Numerical simulation of transition layer at a fluid-porous interface

Transcription

1 International Environmental Modelling and Software Societ (iemss) 2010 International Congress on Environmental Modelling and Software Modelling for Environment s Sae, Fifth Biennial Meeting, Ottawa, Canada David A. Swane, Wanhong Yang, A. A. Voinov, A. Rizzoli, T. Filatova (Eds.) Nmerical simlation of transition laer at a flid-poros interface Carlo Galtieri Hdralic, Geotechnical and Environmental Engineering Department (DIGA), Universit of Napoli "Federico II". Ital. carlo.galtieri@nina.it Abstract: The std of flow phenomena at the sediment-water interface is ver important in the field of environmental hdralics. When a poros laer is overlaid b viscos flid flow, a thin transition zone between the laers is formed, which is responsible for the interfacial momentm and mass transfer. This laer, also referred as Brinman laer, is associated with the depth of penetration of the inflence from the free-flid region. This paper presents the preliminar reslts of nmerical simlation of the flow in a rectanglar geometr representing the region arond the interface between a flid and a poros media. The geometr reprodces that experimentall investigated b Goharzadeh et al. [2005]. The flow was described b the Navier-Stoes eqations in the free region and the Brinman eqation in the poros region. Nmerical simlations were carried ot in laminar stead-state flow sing Mltiphsics 3.5a, a CFD code, in a range of Darc nmber Da from to and of flid-based Renolds nmber Re f from 6 to 21. The thicness of the transition laer, defined as the height below the permeable interface p to which the velocit decreases to Darc scale was measred from the simlated flow field. Nmerical data were generall lower than the experimental data, bt the were in agreement with the observed trends becase the thicness of the transition laer increased with the increasing Da whereas it was not greatl altered b the increasing Re f. Kewords: Environmental hdralics, flid-poros interface, transition laer, CFD, Brinman eqations, experimental validation 1. INTRODUCTION. Flow phenomena and the associated mass and momentm transfer near a poros medim/free flow interface are of great theoretical and practical interest. The are fond in varios fields, sch as environmental hdralics, geophsical flid dnamics, and mechanical engineering among others. In the area of environmental hdralics, the main problem is on clarifing how the free flow in a natral channel will interact with the poros bed, and how the mass, dissolved organic molecles, inorganic ions and flid/gas flx, and momentm will transfer across the free flow/poros bed interface since it ma greatl affect water qalit in both the channel and the bed. For example, the exchange of soltes between the sediment and the water is strongl related to the near-bed shear velocit (Steinberger and Hondzo, [1999]; Higashino et al., [2003]; Galtieri, [2004]; Galtieri, [2005]; Higashino and Stefan [2008]) and near-bed trblence redces the mean hitting time of fine organic particles, reslting in a continos redistribtion of organic particles, which increases the availabilit of resorces near the bed (McNair et al., [1997]). The near-bed trblence also affects the benthic ecolog becase it controls mass transfer processes on permeable river beds (Hettel et al., [2003]; Lore et al., [2003]; Pacman et al., [2004]). Flows over a poros medim are characterized b a hbrid clear flid-poros medim domain that flows both over and throgh the poros medim. The internal flow field of the poros medim remains copled with the overling flid. The complex interaction between the overling flid and the internal flid means the no-slip srface bondar condition is no longer applicable (James and Davis, [2001]; Chan et al., [2007]). Ths, the flid/poros interface problem, the flow interaction above and inside the poros medim, and the transfer mechanisms across the interface, deserve in-depth investigation. This paper

