Application of the Correlation-based g�Reqt Transition Modelto the Spalart-Allmaras Turbulence Model

Shivaji Medida� and James D. Baeder†

Department of Aerospace Engineering, University of Maryland, College Park, MD 20742

The correlation-based g�Reqt transition model was originally developed by Langtry et al. for use with thetwo-equation SST k�w turbulence model. In the present work, the g�Reqt model is modified and coupled withthe one-equation Spalart-Allmaras (S-A) turbulence model. The integrated turbulence-transition equationswill be referred to as the g�Reqt �SA model. New numerical correlations to control the Critical Reynolds num-ber, Reqc, and the transition length function, Flength, are formulated and validated for several two-dimensionaltest cases. Results show that the baseline S-A model is unable to predict boundary layer transition and gen-erated excessive turbulence. Consequently, for steady airfoil problems, the S-A model over-predicted drag atlower angles of attack and lift at higher angles of attack approaching stall. In contrast, the g�Reqt �SA modelpredicted the transition onset location with good accuracy, resulting in significantly improved lift and dragpredictions. Two-dimensional dynamic stall predictions are also significantly improved with the g�Reqt �SAin comparison to fully turbulent S-A model predictions. The performance of the g�Reqt �SA model is com-parable to that of the original g�Reqt model coupled with the SST k�w model.

Nomenclature

d Boundary layer thickness,g Intermittency,µ Molecular viscosity,µt Eddy viscosity,w Specific turbulent dissipation rate,W Vorticity magnitude,r Density,q Momentum thickness,d Nearest wall distance,k Turbulent kinetic energy,M Mach number,Rext Transition onset Reynolds number based on plate length,Req Momentum thickness Reynolds number,Reqt Req at transition onset,Reqt Transported Reqt ,Rev Vorticity Reynolds number,S Strain rate magnitude,Tu Turbulence intensity,U Local flow speed

I. Introduction

A. Background

Laminar-turbulent transition of boundary layers is of practical relevance in various aerodynamic flows such as flowpast wings, airframes, rotor blades, and wind turbine blades. It has a major impact on flow development, separation,

�Graduate Research Assistant, Department of Aerospace Engineering, University of Maryland, College Park. Student Member, AIAA.†Associate Professor, Department of Aerospace Engineering, University of Maryland, College Park. Member, AIAA

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and viscous drag, all of which influence the overall performance of the vehicle/equipment. Therefore, it is important tounderstand the process of transition, and to develop the capability to accurately predict transitional flow phenomena.Conventional turbulence models are not equipped to handle transitional flow problems since they were developed andcalibrated for fully turbulent flows at high Reynolds numbers. Many computational studies have concluded that theexisting turbulence models predict transition too early and generate excessive turbulence resulting in over-prediction ofdrag. Therefore, several methodologies such as the eN method based on linear stability theory, the laminar fluctuationenergy approach, low Re closure, Large Eddy Simulation (LES), and Direct Numerical Simulation (DNS) have beensuccessfully used by researchers to model transitional flows. A comprehensive review of the various approachesadopted to identify transition criteria, and estimate transition onset location and transition extent, is provided byPasquale et al.1 However, most of these methods are either too expensive or not generic enough to be applied to a widerange of applications, especially to problems involving highly unsteady flows that require time-accurate solutions suchas the dynamic stall problem. For example, the eN method needs the RANS flow solver to be coupled to a boundarylayer code that solves the stability equations on a much finer computational grid, since RANS-based grids are toocoarse to evaluate the stability criteria. For unsteady, time-accurate computations of flows which involve moving grids(such as dynamic stall), the computational expense of this procedure is prohibitive. In addition, the eN method byitself is not sufficient to predict separation-induced transition of boundary layers under an adverse pressure gradient,which is often the primary mode of transition in airfoils. More recently, a new class of transition models basedon the concept of intermittency transport have shown promise in solving complex transitional flows within reasonablecomputational expense. Intermittency (g) can be thought of as the probability of a point in the flow field being turbulentat a given instant. Intermittency-based transition models typically rely on experimental correlations to predict thetransition onset location. These experimental correlations provide information on the transition onset location basedon the freestream turbulence intensity and the local momentum-thickness Reynolds number (Reqt ). In order to use theempirical correlations in a CFD code, boundary layer quantities need to be integrated to obtain local Reqt , followedby a search procedure to determine the location where Reqt first exceeds the value predicted by the experimentalcorrelation. The turbulence model is then gradually activated downstream of the identified transition location throughthe use of an intermittency function (g) which varies between 0 and 1. The primary disadvantage of this approach isthe use of search and integration procedures which can become very expensive for large scale 3-D flow problems. Asuitable transition model is therefore one which can be readily integrated into an existing Navier-Stokes CFD solverwithout any dependence on non-local parameters and operations (such as searching or integrating along streamlines),or geometry of the flow domain. While many of the existing methods suffer from some form of the above mentioneddrawbacks, Langtry and Menter2 have developed a local correlation-based transition model, called the g�Reqt model,which satisfies most of the requirements for a transition model that is fully compatible with a conventional CFD solver.It has been extensively validated for a wide range of aerodynamic problems including flow past a transonic wing, acompressor cascade, a wind turbine blade, and a helicopter fuselage.3 The g�Reqt model was originally formulatedand calibrated for use with the two-equation SST k�w turbulence model.4 Since the one-equation Spalart-Allmaras(S-A) turbulence model5 is computationally less expensive compared to the SST k�w model, and is known for reliableprediction of external flow problems, it is desirable to appropriately modify and re-calibrate the g�Reqt model for usein conjunction with the S-A model. Additionally, since the S-A turbulence model can be easily modified to includehybrid RANS-LES models for capturing 3-D separated flow physics, it is advantageous to have the capability oftransition modeling integrated into it. Therefore, the present work is a preliminary attempt to integrate intermittency-transport based g�Reqt transition model with the one-equation S-A model, validate it for simple two-dimensionalflow problems including flow past a flat plate and various airfoils.

