Computers and Fluids 151 (2017) 144–158
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Assessment of subgrid-scale modeling for large-eddy simulation of a spatially-evolving compressible turbulent boundary layer O. Ben-Nasr a, A. Hadjadj a,∗, A. Chaudhuri b, M.S. Shadloo a a b
CORIA-UMR 6614, Normandie University, CNRS-University and INSA of Rouen, 76000 Rouen, France Department of Aerospace Engineering and Engineering Mechanics, San Diego State University, 5500 Campanile Drive, San Diego, CA, 92182, United States
a r t i c l e
i n f o
Article history: Received 18 March 2016 Revised 24 June 2016 Accepted 5 July 2016 Available online 9 July 2016 Keywords: Subgrid-scale (SGS) modeling supersonic turbulent boundary layer (STBL) Large-Eddy simulation (LES) Wall-adapting local eddy-viscosity (WALE) Dynamic smagorinsky model (DSM) Coherent structures model (CSM)
a b s t r a c t The performance of three standard subgrid-scale (SGS) models, namely the wall-adapting local eddyviscosity (WALE) model, the Dynamic Smagorinsky model (DSM) and the Coherent Structures model (CSM), are investigated in the case of a spatially-evolving supersonic turbulent boundary layer (STBL) over a flat plate at M∞ = 2 and Reθ ≈ 2600. A high-order split-centered scheme is used to discretize the convective fluxes of the Navier–Stokes equations, and is found to be highly effective to overcome the dissipative character of the standard shock-capturing WENO scheme. The consistency and the accuracy of the simulations are evaluated using direct numerical simulations taken from the literature. It is demonstrated that all SGS models require a comparable minimum grid refinement in order to capture accurately the near-wall turbulence. Overall, the models exhibit correct behavior when predicting the dynamic properties, but show different performances for the temperature distribution in the near-wall region even for cases with satisfactory energy resolution of more than 80%. For a well-resolved LES, the SGS dissipation due to the fluctuating velocity gradients is found to dominate the total SGS dissipation. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Compressible turbulent boundary layer is a basic phenomenon that occurs in a wide range of high-speed applications, such as internal and external aerodynamics of space vehicles. From a physical view-point, this phenomenon is still of primordial interest for fundamental research as well as for numerical modeling. Highly resolved numerical simulations, using Direct Numerical Simulations (DNS) and Large-Eddy Simulations (LES), emerge as a promising tool to help getting more insight into this topic. A recent literature review on DNS of compressible boundary layers can be found in Shadloo et al. [1] Few LES of supersonic turbulent boundary-layer flows have been performed [2–5]. For instance, Spyropoulos & Braisdell [2] reported LES of spatially-evolving supersonic turbulent-boundary layer at Mach M = 2.25. A second- as well as a fourth-order accurate upwind biased finite differences schemes were used for the convective fluxes. In terms of wall-units, all considered grids had resolutions ranging between 59 ≤ x+ ≤ 88 for the streamwise direction, 0.77 ≤ y+ ≤ 0.97 for the wall-normal direction min and 11.4 ≤ z+ ≤ 42.1 for the spanwise direction. It was concluded
∗
Corresponding author. E-mail address:
[email protected] (A. Hadjadj).
http://dx.doi.org/10.1016/j.compfluid.2016.07.004 0045-7930/© 2016 Elsevier Ltd. All rights reserved.
that a decrease of about 20% of the computed skin-friction is found when lower order schemes are employed, mainly a third order upwind scheme for the convective terms and second order for the viscous terms. Because of the low considered Mach number, the modeling of the isotropic part of the shear stresses was not found to have a considerable effect on the skin-friction coefficient, Cf . The insufficient amount of turbulent transport was attributed to the use of the dynamic Smagorinsky model, in which the eddy viscosity is computed using the smallest resolved scales. Using monotonically integrated large-eddy simulation (MILES) approach, Yan et al. [5] have conducted a numerical study of supersonic flat-plate boundary layers in which the numerical dissipation induced by the scheme substitutes the SGS eddy viscosity, mimicking thereby from an energetic view-point the action of the SGS terms on the flow dynamics. The simulated flows evolved at freestream Mach numbers of 2.88 and 4. An adiabatic as well as an isothermal cases with Tw /Tr = 1.1 (where the recovery γ −1 2 temperature Tr T∞ (1 + r 2 M∞ ), r = 0.89 is the recovery factor) were performed. In terms of wall-units, their grid resolutions were x+ = 18, y+ = 1.5 and z+ = 6.5. It was reported min that the mean streamwise velocity profiles using the van-Driest transformation were in good agreement with the viscous sublayer + linear approximation and law-of-the-wall (u+ vd = 2.5 log y + 5.7). The distributions of the streamwise Reynolds stresses scaled by mean density ρ and wall shear stress τ w , were found to be
O. Ben-Nasr et al. / Computers and Fluids 151 (2017) 144–158
very similar except close to the wall, and showed good agreement in the outer region of the boundary layer (y/δ > 0.2). The peak magnitude in the near-wall region (y/δ < 0.2) was supported by both experimental and DNS results, although its location was not consistent with the reference data. The Reynolds shear stresses of both cases showed good agreement with the reference solutions. Finally, the turbulent Prandtl number Prt was found to be in good agreement with the experimental value of 0.89. Urbin & Knight [6] performed an LES of an adiabatic Mach 3 boundary layer. A detailed grid refinement study was performed to assess the required grid resolution in the viscous sublayer, the logarithmic and the outer regions of the boundary layer. They emphasized that the subgrid-scale effects can be modeled using MILES without Smagorinsky model. On the other hand, Kawai & Lele [7] proposed a simple mesh-resolution-dependent dynamic wall model for LES of compressible supersonic turbulent-boundary layer over a flat plate. Recently, Hadjadj et al. [8] performed a series of LES computations of supersonic boundary layers with detailed statistical analysis of the unsteady flow-field using the WALE SGS model. The focus of their work was on the effects of wall temperature on the near wall turbulence behavior, while the effects of the SGS modeling was left for future investigations. To the authors’ best knowledge such investigations related to STBL has not completely been addressed in the literature so far. In LES, the accuracy of the resolved scales highly relies on the mesh size. Locally refined grids usually lead to more resolved turbulent energy but will definitely be more costly in terms of CPU time and memory requirements. The strategy in LES is then to make the best compromise between accuracy and computational cost. Dissipation of a given SGS model may originate, in different proportions, either from the resolved velocity fluctuations or from the mean-averaged velocity gradients. The present work aims to assess the prediction quality of three popular SGS models on the near-wall asymptotic behavior of a supersonic turbulent boundary layer (STBL). Some of the turbulence statistics are reported in this paper in order to assess their effects in conjunction with the numerical scheme and the mesh resolution. The obtained results are compared with available directnumerical simulation (DNS) data and showed an overall good agreement. The paper is organized as follows: the governing equations and the numerical discretization are presented in Section 2, where the filtered Navier–Stokes equations and the SGS modeling are presented. The results are presented and discussed in Section 3. The issue of the SGS activity is addressed in the same section before drawing the concluding remarks in Section 4.
