EXPERIMENTAL AND NUMERICAL STUDIES ON INERTIAL EFFECT IN POROUS MEDIA FLOW

7th ASEAN ANSYS Conference 1 EXPERIMENTAL AND NUMERICAL STUDIES ON INERTIAL EFFECT IN POROUS MEDIA FLOW Zhiyong Hao 1, Nian-Sheng Cheng 2, and Soon Keat Tan 3 1 Maritime Research Centre, Nanyang Technological University (NTU), Singapore 2 School of Civil and Environmental Engineering, NTU, Singapore 3 DHI-NTU Water & Environment Research Centre and Education Hub, Singapore Abstract Flows in porous media appear in many fields of science and engineering applications. In this paper, the inertial effect on fluid transport in a porous medium is investigated both experimentally and numerically by employing a simplified ordered porous medium model. Transition from linear to nonlinear behavior is evaluated empirically using a friction factor Reynolds number correlation. The numerical model is first validated by comparing computed macroscopic flow characteristics with experimental observation, and then the validated model is used to further explore localized microscopic flow patterns in detail. The results show that the highvelocity flow localized in a narrow tube dominates the bulk characteristics of the fluid transport. The dissipative pressure losses in the recirculation zones around flow separation play a significant role in the nonlinear behavior of porous media flow. Keywords Porous media; Darcy law; Forchheimer equation; Inertial effect; Nonlinear flow 1 INTRODUCTION Flows in porous media are associated with many fields of science and engineering. Classical treatment of pressure losses in groundwater is described by Darcy s law, which indicates that the pressure drop P is linearly proportional to the bulk seepage velocity V (Dullien, 1992), i.e.: P k V, (1) L where k is the coefficient of permeability, is the dynamic viscosity of fluid and L is the length of the segment in the flow direction. The Darcy flow description is valid in the viscous flow regime for which Reynolds number Re << 1. Here Re vd p /, is the fluid density, v is the microscopic velocity, and effective particle diameter. D p is the microscopic Darcy s law is not valid when Re exceeds its critical value, Re 0. By then the porous media flow is characterized by local nonlinear interactions among inertia, viscous and pressure forces (Skjetne and Auriault, 1999a,b). These nonlinear effects lead to higher pressure loss than that predicted by Equation (1) and additional nonlinear terms have been thus introduced, for example, the Forchheimer equation

2 7th ASEAN ANSYS Conference (Dullien, 1992): P 2 V V (2) L where the coefficient corresponds to the reciprocal permeability and is usually called the inertial parameter, or non-darcy flow coefficient. Both and depends on the porosity of the porous material. Although it appears to be empirical, Eq. (2) could actually be derived by performing an appropriate averaging of the Navier-Stokes equation for a one-dimensional, steady incompressible laminar flow of a Newtonian fluid in a rigid porous medium (e.g. Ruth and Ma, 1992). The quadratic nature of the nonlinearity in the Navier-Stokes equations for the microscopic flow vaguely suggests that the average relationship should be quadratic as well (Rojas and Koplik, 1998). A third order term may exist, at least within the transition regime, as shown using numerical simulations, e.g. (e.g. Rojas and Koplik, 1998; Koch and Ladd, 1997), or rigorous theoretical upscaling techniques, e.g. Mei and Auriault (1991). These authors argued that the Forchheimer equation may be valid only at high Re, but before fully developed turbulent flows. Certainly, as pointed out by Andrade et al. (1999), a better representation of experimental data in the non-darcy flow regime can be obtained with a third order velocity correction to the Forchheimer equation. However, the theoretical interpretation for this type of correction is still incomplete. Cheng et al. (2007) investigated the quadratic and power laws using a simplified porous model of ordered glass beads. Their results show that the quadratic law generally represents well (but not exactly) the resistance of the seepage flow covering the Darcy and non-darcy regimes, while the power law is applicable only when the seepage velocity varies within a limited range. They also found that the two parameters included in the power law are generally interrelated, and their relationship derived based on the quadratic law compares well with the experimental results. Turbulence as the cause of the non-linearity has generally been disregarded (Churchill, 1988). Instead, Hassanizadeh and Gray (1987) reported that the plausible source is the viscous force at pore level. Numerical investigations show that the contribution of inertia to the flow in the pore space should be examined first in the framework of the laminar flow regime before taking into account effects of fully developed turbulence (e.g. Edwards, 1990; Koch and Ladd, 1997, Andrade et al., 1999). Ma and Ruth (1993) conducted a simulation of high Forchheimer number flow through a rigid porous medium by means of the volumetric averaging concept. Their work showed that inertial forces distort the streamlines and consequently increase the velocity gradient and also pressure drop. A rigorous treatment of pressure loss related to boundary layers is provided by Skjetne and Auriault (Skjetne and Auriault, 1999b). The authors suggested that the square pressure loss in the laminar Forchheimer equation is caused by development of strong localized dissipation zones around flow separation. Fourar et al. (2004) further showed that the inertial effects lead to the formation of eddies with sizes and forms depending on the Reynolds number. The presence of these eddies reduces the flow-section with consequential increase of the total viscous dissipation. This present study presents the writers investigation of the inertial effect of seepage flow under different flow conditions using a simplified ordered porous media model. Transition from linear to nonlinear conditions was first examined with the correlation between the friction factor and Reynolds number based on the Forchheimer equation. Numerical simulation was then carried out to elucidate the local flow mechanism associated with the inertial effect. The numerical model is first validated by comparing the macroscopic

