376
12. Special Topics
Under the assumptions made above, the difference in grid velocities at the
opposite CV sides can be expressed as (see Fig. 12.1):
By substituting these expressions into Eq. (12.10) and noting that ( d o ) " + ' =
(AxAy)"+' and
= AX)^+' - 6 ~ ] [ ( A y ) ~ + '
- 6y], we find that the
discretized mass conservation equation is not satisfied - there is a mass source
The same error (with opposite sign) is obtained with the explicit Euler
scheme. For constant grid velocities it is proportional to the time step size,
i.e. it is a first-order discretization error. One might think that this is not
a problem, since the scheme is only first-order accurate in time; however,
artificial mass sources may accumulate with time and cause serious problems.
The error disappears if only one set of grid lines moves, or if the grid velocities
are equal at opposite CV sides.
Under the above assumptions, both the Crank-Nicolson and three-timelevel implicit scheme satisfy the continuity equation exactly. More generally,
when the fluid and/or grid velocities are not constant, these schemes can also
produce artificial mass sources.
Mass conservation can be obtained by enforcing the so-called space conservation law (SCL) which can be thought of as the continuity equation in
the limit of zero fluid velocity:
This equation describes the conservation of space when the CV changes its
shape and/or position with time.
In what follows the implicit Euler scheme for time integration is used for illustration; the extension to higher-order schemes is straightforward, but more
complicated. For spatial integration we use the midpoint rule and centraldifference schemes.
Equation (12.13) reads, in discretized form:
The difference between the 'new' and the 'old' CV volume can be expressed
as the sum of volumes 60, swept by the CV faces during the time step, see
Fig. 12.2:
12. Special Topics
Under the assumptions made above, the difference in grid velocities at the
opposite CV sides can be expressed as (see Fig. 12.1):
By substituting these expressions into Eq. (12.10) and noting that ( d o ) " + ' =
(AxAy)"+' and
= AX)^+' - 6 ~ ] [ ( A y ) ~ + '
- 6y], we find that the
discretized mass conservation equation is not satisfied - there is a mass source
The same error (with opposite sign) is obtained with the explicit Euler
scheme. For constant grid velocities it is proportional to the time step size,
i.e. it is a first-order discretization error. One might think that this is not
a problem, since the scheme is only first-order accurate in time; however,
artificial mass sources may accumulate with time and cause serious problems.
The error disappears if only one set of grid lines moves, or if the grid velocities
are equal at opposite CV sides.
Under the above assumptions, both the Crank-Nicolson and three-timelevel implicit scheme satisfy the continuity equation exactly. More generally,
when the fluid and/or grid velocities are not constant, these schemes can also
produce artificial mass sources.
Mass conservation can be obtained by enforcing the so-called space conservation law (SCL) which can be thought of as the continuity equation in
the limit of zero fluid velocity:
This equation describes the conservation of space when the CV changes its
shape and/or position with time.
In what follows the implicit Euler scheme for time integration is used for illustration; the extension to higher-order schemes is straightforward, but more
complicated. For spatial integration we use the midpoint rule and centraldifference schemes.
Equation (12.13) reads, in discretized form:
The difference between the 'new' and the 'old' CV volume can be expressed
as the sum of volumes 60, swept by the CV faces during the time step, see
Fig. 12.2: