The effect of an unsteady drag force on the structure of a non-equilibrium region behind a shock wave in a gas-particle mixture

For numerical analysis of shock wave propagation in gas-particle mixtures, drag coefficients of a sphere in steady flows are generally used. However, it is shown both experimentally and numerically that a shock loaded solid sphere experiences unsteady drag forces. The paper describes a model of unsteady drag force and its effect on the structure of the non-equilibrium region behind a shock front traveling in a dusty gas. The results are compared with those obtained by using a steady drag coefficient and are discussed. It is demonstrated that the large drag force at the early stage of the interaction between shock-wave induced flow and a solid particle affects the flow structure that is obtained with a steady drag force.      

Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid Past a Vertical Plate Immersed in a Porous Medium

The effects of viscous dissipation on unsteady free convection from an isothermal vertical flat plate in a fluid saturated porous medium are examined numerically. The Darcy–Brinkman–Forchheimer model is employed to describe the flow field. A new model of viscous dissipation is used for the Darcy–Brinkman–Forchheimer model of porous media. The simultaneous development of the momentum and thermal boundary layers are obtained by using a finite difference method. Boundary layer and Boussinesq approximation have been incorporated. Numerical calculations are carried out for various parameters entering into the problem. Velocity and temperature profiles as well as local friction factor and local Nusselt number are shown graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficient approach steady state.     

Pressure drop and friction factor of steady and oscillating flows in open-cell porous media

Open-cell metal foams are often used in heat exchangers and absorption equipment because they exhibit large specific surface area and present tortuous coolant flow paths. However, published research works on the characteristics of fluid flow in metal foams are relatively scarce, especially for the flow oscillation condition. The present experimental investigation attempts to uncover the behavior of steady and oscillating flows through metal foams with a tetrakaidecahedron structure. In the experiments, steady flow was supplied by an auto-balance compressor and flow oscillation was provided by an oscillating flow generator. The pressure drop and velocity were measured by the differential pressure transducer and hot-wire sensor, respectively. The friction factor of steady flow in metal foam channel was analyzed through the permeability and inertia coefficient of the porous medium. The results show that flow resistance in the metal foams increases with increasing form coefficient and decreasing permeability. The empirical equation obtained by the present study indicates that the maximum friction factor of oscillating flow through the tested aluminum foams with specific structure is governed by the hydraulic ligament diameter-based kinetic Reynolds number and the dimensionless flow amplitude.     

幂律流体圆柱绕流的格子波尔兹曼模拟简

基于插值补充格子波尔兹曼方法和幂律流体的本构方程,建立了贴体坐标系下适用于幂律流体的格子波尔兹曼模型,模拟了幂律流体的圆柱绕流问题,采用非平衡外推格式处理圆柱表面的速度无滑移边界,利用应力积分法确定曳力系数和升力系数,并与基于标准的格子波尔兹曼方法和有限容积法获得的数值数据进行对比,吻合良好. 进行了网格无关性验证之后,分析了稳态流动时,不同雷诺数下幂律指数对于尾迹长度、分离角、圆柱表面黏度分布、表面压力系数及曳力系数的影响,以及非定常流动中,幂律指数对于流场、曳力系数、升力系数和斯特劳哈尔数的影响. 获得的变化规律与基于其他数值模拟方法得到的结果相一致,充分验证了模型的有效性和正确性. 结果表明：插值补充格子波尔兹曼方法可以用来模拟幂律流体在具有复杂边界流场内的流动问题,通过引入不同的非牛顿流体本构方程,该方法还可以进一步应用于其他类型的非牛顿流体研究中.     

