LATTICE BOLTZMANN FLOW SIMULATION OF POROUS MEDIUM-CLEAR FLUID INTERFACE REGIONS

Transcription

1 LATTICE BOLTZMANN FLOW SIMULATION OF POROUS MEDIUM-CLEAR FLUID INTERFACE REGIONS Kazuhiko Suga 1 Yoshiumi Nishio 1 1 Department o Mechanical Engineering Osaka Preecture University Sakai JAPAN ABSTRACT The eects o a porous medium on the low in the interace region between the porous wall and clear luid are discussed. Laminar lows in the interace regions o oamed porous walls are microscopically simulated by the lattice Boltzmann method. The chosen porous structure is the body-centered-cubic or the unit cube structure whose porosity ranges The velocity distribution in the interace regions show that the low penetration is very little and it decays until one porediameter depth rom the interace. The results also show that the coeicients o the stress jump condition across the interace are negative and the total riction o the porous interace is slightly lower than that o a solid smooth surace. INTRODUCTION Viscous luid lows through porous media have been studied or many years ater Darcy (1856) since porous materials are so common in engineering ields as well as in the earth science. Once such porous materials are applied to luid lows or guiding iltering or controlling the luid the characteristics o lows over/around porous suraces are also very important or designing the lowdevices. Accordingly boundary layer lows over permeable walls have been studied by many researchers as well. Beavers and Joseph (1967) studied on the eects o a porous medium on the low in the interace region between the porous wall and clear luid (named interace region hereater). They measured mass low rates over permeable beds in laminar low conditions and ound that the mass low rates increased compared with those in impermeable cases. Many other ollowing studies (e.g. Rudraiah 1985; Larson and Higdon 1986; Sahraoui and Kaviany 1992; Ochoa-Tapia and Whitaker 1995; Gupte and Advani 1997; James and Davis 2001; Breugem et al.2005) also ocused on the laminar low regime experimentally and numerically since the low physics is rather complicated and controversial even in laminar lows. In act the eects o low penetration into porous media are still not ully understood. Gupte and Advani (1997) perormed mean low measurements in the interace region o ibrous mats using a laser Doppler anemometry. They reported that the low penetration into the porous layer was stronger than that estimated semitheoretically in the laminar low regime. On the contrary James and Davis (2001) reported numerically that the external low penetrated the ibrous porous medium very little. The slip velocity at the interace is also one o the issues. On the porous interace the luid motion is not totally damped having non-zero statistical tangential velocity component. Since this slip velocity is ruired to solve the momentum uations over the interace as a boundary condition Beavers and Joseph (1967) proposed a condition that is a unction o the permeability and the Darcy velocity o the porous media. This condition was supported by many studies such as Saman (1971) Larson and Higdon (1986) and Sahraoui and Kaviany (1992). Ochoa-Tapia and Whitaker (1995) proposed another relation introducing a jump condition o the stress distribution across the interace. Both o the conditions ruire experimental measurements or determining their coeicients. Another important issue is the riction o the porous suraces. In laminar low regimes Beavers and Joseph (1967) reported that the riction over the porous walls reduced due to the porosity. However Zagni and Smith (1976) and Zippe and Gra (1983) experimentally ound that the riction actors o turbulent lows over permeable beds became higher than those over impermeable walls with the same surace roughness. Kong and Schetz (1982) observed that the increase o the skin riction was due to the combined eects o roughness and porosity. In order to understand the low physics causing the dierence between the laminar and the turbulent regimes it is necessary to discuss the detailed low physics in the interace region. However measuring low proiles across the interace region is almost impossible even with the latest laser techniques. Thus we expect that numerical analyses can provide useul inormation instead. Recently the LBM (lattice Boltzmann method) (McNamara and Zanetti 1988; Higuera and Jimenez 1989) has become powerul tool or simulating lows inside porous media (e.g. Martys and Chen 1996; Keehm et al. 2004; Pan et al. 2006) due to its coding simplicity o the method or complicated low geometries. Thereore in order to understand the low characteristics such as the low penetration the slip velocity and the riction o the porous interaces in laminar low regimes the present study has perormed numerical simulations using the LBM. LATTICE BOLTZMANN METHOD The presently used LBM employs the single-relaxationtime (SRT) and multiple-relaxation-time (MRT) Bhatnagar-Gross-Krook (BGK) models (Bhatnagar et al. 1954) which are briely described below. Single relaxation time LBM The lattice Boltzmann method or low simulation by the SRT BGK model may be written as () r t () r t ( r+ eδt t + δt ) = ( r t) τ (1) ( = 012 Q 1)

