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Prediction of Laminar/Turbulent Transition in Airfoil Flows

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Prediction of Laminar/Turbulent Transition in Airfoil Flows 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$,tr is the Reynolds number based on momentum thickness, and ReXfr is the Reynolds number, based on the distance measured from the stagnation point. The second model is based on linear stability theory and is referred to as the en model by Smith 14 and van Ingen ls. Linear stability theory suggests that the unperturbed steady and parallel mean flow is superimposed with a time-dependent sinusoidal perturbation - the Tollmien-Schlichting waves. This results in the well known Orr-Sommerfeld equation, which is a 4th order linear eigenvalue problem in <j>, where <j> is the amplitude of the perturbation. This equation determines whether spatially developing waves will be stable or unstable due to the amplification factor a,-, which is the imaginary part of the spatial wave number. For positive a; the waves are damped and for negative a, the waves are growing and the flow becomes unstable. In this way the point of instability can be determined. This is defined as where a,- = 0, i.e. neutral stability. The en model predicts turbulence when the amplitude of the most unstable frequency exceeds the initial unstable amplitude by a factor en. The n factor is empirically determined from several experimental data, and can vary from one flow situation to another. It is usually set at a value around 8-10. In the present work it is set at 9. For further details about the e” model see 2’14, and l8 . In the present work a database on stability, with integral boundary layer parameters as input, has been established, as suggested by Stock and De- genhart 17. The approach is based on the idea that a discrete set of results to the Orr- Sommerfeld equation is representative for all possible laminar velocity pro­ files and for all relevant disturbance frequencies. The database consists of stability data on laminar velocity profiles represented by laminar boundary layer parameters, i.e. displacement thickness, <5*, momentum thickness, 8, ki­ netic energy thickness, <$ 3, and velocityat the edge of the boundary layer, The velocity profiles included in the database varies from velocity profiles with strong favorable pressure gradient to separated velocity profiles. For each of these profiles a number of frequencies are investigated by solving the Orr-Sommerfeld equation, and the resulting stability data are stored in the database together with the corresponding boundary layer parameters. The relevant stability information can then be extracted by interpolation. In figure 1 the process of determining the n factor is depicted. 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).
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