An Entropy-Assisted Shielding Function in DDES Formulation for the SST Turbulence Model

Abstract

The intent of shielding functions in delayed detached-eddy simulation methods (DDES) is to preserve the wall boundary layers as Reynolds-averaged Navier–Strokes (RANS) mode, avoiding possible modeled stress depletion (MSD) or even unphysical separation due to grid refinement. An entropy function f_s is introduced to construct a DDES formulation for the k-ω shear stress transport (SST) model, whose performance is extensively examined on a range of attached and separated flows (flat-plate flow, circular cylinder flow, and supersonic cavity-ramp flow). Two more forms of shielding functions are also included for comparison: one that uses the blending function F₂ of SST, the other which adopts the recalibrated shielding function f_{d_cor} of the DDES version based on the Spalart-Allmaras (SA) model. In general, all of the shielding functions do not impair the vortex in fully separated flows. However, for flows including attached boundary layer, both F₂ and the recalibrated f_{d_cor} are found to be too conservative to resolve the unsteady flow content. On the other side, f_s is proposed on the theory of energy dissipation and independent on from any particular turbulence model, showing the generic priority by properly balancing the need of reserving the RANS modeled regions for wall boundary layers and generating the unsteady turbulent structures in detached areas.

1. Introduction

Detached-eddy simulation (DES) takes advantage of the Reynolds-averaged Navier–Strokes (RANS) method where mean flow is attached and steady (e.g., walls), while offering, like large-eddy simulation (LES), the sensitivity to capture unsteady flow phenomena in areas of physical interest such as wakes or recirculation zones [1,2,3,4,5]. Although this strategy is beyond the computational cost of a steady RANS calculation, it reveals nearly as much information about the flow dynamics as LES. For this reason, DES has been serving as a promising way out of the limitation detaining LES from being applied to high Reynolds numbers in the past decades. While the idea of the original DES model is straightforward, DES is nevertheless one of the more difficult models to use in complex applications. A major concern is that the interface between the RANS and LES mode greatly depends on the grid spacing. The transition from RANS to LES mode would be located within the boundary layer, if the mesh is refined with grid spacing is much smaller than the boundary-layer thickness. The premature switch from RANS to LES mode will provide insufficient modeled Reynolds stresses, resulting in modeled stress depletion (MSD) and even non-physical separation [6,7]. To alleviate this deficiency, Menter and Kuntz [8] used the blending function F₂ of the k-ω shear stress transport (SST) model [9] to “shield” the boundary layer, by which they implied “preserve RANS mode”, or “delay LES function” in 2004 (SST-DDES-F₂). As a derivative of this proposal, Spalart et al. [7] proposed a DDES variant based on the Spalart–Allmaras model [10] in 2006 (SA-DDES), by constructing a generic shielding function f_d to detect the boundary-layer region and “preserve RANS mode”. In turn, Gritskevich et al. [11,12] employed f_d to consolidate a “standard” SST-DDES approach (SST-DDES-f_{d_cor}) since the blending function F₂ is found to be relatively conservative. In their work, f_d was simply modified with a constant C_d1 increasing from 8 to 20, based on recalibrations in several flow cases rather than adequate physical negotiations. Recently, Zhao et al. [13] deduced an entropy function f_s to distinguish the turbulent boundary layer from the external flow. This function is concluded to be general, independent of inflow conditions or any specific turbulence model. With this function, a new version of SA-based DDES (SA-SDES) is proposed. As aforementioned, due to the different combinations of baseline RANS model and shielding function, those DDES variants have been proposed with rather different characteristics, making model selection and interpretation of results challenging.

As a first step, this article is aimed at avoiding the ambiguity of numerous shielding functions for the SST-based DDES method. In particular, the performance of entropy function f_s is evaluated by promoting a novel SST-SDES method. Focusing on this ambition, this paper is organized as follows: Section 2 presents constructions of the two-equation SST model and SST-based DDES methods employed in the current study. Especially, the newly developed SST-SDES is described in detail. Section 3 is dedicated to the validation of the SST-SDES method. Comparisons with SST-DDES-F₂ and -f_{d_cor} are also provided, along with the merits and deficiencies of the above methods discussed. In this part, the effects of the baseline RANS model are preliminarily discussed. Finally, Section 4 gives the conclusions and future directions of this research.

