Monika Neda

NUMERICAL ANALYSIS OF FILTER-BASED STABILIZATION FOR EVOLUTION EQUATIONS ∗

We consider filter-based stabilization for evolution equations (in general) and for the Navier--Stokes equations (in particular). The first method we consider is to advance in time one time step by a given method and then to apply an (uncoupled and modular) filter to get the approximation at the new time level. This filter-based stabilization, although algorithmically appealing, is viewed in the literature as introducing far too much numerical dissipation to achieve a quality approximate solution. We show that this is indeed the case. We then consider a modification: Evolve one time step, filter, deconvolve, then relax to get the approximation at the new time step. We give a precise analysis of the numerical diffusion and error in this process and show it has great promise, confirmed in several numerical experiments.

On an Efficient Finite Element Method for Navier-Stokes-ω with Strong Mass Conservation

Abstract We study an efficient finite element method for the NS-ω model, that uses van Cittert approximate deconvolution to improve accuracy and Scott-Vogelius elements to provide pointwise mass conservative solutions and remove the dependence of the (often large) Bernoulli pressure error on the velocity error. We provide a complete numerical analysis of the method, including well-posedness, unconditional stability, and optimal convergence. Additionally, an improved choice of filtering radius (versus the usual choice of the average mesh width) for the scheme is found, by identifying a connection with a scheme for the velocity-vorticity-helicity NSE formulation. Several numerical experiments are given that demonstrate the performance of the scheme, and the improvement offered by the new choice of the filtering radius.

Discontinuous Time Relaxation Method for the Time-Dependent Navier-Stokes Equations

A high-order family of time relaxation models based on approximate deconvolution is considered. A fully discrete scheme using discontinuous finite elements is proposed and analyzed. Optimal velocity error estimates are derived. The dependence of these estimates with respect to the Reynolds number Re is 𝒪(Re𝑒Re), which is an improvement with respect to the continuous finite element method where the dependence is 𝒪(Re𝑒Re3).

On the Vreman filter based stabilization for the advection equation

This report investigates a stabilization method for first order hyperbolic differential equations. If the usual unstabilized finite element method is used to solve numerically first order hyperbolics on unstructured meshes using general elements, it is expected a loss of accuracy by one power of the mesh size and also for non-smooth solutions the computed approximation might contain spurious oscillations.To stabilize the finite element method the authors consider adding to the first order hyperbolic system, as a stabilization term, the Vreman filtering analog of the VMS (variational multiscale method), Vreman (2003) 22.The resulting model is shown to have the expected highest attainable convergence rate, O ( h k + 0.5 ) , when the filter radius ? is optimally chosen. The theoretical rates are also checked numerically and comparison with another known scheme is provided.

Numerical analysis and computations of a high accuracy time relaxation fluid flow model

We present a numerical study based on continuous finite element analysis for a time relaxation regularization of Navier–Stokes equations. This regularization is based on filtering and deconvolution. We study the convergence of the regularized equations using a fully discretized filter and deconvolution algorithm. Velocity and pressure error estimates and the L 2 Aubin–Nitsche lift technique are proved for the equilibrium problem, and this analysis is accompanied by the velocity error estimate for the time-dependent problem, too. Thus, optimal error estimates in L 2 and H 1 norms are derived and followed by their computational verification. Also, computational results of the vortex street are presented for the two-dimensional cylinder benchmark flow problem. Maximum drag and lift coefficients and difference in pressure between the front and back of the cylinder at the final time were investigated as well, showing that the time relaxation regularization can attain the benchmark values.

Numerical Study of the Navier-Stokes-α Deconvolution Model with Pointwise Mass Conservation

ABSTRACT This paper presents an efficient, universally stable finite-element scheme for the NSα deconvolution model. Accuracy is enhanced by van Cittert approximate deconvolution, as well as through the choice of pointwise divergence-free discrete spaces. Finite-element analysis is provided, which includes results for stability, well-posedness, and optimal convergence of both velocity and pressure solutions. Finally, several numerical experiments are presented which demonstrate the performance of NSα, as well as illustrate the advantages of pointwise divergence-free finite elements.

Sensitivity analysis of the grad-div stabilization parameter in finite element simulations of incompressible flow

Abstract We present a numerical study of the sensitivity of the grad-div stabilization parameter for mixed finite element discretizations of incompressible flow problems. For incompressible isothermal and non-isothermal Stokes equations and Navier-Stokes equations, we develop the associated sensitivity equations for changes in the grad-div parameter. Finite element schemes are devised for computing solutions to the sensitivity systems, analyzed for stability and accuracy, and finally tested on several benchmark problems. Our results reveal that solutions are most sensitive for small values of the parameter, near obstacles and corners, when the pressure is large, and when the viscosity is small.

A sensitivity study of the Navier–Stokes-α model

Abstract We present a sensitivity study of the Navier Stokes- α model with respect to perturbations of the differential filter length α . The parameter-sensitivity is evaluated using the sensitivity equations method. Once formulated, the sensitivity equations are discretized and computed alongside the NS α model using the same finite elements in space, and Crank–Nicolson in time. We provide a complete stability analysis of the scheme, along with the results of several benchmark problems in both 2D and 3D. We further demonstrate a practical technique to utilize sensitivity calculations to determine the reliability of the NS α model in problem-specific settings. Lastly, we investigate the sensitivity and reliability of important functionals of the velocity and pressure solutions.

A mixed finite element method with time relaxation for recirculating flows: the slip with friction boundary condition

We study the effect of the slip with friction on the boundary where vortices are generated coupled with time relaxation regularization of the Navier‐Stokes equations. The aim of the regularization term is to drive the unresolved fluctuations in a simulation to zero exponentially fast by an appropriate and often problem dependent choice of its coefficient. The family of models is coupled with slip with friction boundary conditions, discretized with mixed finite element method, and tested on the 2D benchmark step problem. The numerical experiments show a successful shedding of the eddies behind the step, which is the essential feature of the step flow problem.

Architecture of Approximate Deconvolution Models of Turbulence

This report presents the mathematical foundation of approximate deconvolution LES models together with the model phenomenology downstream of the theory. This mathematical foundation now begins to be complete for the incompressible Navier–Stokes equations. It is built upon averaging, deconvolving and addressing closure so as to obtain the physically correct energy and helicity balances in the LES model. We show how this is determined and how correct energy balance implies correct prediction of turbulent statistics. Interestingly, the approach is simple and thus gives a road map to develop models for more complex turbulent flows. We illustrate this herein for the case of MHD turbulence.