Turbulent kinetic energy distribution across the interface between a porous medium and a clear region

For hybrid media, involving both a porous substrate and an unobstructed flow region, difficulties arise due to the proper mathematical treatment given at the macroscopic interface. The literature proposes a jump condition in which shear stresses on both sides of the interface are not of the same value. This paper presents numerical solutions for such hybrid medium, considering here a channel partially filled with a porous layer through which an incompressible fluid flows in turbulent regime. Here, diffusion fluxes of both momentum and turbulent kinetic energy across the interface present a discontinuity in their values, which is based on a certain jump coefficient. Effects of such parameter on mean and turbulence fields around the interface region are numerically investigated. Results indicate that depending on the value of the stress jump parameter, a substantially different structure for the turbulent field is obtained.     

Turbulent flow over a layer of a highly permeable medium simulated with a diffusion-jump model for the interface

Flow over a finite porous medium is investigated using different interfacial conditions. In such configuration, a macroscopic interface is identified between the two media. In the first model, no diffusion-flux is considered when treating the statistical energy balance at the interface. The second approach assumes that diffusion fluxes of turbulent kinetic energy on both sides of the interface are unequal. Comparing these two models, this paper presents numerical solutions for such hybrid medium, considering here a channel partially filled with a porous layer through which fluid flows in turbulent regime. One unique set of transport equations is applied to both regions. Effects of Reynolds number, porosity, permeability and jump coefficient on mean and turbulence fields are investigated. Results indicate that depending on the value of the stress jump parameter, substantially dissimilar fields for the turbulence energy are obtained. Negative values for the stress jump parameter give results closer to experimental data for the turbulent kinetic energy at the interface.     

Turbulent flow in a channel occupied by a porous layer considering the stress jump at the interface

For hybrid media, involving both a porous structure and a clear flow region, difficulties arise due to the proper mathematical treatment given at the interface. The literature proposes a jump condition in which shear stresses on both sides of the interface are not of the same value. This paper presents numerical solutions for such hybrid medium, considering here a channel partially filled with a porous layer through which fluid flows in turbulent regime. One unique set of transport equations is applied to both regions. Effects of Reynolds number, porosity, permeability and jump coefficient on mean and turbulence fields are investigated. Results indicate that depending on the value of the stress jump parameters, a substantially different structure for the turbulent field is obtained.     

EFFECTS OF THERMAL DISPERSION AND TURBULENCE IN FORCED CONVECTION IN A COMPOSITE PARALLEL-PLATE CHANNEL: INVESTIGATION OF CONSTANT WALL HEAT FLUX AND CONSTANT WALL TEMPERATURE CASES

In this article, a composite parallel-plate channel whose central portion is occupied by a clear fluid and whose peripheral portion is occupied by a fluid saturated porous medium, is considered. The flow in the porous region of the channel is assumed to be laminar, governed by the Brinkman-Forchheimer-extended Darcy equation, while the flow in the clear fluid region of the channel is assumed to be turbulent. The validity of this laminar/turbulent assumption is validated by estimating Reynolds numbers in the clear fluid and porous regions of the channel. Although the flow in the porous region remains laminar, it is still fast enough for the quadratic drag (Forchheimer) effects to be important. In this situation, hydrodynamic mixing of the interstitial fluid at the pore scale becomes important and may cause significant thermal dispersion. It is shown that thermal dispersion may result in some counterintuitive effects, such as the increase of the Nusselt number when the width of the clear fluid region in the center of the channel is decreased.     

岛屿地貌单元的湍流能耗物理模型试验

岛屿地貌单元是珠江三角洲发育演变过程中的沉积核心,研究其消能机制,对理解河口动力过程及三角洲发育演变有重要意义。通过建立岛屿地貌单元的湍流能耗特性概化物理模型,基于16 MHz ADV采集高频流速数据,统计了时均及湍流特征量,并利用惯性耗散法分析了岛屿地貌单元的湍流动能耗率。结果表明,相同控制条件下岛屿地貌单元的形态阻力致使尾流中紊动强度量值为明渠的2~3倍,湍流剪切应力及湍流动能较明渠水流的大近1个数量级,湍流动能耗散率比明渠水流湍流动能耗散率大1~2个数量级。岛屿地貌单元的局部形态阻力导致尾流时均流速的空间梯度、切应力增大是湍流能耗率增大的原因。岛屿地貌单元的汇流作用增加了下游尾流区的水流掺混,并在尾流区域形成大量微尺度涡,导致区域湍流能耗作用增强,有利于岛屿沉积核心发育。研究成果有助于理解河口动力及三角洲的发育演变过程。     

