Prediction of the Flow over an Airfoil at Maximum Lift

Rupesh B. Kotapati-Apparao; Kyle D. Squirest M A E Department, Arizona State University, Tempe, A Z

James R. Forsythet Cobalt Solutions LLC, 4636 New Carlisle Pike, Springfield, OH

Predictions of the flow around the Aerospatiale A-airfoil at maximum lift, a = 13.3O, and Re = 2 x lo6 are obtained using solutions of the steady Reynolds-averaged Navier- Stokes (RANS) equations and using Detached-Eddy Simulation (DES). RANS predictions of the two-dimensional flow are computed using the Spalart-Allmaras and SST turbulence models. Solutions of the fully-turbulent flow are computed using both RANS models in addition to prediction of the flow with laminar separation near the leading edge using the tripless approach within the Spalart-Allmaras model. For the tripless case, transition from laminar-to-turbulent flow along the lower (pressure) surface was fixed by tripping the boundary layer at x/C = 0.3. The pressure coefficient in the RANS solutions are in reasonable agreement with measurements prior to the trailing edge separation. The trip- less solution using the Spalart-Allmaras model predicts laminar separation at x/C = 0.088 with turbulent reattachment at x/C = 0.124 and followed by separation of the turbulent boundary layer near the trailing edge at x/C = 0.96. Trailing edge separation in the fully turbulent solutions is also further aft compared to the location reported in experiments, predicted at x/C = 0.88 using the Spalart-Allmaras model and a t x/C = 0.92 using SST. Comparison of the mean velocity predictions to measurements shows a behavior consis- tent with the mismatch in separation location compared to measurements. Preliminary DES predictions are presented and aspects requiring future research are identified.

Introduction

P REDICTION of complex flows that include laminar-to-turbulent transition, adverse pressure

gradient, streamline curvature, and boundary layer separation rernain among the most challenging for tur- bulence simulation strategies. A prototypical example that is the focus of the present investigation is the flow over an airfoil at maximum lift. Flow regimes are sensitive to the airfoil geometry, angle of attack, and Reynolds number and motivate various hierarchies of simulation strategies.

Reynolds-averaged Navier-Stokes (RANS) a p proaches remain the most widely applied simulation technique for complex flows. For flows that are not far from the thin shear layers used to calibrate the models, RANS approaches are often sufficient. In other regimes, e.g., flows with significant effects of sep- aration and which are characterized by flow structures atypical compared to the thin (and usually attached) shear layers predicted by RANS models, techniques such as Large-Eddy Simulation (LES) are attractive. While LES carries a prohibitive computational cost for resolving boundary layer turbulence at high Reynolds

'Research Assistant, present affiliation: Mechanical and Aerospace Engineering Department, The George Washington University

t~rofessor , AIAA Member t ~ i r e c t o r of Research, Senior Member AIAA Copyright @ 2001 by the American Institute of Aeronautics

and Astronautics, Inc. All rights reserved.

numbers, the technique offers many advantages over RANS methods for predicting separated flows. This in turn provides strong incentive for the merging of these techniques in hybrid RANS-LES approaches.

Detached-Eddy Simulation (DES) is among the more well known and actively applied hybrid RANS-LES strategies. The method was proposed by Spalart et al. [I] as a numerically feasible and plausibly accurate approach for prediction of mas- sively separated flows at high Reynolds numbers. To date, there have been a range of flows predicted using DES. These investigations have been largely successful, yielding predictions superior to those ob- tained using RANS approaches while resolving three- dimensional, time-dependent features because of the LES treatment of separated regions (e.g., [2], 131, 141). In "natural" applications of the technique, boundary layer growth and separation is under control of the RANS model, the method becomes an LES in regions away from solid surfaces provided the grid density is sufficient. The approach has proven especially effec- tive for prediction of massively separated flows, in turn motivating extension of the method to the accurate prediction of other complex flow regimes.

The work reported in this manuscript constitutes part of a longer-term project aimed at improving DES for predicting flows outside the design-range of mas- sive separations. The specific flow of interest is that over the Aerospatiale-A airfoil at an angle of attack

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AIAA 2004-259

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of 13.3" and Reynolds number of 2 x lo6 , correspond- ing to maximum lift. The flow has been measured in separate experiments [5], [6] and was the subject of an coordinated set of investigations through the LES- FOIL project (e.g., see Mellen et al. [7] for a recent overview).

