6th uropean Conference on Computational echanics (CC 6) 7th uropean Conference on Computational Fluid Dynamics (CFD 7) 5 June 8, Glasgow, UK ODLING AND SIULATION OF FR FLUID OVR A POROUS DIU BY SLSS TOD AGDALNA IRZWICZAK¹ ¹Poznan University of Technology 5. Słodowsa-Curie Square magdalena.mierzwicza@wp.pl, www.put.poznan.pl/en Key words: the Darcy equation, Brinman equation, effective viscosity, meshless method Astract. The effective viscosity in the Brinman filtration equation is determined y a numerical simulation. The porous medium is modelled y a undle of parallel fiers arranged in a regular square array. In the first step, in order to determine the permeaility, the flow driven y the pressure gradient in an unounded porous medium is considered. The flow is longitudinal with respect to fiers. In the second step, to determine the effective viscosity, the Poisseulle flow in a flat layer of porous medium is considered. Comparison of the tangential stress allows to determine the effective viscosity in filtration equation. In the third step, to determine the length of the area where the Brinman equation is applicale, the flow through a porous medium connected with a pure fluid region is considered The Trefftz method with the special purpose Trefftz functions is applied in numerical simulations. INTRODUCTION In the surrounding of a oundary of a porous region with a high porosity adjacent to the free flow area, two modeling methods are nown. In the first method, the Darcy filtration equation is used to incompressile fluid flow in porous region µ P q () where q is the macroscopic velocity, P is the pressure, µ is the dynamic viscosity of the fluid, is the permeaility of the porous medium. In the free flow area the Navier - Stoes equations are used. The Beavers - Joseph oundary condition is applied for the tangent component of the velocity vector etween this regions []. In such modeling a slip constant in the oundary condition appears. In paper [] this constant was determined in a physical experiment, however, in some papers this constant is determined in the numerical experiment. In the second modeling method, the Brinman filtration equation [] is used for a porous medium with very high porosity in the presence of a free fluid region, or for a wall-ounded porous medium µ q ~ µ P q ()
agdalena ierzwicza where ~ µ is the effective viscosity of the porous medium. As a matter of fact, this is the Stoes equation for creeping flow with the Darcy resistant term. In contrast to Darcy s equation, q. () includes the term related to the viscous transfer of momentum. The continuity of the filtration and the free flow velocity is assumed at the contact oundary of the considered areas []. In this modeling case, an effective viscosity in the Brinman equation appears. Almost all authors using the Brinman equation have supposed that the effective viscosity ~ µ is equal to the viscosity μ of a pure fluid. owever, there are some papers in which this viscosity is determined y numerical simulation [3]. The Beavers-Joseph oundary condition appiers in the firs method can e deduced as a consequence of what is now commonly called the Brinman equation. In the prolem of flow in a channel ounded y a thic porous wall one gets the same solution with the Brinman equation as with the Darcy equation together with the Beavers-Joseph condition, provided that one identifies the constant α in this condition as ~ µ µ. In this way, experimental or computational data for the constant α can e used for determination of the effective viscosity. immovale wall high porosity area free flow area immovale wall l l Figure. Porous medium with very high porosity region in the presence of free fluid region etween two impermeale walls. In this paper, a