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LES approach coupled with stochasticforcing of subgrid acceleration in ahigh-Reynolds-number channel flowRémi Zamansky a , Ivana Vinkovic a & Mikhael Gorokhovski aa Laboratoire de Mécanique des Fluides et d'Acoustique , ÉcoleCentrale de Lyon–CNRS–Université Claude Bernard Lyon 1–INSALyon , F-69134, Écully, FrancePublished online: 27 Jul 2010.

To cite this article: Rémi Zamansky , Ivana Vinkovic & Mikhael Gorokhovski (2010) LES approachcoupled with stochastic forcing of subgrid acceleration in a high-Reynolds-number channel flow,Journal of Turbulence, 11, N30, DOI: 10.1080/14685248.2010.496787

To link to this article: http://dx.doi.org/10.1080/14685248.2010.496787

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Journal of TurbulenceVol. 11, No. 30, 2010, 1–18

LES approach coupled with stochastic forcing of subgrid accelerationin a high-Reynolds-number channel flow

Remi Zamansky, Ivana Vinkovic and Mikhael Gorokhovski∗

Laboratoire de Mecanique des Fluides et d’Acoustique, Ecole Centrale de Lyon–CNRS–UniversiteClaude Bernard Lyon 1–INSA Lyon, F-69134 Ecully, France

(Received 5 January 2010; final version received 23 May 2010)

In this paper, the high-Reynolds-number channel flow is simulated by numerical ap-proach at coarse resolution, in which the instantaneous acceleration is decomposed intofiltered and subgrid parts, and then both components are modeled. The filtered accel-eration is modeled in the framework of the large-eddy simulation approach. The modelfor the subgrid acceleration is based on two stochastic processes. The first is for itsnorm and is based on statistical universalities in fragmentation under scaling symme-try, providing correlation of subgrid forcing across the channel. The second is for itsorientation and is based on the Brownian motion on a unit sphere in order to representa stochastic relaxation toward full isotropy away from the wall. Two main parametersof the stochastic process include the Reynolds number based on the friction velocityand the channel half-width. In order to assess the capability of the model proposed, thepaper illustrates its application versus recent high-Reynolds-number direct numericalsimulations, including direct numerical simulations performed in this paper.

Keywords: Turbulent channel flow; intermittency; large-eddy simulation; subgrid-scalemodel

1. Introduction

Application of large-eddy simulation (LES) in turbulent channel flow becomes too ex-pensive when the Reynolds number is high and the resolution of strong gradients in thenear-wall region is needed. Starting from the earlier study of Moin and Kim [1], differentmodifications of this approach have been proven successful (see [2–6]). Among them oneis to incorporate direct simulation of subgrid-scale (SGS) turbulence into LES techniqueson the basis of estimation of the unresolved velocity field. Approaches proposed by Do-maradzki and Adams [7], Park and Mahesh [8], and Westbury et al. [9] are focused onsimulation of the residual-stress tensor. The model proposed by Schmidt et al. [10] is basedon the LES–1-D turbulence coupling [11, 12] and provides stochastically the unresolvedvelocity field as the wall is approached. The approach of Kemenov and Menon [13] is basedon the simulation of a turbulent velocity field, which is defined by superposition of large-scale and small-scale components. The latter is computed directly on SGSs by a simplified1-D motion equation (in three directions). It is worth noting that the above reported SGSmodels are invariant to the Reynolds number. Thereby, in these models, the intermittencyeffects are discarded on SGSs.

∗Corresponding author. Email: mikhael.gorokhovski@ec-lyon.fr

ISSN: 1468-5248 online onlyC© 2010 Taylor & Francis

DOI: 10.1080/14685248.2010.496787http://www.informaworld.com

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2 R. Zamansky et al.

In the approach proposed by Sabel’nikov et al. in [14], the key element of the SGS modelis acceleration in the residual flow. This acceleration has been modeled stochastically in thecase of 3-D box turbulence, with statistical properties depending on the Reynolds number.The general idea of this approach, referred to as LES–stochastic subgrid acceleration model(SSAM)), consists of the following. The total instantaneous acceleration

dui

dt= − 1

ρ

∂p

∂xi

+ ν∂2ui

∂xk∂xk

, (1)

∂uk

∂xk

= 0,

is composed of two contributions:

dui

dt=

(dui

dt

)+

(dui

dt

)′, (2)

where dui

dt= ∂ui

∂t+ uk

∂ui

∂xk. The first component is the filtered total acceleration

(dui

dt

)= dui

dt+ ∂ (uiuk − uiuk)

∂xk

,

= − 1

ρ

∂p

∂xi

+ ν∂2ui

∂xk∂xk

, (3)

∂uk

∂xk

= 0,

where the overbar denotes the filtering operation and ui denotes the filtered velocity. Thesecond component represents the total acceleration in the residual field

(dui

dt

)′= ∂u′

i

∂t+ ∂ (ukui − ukui)

∂xk

,

= − 1

ρ

∂p′

∂xi

+ ν∂2u′

i

∂xk∂xk

, (4)

∂u′k

∂xk

= 0.