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where ρ , p and ui denote density, pressure and velocity vector, respectively. The equation of state, the total energy, the viscous shear stress and the heat flux are given by:
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p 1 ⎪ ρ Eˇ = + ρ ui ui ⎪ ⎪ ⎪ γ −1 2 ⎬
∂ uj 2 ∂ ui uk ∂ ⎪ σˇ i j = μ + − μ δ ⎪ ⎪ ∂ xi ∂ x j 3 ∂ xk i j ⎪ ⎪ ⎪ ⎪ ⎪ C p ∂ μ T ⎪ ⎪ ⎭ qˇ j = − Pr ∂ x j p = ρ r T
(2)
is the dynamic viscosity obeying to the Sutherland’s law with μ and Pr is the Prandtl number equal to 0.72 (for air with γ = C p /Cv = 1.4). Unlike the (. ) and the ( . ) symbols, the (ˇ. ) symbol does not denote a filter operation but indicates that the quantity is based on primitive filtered variables. Thus, Eˇ refers to the resolved total energy, which is not equal to the filtered total energy. Note that, following [9–11], the unclosed sub-grid scale (SGS) terms are neglected in both momentum and energy equations. In this study, different LES models are used to model the action of the subgrid-scales (SGS) on turbulence: the Dynamic Smagorinsky procedure of Germano et al. [12], Moin et al. [9] and Lilly [13], the coherent structures model proposed by Kobayashi [14] and the Wall-Adapting Local Eddy-viscosity model by Nicoud and Ducros [15]. In order to better quantify the real contribution of the SGS modeling, an Implicit LES is also performed, in which the numerical dissipation mimics the action of the small scales on turbulence. 2.1.1. Modeling the SGS stress tensor The SGS stress tensor, τ ij , in Eq. (1) is defined by:
τi j = ρ u ui uj iu j −
(3)
It is modeled via the definition of a SGS eddy viscosity, μsgs , which yields:
1 1 τi j − τkk δi j = −2μsgs Si j − Skk δi j 3
3
(4)
where Si j = 1/2 ∂ ui /∂ x j + ∂ u j /∂ xi is the strain rate tensor of the resolved scales. The SGS viscosity, μsgs , is given by:
μsgs = ρ Cs 2 | S| where | S| =
(5)
2 Si j Si j is the second invariant of the strain rate ten-
sor, and Cs is a dynamically-retrieved modeling constant. For compressible flows, Yoshisawa [16] proposed a closure for the isotropic part of the SGS stress tensor, τ kk , defined by:
2. Methodology
τkk = 2ρ CI 2 | S |2
2.1. Large-eddy simulation
The model constant, CI , is dynamically retrieved for the DSM procedure, or set equal to 0.005 for the CSM (Moin et al. [9]). Unless stated, the isotropic part of the SGS stress tensor, τ kk , is not modeled for both CSM and WALE model.
The filtered compressible Navier–Stokes equations expressed in a conservative form are written:
⎫ ∂ ρ ∂ ρ ui ⎪ + = 0⎪ ⎪ ⎪ ∂t ∂ xi ⎪ ⎪ ⎪ ⎪ ∂ ρ ui uj ∂τi j ⎪ ∂ ρ ui ∂ p ∂ σˇ i j ⎪ ⎪ + + − ≈− ⎬ ∂t ∂xj ∂ xi ∂ x j ∂xj ⎪ uj ∂ ui σˇ i j ∂ qˇ j ⎪ ∂ ρ Eˇ ∂ ρ Eˇ + p ⎪ ⎪ + − + ⎪ ∂t ∂xj ∂xj ∂xj ⎪ ⎪ ⎪ ⎪ ⎪ uj ∂τi j ⎪ ⎪ 1 ∂ pu j − p ⎪ ⎭ ≈− − uj γ −1 ∂xj ∂xj
(1)
(6)
Dynamic Smagorinsky model In the Dynamic Smagorinsky procedure, the model’s constants, Cs and CI , are dynamically extracted from the resolved flowfield quantities. A test filter, denoted as ( . ), whose width is larger than the grid-filter width, is applied to the grid-filtered quantities. The model’s constants are then calculated at the test-filter wavenumber, and are assumed to remain about the same within [ktest , kc ] as the test-filter width and is wavenumbers range. Denoting / = 2. the grid-filter width, it is common to define After dynamically retrieving Cs and CI , and to avoid any numerical instability due to negative values, both constants are averaged
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in the homogeneous direction (z), and clipped within [0, 0.08] and [0, 0.02], respectively.
