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 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.     

Turbulent flow in a composite channel

Numerical solutions for turbulent flow in a composite channel are presented. Here, a channel with a centered porous material is considered. The interface between the porous medium and the clear flow was assumed to have different transversal positions and the porous matrix was simulated with distinct permeabilities. Governing equations were discretized and solved for both domains making use of one unique numerical methodology. Increasing the size of the porous material pushes the flow outwards, increasing the levels of turbulent kinetic energy at the macroscopic interface. For high permeability media, a large amount of mechanical energy is converted into turbulence inside the porous structure.     

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.     

A new turbulence model for porous media flows. Part II: Analysis and validation using microscopic simulations

A new model for turbulent flows in porous media developed in an accompanying paper [F.E. Teruel, Rizwan-uddin, A new turbulence model for porous media flows. Part I: Constitutive equations and model closure, Int. J. Heat Mass Transfer (2009), doi:10.1016/j.ijheatmasstransfer.2009.04.017.], in which new definitions of the macroscopic turbulence quantities are introduced, is analyzed and validated. The model is validated using a simple but often used porous medium consisting of a staggered arrangement of square cylinders. Theoretically predicted values of the newly defined turbulence variables, under fully developed conditions, are compared with corresponding variables used in existing turbulence models. Additionally, evolution of the macroscopic turbulence quantities obtained numerically using the model developed here are compared with reference results, obtained by averaging over space the microscopic level solution of the RANS equations. Comparison exercise for the 75% porosity case is carried out for a range of turbulence intensity at the entrance of the porous medium. Comparison of results shows very good agreement. The spatial evolution of the dispersive kinetic energy, which is included in the definition of the macroscopic turbulent kinetic energy introduced here, is computed using the microscopic solution. Its magnitude relative to the conventional turbulent kinetic energy shows the importance of this quantity in the representation of turbulence effects in porous media flows.     

Heat- mass transfer and flow characteristics of two-phase countercurrent annular flow in a vertical pipe

Experimental and theoretical results on flow, heat and mass transfer characteristics for the countercurrent flow of air and water in a vertical circular pipe are compared. An experimental setup was designed and constructed. Hot water is introduced through a porous section at the upper end of a test section and flows downward as a thin liquid film on the pipe wall while the air flows countercurrently. The air and water flow rates used in this study are those before the flooding is reached. A developed mathematical model is separated into three parts: A high Reynolds number turbulence model, in which the local state of turbulence characteristics consists of the turbulent kinetic energy (k) and its dissipation rate (ϵ).The transport equations for both k and s are solved simultaneously with the momentum equation to determine the kinetic turbulence viscosity, the pressure drop, interfacial shear stress and then the friction factor at the film/core interface; Heat and mass transfer models are proposed in order to estimate the distribution of the temperature and the mass fraction of water vapor in gas core. The results from the model are compared with the present experimental ones. It can be shown from the present study that the influence of the interfacial wave phenomena is significant to the pressure loss, and the heat and mass transfer rate in the gas phase.     

Boundary conditions at a planar fluid–porous interface for a Poiseuille flow

The velocity boundary condition that must be imposed at an interface between a porous medium and a free fluid is investigated. A heterogeneous transition zone characterized by rapidly varying properties is introduced between the two homogeneous porous and free fluid regions. The problem is solved using the method of matched asymptotic expansions and boundary conditions between the two homogeneous regions are obtained. The continuity of the velocity is recovered and a jump in the stress built using the viscosity (and not the effective viscosity) appears. This result also provides an explicit dependence of the stress jump coefficient to the internal structure of the transition zone and its sensitivity to this microstructure is recovered.     

Fluid dynamics and oxygen transport in a micro-bioreactor with a tissue engineering scaffold

The flow environment in the micro-bioreactor with a tissue engineering scaffold was numerically modeled. The finite volume method based on multi-block grid was applied to simulate the coupled flow and oxygen transport in both porous medium and homogenous fluid regions. At porous–fluid interface, a stress jump condition that includes both viscous and inertial effects was imposed. A parametric study was performed to investigate the effects of Reynolds, Darcy, and Damkohler numbers on the flow and oxygen concentration fields inside and outside the scaffold. The minimum oxygen concentration in the scaffold and its location under different conditions were summarized.     

Momentum transport at a fluid-porous interface

The momentum balance at the interface between a liquid and a porous substrate is investigated for a configuration with forced flow parallel to the interface. An heterogeneous continuously varying transition layer between the two outer bulk regions is introduced. The stress jump coefficient earlier introduced in the jump interface condition is here derived as an explicit function of the variations of the velocity and effective properties of the transition layer. Agreement is found between our numerical results based on the single-domain approach and the existing ad hoc estimates in the literature. Further advantages of this non-homogeneous analysis are also provided.     

