MS172 - Numerical Methods for Euler Flows: Dissipative Weak Solutions and Singular Behavior
Keywords: Fluid Dynamics, Hyperbolic Transport, Onsager, Singularities, Turbulence, Euler
The dynamics of nonlinear hyperbolic transport in fluid dynamics fascinate not only mathematicians and physicists due to the open Clay Millennium Navier-Stokes problem or the hidden mysteries of turbulence, but also CFD method developers and engineers due to the highly nonlinear nature of the model problem involving complex, fine-scale physical processes and the need for robust, computationally efficient numerical methods. Turbulent flows are characterized by a rich spectrum of scales and a positive dissipation rate even in the limit of vanishing viscosity (infinite Reynolds number), a phenomenon termed anomalous energy dissipation or the zeroth law of turbulence. Onsager conjectured that the occurrence of anomalous dissipation is linked to singular behavior, a conjecture that has been proven recently. This novel mathematical insight might open new paradigms in the development of numerical methods for Euler (or high-Re Navier-Stokes) flows, e.g. by taking into account the mathematical regularity of the problem (involving singular behavior) in the analysis and design of a discretization scheme. This mini-symposium aims to bring together researchers from various disciplines engaged in the fields of Euler equations, Onsagerâ€™s conjecture, turbulence, or large-eddy simulation. Due to the interdisciplinarity of the topic, we invite contributions with a numerical but also mathematical focus, from the compressible or incompressible Navier-Stokes/Euler communities. An open problem is to prove convergence of a numerical scheme to dissipative weak solutions. Emerging structure-preserving or physics-compatible numerical methods aiming at low-dissipation, scale-resolving simulations face the challenge to treat anomalous dissipation and singularities in a generic, numerically robust, and physically consistent way.