Table 1 Lines and symbols of the different cases. Case
Coherent structures model For the coherent structures model, Cs is dynamically calculated using a function of the velocity gradient tensors. This function is based on the assumption which states that, for a well-resolved DNS grid, the SGS dissipation is small at the center of a coherent fine-scale eddy, and that the energy transfer between resolved scales and SGS is located around this coherent eddy (Kobayashi [17]; Kobayashi [18]; Onodera et al. [19]). The model’s constant, Cs , is defined by:
Cs = Ccsm |Fcs |3/2
Fcs =
Q E
(7)
where Ccsm is a model’s parameter (by default equal to 1/30) and and E are respectively the Fcs is the coherent structures function. Q second invariant of the resolved velocity gradient and the magnitude of a resolved velocity gradient tensor, given by:
= 1 W = 1 W i j W i j − Si j Si j i j W i j + Si j Si j Q E 2 2
(8)
i j are the velocity-strain tensor and the vorticity tenwith Si j et W sor in a grid scale flowfield, respectively. It follows that:
= − 1 ∂ u j ∂ ui Q 2 ∂ xi ∂ x j
= 1 ∂uj ∂uj E 2 ∂ xi ∂ xi
(9)
One should note that −1 ≤ Fcs ≤ 1. It follows that the model’s constant is bounded (0 ≤ Cs ≤ 0.05) and admits a weak variance. Wall-adapting local Eddy-viscosity model The WALE model calculates the eddy viscosity, μsgs , based on the invariants of the velocity gradient. It is defined by:
μsgs = ρ 2Cw2
Si j Si j
Si∗j Si∗j
5 / 2
3 / 2
+ Si∗j Si∗j
5 / 4
g2i j = gik gk j
(11)
2.1.2. Modeling the SGS heat flux By analogy to the SGS stress tensor modeling, the SGS heat flux is modeled using an eddy viscosity formulation, which is written:
γ
DNS DSM CSM WALE ILES LES-L1 LES-L2 LES-N2 LES-P2 LES-Q2 LES-R2 LES-S2
Table 2 Grid resolution sensitivity study using the CSM. Subscript (+) denotes the normalization by the friction velocity, with y+ = ρw yuτ /μw . Case
Nx
Ny
Nz
x+
y+min
z+
β
DNS [24] LES-L1 LES-L2 LES-N2 LES-P2 LES-Q2 LES-R2 LES-S2
4160 1152 768 768 768 768 512 768
221 180 180 180 90 90 90 45
440 96 96 64 64 48 48 64
5.7 22.7 35.8 35.8 35.9 35.4 56.0 36.2
0.7 0.95 1.00 1.00 1.24 1.22 1.28 1.29
4.9 8.19 8.60 12.9 12.9 16.9 17.9 13.0
NC 4.85 4.85 4.85 5.45 5.45 5.45 5.85
is used, while at the inlet, an inflow turbulent boundary condition, relying on a modified version (Pirozzoli et al.) [22] of the original recycling/rescaling procedure (Lund et al.) [23] is employed. The configuration is parallelized in the stream- and spanwise directions using MPI libraries. 2.3. Problem setup
Cw is a model’s constant, by default taken equal to 0.5 (Nicoud & Ducros) [15] and gi j = ∂ ui /∂ x j .
uj μsgs ∂ 1 ∂ pu j − p T =− Cp −1 ∂xj P rsgs ∂ x j
Symbol
(10)
with
1 2 1 2 Si∗j = g + g2ji − g δi j 2 ij 3 kk
Line
(12)
The SGS Prandtl number, Prsgs , is taken constant and equal to 0.9. 2.2. Numerical method The convective fluxes are discretized using a sixth-order locallyconservative skew-symmetric split-centered formulation (Pirozzoli [20]). A fourth-order split-centered scheme as well as a Bandwidth Optimized WENO scheme (hereafter denoted as WENOBWO) proposed by Martin et al. [21] are also used and a comparison between the different schemes will be discussed. The viscous fluxes are discretized using a fourth-order compact central differences scheme. Time advancement is assessed by a standard explicit Runge–Kutta algorithm of third-order. At the bottom of the computational domain, an adiabatic noslip boundary condition is imposed, while a freestream border is set at the upper boundary. At the outlet, an extrapolation condition
The incoming boundary layer is spatially evolving at a freestream Mach number, M∞ = 2, and an inlet Reynolds number, Reτin = ρw uτ δin /μw ≈ 245 or Reθin = ρ∞ u∞ θin /μ∞ ≈ 1150 (where uτ is the friction velocity, δ in is the inflow boundary layer thickness [24], and θ in is the momentum thickness at the inlet). The computational domain used in this study has a size of Lx × Ly × Lz = 106 δin × 9.13 δin × 3.18 δin in the streamwise (x), wall-normal (y) and spanwise (z) directions, respectively. As shown in Table 2, different grid resolutions are used with uniformly spaced grid in the streamwise and spanwise directions, and clustered grid in the wall-normal direction based on Ly sinh (by η)/sinh (by ), where Ly is the box size in the y-direction and the stretching factor by = 5.5. The mapped coordinate η is equally spaced and runs from 0 to 1. The flowfield is initialized using a digital filter procedure based on Klein’s method (Klein et al. [25]) where the r.m.s. velocity profiles are extracted from the DNS of Bernardini and Pirozzoli [24]. A series of approximately 140 characteristic times, τc = δin /u∞ , is achieved to sweep the initial transient solution. Then, turbulence statistics are sampled and extracted each time step from time series covering τ ≈ 300τ c . By plotting the time evolution of the main boundary layer statistics, such as the boundary layer thickness and the friction velocity, this sampling time is judged to be sufficient to reach a statistical convergence of the considered quantities. A reference simulation (e.g. LES-P2 case) is performed over about 40 h using 64 processors, for a total of about 2560 CPU hours. The First-half of the computational domain is dedicated to the recycling/rescaling procedure, and the second-half is used for data analysis. In the latter domain, Reτ ranges from 510 to 640 and Reθ