The objective of the LESFOIL project was to ad- vance Large-Eddy Simulation in particular, and Com- putational Fluid Dynamics in general, for predicting complex flows. Advantages of LES for the general case include the reduced sensitivity to modeling er- rors than occurring in RANS methods and the use of grid refinement as a means for increasing the fi- delity of the predictions. Such features are quite unlike those in RANS approaches given the fixed mod- eling errors that exist, even in the fine-grid limit. The flow over an airfoil near maximum lift strongly challenges RANS methods, the inter-mingling of ef- fects such as laminar-to-turbulent transition, adverse pressure gradient, streamline curvature, shallow sep- aration, and reattachment complicating analysis and modeling. Unfortunately, one of the findings of the LESFOIL project is that even a narrow section of the airfoil, at a moderate Reynolds number and unswept, continues to pose an extreme cost challenge to LES. hlellen et al. [7] also noted that other aspects such as the numerical method, grid generation and mesh reso- lution requirements, especially in the near-wall region, coupled with considerations of the spanwise period in three-dimensional eddy-resolving computations stress essentially all elements of any simulation strategy.

Reported below are RANS predictions of the steady- state solution and DES of the time-dependent and three-dimensional flow. RANS predictions are ob- tained using two turbulence models - the Spalart- Allmaras one-equation model [8] (referred to as S- A throughout) and the two-equation SST model of Menter [9]. As described below, a laminar separa- tion (and subsequent transition to turbulence) along the suction surface occurs not far from the leading edge in the experiments. Using the S-A model, the tripless approach of Travin et al. [lo], which has the effect of disabling the turbulence model upstream of separation is employed for prediction of the flow with laminar separation. Both S-A and SST models are also used in fully turbulent simulations of the domain, i.e., for which the boundary layers from the leading edge of the airfoil are turbulent. Inter-comparisons are shown of the models and against experimental measurements of the mean flow. The S-A model forms the base for the DES predictions shown in the following sections.

Presented in the next section is a summary of the flow features over the Aerospatiale-A airfoil at maxi- mum lift, followed by an outline of the computational approach, including grid construction and turbulence models. Properties of the RANS and DES predic- tions are t hen evaluated against measurements of t he

Laminar Boundary Layer

Separation Bubble. Transnion

Attached Turbulent Boundary Layer

'.~ / Turbulent Separation

.'-A<.- "\

Wake

Tripped Transition

Fig. 1 Flow regimes on the Aerospatiale-A airfoil at 13.3" (adapted from Mellen et al. [7]).

velocity field, surface pressure, and skin friction distri- butions.

Approach Flow Description

The flow considered is the same as used as the test case for the LESFOIL project: the Aerospatiale A- airfoil at an angle of attack a = 13.3", corresponding to the maximum lift configuration, and at a chord- based Reynolds number of 2 x lo6. The general char- acteristics of the flow are illustrated in Figure 1. As summarized in Mellen et al. [7], along the upper suc- tion surface the laminar boundary layer experiences a free transition in a small laminar separation bubble with turbulent reattachment at around x / C = 0.12. The turbulent boundary layer grows along the suction side of the airfoil, separating at x /C = 0.83, result- ing in a recirculation that extends to the trailing edge. Measurements show that the reverse-flow region ex- tends to roughly a wall-normal location y/C = 0.016 where y defines the coordinate normal to the suction surface and that the location of maximum reverse flow occurs at y / C = 0.003. Transition from laminar-to- turbulent flow along the lower (pressure) surface was fixed in experiments by tripping the boundary layer at x /C = 0.3. Measurements were acquired in two different wind tunnels, the airfoil lift and drag were measured along with the skin friction and pressure dis- tributions as well as mean velocity and Reynolds stress profiles at several stations along the airfoil and into the near-wake [5] , [6].

One of the more difficult aspects of the prediction is the relatively high sensitivity of such flows with incip- ient and then shallow separation to details of the flow development. Small differences between simulations and experiments in the upstream flow can amplify with downstream development. Consequently, the flow is a more difficult case compared to other flows experienc- ing massive separation (e.g., airfoils at high angle of attack) or where separation is fixed by the geometry, e.g., as occurs at a sharp edge.

Turbulence Models Shear Stress Transport Spalart- Allmaras

The Spalart-Allmaras RANS model solves an equa- tion for the variable fi which is dependent on the turbulent viscosity [8]. The model is derived based on empiricism and arguments of Galilean invariance, dimensional analysis and dependence on molecular vis- cosity. The model includes a wall destruction term that reduces the turbulent viscosity in the laminar sub- layer and trip terms to provide smooth transition to turbulence. The transport equation for the working variable D used to form the eddy viscosity is written as,

where i; is the working variable. The eddy viscosity vt is obtained from,

where v is the molecular viscosity. The production term is expressed as,

where S is the magnitude of the vorticity. The function f, is given by,

with large values of r truncated to 10. The trip func- tion ft2 is defined as,

The trip function ftl is specified in terms of the dis- tance dt from the field point to the trip, the wall vorticity wt at the trip, and AU which is the differ- ence between the velocity at the field point and that at the trip,

where gt = min(0.1, AU/wtAx) where Ax is the grid spacing along the wall at the trip.