2. Numerical Methods

The numerical algorithm for solving the mean flowfield is essentially the same as presented in Reference [14]. The time-dependent, compressible Reynolds-averaged Navier–Stokes equations are formulated in a generalized coordinate system. The 5th order weighted essentially nonoscillatory (WENO) scheme [15] is used to discretize the inviscid components, while the 4th order central differencing [16] is employed for the viscous terms. Time integration is achieved by dual-time stepping with sufficient sub-iterative convergence, which results in a second-order accuracy. The two-equation SST turbulence model is chosen as the base for the construction of the following DDES variants. This model uses a parameter F₁ to switch from k-ω to k-ε, which is considered as one of the popular two-equation RANS models, particularly for moderate separation prediction [9].

$$\frac{\partial \rho k}{\partial t}+\frac{\partial \rho u_{j}k}{\partial x_j}=P_K-D_k+\frac{\partial }{\partial x_j}\left[\left(\mu +{\sigma }_k{\mu }_t\right)\frac{\partial k}{\partial x_j}\right]$$

(1)

$$\frac{\partial \rho \omega }{\partial t}+\frac{\partial \rho u_j\omega }{\partial x_j}=P_{\mathsf{\omega }}-D_{\omega }+\frac{\partial }{\partial x_j}[(\mu +{\sigma }_{\mathsf{\omega }}{\mu }_t)\frac{\partial \omega }{\partial x_j}]+2(1-F_1)\frac{\rho {\sigma }_{{\mathsf{\omega }}_2}}{\mathsf{\omega }}\frac{\partial K}{\partial x_j}\frac{\partial \omega }{\partial x_j},$$

(2)

where the blending function F₁ is defined by

$$\begin{array}{l}{F}_1=\tanh \left\{{\left\{\min \left[\max \left(\frac{\sqrt{k}}{\beta \omega y},\frac{500v}{y^2\omega }\right),\frac{4\rho {\sigma }_{\mathsf{\omega }2}k}{C{D}_{\mathrm{k}\mathsf{\omega }}{y}^2}\right]\right\}}^4\right\}\\ C{D}_{\mathrm{k}\mathsf{\omega }}=\max \left(\frac{2\rho {\sigma }_{\mathsf{\omega }2}}{\omega }\frac{\partial k}{\partial x_i}\frac{\partial \omega }{\partial x_i},{10}^{-20}\right).\end{array}$$

(3)

F₁ is equal to 0 away from the surface (k-ε model), and switches over to 1 inside the boundary layer (k-ω model).

The turbulent eddy viscosity is defined as follows

$${\mu }_{\mathrm{t}}=\frac{\rho k}{\max \left(\omega ,\mathrm{\Omega }{F}_2/a_1\right)},$$

(4)

in which F₂ is a second blending function and behaves similarly to F₁, defined by

$$F_2=\tanh \left\{{\left[\max \left(\frac{2\sqrt{k}}{\beta \omega y},\frac{500v}{y^2\omega }\right)\right]}^2\right\}.$$

(5)

The source terms in the model are given by

$$\begin{array}{l}{P}_{\mathrm{k}}={\tau }_{ij}\frac{\partial u_i}{\partial x_j},D_{\mathrm{k}}=k^{3/2}/l_{\mathrm{k}-\mathsf{\omega }},l_{\mathrm{k}-\mathsf{\omega }}=k^{1/2}/({\beta }^{\ast }\omega )\\ P_{\mathsf{\omega }}=\frac{\alpha }{{\nu }_t}{\tau }_{ij}\frac{\partial u_i}{\partial x_j},D_{\mathsf{\omega }}=\rho \beta {\omega }^2,\end{array}$$