2 where () r t is the density distribution unction along the direction at the lattice site r at time t e is the discrete velocity δ t is the time step and τ is the dimensionless relaxation time ( τ = 1 is applied in this study). The uilibrium distribution unction determined by the luid density and momentum is ( ) c S wρ e u e u u = 1+ + (2) 2 4 cs 2c S where w is the weight coeicient and c S is the sound speed. Once the density distribution unction is known the density ρ the velocity u the pressure p and the kinematic viscosity ν are obtained rom the conservation laws and the uation o state: ρ= ρ u = c p =ρcs ν= csτ δt. (3) 2 In the present study the D3Q19 discrete velocity model whose parameters (Chen 1992) are listed in Table 1 is employed with the so-called hal-way bounce back condition or solid boundaries. Multiple relaxation time LBM For the D3Q19 MRT LBM (d Humieres et al. 2002) a set o distribution unctions is deined on each lattice node x i. The collision is executed in the moment space and the advection is perormed in the velocity space. The evolution uation is thus written as ( ) () 1 r + e δt t + δt = r t M S m() t m () t r r (4) where φ() r t ( φ0() r t φ1() r t φ 1() ) T Q r t. The matrix M is a Q Q matrix which linearly transorms the distribution unctions to the velocity moments m : m = M (5) M = (6) The collision matrix S = M S M 1 is diagonal: S diag( s0 s1 s Q 1) (7) = (0 s1 s20 s40 s40 s4 s9 s2 s9 s2 s9 s9 s9 s16 s16 s16). whose components are given as ν = s 1 = 1.19 s 2 = 1.4 s 4 = 1.2 s 16 = (8) 3 s9 2 The uilibria o the moments are the unctions o the conserved moments and given as 2 S Table 1 Parameters o the discrete velocity model o D3Q19. c e w 1/3 (000) ( ± 1 0 0)(0 ± 1 0) (00 ± 1 ) ( ± 1 ± 1 0)( ± ± 1 ± 1 ± )(0 ) 12/36( = 0) 2/36( = 1 6) 1/36( = 7 18) m0 = ρ m1 = 11 ρ + ( jx + jy + jz) (9) ωε j m2 = ωρ ε + ( jx + jy + jz) m357 = j x yz (10) m468 = j x yz m9 = (2 jx jy jz ) 3 (11) m10 = ωxxm9 11 ( ) m = jy jz m12 = ωxxm11 (12) m13 = jx jy m14 = j y j z m 15 = jz jx (13) m = 0 (14) where j = ( jx jy jz) = u ρ 0 is the mean density in the system which is set to be unity and the optimized parameters are ωε = ωxx = 0 ω ε j = 475/ 63. RESULTS AND DISCUSSIONS Geometry o oamed porous media Fig.1 shows a oamed porous medium which has the open-cell oam structure. Due to the oaming process the bubbles o the oam attain an uilibrium state where the surace energy is minimum. In the present study the body-centered-cubic (BCC) or the unit cube (UC) structure is chosen to represent the oamed porous media since they are simple and easily generated. For the BCC oam geometry creation the shapes o the pores are assumed to be spherical and spheres o ual volume are located at the vertices and the center o the unit cell as shown in Fig2(a). For the UC oam creation the spherical pores are located at the center o the unit cell as shown in Fig2(b). The periodic oam geometry is then obtained by subtracting the spheres rom the unit cell cube. For the open-cell structure the pore diameter d p should be larger than the height o the cube (:cell height) A. The pore diameter also has a maximum limit to orm the structure. Thus the BCC structure has the range o the porosity as < < while the UC has < < Fig.1. Foamed porous medium.

