Development and Validation of a New Formulation of Hybrid Temporal Large Eddy Simulation

Abstract

Hybrid RANS–LES approaches have aroused interest for years since they provide unsteady information at a reduced numerical cost compared to LES. In the hybrid context, the use of temporal filtering, to control the energy partition between resolved and modeled scales, ensures a consistent bridging between RANS and LES models. In this regard, a new formulation of Hybrid Temporal Large Eddy Simulation (HTLES) is developed, aiming at improving the theoretical foundation of the model associated with an eddy-viscosity closure. The analytical development is performed, applying the Hybrid-Equivalence criterion, and the model is calibrated in decaying isotropic turbulence. In addition, an upgraded version of the approach is proposed to improve the behavior of the model in near-wall regions, introducing a two-fold shielding function and an internal consistency constraint to provide a suitable control of the RANS-to-LES transition. Applying HTLES to the k–$\omega$ SST model, the validation process is carried out on channel and periodic-hill flows, over a range of grids and Reynolds numbers. The predictive accuracy and the robustness to grid coarsening are assessed in these cases, ensuring that HTLES offers a cost-saving alternative to LES.

Appendix

1.1 Perturbation Analysis to Ensure H-Equivalence Between TPITM and HTLES

Postulating that Two hybrid approaches based on the same closure, but using different method of control of the energy partition, yield similar low-order statistics of the resolved velocity fields provided that they yield the same level of subfilter energy (Friess et al. 2015), the H-equivalence between the TPITM system Eq. (9, 10) and the HTLES system Eq. (12, 18) is ensured if the two methods tend to the same closure when the filter width is going to infinity and lead to the same energy partition for a particular situation. Hence, using a k–$\varepsilon$ closure herein, the objective of the analysis is to identify the relation between the hybridization functions $C_{\varepsilon 2}^P(r)$ and $\psi (r)$, to provide the same level of modeled energy $k_\text {m}$.

Departing from the RANS limit where the H-equivalence is ensured ($k_\text {m}=k$), infinitesimal perturbations of the hybridization functions $\delta C_{\varepsilon 2}^P$ and $\delta \psi$ are introduced into the TPITM and HTLES systems, which remain H-Equivalent as long as the same infinitesimal variation $\delta k_\text {m}$ of the modeled energy is obtained. The perturbation analysis can be carried out in some particular situations, following (Friess et al. 2015). Calculations are conducted herein in the less restrictive situation: inhomogeneous turbulence in straight duct flows. In this case, both modeled energy and dissipation are in equilibrium along streamlines, which leads to the following system of equations for HTLES,

$$\begin{aligned} \nu _\text {m} = C_\mu \frac{k_\text {m}^2}{\psi (r) \varepsilon _\text {m}^*}, \qquad \left\{ \begin{array}{l} 0 = \displaystyle P_\text {m} + D_{k \text {m}} - \psi (r) \varepsilon _\text {m}^*, \\ 0 = \displaystyle C_{\varepsilon 1} \frac{\varepsilon _\text {m}^*}{k_\text {m}} P_\text {m} + D_{\varepsilon \text {m}}^* - C_{\varepsilon 2} \frac{{\varepsilon _\text {m}^*}^2}{k_\text {m}}. \end{array} \right. \end{aligned}$$

(41)

Introducing an infinitesimal perturbation $\delta \psi$ of the coefficient $\psi (r)$ in the equation results in infinitesimal variations of the solution, such that the following relation is obtained,