(6)

where l_k-ω acts as the turbulent length scale of the SST model. The model constants used in SST model are defined as a₁ = 0.31 and β^* = 0.09. The remaining variables are obtained by blending the coefficients in the k-ω model (φ₁) with those of the k-ε model (φ₂) as follows

$$\begin{array}{c}\phi =F_1{\phi }_1+\left(1-F_1\right){\phi }_2\\ {\sigma }_{\mathrm{k}1}=0.85,{\sigma }_{\mathsf{\omega }1}=0.500,{\beta }_1=0.0750,{\alpha }_1=0.553\\ {\sigma }_{\mathrm{k}2}=1.00,{\sigma }_{\mathsf{\omega }2}=0.856,{\beta }_2=0.0828,{\alpha }_2=0.440\end{array}$$

(7)

2.1. SST-DDES-F₂

For fine grids, the switch from RANS to LES mode in the pure DES strategy is found to take place somewhere inside the boundary layer and produce a premature (grid-induced) separation. In order to reduce grid influence, SST-DDES-F₂ was first proposed with the help of underlying zonal formulation of the SST model. The turbulent length scale l_k-ω is replaced by $\tilde{l}=\min (l_{\mathrm{k}-\mathsf{\omega }},C_{\mathrm{DES}}{\Delta }_{\max }/(1-F_{\mathrm{SST}}))$ with F_sst = F₁ or F₂. In this work, we chose F₂ as the shielding function following Reference [6,17]. Since SST model is based on a blending of k-ω and k-ε, Strelets [18] calibrated the model by running both the k-ω and k-ε models on isotropic turbulence. This leads to a blending constant as below

$$C_{\mathrm{DES}}=(1-F_1)\times 0.61+F_1\times 0.78.$$

(8)

Δ_max is the largest grid spacing defined by Δ_max = max(Δx, Δy, Δz).

The default of SST-DDES-F₂ is the relatively conservative F₂ function, which would suppress the formation of resolved turbulence in detached flow regions not sufficiently removed from walls (e.g., backward facing step flow, tip gap flows in axial turbines, etc.). This is the motivation for the development of SST-DDES-f_{d_cor}.

2.2. SST-DDES-f_{d_cor}

With the same “shielding” purpose, Spalart et al. [7] designed a function f_d to ensure that the attached boundary layers are treated in RANS regardless of the grid resolution by using the quantity

$$r_{\mathrm{d}}=\frac{v_{\mathrm{t}}+v}{\sqrt{U_{i,j}{U}_{i,j}}{\kappa }^2{d}^2},$$

(9)

where v_t is the kinematic eddy viscosity, v is the molecular viscosity, U_i,j represents the velocity gradients, κ = 0.41 is the von Karman’s constant, and d is the distance to the wall. The parameter r_d is slightly modified relative to the SA definition, whose value equals 1.0 in the logarithmic layer, and falls to 0 gradually towards the exterior edge of the boundary layer. This quantity is used in the function:

$$f_{\mathrm{d}}=1-\tanh ({[8r_{\mathrm{d}}]}^3),$$

(10)

which is designed to be 0 in the boundary layer and 1.0 elsewhere.

Since f_d depends only on the eddy viscosity and the wall distance, it can therefore, in principle, be applied to any eddy viscosity based DDES model. Whereas the shielding function f_d was considered generic, it was essentially calibrated for a one-equation SA model. Gritskevich et al. [11,12] proved that a recalibration is required if the same function is to be applied to a two-equation SST model. The original shielding function f_d is modified as below:

$$f_{\mathrm{d}\_\mathrm{cor}}=1-\tanh ({[20r_{\mathrm{d}}]}^3).$$

(11)

In addition, the SST-DDES-f_{d_cor} approach was consolidated with the length scale

$$\tilde{l}=l_{\mathrm{k}-\mathsf{\omega }}-f_{\mathrm{d}\_\mathrm{cor}}\max (0,l_{\mathrm{k}-\mathsf{\omega }}-C_{\mathrm{DES}}{\Delta }_{\max }).$$

(12)

From the comparison between Equations (10) and (11), only one empirical constant is increased from 8 to 20. It may be questionable since the modification depends on recalibrations rather than physical negotiations. The essential parameter r_d is not changed, which is relative to the SA definition. For the above reasons, the performance of SST-DDES-f_{d_cor} will be further investigated in our work.

