Projection of the solution of the linearized navier-stokes equations in reacting high speed boundary layers onto discrete modes

The problem of multimode decomposition of small perturbations in high-speed boundary layers in chemical nonequilibrium is addressed using the discretized adjoint approach. The solution of the linearized Navier-Stokes equations is considered in the quasi-parallel flow approximation using the normal mode analysis, and the ordinary differential equations for the amplitude functions are discretized using fourth-order finite differences. The discretization leads to a system of linear algebraic equations in the form of the generalized eigenvalue problem. It is straightforward to define the left eigenvectors (eigenvectors of the adjoint problem) and to formulate the biorthogonality condition for the discrete modes. Assuming that there is a complete system of eigenfunctions of the discrete and continuous spectra, the biorthogonality condition allows for the finding of the projection of a solution onto the discrete modes. The biorthogonality condition is also utilized for solving the receptivity problem with perturbations introduced at the wall.