On the non-normality and sensitivity of the linear stability equations in boundary layer flow

Abstract

Recall that the first aim of this thesis was to assess whether we can derive bi-orthogonality relations demonstrating that the adjoint LST eigenvectors are orthogonal to the direct eigenvectors, for both the compressible and incompressible LST equations; thereby resolving the non-orthogonality complication. It is to that end that we derived the adjoint operators for the incompressible and compressible LST equations using both the discrete and the continous approach in chapter 3. Subsequently, we derived, implemented and assessed bi-orthogonality relations using the adjoint eigenvectors resulting from the discrete and continous approaches in chapter 3. It was found that theoretically relations can be derived for which a weight matrix ensures bi-orthogonal sets of direct and adjoint eigenvectors. Numerical implementation of these biorthogonality relations for both the incompressible and compressible LST yielded diagonal biorthogonality coefficient matrices for the discrete case. Results for the continous approach have shown that the coefficient matrices are diagonally dominant, but do contain substantial offdiagonal terms. As per the theoretical bi-orthogonality derivation, the complex conjugates of the adjoint eigenvalues were shown to match the direct eigenvalues in the physically interesting range, for both the incompressible and compressible LST spectra. The discrete and continous adjoint eigenmodes were found to be similar in shape and location, but the modes did not coincide. A potential explanation for the discrepancy might be the use of the Chebyshev collocation nodes in the continous adjoint EVP. The adjoint EVP may require different stretching of the mesh. Alternatively, closing the continuous adjoint system using the ’adjoint wall-normal equation’ as a compatiblity condition for the adjoint pressure amplitude might possibly adversely affect the continous adjoint eigenmodes. An interesting observation was made when considering the spatial structure defining the overlap between direct eigenmodes and discrete adjoint eigenmodes. For both a wall mode, and a ’middle region’ mode, the location of the maximum amplitude of the direct eigenmode was found to differ spatially from that of the maximum amplitude of the discrete adjoint eigenmode.