2.3. SST-SDES

Different from F₂, f_d and f_{d_cor}, which originate from turbulence models, the entropy function f_s is initially proposed to distinguish the turbulent boundary layer from the point of energy dissipation [13]. The basic hypothesis is that the turbulent boundary layer could be defined as the region where the local entropy generation rate caused by viscous dissipation is the most significant [19]. A novel entropy concept, named entropy increment ratio $s_{\mathrm{vis}}$, was proposed as follows:

$$s_{\mathrm{vis}}=\frac{\Delta s_{\mathrm{vis}}}{\Delta s_{\max }}=\frac{\Phi }{\Phi +\alpha \psi }\times \frac{\Delta s}{\Delta s_{\max }},$$

(13)

where $\Delta s_{\mathrm{vis}}$ means the entropy increment caused by the viscous dissipation. The derivation process of $\Delta s_{\mathrm{vis}}$ could be referred to [13,20], and the final form is presented as below:

$$\Delta s_{\mathrm{vis}}=\frac{\left(1+{\mu }_{\mathrm{t}}/\mu \right)\Phi }{\left(1+{\mu }_{\mathrm{t}}/\mu \right)\Phi +\left(1+k_{\mathrm{t}}/k\right)\psi }\times \Delta s=\frac{\Phi }{\Phi +\alpha \psi }\times \Delta s,$$

(14)

where $\Phi =\frac{\mu }{2}{\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)}^2$, $\psi =\frac{k}{T}\left(\frac{\partial T}{\partial x_j}\frac{\partial T}{\partial x_j}\right)$, and $\alpha =\frac{\mu P{r}_{\mathrm{t}}+{\mu }_{\mathrm{t}}Pr}{\mu P{r}_{\mathrm{t}}+{\mu }_{\mathrm{t}}P{r}_{\mathrm{t}}}$. It should be noted that the entropy increment caused by wall-heat convection has been neglected in Equation (14) for numerical consideration, which may reduce the precision of $\Delta s_{\mathrm{vis}}$ at the isothermal wall. $\Delta s$ is the state function of entropy obtained from Gibbs equations. For compressible flows, $\Delta s$ has the formation as follows:

$$\Delta s=c_{\mathrm{v}}\ln \frac{T}{T_{\infty }}-R\ln \frac{\rho }{{\rho }_{\infty }}=c_{\mathrm{v}}\ln \left[\frac{p}{p_{\infty }}{\left(\frac{{\rho }_{\infty }}{\rho }\right)}^{\gamma }\right],$$

(15)

and, in incompressible flows, $\Delta s$ can be expressed by

$$\Delta s=c_{\mathrm{v}}\ln \frac{T}{T_{\infty }}.$$

(16)

When the potential flows pass the wall, the mechanical energy is dissipated to zero due to the viscous frication. Therefore, the value of $\Delta s_{\mathrm{vis}}$ varies by orders of magnitude from the low-speed flows to hypersonic flows at the wall. Based on the modeling convenience, $\Delta s_{\mathrm{vis}}$ is normalized by the maximum entropy increment $\Delta s_{\max }$, which is approximated in adiabatic boundary-layer flows and given by

$$\Delta s_{\max }=c_{\mathrm{v}}\ln {\left(1+\frac{\gamma -1}{2}M{a}_{\infty }^2\right)}^{\gamma }.$$

(17)

The remaining variables in Equations (14)–(17) are c_ν = R/(γ − 1) the specific heat at constant volume, R the gas constant, γ = 1.4 the specific heat ratio, and T, p and ρ the local temperature, pressure and density, respectively. $k=\frac{\mu \gamma R}{(\gamma -1)Pr}$ is the thermal conductivity and Pr = 0.7 is Prandtl number in laminar flow, while $k_{\mathrm{t}}=\frac{{\mu }_{\mathrm{t}}\gamma R}{(\gamma -1)Pr{}_{\mathrm{t}}}$ and Pr_t = 0.9 are variables in turbulent flows. Subscript ∞ means the quantity in the far field.

