Journal of Applied Fluid Mechanics, Vol. 14, No. 2, pp. 459-472, 2021. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645. DOI: 10.47176/jafm.14.02.31381

Natural of Power Law Fluid through a Porous Deposit: MRT-LBM Approach

A. Bourada1†, A. Boutra1, 2, K. Bouarnouna1, D. E. Ameziani3 and Y. K. Benkahla1

1 Laboratory of Transfer Phenomena, RSNE Team, FGMGP, USTHB, Bab Ezzouar, Algiers, 16111, Algeria 2 Superior School of Applied Sciences, Casbah, Algiers,16001, Algeria 3 Laboratory of Multiphase Transport and Porous Media, FGMGP, USTHB, Bab Ezzouar, Algiers, 16111, Algeria

†Corresponding Author Email: [email protected]

(Received January 23, 2020; accepted July 25, 2020)

ABSTRACT

In this research, natural convection of power law fluid in a square cavity, with a porous deposit in the shape of a semi-cylinder is studied numerically, using the multiple-relaxation-time lattice Boltzmann method. The modified Darcy-Brinkman model is applied for modelling the momentum equations in porous medium and the Boussinesq assumption is adopted to model the buoyancy force term. The influences of power law index (0.6 ≤ n ≤ 1.4), Darcy number (10−5 ≤ Da ≤ 10−2), (103 ≤ Ra ≤ 106) and the radius ratio of the semi-cylindrical porous deposit (0.05 ≤ R ≤ 0.5) on hydrodynamic and are studied. The obtained results show that these parameters have an important effect, on the structure of hydrodynamic and thermal transfer. The improvement of the power law index leads to a decrease in the heat transfer rate, illustrated by the average , and the augmentation in Darcy number induces an increase in that rate. Moreover, the variation of the Rayleigh number and the porous deposit radius has a significant effect on the transfer rate and convective structure. Besides, an unusual phenomenon is noticed for high Rayleigh numbers, where a better heat evacuation from the porous deposit is noticed for the dilatant fluid compared to the pseudoplastic one.

Keywords: Modified Darcy-Brinkman model; Square cavity; Semi-cylinder.

NOMENCLATURE

Ct tortuosity factor ε porosity of porous medium cp specific heat of fluid  dimensionless temperature Da Darcy number μa apparent dynamic viscosity ei discrete particle velocity  fluid density F total body force  porous matrix coefficient fi density distribution function  kinematic viscosity G driving force e effective kinematic viscosity g acceleration of gravity R dimensionless radius of porous deposit K permeability of the porous medium r dimensional radius of porous deposit k thermal conductivity Ra Rayleigh number K* modified permeability of the porous si relaxation rate medium Sα  strain rate tensor  K consistency coefficient T temperature L length and height of the cavity t time LBM Lattice Boltzmann Method vx dimensional velocity component in x- MRT Multiple-Relaxation-Time direction n power law index vy dimensional velocity component in y-  n modified power law index direction Nu local Nusselt number Vx dimensionless velocity component in x- Nuavg average Nusselt number direction p pressure Vy dimensionless velocity component in y- Pr direction  local shear rate w size of heat source A. Bourada et al. / JAFM, Vol. 14, No. 2, pp. 459-472, 2021. x, y dimensional Cartesian coordinates Subscripts X, Y dimensionless Cartesian coordinates avg average C cold e effective α thermal conductivity eq equilibrium  thermal expansion coefficient H hot α,  cartesian coordinates i direction index  collision operator

