Wall-Modeled Large Eddy Simulation and Detached Eddy Simulation of Wall-Mounted Separated Flow via OpenFOAM

Abstract

Considering grid requirements of high Reynolds flow, wall-modeled large eddy simulation (WMLES) and detached eddy simulation (DES) have become the main methods to deal with near-wall turbulence. However, the flow separation phenomenon is a challenge. Three typical separated flows, including flow over a cylinder at $R{e}_D$ = 3900 based on the cylinder diameter, flow over a wall-mounted hump at $R{e}_c$ = 9.36 × 10⁵ based on the hump length, and transonic flow over an axisymmetric bump with shock-induced separation at $R{e}_c$ = 2.763 × 10⁶ based on the bump length, are used to verify WMLES, shear stress transport k-$\omega$ DES (SST-DES), and Spalart–Allmaras DES (SA-DES) methods in OpenFOAM. The three flows are increasingly challenging, namely laminar boundary layer separation, turbulent boundary layer separation, and turbulent boundary layer separation under shock interference. The results show that WMLES, SST-DES, and SA-DES methods in OpenFOAM can easily predict the separation position and wake characteristics in the flow around the cylinder, but they rely on the grid scale and turbulent inflow to accurately simulate the latter two flows. The grid requirements of Larsson et al. ($\left(\delta /\Delta x,\delta /\Delta y,\delta /\Delta z\right)\approx (12,50,20)$) are the basis for simulating turbulent boundary layers upstream of flow separation. A finer mesh ($\left(\delta /\Delta x,\delta /\Delta y,\delta /\Delta z\right)\approx (40,75,40)$) is required to accurately predict the separation and reattachment. The WMLES method is more sensitive to grid scales than the SA-DES method and fails to obtain flow separation under a coarser grid, while SST-DES method can only describe the vortices generated by the separating shear layer, but not within the turbulent boundary layer, and overestimates the separation-reattachment zone based on the grid system in this paper.

1. Introduction

In 1963, the large eddy simulation (LES) technique was initially proposed by Smagorinsky to simulate atmospheric current [1]. LES has been applied in unsteady, multiscale, and multi-physics turbulent flows including atomization, combustion, acoustics, and atmospheric boundary layer [2,3,4,5,6]. However, turbulent boundary layers with high Reynolds numbers would incur huge computational costs for wall-resolved LES (WRLES) that almost match the direct numerical simulation (DNS) method. Choi and Moin [7] estimated the grid requirements for a turbulent flat plate flow for DNS, wall-resolved LES, and wall-modeled LES (WMLES), which are of the order of $R{e}_{L_x}^{37/14}$, $R{e}_{L_x}^{13/7}$ and $R{e}_{L_x}$, respectively. Here $L_x$ is the length of the flat plate. Various wall-modeled approaches for the turbulence in the inner part of the boundary layer are designed to remove the need to resolve any turbulent eddies here. These wall-modeled approaches are mainly divided into hybrid Reynolds-averaged Navier–Stokes (RANS)/LES and WMLES based on wall shear stress [8,9,10].

To keep the multiscale nature in the turbulent boundary layer, these WMLES methods are more demanding on the grid requirements than RANS methods. The grid requirements are separated in LES domain and near-wall modeled domain. Larsson [9] estimated grid requirements for the LES domain in the outer boundary layer (or $y/\delta >0.2$) as $\left(\Delta x/\delta ,\Delta y/\delta ,\Delta z/\delta \right)\approx (0.08,0.02,0.05)$, where $\delta$ is the boundary layer thickness. Menter [11] estimated a lower requirement, $\left(\delta /\Delta x,\delta /\Delta y,\delta /\Delta z\right)\approx (10,30,20)$. The grid requirements for the inner layer (i.e., the viscous and logarithmic layers) are further adapted to the RANS method or the modeled method based on wall shear stress. Figure 1 shows turbulence structures in a channel flow at $R{e}_{\tau }=(u_{\tau }·0.5H)/\nu =2320$ based on three grid systems, where H is the height of the channel, $\nu$ is the kinematic viscosity, and $u_{\tau }=\sqrt{{\tau }_w/\rho }$ is the friction velocity defined from the wall stress ${\tau }_w$ and the density $\rho$. The first grid is commonly used in the RANS method, which relaxes the requirements on the streamwise grid size and does not obtain the vortex size. The second set of grids is obtained by refining its streamwise grid size, and the large-scale vortices in the outer layer can be obtained. Finer vortex structures can be obtained by further adopting a near-wall refined cubic mesh. Therefore, due to the requirements of WMLES on the streamwise and spanwise grid size of the region of interest, including boundary layers, separation layers and wakes, the structural grid with local refinement is an effective choice to control the total grid count.

OpenFOAM, as a free and open-source computational fluid dynamics (CFD) software, includes several common hybrid RANS/LES methods, including Spalart–Allmaras detached eddy simulation (SA-DES) [12], Spalart–Allmaras delayed DES (SA-DDES) [13], Spalart–Allmaras Improved DDES (SA-IDDES) [14], k-$\omega$ SST Detached Eddy Simulation (SST-DES) [15]. OpenFOAM already has an explicit LES method with various sub-grid scales (SGS) models, including classical Smagorinsky [1], wall adapting local eddy viscosity (WALE) [16], and k-equation eddy viscosity [17] models. However, there is no effective Wall-modeled LES (WMLES) method in OpenFOAM. Recently, a wall-modeled library was developed by Mukha et al. [18] and can be combined with the SGS models in OpenFOAM to form WMLES, but it only works with incompressible flows. These techniques in OpenFOAM are being successfully applied to turbulent boundary layers and simple flows at low Reynolds numbers.

