A computational fluid dynamics (CFDs) method
utilizing unstructured grid technology has been
employed to compute vortical flow around a
In recent years, computational fluid dynamics (CFDs) has matured as a routine tool to predict complex genuine flows at designed condition. However, it is still difficult to deal with complex flows at off-design conditions where flow separations and vortices are dominated. For these flows, inappropriate grid resolution has been recognized as one of the dominant reasons for inaccurate predictions of flow features and aerodynamic forces. Actually, in general, the computational grid may become rapidly coarser as it becomes far away from the body surface. In this case, the vortices would be highly diffused due to the numerical discretization error. A possible way to minimize the numerical dissipation of the vortices is to use a highly dense grid. But flow computations around three-dimensional complex bodies with large-scale separations and vortices are difficult within the realistic number of grid points. In order to obtain numerical results for such flows with a given level of accuracy by a limited capacity of the computer, the method for grid adaptation is very useful for practical computations, which offers the possibilities to avoid the use of overly refined grids to guarantee accuracy. The basic premise is to locally enrich the computational grids in regions which most adversely affect the accuracy of the final solution. Its emphasis is to define a proper adaptive criterion (also known as the adaptive sensor) combined with distinct methods to automatically generate an adaptive mesh.
Currently, many sensors for unstructured grid adaptation are available, for example, the wake sensor based on difference in velocity, the shock sensor based on ratio of total pressure, the vortex sensor based on creation of entropy, and the vortex sensor based on eigenvalue analysis of the velocity-gradient tensor, and so forth [
In this paper, an adjoint-based adaptation with isotropic h-refinement for unstructured grid is introduced to catch the vortex-dominant flow. These kinds of methods are firstly proposed in finite element communities [
The flow solver used in this paper is WoF90, which is developed by ACTRI. It is a solver employing hybrid unstructured grid both for Euler and Reynolds-averaged Navier-Stokes (RANS) equations. The solver can deal with arbitrary types of unstructured elements including tetrahedrons, hexahedrons, prisms, and pyramids. It is based on an edge-based formulation and uses a node-centered finite volume technique to solve the governing equations. The control volumes are nonoverlapping and are formed by a dual grid, which is computed from the control surfaces for each edge of the primary input mesh. The equations are integrated explicitly towards steady state with Runge-Kutta time integration. The spatial discretization is either central with artificial dissipation or upwind; both approaches are second order accurate. The solver also adopts an agglomeration multigrid algorithm and an implicit residual smoothing algorithm to accelerate the convergence of a simulation.
In this current study, Euler equation is solved by WoF90 for conserved variables
Let
The adjoint solutions can provide an efficient way to estimate error of the objective output function
As shown in (
A general introduction to the flow chart of grid adaptation can be found in [
To use the adjoint-based adaptive sensor defined in (
Inviscid transonic flow over RAE2822 airfoil is chosen to validate the rationality of adjoint-based adaptive method for two-dimensional test cases. This test case is a classical validation case, which has been used worldwide, especially in EUROVAL project and by AGARD. The simulation is performed at the flow condition as
The initial mesh has a total of 15,852 nodes and 31,388 triangles, which is shown in Figure
Initial (unadapted) RAE2822 airfoil mesh.
Initial Mach contours for RAE2822 airfoil at
To apply adjoint-based adaptation method, we choose the lift coefficient as the objective output function for adjoint solutions. The final adapted mesh using adjoint-based sensor and isotropic adaptation with h-refinement is shown in Figure
Adapted RAE2822 airfoil mesh.
Inviscid transonic flow over ONERA M6 wing is chosen to validate the rationality of adjoint-based adaptive method for three-dimensional test cases. This configuration is simulated at the flow condition at
Initial (unadapted) surface mesh of ONERA M6 wing.
Again, we choose the lift coefficient as the objective output function for adjoint solutions. After grid adaptation using adjoint-based sensor and isotropic adaptation with h-refinement, the adapted mesh contains a total of 128,386 nodes and 740,366 tetrahedrons, where there are 8,041 boundary nodes on the wing surface. The adapted wing surface mesh is shown in Figure
Adapted surface mesh of ONERA M6 wing.
The Mach contours on the adapted upper wing surface are shown in Figure
Mach contours on adapted surface of ONERA M6 wing at
Comparisons of spanwise pressure coefficient distributions for ONERA M6 wing between before and after adaptation.
As mentioned above, the test case of
The detailed description of the model can be found in [
The initial mesh contains a total of 64,835 nodes and 365,681 tetrahedrons, where there are 9,464 boundary nodes on the wing surface. No attempt has been made to cluster extra grid points in the field at the vortex location. The initial surface mesh on the upper surface is shown in the upper portion of Figure
Initial (unadapted) upper surface mesh and Euler inviscid solution on the sharp leading-edge delta wing at
To apply the adjoint-based adaptation, we also choose the lift coefficient as the output function. After the application of grid adaptation, the adapted mesh contains a total of 139,682 nodes and 766,948 tetrahedrons, where there are 13,333 nodes boundary nodes on the wing surface. The adapted upper surface mesh and the corresponding Euler solution on the adapted mesh are shown in Figure
Adapted upper surface mesh and Euler inviscid solution on the sharp leading-edge delta wing at
Comparisons of the grid resolution in space from two typical sections at
In Figure
Comparisons of spanwise pressure coefficient distributions for delta wing with sharp leading edge between before and after mesh adaptation.
An adjoint-based adaptation on unstructured grid has been implemented and is applied to simulate the complex vortex flow of a
It should be pointing out that although all simulations in this study are limited to inviscid Euler computations, the adjoint-based adaptive method itself can be extended to viscous calculations by developing a viscous adjoint solver and using anisotropic refinement for grid within boundary layers [
This work is supported by the State Key Development Program of Basic Research of China (973) under Grant no. 2009CB723804. The authors would like to thank the German Aerospace center (DLR) for proving and authorizing to use the experimental data of delta wing in VFE-2 project. Mr. YANG Zhenhu, one of our colleagues at ACTRI, is also appreciated due to his part of work on adjoint-based mesh adaptive method.