PREDICTION OF LAMINAR/TURBULENT TRANSITION IN AIRFOIL FLOWS
by
Jeppe Johansen and Jens Norkaer Sorensen
DTU FLUID MECHANICS H
ENERGY ENGINEERING TECHNICAL UNIVERSITY OF DENMARK / DANISH CENTER FOR APPLIED MATHEMATICS AND MECHANICS
Scientific Council
Poul Andersen Dept, of Naval Architecture and Offshore Engineering Martin P. Bends0e Dept, of Mathematics Ove Ditlevsen Dept, of Structural Engineering and Materials Ivar G. Jonsson Dept, of Hydrodynamics and Water Resources Wolfhard Kliem Dept, of Mathematics Steen Krenk Dept, of Structural Engineering and Materials P. Scheel Larsen Dept, of Energy Engineering Frithiof I. Niordson Dept, of Solid Mechanics Pauli Pedersen Dept, of Solid Mechanics P. Temdrup Pedersen Dept, of NavalArchitecture and Offshore Engineering- Dan Rosbjerg Dept. of Hydrodynamics and Water Resources Jens Nprkser Sprensen Dept, of Energy Engineering P. Grove Thomsen Dept, of Mathematical Modelling Hans True Dept, of Mathematical Modelling Viggo Tvergaard Dept, of Solid Mechanics
Secretary
Pauli Pedersen, Docent, Dr.techn. Department of Solid Mechanics, Building 404 Technical University of Denmark DK-2800 Lyngby, Denmark DISCLAIMER
Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. Prediction of Laminar/Turbulent Transition in Airfoil Flows
Jeppe Johansen* and Jens N. Sprensen* * Ris0 National Laboratory, Denmark and * Department of Energy Engineering, Technical University of Denmark
Presented as Paper 98-0702 at the AIAA 36th Aerospace Sciences Meeting Sc Exhibit, Reno, NV, January 12*15, 1998
tphD student, Wind Energy and Atmospheric Physics Department, Rise National Laboratory, DK-4000 RoskUde, Denmark * Associate Professor, Department of Energy Engineering, Technical University of Denmark, DK- 2800 Lyngby, Denmark
1 Abstract
The prediction of the location of transition is important for low Reynolds number airfoil flows. The laminar/turbulent properties of the flow field have an important influence on skin friction and separation and therefore on lift and drag characteristics. In the present study transition is predicted using the more general transition prediction method, the e" model, and compared to the simple empirical Michel criterion. The flow is computed using an incompressible Navier-Stokes solver and the turbulent region is computed with the two-equation k — u, SST model. The en method is based on lin ear stability analysis employing the Orr-Sommerfeld equation to determine the growth of spatially developing waves. In order not to compute growth rates for each velocity profile, a database on stability has been established. The problem of determining boundary layer properties usinga Navier-Stokes solver, is solved using a two-equation integral formulation, which is solved using a direct/inverse Newton-Raphson method. The test cases under in vestigation are incompressible transitional flows around airfoils at low and moderate Reynolds numbers, at fixed angles of attack, varying from attached flow through light stall. At high Reynolds numbers no large difference is ob served between the two transition models. But for lower Reynolds numbers, the en method shows better agreement with experiments. Furthermore it has shown to be more stable. It is therefore preferable to the empirical transition model. Introduction
Computation of flows over airfoils is a challenging problem due to the various complex phenomena connected with the occurrence of separation bubbles and the onset of turbulence. In many engineering applications involving weak stream wise pressure gradients and small curvature effects, turbulent quantities can be well predicted assuming fully turbulent flow. In the case of low Reynolds number airfoil flows (Re < o(106)), proper modeling of the transition point is crucial for predicting leading edge separation. The transition prediction algorithm must be reliable since the transition point may affect the termination of a transitional separation bubble and hence determine bubble size and associated losses. This again has a strong influence on airfoil characteristics, with drag being the most affected. A popular transition prediction model is the empirical criterion by Michel10. As shown by e g. Ekaterinaris et al. 6 and Mehta et al. 8 , this model gives
