Sultan Sial

ss@look1.apmaths.uwo.ca

http://look1.apmaths.uwo.ca

The modeling of shock waves illustrates themes that are common in the field of Applied Mathematics:

- The problem is
too difficult to be solved in general by analytic means.
- One can make simplifying physical assumptions that allow
the problem to be model-led on a computer.
- This means that the modeler must have physical insight into the problem as well as a knowledge of computational techniques.

**Pictures for the impatient**

We will illustrate these themes by looking at a simple case - the Sod shock tube. Click here to see the mass density in the tube after the shocks have developed. Click here to see the velocities in the shock tube after the shocks have developed.

**Probabilistic Viewpoint**

There are a number of approaches to thinking about and modeling a fluid like air or water.

One way is to try and describe the motion of each
individual particle of the fluid. This leads to the field
of *molecular dynamics*. A problem with this method is that
it is very computationally intensive to track even a few
thousand molecules let alone the millions in even a very
small volume.

Another approach is to assign a *probability
density* function *f* to the fluid that
describes fluid.

In statistical mechanics such a function takes a form
*f(u,v,w)* where *u*, *v*, *w*
are the *x*-, *y*-, and
*z*-velocities of individual molecules. *f*
tells us what fraction of the molecules of the fluid have
the velocities *u*, *v* and *w*.

What form should the function *f* take? This
questions leads to our first assumption.

**Maxwell-Boltzmann Distribution**

It is a principle of statistical mechanics that a system
will seek to maximize 'disorder' or entropy. For an
ideal gas, the distribution that maximizes the entropy is the
Maxwell-Boltzmann distribution. So we could assume that,
at any point in fluid, the fluid can be described by the
Maxwell-Boltzmann distribution. This is of course
**false**. However, we have to start
somewhere and the Maxwell-Boltzmann distribution may be a
good approximation in some cases.

**Finite-Volume Method**

There are several methods for describing the evolution of a fluid. The Finite-Volume methods take the volume of interest and divide it up into a number of smaller volumes.

We make the assumption that within each small finite volume the fluid will have the same mass, momentum, and energy. Again this is false, but with enough finite volumes is often a good approximation.

**Numerical Interpolations**

We can specify the mass, momentum and energy in each cell at
the beginning of the simulation. But what about the
values at the interfaces of the cells? At this point some
sort of *numerical interpolation* is required.
Numerical interpolation techniques attempt to make
estimates for the value of a function when the exact value
is difficult, expensive, or impossible to determine.
Using the values in neighbouring finite volumes one can
estimate the values at th interfaces between the finite
volumes.

**Fluxes**

The rate at which any quantity *s* is flowing in
the *x*-direction is given by integrating the function
*usf* over all values of __ from minus infinity to
infinity. The fluxes in the other two directions are
given by integrating vsf and wsf.
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We require one more false assumption. The function
f, which we assume is the Maxwell-Boltzmann
distribution, is a function of the average momentum at the
point of interest. Assume that over a small time step
T the average momentum does not change. Then,
the amount of mass, momentum or energy transferred between
two finite volumes will be given by the flux of that
quantity times T.
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Algorithm
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With these assumptions it is possible to watch a fluid
evolve. We begin a simulation by specifying the mass,
momentum and energy in each finite volume. We then
interpolate the values of these quantities at the
interfaces between the finite volumes. These values give
us the Maxwell-Boltzmann distribution at the interfaces
from which the fluxes can be calculated. The fluxes tell us
how much mass, momentum, and energy is transferred between
the finite volumes in each time step. At the end of the
time step we can add these fluxes to each finite volume to
get new values for the mass, momentum, and energy. We
then repeat the entire process over and over again to see
how the fluid evolves.
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Simulation
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An illustration of a finite volume technique is
the simulation of the Sod shock tube. This is a
standard test case for which analytical techniques
give an answer against which numerical techniques
can be tested.
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The Sod test case consists of a one dimensional
tube in which one half of the tube has different
initial conditions than the other half. initially,
the left half of the tube has a mass density of 1
and an energy of 2.5. The right side of tube
initially has a mass density of 0.125 and an energy
of 0.25.
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As the flow develops shock waves form.
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The results shown below are from a FORTRAN90 program
run on an IBM RISC6000 workstation. Since the
code is essentially parallel it runs well on
super-computers also.
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Initially, the shock tube has a high and a low
density region. As the flow develops, regions of
intermediate density develop as well. This is
shown in the digram below.
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Initially, the fluid is at rest. Then a flow
develops in the middle of the tube which moves
towards the right.
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