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.     

Turbulent flow with combustion in a moving bed

This paper presents a mathematical model for treating turbulent combusting flows in a moving porous bed, which might be useful to design and analysis of modern and advanced biomass gasification systems. Here, one explicitly considers the intra-pore levels of turbulent kinetic energy and the movement of the rigid solid matrix is considered to occur at a steady speed. 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. The rate of fuel consumption is described by an Arrhenius expression involving the product of the fuel and oxidant mass fractions. Results indicate that fixing the gas speed and increasing the speed of the solid matrix pushes the flame front towards the end of the reactor. Also, since the rate of production of turbulence is dependent on the relative velocity between phases, as the solid velocity approaches that of the gas stream, the level of turbulence in the flow is reduced.     

Numerical computation of macroscopic turbulence quantities in representative elementary volumes of the porous medium

In this study, fully developed macroscopic turbulence quantities—based on their definitions in some existing turbulence models for porous media as well as those based on definitions introduced in a recently developed model [F.E. Teruel, Rizwan-uddin, A new turbulence model for porous media flows. Part I: Constitutive equations and model closure, Int. J. Heat Mass Transfer (2009)]—are computed and analyzed for different Reynolds numbers as well as for different porosity levels. When computed based on the definition introduced in the new model, these numerically computed, pore-level turbulent quantities provide closure to the formulation. A large set of microscopic turbulent flow simulations of the REV of a porous medium, formed by staggered square cylinders, is carried out to achieve these tasks. For each Reynolds number selected, ten different porosities are simulated in the 5–95% range. The Reynolds number is varied from Re = 10³ to Re = 10⁵, covering four different cases of the turbulence flow regime. Numerical results obtained for the macroscopic turbulent kinetic energy based on the new definition show that the spatial dispersion of the mean flow is the main contributor to this quantity at low porosities. Additionally, it is found that for high porosities, the spatial average of the turbulent kinetic energy is the main contributor but the spatial dispersion of the mean flow cannot be neglected. The new definition of the macroscopic dissipation rate is found to asymptotically approach the volume average of this quantity at high Reynolds numbers. It is confirmed that microscopic numerical simulations are consistent with the macroscopic law that states that the macroscopic dissipation rate is determined by the pressure-drop through the REV.     

Zone conditional assessment of flame-generated turbulence with DNS database of a turbulent premixed flame

Zone conditional two-fluid equations are derived and validated against a DNS database for a turbulent premixed flame. The conditional statistics of major flow variables are investigated to understand the mechanism of flame-generated turbulence. The flow field in the burned region shows substantially increased, highly anisotropic turbulence to conserve mass through a flamelet surface. The transverse component may be larger than the axial component for a distributed pdf of the flamelet orientation angle in the middle of the flame brush. The opposite occurs due to redistribution of turbulent kinetic energy and flamelet orientation mostly normal with respect to the mean flow at the end of the flame brush. The major source or sink terms of turbulent kinetic energy are the interfacial transfer by the mean reaction rate and the work terms induced by fluctuating pressure and velocity on the flame surface. Ad hoc modeling of some interfacial terms may be required for further application of the two-fluid model for modeling turbulence in turbulent premixed combustion simulations.     

Analysis of turbulent combustion in inert porous media

The objective of this paper is to present an extension of a simplified reaction kinetics model that, combined with a thermo-mechanical closure, entails a full-generalized turbulent combustion model for flow in porous media. In this model, one 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. The rate of fuel consumption is described by an Arrhenius expression involving the product of the fuel and oxidant mass fractions. These mass fractions are double decomposed in time and space and, after applying simultaneous time-and-volume integration operations to them, distinct terms arise, which are here associated with the mechanisms of dispersion and turbulence. Modeling of these extra terms remains an open question and the derivations herein might motivate further development of models for turbulent combustion in porous media.     

Modeling of turbulent heat transfer and thermal dispersion for flows in flat plate heat exchangers

