Adjoint-based Grid Adaptation For Viscous Flow Simulation

This paper develops an adjoint based grid adaptation method to improve the accuracy of output functions. The adjoint solution represents the sensitivity of the objective function respect to local residuals of viscous flow equations. Error of the objective function can be expressed as summation of computable correction and remaining error. The computable correction can be estimated by inner product of adjoint solutions and local residuals of viscous flow equations. The remaining error term is adopted in constructing a reliable mesh adaptation sensor for simulating viscous flows. More accurate computable correction can be acquired by decreasing the remaining error, and as a result the objective function reaches higher precision. Fine grid values needed in computing the adaptation sensor are acquired by interpolation. The two-dimensional compressible laminar flow equations are solved by finite volume scheme based on median-dual cells. The centered Jameson scheme is applied in convection flux computation. And the boundary conditions are non-reflecting condition for far–field boundary, no slip and isothermal condition for the wall. The five-step explicit Runge-Kutta scheme is adopted for time discretization. The adaptation process is implemented with discrete adjoint approach. A time like derivative is added to the adjoint equation, and the solution is obtained by marching in time, as done in the flow solver. Algorithms of computing adjoint flux for convection and artificial dissipation term are derived. Thin shear layer approximation is adopted to simplify adjoint diffusion flux calculation. Due to the imposed strong boundary conditions, the discrete adjoint takes on a different character on the boundary relative to the interior. The adjoint on strong boundary is calculated in a post processing step once the interior adjoint has been obtained. A conservative method to obtain the linear sensitivity of objective function on the right hand side of the adjoint equation is presented. The adaptation approach and function correction technique are tested on simulating flows over NACA0012 airfoil and cylinder with lift and drag coefficients as target functions, which are meaningful in engineering applications. Numerical results show that the presented adaptation approach is capable of effectively improving the accuracy of objective function.