2 presents some reslts of nmerical simlation of the flow in a 2D geometr representing the region arond the interface flid-poros interface. The geometr reprodces that experimentall investigated b Goharzadeh et al. [2005]. The flow was described b the Navier-Stoes eqation in the free region and the Brinman eqations in the poros region. Nmerical simlations were carried ot in laminar stead-state flow sing Mltiphsics 3.5a in a range of Darc nmber Da from to and of flid-based Renolds nmber Re f from 6 to to point ot the inflence of these parameters on the flow field characteristics. Also, the inflence of the flid height on the flow field was evalated. The thicness of the transition laer, defined as the height below the permeable interface p to which the velocit decreases to Darc scale, was finall obtained from the nmerical velocit otpts and compared with the experimental findings of Goharzadeh et al. [2005]. 2. FLOW PROCESSES AT THE SEDIMENT-WATER INTERFACE. The flow of viscos flids in a channel can be described b Navier-Stoes eqations. Inside a poros medim, the global transport of momentm b shear stresses in the flid is often negligible becase the pore walls impede momentm transport to the flid otside the individal pores. Also, althogh at the fndamental microscopic level these eqations still appl, it is qite difficlt to obtain their soltion considering the complexit and often onl statisticall nown geometr of the solid srfaces in the medim. Hence, a homogenization of the poros and flid media into a single medim is a common alternative approach and the flow in a poros medim is sall described b volme averaged eqations, sch as the empirical Darc s law, which states that the volme averaged velocit field is determined b the pressre gradient, the flid viscosit and the strctre of the poros medim: r V = µ p where µ is flid dnamic viscosit [M L - ¹ T - ¹], is poros medim permeabilit [L²], which is a scalar for an isotropic poros medim, and p is the pressre [M L - ¹ T - ²]. These averaged or macroscopic properties are defined at an point inside the medim b means of averaging of the actal flid properties over a certain volme srronding the point. The volme is small compared to the tpical macroscopic dimensions of the problem, bt it is large enogh to contain man pores and solid matrix elements. Eq. 1 together with the continit eqation and eqation of state for the pore flid provide a complete mathematical model sitable for a wide variet of applications involving poros media flows, where the pressre gradient is the major driving force. (1) (a) (b) (c) Free-flid Free-flid Flid-poros interface Interface laer Transition laer Poros medim 0 ε Interface Poros medim Freeflid and poros medim Figre 1. Macroscopic flow above and within a poros medim. If the poros medim is bonding an open channel flow two intitivel clearl distinct flow regions exist and the no-slip condition at the srface of the poros medim generall does not appl (Fig.1). The free-flid region above the permeable wall, where there is no solid matrix so porosit is one, and the poros medim region, where porosit is less than one

3 (Fig.1a) (Porajac and Manes, [2008]). Note that porosit ε is defined as the fraction of the control volme within the poros medim, which is occpied b pores. In the free-flid region flow featres resemble those of bondar laer flows, whereas below the flidporos interface the flow has the featres of poros media flows. Across the flid-poros interface there is the need to establish continit of flid velocities and stresses. Two possibilities hold. The transition between the flow above and inside the permeable wall ma be sharp (Fig.1b), so the flow variables ma have a step change across the macroscopic bondar. Alternativel, if there is a sfficient degree of continit, flow variables gradall change between the two regions and the transition is smooth (Fig.1c). In the literatre, sharp transition is associated with the two-domain simlation models and the smooth transition with the single-domain models, as below described. W Free-flid int Poros medim Darc slip Figre 2. Velocit profile for pressre-driven planar flow in a channel/poros medim. Since there is effectivel a slip velocit at the srface, previos stdies of interfacial flow have focsed on the dependence of this velocit on the characteristics of the medim and the external flow. Consider a laminar flow in the channel which is driven b pressre, where W is channel width (Fig.2). The velocit profile awa from the interface is parabolic and approaching the flid-poros interface the velocit is decreasing down to int, which is the velocit at the interface. Herein, the higher free-flid momentm is transferred into the pores via viscos stress. Below the interface, inside the poros medim, there is a laer where the flow is inflenced b the free-flow field and the velocit of flid particles decreases drasticall, ntil it reaches an averaged constant velocit which is predicted b Darc eqation Darc. This laer is sall termed transition laer or Brinman laer (Fig.1c). This thin laer is responsible for the interfacial momentm and mass transfer and it acts as transition laer between the realms of the Navier-Stoes eqations and the Darc eqation (Goharzadeh et al., [2005]). Besides the transition laer, a space necessar for the solid matrix to achieve the geometrical properties of the poros medim mst be considered (Fig.1b), namel ε (Porajac and Manes, [2008]). This is termed interface laer, which is hence defined b the geometr of the pore space while the transition laer is related to flow conditions. Ths, these two laers have clearl defined and different phsical properties. The above discssion pointed ot that there are two general approaches to describe macroscopic flow within the transition laer, both developed for laminar bondar laer flows: sharp interface and smooth interface approach (Porajac and Manes, [2008]). The first approach consists of two homogeneos regions divided b a sharp interface, which is selected somewhere within the interface region (Fig.1b). At this interface an appropriate conditions, sall expressed as a step change in some spatiall averaged flow variable, is adopted. Beaver and Joseph [1967] introdced a slip velocit slip, which was defined as the difference between the free-flow velocit at the interface int and the Darcian velocit Darc far awa from the interface, below the transition laer (Fig.2). The proposed that, for the for the planar geometr shown in Fig. 2, the slip velocit is proportional to the velocit gradient in the open channel at the interface: U slip 1 2 = α d d = 0 + (2)

