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forcing.doc
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Notes on Isotropic Homogenious Forcing
Daniel Livescu and Mark Taylor
last revised on: 6/19/20002
There seem to be two types of forcing:
1. Deterministic.
Examples include:
Chen, Doolen, Kraichnan and She, Phys. Fluids A, 1993
Sullivan, Mahalingam, and Kerr, Phys. Fluids 1994
Overholt and Pope, Computers and Fluids, 1998
All of these forcings are actually quite similar. The equations are
forced in the form:
u_t(K) + navier_stokes_terms = f(k) u(K)
Where k is the spherical shell wave number index, and K is the full
3-component wave number, K=(k1,k2,k3) and k=|K|. You specify a model
spectrum, Em(k). The forcing f(k) is a function of the energy at the
current time E(k) and the model spectrum Em(k), and is chosen to relax
E(k) back to Em(k) for some values of k. The different forcing
techniques cited above just differ in the details of how f(k) is
chosen at each time step. Once equillibrium is obtained, they all
give similar resutls.
Isotropy problems: The isotropy problems we have been seeing are
related not to the forcing methodology, but instead to the choice of
model spectrum Em(k). It is appealing to use the following for the
model spectrum:
Em(k) = k^{-5/3} k <= 2.5
Em(k) = 0 k > 2.5
because it will make for the largest possible inertial range. But in
practice this produces poor results.
The energy is peaked in wave numbers k=1 and 2. These scales
are inherently anisotropic since they feel the effects of the
anisotropic geometory (a periodic box).
In an unforced flow, these large scales evolve on an extremely slow
time scale. This type of deterministic forcing has no effect on the
phase, it only relaxes the amplitude (with the same coefficient in
each shell). Thus the time scale for the evolution of any anistropy
is not effected by the forcing. In any single forced simulation,
anisotropy in the k=1 and k=2 modes will be retained for the entire
simulation. The only way to obtain isotropic results is to repeat the
simulation with many different initial conditions and average over all
realizations.
A better approach is to use a more realistic spectrum peaked at wave
number between 4 and 10 (Daniel recommends 10), with a steep increase
(for exampe: k^4) from wave number 1 up to the peak. This is what is
used in Overholt and Pope, and they show very good isotropy results.
With this type of model spectrum, there is very little energy in the
large scales which minimizes their influence on the statistics.
advantages: smooth forcing, less variablity in statistics
disadvantages: some loss of resolution since spectrum cannot be peaked
at k=1
2. Stochastic forcing:
Ref: Gotoh, Fukayama, Nakano Phys. Fluids 2002
They use a stochastic forcing of the form:
u_t(K) + navier_stokes_terms = f(K)
(as before, K=(k1,k2,k3) and k=|K|). To make f divergence free,
we define f= curl(psi), and w = -laplacian(psi) and w(K) is
chosen randomly and uncorrelated in space
and time for k<=2.5. The forcing is normalized by specifing
the variance in the first two shells:
F(1) = sum over k=1 of <w(K),w(K)>
F(2) = sum over k=2 of <w(K),w(K)>
The strong, uncorrelated nature of the forcing dramatically lowers the
time scale on which the k=1 and k=2 modes evolve. The results are
very isotropic, but the statistics have a higher variability when
compared to deterministic forcing, and thus longer time integrations
are needed to obtain meaningfull averages.
advantages: largest possible inertial range
disadvantages: non-smooth forcing, longer integrations needed for good
statistics