If both parts are modeled, their sum gives an approximation to (1).The first assumption is to replace (3) by a classical LES equation with a simple eddy-

viscosity model

(dui

dt

)�

= dui

dt− ∂

∂xk

[νt

(∂ui

∂xk

+ ∂ui

∂xk

)],

= − 1

ρ

∂p

∂xi

+ ∂

∂xk

[ν

(∂ui

∂xk

+ ∂ui

∂xk

)], (5)

∂uk

∂xk

= 0,

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Journal of Turbulence 3

where νt denotes the eddy viscosity and subscript � is the length scale associated with thefilter size.

The second assumption is to replace (4) by

(dui

dt

)′

S

= − 1

ρ

∂p∗∂xi

+ a′i , (6)

∂u′k

∂xk

= 0,

where p∗ is the pressure providing the subgrid velocity model field to be solenoidal and ai

is attributed to the stochastic acceleration of the residual flow.The third assumption is to replace the sum of (5) and (6) by the model equation for a

surrogate field, denoted hereafter by the symbolˆ

dui

dt= − 1

ρ

∂p

∂xi

+ ∂

∂xk

[(ν + νt )

(∂ui

∂xk

+ ∂uk

∂xi

)]+ a′

i . (7)

In this equation, the new pressure field p is such that

∂uk

∂xk

= 0. (8)

Equations (7) and (8) will be applied here to a high-Reynolds-number channel flow, with anew subgrid acceleration model for a′

i .In high-Reynolds-number channel flows, the acceleration is a highly intermittent vari-

able. It has been reported by direct numerical simulations (DNS) [15] that this intermittencyis mostly related to quasi-coherent vortical structures in the wall layer. A fluid particle,trapped in such a vortex, is subjected to an intense centripetal acceleration. It was re-vealed [15] that these fluid particles preserve the magnitude of their acceleration longerthan its direction. This finding is consistent with the experimental studies [16–18] of ahigh-Reynolds-number “free” turbulence. In wall-bounded turbulent flow, the intermittencyeffects are manifested by deep cross-channel interaction between energy containing eddiesof very different length scales [19]. Recent DNS studies identified mechanisms, which maybe responsible for such turbulent correlations across the channel. In the near-wall region,there are upward ejections of vortical fluid, as a part of vortices–streaks regeneration cy-cle [20–25]. Above the buffer layer, where the Reynolds-number-dependent flow [26, 27]has intermittent streaky structure as well [28–30], the pressure fluctuations acting on theflow in the buffer zone and in the viscous sublayer may generate eruptions of small-scalewall structures toward outer flow [31–33]. Besides, the unsteady bursting events, when thelarge outer-flow eddies appear closely to the wall, have been reported in [34]. ConcerningLES of such flow, the significant role of the “under-resolved” momentum transfer acrossthe channel was emphasized in [35].

The question raised in this study is as follows. Despite the huge number of very differentturbulent structures produced by different random realizations in the wall-bounded flowand interacting across the scales, is it possible to match recent DNS statistics by a rathersimple model in the framework of LES-SSAM approach. The purpose of this paper is topropose such a model with assessment of its capability by comparison with DNS. Thesubgrid acceleration model proposed here is based on the following assumptions: (1) thereare two independent stochastic processes, one for the norm, and another for the orientation;

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4 R. Zamansky et al.

(2) both processes are correlated over the channel height; (3) the stochastic process forthe norm of acceleration is given by the statistical universalities of fragmentation underscaling symmetry; (4) the stochastic process for the orientation relaxes toward isotropy inthe channel centerline. Two main parameters of the stochastic process include the Reynoldsnumber based on the friction velocity and the channel half-width.

This paper is organized as follows. The stochastic model for the nonresolved accelera-tion will first be presented. The models for the norm and the orientation of the nonresolvedacceleration will be described. Then, details about the performed numerical simulationswill be given. Finally, the comparison between the classical LES, DNS, and the LES-SSAMapproach will be presented and discussed.

