ME 523

Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems

# Application of Difference Methods to a Full Set of Flow Equations

So far we've just tinkered with a simple advection-diffusion equation.  Now I'm going to discuss issues related to a full set of fluid flow equations (conservation of mass, momentum, and energy).   I've provided summary pages of some standard time level selections for flow equations in a PDF file.  When implicit methods are selected we are faced with a set of coupled non-linear equations.  Over the years, a number of iterative strategies have been developed to solve these equations.  One of the most popular has been the SIMPLE methodology, which is a simple iteration added around the older "semi-implicit"  family of methods.  However, my experience has been that the best approach is to structure the solution as a Newton Iteration, and then deal with the resulting Jacobian Matrix with appropriate linear system solvers.  For those out of practice, I've provided a basic summary of a Newton iteration.  I have also provided an illustration of use of Newton iteration to solve a "semi-implicit" set of equations.

As you look at the description of the semi-implicit method and other material that I provide on approximation of fluid equations you will see that I am working on a staggered mesh.  This simply means that mass and energy equations and associated thermodynamic equations are evaluated at volume centers, and momentum equations with associated velocities are evaluated at the edges of the mass and energy volumes.  Digging deeper you will find that the momentum equations in this approach are derived from a different set of volumes that are staggered with respect to the mass and energy volumes.  This staggered approach has been used historically because of its relative ease of implementation and overall robustness.  The staggered approach introduces complications when dealing with fully conservative energy equations, and it is now common to have momentum equations and associated velocities evaluated at the centers of the same volumes used for mass and energy equations (coincident mesh). However, if you use a coincident mesh, you need to know that the obvious evaluation of the momentum equation is subject to robustness problems, and you should implement a method based upon the work of Rhie and Chow (C.M.Rhie and W.L.Chow., "Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation",AIAA  Journal, 21(11), 1983, pp1525-1532 ).

When I introduced von Neumann stability analysis, I gave a vague indication of how it is extended beyond a single PDE.  Now that we've reached the flow equations, I've got a specific example for stability of the semi-implicit equations.  I've selected this example, because it illustrates key steps in the process, not because you are likely to employ a semi-implicit approach in your careers.

## Other Lectures / Home

Maintained by John Mahaffy : jhm@psu.edu