Numerical simulation of the flow around a porous covering square cylinder in a channel via lattice Boltzmann method

In this paper, a detailed investigation on the flow past a porous covering cylinder is presented through the lattice Boltzmann method. The Brinkman‐Forchheimer‐extended Darcy model is adopted for the entire flow field with the solid, fluid, and porous medium. The effects of several parameters, such as porous layer thickness, Darcy number, porosity, and Reynolds number on flow field are discussed. Compared with the case of a solid cylinder, the present work shows that the porous layer may play an important role on the flow, the lift and drag force exerted on the cylinder. The numerical results indicate that the maximal drag coefficient C_d and maximal amplitude of lift coefficient C_l exist at certain Darcy number which is in the range of 10^?6–10^?2. Copyright © 2010 John Wiley & Sons, Ltd.     

多孔介质壁面剪切湍流速度时空关联的研究

作为一个基础统计量,时空关联函数在湍流问题的研究中有着广泛的应用,是研究湍流噪声、湍流中物质扩散和大涡模拟亚格子模型等问题的重要参考.本文通过建立三维多孔结构壁面剪切湍流模型,采用含Darcy-Brinkman-Forchheimer作用力项的格子Boltzmann方程对无穷大多孔介质平行板之间壁湍流进行了数值模拟,进而研究其速度脉动时空关联函数的统计特性.一方面,根据计算得到的流场数据,对比分析了常规槽道湍流与多孔介质壁面槽道湍流的时间关联函数.另一方面,计算并讨论了不同孔隙率和渗透率的多孔介质壁面对速度脉动时空关联性的影响.通过研究表明:多孔结构壁面剪切湍流的时空关联函数等值线与椭圆理论相符;在研究参数范围内,多孔介质壁面的速度时空关联系数随着孔隙率增大而增大,随着渗透率增大而减小.同时发现在槽道壁面的近壁区、过渡区、对数律区和中心区等不同位置处,速度时空关联呈现较大差异性:越远离壁面位置(对数律区和中心区),其时空关联函数所呈现的关联等值线椭圆越细长,高值相关等值线越集中.多孔介质主要改变速度时空关联椭圆图像的椭圆率,说明多孔介质壁面主要影响湍流横扫速度.     

Lattice Boltzmann simulation of MHD natural convection in a cavity with porous media and sinusoidal temperature distribution

The lattice Boltzmann method (LBM) is used to simulate the effect of magnetic field on the natural convection in a porous cavity. The sidewalls of the cavity are heated sinusoidally with a phase derivation, whereas the top and bottom walls are thermally insulated. Numerical simulation is performed, and the effects of the pertinent parameters, e.g., the Hartmann number, the porosity, the Darcy number, and the phase deviation, on the fluid flow and heat transfer are investigated. The results show that the heat transfer is affected by the temperature distribution on the sidewalls clearly. When the Hartmann number is 0, the maximum average Nusselt number is obtained at the phase deviation 90°. Moreover, the heat transfer enhances when the Darcy number and porosity increase, while decreases when the Hartman number increases.     

Direct numerical simulation of turbulent heat transfer over fully resolved anisotropic porous structures

Lattice Boltzmann direct numerical simulations of turbulent heat transfer over and inside anisotropic porous media are performed. This study considers turbulent plane channel flows whose bottom walls are made from the porous media at the bulk Reynolds number of 2900 with isothermal and conjugate heat transfer wall conditions. Four different porous walls are considered. They are walls with only the wall-normal permeability, with the wall-normal and spanwise permeabilities, with the wall-normal and streamwise permeabilities, and with the isotropic wall-normal, spanwise and streamwise permeabilities. The porosity of the porous walls ranges from 0.6 to 0.8. Discussions on the effects of the anisotropic permeability on turbulent thermal fields are carried out by the instantaneous flow visualizations and the statistical quantities. In particular, temperature fluctuations, turbulent and dispersion heat fluxes are examined both inside and outside the porous walls. Finally, the heat transfer performance is discussed considering the effects of the anisotropic permeability.     