The wall boundary condition is i7 = 0. The con- stants are cbl = 0.1355, u = 213, cb2 = 0.622, fC = 0.41, cwl = cbl//C2 + (1 + cb2)/u, G 2 = 0.3, cU3=2, ~ ~ l = 7 . 1 , ~ ~ 2 = 5 , C ~ I = 1 , ~ t 2 = 2 , ~ t 3 = 1.1, and ct4 = 2.

The Shear Stress Transport (SST) model was devel- oped by Menter [9] to improve the accuracy of the k-w model for prediction of separated flows. The baseline form combines k-E and k-w formulations, using a pa- rameter Fl to bridge from I;-w near the wall to k-E in the freestream. The transport equations governing k and w take the form,

where ~ i j above is the (modeled) turbulent shear stress. The switching function Fl is given by,

Fl = tanh (argt) , (9)

The switching function Fl is also used to determine the values of the model constants. If represents a generic constant in the k-w equations and @2 r e p resents the same constant in the k-E equations, then the model constants employed in (7) and (8) are de- termined by,

4 = Fld1 + (1 - F1)d2. (11)

In the baseline version of the model the turbulent eddy viscosity is determined as pt = pklw. The SST model limits the turbulent shear stress to palk where a1 = 0.31. This in turn leads to an expression for the eddy viscosity,

palk pt =

max (alw; RF2) ' where R is the absolute value of vorticity. The func- tion F2 is included to prevent singular behavior in the freestream where R goes to zero and is given by,

F2 = tanh (ar& , (13)

The k-w model constants are given by ukl = 0.85, 0,1 = 0.5, P1 = 0.0750, Pt = 0.09, K = 0.41, 71 = Pl/,!Y - u , l l i 2 / m . The values of the k-E model constants are a k 2 = 1.0, uw2 = 0.856, Dl = 0.0828, p* = 0.09, K. = 0.41, 7 2 = P2/P* - u,&n2/fl.

3 OF 13

AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-0259

Detached-Eddy Simulation

The DES formulation is obtained by replacing in th_e S-A mcdel the distance to the nearest wall, d, by d, where d is defined as,

with the value of the additional model constant CDEs = 0.65 set in simulations of homogeneous turbu- lence [ l l ] . In (14), A is the largest distance between the cell center under consideration and the cell cen- ter of the neighbors (i.e., those cells sharing a face with the cell in question). In "natural" applications of DES, the wall-parallel grid spacings (e.g., streamwise and spanwise) are on the order of the boundary layer thickness, the S-A RANS model is retained through- out most or all of the boundary layer, i.e., d = d, and prediction of boundary layer separation is deter- mined in the "RANS mode" of DES. Away from solid boundaries, the closure is a one-equation model for the subgrid scale eddy viscosity. When the production and destruction terms of the model are balanced, the length scale d = CDEsA in the LES region yields a Smagorinsky-like eddy viscosity i? oc SA2. Analogous to classical LES, the role of A is to allow the energy cascade down to the grid size.

While most natural applications of DES treat the entire boundary layer in RANS mode, grid refinement in the wall-parallel directions (both streamwise and spanwise) layer will cause the RANS-LES interface to move nearer the wall, activating the DES limiter and reducing the eddy viscosity below its RANS levels. This process can degrade predictions if mesh densi- ties are insufficient to support eddy content within the boundary layer, resulting in lower Reynolds stress lev- els compared to that provided by the RANS model [I].

Flow solver and grid

In this work, the compressible Navier-Stokes equa- tions are solved on unstructured grids using Cobalt. The numerical method is a cell-centered finite vol- ume approach applicable to arbitrary cell topologies (e.g, hexahedra, prisms, tetrahedra) and described in Strang et al. [12]. The spatial operator uses the ex- act Riemann Solver of Gottlieb and Groth [13], least- squares gradient calculations using QR factorization to provide second order accuracy in space, and TVD flux limiters to limit extremes at cell faces. A point implicit method using analytic first-order inviscid and viscous Jacobians is used for advancement of the dis- cretized system. For time-accurate computations, a Newton sub-iteration scheme is employed, the method is second order accurate in time. The domain decom- position library ParMETIS [14] is used for parallel implementation and provides optimal load balancing with a minimal surface interface between zones. Com- munication between processors is achieved using Mes- sage Passing Interface.

Fig. 2 Grid in the vicinity of the Aerospatiale-A airfoil, 946 x 117 points in the surface-tangent and surface-normal directions, respectively.

