A new class of finite element variational multiscale turbulence models for incompressible magnetohydrodynamics

Abstract

New large eddy simulation (LES) turbulence models for incompressible magnetohydrodynamics (MHD) derived from the variational multiscale (VMS) formulation for finite element simulations are introduced. The new models include the variational multiscale formulation, a residual-based eddy viscosity model, and a mixed model that combines both of these component models. Each model contains terms that are proportional to the residual of the incompressible MHD equations and is therefore numerically consistent. Moreover, each model is also dynamic, in that its effect vanishes when this residual is small. The new models are tested on the decaying MHD Taylor Green vortex at low and high Reynolds numbers. The evaluation of the models is based on comparisons with available data from direct numerical simulations (DNS) of the time evolution of energies as well as energy spectra at various discrete times. A numerical study, on a sequence of meshes, is presented that demonstrates that the large eddy simulation approaches the DNS solution for these quantities with spatial mesh refinement.

On the performance of Krylov smoothing for fully coupled AMG preconditioners for VMS resistive MHD

“…The first transient MHD test case is a Taylor‐Green vortex generalized to MHD as described in the work of Pouquet et al The Taylor‐Green MHD vortex problem is useful for studying characteristics of the turbulent total energy decay spectrum, and the interchange of kinetic and magnetic energy for the system (see, eg, the work of Pouquet et al and the references contained therein). We employ the same domain and initial conditions as described in the work of Sondak et al This MHD flow is initialized with a large fluid vortex and associated magnetic field structure that decays into a fine scale turbulent MHD flow. This problem begins with smooth initial conditions for the velocity field and magnetic induction The domain is a periodic box of size [− π , π ] 3 with Reynolds number Re =1800 and magnetic Reynolds number Re m =1800.…”

“…This construction implies that the unresolved scales are driven by the resolved scale residuals and therefore the exact continuous solution satisfies the system of partial differential equations (PDEs) and is for this reason consistent in this sense . The benefit of this representation is that VMS methods provide stabilization of the Galerkin FE weak form for convective‐type operators, the stabilization of the saddle‐point coupling between ( u , P ) and ( B , ψ ) that allows the use of trilinear hexahedral FE basis functions for all of the variables, and in the context of turbulence modeling provides a means of dynamically modeling subgrid scale interaction terms …”

“…In the formulation of the variational problem for Equations to , the weak form solution that is developed seeks solutions in the function space defined as as described in detail in the works of Shadid et al and Sondak et al Corresponding to the additive decomposition of the solution, U is a splitting of the function space into a finite‐dimensional resolved‐scale subspace and the infinite dimensional unresolved‐scale subspace. The most common practice assumptions for VMS localize the approximation of the unresolved scales and define a computable approximation of the subgrid scale unknowns and the corresponding residuals as where τ is a diagonal matrix of stabilization parameters …”

“…as described in detail in the works of Shadid et al 4 and Sondak et al 21 Corresponding to the additive decomposition of the solution, U is a splitting of the function space  =  h +  ′ into a finite-dimensional resolved-scale subspace and the infinite dimensional unresolved-scale subspace. The most common practice assumptions for VMS localize the approximation of the unresolved scales and define a computable approximation of the subgrid scale unknowns and the corresponding residuals as…”

Dimension reduction in magnetohydrodynamics power generation models: Dimensional analysis and active subspaces

“…Our recent work has developed scalable simulations for resistive MHD models . To incorporate such simulations into generator design, a designer must understand the sensitivities of model predictions to changes in model input parameters.…”