Approximation Methods for Statistical Conservation Laws of Hyperbolically Dominated Flow

We consider turbulent flow fields that are governed by the incompressible Navier-Stokes system and  focus on convection-dominated scenarios when nonlinear hyperbolic transport starts to prevail. This low-viscosity regime gives rise to the dissipative anomaly which is classically expressed in the form of statistical scaling laws connecting the energy dissipation rate with the mean variation of randomly forced flow fields. Most of these famous scaling laws cannot be linked rigorously to  the underlying hydromechanical equations but still remain hypotheses. To describe the dynamics of ensemble averages for turbulent observables on rigorous grounds, statistical turbulence aims at deriving equations for the associated probability density distributions. This leads to the Lundgren-Monin-Novikov (LMN) hierarchy, i.e. an infinite family of linear statistical conservation laws in the form of kinetic Fokker-Planck equations. To analyze the LMN hierarchy we pursue in this project two approaches exploiting the mutual relations between them.

The first approach exploits statistical symmetries to get a less complex hierarchy, and aims then to approximate the statistical conservation laws via tailored finite-dimensional ansatz spaces. More precisely, the LMN hierarchy for isotropic turbulence is reduced and projected via a product/sum ansatz to a finite-dimensional nonlinear eigenvalue problem for which we expect anomalous scaling. For vanishing viscosity, a singular asymptotic presumably leads to dissipation as an invariant. A corresponding approach for the log-law of shear flows gives rise to an analogous nonlinear eigenvalue problem, where the von Kármán constant appears as an eigenvalue.

The second approach targets at a hybrid numerical approximation method. It relies on truncating the LMN hierarchy for one-point correlations which results in an unclosed statistical conservation law. Based on a structure-preserving  discretisation for this linear equation the unclosed transport terms are  determined via precision-controlled uncertainty quantification methods for the underlying hyperbolically dominated evolution equations. Scalar viscous balance laws  provide a testing ground for the entire method. To address the  the Navier-Stokes system we use reduced hierarchies for isotropic turbulence and new hierarchies derived from hyperbolic relaxation systems approximating the flow equations.

Both new approaches to the LMN hierarchy will be validated, and then jointly used to verify or falsify selected information from scaling laws in low-viscosity regimes.