Semi‐implicit schemes for transient Navier–Stokes equations and eddy viscosity models

This study presents two computational schemes for the numerical approximation of solutions to eddy viscosity models as well as transient Navier–Stokes equations. The eddy viscosity model is one example of a class of Large Eddy Simulation models, which are used to simulate turbulent flow. The first approximation scheme is a first order single step method that treats the nonlinear term using a semi‐implicit discretization. The second scheme employs a two step approach that applies a Crank–Nicolson method for the nonlinear term while also retaining the semi‐implicit treatment used in the first scheme. A finite element approximation is used in the spatial discretization of the partial differential equations. The convergence analysis for both schemes is discussed in detail, and numerical results are given for two test problems one of which is the two dimensional flow around a cylinder. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Stabilized Crouzeix‐Raviart element for the Darcy‐Stokes problem

We stabilize the nonconforming Crouzeix‐Raviart element for the Darcy‐Stokes problem with terms motivated by a discontinuous Galerkin approach. Convergence of the method is shown, also in the limit of vanishing viscosity. Finally, some numerical examples verifying the theoretical predictions are presented. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 21, 2005.     

Numerical time schemes for an ocean‐related system of PDEs

In this article we consider a system of equations related to the δ‐primitive equations of the ocean and the atmosphere, linearized around a stratified flow, and we supplement the equations with transparent boundary conditions. We study the stability of different numerical schemes and we show that for each case, letting the vertical viscosity δ go to 0, the stability conditions are the same as the classical CFL conditions for the transport equation. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Web‐spline‐based finite element approximation of some quasi‐newtonian flows: Existence‐uniqueness and error bound

This article deals with the web‐spline‐based finite element approximation of quasi‐Newtonian flows. First, we consider the scalar elliptic p‐Laplace problem. Then, we consider quasi‐Newtonian flows where viscosity obeys power law or Carreau law. We prove well‐posedness at the continuous as well as the discrete level. We give some error bounds for the solution of quasi‐Newtonian flow problem based on the web‐spline method. Finally, we provide the numerical results for the p‐Laplace problem. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 54–77, 2015     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

A linearized heat‐conducting fluid and its quasi‐static approximation

A linearized flow of a compressible inviscid heat‐conducting fluid is considered and a comparison is made with its coupled/quasi‐static approximation. Copyright © 2005 John Wiley & Sons, Ltd.     

Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

The r‐Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two‐level finite element approximation schemes applied to the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity, where r is the order of the power‐law artificial viscosity term. In the two‐level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single‐step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two‐level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide rigorous numerical analysis of the two‐level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H, fine mesh size h, and filter radius δ. We also investigate the two‐level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward‐facing step at Reynolds number Re = 5100. In all numerical tests, the two‐level algorithm was proven to achieve the same order of accuracy as the standard one‐level algorithm, at a fraction of the computational cost. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Pedestrian flows and non‐classical shocks

We present a model for the flow of pedestrians that describes features typical of this flow, such as the fall due to panic in the outflow of people through a door. The mathematical techniques essentially depend on the use of non‐classical shocks in scalar conservation laws. Copyright © 2005 John Wiley & Sons, Ltd.     

Invariant solution for an axisymmetric turbulent free jet using a conserved vector

An axisymmetric turbulent free jet described by an effective viscosity, which is the sum of the kinematic viscosity and the kinematic eddy viscosity, is investigated. The conservation laws of the jet are derived using the multiplier method. A second conserved vector, in addition to the elementary conserved vector, exists provided the effective viscosity has a special form. The Lie point symmetry associated with the elementary conserved vector is obtained and used to generate the invariant solution. The analytical solution is derived when the effective viscosity depends only on the distance along the jet. The numerical solution is obtained when the effective viscosity depends also on the distance across the jet. The eddy viscosity causes an apparent increase in the viscosity of the mean flow which produces an increase in the width of the jet due to an increase in diffusion and also a decrease in the maximum mean velocity along the axis of the jet.     

On measure‐valued solutions to a two‐dimensional gravity‐driven avalanche flow model

This paper concerns measure‐valued solutions for the two‐dimensional granular avalanche flow model introduced by Savage and Hutter. The system is similar to the isentropic compressible Euler equations, except for a Coulomb–Mohr friction law in the source term. We will partially follow the study of measure‐valued solutions given by DiPerna and Majda. However, due to the multi‐valued nature of the friction law, new more sensitive measures must be introduced. The main idea is to consider the class of x‐dependent maximal monotone graphs of non‐single‐valued operators and their relation with 1‐Lipschitz, Carathéodory functions. This relation allows to introduce generalized Young measures for x‐dependent maximal monotone graph. Copyright © 2005 John Wiley & Sons, Ltd.     