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Table 3 Boundary layer properties using LES-P2 grid for different subgrid models. Reτ = ρw uτ δ /μw ; Reθ = ρ∞ u∞ θ /μ∞ ; C f = 2τw /ρ∞ u2∞ ; H = δ ∗ /θ ; Mτ = uτ /(γ RTw )1/2 , δ ∗ is the displacement thickness. √ 2 p w Case Reτ Reθ 103 Cf δ ∗ /δ 102 θ /δ H Tw /T∞ Mτ τw DNS [24] DSM CSM WALE ILES
583.9 584.4 590.7 572.1 567.1
2865.9 2423.2 2597.5 2560.8 2542.0
2.53 2.78 2.98 2.94 2.94
0.259 0.245 0.282 0.297 0.301
8.50 8.56 9.08 9.03 8.97
2.96 2.86 3.11 3.29 3.36
1.717 1.620 1.666 1.664 1.703
0.0712 0.0745 0.0773 0.0767 0.0767
2.62 2.89 3.40 3.10 3.45
3.1.2. First- and second-order statistics Distributions of the mean-averaged quantities as a function of y/δ are plotted in Fig. 2. Overall, the profiles collapse well with the DNS data, and the CSM seems best estimating the averaged quantities. The value of the normalized temperature at the wall, Tw /T∞ , is estimated at 1.717 for the DNS, 1.666 for the CSM, 1.664 for the WALE model and 1.62 for the DSM. The normalized mass flux, ρu/ρ ∞ u∞ , is best estimated by the CSM. However, the under-estimation of the wall-temperature by the DSM could originates from a very high damping of the turbulent fluctuations in the near-wall region, probably caused by a wrong prediction of the near-wall asymptotic behavior of the SGS viscosity μsgs . We can also observe that coarsening the grid seems weakly affecting these quantities (see Fig. 2b). The mean total temperature Tt is defined as:
Tt = T + Fig. 1. Instantaneous auto-correlation functions at y+ 30 for CSM model using LES-N2 grid.
from 2090 to 2900. Table 3 reports some statistical properties of the considered test-cases at a given station xres ࣃ 84.8δ in , which corresponds to Reτ ≈ 580. 3. Results and discussions
ϕ (x, y ) =
1 1 Lz Ts
Lz
Ts
ϕ (x, y, z, t ) dt dz
(13)
3.1. Turbulence statistics 3.1.1. Auto-correlation coefficients To ensure that the computational domain is sufficiently wide in the spanwise direction (z), we analyzed the two-point correlation functions given by:
Nz/2
Rϕ ϕ ( rz ) =
ϕk ϕk +kr H kr = 0, 1 . . . , Nz /2
(14)
k=1
where rz = kr z , ϕ represents the fluctuations of flow variables, and the angled brackets .H represent an averaging operation over homogeneous directions. As shown in Fig. 1, the auto-correlation distributions decrease to zero, which means that two points of the domain separated by Lz /2 are completely independent, and thus the domain is wide enough so that the principle turbulence mechanisms are not inhibited.
(15)
As it can be seen from Fig. 3, the total temperature is not constant throughout the layer, showing an overshoot that does not exceeds 2% for all cases. This overshoot is expected to increase with positive heat flux, namely heated wall (Smits & Dussauge [26]). The van-Driest transformed mean streamwise velocity u+ vd , which accounts for the variation of the mean flow properties aiming to collapse a compressible velocity profile with its incompressible counterpart, is defined as:
u+vd =
In what follows, unless stated, only the resolved part of a given i ≡ quantity is considered for the results’ discussion. It yields to ϕ ϕi and ϕi ≡ ϕi . Furthermore, each statistical quantity is obtained by time-averaging over a sampling period Ts and spatial-averaging in the homogeneous direction (z). A mean quantity noted ϕ is defined as:
1 γ − 1 ui 2 + u i u i 2 γR
u+ 0
ρ/ρw du+
(16)
The distribution of the van-Driest transformed mean stream+ wise velocity u+ vd as a function of y is reported in Fig. 4. Except a very good estimation of u+ by the DSM, the results show a vd slight under-estimation of the distribution in the Log-region compared to the DNS data, for the remaining SGS models (Fig. 4a). In the viscous sub-layer and up to the buffer region (y+ < 30), all models collapse with the DNS. As expected for y+ < 5, the velocity evolves linearly with y+ . As observed before, a weak underestimation of u+ vd by both WALE model and CSM (5% and 10%, respectively) is observed in the region 30 < y+ < 100, even if the slope is correctly reproduced, and all models exhibit an underestimation of u+ vd in the wake region and in the outer part of the boundary layer. In Fig. 4b, u+ vd shows sensitivity to grid resolution, and the noted under-estimation decreases as the grid is refined. The spanwise resolution is found to further influence this under-estimation compared to streamwise and wall-normal resolutions. Fig. 5 depicts the van-Driest transformed mean streamwise velocity deficit as a function of y/δ . The models are found to exhibit overall acceptable results. While the CSM fits well with the DNS, both DSM and WALE models under-estimate the velocity deficit, while the Implicit LES over-estimate it. Fig. (6a and b) show the normalized velocity fluctuations as a function of y/δ and Fig. (6c and d) depict the velocity fluctuations in Morkovin’s scaling as a function of y+ . Except a good fitting in the near-wall region (y/δ < 0.05 and y+ < 10), all models present