2 fairly good results for many airfoil flows. In the present study the more general e” method, proposed originally by Smith 14 and van Ingen 1S, is com pared to the Michel criterion. The e” method is based on linear stability analysis using the Orr-Sommerfeld equation to determine the growth of spa tially developing waves. There have been severalattempts to apply simplified versions of the en method in combination with viscous-inviscid interaction procedures (e.g, Drela and Giles 5 and Cebeci 3). In the present work a database on stability, with integral boundary layer parameters as input, has been established. This database avoids the need for computing growth rates for each velocity profile. Furthermore, in order not to determine boundary layer parameters such as displacement thickness, momentum thickness, and boundary layer edge velocity usinga Navier-Stokes (N-S) solver, the N-S solver is combined with the solution of the integral boundary layer equations. Methods Flow Solver The results determined in the present study are computed using EllipSys2D, a general purpose 2-D incompressible Navier-Stokessolver. It is developedby Michelsen and Sarensen 11,12,16, and is a multiblock finite volume discretiza tion of the Reynolds averaged Navier-Stokes equations in general curvilinear coordinates. The code uses primitive variables (u,v, and p). The pres sure/velocity coupling is obtained with the SIMPLE method by Patankar 13 for steady state calculations. Solution of the momentum equations is ob tained using a second order upwind scheme. The steady state calculations are accelerated by the use of local time stepping and a three level grid sequence. The turbulence model employed in the present work is the two-equation k—ui Shear Stress Transport (SST) turbulence model by Menter 9, who obtained good predictions in flows with adverse pressure gradients.
Transition Prediction Models In the present study, two different transition prediction models are used. These are the empirical one-step model of Michel 10, and a semi-empirical e" model based on linear stability in the form of a database, as suggested by Stock and Degenhart 17 . The Michel criterion is a simple model based on experimental data which correlates local values of momentum thickness with position of the transition point. It simply states that transition onset location takes place where Rea.tr = 2.9 Re°x% (1)
3 where Re$,tr is the Reynolds number based on momentum thickness, and ReXfr is the Reynolds number, based on the distance measured from the stagnation point. The second model is based on linear stability theory and is referred to as the en model by Smith 14 and van Ingen ls. Linear stability theory suggests that the unperturbed steady and parallel mean flow is superimposed with a time-dependent sinusoidal perturbation - the Tollmien-Schlichting waves. This results in the well known Orr-Sommerfeld equation, which is a 4th order linear eigenvalue problem in
4 Figure 1: Diagrammatic sketch of the neutral curve, the amplification factor o,-, and the integrated N factor for a boundary layer with a constant value of shape factor H. From 17
5 In the top graph of figure 1 the neutral curve obtained from the Orr-Sommerfeld equation is shown (a, = 0). F is the reduced frequency, Res- is the Reynolds number based on displacement thickness, and o, is the amplification factor of the perturbations in spatial stability theory. Sweeping through the neutral curve for different values of F, Fi, the N factor can be determined for each frequency using
(2) resulting in a number of N curves as the one depicted in the lower graph of figure 1. The envelope of all N curves results in a nmol curve, which can be used to determine the point of transition i.e. where the nmax curve reaches the value of 9. Having computed a velocity field using the N-S solver, it is possible to de termine laminar boundary layer integral parameters such as displacement thickness, 5*, momentum thickness, 6, and kinetic energy thickness, S3. These values are computed by an additional integral boundary layer algorithm and used as input to the database. The stability characteristics are subsequently evaluated by interpolation. The database was originally implemented by us ing Falkner-Skan velocity profiles and has recently been extended to include separated velocity profiles described by hyperbolic tangent functions.
Transition region
The extension of the transition region is obtained by an empirical model, suggested by Chen and Thyson 4. This is a conceptually simple model that scales the eddy viscosity by an intermittency function, varying from zero in the laminar region and progressively increases in the transitional region until it reaches unity in the fully turbulent region. The intermittency function, 7(r, is given by
7,r(x) = 1 - exp (3 )
The modeling constant, G-,tr was originally suggested to be 1200 for high Reynolds number flows. In order to take into account separation, especially for low Reynolds number flows, it was modified by Cebeci 3 to take the form.
Gy,r = 213[log(ReItr)-4.732]/3 (4)
The range at which this modification is valid is Rec = 2.4 x 105 to 2 x 10s.
