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Nav~er-Stokes Analysis of Turbulent Boundary Layer and Wake for Tw - Dimensional Lifting Bodies P. Nguyen, I. Gorski (David Taylor Research Center, USA) ABSTRACT Navier-Stokes calculations were per- formed on two 2-D lifting foils which have been tested in a wind tunnel. In the experi- ment, the angles of attack for the two foils were set up to yield approximately the same lift at a Reynolds number of 2.~6xio6 ~ based on chord). One foil has a thicker trailing edge than the other, and has mild flow separation on the last No chord of the suc- tion side. The flow solver, called the Davic! Taylor Navier-Stokes (DTNS) code, is for- mulatec} with artificial compressibility and upwind differencing. The Launder-Spalding k-e turbulence mode! is used. Predictions of the turbulent flow quantities of the boundary layer and wake are compared with the experimental data for both foils. These predictions, including flow separation location, agree reasonably well with the data. After these validation predictions, the Navier-Stokes analysis method and a design technique baser] on conformal mapping are combined to develop new 2-D foil sections. Since the turbulent kinetic energy is the dynamic pressure, and the ReynoIcis shear stresses are related to the turbulence pro- duction, these quantities are used to develop new 2-D sections with desirable turbulent boundary layer characteristics. The characteristics of one new section are presented as results of the new foil design process. INTRODUCTION In this paper, with the aid 0 f a Navier-Stokes ~ N-S) analysis method we explore the potential of tailored blade 633 sections instead of stan(lar(1 NACA sections for optimization of propeller performance. Propeller designers normally use sections with NACA 16 or NACA 66 thickness dis- tributions and an a 0.S meanline. Due to recent improvement in computational capa- bility, it is now feasible to shape the section to achieve a specific design goal, whether it . ~ ~ ~ · · · · · - ~e maximizing e~nclency, mlnlmlzlng cavl- tation, or boundary layer control. Also, for some applications, it is desirable to maxim- ize the section thickness without degrading the propeller performance by massive flow separation. The motivation for this N-S analysis is due to the experimental results of Gershfeld et al. A, and Huang et al. {2i, which have shown that the pressure spectra on the trailing edge are related to the turbulent flow characteristics in the near-wake region. The turbulent flow data in Ref. t2] are used for validation of the N-S analysis. This paper presents the vali- dation results, and the calculated flow characteristics for a new section developed with the aid of the N-S analysis. The mean momentum balance for viscous flow at high Reynolds number yields the time-averaged N-S equations. The full N-S formulation is used here as separate(1 flow is analyzecl. There are two fundamental difficulties in using the N-S equations to predict the flows: 1 ) numerical instability clue to the convection terms, anti 2) mo(lelling of turbulence. The instability problem has been attacked by various numerical techniques such as 1 ) central differencing with artificial damping ~ 34, and 2) upwind (1ifferencing with Total

Variational D iminishing ~ TVD ~ schemes A. These techniques, however, only address the mean flow. The nonlinearity of the fluctuating flow yields turbulence, which is a more challenging problem. Little progress has been made in the development of a general theory for com- plex turbulent flows TV. Most of the funda- mental understanding of turbulent flow has been acquired through experimentation, and just recently through direct numerical simulation. In practice, turbulence models have evolved from the simple m~xing- length models, to the more physically real- istic models such as Mean Vorticity and Covariance ~ 6] . Most turbulence models are based on the eddy-viscosity concept which, although not very rigorous, has been widely used since it was proposed. The turbulence models used in this paper are baser! on this approach. OUTLINE OF ANALYTICAL METHOD The objective is to maximize the thickness of a 2-D lifting foil and to reduce trailing edge turbulent kinetic energy without incurring significant flow separa- tion. This