Separated flows occur in a variety of applications, such as buildings, ships, vehicles, and aircraft, from low-speed incompressible flows to supersonic compressible flows. Accurate simulation of separations is critical to their design and analysis. However, separated flows over a smooth surface present a challenge to numerical simulation methods. The flow over a circular cylinder at $R{e}_D=3900$ based on the cylindrical diameter, the flow over a wall-mounted hump and the transonic flow over an axisymmetric bump are used to test various wall-modeled approaches in OpenFOAM. The flow over a circular cylinder is a simple laminar separation and has a large amount of experimental and simulation data [19,20,21,22,23,24,25,26,27,28,29,30,31,32]. Lysenko et al. [27] applied LES methods with classical Smagorinsky and dynamic k-equation eddy viscosity SGS models in OpenFOAM to the flow over a circular cylinder at $R{e}_D=3900$. Jiang and Cheng [32] used the WALE SGS model. D’Alessandro et al. [28] used the SA-DES method and a newly developed DES model based on the $\overline{v^2}$-f approach. The flow over a wall-mounted hump experiment of Greenblatt et al. [33] has been chosen as a test case for the RANS models in the NASA CFDVAL2004 Workshop [34]. The transonic flow over an axisymmetric bump experiment of Bachalo et al. [35] is a test case for the NASA Revolutionary Computational Aerosciences (RCA) challenge under the Transformational Tools and Technologies (TTT) project. The latter two flows belong to NASA Turbulence Modeling Resource (TMR) website. [36] They have fully developed a turbulent boundary layer before flow separation, and even shock/boundary layer interference occurs. This study uses SA-DES, SST-DES and WMLES in OpenFOAM to simulate these three flows to assess the capability for various separated flows. The sensitivities of these methods to grid scale and turbulent inflow are also tested.

The work is organized as follows: In Section 2, SA-DES, SST-DES and WMLES based on wall shear stress in OpenFOAM are described. In Section 3, three flows over a circular cylinder, a wall-mounted hump and an axisymmetric bump and their physics models are introduced. In Section 4, the simulation results of several WMLES methods on these three flows are compared. Conclusions are listed in Section 5.

2. Numerical Methods

This section describes the WMLES and DES methods in OpenFOAM. The WMLES method uses the flow information from the classical LES near the wall to correct the wall shear stress. The DES method introduces switch functions into the RANS equations to switch the domain between RANS and LES. Their implementations on OpenFOAM framework are shown in Figure 2.

2.1. WMLES

In the WMLES method, the values of quantities such as pressure, density, and velocity are sampled from a specified cell center in the LES domain, located at a distance from the wall, to calculate the wall shear stress at the given surface using the wall model. Therefore, it is divided into two parts, the LES method and the wall model.

2.1.1. Governing Equations for LES

The governing equations for LES are obtained by spatial filtering of the incompressible Navier–Stokes (N-S) equations [4,5]

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\overline{u}}_j}{\partial x_j}}& =0\hfill \\ \hfill {\displaystyle \frac{\partial }{\partial t}}\left({\overline{u}}_i\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left({\overline{u}}_i{\overline{u}}_j\right)-\nu {\displaystyle \frac{{\partial }^2{\overline{u}}_i}{\partial x_j\partial x_j}}& =-{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}-{\displaystyle \frac{\partial {\tau }_{ij}}{\partial x_j}}\hfill \end{array}$$

(1)

where t is the time, $\left(x_1,x_2,x_3\right)$ are Cartesian coordinates, $u_i$ is the velocity component in the direction $x_i$, p and $\nu$ are the kinematic pressure (i.e., pressure divided by fluid density) and kinematic viscosity, respectively. The superscript $\overline{\left(·\right)}$ denotes the resolved-scale quantity after spatial filtering, $\overline{f}\left(x,t\right)={\int }_{-\infty }^{\infty }G\left(x-x^{\prime }\right)f\left(x^{\prime },t\right)\mathrm{d}{x}^{\prime }$, and $G\left(x-x^{\prime }\right)$ is the filter function. Compared with the classic N-S equations, a term with the SGS stress tensor (${\tau }_{ij}$) is added [1]. There are various methods to calculate it. Several implementations in OpenFOAM are summarized here.

Based on the assumption of the balance between the SGS energy production and dissipation (local equilibrium), Smagorinsky defined it as [1]

$${\tau }_{ij}={\displaystyle \frac{2}{3}}{k}_{\mathrm{sgs}}{\delta }_{ij}-2{\nu }_{\mathrm{sgs}}\mathrm{dev}{\left({\overline{S}}_{ij}\right)}_{ij}$$

(2)

where ${\nu }_{\mathrm{sgs}}$ is the SGS eddy viscosity, ${\nu }_{\mathrm{sgs}}=C_k\Delta \sqrt{k_{\mathrm{sgs}}}$ with $C_k=0.094$. Here ${\nu }_{\mathrm{sgs}}$ is determined by the wall adapting local eddy viscosity (WALE) model in OpenFOAM that is an algebraic eddy viscosity model taking into the rotation rate and is able to handle transition [16,37].

$${\nu }_{\mathrm{sgs}}={\left(C_w\Delta \right)}^2{\displaystyle \frac{{\left(S_{ij}^d{S}_{ij}^d\right)}^{3/2}}{{\left({\overline{S}}_{ij}{\overline{S}}_{ij}\right)}^{5/2}+{\left(S_{ij}^d{S}_{ij}^d\right)}^{5/4}}}$$

(3)

where $S_{ij}^d$ is the traceless symmetric part of the square of the velocity gradient tensor $S_{ij}^d={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial {\overline{u}}_k}{\partial x_i}}{\displaystyle \frac{\partial {\overline{u}}_j}{\partial x_k}}+{\displaystyle \frac{\partial {\overline{u}}_k}{\partial x_j}}{\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_k}}\right)-{\displaystyle \frac{1}{3}}{\delta }_{ij}{\displaystyle \frac{\partial {\overline{u}}_k}{\partial x_l}}{\displaystyle \frac{\partial {\overline{u}}_l}{\partial x_k}}$. $C_w=0.325$ is the model constant.

