In [1], a robust and efficient three-dimensional automatic mesh adaptation methodology, with CAD integrity, OptiGrid, has been developed. The adaptation process, however, can take several solution-adaptation cycles depending on the quality of the initial grid and the complexity of the test case. To improve the performance of the mesh adaptation procedure and reduce the number of adaptation cycles, we propose a new variable for adaptation that better represents global physical features. It is commonly accepted that the Mach number is an appropriate adaptation variable in many situations. However, an increasing number of mesh adaptation iterations may be required to obtain a converged solution for low subsonic flows where Mach number gradients are less dominant (than in transonic flows). Furthermore, for viscous flows, the Mach number is theoretically zero at the surface and therefore an optimal surface grid cannot be achieved when using the Mach number to drive the mesh optimization process. On the other hand, the Mach number does capture a strong gradient in the boundary layer, which needs to be preserved. It would also be undesirable to use a flow parameter such as pressure to drive the adaptation, as the lack of a gradient in the boundary layer will destroy the fine grid spacing throughout the boundary layer. To take the advantage of both variables, we propose to combine the pressure and the Mach number via their corresponding adaptation metrics. Another proposed improvement is a modification of the metric taking account of the first derivatives which have so far been neglected in the error indicator computation. The commonly used metric uses the second derivatives to compute the error indicator. It is obvious that to capture high gradients, a certain level of node concentration is necessary. In regions of high gradient the solution could be linear or close to linear, causing the second derivatives to vanish. Conversely, the gradient vanishes in regions of extrema, which correspond to regions of maximum of curvature. To improve the quality of the adapted mesh, therefore, we need to combine both the Hessian and the gradient to compute the error indictor. Since Hessian and gradient generally lie in different ranges, some normalization and balancing between the two associated metrics is necessary. This is done using the maximum and minimum spectral radius of both metrics.

The NASA semi-span flap [2] test case was chosen for evaluating the impact of the mesh adaptation on the prediction of the lift coefficient. First, the previous parameters are used for the adaptation, the obtained results match well the experimental results, and as the adaptation proceeds, the lift coefficient converges to the experimental value. As a conclusion, this test case clearly demonstrates that mesh adaptation strategy is an efficient and robust method to ensure the convergence of the solution (to experimental) and to correctly capture physical features that are of a high importance in industrial applications. However, 8 cycles were necessary to achieve convergence, which could be considered somewhat costly in terms of time, especially when we deal with large test cases. This test case is repeated using the new proposed parameters. The same results are obtained with a reduction of the number of cycles to almost a half, and moreover, the required number of nodes is also reduced to almost a half. This demonstrates clearly the efficiency of the new variable of adaptation and the modified metric. To show the effect of the mesh adaptation on the drag coefficient, two transonic turbulent flow cases are selected, the RAE2822 [3] and the NACA0012 [4]. Using the new adaptation parameters, results show the sensitive improvement of the drag coefficient with adaptation.

As a conclusion, this paper shows clearly the impact of mesh adaptation on the convergence of numerical calculation of lift and drag to experimental results. It has been demonstrated that, by generating a metric that combines pressure and Mach number in a unique way and introducing a new metric that takes into account the information from the gradient, the convergence of the mesh optimization process can significantly be accelerated.

1

W.G. Habashi, J. Dompierre, Y. Bourgault, M. Fortin, M.G. Vallet, "Certifiable Computational Fluid Dynamics Through Mesh Optimization", Special Issue on Credible Computational Fluid Dynamics Simulation, AIAA Journal, Vol. 36, No. 5, pp. 703-711, 1998. doi:10.2514/2.458

2

B.L. Storms, T. Takahashi, J.C. Ross, "Aerodynamic Influence of a Finite-Span Flap on a Simple Wing", SAE paper 951977, September 1998.

3

S.S. Davis, "NACA 64A010 (NASA Ames model) oscillatory pitching", AGARD Report 702, August 1982. In Compendium of Unsteady Aerodynamic Measurements

4

D.H. Harris, "Two-Dimensional Aerodynamic Characteristics of the Naca0012 Airfoil in the Langley 8-foot Transonic Pressure Tunnel", NASA TM 81927, April 1981.

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