Hot Dates

For this article, an analytical study has been conducted on the flow and energy transfer of unsteady compressible oscillating flow through channels filled with porous medium representing stack in thermoacoustic engines/refrigerators. The flow in the porous material is described by the Darcy momentum equation. Analytical expressions for oscillating velocity, temperature in the porous layer, complex Nusselt number, and energy flux density are obtained after simplifying and solving the governing differential equations with reasonable approximations (such as long wave, short stack, small amplitude oscillation, etc.). The result for heat transfer between the porous medium and the channel wall is expressed as a dimensionless Nusselt number. For the limiting case of nonporous medium, the Nusselt number obtained in the present study matches quantitatively with the expression available in the existing literature. The results reveal that the Nusselt number in oscillating flow is significantly enhanced (almost an order of magnitude) by employing sufficiently large thermal conductivity of porous media in a channel. The system of equations developed in the present study is a helpful tool for thermal engineers to design porous stacks for thermoacoustic devices.     

Investigation of a Confined Laminar Impinging Jet on a Plate with a Porous Layer Using the Preconditioned Density-Based Algorithm

The preconditioned density-based algorithm and two-domain approach were used to investigate the fluid flow and heat transfer characteristics of a confined laminar impinging jet on a plate covered with porous layer. In the porous zone, the momentum equations were formulated by the Darcy-Brinkman-Forchheimer model; the thermal nonequilibrium model was adopted for the energy equation. At the porous/fluid interface, the applicability and influence of different hydrodynamic and thermal interfacial conditions were analyzed for the problem. The governing equations were solved by the preconditioned density-based finite-volume method, with preconditioning matrix for equations of porous domain adopted, aiming to eliminate the equation stiffness of porous seepage flows. The effects of Reynolds number, porosity, Darcy number, thermal conductivity ratio, Biot number, and porous layer thickness on the flow pattern and local heat transfer performance were studied. Results indicate that the Reynolds number and porosity don't strongly influence the flow pattern of porous channel, while the Darcy number and porous layer thickness have obvious influence on the flow pattern. The heat transfer performance are greatly influenced by the parameters studied.     

Transient and non-Darcian effects on natural convection flow in a vertical channel partially filled with porous medium: Analysis with Forchheimer–Brinkman extended Darcy model

The present study addresses the transient as well as non-Darcian effects on laminar natural convection flow in a vertical channel partially filled with porous medium. Forchheimer–Brinkman extended Darcy model is assumed to simulate momentum transfer within the porous medium. Two regions are coupled by equating the velocity and shear stress in the case of momentum equation while matching of the temperature and heat flux is taken for thermal energy equation. Approximate solutions are obtained using perturbation technique. Variations in velocity field with Darcy number, Grashof number, kinematic viscosity ratio, distance of interface and variations in temperature distribution with thermal conductivity ratio, distance of interface are obtained and depicted graphically. The skin-friction and rate of heat transfer at the channel walls are also derived and the numerical values for various physical parameters are tabulated.     

Forced convection in a channel partly occupied by a bidisperse porous medium: Asymmetric case

This paper presents an analytic investigation of forced convection in parallel-plate channel partly occupied by a bidisperse porous medium and partly by a fluid clear of solid material, the distribution being asymmetrical. The walls of the channel are subject to an uniform heat flux; the flow is assumed to be hydrodynamically and thermally fully developed. The layer of a bidisperse porous medium is attached to one of the channel walls; it is modeled utilizing a two-velocity two-temperature formulation using Darcy’s law. The Beavers–Joseph boundary condition is employed at the bidisperse porous medium/clear fluid interface. The dependences of the Nusselt number on a conductivity ratio, a velocity ratio, a volume fraction, internal heat exchange parameter, and the position of the porous-fluid interface are investigated. Both cases of symmetric and asymmetric heating are investigated, which is specified by the asymmetry heating parameter introduced here. For the case of asymmetric heating, a singular behavior of the Nusselt number is found and explained.     

Vortices within the Separation Region in a Decelerating Channel Flow （Effect of Inlet Velocity Gradient）

An experimental investigation was made into three-dimensional separated flow and the vortices within the flow separation in a decelerating channel flow generated by the suction from a porous side wall. The flows along the side and bottom walls were visualized by the surface tuft method. The turbulent internal flow was measured by the split-film probe to investigate the turbulent flow including the reverse flow. In the flow visualization for the strong decelerating flow (the suction flow ratio:0.8), two typical flow patterns appear alternatively. One is that the flow near the bottom wall separates more upstream than the flow near the top wall and a clockwise vortex can be seen in the separation region. Another is the reversal flow pattern with a counterclockwise vortex. By the turbulent flow measurement using the split-film probe, two peaks of turbulence level are observed for the strong decelerating flow case. These peaks can be related with two flow patterns mentioned above.     