$$\begin{aligned} \left\{ \begin{array}{l} 0 = \displaystyle \delta P_\text {m} + \delta D_{k \text {m}} - \left( \frac{\delta \psi }{\psi (r)} + \frac{\delta \varepsilon _\text {m}^*}{\varepsilon _\text {m}^*} \right) \psi (r) \varepsilon _\text {m}^*, \\ 0 = \displaystyle C_{\varepsilon 1} \frac{\varepsilon _\text {m}^*}{k_\text {m}} P_\text {m} \left( \frac{\delta P_\text {m}}{P_\text {m}} + \frac{\delta \varepsilon _\text {m}^*}{\varepsilon _\text {m}^*} - \frac{\delta k_\text {m}}{k_\text {m}} \right) + \delta D_{\varepsilon \text {m}}^* - C_{\varepsilon 2} \frac{{\varepsilon _\text {m}^*}^2}{k_\text {m}} \left( 2\frac{\delta \varepsilon _\text {m}^*}{\varepsilon _\text {m}^*} - \frac{\delta k_\text {m}}{k_\text {m}} \right) . \end{array} \right. \end{aligned}$$

(42)

The diffusion terms are given by the gradient-diffusion hypothesis with the usual coefficients $\sigma _k$ and $\sigma _\varepsilon$,

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle D_{k \text {m}} = \frac{\partial }{\partial x_k} \left( \frac{C_\mu }{\sigma _k} \frac{k_\text {m}^2}{\psi (r) \varepsilon _\text {m}^*} \frac{\partial k_\text {m}}{\partial x_k} \right) , \\ \displaystyle D_{\varepsilon \text {m}}^* = \frac{\partial }{\partial x_k} \left( \frac{C_\mu }{\sigma _\varepsilon } \frac{k_\text {m}^2}{\psi (r) \varepsilon _\text {m}^*} \frac{\partial \varepsilon _\text {m}^*}{\partial x_k} \right) , \end{array} \right. \quad \text {such that} \quad \left\{ \begin{array}{l} \displaystyle \frac{\delta D_{k \text {m}}}{D_{k \text {m}}} = 3\frac{\delta k_\text {m}}{k_\text {m}} - \frac{\delta \varepsilon _\text {m}^*}{\varepsilon _\text {m}^*} - \frac{\delta \psi }{\psi (r)}, \\ \displaystyle \frac{\delta D_{\varepsilon \text {m}}^*}{D_{\varepsilon \text {m}}^*} = 2\frac{\delta k_\text {m}}{k_\text {m}} - \frac{\delta \psi }{\psi (r)}. \end{array} \right. \end{aligned}$$

(43)

Moreover, in eddy-viscosity models the production term is given by,

$$\begin{aligned} P_\text {m} = 2 \gamma C_\mu \frac{k_\text {m}^2}{\psi (r)\varepsilon _\text {m}^*} S^2, \end{aligned}$$

(44)

where S is the statistical averaged of the rate of strain, and $\gamma =\overline{P_\text {sfs}}/P_\text {m}$ is a correlation coefficient. Then, the perturbation analysis is applied, leading to

$$\begin{aligned} \frac{\delta P_\text {m}}{P_\text {m}} = \frac{\delta \gamma }{\gamma } + 2 \frac{\delta S}{S} + 2 \frac{\delta k_\text {m}}{k_\text {m}} - \frac{\delta \varepsilon _\text {m}^*}{\varepsilon _\text {m}^*} - \frac{\delta \psi }{\psi (r)}. \end{aligned}$$

(45)

Finally, the previous Eqs. (43, 45) are introduced into the system of Eq. (42), which yields

$$\begin{aligned} \delta \psi = - \frac{\left( C_{\varepsilon 2} - C_{\varepsilon 1} \psi (r) \right) \psi (r)}{C_{\varepsilon 2}} \frac{P_\text {m}}{\psi (r) \varepsilon _\text {m}^*} \left( \frac{\delta k_\text {m}}{k_\text {m}} - \frac{\delta \gamma }{\gamma } - 2 \frac{\delta S}{S} \right) . \end{aligned}$$

(46)

A similar analysis can be conducted for TPITM (see Friess et al. (2015) for details), and the following relation is obtained

$$\begin{aligned} \delta C_{\varepsilon 2}^P = (C_{\varepsilon 2}^P(r) - C_{\varepsilon 1}) \frac{P_\text {m}}{\varepsilon _\text {m}} \left( \frac{\delta k_\text {m}}{k_\text {m}} - \frac{\delta \gamma }{\gamma } - 2 \frac{\delta S}{S} \right) . \end{aligned}$$