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Fig. 2. Distributions of the time-averaged mean quantities as a function of y/δ . (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
Fig. 3. Distribution of the normalized time-averaged total temperature, Tt /Tt∞ , as a function of y/δ . ◦ DNS Pirozzoli et al. [27] at Reθ = 4260. (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
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+
Fig. 4. Distribution of the van-Driest transformed mean streamwise velocity uvd as a function of y+ . C = 5.2; κ = 0.41. (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
Fig. 5. Normalized mean velocity deficit uvd∞ − uvd /uτ as a function of y/δ . (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
under-estimated magnitudes compared to the DNS, especially for the u+ rms component (5% for the DSM, 8% for the CSM and 10% for the WALE model). The location of the streamwise component peak is well estimated by the models (0.02δ ). It is also found that the wall-normal fluctuations, v+ rms , as well as the spanwise fluctuations,
+ w+ rms , along with the Reynolds shear stress u v are better estimated by both CSM and WALE model. In Fig. 6b, the normalized velocity fluctuations are found to be weakly sensitive to the grid resolution. Fig. (6e and f) shows the normalized velocity fluctuations in Morkovin’s scaling as a function of y+ in Log-scale, where results, taken at Reδ2 ≈ 1800 in the region 50 < y+ < 300, are compared to: - The curve-fitting of DNS data (Pirozzoli et al. [27]) at Reδ2 ≈ 2400:
ρ u 2 ≈ 1.500 − 1.086 log(y/δ ) − v(y+ ) ρw u2τ ρ v 2 ≈ 1.526 − v(y+ ) ρw u2τ
ρ w 2 ≈ 1.243 − 0.510 log(y/δ ) − v(y+ ) ρw u2τ
(17)
- The subsonic experimental results by Perry & Li [28] at Reδ2 = 50 0 0:
u 2i u2τ
= Bi − Ai log(y/δ ) − v(y+ )
(18)
where B1 = 2.39, A1 = 1.03, B2 = 1.6, A2 = 0, B3 = 1.20, A3 = 0.475. In both Eqs. (17) and (18), v(y+ ) represents a correction of the deviation from the logarithmic profile due to viscous action in the turbulent wall-region. The function increases with y+ as: −1/2
v(y+ ) = 5.58y+
−1
− 22.4y+
−5/4
+ 22.0y+
−2
− 5.62y+
−11/4
+ 1.27y+
(19)
It can be seen from Fig. (6e and f) that although the streamwise and the spanwise components show an overall weak under-estimation in the considered region (50 < y+ < 300), the slope is well reproduced for almost all considered cases. The wall-normal component collapses well with the curve-fitting, in terms of slope and magnitude. When refining 2 the mesh, u+ rms ρ/ρw shows a small shift towards the reference, while
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Fig. 6. (a–b) r.m.s of velocity fluctuations components ui, rms /uτ as a function of y/δ ; (c-d) r.m.s velocity fluctuations in Morkovin’s scaling as a function of y+ ; (e-f) r.m.s velocity fluctuations in Morkovin’s scaling as a function of y+ in Log-scale. (· − ·) Perry & Li [28] in Eq. (18). (−−) Pirozzoli et al. [27] in Eq. (17). For legend, see Table 1.
the two other components remain unaffected by the grid refinement. It is also found that the results are more compatible with the DNS data rather than the subsonic experimental measurements. This difference can be attributed to the effect of Reynolds number. The resolved turbulent kinetic energy, K = u u /2, is shown in i i
Fig. 7. It is found that the peak position of K is overall well retrieved by all SGS models, and is located around y+ 13 for the DSM and y+ 12 for the remaining models. This location is found to be weakly sensitive to the grid resolution, where it is y+ ≈ 14 for the LES-L1 grid, which constitutes a classical finding. For incompressible flows, the structure parameter is nearly constant and lies between 0.14 and 0.17 (Smits & Dussauge [26]). The structure parameter shows a constant value ≈ 0.15 as predicted by the models at 0.2 < y/δ < 0.8. This value seems to be weakly sensitive to the grid resolution since all used grids exhibit values that lie be-
tween 0.14 and 0.17 (see Fig. 8). Duan et al. [29] also showed in a DNS study that this parameter is between 0.14 and 0.16 at 0.1 < y/δ < 0.9 when varying the Mach number from 3 to 5. The distribution of the r.m.s. vorticity components as a function of y+ are plotted in Fig. 9, where the ith component of the r.m.s of vorticity is defined as:
1/2 i = ωi 2
(20)
Both CSM and WALE model are seen to give better results compared to the DSM, while in the near-wall region, the r.m.s. vorticity components are overall well estimated and fit well with the references (y+ ≤ 10). They are however under-estimated away from the wall, especially for the streamwise component. As expected, the vorticity fluctuations are not isotropic for y+ ≥ 30 and only ω2 ≈ ω3 away from the wall.
