Fluid flow in a channel partially filled with porous material

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1 Fluid flow in a channel partially filled with porous material M. Hriberkk', R. Jec12& L. Skerget' 'Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia 2Facultyof Civil Engineering, University of Maribor, Smetanova 17, SI Maribor, Slovenia Abstract The paper deals with computation of flows in domains partially filled with porous material. The aim of the research is to investigate the influence of porous domains with relatively high values of permeability on the flow field inside a narrow channel. Boundary-domain integral method is used to solve the resulting Navier-Stokes equations, where for porous domain Brinkman extension to Darcy model is applied. As a test case a narrow channel with a porous domain of half the width of the channel is computed. Influence of different values of permeability of the porous medium on flow characteristics is studied. 1 Introduction sistems or flows through filtration media. In these cases, porosity and per- meability of domains are often very low due to microscopic porous structure and the use of Darcy model for numerical investigation of flow inside such Computations of flow field in heterogenous materials is a very important engineering research field. A well known examples are flows in geothermal domains is accurate enough. However, quite often one encounters also configurations where part of a bounded domain, where fluid flows, has internal flow obstacles of a size only an order of magnitude lower than the charac- teristic dimensions of the domain. One of such examples is a flow inside a compact heat exchanger [l],where additional fins inside the channels ensure

2 466 Advmces irl Fluid Mechunks W higher heat transfer rates than in case of a clear channel. It is of course possible to numericaly discretize dl the solid boundaries, i.e. fins, inside such domain, but the computational cost could quickly become too large for fast engineeringcomputations. In such cases a better option is to try to model such a domain as porous medium and try todefine its physical prop- erties in a way to get at the end the same integral results (pressure drop, heat transfer rate) as would be obtained by a direct numerical approach. As such configurations have large porosities, from and also quite high values of permeability, the Darcy model no longer gives accurate enough re- sults. Since inertial terms become important, the use of Brinkman term in the momentum equation ensures a more accurate mathematical model. The paper deals with the numerical simulation of a fluid flow in domain partially filled with porous medium by using the Boundary-Domain Integral Method (BDIM). This method has alredy been developed for natural convectionflows inside a porous cavity [4]and is here extended to heteroge- nous domains. Brinkman extended Darcy model is applied. Particular attention is given to formulation of appropriate integral representations for all field functions. For the kinematics the parabolic diffusion fundamental solution is used, while the integral statements of all kinetic transport equa- tions are based on elliptic diffusion-convectionfundamental solution. The finite difference approximation is considered for the time derivatives of field functions. 2 Navier-Stokesequations-velocity-vorticity formulation As mathematical model, describing fluid flow in clear and porous domains, the Navier-Stokes equations set for viscous incompressible fluid flowwas used. Planar geometry was considered and the governing set of equations consisted of: e continuity equation, e momentum equation for fluid domain, momentum equation for porous domain - Brinkman equation 1dVi 1avjvi - 1 dp q5 dt 4 ax3 P 8% -_ +-- U Fvi+U~8x3 8x3 (2.3)

3 A d m c c ~ sill Fluid Mdxznics IV 467 where uz is the i velocity component, xi is the i coordinate, DIDt represent the substantial derivative, P = p - pgjrj is the modified pressure and p is the static pressure. The material properties such as mass density p, kinematic viscosity v and dynamic viscosity q,porosity I$ and permeabiliy K are assumed to be constant parameters. The fluid is Newtonian. In the Brinkman extension to momentum equation the effective viscosity v' may have a different value than the fluid viscosity v, therefore parameter R, denoting viscosity ratio, is introduced: A= -. v' U With this definition the equation (2.3) becomes 1avi+L*- 1dP v +Av% 4 at p axj pa~:i K axj axj ' (2.5) In general, h is a function of porosity. In cases with a high value of porosity (near l)a reasonable aproximation is R = 1. With the introduction of vorticity vector wi the fluid motion compu- tation scheme is partitioned into its kinematics and kinetics. The kinetic aspect is governed by the vorticity transport equation obtained as a curl of the momentum equations (2.2) and (2.5),which for the planar case read as fluid domain porous domain a w + y i a w i d2w - = v ~ at axjaxj ' (2.6) ei3 being the permutation unit symbol and r = 5. The kinematics can be formulated by the elliptic equation for the vector potential or in the form of an elliptic equation for the velocity vector, [2]. To accelerate convergency and stability of the coupled velocity-vorticity iterative scheme, the false transient approach is applied [2], resulting in the following parabolic kinematic statement: where a is a relaxation parameter. It is obvious that the governingvelocity equation (2.8) is exactly satisfied only at the steady state (t+ m), when the artificial time derivative term vanishes. With the new basic set of equations for BDIM, (2.6), (2.7) and (2.8), all equations can be represented by the parabolic diffusion-convectiveequation, which accurate integral representation is of key importance to the success of the BDIM.

