LES Modelings based on the Lagrangian Renormalized Approximation

LES Modelings based on the Lagrangian Renormalized Approximation Kyo Yoshida, Takashi Ishihara, Daishi Fujita, Tomomichi Yamahira, and Yukio Kaneda Graduate School of Engineering, Department of Computational Science and Engineering, Graduate School of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan A systematic way for formulating large-eddy simulation (LES) models based on the Lagrangian renormalized approximation is presented. A spectral LES to simulate the energy spectrum and a probabilistic LES to simulate both the energy spectrum and the error spectrum are formulated for 3-dimensional turbulence and 2dimensional turbulence with the inverse energy cascading range. The derived models are free from any ad hoc parameters. The models are veri ed using high resolution numerical simulations with 5123 grid points for 3-dimension and 10243 for 2-dimension. Abstract.

1

Introduction

Environmental and industrial ows, in general, have huge degrees of freedoms and wide ranges of time and length scales. There are practical needs to solve problems concerning such ows; to predict their time evolution with a given initial condition, to obtain their mean pro les averaged over long time period, etc. . With recent development of the computer hardware, the numerical simulations are acquiring the potential to cope with such problems. However, the most powerful supercomputers today still can not take account of the all degrees of freedoms involved in such ow systems. Therefore the number of degrees of freedoms must be reduced by some mean in the simulation . One of such methods is a large-eddy simulation (LES) in which small scale eddies which fall into subgrid scale are modeled, while large scale eddies are directly solved. Note that the sub-grid scale eddies whose individual information is lost are strongly coupled with the resolved scale eddies in general turbulent ows. Therefore one can hardly expect LES to simulate the individual ows exactly at resolved scales over a long time period. LES may only simulate some averaged quantities, or in other words statistical quantities, where the average may be taken over a spatial domain, a time period or some ensemble, depending on the ow under consideration. What statistical quantities LES can simulate should depend on the details of the modeling. In this paper, a systematic way of developing LES models based on the Lagrangian renormalized approximation (LRA) is presented. The LRA is the simplest 2-point closures among few ones free from any ad hoc parameters and gives the proper Kolmogorov energy spectrum E (k ) = Ko 2=3 k05=3 as its prediction up to the constant factor (Ko = 1:72) [1,2]. Here, k is the wavenumber and

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 is the energy dissipation rate per unit mass. The guiding principle to construct

the LES models is the universality assumption, that is, although the large scale eddies of turbulence are strongly in uenced by external conditions (geometry, type of boundary conditions and forcing), the statistics of small-eddies at suf ciently high Reynolds number are insensitive to the details of these external conditions. The assumption is widely supported by experiments and observations. The LES models are derived for when subgrid scales are in the universal scale range. The statistical quantities to simulate are speci ed rst and then the form of the LES model is determined accordingly. The model parameters are estimated systematically and without any ad hoc parameters. In Section 2, a spectral LES model which aims at simulating the energy spectrum is formulated and the result of the LES is compared with that of a high resolution direct numerical simulation (DNS). In Section 3, the concept of an error due to the uncertainty in the subgrid scales is introduced and a probabilistic LES (PLES) model which aims at not only simulating the energy spectrum but also estimating the error is formulated. The PLES result is compared with the corresponding result of high resolution DNS. In Section 4, LES and PLES models are formulated for 2-dimensional turbulence with inverse energy cascading scale range. In Section 5, further application of the present method is discussed. 2

Spectral LES model

We consider 3-dimensional statistically homogeneous turbulent velocity eld which obeys the incompressible Navier-Stokes equations. The equations are given in spectral form by

@

@t

+ k

2



ui (k) = Mimn (k)

kiui(k) = 0 with

X

k=p+q

um(p)un(q) + fi (k; t);

(1) (2)

Mimn (k) = 0 2i [kmPin (k) + knPim (k)]; Pij (k) = ij 0 kkikj : 2

(3)

Here, u(k);  , and f (k) are a velocity eld, kinematic viscosity and an external solenoidal force eld respectively. Here and hereafter, we use the summation convention for repeated indices. Throughout this paper, we consider the sharp spectral lter. The ltered eld u< and the subgrid eld u> in wavevector space are de ned by



u< (k) = u(0k)

k < kc k  kc ;



u> (k) = u0(k; )

k < kc ; k  kc

(4)

where kc is a cut-o wavenumber. The model eld in LES, which we will denote ~ , is supposed to simulate u< in some statistical sense. In this section, we by u

LES Modelings based on the Lagrangian Renormalized Approximation

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formulate a LES model equation which simulates the energy spectrum. The energy spectrum of the velocity eld u is de ned by

E(k) = 12

X

hu(k0) 1 u(0k0 )i;

(5)

k01=2 to ltered eld u< . The LRA estimates the Kolmogorov constant to be o(2D) = 7 41. Substituting (59) with the LRA estimate of o(2D) in (53), (54) and (55) yields

K

K

the following expressions:

e D (kjkc ) =  = kc0 = e D 3 (k=kc ); e D (kjkc ) =  = kc0 = e D 3(k=kc ); Fe D (kjkc ) = 2Ko D kc = f D 3 (k=kc); (2

)

1 3

4 3 (2

)

(2

)

1 3

4 3

)

(2

)

(2

)

(2

5 3

(2

)

:

(60) (61) (62)

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HNS LES

0.001 ∝k−5/3

E(k)

0.0001 1e-05 1e-06 1e-07 1

10

100

k: wavenumber

Fig. 8.

The energy spectrum E (k) of HNS and E~ (k) of LES at t = 20 = 170T

0.01

0.001 ∝k−5/3

0.0001 1e-05 1e-06

~ ~ E(k),∆(k)

0.001

E(k),∆(k)

~ ~ E(k)(t=20) ∆(k)(t=1−20)

0.01

E(k)(t=20) ∆(k)(t=0−20)

∝k−5/3

0.0001 1e-05 1e-06

1e-07

1e-07 1

10 k: wavenumber

100

1

10

100

k: wavenumber

Energy spectrum E (k) and error spectra 1(k) of the HNS (left) and E~ (k) and 1~(k) of the PLES (right). The error spectra are plotted with the time interval 1.

Fig. 9.

where the normalized eddy viscosities e(2D)3 and e(2D)3 and the normalized D)3 are dimensionless functions. These functions were numerically eddy force (2 f computed and are shown in Fig. 6. Note that both eddy viscosities e(2D) (k jkc ) D) and (2 e (k jkc ) are negative for small k=kc . This means that the eddies whose scales are much larger than 1=kc receive energy from the subgrid scale eddies. Since the energy is transfered from the subgrid scale to the resolved scale, the transfer  is determined by the subgrid eld u> rather than by the ltered eld u< . We assume statistical stationarity in the subgrid scale and put e(2D) (kjkc), (2e D)(kjkc) and Fe(2D)(kjkc ) to be constant in time. We replace  in (60), (61) and (62) by [E (kc )=(Ko(2D) )]3=2 kc5=2 with E (kc ) obtained from the initial data of the high resolution numerical simulation. The LES and PLES of 2-dimensional turbulence with the inverse energy cascade range were performed with 2562 grid points (kc = 120) and the results were compared with the corresponding high resolution numerical simulations (HNS) with 10244 grid points. By using the similar procedures as those in Section 2 and 3, the initial conditions of HNS, LES and PLES were generate from an almost statistically stationary state of forced 2-dimensional turbulence with the energy inverse cascading range obtained in Ref. [12] ( = 0 in Series E). See

LES Modelings based on the Lagrangian Renormalized Approximation 0.7

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HNS PLES

0.6

∆total/E