II. Modified g�Reqt Transition Model

A. Summary of g�Reqt Transition Model for SST k�w Turbulence Model

The g�Reqt transition model uses the concept of intermittency (g), in order to trigger transition locally, by controllingthe production of turbulent kinetic energy. The value of g varies between 0 (always laminar) and 1 (always turbulent).The local intermittency value is used to turn on the production of turbulent kinetic energy (TKE) in the k-equation ofthe SST k�w turbulence model as follows:

D(rk)Dt

= Pk � Dk +¶

¶x j[(µ+skµt)

¶k¶x j

] (1)

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where, the production and destruction terms of the original k-equation from SST k�w turbulence model (Pk and Dk),are modified using the local intermittency value as shown in Eq. 2. It should be noted that the governing equation forspecific turbulence dissipation rate, w, is not modified in this formulation.

Pk = geffPk; Dk = min(max(geff;0:1);1:0)Dk (2)

A transport equation based only on the local flow field variables and gradients, along with the nearest wall distance, issolved to obtain the intermittency field. The intermittency transport equation as proposed by Langtry3 is given below:

D(rg)

Dt= Pg �Dg +

¶

¶x j

�(µ+

µt

s f)

¶g

¶x j

�(3)

The production and destruction terms of intermittency are given by:

Pg = Flengthca1rS[gFonset]0:5 (1:0� ce1g) ; (4)

andDg = ca2rWgFturb(ce2g�1:0); (5)

Fonset = max(Fonset2 �Fonset3;0) (6)

Fonset1 =Rev

2:193:Reqc(7)

Fonset2 = min�max

�Fonset1;F4

onset1�;2:0

�(8)

Fonset3 = max

1��

RT

2:5

�3

;0

!(9)

Fturb = e

�

RT

4

!4

(10)

Rev =rd2S

µ; RT =

rkµw

(11)

The onset of transition is determined by the ratio of the Vorticity Reynolds number and the critical Reynolds numbergiven by Eq. (7). The intermittency production term depends on two parameters: (1) Critical Reynolds number Reqc,and (2) Transition length function Flength. Critical Reynolds number determines the transition onset location, and isdefined as the point at which intermittency first starts to grow in the boundary layer. This occurs slightly upstream ofthe transition momentum thickness Reynolds number, Reqt . The Fonset function uses Reqc and is designed to rapidlyswitch from 0 in a laminar boundary layer to 1 in a turbulent boundary layer. The Flength function determines thelength of the transition regime, during which the boundary layer becomes fully turbulent. These two parameters arecalculated based on correlations that are functions of the local transition momentum thickness Reynolds number Reqt .The correlations can be developed based on a few benchmark cases of transitional flow past a flat plate. The value ofintermittency, g, is set to unity in the freestream, and to zero at a wall boundary.

A second equation governs the transport of Reqt as follows:

D(rReqt)

Dt= Pqt +

¶

¶x j

�sqt(µ+µt)

¶Reqt

¶x j

�(12)

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The source term for Reqt is defined as:

Pqt = cqtr

t(Reqt �Reqt)(1:0�Fqt); (13)

Fqt = min

max

Fwake:e�(

dd)

4;1:0�

�g�1=ce2

1:0�1=ce2

�2!;1:0

!(14)

qBL =ReqtµrU

; dBL = 7:5qBL; d =50wd

U:dBL (15)

Fwake = e(Rew

1E+5 )2

; Rew =rwd2

µ(16)

where t is a timescale present for dimensional reasons, and Fqt is a blending function that turns off the source termsinside a boundary layer. The source term (Pqt ) is designed to maintain the freestream value of Reqt outside the boundarylayer (Reqt = Reqt¥), and is turned off inside the boundary layer allowing for the convection and diffusion of theconserved quantity.The motivation behind solving the Reqt equation is that, since the turbulence intensity strongly varies in a flow domain,it is not appropriate to use the freestream value of Reqt to determine transition onset and extent. Instead, the freestreamturbulence effects need to be transported into the boundary layer. Additionally, it was shown by Langtry3 that thisgoverning equation accounts for history effects of pressure gradient on the onset of transition, since transition onset isinfluenced not just by the local value of the pressure gradient.The local value of Reqt that is needed to compute the production term is obtained from the experimental correlationsformulated as a function of freestream turbulence and the local pressure gradient as follows:

Reqt =

8<: (1173:51�589:428 Tu+0:2196

Tu2 )F(lq); Tu � 1:3

331:50[Tu�0:5658]�0:671F(lq); Tu > 1:3(17)

F(lq) =

(1� [�12:986lq �123:66l2

q�405:689l3

q]e�[ Tu

1:5 ]1:5; lq � 0

1+0:275[1� e[�35:0lq]]e�[ Tu0:5 ]; lq > 0

(18)

The pressure gradient parameter, lq, is given by:

lq =rq2

µdUds

(19)

wheredUds

represents the streamwise acceleration. The freestream value of Reqt is set to Reqt¥, which can be computed

from the above experimental correlations under a zero pressure gradient condition by setting lq to zero. At a wallboundary, Reqt is set to zero.The model constants are:

ce1 = 1:0; ca1 = 2:0; ce2 = 50:0; ca2 = 0:06; s f = 1:0

cqt = 1:0; sqt = 2:0

A correction to the intermittency function was included to improve prediction of transition induced by a laminarseparation bubble and is given by:

gsep = min�

s1max�

0;�

Rev

3:235Reqc

��1�

Freattach;2:0�

Fqt (20)

geff = max(g;gsep) (21)

where, Freattach = e��

RT20

�4

, and s1 = 2.0.

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B. Integration of g�Reqt Transition Model with Spalart-Allmaras Turbulence Model

The one-equation Spalart-Allmaras turbulence model5 solves for the modified Eddy viscosity, n, which relates to thekinematic Eddy viscosity nt as follows:

nt = n fv1; fv1 =c3

c3 + c3v1

; c =n

n(22)

The governing equation of the S-A model without the trip terms is given by:

Dn

Dt= Pn �Dn +

1s

�Ñ:((n+ n)Ñn)+ cb2(Ñn)2� (23)

where, Pn and Dn are the production and destruction terms of n, given by:

Pn = cb1Wn and Dn = cw1 fw[n

d]2 (24)

W is a function of the vorticity magnitude, W, and is defined as:

W = W+n

k2d2 fv2; fv2 = 1� c

1+c fv1(25)

The function fw is defined as:

fw = g�

1+ c6w3

g6 + c6w3

� 16

; g = r+ cw2(r6 � r); r =n

Wk2d2(26)

The model constants are: cb1 = 0:1355;s = 2=3;cb2 = 0:622;k = 0:41;cw1 = cb1=k2 +(1+ cb2)=s;cw2 = 2:0;cv1 =7:1: The current implementation of the S-A turbulence model uses the rotational correction20 which reduces the pro-duction of turbulence in regions where vorticity exceeds strain rate, such as in vortex core regions, where pure rotationshould not produce turbulence. In this version of the S-A model, the magnitude of vorticity, W, used to calculate W isreplaced with W+2:min(0;S�W) in Eq. 25, which is then used to compute the production term.