comination of these two modeling approaches is proposed. For this purpose, it is assumed that in the porous area with high porosity there is a layer near the oundary in which the flow is governed y the Brinman filtration equation (), while in the rest of the area the flow is governed y the Darcy filtration equation (). Then we consider three suareas. At the oundaries the continuity of the velocity vector is assumed. The prolem in such a modeling is connected with the width of this layer l and with the unnown effective viscosity µ ~. To demonstrating of the proposed new model the following numerical simulation is carried out. The longitudinal laminar flow (the Poisseulle flow) in a parallel-plate conduit is considered (Fig. ). The first half of the considered region is a porous medium and the second one pure fluid region. The porous medium is modeled as a undle of parallel fires arranged in a square array. The purpose of the present consideration is determination of the effective viscosity µ ~ in the Brinman equation and the width of the layer l in which this equation is to e used. But for do this; the permeaility of the porous medium is required. Then to determine the permeaility the flow with the pressure gradient in unounded porous medium was
agdalena ierzwicza considered. Numerical simulations are conducted using the meshless methods: the special purpose Trefftz functions (SPTF) []. Let us consider a plane infinite channel adjoining with a plane infinite porous region in which flow is governed y the Darcy filtration equation dp µ q D for x l (3) dz the Brinman filtration equation dp µ ~ d qb qb µ for l x () dz dx and the Stoes equation dp d w µ for x (5) dz dx For the dimensionless variales w x qb( D) ~ µ W,, Q B( D), K, α (6) dp dp µ µ dz µ dz the governing q. (3-5) otain the form Q D for L (7a) K d QB α Q B for L (7) d K d W for d (7c) which is solved with following oundary conditions Q D Q B for L (8a) Q W dq dw for d d (8) W for (8c) The exact solutions of the prolem (7-8) taes the form Q B ( ) D exp D exp K αk αk (9a) W ( ) C C (9) L where D exp ( K ) ; D exp ( K ) ; B αk B αk ( K ) L L K C exp exp ; C exp exp ; B αk αk αk B αk αk αk L B exp exp. αk αk αk αk 3
agdalena ierzwicza DTRINATION OF T PRABILITY One of the method for determination of the permeaility of the porous medium ased on the mesurment filtration velocity at the nown pressure gradient. In the present paper the longitudinal flow prolem is consider y using the special purpose Trefftz functions. The method is semi-analytical. Application of the method gives analytical form of the dimensionless permeaility of the porous medium. Only the unnown coefficients of the solution are otained numerically. a) a W / Figure : Unounded porous medium (a), repeated element Y ) n Ω F (). Let us consider steady, fully developed, laminar, incompressile viscous fluid flow driven y a constant pressure. A porous medium is modelled y a regular array of parallel fiers. The flow is longitudinal with respect to fiers. which are arranged in a regular square (Fig. a) arrays. The equation of motion is given y dp w, in Ω F () µ dz where w is the axial velocity, p is the pressure, μ is the viscosity of the fluid, and Ω F is the fluid domain. For the dimensionless variales w r a W, R,. () dp µ dz the governing q. () otain the form W. () quation () is solvd with the oundary conditions (no slip and symmetry conditon) presented on Fig.. The exact solution of q. () can e expressed using the special purpose Trefftz functions N 8 (, ) ( ) R W R θ R B cos( ) ln, R θ B N R (3) The unnown coefficients B (,, N) are determined y solving the system of linear equations resulting from satisfying of the oundary condition for, using the oundary collocation technique [5].
