NAMColloquium On numerical evidence of anomalous energy dissipation for incompressible turbulent flows |  |
Speaker:Dr. Niklas Fehn, Technische Universität München
Organizer:Institut für Numerische und Angewandte Mathematik
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Turbulent flows have long been suspected to dissipate kinetic energy in the inviscid limit, a phenomenon known as “anomalous energy dissipation”. Onsager conjectured that this phenomenon is related to singularities of the velocity field, i.e., a velocity field losing regularity does not necessarily conserve kinetic energy in the absence of viscosity. While energy cannot be dissipated in the absence of viscosity if the velocity is Hölder continuous with exponent >1/3, “anomalous” dissipation of energy might occur if the velocity is Hölder continuous with exponent ≤1/3. To date, Onsager’s conjecture has been proven mathematically. However, numerical evidence of anomalous dissipation is still outstanding. State-of-the-art discretization schemes for turbulent flows are traditionally designed to be kinetic-energy-conserving in the discrete case. The one-dimensional inviscid Burgers equation with formation of shock is a suitable example to internalize that such schemes are, however, not suitable to investigate the phenomenon of anomalous energy dissipation. By construction, they rule out the occurrence of anomalous dissipation. Since no energy can leave the system, energy-conserving numerical methods lead to an accumulation of energy in small scales, a phenomenon called “thermalization”. At the heart of the present contribution is the insight that schemes with purely numerical mechanisms of kinetic energy dissipation can indeed be a powerful tool to predict physical dissipation. Out talk is based on an article by Fehn et al. (2022) published recently in the Journal of Fluid Mechanics. High-resolution numerical simulations have been conducted for the inviscid three-dimensional Taylor-Green vortex problem. The discretization scheme of the incompressible Navier-Stokes equations is based on a novel discontinuous Galerkin formulation with suitable inbuilt dissipation mechanism. The flow solver is realized in the open-source CFD software ExaDG. By monitoring the temporal evolution of the kinetic energy and its dissipation rate, we show that our numerical results for three-dimensional inviscid turbulent flows exhibit a dissipative behaviour under mesh refinement, which is consistent with anomalous energy dissipation. Due to the relation between anomalous dissipation and finite-time Euler singularities, we elaborate whether such novel results might be indicative of finite-time singularities in incompressible Euler flows.
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Type:Colloquium
Language:English
Category:Research
Host:Prof. Dr. Gert Lube
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Direct link to event:https://events.goettingen-campus.de/event?eventId=844214
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