不同壁面绕流特性的格子Boltzmann模拟研究

为了探讨不同壁面的绕流特性,针对粘性流场中,不同壁面诱导的涡脱落现象以及升阻力系数等流场特性进行了格子Boltzmann数值研究。利用基于分子动理论的格子Boltzmann方法(LBM)求解Navier-Stokes方程,实现对流体运动的描述,针对不同的壁面条件,分别采用不同的格子Boltzmann流-固壁面处理方法。采用Half-way反弹边界条件来处理平直壁面,而曲壁面则采用LBM与有限差分法相结合的形式进行处理,计入了壁面与标准网格不重合对结果造成的影响。开发相应的计算程序,计算结果与已发表文献结果吻合良好,验证了数值模型的正确性。同时,探讨了进出口边界与钝体中心的距离对结果的影响。对比分析了不同壁面的绕流模型中升阻力系数、斯托罗哈数和涡量云图等,并进一步研究了雷诺数条件的影响。结果表明,不同壁面的绕流特性具有明显差异,且同时受雷诺数的显著影响;一般地,平直壁面对于来流作出的响应更迅速。     

Mixed Convection in a Vertical Porous Channel

A numerical study of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman–Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left and right walls of the channel and kept either at equal or at different temperatures. The governing equations are solved numerically by finite difference method with Southwell–Over–Relaxation technique for extended Darcy model and analytically using perturbation series method for Darcian model. The velocity and temperature fields are obtained for various porous parameter, inertia effect, product of Brinkman number and Grashof number and the ratio of Grashof number and Reynolds number for equal and different wall temperatures. Nusselt number at the walls is also determined for three types of thermal boundary conditions. The viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. The Darcy and inertial drag terms suppress the flow. It is found that analytical and numerical solutions agree very well for the Darcian model. An erratum to this article is available at .     

Flow instability in a curved porous channel formed by two concentric cylindrical surfaces

Stability of laminar flow in a curved channel formed by two concentric cylindrical surfaces is investigated. The channel is occupied by a fluid saturated porous medium; the flow in the channel is driven by a constant azimuthal pressure gradient. The momentum equation takes into account two drag terms: the Darcy term that describes friction between the fluid and the porous matrix, and the Brinkman term, which allows imposing the no-slip boundary condition at the channel walls. An analytical solution for the basic flow velocity is obtained. Numerical analysis is carried out using the collocation method to investigate the onset of instability leading to the development of a secondary motion in the form of toroidal vortices. The dependence of the critical Dean number on porosity and the channel width is analyzed.     

On the flow of fluids through inhomogeneous porous media due to high pressure gradients

Most porous solids are inhomogeneous and anisotropic, and the flows of fluids taking place through such porous solids may show features very different from that of flow through a porous medium with constant porosity and permeability. In this short paper we allow for the possibility that the medium is inhomogeneous and that the viscosity and drag are dependent on the pressure (there is considerable experimental evidence to support the fact that the viscosity of a fluid depends on the pressure). We then investigate the flow through a rectangular slab for two different permeability distributions, considering both the generalized Darcy and Brinkman models. We observe that the solutions using the Darcy and Brinkman models could be drastically different or practically identical, depending on the inhomogeneity, that is, the permeability and hence the Darcy number.     

Numerical study of Prandtl effect on MHD flow at a lid‐driven porous cavity

In this paper, the lattice Boltzmann method is used to study the Prandtl number effect on flow structure and heat transfer rates in a magnetohydrodynamic flow mixed convection in a lid‐driven cavity filled with a porous medium. The right and left walls are at constant but different temperatures (θ_h and θ_c), while the other walls are adiabatic. Gallium and salt water (0.02 < Pr < 13.4) are used as samples of the electroconducting fluids in the cavity. Typical sets of streamlines and isotherms are presented to analyze the flow patterns set up by the competition among the forced flow created by the lid‐driven wall, the buoyancy force of the fluid and the magnetic force of the applied magnetic field. Mathematical formulations in the porous media were constructed based on the Brinkman–Forchheimer model, while the multidistribution‐function model was used for the magnetic field effect. Numerical results were obtained and the effects of the Prandtl number and the other effective parameters such as Richardson, Hartman, and Darcy numbers were investigated. It was found that the fluid fluctuations within the cavity were reduced by increasing the Hartman number. A similar pattern was observed for the Darcy number reduction. Heat transfer was essentially dominated by the conduction for the low Prandtl number and forced convection dominated as the Prandtl number increased. Also, the average Nusselt number was raised by increasing the Prandtl number. It was discovered that a remarkable heat transfer enhancement of up to 28% could be reached by increasing the Prandtl number (from 0.02 to 13.4) at constant Richardson and Darcy numbers. Copyright © 2011 John Wiley & Sons, Ltd.     

Vortex structure of turbulence over permeable walls

In order to understand the effect of the wall permeability on the turbulent vortex structure near porous walls, based on PIV experimental data, a probability density analysis of fluctuating velocities, statistical quadrant and quadrant-hole analyses of the Reynolds shear stress are performed. The investigated flow fields are turbulent channel flows whose bottom walls are made of porous media. The porous media used are three kinds of foamed ceramics which have almost the same porosity (∼0.8) but different permeability. From the discussions on those analyses, a conceptual scenario of the development of the vortex structure near a permeable wall is proposed for a moderate permeability Reynolds number case. It explains the reason why the near-wall long streaky structure tends to vanish near a porous wall with increasing wall permeability.     