is achieved by control of the tur- bulent boundary layer and wake charac- teristics through careful shaping of the 2-D section. The N-S analysis is used in the last stage of a foil design process to calculate the turbulent flow characteristics. The design parameters are the turbulent kinetic energy, and the turbulent shear stress. There are two steps in obtaining a N- Rig I S so lutio n: 1 ) geo m etry preparatio n, inclu(l ing grid generation, and 2) flow calculation. For this study, the grid generation is basest on the work of Coleman Hi, which uses partial differential equations to define a body-fitted grid. Here, a multi-zone grid is used for better control of the grid struc ture, which is especially useful when com bined with a multi-zone flow code such as the D avid Taylor Navier-Stokes (D TNS) code developed by Gorski Gil. For a high Reynolds number flow ~ greater than io6) the first grid point should be as close as in- 5 chord length away from the body, approximately y+ of 5, so that the sub-layer can be reso Ived. Fig. ~ sh o ws a 3- z o n e grid used for computing the flow over a foil with Reynolds number of ~ x io6. The flow 634 calculation steps are described in the fol- lowing sections. Using the idea of artificial compressi- bility developed by Chorin [9] allows the N-S equations for an incompressible fluid, in cartesian coordinates, to be written in the following conservative form: in + 0(f~+9~) + (~3+g2) ° (1) where the subset ~q oft ~f2 _O (2) constitutes the corresponding inviscid flow equations. For 2-D flow, the dependent variable q and the inviscid fluxes fir and f2 are given by: q [I/], /} j:2+p~, /2-t UV ~ V US Lv2+p where p is pressure, and u and v are the the x and y directions, respectively. The term p/,3 is the pseudocompressibility which should approach zero as the solution converges. controls the convergence rate of the scheme with a value of ~ being used for the present calculations. The viscous "fluxes" 91 and 92 are given by: Cartesian velocity components in ~ Bu 0~ ~ 92 -~ By REV . ~ REV . . where Re is the Reynolds number and ,u is the molecular viscosity. The equations (1) and ( 2) form a hyperbolic system which can be marched in time using implicit tech- niques. k- ~ Equations The k- ~ mode} used here is developed by Launder and Spaiding 0. The mode! equations already have a time derivative term and can be written in a form similar to the N-S equations ( 1). ark + Bf Jk l+9k 1) + Bt Jk 2+9k 2) at ox By (3)

where qk t6 ~ ~ fk} ~6 ) ~ Jk2 = EVEN 1 ski R e ok Ink 0~ 1 SE ~ 9k2=R 8z e 1 P- cRe S ~ Re C1 kP - C2 k Re and ok ilk By ~ By . ~ lark (1~ + fitly) ~ Ice ~ (1] + lastly) where k is turbulent kinetic energy, ~ is turbulent dissipation, and ,u t is the eddy viscosity. P represents the production of kinetic energy and the following form of it is used here: P pt(Uy2+Vx2+2nyVx) Here fk~ and fk2 are convective terms and Ski and 9k2 are viscous diffusion terms. S is a source term added to the equations which moclels the production and dissipation of turbulent kinetic energy. The k- ~ model still employs the eddy viscosity/diffusivity concept as it relates eddy viscosity to the kinetic energy and dis- sipation by k2 ~ t-Cp-R e -This eddy viscosity is then used to create all effective viscosity (,u +,u`) which replaces ,u in the N-S equations ( 14. To implement the above turbulence mode! the following constants are specified as given in Ref. ~ 104: ~ k 1.0, ~ ~ = 1.3, Ci-1.44, c2 i.g2, and c~=o.os. Because the N-S and k- ~ equations are similar the same numerical technique has been used for both sets of equations. Solution Procedure The N-S and the k-e equations con- tain both first derivative convective terms and second derivative viscous terms. The viscous terms are numerically well-behaved terms and central di~erencing is used. An upwind differenced TVD scheme was used for differencing the convective part of the equations. This upwind differenced scheme gives third-order accuracy without any artificial dissipation terms being added to the equations. Details of how this discreti zation method is applied to the N-S equa tions for incompressible flows can be found in Gorski iS]. The equations are solved in an impli cit coupled manner using approximate fac torization. The implicit sides of the equa tions (the sides in which the values at the grid points are unknown) are discretized with a firsLorder accurate upwind scheme for the convective terms. This creates a iagonally (lominant system which requires the inversion of block tri-cliagonal matrices. The implicit sides of the equations are only firsLorder accurate but the final converged solution has the higher order of accuracy of the explicit sides of the equations (the sides in which the values at the grid points are known already). An important quality of any scheme is its convergence rate. The (liagonal domi nance of the present method allows large time steps to be used for fast convergence. A spatially varying time step was also implemented but not used in this case. The solution starts with intial esti mates for the kinetic energy and dissipation fields. Here, a calculation with the ; 4y Baldwin-Lomax t Il] turbulence mode} pro vides the estimates. The N-S equations and the k- ~ mode} equations are iterated in pseudo-time until convergence is obtained. The N-S equations are solved to the wall with proper no-sTip boundary conditions for all cases. With the Baldwin-Lomax tur bulence model, the Van D riest m~xing length mode! takes care of the near-wall region. The k- ~ model however needs approximation as the flow physics in the near-wall sublayer ( y+< 15) is not well represented by the standard k- ~ equations. This DINS code has a novel near-wall cal culation technique 112] in which the N-S equations are solved to the wall, but near wall empirical algebraic equations are used to calculate the kinetic energy and dissipa tion in this region. 635

RESULTS AND DISCUSSION The experimental data A] for two 2- dimensional lifting foils are used as bench- marks for the N-S analysis. Both foils have an a 0.8 meanline and NACA 16 Type IT thickness distribution. These foils are plot lied in Figs. 2.a and 3.b. The thick foil (also referred to as Ti) has acIditional thickness from the midchord to the trailing edge as compared to the thin foil (also referred to as TNO). The wedge angle at the trailing edge of the TNO foil is approximately 20 deg. That angle for the T! foil is about 4S deg which results in a bevel shape on the suction side. The geometrical details are carefully documented in Ref. ~ 2] . Precision of Validation Data and Accuracy of Calculations The data from Ref. [2] include: sur- face pressure distribution, wake mean velo- city profiles, and turbulence characteristics such as Reynolds stresses, and power spec- tral density. Surface pressures were meas- ured with a scanning valve system and a precision pressure transducer. Repeater! measurements of the streamwise and nor- mal velocity components yielded precision within two of the measured free-stream velocity at any position. An(l, the precision of the measured turbulence intensities and Reynolds shear stress was within No of the maximum measured values of a given wake profile. Wall shear stress measurement across the span, and hot-fiIm measurement across the wake indicated that the mean flow approximates a 2-dimensional flow field well. Since the acoustic measurements of Ref. ~ I~ were also performed with exactly the same foil models and setup, the models had to be located with the aft 1/3 foil sticking outside of the test tunnel. Later pressure measurements in free 2-D jet agrees! with the earlier measurements at most positions. The geometrical angle of attack for both foils was set at 0.68 deg. D ue to the effect of the 2-D jet configuration on the foils, the corresponcI- ing free-field angles of attack were calcu- lated in Ref. t2] to be -1.01 cleg for foil TNO, and -~.54 deg for T1 foil. To calcu- late these angles, iterations were performed on the free-field lift coefficient using the free 2-D jet correction formula of Rae and Pope t13] and the boundary layer program of Cebeci et al. t 14~ . All experimental measurements were reported without any "correction". The above free-field angles of attack were used only for analytical calcula- tions to-compare with the data. For the N-S calculations, several grid sizes were used to establish computational accuracy. First of all, the boundaries were established by preliminary calculations to be at least 9 to t0 chord lengths upstream and normal to the foil so that free-stream con- dition