Optimal eddy viscosity in closure models for two-dimensional turbulent flows

We consider the question of fundamental limitations on the performance of eddy-viscosity closure models for turbulent flows, focusing on the Leith model for two-dimensional large-eddy simulation. Optimal eddy viscosities depending on the magnitude of the vorticity gradient are determined subject to minimum assumptions by solving PDE-constrained optimization problems defined such that the corresponding optimal large-eddy simulation best matches the filtered direct numerical simulation. First, we consider pointwise match in the physical space, and the main finding is that with a fixed cutoff wave number $k_c$, the performance of the large-eddy simulation systematically improves as the regularization in the solution of the optimization problem is reduced, and this is achieved with the optimal eddy viscosities exhibiting increasingly irregular behavior with rapid oscillations. Since the optimal eddy viscosities do not converge to a well-defined limit as the regularization vanishes, we conclude that in this case the problem of finding an optimal eddy viscosity does not in fact have a solution and is thus ill-posed. We argue that this observation is consistent with the physical intuition concerning closure problems. The second problem we consider involves matching time-averaged vorticity spectra over small wave numbers. It is shown to be better behaved and to produce physically reasonable optimal eddy viscosities. We conclude that while better behaved and hence practically more useful eddy viscosities can be obtained with stronger regularization or by matching quantities defined in a statistical sense, the corresponding large-eddy simulations will not achieve their theoretical performance limits.

Figure 1

(a) Dependence of the functional ${\mathcal{J}}_1\left({\phi }^{\left(n\right)}\right)$ on the iteration $n$ and (b) dependence of the corresponding optimal eddy viscosity $\check{\nu }$ on $\sqrt{s}$ for cases A, B, and C; cf. Table 1. Panel (b) also shows the PDF of $\sqrt{s}$ in case C.

Figure 2

(a) Dependence of the normalized functional ${\mathcal{J}}_1\left({\phi }^{\left(n\right)}\right)/{\mathcal{J}}_1\left({\phi }_0\right)$, with ${\mathcal{J}}_1\left({\phi }_0\right)$ from case C, on the iteration $n$ and (b) dependence of the corresponding optimal eddy viscosity $\check{\nu }$ on $\sqrt{s}$ for cases C, D, and E; cf. Table 1. The inset in panel (b) shows magnification of the region $\sqrt{s}\in [0,25]$. Panel (b) also shows the Leith model with $k_c=20$ and the eddy viscosity ${\nu }_L\left(s\right)$; cf. (6).

Figure 3

Adjusted normalized correlations (28) for the LES with (a) no closure and the optimal eddy viscosity in cases A, B, and C, and (b) no closure and the optimal eddy viscosity in cases C, D, and E. The correlation is also shown for the Leith model with $k_c=20$ and the eddy viscosity ${\nu }_L\left(s\right)$ [ cf. (6)] in (a) and for an optimal closure model based on the stochastic estimator [37] in (b). Thick and thin lines correspond to, respectively, time in the “training window” ($t\in [0,T]$) and beyond this window ($t\in (T,2T]$).

Figure 4

For case E we show (a) the vorticity field $\tilde{\omega }(T,\mathbf{x})$, $\mathbf{x}\in \mathrm{\Omega }$, (b) the corresponding state variable $s(T,\mathbf{x})$ [cf. (7)], and the spatial distribution of (c) the eddy viscosity ${\nu }_L(s(T,\mathbf{x}))$ in the Leith model [cf. (6)] with $\delta =0.02$, and (d) the optimal eddy viscosity $\check{\nu }(s(T,\mathbf{x}))$ [cf. Fig. 2], all shown at the end of the training window for $t=T$. For better comparison the same color scale is used in panels (c) and (d).

Figure 5

(a) Dependence of the normalized functional ${\mathcal{J}}_2\left({\phi }^{\left(n\right)}\right)/{\mathcal{J}}_2\left({\phi }_0\right)$ on the iteration $n$ and (b) dependence of the corresponding optimal eddy viscosity $\check{\nu }$ on $\sqrt{s}$ for cases F and G; cf. Table 2. Panel (b) also shows the PDF of $\sqrt{s}$ in case G.

Figure 6

The difference between time-averaged vorticity spectra (11) in the filtered DNS and in the LES with no closure and with the optimal eddy viscosity $\check{\nu }$ obtained in cases F and G (cf. Table 2) as function of the wave number $k$.

Figure 7

Adjusted normalized correlations (28) for the LES with no closure and the optimal eddy viscosity in cases F and G. Thick and thin lines correspond to, respectively, time in the “training window” ($t\in [0,T]$) and beyond this window ($t\in (T,2T]$).

Figure 8

For case G we show (a) the vorticity field $\tilde{\omega }(T,\mathbf{x})$, $\mathbf{x}\in \mathrm{\Omega }$, (b) the corresponding state variable $s(T,\mathbf{x})$ [cf. (7)], and the spatial distribution of (c) the eddy viscosity ${\nu }_L(s(T,\mathbf{x}))$ in the Leith model [cf. (6)] with $\delta =0.02$, and (d) the optimal eddy viscosity $\check{\nu }(s(T,\mathbf{x}))$ [cf. Fig. 5], all shown at the end of the training window for $t=T$. For better comparison the same color scale is used in panels (c) and (d).