Eigenmode Analysis of Boundary Conditions for the One-Dimensional Preconditioned Euler Equations

The effect of local preconditioning on boundary conditions is analyzed for the subsonic, one-dimensional Euler equations. Decay rates for the eigenmodes of the initial boundary value problem are determined for different boundary conditions and different preconditioners whose intent is to accelerate low Mach number computations. Riemann invariant boundary conditions based on the unpreconditioned Euler equations are shown to be reflective when used with preconditioning, and asymptotically, at low Mach numbers, initial disturbances do not decay. Other boundary conditions are shown to be perfectly non-reflective in conjunction with preconditioning. Two-dimensional numerical results confirm the trends predicted by the one-dimensional analysis

Eigenmode analysis for turbomachinery applications

This paper discusses the numerical computation of unsteady eigenmodes superimposed upon an annular mean flow which is uniform axially and circumferentially, but non-uniform in the radial direction. Both inviscid and viscous flows are considered, and attention is paid to the separation of the eigenmodes into acoustic, entropy and vorticity modes. The numerical computations are validated by comparison to analytic test cases, and results are presented for more realistic engineering applications, showing the utility of the approach for post-processing and for the construction of non-reflecting boundary conditions.\ud \ud This work was supported by Rolls-Royce and DTI funding

Eigenmode Analysis for Turbomachinery Applications

This paper discusses the numerical computation of unsteady eigenmodes superimposed upon an annular mean ow which is uniform axially and circumferentially, but non-uniform in the radial direction. Both inviscid and viscous ows are considered, and attention is paid to the separation of the eigenmodes into acoustic, entropy and vorticity modes. The numerical computations are validated by comparison to analytic test cases, and results are presented for more realistic engineering applications, showing the utility of the approach for post-processing and for the construction of non-reecting boundary condition

Preconditioned Euler and Navier-Stokes calculations on unstructured meshes

Multigrid techniques for unstructured meshes have proven to be very successful for both 2D and 3D inviscid problems, and recent advances concerning unstructured mesh generation, flow solvers and parallel computing techniques have made the predictions of these flows for complex geometries a rapid and robust procedure [3, 8, 11]. However, for an accurate aerodynamic analysis, viscous effects must be considered and to capture these, turbulence and transition modelling are required. The highly stretched computational cells, that are needed to efficiently resolve a high Reynolds number boundary layer limit the effectiveness of multigrid procedures which can not eliminate all the error modes which can exist in the solution. To overcome this drawback, different methods have been proposed. One is a semi-coarsening multigrid strategy, in which the mesh is not coarsened in every direction simultaneously [10], while others are based on the use of a preconditioner [1] which has the effect of moving the eigenvalues away from the origin of the Fourier complex plan providing, within an optimised Runge-Kutta update, a very good damping of the high-frequency error modes.\ud \ud Recently, Pierce and Giles [12] have analysed different combinations of preconditioner and multigrid method for both inviscid and viscous flow applications on structured grids. It has turned out that for turbulent Navier-Stokes calculations, a block-Jacobi preconditioner and a J-coarsened multigrid method provide an effective damping of all modes inside the boundary layer. The preconditioner damps all the convective modes, while the multigrid strategy, in which the grids are coarsened only along the normal to the boundary layer, ensures that all acoustic modes disappear. Thus, they have demonstrated that considerable speed-up can be achieved when using stretched structured meshes.\ud \ud The present work follows this idea, but uses an unstructured grid approach. The same preconditioner has been implemented in a multigrid solver which has proven to be highly successful for inviscid meshes [4, 2], and has been modified to treat the highly stretched grids required for high Reynolds number flows [5] so that the equivalent of a J-coarsening strategy is employed. In the following sections we describe how the local preconditioner is constructed, how the boundary conditions are treated, and exemplify the resulting method through 2D and 3D test cases

Stability Analysis of Preconditioned Approximations of the Euler Equations on Unstructured Meshes

This paper analyses the stability of a discretisation of the Euler equations on 3D unstructured grids using an edge-based data structure, first-order characteristic smoothing, a block-Jacobi preconditioner and Runge-Kutta time-marching. This is motivated by multigrid Navier-Stokes calculations in which this inviscid discretisation is the dominant component on coarse grids. The analysis uses algebraic..