Prediction of Local Losses of Low <Emphasis Type="Italic">Re</Emphasis> Flows in Non-uniform Media Composed of Parrallelpiped Structures

A method is presented to predict the local losses of low Re flow through a porous matrix composed of layers of orthogonally oriented parallelepipeds for which the local geometry varies discreetly in the direction of bulk flow. In each layer, the variations in the pore lengths perpendicular to and parallel to the direction of bulk flow are restricted to be proportional to one another so that the variation in the geometry of each layer may be characterized by a single parameter, \(\beta \). The solutions to the Navier–Stokes equations are determined for flows through geometries that vary in a forward expansion about this parameter. These provide the data used in the development of a correlation that is able to directly relate local hydraulic permeability to the variation in local pore geometry. In this way, the local pressure losses (as well as the relationship between the volumetric flow rate and the total pressure drop) may be determined without requiring the explicit solution of the entire flow field. Test cases are presented showing that the correlation predicts the local pressure losses to be within 0.5% of the losses determined from the numerical solution to the Navier–Stokes equations. When the magnitude of the variation to the geometry is such that the change in the parameter \(\beta \) between layers is constant throughout the medium, a reduced form of the correlation (requiring the evaluation of only three constants) is able to provide predictions of flow rate and interface pressures that agree to within about 1% with the results of the numerical solutions to the Navier–Stokes equations.     

Viscous fluid flow induced by rotational-oscillatory motion of a porous sphere

Viscous fluid flow induced by rotational-oscillatorymotion of a porous sphere submerged in the fluid is determined. The Darcy formula for the viscous medium drag is supplementedwith a term that allows for the medium motion. The medium motion is also included in the boundary conditions. Exact analytical solutions are obtained for the time-dependent Brinkman equation in the region inside the sphere and for the Navier–Stokes equations outside the body. The existence of internal transverse waves in the fluid is shown; in these waves the velocity is perpendicular to the wave propagation direction. The waves are standing inside the sphere and traveling outside of it. The particular cases of low and high oscillation frequencies are considered.     

Implicit application of non‐reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd.     

On the Inviscid Limit for the Compressible Navier–Stokes System in an Impermeable Bounded Domain

In this paper we investigate the issue of the inviscid limit for the compressible Navier–Stokes system in an impermeable fixed bounded domain. We consider two kinds of boundary conditions. The first one is the no-slip condition. In this case we extend the famous conditional result (Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on nonlinear partial differential equations, vol. 2, pp. 85–98. Math. Sci. Res. Inst. Publ., Berkeley 1984) obtained by Kato in the homogeneous incompressible case. Kato proved that if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes then the solutions of the incompressible Navier–Stokes equations converge to some solutions of the incompressible Euler equations in the energy space. We provide here a natural extension of this result to the compressible case. The other case is the Navier condition which encodes that the fluid slips with some friction on the boundary. In this case we show that the convergence to the Euler equations holds true in the energy space, as least when the friction is not too large. In both cases we use in a crucial way some relative energy estimates proved recently by Feireisl et al. in J. Math. Fluid Mech. 14:717–730 (2012).     

Three-dimensional viscous jet flow with plane free boundaries

The problem of steady three-dimensional viscous flow with plane free boundaries, induced by a linear source or sink, is solved. The nonuniqueness of the solution in the case of a source and its vanishing in the case of a sink, as the Reynolds number reaches a certain critical value, is proved. The problem is investigated within the framework of the known class of the exact solutions of Navier–Stokes equations generalized in this study.     

Three-dimensional reduced Navier–Stokes solutions for subsonic separated and non-separated flows,using a global pressure relaxation procedure

The reduced Navier–Stokes and thin layer approximations to the Navier–Stokes equations are used to obtain solutions for viscous subsonic three-dimensional flows. A spatial marching method is combined with a direct sparse matrix solver to obtain successive solutions in a global relaxation process. Results have been obtained for flow fields with and without regions of flow reversal.     

Fourth‐order compact formulation of Navier–Stokes equations and driven cavity flow at high Reynolds numbers

A new fourth‐order compact formulation for the steady 2‐D incompressible Navier–Stokes equations is presented. The formulation is in the same form of the Navier–Stokes equations such that any numerical method that solve the Navier–Stokes equations can easily be applied to this fourth‐order compact formulation. In particular, in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601 × 601. Using this formulation, the steady 2‐D incompressible flow in a driven cavity is solved up to Reynolds number with Re = 20 000 fourth‐order spatial accuracy. Detailed solutions are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

L'équation de Navier–Stokes avec mémoire et le transfert de quantité de mouvement dans un milieu poreuxThe Navier–Stokes equation with memory,and momentum transfer in a porous medium

This paper is concerned with the momentum transfer in a porous medium. The equations for the continuous equivalent medium are written by the averaging method but the closure is obtained using an extended thermodynamics. The resulting model corresponds to the Navier–Stokes equation, in which the force exerted by the solid matrix on the fluid satisfies of first order differential equation. This Navier–Stokes equation model with memory generalises the Darcy model with an integro-differential term. To cite this article: O. Séro-Guillaume, D. Calogine, C. R. Mecanique 330 (2002) 383–389.     