6 Integral Boundary Layer Formulation
The input parameters for the database are the laminar boundary layer pa rameters, <$", 9, and 63 together with the velocity at the boundary layer edge, ue, and the Reynolds number based on chord length, Rec. This results in some difficulties. Firstly, the determination of boundary layer parameters using the N-S solver is not accurate, since the boundary layer thickness is not a trivial parameter to determine. Various attempts include definitions of J based on vorticity, see e.g. 16. Secondly, the turbulence starting from the transition point influences the flow upstream due to the elliptic nature of the flow. This results in boundary layer parameters differing from their fully laminar value resulting in erroneous interpolation in the database. An alternative procedure is thus required for calculating these parameters. The procedure chosen in the present study solves the integral boundary layer equations using a N-S solver as boundary condition for freestream velocity. Since the transition procedure should be able to take separation into account the necessity of a N-S solver is clear. The boundary formulation is a two equation integral model based on dissipation closure. The two equations are the von Karman integral relation given by:
6 due £i (5) g + C + ue d£ 2 ’ where $ is the momentum thickness, H is the shape factor, ut is the velocity at the boundary layer edge, £ is the streamwise coordinate, and Cj is the skin friction coefficient. The second equation is a combination of eq (5) and the kinetic energy thickness equation, and is given by:
where H* is the kinetic energy shape parameter and Co is the dissipation coefficient. For laminar flow, the two ordinary first order differential equa tions can be solved with the following closure relationships for H", C/, and Co respectively.
ff* = 1.515 + 0.076&^, H < 4 H' = 1.515 + 0.040*2^, H >4 (7)
= -0.067 + 0.01977% —^,# < 7.4 (8) l [n — 1)
7 2Co fieg 0.207 + 0.00205(4 - R)5-5, H < 4 R* 2 Cp Reg 0.207-0.003 (1J£#l4)t),g>4 (9) H* This model has successfully been used by Drela and Giles and for further details is referred to 5. To solve the two equations, (5) and (6), a third relation is necessary. By using the N-S solver to compute the pressure at the surface and assuming no pressure variation across the boundary layer (pwait = pe), can be de termined from Uoo using the Bernoulli equation along a streamline. The two equations are then solved by a Newton-Raphson method. Close to stagnation point, where the skin friction varies dramatically and a small variation in skin friction factor causes a large variation in edge velocity, the equations are solved using a direct procedure (ue given, solving for 9 and H). When approaching separation, the direct procedure becomes ill- conditioned because a single edge velocity corresponds to two different skin friction factors. By computing C/ using the N-S solver, H can be computed using the closure relation, eq.(8), and eqs. (5) and (6), can be solved inversely with 9 and ue as variables. Results and Discussion
Two airfoils were tested. The thin NACA0012 airfoil at fie = 3 x 106 was chosen as a high Reynolds number case, and the 19% thick FX 66-S-196 VI Wortmann airfoil at fie = 1.5 x 10s was chosen where a larger transitional influence should be expected. The computations were carried out assuming steady state conditions in order to take advantage of the local time stepping procedure. For each airfoil a grid refinement study was carried out resulting in a 288 x 96 grid for both airfoils. The grid spacing to the first gridline off the surface is y+ % 1, Figures 2 to 5 shows the computational results for the NACA0012 airfoil. As seen from figure 3 the transition point has no large effect on the pressure distribution, except at the leading edge separation bubble at a = 12°. This indicates that the computed lift coefficient is hardly influenced. The skin friction shown in figures 2 and 4 shows a clear difference between fully turbulent and transitional computation. This underlines the importance of transition point prediction when computing drag characteristics. Looking at the skin friction for a = 12°, figure 4, it is seen that the linear stability model predicts transition well into the laminar separation bubble, which corresponds well with experimental data, where the separation occurs in the laminar region and reattaches in the turbulent region.
8 0.012 Fully turbulent comp. ----- Transitional corap. en----- Comp. Xq, upper side Comp. x m lower side
Cf 0.004
0.002
-0.004 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x/c
Figure 2: Skin friction for fully turbulent flow compared with transitional flow and computed transition point location, NACA0012, a = 3°, Re = 3 x 10®
Fully turbulent comp. ----- Transitional comp. en ----- Comp. x m upper side
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 x/c
Figure 3: Pressure distribution for fully turbulent flow compared with tran sitional flow and computed transition point location, NACA0012, a = 12° Re = 3 x 10®
9 Fully turbulent comp. ----- Transitional comp., en ----- Comp. X*. upper side
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 01 x/c
Figure 4: Skin friction for fully turbulent flow compared with transitional flow and computed transition point location, NACA0012, a = 12° Re = 3 x 106
Fully turb.
