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Write a CFD code to solve for the flow in a square lid-driven cavity with the top wall
moving at velocity Ulid. Use the 2D incompressible Navier-Stokes equations with time
derivative preconditioning (assuming constant temperature and viscosity) given by
where |λx|max is the magnitude of the largest eigenvalues in (x, t) space, |λy|max is the
magnitude of the largest eigenvalues in (y, t) space, and C(4) is a constant that generally
lies in the range
In the above equation, β is the time-derivative preconditioning term given by
(
)
2
β2 = max u 2 + v 2 ,κUlid
2
Use the simple explicit method (i.e., a point Jacobi method) with second-order accurate
central differences to advance the discrete equations in pseudo-time until you reach the
steady-state solution. For students enrolled in the graduate course (AOE/ME 5434), also
implement an explicit (point) symmetric Gauss-Seidel scheme. Monitor iterative
convergence using the steady-state iterative residuals (i.e., the steady portion of your
discretization evaluated all at the same time level). A relative iterative convergence (i.e.,
the ratio of iterative residuals to initial iterative residuals at step 1) of at least 8 orders of
magnitude is strongly recommended. Use the stability criteria we discussed in class to
determine the time step; you may choose local or global time stepping, but local time
stepping is recommended (i.e., take the largest allowable time step at each node).
The Fortran, C, C++, MATLAB, and Python code templates you will be given have the
capability to run manufactured solutions (by setting the imms input flag to one).