Effect of Confining Wall on Properties of Gas Flow Through Metal Foam: An Experimental Study

While the Darcy and Forchheimer relations are widely applied to determine the permeability and the form drag coefficient of open-cell metal foam, they both assume that the porous medium is infinite in all directions, i.e., large enough so that the effect of any confining walls is negligible. Many researchers, however, pay little or no attention to the size of metal foam samples in pressure-drop studies. The size of a foam sample perpendicular to the flow direction may be small enough such that wall effects are significant. This article experimentally investigates the wall effect on the permeability and form drag coefficient for two types of open-cell aluminum foam subjected to airflow entering the foam in the Forchheimer regime. The Forchheimer equation was recast in two different manners, which resulted in new non-dimensional numbers that correlated very well with the diameter of the foam samples measured in cells. The correlations are valid for a confining-tube-diameter-based Reynolds number ranging from approximately 13,000 to 105,000, and for diameters ranging from 12 to 36 cells and 24 to 60 cells for 10- and 20-pore per inch foam, respectively. The obtained correlations allow for determining pressure drop given only the velocity and the diameter of an aluminum foam sample.     

幂律流体圆柱绕流的格子波尔兹曼模拟

基于插值补充格子波尔兹曼方法和幂律流体的本构方程,建立了贴体坐标系下适用于幂律流体的格子波尔兹曼模型,模拟了幂律流体的圆柱绕流问题,采用非平衡外推格式处理圆柱表面的速度无滑移边界,利用应力积分法确定曳力系数和升力系数,并与基于标准的格子波尔兹曼方法和有限容积法获得的数值数据进行对比,吻合良好. 进行了网格无关性验证之后,分析了稳态流动时,不同雷诺数下幂律指数对于尾迹长度、分离角、圆柱表面黏度分布、表面压力系数及曳力系数的影响,以及非定常流动中,幂律指数对于流场、曳力系数、升力系数和斯特劳哈尔数的影响. 获得的变化规律与基于其他数值模拟方法得到的结果相一致,充分验证了模型的有效性和正确性. 结果表明:插值补充格子波尔兹曼方法可以用来模拟幂律流体在具有复杂边界流场内的流动问题,通过引入不同的非牛顿流体本构方程,该方法还可以进一步应用于其他类型的非牛顿流体研究中.      

多孔介质中稳定泡沫的流动计算及实验研究

将多孔介质简化为一簇变截面毛管束,根据多孔介质的颗粒直径、颗粒排列方式、孔喉尺度比及束缚水饱和度,计算出变截面毛细管的喉道半径和孔隙半径. 在考虑多孔介质喉道和孔隙中单个气泡的受力和变形基础上,利用动量守恒定理,推导出单个孔隙单元内液相的压力分布和孔隙单元两端的压差计算公式,最终得到多孔介质的压力分布计算公式. 利用长U型填砂管对稳定泡沫的流动特性进行了实验研究. 研究结果表明:稳定泡沫流动时多孔介质中的压力分布呈线性下降,影响泡沫在多孔介质中流动特性的因素包括:多孔介质的孔喉结构、泡沫流体的流量和干度、气液界面张力、气泡尺寸,其中孔喉结构和泡沫干度是影响泡沫封堵能力的主要因素.关键词: 稳定泡沫;多孔介质;变截面毛管;流动;表观粘度;压力分布;实验研究      

Drag and Separation Flow Past Solid Sphere with Porous Shell at Moderate Reynolds Number

Axisymmetric viscous, two-dimensional steady and incompressible fluid flow past a solid sphere with porous shell at moderate Reynolds numbers is investigated numerically. There are two dimensionless parameters that govern the flow in this study: the Reynolds number based on the free stream fluid velocity and the diameter of the solid core, and the ratio of the porous shell thickness to the square root of its permeability. The flow in the free fluid region outside the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by a Darcy model. Using a commercially available computational fluid dynamics (CFD) package, drag coefficient and separation angle have been computed for flow past a solid sphere with a porous shell for Reynolds numbers of 50, 100, and 200, and for porous parameter in the range of 0.025–2.5. In all simulation cases, the ratio of b/a was fixed at 1.5; i.e., the ratio of outer shell radius to the inner core radius. A parametric equation relating the drag coefficient and separation point with the Reynolds number and porosity parameter were obtained by multiple linear regression. In the limit of very high permeability, the computed drag coefficient as well as the separation angle approaches that for a solid sphere of radius a, as expected. In the limit of very low permeability, the computed total drag coefficient approaches that for a solid sphere of radius b, as expected. The simulation results are presented in terms of viscous drag coefficient, separation angles and total drag coefficient. It was found that the total drag coefficient around the solid sphere as well as the separation angle are strongly governed by the porous shell permeability as well as the Reynolds number. The separation point shifts toward the rear stagnation point as the shell permeability is increased. Separation angle and drag coefficient for the special case of a solid sphere of radius r = a was found to be in good agreement with previous experimental results and with the standard drag curve.     