1. INTRODUCTION simulated the natural convection of power law fluid in an annulus. Turan et al. (2011) studied The problem of natural convection in a medium numerically the free convection in differentially with porous matrix has attracted the attention of heated cavities, saturated by power law fluids. numerous researchers, due to the wide range of Gangawane and Manikandan (2017) have relevant applications in mechanical, technical and compared, in natural convection, the isothermal chemical fields, such as electronic cooling, fibrous heating to the uniform heat flow of a hexagonal insulation, packed-bed reactors, fluid flow in block attached to the centre of a square enclosure. geothermal reservoirs, crude oil production, grain The latter is filled with power law fluid. Dash and storage system, food processing, drying,…etc. lee (2014) used a new flexible forcing immersed boundary LBM to study numerically the natural Over the last few decades, the free convection of convection in an enclosure, equipped with inclined Newtonian fluids in porous medium has been square hot cylinder. Moreover, Dash and Lee considered by many authors. Basak et al. (2006) (2015) studied free convection within a square have compared, in natural convection, the impact of enclosure with different horizontal and diagonal uniform versus non-uniform heating, in an entirely eccentric square heat sources. Khezzar et al. (2012) porous square cavity, employing the Darcy- have treated natural convection in inclined cavities, Forchheimer model. Jafari et al. (2018) considered saturated by power law fluids, for various aspect natural convection of nanofluid for five different ratios. configurations of square porous cavities, equipped with cylindrical hot pins, using the lattice Nevertheless, natural convection of Ostwald-de Boltzmann method. Ragui et al. (2017) have Waele fluids in porous media has not been treated simulated free convection in a partitioned porous considerably. Jecl and Skerget (2003) have cavity, under the effect of thermosolutal buoyancy presented free convection in totally porous square forces. Habbachi et al. (2017) have examined heat enclosures, containing power law and Carreau transfer in natural convection inside cubic fluids. The work was done using the modified enclosure, equipped with a cubic porous obstacle. Darcy-Brinkman model for flow in porous medium Numerical investigation on the influence of porous and considering the non-Darcy viscous effects. layer thickness was proposed by Chen et al. (2009). Abdel-gaied and Eid (2011) have performed a study Heidary et al. (2016) analysed natural convection in of thermosolutal natural convection of Ostwald-de porous inclined enclosures, equipped with one or Waele fluids, flowing on an axisymmetric body of two obstacles, with the presence of sinusoidal arbitrary form in a porous media. The work was heated wall and magnetic field. A numerical done utilizing the modified Darcy model. In natural investigation on natural convection was performed convection mode, Getachew et al. (1996) have used by Astanina et al. (2018) in a partially porous the modified Darcy equation to model the power square cavity. The latter contains a conductive and law fluid flow in a porous cavity. Kefayati (2016) heat-generating block. The simulations were run used finite difference LBM to simulate using Darcy-Brinkman equation and considering a thermosolutal natural convection of Ostwald-de temperature-dependent viscosity. Omara et al. Waele fluids within inclined porous enclosure. (2016) performed a numerical study on natural Zhuang et al. (2017) have reported a numerical convection in a square porous enclosure with study on thermosolutal natural convection of power partially heated left wall and partially cooled right law fluids within a porous cubic enclosure by one. The study was realized using thermal non adopting the generalized non-Darcy model and equilibrium model and Darcy-Brinkman- considering a chemical reaction. Zhuang and Zhu Forchheimer equation. (2018) have investigated the buoyancy-Marangoni convection of power law nanofluids saturated The investigation of the natural convection of porous cubic enclosure by adopting the Darcy- Newtonian and power law fluids also attracted Brinkman model. Raizah Abdelraheem et al. (2018) considerable attention in recent decades. Dash et al. have presented a numerical investigation of laminar (2014) introduced a novel flexible forcing flow of power law nanofluid, saturated porous immersed boundary LBM for simulating natural media, inside an inclined open shallow enclosure, convection in an annulus of hot eccentric square where the Darcy model is applied. inner cylinder. Habibi Matin et al. (2013) have