2. Stochastic model for the nonresolved acceleration

The stochastic behavior of the subgrid acceleration is prescribed to its norm and to itsorientation. It is assumed that the orientation of subgrid acceleration is mostly controlledby orientation of vortex sheets and tubes, lying mostly in the streamwise–spanwise planenear the wall. These structures become randomly distributed with increasing distance fromthe wall. As to the subgrid acceleration norm, it is constructed in terms of wall parameters,following DNS observation [15], in which the significance of the solenoidal component ofacceleration very near to the wall was shown. In this study, the suggestion is to introduce thestochastic processes for this norm with correlation across the channel. The subgrid forcinga′ in (7) is therefore given by two independent stochastic processes, one for its norm |a|and another for its orientation ei , i.e.

a′i = |a|ei . (9)

This model will be different from the model proposed in [14], where authors introduceda model for the acceleration in homogeneous and isotropic turbulent flow given also bytwo independent stochastic processes, one for the norm of the acceleration and another onefor its orientation. In [14], the model for the acceleration norm has been derived from thelognormal model for the energy dissipation [36–40] (see also recent paper advocating thelognormal model, such as [41, 42]). Simultaneously, its orientation was generated randomlyin each time step of the order of the Kolmogorov time scale.

2.1. Model for the norm of the subgrid acceleration

We represent the norm of the nonresolved acceleration by a typical velocity increment at

the wall expressed as �u2

∗ν

and multiplied by a frequency f , which is a random variable

|a| = �u2

∗ν

f. (10)

Here u∗ is the friction velocity. High values of f correspond to strong velocity gradients.Usually, the stochastic process evolves with time. In order to introduce the cross-channelcorrelation in subgrid forcing by a stochastic process for f , we shall replace time by distancefrom the wall. Thus the random frequency f will evolve stochastically from the wall to theouter flow. To this end, we introduce the nondimensional evolution parameter τ , which is

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Journal of Turbulence 5

defined as follows:

τ = −ln

(h − y

h

), (11)

where h is the channel half-width and y is the wall distance. It can be seen from (11) thatwhen y = 0 then τ = 0, and when y → h then τ → ∞. Hence f evolves with τ , increasingfrom zero to infinity, correspondingly to y increasing from zero to the channel half-width.

The near-wall region is characterized by strong velocity gradients (high values of f ),which decrease in mean toward the outer flow through a highly intermittent dynamics. Weassumed that with increasing distance from the wall (increasing τ ), the frequency f ischanging by a random independent multiplier α (0 < α < 1), f → αf ; α is governed bydistribution q(α),

∫ 10 q(α)dα = 1. This multiplicative process, often referred to as frag-

mentation under scaling symmetry, requires knowledge of q(α), which is, in principle,unknown. However, if relaxation of f is fast, comparing with advancement from the wallto the outer region, the population balance equation, describing continuous evolution ofdistribution G(f ; τ ), can be reduced exactly to the Fokker–Planck equation [43]

∂G(f ; τ )

∂τ= −〈ln α〉 ∂

∂f(f G) + 〈ln2 α〉

2

∂

∂f

(f

∂

∂f(f G)

). (12)

Here G(f ; τ ) is the normalized distribution function,∫ ∞

0 G(f )df = 1. It should be noted

that only the first two logarithmic moments of α, denoted here as 〈lnk α〉 = ∫ 10 q(α) lnk αdα,

k = 1, 2, appear in the evolution of G(f ; τ ) and 〈ln α〉/〈ln2 α〉 = 〈ln f 〉/〈(ln f − 〈ln f 〉)2〉,with 〈ln f 〉 = ∫ ∞

0 G(f ) ln f d f.The solution of Equation (12) and its properties are described in [43]. It is shown

that with increasing time (here with increasing of parameter τ , O < τ < ∞ instead ofO < t < ∞), the initial distribution G(f ; τ = 0) goes first to the lognormal distribution

G(f ; τ ) = 1

F√

2π〈ln2 α〉τexp

⎛⎜⎝−

(ln f

F

)2

2〈ln2 α〉τ

⎞⎟⎠

(f

F

)1−〈ln α/〉〈ln2 α〉, (13)

where F is the initial scale of frequency, which is introduced below. It is worth noting thatoriginally the lognormal distribution has been identified by Kolmogorov [44] in his discretemodel of cascade breakage events. However, the Kolmogorov’s lognormality was derivedin the context of the central limit theorem. The analytical solution (13) of the populationbalance equation, obtained without appealing to the central limit theorem, shows that byfurther increase of τ , the “lognormal” multiplier in Equation (13), exp(−(ln f

F)2/2〈ln2 α〉τ ),

tends to unity, and the long-time limit distribution is determined by the power law withone universal parameter, 〈ln α〉/〈ln2 α〉. In our paper, instead of using those universalasymptotics, we use Equation (12) in order to derive the stochastic equation. In the Itointerpretation, it reads

df = (〈ln α〉 + 〈ln2 α〉/2)f dτ +√

〈ln2 α〉/2f dW (τ ), (14)

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6 R. Zamansky et al.

where dW (τ ) is a Wiener process (〈dW (τ )〉 = 0 and 〈dW (τ )2〉 = 2dτ , here and below thebrackets denote the ensemble average). Parameters are chosen in the following form:

〈ln α〉 = −Re1/3+ ,

〈ln2 α〉 = −〈ln α〉, (15)

where Re+ = u∗h/ν.The main properties of the stochastic process (14) are: (1) it provides a subgrid forcing

which is correlated across the channel height; (2) as the wall distance increases, it generatesevolution of frequency from high to low mean values through two intermediate universalasymptotics: the former is lognormal distribution, while the latter is power law, with highprobability of zero frequency and stretched tail of high frequency. This stretched tail isdetermined by Re+ and induces rare events of strong subgrid forcing in the outer flow,thereby mimicking bursting effects.