A numerical study of fluids with pressure‐dependent viscosity flowing through a rigid porous medium

Much of the work on flow through porous media, especially with regard to studies on the flow of oil, are based on ‘Darcy's law’ or modifications to it, such as Darcy–Forchheimer or Brinkman models. While many theoretical and numerical studies concerning flow through porous media have taken into account the inhomogeneity and anisotropy of the porous solid, they have not taken into account the fact that the viscosity of the fluid and drag coefficient could depend on the pressure in applications, such as enhanced oil recovery (EOR). Experiments clearly indicate that the viscosity varies exponentially with respect to the pressure and the viscosity can change, in some applications, by several orders of magnitude. The fact that the viscosity depends on pressure immediately implies that the ‘drag coefficient’ would also depend on the pressure. In this paper we consider modifications to Darcy's equation wherein the drag coefficient is a function of pressure, which is a realistic model for technological applications, such as EOR and geological carbon sequestration. We first outline the approximations behind Darcy's equation and the modifications that we propose to Darcy's equation, and derive the governing equations through a systematic approach using mixture theory. We then propose a stabilized mixed finite element formulation for the modified Darcy's equation. To solve the resulting nonlinear equations we present a solution procedure based on the consistent Newton–Raphson method. We solve representative test problems to illustrate the performance of the proposed stabilized formulation. One of the objectives of this paper is to show that the dependence of viscosity on the pressure can have a significant effect both on the qualitative and quantitative nature of the solution. Copyright © 2010 John Wiley & Sons, Ltd.     

Dynamic Effects Accompanying the Flow Past Oscillating Bodies Mounted with a Narrow Clearance in Pipes

The unsteady and steady flow past a sphere mounted with a very narrow clearance in a cylindrical pipe is experimentally investigated. The unsteady flow is studied for the case of regular transverse self-oscillations of the sphere accompanied by its impact interaction with the pipe wall. In the steady flow regime the center of the sphere is fixed on the pipe axis. The dependence of the local resistance due to the presence of the sphere and of the body drag coefficient on the relevant dimensionless parameters is determined. The dynamic characteristics for the steady and unsteady regimes are compared.     

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.     

考虑气体压缩性的多孔材料渗透率和惯性系数的测定

多孔介质材料的渗透率和惯性系数是决定多孔介质中流体流动特性的重要参数，一般需要通过实验进行测定，在测定渗透律和惯性系数量，选用气体作为工作介质可以为实验带来极大的方便，然而通常的实验都将气体看作不可压缩流体，直接根据Darcy-Forchheimer定律得到这两个参数，这种近似对实验条件如样品厚度、工作压力等提出了很多要求，本文提出了在考虑气体压缩性的情况下测定渗透率和惯性系数的方法，该方法可以大大降低实验时对样品厚度、工作压力等条件的要求。本文还根据该方法对多孔材料PVF进行了渗透率和惯性系数的测定，并对测量结果进行了验证。     

Effect of momentum transfer condition at the interface of a model of creeping flow past a spherical permeable aggregate

Creeping flow past an isolated, spherical and permeable aggregate has been studied adopting the Stokes equation to model the fluid external to the aggregate and the Brinkman equation for the internal flow. At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used along with the continuity of velocity components and continuity of the normal stress. Using Faxen’s laws, drag and torque are calculated for different flow conditions and it is observed that drag and torque not only change with the permeability of the porous region, but as stress jump coefficient increases, the rate of change in behavior of drag and torque increases.     

Identification of Drag Parameters of Flow in High Permeability Materials by U-Tube Method

The identification procedure of linear and nonlinear drag parameters of flow of liquid in high permeability materials by U-tube method is presented. The experimental technique is based on control of pressure in liquid oscillating in the U-tube including porous material and direct computer data acquisition. The macroscopic model which takes into account inertial forces, gravity, and interaction of oscillating liquid with porous material and U-tube walls is elaborated. The drag parameters are determined numerically for porous foams by fitting model predictions to experimental data. The methodology incorporates calibration of the U-tube system without sample of porous material, which is a necessary step to determine independently parameters of interaction of liquid with tube walls.     

