# 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 [11] and [2] for a similar point of view).

There is another, complementary point of view pre-

sented in [7]. 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.

111.2 Extensions

The generalization of the analysis in [7] 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)

k=0

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 inthis "cell":

1 Jx+Ax/2

u(x) u(x')dx' .

x-Ax/2

An elementary calculation leads to the result:00

u(x) = E [akcos

k=o(2rkx )

(2)

(3)

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

written asE(k) = (a2 + b2)f2 (7rkAx)

L )

= E(k) f2 L7~x .(5)

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-

tions.

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 termE(k)

f2 (irk/N)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 N2E1(k) _ E2(k)

f2(7k/N1) f2(7k/N2)(7)

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

1/N2.3

(6)

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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.