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Fig. 7. Normalized turbulent kinetic energy K/u2τ as a function of y/δ . (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
Fig. 8. Structure parameter −u v /2K as a function of y/δ . (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
Fig. 10 shows different normalized shear stresses as a function of y+ . The CSM and the WALE model show higher total shear stress compared to the DSM (Fig. 10a). Apart from the sublayer, where viscous effects are dominant, the Reynolds shear stress accounts for almost all the total shear stress. As pointed out by Pirozzoli et al. [27], the Reynolds shear stress does not exceed unity for all models. However, the total normalized shear stress is not constant throughout the viscous layer and in the lower part of the log-layer (y+ < 50) as found by Pirozzoli et al. [27] and Duan et al. [29], and it decreases from 1 to 0.85 in the so-called region. Fig. 10b shows that the different shear stresses are insensitive to the grid resolutions.
maximum of about 25% in the Log-region. The streamwise velocity fluctuations shows also the same tendency, where the peak is over-predicted by about 25% compared to the reference data. The wall-normal and the spanwise fluctuations are however underestimated. The observed trends are the footprint of an excessive numerical dissipation due to the WENO scheme. The the fourthand the sixth-order split-centered schemes show similar results, which are slightly under-estimated compared to the DNS data. However, the relative difference is much lower compared to the results of the WENO scheme.
Numerical schemes assessment Fig. 11 depicts a comparison between the fifth-order WENOBWO scheme, the fourth- and the sixth-order split-centered schemes, through the van-Driest transformed mean streamwise velocity, u+ vd , and the velocity fluctuations in Morkovin’s scaling, as a function of y+ . Except the viscous sublayer, u+ vd exhibits an over-estimation compared to the DNS data, which reaches a
The objective here is to assess the evaluation of the LES resolution by investigating the contribution of the SGS model to the turbulent energy dissipation.
3.2. Subgrid-scale analysis - On evaluating grid resolution
3.2.1. Near-wall asymptotic behavior of the SGS viscosity For a refined mesh the SGS model should exhibit the correct 3 asymptotic behavior in the vicinity of a solid wall (νsgs ∝ y+ ). If
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Fig. 9. Distribution of r.m.s. vorticity components as a function of y+ . (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
this assumption does not hold for a given SGS model, it results in a very high damping of turbulent fluctuations in this region, and in a wrong prediction of the skin friction, for instance (Garnier et al. [30]). Fig. 12 shows the ratio of the SGS viscosity to the molecular viscosity, μsgs /μ, as a function of y+ . In Fig. 12a, the DSM does not exhibit the correct slope in the near-wall region, while the CSM and the WALE model retrieve the correct slope. Also, the DSM seems not properly vanishing at the wall, although the flow is well resolved in this region (y+ 1). However, the CSM has a min lower amount of μsgs , that monotonically increases when the grid is coarsen (Fig. 12b). The CSM is found to be less dissipative than the DSM, and does preserve the good asymptotic behavior of the flow at the wall. For the DSM, the combined effect of a relatively high amount of SGS viscosity and a wrong asymptotic behavior
near the wall can be a reason to its under-estimation of the mean temperature distribution (see Fig. 2). 3.2.2. Ratio of turbulent kinetic energy The parameter ϑ, initially introduced by Pope [31] and used by Davidson [32], is defined as:
ϑ=
K k sgs + K
(21)
where ksgs = τii /(2ρ¯ ) is the modeled turbulent kinetic energy and K = u i u i /2 is the resolved turbulent kinetic energy, with u i the velocity fluctuation of the ith component of the resolved filtered field. It represents the ratio of the resolved turbulent kinetic energy, K , to the total kinetic energy, ksgs + K . The isotropic part of the stress tensor is modeled according to Yoshi-
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Fig. 10. Turbulent shear stress −ρ u v , mean viscous shear stress μ∂ u/∂ y and total shear stress −ρ u v + μ∂ u/∂ y as a function of y+ . (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
+
Fig. 11. (a) Distribution of the van-Driest transformed mean streamwise velocity uvd as a function of y+ ; (b) r.m.s velocity fluctuations in Morkovin’s scaling as a function ) WENO-BWO scheme; ( ) 4th split-centered; ( ) 6th split-centered. of y+ . (
sawa, where CI is either retrieved using the dynamic procedure in the DSM, or set constant for the CSM and the WALE model. Based on the parameter defined in Eq. (21), Pope [31] suggests that when 80% of the kinetic energy is accounted for, the LES can be considered as well-resolved. In order to check this criterion, the parameter ϑ is plotted as a function of y+ for SGS models (Fig. 13a) and grid resolutions (Fig. 13b). At the wall, the DSM shows higher value of ϑ than the CSM, with 55% of the kinetic energy being resolved for the DSM against 45% for the CSM. However, one should remember that in the vicinity of the wall, the viscous effects are dominant, and the action of the SGS model on the turbulence structure is weak. At the frontier of the viscous sublayer, this tendency is inverted, and up to y+ 100, the CSM better predicts ϑ, until it reaches 1 for both SGS models. Fig. 13b shows that ϑ is sensitive to the grid resolution at the wall, showing a monotone decrease while coarsening the grid, until reaching ࣃ 1 at y+ 20 for all grids. Apart from a small difference in a region very close to the wall, ϑ reaches
quickly its maximum value. However, because the SGS model is not supposed to act in the viscous sublayer, where the grid is refined enough, and based on Pope’s assumption, the parameter ϑ seems not to be sufficient to fully assess the quality of the LES resolution. 3.2.3. Subgrid-scale dissipation In order to quantify the contribution of the SGS model to the production/destruction of turbulence, it is interesting to consider the contribution of the SGS terms to turbulence. We can
and ε define εsgs sgs , which are the production/destruction term of the resolved turbulent kinetic energy, K , and the production/destruction term in the modeled turbulent kinetic energy, ksgs , respectively. Following the developments by Davidson [33], these terms can be explicitly formulated so that,