With $\Delta s_{\mathrm{vis}}$ normalized by $\Delta s_{\max }$ as Equation (13) shows, $s_{\mathrm{vis}}$ represents the viscous dissipation rate per unit mechanical energy, whose value approaches unity towards the wall with a consistent trend. The range of boundary layer is well represented by $s_{\mathrm{vis}}$ > 0 [13]. Moreover, in order to avoid the disturbance of entropy increase caused by shocks and detached vortex in complex flows, the entropy function f_s is proposed to confine the predicted turbulent boundary layer near the wall,

$$f_{\mathrm{s}}=1.0-\tanh \left(s_{\mathrm{vis}}/l_{\mathrm{s}}^3\right)$$

(18)

where l_s is the length-scale ratio, which is designed to be less than 1.0 in the boundary layer and increase quickly in the external flows. The formation of l_s is as below

$$l_{\mathrm{s}}=\{\begin{array}{c}{C}_{\mathrm{s}}f(a_1,a_2)d/C_{\mathrm{DES}}{\Delta }_{\max },s_{\mathrm{vis}}>0.05\\ d/C_{\mathrm{DES}}{\Delta }_{\max },\mathrm{otherwise}\end{array},$$

(19)

in which C_s = 0.12 and C_DES = 0.65 are empirical constants. f(a₁,a₂) is an anisotropic function recommended by Lilly [21], which is a function of grids aspect ratios

$$f(a_1,a_2)=\cosh \sqrt{\frac{4}{27}\left[{\left(\ln a_1\right)}^2-\ln a_1\ln a_2+{\left(\ln a_2\right)}^2\right]},$$

(20)

where $a_i={\Delta }_i/{\Delta }_{\max },i=1,2$ and Δ_i is one of the two shorter edges in the three directions.

Then, the SST-SDES approach could be designed with the length scale

$$\tilde{l}=l_{\mathrm{k}-\mathsf{\omega }}-f_{\mathrm{s}}\max (0,l_{\mathrm{k}-\mathsf{\omega }}-C_{\mathrm{DES}}{\Delta }_{\max }).$$

(21)

As mentioned above, the turbulent length scales of the above DDES methods are listed in

Table 1. The turbulent length scales in SST-based DDES methods.

Strategies	Turbulent Length Scale
	RANS Mode	Transition Mode	LES Mode
SST-DDES-F2	lk-ω	-	CDESΔmax/(1 − F2)
SST-DDES-fd_cor	lk-ω	lk-ω − fd_cor max(0, lk-ω − CDESΔmax)	CDESΔmax
SST-SDES	lk-ω	lk-ω − fs max(0, lk-ω − CDESΔmax)	CDESΔmax

. Note that, SST-DDES-F₂ would automatically choose the smaller one by comparison of the length scales of the corresponding RANS and LES mode. For the left, they may adopt a combination of the length scales of the two modes, where the value of the corresponding shielding function lies in the range 0–1. Moreover, each shielding function could only decide the RANS modeled region near the wall. In the farfield where its value equals 1, the alternation of RANS and LES mode depends on the magnitudes of l_k-ω and C_DESΔ.

