Showing papers on "Navier–Stokes equations published in 2018"
TL;DR: In this paper, a family of virtual element methods for the Navier-Stokes equations is proposed and analyzed, which provide a discrete velocity field which is pointwise divergence-free.
Abstract: A family of virtual element methods for the two-dimensional Navier--Stokes equations is proposed and analyzed. The schemes provide a discrete velocity field which is pointwise divergence-free. A ri...
150 citations
TL;DR: In this article, the authors prove global existence of appropriate weak solutions for the compressible Navier-Stokes equations for more general stress tensors than those covered by P. Lions and E. Feireisl's theory.
Abstract: We prove global existence of appropriate weak solutions for the compressible Navier–Stokes equations for more general stress tensor than those covered by P.–L. Lions and E. Feireisl’s theory. More precisely we focus on more general pressure laws which are not thermodynamically stable; we are also able to handle some anisotropy in the viscous stress tensor. To give answers to these two longstanding problems, we revisit the classical compactness theory on the density by obtaining precise quantitative regularity estimates: This requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation. These two cases open the theory to important physical applications, for instance to describe solar events (virial pressure law), geophysical flows (eddy viscosity) or biological situations (anisotropy).
136 citations
TL;DR: In this article, a stabilised and reduced Galerkin projection of the incompressible unsteady Navier-Stokes equations for moderate Reynolds number is presented, based on a finite volumes approximation.
132 citations
TL;DR: It is shown that with modifications of the function spaces in the method of Labeur and Wells it is possible to formulate a simple method with pointwise divergence-free velocity fields which is momentum conserving, energy stable, and pressure-robust.
Abstract: We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier---Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells (SIAM J Sci Comput 34(2):A889---A913, 2012). We show that with modifications of the function spaces in the method of Labeur and Wells it is possible to formulate a simple method with pointwise divergence-free velocity fields which is momentum conserving, energy stable, and pressure-robust. Theoretical results are supported by two- and three-dimensional numerical examples and for different orders of polynomial approximation.
80 citations
TL;DR: The AFiD code, an open source solver for the incompressible Navier–Stokes equations, has been ported to GPU clusters to tackle large-scale wall-bounded turbulent flow simulations and the results are in excellent agreement with the experimental and computational data.
79 citations
TL;DR: In this paper, a numerical approach was established to assess the effects of shear on the hydraulic properties of 2-D discrete fracture networks (DFNs) in both linear and nonlinear regimes.
77 citations
TL;DR: In this paper, a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus g(S) is proposed, based on a reformulation of the equation in Cartesian coordinates of the embedding R3.
Abstract: We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus g(S). The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding R3, penalization of the normal component, a Chorin projection method, and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincare-Hopf theorem on n-tori.
76 citations
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TL;DR: In this article, a class of weak Navier-Stokes equations with bounded kinetic energy, integrable vorticity, and smoothness outside a fractal set of singular times with Hausdorff dimension strictly less than 1 were studied.
Abstract: We prove non-uniqueness for a class of weak solutions to the Navier-Stokes equations which have bounded kinetic energy, integrable vorticity, and are smooth outside a fractal set of singular times with Hausdorff dimension strictly less than 1.
75 citations
TL;DR: A new embedded finite element method for the linear advection–diffusion equation and the laminar and turbulent incompressible Navier–Stokes equations, based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary.
68 citations
TL;DR: In this article, the authors employed partially-averaged Navier-Stokes (PANS) equations to simulate the flow around a smooth circular cylinder at Reynolds number 3900 and evaluated the importance of discretization and modelling errors on the accuracy of this mathematical model.
64 citations
TL;DR: The results show that the 5th-order GKS scheme can achieve the theoretical order of accuracy for the Euler equations, and present accurate Navier–Stokes solutions as well due to the coupling of inviscid and viscous terms in the GKS formulation.
TL;DR: A mixed virtual element method (mixed-VEM) for a pseudostress-velocity formulation of the two-dimensional Navier-Stokes equations with Dirichlet boundary conditions is proposed and analyzed in this paper.
Abstract: A mixed virtual element method (mixed-VEM) for a pseudostress-velocity formulation of the two-dimensional Navier–Stokes equations with Dirichlet boundary conditions is proposed and analyzed in this...
TL;DR: In this paper, a Nitsche-based cut finite element formulation for the Oseen problem is proposed, which allows the boundary of the domain to cut through the elements of an easy-to-generate background mesh.
TL;DR: In this article, the stability of boundary layer solutions of the 2-dimensional incompressible Navier-Stokes equations was investigated, and it was shown that if UBL is monotonic and concave in Y =y/ν, then uν is stable over some time interval (0,T), T independent of ν, under perturbations with Gevrey regularity in x and Sobolev regularality in y.
