A Study of the Time Constant in Unsteady Porous Me

Authors: Tao Zhu Christian Waluga Barbara Wohlmuth Michael Manhart

Publish Date: 2014/05/21

Volume: 104, Issue: 1, Pages: 161-179

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Abstract

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 volumeaveraging of the Navier–Stokes equations together with a closure for the interaction term Two different closures can be found in the literature a steadystate 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 volumeaveraging 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 volumeaveraging the Navier–Stokes equation using the steadystate closure for the flow resistance termUnsteady flow in porous media can arise from unsteady boundary conditions or unsteady pressure gradients Such flows can be found in many fields of environmental technical and even biomechanical background Some examples are mass transfer between the turbulent atmospheric boundary layer and a forest eg Finnigan 2000 at the soil or snow/atmosphere interface eg Bowling et al 2011 Maier et al 2011 and between a turbulent or wavy water stream and a plant canopy eg Lowe et al 2005 Blood flow through organs can also be regarded as unsteady flow through a porous media eg Fan and Wang 2011 So far a unique description of these flow problems cannot be found in the literature and different concepts on how to treat unsteady porous media flow existFor steady porous media flows it is generally accepted and confirmed by numerous experimental results that the pressure drop or hydraulic gradient on a scale considerably larger than the pore scale can be represented by two terms a linear and a quadratic one in the spaceaveraged velocity For low Reynolds numbers based on pore diameter the pressure drop increases linearly with the flow velocity as it is dominated by viscous forces creeping flow This regime textitRe mathrmpore1 is called the Darcy regime as it was first discussed by Darcy 1857 At larger pore Reynolds numbers the quadratic term first gains weight in the force balance and then at textitRe mathrmpore300 the flow becomes turbulent For the nonlinear regime Forchheimer 1901 proposed a quadratic correction to the relation between pressure drop and macroscopic flow velocity The resulting expression can be derived either by a dimensional analysis or by rigorous averaging of the Navier–Stokes equations over a representative elementary volume see Whitaker 1986 1996Whitaker 1986 1996 proposed using the volumeaveraged Navier Stokes equations VANS to predict flow in porous media By volumeaveraging the momentum equation over a representative control volume he derived a superficial averaged form of the Navier–Stokes equations The interaction term at the pore/grain interface is closed by the steadystate Darcy and Forchheimer approximations Under the condition of unidirectional flow through isotropic homogeneous material this equation can be written with a structure identical to that of Eq 1 This approach has been adopted by several authors because it offers a mathematically sound framework for applications in which strong changes in material properties such as porosity and permeability are present Examples are the investigations of turbulent flow/porous media interaction by Breugem et al 2006 the study of instabilities in Poiseuille flow over a porous layer Hill and Straughan 2008 Tilton and Cortelezzi 2008 and analysis of bioheat transport Fan and Wang 2011 A similar formulation was used by Kuznetsov and Nield 2006 Wang 2008 and Habibi et al 2011 to directly derive an analytical solution for unsteady flow in a porous channelAnother way to derive an unsteady Darcy equation in the form of 1 was presented by Rajagopal 2007 within the context of mixture theory He also discussed possible implications of the unsteadiness of the flow field suggesting consideration of a virtual mass term if inertial nonlinearities cannot be neglected This term was proposed by Sollitt and Cross 1972 to account for the inhomogeneity of the flow field surrounding individual structures in the porous material during transient flow It results in a larger time scale in unsteady porous media flow compared to the VANS approach with steadystate closure To the authors’ best knowledge a systematic comparison and assessment of the different formulations of the unsteady Darcy equation has not been done so farIn this paper we investigate the time scale tau =fracca in Eq 1 for unsteady porous media flow First we summarize the resulting forms of the unsteady Darcy equation from the VANS and the virtual mass approaches Sect 2 Then we propose an alternative expression for the time scale in unsteady porous media flow by volumeaveraging the equation of the kinetic energy Sect 3 Finally we use fully resolved direct numerical simulation DNS of the flow in the pore space to verify this expression Sect 4 By keeping the Reynolds number small the flow remains linear and the nonlinear Forchheimer term can be neglected The time scales obtained from the simulation results are compared to those obtained by the VANS the virtual mass and our new kinetic energy approachesFor consistency with the prior work of Whitaker 1986 we denote the fluid phase by beta and the solid phase by sigma The total control volume is denoted by V and V beta denotes the volume occupied by the fluid On the fluidsolid interface A beta sigma we define the normal vector mathbf n beta sigma The aim of this paper is to assess the validity of the unsteady Darcy Eq 7 with constant coefficients This can be transformed into the question of whether the permeability and the time constant can be treated as timeindependent in the general unsteady case Furthermore we address the question of how these values can be determined To achieve this aim we first discuss two wellknown approaches for deriving the unsteady Darcy equation which are based on the volumeaveraging of the Navier–Stokes Eq 2 In Sect 3 we shall derive and discuss an alternative way based on the volumeaveraging of the kinetic energy equation In our discussion we consider that we have a control volume V with periodic boundary conditions on partial V This assumption is reasonable for homogeneous porous media in regions far from the boundaries where the control volume V ideally repeats itself Furthermore we restrict ourselves to unidirectional flow through this representative elementary volume REV Finally we assume a small Reynolds number and uniform macroscopic flow

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