4 468 Advmces irl Fluid Mechunks W 2.1 Integral representations for flow kinematics The velocity equation (2.8) can be diffusion equation of the form given as a nonhomogenous parabolic By assuming a linear variation of field quantities within the individual false transient time increment At = tf - tf-1 (F denotes false transient time step) and the use of the parabolic diffusion fundamental solution, given by the expression (2.10) the boundary-domain integral statement for the false transient kinematics equation (2.8) for the plane motion reads as S ViF-lU$-ldfl, (2.11) R where U* is time integrated fundamental solution (2.10). After discretisation of computational domain, assembly of all influence matrices and incor- poration of boundary and initial conditions, the following matrix form of equation appears: + + -

5 2.2 Integral representations for vorticity kinetics A d m c c ~ ill s Fluid Mdxznics IV 469 The formulation of the integral representation for the kinetics is based on linear differential operator of the elliptic diffusion-convective type with the constant reaction term. By using a second order finite difference approx- imation of the field function time derivative for a time increment At and elliptic diffusion-convective fundamental solution - 1 U; = - K o ( p r ) e x p ( F )(2.13), 27r where the decomposition of velocity field uj = vj + C3 into vj an average constant vector and 83 a perturbated part is applied and parameter p is defined by with p = m p2= (;)2+P> (2.14), the integral representation for flow kinetics now reads as: e fluid part of domain c(e)w(() + D J uxdi'= D J gur]dl'- J wtijnjur]di' dn r r r porous part of domain (2.15) where N and N - 1 denote previous time step values. The corresponding matrix form of vorticity transport now reads as: fluid part of domain (2.17)

6 470 Advmces irl Fluid MechmksW o porous part of domain (2.18) 3 Boundary conditions Matrix form of governing equations (2.12), (2.17) and (2.18) applies to one subdomain. As BDIM uses a macro element approach where each subdomain is represented by one internal cell and four boundary elements, bound- ary conditions in form of compatibility and equilibrium conditions [a], [3] has to be applied in order to connect all the subdomains. Sincein the present case we have an additional boundary between porous domain and surrounding fluid domain, one has to prescribe boundary conditions also on this boundary. This is in general not an easy task as the porous boundaries do not act like solid ones and the no-slip conditions are not valid here. A general interface condition is a continuity of total flux of a quantity across the interface [5]. Since we are dealing with very thin porous domains with q5 M 0.9 continuity conditions for velocities and vorticities at the interface, equal to boundary conditions between BDIM subdomains, can be used, what greatly simplifies numerical implementation. 4 Test example As a test example a narrow channel with a porous insert was selected. The geometry of the problem is presented in Fig.4.1. The computational domain consists of the inlet and outlet parts, where no porous media is present. In between, the channel is divided into two halfs, the upper one being the porous medium. At the inlet, fully developed laminar flow in a parallel plate channel was prescribed. At the outlet, fullydeveloped flow conditions were imposed. Computational mesh consisted of 8 subdomains in y and 20 in z direction, a total of 160 equally spaced subdomains. The Darcy number, defined on basis of the porous insert height 5 (H is the channel height) 4K Da= ~2 (4.19) had the values for the first 0.1 and 0.01 for the second test computation. Figure 4.2 shows velocity vectors for Re number (Re = F)value 100 and Da = The influence of additional resistance to fluid flow in the

7 Admcc.sill Fluid Mdxznics IV 47 1 inlet solid wall porous medium outlet solid wall Figure 4.1: Geometry and physical layout fo the problem porous part of the channel is clearly recognizable. This is also presented in Fig. 4.3, where velocity profiles through the middle (between the inlet and outlet) of the channel are plotted for both values of Da number. This resistance causes the maximum velocity in the channel to move towards the centerline of the fluid part of the cross section. In case of Da= 0.01 this shift is significant, whereas for Da= 0.1 this shift is still present but less noticable. This is also in agreement with analitical results for a similar physical problem, dealt with in the works [6] and [7]. Figure 4.2: Velocity field for Da=O.Ol 5 Conclusions A numerical method, based on Boundary-domain integral method, was presented for computation of flows in domains partially filled with porous medium. Darcy-Brinkman model was applied to account for additional flow resistance in porous part of domain. As porous medium with very high porosity was selected continuity conditions for velocities and vortici- ties boundary conditions could be prescribed at the fluid-porous interface. The computed results for the channel with porous insert showed that the presented method is capable of dealing with fluid flow in different types of physical domains. References [l]hribersek, M., BaSiF, S., Skerget, L, Sirok, B.: Numerical modeling of heat and flow conditions of laundry dryer condenser. Proceedings of

8 472 Advmces irl Fluid Mechunks W c -I Figure 4.3: Velocity profiles for the middle part of the channel icheap-5, VoLl,, Florence, Italy, pp , [2] Skerget, L., HriberSek, M., Kuhn, G.: Computational Fluid Dynamics by Boundary Domain Integral Method; Int. J. Num. Meth. Eng., 46, pp , [3] HriberSek, M., Skerget, L.: Fast Boundary-Domain Integral Algorithm for Computation of Incompressible Fluid Flow Problems; kt. J. Num. Meth. Fluids, 31, pp , [4]Jecl, R., Skerget, L., PetreSin, E.: Boundary domain integral method for transport phenomena in porous media; Int. J. Num. Meth. Fluids, 2001, 35, pp [5] Bear, J., Bachmat, Y.: Introduction to modeling of transport phenomena in porous media. Kluwer academics publishers, Dordrecht, [6] Abu-Hijleh, B. A., Al-Nimr, M.A.: The effect of the local inertial term on the fluid flow in channels partially filled with porous material; int. J. Heat and Mass Transfer, 44, pp , [7]Al-Nimr, M.A., Alkam, M. K.: Unsteady non-darcian fluid flow in parallel-plates channel partially filled with porous material; Heat and Mass Transfer, 33, pp , 1998.

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