The starting condition (first grid cell, τ = 0) is given as follows. First, the mean valueof the frequency at the wall is introduced as f+ = u∗/λ, where λ is a Taylor-like scale. Itis estimated by Kolmogorov’s scaling in terms of wall parameters. The Reynolds number,based on friction velocity, is: Re+ = u∗h/ν = h/y0 ≈ Re

3/4h , where y0 is the thickness of

the viscous layer and Reh is the Reynolds number based on the centerline velocity. Onethen yields: λ ≈ hRe

−1/2h ≈ hRe

−2/3+ . Second, the starting value for the random path (14)

is sampled from the stationary lognormal distribution of f/f+.The probability density function (PDF) of the frequency f has been computed for

different wall distances by means of DNS in a channel flow with Re+ = 590. Detailsabout this simulation are given in Table 1. First, the norm of the acceleration was com-puted and then f was estimated by using Equation (10). As it can be seen from Figure 1,surprisingly, the model (Equations (14) and (15)) matches well with the results of DNS.In agreement with DNS data, the stochastic process provides a broad frequency distribu-tion in the wall region and relaxes toward low frequencies when distance from the wallincreases.

0.001

0.01

0.1

1

10

0 2 4 6 8 10

f/f+

y+= 1y+= 50

y+= 100

Figure 1. Distribution of f/f+ from SSAM (cross) and comparison with DNS (line) at Re+ = 590,for several distances from the wall.

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Journal of Turbulence 7

Tabl

e1.

Sum

mar

yof

para

met

ers

used

for

num

eric

alsi

mul

atio

ns.

Nam

eR

e +R

e cN

x×

Ny×

Nz

Lx×

Ly×

Lz

�x

+×

�y

+×

�z

+dt+a

Cs

A/h

DN

S18

032

8019

2×

193

×19

23π

h×

2h×

4 3π

h9.

0×

(0.0

2∼

3.0)

×4.

00.

030

——

DN

S58

712

,490

384

×25

7×

384

3 2π

h×

2h×

3 4π

h7.

2×

(0.0

4–7.

2)×

3.6

0.03

3—

—D

NS

b58

712

,547

384

×25

7×

384

2πh

×2h

×π

h9.

7×

(0.0

4–7.

2)×

4.8

——

—L

ES

587

14,1

6064

×65

×64

3πh

×2h

×π

h87

×(0

.71–

29)×

290.

10.

160.

015

LE

S-S

SA

M58

712

,760

64×

65×

643π

h×

2h×

πh

87×

(0.7

1–29

)×

290.

10.

160.

015

DN

S10

0022

,250

512

×38

5×

512

4 3π

h×

2h×

2 3π

h8.

2×

(0.0

3–8.

3)×

4.1

0.03

4—

—D

NS

c93

420

,960

3072

×38

5×

2304

8πh

×2h

×3π

h7.

6×

(0.0

6–7.

6)×

3.8

——

—L

ES

1000

25,4

3096

×97

×96

3πh

×2h

×π

h99

×(0

.53–

33)×

330.

10.

160.

009

LE

S-S

SA

M10

0023

,380

96×

97×

963π

h×

2h×

πh

99×

(0.5

3–33

)×

330.

10.

160.

009

LE

S10

0025

,500

64×

65×

643π

h×

2h×

πh

147

×(1

.2–4

9)×

490.

10.

20.

015

LE

S-S

SA

M10

0023

,700

64×

65×

643π

h×

2h×

πh

147

×(1

.2–4

9)×

490.

10.

20.

015

DN

Sc

2003

48,6

8061

44×

633

×46

088π

h×

2h×

3πh

8.2

×8.

9×

4.1

——

—L

ES

2000

49,3

5012

8×

129

×12

83π

h×

2h×

πh

147

×(0

.60–

49)×

490.

10.

160.

006

LE

S-S

SA

M20

0048

,950

128

×12

9×

128

3πh

×2h

×π

h14

7×

(0.6

0–49

)×

490.

10.

160.

006

LE

S20

0052

,640

64×

65×

643π

h×

2h×

πh

295

×(2

.4–9

8)×

980.

10.

20.

015

LE

S-S

SA

M20

0049

,050

64×

65×

643π

h×

2h×

πh

295

×(2

.4–9

8)×

980.