Implementation of the original g�Reqt model requires an explicit knowledge of the local values of turbulence in-tensity (calculated from local TKE, k), and the specific dissipation rate (w) in the flow domain, which are readilyavailable when using the SST k�w model. Local value of Tu was used by Langtry3 to compute the value of Reqtfrom the experimental correlations in Eq. (17), while retaining the pressure gradient function F(lq) calculated fromEq. (18). However, Suluksna et al6 have shown that it is redundant to use the local value of Tu as well as the pressuregradient parameter, since the local variation in turbulence intensity is a direct consequence of a non-zero pressuregradient experienced by the boundary layer. An adverse pressure gradient in the boundary layer causes the local Tuto increase, whereas a favorable pressure gradient causes the local Tu to decrease. Therefore, it should be sufficientto use either a local value of Tu and discard the pressure gradient term, or use the freestream value of Tu along withthe pressure gradient term. Since the S-A turbulence model solves directly for the eddy viscosity, it is not possibleto obtain local values of Tu or w. In order for this transition model to be applicable to the S-A model, the presentformulation uses the freestream value of Tu along with the pressure gradient parameter to compute Reqt in Eq. (17).Successful application of this modification has been demonstrated through various test cases that are presented in thiswork. The new formulation will be referred to as the g�Reqt �SA model. In the original transition model formula-tion, the local value of w was used to calculate the Fwake function, which deactivates the blending function Fqt in thewake regions of an airfoil (Eq. (14)). The purpose of Fwake is to distinguish between a wall boundary layer and a freeshear layer, so that the production term of Reqt could be turned off only in the wall boundary layer and not in a freeshear layer. The airfoil calculations in the present work have been performed by manually turning off the blendingfunction. The parameter RT given in Eq. 11 is redefined as

µt

µ, thus eliminating the need for knowing w in the flowfield.

With the above modifications, the g�Reqt transition model is now fully compatible with the S-A turbulence modelwithout any dependence on k and w. The solution of the intermittency transport equation is then used to control theproduction and destruction terms of eddy viscosity in the S-A model similar to Eq. (2):

Dn

Dt= Pn � Dn +

1s

�Ñ:((n+ n)Ñn)+ cb2(Ñn)2� (27)

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Case Tu¥(%) Rext �10�6 (Expt)SK 0.03 2:678SS1 0.042 2:655SS2 0.10 2:569SS3 0.20 2:048SS4 0.26 1:596SS5 0.34 1:305

Table 1. Zero pressure-gradient flat plate experimental data

X

Y

­0.4 ­0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8­0.08

0

0.08

0.16

0.24

0.32

Symmetry

Leading EdgeSymmetry Viscous Wall

Velocity Inlet Pressure Outlet

Figure 1. Typical mesh used for flat plate simulations (300 � 150 cells)

Pn = geff:Pn and Dn = max(min(g;b) ;1:0) :Dn; b = 0:5 (28)

where, Pn and Dn are the production and destruction terms of the original S-A model (Eq. (23)). The significance ofthe parameter b in the modified destruction term will be discussed in the following section.

C. Calibration of g�Reqt �SA Transition Model

The proposed transition model is calibrated primarily for natural transition applications, where the freestream turbu-lence intensity is much less than 1.0 %. As mentioned earlier, the numerical correlations for Reqc and Flength parametersare obtained using a series of zero-pressure-gradient flat plate test cases,7, 8 which are summarized in Table 1. TheCartesian mesh of size 300�150 cells that is used to simulate the flat plate cases is shown in Fig.1. Recall that the crit-ical Reynolds number, Reqc, as described by Langtry,2 is the location at which intermittency first starts to grow, whichis slightly upstream of the actual transition onset location. Therefore, Reqc should always be less than Reqt . Addition-ally, Flength controls the length of the transition zone, once the intermittency production term is activated through theFonset function. The numerical correlations for both Reqc and Flength proposed by Langtry2 were polynomial functionsof Reqt and were applicable for any value of Tu¥. Since the transition model is now coupled with the S-A model, anew set of correlations needed to be developed and used in this work. The following procedure was adopted to developthe correlations:

1. For a given value of Tu¥, assume a linear relationship between Reqc and Reqt , Reqc = aReqt , and assume Flengthto be constant.

2. Identify unique values of a and Flength to match the transition onset location and length for each of the flat platecases

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Re_x

Cf

0E+00 1E+06 2E+06 3E+06 4E+06 5E+060

0.002

0.004

0.006

0.008

0.01

SK

SS1

SS2

SS3

SS4

SS5

SK Expt

Figure 2. Skin-friction profiles of zero pressure-gradient flow past flat plate

3. Obtain expressions for a and Flength as a function of Tu¥

The procedure outlined above is relatively simple and since the focus of this work is natural transition where Tu¥ ismuch less than unity, it was not necessary to obtain a single set of correlations for a wide range of Tu¥ values. It isimportant to rememeber that the outlined procedure is only a guideline to calibrate the transition model and that theexact form of the correlations is not unique. Several forms of correlations were developed and used successfully aspart of this work. The numerical correlations used to obtain the presented results are given below:

Reqc =�4:45Tu3

¥ �5:7Tu2¥ +1:37Tu¥ +0:585

�Reqt (29)

Flength = 0:171Tu2¥ �0:0083Tu¥ +0:0306 (30)

The above correlations are valid only for Tu¥ < 1:0%, where natural transition is expected to occur. Similar expres-sions were formulated in the bypass transition regime ( Tu¥ > 1:0% ) using the T3-series flat plate simulations,9 butare not presented here since they are yet to be validated for airfoil problems. It should be noted that the correlation forFlength given in Eq. 30 is dependent only on the freestream turbulence intensity and not on the local value of Reqt asformulated by Langtry.2 An alternate form of the correlation for Flength parameter that depends on the the local Reqt isproposed:

Flength =fn(Tu¥)

Reqt(31)

The proposed inverse relationship between Flength and Reqt is consistent with the observation that the transition lengthis greater at lower values of Tu¥, which corresponds to a higher value of Reqt . Similar calibration procedure as de-scribed earlier is adopted to obtain a polynomial expression for fn(Tu¥). Simulations were performed using bothforms of correlation for Flength and a good agreement was observed between the two forms. The advantage of for-mulating Flength as a function of both Reqt and Tu¥ as opposed to only Tu¥ is thought to be the inclusion of pressuregradient effects on the transition length, although this assumption needs to be further verified. All the results presentedin this paper are obtained using Eqs. 29 and 30.

