On Some Weighted Stokes Problems. Application on Smagorinsky Models

In this paper we study existence and uniqueness of weak solutions for some non-linear weighted Stokes problems using convex analysis. The characteri- zation of these considered equations is that the viscosity depends on the strain rate of the velocity field with a weight being a positive power of the distance to the boundary of the domain. These non-linear relations can be seen as a first approach of mixing-length eddy viscosity from turbulent modeling. A well known model is von Karman’s on which the viscosity depends on the square of the distance to the boundary of the domain. Numerical experiments conclude the work and show prop- erties from the theory.

On Von Karman Modeling For Turbulent Flow Near A Wall

Jacques Rappaz, Jonathan Rochat

Mixing-length models are often used by engineers in order to take into account turbulence phenomena in a flow. This kind of model is obtained by adding a turbulent viscosity to the laminar one in Navier-Stokes equations. When the flow is confined between two close walls, von Karman's model consists of adding a viscosity which depends on the rate of strain multiplied by the square of distance to the wall. In this short paper, we present a mathematical analysis of such modeling. In particular, we explain why von Karman's model is numerically ill-conditioned when using a finite element method with a small laminar viscosity. Details of analysis can be found in [1], [2].

2019

Numerical simulation and flow analysis of an elbow diffuser

Sebastiano Mauri

Numerical simulation of the unsteady turbulent flow in a three-dimensional elbow diffuser is performed. The investigation is carried out with a commercial finite volume solver implementing the Reynolds averaged Navier-Stokes equations. Against the background of current research in DNS and LES, the modeling of most practically relevant turbulent flows continues to be based on this system of equations. For this reason it is important to evaluate the limitations of the Reynolds averaging approach with the associated turbulence modeling, in particular for the prediction of time-dependent flows. Verification and validation are presented; detailed measurements are compared with computations. While a great deal of research has focused on draft tube design, relatively little is known about the complex flow features present. The flow is analyzed over a wide range of operating conditions including part load. Topological changes in the flow patterns with the global characteristics of the diffuser are presented. Visualization provides extra insight into the complex flow. Forced and self-sustained time-dependent flow phenomena are captured. Falling into these categories are flow field fluctuations introduced by the runner, self-sustained vortex shedding phenomena, and the typical rotating helical vortex observed at part load. Additionally, the linear stability of measured inlet profiles is investigated, providing a fuller understanding of the basic instability mechanism.

EPFL2002

Simulation of shear-driven flows

Roland Bouffanais

The research work reported in the present dissertation is aimed at the analysis of complex physical phenomena involving instabilities and nonlinearities occurring in fluids through state-of-the-art numerical modeling. Solutions of intricate fluid physics problems are devised in two particularly arduous situations: fluid domains with moving boundaries and the high-Reynolds-number regime dominated by nonlinear convective effects. Shear-driven flows of incompressible Newtonian fluids enclosed in cavities of varying geometries are thoroughly investigated in the two following frameworks: transition with a free surface and confined turbulence. The physical system we consider is made of an incompressible Newtonian fluid filling a bounded, or partially bounded cavity. A series of shear-driven flows are easily generated by setting in motion some part of the container boundary. These driven-cavity flows are not only technologically important, they are of great scientific interest because they display almost all physical fluid phenomena that can possibly occur in incompressible flows, and this in the simplest geometrical settings. Thus corner eddies, secondary flows, longitudinal vortices, complex three-dimensional patterns, chaotic particle motions, nonuniqueness, transition, and turbulence all occur naturally and can be studied in the same geometry. This facilitates the comparison of results from experiments, analysis, and computation over the whole range of Reynolds numbers. The flows under consideration are part of a larger class of confined flows driven by linear or angular momentum gradients. This dissertation reports a detailed study of a novel numerical method developed for the simulation of an unsteady free-surface flow in three-space-dimensions. This method relies on a moving-grid technique to solve the Navier-Stokes equations expressed in the arbitrary Lagrangian-Eulerian (ALE) kinematics and discretized by the spectral element method. A comprehensive analysis of the continuous and discretized formulations of the general problem in the ALE frame, with nonlinear, non-homogeneous and unsteady boundary conditions is presented. In this dissertation, we first consider in the internal turbulent flow of a fluid enclosed in a bounded cubical cavity driven by the constant translation of its lid. The solution of this flow relied on large-eddy simulations, which served to improve our physical understanding of this complex flow dynamics. Subsequently, a novel subgrid model based on approximate deconvolution methods coupled with a dynamic mixed scale model was devised. The large-eddy simulation of the lid-driven cubical cavity flow based on this novel subgrid model has shown improvements over traditional subgrid-viscosity type of models. Finally a new interpretation of approximate deconvolution models when used with implicit filtering as a way to approximate the projective grid filter was given. This led to the introduction of the grid filter models. Through the use of a newly-developed method of numerical simulation, in this dissertation we solve unsteady flows with a flat and moving free-surface in the transitional regime. These flows are the incompressible flow of a viscous fluid enclosed in a cylindrical container with an open top surface and driven by the steady rotation of the bottom wall. New flow states are investigated based on the fully three-dimensional solution of the Navier-Stokes equations for these free-surface cylindrical swirling flows, without resorting to any symmetry properties unlike all other results available in the literature. To our knowledge, this study delivers the most general available results for this free-surface problem due to its original mathematical treatment. This second part of the dissertation is a basic research task directed at increasing our understanding of the influence of the presence of a free surface on the intricate transitional flow dynamics of shear-driven flows.

EPFL2007

Simulation of shear-driven flows

Roland Bouffanais

EPFL2007