Chapter 8 - Channel flow

1. Problems

(i) Use numerical quadrature of the analytic solution (8.125) to compute the coefficient for corner vorticity in (8.139) for the rectangular finite volume in Figure 8.25, (b).

(ii) The spectral method of Burggraf and Duck (1982) would seem to have ideal properties for calculating interactive boundary layer solutions for channels with periodic disturbances. Implement such a scheme and compare its resolution and computational requirement with the simple interactive boundary layer solver provided on the website.

(iii) One almost curious aspect of finite volume methods is to use differences between volume averaged values to represent derivatives or more precisely, fluxes. If Weinbaum's solution is used to calculate the average value of $\psi$ at the corner and to calculate the exact flux across the boundary of the finite volume, by how much does the difference between the average values of $\psi$ at the corner and at say $(0,h)$ differ from the calculated flux across the line $y=h/2, -h/2<x<h/2$?

2. Software

(i) coord.c generate conformal mesh for a class of furrowed periodic channels. The method of solving for the conformal transformation is simple relaxation - the program could be accelerated by using conjugate gradient for example. The iteration for the non-linear boundary conditions (orthogonality) is also prone to instability if the mesh deformation is changed too rapidly. One strategy is to `grow' the bounday slowly, obtaining a converged solution between changes in the boundary deformation. The user interacts with this program through an X window interface and the program can save the conformal mesh to a file which is suitable for being read by the Navier-Stokes solver unsteady12. A makefile for the program coord program is in makecoord

(ii) unsteady12.c time marching Navier-Stokes solver for channel using conformal mesh generated by coord.c. Uses X window interface and provides streamline contours as solution evolves. Uses multigrid to solve stream-function-vorticity equation. A makefile for the program unsteady12 program is in makeunsteady12

steady12.c is a steady version with a makefile for this program in makesteady12

(iii) channel.f `triple deck' solution for channel with no cross channel pressure variation

(iv) lcw.f Implementation of Woods' method to calculate free streamline solution in a channel for various wall conditions

(v) corner.m contours near a corner using Weinbaum's solution

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