Computed skin friction profiles for the flat plate cases listed in Table 1 are plotted in Fig. (2). The transition onsetlocation is indicated by a sharp rise in the skin friction value. The effect of Reqc and Flength on the transition onsetlocation and transition extent is demonstrated in Figs. 3 (a), (b). Increasing the value of Reqc delays the transitiononset location, whereas increasing the value of Flength results in a shorter transition length. The non-dimensionalfreestream value of nt was set to 1.e-8. For small values of Tu¥ (< 1:0%), using large values of freestream nt causesthe skin friction profile to deviate from a purely laminar behavior upstream of the transition onset location, as seen in

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Re_x

Cf

0E+00 1E+06 2E+06 3E+06 4E+06 5E+060

0.002

0.004

0.006

0.008

0.01

Re_tc = 0.50 Re_t

Re_tc = 0.55 Re_t

Re_tc = 0.59 Re_t

Re_tc = 0.65 Re_t

Re_tc = 0.70 Re_t

SK Expt

(a) Effect of Reqc

Re_x

Cf

0E+00 1E+06 2E+06 3E+06 4E+06 5E+060

0.002

0.004

0.006

0.008

0.01

F_length = 0.01

F_length = 0.02

F_length = 0.03

F_length = 0.04

F_length = 0.05

SK Expt

(b) Effect of Flength

Re_x

Cf

0E+00 1E+06 2E+06 3E+06 4E+06 5E+060

0.002

0.004

0.006

0.008

0.01

nut_inf = 1.e­8

nut_inf = 1.e­6

nut_inf = 1.e­4

nut_inf = 1.e­2

nut_inf = 1.e­1

SK Expt

(c) Effect of nt¥

Re_x

Cf

0E+00 1E+06 2E+06 3E+06 4E+06 5E+060

0.002

0.004

0.006

0.008

0.01beta = 0.1

beta = 0.2

beta = 0.3

beta = 0.4

beta = 0.5

beta = 1.0

SK Expt

Laminar

Turbulent

(d) Effect of b

Figure 3. Calibration of g�Reqt transition model

Fig. 3(c). However, for higher values of Tu¥, experiments show that the skin friction profile deviates from a purelylaminar behavior and therefore higher values of freestream nt are required when developing correlations in the bypasstransition regime. The experimental skin friction profile indicates an overshoot beyond the fully turbulent value (Fig.3(d)). In the current formulation, the amount of overshoot can be controlled by the value of b in the modified S-Amodel destruction term, given by Eq. (28). The effect of b variation on the amount of skin friction overshoot is shownin Fig. 3(d). At b = 1:0, the original destruction term of the S-A model is recovered from Eq. (28), which is alsoreflected in the turbulent section of the skin friction profile plotted for b = 1:0. It is also seen that the value of b

influences the transition length to some extent.

III. Numerical Methodology

A. Baseline Flow Solver

In the present work, a structured, finite-volume, compressible, Reynolds-averaged Navier-Stokes solver is developedand used to perform all the simulations. The fifth-order mapped-weighted essentially non-oscillatory scheme11 is usedfor spatial reconstruction, and the implicit second-order backward difference scheme is used for time marching. Thediagonalized ADI scheme12, 13 is used for implicit inversion of the linear system of equations. The Roe flux-differencescheme14 is used for calculating inviscid fluxes and a second-order central scheme is used to discretize the viscous

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(a) Aerospatiale-A (b) VA-2 (c) S809

(d) S6 leading-edge slat configuration (e) OA209

Figure 4. Airfoil profiles used for simulations

terms. The convection terms of turbulence and transition model equations are discretized using a second-order upwindscheme. Ghost cells are used to enforce all boundary conditions (viscous wall, wake-cut, and farfield boundaries). Thefarfield boundary conditions are implemented using the 1-D characteristic-based Riemann invariants to avoid spuriouswave reflections at the computational domain boundaries.

B. Computational Mesh

Structured, body-fitted C-mesh topology is used for airfoils in all the test cases presented in this work. In the wrap-around direction of the C-mesh, 300 cells are used on the airfoil surface alone. When using a fifth-order spatialreconstruction scheme, this mesh resolution is slightly finer than the typical grids used in fully turbulent simulations,which is necessary to capture small regions of laminar separation bubbles (LSB) that may form in transitional flows,while ensuring a grid-converged solution. In the wall normal direction, the wall grid spacing is chosen such that ay+ value of less than 1 is maintained based on the Reynolds number for each test case. For airfoil simulations infreestream, the outer boundary of the C-mesh is placed 20 chords away from the airfoil surface to avoid any spuriouswave reflections.

For dynamic stall simulations, Duraisamy et al.15 have shown that it is essential to model the wind tunnel wallsin order to obtain good agreement with experimental data. Without the wind tunnel walls, the upstroke lift magni-tude is generally under-predicted and the lift-curve slope during upstroke is not well captured causing the stall eventto initiate at an incorrect angle of attack. An overset mesh system consisting of a C-mesh around the airfoil and arectangular background mesh for the wind tunnel is used for all dynamic stall simulations. Data transfer between theairfoil mesh and the wind tunnel mesh is handled using the improved Implicit Hole Cutting (IHC) methodology.16, 17 ACartesian mesh is used to model the wind tunnel walls in the background of the airfoil mesh, with sufficient clusteringaround the airfoil in both streamwise and normal directions. The height of the wind tunnel mesh is usually knownfrom experiments. The upstream and downstream boundaries of the wind tunnel mesh are placed around 20 chordsaway from the airfoil.

