Dynamic one-equation-based subgrid model for large-eddy simulation of stratified turbulent flows

In this study, the dynamic one-equation-based subgrid model for large-eddy simulation (LES) of turbulent flows is extended to account for the effects of density stratification under the Boussinesq approximation. The model utilizes an eddy viscosity closure by solving a modeled transport equation for the subgrid-scale (SGS) turbulent kinetic energy ($k^{\mathrm{sgs}}$) to obtain the characteristic velocity scale. The closure of the SGS density flux is attained through an algebraic eddy diffusivity-based approach using the eddy viscosity and the turbulent Prandtl number. The transport equation for $k^{\mathrm{sgs}}$ gets modified due to density stratification, which leads to a coupling of the SGS stress and the SGS density flux. All the model coefficients are determined locally in a dynamic manner. The dynamic model is evaluated by simulating a fully developed turbulent flow in a periodic channel under neutral and stratified conditions at frictional Reynolds number (${\mathrm{Re}}_{\tau }$) of 180, 550, and 950 and frictional Richardson number (${\text{Ri}}_{\tau }$) of 0, 60, 120, and 240. The LES predictions are compared with direct numerical simulation results. A comprehensive assessment is performed by examining the behavior of the model coefficients, comparison of static and dynamic approaches for the model coefficients, along with comparison with other well-established algebraic closures for the SGS stress and the SGS density flux at ${\text{Re}}_{\tau }=550$. The results show that the proposed model appropriately responds to the changes in the behavior of the flow due to stratification, at both the resolved and the SGS levels, particularly in the regions away from the wall where internal waves tend to coexist with the shear-generated turbulence. An explicit coupling of the SGS buoyancy flux with the SGS kinetic energy in the dynamic model, which is absent in the other algebraic models, allows improved accuracy in capturing the instantaneous and the statistical features of both the resolved and the SGS quantities.

Figure 1

Schematics of the computational domain for fully developed turbulent flow in a channel (a) and the density stratification along the vertical direction (b). In panel (b), ${\rho }_{\mathrm{b}}$ denotes deviation in the density field about the reference value ${\rho }_0$ along the vertical direction.

Figure 2

Instantaneous contours of ${\rho }^{*}$ in the central vertical plane (a), (b), isosurface of ${\rho }^{*}=0$ colored by the contours of the normalized streamwise velocity fluctuations (c), (d), PDF of the surface (${\rho }^{*}=0$) streamwise velocity fluctuations (e), and PDF of the vertical location of points corresponding to the isosurface of ${\rho }^{*}=0$ at ${\text{Re}}_{\tau }=550$ for the neutral (Case B0) and the stratified (Case B1) cases. In panels (a) and (b) positive and negative isocontours are represented by solid and dashed curves, respectively.

Figure 3

Streamwise spectrum of the phase angle ${\theta }_{w\rho }$ for the neutral (Case B0) and the stratified (Case B1) cases at different vertical locations measured from the bottom surface. (a) $z^{+}=10$, (b) $z^{+}=450$, and (c) $z^{+}=550$.

Figure 4

Streamwise spectrum of the vertical component of the resolved turbulent kinetic energy (a) and the density field (b) at different vertical locations for the neutral (Case B0) and the stratified (Case B1) cases. A reference curve indicating ${\kappa }_x^{-5/3}$ is also included in both panels to show the presence of an inertial range. The spectra at locations $z^{+}=10$, 450, and 550, are scaled by a factor of ${10}^6$, ${10}^3$, and 1, respectively, for clarity.

Figure 5

PDF of the model coefficients $C_{\nu }$ (a) and ${\text{Pr}}_t^{-1}$ (b) used by the closure expressions for ${\tau }_{ij}^{\mathrm{sgs}}$ and ${\lambda }_j^{\mathrm{sgs}}$ in Eq. (3) and Eq. (4), respectively, for the neutral (Case B0) and the stratified (Case B1) cases.

Figure 6

Variation of vertical component of the normalized SGS density flux $[{\lambda }_3^{\mathrm{sgs}}/\left({\rho }_{\tau }{u}_{\tau }\right)]$ with respect to the normalized SGS turbulent kinetic energy ($k^{\mathrm{sgs}}/u_{\tau }^2$) for the neutral (Case B0) (a) and the stratified (Case B1) (b) cases. Dashed curve denotes the conditional average of ${\lambda }_3^{\mathrm{sgs}}/\left({\rho }_{\tau }{u}_{\tau }\right)$ with respect to $k^{\mathrm{sgs}}/u_{\tau }^2$.