Large eddy simulation of turbulent flow in porous media

An LES (large eddy simulation) study was conducted using one of standard numerical models for a porous medium, namely, a flow through a periodic array of square cylinders. The LES results were processed to extract macroscopic results such as the macroscopic turbulent kinetic energy and the macroscopic pressure gradient. These macroscopic results are compared against those obtained using conventional models of turbulent kinetic energy and its dissipation rate, so as to examine the validity of extending the conventional two equation models of turbulence to the flow in porous media. The spectrum of turbulence was also examined to appreciate the onset of turbulence.     

Self-heating in a bioreactor: Coupling of heat and mass transfer with turbulent convection

A tridimensional mathematical model that considers the turbulent flow inside a bioreactor of municipal waste material (MWM) is studied. The MWM is self-heating by the biological activity of micro-organisms. The model includes the unsteady solution of the turbulent flow field inside the reactor, the energy transport in the flow and the ensuing heat and mass transfer by convection over the MWM load. The MWM is treated as a porous medium and the oxygen depletion and heat generation produced by aerobic bacteria are also included. Turbulence treatment is done with the κ–ε model. The numerical simulation shows good agreement with experimental measurements available. The results also shed light over some design criteria that can be explored in order to increase the efficiency of the process.     

Investigations of Heat Transfer Enhancement in a Channel with Staggered Porous Ribs by the Preconditioned Density-Based Algorithm

The preconditioned density-based algorithm and two-domain approach were used to investigate the fluid flow and heat transfer characteristics in a channel with staggered porous/solid ribs. In the porous zone, the momentum equations were formulated by the Darcy–Brinkman–Forchheimer model; and the local thermal equilibrium (LTE) model was adopted for energy equation. At the porous/fluid interface, the stress–continuity interfacial condition was utilized. The governing equations are solved by the preconditioned density-based control-volume method, with preconditioning matrix for equations of porous domain adopted, aiming to eliminate the equation stiffness of the porous seepage flow. The effects of Reynolds number, geometry parameters of ribs (rib length and thickness), and physical property of porous media (permeability and porosity) on the flow pattern and heat transfer performance were analyzed. Results indicate that, compared with that of solid ribs, the recirculating bubble behind the porous ribs is completely detached from it because of the permeability of porous media, and the size of the recirculating bubble is suppressed. The parameters that would affect the mass flow of fluid penetrating the porous ribs, including permeability, Reynolds number, baffle length and thickness, have remarkable influence on the flow pattern. All the aforementioned parameters would affect the local heat transfer performance.     

Analysis of 2D flow and heat transfer modeling in fracture of porous media

Heat and mass transfer between porous media and fluid is a complex coupling process,which is widely used in various fields of engineering applications,especially for natural and artificial fractures in oil and gas extraction.In this study,a new method is proposed to deal with the flow and heat transfer problem of steady flow in a fracture.The fluid flow in a fracture was described using the same method as Mohais,who considered a fracture as a channel with porous wall,and the perturbation method was used to solve the mathematical model.Unlike previous studies,the shear jump boundary condition proposed by Ochoa-Tapia and Whitaker was used at the interface between the fluid and porous media.The main methods were perturbation analysis and the application of shear jump boundary conditions.The influence of permeability,channel width,shear jump degree and effective dynamic viscosity on the flow and heat transfer in the channel was studied by analysing the analytical solution.The distribution of axial velocity in the channel with the change of the typical parameters and the sensitivity of the heat transfer was obtained.     

Simulation of a Turbulent Impinging Jet into a Layer of Porous Material Using a Two–Energy Equation Model

This article presents numerical results for a turbulent jet impinging against a flat plane covered with a layer of permeable and thermally conducting material. Distinct energy equations are considered for the solid porous material attached to the wall and for the fluid that impinges on it. Parameters such as Reynolds number, porosity, permeability, thickness, and thermal conductivity of the porous layer are varied in order to analyze their effects on the local distribution of Nu. The macroscopic equations for mass, momentum, and energy are obtained based on volume-average concept. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted nonorthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Results indicate that inclusion of a porous layer eliminates the peak in Nu at the stagnation region. For highly porous and highly permeable material, simulations indicate that the integral heat flux from the wall is enhanced when a thermally conducting porous material is attached to the surface.     

