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.
Maintained by John
Mahaffy
: jhm@psu.edu