Abstract
The linearized pressure Poisson equation (LPPE) is used in two and three spatial dimensions in the respective matrix-forming solution of the BiGlobal and TriGlobal eigenvalue problem in primitive variables on collocated grids. It provides a disturbance pressure boundary condition which is compatible with the recovery of perturbation velocity components that satisfy exactly the linearized continuity equation. The LPPE is employed to analyze instability in wall-bounded flows and in the prototype open Blasius boundary layer flow. In the closed flows, excellent agreement is shown between results of the LPPE and those of global linear instability analyses based on the time-stepping nektar++, Semtex and nek5000 codes, as well as with those obtained from the FreeFEM++ matrix-forming code. In the flat plate boundary layer, solutions extracted from the two-dimensional LPPE eigenvector at constant streamwise locations are found to be in very good agreement with profiles delivered by the NOLOT/PSE space marching code. Benchmark eigenvalue data are provided in all flows analyzed. The performance of the LPPE is seen to be superior to that of the commonly used pressure compatibility (PC) boundary condition: at any given resolution, the discrete part of the LPPE eigenspectrum contains converged and not converged, but physically correct, eigenvalues. By contrast, the PC boundary closure delivers some of the LPPE eigenvalues and, in addition, physically wrong eigenmodes. It is concluded that the LPPE should be used in place of the PC pressure boundary closure, when BiGlobal or TriGlobal eigenvalue problems are solved in primitive variables by the matrix-forming approach on collocated grids.
Highlights
Global linear instability theory is enjoying increasing acceptance in the fluid mechanics community, as witnessed by the contents of this volume
The BiGlobal EVP has been solved by several authors with mixed degrees of success and it is interesting to contribute here to the related discussion in the literature, by separating issues arising due to the pressure boundary condition imposed at the wall boundary from those related with the open boundary treatment
Steady laminar flows in closed domains have been analyzed in two spatial dimensions, where results of the linearized pressure Poisson equation (LPPE) boundary closure have been compared with those obtained by the FreeFEM++ matrix-forming and the nektar++ and Semtex timestepping codes; excellent agreement was obtained in all three classes of flow instability problems considered
Summary
Global linear instability theory is enjoying increasing acceptance in the fluid mechanics community, as witnessed by the contents of this volume. Using spectral collocation on staggered grids, the momentum equations are collocated and solved on two-dimensional tensor-product grids based on extremum (e.g., Gauss–Lobatto) gridpoints, continuity is solved on interior (e.g., Gauss) gridpoints and spectral interpolation operators are used in order to transfer velocity from the extremum onto the interior grid and vice versa [40] This approach has been used successfully in the context of global instability analysis by Theofilis and Colonius [62] to obtain eigenvalue problem results in a compressible flat plate boundary layer as limiting validation cases of the algorithm used for the solution of the BiGlobal eigenvalue problem in compressible open cavity flow.
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