Accurate numerical solution of Euler equations by optimal control of dissipation

摘要

Kinetic Flux Vector Splitting (KFVS) has been extensively used to compute inviscid as well as viscous flows over subsonic, transonic and supersonic speeds over the last two decades. Recently, modified KFVS approach with dissipation control parameter (called m-KFVS) has been developed primarily to reduce inherent numerical diffusion. Generally second order upwind method requires a five point stencil (in case of 1D problem) but with m-KFVS it is possible to achieve nearly second order accuracy with a 3 - point stencil with a suitable choice of the control parameter alpha. The parameter alpha in general will depend on coordinates of grid points thus giving us distributed control on numerical diffusion. A possibility therefore arises about obtaining optimal distributed control by minimising cost function such as numerical entropy produced, maximising total pressure behind a normal shock or for minimising the spread of shock. First, some preliminary results for 1D nozzle problem, subsonic, transonic and supersonic flows around 2D airfoil are presented. Next the m-KFVS CFD Euler solver is coupled with adjoint method to obtain optimal solution. For this purpose TAPENADE software developed by INRIA has been found to be very useful in calculating the gradients of residue vector with respect to conserved vector U and control parameter alpha. It is shown that by combination of the above tools, the suction peak near the leading edge of subsonic flow around 2D airfoil can be very accurately captured.