0 0.25 0.5 0.75 10

0.002

0.004

0.006

0.008

0.01

Figure 4.44: Predicted distribution of Cf from the �nal calculation. CAST10 aerofoil,M0 = 0:725, � = �0:35Æ, Re = 3:9:0 106

locations predicted by other authors.Transition on the upper surface in the third calculation resulted from laminar

separation indicated by the boundary layer code and occurred at 38% chord. This agreeswith the value predicted by INTA using Drela's criterion and is close to the value observedexperimentally. However, the weak shock wave is not observed experimentally thoughsimilar features can be seen in other computed solutions. The location of transition onthe lower surface from this calculation also resulted from laminar separation indicated bythe boundary layer code and occurred at 52.1% chord, about 5% upstream of the valuepredicted by INTA. The variation of skin friction coe�cient from the third calculation isshown in �gure 4.44.

Chapter 5

Conclusion and perspectives

The �rst step of the study has shown that all the Navier-Stokes codes used by the partnersgive poor results regarding the boundary layer characteristic required for transition predic-tion, i. e. large underestimation of the shape factor and overprediction of the momentumthickness. This simply con�rms the conclusions of previous works such as in EURO-TRANS project [19] or at DLR [52]. If progress can be expected with the improvement ofthe numerical techniques (arti�cial viscosity damping in viscous layers, mesh re�nement,embedded meshes ...), the accuracy problem will not be solved in a near future. To copewith the present available techniques, two di�erent strategies have been followed by thepartners:

� use of boundary layer codes coupled with Navier-Stokes solvers,

� direct use in Navier-Stokes solvers of existing transition criteria after minor adaptation.

The �rst strategy is relatively simple to develop for 2D �ows. From the wall pressuredistribution given by the Navier-Stokes solver, a boundary layer computation providesthe transition location which is used to impose the laminar region in the NS solver. Theprocess is repeated until convergence is reached. This technique allows a high precisionlevel thanks to the quality of the transition prediction in the boundary layer computation.The extension to simple, but highly interesting, 3D con�gurations such as wings needssome relatively minor assumptions for the cross-�ow. DLR, INTA, and QinetiQ havechosen this coupling technique. The obtained results clearly prove the e�ciency of themethod and the high precision level which can be obtained.

For complex con�gurations, the extension of the previous technique may be im-possible. This is evident for regions where the boundary layer approach itself is question-able, such as corner �ows, but no transition criteria exist for these �ows. Without goingso far, the coupling approach may reach practical limitations even for relatively simplecon�gurations when multi-block meshes are used by the NS solvers, which is more andmore the case. The second strategy overcomes these problems. Although it has beenclearly con�rmed that the precision obtained in the NS codes regarding the boundarylayer parameters needed to apply transition criteria is not acceptable, at least when tran-sition is induced by instability mechanisms such as Tollmien Schlichting waves, the second

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strategy has to be considered. In some situations, the transition process is much less sen-sitive to the boundary layer parameters, such as in �by-pass� process. This is particularlythe case in turbomachinery applications for which the external turbulence level may bevery high. Even in the case of small Tu, large adverse pressure gradient e�ects reducethe sensitivity to the accuracy of the boundary layer parameters. This is illustrated bythe success obtained at NLR for hight-lift con�gurations with the Granville's criterionassociated to some other constraints based on physical properties (transition imposed atlaminar separation point or due to wall roughness or external turbulence).

The di�culty of introducing transition criteria in NS codes is exempli�ed by theuse of the shape parameter of the velocity pro�le to characterize the pressure gradiente�ect in the criteria. As this parameter is obtained with the poorest precision whereas agreat precision is needed, it is tempting to replace the use of Hi by the pressure gradientparameter itself (�2=� due=ds). This is the way followed by ONERA with the elsAsolver. Regarding the di�cult CAST10 test case, the transition location obtained withthis method is not as good as which is achieved using coupling with a boundary layer code.However, improvement can be expected by taking into account compressibility e�ects. Forless di�cult cases such as the �B-airfoil� or the RAE2822 at incidence, a correct precisionlevel is reached.

Perspectives

The encouraging results obtained in 2D con�gurations are worth being extended to rela-tively simple 3D con�gurations such as wings. To this end, the eN criterion or empiricalcriteria taking into account cross-�ow instability modes could be used in both approaches,boundary layer coupling or direct introduction in Navier-Stokes solvers.

The correct prediction of the transition onset is only one part of the problem.Presently, the intermittency region is correctly modelled only with algebraic turbulencemodels such as the mixing length or the Baldwin-Lomax formulations. The modeling ofthe intermittency region must be improved when using turbulence transport equations.In these equations, the objective of the so called �low Reynolds number e�ects� is moreto take into account the wall e�ects rather than the development of the turbulence in thetransitional region. The intermittency functions used for algebraic models are not directlyuseful for transport equations models.

In the past, attempts have been made to predict transition using turbulent trans-port equation models with low Reynolds terms. Although this approach has not beensuccessful, at least for low Tu levels, it could be interesting to reconsider the question inthe light of recent developments, as proposed by NLR. The use of a transport equationfor the intermittency, or a related quantity, could also be considered.

At low Reynolds number, transition can occur through a laminar separation bub-ble. Presently, transition is imposed at the separation point. This is coherent with thepoor mesh resolution of the NS computations in the �ow direction, rarely better than theboundary layer thickness, but this is not satisfactory. Speci�c empirical transition crite-ria exist for separation bubble and could be put in NS solvers (but not in a direct-modeboundary layer solver). This would probably be an improvement providing the separation

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bubble is correctly resolved in the �ow direction.The few perspectives summarised above will be speci�ed and discussed more

deeply in the new GARTEUR Exploratory Group EG52 with the objective of creating anew GARTEUR Action Group to extend the AG35 work.

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