3. Results and Discussion

3.1. Flat Plate Flow

The performances of the above DDES methods are preliminarily investigated on a zero-pressure-gradient boundary-layer flow with Re = 2 × 10⁶/m. All cases are computed in RANS mode with the DDES option activated. Similar to the procedure adopted by Reference [7], we also present three types of grids with different mesh resolutions in order to evaluate the grid sensitivity of the above DDES methods. Figure 1 displays the sketch maps of three grid densities in a boundary layer. In a Type I grid, the wall-parallel spacing Δx and Δz set Δ via the max formula and exceed δ, so that the DES length scale is on the “RANS branch” throughout the boundary layer. The shielding functions of DDES methods have no effect in a Type I grid, and all the results are consistent with those of SST (Figure 2). However, the modified shielding function f_{d_cor} turns out to be overly conservative as it covers double the boundary-layer thickness, while both shielding functions F₂ (actually 1 − F₂ in current notation) and f_s accurately denote the whole layer (Figure 2a). Such a conservative shielding function f_{d_cor} will inhibit the main DES functionality by suppressing the LES mode for resolved turbulence. The resolution of Type II grid ranges between the classical values used in LES and RANS simulations (with a target value of the grid-spacing equal to one tenth of the boundary-layer thickness). With this ambiguous grid, the RANS-modeled range (d_w < C_DESΔ) in SST-DES [18] only holds 18% of the boundary layer (Figure 3a), resulting in underestimating the eddy viscosity by almost 60% (Figure 3b) and the velocity profile slightly departing from that of SST at the log-law region (Figure 3a). This premature switching inside the boundary layer to the LES mode is completely eliminated for all the DDES methods, whereas the performances of the inherent shielding functions are similar to those in a Type I grid (Figure 2a). For a Type III LES grid, all spacings are much smaller than the boundary-layer thickness δ. Since the LES region where C_DESΔ_max < l_k-ω occupies the bulk of the boundary layer, the velocity profile of SST-DES further departs from that of SST, along with the eddy viscosity being underestimated by 87% (Figure 4). All DDES methods could predict consistent results with those of SST. Compared with the performances in Type I and II grids, the modified f_{d_cor} predicts a more accurate range of δ (f_{d_cor} = 0), but rises to 1 more slowly than F₂ and f_s. Additionally, the original f_d is introduced in SST-DDES (SST-DDES-f_d) and the results are also shown in Figure 5. The original one increased to be 1 in a much narrower domain (Figure 5a), which results in a less-reliable shielding capability. The SST-DDES-f_d underestimates the eddy viscosity by about 40% (Figure 5b). This deficiency was also revealed in References [11,12].

From the discussion above, while the shielding function f_d of SA-DDES was considered generic [7], it is essentially calibrated for SA model. When it is applied into the SST-based DDES, the original f_d proves to be less reliable, while the recalibrated f_{d_cor} recommended in [11,12] turns out to be much too conservative. Compared with the performance of F₂, the entropy function f_s increases to 1 more quickly towards the edge of boundary layer, which is favorable for the safe protection of the LES resolved region.

3.2. Circular Cylinder Flow

The flow past a circular cylinder at Reynolds number 3900 based on the cylinder diameter D is chosen for the intended investigations. This kind of large separated flow is considered to be the primary application of DES variants. The size of the integration domain for the O-type grid is 20D in the cross-section plane [22], and the grid extends πD/2 in the z direction (Figure 6). The grid is clustered near the cylinder and the spacing is increased in a proper ratio. The distance of the first grid line to the wall is 10⁻⁵, which corresponds to a y⁺ less than 1.0. The dimensions in the order “streamwise × transverse × spanwise” are 137 × 137 × 41, which had been proved to be refined enough for DES simulations [23]. Periodic boundary condition was employed at the boundaries in the spanwise direction and no-slip boundary condition was prescribed at the surface of cylinders.

Three calculations were carried out on the same grid. The time-averaged distributions of shielding functions in the three DDES methods are compared in Figure 7. As pointed out by Zdravkovich [24], when the Reynolds number varies from 350 to 2 × 10⁵, the flow past a circular cylinder is in the transition-in-shear-layers region, in which the separated boundary layer remains laminar, while a transition takes place along the free-shear layers with shedding vortexes leaving the body as large-scale turbulent vortices. Therefore, a little RANS-modeled region is needed before the separation point. However, both SST-DDES-F₂ and -f_{d_cor} preserve an obvious RANS region in front of the cylinder which seems redundant (Figure 7a,b). Comparatively, the shielding function f_s employed by SST-SDES is sensitive to the local flow topology and could detect the boundary layer more physically (Figure 7c).

Table 2. Global flow quantities computed by four DES strategies.