agdalena ierzwicza 5 Using the Darcy law () the longitudinal component of the filtration velocity can e related to the average velocity through the repeated element of the fier system dz dp F q z ϕ µ, () where Ω Ω sec tan, θ θ β β ϕ d RdR W ddy Y W F T F (5) tan β is parameter of the porous medium. The dimensionless component of the permeaility tensor in the direction parallel to the fiers is a function of the numer of collocation points N and can e calculated from 3 cos ln tan 3 tan cos 8 B G B F N N where cos sin,. cos sin G The value of the permeaility parameter is calculated as F β σ. (6) Figure 3. The non-dimensional component of the permeaility tensor in the direction parallel to the fiers and the permeaility parameter. The numerical results (mars) compared with the analytical solution (line)[9] The value of the dimensionless permeaility as a function of the fiers volume fraction is presented in Fig. 3. The prolem was solved for N 7 collocation points on the oundary.......6.8 permeaility; F(φ) φ - 3 5 7 9 3 5...6.8 permeaility parameter; σ(φ) φ
agdalena ierzwicza The numerical results (mars) are compared with the analytical presented y Drummond and Tahir [6] (line). Compatiility etween the results is great. 3 DTRINATION OF T FFCTIV VISCOSITY In the rotational viscometer the fluid viscosity is determined y measurement of the torque of the rotating cylindrical surface. The effective viscosity in the Brinman equation can e determined in a similar way. In this case the imaginary physical experiment is realized y a numerical simulation. The porous region is placed etween two parallel plates. One plate is fixed ( q z ) whereas the other moves with a constant velocity ( q z u ). The determination of the effective viscosity is possile y calculation of the shear stress on the moveale plate. The Brinman equation for one-dimensional shear flow in the asence of pressure gradient through the porous layer has the following form ~ d qz µ µ q, z in Ω F. (7) dx where q z is the filtration velocity in direction of fier axis, ~ µ is the effective viscosity of the fluid. The q. (7) can e written in the non-dimensional form d Q Q in d αβf q x ~ z µ h where Q,, α, F,. u µ The exact solution taes the form Ω F. (8) exp exp sinh αβf αβf αβf Q exp exp sinh αβf αβf αβf (9) The tangential stress on the movale plate can e expressed as dq z u τ ~ µ ~ µ coth dx x αβf αβf () On the other hand, the microstructural shear flow prolem etween two plates W W R R R R θ () with no-slip oundary conditions: W on the immovale wall and fiers and W on the movale wall is solved y means of the Trefftz method. The velocity of the flow is approximated y a linear comination of special purpose Trefftz functions. 6
agdalena ierzwicza a) y immovale wall x a LR Figure.. Wall-ounded porous medium. movale wall Since the array of fiers is streaed and periodic in one direction it is sufficient to consider the prolem only in one repeated strip. The repeated strip is divided into smaller elements associated with each of the fiers which are called large finite elements. W W W Y R θ W W Y R Figure. 5. The symmetry lane of porous medium Θ W For each large finite element the approximate solution is express as N ( ) R ( ) W ( R, θ ) C ln CR cos[ θ ] R which satisfies exactly the governing q. () and some of the oundary conditions. The unnown coefficients C (,,,N) are determined using the oundary collocation technique y satisfying the remaining oundary conditions (in particular the splitting oundary conditions etween the large finite elements). The tangential stress on the movale wall can e determined from u τ µ dw d Comparing qs. () and (3) the following relationship can e written ~ µ c( ϕ, LR, N ) αβf α µ coth αβf dy () (3) () 7
agdalena ierzwicza where ( LR, N ) dw c ϕ, dy and LR denotes the numer of fiers rows. d The value of the constant c and the dimensionless effective viscosity as a function of the fier volume fraction are presented in Fig. 6. The dimensionless effective viscosity was calculated for 5 collocation points on each oundary and 8 rows of fiers. The effective viscosity is smaller than the pure fluid viscosity. The ratio of the effective viscosity to the viscosity of the pure fluid; α at the eginning is decreasing, otains minimum and then is growing. the effective viscosity ; α(φ).9.8.7.6.5..3... φ..6.8 Figure 6. Dimensionless effective viscosity α, and the constant c. T ICROSTRUCTURAL SAR FLOW PROBL Let us consider a layer of the porous medium located etween two fixed plates (Fig. ). The microstructural flow in the layer of porous medium is governed y the dimensionless Poisson equation W (5) with the no slip oundary conditions: W on the immovale wall. the constant c(φ).. φ..6.8 fixed plate a free flow area fixed plate LR Figure 7. The repeating part of the considered channel 8