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Lattice Boltzmann is a relatively new method compared to conventional approaches in numerical simulation. Yet, between the foundation of statistical physics, on which LBM is based, and its theoretical completion, more than a century has passed. The construction of the lattice Boltzmann method can be summarized in two paradoxically independent steps: the development of statistical physics of one side, and the appearance of cellular automatons on the other side. LBM is one of the most powerful and most used approaches for the simulation of transfer phenomena within a porous medium. This may be relative to its simplicity and its ability to manage complex geometries and boundary conditions. This tool has also been used as a computational approach to simulate flows of non-Newtonian fluids. The scarcity of published works treating the natural convection of non-Newtonian power law fluid in medium with a porous matrix has motivated us to Fig. 1. Physical problem scheme with boundary continue in this field. Such fluid flows are abundant conditions. in many applications both technologically and theoretically. In this article, the MRT-LBM is developed to solve the general equations governing 3. MACROSCOPIC GOVERNING the convective heat transfer of power law fluid in a EQUATIONS square cavity containing a semi-cylindrical porous deposit. The main aim of this study is to determine the combined effect produced by the coexistence of Based on the assumptions mentioned above: the power law fluid and porous matrix, on the governing equations of heat transfer by convection hydrodynamic and heat transfer characteristics. This in their dimensional form, using momentum and effect was computed by implementing the modified energy conservation, are written, in porous medium Darcy-Brinkman model for power law fluid in the and fluid region, as follows (Nebbali and Bouhadef Boltzmann equation. The influence of diverse 2011, Shenoy 1994): parameters such as power law index, Darcy number, Continuity equation Rayleigh number and radius ratio of the semi- cylindrical porous deposit were studied. v v x  y  0 (1) x y 2. PHYSICAL PROBLEM x momentum equation: Two-dimensional natural convection of power law v v v v v  p fluid is studied in a partially heated square x  x x  y x    enclosure (Fig. 1). The hot source of size w (w / L t  x  y 0 x = 0.6), is placed at the central part of the bottom   2v 2v  wall, and kept at a uniform temperature (TH). The e  x  x   F n1  2 2  x (2) side walls are maintained at a constant cold   x y  temperature (TC), while the other walls are adiabatic. The cavity contains a semi-cylindrical y momentum equation: porous deposit of radius r and constant porosity  vy v vy vy vy  p = 0.6, located on the centre of the bottom wall. The  x     porous matrix is considered as homogeneous and t  x  y 0 y isotropic. Also, the fluid and the porous matrix are  2v 2v  thermally in equilibrium. We have taken advantage e  y y    F (3) of the modified Darcy-Brinkman model for n1  2 2  y   x y  modelling the momentum equations in the porous deposit. Energy equation: The fluid is considered as incompressible with In the fluid region: constant thermo-physical properties. Only in the buoyancy term, the density is described by the  T T T   2T 2T   c   v  v   k    Boussinesq approximation (Bejan 2004). The fluid p  x y   2 2  (4) viscosity is equivalent to the effective viscosity and  t x y   x y  the Prandtl number is equal to 10 in all the study. In the porous medium:

