Segregated Runge–Kutta time integration of convection-stabilized mixed finite element schemes for wall-unresolved LES of incompressible flows

In this work, we develop a high-performance numerical framework for the large eddy simulation (LES) of incompressible flows. The spatial discretization of the nonlinear system is carried out using mixed finite element (FE) schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint. These additional terms introduced at the discrete level have been proved to act as implicit LES models. In order to perform meaningful wall-unresolved simulations, we consider a weak imposition of the boundary conditions using a Nitsche’s-type scheme, where the tangential component penalty term is designed to act as a wall law. Next, segregated Runge–Kutta (SRK) schemes (recently proposed by the authors for laminar flow problems) are applied to the LES simulation of turbulent flows. By the introduction of a penalty term on the trace of the acceleration, these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms. SRK schemes are excellent for large-scale simulations, since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level, leading to two uncoupled systems. The pressure system is a Darcy-type problem that can easily be preconditioned using a traditional block-preconditioning scheme that only requires a Poisson solver. At the end, only coercive systems have to be solved, which can be effectively preconditioned by multilevel domain decomposition schemes, which are both optimal and scalable. The framework is applied to the Taylor–Green and turbulent channel flow benchmarks in order to prove the accuracy of the convection-stabilized mixed FEs as LES models and SRK time integrators. The scalability of the preconditioning techniques (in space only) has also been proven for one step of the SRK scheme for the Taylor–Green flow using uniform meshes. Moreover, a turbulent flow around a NACA profile is solved to show the applicability of the proposed algorithms for a realistic problem.Peer ReviewedPostprint (author's final draft

Finite element LES and VMS methods on tetrahedral meshes

AbstractFinite element methods for problems given in complex domains are often based on tetrahedral meshes. This paper demonstrates that the so-called rational Large Eddy Simulation model and a projection-based Variational Multiscale method can be extended in a straightforward way to tetrahedral meshes. Numerical studies are performed with an inf–sup stable second order pair of finite elements with discontinuous pressure approximation

A variational multiscale method for turbulent flow simulation with adaptive large scale space

In turbulent flows it is only feasible to simulate large flow structures. Variational multiscale (VMS) methods define these flow structures by projections into appropriate function spaces. This paper presents a finite element VMS method which defines the large scale projection space adaptively. The adaption controls the influence of an eddy viscosity model and it is based on the size of the so–called resolved small scales. The adaptive procedure is described in detail. Numerical studies at a turbulent channel flow and a turbulent flow around a cylinder are presented. It is shown that the method selects the large scale space in a correct way and that appropriately chosen parameters improve the results compared to the basic method, which uses the same local large scale space in the whole domain and for all times

On the performance of a high-order multiscale DG approach to LES at increasing Reynolds number

The variational multiscale (VMS) approach based on a high-order discontinuous Galerkin (DG) method is used to perform LES of the sub-critical flow past a circular cylinder at Reynolds 3 900, 20 000 and 140 000. The effect of the numerical flux function on the quality of the LES solutions is also studied in the context of very coarse discretizations of the TGV configuration at Re = 20 000. The potential of using p-adaption in combination with DG-VMS is illustrated for the cylinder flow at Re = 140 000 by considering a non-uniform distribution of the polynomial degree based on a recently developed error estimation strategy. The results from these tests demonstrate the robustness of the DG-VMS approach with increasing Reynolds number on a highly curved geometrical configuration

Variational Multiscale Modeling and Memory Effects in Turbulent Flow Simulations

Effective computational models of multiscale problems have to account for the impact of unresolved physics on the resolved scales. This dissertation advances our fundamental understanding of multiscale models and develops a mathematically rigorous closure modeling framework by combining the Mori-Zwanzig (MZ) formalism of Statistical Mechanics with the variational multiscale (VMS) method. This approach leverages scale-separation projectors as well as phase-space projectors to provide a systematic modeling approach that is applicable to complex non-linear partial differential equations. %The MZ-VMS framework is investigated in the context of turbulent flows. Spectral as well as continuous and discontinuous finite element methods are considered. The MZ-VMS framework leads to a closure term that is non-local in time and appears as a convolution or memory integral. The resulting non-Markovian system is used as a starting point for model development. Several new insights are uncovered: It is shown that unresolved scales lead to memory effects that are driven by an orthogonal projection of the coarse-scale residual and, in the case of finite elements, inter-element jumps. Connections between MZ-based methods, artificial viscosity, and VMS models are explored. The MZ-VMS framework is investigated in the context of turbulent flows. Large eddy simulations of Burgers' equation, turbulent flows, and magnetohydrodynamic turbulence using spectral and discontinuous Galerkin methods are explored. In the spectral method case, we show that MZ-VMS models lead to substantial improvements in the prediction of coarse-grained quantities of interest. Applications to discontinuous Galerkin methods show that modern flux schemes can inherently capture memory effects, and that it is possible to guarantee non-linear stability and conservation via the MZ-VMS approach. We conclude by demonstrating how ideas from MZ-VMS can be adapted for shock-capturing and filtering methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145847/1/parish_1.pd

A review of variational multiscale methods for the simulation of turbulent incompressible flows

Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier-Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.Ministerio de Economía y CompetitividadV Plan Propio de Investigacion (niversidad de Sevilla)Fondation Sciences Mathematiques de Pari

An assessment of two classes of variational multiscale methods for the simulation of incompressible turbulent flows

A numerical assessment of two classes of variational multiscale (VMS) methods for the simulation of incompressible flows is presented. Two types of residual-based VMS methods and two types of projection-based VMS methods are included in this assessment. The numerical simulations are performed at turbulent channel flow problems with various friction Reynolds numbers. It turns out the the residual-based VMS methods, in particular when used with a pair of inf-sup stable finite elements, give usually the most accurate results for second order statistics. For this pair of finite element spaces, a flexible GMRES method with a Least Squares Commutator (LSC) preconditioner proved to be an efficient solver