3 Grid and relaxation-time model dependency The computational domain includes a channel over a porous layer whose thickness H p is the same as that o the channel H c as shown in Fig.3. The porous layer consists o 7 cells o the BCC or UC structure in the y direction and whose porosity ranges Periodical boundary conditions are imposed in the low ( x ) and spanwise ( z ) directions and thus lattice nodes are used or the main computations whose bulk Reynolds number is Re = 80. Since the simulated lows are well in the laminar low regime the normalized values o low characteristics such as permeability K and the Darcy velocity U d do not vary by the Reynolds number. (It has been conirmed by comparing results o dierent Reynolds numbers.) Grid dependency tests have been perormed comparing the results by lattice nodes. Fig.4 compares sectionally supericial averaged streamwise velocity U proiles in the interace regions. (The location o the interace is y = 0 and the channel region is in y 0 while the porous region is in y 0.) Figs. 4(a) and (b) respectively corresponding to the BCC and the UC structure cases show virtually the same distribution proiles o the two lattice node cases and conirm that the results o lattice nodes are well grid independent. Pan et al. (2006) reported that the MRT model was suitable or predicting porous medium lows. Hence Fig.4 also compares the results by the two relaxationtime (SRT and MRT) models o the LBM. It is clear that the dierence between the proiles o the SRT and MRT models are marginal. This conirms that discussions on the low characteristics such as the slip velocity and the low penetration by the SRT model are reliable as well. However as in the report o Pan et al. (2006) the predicted permeability distribution slightly diers as 2 shown in Fig.5. The normalized permeability K / d is deined ollowing Bhattacharya et al. (2002). The microscopic characteristic length d o Du Plessis et al. (1994) is determined with the pore diameter d p as d = 2 1/(3 π) d p. (15) The permeability is estimated by the ollowing relation using the results o several dierent pressure gradients (a) A d (a) cube spheres BCC model A (b) cube sphere UC model Fig.2. Models o oamed materials. dp (b) Fig.4. Grid and relaxation-time model dependency. Hc low Hp z y porous medium x Fig.3. Flow ield geometry. Fig.5. Comparison o permeability distribution.

4 Δ P/ A as 2 Δ P / A = aud + bu d (16) K = μ / a (17) where μ is the luid viscosity. A least-squares it is used to determine the coeicients a and b. (Eq.(16) can be derived rom the momentum uation assuming that low is ully developed and the viscous eects are small.) Although the MRT model predicts in better agreement with the experiments the results o the SRT model also distribute in a reasonable range compared to the experimental deviation. Thereore the results o the SRT Fig.6. Velocity proile. model are used in the ollowing discussions. General low characteristics For the general idea o the simulated low proiles Fig.6 shows a simulated velocity proile normalized by the bulk mean velocity U b. In the channel region the velocity distributes parabolically while one can observe a slip velocity at the interace. (This slip velocity U w is discussed in the latter section in detail.) A sectionally supericial averaged velocity proile inside a porous layer is displayed in Fig.7 along with the proile o the Brinkman uation s (Brinkman 1949) solution: U = ( Ud Uw)exp y / K. (18) In Fig. 7 K U d and U w o the simulation are applied to Eq. (18) to obtain the dotted line. Due to the structure whose void-raction changes in y direction a sinusoidal velocity proile is observed. Fig.8 shows the permeability distribution o the presently simulated porous layers. Although both the BCC and UC cases show the increase o the permeability with the increase o the porosity one can see an obvious dierence between the structure cases. This clearly implies that the permeability is subjected to both the porosity and the structure. Flow penetration and slip velocity As discussed with Fig.7 the low distribution depends on the phase o the local void-raction. In the simulations the porous media have perectly regular structure while the real porous media usually have some randomness. Thus in order to obtain reasonably general low characteristics o the structure averaging several Fig.7. Sectionally supericial averaged velocity proile in the porous region. Fig.9. Interace positions. Fig.8. Computed permeability. Fig.10. Averaged velocity proile o 4 dierent phases o the UC structure.