Abstract: We investigate the stability of boundary layer solutions of the 2-dimensional incompressible Navier–Stokes equations. We consider shear flow solutions of Prandtl type: uν(t,x,y)=(UE(t,y)+UBL(t,yν),0), 0<ν≪1. We show that if UBL is monotonic and concave in Y=y/ν, then uν is stable over some time interval (0,T), T independent of ν, under perturbations with Gevrey regularity in x and Sobolev regularity in y. We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in x and y). Moreover, in the case where UBL is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr–Sommerfeld operator.
TL;DR: Taking into account the loss of regularity suffered by the solution of the Navier–Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order O(h2)$\mathcal O( h^{2})$ in space are proved.
Abstract: This paper studies inf-sup stable finite element discretizations of the evolutionary Navier–Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier–Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order $\mathcal O(h^{2})$
in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. Both the continuous-in-time case and the fully discrete scheme with the backward Euler method as time integrator are analyzed.
13 Apr 2018
TL;DR: In this paper, robust estimates for the kinetic and dissipation energies of Navier-Stokes flows are considered. But the focus lies on robust estimates of the energy in a twofold sense: pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability.
Abstract: Inf-sup stable FEM applied to time-dependent incompressible Navier–Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure–robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption $$
abla u \in L^1(0,T;L^\infty (\varOmega ))$$
which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semi-robust estimates for pointwise divergence-free $$H^1$$
-conforming FEM (like Scott–Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.
TL;DR: In this article, a fourth-order compact gas-kinetic scheme (GKS) is developed for the compressible Euler and Navier-Stokes equations under the framework of two-stage four-order temporal discretization and Hermite WENO (HWENO) reconstruction.
TL;DR: It is proved that any weak space-time vanishing viscosity limit of a sequence of strong solutions of Navier–Stokes equations in a bounded domain of R2 satisfies the Euler equation if the solutions’ local enstrophies are uniformly bounded.
Abstract: We prove that any weak space-time \(L^2\) vanishing viscosity limit of a sequence of strong solutions of Navier–Stokes equations in a bounded domain of \({\mathbb R}^2\) satisfies the Euler equation if the solutions’ local enstrophies are uniformly bounded. We also prove that \(t-a.e.\) weak \(L^2\) inviscid limits of solutions of 3D Navier–Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second-order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.
TL;DR: In this article, the sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for binary fluids with moving contact lines are studied by asymptotic analysis and numerical simulations.
Abstract: The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for binary fluids with moving contact lines are studied by asymptotic analysis and numerical simulations. The effects of the mobility number as well as a phenomenological relaxation parameter on the boundary condition are considered. In asymptotic analysis, we consider both the cases that the mobility number is proportional to the Cahn number and the square of the Cahn number, and derive the sharp-interface limits for several set-ups of the boundary relaxation parameter. It is shown that the sharp-interface limit of the phase-field model is the standard two-phase incompressible Navier–Stokes equations coupled with several different slip boundary conditions. Numerical results are consistent with the analysis results and also illustrate the different convergence rates of the sharp-interface limits for different scalings of the two parameters.
TL;DR: In this paper, a Nitsche-based imposition of generalized Navier conditions on cut meshes for the Oseen problem is presented, which remains wellposed and stable for arbitrary choice of slip length, including zero.
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TL;DR: For the 2D steady Navier-Stokes equations in the region of the Blasius boundary layer, this paper showed that the viscosity of the 3D Prandtl layer can be approximated in constant varepsilon with no slip boundary conditions.
Abstract: Let the viscosity $\varepsilon \rightarrow 0$ for the 2D steady Navier-Stokes equations in the region $0\leq x\leq L$ and $0\leq y<\infty$ with no slip boundary conditions at $y=0$. For $L<<1$, we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in $\varepsilon$ are achieved through a fixed-point scheme: \begin{equation*} [u^{0}, v^0] \overset{\text{DNS}^{-1}}{\longrightarrow }v\overset{\mathcal{L}^{-1}}{ \longrightarrow }[u^{0}, v^0] \label{fixedpoint} \end{equation*} for solving the Navier-Stokes equations, where $[u^{0}, v^0]$ are the tangential and normal velocities at $x=0,$ DNS stands for $\partial _{x}$ of the vorticity equation for the normal velocity $v$, and $\mathcal{L}$ the compatibility ODE for $[u^{0}, v^0]$ at $x=0.$
TL;DR: In this article, a modified cubic law (MCL) was proposed for single-phase saturated laminar flow prediction in rough rock fractures, which incorporates modifications to the aperture field by considering flow tortuosity, aperture variation and local roughness effects.
