The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which potentially might pollute the computed velocity. Mathematically, these methods are not robust in the sense that a contribution from the righthand side which influences only the pressure in the continuous equations possesses an impact on both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergencefree solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, H(div)-conforming finite elements, and mixed methods which an appropriate projection of the test function. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained with utilizing robust discretizations.

We propose a data-driven filtered reduced order model (DDF-ROM) framework for the numerical simulation of fluid flows. The novel DDF-ROM framework consists of two steps: (i) In the first step, we use explicit ROM spatial filtering of the nonlinear PDE to construct a filtered ROM. This filtered ROM is low-dimensional, but is not closed (because of the nonlinearity in the given PDE). (ii) In the second step, we use data-driven modeling to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved modes. To this end, we use a quadratic ansatz to model this interaction and close the filtered ROM. To find the new coefficients in the closed filtered ROM, we solve an optimization problem that minimizes the difference between the full order model data and our ansatz. We emphasize that the new DDF-ROM is built on general ideas of spatial filtering and optimization and is independent of (restrictive) phenomenological arguments.We investigate the DDF-ROM in the numerical simulation of a 2D channel flow past a circular cylinder at Reynolds number Re = 100. The DDF-ROM is significantly more accurate than the standard projection ROM. Furthermore, the computational costs of the DDF-ROM and the standard projection ROM are similar, both costs being orders of magnitude lower than the computational cost of the full order model. We also compare the new DDF-ROM with modern ROM closure models in the numerical simulation of the 1D Burgers equation. The DDF-ROM is more accurate and significantly more efficient than these ROM closure models.where a is the vector of unknown ROM coefficients and A ∈ R r×r , B ∈ R r×r×r are ROM operators. (vi) In an offline stage, compute the ROM operators. (vii) In an online stage, repeatedly use the Proj-ROM (1.1) (for various parameter settings

We study conservation properties of Galerkin methods for the incompressible NavierStokes equations, without the divergence constraint strongly enforced. In typical discretizations such as the mixed finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier-Stokes equations dictate that they should. We aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convective formulation does not conserve most of these quantities). Several numerical experiments are performed, which verify the theory and test the new formulation.

We study a computationally attractive algorithm (based on an extrapolated CrankNickolson method) for a recently proposed family of high accuracy turbulence models (the Leray-deconvolution family). First we prove convergence of the algorithm to the solution of the Navier Stokes equations (NSE) and delineate its (optimal) accuracy. Numerical experiments are presented which confirm the convergence theory. Our 3d experiments also give a careful comparison of various related approaches. They show the combination of the Leray-deconvolution regularization with the extrapolated CrankNicolson method can be more accurate at higher Reynolds number that the classical extrapolated trapezoidal method of Baker [6]. We also show the higher order Leraydeconvolution models (e.g. N = 1, 2, 3) have greater accuracy than the N = 0 case of the Leray-alpha model. Numerical experiments for the 2-dimensional step problem are also successfully investigated, showing the higher order models have a reduced effect on transition from one flow behavior to another. To estimate the complexity of using Leraydeconvolution models for turbulent flow simulations we estimate the models' microscale.

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