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IV. Results and Discussion

A. Aerospatiale-A Airfoil

The Aerospatiale-A airfoil (Fig. 4(a)) test case results are presented in this section to demonstrate the capability of themodified g�Reqt �SA transition model to predict separation-induced transition. Flow past the airfoil is simulated ata Mach number of 0.15, Reynolds number of 2.1�106, and at an angle of attack (AoA) of 13:1� which is its maxi-mum lift configuration. The freestream turbulence intensity is specified as 0.05%, which is typical of a low-turbulencewind tunnel. Under these conditions, experimental data10 shows that the flow undergoes free transition through theformation of a laminar separation bubble with turbulent reattachment at around 12% of chord on the airfoil upper sur-face. The skin friction profile plotted in Fig. 5(a) shows that the g�Reqt �SA transition model is able to capture thisseparation-induced transition, indicated by a sharp rise in the skin friction value. The baseline S-A model produced afully turbulent boundary layer and hence is unable to capture the separation-induced transition. The surface pressurecoefficient plotted in Fig. 5(b) shows the presence of a laminar separation bubble around 12% of the chord, indicatedby a small region of constant pressure. It can be seen in Fig. 6(a) that the intermittency production in the boundarylayer starts near 12% of the chord. The corresponding eddy viscosity contours in Fig. 6(b) verify that the baselineS-A model is activated once the intermittency starts to increase beyond a certain value above zero. Compared to thefully turbulent S-A model, the g�Reqt � SA model significantly reduces the turbulence production as seen in Fig.7. The experimentally measured values of lift and drag coefficients are 1.56 and 0.0208 respectively. As expected,the fully turbulent S-A model predicted these values to be 1.52 and 0.027 with an over-prediction of drag, while theg�Reqt �SA model predicted them to be 1.60 and 0.0203 respectively.

The original g�Reqt model coupled with the two-equation SST k�w turbulence model (g�Reqt �kw) is also im-plemented in the current work. Figure 8 shows a comparison of the Reqt contours obtained using the two versions ofthe transition model. Besides showing a good agreement between the two models, the Reqt contours verify the imple-mentation of the experimental correlations for Reqt (Eq. 17) through the transport equation for Reqt (Eq. 12). Recallthat the experimental correlations for Reqt are a function of the local pressure gradient and the freestream turbulenceintensity (Tu¥). The value of Reqt is expected to be lower in the presence of higher pressure gradient, since transitiononset occurs earlier for a larger pressure gradient. This trend is reflected well in Fig. 8, showing reduced levels of Reqton the upper surface of the airfoil with a larger pressure gradient, compared to the lower surface. Another observationis that, reduced levels of Reqt are seen in the wake region of the airfoil when using the g�Reqt �kw model, whichis not seen in the case of the g�Reqt �SA model. This decrease in local Reqt is a consequence of increased levelsof local turbulence intensity, since the implementation of g�Reqt �kw model uses the local value of Tu as opposedto the freestream Tu used by the g�Reqt �SA model. One aspect of improvement in the g�Reqt �SA model is toaccount for this increase in local turbulence levels outside the boundary layer, which may be important for problemswhere wake turbulence needs to be modeled accurately. There are many practical applications where wake-inducedtransition greatly affects the transitional nature of the boundary layer on an aerodynamic surface (such as blades of aturbine-cascade, multi-element airfoils, or surfaces operating in the wake of bluff bodies).

Chordwise position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.005

0

0.005

0.01

0.015

0.02

Baseline S­A

S­A + Transition

Experiment

(a) Skin friction distribution

Chordwise position, x/c

Pressure

Coefficient,­C

p

0 0.2 0.4 0.6 0.8 1­1

0

1

2

3

4

5

Baseline S­A

S­A + Transition

Experiment

(b) Surface pressure distribution

Figure 5. Aerospatiale-A predictions using g�Reqt �SA transition model

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(a) Intermittency, g (b) Non-dimensional Eddy viscosity, µt

Figure 6. Aerospatiale-A: Production of Eddy viscosity triggered by increase in intermittency

(a) Baseline S-A model (b) g�Reqt �SA model

Figure 7. Aerospatiale-A: Comparison of non-dimensional turbulent viscosity levels

B. VA-2 Supercritical Airfoil

The VA-2 supercritical airfoil18 (Fig. 4 (b)) provides for an interesting test case, since its upper surface has very littlecurvature and almost simulates a flat plate at an angle of attack of -0:5� . Therefore, at -0:5� AoA, the upper surfacetransition onset location is determined primarily by freestream turbulence intensity and Reynolds number, without anypressure gradient effect. Two sets of results are presented in this section: (1) Variation of angle of attack at Re =2�106 and (2) Variation of Reynolds number at AoA = -0:5�. The freestream conditions for this test case are: M =0.2 and Tu¥ = 0.5%. The computed skin friction profiles for AoA variation with and without the transition model arecompared with experimental data in Fig. 9. It is seen that the transition model is able to capture the onset location andextent of transition with reasonable accuracy, whereas the baseline S-A model predicted a fully turbulent boundarylayer on both upper and lower surfaces. As the angle of attack increases, the transition onset location moves upstreamon the upper surface and downstream on the lower surface. In most cases, the transition model tends to over-predictthe turbulent skin friction value compared to the experimental data. The second set of results showing the effect ofReynolds number variation at an AoA = -0:5� are plotted in Fig. 10 (see Fig. 9(a) for Re = 2�106 case). As expected,the upper surface onset location moves upstream with increasing Reynolds number, and the behavior of skin frictionprofile on the upper surface is very similar to that of a flat plate. It is also evident that the baseline S-A model istreating the boundary layer as fully turbulent for all Reynolds numbers and AoA values.

