Vanishing Viscosity Research Articles

Gradient jump penalty stabilisation of spectral/[formula omitted] element discretisation for under-resolved turbulence simulations

One of the strengths of the discontinuous Galerkin (DG) method has been its balance between accuracy and robustness, which stems from DG’s intrinsic (upwind) dissipation being biased towards high frequencies/wavenumbers. This is particularly useful in high Reynolds-number flow simulations where limitations on mesh resolution typically lead to potentially unstable under-resolved scales. In continuous Galerkin (CG) discretisations, similar properties are achievable through the addition of artificial diffusion such as spectral vanishing viscosity (SVV). However although SVV is recognised as very useful in CG-based high-fidelity turbulence simulations, this approach has been observed to be sub-optimal when compared to DG at intermediate polynomials orders (P≈3). In this paper we explore an alternative stabilisation approach through the introduction of a continuous interior penalty on the gradient discontinuity at elemental boundaries, which we refer to as a gradient jump penalisation (GJP). Analogous to DG methods, this introduces a penalisation at the elemental interfaces as opposed to the interior element stabilisation of SVV. Detailed eigenanalysis of the GJP approach shows its potential as equivalent (sometimes superior) to DG dissipation and hence superior to previous SVV approaches. Through eigenanalysis, a judicious choice of GJP’s P-dependent scaling parameter is made and found to be consistent with previous a-priori error analysis. The favourable properties of the GJP stabilisation approach are also supported by turbulent flow simulations of the incompressible Navier–Stokes equation, as we achieve higher quality flow solutions at P=3 using GJP, whereas SVV performs marginally worse at P=5 with twice as many degrees of freedom in total.

Limit of a Consistent Approximation to the Complete Compressible Euler System

The goal of the present paper is to prove that if a weak limit of a consistent approximation scheme of the compressible complete Euler system in full space mathbb {R}^d,; d=2,3 is a weak solution of the system, then the approximate solutions eventually converge strongly in suitable norms locally under a minimal assumption on the initial data of the approximate solutions. The class of consistent approximate solutions is quite general and includes the vanishing viscosity and heat conductivity limit. In particular, they may not satisfy the minimal principle for entropy.

Dissipation Mechanisms for Fluids and Objects in Relative Motion Described by the Navier-Stokes Equation.

This work demonstrates that an additional resistance term should be included in the Navier–Stokes equation when fluids and objects are in relative motion. This is based on an observation that the effect of the microscopic molecular random velocity component parallel to the macroscopic flow direction is neglected. The two components of the random velocity perpendicular to the local mean flow direction are accounted for by the viscous resistance, e.g., by Stokes’ law for spherical objects. The relationship between the mean- and the random velocity in the longitudinal direction induces differences in molecular collision velocities and collision frequency rates on the up- and downstream surface areas of the object. This asymmetry therefore induces flow resistance and energy dissipation. The flow resistance resulting from the longitudinal momentum transfer mode is referred to as thermal resistance and is quantified by calculating the net difference in pressure up- and downstream the surface areas of a sphere using a particle velocity distribution that obeys Boltzmann’s transport equation. It depends on the relative velocity between the fluid and the object, the number density and the molecular fluctuation statistics of the fluid, and the area of the object and the square root of the absolute temperature. Results show that thermal resistance is dominant compared to viscous resistance considering water and air in slow relative motion to spherical objects larger than nanometer-size at ambient temperature and pressure conditions. Including the thermal resistance term in the conventional expression for the terminal velocity of spherical objects falling through liquids, the Stokes–Einstein relationship and Darcy’s law, corroborates its presence, as modified versions of these equations fit observed data much more closely than the conventional expressions. The thermal resistance term can alternatively resolve d’Alembert’s paradox as a finite flow resistance is predicted at both low and high relative fluid–object velocities in the limit of vanishing fluid viscosity.