10.

20.

015

a dt+

=dt/

t ∗is

the

tim

est

epof

the

sim

ulat

ion;

t ∗=

ν/u

2 ∗is

avi

scou

sti

me

ofth

eor

der

ofth

eK

olm

ogor

ovti

me

scal

eat

the

wal

l.b[4

7].

c [48]

.

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8 R. Zamansky et al.

y, v

x, u

z, w

Flow

φ

θ

e1 = (θ1↪ φ1)

e2 = (θ2↪ φ2)

βγ

Figure 2. Coordinate system and definition of the angles φ, θ , γ, and β.

2.2. Model for the orientation of the acceleration vector

We assumed that with increasing distance from the wall, the orientation of subgrid ac-celeration relaxes stochastically from the streamwise–spanwise alignment (with randomazimuth) to full isotropy.

First, this assumption was assessed by DNS performed in this paper. The orientationvector ei in Equation (9) is determined by longitude φ and latitude θ

ei = a′i

|a| =

⎧⎪⎨⎪⎩

ex = cos(θ ) cos(φ),

ey = sin(θ ),

ez = cos(θ ) sin(φ),

(16)

where −π ≤ φ ≤ π characterizes orientation in the streamwise–spanwise (x, z) plane, and−π/2 ≤ θ ≤ π/2 defines orientation relatively to the normal-to-wall direction (θ = 0 andθ = ±π/2 will correspond to accelerations, which are parallel to the wall and normal to thewall, respectively). The schematic representation is given in Figure 2. If φ and θ are random,their PDF’s corresponding to full isotropy, have the following forms Pisotropic(φ) = 1/2π

and Pisotropic(θ ) =|cos(θ )|/2, respectively. The mean value for both distributions is zero,and computation of variance yields

< φ2 >isotropic =∫ π

−π

φ2Pisotropic(φ)dφ = π2

3, (17)

< θ2 >isotropic =∫ π/2

−π/2θ2Pisotropic(θ )dθ = π2

4− 2. (18)

By DNS, at Re+ = 180, 590, and 1000, one can obtain the statistics of φ and θ (seeSection 3, and especially, Table 1 for details about the resolution used in the simula-tion). Then 〈φ2〉 and 〈θ2〉 can be computed and compared with Equations (17) and (18),respectively. To this end, large-scale acceleration obtained by our DNS has been removedby using sharp spectral filtering in homogeneous directions and Gaussian filtering in thenormal to the wall direction. In such filtering operation, the mean of these two angles isnegligibly small. The results of computations are given in Figure 3 for different distancesfrom the wall. It can be seen that the variance of φ (Figure 3a) is close to the varianceof the isotropic distribution (Equation 17) for almost the entire range of y+ and for allthree Reynolds numbers. As to variance of θ (Figure 3b), it is negligibly small at the wall,and then it grows almost linearly with the distance from the wall up to y+ ≈ 30–50, after

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Journal of Turbulence 9

0

0.2

0.4

0.6

0.8

1

1.2

0 200 400 600 800 1000

<φ2 >

/<φ2 >

isot

ropi

c

y+

0.8

1

1.2

0 20 40 60 80 100

(a)

0

0.2

0.4

0.6

0.8

1

1.2

0 200 400 600 800 1000

<θ2 >

/ <

θ2 >is

otro

pic

y+

0 0.2 0.4 0.6 0.8

1 1.2

0 20 40 60 80 100

(b)

Figure 3. Variance of the angles (a) φ and (b) θ for the orientation of the acceleration at small scaleobtained from DNS, for Re+ = 180 (full line) Re+ = 590 (dotted line) and Re+ = 1000 (dashedline). The variances are normalized by the value of the variance for an isotropic distribution of theorientation vector (Equations 17 and 18). Insert: zoom near the wall.

which it tends toward a constant value. This value is relatively close to the variance of theisotropic distribution of θ (Equation 18). From this observation, we assume that the subgridacceleration vector relaxes toward isotropy, reaching a nearly isotropic distribution at abouty+ ≈ 50.

In order to simulate this tendency toward isotropy, the stochastic model for the orien-tation (16) is constructed as a Brownian random walk on the surface of a sphere of unityradius. This random walk takes place with increasing distance from the wall. The evolutionof the unit vector ei is defined by the following stochastic process:

{γ = 2DdW,

0 ≤ β < 2π,(19)

where γ is the path length between two successive positions on the sphere and β is theinitial direction from point 1 to point 2 (see Figure 2). β is chosen randomly from theuniform distribution. D is a diffusion coefficient and dW is the standard Wiener processdefined by 〈dW 〉 = 0, 〈dW 2〉 = 2dy+, with dy+ representing the cell size (in wall units)in the wall-normal direction. The rules of this random walk are given by geodesic calculus

⎧⎪⎪⎨⎪⎪⎩

θk+1 = sin−1(sin θk cos γ + cos θk sin γ cos β),φk+1 = φk + arg(ξ ),�(ξ ) = sin β sin γ cos θk,

�(ξ ) = cos γ − sin θk sin θk+1,

(20)

where θk and φk are two angles corresponding to the node k, and ξ is a complex number,with real and imaginary part �(ξ ) and �(ξ ), respectively.