P/L 7th ASEAN ANSYS Conference 3 characteristics with experimental observation (Section 4), and then used to explore localized flow patterns in detail (Section 5). 2 EXPERIMENTAL SETUP The present experimental setup was the same as those reported by Cheng et al. (2007). The porous medium employed was constructed by placing glass beads into a square steel pipe of 3 m in length (Fig. 1). The diameter of the glass bead, D, was 16 mm and the square pipe measured 16.4 mm 16.4 mm. The glass beads aligned closely in the pipe and the resulted cross-sectional porosity varied from 1 to 0.25. This simplified model was free of irregularities induced by granular configuration and also facilitated control of flows. In particular, its flow boundary was well defined, which is convenient to explore details of fluid flow patterns in the pores using computational fluid dynamics (CFD) techniques. As shown later in this paper, the ordered porous model also provided insight into the transition behaviour and thus local flow mechanism due to the inertial effect. Flow direction measured using a viscometer. Altogether four series of experiments with 242 tests were conducted. The variations of the pressure losses with seepage velocity are plotted in Fig. 2. The average flow velocity V varied from 0.004 to 0.282 m/s. The use of glycerine-water mixture covered a wide range of kinematic fluid viscosity varying from 0.7 10-6 to 35 10-6 m 2 /s. As a result, the Reynolds number, defined as VD / varied from 2 to 5550. The observed flows covered the Darcy-type linear regime and also inertia-dominant nonlinear regime (which will be further discussed in Section 4). 20 15 10 5 Series G00 Series G50 Series G70 Series G80 0 0 0.1 0.2 0.3 Fig. 2 Measured pressure losses varying with seepage velocity. V Glass bead 3 NUMERICAL MODEL Square steel pipe Fig. 1 Schematic diagram of a segment for the packed bed. Pure water and water-glycerin mixture were used as the fluid medium in the experiment. Four kinds of water-glycerin mixtures (G00, G50, G70, and G80) were prepared by setting the concentration of glycerine roughly as 0%, 50%, 67% and 80%, respectively. The viscosity of the mixture which varied with temperature was To understand the non-linear effects, numerical simulations by ANSYS CFX (ANSYS, 2008) were performed to elucidate the phenomena of confined micro-flow in the pore space. The Navier-Stokes equations were solved using translational periodicity interface technique with approximately 2 10 6 control volumes. The solution domain is shown in Fig. 3. The domain interface model is defined as translational periodicity. The simulations converged well with the RMS-residuals less than 10-5 and the maximum residuals less than 10-3. The geometries of the glass bead and steel square pipe used in the laboratory experiment was