Quasi-neutral limit of the Isothermal Naiver–Stokes–Poisson with boundary

The quasineutral limit of the isothermal Navier–Stokes–Poisson system is rigorously proved when the combined quasineutral and vanishing viscosity limit is considered in a domain with boundary. The convergence of the global weak solution for Navier–Stokes–Poisson system to the strong solution for incompressible Euler equations is obtained.

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

A fractional degenerate parabolic-hyperbolic Cauchy problem with noise

We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy solution and provide a new technical framework to prove the uniqueness. The existence proof relies on the vanishing viscosity method. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities and derive error estimate for the stochastic vanishing viscosity method. In addition, we develop uniqueness method “à la Kružkov” for more general equations where the noise coefficient may depend explicitly on the spatial variable.

High-fidelity simulations of gravity currents using a high-order finite-difference spectral vanishing viscosity approach

This numerical work investigates the potential of a high-order finite-difference spectral vanishing viscosity approach to simulate gravity currents at high Reynolds numbers. The method introduces targeted numerical dissipation at small scales through altering the discretisation of the second derivatives of the viscous terms in the incompressible Navier-Stokes equations to mimic the spectral vanishing viscosity (SVV) operator, originally designed for the regularisation of spectral element method (SEM) solutions of pure advection problems. Using a sixth-order accurate finite-difference scheme, the adoption of the SVV method is straightforward and comes with a negligible additional computational cost. In order to assess the ability of this high-order finite-difference spectral vanishing viscosity approach, we performed large-eddy simulations (LES) of a gravity current in a channelised lock-exchange set-up with our SVV model and with the well-known explicit static and dynamic Smagorinsky sub-grid scale (SGS) models. The obtained data are compared with a direct numerical simulation (DNS) based on more than 800 million mesh nodes, and with experimental measurements. A framework for the energy budget is introduced to investigate the behaviour of the gravity current. First, it is found that the DNS is in good agreement with the experimental data for the evolution of the front location and velocity field as well as for the stirring and mixing inside the gravity current. Secondly, the LES performed with less than 0.4% of the total number of mesh nodes compared to the DNS, can reproduce the main features of the gravity currents, with the SVV model yielding slightly more accurate results. It is also found that the dynamic Smagorinsky model performs better than its static version. For the present study, the static and dynamic Smagorinsky models are 1.8 and 2.5 times more expensive than the SVV model, because the latter does not require the calculation of explicit SGS terms in the Navier-Stokes equations nor spatial filtering operations.

Implicit large-eddy simulations of turbulent flow in a channel via spectral/hp element methods

The spectral/hp element method [which is the hp-version finite element method, where h denotes the h-version finite element method and p denotes the p-version finite element method (or the spectral element method) with elementwise expansion based on (modified) orthogonal polynomials up to pth-order in each element] together with the regularized spectral vanishing viscosity (SVV) is employed to perform implicit large eddy simulation (iLES) of the turbulent separated flows in a channel with streamwise periodic hill-shaped constriction. The simulations are conducted at a Reynolds number of 10 595 based on the hill height and the bulk velocity magnitude above the crest, where the standard benchmark was presented with abundant experimental and numerical data. The flow statistical properties are discussed in detail, including mean velocities, Reynolds stresses, anisotropy measures, and spectra, which are in good agreement with the available numerical and experimental data in the literature. It is demonstrated that the SVV-iLES model performs at least as well as the established explicit models and therefore, the high-order spectral/hp element method via the calibrated model-free iLES is well-prepared for highly resolved wall-bounded turbulent simulations with large-scale separations and certainly for industrial complex flows.

Weakly nonlinear instability of annular viscous sheets

A second-order perturbation analysis has been performed on the nonlinear temporal instability of para-sinuous disturbances on annular viscous sheets moving in an inviscid stationary gas medium. The mathematical expressions of second-order interface disturbances, velocity, and pressure have been derived. The nonlinear instability of annular viscous sheets has several characteristics which differ from that of planar viscous sheets: (1) both the first-order interface disturbances and the second-order interface disturbances contribute to breakup; (2) the zero-wavenumber component of interface disturbances in the second-order solution is nonzero; (3) the second-order interface disturbance is para-varicose in most cases, but para-sinuous for some cases. As with planar viscous sheets, it was found that viscosity plays a dual role in the nonlinear instability of annular viscous sheets. However, with the decrease in the ratio of inner radius to sheet thickness, the interval between the upper and the lower critical Reynolds numbers shrinks, and when the ratio of inner radius to sheet thickness is less than a certain value, the dual effect of viscosity vanishes.