Calculations were carried out on a domain that ex- tends approximately eight chord lengths upstream of the leading edge of the airfoil. The domain down- stream of the trailing edge extends about 20 chord lengths. In the cross-stream direction the grid ex- tends ten chord lengths both above and below the airfoil. The computations reported in this manuscript were performed using structured C-type grids gener- ated using Gridgen [15]. A view of the mesh in the vicinity of the airfoil is shown in Figure 2. The grid is comprised of 945 points along the airfoil surface and 117 points normal to the wall. Of the 945 points along the airfoil, 693 are on the leading edge and up- per (suction) surface with the remainder along the branch cut and the lower (pressure) surface. The mesh was designed to possess sufficient density to support boundary layer turbulence and the coordinate along the airfoil surface is resolved with approximately 10- 12 points per boundary laver thickness beginning at roughly the half chord ( x / C x 0.50) - this determines the DES length scale (A = 0.0015C). The grids are clustered near the airfoil surface with the average dis- tance from the wall to the first cell center less than one viscous unit (AylC = 4.7~10-~) . The cells were stretched at a geometric growth rate of 1.2 for the first 32 layers. At this point the wall normal spacing was about 0.0011C, slightly below the target grid spacing of 0.0015C. This wall normal spacing was maintained until outside the expected boundary layer thickness and separation bubble. After that, the wall normal spacing was stretched at a geometric growth rate of 1.1.

While possessing finer mesh spacing than typical for RANS calculations, the two-dimensional FUNS pre- dictions shown below were computed using the grid possessing 945 x 117 points. As also described be- low, for the DES the spanwise grid spacing is specified to yield comparable resolution. i.e., 10-12 points per boundary layer thickness beginning at x/C x 0.50.

Results RANS

Measurements show that the flow over the AerospatialeA airfoil experiences a laminar separation in the vicinity of the leading edge region, just down- stream of the peak negative pressure along the suction side. Transition occurs in the separated shear layer with the reattached turbulent boundary layer evolv- ing further along the suction side prior to a subsequent separation near the trailing edge. The laminar sepa- ration and transition is accounted for in the present S-A predictions using the "tripless" approach outlined by Travin et al. [lo]. The tripless approach provides a means to accommodate the laminar separation and transition in the separated shear layer, in the present calculations represented by an activation of the tur- bulence model. The eddy viscosity upstream of the airfoil is zero, non-zero values are seeded into the suc- tion side of the airfoil using a boundary layer trip at x/C = 0.2, which is downstream of the location of the laminar separation. In the initial stages before the calculation reaches steady state, reversing flow sweeps upstream eddy viscosity from regions downstream of the trip, bringing this fluid into contact with regions of zero eddy viscosity. The recirculation in the separa- tion bubble near the leading edge then sustains the model and a t steady state the turbulence model is active slightly downstream of the point of flow s e p aration. A second boundary layer trip is located at x/C = 0.3 along the pressure surface, the same loca- tion as used in the experiments.

Predictions are also obtained of the fully-turbulent flow using the S-A and SST turbulence models. The fully-turbulent solutions are computed by seeding a small level of eddy viscosity into the upstream flow, sufficient to ignite the turbulence model as the fluid enters the boundary layers. As shown in Figure 4, the initial (laminar) separation near the leading edge does not occur in the fully turbulent solutions. Other aspects of these calculations are also different from the tripless solutions and are described below.

Streamlines from the tripless S-A RANS predic- tions are shown in Figure 3. The zoomed view in the lower frame highlights the small separation bubble which is initiated at x/C = 0.0088, the figure showing velocity vectors colored by the eddy viscosity ratio. The turbulent flow then reattaches at x/C = 0.124, in good agreement with experimental measurements (e.g., see Mellen et al. [7]). A second separation is pre- dicted in the tripless S-A result near the trailing edge, beginning at at x/C = 0.96. The trailing edge (turbu- lent) separation is further aft of the value reported in experiments, x/C = 0.83 [7].

Streamlines and contours of the velocity magnitude from each of the RANS predictions are shown in Fig- ure 4. The airfoil has been extruded in the direction normal to the plane showing the velocity magnitude in

Fig. 3 Streamlines over the Aerospatiale-A airfoil at 13.3" angle of attack and chord Reynolds number of 2 x lo6. Result shown in the figure from the tripless S-A prediction. Lower frame is an enlarged view showing the laminar separation bubble near the leading edge (x /C M 0.1).

Fig. 4 Streamlines and contours of the velocity magnitude. The isosurface of zero streamwise ve- locity is also shown on the airfoil surface. Upper frame: tripless S-A. Middle frame: fully-turbulent S-A. Lower frame: fully-turbulent SST.

AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004--0259

order to also plot the contour (isosurface in the figure) of zero streamwise velocity. This view shows the sepa- ration bubble beginning at x/C = 0.088 in the tripless S-A prediction (upper frame). Figure 4 also shows that there is no leading edge separation in the fully tur- bulent S-A and SST predictions, the turbulent state of the boundary layer in these simulations possessing sufficient momentum to resist leading-edge separation. The fully-turbulent S-A result (middle frame) predicts separation at x /C = 0.88 and SST (lower frame) pre- dicts boundary layer separation at x /C = 0.92. The separation locations in the fully turbulent runs are both closer to that deduced in experiments compared to the tripless result. Figure 4 also indicates that the displacement effects deduced from the velocity magni- tude in the tripless result (top frame) appear slightly smaller than in either of the fully turbulent solutions, with the largest differences visible between the tripless S-A and fully turbulent SST predictions. As tabulated below, the lift and drag predictions from the tripless S-A calculation are consistent with a smaller effect of the separation compared to the fully turbulent runs.