Turbulent heat transfer past a sudden expansion with a porous insert using a nonlinear model

This work presents numerical investigations for turbulent flow and heat transfer in a backward-facing step with and without porous inserts. Two classes of the model were employed, namely linear and nonlinear turbulence closures. The entire set of transport equations was discretized by means of the control volume method and the system of algebraic equations obtained was relaxed using the SIMPLE (Semi Implicit Pressure-Linked Equations) method. Results were first validated against the experimental data and the simulations follow experimental values and trends. Computations further indicated that when using the porous insert, the size, shape, and length of the recirculating region were drastically reduced in addition to being pushed toward the channel exit, leading eventually to a complete bubble suppression for thicker inserts. A more permeable medium gave better results in quickly suppressing the circulatory motions. By including porous inserts in the channel, turbulence generated due to the shear inside the recirculating region was damped, whereas high levels of k were concentrated within the permeable structure. Large variations for the skin friction factor along the bottom wall were also smoothed out by placing inserts, spanning from a typical distribution for an unobstructed back-step flow to a standard parallel channel flow distribution as the inserts got ticker. On the other hand, at the upper wall, flow pushed toward the top surface gave rise to a sudden increase of the skin friction factor, which was later stabilized downstream the flow. Heat transfer analysis followed showing damping for Nu at the bottom wall as the thickness of the porous substrate was increased. Overall, the thickness of the insert played a dominant role in changing the final flow and heat transfer characteristics rather than the porosity or permeability of the porous material. Finally, this work indicated that the sudden increase of Nu around the reattachment point, known to be undesirable in many practical situations for causing additional thermomechanical loads on the surface, may by avoided by the use of a porous obstacle past the back-step.     

A rule of minimal energy dissipation in modeling of channel flow turbulence

ABSTRACT

A model of turbulence is proposed to solve Reynolds equations for fully developed flow in a wall-bounded straight channel. We show that for the channel flow the Reynolds number can be defined as a ratio of flow kinetic energy to the work of friction/dissipation forces. Then, we introduce a turbulent Reynolds number as a balance between energy losses due to the momentum exchange by turbulent vortices traveling from lowto high-velocity areas and wall friction. The main idea of the model is expressed in the following phenomenological law: The minimal energy dissipation rule requires that a local deformation of the axial velocity profile can and, in the presence of finite-size instabilities, should generate turbulence with such intensity that it keeps the local turbulent Reynolds number below the critical value. Thus, the only empirical parameter in the model is the critical Reynolds number.

The model is applied to several basic channel flows such as the fully developed flow in a circular tube, in an infinite plane channel, and in an annulus. The application of the minimal energy dissipation rule requires an additional integral equation, and this can be considered as an integral-equation algebraic model of turbulence.     

On the intrinsic nature of jump coefficients at the interface between a porous medium and a free fluid region

In this paper, we discuss the physical nature of the jump parameters that generally appear in the expression for the jump conditions at a fluid/porous interface. These jump parameters are generally thought of as intrinsic interfacial properties, just like surface tension in the case of fluid/fluid interfaces. Based on a two-step up-scaling analysis, we show that jump parameters can be interpreted as surface-excess quantities. The value of a surface-excess quantity is shown to depend linearly on the position of the discontinuous interface and is therefore not an intrinsic property. We propose a theoretical approach that allows to introduce genuine intrinsic interfacial properties and to propose a best choice for the position of the discontinuous interface. We show that these properties are tightly related to the definition of the interfacial zone. This theoretical approach is successfully assessed on three important cases: a laminar flow parallel to a fluid/porous interface, a turbulent flow perpendicular to a porous/fluid interface and heat transfer perpendicular to a fluid/porous interface. It is believed that this approach is general enough to be applied to any interfacial transport phenomenon.     

The effect of porous insert position on the enhanced heat transfer in partially filled channels

In the present work, the effect of porous insert position on enhanced heat transfer in a parallel-plate channel partially filled with a fluid-saturated porous medium was studied. The fully-developed laminar flow and convective heat transfer in the channel were simulated using Lattice Boltzmann Method (LBM). The walls of the channel were subject to a uniform constant temperature. The flow field and thermal performance of the channel were investigated and compared for two configurations: first the porous insert was attached to the channel walls, and second the same amount of the porous material was positioned in the channel core. Comparing the results of the present study to the analytical solutions, a reasonable agreement was observed. The effects of various parameters like Darcy number, porous medium thickness, etc. on the conduit thermal performance were investigated in both channel configurations. It was found that the position of the porous insert has significant influence on the thermal performance of the channel.     

Numerical simulation of turbulent combustion in porous materials

This paper presents one-dimensional simulations of combustion of an air/methane mixture in porous materials using a model that explicitly considers the intra-pore levels of turbulent kinetic energy. Transport equations are written in their time-and-volume-averaged form and a volume-based statistical turbulence model is applied to simulate turbulence generation due to the porous matrix. Four different thermo-mechanical models are compared, namely Laminar, Laminar with Radiation Transport, Turbulent, Turbulent with Radiation Transport. Combustion is modeled via a unique simple closure. Preliminary testing results indicate that a substantially different temperature distribution is obtained depending on the model used. In addition, for high excess air peak gas temperature is reduced and the flame front moves towards the exit of the burner. Also, increasing the inlet flow rate for stoichiometric mixture pushes the flame out of the porous material.