+ε εsgs = εsgs sgs
(22)
Subgrid-scale dissipation due to resolved fluctuations The term of production/destruction of the resolved turbulent kinetic energy, K , by the SGS turbulence, which can also be inter-
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Fig. 12. Normalized SGS viscosity μsgs /μ as a function of y+ . (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
Fig. 13. Ratio of the turbulent kinetic energy ϑ as a function of y+ . (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
preted as the SGS dissipation term due to the resolved fluctuations, was defined by Davidson [33] as:
= εsgs
τi∗j − τi∗j
∂ u
i
∂xj
(23)
where τi∗j = τi j − τkk δi j /3 is the modeled SGS stress tensor, and u i is the resolved fluctuations of the ith component of the filtered velocity field. Simplifying the terms between the square-brackets yields:
∂ uj ui ∂ + ∂ x j ∂ xi
∂ u j 1 ∂ ui
si j = + 2 ∂xj ∂ xi
si j =
1 2
s i∗j = s i j −
1
s δi j 3 kk
(25)
Then, it follows:
τi∗j − τi∗j = τi j∗ = −2 μsgs s∗i j − μsgs s∗i j −2μsgs s i∗j
tuating tensor of s∗i j :
(24)
where s∗i j = si j − skk δi j /3 is the deviatoric part of the strain-rate
tensor computed from the filtered velocity field, and s i∗j is the fluc-
∂ u i
∗ −2 μsgs si ∗j s i j + ωi j sgs = τi j ∂xj −2 μsgs s i∗j s i j
ε
(26)
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155
/ε + Fig. 14. (a and d) Ratio of the SGS dissipation due to the fluctuating flowfield to the total SGS dissipation εsgs sgs as a function of y ; (b and e) Ratio of the SGS dissipation due to the mean-averaged flowfield to the total SGS dissipation ε sgs /ε sgs as a function of y+ ; (c and f) Ratio of the SGS dissipation due to the fluctuating flowfield in a
as a function of y+ . (a-c) SGS models study using LES-P2 grid; (d-f) Grid sensitivity /εsgs wall-parallel plane to the SGS dissipation due to the fluctuating flowfield εsgs x+z study using CSM. For legend, see Table 1.
where s i∗j ωi j is equal to zero, since it is the product of a symmetric and anti-symmetric tensors, s i j ωi j :
i j = ω
1 2
∂ uj ui ∂ − ∂ x j ∂ xi
1 s i∗j ωi j = s i j ωi j − s kk δi j ωi j 3 1
= si j ωi j − skk ωkk 3
∂ ui i j εsgs = τi∗j = −2μsgs s∗i j si j + ω ∂xj −2μsgs s∗i j si j (27)
It is interesting to quantify the ratio of the SGS dissipation term due to resolved velocity fluctuations in a wall-parallel plane
εsgs , since the streamwise and spanwise components of the SGS x+z dissipation term make an important contribution to the total SGS dissipation (Davidson [32]). It is defined as:
εsgs = −2 μsgs s i∗j s i j x+z x+z
∗
∗
∗
= −2 μsgs s11 s11 + s 33 s33 + 2s 13 s13 x+z
Subgrid-scale dissipation due to time-averaged field The production/destruction term, εsgs , in the modeled turbulent kinetic energy ksgs , also interpreted as the SGS dissipation due to the time-averaged velocity field, is (Davidson [33]):
where is equal to zero, ui is the time-averaged velocity field of the ith component, and:
si j =
1 2
i j = ω
1 2
∗
∂ u j ui ∂ + ∂xj ∂ xi ∂ u j ui ∂ − ∂xj ∂ xi
si j = si j −
(28)
(29)
i j s∗i j ω
1 s δi j 3 kk
(30)
Fig. 14a and d depict the ratio of the SGS dissipation due to the fluctuating flowfield to the total SGS dissipation. David-
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Fig. 15. SGS Activity parameter ς as a function of y+ . (a-b-c) Different SGS models using LES-P2 grid; (d-e-f) Grid sensitivity study using CSM. For legend, see Table 1.
Fig. 16. Ratio of the SGS dissipation to the viscous dissipation as a function of y+ . (a) Different SGS models using LES-P2 grid; (b) Grid sensitivity study using CSM. For legend, see Table 1.
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son [32] showed that for higher resolution, higher is the value of
/ε . Fig. 14a shows that, for the same given grid and above the εsgs sgs log layer, both DSM and WALE model give marginally higher esti /ε + mation of this ratio than CSM. εsgs sgs is very low up to y ≤ 10 and increases steeply to reach its maximum at y+ 50 for all models. However, it is shown in Fig. 14d that this ratio tends to monotonically increase when the grid is coarsen. For all grids, the ratio
/ε , ε varies between 0.85 and 0.95. In contrast to εsgs sgs sgs /ε sgs increases when refining the grid (Fig. 14e). Fig. 14c and f show the ratio of the SGS dissipation due to the fluctuating flowfield in a
wall-parallel plane εsgs to the SGS dissipation due to the flucx+z
tuating flowfield εsgs . The main contribution of the resolved fluctuations in a given wall-parallel plane varies between 0.4 and 0.6 depending on the model. This contribution appears to be barely af
fected by the grid resolution. The ratio εsgs /ε is monotonically x+z sgs decreasing when the grid is coarsen (for y+ ≤ 50), and the refinement in the spanwise direction is more important. 3.2.4. Subgrid-scale activity The total SGS dissipation of a LES model, ε sgs , is defined as the sum of both the SGS dissipation due to the fluctuating flowfield,
, and the one due to the mean-averaged flowfield, ε εsgs sgs . The amount of SGS turbulent dissipation is the central quantity used to assess the importance of the SGS model, i.e., to quantify the amount of modeling in a LES compared to a DNS (Geurts & Fröhlich [34]) . The SGS activity parameter is defined as:
ς=
εsgs εsgs + ε
(31) ∂ u
where ε = τik ∂ x i is the resolved molecular dissipation. It comes k that 0 ≤ ς < 1 by definition, with ς = 0 corresponding to DNS and ς = 1 to LES at infinite Reynolds number (Geurts & Fröhlich [34]). Fig. 15 shows the SGS activity parameter ς as a function of y+ , for different SGS models (Fig. 15a, b and c) and grid resolutions (Fig. 15d, eand f). In the viscous sublayer and up to y+ 5, the SGS activity parameter ς is about 0.8 for all models. It is slightly over-predicted by the DSM, compared to CSM and WALE model, reaching a maximum of ≈ 0.84 at the wall. In this region, ς is weakly affected by the grid resolution, and all studied grids predict nearly similar value at the wall. In the transition region and up to y+ 300, ς decreases to different values for different SGS models. In this region, ς reaches a minimum of 0.5 for DSM at y+ 20, 0.4 for the WALE model at y+ 20 and 0.2 for the CSM at y+ 30. Similar behavior is observed for different meshes. ς increases in the outer region of the boundary layer (Fig. 15a), reaching a maximum of about 0.6 for the DSM, 0.6 for the WALE model and 0.4 for the CSM, to decrease again at the edge of the boundary layer. In this region, ς shows a monotone increase when coarsening the grid, with a maximum of 0.2 for LES-L1 grid to 0.5 for LES-S2 using the CSM.