(47)

Applying the H-equivalence postulate (Friess et al. 2015), i.e. the same modification $\delta k_\text {m}$ of the modeled energy leads to the same infinitesimal variation of the low-order statistics ($\delta S$, $\delta \gamma$), and recalling the equality $\varepsilon _\text {m}=\psi (r) \varepsilon _\text {m}^*$ (Eq. 17), then if HTLES and TPITM are H-Equivalent for some initial values of $\psi (r)$ and $C_{\varepsilon 2}^P(r)$, they remain equivalent if the infinitesimal perturbations of their hybridization functions expressed by Eq. (46) and (47) satisfy the relation

$$\begin{aligned} \frac{\delta C_{\varepsilon 2}^P}{\left( C_{\varepsilon 2}^P(r) - C_{\varepsilon 1}\right) } = -\frac{C_{\varepsilon 2} \delta \psi }{\left( C_{\varepsilon 2} - C_{\varepsilon 1}\psi (r)\right) \psi (r) }. \end{aligned}$$

(48)

Lastly, the previous formula is integrated between the RANS state ($r=1$, $C_{\varepsilon 2}^*(1)=C_{\varepsilon 2}$, $\psi (1)=1$) and some arbitrary LES state, such that

$$\begin{aligned} \int _{C_{\varepsilon 2}}^{C_{\varepsilon 2}^P} \frac{1}{x-C_{\varepsilon 1}} \text {d}x = \displaystyle - \int _{1}^{\psi } \frac{C_{\varepsilon 2}}{(C_{\varepsilon 2}- C_{\varepsilon 1} y) y} \text {d}y, \end{aligned}$$

(49)

and simple algebra leads to

$$\begin{aligned} \psi (r) = \frac{C_{\varepsilon 2}}{C_{\varepsilon 2}^P(r)} = \frac{C_{\varepsilon 2}}{C_{\varepsilon 1} + r (C_{\varepsilon 2} - C_{\varepsilon 1})}. \end{aligned}$$

(50)

In the other situations considered in Friess et al. (2015), it is assumed that the modeled dissipation rate is not affected by the modification of the energy partition, such that $\delta \varepsilon _\text {m}=\delta (\psi (r) \varepsilon _\text {m}^*)=0$ (Eq. 17). Calculations are much simpler in this way and the same formulation of $\psi (r)$ is obtained (not shown here) .

1.2 H-Equivalence Between TPITM and HTLES Based on SST Closure

In this section, the k–$\omega$ SST HTLES formulation is derived from k–$\omega$ SST TPITM, while preserving the H-equivalence criterion. In the k–$\omega$ SST TPITM model, developed by Bentaleb and Manceau (2011), the modeled viscosity $\nu _\text {m}$, energy $k_\text {m}$ and specific dissipation $\omega _\text {m}$, satisfy

$$\begin{aligned} \nu _\text {m} = \frac{k_\text {m}}{\omega _\text {m}}, \qquad \left\{ \begin{array}{ccl} \displaystyle \frac{D k_\text {m}}{D t} &{} = &{} \displaystyle P_\text {m} + D_{k \text {m}} - C_\mu k_\text {m} \omega _\text {m}, \\ \displaystyle \frac{D \omega _\text {m}}{D t} &{} = &{} \displaystyle \gamma _\omega \frac{\omega _\text {m}}{k_\text {m}} P_\text {m} + D_{\omega \text {m}} - \beta _\omega ^P {\omega _\text {m}}^2 + C_{\omega \text {m}}, \end{array} \right. \end{aligned}$$