An exact non‐linear Navier–Stokes compressible‐flow solution for CFD code verification

An exact similarity solution of the compressible‐flow Navier–Stokes equations is presented, which embeds supersonic, transonic, and subsonic regions. Describing the viscous and heat‐conducting high‐gradient flow in a shock wave, the solution accommodates non‐linear temperature‐dependent viscosity as well as heat‐conduction coefficients and provides the variation of all the flow variables and their derivatives. Also presented are methods to obtain time‐dependent and/or multi‐dimensional solutions as well as verification benchmarks of increasing severity. Comparisons between the developed analytical solution and CFD solutions of the Navier–Stokes equations, with determination of convergence rates and orders of accuracy of these solutions, illustrate the utility of the developed exact solution for verification purposes. Copyright © 2012 John Wiley & Sons, Ltd.     

From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations,Existence of Global Weak Solution for the Quasi-Solutions

We consider the compressible Navier–Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier–Stokes equations using solutions of the pressureless Navier–Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667–674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are \({C^{\infty}}\) on \({(0,T)\times \mathbb{R}^{N}}\) for any \({T > 0}\). Finally we show the convergence of the global weak solution of compressible Navier–Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to \({\mu(\rho)=\rho^{\alpha}}\) with \({\alpha > 1}\). Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density \({\rho_{0}}\).     

Numerical investigation of premixed combustion in a porous burner with integrated heat exchanger

In this paper, we perform a numerical analysis of a two-dimensional axisymmetric problem arising in premixed combustion in a porous burner with integrated heat exchanger. The physical domain consists of two zones, porous and heat exchanger zones. Two dimensional Navier–Stokes equations, gas and solid energy equations, and chemical species transport equations are solved and heat release is described by a multistep kinetics mechanism. The solid matrix is modeled as a gray medium, and the finite volume method is used to solve the radiative transfer equation to calculate the local radiation source/sink in the solid phase energy equation. Special attention is given to model heat transfer between the hot gas and the heat exchanger tube. Thus, the corresponding terms are added to the energy equations of the flow and the solid matrix. Gas and solid temperature profiles and species mole fractions on the burner centerline, predicted 2D temperature fields, species concentrations and streamlines are presented. Calculated results for temperature profiles are compared to experimental data. It is shown that there is good agreement between the numerical solutions and the experimental data and it is concluded that the developed numerical program is an excellent tool to investigate combustion in porous burner.     

A kinetic theory solution method for the Navier–Stokes equations

The kinetic-theory-based solution methods for the Euler equations proposed by Pullin and Reitz are here extended to provide new finite volume numerical methods for the solution of the unsteady Navier–Stokes equations. Two approaches have been taken. In the first, the equilibrium interface method (EIM), the forward- and backward-flowing molecular fluxes between two cells are assumed to come into kinetic equilibrium at the interface between the cells. Once the resulting equilibrium states at all cell interfaces are known, the evaluation of the Navier–Stokes fluxes is straightforward. In the second method, standard kinetic theory is used to evaluate the artificial dissipation terms which appear in Pullin's Euler solver. These terms are subtracted from the fluxes and the Navier–Stokes dissipative fluxes are added in. The new methods have been tested in a 1D steady flow to yield a solution for the interior structure of a shock wave and in a 2D unsteady boundary layer flow. The 1D solutions are shown to be remarkably accurate for cell sizes large compared to the length scale of the gradients in the flow and to converge to the exact solutions as the cell size is decreased. The steady-state solutions obtained with EIM agree with those of other methods, yet require a considerably reduced computational effort.     

Interactive boundary layer models for channel flow

In this paper, the aim is to present the results of a new approach for the asymptotic modeling of two-dimensional steady, incompressible, laminar flows in a channel. More precisely, for high Reynolds numbers, the walls of the channel are deformed in such a way that separation is possible. Of course, numerical solutions of Navier–Stokes equations can be calculated but it is believed that an asymptotic analysis helps in the understanding of the flow structure. Numerical solutions of Navier–Stokes equations are compared with solutions of asymptotic models included in a more general model called global interactive boundary layer.     

Reduction in drag and vortex shedding frequency through porous sheath around a circular cylinder

A numerical study on the laminar vortex shedding and wake flow due to a porous‐wrapped solid circular cylinder has been made in this paper. The cylinder is horizontally placed, and is subjected to a uniform cross flow. The aim is to control the vortex shedding and drag force through a thin porous wrapper around a solid cylinder. The flow field is investigated for a wide range of Reynolds number in the laminar regime. The flow in the porous zone is governed by the Darcy–Brinkman–Forchheimer extended model and the Navier–Stokes equations in the fluid region. A control volume approach is adopted for computation of the governing equations along with a second‐order upwind scheme, which is used to discretize the convective terms inside the fluid region. The inclusion of a thin porous wrapper produces a significant reduction in drag and damps the oscillation compared with a solid cylinder. Dependence of Strouhal number and drag coefficient on porous layer thickness at different Reynolds number is analyzed. The dependence of Strouhal number and drag on the permeability of the medium is also examined. Copyright © 2009 John Wiley & Sons, Ltd.     