Advection Diffusion Equations with Sobolev Velocity Field

In this note we study advection-diffusion equations associated to incompressible $$W^{1,p}$$ velocity fields with $$p>2$$ . We present new estimates on the energy dissipation rate and we discuss applications to the study of upper bounds on the enhanced dissipation rate, lower bounds on the $$L^2$$ norm of the density, and quantitative vanishing viscosity estimates. The key tools employed in our argument are a propagation of regularity result, coming from the study of transport equations, and a new result connecting the energy dissipation rate to regularity estimates for transport equations. Eventually we provide examples which underline the sharpness of our estimates.

Vanishing viscosity limit of the 3D incompressible Oldroyd-B model

Consider the vanishing viscosity limit of the 3D incompressible Oldroyd-B model. It is shown that this set of equations admits a unique global solution with small analytic data uniformly in the coupling parameter ω close to 1 that corresponds to the inviscid case. We justify the limit from the Oldroyd-B model to the inviscid case ω=1 for all time. Moreover, if the nonlinear term gα(τ,∇u) is ignored, similar results hold without resorting to the analytic regularity.

Stability of planar rarefaction waves under general viscosity perturbation of the isentropic Euler system

We consider the vanishing viscosity limit for a model of a general non-Newtonian compressible fluid in Rd, d=2,3. We suppose that the initial data approach a profile determined by the Riemann data generating a planar rarefaction wave for the isentropic Euler system. Under these circumstances the associated sequence of dissipative solutions approaches the corresponding rarefaction wave strongly in the energy norm in the vanishing viscosity limit. The result covers the particular case of a linearly viscous fluid governed by the Navier–Stokes system.

Second order differentiation formula on RCD*$(K,N)$ spaces

The aim of this paper is to prove a second order differentiation formula for H^{2,2} functions along geodesics in RCD* (K,N) spaces with K ıin \mathbb R and N < \infty . This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity. We establish this result by showing that W_2 -geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolations. In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain: Finally, the techniques adopted in this paper can be used to show that in the RCD setting the viscous solution of the Hamilton–Jacobi equation can be obtained via a vanishing viscosity method, as in the smooth case. With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality.

Global Bounded Weak Entropy Solutions to the Euler--Vlasov Equations in Fluid-Particle System

In this paper, we consider a fluid-particle system which describes the evolution of a two-phase flow. The system consists of the compressible Euler equations for the fluid (fluid phase) coupled with the Vlasov equation for the particles (disperse phase) through the drag force. We obtain a global bounded weak entropy solution for such one-dimensional Euler--Vlasov equations with arbitrarily large initial data for the whole range of physical adiabatic exponents $\gamma>1$. To achieve this, we apply the vanishing viscosity method and compensated compactness theory. We construct globally defined approximate solutions by adding our novel viscosity terms to the Euler equations, which together with our key observation on relative velocity plays a fundamental role in the hardest part of our proof: the uniform $L^\infty$ estimate. After deriving an entropy dissipation estimate, we prove the convergence of approximate solutions by the compensated compactness argument.

Asymptotic stability of shock profiles and rarefaction waves under periodic perturbations for 1-D convex scalar viscous conservation laws

This paper studies the asymptotic stability of shock profiles and rarefaction waves under space-periodic perturbations for one-dimensional convex scalar viscous conservation laws. For the shock profile, we show that the solution approaches the background shock profile with a constant shift in the $ L^\infty(\mathbb{R}) $ norm at exponential rates. The new phenomena contrasting to the case of localized perturbations is that the constant shift cannot be determined by the initial excessive mass in general, which indicates that the periodic oscillations at infinities make contributions to this shift. And the vanishing viscosity limit for the shift is also shown. The key elements of the poof consist of the construction of an ansatz which tends to two periodic solutions as $ x \rightarrow \pm\infty, $ respectively, and the anti-derivative variable argument, and an elaborate use of the maximum principle. For the rarefaction wave, we also show the stability in the $ L^\infty(\mathbb{R}) $ norm.