Michel Exp. data
Figure 5: Drag curve for fully turbulent flow compared with transitional flow and experimental data 7, NACA0012, Re = 3 x 10s
Figure 5 shows the drag curve for the NACA0012 airfoil, computed with fully turbulent flow and using the two different transition prediction models
10 compared with experimental data 7 . A better prediction is obtained for the drag using the transition models. Again the importance of transition point prediction when computing drag characteristics is emphasized. There is no significant difference between the two transition models. In Table 1 computed transition point locations are compared with experimen tal data. For small angles of attack both models exhibits good agreement with experimental data. For higher angles of attack (a > 5°) the en model gives a better prediction of the location of the transition point. It should be stated here that the transition point predicted by the Michel criterion showed a more fluctuating behavior than that predicted by the en model, resulting in larger fluctuations in lift and drag coefficients. The transition point predicted using the e” model stabilized in just a few iterations and the converged solution was obtained faster.
Table 1: Transition point locations for the NACA0012 airfoil at Re = 3 x 106
a xtruv exp e" Michel exp en Michel 0 0.45 0.44 0.43 0.45 0.44 0.43 3 0.20 0.19 0.21 0.66 0.68 0.72 5 0.085 0.060 0.13 0.79 0.84 0.88 8 0.024 0.025 0.070 0.92 0.99 1.00 10 0.013 0.014 0.040 1.00 1.00 1.00 12 - 0.012 0.014 1.00 1.00 1.00
The lift and drag curves of the 19% thick FX 66-S-196 VI airfoil are shown in figure 6. The fully turbulent computation is obviously unacceptable; using the transition prediction models gives better results. Both models predict lift and drag well up to maximum lift, but at light stall the models predict transition before the experimentally determined value. At high angles of at tack the transition point approaches the leading edge, i.e. the lift approaches the fully turbulent computation. From the pressure distribution in figure 7, for a = 8°, it is seen that the transition point affects Cp globally and not locally as seen for the NACA0012 airfoil. The reason for this is that the Wortmann airfoil is a laminar airfoil, where transition takes place further downstream. As seen on figures 7 and 8, the transition on the suction side takes place at xtr = 0.45 which causes a large part of the flow around the airfoil being laminar, and therefor a large deviation from fully turbulent flow is obtained. The skin friction distribution in figure 8 shows a small separation where transition takes place. The large effect of transition causes the fully turbulent flow to under predict lift and over predict drag, figure 6.
11 Fully turb. e° Michel Exp. data
LOO -
Figure 6: Lift and drag curves for fully turbulent flow compared with tran sitional flow and experimental data FX 66-S-196 VI, Re = 1.5 x 10s
Fully turbulent comp. ----- Transitional comp., e11-----
Figure 7: Pressure distribution for fully turbulent flow compared with transi tional flow and computed transition point location, FX 66-S-196 VI, a = 8°, Re = 1.5 x 106
12 0.02
-0.01 ------'------'------'------'------0 0.2 0.4 0.6 0.8 1 x/c
Figure 8: Skin friction for fully turbulent flow compared with transitional flow and computed transition point location, FX 66-S-196 VI, a = 8°, Re = 1.5 x 106
The computed transition points are shown in Table 2 and both models give comparable predictions. Again using the Michel criterion resulted in a fluc tuating transition point location leading to a longer convergencetime.