Figure 7

PDF of the normalized SGS density flux $[{\lambda }_3^{\mathrm{sgs}}/\left({\rho }_{\tau }{u}_{\tau }\right)]$ at fixed value of the normalized SGS turbulent kinetic energy ($k^{\mathrm{sgs}}/u_{\tau }^2$) for the neutral (Case B0) and the stratified (Case B1) cases. The PDF is obtained at $k^{\mathrm{sgs}}/u_{\tau }^2=1.0\pm 0.02$ for (a) $z^{+}=10$, and at $k^{\mathrm{sgs}}=0.2\pm 0.02$ for (b) and (c) $z^{+}=450$ and 550.

Figure 8

Time evolution of ${\text{Re}}_{\tau }$ (a) and Nu (b) for the neutral (Case B0) and the stratified (Case B1) cases. Here $t^{*}=(t-t_0)u_{\tau }/h$ is the nondimensional time, with $t_0$ denoting an arbitrary time instant at which Case B0 exhibits a statistically stationary state of turbulence. The dotted line in panel (a) corresponds to ${\text{Re}}_{\tau }=550$, and in panel (b) they correspond to $\text{Nu}=17.5$ and $\text{Nu}=8.3$ for cases B0 and B1, respectively.

Figure 9

Profile of the normalized mean streamwise velocity (a) and density along the vertical direction compared with the reference DNS [13] results.

Figure 10

Profile of the normalized turbulence intensity of the streamwise (a), vertical (b), and spanwise (c) velocity components, and the density fluctuations (d) along the vertical direction compared with the reference DNS [13] results. Note that reference DNS results for $v_{\mathrm{rms}}$ are not available and, therefore, are not included in panel (c).

Figure 11

Profile of the mean normalized vertical turbulent momentum flux (a) and the buoyancy flux (b) along the vertical direction compared with the reference DNS [13] results.

Figure 12

Profile of the mean normalized vertical turbulent buoyancy flux (total, resolved, and SGS) for the neutral (a) (Case B0) and the stratified (b) (Case B1) compared with the reference DNS [13] results.

Figure 13

Profile of the mean normalized eddy viscosity (a), the inverse of turbulent Prandtl number (b) along the vertical direction.

Figure 14

Profile of the mean model coefficients $\langle C_{\nu }\rangle$ (a), (c) and $\langle C_{\epsilon }\rangle$ (b) along the vertical direction. The variation of $\langle C_{\nu }\rangle$ in inner coordinates is shown in panel (c). The symbols ($\blacksquare$) in panels (a) and (c) correspond to the reference analytical expression given by Eq. (13).

Figure 15

Time evolution of ${\text{Re}}_{\tau }$ (a) and Nu (b) obtained from cases employing different SGS modeling approaches. The baseline cases B0 and B1 correspond to neutral and stratified conditions, respectively. The cases ${\mathrm{B}1}^{\mathrm{nm}}$ and ${\mathrm{B}1}^{\mathrm{dsm}}$ employ the no-model and the dynamic Smagorinsky approach, respectively, for the SGS closure and correspond to stratified condition. The dotted line in panel (a) corresponds to ${\text{Re}}_{\tau }=550$.

Figure 16

Vertical variation of $(\langle \rho \rangle +\mathrm{\Delta }\rho /2)/\mathrm{\Delta }\rho$ (a), (b) and ${\rho }_{\mathrm{rms}}/\mathrm{\Delta }\rho$ (c), (d) obtained from LES cases employing different modeling strategies and compared with the reference DNS [13] cases for neutral (a), (c) and stratified (b), (d) conditions. The cases B0 and B1 correspond to the baseline LES cases under neutral and stratified conditions, respectively. The cases labeled with superscripts “nm” and “dsm” correspond to LES cases employing the no-model and the dynamic Smagorinsky approach, respectively.