2.1.2. Wall-Modeled Method

The wall stress in the inner boundary layer could be described with thin boundary layer equations [9,10]

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}y}}\left[\left(\mu +{\mu }_t\right){\displaystyle \frac{\mathrm{d}u}{\mathrm{d}y}}\right]=0$$

(4)

where the turbulent eddy viscosity ${\mu }_t$ can be taken as ${\mu }_t=\kappa \mu y^{+}{\left[1-\exp \left(-y^{+}/A^{+}\right)\right]}^2$, and $y^{+}=y{u}_{\tau }/\nu$ is the normalized distance to the wall with the kinematic viscosity. The thin boundary layer equation is solved using physical quantities from a cell center in the LES domain near a wall and returns the wall shear stress to the wall. Mukha et al. [18] have implemented it for incompressible flow on the OpenFOAM framework in the form of ordinary differential equations. The current program needs to manually specify the grid point within the inner boundary layer. It should be noted that the information in the first cell near the wall cannot be sampled. When using the information of the first node as the input of the wall model, Mukha et al. [18] found deviations when simulating the channel flow, and Iyer and Malik [38] did not accurately predict the separation and reattachment. Therefore, the information from the third cell is sampled in this paper.

2.2. Detached Eddy Simulation (DES) Method

The SA-DES and SST-DES method add a switch function in the eddy viscosity ${\nu }_t$-equation to switch the domain between RANS and LES, while the SST-DES method adds another switch function in the turbulent kinetic energy k-equation.

2.2.1. Spalart–Allmaras DES

Based on the Spalart–Allmaras RANS model, the SA-DES model modifies the diffusion term in the eddy viscosity ${\nu }_t$-Equation [12].

$$\begin{array}{cc}\hfill \frac{\partial {\nu }_t}{\partial t}+{\displaystyle \frac{\partial }{\partial x_j}}\left({\nu }_t{u}_j\right)& -\frac{1}{{\sigma }_{\nu }}\left\{{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\nu +{\nu }_t\right){\displaystyle \frac{\partial {\nu }_t}{\partial x_i}}\right]+c_{b2}{\displaystyle \frac{\partial {\nu }_t}{\partial x_j}}{\displaystyle \frac{\partial {\nu }_t}{\partial x_j}}\right\}\hfill \\ \hfill & =c_{b1}{\nu }_t\left[S+f_v\left({\displaystyle \frac{{\nu }_t}{\nu }}\right)\frac{{\nu }_t}{{\kappa }^2{\tilde{d}}^2}\right]-c_{w1}{f}_w\left(\frac{{\nu }_t}{{\kappa }^2{\tilde{d}}^2}\right){\left(\frac{{\nu }_t}{\tilde{d}}\right)}^2\hfill \end{array}$$

(5)

Model constants ${\sigma }_{\nu }$, $c_{b1}$, $c_{b2}$, $c_{w1}$, $c_{w2}$ and $\kappa$ are the same as the Spalart–Allmaras RANS model. The two terms on the right side represent the production and destruction for the eddy viscosity ${\nu }_t$ and are related to the length scale $\tilde{d}$. In DES model, it is defined as

$$\tilde{d}=\left\{\begin{array}{c}\min \left(d,C_{DES}\Delta \right),\mathrm{for}\mathrm{DES}\hfill \\ d-f_d\max \left(0,d-C_{DES}\Delta \right),\mathrm{for}\mathrm{DDES}\hfill \end{array}\right.$$

(6)

where $f_d=1-\tanh \left[{\left(8r_d\right)}^3\right]$ with $r_d=\left({\nu }_{SGS}+\nu \right)/\left(\left|\nabla \mathit{u}\right|{\kappa }^2{d}^2\right)$. $C_{DES}\Delta$ is the LES scale. $C_{DES}$ is a constant and its default value is 0.65. Please note that $\tilde{d}=d$ near a wall, and Equation (5) degenerates to the Spalart–Allmaras RANS model.

2.2.2. k-ω SST DES

Based on the SST k-$\omega$ RANS model, The SST-DES model modifies the diffusion term in the turbulent kinetic energy k-Equation [15].

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial k}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(k{u}_j\right)-{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\nu +{\displaystyle \frac{{\nu }_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_i}}\right]& =P_k-{\beta }^{*}k\omega F_{\mathrm{DES}}\hfill \\ \hfill {\displaystyle \frac{\partial \omega }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\omega u_j\right)-{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\nu +{\displaystyle \frac{{\nu }_t}{{\sigma }_{\omega }}}\right){\displaystyle \frac{\partial \omega }{\partial x_i}}\right]& =P_{\omega }-\beta {\omega }^2+2\left(1-F_1\right){\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_i}}\hfill \end{array}$$

(7)

where k and $\omega$ are the turbulent kinetic energy and specific dissipation rate, respectively. The production terms ($P_k$ and $P_{\omega }$) and switching functions ($F_1$ and $F_2$) are the same as in the SST k-$\omega$ RANS model [15].