Spectral and modal energy transfer analyses of LES using the discontinuous Galerkin method and their application to the Variational Multiscale approach

In this work we perform a-priori analyses of the Discontinuous Galerkin (DG) Variational Multiscale (VMS) method for Large Eddy Simulation (LES). An analytical framework is introduced to study the ideal energy transfer between resolved and unresolved scales. The proposed framework is consistent with the discretization employed for the DG-LES simulations. The concept of modal eddy viscosity is also introduced which can be employed for the a-priori analysis of the DG-VMS method or spectral vanishing viscosity approaches. The developed framework is then applied to the analysis of the energy transfer in DG-LES by employing a DNS database of the Taylor-Green Vortex (TGV) at Re = 5000, 20000 and 40000. A-priori analyses are carried out for the Vreman [1] and all-all [2] variants of the DG-VMS approach. The performed analysis demonstrates that when the DG-LES resolution limit falls at the beginning of the dissipation range the assumption of large scales free of interaction with the unresolved scales is valid and the DG-VMS approach can replicate the ideal SGS dissipation spectrum. For coarser resolutions, typical of LES at high Reynolds numbers, the DG-VMS approach is unable to replicate the ideal energy transfer mechanism at the large-resolved scales. It is shown, a-priori, that a more accurate agreement can be obtained by employing a mixed Smagorinsky and DG-VMS approach with a fixed value of the scale-fraction parameter.

Existence and Large Time Behavior of Entropy Solutions to One-Dimensional Unipolar Hydrodynamic Model for Semiconductor Devices with Variable Coefficient Damping

In this paper, we investigate the global existence and large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Possion equations with time and spacedependent damping in a bounded interval. Firstly, we prove the existence of entropy solutions through vanishing viscosity method and compensated compactness framework. Based on the uniform estimates of density, we then prove the entropy solutions converge to the corresponding unique stationary solution exponentially with time. We generalize the existing results to the variable coefficient damping case.

Vanishing viscosity limit for the compressible Navier\u2013Stokes system via measure-valued solutions

We identify a class of measure-valued solutions of the barotropic Euler system on a general (unbounded) spatial domain as a vanishing viscosity limit for the compressible Navier–Stokes system. Then we establish the weak (measure-valued)–strong uniqueness principle, and, as a corollary, we obtain strong convergence to the Euler system on the lifespan of the strong solution.

Uniqueness and stability of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large family of Navier–Stokes systems

We prove the uniqueness and stability of entropy shocks to the isentropic Euler systems among all vanishing viscosity limits of solutions to associated Navier–Stokes systems. To take into account the vanishing viscosity limit, we show a contraction property for any large perturbations of viscous shocks to the Navier–Stokes system. The contraction estimate does not depend on the strength of the viscosity. This provides a good control on the inviscid limit process. We prove that, for any initial value, there exist a vanishing viscosity limit to solutions of the Navier–Stokes system. The convergence holds in a weak topology. However, this limit satisfies some stability estimates measured by the relative entropy with respect to an entropy shock. In particular, our result provides the uniqueness of entropy shocks to the shallow water equation in a class of inviscid limits of solutions to the viscous shallow water equations.

A generalized one phase Stefan problem as a vanishing viscosity limit

We study the vanishing viscosity limit of a nonlinear di¤usion equation describing chemical reaction interface or the spatial segregation interface of competing species, where the di¤usion rate for the negative part of the solution converges to zero. As in the standard one phase Stefan problem, we prove that the positive part of the solution converges uniformly to the solution of a generalized one phase Stefan problem. This information is then employed to determine the limiting equation for the negative part, which is an ordinary differential equation.