The pressure coefficient over the airfoil on both the pressure and suction sides and an enlarged view of the trailing edge region is shown in Figure 5. The fig- ures show that compared to the experimental measure- ments, the overall pressure distribution is adequately captured by both fully turbulent solutions over the pressure side and for the suction side upstream of the region strongly influenced by the trailing edge separa- tion (x/C > 0.8). The enlarged view of the pressure coefficient distribution in the lower frame of Figure 5 shows the fully turbulent solutions are closer to the ex- perimental measurements than the tripless prediction. The mismatch between the RANS and measurements depicted in the lower frame of the figure illustrates one of the outcomes of the differences in the separation pre- diction. Differences in the location of separation and in the characteristics of the separated flow region both contributing to the less negative C, predicted by the RANS models along the suction surface and illustrated in Figure 5. The tripless S-A result shown in Figure 5 also yields a larger peak pressure near the leading edge of the suction side than measured or predicted by ei- ther of the fully turbulent S-A or SST runs. The larger peak suction is similar to the behavior observed in pre- vious computations which also showed larger suction pressures in predictions with delayed trailing edge sep- aration, a result attributed to changes in the airfoil circulation [7].

A notable difference between the fully turbulent and tripless predictions is apparent for the suction-side pressure distribution near the airfoil leading edge. In the vicinity x /C FZ 0.1 the tripless prediction shows a slight plateau in the pressure distribution, a result of the laminar separation bubble captured in this cal- culation (c.f., Figure 4), though not in either of the

S-A FT - . .- . . -. . - S-A tripless ---- SST FT

0 Measurements

S-A FT - . . - . . - . . - S-A tripless ---- SST FT

0 Measurements

Fig. 5 Pressure coefficient. Upper frame shows the distribution over both the pressure and suc- tion sides of the airfoil. The lower frame shows an enlarged view of the trailing edge region. Measure- ments from Gleyzes [6].

fully turbulent runs. Previous LES calculations of the flow were also able to resolve the laminar separation, in addition to details of the transition process, though at a substantially increased cost [lG].

The skin friction coefficient over the suction side of the airfoil is shown in the upper frame of Fig- ure 6 and with an enlarged view of the trailing edge distribution in the lower frame. Analogous to the be- havior observed in the pressure coefficient, the fully turbulent calculations yield similar predictions of the skin friction, both S-A and SST in reasonable agree- ment with measurements for chordwise stations past x/C = 0.4. The tripless S-A result predicts a lam- inar boundary layer prior to separation and yields a substantially lower Cf compared to the fully tur- bulent runs upstream of x/C FZ 0.1. The negative

A ~ ~ E R I C A N INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-0259

1 X I , , ---- s-A FT

i ,: +!:, S-A tripless . -- - - SST FT I k ! I

o Measurements. F2, Re = 2.0 x l o6 I o Measurements, Fl , Re = 2.1 x 10'

9 : 0

Fig. 6 Skin friction coefficient. Upper frame shows the distribution over the suction side of the airfoil. The lower frame shows an enlarged view of the trailing edge region. Measurements "Fl" from Huddeville et al. [5], "F2" from Gleyzes [6].

0 005

0 0025

u-

0

region in Cf near x/C = 0.1 identifies the separation point and length of the reverse-flow region. Following reattachment, the skin friction in the tripless solution sharply increases to values slightly higher than in the fully turbulent predictions. The trailing edge distri- bution shown in the lower frame of the figure shows a better agreement between experimental measurements and the fully turbulent predictions as compared to the tripless result. The later separation in the tripless pre- diction compared to both fully turbulent solutions and the experimental measurements is also evident in the Cf distributions.

- - -- S-A FT - - - - S A trtpless

- SSTFT o Measurements, F2, Re = 2 0 x lo6 o Measurements, F1, Re = 2 I x 10' . . . - . . . .

% . . .. 0 . - --. . -ls,

. . . o * 0 . . . 0 . .

I - 0 a - v-- 2. +$.-,J

4

0 0 0 2 ~ 6 ~ " ' ~ ~ ' ~ ~ 1 " " ~ ~ ~ " ~ 0 7 0 8 0 9 1

The mean velocity components in coordinates lo- cally tangent and normal to the airfoil surface were measured in experiments [6], a representative coni- parison of the simulations to the data are shown in

XIC

xIC = 0.400 !I! $i

- . . - . . - . . - S-A tripless . . . . . . . . . . . . . . . . . . . . . Q

S-A FT - - i / l

SST FT 0 Measurements

v. I i! I! Ij j ij! i i i

@I q ' 1 d!