Strategies	Global Flow Quantities
	${\overline{\mathit{L}}}_{\mathbf{r}}/\mathit{D}$	${\overline{\mathit{C}}}_{\mathbf{D}}$	St	$-\overline{\mathit{C}}{\mathit{p}}_{\mathbf{b}}$
SST-DDES-F2	0.92	1.18	0.2031	1.144
SST-DDES-fd_cor	1.05	1.14	0.2042	1.079
SST-SDES	1.32	1.12	0.2048	0.957
Experiment [27]	1.33 ± 0.05	0.99 ± 0.05	0.215 ± 0.005	0.88 ± 0.05

${\overline{L}}_{\mathrm{r}}$: the time-averaged formation length based on the location of zero averaged-velocity, D: cylinder diameter, ${\overline{C}}_{\mathrm{D}}$: time-averaged drag coefficient, St: Strouhal number, $-\overline{C}{p}_{\mathrm{b}}$: back-pressure coefficient.

presents the values of global flow quantities in all cases and the experiment. According to Bearman [25], the mean recirculation length ${\overline{L}}_{\mathrm{r}}/D$ of circular cylinders is inversely proportional to the mean base pressure coefficient $-\overline{C}{p}_{\mathrm{b}}$, whereas the mean drag coefficient ${\overline{C}}_D$ is proportional to $-\overline{C}{p}_{\mathrm{b}}$ at subcritical Reynolds numbers. Following this rule, one can easily understand the relationship between Figure 8, Figure 9 and Table 2. As Figure 8 shows, the profile of SST-DDES-F₂ is notably below the experimental result, related with the shortest recirculation area (Figure 9) and the largest mean drag coefficient among the three methods (Table 2). Compared with other results, the behavior of SST-DDES-F₂ tends to be that of unsteady RANS (URANS) in some sense [18], as it reserved more regions for RANS mode. In contrast, SST-SDES predicts the most convenient results with the experimental data, with the longest recirculation area and the largest Strouhal number. After all, considering the experimental data and LES results with 961 × 960 × 48 grid resolution [26], all three DDES strategies could give convenient mean and fluctuating velocity distributions in the wakes, proving the capability to solve the large-separated flows (Figure 10 Figure 11).

3.3. Cavity-Ramp Flow

The cavity-ramp configuration could be considered as a simplified scramjet or ramjet for the next-generation hypersonic vehicles, which may be used to provide flame stabilization. The dominant features of the flow are the free-shear layer over the cavity and a large recirculation zone behind the cavity leading edge, which should be resolved in LES mode of DES variants. After reattachment, the flow forms the turbulent boundary layer on the ramped portion of the cavity, where the RANS mode is needed (Figure 12). Settles et al. [29] had carried out corresponding experiments, and the nominal inflow Mach number, pressure, and temperature are 2.92, 21,240 Pa, and 95.37 K, respectively. As Figure 13 shows, the three-dimensional grid used for this case consists of two blocks, containing 37 × 85× 33 points upstream of the cavity and 154 × 108 × 33 points downstream of the leading edge of the cavity. The grid was clustered to all solid surfaces, while the free-shear layer and reattachment regions were paid particular attention. It should be noted that this grid resolution is almost the same as Reference [30], which was proved to be refined enough for DES-like methods but were too coarse to properly capture the eddy structure by LES. A no-slip adiabatic condition was applied to the surface, and periodic boundary conditions were employed in the z direction. Initial conditions for DDES simulations are obtained by solving the flowfield with the corresponding RANS model. This case is also employed to evaluate the performances of SA-based DES and SDES [13].

Figure 14 presents the time-averaged distribution of the shielding function inherent in corresponding DDES method. All of the three DDES methods could preserve a visible RANS-modeled region along the wall, whereas both SST-DDES-F₂ and -f_{d_cor} mistakenly shield the whole cavity as the boundary layer. Moreover, the inlet distribution of f_{d_cor} fluctuates due to U_i,j ≈ 0 outside of the inflow boundary layer. On the other side, SST-SDES protects a reasonable extent of the boundary layer as RANS mode in the cavity and after the reattachment along the ramp (Figure 14c). Specially, the distribution of entropy function f_s of SST-SDES is almost the same as the one inherent in SA-SDES, proving the independence of f_s on any particular turbulence model.