C. S809 Laminar Airfoil

The S809 airfoil is a 21% thick, laminar-flow profile (Fig. 4 (c)) designed specifically for horizontal-axis wind turbineapplications.19 This airfoil is chosen as a test case since it has laminar flow up to 50% of the chord on both upperand lower surfaces at lower angles of attack. Steady simulations in freestream are carried out over a range of angle ofattack (AoA) values, at a Mach number of 0.1 and a Reynolds numbers of 2�106. The freestream turbulence intensitywas set to 0.05%. Computationally predicted static lift and drag coefficients for the S809 airfoil are compared to theexperimental values in Fig. 11. It is seen that the g�Reqt �SA model shows improved prediction of the magnitudeof lift and drag coefficients at lower AoA values, compared to the baseline S-A turbulence model. A closer look at

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(a) g�Reqt �k�w Model (b) g�Reqt �SA Model

Figure 8. Aerospatiale-A: Comparison of Reqt contours

Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.01

­0.005

0

0.005

0.01Baseline S­A

S­A + Transition

Experiment

(a) AoA = -0:5�

Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.01

­0.005

0

0.005

0.01Baseline S­A

S­A + Transition

Experiment

(b) AoA = 3:5�

Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.01

­0.005

0

0.005

0.01Baseline S­A

S­A + Transition

Experiment

(c) AoA = 7:5�

Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.01

­0.005

0

0.005

0.01Baseline S­A

S­A + Transition

Experiment

(d) AoA = 11:5�

Figure 9. Comparison of skin-friction profiles at different AoA for VA-2 airfoil, Re = 2�106

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Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.01

­0.005

0

0.005

0.01Baseline S­A

S­A + Transition

Experiment

(a) Re = 0.6�106

Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.01

­0.005

0

0.005

0.01Baseline S­A

S­A + Transition

Experiment

(b) Re = 6� 106

Figure 10. Comparison of skin-friction profiles at different Reynolds numbers for VA-2 airfoil, AoA = -0.5�

the data from lower AoA values in Fig. 11(c) reveals that the baseline S-A model over-predicts drag due to the fullyturbulent treatment of the boundary layer. The computed transition onset locations shown in Fig. 11(d) compare wellwith the experimental data. The surface pressure distributions are plotted at 14� and 20� AoA in Fig. 12. The transitionmodel prediction is closer to the experimental data compared to the fully turbulent prediction at these higher angles.At 14� AoA the baseline S-A model fails to predict the full extent of trailing-edge flow separation on the upper surfaceas seen in the experiment. This is also evident from the streamlines around the airfoil at 14� AoA, shown in Fig. 13.

The present results were obtained using the S-A model with rotational correction, which reduces turbulence produc-tion in vortex cores, as mentioned earlier. The S809 test cases were also run without using the rotational correction inthe S-A model. The effect of rotational correction on the lift and drag predictions (when using the transition model) isshown in Fig. 14 . The rotational correction is able to improve the lift prediction at higher angles of attack with largeregions of trailing-edge separated flow through reduced levels of turbulence. However, the drag prediction did notshow visible improvement with the use of rotational correction. It is well known that large regions of separated flowat higher AoA values are not modeled accurately in 2-D simulations. In order to further improve the predictions nearstall onset, 3-D simulations including hybrid RANS-LES methods such as the Detached-Eddy Simulation (DES)22 andDelayed-DES23, 24 are necessary, along with the transition model. These methods reduce the turbulence productionoutside the RANS region where large vortical structures are present in massively separated flows.

Figure 15 shows the lift and drag predictions obtained using original as well as the modified transition models for theS809 airfoil case. It is seen that the performance of the g�Reqt � SA model is comparable to that of the g�Reqt�kw model at lower AoA. Although the current implementation of the g�Reqt �kw model shows poorer predictionat higher AoA due to poor convergence, it is believed that this is not indicative of the performance of the transitionmodel itself and is likely due to some implementation issues specific to the CFD solver in use. The RANS-based SSTk�w turbulence model is not sufficient to predict separated flow at higher AoA (similar to the S-A model) and asmentioned earlier, 3-D simulations with a DES-type model is necessary to further improve predictions at higher AoA.

Previously published results for the S809 test case21 using the g�Reqt �SA model showed a more drastic improve-ment of lift and drag predictions at higher angles of attack. However, those results were obtained using correlationsthat resulted in lower values for Flength and higher values for Reqc in the boundary layer. Using these correlations forflat plate simulations results in delayed transition onset and a longer transition length compared to the experimentaldata. Additionally, b = 0:1 was used previously in the destruction term of the modified S-A model (Eq. 28), whichcauses excessive skin friction overshoot in the turbulent boundary layer for a zero pressure gradient flat plate case asshown in Fig. 3(d). Figure 16 shows the lift and drag predictions for the S809 test case using the improved calibrationparameters as mentioned above. It is seen that the transition model performance at higher AoA values is significantlyimproved when the model is calibrated to obtain a more laminar boundary layer due to lower Flength and higher Reqc

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Angle of Attack (degrees)

LiftCoefficient

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Baseline S­A

S­A + Transition

Experiment

(a) Lift variation with AoA

Angle of Attack (degrees)

DragCoefficient

0 5 10 15 200

0.05

0.1

0.15

0.2

Baseline S­A

S­A + Transition

Experiment

(b) Drag variation with AoA

Angle of Attack (degrees)

DragCoefficient

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

Baseline S­A

S­A + Transition

Experiment

(c) Drag prediction at lower AoA

Angle of Attack (degrees)

TransitionOnsetLocation

0 3 6 9 12 150

0.2

0.4

0.6

0.8

1

Lower Surface ­ CFD

Upper Surface ­ CFD

Lower Surface ­ Expt

Upper Surface ­ Expt

(d) Transition onset location variation with AoA

Figure 11. g�Reqt �SA transition model predictions for S809 airfoil, Re = 2� 106

values. Additionally, recall that b = 0:5 gave the best agreement of turbulent skin friction profile with experimentaldata for flat plate test cases as opposed to b = 0:1 used to obtain the improved S809 predictions shown here. Thismay indicate that there is a stronger dependence of Flength, Reqc, and b on the history of pressure gradient, which maynot be accounted for when the model is calibrated using zero-pressure-gradient flat plate data. The effect of pressuregradient history on the values of Flength, Reqc, and b will be further investigated. While it is clear that the choice ofFlength, Reqc, and b values has a large impact on the nature of the boundary layer, a more thorough investigation of themodel behavior is necessary to understand the effect of each of the model constants.

