Difference Equations and Conservation
Assignment :
Study and understand the Equations in the PDF file
This lecture is an extension of the last one, as I look at
implications of finite volume and finite difference approximations. We
will begin with a review of the equations, expand discussion of the
Courant stability limit, and discuss positivity of cell mass and
conservation of system mass.
Look at the Difference Equation Summary
(Postscript), be sure you learn the difference between Fully Implicit,
Fully Explicit, Crank-Nicholson, and Semi-Implicit. Remember to think
in terms of a staggered mesh with thermodynamic variables defined at
volume centers and velocities at volume edges. Pay careful attention to
the definition of the finite volume divergence operator. We will use
it shortly in discussions of Conservative difference equations.
Memorize the definitions of averages for Donor-Cell (Upwind) and
Central
difference methods. Finally take note of the pattern of use of new and
old time values in the Semi-Implicit form of the Euler flow equations.
Conservation of mass says that the integral of density over the
entire volume of the system should only change by the net amount fluxed
into or out of the system. This can be obtained by integrating the
differential equation for conservation of mass over the full system.
For our types of systems results are best when a similar result can be
obtained by summing the finite volume equation over all volumes in the
system.
Stability of Equations
The Courant stability limit applies to an Explicit scheme because it
can't propagate any information, including sound wave more than one
cell
in one step. For example, when you solve the explicit finite volume
mass equation for new time density, the right hand side of the equation
only contains information about density in the adjacent cell. Assume a
straight pipe ( cell volume equal to cell length times cell area), and
assume a positive flow velocity. Think about a situation where the
time step is greater than the cell length divided by the sum of sound
speed and magnitude of the flow velocity. In the real system this is
enough time for information (pressure wave) to travel from cell j-2 to
cell j. However the equations for change in properties in cell j,
contain no information about cell j-2. You expect strange things to
happen, and they do (instabilities as in your home work).
The Semi-Implicit scheme propagates sound waves over the entire
system in each step. So we don't worry about sound speed in stability
considerations. However a Material Courant stability limit applies,
because the method can't propagate mass, energy or momentum flux
information more than one cell in one step. Time step must be less than
cell length divided by the magnitude of velocity.
Positive Density (and Energy)
In addition to basic stability, one would generally like a numerical
solution to preserve basic physical ranges for properties, such as
positive cell mass and internal energy. As long as the material Courant
stability limit is obeyed, a rearrangement of a Donor-Cell difference
equation shows that density will stay positive (assume positive
velocity)
If velocity changes sign, across a cell (say negative at face j-1/2
and positive at j+1/2), then problems could arise with negative
densities.
A central difference method also has potential problems with
negative densities if a sharp enough density gradient exists.
Two-Phase Considerations
- What happens when solving for void fraction when phase change is
present (look at the liquid mass equation)
- May predict void < 0 or > 1, because phase change is not
properly limited
- Standard to round to proper limit, but this results in mass
errors
- When flow switches from two-phase to single-phase, iterative
solutions permit replacement of the vapor mass equation
with an equation that simply states void fraction is zero or one.
The vapor energy equation is replaced with the condition
that vapor temperature equals saturation temperature at the pure liquid
point (and an analogous pair at the all vapor end). Continued use
of the mean mass and energy equations provides proper conservation of
mass and energy.
Energy Conservation
- There are no conservation guarantees on the cancellation of terms
in
the
sum of the pressure work term, when doing an equation summation over
all
cells
- Definite problems with hot rods and all vapor system, energy
balance problems can be noticed, particularly in the steady state
- This will also cause problems at an abrupt area expansion.
Non-conservative energy equation can not predict both the pressure and
temperature correctly downstream of the area change.
- Introduce the conservative Energy
equation
- Derivation from previous equations tells you something is
missing. The non-conservative energy equation on the handout does not
have frictional heating terms. RELAP5 and TRACE actually include these
terms, because they can be significant in long duration transients.
- Tells you something about enthalpy in a moving fluid system.
Specific Flow enthalpy (stagnation enthalpy) is defined as static
specific enthalpy plus specific kinetic energy.
Maintained by John Mahaffy : jhm@psu.edu