4 7th ASEAN ANSYS Conference modelled exactly in the numerical model. No-slip condition was applied over all solid surfaces of the domain. A total of 11 simulation cases were carried out. Table 1 lists the parameters used in the simulations. In the table, V m is the measured bulk seepage velocity and V cal is the computed value. The same fluids, G00, G50, G70, and G80, were also used in the simulations. The error between the measured and simulated seepage velocity was less than 16%, which was deemed acceptable for general CFD studies in view of the inherent uncertainties associated with both numerical simulations and laboratory measurements. Domain interface (inlet) Glass bead Flow direction Wall Domain interface (outlet) Fig. 3 Solution domain, only a quarter being shown. Table 1 Comparison of the selected simulation cases with laboratory measurements. 4 TRANSITION FROM LINEAR TO NONLINEAR REGIME The Forchheimer equation, i.e. Equation (2) can be re-written as where f P / L V 1 f 1 (3) Re' 2 is the generalized friction factor and Re' V / is the modified Reynolds number. Equation (3) which is a friction factor Reynolds number type of correlation may be considered to be universal since it has been used successfully to correlate experimental data for a wide range of porous media and flow conditions (Churchill, 1988). To evaluate the coefficients, and, the measured pressure drops and bulk seepage velocities at different Reynolds numbers were fitted using Equation (3). As shown in Fig. 4, the solid line is the best-fit curve to the Forchheimer equation while the dashed line is that for Darcy s law. The four pairs of and were estimated individually for the four fluids (G00, G50, G70, and G80). A gradual and smooth transition from viscous to inertial regime can be identified readily in Fig. 4. The point of departure from linear to nonlinear behavior is found to lie in the range of 2 1 10 Re' 10, which are consistent with previous observations, e.g. Andrade et al. (1999). While the present experimental results agree favourably with the Forchheimer equation, the numerical model slightly over-predicts the friction factor. In Fig. 4, the generalized friction factor is observed to approach 1 as Re increases, suggesting that the pressure drop varies linearly with the square of the velocity and is independent of viscosity for Re >>1. This trend indicates that the pressure loss should be mainly due to inertial effect

f 7th ASEAN ANSYS Conference 5 rather than viscous effect for high Re, which is consistent with previous conclusions (e.g. Skjetne and Auriault, 1999b; Andrade et al., 1999) 10 2 10 1 10 0 Darcy's law Forchheimer law Simulation results G80 G70 G50 G00 pipe flow that represents the porous media (Cheng et al., 2007). Three values of Re p are selected to cover the transition regime. It can be seen that the closed recirculation zones, caused by the inertial effect, increase gradually in size as Re p increases from 3.5 to 600.8. The closed recirculation zones are characterized largely by low velocity and constant pressure distribution as indicated in Fig. 6. Their contributions to the longitudinal transport are insignificant. 10-2 10-1 10 0 10 1 Fig. 4 Re' Dependence of the generalized friction factor f on the modified Reynolds number Re. 5 LOCALIZED FLOW PHENOMENA Due to their empirical and macro-averaged nature, both Darcy law and Forchheimer equation preclude in-depth physical understanding of flows through porous media. To explore the interplay between the porous structure and fluid flow, it is necessary to relate the local aspects of the pore morphology to relevant mechanisms of momentum transfer, e.g. (e.g. KoponenAet al., 1997; Skjetne and Auriault, 1999b; Andrade et al., 1999; Amiri, 2000; Guo and Zhao, 2002; Fourar et al., 2004). In the present numerical study, a longitudinal vertical plane at z = 6 mm (Fig. 5a) is chosen to investigate the phenomenological flow patterns. This plane passes through the core flow (see the velocity contour on spanwise vertical plane at x = 0 mm for Re p = 600.8), i.e. the high-velocity flow tube as discussed next. The associated streamlines are plotted in Figs. 5(b-d) for different Re p. Here, the seepage Reynolds number Re p is defined as Re DV/ (1 ), where is the average p p porosity. Re p can be understood as a measure of the ratio of inertial to viscous effect in a simplified Fig. 5 Streamlines for different Re on plane z = 6 mm. Fig. 6 Pressure contour for Re p = 600.8 The viscous dissipation in the re-circulation zones is also negligible even for high Reynolds numbers (see Fig. 7 for Re p = 600.8). Even though the dissipative effect of these eddies is negligible, the formation of the re-circulation zones with flow separation reduces the whole flow-section. Skjetne

