Keywords: computational fluid dynamics, mesh adaptation, anisotropy, unstructured meshes, turbulence models, boundary layer flow.

In the work presented here, a mesh adaptation module is extended to account for the requirements of turbulence models on 3-D unstructured tetrahedral meshes. adaptation is implemented by modifying the error metric of near-wall elements to obtain an appropriate grid point distribution in the boundary layer. In the final paper, results will be presented to demonstrate the improvements in the meshes that result from this new function and the corresponding improvements in the CFD solutions.

Most turbulence models require that the distribution of grid points in the boundary layer fall within a specific range of values. To achieve this, adaptation is incorporated into the standard mesh adaptation process to adjust the thickness of elements in the near-wall region. The need for and importance of adaptation has been documented in recent literature [1,2,3].

Many turbulence models require that near-wall elements be orthogonal to the wall, necessitating the use of hexahedral meshes or hybrid tetrahedral meshes with layers of prisms on the walls. adaptation on these types of meshes has already been implemented [1]. Alternately, certain turbulence models such as Spalart-Allmaras may be used with entirely unstructured tetrahedral meshes. This can be advantageous when dealing with complex geometries that often present challenges for generating semi-structured meshes. Therefore, adaptation for tetrahedral meshes is implemented to facilitate the generation of initial meshes and improve the quality of the solutions obtained following adaptation.

It should be noted that the capacity to produce anisotropic meshes is particularly important in this case, because the turbulence model requires an extremely high mesh density in the direction normal to the wall. The adaptation software produces elements that are stretched in the directions parallel to the wall, so the adapted mesh has a much smaller number of nodes than an equivalent isotropic mesh with the same boundary layer resolution. However, adapting these stretched elements presents additional difficulties, especially for 3-D meshes.

The standard mesh adaptation process is governed by the error metric, in the form of a Hessian matrix H, which is computed from the second derivatives of the adaptation scalar. adaptation for tetrahedral meshes is implemented by modifying the error metric in the near-wall regions and then proceeding with the adaptation in the entire domain in the standard fashion. A similar approach for 2-D meshes has been documented in the literature [3,4].

In order to modify the metric in the direction normal to the no-slip wall, the Hessian is decomposed into its eigenvectors such that , and the eigenvectors are rotated so that one of them points in the normal direction. The eigenvalue corresponding to the normal eigenvector is then set to to achieve the local element thickness . The thickness at a distance from the wall is computed using the target element thickness at the wall and the growth ratio set by the user: .

For adaptation, is calculated from the value in the solution and the target set by the user. For mesh smoothing without a solution, is set directly by the user. The distance to the no-slip walls is computed everywhere in the domain by solving a Poisson equation. The normal to the nearest no-slip wall is approximated by the gradient of the wall distance scalar.

adaptation for unstructured 3-D tetrahedral meshes allows users to produce high-quality anisotropic meshes for turbulent solutions without the difficulties of generating structured meshes for complex geometries. The initial mesh obtained from the mesh generation software may be first smoothed to increase the density in the boundary layer for the initial solution and then adapted with a target value to obtain the optimal mesh for turbulent solutions.

1

C.Y. Lepage, F. Suerich-Gulick, and W.G. Habashi, "Anisotropic 3-D Mesh Adaptation on Unstructured Hybrid Meshes", AIAA Paper 2002-0859, 2002.

2

A. Khawaja, T. Minyard, and Y. Kallinderis, "Adaptive Hybrid Grid Methods", Comput. Methods Appl. Mech. Eng., 189, 1231-1245, 2002. doi:10.1016/S0045-7825(99)00375-8

3

G. Xia, D. Li, and C.L. Merkle, "Anisotropic Grid Adaptation on Unstructured Meshes", AIAA Paper 2001-0443, 2001.

4

M.J. Castro-Diaz, F. Hecht, B. Mohammadi, and O. Pironneau, "Anisotropic Unstructured Mesh Adaptation for Flow Simulations", Int. J. Numer. Meth. Fluids, 25, 475-491, 1997. doi:10.1002/(SICI)1097-0363(19970830)25:4<475::AID-FLD575>3.0.CO;2-6

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