Approximation methods for statistical conservation laws of hyperbolically dominated flow

PN 1 A-7

Project description

We consider turbulent flow fields that are governed by the 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. Our approach targets at a hybrid numerical approximation method. It relies on truncating the statistical hierarchies 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.

Project information

Project title Approximation methods for statistical conservation laws of
hyperbolically dominated flow
Project leader Christian Rohde (jointly with Prof. Martin Oberlack, TU Darmstadt)
Project staff Qian Huang, postdoctoral researcher
Group webpage https://www.spp2410.uni-stuttgart.de/SPPProjects/16_oberlack-rohde/
Project duration September 2023 - August 2026
Project number PN 1 A-7
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