4 where α is a dimensionless slip coefficient, which was assmed to be the propert of the poros medim architectre and independent of the flid viscosit. Several researchers investigated the slip coefficient α, pointing ot that it is ver sensitive to the adopted location of the interface (James and Davis, [2001]; Porajac and Manes, [2008]). Other athors assmed continit of the velocit at the flid-poros interface and proposed different shear stress condition (Vafai and Kim, [1990]; Ochoa-Tapia and Whitaer, [1995]). The second approach assmes a continos transition between the free-flid and the homogeneos poros medim (Fig.1c). The phsical properties of the flid are defined as intrinsic volme averages that corresponds to a nit volme of pores and are assmed to be continos over the interface region. The flow velocities are defined as sperficial volme averages and the correspond to a nit volme of the medim inclding both pores and matrix. Ths, the velocit field is continos across the flid-poros interface. This approach eliminates the need for explicit consideration of interfaces and the flow can be modeled b sing the same variables for the entire domain, bt flow eqations need to be properl adjsted to reflect the change of phsics above introdced. This approach was first proposed b Brinman [1947], who added a macroscopic viscos shear stress term, with flid viscosit, to Eq. (1), in order to accont for the velocit gradient in the transition laer: p µ r V ' + µ 2 r V = 0 where µ' is the apparent or effective viscosit. Several theoretical and nmerical stdies were proposed to qantif the dependenc of µ' from both poros medim characteristics and flid viscosit (Tan and Pillai, [2009]). Often µ'=µ for simplicit, bt, if porosit is high, this assmption leads to an overestimation of the interfacial velocit and to high flowrate throgh the free-flid domain. Comparing Eq. (1) and Eq. (3), if the pressre gradient is applied in the x-direction onl and the sperficial velocit gradients exist in the -direction onl, the last term in Eq. (3) ma be considered as interpolating the interfacial velocit int and the Darcian velocit Darc and this deca is (Gpte and Advani, [1997]): ( ) ( ) = Darc + int Darc exp (4) 1 2 ' 1 2 µ µ This is a soltion of Eq. (3) which meets the following bondar conditions: = = Darc int as as = 0 Interestingl, Eq. (1) and Eq. (3) ield an identical vale of int if the slip coefficient α introdced in Eq. (1) is assmed to be eqal to (µ'/µ)¹ / ², that is the sqare root of the ratio between the effective viscosit and the flid viscosit (Gpte and Advani, [1997]). Original Brinman eqation was modified b Ochoa-Tapia and Whitaler [1995] as: p µ r V µ + ε 2 r V = 0 which is identical to Eq. (1) except µ'=µ/ε and is termed modified Brinman eqation. Several stdies were carried ot assming a variable permeabilit and a combination of variable permeabilit and viscosit (Porajac and Manes, [2008]). The approach based on Brinman eqation has the advantage to predict a transition laer thicness which is on the order of ¹ / ², as sggested in man theoretical stdies, bt the variable parameters that give accrate prediction of the velocit profiles cannot be given in a general form. 3. NUMERICAL MODELLING OF FLOW PROCESSES AT THE FLUID- POROUS INTERFACE. As above explained, the smooth or single-domain approach allows to consider the same nnown variables for the entire domain inclding both free-flow and poros medim regions. The distinction is made b switching on and off certain terms in the governing eqations. This approach was applied in this nmerical std. Free-flid flow was simlated (3) (5) (6)