D. SC2110 Airfoil with Leading-edge Slat

Multi-element airfoil (MEA) configurations with a leading-edge slat and/or a trailing edge flap are commonly usedin fixed-wing aircraft in order to increase maximum lift and delay the onset of stall at low speeds. Similar conceptshave been investigated for rotorcraft applications to alleviate the onset of dynamic stall. The amount of turbulence inthe wake from the flow past a leading-edge slat strongly influences the transition to turbulence of the main elementflow. Thus, it is important to accurately model the transitional flow on the slat for good overall force and momentpredictions. Steady flow simulation results of the S-6 leading-edge slat configuration developed by Lorber et al.25 arepresented here. Flow conditions for this test case are as follows: Mach Number = 0.3, Reynolds Number (based on

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Chordwise Position, x/c

­Cp

0 0.2 0.4 0.6 0.8 1­2

0

2

4

6

8

Baseline S­A

S­A + Transition

Experiment

(a) AoA = 14�

Chordwise Position, x/c

­Cp

0 0.2 0.4 0.6 0.8 1­2

0

2

4

6

8

Baseline S­A

S­A + Transition

Experiment

(b) AoA = 20�

Figure 12. Comparison of surface pressure profiles for S809 airfoil

(a) Baseline S-A, AoA = 14� (b) S-A with transition, AoA = 14�

Figure 13. Comparison of streamlines for S809 airfoil at 14� AoA

main element chord) = 4.14�106, and Tu¥ = 0.5%. Since the experimental data is not corrected for wind tunnel walleffects, the wind tunnel walls are modeled in this test case. The height of the wind tunnel in the experiments is fourtimes the main element chord length. The overset mesh system used to simulate this test case is shown in Fig. 4(d).The main element has 300 cells on its surface as mentioned earlier, and the slat has 200 cells on its surface. Individualvariation of lift and drag coefficients obtained with and without the transition model are plotted against experimentaldata in Figs. 17 (a) and (b) respectively. While the lift predictions at lower angles of attack are acceptable even withoutthe transition model, the baseline S-A model fails to predict the reduced drag owing to a significant region of laminarflow on both the main element and the slat. The comparison of skin friction profiles at the maximum lift-to-drag ratio(obtained at an AoA of 10� ) plotted in Fig. 18 confirms this observation. The skin friction profile of the slat indicatesthat the boundary layer remains laminar up to half of the slat chord length. It is also seen that the flow past the mainelement remains laminar up to 9% of the chord from the leading edge. Eddy viscosity contours in Fig. 19 show thereduced production of turbulence on both the main element and the slat with the use of the transition model.

E. OA209 Dynamic Stall

The present work is part of a larger effort to improve 2-D dynamic stall predictions using transition modeling. Theresults of the OA209 pitching airfoil undergoing dynamic stall are presented in this section. Experimental data for thistest case was obtained from results published by researchers at ONERA.26 An overset mesh system was used to modelwind tunnel walls as shown in Fig. 4(e). The experimental conditions are summarized below:

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Angle of Attack (degrees)

LiftCoefficient

8 10 12 14 16 18 200.8

1

1.2

1.4

1.6Without Rotation

With Rotation

Experiment

(a) Lift variation with AoA

Angle of Attack (degrees)

DragCoefficient

8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

Without Rotation

With Rotation

Experiment

(b) Drag variation with AoA

Figure 14. Effect of rotational correction on force prediction at higher AoA for S809 airfoil (results obtained using g�Reqt �SA model)

Angle of Attack (degrees)

LiftCoefficient

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

SST k­w + Transition

S­A + Transition

Experiment

(a) Lift variation with AoA

Angle of Attack (degrees)

DragCoefficient

0 5 10 15 200

0.05

0.1

0.15

0.2SST k­w + Transition

S­A + Transition

Experiment

(b) Drag variation with AoA

Figure 15. Comparison of g�Reqt �SA and g�Reqt �kw transition models for S809 airfoil

Reynold number = 1.8 �106, Mach number = 0.16Reduced frequency = 0.1, Mean AoA = 13�, Pitching amplitude = 5�

Under these conditions, the stall event is initiated primarily through a trailing edge separation zone that propagatesupstream with increasing angle of attack. During the upstroke, lift and moment predictions from computations withand without the transition model are in good agreement with experimental measurements (see Figs.20 (a), (b)). How-ever, during the downstroke, the baseline S-A model fails to predict a deep stall and results in reattachment earlierthan what is seen in the experiments. This is attributed to the failure of the fully turbulent S-A model to predict transi-tion through the formation of a leading-edge separation bubble (seen in Fig. 21) that makes the boundary layer moresusceptible to separation at higher angles of attack. It can be seen that the g�Reqt �SA transition model successfullypredicts the stall event (although, under-predicting the onset of stall), followed by a lift and moment recovery path thatqualitatively follows the experimental data. During the dynamic stall cycle, the overshoot of lift and the large negativepitching moment at the maximum angle of attack seen in experiments is typically indicative of a leading-edge vortexformation and convection along the airfoil surface. However, it appears that the current transition model fails to predicta leading-edge vortex in this case. This is evident from an under-prediction of the negative pitching moment during the

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Angle of Attack (degrees)

LiftCoefficient

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Flat Plate Calibration

Improved Calibration

Experiment

(a) Lift variation with AoA

Angle of Attack (degrees)

DragCoefficient

0 5 10 15 200

0.05

0.1

0.15

0.2

Flat Plate Calibration

Improved Calibration

Experiment

(b) Drag variation with AoA

Figure 16. Force predictions using improved calibration parameters for S809 airfoil

Angle of Attack (degrees)

LiftCoefficient

­3 0 3 6 9 12 15 18 21 240

0.5

1

1.5

2

2.5

Baseline S­A

S­A + Transition

Experiment

(a) Lift variation with AoA

Angle of Attack (degrees)

DragCoefficient

­3 0 3 6 9 12 15 18 210

0.01

0.02

0.03

0.04

0.05

0.06

Baseline S­A

S­A + Transition

Experiment

(b) Drag variation with AoA

Figure 17. Transition model predictions for S6 leading-edge slat configuration, Re = 4.14�106

downstroke by the transition model. Similar to the S809 test case results, when the transition model was calibrated toobtain a more laminar boundary layer, the lift and pitching moment predictions for the OA209 dynamic stall case weresignificantly improved with the prediction of a leading-edge vortex formation resulting in a stronger flow separationand unsteadiness during downstroke (published earlier21). This reiterates the observation that calibration based on zeropressure gradient flat plate data may not be suitable for airfoil predictions at high angles of attack at which pressuregradient effects are dominant.

