Implicit turbulence modeling for high reynolds number flows. Page: 4 of 9
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sive terms similar to those in the MPDATA modified
equation. The rationale for implicit turbulence mod-
eling then is the more accurate approximation of the
finite-volume governing equations by MDPATA (and
more generally, by NFV algorithms).
The finite scale Burgers' equation merits further dis-
cussion. In particular, one might be tempted to inter-
pret the nonlinear dispersive terms as a model for the
effects of the unresolved scales of motion - i.e., as a sub-
grid scale stress. However a careful examination of the
derivation shows that this is not the case, a point that
is emphasized by the authors. In fact, the finite scale
equation governs the evolution of the volume-averaged
velocity independent of the details of the subgrid scale
velocity field. In other words, the nonlinear dispersive
term regularizes the flow in much the same way that
artificial viscosity regularizes shocks in high speed flows
(see  and  for a similar point of view).
There is another, complementary point of view pre-
sented in . The finite scale equation can be consid-
ered as a model for the measurements (experimental or
computational) made by some observer. Experimental
devices and simulations both have finite scales of length
and time, implying that information about unresolved
scales is lost and the measurements do not exactly cor-
respond to "the flow". Since we are in a regime of classi-
cal physics, we are not concerned that the measurement
process alters the flow, and so we should expect that
the measurements accurately reflect the resolved scales.
In this sense, the finite scale equation is a better model
of the observations than the point equation. One of
our principal results in section IV. is to verify that the
MPDATA simulations do accurately reproduce the large
scales of the turbulence.
The generalization of the analysis in  to 3D Navier-
Stokes is beyond the scope of this paper. Instead, we
shall assume that the basic result of the Burgers' equa-
tion analysis - that MPDATA estimates the volume-
averaged velocity -remains true for more general equa-
tions. This simple and reasonable assumption will allow
us to predict the relationship of turbulent energy spec-
tra produced at different resolutions. The verification
of this relationship then reinforces our understanding of
the performance of MPDATA.
Consider a 1D periodic domain of length L. A measur-
able velocity u(x) can be expanded in a Fourier series:
u(x) = E [akcos (2Lx + bksin (2.x (1)
Now consider a small segment [x- Ax/2, x+Ax/2]. In
a 1D simulation, this would represent a computational
cell, and Ax/L - 1/N where N is the total number of
cells. We now compute the averaged component t in
u(x) u(x')dx' .
An elementary calculation leads to the result:
u(x) = E [akcos
bksi. (2xrk) f (7rkAx)
where the function
f(X) = SinX
attenuates each of the spectral coefficients of the original
velocity in a wavenumber dependent fashion. The 1D
volume-averaged energy associated with it can now be
E(k) = (a2 + b2)f2 (7rkAx)
= E(k) f2 L7~x .
These calculations can be easily extended to 3D; how-
ever using the anticipated isotropy of the velocity field,
the above result can be applied directly in our simu-
lations. In fact we have calculated the energy spectra
shown later in Figs. 2, 3, and 4, by averaging the 1D
spectra calculated in each of the three coordinate direc-
Let us now identify u as the discrete values of the
velocity components in an MPDATA simulation. Equa-
tion (5) can be used to estimate the underlying energy
spectrum in part - up to the finite wavenumber deter-
mined by the resolution of the simulation. We term
the asymptotic spectrum. Furthermore, since the simu-
lated spectrum at each resolution has the same asymp-
totic spectrum, we can relate the energy spectral coef-
ficients of two simulations at resolutions N1 and N2
E1(k) _ E2(k)
where this relation holds up to the largest wavenumber
of the more coarsely resolved simulation.
We close this section with these remarks about our
result. First, the derivation of eqs. (6) and (7) does not
depend on the form of the governing equations, and are
not specific to Navier-Stokes. Second, these equations
allow us to estimate an asymptotic spectrum, given a
simulated spectrum at finite resolution, but do not pre-
dict any universal form for this spectrum. Third, for
small values of its argument, eq. (6) implies convergence
of the simulated spectra to the asymptotic spectrum as
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Margolin, L. G.; Smolarkiewicz, P. K. (Piotr K.) & Wyszogrodzki, A. A. (Andrzej A.). Implicit turbulence modeling for high reynolds number flows., article, January 1, 2001; United States. (digital.library.unt.edu/ark:/67531/metadc926043/m1/4/: accessed February 16, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.