applies. And the downstream boun- dary was set at 10 chords from the trailing edge to assume negligible streamwise gra- dients there. With the wake streamwise grid fixed at 40 points;, different grid sizes were use(l: 121x40, 121x60, ISIx60, 211x60, and 241x60. Convergence of the calculation with respect to the grid was established when the foil surface pressures changes within 0.2~o of the free-stream pressure. Validation Results The bench-mark cases were simulated as free-fiel(1 2-D flows with the following conditions: - Reynolds number 2.25x io6 - The angles of attack use(1 were the same as the "corrected angles" use(1 for boun- dary layer calculations in Ref. t2] (-~.01 deg for foil TNO, en cl -~.54 deg for foil TI). - A 3-zone C-gricl was used with 241 points; around the bo(ly and 60 points nor- mal to the body. The upstream boundary is 14 chord lengths from the leading edge. Top and bottom boundaries are 10 chords from the bo(ly. The downstream boundary is 10 chords from the trailing edge. - The first grill point normal to body locatecl at the trailing edge region is about i.4xi0- 5 chord away which translates to y+<s . - The first axial grid point in the wake is 1.5xio-4 chor(1 from the trailing edge which is about the same as the value used in Ref. t15] to resolve the streamwise gradient of mean velocity near the trailing edge. Figs. 2.a and 2.b show the computed anti measured pressure (distributions for the TNO foil, and the T! foil, respectively. The 636

calculated pressure distributions are in rea- sonable agreement with the experimental data of Ref. t2], more so for the thin trailing edge foil than for the thick one. Note that the predicted loading for foil T! is higher than the data (see Fig. 2.b), espe- cially in the aft region. Therefore, the 2-D jet correction to the angles of attack may not be adequate. Several different angles of attack were tried for both foils and the resulting pressure distributions were not any better. One of the reasons for the discrepancy between the calculations and the data may be clue to the wall effect. The foil chord length is 0.9144 m (3 fig, and the walls are about 2.667 chords away from the foil. And according to Rae Knot Pope t13], the walI/chorc} ratio should be at least 4 so that the measured lift coefficient is negligi- bly different from the free-field value. How the 2-D jet arrangement affects the foil surface pressure distribution is unknown. Also, recall in the previous section that the foils were sebup to be 2/3 in the tunnel test section and 1/3 outside. It is probable that this arrangement changes the pressure field of the foils more than the usual free 2-D jet arrangement. No rigorous reason can be found at this time to explain the above discrepancy. Other than the calcu- latecI pressure, the calculated flow separa- tion location for the thick foil agrees well with the experimental value, which is around 96~% chord on the suction side. Figs. 3.a and 3.b show good match between the computed and measured velo- city vectors for foil TN0, and foil TI, respectively. This match is relatively better than that of the foil surface pressure distri- bution. A possible explanation is that the surface pressure is more sensitive than the boundary layer flow, and the presumed wall effect is not significant for the boundary layer development. These velocity vectors are in the near-wake region, from To to 10~o chord length downstream of the trail- ing edge. The wake deficit for the thick foil is larger than that for the thin foil. The thick foil also shows larger normal velocity component than the thin foil, which is clue to the flow separation on the suction side of the thick foil. This flow separation, even though very mild, results in a recirculating region which the N-S code does not calcu late very well as seen in Fig. 3.b for the x/C=1.02 station. A possible reason is that in the turbulence models local isotropy is assumed, i.e. all three velocity components contribute equally to the turbulent kinetic energy. This assumption is not met when there is flow separation. Further down- stream in the wake, however, the calcu- late(1 velocity vectors agree well with the data. The N-S calculations of the turbulent kinetic energy k also match the data rea- sonably well. Figs. 4.a and 4.b show the