6 7th ASEAN ANSYS Conference and Auriault (1999b) also found that, for moderate Reynolds numbers, the high-velocity dissipation is highly localized around flow separation region. This is a strong indication that the dissipation would be the direct cause of the square term in the Forchheimer Equation. The present observed phenomena also support Nield s explanation (Nield, 2004) on how the inertial term in the Forchheimer equation contributes to the viscous drag and pressure loss. Nield s explanation of the apparent d Alembert s paradox lies in the recognition that the Forchheimer drag term models essentially the various effects of form drag, boundary layer separation and wake formation behind solid obstacles. Fourar et al. (2004) further suggested that, for periodic flow through porous media, the macroscopic pressure drop results from the contribution of the pressure drag and the viscous drag terms. Fig. 7 Turbulence eddy dissipation contour for Re p = 600.8. According to Skjetne and Auriault (1999b), the lowest layer within the triple deck is a viscous boundary layer (fluid/solid surface), which is governed by Prandtl s boundary layer equations. However, the pressure gradient within this lowest layer is due to the interactions between the viscous boundary layer and the outer irrotational flow. The middle layer (main deck) is the continuation of the wall boundary layer. The flow is accelerated and becomes essentially inviscid and rotational. The outer deck has inviscid irrotational flow. In the present numerical study, the triple deck cannot be resolved completely due to the limitation of computing capability. However, the variation of the pressure P and shear stress along the fluid/solid surface can still be examined in detail. The pressure and shear stress are calculated along the half-upper part of the intersection between the bead and the longitudinal vertical plane at z = 6 mm (see Fig. 5a) and then normalized as in P * and *, respectively, with the value of the external pressure drop P. The results are shown in Fig. 8. For low Reynolds numbers, the inertial forces are comparable to the viscous forces and the streamlines follow the bead surface, as shown in Fig. 5(b). Such flows display symmetrical characteristics, which can also be observed from the distribution of P * and * in Fig. 8(a). This corresponds to Darcy s regime. Fig. 8(a) shows that P * reaches its maximum at the upstream (around x = -0.005mm) while * is nearly zero. Then, the pressure decreases whereas the shear stress increases up to its maximum at x = 0. Therefore, from x = -0.005 to 0mm, the pressure gradient is negative and thus favorable to the flow while the shear stress is unfavorable. As a result, the velocity increases quickly from zero (no-slip condition) to the maximum value in a thin layer, i.e. the viscous boundary shear layer of the triple deck attached to the bead surface, leading to the maximum value of *. After x = 0, both P * and * decease until they reach a local minimum at x = 0.005 mm. It should be noted that, for the whole process, the pressure effect is more significant than that of the shear stress. As Re p increases, inertia forces become more prominent compared to viscous forces and the flow becomes asymmetric, see Figs. 8(b, c). For Re p = 25.8 in Fig. 8 (b), positive pressure gradient (adverse pressure gradient) appears after x = 0.004mm. As a result, the fluid flows countercurrent-wise downstream close to the bead and two eddies which rotate in opposite directions

7th ASEAN ANSYS Conference 7 are formed behind the solid as illustrated in Figs. 5(c). As a consequence of the formation of eddies, the flow-section is reduced, leading to modifications of both pressure and velocity fields. For high Reynolds number at Re p = 600.8, two other eddies are also formed upstream of the bead as shown in Fig. 5(d). The size of eddies also increases with Re. The eddies elongate, coalesce and occupy the whole region afore the bead. Thus, the flow-section is considerably reduced and adverse pressure gradient then occurs upstream of the bead. This phenomenon corresponds to the strong inertial regime. Fig. 8 Evolution of dimensionless pressure and shear stress at the fluid/solid surface in the macroscopic flow direction. The flow is from left to right: (a) very low Re; (b) low Re; (c) high Re. The localized inertial effects in the transition regime for Re p = 600.8 can also be examined using the velocity-isosurface as shown in Fig. 9, which is related to the light yellow (i.e. contour above 0.46 m/s) region of the velocity contour in Fig. 10. The shape of the iso-velocity field resembles a narrow tube, the diameter of which is in the order of the smallest effective width of the pore. These tubes are almost piecewise linear in open pores but their directions change after impinging the pore walls. Generated by the strong inertial motion of the porous media flow, this kind of high-velocity flow tube dominates the fluid transport. Skjetne et al. (1999) suggested that in terms of macroscopic scale, the extra pressure loss is induced by two factors. One is the increase in the wall friction at locations where the impinging of a flow tube yields a boundary layer such as those shown in Fig. 11 for very high Reynolds number (Re p = 10720.2). The high-velocity flow tube is essentially a jet like flow, which tends to bend when it impinges on the pore wall leading to a high viscous dissipation effect as shown in Fig. 12. Secondly, there is a strong segregation of velocities with increasing Reynolds number, and the velocity in the flow tube increases much faster than the interstitial velocity. For example, in Fig. 9, the averaged seepage velocity for Re p = 600.8 is only 0.103 m/s; however, the isovelocity of the flow tube is found to be 0.46 m/s. The physical process behind such velocity segregation is highly nonlinear and may lead to the second order velocity term. Fourar et al. (2004), however, argued that while the above phenomenological arguments are true for an ideal, regularly packed porous media, the same may not be applicable to real porous media. In real porous media with high tortuosity, the flow field will be random and therefore stable recirculation zones are less likely to form. As a result, viscous dissipation may play a more direct role. In fact, high velocity with fully developed turbulence is always dissipative (see Fig. 12) and needs a continuous supply of energy to make up for the viscous losses. Therefore, high velocity porous media flow is dominated by the interactions between pressure and inertial forces by which the energy supply from large scale turbulent structures to the smaller dissipative structures can be sustained. 6 CONCLUSIONS Smooth transition from viscous to inertial regime is identified based on a simplified porous model made up of ordered glass beads. Successful representation of the pressure loss and its