With increasing distance from the wall (y+), the diffusive equilibrium is attained as afinal state, which corresponds to the isotropic PDF’s

{Pθ → Pisotropic(θ ),Pφ → Pisotropic(φ),

(21)

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Figure 4. Realization of the random walk on the sphere given by Equations 19 and 20.

where, at the distance from the wall y+, Pθ (θ, y+) and Pφ(φ, y+) are the PDF’s of θ andφ, respectively. An example of one realization of such a Brownian motion on a sphere isgiven in Figure 4.

Correspondingly to DNS, the process starts on the wall (y+ = 0) with

{Pθ (θ, y+ = 0) = δ(θ ),Pφ(φ, y+ = 0) = Pisotropic(φ),

(22)

where δ is the Dirac distribution. This implies that the SGS acceleration at y+ = 0 is takento be parallel to the wall with random azimuth.

In our case, the diffusion coefficient D controls the relaxation rate toward isotropy.A linear regression of DNS data (see Figure 5 for y+ < 20) yields D = 0.01. Figure 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100

<θ2 >

y+

Figure 5. Evolution of the variance of θ . Comparison between DNS for Re+ = 180 (full line),Re+ = 590 (dotted line) and Re+ = 1000 (dashed line) and the stochastic model defined by Equa-tions 19, 20 and 22 (crosses). The straight line corresponds to the variance for an isotropic distribution.

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Journal of Turbulence 11

0

10

20

30

40

50

0.1 1 10 100 1000

U+

y+

Figure 6. Streamwise mean velocity, for Re+ = 590, Re+ = 1000, and Re+ = 2000 from bottomto top, respectively, shifted by 10 wall units upward. Square: LES; cross: LES-SSAM; dash: DNS(only for Re+ = 590 and Re+ = 1000); dots: DNS from [47] for Re+ = 590 and from [48] forRe+ = 1000 and Re+ = 2000. Long dashed line: logarithmic law of the wall.

shows statistics of θ computed from the model proposed here (Equations 19, 20, and 22,and D = 0.01) and from our DNS. It can be seen that the model matches well the DNSdata. For φ, with the prescribed wall condition (22), the model provides exactly 〈φ2〉 =〈φ2〉isotropic.

3. Characteristics of the numerical simulations

For assessment of the SGS acceleration model, we ran simulations of a pressure-driventurbulent channel flow. A pseudospectral computational code is used. The code uses thespectral approximation (Fourier Chebyshev) and a variational projection method on adivergence free space following the original idea of Moser [45]. The time integration usesthe explicit Adam–Basforth algorithm for nonlinear terms and the semi-implicit algorithmfor diffusion terms. A rotational form is used for convective terms in order to ensure energyconservation. Periodic boundary conditions were applied along the streamwise (x) andthe spanwise (z) directions, whereas the no-slip condition is imposed on the walls. Thecomputer code was developed in the scientific group of M. Buffat [46].

A posteriori tests have been done for three Reynolds numbers: Re+ = 590, 1000 and2000. Results of LES-SSAM approach were compared with the standard LES and the DNSgiven in the literature [47, 48], and also with the DNS performed specifically in the presentwork. The conditions for these three approaches were exactly the same, including the samecomputational mesh for LES and LES-SSAM. The parameters taken for these simulationsare summarized in Table 1.

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0

0.5

1

1.5

2

2.5

3

3.5

0 100 200 300 400 500 600

u i+

rms

y+

(a)

0

0.5

1

1.5

2

2.5

3

3.5

0 200 400 600 800 1000

u i+

rms

y+

(b)

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000

u i+

rms

y+

(c)

Figure 7. Standard deviation of streamwise (u), spanwise (w) and normal (v) velocity (in wallunites) for (a) Re+ = 590, (b) Re+ = 1000, and (c) Re+ = 2000, from top to bottom, respectively.Square: LES; cross: LES-SSAM; dash: DNS (only for Re+ = 590 and Re+ = 1000); dots: DNSfrom [47] for Re+ = 590 and from [48] for Re+ = 1000 and Re+ = 2000.