- . . - . . - . . - S-A tripless !I S-A FT

---- SST FT o Measurements

Fig. 7 Mean streamwise (upper frame) and wall- normal (lower frame) velocity at x /c = 0.40. Mea- surements from Gleyzes [6].

Figures 7-13. Figures 7-9 show the comparison be- tween RANS and experimental measurements at three streamwise locations - x/C = 0.4, 0.5, and 0.7 - prior to trailing edge separation. The tripless calculation has reattached following the laminar separation, ini- tiating a turbulent boundary layer from z/C = 0.124 while the fully turbulent solutions initiate boundary layers from the leading edge (without any separation until the trailing edge region). Each of Figures 7-9 show that the fully turbulent S-A and SST predic- tions of the mean streamwise velocity are in reasonable agreement, with measurements: with larger differences visible between the runs at x/C = 0.7 and the SST result closer to the measured profile than S-A. The figures also show that for these chordwise stations the mean streamwise velocity predicted in the tripless cal- culation leads that of the fully turbulent solutions and also exceeds the experimentally measured profiles. At

dC = 0.500 !j - . . - . . - . . - i :!

S-A tripless . ,, i.$

S-A FT : *. SST FT , A !.$

0 Measurements ;/ . ,.

x/C = 0.700 j j/ - . . - . . - . . - S-A tripless 9'

.. . . S-A FT ---- i;.!

SST FT 0 Measurements i!

0 jr;

Fig. 8 Mean streamwise (upper frame) and wall- Fig. 9 Mean streamwise (upper frame) and wall- normal (lower frame) velocity at x / c = 0.50. Mea- normal (lower frame) velocity at x /c = 0.70. Mea- surements from Gleyzes [6]. surements from Gleyzes [6].

0.12

0.1

0.08

x/C = 0.4 and x/C = 0.5, the tripless S-A prediction of the mean streamwise velocity show an overshoot that is not apparent in the experimental measurements and less pronounced in the fully turbulent S-A and SST predictions. This difference in the streamwise mean velocities is in turn consistent with a weaker trailing edge separation and larger suction near the leading edge as discussed above, aspects that slightly increase the circulation of the airfoil.

At x/C = 0.4 and x/C = 0.5, Figure 7 and Fig- ure 8 show good agreement between the fully turbulent RANS predictions and measurements for most of the wall-normal profile, the SST results yielding the clos- est match to the data. At x/C = 0.7, the wall-normal mean flow in Figure 9 (lower frame) depicts a larger difference between e x h of the RANS predictions and the measured profile. The under-prediction of the RANS solution indicative of a relatively weaker dis-

- , * ' - x/C= 0.700 j !!

-. . - . . - . . - S-A tripless / / i - S-A FT - ---- SST FT i / ! / / i - 0 Measurements : :

! 0 ; i: - / / / 0

/ /; 0 j i ! 0 . :;

placement of the mean flow under the action of the adverse pressure gradient developing over the suction side of the airfoil.

The remaining profiles, beginning with those mea- sured and predicted at x/C = 0.825 in Figure 10 show larger differences between the predictions and mea- surements and more substantial differences amongst the RANS models. This is anticipated based on the pressure and skin friction distributions, showing that the RANS and measurements are increasingly differ- ent towards the trailing edge region. The profiles at x/C = 0.825 are close to the location of boundary layer separation in the experiments. Both the mean stream- wise and wall-normal velocities measured in the exper- iments and depicted in Figure 10 show the boundary layer on the verge of separation. The RANS predic- tions, with separation further aft, yield streamwise mean velocities that lead the measurements and wall-

A

0.04 -

0.02 -

0 -

A~IERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-0259

x/C = 0.825 - . . - . . - . . - S-A tripless

S-A FT I!

..... -. . . . .. SST FT 'pi 0 Measurements 1

0.12 - x/C=O.825

j j;

!

1 ,,; 0 1 5 0

- - - - S-A tnples -- -

/ S-A FT ---- SST FT

o Measurements

0.12

0.1

0.08

0.06

R

0.04

0.02

0

-0.3

0.1

0.08

Fig. 10 Mean streamwise (upper frame) and wall- Fig. 11 Mean streamwise (upper frame) and wall- normal (lower frame) velocity at x/c = 0.825. Mea- normal (lower frame) velocity at x/c = 0.87. Mea- surements from Gleyzes [6]. surements from Gleyzes [6].

- - x/C = 0.870 - . . - . . - . . - S-A tripless 4

S-A FT : .......... ........ SST FT - o Measurements

-

- B

0 A - o / / . ,

0 .,/ \ 0 ,. '/ !

