Introduction to Finite Volume and Difference Methods
Assignment : Start HW3 ,
also read the TRACE Theory Manual description of the finite volume mass equation on pages
21-24 (available on ANGEL).
We are going to start the process of generating a useful numerical
model for the differential equations studied in the last lecture. You
a general background in the various finite volume methods used for
power plant simulation. I will point out several limitations inherent
in averaging required for numerical solutions. The material below is
just an outline of the lecture. For a more detailed discussion
read the Field Equations chapter in the TRACE theory manual available
on ANGEL or here.
- Finite volume is closely related to the process of deriving the
differential equations for flow. You can just run the argument
chain in reverse.
- Consider flow in a pipe, and break it into a finite number of
- cell centers indexed j-1, j, j+1
- cell edges labeled j-1/2, j+1/2
- Integrate the differential equation over volume j and apply
Gauss' theorem to convert the integral of the divergence to a surface
integral of mass fluxes. The result is that this integral is the
difference between the mean mass flows at cell edges j+1/2 and j-1/2.
- Discuss area averages, average of products
- Recall discussion of results for laminar and turbulent flow
- Stratified flow is not a major issue for the two-fluid model
because of the division of equations. However, an extra force
term is required proportional to the spactial derivative of void
fraction along the horizontal direction.
- Non-uniform distribution of bubbles and drops in flow can cause
problems, as can a liquid flow that is a mix of drops and liquid film
on the wall. This is addressed in codes like COBRA-TF by adding a
second liquid field. In this case one liquid field follows the
wall film and the second follows the droplets. You can imagine
scenarios where more than two liquid fields are needed. Detailed
models of bubbly flow need at least two gas fields if you are going to
naturally model the evolution from bubbly to slug flow.
- The volume integral of the time derivative becomes the time
derivative of the volume average. The time derivative must be
generated by dividing time into chunks . The equations, that you are
likely to use, only deal with information at two points in time for
each step forward in time. These are generally referred to as the old
time level (stuff you already know as a result of initial conditions or
the last step forward in time) and the new time level (stuff you don't
know until you solve the coupled set of equations). Old time
information is generally marked in an equation with a superscript of
and new time information marked with a superscript of "n+1". The
approximation for the time derivative of the volume average density is
then the difference to the new and old time average densities, divided
by the time increment (step) between the two points in time.
- Recall location of variables on the staggered mesh
- Look at the original division of the pipe
- Remember that the cell center is taken to be the location of
the average cell values for thermodynamic variables.
- Velocities are evaluated at the cell face
- Mass flux at the cell face is the product of the area of the
face (input), velocity at the face (calculated directly), and density
at the cell face (not calculated directly since density is a cell
- Spatial average to obtain density at a cell face
- The obvious choice is the average of cell center values
- This doesn't work well for two phase flow especially with
standard choices for time level evaluation.
- A simple but inaccurate average called the donor-cell gives
good stability behavior, but is generally inaccurate. An exception to
the accuracy problems is evident in the homework.
- When selecting time levels for terms in
the cell edge flux, we
generally deal with one of three well defined forms.
- The Fully Implicit method evaluates both density and
velocity at the new time level. This results in a rather complicated
coupled set of non-linear equations for the system variables. At
the moment only CATHARE uses this procedure rigourously, although
RETRAN is close, and TRACE will have the option within a year.
- The Fully Explicit method evaluates both the density
and velocity at the old time level. This produces a simple uncoupled
set of equations. However, this method is not used in standard
power plant simulation tools because of time step limitations.
- The Semi-Implicit method evaluates the density at the
old time level and the velocity at the new time level. It results in a
coupled set of equations that is much simpler to solve than those
produced by a fully implicit method. This is the standard method
for RELAP5 and an optional method (NOSETS=1) in TRACE.
- The Stability Enhancing Two-Step (SETS)
or "Nearly Implicit" (a variation used in RELAP5) methods add second
stabilizing evaluation of each equation (mass, energy, momentum) to the
basic semi-implicit equation set to produce results more like a fully
implicit method with less computational effort (see the TRACE Theory
Manual for more details)
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