A Hybrid High-Order method for incompressible flows of non-Newtonian fluids with power-like convective behaviour

In this work, we design and analyze a Hybrid High-Order (HHO) discretization method for incompressible flows of non-Newtonian fluids with power-like convective behaviour. We work under general assumptions on the viscosity and convection laws, that are associated with possibly different Sobolev exponents r > 1 and s > 1. After providing a novel weak formulation of the continuous problem, we study its well-posedness highlighting how a subtle interplay between the exponents r and s determines the existence and uniqueness of a solution. We next design an HHO scheme based on this weak formulation and perform a comprehensive stability and convergence analysis, including convergence for general data and error estimates for shear-thinning fluids and small data. The HHO scheme is validated on a complete panel of model problems.Comment: 33 pages, 3 figures, 3 table

A Hybrid High-Order method for creeping flows of non-Newtonian fluids

In this paper, we design and analyze a Hybrid High-Order discretization method for the steady motion of non-Newtonian, incompressible fluids in the Stokes approximation of small velocities. The proposed method has several appealing features including the support of general meshes and high-order, unconditional inf-sup stability, and orders of convergence that match those obtained for Leray-Lions scalar problems. A complete well-posedness and convergence analysis of the method is carried out under new, general assumptions on the strain rate-shear stress law, which encompass several common examples such as the power-law and Carreau-Yasuda models. Numerical examples complete the exposition.Comment: 26 pages, 3 figure

Numerical dynamo action in cylindrical containers

The purpose of this paper is to present results from numerical simulations of dynamo action in relation with two magnetohydrodynamics (MHD) experiments using liquid sodium in cylindrical containers. The first one is the von Kármán sodium (VKS) experiment from Cadarache (France), the second one is a precession-driven dynamo experiment from the DREsden sodium facility for DYNamo and thermohydraulic studies (DRESDYN)

Solving the MHD Equations in the Presence of Non-Axisymmetric Conductors Using the Fourier-Finite Element Method

The research presented in this dissertation focus on the numerical approximation of the magnetohydrodynamic (MHD) equations in the von K´arm´an Sodium (VKS) set-up. These studies are performed with the SFEMaNS MHD code developed by J.-L. Guermond and C. Nore since 2002 for axisymmetric geometries. SFEMaNS is based on a spectral decomposition in the azimuthal direction and a Lagrange finite element approximation in a meridian plane. To overcome the axisymmetric restrictions, we propose a novel numerical method to solve the Maxwell part of the MHD equations, and use a pseudo-penalty method to model the rotating impellers. We then present hydrodynamic and MHD simulations of the VKS set-up. Hydrodynamic results compare well with the experimental data in the same range of kinetic Reynolds numbers: at small impeller rotation frequency, the flow is steady; at larger frequency, the fluctuating flow is characterized by small scales and helical vortices localized between the blades. MHD computations are performed for two different flows. One with small kinetic Reynolds number, and the other with a larger one. In both cases, using a ferromagnetic material for the impellers decreases the dynamo threshold and enhances the predominantly axisymmetric magnetic field: the resulting dynamo is a mostly axisymmetric axial dipole with an azimuthal component concentrated in the impellers as observed in the VKS experiment

Solving the MHD Equations in the Presence of Non-Axisymmetric Conductors Using the Fourier-Finite Element Method

The research presented in this dissertation focus on the numerical approximation of the magnetohydrodynamic (MHD) equations in the von K´arm´an Sodium (VKS) set-up. These studies are performed with the SFEMaNS MHD code developed by J.-L. Guermond and C. Nore since 2002 for axisymmetric geometries. SFEMaNS is based on a spectral decomposition in the azimuthal direction and a Lagrange finite element approximation in a meridian plane. To overcome the axisymmetric restrictions, we propose a novel numerical method to solve the Maxwell part of the MHD equations, and use a pseudo-penalty method to model the rotating impellers. We then present hydrodynamic and MHD simulations of the VKS set-up. Hydrodynamic results compare well with the experimental data in the same range of kinetic Reynolds numbers: at small impeller rotation frequency, the flow is steady; at larger frequency, the fluctuating flow is characterized by small scales and helical vortices localized between the blades. MHD computations are performed for two different flows. One with small kinetic Reynolds number, and the other with a larger one. In both cases, using a ferromagnetic material for the impellers decreases the dynamo threshold and enhances the predominantly axisymmetric magnetic field: the resulting dynamo is a mostly axisymmetric axial dipole with an azimuthal component concentrated in the impellers as observed in the VKS experiment

A pressure-robust HHO method for the solution of the incompressible Navier-Stokes equations on general meshes

In a recent work [11], we have introduced a pressure-robust Hybrid High-Order method for the numerical solution of the incompressible Navier-Stokes equations on matching simplicial meshes. Pressure-robust methods are characterized by error estimates for the velocity that are fully independent of the pressure. A crucial question was left open in that work, namely whether the proposed construction could be extended to general polytopal meshes. In this paper we provide a positive answer to this question. Specifically, we introduce a novel divergence-preserving velocity reconstruction that hinges on the solution inside each element of a mixed problem on a subtriangulation, then use it to design discretizations of the body force and convective terms that lead to pressure robustness. An in-depth theoretical study of the properties of this velocity reconstruction, and their reverberation on the scheme, is carried out for polynomial degrees $k \geq 0$ and meshes composed of general polytopes. The theoretical convergence estimates and the pressure robustness of the method are confirmed by an extensive panel of numerical examples

A Hybrid High-Order method for the incompressible Navier--Stokes problem robust for large irrotational body forces

We develop a novel Hybrid High-Order method for the incompressible Navier--Stokes problem robust for large irrotational body forces. The key ingredients of the method are discrete versions of the body force and convective terms in the momentum equation formulated in terms of a globally divergence-free velocity reconstruction. Denoting by $\lambda$ the $L^2$-norm of the irrotational part of the body force, the method is designed so as to mimic two key properties of the continuous problem at the discrete level, namely the invariance of the velocity with respect to λ and the non-dissipativity of the convective term. The convergence analysis shows that, when polynomials of total degree $\le k$ are used as discrete unknowns, the energy norm of the error converges as $h^{k+1}$ (with $h$ denoting, as usual, the meshsize), and the error estimate on the velocity is uniform in $\lambda$ and independent of the pressure. The performance of the method is illustrated by a complete panel of numerical tests, including comparisons that highlight the benefits with respect to more standard HHO formulations

A Hybrid High-Order method for creeping flows of non-Newtonian fluids

In this paper, we design and analyze a Hybrid High-Order discretization method for the steady motion of non-Newtonian, incompressible fluids in the Stokes approximation of small velocities. The proposed method has several appealing features including the support of general meshes and high-order, unconditional inf-sup stability, and orders of convergence that match those obtained for Leray-Lions scalar problems. A complete well-posedness and convergence analysis of the method is carried out under new, general assumptions on the strain rate-shear stress law, which encompass several common examples such as the power-law and Carreau-Yasuda models. Numerical examples complete the exposition