5 b sing the well-nown mass conservation and momentm balance eqations. For a planar, stead-state and incompressible flow, the are: v + = 0 x p 2 ρ + v = + µ + F x x v v p 2 ρ + v = + µ v + F x where in the Navier-Stoes eqations, ρ is flid densit, p is flid pressre, F x and F are force terms acconting for gravit or other bod forces, and, v are velocit components in the x and directions, respectivel. In the poros medim, the governing eqations are again continit and momentm balance eqations. For a planar, stead-state and incompressible flow, continit eqation is Eq. (7), while momentm balance eqations in a poros medim are: µ p µ 2 = + + Fx x ε µ p µ 2 v = + v + F ε Note that Eq. (9), which is formall identical to Eq. (6), can be obtained from Eq. (8) replacing in the left-hand side the advective term b the Darcian term and assming the porosit different from 1. x (7) (8) (9) Figre 3. The investigated geometr. These eqations were solved sing Mltiphsics 3.5a modeling pacage, which is a commercial mltiphsics modeling environment (Mltiphsics, [2008]). It solved Eqs. from (7) to (9) for the pressre p and the velocit vector components and v within the entire domain of the flow. Mltiphsics 3.5a was applied to the geometr presented in Fig.3, which is the lateral view of the channel sed b Goharzadeh et al. in their experimental wors in laminar flow over a poros medim (Goharzadeh et al., [2005]). The condcted their experiments in a rectanglar flme 0.50 m long, 0.10 m wide and 0.05 m high. The central region of the extent m was filled b a random pacing of a poros sample. Experiments were condcted sing five different tpes of transparent borosilicate monodisperse glass beads and poldisperse granlates with phsical properties listed in Table 1, where d is the grain size. The porosit and permeabilit were measred separatel b laborator experiments. The poros material was embedded between two m filters. The pper srface of the poros bed was flattened with a solid object to ensre a horizontal interface. The flme was then filled with a silicon oils mixtre with densit ρ=1006 Kg/m³, dnamic viscosit µ= ² Kg/m s and inematic viscosit ν=

6 6 m²/s. The height of the poros laer was constant for all experiments h p =0.04 m. A niform pressre gradient was maintained in the longitdinal direction of the channel b a pmp with an approximate maximal flow capacit of Q=5 L/min (Goharzadeh et al., [2005]). Using particle image velocimetr (PIV) and refractive index matching (RIM), twodimensional velocit measrements in the interfacial region were performed. The thicness of the Brinman or transition laer δ Brin was evalated from the experimental data sing two approaches. The first criterion was based on a statistical test and the second one assmed bondar laer thicness definitions, as below explained in details (Goharzadeh et al., [2005]). Table 1. Phsical properties of the poros medim: glass beads (GB) and granlates (GL) Samples d - cm - m² ε GL ± ±0.01 GL ± ±0.01 GB ± ±0.01 GB ± ±0.01 GB ± ±0.01 In the experiments, flow conditions were characterized b sing a nmber of different Renolds nmbers. First, a classical flid-based Renolds nmber was defined as max h f Re f = (10) ν where max was the maximm flow velocit at the flid srface in the x direction and h f the height of the flid laer over the poros medim. Goharzadeh et al. [2005] arged that Re f was based on a velocit in the free-flid and hence, was not characteristic for the velocit in the entire domain. So, as it is cstomar in the literatre, the introdced a Renolds nmber Re based on the permeabilit of the poros medim and the Darcian velocit: Re = Darc ν (11) Third, a particle Renolds nmber was defined b sing the grain size: d Re = Darc p ν (12) Finall, the considered a Renolds nmber for the interfacial region: int d Reint = (13) ν Another relevant parameter of a flow over a poros medim is the Darc nmber Da: Da = (14) ( h ) 2 f + h p Twelve nmerical simlations were carried ot sing Mltiphsics 3.5a to investigate the inflence on the flow of (1) the permeabilit of the poros medim, (2) the flid-based Renolds nmber Re f and (3) the height of flid laer above the poros medim h f. The reprodced the experiments performed b Goharzadeh et al. [2005]. The first grop of simlations, i.e. Rn 1 5, was with Re f =20.99, h f =0.04 m and Da in the range from to The second grop, i.e. Rn 6 9, was with Re f from 6 to 17.04, h f =0.04 m and Da= The third grop, i.e. Rn 10 12, was with Re f arond 20.96, h f from 0.02 to 0.05 m and Da= Table 2 lists all the operative conditions, where all of the Renolds nmbers defined for the poros medim are mch below 1. Bondar conditions were assigned at the inlet, the otlet and at the walls of the domain: to obtain the parabolic velocit profile, the bondar condition at the inlet was in =(6 avg s (1-s), where avg is the average velocit and s is a bondar parameterization variable that rns from 0 to 1 along the bondar;