5 dierent cases o the surace phase-position is applied. Although there are an ininite number o phases to determine the porous interace 2 types o the interace position are applied or the BCC structure while 4 types are applied or the UC structure as shown in Fig.9. With this averaging the sinusoidal velocity proile is smoothed out as shown in Fig.10. In act the velocity proile reasonably accords with the proile o the solution o the Brinkman uation. (In the ollowing discussions all velocity proiles are obtained by this averaging process.) The low penetration is deined as the region where the averaged velocity is 5% greater than the Darcy Fig.11. Flow penetration length. velocity U d in this study. The obtained penetration length l p is plotted in Fig.11. It is clear that the penetration length generally increases with the porosity. However the maximum penetration length is less than the pore diameter d p. This tendency is also observed in a iner porous layer which has the same thickness but 28 layers o the cells. (Extra simulations by lattice nodes are also perormed.) Fig. 12(a) shows the slip velocity against the porosity. Although the slip velocity increases with the porosity it strongly depends on the porous structure. Since the slip velocity also depends on the pore size the parameter : ( Uw/ Ud)( dp/ H c) is introduced. As shown in Fig.12(b) this parameter distributes airly monotonically with the porosity. (It is also conirmed that the parameter o the slip velocity in the iner porous layer accords well with the line in Fig. 12(b).) Stress conditions Beavers and Joseph (1967) proposed a clear luid low boundary condition at the interace as U = ( Uw Ud). (19) 0+ K This BJ condition is also derived rom the Brinkman uation as u 0 = ( Uw Ud) (20) K (a) Fig.13. Coeicients o the BJ condition. (b) Fig.12. Slip velocity distribution. Fig.14. Coeicients o the stress jump condition.

6 where u is the luid phase velocity (intrinsic averaged velocity). With the assumption: U 0+ u = 0 (21) the coeicient is comparable to 1/. The distribution o obtained rom the present simulations is shown in Fig.13. Although 1/ decreases against the porosity increases. It is thus obvious that the assumption o Eq.(21) is not very reasonable. Ochoa-Tapia and Whitaker (1995) improved the BJ condition and proposed the stress jump condition: u U β = U K 0 0+ w. (22) In their report the stress jump coeicient β ranged rom 1 to 1.47 depending on the porous materials. In the present study however the coeicients are always negative as shown in Fig.14 since the intrinsic averaged velocity gradient underneath the interace is always smaller than that over the interace. Wall riction In this study the wall shear stress on the porous wall is deined as H τ c p =ΔP τ s (23) A where ΔP is the pressure loss along the length A in the channel and τ s is the wall shear stress on the smooth solid wall. Fig.15 shows the distribution o τ p normalized by τ s. Obviously τ p generally decreases with the increase o the porosity. However at the present Reynolds number (Re=80) the ratios are only slightly smaller than unity though the solid surace areas at the interaces o the simulations are rather small. This is because o the drag orce caused by the surace structure. In act as shown in Fig. 16 at the edges o the porous structure the surace pressure increases due to the low penetration. Thus this implies that the wall riction on the porous interace would be greater than that o the solid smooth wall at higher Reynolds number cases such as those in turbulent low regimes. CONCLUSIONS In the present study several low characteristics in the porous medium-clear luid interace regions are discussed by lattice Boltzmann low simulations. The simulated porous structure is the body-centered-cubic or the unit cube structure whose porosity ranges From the comparison o the results by two relaxation-time (SRT and MRT) models o the LBM discussions on the low characteristics such as the slip velocity and the low penetration by the SRT model are reliable though the MRT model has better accuracy. The simulation results imply that the permeability is subjected to both the porosity and the structure. It is conirmed that the penetration length generally increases with the porosity while the maximum penetration length is less than the pore diameter. The proposed parameter which Fig.15. Porous-wall riction. Fig.16. Surace pressure distribution and velocity vectors in the interace region. is a product o the normalized slip velocity by the Darcy velocity and the normalized pore diameter distributes airly monotonically with the porosity. The predicted stress jump coeicients are always negative since the intrinsic averaged velocity gradient underneath the interace is always smaller than that over the interace. Finally at the bulk Reynolds number Re=80 the simulated wall riction on the porous interace is slightly smaller than that o the solid smooth surace due to the drag orce caused by the surace structure and the low penetration. ACKNOWLEDGMENTS This work was partly supported by the ollowing research unds. Core Research or Evolutional Science and Technology (CREST) o Japan Science Technology (JST) Agency (No R) The Japan Society or the Promotion o Science through a Grantin-Aid or Scientiic Research (B) (No ) and Strategic Development o PEFC Technologies or Practical Application: Development o Technology or Next-generation Fuel Cells o New Energy and

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