For LES and LES-SSAM simulations, the classical Smagorinsky model is used withwall damping function for the turbulent viscosity [4]

νturb = (Cs�fV D)2|S|,|S| = (

2SijSij

)1/2,

fV D = 1 − e−y/A,

(23)

where A is specified below. Here Cs is the Smagorinsky constant, � = (�x × �y × �z)1/3

is the characteristic grid size, Sij = 12 ( ∂ui

∂xj+ ∂uj

∂xi) is the resolved strain-rate tensor, fV D

is the Van Driest function, y is the distance from nearest wall, and A is the constantcontrolling the damping function fV D . As suggested by [49], for a detached-eddy simu-lation, the turbulent length scale in the wall layer is defined as � = min(y,�). In orderto fulfill this suggestion, in the Van Driest damping function, we choose A such that� × fV D ∼ min(y,�).

It should be noted that in this code, the Reynolds number is given by choosing ν andthe pressure gradient− 1

ρ

∂Pt

∂x. In order to attain the required Reynolds number, we used the

Dean’s correlation [50]: ν = 0.110UchRe−1.1296+ and− 1

ρ

∂Pt

∂x= Re2

+ν2/h3, where Uc is the

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Journal of Turbulence 13

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60

τ / τ

tot

y+

τturb / τtot

τvisc / τtot

Figure 8. Ratios of turbulent τturb = −ρ〈u′v′〉 and viscous τvisc = −ρν〈 ∂u

∂y〉 stresses compared

with the total one τtot = τvisc + τturb. Re+ = 1000. Square: LES; cross: LES-SSAM; dash: DNS.

centerline velocity. It can be noticed from Table 1 that the LES-SSAM matches better theDNS Reynolds number than the standard LES. Thereby, for a given set of parameters (ν and− 1

ρ

∂Pt

∂x), both the centerline velocity and the mass flow rate are improved by LES-SSAM

model.

1e-05

0.0001

0.001

0.01

0.1

1

0.0001 0.001 0.01 0.1 1

E(v

+)

k+

y+= 5

(a)

0.001

0.01

0.1

1

10

100

0.0001 0.001 0.01 0.1 1

E(v

+)

k+

y+= 20

(b)

0.0001

0.001

0.01

0.1

1

10

1e−05 0.0001 0.001 0.01 0.1 1

E(v

+)

k+

y+= 5

(c)

0.01

0.1

1

10

100

1e−05 0.0001 0.001 0.01 0.1 1

E(v

+)

k+

y+= 20

(d)

Figure 9. Normalized longitudinal 1-D spectra of normal to wall velocity for two distances to thewall : y+ = 5 and y+ = 20, for Re+ = 1000 (a and b) and Re+ = 2000 (c and d). Square: LES;cross: LES-SSAM; dash: DNS; dots: DNS from [48].

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 500 1000 1500 2000

ρ uu

∆ x+

y+= 5

(a)

-0.2

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

ρ vv

∆ x+

y+= 5

(b)

-0.2

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

ρ ww

∆ x+

y+= 5

(c)

Figure 10. Longitudinal autocorrelation coefficient of (a) streamwise, (b) normal, and (c) spanwisevelocity component, at y+ = 5, Re+ = 1000. Square: LES; cross: LES-SSAM; dash: DNS.

4. Results and discussion

For the sake of simplicity, we will only present simulations for the 64 × 65 × 64 grid forall three Reynolds numbers. For finer resolutions, the difference between LES-SSAM andstandard LES is less pronounced, but still present. Figure 6 shows the evolution of the meanvelocity across the channel. The LES-SSAM reproduces well the logarithmic law of thewall predicted by DNS, which is not the case with standard LES.

Figure 7 also shows the improvement in the prediction of the velocity variance whenLES-SSAM is used. This can be seen especially for the streamwise velocity compared withthe DNS of [48]. For the other components, the profile is also better predicted by LES-SSAM than by the standard LES. It should be noted that the variance of the streamwisevelocity at Re+ = 1000 computed by our DNS deviates slightly from the DNS in [48].This may be due to a smaller computational domain and a higher Reynolds number, incomparison with [48].

Figure 8 illustrates the vertical profiles of the turbulent and viscous stresses,τturb = −ρ〈u′v′〉 and τvisc = −ρν

∂〈u〉∂y

, respectively. The results are presented as ratiosτturb/(τturb + τvisc) and τvisc/(τturb + τvisc). Once again, the advantage of the LES-SSAM approach versus the classical LES is explicitly shown. Both the viscous and theturbulent stresses are better predicted by LES-SSAM.

Velocity spectra are illustrated in Figure 9. For both Reynolds numbers and both y+

shown in the figure, it can be clearly seen that the high wave number part of the spectrumis much better resolved by LES-SSAM than by LES.

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Journal of Turbulence 15

1e−05

0.0001

0.001

0.01

0.1

1

−15 −10 −5 0 5 10 15(az ––az)/az rms

y+= 5

(a)

1e−05

0.0001

0.001

0.01

0.1

1

−15 −10 −5 0 5 10 15(az–

–az)/az rms

y+= 20

(b)

1e−05

0.0001

0.001

0.01

0.1

1

−15 −10 −5 0 5 10 15(az–

–az)/az rms

y+= 950

(c)

Figure 11. Distribution of spanwise acceleration at three distances from the wall: y+ = 5, y+ = 20,y+ = 950. Re+ = 1000. Square: LES; cross: LES-SSAM; dash: DNS.