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y direction.  T T T   2T 2T   c   v  v   k    (5) p  x y  e  2 2  The shear stress for a power law fluid is depicted in  t x y   x y  tensor form as (Habibi Matin et al. 2013): where vx and vy are respectively the transversal and  v v  longitudinal constituents of the averaged volume  x y   xy  a    (10) velocity, 0 is the average fluid density, p the  y x  pressure and T the temperature,  is the porosity of the porous deposit (this parameter is equal to a is the apparent viscosity defined by: unity in the fluid region), e is the effective n  1 2 2 2 kinematic viscosity,  = [ ( cp)+(1  ) (s cps)] /   2   '   vx   vy   vy vx   a  K 2            (11) ( cp) is the ratio of thermal capacity,  and s are x  y   x y           densities of the fluid and the solid, respectively, cp   and cps are respectively the fluid and solid specific where K  is the consistency coefficient. heat capacities, k and ke are the thermal conductivities of the fluid and the porous matrix, The non-dimensional parameters of this study are: respectively. In momentum equations, n is the 3 power law index, F = (Fx, Fy) refers to the body's x y vx L vy L g TH  TC L X  ;Y  ;Vx  ;Vy  ; Ra  ; total force, caused by the existence of the porous L L e e  e deposit, and the Buoyancy term, given by (Nebbali K *   T  T r and Bouhadef 2011, Shenoy 1994): Da  ;Pr ; J  e ;  0 ; R  2 L e  TH  TC L  n1 F   e v v  G (6) In which e is the effective thermal diffusivity, Ra K* is the Rayleigh number, Da is the number of Darcy, Pr is the number of Prandtl, L is the where v = (vx, vy) is the velocity in the x and y characteristic length,  is the fluid kinetic 2 2 directions, v  vx  vy ,  is the porous matrix viscosity, J is the viscosity ratio and R is the radius ratio of the semi-cylindrical porous deposit. coefficient which is equal to 1 in the porous region and 0 in the fluid region. 4. LATTICE BOLTZMANN APPROACH K* is the modified permeability of Ostwald-de Waele fluid, defined by Christopher and middleman The LBM may be applied to solve the problems of (1965), it relies both on the power law index and the fluid flow, heat transfer and mass transport existence of porous medium: phenomena. In this approach, the fluid is considered n n1/ 2 as a set of particles, which move (stream) with 1  n   50K  K*      (7) discrete velocity in specified directions, depending 2Ct  3n 1  3  on the lattice structure, and collide (interact) with each other on lattice nodes. The distribution function K refers to the intrinsic permeability and Ct is the tortuosity factor defined thus: f is the probability of finding a particle at a given node with a certain velocity. Collision and particle 25 advection are driven by the LB equation, which  Christofer and  12 Middelman (1965) reflects the development of these functions and  external forces introduced by a source term (Boutra  n et al. 2017, Kumar et al. 2017, Shah et al. 2017,   5  Kemblowski and    2(1n)/ 2 Sullivan et al. 2006): Ct    2  Michiewicz (1979)   fi (x  eit ,t t )  fi (x,t)    fi t Fi (12)  ' n' 3(10n 3) ' '  2  8n  10n  3  75 (10n' 11) Dharmadhikari and        '   '  Kale (1985) where fi is the function of distribution with  3  9n  3   6n 1  16 velocity ei at lattice node x at time t, t is the discrete time step, (fi) is the discrete collision (8) operator and Fi is the external forces term. In this work, the Dharmadhikari and Kale (1985) The lattice Bhatnagar-Gross-Krook (BGK) is the expression is used. For this expression, the power most simple of the LB methods, approximating the law index must be changed to a new power index: collision operator with single relaxation time n = n + 0.3(1n). (Bhatnagar et al. 1954, Khali et al. 2013). By adopting this approximation, the collision operator eq The driving force G described by the Boussinesq is linearized around an fi equilibrium state ; it is approximation is given by: thus written in the following matrix form:

G  g  T T0 j (9) fi (x  eit ,t  t )  fi (x,t)  1 eq in which g is the gravity acceleration, β is the  fi (x,t)  fi (x,t) t Fi (13) thermal expansion coefficient, T0 = (TH + TC)/2 is  the reference temperature, j is the unit vector in the eq where fi is the equilibrium distribution function

