Implicit turbulence modeling for high reynolds number flows. Page: 2 of 9
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Implicit Turbulence Modeling for High Reynolds Number Flows *
L.G. Margolin t
P.K. Smolarkiewicz, I
We continue our investigation of the implicit tur-
bulence modeling property of the nonoscillatory fi-
nite volume scheme MPDATA. We start by compar-
ing MPDATA simulations of decaying turbulence in
a triply periodic cube with analogous pseudospec-
tral studies. In the regime of direct numerical sim-
ulation, MPDATA is shown to agree closely with
the pseudospectral model. As viscosity is reduced,
the two model results diverge. We study the MP-
DATA results in the inviscid limit, using a combi-
nation of mathematical analysis and computational
experiment. We validate the inviscid MPDATA re-
sults as representing the turbulent flow in the limit
of very high Reynolds number.
There is a kind of magic about nonoscillatory meth-
ods for numerical simulation of complex fluid flows. Be-
yond the obvious benefits of avoiding unphysical oscilla-
tions by preserving positivity and/or monotonicity, and
of ensuring nonlinear stability, high-order upwind meth-
ods appear to correlate well with the underlying physics,
*This work performed under the auspices of the U.S. Depart-
ment of Energy by Los Alamos National Laboratory under Con-
tract W-7405-ENG-36. This paper is declared work of the U.S.
Government and is not subject to copyright protection in the
nt Cater for Nonlinear Studies, Theoretical Division, Los
Alamos National Laboratory, MS B258, Los Alamos, NM 87545,
t Microscale and Mesoscale Meteorology Division, National
Center for Atmospheric Research, Boulder, CO, 80307,
5Institute for Geophysics and Planetary Physics, Earth and
Space Sciences Division, Los Alamos National Laboratory, MS
C305, Los Alamos, NM 87545, wyszogekokopelli.lanl.gov; on
leave from Institute of Geophysics, Warsaw University, Warsaw,
leading to simulations that are more physically realiz-
As a recent and unexpected example of realizability,
nonoscillatory methods have demonstrated the ability
to simulate turbulent flows without need for explicit
subgrid scale models, a property that we refer to as
"implicit turbulence modeling." This property has been
validated in direct comparisons with experimental data
and with high resolution DNS (direct numerical simula-
tion) for a variety of flows, and for a variety of nonoscil-
latory algorithms (see for example , , , ) over
the past ten years. In our own research, we have em-
ployed the nonoscillatory algorithm MPDATA (for Mul-
tidimensional Positive Definite Advection Transport Al-
gorithm; see  and references therein) to model all-
scale meteorological flows including atmospheric bound-
ary layers , gravity-wave dynamics , and global
More recently, the beginnings of a theoretical frame-
work has been proposed for implicit turbulence mod-
eling in . These authors derived a finite-scale (i.e.,
coarse-grained) version of the point-wise Burgers' equa-
tion - a version appropriate for describing the dynam-
ics of finite volumes of (Burgers') fluid. They compared
this finite-scale equation to the MPDATA approxima-
tion of the point-wise equation, and showed that MP-
DATA already accounts for the finite-scale effects. Since
each computational cell is a finite volume, they rational-
ized that the success of MPDATA in modeling turbu-
lent flows results from its accurate representation of the
coarse-grained equations of motion.
The theory in  and the computational examples
that support it are suggestive. However there are sig-
nificant differences between Burgers' and Navier-Stokes
equations. Perhaps the most important of these is
that the solutions of Burgers' equation are deterministic
while those of Navier-Stokes are stochastic. This differ-
ence may be readily appreciated by considering simula-
tions at two distinct resolutions. In the case of Burg-
ers' equation the two solutions will be close, while for
Navier-Stokes equations it is only the statistics of the
solutions that will be close .
In this paper, we extrapolate the finite-volume theory
of  to analyze nonoscillatory simulations of a turbu-
lent flow governed by 3D Navier-Stokes equations. We
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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/m1/2/: accessed November 16, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.