Engineering topics
Publications and source records attributed to Hartwich, P.-M..
Navier-Stokes computations of three-dimensional vortical flows over a round-edged double-delta wing and a tangent-ogive forebody are performed using an implicit upwind finite-difference scheme. Computed particle traces are compared with experimental oil-flow streaks.
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Two hybrid upwind models are defined for solving the Euler equations. The algorithms both employ approximate factorization (AF) in crossplane and symmetric block Gauss-Seidel relaxation in the third direction. One approach adds an additional factorization step to lower the number of required grid point operations for inversion of the block tridiagonal matrices; however, the move permits only one third of the operations to be vectorized. Finite difference solutions are calculated on a C-H-type grid, in this case enveloping a slender, sharp-edged delta wing. Sample data are provided for the calculated vortex flow for Re of 10,000, at a 20.5 deg angle of attack, represented in a crossflow velocity vector plot and in a spanwise pressure coefficient distribution. The AF scheme, without additional factorization, when used with a grid covering 51 x 51 x 72 points provides a convergent solution with no time step lasting longer than 0.00001 sec.
Implicit hybrid algorithms employing symmetric planar Gauss-Seidel (SPGS) relaxation and either block-tridiagonally structured coefficient matrices (AF-SPGS) or block-triangular coefficient matrices (LU-SPGS) are derived to solve the flux-difference-split Navier-Stokes equations for three-dimensional incompressible flow in an upwind scheme. The physical basis of the approach is discussed, and results for problems involving vortex flow around a thin delta wing at Reynolds numbers 900,000 and 10,000 are presented graphically. It is found that AF-SPGS converges faster on vector computers which depend on long vector lengths to achieve optimum performance, whereas LU-SPGS is preferable on sequentially operating machines and vector computers using shorter vector lengths.
An elliptic grid generation scheme is described which produces a curvilinear, boundary-fitted grid for highly swept wings with sharp leading edges. The three-dimensional integration domain is segmented in successive crossflow sections wherein the physical coordinates are transformed to computational coordinates by solving an elliptic set of two quasi-linear differential equations. Grid points are clustered in the vicinity of the wing surface, especially in the neighborhood of the leading edges. The formulation of the source terms in the equations governing the transformation contains adjustable parameters which are determined from limiting forms of the elliptic equations along the boundaries. The source terms are assigned to each grid point by interpolation of these parameters into the interior domain. The formulation does not depend on the boundary shape or on the distribution of the grid points along the boundaries. For the first time, the discretized governing equations have been solved using a fast AF1 iteration scheme.
A new, implicit finite-difference scheme designed to solve the conservative, flux-difference split Navier-Stokes equations is used to compute incompressible vortex flows around delta wings. The completely vectorizable hybrid algorithm is constructed in delta form for steady state solutions independent of the time-step sizes. The scheme combines approximate factorization in crossflow planes with a symmetric planar Gauss-Seidel relaxation in the remaining spatial direction. The governing equations are solved in curvilinear, body-fitted coordinates for treating complex geometries. The computed flow field results are compared with other theoretical and experimental data.
Attempts have been made to explain why finite difference solutions of the Euler equations can describe flows with large vortical structures around sharp-edged bodies. The present paper is concerned with the influence of a singular sharp edge on the truncation error for a set of discretized Euler equations. An analysis is conducted of the distribution of the truncation error of one finite difference approximation of the Euler equations near a sharp edge of a thin plate. The analysis leads to a determination of the size of the region of the neighborhood of such a singularity. Attention is given to the consistency of a discretization of the Euler equations, and numerical experiments.