Table 2: Transition point locations for the FX 66-S-196 VI airfoil at Re = 1.5 x 10s ______a xtrnp exp en Michel exp e" Michel -4 - 0.50 0.56 - 0.27 0.25 0 0.53 0.46 0.50 0.50 0.46 0.50 4 0.50 0.45 0.46 0.54 0.50 0.57 8 0.46 0.43 0.40 0.60 0.54 0.73 9 0.45 0.35 0.32 0.62 0.55 0.76 10 0.27 0.17 0.18 0.66 0.57 0.93 11 - 0.11 0.12 - 0.61 1.00 12 - 0.05 0.09 - 0.64 1.00
13 Conclusions
This study has described the coupling of a simplified version of the en tran sition model together with a Navier-Stokes solver applied to low Reynolds number airfoil flows. A two-equation integral laminar boundary layer for mulation was solved using a coupled direct/inverse formulation in order to determine laminar integral boundary layer parameters. The inverse proce dure was chosen in order to obtain integral parameters well into the sepa rated region, and also to overcome the difficulty of defining the velocity at the edge of the boundary layer, uc. Comparisons with the simple empirical Michel criterion and with experimentally determined transition point loca tions were made. The results obtained in the present study indicate that proper transition point prediction is crucial, especially when considering the drag characteristics. The NACA0012 airfoil at Re = 3 x 106 has only a minor effect on the lift prediction while the drag characteristics are influenced to a greater extent. The FX 66-S-196 VI airfoil at Re = 1.5 x 10s shows clearly the importance of transition prediction for both lift and drag characteristics. No particular difference in integrated flow characteristics was obtained be tween the two models but a faster convergence was obtained using the e” model due to stable transition point location. This model is thus superior to the empirical model and is therefore preferable.
References
[1] Althaus D. and Wortmann F. X. Stuttgarter Profilkatalog, Technical Report, F. Wieweg, Braunschweig, 1981. [2] Arnal D. Boundary Layer Transition: Predictions based on Linear Theory. Technical Report, AGARD-R-793, AGARD, 1994. [3] Cebeci T. Essential Ingridients of a Method for Low Reynolds-Number Airfoils. AIAA Journal, 27 (12), 1988. [4] Chen K. K. and Thyson N. A. Extension of Emmons Spot Theory to Flows on Blunt Boddies. AIAA Journal, 9 (5), 1971. [5] Drela M., and Giles M. B. Viscous-Inviscid Analysis of Transonic and Low Reynolds Number Airfoils. AIAA Journal, 25 (10), 1986. [6] Ekaterinaris J. A., Chandrasekhara M. S., and Platzer M. F. Analysis of Low Reynolds Number Airfoil Flows. Journal of Aircraft, pp. 625-630, 1995.
14 [7] Gregory N. and O’Reilly C. L. Low-Speed Aerodynamic Characteristics of NACA0012 Airfoil Section, Including the effects of Upper-Surface Roughness Simulating Hoar Frost. Technical Report, NPL AERO Rept. 1308, 1970. [8] U. Mehta and K. C. Chang and T. Cebeci Relative Advantages of Thin- Layer Navier-Stokes and Interactive Boundary-Layer Procedures. Technical Memorandum, NASA, 1985. [9] Menter F. R. Zonal Two Equation k-o; Turbulence Models for Aerody namic Flows. AIAA-paper-932906, 1993. [10] Michel R. Etude de la Transition sur les Profiles d ’Aile;Etablissement d ’un Critere de Determination de Point de Transition et Calcul de la Trainee de Profile Incompressible. Technical Report, ONERA, Report 1/1578A, 1951. [11] Michelsen J. A. Basis3D - a Platform for Development of Multiblock PDE Solvers. Technical Report AFM 92-05, Technical University of Denmark, 1992. [12] Michelsen J. A. Block structured Multigrid solution of 2D and 3D elliptic PDE’s. Technical Report AFM 94-06, Technical University of Denmark, 1994. [13] Patankar S. V. "Numerical Heat Transfer and Fluid Flow." Hemisphere Publishing Corporation, 1980 [14] Smith A. M. O. Transition, Pressure Gradient, and Stability The ory. Proceedings of the IX International Congress of Applied Mechanics, 4, pp.234-244, 1956. [15] Spalart P. R. and Watmuff J. H. "Experimental and numerical study of a turbulent boundary layer with pressure gradient. ” J. Fluid Mechanics, Vol. 249, april, 1993. [16] Sprensen N. N. General Purpose Flow Solver Applied to Flow over Hills. Ris0-R- 827-(EN), Rise National Laboratory, Roskilde, Denmark, June 1995.
[17] Stock H. W. and Degenhart E. A simplified en method for transition prediction in two-dimensional, incompressible boundary layers. Z. Flugwiss., Weltraumforsch. 13, 1989. [18] Van Ingen J. L. A Suggested Semi-empirical Method for the Calculation of the Boundary-Layer Region. Technical Report, Rept. no VTH71, VTH74, Delft Holland, 1956.
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