0 , .;/ ' /./. - od'O,,;,/' ,.. -- .,"

; &k<./ r a ~ ~ ~ l ~ ~ ~ ~ l ~ ~ ~ t l ~ ~ t ~ 1 ~ ~ ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ ~ ~ 1

0 0.3 0.6 0.9 1.2 1.5 1.8

; j j ' 0

0 -

- 0

normal means that lag the data in the outer region of the boundary layer. The tripless S-A prediction shows the largest difference compared to measurements in both profiles, the fully turbulent SST solution is clos- est to the measured distributions.

At x / C = 0.87 in Figure 11, the measured profiles of the mean streamwise velocity indicate a weak reverse flow. There is also relatively mild change in the mean wall-normal velocity compared to the measurements at x / C = 0.825. The RANS predictions are also sim- ilar to those at the upstream station x / C = 0.825 - none of the RANS predictions indicate boundary layer separation and the mean wall-normal velocity lags the measured values in the outer region.

Compared to the flow upstream, an increase in the peak reverse velocity in the measurements of the streamwise mean is observed at x / C = 0.96 in Fig- ure 12. The fully turbulent S-A and SST prediction

u/U_

indicate flow separation prior to this streamwise loca- tion. Figure 12 shows that the reversed flow in the streamwise component predicted by both models is much smaller than observed in the measured profile, consistent with the later separation predicted by the models compared to the experiments. As also observed in the profiles upstream of this location, the tripless S- A result over-predicts the measured profile over nearly all of the boundary layer. The mean wall-normal veloc- ities in the lower frame of Figure 12 show the closest agreement between the measurements and fully tur- bulent RANS solutions, though the profiles show less peak-to-peak variation than indicated in the measure- ments. The last streamwise station along the airfoil for which measurements were acquired, x / C = 0.99, in Figure 13 shows the SST prediction closest to the mea- sured profile in both the streamwise and wall-normal components.

0

X

0.04

0.02

0

- - . . - . . - . . - S-A triples

./ 6 S-A FT - SST FT o Measurements

- I , i , , l l l ~ # l ~ , l ~ l ~ ~ ~ ~ I ~ ~ ~ t I ~ ~ c ~ I

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 v/u_

S-A tripless

o Measurements

S-A tripless S-A FT SST FT Measurements

Fig. 12 Mean streamwise (upper frame) and wall- normal (lower frame) velocity at x/c = 0.96. Mea- surements from Gleyzes [6].

Table 1 shows that the fully turbulent solutions yield similar predictions of the lift and drag coefficients, a useful result given the differences in the modeling (e.g., one equation for S-A compared to two equations for SST). All of the simulations yield lift coefficients higher than the values reported in Huddeville et al. [5] and Gleyzes [6]. The differences in the lift and drag predictions compared to the measured values highlight differences in the flow evolution, especially the differ- ences in the details of flow separation near the trailing edge. The tripless prediction yields the highest lift coefficient, consistent with the later separation than observed in the experiments. The drag coefficients summarized in Table 1 are below the measured values, though the difference in the measured drag coefficient between the two facilities is not small.

XJC = 0.990 S-A tripless S-A FT SST FT

0 Measurements

0.08

XJC = 0.990 S-A tripless S-A FT

0 Measurements

0.08

Fig. 13 Mean streamwise (upper frame) and wall- normal (lower frame) velocity at x/c = 0.99. Mea- surements from Gleyzes [6].

Case Cr. Cn - - FT S-A 1.58 0.0162 FT SST 1.52 0.0176

tripless S-A 1.68 0.0095 Exp. F1 [5] 1.56 0.0204 Exp. F2 [6] 1.52 0.0308

Table 1 Lift coefficient CL and drag coefficient CD.

AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-0259

Fig. 14 Contours of the instantaneous vorticity in the DES. Cutting planes at x /C = 0.4, 0.65, 0.8, and 0.9.

DES

For t>he t,hree-dimensional, time-dependent and fully turbulent DES predictions the grid (c.f., Figure 2) was extruded into the spanwise ( z ) direction using 68 points arid with a constant spanwise spacing A,/C = 0.0015. Periodic boundary conditions are a,pplied, the spanwise period being 0.1C for the current grid spac- ing. The tirnestep used in the DES when made dimen- sionless using the chord length and freestream speed is 7.8 x lop4, yielding a CFL number based on the spanwise (target) grid spacing of approximately 0.5.

As summarized above for the current grid, the wall- parallel directions near the airfoil surface are resolved with 10-12 points per boundary layer thickness begin- ning at roughly the half-chord, x/C' = 0.5. Previous applications of DES in channel flow indicate such reso- lution corresponds t,o t,he coarser range of grid spacings ca.pable of supporting eddy content in the bounda.ry layer [17], [18]. Grids with sufficient density to re- solve boundary layer turbulence are not the norm for the design range of applications aimed at niassively separated flows. For the current niesh the interface between tlie RANS and LES regions is within the boundary layer, activating the DES limiter arid lower- ing t,he Reynolds stresses near the interface compared t,o the levels that would be given by t,he RANS model.