The switching function $F_{\mathrm{DES}}$ incorporates DES features into Equation (7).

$$F_{\mathrm{DES}}=\max \left[\left(1-F_2\right){\displaystyle \frac{L_{\mathrm{RANS}}}{C_{\mathrm{DES}}\Delta }},1\right]$$

(8)

where the turbulent length scale is $L_{\mathrm{RANS}}=\sqrt{k}/\left({\beta }^{*}\omega \right)$. Please note that when $F_{\mathrm{DES}}=1$, Equation (7) degenerates to the SST k-$\omega$ RANS model.

The incompressible solver pisoFoam using pressure implicit with splitting of operator (PISO) algorithm in OpenFOAM is used for the first two separation flows below, while the third uses the compressible solver myLusgsFoam developed by Fürst [39], which is a lower-upper symmetric Gauss–Seidel (LU-SGS) matrix-free solver. Compared to several compressible solvers in OpenFOAM, myLusgsFoam allows the use of larger time steps and captures shock waves using the HLLC flux scheme. The gradient and divergence terms all adopt the second-order schemes. The time terms adopt the second-order backward format. HLLC flux scheme. The HLLC fluxes for convective terms are considered in myLusgsFoam for the third transonic flow.

3. Physic Models

Three separation flows, including a flow over a circular cylinder at $R{e}_D=3900$ based on the diameter, a flow over a wall-mounted hump at $R{e}_c=9.36\times {10}^5$ based on the length of the hump, and a flow over an antisymmetric bump at $R{e}_c=2.763\times {10}^6$ and $Ma=0.875$ are simulated by the SA-DES, SST-DES and WMLES methods in OpenFOAM.

The computational domains for the three flows are a circular cylinder, a rectangle, and a wedge, respectively. Their geometric structure and computational domain are shown in Figure 3. Their flow conditions are summarized in

Table 1. Conditions for three flows.

		Cylinder	Wall-Mounted Hump	Axisymmetric Bump
Reynolds number		3900	9.36 × 105	2.763 × 106
Mach number		0.2	0.1	0.875
Characteristic length		$D=8.37\times {10}^{-4} \mathrm{m}$	$c=0.42 \mathrm{m}$	$c=0.2032 \mathrm{m}$
Spatial domain		$30D\times 30D\times \pi D$	$10.4D\times 0.91D\times 0.3D$	$7.63D\times 3.62D\times {30}^{\circ }$
NASA-Mesh	$N_{\mathrm{cell}}$	1.44 million	10.58 million	13.82 million
		$150\times 200\times 48$	$816\times 216\times 60$	$720\times 320\times 60$
	${\Delta }_{y,1}$	$4.0\times {10}^{-3}D$	$6.8\times {10}^{-6}c$	$8.0\times {10}^{-6}c$
	$N_{\mathrm{cell}}$	-	17.04 million	24.90 million
NewMesh	${\Delta }_{y,1}$	-	$4.7\times {10}^{-4}c$	$9.2\times {10}^{-4}c$

. The computational domain and mesh structure for the cylinder case is consistent with the literature [27] that presented the WRLES of the flow field. The spatial domain is $30D\times 30D\times \pi D$. The grid number is $150\times 200\times 48=$ 1.44 million, which is one-fourth of the literature [27].

The length of the hump is $c=420 \mathrm{mm}$, and its height is $h=53.7 \mathrm{mm}$. The size of the computational domain corresponds to $\left[L_x,L_y,L_z\right]=\left[10.4c,0.91c,0.3c\right]$ Note the geometry is two-dimensional and does not vary in the span in the simulation. Here the same spanwise dimensions as Iyer and Malik [38]. A contoured top wall is set upon the hump in the simulations based on the recommendations of the NASA CFDVAL2004 Workshop [34] to account for the effect of the side-mounted end plates in the experimental test. Two grid systems are used for the wall-mounted hump case. The first gird is from the Turbulence Modeling Resource Web of Langley Research Center, and is defined as NASA-Mesh in Figure 3. This grid has been verified to satisfy $y^{+}<1$ for various RANS models, but its streamwise grid size does not meet the requirements of LES. Another Cartesian grid system with prismatic boundaries is set for the wall-mounted hump case, namely NewMesh. Specific details and independent verification of NewMesh are detailed in the next section.

The transonic flow over an axisymmetric bump experiment of Bachalo et al. [35] is simulated. The length of the bump is $c=203.2 \mathrm{mm}$. The domain consists of a 30° wedge with inner radius $r_{\mathrm{inner}}/c=0.375$ and outer radius $r_{\mathrm{outer}}/c=4.0$. The bump height is $h/c=0.09375$. This case also uses two grid systems similar to the wall-mounted hump case.

The first two cases have similar boundary conditions. A no-slip boundary condition is imposed for the velocity at the wall. The inlet boundary condition (on the left) is a fixed-value velocity with zero-pressure gradient, while the outlet boundary condition (on the right) adopts a wave transmissive boundary with a fixed-value pressure. The boundaries on both sides of the computational domain are periodic. The third case has different settings at the inlet and outlet. There is a fixed-value velocity, total temperature, and total pressure at the inlet boundary, while there is a zero-gradient velocity, temperature and pressure at the outlet boundary. Furthermore, the incoming flow of the latter two cases is divided into laminar incoming flow and turbulent incoming flow. For turbulent incoming flow, a composite setting is used at the inlet boundary, i.e., the velocities within the turbulent boundary layer are mapped directly from a certain distance downstream, while the upper layer velocities are directly set to a fixed value, as shown in Figure 4.