calculated and measured k for the TN0 foil, and the T! foil, respectively. The data are actually approximated because the z- component was not measured en c! only the x- and y- components were obtained from Huang et al. in Ref. [2~. According to the boundary layer data of Klebanoff t16], the z-component can be assumed to be approx- imately equal to the average of the x- component and the y-component. For the TN0 foil, the N-S calculation tends to over-predict k on the suction side con- sistently for all three wake stations. The pressure side, which looks almost flat, has better agreement. Therefore, this observa- tion could signal that the turbulence mo(lels do not work well with a highly curved wall or large adverse pressure graclient. For the thick T1 foil, the miTcl flow separation on the suction sicle causes large over- prediction of the (lata for the wake station closest to the trailing edge, x/C 1.02. Further downstream in the wake, however, the calculate(1 k agrees relatively better with the data. As discussed earlier in the previ- ous paragraph for the velocity profiles, this observation could mean that the turbulence models do not work well even for mild flow separation. Also, this observation illustrates the elliptic nature of the N-S simulation, i.e. errors upstream do not neccessarily pro- pagate (lownstream anti increase as seen in the boundary layer simulation in Ref. [24. The N-S calculations of the Reynolds shear stress uv also match the data reason- ably well. Figs. 5.a and 5.b show the caTcu- late(1 and measured uv for foil TNO, and foil T1, respectively. The magnitude of uv is slightly over-predicted on the suction side for the TN0 foil. But the distribution shape is well predicte(1 for both foils, i.e. the cal 637

culated locations of the two extreme of uv agree well with the data. This could be because the modelled uv term is propor- tional to the local velocity gradient which is predicted very well. Again, we observe that the pressure side is predicted better than the suction sicle because it is almost flat. Also, the data for foil TI in the closest wake station x/C-1.02 are over-predicted due to the mild flow separation on the suc- tion sicle. Both of these observations corre- late with the previous ones for the velocity vectors and the turbulent kinetic energy. Application of N-S Analysis to Section Design With this successful validation of the N-S analysis, we can have confidence in using such a too! to develop new section shapes. Here, the design goal is to maxim- ize the thickness of the section without too much flow separation. The baseline sec- tion, from an existing design, has an a-0.S meanline an] a NACA 16 thickness distri- bution with thickness of 17.16 To chord en cl camber of 4.79 ~ chord. The design lift coefficient is approximately 0.68. For this study, the chosen Reynolds number is 5xio6 to match the conditions for 1/4-scale tests of naval propellers. The new section, shown in Fig. 6, is initially designed with the conformal mapping technique of Eppler-Somers t 17~ . The (resign approach is: I) move the minimum pressure on the suction side further upstream, 2) start recovering the pressure with a steep gra- dient because the boundary layer is still strong after the minimum pressure point, and 3) decrease the adverse pressure gra- dient as the trailing edge is approached to avoid flow separation. After the initial design, a thin section is produced which has the three features stated above. As the desired pressure distribution is not input to this Eppler-Somers code, a final design is not easily obtained at this step. Also, the conformal mapping technique is based on the potential flow model and therefore can not account for the thickness effect accu- rately. The N-S analysis is used iteratively to obtain the final design. Two parameters are used in the iteration with the N-S analysis: thickness, and angle of attack. For simplicity, thin airfoil theory is used to cal- culate the "camber-versus-lif~coefficient" behavior of the new section. The criterion for the final design is "no-separation" in the ~ 4 (leg around the design angle of attack with the thickness as high as possi- ble. The ~ 4 deg range is normally the fluctuation of angle of attack that a pro- peller section sees in straighLahea`;1 opera- tion. After some iterations with the N-S analysis, the final design is produced, with the pressure distribution at design angle of attack (2.5 deg) shown in Fig. 6. The lift coefficient from this pressure distribution is approximately 0.6S, the same as the base- line. The desirable characteristics for the boundary layer development are presented in this pressure distribution. Maximum suc- tion peak is around! 