7 at the otlet, a zero pressre tpe condition was assigned; at the wall above the free flid, a smmetr tpe condition was assigned; at the walls besides and below the poros medim, a no-slip tpe condition was assigned; the flid-poros interface was treated as an internal bondar and a continit tpe condition was assigned. Table 2. Ke parameters of the nmerical simlations Rn - m² ε in m/s Da Re f Re Re int E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-03 Different mesh characteristics were tested. After that, the mesh generation process was made assming, among the others, as element growth rate and resoltion of narrow regions 1.3 and 1.00, respectivel. The element growth rate determines the maximm rate at which the element size can grow from a region with small elements to a region with larger elements. The vale mst be greater or eqal to 1. In the resoltion of narrow regions field the nmber of laer of elements created in narrow regions is controlled. Finall, different vales for the maximm element size were selected for the free flid, the poros medim and the flid-poros interface. Maximm elements sizes were 0.002, and m for the free flid, the poros medim and the flid-poros interface, respectivel. Overall, the sed mesh had elements, with a minimm element qalit of (Fig.4). This parameter is related to the element aspect ratio, which means that anisotropic elements can get a low qalit measre even thogh the element shape is reasonable. It is a scalar from 0 to 1. Mesh qalit visalization demonstrated a niform qalit of the elements of the mesh. Figre 4. The generated mesh. Abot the solver settings, stationar segregated solver with non-linear sstem solver was sed, where the relative tolerance and the maximm nmber of segregated iterations were set to ³ and 100, respectivel. The segregated solver allows to split the soltion steps into sbsteps. These are defined b groping soltion components together. This procedre can save both memor and assembl time. 4. NUMERICAL RESULTS. DISCUSSION Nmerical simlations provided velocit field and pressre vales throghot the flow domain. The analsis of reslts was first focsed to compare nmerical velocit profiles with the experimental data. Fig.5 compares the vertical profiles for Rn 5. Fig.5a presents the profile in the entire domain, while Fig.5b is zoomed inside the poros region. Nmerical

8 vales tended to nderestimated vales both in the free-flid region and in the transition laer, whereas there is a reasonable agreement inside the poros medim Rn Rn Goharzadeh et al. [2005] Mltiphsics 3.5a Goharzadeh et al. [2005] Mltiph sics 3.5a 0.0E E E-05 Figs. 5a/5b. Vertical profiles. Overall plot (left), zoom inside the poros medim (right). Second, nmerical vales of the thicness of the Brinman laer δ Brin were compares with the available experimental data. This thicness was obtained from the simlated velocit profile sing the bondar laer thicness definition alread applied b Goharzadeh et al. [2005]. Ths δ Brin was defined as the distance below the flid-poros interface at which the velocit first approaches to within 1% of the Darc velocit D : = 1.01 Darc (15) = δ Brin Table 3 lists these vales for Rn 1 5. Comparison confirmed that nmerical reslts were in the order of magnitde of the experimental data bt nderestimated the depth of the transition region. Table 3. Nmerical and experimental data of the thicness of the Brinman laer δ Brin m Rn Mltiphsics 3.5a Goharzadeh et al. [2005] Third, following what alread carried ot b Goharzadeh et al. [2005] sing their experimental data, the analsis of the nmerical reslts was addressed to investigate the inflence of the permeabilit of the poros medim, the flid-based Renolds nmber Re f and the height of flid laer above the poros medim h f on the flow characteristics. Figs. 6a/6b. Flow field. Rn 1(left), Rn 5 (right). Rn 1 5 provided information abot the inflence of the permeabilit. Fig.6 compares the flow field for the lowest and the largest permeabilit, i.e. Rn 1 and Rn 5. Larger

