Weighted averaged equations for modeling velocity profiles of 1D steady open channel flows

A new mathematical algorithm is proposed to address the essential details of vertical distributions of horizontal velocity for one‐dimensional steady open‐channel flow. This new algorithm comprises a system of weighted averaged equations developed from corresponding Reynolds equations by performing weighted average operations instead of conventional depth average operations. It is the system of weighted averaged equations, instead of the vertical grids, that allows for more hydraulic coefficients identifiable. It can be thought of as an extension of the St. Venant equations to address the vertical distributions of horizontal velocities, as well as the water surface profiles. To avoid the difficult expansion of governing partial differential equations in high order, an indirect scheme is proposed to solve hydraulic variables through their weighted average values. The governing partial differential equations are generated by using a variety of weight functions, and the weighted averages of relevant hydraulic variables are taken as the unknown independent variables to be solved first. Then, on the basis of the values and polynomial expansions of these weighted averaged velocities, a system of linear algebraic equations is generated and the unknown hydraulic variables or their coefficients are easily solved. Note that the new model is not proposed to compete with any three‐dimensional models in modeling accuracy or accommodation ability to all conditions. It just provides a valuable option to study the vertical structure of flow in open channels where only essential detail and reasonable accuracy of vertical distributions are required, and the data availability and other conditions limit the application of fully three‐dimensional models. The performance of the model is evaluated with experimental data of flows in two different flumes. It is shown that the model well predicted the velocity profiles of sections along the centerlines of these flumes with reasonable accuracy and essential details of vertical distributions of horizontal velocity. Copyright © 2013 John Wiley & Sons, Ltd.     

Predicting pressure drop in open-cell foams by adopting Forchheimer number

Interconnected struts arranged in 3-D foam structures pose a challenge in understanding fluid flow, which is significantly different from that in traditional porous media. Different flow regimes (Darcy, transition and weak inertia regimes) and thus, different flow laws in open-cell foams are used. The impact of characteristic lengths’ choices based on both, morphological and hydraulic parameters on flow law formulation has been studied. Ambiguities in definitions and measurements of several key parameters have been shown and limitations in the use of some parameters have been pointed out.An equivalent Reynolds number in the form of Forchheimer number (F_o) has been proposed to establish the friction factor relationship in order to avoid any morphological ambiguities. This number takes into account hydraulic characteristics of viscous and inertia regimes simultaneously. It has been observed that when F_o < 0.1, the flow through open-cell foams remains in the Darcy regime while the occurrence of weak inertia regime dominates when F_o > 1. Transition regime occurs in a narrow range of flow velocity when 0.1 < F_o < 1. The limits of transition for regime identification are found to be independent of foam morphologies. The form drag coefficient varies in relation with foam morphological parameters and is not a “universal” constant.Empirical correlations have been derived to predict hydraulic characteristics and friction factor data for different strut shapes and porosities. An excellent agreement has been obtained between predicted and numerical/experimental flow data.     

Filtration Behaviour of Cement-Based Grout in Porous Media

Filtration behaviour of cement particles, especially under the high grouting pressure with a rapid grout flow velocity, has a significant effect on the grout injection. However, there have been few studies on this field where the governing equation of this behaviour remains unclear. In the present study, a novel experimental procedure for grout injection was adopted to acquire the spatial and temporal variations in porosity and viscosity of high-speed grout flow in coarse sand. Experimental observations showed that there were dramatic variations in viscosity and porosity during the grout penetration within the first 50 s, suggesting that the high velocity had a significant influence on the distribution of the filtration coefficient. A model based on the Stokes–Brinkman (S–B) equation and advection–filtration equations was established to describe the filtration of grout flow in porous media. Meanwhile, numerical solutions from both the proposed model and traditional Darcy’s law were compared with experimental results. The comparative results showed that the proposed approach can match the laboratory tests well; the analysis indicated that Darcy’s law was unable to properly describe high-speed grout flow in porous media due to the lack of a shear force and the inertial term. Nonuniform filtration behaviour of cement grout flowing in porous media was revealed. Due to the nonuniform distribution of the pore velocity isoline caused by Poiseuille flow, it led to a heterogenous distribution of porosity as well. Parametric studies on the applicability of Darcy’s law and S–B equation for grout flow were discussed, in which an error of less than 10% was calculated when the Reynolds number was less than 2.5.     