One time‐step finite element discretization of the equation of motion of two‐fluid flows

We discretize in space the equations obtained at each time step when discretizing in time a Navier‐Stokes system modelling the two‐dimensional flow in a horizontal pipe of two immiscible fluids with comparable densities, but very different viscosities. At each time step the system reduces to a generalized Stokes problem with nonstandard conditions at the boundary and at the interface between the two fluids. We discretize this system with the “mini‐element” and establish error estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A dynamic subgrid-scale model for the large eddy simulation of stratified flow

A new dynamic subgrid-scale (SGS) model, including subgrid turbulent stress and heat flux models for stratified shear flow is proposed by using Yoshizawa’s eddy viscosity model as a base model. Based on our calculated results, the dynamic subgrid-scale model developed here is effective for the large eddy simulation (LES) of stratified turbulent channel flows. The new SGS model is then applied to the large eddy simulation of stratified turbulent channel flow under gravity to investigate the coupled shear and buoyancy effects on the near-wall turbulent statistics and the turbulent heat transfer at different Richardson numbers. The critical Richardson number predicted by the present calculation is in good agreement with the value of theoretical analysis     

A two‐level method in space and time for the Navier‐Stokes equations

A two‐level method in space and time for the time‐dependent Navier‐Stokes equations is considered in this article. The approximate solution u_M∈H_M is decomposed into the large eddy component v∈H_m(m < M) and the small eddy component w∈H. We obtain the large eddy component v by solving a standard Galerkin equation in a coarse‐level subspace H_m with a time step length k, whereas the small eddy component w is derived by solving a linear equation in an orthogonal complement subspace H with a time step length pk, where p is a positive integer. The analysis shows that our two‐level scheme has long‐time stability and can reach the same accuracy as the standard Galerkin method in fine‐level subspace H_M for an appropriate configuration of p and m. Moreover, some numerical examples are provided to complement our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Lorenz‐gauged vector potential formulations for the time‐harmonic eddy‐current problem with L∞‐regularity of material properties

In this paper, we consider some Lorenz‐gauged vector potential formulations of the eddy‐current problem for the time‐harmonic Maxwell equations with material properties having only L^∞‐regularity. We prove that there exists a unique solution of these problems, and we show the convergence of a suitable finite element approximation scheme. Moreover, we show that some previously proposed Lorenz‐gauged formulations are indeed formulations in terms of the modified magnetic vector potential, for which the electric scalar potential is vanishing. Copyright © 2007 John Wiley & Sons, Ltd.     

Effect of variable viscosity and viscous dissipation on the Hagen‐Poiseuille flow and entropy generation

In this article, the steady‐state flow of a Hagen‐Poiseuille modelin a circular pipe is considered and entropy generation due tofluid friction and heat transfer is examined. Because of variationin fluid viscosity, the entropy generation in the flow varies. Inhis model, Arrhenius law is applied for temperature equation‐dependent viscosity, and the influence of viscosity parameters on the entropy generation number and distribution of temperature and velocity is investigated. The governing momentum and energy equations, which are coupled due to the dissipative term in the energy equation, were solved by analytical techniques. The solutions of equations via perturbation method and homotopy perturbation method are obtained and then compared with those of numerical solutions. It is found that the fluid viscosity influences considerably the temperature distribution in the fluid close to the pipe wall, and increasing pipe wall temperature enhances the rate of entropy generation. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 529–540, 2011     

A steady stratified purely azimuthal flow representing the Antarctic Circumpolar Current

We construct an explicit steady stratified purely azimuthal flow for the governing equations of geophysical fluid dynamics. These equations are considered in a setting that applies to the Antarctic Circumpolar Current, accounting for eddy viscosity and forcing terms.     

Viscous two‐fluid flows in perturbed unbounded domains

Two stationary plane free boundary value problems for the Navier‐Stokes equations are studied. The first problem models the viscous two‐fluid flow down a perturbed or slightly distorted inclined plane. The second one describes the viscous two‐fluid flow in a perturbed or slightly distorted channel. For sufficiently small data and under certain conditions on parameters the solvability and uniqueness results are proved for both problems. The asymptotic behaviour of the solutions is investigated. For the second problem an example of nonuniqueness is constructed. Computational results of flow problems that are very close to the above problems are presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of V‐ and F‐cycle multigrid methods for the biharmonic problem using the Hsieh‐Clough‐Tocher element

Multigrid V‐ and F‐cycle algorithms for the biharmonic problem using the H‐C‐T element are studied in the article. We show that the contraction numbers can be uniformly improved by increasing the number of smoothing steps. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Global C∞‐solutions to 1D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we consider the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries. The initial density ρ₀∈W^1,2n is bounded below away from zero and the initial velocity u₀∈L²ⁿ. The viscosity coefficient µ is proportional to ρ^θ with 0<θ?1, where ρis the density. The existence and uniqueness of global solutions in Hⁱ([0,1])(i = 1,2,4) have been established in (J. Math. Phys. 2009; 50 :023101; Meth. Appl. Anal. 2005; 12 :239–252; J. Differ. Equations 2008; 245:3956–3973; Commun. Pure Appl. Anal. 2008; 7 :373–381). By mathematical induction method, we will establish the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries when the initial data ρ₀ and u₀ are smooth. Copyright © 2011 John Wiley & Sons, Ltd.