(51)

where $D_{\omega \text {m}}$ stand for the diffusion of $\omega _\text {m}$, $C_{\omega \text {m}}$ is the cross-diffusion term, $\gamma _\omega =C_{\varepsilon 1}-1$ and $\beta _\omega =C_\mu (C_{\varepsilon 2}-1)$ are the usual coefficients, and the hybridization term $\beta _\omega ^P$ is defined by

$$\begin{aligned} \beta _\omega ^P(r) = C_\mu \gamma _\omega + r (\beta _\omega - C_\mu \gamma _\omega ). \end{aligned}$$

(52)

The k–$\omega$ SST HTLES model is derived by transferring the new hybridization function $\psi '(r)$ in the modeled energy equation and introducing a new variable $\omega _\text {m}^*$ transported by the second equation, in the same way as for the derivation of the k–$\varepsilon$ HTLES in Sect. 3.1, such that

$$\begin{aligned} \nu _\text {m} = \frac{k_\text {m}}{\psi '(r) \omega _\text {m}^*}, \qquad \left\{ \begin{array}{ccl} \displaystyle \frac{D k_\text {m}}{D t} &{} = &{} \displaystyle P_\text {m} + D_{k \text {m}} - C_\mu k_\text {m} \psi '(r) \omega _\text {m}^*, \\ \displaystyle \frac{D \omega _\text {m}^*}{D t} &{} = &{} \displaystyle \gamma _\omega \frac{\omega _\text {m}^*}{k_\text {m}} P_\text {m} + D_{\omega \text {m}}^* - \beta _\omega {\omega _\text {m}^*}^2 + C_{\omega \text {m}}^*, \end{array} \right. \end{aligned}$$

(53)

where $D_{\omega \text {m}}^*$ stands for the diffusion of $\omega _\text {m}^*$.

Similar to the case of the k–$\varepsilon$ model, the perturbation analysis is carried out to ensure H-equivalence. The calculations (not shown here) are very similar to those presented in Appendix 7.1, leading to the new expression of the hybridization function

$$\begin{aligned} \psi '(r) = \frac{\beta _\omega }{\beta _\omega ^P(r)} = \frac{\beta _\omega }{C_\mu \gamma _\omega + r (\beta _\omega - C_\mu \gamma _\omega )}. \end{aligned}$$

(54)

Again, it is noted that the same formulation of $\psi '(r)$ is obtained, irrespective of the situations proposed by Friess et al. (2015) to conduct the perturbation analysis (not shown here).

1.3 The k–$\omega$ SST HTLES Model

The k–$\omega$ SST HTLES formulation is given by :

$$\begin{aligned}&\nu _\text {sfs} = \frac{a_1 k_\text {sfs}}{\text {max}\left[ a_1 \psi '(r) \omega _\text {sfs}^*, F_2 \widetilde{S} \right] }, \end{aligned}$$

(55)

$$\begin{aligned}&\left\{ \begin{array}{ccl} \displaystyle \frac{\partial k_\text {sfs}}{\partial t} + \widetilde{U}_j \frac{\partial k_\text {sfs}}{\partial x_j} &{} = &{} \displaystyle P_k + \frac{\partial }{\partial x_j} \left[ \left( \nu + \frac{\nu _\text {sfs}}{\sigma _k} \right) \frac{\partial k_\text {sfs}}{\partial x_j} \right] - \frac{k_\text {sfs}}{T_\text {m}}, \\ \displaystyle \frac{\partial \omega _\text {sfs}^*}{\partial t} + \widetilde{U}_j \frac{\partial \omega _\text {sfs}^*}{\partial x_j} &{} = &{} \displaystyle \gamma _\omega \frac{1}{\psi '(r)}\widetilde{S}^2 + \frac{\partial }{\partial x_j} \left[ \left( \nu + \frac{\nu _\text {sfs}}{\sigma _\omega } \right) \frac{\partial \omega _\text {sfs}^*}{\partial x_j} \right] - \beta _\omega {\omega _\text {sfs}^*}^2 \\ &{} &{} \displaystyle \qquad + (1-F_1) 2 \frac{1}{\sigma _{\omega 2}} \frac{1}{\psi '(r) \omega _\text {sfs}^*} \frac{\partial \omega _\text {sfs}^*}{\partial x_j} \frac{\partial k_\text {sfs}}{\partial x_j}, \end{array} \right. \end{aligned}$$

(56)

where the production limiter is defined as :

$$\begin{aligned} P_k = \text {min} \left[ \nu _\text {sfs} \widetilde{S}^2 , a_2 C_\mu k_\text {sfs} \psi '(r) \omega _{\text {sfs}}^* \right] , \end{aligned}$$

(57)

and the function $F_1$ and $F_2$ are the blending functions (not modified in the hybrid context), and $\gamma _\omega$, $\beta _\omega$, $a_1$, $a_2$, $\sigma _k$, $\sigma _\omega$ are the usual coefficients dependent of the branch of the k–$\omega$ SST model.

The hybridization of the model is carried out by the energy ratio r, via the function $\psi '(r)$ and the time scale $T_m$, applying the shielding functions and the internal consistency constraint, such that

$$\begin{aligned} T_\text {m}= & {} \frac{r}{\psi '(r)} \frac{k_\text {m} + c_r k_\text {r}}{C_\mu k_\text {m} \omega _\text {m}^*} , \qquad \text {where} \ \ c_r = \left\{ \begin{array}{cc} 0 &{} \text {if} \ r = 1, \\ f_s &{} \text {if} \ r < 1, \end{array} \right. \end{aligned}$$

(58)

$$\begin{aligned} \psi '(r)= \,& {} \frac{\beta _\omega }{C_\mu \gamma _\omega + r (\beta _\omega - C_\mu \gamma _\omega )}, \end{aligned}$$

(59)

$$\begin{aligned} r\,=\, & {} (1-f_s) \times 1 + f_s \times \text {min}[1,r_K], \end{aligned}$$

(60)

$$\begin{aligned} r_K\,=\, & {} \frac{1}{\beta _0} \left( \frac{U_s}{\sqrt{k}} \right) ^{2/3} \left( \omega _c \frac{k}{\varepsilon } \right) ^{-2/3} \quad \text {where} \ \ \omega _c = \text {min} \left[ \frac{\pi }{dt}, \frac{U_s \pi }{\varDelta } \right] , \ \ \nonumber \\ U_s\,=\, & {} U + \gamma \sqrt{k} \ \ \text {and} \ \ \varDelta = \varOmega ^{1/3}, \end{aligned}$$

(61)

where $\varOmega$ is the volume of the cell,

$$\begin{aligned} f_s= & {} 1 - \text {tanh}\left[ \text {max}\left[ \xi _K^{p_1} , \xi _D^{p_2} \right] \right] , \end{aligned}$$

(62)

$$\begin{aligned} \xi _K= & {} C_1 \frac{(\nu ^3/\varepsilon )^{1/4}}{d_w} \quad \text {and} \quad \xi _D = C_2 \frac{\varDelta _\text {max}}{d_w}, \quad \text {where} \ \ \varDelta _\text {max} = \text {max} \left[ \varDelta _x, \varDelta _y, \varDelta _z \right] , \end{aligned}$$

(63)

and the statistically-averaged energy and dissipation terms are estimated as

$$\begin{aligned} k = k_\text {m} + k_\text {r} \quad \text {and} \quad \varepsilon = \varepsilon _\text {m} = C_\mu k_\text {m} \psi '(r) \omega _\text {m}^*. \end{aligned}$$

(64)

The estimates of statistically-averaged quantities $\overline{f}$ are provided by an exponentially-weighted average (Pruett et al. 2003), with a time filter width $\varDelta _T$ corresponding to several tens of flow-through times herein, such that

$$\begin{aligned} \frac{\partial \overline{f}(t,\varDelta _T)}{\partial t} = \frac{\widetilde{f}(t) - \overline{f}(t,\varDelta _T)}{\varDelta _T}. \end{aligned}$$

(65)

The main coefficients of the upgraded hybrid method are summarized in Table 3.

Table 3 Coefficients of the k–$\omega$ SST HTLES model