Figure 10 represents the evolution of the longitudinal autocorrelation coefficient of thevelocity components at y+ = 5, representing the near-wall region. The improvement of thecorrelation length because of the better resolution of the small-scale part of the spectrumby LES-SSAM is clearly seen in this figure. Another observation is that for small walldistances, the streamwise longitudinal autocorrelation is more spatially correlated than theremaining two components, because of the alignment of streaky structures with the wall.

Figure 11 illustrates the PDF of the spanwise acceleration. At three very different dis-tances to the wall, the PDF obtained by the DNS displays stretched tails, as a manifestationof intermittency. The same evidence of intermittency is obtained by LES-SSAM, whilein the case of the standard LES, these tails are much less developed. This effect is alsoobserved for the PDF of the other two components of the acceleration.

5. Conclusion

In numerical simulation of a high-Reynolds-number channel flow at moderate resolution,the instantaneous acceleration has been split into filtered and subgrid parts. Then, bothcomponents were modeled in the framework of LES combined with the stochastic subgridacceleration model.

The aim of the new model for subgrid acceleration was to account for intermittency inthe residual (unfiltered) wall-bounded flow. This model was constructed in accordance withDNS observations, reported in the literature, and DNS performed specifically in the present

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paper. Analogously to proposal in [51] and [14], the model is based on two independentstochastic processes: one for the norm of the subgrid acceleration and the other for itsorientation.

The norm is given by a characteristic velocity increment, in terms of wall-parameters,multiplied by stochastic frequency. The stochastic process, constructed in lines of statisticaluniversalities in fragmentation under the scaling symmetry, evolves with distance from thewall toward outer flow. To this end, the nondimensional evolution parameter was introduced,which increases from zero to infinity, as the distance to the wall increases from the wallto the channel half-width. The multiplicative stochastic process provides correlation ofthe subgrid forcing across the channel. With increasing wall distance and, consequently,decreasing mean frequency, this process has two intermediate universal asymptotics. Theearlier is the lognormal distribution, while the later is the power distribution. The power lawhas a high peak of probability for zero frequency and a stretched tail for high frequencies,which depends on the Reynolds number. Sampling from that stretched tail induces rareevents of strong subgrid forcing in the outer flow, mimicking thereby bursting events.

It was assumed that orientation of the subgrid acceleration is controlled by orientationof vortices, which become randomly distributed, if distance from the wall increases. In orderto fulfill this tendency toward full isotropy, the stochastic model for subgrid accelerationorientation is based on a Brownian random walk on the surface of a sphere of unity radius.Starting close to the wall, from the direction in the streamwise–spanwise plane with randomazimuth, the random walk takes place with increasing distance from the wall and tends tothe diffusive equilibrium as a final state, which corresponds to isotropic PDF’s for the twogeodesic angles, longitude and latitude. The diffusion coefficient was computed by DNSand found to be almost independent of the Reynolds number.

In the case of turbulent channel flow, the results of the LES-SSAM approach werecompared with DNS data (Re+ = 590, 1000, and 2000) and with standard LES. The LES-SSAM model was proven successful in predictions of DNS results: (1) resolution of thesmall-scale part of the energy spectrum and simulation of spatial correlations are close toDNS; (2) stretched tails in the PDF of acceleration are predicted in agreement with DNS;(3) computation of mean and of variance of the velocity components as well as of turbulentand viscous stresses are explicitly improved. Along with fundamental interest, the LES-SSAM approach has a practical relevance, when significant physics take place on SGSs. Insuch conditions, simulations at a high Reynolds number warrant advanced SGS models thataccount for intermittency effects on small spatial scales. Examples include scalar mixingand turbulent combustion, dispersion, vaporization, and combustion in two-phase flows.This requires the further development of this approach with many associated questions,concerning complex geometry, correlation between norm and orientation of the subgridacceleration, time correlations, and so on. However, the approach given in [14] and in thepresent paper is very simple and easily realizable for any LES code without supplementaryCPU requirement.

AcknowledgmentsM. Buffat is acknowledged for the development of the computational code. The authors expresstheir gratitude to F. Laadhari who kindly provided his initial fields for the DNS. It is a pleasure tothank V. Sabel’nikov, P. Moin, and J. Jimenez for discussions on this subject with many benefitsfor us. This work was granted access to the High Performance Computing resources of CentreInformatique National de l’Enseignement Suprieur under the allocation 2009-c200902560 made byGrand Equipement National de Calcul Intensif.

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