10:40
MS23: Recent Advances in Numerical Simulation of Incompressible Flows (Part 3)
Chair: Christoph Lehrenfeld
10:40
25 mins
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Hydrodynamic stability and uncertainty quantification
David Silvester
Abstract: This talk highlights some recent developments in the design of robust solution methods for the Navier-Stokes equations modelling incompressible fluid flow. Our focus is on uncertainty quantification.
We discuss stochastic collocation approximation of critical eigenvalues of the linearised operator associated with the transition from steady flow in a channel to vortex shedding behind an obstacle. Our computational results confirm that classical linear stability analysis is an effective way of assessing the stability of such a flow.
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11:05
25 mins
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Thin Liquid Films - Numerical Simulations, Geometric Interactions and Extensions
Sebastian Reuther
Abstract: We consider the mathematical formulation of thin liquid films, which are approximated by two-dimensional evolving surfaces. The resulting equation, which describes the fluid flow on such fluid films, is the incompressible surface Navier-Stokes equation. We present a numerical approach in order to solve this equation, which is based on the reformulation in the Euclidean basis, the Chorin projection method and spatial discretization with the standard surface finite element method. In various examples the highly nonlinear coupling of the interfacial hydrodynamics and the geometric properties is analyzed. Additionally, we present an extension of the proposed method, which describes polar liquid crystals on evolving surfaces.
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11:30
25 mins
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On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond
Alexander Linke, Philipp Schroeder, Nicolas Gauger
Abstract: An improved understanding of the divergence-free constraint for the incompressible Navier-Stokes equations leads to the observation that a semi-norm and corresponding equivalence classes of forces are fundamental for their nonlinear dynamics. The recent concept of pressure-robustness allows to distinguish between space discretisations that discretise these equivalence classes appropriately or not. This contribution compares the accuracy of pressure-robust and non-pressure-robust space discretisations for transient high Reynolds number flows, starting from the observation that in generalised Beltrami flows the nonlinear convection term is balanced by a strong pressure gradient. Then, pressure-robust methods are shown to outperform comparable non-pressure-robust space discretisations. Indeed, pressure-robust methods of formal order k are comparably accurate than non-pressure-robust methods of formal order 2k on coarse meshes. Investigating the material derivative of incompressible Euler flows, it is conjectured that strong pressure gradients are typical for non-trivial high Reynolds number flows. Connections to vortex-dominated flows are established. Thus, pressure-robustness appears to be a prerequisite for accurate incompressible flow solvers at high Reynolds numbers. The arguments are supported by numerical analysis and numerical experiments.
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11:55
25 mins
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A mass conserving mixed stress formulation for incompressible flows
Jay Gopalakrishnan, Philip L. Lederer, Joachim Schöberl
Abstract: One of the main difficulties in computational fluid dynamics lies in the proper treatment of the incompressibility condition. In particular, it is known that a weak treatment of this constraint can lead to bad velocity approximations if the viscosity tends to get very small.
Recent developments show that H(div)-conforming finite elements for the approximation of the velocity provide major benefits such as exact mass conservation, pressure-independence and polynomial robust error estimates. By introducing a new variable which approximates the gradient of an H(div)-conforming velocity we derive the mass conserving mixed stress formulation (MCS) of the incompressible Stokes equations. For the analysis a new function space, the H(curldiv), is defined, in which we can show well posedness.
In the discrete setting two different approaches lead to a stable analysis. We present the construction of proper Finite elements, discuss solvability and verify our method with several numerical examples implemented in NGSolve (www.ngsolve.org) with the new NGS-Py interface. We conclude the talk by presenting a weakly symmetric version of the MCS method for the discretization of the symmetric Stokes equations
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