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eq and  is the dimensionless relaxation time. The local equilibrium distribution function fi for the hydrodynamic field in porous medium is Numerical instability, limitation in representation of calculated by Eq. (17) (Fallah et al. 2012, Kumar et certain flow problems and fixing of Prandtl number al. 2017): and the report of kinematic viscosity to apparent viscosity at the unit values, appear as default of this  9e v2 3v2  method and give limits to its use. f eq  w 1 3e v  i   (17) i i i 2 2 In order to overcome these shortcomings,   d’Humières (1992) proposed the moment approach known as the multiple-relaxation-time model, where the weighting factors wi are given as: employing different relaxation times to simulate the evolution of macroscopic quantities. The execution w0  4/9 ; w1-4 1/9 ; w5-8 1/36 (18) of the collision in an orthogonal moment space leads to an efficient and flexible scheme. This M is the transformation matrix, which projects fi eq model presents an optimal stability compared to the and fi into the moment space via m = Mf and eq eq eq BGK model. In the MRT scheme, the collision term m = Mf , in which m and m are the is defined in the following way (Bouarnouna et al. macroscopic and equilibrium macroscopic variables 2019, Shah et al. 2017): vectors, respectively. The functions of distribution in moment space are described below (Fallah et al. 1 eq 2012, Kumar et al. 2017): i  M Ci mi (x,t)  mi (x,t) (14)    eq where mi and mi are macroscopic and    1 1 1 1 1 1 1 1 1   f0   e       4 1 1 1 1 2 2 2 2  f1  equilibrium macroscopic variables, respectively.           f   4  2  2  2  2 1 1 1 1  2   j  t F     x x   0 1 0 1 0 1 1 1 1   f3  4.1 D2Q9 MRT-LB Equation for the flow 2   m   q    0  2 0 2 0 1 1 1 1  f  Mf Field in Porous Media  x     4  (19) t    j  F   0 0 1 0 1 1 1 1 1 f5  y 2 y      0 0  2 0 2 1 1 1 1  f6  For flow field, the two-dimensional nine velocities  qy       0 1 1 1 1 0 0 0 0   f  model D2Q9 is used in this study (Fig. 2). The p  7   xx         0 0 0 0 0 1 1 1 1  f8  MRT-LB equation (Higuera et al. 1989, Khali et al.  pxy  2013) with a specific treatment of the force term (Kumar et al. 2017, Li et al. 2010) is employed: where  is the fluid density, e is related to energy,  is related to the square of the energy, jx,y f (x  e  ,t  )  f (x,t)  i i t t i are the momentum components j = (jx, jy) = (vx, 1 eq 1  Ci  vy), qx,y are the energy flux in two directions, and  M Ci mi (x,t)  mi (x,t) M t  I  Di (15)  2  pxx,xy refer to the diagonal and off-diagonal strain- rate tensor components.

Density  and momentum jx,y are conserved quantities, while the six other moments of velocity are non-conserved. meq for the non-conserved moments are described as (Liu et al. 2014):

2 2 3 v 3 v eeq  2  0 ;  eq    0 ;   eq eq qx  0vx ; qy  0vy ; (20)  v2  v2   v v peq  0 x y ; peq  0 x y xx  xy  Fig. 2. The D2Q9 lattice structure. In the equilibrium moments above, the incompressibility approximation was used. The The nine discrete velocities ei are given by (Liu et al. 2014, Ghatreh Samani and Meghdadi Isfahani fluid density is written in the following manner  = 2019): (  )   ( is the density fluctuation), and j = (jx, jy)  v. The mean fluid density  is taken  equal to 1 for simplicity. 0,0 i  0  I is the identity matrix and C is the diagonal   ei  cos,sinc   i 1 i  1,2,3,4 (16) relaxation matrix given by:  2   C  diags ,s ,s ,s ,s ,s ,s ,s ,s  (21) cos,sin 2c   2i  9 i  5,6,7,8 0 1 2 3 4 5 6 7 8  4 si are the relaxation rates. Their values must be where c = (x/t) = (y/t) is the lattice speed, and between 0 and 2 to maintain stability. In this x and y are the lattice cells dimensions, these simulation, they are selected as follows: quantities are chosen equal to the unit x = y = t = C  diag1,1.2,1.4,1,1.2,1,1.2,1/,1/  (22) 1, thus c 1. The non-conserved moments relax in linear terms

463 A. Bourada et al. / JAFM, Vol. 14, No. 2, pp. 459-472, 2021. into their equilibrium values (Mezrhab et al. 2010). 8  In the moment space, the collision step is performed  v  e f  t  F (32) 0  i i 2 0 as: i0 * eq mi (x,t)  mi (x,t)  Ci mi (x,t)  mi (x,t) 4.2 D2Q5 MRT-LB Equation for the Temperature Field  Ci  t  I  Di (23)  2  The two-dimensional D2Q5 model (Fig. 3) with five velocities is used in this work for the In the velocity space, the streaming step is temperature field. The MRT-LB equation may be performed by: written as: * fi (x  eit ,t t )  fi (x,t) (24) gi (x  eit ,t t )  gi (x,t)  where f* = M1m*. 1 eq  N Ei ni (x,t)  ni (x,t) (33) m* is the moment after collision.

