Ensemble-variational assimilation of statistical data in large-eddy simulation

A nonintrusive data assimilation methodology is developed to improve the statistical predictions of large-eddy simulations (LES). The ensemble-variational (EnVar) approach aims to minimize a cost function that is defined as the discrepancy between LES predictions and reference statistics from experiments or, in the present demonstration, independent direct numerical simulations (DNS). This methodology is applied to adjust the Smagorinsky subgrid model and obtain data assimilated LES (DA-LES) which accurately estimate the statistics of turbulent channel flow. To separately control the mean and fluctuations of the modeled subgrid tensor, and ultimately the first- and second-order flow statistics, two types of model corrections are considered. The first one optimizes the wall-normal profile of the Smagorinsky coefficient, while the second one introduces an adjustable steady forcing in the momentum equations to independently act on the mean flow. Using these two elements, the data assimilation procedure can satisfactorily modify the subgrid model and accurately recover reference flow statistics. In a posteriori testing, the retrieved subgrid model significantly outperforms more elaborate baseline models such as dynamic and mixed models. The robustness of the present data assimilation methodology is assessed by changing the Reynolds number and considering grid resolutions that are away from usual recommendations. Taking advantage of the stochastic formulation of EnVar, the developed framework also provides the uncertainty of the retrieved model.

Figure 1

Smagorinsky coefficient $C_{\mathrm{s}}$ and subgrid dissipations ${\varepsilon }^{\mathrm{ms}}$ and ${\varepsilon }^{\mathrm{fs}}$ in Eq. (9) evaluated using the Smagorinsky model ($‐‐‐$), the dynamic model ($\_\_\_$), the mixed model ($‐·‐·$), and filtered DNS ($•$) at ${\text{Re}}_{\tau }=590$. Grid LES590c is used for the LES calculations.

Figure 2

Control vector and flow statistics of case 1 for DNS ($•••$), the Smagorinsky model ($‐‐‐$), and DA-LES ($\_\_\_$, from thin to thick for increasing iteration of the data assimilation procedure).

Figure 3

Statistics of the first ensemble of case 1: (a) standard deviation $s_b$ of a quantity $b$ normalized by its magnitude at the same $y$ location, where $b$ refers to $C_{\mathrm{s}}^2$ ($\_\_\_$), $\langle {\tau }_{12}\rangle$ ($......$), $\langle {\overline{u}}^{\prime \prime }{\overline{v}}^{\prime \prime }\rangle$ ($‐‐‐$), or $U$ ($‐·‐·$); (b–e) nondimensional gradient $g[b\left(y_2\right),C_{\mathrm{s}}^2\left(y_1\right)]$ in Eq. (29), where $b$ refers to $C_{\mathrm{s}}^2$ (b), $(-\langle {\tau }_{12}\rangle )$ (c), $(-\langle {\overline{u}}^{\prime \prime }{\overline{v}}^{\prime \prime }\rangle )$ (d), or $U$ (e).

Figure 4

Control vector and flow statistics of case 2 for DNS ($•••$), the Smagorinsky model ($‐‐‐$), and final DA-LES ($\_\_\_$).

Figure 5

Control vector and flow statistics of case 3 for DNS ($•••$), the Smagorinsky model ($‐‐‐$), and final DA-LES ($\_\_\_$).

Figure 6

Control vector and flow statistics of case 3c for DNS ($•••$), the Smagorinsky model ($‐‐‐$), and final DA-LES ($\_\_\_$).

Figure 7

Control vector and flow statistics of case 4 for DNS ($•••$), the Smagorinsky model ($‐‐‐$), and final DA-LES ($\_\_\_$).

Figure 8

Second-order statistics of case 4 for DNS ($•••$), the Smagorinsky model ($‐‐‐$), and final DA-LES ($\_\_\_$).

Figure 9

Production of (a) resolved and (b) total turbulent kinetic energy for DNS ($•••$), the Smagorinsky model ($‐‐‐$), and final DA-LES of cases 4 ($‐·‐·$) and 5 ($\_\_\_$). Results in (a) are normalized by $P_r^{\mathrm{smag}}$, which refers to the production of resolved turbulent kinetic energy as predicted by the Smagorinsky model.

Figure 10

Control vector and flow statistics of case 5 for DNS ($•••$), the dynamic model ($\_\_\_$), the mixed model ($‐·‐·$), the Smagorinsky model ($‐‐‐$), and final DA-LES ($\_\_\_$).

Figure 11

Second-order statistics of case 5 for DNS ($•••$), the dynamic model ($\_\_\_$), the mixed model ($‐·‐·$), the Smagorinsky model ($‐‐‐$), and final DA-LES ($\_\_\_$).

Figure 12

Control vector and flow statistics of case 5c for DNS ($•••$), the dynamic model ($\_\_\_$), the mixed model ($‐·‐·$), the Smagorinsky model ($‐‐‐$), and final DA-LES ($\_\_\_$).

Figure 13

One-dimensional spanwise spectra of the resolved streamwise velocity fluctuations $E_{{\overline{u}}^{\prime \prime }{\overline{u}}^{\prime \prime }}\left(k_z\right)$ of cases 5 and 5c at $y^{*}=19$ and $y^{*}=59$. DNS ($•••$); Smagorinsky model ($‐‐‐$); DA-LES ($\_\_\_$).

Figure 14

Control vector and flow statistics of case 6 for DNS ($•••$), the dynamic model ($\_\_\_$), the mixed model ($‐·‐·$), the Smagorinsky model ($‐‐‐$), and final DA-LES ($\_\_\_$).

Figure 15

Flow statistics of cases 5, 5c, and 6 for DNS ($•••$), no-model calculations ($\_\_\_$), the Smagorinsky model ($‐‐‐$), and DA-LES results obtained through $N_{\mathrm{CFD}}=11$ ($‐·‐·$) or $N_{\mathrm{CFD}}=42$ ($\_\_\_$).

Figure 16

Ratio between the standard deviation at the end of the data assimilation procedure $s_b^{\mathrm{a}}$ and at its beginning $s_b^{\mathrm{f}}$ for a quantity $b$ that refers either to the coefficient $C_{\mathrm{s}}^2$ ($\_\_\_$) or the forcing $\sigma$ ($‐·‐·$) for cases 5 (a), 5c (b), and 6 (c).