The simulation of turbulent particle‐laden channel flow by the Lattice Boltzmann method

We perform direct numerical simulation of three‐dimensional turbulent flows in a rectangular channel, with a lattice Boltzmann method, efficiently implemented on heavily parallel general purpose graphical processor units. After validating the method for a single fluid, for standard boundary layer problems, we study changes in mean and turbulent properties of particle‐laden flows, as a function of particle size and concentration. The problem of physical interest for this application is the effect of water droplets on the turbulent properties of a high‐speed air flow, near a solid surface. To do so, we use a Lagrangian tracking approach for a large number of rigid spherical point particles, whose motion is forced by drag forces caused by the fluid flow; particle effects on the latter are in turn represented by distributed volume forces in the lattice Boltzmann method. Results suggest that, while mean flow properties are only slightly affected, unless a very large concentration of particles is used, the turbulent vortices present near the boundary are significantly damped and broken down by the turbulent motion of the heavy particles, and both turbulent Reynolds stresses and the production of turbulent kinetic energy are decreased because of the particle effects. We also find that the streamwise component of turbulent velocity fluctuations is increased, while the spanwise and wall‐normal components are decreased, as compared with the single fluid channel case. Additionally, the streamwise velocity of the carrier (air) phase is slightly reduced in the logarithmic boundary layer near the solid walls. Copyright © 2015 John Wiley & Sons, Ltd.     

Analysis of fluid flow and heat transfer in a channel with intermittent heated porous blocks

A numerical study of forced convection enhancement in a channel intermittently heated is presented in this work. The use of porous blocks mounted on the heated parts of the channel to improve thermal performance is investigated. In order to account for the inertia, drag and boundary effects, the Brinkman-Forchheimer-extended Darcy model is adopted for the flow inside the porous regions. The effects of several parameters such as Darcy number, the block dimensions, the number of blocks and the thermal conductivity ratio are documented. The results show that the blocks may alter substantially the flow pattern depending on the permeability of the porous medium, and may improve the heat transfer and reduce the wall temperature under certain circumstances. Received on 9 June 1997     

Direct numerical simulation of turbulence over resolved and modeled rough walls with irregularly distributed roughness

To simulate turbulent flow over a rough wall without resolving complicated rough geometries, a macroscopic rough wall model is developed based on spatial (plane) averaging theory. The plane-averaged drag force term, which arises through averaging the Navier–Stokes equations in a plane parallel to a rough wall, can be modeled using a plane porosity and a plane hydraulic diameter. To evaluate the developed model, direct and macroscopic model simulations for turbulence over irregularly distributed semi-spheres at Reynolds number of 300 are carried out using the D3Q27 multiple-relaxation time lattice Boltzmann method. The results show that the developed model can be used to predict rough wall skin friction. The results agree quantitatively with standard turbulence statistics such as mean velocity and Reynolds stress profiles with the fully resolved DNS data. Since velocity dispersion occurs inside the rough wall and is found to contribute to turbulence energy dissipation, which the developed model cannot account for, the developed model fails to reproduce dispersion-related turbulence energy dissipation. However, it is found that the plane-averaged drag force term can successfully recover the deficiency of dispersion-related turbulence energy dissipation.     

Free-convection between vertical walls partially filled with porous medium

This paper presents an analytical study of laminar fully developed free-convection flow between two vertical walls partially filled with porous matrix and partially with a clear fluid having interface vertically. The momentum transfer in porous medium is described by the Brinkman-extended Darcy model and the two regions are coupled by equating the velocity and shear stress at the interface. The governing equations having non-linear nature have been solved by using perturbation method. It has been found that effect of Brinkman term is in entire porous domain for large values of Darcy number while its effect is confined nearer to interface and wall for small values of Darcy number. Received on 19 March 1997     

Non-Darcy double-diffusive natural convection in axisymmetric fluid saturated porous cavities

Double-diffusive natural convection in a fluid saturated porous medium has been investigated using the finite element method. A generalised porous medium model is used to study both Darcy and non-Darcy flow regimes in an axisymmetric cavity. Results indicate that the Darcy number should be a separate parameter to understand flow characteristics in non-Darcy regime. The influence of porosity on heat and mass transfer is significant and the transport rates may differ by 25% or more, at higher Darcy and Rayleigh numbers. When compared with the Darcy and other specialised models of Brinkman and Forchheimer, the present generalised model predicts the least heat and mass transfer rates. It is also observed that an increase in radius ratio leads to higher Nusselt and Sherwood numbers along the inner wall.     