agdalena ierzwicza W W Y W Y R θ W Y R Θ W W W Y Figure 8. Large finite element with oundaty conditions Since the array of fiers is streaed and periodic in one direction it is sufficient to consider the prolem only in one repeated strip (Fig. 7). In all cases the repeated strip is divided into smaller elements associated with each of the fiers (Fig. 8). The flow prolem is solved y means of the Trefftz method. For each large finite element the special purpose Trefftz functions are used to express the approximate solution N (, ) ( ) R W R θ R B cos ln, R θ B N R (6) q. (6) satisfies exactly the governing q. (5) and the oundary conditions W for R and θ for θ {,}. For the free flow area the approximate solution is express as Y W (, Y ) cnfn (, Y ) d ngn (, Y ) n n (7) where F n (,Y) and G n (,Y) are the trial functions F (, Y ) n n n Y ( n )! ( ) G (, Y ) n n n ( n )! ( )! ;!. (8) The unnown coefficients B (,,,N), c n, d (,,,) are determined using the oundary collocation technique y satisfying the oundary conditions (Fig. 8). After solution of the microstructural oundary value prolem the average value of the velocity derivative can e calculated ω ( ) (9) The average velocity derivative ω() for the different value of the fier volume fraction φ are presented in Fig. 9. The prolem was calculated for 5 collocation points on each oundary and 8 rows of fiers. Thus, the width of the area 6. The width of the layer with the Brinman equation can e determined on the asis of the otained results, ω(). It depends on the fier volume fraction. The Darcy filtration equation is valid as long as ω() is close to zero. For greater porosity, this area is wider and approximately equals to half of the width of the porous area. The width of the Brinman area decreases as the the fier volume fraction increases. dw d dy Y 9
agdalena ierzwicza ω() 8 6 - - -6-8 - φ 3-7 -5...8....6.8....6-3 8 6 Figure 9. The average of velocity derivative for different value of the fier volume fraction. ω() 7 6 5 CONCLUSIONS In this paper to avoid calculation of the slip constant in the Beavers - Joseph oundary condition the Brinman filtration layer etween porous medium and pure flow fluid area was introduced. In the Brinman filtration equation there are two parameters of the porous medium: the permeaility and the effective viscosity. These two parameters were determined y numerical simulation of an imaginary physical experiment. The porous medium was modeled y a parallel undle of straight fiers arranged in a regular square array. In order to determine the permeaility, the flow driven y the pressure gradient in an unounded porous medium was considered. To determine the effective viscosity, the shear flow in a flat layer of porous medium was considered. To determined the width of the Brinman filtration area the microstructural shear flow was considered. In numerical simulations, the Trefftz method with the special-purpose Trefftz functions was applied. The permeaility of the porous medium decreases as the the fier volume fraction increases.the effective viscosity is lower than the viscosity of the pure fluid. The ratio of the effective viscosity to the viscosity of the pure fluid decreases as the porosity decreases.the width of the Brinman area decreases as the the fier volume fraction increases. ACLOWLGNTS This wor was supported y the grant 5/9/D/ST8/753 from the National Science Center, Poland. RFRNCS [] Beavers G. S. and Joseph D. O. Boundary conditions at a naturally permeale wall, J. Fluid ech. (967) 3:97
agdalena ierzwicza [] Brinman. C. A calculation of the viscous force exerted y a flowing fluid on a dense swarm of particles, Appl. Sci. Res. (97) A:7 [3] Kolodziej J.A., ierzwicza. and Grasi J.Computer simulation of the effective viscosity in rinman filtration equation using the Trefftz method, J. ech. ater. Struct. (7) (): 93 [] ierzwicza. and Kolodziej J.A. Comparison of different methods for choosing the collocation points in the oundary collocation method for D-harmonic prolems with special purpose Trefftz functions, ngineering Analysis with Boundary lements () 36: 883 [5] J.A. Kołodziej, A.P. Zielińsi, Boundary Collocation Techniques and their Application in ngineering, WIT Press, Southampton (9). [6] Drummond, J.., Tahir,.I.,. Laminar viscous flow through arrays of parallel solid cylinders. Int. J. ultiphase Flow (98) : 55