12.4 Moving Grids
379
For incompressible flows, the density is constant so pF = p = const. Compressible flows require special care in order to determine p; such that both
space and mass conservation equations are satisfied. The problem is that the
mass conservation equation (12.19) contains the cell-center density in the unsteady term and cell-face densities in the mass fluxes, so that the evaluation of
pF is not trivial if the density varies rapidly in both space and time. If it does,
small time steps are necessary so that the approximation p; = 1/2(pr+pY+')
is sufficiently accurate.
The importance of taking the space (or geometric) conservation law into
account in unsteady flows with moving boundaries has been recognized by
many authors; see Thomas and Lombard (1979) and DemirdiiC and PeriC
(1988) for a more detailed discussion. Using coordinate transformations or
calculating the grid velocities from the motion of the cell-face center (see Eq.
(12.17)) leads to artificial mass sources or sinks. The error depends on the
time step and grid velocity, (see Eq. (12.12)). When the time step is small
(which is usually the case with explicit schemes), the error is also small and
is often neglected. However, care needs to be taken to avoid accumulation of
the mass-imbalance error.
When the grid location is known at each time level, inclusion of grid
movement in the solution procedure is simple; see DemirdiiC and PeriC (1990)
for more details and an example. When the boundary movement is not known
in advance, an iterative procedure has to be used at each time step (or outer
iteration).
In some implicit time-integration methods (so-called fully-implicit methods), in which fluxes and source terms are computed only at the newest time
level, grid motion can be ignored everywhere except near boundaries. Examples of such methods are the implicit Euler scheme and the three-time-level
scheme, see Chap. 6. Since the fluxes are computed at time level tn+l, we do
not need to know where the grid was (or what shape the CVs had) at the
previous time level t,: instead of Eq. (12.8) we can use the usual equation
for a space-fixed CV:
The two equations differ in the definition of the rate-of-change and convective
terms: for a space-fixed CV convective fluxes are computed using fluid velocity, and time derivative represents the local rate of change at a fixed point
in space (e.g. CV-center). On the other hand, for a moving CV convective
fluxes are computed using relative velocity between fluid and CV-surface, and
the time derivative expresses the rate of change in a volume whose location
changes. If the CV-surface moves with the fluid velocity the same fluid re-
379
For incompressible flows, the density is constant so pF = p = const. Compressible flows require special care in order to determine p; such that both
space and mass conservation equations are satisfied. The problem is that the
mass conservation equation (12.19) contains the cell-center density in the unsteady term and cell-face densities in the mass fluxes, so that the evaluation of
pF is not trivial if the density varies rapidly in both space and time. If it does,
small time steps are necessary so that the approximation p; = 1/2(pr+pY+')
is sufficiently accurate.
The importance of taking the space (or geometric) conservation law into
account in unsteady flows with moving boundaries has been recognized by
many authors; see Thomas and Lombard (1979) and DemirdiiC and PeriC
(1988) for a more detailed discussion. Using coordinate transformations or
calculating the grid velocities from the motion of the cell-face center (see Eq.
(12.17)) leads to artificial mass sources or sinks. The error depends on the
time step and grid velocity, (see Eq. (12.12)). When the time step is small
(which is usually the case with explicit schemes), the error is also small and
is often neglected. However, care needs to be taken to avoid accumulation of
the mass-imbalance error.
When the grid location is known at each time level, inclusion of grid
movement in the solution procedure is simple; see DemirdiiC and PeriC (1990)
for more details and an example. When the boundary movement is not known
in advance, an iterative procedure has to be used at each time step (or outer
iteration).
In some implicit time-integration methods (so-called fully-implicit methods), in which fluxes and source terms are computed only at the newest time
level, grid motion can be ignored everywhere except near boundaries. Examples of such methods are the implicit Euler scheme and the three-time-level
scheme, see Chap. 6. Since the fluxes are computed at time level tn+l, we do
not need to know where the grid was (or what shape the CVs had) at the
previous time level t,: instead of Eq. (12.8) we can use the usual equation
for a space-fixed CV:
The two equations differ in the definition of the rate-of-change and convective
terms: for a space-fixed CV convective fluxes are computed using fluid velocity, and time derivative represents the local rate of change at a fixed point
in space (e.g. CV-center). On the other hand, for a moving CV convective
fluxes are computed using relative velocity between fluid and CV-surface, and
the time derivative expresses the rate of change in a volume whose location
changes. If the CV-surface moves with the fluid velocity the same fluid re-