Figure 15 compares the time-averaged eddy viscosity distributions of the three DDES methods. Since the cavity is treated in RANS mode by both SST-DDES-F₂ and -f_{d_cor}, there are more prominent levels of eddy viscosity than the result of SST-SDES. The entropy function f_s reliably indicates the development of the reattached boundary layer along the ramp, resulting in a more reasonable eddy viscosity distribution. The instantaneous vortex structures are visualized in Figure 16, using the Q-criterion. The initial growth of the shear layer is dominated by Kelvin–Helmholtz structures, which are initially two-dimensional, and quickly break down into small vortices in the cavity. After reattachment, elongated horseshoe vortices are observed in the ramp portion, the size of which is on the order of half the domain width in the z direction [30]. Due to the abundant eddy viscosity in the cavity (Figure 15a,b), the turbulent fluctuations inherent in the separated flow are greatly suppressed, leading to fewer vortex structures (Figure 16a,b). However, for SST-SDES, the turbulence-resolving capability in the separation region is not impaired (Figure 16c), as the cavity is indicated as LES mode by entropy function f_s.

Figure 17a presents the time-averaged distributions of skin friction coefficient in the recovery region. In addition, the results of SA-based DES and SDES in Reference [13] are also included for comparisons. The x-axis is parallel to the ramp, measured from the juncture of the cavity and the ramp (Figure 12). Among the three DDES methods, the results of SST-SDES are the most consistent with the experiment. Since the rise of skin frication coefficient profile indicates the rate of recovery of the boundary layer downstream of reattachment, f_s could denote a comparatively accurate development of the reattached boundary layer. As aforementioned, the friction coefficient distributions are underestimated by SST-DDES-F₂ and -f_{d_cor}, due to the lack of turbulent fluctuations resolved by LES mode. Accordingly, the recovering streamwise velocity predicted by SST-SDES develops more quickly than other results, which agrees well with experimental data (Figure 17b). Compared with the three above SST-based DDES methods, SA-DES and SA-SDES tend to predict much more convenient results with the experimental data. The discrepancies may lie in the variant performances of RANS models in adverse-pressure flows [31].

4. Conclusions

A novel version of SST-based DDES, named SST-SDES, was proposed by employing the entropy function f_s to shield the turbulent boundary layer as RANS mode. For comparisons, the original SST-DDES-F₂ and the recalibrated SST-DDES-f_{d_cor} were also included to interpret their different performances. In the case of turbulent boundary-layer flow, all of the three DDES methods could avoid MSD phenomena. The entropy function f_s in SST-SDES accurately indicates the layer range from the point of energy dissipation, while the modified shielding function f_{d_cor} in SST-DDES-f_{d_cor} turns out to be overly conservative as it covers double the boundary-layer thickness. For the circular cylinder flow with fully separation, all three methods could predict consistent results, regarding their mean and instantaneous performances. However, SST-DDES-F₂ preserves an obviously redundant RANS region in front of the cylinder. For the cavity-ramp flow with both separation and reattachment, the entropy function f_s of SST-SDES protects a reasonable extent of the boundary layer as RANS mode, resulting in the most convenient result with the experiment. Moreover, f_s depicts a consistent distribution as the one inherent in SA-SDES, proving its independence on any particular turbulence model. On the downside, both shielding function F₂ and f_{d_cor} are so conservative that they suppressed the formation of large-scale turbulent structures in the cavity.

In conclusion, all three DDES methods are capable of dealing with fully-separated flows past bluff bodies. However, for the engineering applications containing flow separations and reattachment, SST-SDES is preferred for the confident shielding function f_s, which successfully balances the needs of avoiding the MSD problem and the desire of not impairing the turbulence-resolving capability. Nevertheless, a further recalibration of f_d is suggested to be carried out on a more physical basis, when used in an SST-based DDES model.