Solutions include development of techniques that seed the bouuda.ry layer with eddy content, in an attempt t,o develop resolved Reynolds st,ress to sup- plement the modeled values that are decreased on fine meshes when the RANS-LES interface is close to the wall. One example is the inclusion of' backscatter mod- els, e.g., t.hrough stochastic forcing of tlie monlentum equations, which has been applied in DES of turbu- lent cha.111iel flow [19]. Cont,rol over the location of the RANS-LES interface is an alternative approach and tested in the current simulations. The objective is to maintain RANS behavior in specific regions of the flow, irrespective of the mesh density.

In the present effort, a simple approach is applied by n~air~tainirig the RANS model to x / C = 0.5 since the

Fig. 15 Isosurfaces of the instantaneous fluctu- ating pressure in the DES prediction. Alternating signs in the pressure fluctuations indicated by the red (positive) and gray (negative) isosurfaces.

grid density is not sufficient to support eddy content prior to this chordwise station. Shown in Figure 14 are contours of the instantaneous vorticity in four planes along the airfoil. At x/C = 0.4, the RANS model is retained and the figure illustrates that the solution possesses weak spanwise variation. At the subsequent planes a range of scales is resolved as the flow de- velops eddies in the separating shear layer. Another view is provided in Figure 15 which shows isosurfaces of the instantaneous pressure, the red isosurface indi- cating positively signed pressure fluctuations arid gray isosurface indicating regions of negative pressure fluc- tuations. The figure shows a relatively large spanwise structure and a range of three-dimensional structures developing downstream.

For the present configuration, unlike flows exhibit- ing massive separation, the development of three- dimensional eddy content is not rapid as the solution convects downstream from the structureless RANS re- gion maintained upstream of x /C = 0.5. The lack of eddy content leads to lower Reynolds stress levels and, in the current case, to a separation of the boundary layer upstream of the location observed in experiments (or predicted in the RANS calculations). This feature highlights the need to seed the flow in order to de- velop eddy content to replace Reynolds stress reduced via the action of the DES limiter, in addition to more automated means to monitor and control the location of the RANS-LES interface.

Summary RANS and DES have been applied to the prediction

of the flow over the Aerospatiale-A airfoil at maximum lift. Experimental measurements show that the flow is characterized by a laminar separation followed by tur- bulent reattachment near the leading edge and with a subsequent separation in the trailing edge region. The laminar separation and turbulent rea.ttachment were accurately predicted using the tripless approach within the S-A model, wit,h the reattachnlent location at x/C = 0.124 in the simulations agreeing well with experimental measurements. Subsequent development of the tripless solution, however, was farther from ex- perimental measurements of the pressure coefficient,

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skin friction, and velocity distributions compared to fully turbulent predictions obtained using the S-A and SST models.

All of the RANS models predict separation aft of the location indicated in the experimental measure- ments. Consequently, the mean velocities in the sep- arated region is not as accurately re-produced by any of the RANS models, e.g., differences are noted in the pressure distribution and mean flow near the trail- ing edge. Measurements and predictions of the wall- normal mean velocity indicate the displacement of the flow by the adverse pressure gradient upstream is not similar in the computations. The delayed separation effectively increases the circulation in tripless S-A pre- diction, leading to larger suction near the leading edge and higher lift.

Preliminary DES predictions were obtained by maintaining the RANS model over the airfoil to x/C = 0.5. which constitutes the region upstream of that with sufficient grid spacing to adequately resolve eddy con- tent within the boundary layer. The calculations did not utilize any means of seeding fluctuations within the boundary layer that would more efficiently develop resolved Reynolds stress. Such strategies are not re- quired in natural applications of the technique to mas- sively separated flows since the entire boundary layer is usually treated by the RANS model and the explosive growth of instabilities in the separated region provides an efficient mechanism for creating three-dimensional eddy structure. The process of generating eddy struc- ture within the boundary layer represents a steeper challenge than encountered in natural DES applica- tions and is an important area of future work. In addition, the flow places many other demands on the modeling and numerical treatments, all of which re- quire additional research.

Acknowledgments

The authors gratefully acknowledge the support of the Air Force Office of Scientific Research (Grant F49620-02-1-0117, Program Officers: Dr. Thomas Beutner and Dr. John Schmisseur). Mr. Kul- winder Singh and Mr. Vivek Krishnan assisted with post-processing and analysis. The work was performed as part of a DoD Challenge project, on the Aeronau- tical Systems Center Major Shared Resource Center and the Maui High Performance Computing Center.

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-0259