8 7th ASEAN ANSYS Conference dependence on velocity as expressed in the Forchheimer equation is demonstrated using a friction factor Reynolds number type of correlation. Numerical simulation indicates that fluid with high velocity localized in a narrow flow tube dominates the fluid transport in the transition regime, and the dissipative pressure loss in the recirculation zones around flow separation might be the direct cause of the nonlinear term in the Forchheimer equation. Fig. 11 Flow tube of u = 1.48 m/s for Re p = 10720.2. Fig. 9 Flow tube of u = 0.46 m/s for Re p = 600.8. Fig. 12 Turbulence eddy dissipation contour at Re p = 10720.2. REFERENCES Amiri A. (2000) Analysis of momentum and energy transfer in a lid-driven cavity filled with a porous medium. Int J Heat Mass Transfer, 43:3513-3527. Andrade JS, Costa MS, Almeida MP, Makse MA, Stanley HE. (1999) Inertial effects on fluid flow through disordered porous media. Phys Rev Lett, 82(26):5249-5252. ANSYS CFX http://www.ansys.com/products/cfx.asp Fig. 10 u-contour for Re p = 600.8 Cheng N.S., Hao. Z. and Tan S.K. (2008) Comparison of Quadratic and Power Law for Nonlinear Flow through Porous Media Experimental Thermal and Fluid Science, (doi:10.1016/j.expthermflusci.2008.04.007). Churchill SW. Viscous flows - the practical use of theory. Butterworths series in chemical engineering. 1988. Dullien FAL (1992) Porous media: fluid transport and pore structure. San Diego, CA: Academic Press.

7th ASEAN ANSYS Conference 9 Edwards DA, Shapiro M, Bar-Yoseph, P, Shapira M. (1990) The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders. Phys Fluids, 2:45-55. Fourar M, Radilla G, Lenormand R, Moyne C. (2004) On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media. Advances in Water Resources, 27:669-677. Guo ZL, Zhao TS. (2002) Lattice Boltzmann model for incompressible flows through porous media. Phys Rev E, 66:036304. Hassanizadeh SM, Gray WG. (1987) High velocity flow in porous media. Transport Porous Media, 2:521 31. Koch DL, Ladd AJC. (1997) Moderate Reynolds number flows through periodic and random arrays of aligned cylinders J Fluid Mech., 349:31-66. KoponenA, Kataja M, Timonen J. (1997) Permeability and effective porosity of porous media Phys Rev E, 56: 3319 3325. Ma H, Ruth DW. (1993) The microscopic analysis of high Forchheimer number flow in porous media Transport Porous Media, 13:139 60. Mei CC, Auriault JL. (1991) The effect of weak inertia on flow through a porous medium. J Fluid Mech, 222: 647-663. Nield DA. (2000) Resolution of a paradox involving viscous dissipation and nonlinear drag in a porous medium. Transport in Porous Media, 41:349 357. Rojas S, Koplik J. (1998) Nonlinear flow in porous media. Phys. Rev., 58:4776 4782. Ruth D, Ma H. (1992) On the derivation of the Forchheimer equation by means of the averaging Theorem. Transport in Porous Media, 7:255 264. Skjetne E, Auriault JL. (1999a) New insights on steady, nonlinear flow in porous media. European Journal of Mechanics - B/Fluids, 18:131-145. Skjetne E, Auriault JL. (1999b) High-velocity laminar and turbulent flow in porous media Transport in Porous Media, 36:131 147. Skjetne E, Hansen A, Gudmundsson JS. (1999) Highvelocity flow in a rough fracture. J Fluid Mech, 383:1-28.