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 .     

Finite Element Modelling of Flow Through a Porous Medium Between Two Parallel Plates Using The Brinkman Equation

Despite the widespread use of the Darcy equation to model porous flow, it is well known that this equation is inconsistent with commonly prescribed no slip conditions at flow domain walls or interfaces between different sections. Therefore, in cases where the wall effects on the flow regime are expected to be significant, the Darcy equation which is only consistent with perfect slip at solid boundaries, cannot predict velocity and pressure profiles properly and alternative models such as the Brinkman equation need to be considered. This paper is devoted to the study of the flow of a Newtonian fluid in a porous medium between two impermeable parallel walls at different Darcy parameters (Da). The flow regime is considered to be isothermal and steady. Three different flow regimes can be considered using the Brinkman equation: free flow (Da > 1), porous flow (high permeability, 1 > Da > 10⁻⁶) and porous flow (low permeability Da < 10⁻⁶). In the present work the described bench mark problem is used to study the effects of solid walls for a range of low to high Darcy parameters. Both no-slip and slip conditions are considered and the results of these two cases are compared. The range of the applicability of the Brinkman equation and simulated results for different cases are shown.     

Hydromagnetic flow through uniform channel bounded by porous media

The combined effects of the magnetic field, permeable walls, Darcy velocity, and slip parameter on the steady flow of a fluid in a channel of uniform width are studied. The fluid flowing in the channel is assumed to be homogeneous, incompressible,and Newtonian. Analytical solutions are constructed for the governing equations using Beavers-Joseph slip boundary conditions. Effects of the magnetic field, permeability,Darcy velocity, and slip parameter on the axial velocity, slip velocity, and shear stress are discussed in detail. It is shown that the Hartmann number, Darcy velocity, porous parameter, and slip parameter play a vital role in altering the flow and in turn the shear stress.     

Non-Darcy Mixed Convection in a Vertical Porous Channel with Boundary Conditions of Third Kind

Combined free and forced convection flow in a parallel plate vertical channel filled with porous matrix is analyzed in the fully developed region with boundary conditions of third kind. The flow is modeled using the Brinkman?CForchheimer-extended Darcy equations. The plates exchange heat with an external fluid. Both conditions of equal and different reference temperatures of the external fluid are considered. Governing equations are solved numerically by shooting technique that uses classical explicit Runge?CKutta scheme and Newton?CRaphson method as a correction scheme and analytically using perturbation series method for Darcy model. The velocity field, the temperature field and Nusselt numbers are obtained for governing parameters such as porous parameter, inertia term and perturbation parameter for equal and unequal Biot numbers and are displayed graphically. The dimensionless mean velocity and bulk temperature are also determined. It is found that the numerical solutions agree for small values of the perturbation parameter in the absence of the inertial forces.     

Melting heat transfer effects on stagnation point flow of micropolar fluid saturated in porous medium with internal heat generation （absorption）

The effect of melting heat transfer on the two dimensional boundary layer flow of a micropolar fluid near a stagnation point embedded in a porous medium in the presence of internal heat generation/absorption is investigated. The governing non-linear partial differential equations describing the problem are reduced to a system of non-linear ordinary differential equations using similarity transformations solved numerically using the Chebyshev spectral method. Numerical results for velocity, angular velocity and temperature profiles are shown graphically and discussed for different values of the inverse Darcy number, the heat generation/absorption parameter, and the melting parameter. The effects of the pertinent parameters on the local skin-friction coefficient, the wall couple stress, and the local Nusselt number are tabulated and discussed. The results show that the inverse Darcy number has the effect of enhancing both velocity and temperature and suppressing angular velocity. It is also found that the local skin-friction coefficient decreases, while the local Nusselt number increases as the melting parameter increases.     

Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

A stochastic numerical method is developed for simulation of flows and particle transport in a 2D layer of porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modeled in the layer with prescribed boundary conditions. Numerical experiments are carried out by solving the Darcy equation for each sample of the hydraulic conductivity by a direct solver for sparse matrices, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, longitudinal dispersion coefficient, and the mean displacement of Lagrangian trajectories. We discuss the effect of long-range correlations of the longitudinal velocities which we have found in our numerical simulations. The related anomalous diffusion is also analyzed.     

A Study of the Time Constant in Unsteady Porous Media Flow Using Direct Numerical Simulation

We consider unsteady flow in porous media and focus on the behavior of the coefficients in the unsteady form of Darcy’s equation. It can be obtained by consistent volume-averaging of the Navier–Stokes equations together with a closure for the interaction term. Two different closures can be found in the literature, a steady-state closure and a virtual mass approach taking unsteady effects into account. We contrast these approaches with an unsteady form of Darcy’s equation derived by volume-averaging the equation for the kinetic energy. A series of direct numerical simulations of transient flow in the pore space of porous media with various complexities are used to assess the applicability of the unsteady form of Darcy’s equation with constant coefficients. The results imply that velocity profile shapes change during flow acceleration. Nevertheless, we demonstrate that the new kinetic energy approach shows perfect agreement for transient flow in porous media. The time scale predicted by this approach represents the ratio between the integrated kinetic energy in the pore space and that of the intrinsic velocity. It can be significantly larger than that obtained by volume-averaging the Navier–Stokes equation using the steady-state closure for the flow resistance term.     

Stochastic equations for continuum and determination of hydraulic drag coefficients for smooth flat plate and smooth round tube with taking into account intensity and scale of turbulent flow

The stochastic equations of continuum are used for determining the hydraulic drag coefficients. As a result, the formulas for the hydraulic drag coefficients dependent on the turbulence intensity and scale instead of only on the Reynolds number are proposed for the classic flows of an incompressible fluid along a smooth flat plate and a round smooth tube. It is shown that the new expressions for the classical drag coefficients, which depend only on the Reynolds number, should be obtained from these new general formulas if to use the well-known experimental data for the initial turbulence. It is found that the limitations of classical empirical and semiempirical formulas for the hydraulic drag coefficients and their deviation from the experimental data depend on different parameters of initial fluctuations in the flow for different experiments in a wide range of Reynolds numbers. On the basis of these new dependencies, it is possible to explain that the differences between the experimental results for the fixed Reynolds number are caused by the difference in the values of flow fluctuations for each experiment instead of only due to the systematic error in the processing of experiments. Accordingly, the obtained general dependencies for the smooth flat plate and the smooth round tube can serve as the basis for clarifying the results of experiments and the experimental formulas, which used for continuum flows in different devices.     

Analysis of Laminar Flow and Forced Convection Heat Transfer in a Porous Medium

The flow of an incompressible Newtonian fluid confined in a planar geometry with different wall temperatures filled with a homogenous and isotropic porous medium is analyzed in terms of determining the unsteady state and steady state velocities, the temperature and the entropy generation rate as function of the pressure drop, the Darcy number, and the Brinkman number. The one-dimensional approximate equation in the rectangular Cartesian coordinates governing the flow of a Newtonian fluid through porous medium is derived by accounting for the order of magnitude of terms as well as accompanying approximations to the full-blown three-dimensional equations by using scaling arguments. The one-dimensional approximate energy and the entropy equations with the viscous dissipation consisting of the velocity gradient and the square of velocity are derived by following the same procedure used in the derivation of velocity expressions. The one-dimensional approximate equations for the velocity, the temperature, and the entropy generation rate are analytically solved to determine the velocity, the temperature, and the entropy distributions in the saturated porous medium as functions of the effective process parameters. It is found that the pressure drop, the Darcy number, and the Brinkman number affect the temperature distribution in the similar way, and besides the above parameters, the irreversibility distribution ratio also affects the entropy generation rate in the similar way.     

