Data-based approach for time-correlated closures of turbulence models

Developed turbulent motion of fluid still lacks an analytical description despite more than a century of active research. Nowadays, phenomenological ideas are widely used in practical applications, such as small-scale closures for numerical simulations of turbulent flows. In the present paper, we use a shell model of turbulence to construct a closure intended to have a solid theoretical background and to capture intrinsic probabilistic features of turbulence. Shell models of turbulence are dynamical deterministic systems used to model energy cascade and other key aspects of the Navier-Stokes such as intermittency. We rescale the variables of the Sabra model in a way which leads to hidden symmetries and universal distributions. We then use such fine distributions to write closures, i.e., missing expressions for some of the Sabra variables. Our closures rely on approximating probability density functions using a Gaussian mixture model, which makes them probabilistic by nature and allows us to write time-correlated closures. We also provide a framework where other machine learning tools can be employed with reduced black-box aspects.

Figure 1

PDFs of absolute values and phases of ${\mathcal{U}}_{-1},{\mathcal{U}}_0,{\mathcal{U}}_{1}m=8,\cdots ,14$. Statistics are collected in time and rescaled variables are computed from one Sabra simulation.

Figure 2

Figures 2 and 2 show correlation coefficients between ${\mathcal{U}}_0$ and other delayed shells, and Figs. 2 and 2 show correlation coefficients between ${\mathcal{U}}_1$ and other delayed shells for different values of $\mathrm{\Delta }\tau$. Here, $s=12$ and $m=s+1$.

Figure 3

Cutoff shell is $s=12$. Figures 3 present normalized PDFs of real parts for the closure with modules modeling (MJoint) and the closure with modules and phases modeling (PJoint), while Figs. 3 show energy flux PDFs. All are compared to Sabra statistics in black. Figures 3 and 3 show moments of orders 2 up to 6 for the closure with module modeling and with phase modeling, respectively. Both are compared to Sabra moments in solid lines.

Figure 4

Cutoff shell is $s=12$. Figures 4 present normalized PDFs of real parts for the closure with time conditioning to the three closes shells (3Most) and the closure with time conditioning to the closure variables (self), while Figs. 4 show energy flux PDFs. All are compared to Sabra statistics in black. Figures 4 and 4 show moments of orders 2 up to 6. Both are compared to Sabra moments in solid lines.

Figure 5

Cutoff shell is $s=9$. Figures 3 present normalized PDFs of real parts for the closure with time conditioning to the three closes shells (3Most) and the closure with time conditioning to the closure variables (self), while Figs. 5 show energy flux PDFs. All are compared to Sabra statistics in black. Figures 5 and 5 show moments of orders 2 up to 6. Both are compared to Sabra moments in solid lines.