5097O chord on the suc- tion side; steep pressure recovery follows immediately, then the gradient becomes milder to minimize flow separation as the trailing edge is approached. Also, the pres- sure si(le distribution is rather flat over most of the surface; this should reduce the turbulent kinetic energy. The velocity vec- tor plot in Fig. 7 shows attached flow on both the pressure side and suction side at design angle of attack. This attached flow field of the new section produces lower tur- bulent kinetic energy as seen in Fig. 8.a, and lower Reynolds shear stress as seen in Fig. S.b when compared to the baseline sec- tion. This trend is more pronounced for the suction side than the pressure side. At this design lift for the baseline foil, the N-S analysis indicates some flow separation on the suction side which accounts for the high turbulence activity. And the pressure side of the new section has low turbulence activity because the pressure distribution there is almost flat over the entire surface. An undesirable feature of this new section is the thin trailing edge. This could prove harmful when structural analysis is per- formed even though care is taken during the designing to ensure minimum loading in that region. More details of the design process, and the comparison between the new section and the baseline can be found in Ref. t 181 . CONCLUSIONS In this paper, a N-S analysis is per- formed on the turbulent boundary layer and wake flows over lifting surfaces. This 638

analysis is performed as bench-mark calcu- lations for 2 airfoils at high Reynolds number for which turbulent flow data are available. Overall agreement between data and calculations is reasonably good. There is a better match of the mean velocity than the turbulence stresses. A possible reason for this is the inability of the turbulence mode! to simulate accurately flows with strong adverse pressure gradient or flow separation. Since the normal stress data from Refs. t2,14] show that the streamwise turbulence intensity uu is significantly larger (up to a factor of 2) than the transverse component vv, assumptions of local iso- tropy should be reconsidered. A new 2-dimensional airfoil section is developed by combining a conformal maw ping technique with an iterative N-S analysis. Results show that the new section has better boundary layer characteristics, for the same lift coefficient, than the base- line. Since the design goal is to maximize thickness with minimum flow separation, this new section is not recommended for other applications in which high thickness is not needed. This particular new section will certainly have poor cavitation perfor- mance. Nevertheless, this paper illustrates that N-S analysis is very useful in guiding 2-D section design. The N-S analysis, how- ever, can only give insight about the mag- nitude and the spatial distribution of the mean flow, and Reynolds stresses. The spectral behavior is entirely unknown. Until better turbulence models, numerical tech- niques, and computers become more easily accessible, the N-S analysis should only be used for final design fine-tuning or off- design predictions as done in this case. From the results, further work is recom- mended to: I) simulate the wall in the N-S calculation of the same 2-dimensional foils to establish the significance of the wall effect; 2) develop a new section with thicker trailing edge; 3) develop a series of new sections with different locations of the minimum pressure point on the suction side and experimentally evaluate them in the same manner as in Refs. tI,2~; and 4) concentrate on the development of tur- bulence models that can calculate more accurately turbulent flows with strong adverse pressure gradient, and even separa- tion. AC~NOWLE:DGMEN I The authors would like to thank Drs. Tommy Huang, Pat Purtell, and Yu-Tai Lee (DTRC) for making available the vali- dation dicta, and D r. Rod Coleman (DTRC) for the mesh generation code. The guidance and support of Dr. Frank Peter- son (DTRC) for the New Section work is much appreciated. And the New Section work is supported by the Office of Naval Technology under work unit number 1- 1506-060-34 for FY-90. REVERENCES I. Gershfeld, I., W.K. Blake, C.W. Kins- ley, " Trailing Edge Flows and Aero- dynamic Sound, " ATAA, ASME, ASCE, SIAM, APS Ist National Fluid Dynamics Congress, Cincinnati, Ohio, paper S8-3826- CP, July 1988. 