model in comparison to the standard LES and to direct numerical simulation (DNS). Mots clefs : turbulent channel flow, large eddy simulation, stochastic subgrid ...
19 e` me Congr`es Franc¸ais de M´ecanique
Marseille, 24-28 aoˆut 2009
Large eddy simulation of a turbulent channel flow with stochastic modeling of the subgrid acceleration. R. Z AMANSKY, I. V INKOVIC AND M. G OROKHOVSKI LMFA-UMR 5509-CNRS-Ecole Centrale de Lyon, Universit´e Claude Bernard Lyon 1, 69134 Ecully Cedex, France
R´esum´e : Dans cette article, le champ de vitesse non-filtr´e d’un e´ coulements a` grand nombres de Reynolds dans un canal est simul´e par l’approche LES-SSAM (mod´elisation stochastique de l’acc´el´eration de sous-maille)[1]. Un nouveau mod`ele pour l’acc´el´eration de sous-maille des e´ coulements en canal est introduit. Les param`etres de ce mod`ele sont : le nombre de Reynolds bas´e sur la vitesse de frottement et la hauteur du canal. La d´erivation de ce nouveau mod`ele ainsi que les comparaions entre l’approche LES-SSAM avec le mod`ele propos´e, la LES classique et la DNS sont d´evelopp´es.
Abstract : In this paper, the non-filtered velocity field in a well-developed turbulent channel flow is simulated in the framework of LES-SSAM (stochastic subgrid acceleration model) approach [1]. A new stochastic SGS model is proposed. This model introduces explicitly the cross-channel correlation of subgrid velocity gradients and includes two parameters : the Reynolds number based on the friction velocity, and the channel half-width. The objective was to assess the capability of this model in comparison to the standard LES and to direct numerical simulation (DNS).
Mots clefs : turbulent channel flow, large eddy simulation, stochastic subgrid model, intermittency, acceleration
1
Introduction
The structure of well-developed turbulent wall layer in the channel flow is highly intermittent. Close to the wall, the low-speed regions are interleaved with tiny zones of the high-speed motion. It has been shown that LES may reproduce accurately highly intermittent turbulent structures near the wall. However this requires excessively expensive grid resolution. To reroduce accurately highly intermittent turbulent near-wall flow, the large eddy simulation (LES) requires excessively expensive grid resolution. Then for a high Reynolds number channel flow, The LES at moderate resolution has to be combined with subgrid scale (SGS) model for the non-resolved turbulent motion. The majority of SGS models are focused on simulation of turbulent stresses generated by the non-resolved velocity field [2, 3, 4]. In these models the structure of sub-grid flow is supposed to be independent of the Reynolds number, i.e., to be not intermittent. Therefore the approach recently proposed by [1] is focused directly on the stochastic modeling of the sub-grid acceleration (LES-SSAM). It was shown, by Kolmogorov’s scaling, that , for a given filter width ∆, the non-resolved acceleration may be substantially greater than the resolved acceleration : (ak ak )/(a0i a0i ) ≈ (η/∆)2/3 , where ak and a0i represent resolved and 3/4 non-resolved accelerations and η = L/ReL is the Kolmogorov’s lenght scale. This implies that in any SGS model, which is aimed to introduce the intermittency effects, the non-resolved acceleration must be a key variable. This motivated us to set up a new stochastic model for the sub-grid acceleration of wall bounded flow. The aim of this paper is to assess the capability of the new model to reproduce the near-wall behaviour compared to a standard LES and direct numerical simulation (DNS).
2
LES-SSAM approach and model formulation
In the LES-SSAM framework [1] it is consider that the total instantanous accelertion, governed by the NavierStokes equations, can be represented by the sum of two parts : ai = ai +a0i . The first part represents the spatially ∂uk ui i filtered total acceleration : ai = ∂u ∂t + ∂xk , and is equivalent, with spatially filtering of the Navier-Stokes equations, to : dui 1 ∂P ∂uk ai ≡ =− + ν∆ui ; =0 (1) dt ρ ∂xi ∂xk
1
19 e` me Congr`es Franc¸ais de M´ecanique
Marseille, 24-28 aoˆut 2009
with ν the kinematic viscosity. The second part is associated with the total acceleration in the residual field : ∂u0k dui 0 1 ∂P 0 0 ai ≡ =− + ν∆u0i ; (2) =0 dt ρ ∂xi ∂xk 0
∂uk ui −uk ui i . Both equations need to be modeled. In the LES-SSAM approach, eq. (1) is where a0i = ∂u ∂t + ∂xk modeled in the framework of the classical LES approach, and a0i is considered as a stochastic variable. The resulting model-equation, which reconstructs an approximation for the non-filtered velocity field, writes then as : ∂u ˆi 1 ∂ Pˆ ∂ ∂u ˆi ∂u ˆk ∂u ˆk ∂u ˆi =− + (ν + νturb ) + +a ˆ0i ; =0 +u ˆk (3) ∂t ∂xk ρ ∂xi ∂xk ∂xk ∂xi ∂xk