Figure 22 compares the experimental and computed surface pressure (Cp) and upper surface skin friction (C f ) profilesat 18� upstroke, 17.5� downstroke and 13� downstroke. The pressure distribution at 18� upstroke (Fig. 22(a)) is inreasonable agreement with experimental data, both with and without the transition model. However, it is evident fromthe corresponding skin friction profile (Fig. 22(b)) that only the g�Reqt �SA model predicts transition onset close tothe leading-edge which is not predicted by the baseline S-A model. At all three stages, the transition model predictsthe transition onset location near leading-edge, that moves slightly upstream or downstream depending on the AoAvalue. The Cp distribution at 17.5� downstroke (Fig. 22(c)) shows that the transition model is only predicting a trailingedge separation region, whereas the experimental data indicates the formation of a leading-edge vortex, represented bya region of constant pressure up to 30% of the chord on the upper surface. At 13� downstroke (Fig. 22(e)), the baselineS-A model over-predicts lift due to early re-attachment, which was also seen in the lift and moment hysteresis loops. At

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Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

­0.08 ­0.06 ­0.04 ­0.02 0 0.02 0.04 0.06­0.015

­0.01

­0.005

0

0.005

0.01

0.015

0.02

0.025

Baseline S­A

S­A + Transition

Transition on upper surface

(a) Leading-edge slat

Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.01

­0.005

0

0.005

0.01

0.015

Baseline S­A

S­A + Transition

Transition on upper surface

(b) Main element

Figure 18. Skin friction profiles on slat and main element for S6 configuration at 10� AoA

(a) Baseline S-A Model (b) S-A + Transition Model

Figure 19. Comparison of Eddy viscosity contours for S6 configuration at 10� AoA

this AoA, the transition model shows modest improvement over the fully turbulent prediction, which can be attributedto the transition model predicting a less turbulent boundary layer that resulted in a larger trailing-edge separationzone. Experimental data (Fig. 20) also indicates higher uncertainty in measurements during the downstroke, due tothe presence of 3-D, unsteady vortical structures in the separated flow regions, which cannot be accurately captured in2-D simulations. In order to further improve downstroke prediction of dynamic stall, 3-D dynamic stall simulations,that include transition modeling as well as hybrid RANS-LES methods such as DDES23 and IDDES,24 need to beperformed.

V. Conclusions and Future Work

In the present work, the correlation-based g�Reqt transition model was modified and integrated with the one-equation S-A turbulence model. A series of zero pressure-gradient flat plate test cases were used to develop numericalcorrelations for Reqc and Flength parameters that control the transition onset location and transition length, respectively.The effect of various transition model parameters on the skin friction profiles was studied and documented. Themodified transition model was implemented into a 2-D, structured, finite-volume, compressible Navier-Stokes solverenabled with the improved Implicit Hole Cutting methodology to handle overset mesh systems. It was shown that thebaseline S-A model was unable to predict boundary layer transition and generated excessive turbulence. For steadyairfoil problems, the S-A model consistently over-predicted drag at lower angles of attack and lift at higher angles ofattack approaching stall. The g�Reqt �SA model predicted the transition onset location with good accuracy, resulting

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Angle of Attack (degrees)

LiftCoefficient

7 9 11 13 15 17 190.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Baseline S­A

S­A + Transition

Experiment

(a) Lift variation with AoA

Angle of Attack (degrees)

MomentCoefficient

7 9 11 13 15 17 19­0.3

­0.2

­0.1

0

0.1

Baseline S­A

S­A + Transition

Experiment

(b) Moment variation with AoA

Figure 20. Time history of lift and moment coefficients for OA209 airfoil undergoing dynamic stall

(a) Baseline S-A Model (b) S-A with Transition Model

Figure 21. OA209 airfoil: velocity vectors near the leading-edge, showing a laminar separation bubble, at 13� upstroke

in significantly improved lift and drag predictions. It was observed that the use of rotational correction in the S-Amodel production term reduces the turbulence levels and improves predictions for airfoils at higher angles of attack.Static airfoil simulations in the natural transition regime showed that the performance of the g�Reqt � SA modelis comparable to that of the original g�Reqt �kw model. The ability of the g�Reqt �SA model to improve two-dimensional dynamic stall predictions was also demonstrated. However, it was also shown that the calibration of thetransition model using zero pressure gradient flat plate data may not be suitable for airfoil problems at higher anglesof attack, where the effects of pressure gradient are more significant.

Future work includes a more rigorous analysis of the model behavior leading to the development of improved correla-tions for Reqc and Flength parameters. Validation of the g�Reqt �SA transition model for more complicated 3-D flowproblems, including static and dynamic stall is desired. In realistic 3-D flows, cross-flow instabilities due to stream-line curvature are very important in determining transition onset location. However, the current formulation does notaccount for transition onset due to cross-flow effects. Attempt will be made to develop and validate correlations forcross-flow transition phenomenon. Finally, hybrid RANS-LES methods (DDES,23 IDDES24) will be applied in ad-dition to the g�Reqt �SA transition model to simulate and study 3-D, unsteady, separated flows problems such asdynamic stall.

Acknowledgments

This work is sponsored by the Vertical Lift Consortium / National Rotorcraft Technology Center under the ArmyTechnology Investment Agreement W911 W6-06-2-0002.

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Chordwise Position, x/c

Pressure

Coefficient,­C

p

0 0.2 0.4 0.6 0.8 1­2

0

2

4

6

8Baseline S­A

S­A + Transition

Experiment

(a) Cp at 18� upstroke

Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.008

­0.004

0

0.004

0.008

0.012

0.016

Baseline S­A

S­A + Transition

Expt ­ Hot Films

Expt ­ Friction Gauges

(b) C f at 18� upstroke

Chordwise Position, x/c

Pressure

Coefficient,­C

p

0 0.2 0.4 0.6 0.8 1­2

0

2

4

6

8Baseline S­A

S­A + Transition

Experiment

(c) Cp at 17:5� downstroke

Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.008

­0.004

0

0.004

0.008

0.012

0.016

Baseline S­A

S­A + Transition

Expt ­ Hot Films

Expt ­ Friction Gauges

(d) C f at 17:5� downstroke

Chordwise Position, x/c

Pressure

Coefficient,­C

p

0 0.2 0.4 0.6 0.8 1­2

0

2

4

6

8Baseline S­A

S­A + Transition

Experiment

(e) Cp at 13� downstroke

Chordwise Position, x/c

Skin

FrictionCoefficient,Cf

0 0.2 0.4 0.6 0.8 1­0.008

­0.004

0

0.004

0.008

0.012

0.016

Baseline S­A

S­A + Transition

Expt ­ Hot Films

Expt ­ Friction Gauges

(f) C f at 13� downstroke

Figure 22. Surface pressure and skin friction profiles for OA209 airfoil

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