Inviscid Regularization Research Articles

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.

A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler–Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler–Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter $$\alpha >0$$ . Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly, namely by simulating the better-behaved 3D Euler–Voigt equations. The new criteria are only known to be sufficient criterion for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well known to occur.

An observable regularization of compressible two-phase flow

Many fluid flow problems involving turbulence, shocks, and material interfaces create a common issue high wave number irregularity. The non-linear advection term in the governing equations for all of these problems keep generating higher wave modes as k goes to infinity. In this work, we present an inviscid regularization technique, called observable regularization, for the simulation of two-phase compressible flow. In this technique, we use observable divergence theorem to derive an observable equation for tracking material interface (volume fraction). In some one-dimensional test cases, first we show that this method preserves pressure equilibrium at material interface, then we compare our results to exact Euler solutions. At the end we demonstrate a two-dimensional simulation of shock-bubble interaction showing good agreement with available experimental data from literature.

Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations

We prove existence, uniqueness, and higher-order global regularity of strong solutions to a particular Voigt-regularization of the three-dimensional inviscid resistive magnetohydrodynamic (MHD) equations. Specifically, the coupling of a resistive magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid resistive MHD system. The results hold in both the whole space \({\mathbb{R}^3}\) and in the context of periodic boundary conditions. Weak solutions for this regularized model are also considered, and proven to exist globally in time, but the question of uniqueness for weak solutions is still open. Furthermore, we show that the solutions of the Voigt regularized system converge, as the regularization parameter \({\alpha \rightarrow 0}\), to strong solutions of the original inviscid resistive MHD, on the corresponding time interval of existence of the latter. Moreover, we also establish a new criterion for blow-up of solutions to the original MHD system inspired by this Voigt regularization.

An inviscid regularization of hyperbolic conservation laws

This article examines the utilization of a spatial averaging technique to the nonlinear terms of the partial differential equations as an inviscid shock-regularization of hyperbolic conservation laws. A central motivation is to promote the idea of applying filtering techniques such as the observable divergence method, rather than viscous regularization, as an alternative to the simulation of shocks and turbulence in inviscid flows while, on the other hand, generalizing and unifying previous mathematical and numerical analysis of the method applied to the one-dimensional Burgers’ and Euler equations. This article primarily concerns the mathematical analysis of the technique and examines two fundamental issues. The first is on the global existence and uniqueness of classical solutions for the regularization under the more general setting of quasilinear, symmetric hyperbolic systems in higher dimensions. The second issue examines one-dimensional scalar conservation laws and shows that the inviscid regularization method captures the unique entropy or physically relevant solution of the original, non-averaged problem as filtering vanishes.

The Navier-Stokes-Voight model for image inpainting

In this paper we investigate the use of the 2D Navier-Stokes-Voight (NSV) model for use in algorithms and explore its limits in the context of image inpainting. We begin by giving a brief review of the work of Bertalmio et. al. in 2001 on exploiting an analogy between the image intensity function for the image inpainting problem and the stream function in 2D incompressible fluid. An approximate solution to the inpainting problem was then obtained by numerically approximating the steady state solution of the 2D Navier-Stokes vorticity transport equation, and simultaneously solving the Poisson problem between the vorticity and stream function, in the region to be inpainted. This elegant approach allows one to produce an approximate solution to the image inpainting problem by using techniques from computational fluid dynamics (CFD). Recently, the three-dimensional (3D) Navier-Stokes-Voight (NSV) model of viscoelastic fluid, was suggested by Cao, et. al. as an inviscid regularization to the 3D Navier-Stokes equations. We give some background on the NSV mathematical model, describe why it is a good candidate sub-grid-scale turbulence model, and propose this model as an alternative for image inpainting. We describe an implementation of inpainting use the NSV model, and present numerical results comparing the resulting images when using the NSE and NSV for inpainting. Our results show that the NSV model allows for a larger time step to converge to the steady state solution, yielding a more efficient numerical process when automating the inpainting process. We compare quality of the resulting images using subjective measure (human evaluation) and objected measure (by calculating the peak signal-to-noise ratio (PSNR), also known as peak signal-to-reconstructed measure). We also present some new theoretical results based on energy methods comparing the sufficient conditions for stability of the discretization scheme for the two model equations. These theoretical and numerical studies shed some light on what can be expected from this category of approach when automating the inpainting problem. CONTENTS

Viscous and inviscid regularizations in a class of evolutionary partial differential equations

We investigate solution properties of a class of evolutionary partial differential equations (PDEs) with viscous and inviscid regularization. An equation in this class of PDEs can be written as an evolution equation, involving only first-order spatial derivatives, coupled with the Helmholtz equation. A recently developed two-step iterative method (P.H. Chiu, L. Lee, T.W.H. Sheu, A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation, J. Comput. Phys. 228 (2009) 8034–8052) is employed to study this class of PDEs. The method is in principle superior for PDE’s in this class as it preserves their physical dispersive features. In particular, we focus on a Leray-type regularization (H.S. Bhat, R.C. Fetecau, A Hamiltonian regularization of the Burgers equation, J. Nonlinear Sci. 16 (2006) 615–638) of the Hopf equation proposed in alternative to the classical Burgers viscous term. We show that the regularization effects induced by the alternative model can be vastly different from those induced by Burgers viscosity depending on the smoothness of initial data in the limit of zero regularization. We validate our numerical scheme by comparison with a particle method which admits closed form solutions. Further effects of the interplay between the dispersive terms comprising the Leray-regularization are illustrated by solutions of equations in this class resulting from regularized Burgers equation by selective elimination of dispersive terms.