TL;DR: In this paper, an improved discrete velocity method (DVM) is developed for efficient simulation of fluid problems in all flow regimes, which can effectively remove its drawbacks of low accuracy and efficiency in continuum flow regime with no deterioration of its performance in rarefied flow regime.
Abstract: In this paper, an improved discrete velocity method (DVM) is developed for efficient simulation of fluid problems in all flow regimes. Compared with the conventional explicit DVM, the present scheme could effectively remove its drawbacks of low accuracy and efficiency in continuum flow regime with no deterioration of its performance in rarefied flow regime. One of the novel strategies adopted in the new method is to introduce a prediction step for solving the macroscopic governing equation. By using the prediction step, the equilibrium state is first estimated before solving the discrete velocity Boltzmann equation (DVBE). As a result, the collision term in the DVBE can be discretized implicitly to improve the stability and efficiency of the conventional explicit DVM. Another contribution of the new method is to physically reconstruct numerical flux at the cell interface by incorporating the collision effect into the process. To maintain simplicity and efficiency of the conventional DVM, in the present scheme, the collision effect in the flux reconstruction at the cell interface is considered through the solution of the macroscopic governing equation. This can effectively control the effect of numerical dissipation in the process of updating the macroscopic flow variables in the continuum flow regime. Analyses indicate that the prediction step does not contribute to the evolution of distribution functions in the highly rarefied flow regime while dominating the solutions in the continuum flow regime. Accordingly, the improved scheme automatically converges toward the conventional explicit DVM in the free molecular flow regime and approaches the Navier-Stokes solver in the continuum flow regime.
TL;DR: An analytical model for electro-hydrodynamic flow that describes the relationship between the corona voltage, electric field, and ion charge density is presented, shedding new insights into mass, charge, and momentum transport phenomena.
Abstract: We present an analytical model for electro-hydrodynamic flow that describes the relationship between the corona voltage, electric field, and ion charge density. The interaction between the accelerated ions and the neutral gas molecules is modeled as an external body force in the Navier-Stokes equation. The gas flow characteristics are solved from conservation principles with spectral methods. This multiphysics model is shown to match experimental data for a point-to-ring corona configuration, shedding new insights into mass, charge, and momentum transport phenomena, and can be readily implemented in any numerical simulation.
TL;DR: A robust sharp-interface immersed boundary method for numerically studying high speed flows of compressible and viscous fluids interacting with arbitrarily shaped either stationary or moving rigid solids.
TL;DR: In this article, it was shown that if the initial density decays not too slow at infinity, the 2D Cauchy problem of the density-dependent Navier-Stokes equations on the whole space admits a unique global strong solution.
Abstract: We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Navier–Stokes equations with vacuum as far-field density. It is proved that if the initial density decays not too slow at infinity, the 2D Cauchy problem of the density-dependent Navier–Stokes equations on the whole space admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Furthermore, we also obtain the large time decay rates of the spatial gradients of the velocity and the pressure, which are the same as those of the homogeneous case.
TL;DR: A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem is proposed and analyzed that ensures that all the local subproblems on the fine grid can be solved in parallel.
Abstract: In this paper, based on a two-grid method and a recent local and parallel finite element method, a parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem is proposed and analyzed. This method ensures that all the local subproblems on the fine grid can be solved in parallel. Optimal error bounds of the approximate solution are obtained. Finally, numerical experiments are presented to demonstrate the accuracy and effectiveness of the proposed method.
TL;DR: This work considers the stationary Stokes and Navier--Stokes equations for viscous, incompressible flow in parameter dependent bounded domains $\mathrm{D}_T}, subject to homogeneous Dirichlet (``no-slip'')...
Abstract: We consider the stationary Stokes and Navier--Stokes equations for viscous, incompressible flow in parameter dependent bounded domains $\mathrm{D}_T$, subject to homogeneous Dirichlet (``no-slip'')...
TL;DR: In this article, it was shown that there exist nontrivial steady state weak solutions for the Navier-Stokes equations in L 2 (mathbb{T}^d) where L 2 is the space dimension.
Abstract: Consider the unforced incompressible homogeneous Navier-Stokes equations on the $d$-torus $\mathbb{T}^d$ where $d\geq 4$ is the space dimension. It is shown that there exist nontrivial steady-state weak solutions $u\in L^{2}(\mathbb{T}^d)$. The result implies the nonuniqueness of finite energy weak solutions for the Navier-Stokes equations in dimensions $d \geq 4$. And it also suggests that the uniqueness of forced stationary problem is likely to fail however smooth the given force is.