Moving Grid Computations for Turbulent, Aeroelastic Flows

A computational procedure for performing aeroelastic computations for three-dimensional geometries is developed. Specifically, interaction between a rigid, yet deformable, structure and the surrounding flow field is studied. The flow is modeled using a well-validated Navier-Stokes solver and the structure is modeled by using beam finite elements. The fluid grid is a multiblock structured grid with moving grid capability based on master/slave concept and transfinite interpolation techniques. A well-validated κ−e turbulence model with wall function treatment for near wall region is used here to perform turbulent flow calculations. A fully implicit time marching scheme using the Newmark integration method is used for integrating the equations of motion of structure. One of the key aspects of this paper will be to discuss the impact of geometric conservation law (GCL) on computational accuracy and to propose two different methods to satisfy the discrete form of the GCL in order to accurately evaluate Jacobian values for moving meshes. We also discuss the impact of two different parameters associated with the moving mesh module, which determines the rigidity of mesh movement. We demonstrate the overall computational procedure on an AGARD 445.6 wing configuration for turbulent flows with different angles of attack by showing the time varying distribution of aerodynamic parameters. INTRODUCTION In many CFD applications, we come across situations when one or more boundaries move with respect to time. Aeroelasticity is the field of study wherein the interaction between aerodynamic forces and inertial forces are studied within elastic structural systems. This leads to a deformation or deflection of the structure leading to the famous moving boundary problem. Examples of these kinds of problems include airfoil oscillations, wing flutter, reciprocating engines, etc. Computationally, these movements can be large enough to necessitate solving the problem on a dynamic mesh, which moves or deforms in time. A Computational Aeroelasticity (CAE) model needs to be developed to handle such fluidstructure interaction problems involving moving boundaries. A CAE model involves three major components, they being, flow module, structure module and moving grid module. Several CAE models have been proposed over the years by combining different approaches for each of the above-mentioned modules depending on the application looked at. An excellent review of a few models can be found in Kamakoti et al. (2002). It has been proven over the years that GCL is a key component of CAE problems or more specifically, problems involving deforming grids. It can be related to volume conservation and it needs to be satisfied for problems involving deforming grids. It essentially takes care of certain geometric quantities associated with the deformed grid or the new grid. In the numerical perspective, it is called the discrete geometric conservation law (DGCL). DGCL is a law which states that the computation of the geometric quantities associated with a moving grid should be computed in such a way that, independent of the mesh movement, the numerical scheme used for integrating the flow equations must preserve a uniform flow field (Guillard and Farhat, 2000). This is in conjunction with the fact that preserving uniform field implies first order accuracy. Sufficient evidence exist showing that not satisfying the geometric conservation law leads to erroneous solutions or spurious oscillations in the solution (Shyy et al.1996, 2001; Lesoinne and Farhat, 1996; Guillard and Farhat, 2000; Farhat et al., 2001). It has also been proven that satisfying the DGCL improves the time-accuracy of computations on moving grids (Koobus and Farhat, 1999). Farhat et al. (2001) also showed that for ALE schemes, satisfying the DGCL leads to a necessary and sufficient condition for the numerical scheme to preserve non-linear stability on a fixed grid. However, there have been a few cases where satisfying or not satisfying the ___________________________________ * Graduate student, Dept. of Mechanical and Aerospace Engineering. Member AIAA. † Professor and Chairman, Dept. of Mechanical and Aerospace Engineering,. Fellow AIAA. Copyright © 2002 by Authors, Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. 33 AIAA Fluid Dynamics Conference Orlando, FL. 23-26 June, 2003 Paper Number: AIAA-2003-3719 33rd AIAA Fluid Dynamics Conference and Exhibit 23-26 June 2003, Orlando, Florida AIAA 2003-3719 Copyright © 2003 by Authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. 2 American Institute of Aeronautics and Astronautics GCL produced the same results (Morton et al., 1998) though majority of the cases calls for GCL implementation. It should be noted that since GCL arises due to the numerical procedures devised based on grid movement, its implications are expected to be scheme dependent. Alternative forms of the GCL have been implemented over the years to study its impact on solution accuracy. Thomas and Lombard (1979) implemented the GCL for density-based finite difference schemes on structured meshes by updating the value of the Jacobian at each time step. Shyy et al. (1996, 2001) implemented the GCL along the lines of Thomas and Lombard for pressurebased finite volume schemes by updating the Jacobian values after every time step using a first order backward Euler time-integration scheme. Lesoinne and Farhat (1996) developed a first order, time accurate scheme preserving the GCL using the density-based ALE finite volume as well as finite element schemes on unstructured grids. Koobus and Farhat (1999) proposed a GCL scheme for second-order time-accurate density-based arbitrary Lagrangian-Eulerian (ALE) finite volume schemes. Farhat et al. (2001) summarized six different time-integration schemes based on ALE formulation, some of them preserving the DGCL and some of them that do not, and showed the impact the different schemes have on solution accuracy. In this effort, we assess selected approaches for multi-block structured grids based on finite volume formulation and do a comparative study on these methods. Most previously conducted studies employed the density-based fluid flow solver; in the present effort, the pressure-based fluid flow solver (Shyy et al. 1994, 1997; Thakur and Wright, 2002) is utilized. The implications of different implementation of GCL and the fluid flow solver are of main interest. Together with the previously cited references, the present work offers a more complete assessment of the GCL. We will also take a closer look at the moving boundary module and probe into the impact of two different parameters associated with the moving boundary module in determining the rigidity of grid motion, which affects grid re-generation near wall boundaries, which in turn affects turbulence flow situations since we use the wall functions approach to model near wall regions. Our goal is to successfully perform aeroelastic computations on 3D wing geometries for high Reynolds number flows in turbulent flow regimes. In the process, we will look into the computational difficulties associated with developing a robust model, some of which have been, addressed here. In particular, we will look at the AGARD 445.6 wing configuration as numerous results have already been published for this case and hence can be used as a benchmark model to test our CAE model. NUMERICAL FORMULATION