4. Results and Discussion

4.1. Flow over a Circular Cylinder

The flow over a circular cylinder at $R{e}_D=3900$ belongs to the sub-critical regime [28]. The boundary layer is laminar, and turbulent flow only occurs after separation. There is a periodic Kármán vortex street in the wake, as shown in Figure 5. Norberg [21] measured the mean pressure distribution over a cylinder. Lourenco, Parnaudeau, Ong et al. [19,22,26] measured the flow velocity statistics in a cylindrical wake using particle image velocimetry (PIV) and hot wire anemometer (HWA). It is appropriate to use free vortices to validate numerical methods for simulating free vortices. In recent years it has been an indication of the adequacy and accuracy of LES and hybrid RANS/LES methods in OpenFOAM.

Figure 6 shows the mean pressure coefficient distribution on the cylinder, $C_p=\left(p-p_{\infty }\right)/\left(1/2{\rho }_{\infty }{U}_{\infty }^2\right)$. In addition to WMLES, SA-DES, and SST-DES, WRLES, which ignores wall-modeled approach, is also tested. In addition, WMLES also tested other SGS models in OpenFOAM to replace WALE, including classical Smagorinsky, k-equation eddy viscosity and dynamic-k-equation eddy viscosity. The results of several WMLES and DES methods show good consistency with the experimental data from Norberg [21], while that of WRLES is inaccurate.

Figure 7 shows the mean velocity and resolved Reynolds stress profiles at $z/D=\pi /2$ and $x/D=$ 1.06, 1.54, 2.02, 4, 7, 10 in the wake region. Due to the interference of the cylinder, the mean velocity profiles in Figure 7 fluctuate around $y=0$, while in remote zones, they tend to converge toward freestream velocity. The mean streamwise velocity profiles are ’U’- or ’V’-shaped. The mean normal velocity profiles show an antisymmetric distribution. The mean resolved streamwise Reynolds stress profiles present a double saddle shape, and the two saddles correspond to the position of shear layers. The mean resolved normal Reynolds stress profiles present a single saddle shape. Compared with the experimental data, the mean velocity profiles of WMLES and DES methods are similar to the PIV data of Parnaudeau et al. [26] and the HWA data of Ong and Wallace [22]. There is a discrepancy with the PIV data of Lourenco and Shih [19], which may be an error from the experiment. Comparing various WMLES and DES methods, the mean velocity and resolved Reynolds stress profiles are the same.

Figure 8 shows the power spectra of the normal velocity $u_y$ at $\left(3D,0,\pi /2\right)$ in the wake. Here, 100,000 samples are recorded during the periods of $400D/U_{\infty }$. Based on the literature, the dimensionless vortex shedding frequency is $St=f·D/U_{\infty }\approx 0.2$, so there are about 80 vortices shedding cycles to ensure sufficient accuracy for the spectrum analysis. Figure 8 shows that the vortex shedding frequency is $S{t}_{\mathrm{VS}}=0.21$, which is consistent with the data of Parnaudeau et al. ($S{t}_{\mathrm{VS}}=0.208\pm 0.002$) [26].

Table 2. Integrated flow quantities for the flow over the cylinder at $R{e}_D=3900$.

	${\mathit{N}}_{\mathbf{cell}}$	$〈{\mathit{C}}_{\mathit{d}}〉$	${\left({\mathit{C}}_{\mathit{l}}\right)}_{\mathbf{rms}}$	$\mathbf{St}$	$-〈{\mathit{C}}_{\mathit{p},\mathit{b}}〉$	$〈{\mathit{L}}_{\mathit{r}}〉/\mathit{D}$	$〈{\mathit{\theta }}_{\mathbf{sep}}〉$
Lourenco [19]	-	0.99	-	0.22	-	1.19	86°
Cardell [20]	-	-	-	0.215	-	1.33	-
Norberg [21]	-	0.98	0.04–0.15	-	0.90	-	-
Ong [22]	-	-	-	0.21	-	-	-
Parnaudeau [26]	-	-	-	0.208	-	1.51	88°
WRLES [27]	5.76 million	0.97	0.09	0.209	0.91	1.67	88°
WRLES [30]	-	1.02	0.15	-	0.87	1.45	87°
SA-DES [28]	3.96 million	1.024	0.146	0.222	0.878	1.427	87.00°
SA-DES	1.44 million	0.951	0.099	0.216	0.812	1.750	88.00°
SST-DES	1.44 million	0.960	0.077	0.212	0.820	1.771	86.92°
WMLES	1.44 million	1.000	0.104	0.209	0.824	1.662	88.81°

summarizes some flow quantities often compared in the literature. Here $〈·〉$ denotes the mean quantity over time, and ${\left(·\right)}_{\mathrm{rms}}$ denotes the root-mean-square, $C_d=F_x/\left(\pi D^2\right)$ is the drag coefficient, $C_l=F_y/\left(\pi D^2\right)$ is the lift coefficient, $C_{p,b}$ is $C_p$ at stagnation point behind the cylinder, $L_r$ is the recirculation zone length, ${\theta }_{\mathrm{sep}}$ is the separation flow angle, and $N_{\mathrm{cell}}$ is the cell number. The integrated flow quantities obtained by each method are the same as those experimental data of Lourenco, Cardell, Norberg, Ong et al. [19,20,21,22]. In addition, Table 2 presents the simulation results of Lysenko, Tian, D’Alessandro et al. [27,28,30] using WRLES and DES methods in OpenFOAM. The grid count used in this paper is only a quarter of that of [27], but the results are as reliable as theirs. This proves that the SA-DES, SST-DES in OpenFOAM, and the WMLES developed by Mukha et al. [18] can all predict the vortex evolution caused by the laminar separated shear layer.