9 velocities are in red and low velocities in ble. De to the increased permeabilit of the poros medim, the horizontal velocit component at the flid-poros interface int increased from m/s for Rn 1 to m/s for Rn 5. Also, the thicness of the Brinman laer δ Brin increased with the increasing permeabilit from m to m from. This trend was consistent with that observed b Goharzadeh et al. in their experiments [2005] Rn 4 Rn 6 Rn 7 Rn 8 Rn Rn 4 Rn 6 Rn 7 Rn 8 Rn 9 0.0E E E-05 Figs. 7a/7b. Vertical profiles. Overall plot (left), zoom at the interfacial region (right). To std the response of the flow field to the flid-based Renolds nmber Re f five nmerical simlations can be considered, namel Rn 4 and those from the second grop of simlations, i.e. Rn 6 9. Fig.7 shows the vertical profiles of the x-component for these simlations. It can be noted that slopes of the velocit profiles depended on Re f and the increased with the increasing Re f both in the free-flid and in the interfacial region. Frthermore, the thicness of the transition laer was not significantl changed. This reslts were consistent with previos literatre findings (Choi and Waller, [1997]; Goharzadeh et al., [2005] Rn 4 Rn 10 Rn 11 Rn Rn 4 Rn 10 Rn 11 Rn E E E E-04 Figs. 8a/8b. Vertical profiles. Overall plot (left), zoom at the interfacial region (right). For nmerical simlations, namel Rn 4 and Rn 10 12, were sed to investigate the

10 effect of flid height h f on the flow characteristics. Fig.8 reports the vertical profiles of for these simlations. The slope of the profiles increased with the decreasing h f both in the free-flid and in the interfacial region. The same trend was observed for int, whereas a clear trend for δ Brin was missing. Again, these reslts were consistent with the experimental observations b Goharzadeh et al. [2005] Goharzadeh et al. [2005 ] Mltiphsics 3.5 a Goharzadeh et al. [2005] Mltiphsics 3.5a δ Bri n - m E E E E m δ B ri n - m d - m Figs. 9a/9b. Thicness of the Brinman laer vs permeabilit (left), grain size (right). Finall, both the simlated and the experimental thicnesses of the Brinman laer were plotted against the sqare root of the permeabilit and the grain size. After all, nmerical reslts confirmed the experimental finding that the length scale of the transition laer was in the order of grain diameter and mch larger than the sqare root of permeabilit. 5. CONCLUSION. The flow of a flid over a permeable srface is a freqent case in the field of environmental hdralics. It ma be stdied sing the Brinman eqation, which describes the flow in poros media where momentm transport b shear stresses in the flids is of importance. The paper presented the reslts of a nmerical std of the flow in a free flid-poros medim domain. Nmerical reslts pointed ot the strong effect of permeabilit on the flow characteristics and on the thicness of the laer inside the poros medim affected b the free-flid flow. Also, the Renolds nmber and the flid height over the flid-poros interface were fond to affect the gradient of the horizontal velocit component at the interfacial region while the length scale of the transition laer remained approximatel nchanged. These reslts were consistent with literatre experimental finding althogh nmerical reslts nderestimated the thicness of the transition laer. Finall, the confirmed that this laer ma be scaled with the grain size of the poros medim rather that with the sqare root of the permeabilit. REFERENCES Beavers, G.S., and Joseph, D.D., Bondar conditions at a natrall permeable wall., J. Flid Mech., 30, , Brinman, H.C., A calclation of the viscos force exerted b a flowing flid on a dense swarm of particles., Appl. Scient. Res., A1, pp.27 34, Chan, H.C., Hang, W.C., Le, J.M., and Lai, C.J., Macroscopic modeling of trblent flow over a poros medim., International Jornal of Heat and Flid Flow, 28, pp , Choi, C.Y., and Waller, P. M., Momentm transport mechanism for water flow over poros media., Jornal of Environmental Engineering, ASCE, 123, 8, pp , Darc, H., Les fontaines pbliqes de la ville de Dijon., Victor Dalmont, Paris, 1856.

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