Code Manual for CONTAIN 2.0 - Federation of American Scientists

In the VDI model, if MSTABLE is not specified, Fij is determined dynamically, using an initial value
~ = ?/’2.The motion is governed by the rate at which one considers the flow path to become filled ~
with material from the upstream cell:
dFij _ KW..
——
clt OUti!.
lJ
(4-11)
where K is a dimensionless acceleration factor defined below, Wij is the mass flow rate in the path,
as determined from the flow equation, pa is the upstream, or donor, gas density, LJj = max( IH: -
H{1,~j) is the effective filling length of the flow path, and ~j is the inertial length. Equation (4-1 1)
is coupled to the momentum equation through Equation (4-10) above and is solved with the
constraint O < Fij s 1. Note the value K = 1 when L’ = L would correspond to physically filling an
area equal to the actual flow path area ~j over the inertial length L, and K = m corresponds to an
instantaneous flow path filling, or donor, assumption. In practice, for reasons discussed below, large
values of K are used to accelerate the filling. Much of the time the VDI method gives essentially the
same results m a strtightforwmd donor cell approach, which corresponds to setting Fij = 1 if Wij>O
and Fij = O if Wijin Equation (4-10). This occurs whenever flow in one direction has persisted
for a sufficient time to cause Fij to be pinned at its maximum or minimum value. The time required
for pinning to occur is clearly reduced as K is increased.
As discussed above, if MSTABLE is specified, Fij is simply set equal to Yf2. This value for Fij
corresponds to treating stratifications as metastable over the vertical rise spanned by the flow path, ~
since Fij cannot respond to flow to provide a restoring force for a stable stratification or a
destabilizing force for an unstable stratification.
It should be noted that in the derivation of CONTAIN momentum equation in Table 4-2, the
changeover in density within the flow path to that based on the donor cell is assumed to occur
instantaneously. Thus, Equation (4-1 1) is not strictly consistent with the assumptions of the
momentum equation unless K is effectively infinite. In practice, the value used for K is chosen
sufficiently large (z 10) to give results close to donor cell results but not so large that the
discontinuities associated with the donor cell approach result in numerical problems. The value used
is
K=max 1o,-
[1L!.
g(Atfy
where AL is the flow timestep.
(4-12)
If Fij were in fact solved using values of K close to the “physical” value, in contradiction to the
momentum equation assumption, gravity wave behavior would in general be observed in the
presence of stable stratifications. If one views the density difference between cells as representing
a sharp stratification interface, then these gravity waves would cause unphysical mixing across the
Rev. O 4-24 6/30/97