Figure 17

Profile of $u^{+}$ (a), ${\rho }^{+}$ (b), ${\rho }_{\mathrm{rms}}$ (c), $\langle {\rho }^{\prime }{w}^{\prime }\rangle /{\rho }_{\tau }{u}_{\tau }$ (d), $\langle {\nu }_t\rangle /\nu$ (e) and $\langle {\text{Pr}}_t^{-1}\rangle$ along the vertical direction in inner coordinates for different Reynolds number under neutral (A0, B0, C0) and stratified (A1, B1, C1) conditions. The cases A0/A1, B0/B1, and C0/C1, correspond to ${\text{Re}}_{\tau }=180$, 550, and 950, respectively.

Figure 18

Profile of normalized buoyancy frequency ($Nh/u_{\tau }$) (a), and the gradient Richardson number ${\text{Ri}}_{\mathrm{g}}$ (b) along the vertical direction in inner coordinates, and the variation of flux Richardson number (${\text{Ri}}_{\mathrm{f}}$) (c), and the vertical Froude number (${\text{Fr}}_{\mathrm{w}}$) (d) with respect to ${\text{Ri}}_{\mathrm{g}}$ for different Reynolds number under stratified conditions. The cases A1, B1, and C1, correspond to ${\text{Re}}_{\tau }=180$, 550, and 950, respectively. The dashed curve in panel (b) corresponds to ${\text{Ri}}_{\mathrm{g}}=0.25.$

Figure 19

Profile of $\langle \overline{u}\rangle /u_{\tau }$ (a), $(\langle \overline{\rho }\rangle +\mathrm{\Delta }\rho /2)/\mathrm{\Delta }\rho$ (b), ${\rho }_{\mathrm{rms}}/{\rho }_{\tau }$ (c), $\langle {\rho }^{\prime }{w}^{\prime }\rangle /{\rho }_{\tau }{u}_{\tau }$ (d), $\langle {\nu }_t\rangle$ (e), and $\langle {\text{Pr}}_t^{-1}\rangle$ (f) along the vertical direction obtained from LES cases at ${\text{Re}}_{\tau }=550$ for different values of ${\text{Ri}}_{\tau }$. Cases B0, B1, B2, and B3 correspond to ${\text{Ri}}_{\tau }=0$, 60, 120, and 240, respectively.

Figure 20

Profile of $Nh/u_{\tau }$ (a) and ${\text{Ri}}_{\mathrm{g}}$ (b) along the vertical direction, and the variation of ${\text{Ri}}_{\mathrm{f}}$ (c), and ${\text{Fr}}_{\mathrm{w}}$ (d) with respect to ${\text{Ri}}_{\mathrm{g}}$ from LES cases at ${\text{Re}}_{\tau }=550$ for different values of ${\text{Ri}}_{\tau }$. Case B0 correspond to neutral condition and cases B1, B2, and B3 correspond to ${\text{Ri}}_{\tau }=60$, 120, and 240, respectively. Case B1-DNS denotes reference DNS case at ${\text{Re}}_{\tau }=550$ and ${\text{Ri}}_{\tau }=60$.

Figure 21

Profile of $\mathrm{\Lambda }/h$ (a) and $L_0/\mathrm{\Delta }z$ (b) along the vertical direction obtained from the stratified cases at ${\text{Re}}_{\tau }=550$ for different values of ${\text{Ri}}_{\tau }$. Cases B1, B2, and B3 correspond to ${\text{Ri}}_{\tau }=60$, 120, and 240, respectively. The reference horizontal dotted curve in panel (b) corresponds to $L_0/\mathrm{\Delta }z=1$.

Figure 22

Profile of ${\rho }_{\mathrm{rms}}/{\rho }_{\tau }$ (a), (c), and $\langle {\rho }^{\prime }{w}^{\prime }\rangle /{\rho }_{\tau }{u}_{\tau }$ (b), (d), along the vertical direction obtained from LES cases and compared with the reference DNS [13] results. Cases B0 and B1 denote the baseline LES cases under neutral and stratified conditions, respectively, employing fully dynamic approach, whereas cases ${\mathrm{B}0}^{\mathrm{st}}$ and ${\mathrm{B}1}^{\mathrm{st}}$ denote the neutral and stratified cases, respectively, employing the static approach to specify ${\text{Pr}}_t$. Cases ${\mathrm{B}0}_{\mathrm{f}}$ and ${\mathrm{B}1}_{\mathrm{f}}$ use a relatively finer grid resolution.