Unsteady Conjugate Natural Convection in a Vertical Cylinder Containing a Horizontal Porous Layer: Darcy Model and Brinkman-Extended Darcy Model

Transient natural convection in a vertical cylinder partially filled with a porous media with heat-conducting solid walls of finite thickness in conditions of convective heat exchange with an environment has been studied numerically. The Darcy and Brinkman-extended Darcy models with Boussinesq approximation have been used to solve the flow and heat transfer in the porous region. The Oberbeck–Boussinesq equations have been used to describe the flow and heat transfer in the pure fluid region. The Beavers–Joseph empirical boundary condition is considered at the fluid–porous layer interface with the Darcy model. In the case of the Brinkman-extended Darcy model, the two regions are coupled by equating the velocity and stress components at the interface. The governing equations formulated in terms of the dimensionless stream function, vorticity, and temperature have been solved using the finite difference method. The main objective was to investigate the influence of the Darcy number $10^{-5}\le \hbox {Da}\le 10^{-3}$ , porous layer height ratio $0\le d/L\le 1$ , thermal conductivity ratio $1\le k_{1,3}\le 20$ , and dimensionless time $0\le \tau \le 1000$ on the fluid flow and heat transfer on the basis of the Darcy and non-Darcy models. Comprehensive analysis of an effect of these key parameters on the Nusselt number at the bottom wall, average temperature in the cylindrical cavity, and maximum absolute value of the stream function has been conducted.     

Effects of Brownian Motion and Thermophoresis on the Mixed Convection of Nanofluid in a Porous Channel Including Flow Reversal

This article is devoted to combined convection heat transfer of nanofluids through a vertical channel filled with a homogeneous and isotropic porous medium. The flow is assumed to be fully developed and the “Brinkman extended Darcy” model is used for the flow in the porous media and “clear compatible” viscous dissipation model is considered. Also the model utilized for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing momentum, energy, and nanopartices volume fraction equations are solved both analytically and numerically. The effects of the influential dimensionless parameters such as Brownian and thermophoresis parameters, mixed convection parameter (Gr/Re), Brinkman, Darcy and Lewis numbers on dimensionless velocity and temperature distributions and pressure drop are studied. Also, the results of the Nusselt number for the both left and right walls are presented and discussed.     

非稳态分数阶Oldroyd-B流体绕楔形体流动研究

研究了非稳态分数阶Oldroyd-B流体在多孔介质中通过楔形拉伸板的驻点流动问题。基于分数阶Oldroyd-B流体的本构模型建立了动量方程,并在其中引入了浮升力和驻点流动特征。此外,考虑了具有热松弛延迟时间的修正的分数阶Fourier定律,并将其应用于能量方程和对流换热边界条件。接着,采用与L1算法相结合的有限差分法求解控制偏微分方程。最后,分析了相关物理参数对流动的影响。结果表明,随着楔角参数的增加,流体受到的浮升力增大,导致速度加快;达西数越大,介质的孔隙度变大,流体的流动越快;此外,温度分布先略有上升后明显下降,这表明Oldroyd-B流体具有热延迟特性。     

Numerical simulation of an axisymmetric separated and reattached flow over a longitudinal blunt circular cylinder

This article presents a numerical investigation of turbulent flow in an axisymmetric separated and reattached flow over a longitudinal blunt circular cylinder. The governing equations were discretized by the finite-volume method and SIMPLER method was applied to solve the equations on a staggered grid. The turbulent flow was numerically simulated using the standard k–ε, Abe–Kondoh–Nagano (AKN) and Shear Stress Transport (SST) turbulence models. The comparisons made between numerical results and experimental measurements showed that the SST model is superior to other models in the present calculation.Computations were performed for three different Reynolds numbers of 6000, 10 000 and 20 000 based on the cylinder diameter. To our knowledge, this study represents the first numerical investigation of the present flow configuration. The computational results were validated with the available experimental data of reattachment length, mean velocity distribution and wall static pressure coefficient in the turbulent blunt circular cylinder flows. Further, other characteristics of the flow, such as turbulent kinetic energy, pressure, streamlines, and the velocity vectors are discussed.The results show that the main characteristics of the turbulence flow in the separation region, such as reattachment length or velocity profiles, are nearly independent of the Reynolds number. The obtained results showed that a secondary separation bubble may appear in the main separation bubble near the leading edge. Furthermore, it was found that the turbulent kinetic energy has a large effect on the formation of the secondary bubble.     