2. Huang, T.T., I,.P. Purtell and Y.T. Lee, " Turbulence Characteristics of Trailing- Edge Flows on Thick and Thin Hydro- foils, " presented at the 4th Symposium of Numerical and Physical Aspects of Aero- ynamics Flows, CSU Long Beach, CA, Jan 16-19, 1989. 3. Sung, C.-H., "An Explicit Runge-Kutta Method for 3-D Turbulent Incompressible Flows, " DTNSRDC/SHD Report 1244-01, July 1987. 4. Harten, A., "High Resolution Schemes for Hyperbolic Conservation I`aws, " J. Comp. Phys., Vol. 49, pp. 357-393, 1983. 5. Tennekes, H. anti I,umIey, J.~. 1972. A First Course in Turbulence. MIT Press, Cambridge, MA. 6. Bernard, P. S. and Berger B.S., "A Method for Computing Three-Dimensional Turbulent Flows," SIAM J. Appl. Math., Vol. 42, 1982, pp. 453-470. 7. Coleman, R.M., "Tnmesh: An lnterac- tive Program for Numerical Grid Genera- tion, " D TNSRD C-85/054, 1985. S. Gorski, Jot, "Solutions of the Incompressible Navier-Stokes Equations Using an Upwind D ifferenced TVD Scheme," lith Int. Conf. on Num. 639

Methods in Fluid D ynamics, Williamsburg, Va., Ju! 1988. 9. Chorin, A.~. "A Numerical Method for Solving Incompressible Viscous Flow Prob- lems," I. Comp. Phys., Vol. 2, pp. 12-26, 1967. 10. Launder, B.E. and Spalding, D.B., "The Numerical Computation of Turbulent Flows, " Computer Methods in Applied Mechanics and Engineering, Vol. 269-89, 1974. 11. Baldwin, B.S. and Lomax, H., "Thin Layer Approximation and Algebraic Mode! for Separated Turbulent Flows, " AlAA 16th Aerospace Sciences Meeting, Hunts- ville, Al., Jan. 1978. 12. Gorski, J.~., "A New Near-Wall For- mulation for the k-e Equations of Tur- buTence, " ALLA Paper 86-0556, 1986. 14. Cebeci, T., R.W. Clark, K.C. Chang, N.D. Halsey and K. Lee, "Airfoils with Separation and the Resulting Wakes," Jou. Fluid Mech., Vol. 163, 1986, pp.323-47. 15. Mehta, U., K.C. Chang, and T. Cebeci, "A Comparison of Interactive Boundary- Layer and Thin Layer Navier-Stokes Pro- ce(lures, " Num. and Phys. Aspects of Aero. Flows T1:T, Chapter 11, p. 198, 1985. 3, pp. 16. Klebanoff, P., " Characteristics of Turbulence in a Bounciary-Layer with Zero Pressure Gradient, " NACA TN 3178, 1954. 17. Eppler, R. and Somers, D.M., allow Speed Airfoil D esign and Analysis, " Advanced Technology Airfoil Research Conference, Langley Research Center, NASA, Hamton, Va., Mar 1978. 18. Nguyen, P.N., "A Design Method for Boun(lary-l,ayer Control of 2-D Lifting Sur 13. Rae, W.H., Jr., and A. Pope, Low- faces, " DTRC/SHD 1262-04, 1990 (in Speed Wind Tunnel Testing, John Wiley & review). Sons, New York, 1984, p.361. 640

0.6 0~4 0.2 0.0 -0.2 -0.4 -0.8 [~ _ I r '= _~ -0.1 0.1 0.3 0.5 0.7 X/C Figure 1. C-type grid with 3-zone structure for a lifting surface 0.6 l n 0.4 0.2 0~0 -0.2 0 -Cp data (Re=2.25E6, - 1.01 deg.) -- - - -Cp calculation by DTNS section geometry A\ ,°'\6 opt\ P~ =~\ I I I I ~ 0.0 0.2 0.4 0.6 X/C 0.8 1.0 Figure 2.a) Pressure distribution for the thin section (data from Ref. 2) 0 -Cp data (Re=2.25E6, - 1.54 deg.) -Cp calculation by DINS section geometry 0.6 0.4 0.2 0.0 -0.2 o ,p _ of-o_ ~ o >_ ~-o_ ~ ~ o'er 9 I I I I I 0.0 0.2 0.4 0.6 0.8 1.0 X/C \ Figure 2.b) Pressure distribution for the thick section (data from Ref. 2) 64~