A two‐phase adaptive finite element method for solid–fluid coupling in complex geometries

In this paper we present a method to solve the Navier–Stokes equations in complex geometries, such as porous sands, using a finite‐element solver but without the complexity of meshing the porous space. The method is based on treating the solid boundaries as a second fluid and solving a set of equations similar to those used for multi‐fluid flow. When combined with anisotropic mesh adaptivity, it is possible to resolve complex geometries starting with an arbitrary coarse mesh. The approach is validated by comparing simulation results with available data in three test cases. In the first we simulate the flow past a cylinder. The second test case compares the pressure drop in flow through random packs of spheres with the Ergun equation. In the last case simulation results are compared with experimental data on the flow past a simplified vehicle model (Ahmed body) at high Reynolds number using large‐eddy simulation (LES). Results are in good agreement with all three reference models. Copyright © 2010 John Wiley & Sons, Ltd.     

Flow past a cylinder and a sphere in a porous medium within the framework of the Brinkman equation with the Navier boundary condition

Exact analytical solutions of the problem of flow past a sphere and a cylinder in a porous medium are derived within the framework of the Brinkman equationwith the Navier boundary condition. Attention is drawn to the fact that the no-slip condition imposed on the interface between the porous medium and a solid, used, in particular, in the case of the Brinkman equation, must be in the general case replaced by a condition that admits nonzero flow velocity at the boundary.     

A General Approach to Time Periodic Incompressible Viscous Fluid Flow Problems

This article develops a general approach to time periodic incompressible fluid flow problems and semilinear evolution equations. It yields, on the one hand, a unified approach to various classical problems in incompressible fluid flow and, on the other hand, gives new results for periodic solutions to the Navier–Stokes–Oseen flow, the Navier–Stokes flow past rotating obstacles, and, in the geophysical setting, for Ornstein–Uhlenbeck and various diffusion equations with rough coefficients. The method is based on a combination of interpolation and topological arguments, as well as on the smoothing properties of the linearized equation.     

Implementation of high‐order compact finite‐difference method to parabolized Navier–Stokes schemes

The numerical solution to the parabolized Navier–Stokes (PNS) and globally iterated PNS (IPNS) equations for accurate computation of hypersonic axisymmetric flowfields is obtained by using the fourth‐order compact finite‐difference method. The PNS and IPNS equations in the general curvilinear coordinates are solved by using the implicit finite‐difference algorithm of Beam and Warming type with a high‐order compact accuracy. A shock‐fitting procedure is utilized in both compact PNS and IPNS schemes to obtain accurate solutions in the vicinity of the shock. The main advantage of the present formulation is that the basic flow variables and their first and second derivatives are simultaneously computed with the fourth‐order accuracy. The computations are carried out for a benchmark case: hypersonic axisymmetric flow over a blunt cone at Mach 8. A sensitivity study is performed for the basic flowfield, including profiles and their derivatives obtained from the fourth‐order compact PNS and IPNS solutions, and the effects of grid size and numerical dissipation term used are discussed. The present results for the flowfield variables and also their derivatives are compared with those of other basic flow models to demonstrate the accuracy and efficiency of the proposed method. The present work represents the first known application of a high‐order compact finite‐difference method to the PNS schemes, which are computationally more efficient than Navier–Stokes solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

High velocity flow through fractured and porous media: the role of flow non-periodicity

In the present paper we examine the evolution of the macroscopic flow law in a crenellated channel, representing an element of fractured or porous medium and in function of the Reynolds number Re. A numerical analysis based on the Navier–Stokes equations is applied. We focus on the influence of the flow periodicity or non-periodicity upon the macroscopic law. The physical explanation of the non-linear deviation from Darcy's law is still an issue, as the Ergun–Forchheimer law admitted for high Reynolds numbers comes up against some theoretical problems. In the periodic case, three non-linear flow regimes were revealed: a cubic flow with respect to velocity at low Re, an intermediate non-quadratic law, and a self-similar mode independent of Re at very high Re. The Forchheimer law is not confirmed. The case of a non-periodic flow clearly highlights the link between the flow non-periodicity and the quadratic law. The quadratic deviation becomes all the more important as the non-periodicity degree is high.     

An algorithm for rotating axisymmetric flows: model,validation and industrial applications

The study of axisymmetric flows is of interest not only from an academic point of view, due to the existence of exact solutions of Navier–Stokes equations, but also from an industrial point of view, since these kind of flows are frequently found in several applications. In the present work the development and implementation of a finite element algorithm to solve Navier–Stokes equations with axisymmetric geometry and boundary conditions is presented. Such algorithm allows the simulation of flows with tangential velocity, including free surface flows, for both laminar and turbulent conditions. Pseudo‐concentration technique is used to model the free surface (or the interface between two fluids) and the k–ε model is employed to take into account turbulent effects. The finite element model is validated by comparisons with analytical solutions of Navier–Stokes equations and experimental measurements. Two different industrial applications are presented. Copyright © 2005 John Wiley & Sons, Ltd.