Double diffusive convection of anomalous density fluids in a porous cavity

A numerical study has been performed to analyze the combined effect of temperature and species gradients on the buoyancy-driven natural convection flow of cold water near its density extremum contained in a porous cavity. The governing equations are descretized using the finite volume method. The results of the investigation are presented in the form of steady-state streamlines, velocity vectors, isotherms, and isoconcentrationlines. The results are discussed for different porosities, Darcy numbers, and Grashof numbers. The heat and mass transfer rates calculated are found to behave nonlinearly with hot wall temperature. The heat and mass transfer are increased with increasing Darcy number and porosity. It is found that the convective heat and mass transfer rate are greatly affected by the presence of density maximum.     

Symmetries, Invariance and Scaling-Laws in Inhomogeneous Turbulent Shear Flows

An approach to derive turbulent scaling laws based on symmetry analysis is presented. It unifies a large set of scaling laws for the mean velocity of stationary parallel turbulent shear flows. The approach is derived from the Reynolds averaged Navier–Stokes equations, the fluctuation equations, and the velocity product equations, which are the dyad product of the velocity fluctuations with the equations for the velocity fluctuations. For the plane case the results include the logarithmic law of the wall, an algebraic law, the viscous sublayer, the linear region in the centre of a Couette flow and in the centre of a rotating channel flow, and a new exponential mean velocity profile that is found in the mid-wake region of high Reynolds number flat-plate boundary layers. The algebraic scaling law is confirmed in both the centre and the near wall regions in both experimental and DNS data of turbulent channel flows. For a non-rotating and a moderately rotating pipe about its axis an algebraic law was found for the axial and the azimuthal velocity near the pipe-axis with both laws having equal scaling exponents. In case of a rapidly rotating pipe, a new logarithmic scaling law for the axial velocity is developed. The key elements of the entire analysis are two scaling symmetries and Galilean invariance. Combining the scaling symmetries leads to the variety of different scaling laws. Galilean invariance is crucial for all of them. It has been demonstrated that two-equation models such as the k– model are not consistent with most of the new turbulent scaling laws.     

Modeling full-tensor anisotropy in groundwater flow via an iterative scheme for mixed finite elements

This paper presents an iterative scheme for the efficient simulation of groundwater flow in a two-dimensional, heterogeneous aquifer in which the hydraulic conductivity is anisotropic. The scheme is applicable to matrix equations arising from both mixed finite-element and cell-centered finite-difference approximations to the flow equations, and it extends readily to three space dimensions. The scheme, which generalizes an earlier technique for isotropic aquifer, admits a fast multigrid solver for hydraulic heads. Numerical experiments illustrate both the effectiveness of the scheme and the importance of accurately treating anisotropy: Small changes in the off-diagonal terms in the conductivity tensor cause relatively large changes in both the predicted heads and the Darcy velocities.     

Confined viscoplastic flows with heterogeneous wall slip

The steady, pressure-driven flow of a Herschel-Bulkley fluid in a microchannel is considered, assuming that different power-law slip equations apply at the two walls due to slip heterogeneities, allowing the velocity profile to be asymmetric. Three different flow regimes are observed as the pressure gradient is increased. Below a first critical pressure gradient G ₁, the fluid moves unyielded with a uniform velocity, and thus, the two slip velocities are equal. In an intermediate regime between G ₁ and a second critical pressure gradient G ₂, the fluid yields in a zone near the weak-slip wall and flows with uniform velocity near the stronger-slip wall. Beyond this regime, the fluid yields near both walls and the velocity are uniform only in the central unyielded core. It is demonstrated that the central unyielded region tends towards the midplane only if the power-law exponent is less than unity; otherwise, this region rends towards the weak-slip wall and asymmetry is enhanced. The extension of the different flow regimes depends on the channel gap; in particular, the intermediate asymmetric flow regime dominates when the gap becomes smaller than a characteristic length which incorporates the wall slip coefficients and the fluid properties. The theoretical results compare well with available experimental data on soft glassy suspensions. These results open new routes in manipulating the flow of viscoplastic materials in applications where the flow behavior depends not only on the bulk rheology of the material but also on the wall properties.