Inertial and Darcy’s Terms Ratio in Boundary Layer at Fluid–Porous Medium Interface

The study considers the forced boundary-layer flow overlying the Darcy–Brinkman porous medium and gives a quantitative analysis of the nonlinear inertial terms in the Brinkman filtration equation. The inertial terms are shown to be larger than the Darcy’s drag near the porous medium interface. The applicability range of boundary-layer approach is determined. It is suitable in high-permeable media with moderate velocities of an external flow. If it is slow enough, the inertial terms can be omitted in spite of interface effect. On the other hand, fast external flow produces the filtration with large pore-scale Reynolds number; therefore, the Forchheimer’s drag should be taken into account. It is shown the Brinkman term as well as inertial terms have a significant role in boundary-layer formation within the porous medium.     

Analytical quantification of coefficients in the Ergun equation for fluid friction in a packed bed

A new analytical derivation for momentum transport during laminar flow through granular porous media is discussed and some of its implied results described. In the very low Reynolds number regime fully developed laminar flow is assumed and in the higher laminar Reynolds number regime the Forchheimer (non-Darcy) effect is modelled through reference to form drag induced by the solid constituents of the porous medium. The results are compared to the Ergun equation, which is empirically based on experimental measurements, and the correspondence is shown to be remarkably close.     

Hydromagnetic thin film flow of Casson fluid in non-Darcy porous medium with Joule dissipation and Navier's partial slip

In this paper, the effects of viscous and Ohmic heating and heat generation/absorption on magnetohydrodynamic flow of an electrically conducting Casson thin film fluid over an unsteady horizontal stretching sheet in a non-Darcy porous medium are investigated. The fluid is assumed to slip along the boundary of the sheet. Similarity transformation is used to translate the governing partial differential equations into ordinary differential equations. A shooting technique in conjunction with the 4 th order Runge-Kutta method is used to solve the transformed equations. Computations are carried out for velocity and temperature of the fluid thin film along with local skin friction coefficient and local Nusselt number for a range of values of pertinent flow parameters. It is observed that the Casson parameter has the ability to enhance free surface velocity and film thickness, whereas the Forchheimer parameter, which is responsible for the inertial drag has an adverse effect on the fluid velocity inside the film. The velocity slip along the boundary tends to decrease the fluid velocity. This investigation has various applications in engineering and in practical problems such as very large scale integration(VLSI) of electronic chips and film coating.     

Inertia Effects in High-Rate Flow Through Heterogeneous Porous Media

The paper deals with the effects of large scale permeability–heterogeneity on flows at high velocities through porous media. The media is made of a large number of homogeneous blocks where the flow is assumed to be governed by the Forchheimer equation with a constant inertial coefficient. By assuming the validity of the Forchheimer equation at the large scale, an effective inertial coefficient is deduced from numerical simulations. Different media are investigated: serial-layers, parallel-layers and correlated media. The numerical results show that: (i) for the serial-layers, the effective inertial coefficient is independent of the Reynolds number and decreases when the variance and the mean permeability ratio increases; (ii) for the parallel-layers and the correlated media, the effective inertial coefficient is function of the Reynolds number and increases when the variance and the mean permeability ratio increases. Theoretical relationships are proposed for the inertial coefficient as function of the Reynolds number and the characteristics of the media.     

Stokes flow past an assemblage of axisymmetric porous spherical shell-in-cell models: effect of stress jump condition

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous concentric spherical shell-in-cell model is studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin and Cunningham/Mehta-Morse models are considered. At the fluid-porous interfaces, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. The hydrodynamic drag force acting on the porous shell by the external fluid in each of the four boundary conditions on the cell surface is evaluated. It is found that the normalized mobility of the particles (the hydrodynamic interaction among the porous shell particles) depends not only on the permeability of the porous shells and volume fraction of the porous shell particles, but also on the stress jump coefficient. As a limiting case, the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres with jump.