4.2. Flow over a Hump at $R{e}_D=9.36\times {10}^5$

It is well known that the effect of mesh on flows at a high Reynolds number is not negligible, so mesh independence is first studied.

4.2.1. Mesh Independence of WMLES and DES

The instantaneous streamwise velocity over the wall-mounted hump in the vertical plane is shown in Figure 9a. The upstream turbulent boundary layer is accelerated on the windward side of hump, and the flow separation occurs near $x/c=0.7$, forming a complex wake under the action of the strong turbulent shear layer. Figure 9b,c shows the mean streamwise velocity ($〈u_x〉$) and mean resolved streamwise Reynolds stress ($〈u_x^{\prime }{u}_x^{\prime }〉$) in the vertical plane. The circulation region behind the hump is clearly visible in Figure 9b. Figure 9c indicates that there are obvious pulsations in the front side of the hump and in the wake.

Two grid systems mentioned in the physical model are used for simulation. The first grid system is from the NASA website and has been extensively validated for various RANS methods. Its cell number is $816\times 216\times 60=$ 10.58 million.

Table 3. Grid scales near the wall-mounted hump for NASA-Mesh system.

Location	$\mathit{\delta }/\mathit{c}$	$\left\{{\Delta }_{\mathit{x}},{\Delta }_{\mathit{y}},{\Delta }_{\mathit{z}}\right\}/\mathit{c}$	${\Delta }_{\mathit{x}}^{+},{\Delta }_{\mathit{y}}^{+},{\Delta }_{\mathit{z}}^{+}$	$\mathit{\delta }/\left\{{\Delta }_{\mathit{x}},{\Delta }_{\mathit{y}},{\Delta }_{\mathit{z}}\right\}$	${\mathit{N}}_{\mathit{y}}/\mathit{\delta }$
$x/c=-2.14$	0.08	$0.055,8\times {10}^{-6},0.005$	$960,0.14,87$	$1.45,\mathrm{10,000},16$	105
$x/c=0.5$	0.06	$0.004,7\times {10}^{-6},0.005$	$99,0.17,124$	$15,8600,12$	156
$x/c=1.5$	0.2	$0.011,9\times {10}^{-6},0.005$	$110,0.09,49$	$18,\mathrm{22,000},40$	127

lists the grid scales near the wall at the upstream, zenith and downstream of the hump, where $\delta$ is the boundary layer thickness, ${\Delta }^{+}={\displaystyle \frac{{\mu }_{\tau }}{\nu }}\Delta$ is the dimensionless mesh scale, and $N_y/\delta$ is the grid number along the boundary layer. ${\Delta }^{+}$ is It can be seen that its first layer grid thickness satisfies $y^{+}<1$, and there are hundreds of meshes along the boundary layer. However, the streamwise grid scale is large, especially the scale upstream is comparable to the boundary layer thickness. Considering the grid requirements of the WMLES method proposed by Larsson [9] and Menter [11], a Cartesian grid system with three prism cells near a wall is constructed using the snappyHexMesh of OpenFOAM. Its minimum streamwise grid scale is $1/50$ of the boundary layer thickness, and there are 21 grids along the boundary layer. The height of the first layer grid is ${\Delta }_y^{+}=14$, so the third layer grid is also located in the inner layer of the boundary layer to ensure that the wall modeling method is available. Furthermore, to prove the mesh independence of the new mesh system, two coarser and finer meshes are set, respectively, as shown in

Table 4. Grid scales near the wall-mounted hump at $x/c=0.5$ for NewMesh system.

	${\mathit{N}}_{\mathbf{cell}}$	$\left\{{\Delta }_{\mathit{x}},{\Delta }_{\mathit{y}},{\Delta }_{\mathit{z}}\right\}/\mathit{c}$	${\Delta }_{\mathit{x}}^{+},{\Delta }_{\mathit{y}}^{+},{\Delta }_{\mathit{z}}^{+}$	$\mathit{\delta }/\left\{{\Delta }_{\mathit{x}},{\Delta }_{\mathit{y}},{\Delta }_{\mathit{z}}\right\}$	${\mathit{N}}_{\mathit{y}}/\mathit{\delta }$
Standard	17.04 million	$0.0015,0.0008,0.0015$	$28,14,28$	$40,75,40$	20
Coarser	3.41 million	$0.0031,0.0015,0.0031$	$56,28,56$	$20,37,20$	10
Finer	35.25 million	$0.0011,0.0006,0.0011$	$19,9,19$	$55,100,55$	27

. The grid scale of the new mesh system is constant on all parts of the wall. Table 4 only gives relative quantities at $x/c=-2.14$, and relative quantities at other positions can be directly converted according to the boundary layer thickness given in Table 3.

The WMLES, SA-DES, and SST-DES methods are tested on these four grid systems. Figure 10 shows the mean pressure distributions on the wall-mounted hump. Compared to the WMLES result based on NASA-Mesh and coarser NewMesh, the WMLES results based on standard and finer NewMesh can reproduce the pressure step on the leeward side of the hump, which corresponds to the separation-reattachment region. Furthermore, SA-DES can predict the separation-reattachment region on the latter two grids. The prediction error of SA-DES on the first two grids is smaller than that of WMLES. However, the SST-DES results based on coarser and standard NewMesh become worse than that based on NASAmesh. This is because the SST k-$\omega$ method has stricter requirements on the near-wall mesh than the SA method. Therefore, the following WMLES and SA-DES results are obtained based on standard NewMesh, while SST-DES results are based on NASA-Mesh.