In this paper, heat transfer and dispersion for both laminar and turbulent regimes in heat exchangers and nuclear cores are considered. Such hydraulic systems might be seen as spatially periodic porous media. The existence of a turbulent flow within a porous medium structure suggests the use of a spatial average operator, combined to a statistical average operator. Previous works [M.H.J. Pedras, M.J.S. De Lemos, Macroscopic turbulence modeling for incompressible flow through undeformable porous media, Int. J. Heat Mass Transfer 44 (2001) 1081–1093; F. Kuwahara, A. Nakayama, H. Koyama, A numerical study of thermal dispersion in porous medium, J. Heat Transfer 118 (1996) 756–761] have applied a double average procedure to the thermal balance equation, which led to a macroscopic turbulent transport and a subsequent macro-scale equation featuring dynamic dispersion. Considering the heat flux at the solid surfaces as a boundary condition for the fluid energy balance, the model proposed in this paper allows one to take into account this dispersion as the sum of two contributions. The first one is the classical dispersion due to velocity heterogeneities [G. Taylor, Dispersion of solute matter in solvent flowing slowly through a tube, Proc. Roy. Soc. Lond. A 219 (1953) 186–203] and the second one is due to wall heat transfer. Applying Whitaker up-scaling method [S. Whitaker, Theory and applications of transport in porous media: the method of volume averaging, Kluwer Academic Publishers, 1999], a “closure problem” is then derived for a representative elementary volume, using the so-called Boussinesq approximation to account for small scale turbulence. The model is used to compute macro-scale heat transfer properties for turbulent flows inside a flat plate heat exchanger. It is shown that, for such flows, both dispersive fluxes strongly predominate over the macroscopic turbulent heat flux.     

Numerical simulation of 3D turbulent isothermal flow in a vortex combustor

Numerical simulations of strongly swirling turbulent flows in a vortex combustor (VC) are conducted. A comprehensive investigation of a three-dimensional isothermal VC flow using three first-order turbulence models: the standard k–ε turbulence model, Renormalized Group (RNG) k–ε model and shear stress transport (SST) k–ω model; and a second-order turbulence model, Reynolds stress model (RSM) together with a second-order numerical differencing scheme is conducted in the present work. The computation indicates that the RSM is superior to the other turbulence models in capturing the swirl flow effect in comparison with measurements. The numerical results for the VC flow provide the characteristics of the flow in terms of relevant parameters for the VC design and operation, composed of axial and tangential velocities, pressure fields, and turbulence kinetic energy.     

Heat and Moisture Transfer in a Rectangular Cavity Partially Filled with Hygroscopic Porous Media

Abstract

This work deals with turbulent natural convection heat and moisture transfer with thermal radiation in a rectangular cavity partially filled with hygroscopic porous medium. The governing equations for the momentum and heat transfer in both free fluid and hygroscopic porous media and moisture content transfer in hygroscopic porous medium were solved by the finite element method. Comparisons with experimental and numerical results in the literature have been carried out. Effects of thermal radiation, Rayleigh number on natural convection and heat transfer in both free fluid and porous medium and moisture content transfer in porous medium were analyzed. It was found that surface thermal radiation can significantly change the temperature and moisture content fields in the regions of free flow and hygroscopic porous medium. With the increase in Rayleigh number, the temperature of porous medium at the interface increased slightly, and the magnitude of moisture change becomes smaller.     

Calculation of Vertical Mixing in Plane Turbulent Buoyant Jets on the Basis of the Algebraic Model of Turbulence

An algebraic model of turbulence,involving buoyancy forces,is used for calculating velocity and temperaturefields in plane turbulent vertical jets in a non-homogeneous stagnant medium.A new approach to the solution ofthe governing system of partial differential equations (continuity,conservation of momentum,heat (buoyancy),turbulent kinetic energy,dissipation rate and mean quadratic temperature fluctuation) is suggested which isbased on the introduction of mathematical variables.Comparison is made between the results of the presentcalculations with experimental and numerical data of other authors.     

Investigation of the momentum transfer conditions at the porous/free fluid interface: A benchmark solution

In this work, the conformal mapping method is used to obtain the microscopic solution for the velocity distribution in the porous composite system. The corresponding macroscopic solution is determined by using a two-domain approach, in which the momentum transfer at the porous/free fluid interface is governed by the Beavers-Joseph condition or the stress-jump condition. The boundary parameters in the macroscopic solution, including the velocity slip and the stress jump coefficients, are determined by comparing the microscopic and macroscopic solutions. The impacts of the porous structure, flow type and the thickness of the free fluid layer on the velocity slip coefficient and the stress jump coefficient are discussed. In addition, the errors for the interface velocity, predicted by the velocity slip coefficient and the stress jump coefficient, are investigated. The analytical solution obtained in this work can be used as a benchmark for relevant numerical simulation.     

定容燃烧弹内湍流特性的研究

在对定容燃烧弹内的湍流进行循环分辨分析的基础上 ,计算了湍流的强度、自相关系数、标准一维能谱密度以及泰勒微观时间尺度。同时研究了传统集平均湍流研究方法和循环分辨分析法两者具有较大偏差的根源 ,从频率成分和尺度大小两方面刻画了湍流脉动与集平均脉动之间的区别和联系。通过比较不同“时窗”宽度和不同“相关窗”宽度条件下计算出的湍流能谱、自相关系数等湍流参数 ,找到了“时窗”和“相关窗”各自比较合适的折中值。