data from Ref. 2 calculation data from Ref. 2 calculation 0.02 0.00 -0.02 -0.04 -0.06 _ ~ _ _ , _ r~ __ _ __.~, ~ _ ~ ~_ _ a. _ - _ _ _ -_~ _ e ~ _ . - _ _ _ _ _ __..~. 0.04 0.02 0.00 1.00 1.04 1.08 1 .12 1 .16 x/C Figure 3.a) Velocity vector data and DTNS calculations for the thin section o k data at x/C=1.02 o x/C= 1 .04 x/C=1.10 -- -- calculation by DTNS I S ; o~ ~ ~Oo`' [o~ ta&~) ,,pa~ -; ~ I ~ I ~ I 1 0 1 0 1 100 k/U 2 0.05 , ~ _ -0.05 0.03 0.01 -0.01 -0.03 Figure 4.a) Profiles of turbulent kinetic energy for the thin section (data from Ref. 2) 0.05 0.03 0.01 -0.01 -0.03 -0.05 0 5 I ~ I ~ I ~ I 8 ~ ~ t} ~ ~ ~ -' ~'} ,_' , ~ , ~ , 0 5 0 5 0 5 1000 uv/U.2 -0.02 -0.04 1.00 1.04 1.08 x/C 1.12 1.16 Figure 3.b) Velocity vector data arid DTNS calculations for the thick section o k data at x/C=1.02 o 0.05 0.03 0.0'1 -0.01 -0.03 -0.05 o x/C= 1 .05 x/C=l .10 -- -- calculation by DTNS 1 ~ 1 ~ 1 ]\ ~ ~ ~_ ~1 i. I ~ I ~ I 1 0 1 0 1 100 k/U 2 Figure 4.b) Profiles of turbulent kinetic energy for the thick section (data from ReT. 2) 0.05 0.03 0.01 -0.01 -0.03 -0.05 ' ~ ' ~ ~ t ~ b ~ ~ ~W o ~ ~ g ~ 8 ~ . 0 5 0 5 0 5 0 1000 uv/UO2 s ~' Figure 5.a) Profiles of Reynolds shear stress forFigure 5.b) Profiles of ReYnolds shear stress for the thin section (legend in Fig. 4.a)the thick section (legend in Fig. 4.b) 642

DTNS calc (Re=SE6,2.5 deg.) section geometry 0.e my, 0.4 l ._ 0.2 o I I ~ ,' \ r -0.2 ~I I I I 0.0 0.2 0.4 0.6 0.8 1.0 X/C Figure 6. Pressure distribution for the new section 0.07 0.500 0.250 0.05 0.03 0.01 -0.01 o-new section calc, x/C=1.02 a-x/C= 1 .05 x/C= 1 .09 baseline section calc I I I ~ I >~ -0.03 .o 1.5 0.0 1.5 0.0 1.5 a 100 k/U 2 Figure 8.a) Profiles of turbulent kinetic energy for the baseline and the new sectRon 643 0.000 -0.250 _- = = ~ _ o.oo 0.25 0.50 0.75 1.00 x/C Figure 7. \/elocl~ vector calculations for the new section 0.07 0.05 0.03 0.01 -0.01 -0.03 0 new section calc, x/C=1.02 0 x/C= 1 .05 &- x/C= 1 .09 -- -- baseline section calc it' 5 0 5 0 5 0 5 1000 w/U 2 Figure 8.b) Profiles of Reynolds shear stress for the baseline and the new section

DISCUSSION Wolfgang Faller Sulzer Escher Wyss, Germany For your comparison between experiment and 2-D N-S calculation, the computational domain used in infinite. Would a closer modelling of the actual experimental configuration improve the correlation, e.g., Cp distribution and B.L. development? AUTHORS' REPLY Modelling of the foil and wall configuration would possibly lead to better match of the bench-mark calculations to the hydrodynamics data. Plan is underway to implement a grid structure necessary for this study. DISCUSSION All H. Nayfeh Virginia Polytechnic Institute and State University, USA 1. How sensitive is the pressure distribution of the new section to variation in the angle of attack? 2. How sensitive is the designed shape to the turbulence model? 3. How much is the drag reduced by the new section? AUTHORS' REPLY 1. The new section does not have flow separation in the + 4 deg. range around the design angle of attack. 2. Preliminary study indicates that the flow solution converges to approximately the same pressure distribution for both the Baldwin-Lomax and the k-e turbulence models. 3. Drag is not computed in this study as the current project focuses on the turbulence activity. The drag of the new section is included in the plan for future investigation. DISCUSSION Philippe Genoux Bassin d'Essais des Cartnes, France 1. Is your model able to take in account turbulent levels of the incoming flow? 2. What would become of the turbulent energy levels when the profile is placed in incoming flow? AUTHORS' REPLY 1. The code currently does not have a model for the free-stream turbulence. 2. When placed in flow with free-stream turbulence, the turbulent kinetic energy level of the new section would likely increase (as compared to incoming flow without turbulence). The redistribution and the magnitude of the increase in turbulence energy would need to be calculated with a proper turbulence model. DISCUSSION Hyoung-Tae Kim The University of Iowa, USA (Korea) First, I want know is there any reason not to show the distribution of the shear stress on the surface of the foil section? Secondly, I want to point out that the Low turbulence activity simulated in the computation doesn't necessarily mean the new foil section has a lower drag than the baseline section. AUTHORS' REPLY The study focuses on the turbulence activity in the near-wake region of lifting surfaces. The drag itself is, however, included in the plan for future investigation. The authors agree with the discusser on his second point. No claim is made about drag reduction in the paper. The resistance of the new section to flow separation only provides for higher lift at the same angle of attack as the baseline section. And the low turbulence activity provides for lower fluctuating pressure on the trailing edge of the section. 644