4.2.2. Flow Predicted by DES and WMLES Methods

The flow fields predicted using SST-DES, SA-DES, and WMLES methods are compared. In addition to the turbulent inflow conditions, a laminar inflow for SA-DES and WMLES is also considered.

Figure 11 visualizes transient vortex structures using Q-criterion obtained by these methods with/without turbulent inflow. SA-DES and WMLES methods can characterize coherent structures in the incoming turbulent boundary layer, while the SST-DES method cannot. This is due to the difference in the active regions of RANS and LES obtained by SA-DES and SST-DES. Figure 12a shows the distribution of the switch function $\tilde{d}$ for the SA-DES method, and Figure 12b shows that of the switch function $F_{\mathrm{DES}}$ for the SST-DES method. $\tilde{d}$ is determined by the wall distance and grid scale, so the space far from the wall is the LES region for SA-DES. $F_{\mathrm{DES}}$ is determined by the RANS scale, grid scale and wall distance, so only the separation region is divided into the LES region for SST-DES. Therefore, the SST-DES method only models the vortex structure within the separated shear layer and the wake, while the SST-DES method identifies all the turbulent boundary layers upstream of the separation as RANS regions, as shown in Figure 11a,b. The vortex scale of the separated shear layer obtained by SST-DES is significantly larger than those of SA-DES and WMLES methods, especially those based on NASAmesh. When the incoming flow is laminar, the boundary layer is always laminar until it reaches the windward side of the hump, where the inverse pressure gradient promotes the transition. There are quasi two-dimensional vortices and $\Lambda$-shaped vortices on the windward side of the hump in the close-up view of Figure 11d,f.

Figure 13 shows mean pressure distributions along the wall-mounted hump. The pressure distribution between $x/c=0.6-1.2$ corresponds to the separation and re-attachment circulation region behind the hump. The separation locations predicted by several numerical methods with turbulent inflow are consistent with the experimental data ($x/c=0.67$), and the maximum deviation is only $0.0014c$, as listed in

Table 5. Separation and reattachment locations predicted by several numerical methods with turbulent inflow.

	${\Delta }_{\mathit{x}}^{+},{\Delta }_{\mathit{y}}^{+},{\Delta }_{\mathit{z}}^{+}$	${\mathit{N}}_{\mathbf{cell}}$	Separation Location		Reattachment Location
			$\mathit{x}/\mathit{c}$	$\left({\mathit{x}}_{\mathbf{num}}-{\mathbf{x}}_{\exp }\right)/\mathit{c}$	$\mathit{x}/\mathit{c}$	$\left({\mathit{x}}_{\mathrm{num}}-{\mathit{x}}_{\exp }\right)/\mathit{c}$
Exp [34]	-	-	0.665	-	1.110	-
WRLES [40]	80, 1–80, 50 1	3.95 million	0.667	0.002	1.114	0.004
SA-DES [40]	80, 1–80, 150 1	1.73 million	0.663	−0.002	1.121	0.011
WRLES [41]	658, 5, 100	4.2 million	0.642	−0.023	1.174	0.064
WMLES [42]	90, 13, 40	7.5 million	0.677	0.012	1.138	0.028
WMLES [43]	300, 50, 120	12.9 million	0.680	0.015	1.084	−0.026
WMLES [38]	360, 36, 180	11 million	0.655	−0.010	1.105	−0.005
WRLES [44]	25, 0.8, 25	420 million	0.641	−0.024	1.090	−0.020
SST-DES	960, 0.14, 87	10.6 million	0.654	−0.011	1.12	0.01
SA-DES	28, 14, 28	17.0 million	0.679	0.014	1.04	−0.07
WMLES	28, 14, 28	17.0 million	0.658	−0.007	1.05	−0.06

¹ Here the grid scale is only satisfied downstream of the separation, so these simulations did not reproduce the upstream turbulent boundary layer.

. There is a certain deviation in the reattachment locations predicted by them from the experimental data ($x/c=1.11$). The reattachment locations predicted by SA-DES and WMLES are shifted upstream by $0.07c$ and $0.06c$, while that of SST-DES is shifted downstream by $0.01c$. In addition, compared to SA-DES and WMLES with turbulent inflow, they with laminar inflow predict a larger maximum pressure upstream of the hump, a smaller minimum pressure at the zenith of the hump, and a forward shift of the reattachment location.

Figure 14 shows the mean velocity and mean resolved Reynolds stress profiles at $x/c=$ 0.65, 0.8, 0.9, 1.0, 1.1, 1.2, and 1.3 in the wake region. The mean velocity profiles obtained by SA-DES and WMLES is closer to the experimental data than those of SST-DES. The Reynolds mean resolved Reynolds stress profiles obtained by the first two is larger than the experimental data, while that of SST-DES is smaller. This resulted in the separation-reattachment regions predicted by SA-DES and WMLES being smaller than the experimental results.

The WMLES method above uses the WALE SGS model by default. Other SGS models in OpenFOAM are also tested into WMLES, including classical Smagorinsky, k-equation eddy viscosity, and dynamic-k-equation eddy viscosity. When the inflow is turbulent, their four predictions are consistent. When the inflow is laminar, the results obtained by WALE and dynamics-k-equation eddy viscosity are consistent, while the classical Smagorinsky and k-equation models cannot predict the transition upstream of the hump and obtain extra separation bubbles here.