due to Fig. 15b and e) show the SGS activity parameter ςsgs fluctuating flowfield and Fig. (15c and f) the SGS activity parameter ς sgs due to the mean-averaged flowfield, respectively defined as:
ςsgs =
ςsgs
εsgs
εsgs + ε
εsgs =
εsgs + ε
(32)
From Figs. (15b and e) and (15c and 15f), it is obvious that the SGS dissipation is mainly driven by viscous effects in a near-wall region (up to y+ 10), while the fluctuating flowfield mainly acts in the transition region and up to the edge of the boundary layer (10 ≤ y+ ≤ 10 0 0). From Fig. (15a, b and c), it can be deduced that the DSM produces higher subgrid scale dissipation than both the
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CSM and the WALE model, especially in the outer region of the
is very close to that of boundary layer. For the WALE model, ςsgs the DSM throughout the layer. A direct comparison of ν sgs and ν can also be interesting. Davidson [33] suggested that, if ν sgs ν , the SGS dissipation is much larger than the viscous one, and if this is not the case, the grid is more likely to be a DNS one. At y+ > 10 (Fig. 16), the ratio of ε sgs /ε ≤ 1 in a wide range of the boundary layer, and thus ν sgs ≈ ν , which means that the considered grids are suitable for LES study. 4. Conclusion In this paper, large-eddy simulations of an adiabatic supersonic turbulent boundary layer at a freestream Mach number M∞ = 2 and a friction Reynolds number up to Reτ = 600 are performed and compared to DNS and experimental data as well. For the three considered SGS models, the main statistical quantities show an overall acceptable agreement. As a first observation, the CSM and the WALE model show better results compared to the DSM. In fact, both models give the correct rate of decay as the wall is approached, which ensure that the SGS viscosity vanishes properly within the viscous sublayer. The results show that the DSM predicts the main statistical quantities with better accuracy, especially when estimating the peaks of the velocity fluctuations and the r.m.s. of the wall-pressure distribution. However, it shows less confidence in predicting the mean temperature profile especially at the wall, where the WALE model and the CSM perform better. Furthermore, the use of an Implicit LES shows tendencies that do not vary much compared to the explicit LES models, but it seems that the lack of SGS dissipation affected some thermodynamic quantities, such as Trms and ρ rms , where a bump in the outer region of the boundary layer is observed. This bump can express the manifestation of an accumulation of non-dissipated energy that moved away from the wall. By discussing the different components of the subgrid-scale dissipation, an attempt to set a criterion to evaluate the quality of LES grid resolutions is made. Finally, it is worth mentioning that the CSM and the WALE model show similar computational performance in terms of CPU time, using almost 25% less time than the DSM. Acknowledgments The authors gratefully knowledge financial support from the Regional Council of Upper–Normandy (Région Haute Normandie). The use of the DNS database of Prof. S. Pirozzoli and Dr. M. Bernardini from University of Roma, ‘La Sapienza’ is also acknowledged. This work was performed using HPC resources from GENCI [CCRT/CINES/IDRIS] (grant t20162a7544) and from CRIANN (Centre Régional Informatique et d’Applications Numériques de Normandie, Rouen). References [1] Shadloo MS, Hadjadj A, Hussain F. Statistical behavior of supersonic turbulent boundary layers with heat transfer at m=2. Int J Heat Fluid Flow 2015;53:113–34. [2] Spyropoulos ET, Braisdell GA. Large-eddy simulation of a spatially evolving supersonic turbulent boundary layer flow,. AIAA J 1998;36(11):1983–90. [3] Stolz S, Adams NA. Large eddy simulation of high-Reynolds number supersonic boundary layers using the approximate deconvolution model and a rescaling and recycling technique. Phys Fluids 2003;15:2398–412. [4] Urbin G, Knight D. Large-eddy simulation of supersonic boundary layer using an unstructured grid. AIAA J 2001;7(39):1288–95. [5] Yan H, Knight D, Zheltovodov AA. Large-eddy simulation of supersonic flat– plate boundary layers using the monotonically integrated large-eddy simulation MILES technique. J Fluids Eng 2002;124:868–75. [6] Urbin G, Knight D. Compressible large eddy simulation using unstructured grid: supersonic boundary layer. 2nd AFOSR Conference on DNS/LES, Kluwer Academic Publishers, Rutgers University; 1999. p. 443–58.
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