Analysis of Flow and Heat Transfer in a Vertical Rectangular Duct Using a Non-Darcy Model

Numerical investigation of steady natural convection flow through a fluid-saturated porous medium in a vertical rectangular duct is investigated. The Darcy-Forchheimer-Brinkman model is used to represent the fluid transport within the porous medium. One of the vertical walls of the duct is cooled to a constant temperature, while the other wall is heated to constant but different temperature. The other two sides of the duct are insulated. The finite difference method of second-order accuracy is used to solve the non-dimensional governing equations. The results are presented graphically to show the effects of the Darcy number, inertial parameter, Grashof number, Brinkman number, aspect ratio, and viscosity ratio. It is found that an increase in the Darcy number and inertial parameter reduces the flow intensity whereas an increase in the Grashof number, Brinkman number, aspect ratio, and viscosity ratio increases the flow intensity.     

An analysis on forced convection in a channel filled with a Brinkman-Darcy porous medium: Exact and approximate solutions

An analysis was made to investigate non-Darcian fully developed flow and heat transfer in a porous channel bounded by two parallel walls subjected to uniform heat flux. The Brinkmanextended Darcy model was employed to study the effect of the boundary viscous frictional drag on hydrodynamic and heat transfer characteristics. An exact expression has been derived for the Nusselt number under the uniform wall heat flux condition. Approximate results were also obtained by exploiting a momentum integral relation and an auxiliary relation implicit in the Brinkmanextended Darcy model. Excellent agreement was confirmed between the approximate and exact solutions even in details of velocity and temperature profiles.     

Prediction of Non-Darcy Coefficients for Inertial Flows Through the Castlegate Sandstone Using Image-Based Modeling

Near wellbore flow in high rate gas wells shows the deviation from Darcy??s law that is typical for high Reynolds number flows, and prediction requires an accurate estimate of the non-Darcy coefficient (?? factor). This numerical investigation addresses the issues of predicting non-Darcy coefficients for a realistic porous media. A CT-image of real porous medium (Castlegate Sandstone) was obtained at a resolution of 7.57???m. The segmented image provides a voxel map of pore-grain space that is used as the computational domain for the lattice Boltzmann method (LBM) based flow simulations. Results are obtained for pressure-driven flow in the above-mentioned porous media in all directions at increasing Reynolds number to capture the transition from the Darcy regime as well as quantitatively predict the macroscopic parameters such as absolute permeability and ?? factor (Forchheimer coefficient). Comparison of numerical results against experimental data and other existing correlations is also presented. It is inferred that for a well-resolved realistic porous media images, LBM can be a useful computational tool for predicting macroscopic porous media properties such as permeability and ?? factor.     

Effect of an Inserted Porous Layer Located at a Wall of a Parallel Plate Channel on Forced Convection Heat Transfer

A theoretical study is performed on heat and fluid flow in partially porous medium filled parallel plate channel. A uniform symmetrical heat flux is imposed onto the boundaries of the channel partially filled with porous medium. The dimensional forms of the governing equations are solved numerically for different permeability and effective thermal conductivity ratios. Then, the governing equations are made dimensionless and solved analytically. The results of two approaches are compared and an excellent agreement is observed, indicating correctness of the both solutions. An overall Nusselt number is defined based on overall thermal conductivity and difference between the average temperature of walls and mean temperature to compare heat transfer in different channels with different porous layer thickness, Darcy number, and thermal conductivity ratio. Moreover, individual Nusselt numbers for upper and lower walls are also defined and obtained. The obtained results show that the maximum overall Nusselt number is achieved for thermal conductivity ratio of 1. At specific values of Darcy number and thermal conductivity ratio, individual Nusselt numbers approach to infinity since the value of wall temperatures approaches to mean temperature.     

Effects of wall permeability on turbulence

In order to understand the effects of the wall permeability on turbulence near a porous wall, flow field measurements are carried out for turbulent flows in a channel with a porous bottom wall by a two-component particle image velocimetry (PIV) system. The porous media used are three kinds of foamed ceramics which have almost the same porosity (0.8) but different permeability. It is confirmed that the flow becomes more turbulent over the porous wall and tends to be turbulent even at the bulk Reynolds number of Re_b=1300 in the most permeable wall case tested. Corresponding to laminar to turbulent transition, the magnitude of the slip velocity on the porous wall is found to increase drastically in a narrow range of the Reynolds number. To discuss the effects of the wall roughness and the wall permeability, detailed discussions are made of zero-plane displacement and equivalent wall roughness for porous media. The results clearly indicate that the turbulence is induced by not only the wall roughness but the wall permeability. The measurements have also revealed that as Re_b or the wall permeability increases, the wall normal fluctuating velocity near the porous wall is enhanced due to the effects of the wall permeability. This leads to the increase of the turbulent shear stress resulting in higher friction factors of turbulence over porous walls.