On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models

We prove higher-order and a Gevrey class (spatial analytic) regularity of solutions to the Euler-Voigt inviscid $\alpha$-regularization of the three-dimensional Euler equations of ideal incompressible fluids. Moreover, we establish the convergence of strong solutions of the Euler-Voigt model to the corresponding solution of the three-dimensional Euler equations for inviscid flow on the interval of existence of the latter. Furthermore, we derive a criterion for finite-time blow-up of the Euler equations based on this inviscid regularization. The coupling of a magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid irresistive magneto-hydrodynamic (MHD) system. Global regularity of the regularized MHD system is also established.

A New Potential Regularization of the One-Dimensional Euler and Homentropic Euler Equations

This paper examines an averaging technique in which the nonlinear flux term is expanded and the convective velocities are passed through a low-pass filter. It is the intent that this modification to the nonlinear flux terms will result in an inviscid regularization of the homentropic Euler equations and Euler equations. The physical motivation for this technique is presented and a general method is derived, which is then applied to the homentropic Euler equations and Euler equations. These modified equations are then examined, revealing that they share conservative properties, shock speeds, and traveling wave solutions with the original equations. If compactness of the solutions is assumed, then it can be proven that as the averaging is diminished the solutions converge to weak solutions of the original equations. Finally, numerical simulations are conducted showing that the modified equations appear regularized and converge to the entropy solution in shock tube simulations.

Regularization modeling for large-eddy simulation of homogeneous isotropic decaying turbulence

Inviscid regularization modeling of turbulent flow is investigated. Homogeneous, isotropic, decaying turbulence is simulated at a range of filter widths. A coarse-graining of turbulent flow arises from the direct regularization of the convective nonlinearity in the Navier–Stokes equations. The regularization is translated into its corresponding sub-filter model to close the equations for large-eddy simulation (LES). The accuracy with which primary turbulent flow features are captured by this modeling is investigated for the Leray regularization, the Navier–Stokes-α formulation (NS-α), the simplified Bardina model and a modified Leray approach. On a PDE level, each regularization principle is known to possess a unique, strong solution with known regularity properties. When used as turbulence closure for numerical simulations, significant differences between these models are observed. Through a comparison with direct numerical simulation (DNS) results, a detailed assessment of these regularization principles is made. The regularization models retain much of the small-scale variability in the solution. The smaller resolved scales are dominated by the specific sub-filter model adopted. We find that the Leray model is in general closest to the filtered DNS results, the modified Leray model is found least accurate and the simplified Bardina and NS-α models are in between, as far as accuracy is concerned. This rough ordering is based on the energy decay, the Taylor Reynolds number and the velocity skewness, and on detailed characteristics of the energy dynamics, including spectra of the energy, the energy transfer and the transfer power. At filter widths up to about 10% of the computational domain-size, the Leray and NS-α predictions were found to correlate well with the filtered DNS data. Each of the regularization models underestimates the energy decay rate and overestimates the tail of the energy spectrum. The correspondence with unfiltered DNS spectra was observed often to be closer than with filtered DNS for several of the regularization models.

An inviscid regularization for the surface quasi‐geostrophic equation

AbstractInspired by recent developments in Berdina‐like models for turbulence, we propose an inviscid regularization for the surface quasi‐geostrophic (SQG) equations. We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter α tends to 0 for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite‐time blowup testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times. © 2007 Wiley Periodicals, Inc.

Fine Structure of Vortex Sheet Rollup by Viscous and Inviscid Simulation

Numerical simulations of the large amplitude stage of the Kelvin-Helmholtz instability of a relatively thin vorticity layer are discussed. At high Reynolds number, the effect of viscosity is commonly neglected and the thin layer is modeled as a vortex sheet separating one potential flow region from another. Since such vortex sheets are susceptible to a short wavelength instability, as well as singularity formation, it is necessary to provide an artificial “regularization” for long time calculations. We examine the effect of this regularization by comparing vortex sheet calculations with fully viscous finite difference calculations of the Navier-Stokes equations. In particular, we compare the limiting behavior of the viscous simulations for high Reynolds numbers and small initial layer thickness with the limiting solution for the roll-up of an inviscid vortex sheet. Results show that the inviscid regularization effectively reproduces many of the features associated with the thickness of viscous vorticity layers with increasing Reynolds number, though the simplified dynamics of the inviscid model allows it to accurately simulate only the large scale features of the vorticity field. Our results also show that the limiting solution of zero regularization for the inviscid model and high Reynolds number and zero initial thickness for the viscous simulations appear to be the same.