4.3. Transonic Flow over an Axisymmetric Bump

This flow is also tested on two grid systems. Since this flow separation is induced by the adverse pressure gradient of $\lambda$-shock, as shown in Figure 15, the mesh is refined at the shock as well as at the boundary layer and separation zone, as shown in Figure 3c.

Table 6. Grid scales near the antisymmetric bump at $\delta /c=0.04$ for two grid systems.

	${\mathit{N}}_{\mathbf{cell}}$	$\left\{{\Delta }_{\mathit{x}},{\Delta }_{\mathit{y}},{\Delta }_{\mathit{z}}\right\}/\mathit{c}$	${\Delta }_{\mathit{x}}^{+},{\Delta }_{\mathit{y}}^{+},{\Delta }_{\mathit{z}}^{+}$	$\mathit{\delta }/\left\{{\Delta }_{\mathit{x}},{\Delta }_{\mathit{y}},{\Delta }_{\mathit{z}}\right\}$	${\mathit{N}}_{\mathit{y}}/\mathit{\delta }$
NASA-Mesh	20.74 million	$0.014,6.5\times {10}^{-6},0.0011$	$490,0.23,39$	$3,6200,36$	183
NewMesh	24.90 million	$0.0019,0.00092,0.0012$	$67,33,42$	$21,43,33$	20

lists grid scales near the antisymmetric bump at $\delta /c=0.04$ for two grid systems. There is $720\times 320\times 90=$ 20.74 million cells in NASA-Mesh, while NewMesh is a four-level local-refined hexahedral mesh with a total of 24.90 million cells.

Figure 16 shows the mean pressure distribution along the wall-mounted axisymmetric bump obtained by WMLES SA-DES and SST-DES methods. The result of SST RANS is also presented in Figure 16. Since this experiment has been used in the calibration of the SST model coefficients, the result of SST RANS is in agreement with the experimental data [45]. However, the reattachment position predicted by the SST-DES results based on the two grid systems is significantly shifted to the right. The pressure distribution within $x/c=0.7-1.1$ predicted by WMLES is linearly decreasing, indicating that it does not predict the flow separation zone. The SA-DES result is in the best agreement with the experimental data.

Figure 17 shows the mean streamwise velocity and mean resolved Reynolds stress $〈u_x^{\prime }{u}_y^{\prime }〉/U_{\infty }^2$ profiles at $x/c=$ 0.813, 0.938, 1.0, 1.125, and 1.25 in the wake region of axisymmetric bump. The Reynolds stress in the separation zone obtained by SST-DES is almost zero, i.e., the velocity pulsation here cannot be obtained. The velocity fluctuation obtained by WMLES is much higher than the experimental data, especially near the separated shear layer at $x/c=$ 0.813.

5. Conclusions

In this study, the WMLES, SA-DES, and SST-DES methods in OpenFOAM were tested on three separated flows, including flow over a cylinder at $R{e}_D$ = 3900, separated flow over a wall-mounted hump at $R{e}_c$ = 9.36 × 10⁵, and transonic flow over an axisymmetric bump with shock-induced separation at $R{e}_c$ = 2.763 × 10⁶. The complexity of these three flows is gradually increasing. WMLES, SST-DES, and SA-DES methods in OpenFOAM can easily predict the separation position and wake characteristics in the flow around the cylinder, and they rely on the grid scale and turbulent inflow to accurately simulate the latter two flows.

In terms of grid independence, when the grid scales in the boundary layer meet the grid requirements proposed by Larsson et al. [9] ($\left(\delta /\Delta x,\delta /\Delta y,\delta /\Delta z\right)\approx (12,50,20)$), it is possible to simulate the vortex structure in the turbulent boundary layer upstream of the flow separation. However, finer meshes are required to accurately predict the separation and reattachment. The second flow could be reproduced when using a three-level local-refined hexahedral mesh grid system with a near-wall grid scale of $\left(\delta /\Delta x,\delta /\Delta y,\delta /\Delta z\right)\approx (40,75,40)$. The third flow could be reproduced when using a four-level local-refined hexahedral mesh grid system with a near-wall grid scale of $\left(\delta /\Delta x,\delta /\Delta y,\delta /\Delta z\right)\approx (42,127,73)$. WMLES method is more sensitive to grid scales than the SA-DES method. The new grid constructed in this paper cannot meet the grid requirements of SST-DES, so its simulation results based on the structural grid system from NASA are closer to the experimental data.

Table 7. Grid requirements for WMLES, SA-DES and SST-DES in OpenFOAM.

Method	${\Delta }_{\mathit{x}}^{+}$	${\Delta }_{\mathit{y}}^{+}$	${\Delta }_{\mathit{z}}^{+}$	$\mathit{\delta }/\left\{{\Delta }_{\mathit{x}},{\Delta }_{\mathit{y}},{\Delta }_{\mathit{z}}\right\}$
WRLES	∼1	∼1	∼1	-
WMLES	-	O(10)∼O(100)	-	≈$\left(12,50,20\right)$ for turbulent boundary layer [9]; ≈$\left(40,75,40\right)$ for turbulent separation.
SA-DES	-	O(10)∼O(100)	-
SST-DES	-	O(1)	-

summarizes grid requirements for WMLES, SA-DES, and SST-DES in OpenFOAM.

In terms of vortex description, SST-DES can only simulate vortices induced by the separated shear layer, while SA-DES and WMLES can also simulate vortices in the turbulent boundary layer upstream of the flow separation. Furthermore, WMLES with WALE and dynamic-k-equation SGS models and SA-DES can simulate the transition of the laminar boundary layer in the flow over the wall-mounted hump.