Autonomic closure for turbulence simulations

A new approach to turbulence closure is presented that eliminates the need to specify a predefined turbulence model and instead provides for fully adaptive, self-optimizing, autonomic closures. The closure is autonomic in the sense that the simulation itself determines the optimal local, instantaneous relation between any unclosed term and resolved quantities through the solution of a nonlinear, nonparametric system identification problem. This nonparametric approach allows the autonomic closure to freely adapt to varying nonlinear, nonlocal, nonequilibrium, and other turbulence characteristics in the flow. Even a simple implementation of the autonomic closure for large eddy simulations provides remarkably more accurate results in a priori tests than do dynamic versions of traditional prescribed closures.

Figure 1

Test stress fields $T_{11}(\mathbf{x},t)$ (top row), $T_{12}(\mathbf{x},t)$ (middle row), and $T_{13}(\mathbf{x},t)$ (bottom row), showing results for (left column) the true stress $T_{ij}(\mathbf{x},t)$, (middle column) the autonomic stress $T_{ij}^{\mathcal{F}}(\mathbf{x},t)$, and (right column) the stress $T_{ij}^{DS}(\mathbf{x},t)$ from the dynamic Smagorinsky model.

Figure 2

Coarse-grained turbulent stress fields ${\tau }_{11}(\mathbf{x},t)$ (top row), ${\tau }_{12}(\mathbf{x},t)$ (middle row), and ${\tau }_{13}(\mathbf{x},t)$ (bottom row), showing results for (left column) the true stress ${\tau }_{ij}(\mathbf{x},t)$, (middle column) the autonomic closure ${\tau }_{ij}^{\mathcal{F}}(\mathbf{x},t)$, and (right column) the dynamic Smagorinsky model ${\tau }_{ij}^{DS}(\mathbf{x},t)$.

Figure 3

Probability densities of turbulent stress components at the test-filter scale (top row) and coarse-grain scale (bottom row), showing results for $T_{11}$ and ${\tau }_{11}$ (left column), $T_{12}$ and ${\tau }_{12}$ (middle column), and $T_{13}$ and ${\tau }_{13}$ (right column), showing densities for the true stresses from DNS (black lines), stresses from the autonomic closure (red lines), and stresses from the dynamic Smagorinsky model (blue lines).

Figure 4

Kinetic energy flux fields at the test-filter scale (top row) and coarse-grain scale (bottom row), showing results for the true flux fields $\tilde{P}(\mathbf{x},t)$ (left), the fields ${\tilde{P}}^{\mathcal{F}}(\mathbf{x},t)$ resulting from the autonomic closure (center), and the fields ${\tilde{P}}^{DS}(\mathbf{x},t)$ from the dynamic Smagorinsky model. Even this simple implementation of autonomic closure is far more accurate than the prescribed closure in the dynamic Smagorinsky model.

Figure 5

Probability densities of kinetic energy fluxes (a) $\widehat{P}$ at the test-filter scale and (b) $\tilde{P}$ at the coarse-grain scale, showing distributions of the true fluxes from DNS (black lines), fluxes from the autonomic closure (red lines), and fluxes from the dynamic Smagorinsky model (blue lines).