where ˆ• represents a synthetic field and νturb is given by the Smagorinsky subgrid model. For the futher developement of LES-SSAM approch, we propose a new model for the non-resolved acceleration a ˆ0i . We introduce the separation of variables for the non-resolved acceleration ai . On the basis of our DNS for turbulent channel flow (see table 1) and experiences [5, 6, 7], |a|, the modulus of the subgrid acceleration and ei its orientation, are two independent random variables, caracterised by longue memory and rapid decorrelation, respectively. Then the non-resolved acceleration is written as : a ˆ0i = |a|ei
(4)
For |a|, our proposal is to emulate the modulus of the non-resolved acceleration in the following form : |a| = f ∆u2∗ /ν
(5)
2 where ∆ is the characteristic cell size and u∗ the friction velocity, u2∗ /ν ≡ ∂u ∂y |wall ; so ∆uτ /ν will be considered as a typical normal to wall velocity increment in the near to wall region, and f is the subgrid frequency, considered as stochastic variable. The frequency f is supposed to have a stochastic evolution process from the wall to the outer flow driven by the non-dimensional parameter τ defined as follow : h−y τ = −ln (6) h
where h is the channel half-width, and y is the wall distance (y = 0 : τ = 0 and y → h : τ → ∞). The near-wall region is characterized by strong velocity gradients (high values of f ), which are decreasing in mean toward the outer flow through the highly intermittent boundary layer. Thereby we assumed that with increasing of the normal distance from the wall, the frequency f is changing by a random independent multiplier α R1 (0 < α < 1), governed by distribution q(α), 0 q(α)dα = 1, which is principle unknown. In other words, we apply the fragmentation stochastic process under scaling symmetry for the frequency f . From [8], we derive the following stochastic equation corresponding to this process is : p (7) df = hlnαi + hln2 αi/2 f dτ + hln2 αi/2f dW (τ ) R1 where hlnk αi = 0 q(α)lnk αdα ; k = 1, 2, and dW (τ ) is the Wiener process and hdW (τ )2 i = 2dτ . In the present study, parameters are chosen in the following form : 1/3
− hlnαi = hln2 αi = Re+
(8)
where Re+ is the Reynolds number, based on the friction velocity u∗ . The starting condition, τ = 0, for this stochastic process (the first grid cell on the wall) is given as follows. We introduce the mean value of frequency at the wall f+ = λ/u∗ , where λ is determined, as a Taylor-like scale, which can be estimated by the Kolmogorov’s scaling in the framework of definitions of wall parameters. The Reynolds number, based on 3/4 friction velocity, is Re+ u∗ h/ν = h/y0 ≈ Reh where y0 is the thickness of the viscous layer, and Reh is −1/2 −2/3 the Reynolds number based on the center-line velocity. One then yields : λ ≈ hReh ≈ hRe+ . Similar to Kolmogorov-Oboukhov 62 , the starting condition for random path (7) is sampled from the stationary lognormal distribution of f /f+ : (ln(f /f+ )−µ)2 f+ − 2σ 2 √ P0 (f /f+ ) = e (9) f 2πσ 2 with σ 2 = ln 2 and µ = − 12 σ 2 , this imposed hf i = (hf 2 i − hf i2 )1/2 = f+ . Then this stochastic process will relax f from a log-normal distribution at the wall (τ = 0) to the power distribution as the distance to the 2
19 e` me Congr`es Franc¸ais de M´ecanique 10
Marseille, 24-28 aoˆut 2009 10
y+=1 + y =50 y+=100
1
1
0.1
0.1
0.01
0.01
0.001 0
2
4 f/f+
6
8
10
0.001 -π/2
-π/4
0 θ
π/4
π/2
F IG . 1 – a : Distribution of f /f+ from SSAM (cross) and comparison with DNS (line) at Re+ = 590, for several distances from the wall.b : Distribution of θ for small scale acceleration from DNS (line) and from model (cross), for Re+ = 590, and for several distances from the wall (y+ = 3, y+ = 10 and y+ = 30).
wall is increasing (τ → ∞). The evolution through the channel, for distributions of the frequency predicted by the stochastic equation can be compared with the evolution of the frequency computed from DNS, via eq. (5). According to fig. 1a the SSAM ensures a good relaxation of the frequency, as the distance to the wall increases. In order to emulate the orientation vector of the sub-grid scale acceleration, ei , we considered a random walk evolving on the surface of a sphere of unity radius. The direction of this vector is defined by two stochastic variables which are longitude φ and latitude θ : ( ex = cos(θ) cos(φ) ey = sin(θ) (10) ez = cos(θ) sin(φ) It is seen from DNS (fig. 1b) that the orientation vector relaxs toward isotropy with increasing distance from the wall. In order to represent the tending to isotropy, we realise the Kubo oscilator with real coefficient α as random motion of unit vector on the sphere. Each position increment of the random walk is given by ζ = αdW (τ ), with : r ln Re+ (11) α= 2 and the direction β at each time step is chosen randomly from the uniform distribution. For each step the position is moving from (θi , φi ) to (θi+1 = θi + dθ, φi+1 = φi + dφ) : θi+1 dφ