The constituents of the vector D in the moment space are as given below (Liu et al. 2014): 6 vF 6 vF D  0 ; D  0 ; D   0 ; D   F ; 0 1  2  3 0 x D4  0Fx ; D5  0Fy ; D6  0Fy ; (25)

20 vxFx  vy Fy  0 vxFy  vy Fx  D  ; D  7  8  In the MRT-LBM, the fluid kinematic viscosity is related to the relaxation parameter of the flow field Fig. 3. The D2Q5 lattice structure.  (Guo et al. 2010): 1  1  where gi(x,t) is the function of temperature      (26) 3  2  distribution.

In D2Q5 model, ei are given by: For a Newtonian fluid,  is constant in each node, whereas for a power law fluid,  is variable 0,0 i  0 according to the local strain rate (Sullivan et al.  ei    (34) 2006). The form of local viscosity for the power cos,sinc   (i 1) i 1,2,3,4 law fluid is given by:  2 The function of equilibrium distribution is denoted '  n1   K   (27) by:

 is the local shear rate. It is linked to the eq gi  wi T 1 5eiv (35) symmetrical strain rate tensor's second invariant S  in the following way (Sullivan et al. 2006): where the weighting factors wi are given as:

  2 S S (28) ' ' ' ' ' ' w0  3/ 5 ; w14 1/10 (36) with N is the 5×5 orthogonal transformation matrix, eq which projects gi and gi into the moment space v' v ' with n = Ng and neq = Ngeq, where n and neq S' '   (29) x ' x' are the macroscopic and equilibrium macroscopic temperature variables, respectively. In MRT method, the strain rate tensor S may The functions of temperature distribution in be written as (Fallah et al. 2012): moment space are provided below (Mezrhab et al. 8 8 1 1 eq 2010): S' '   e j' e j ' M SMji fi x,t fi x,t (30) 2 C2   S t j0 i0 n0   1 1 1 1 1   g0        n g where CS = c / 3 is the speed of sound.  1   0 1 0 1 0   1      n  n2   0 0 1 0 1 g2  Ng (37) The macroscopic fluid variables are calculated as       follows:  n3   4 1 1 1 1   g3        n4   0 1 1 1 1  g4  8    fi (31) E is the diagonal relaxation matrix: i0 E  diagr0,r1,r2,r3,r4  (38)

464 A. Bourada et al. / JAFM, Vol. 14, No. 2, pp. 459-472, 2021.

Table 1 Comparison of average Nusselt number for various Da and Ra.  = 0.4 ; Pr = 1 Maximum Liu et al. Nithiarasu et Guo and Zhao Da Ra Present work relative (2014) al. (1997) (2005) deviation (%) 103 1.007 1.010 1.008 1.008 0,198 10−2 104 1.362 1.408 1.367 1.357 3,622 105 3.009 2.983 2.998 3.057 2,480 105 1.067 1.067 1.066 1.066 0,093 10−4 106 2.630 2.550 2.603 2.597 1,843 107 7.808 7.810 7.788 7.792 0,230 107 1.085 1.079 1.077 1.077 0,737 10−6 108 2.949 2.970 2.955 2.935 1,178 109 11.610 11.460 11.395 11.770 3,290

For stability reasons, the relaxation rates are chosen f2,n = f4,n , f5,n = f7,n , f6,n = f8,n (46) as follows: left wall: E  diag1,1/T ,1/T ,1.25,1.25 (39) f1,n = f3,n , f5,n = f7,n , f8,n = f6,n (47) where T is the relaxation parameter for right wall: temperature field. f3,n = f1,n , f7,n = f5,n , f6,n = f8,n (48) The D2Q5 MRT-LB collision process is accomplished as follows: Boundary conditions for Temperature:

* eq For the thermal field (D2Q5 model): ni (x,t)  ni (x,t)  Eni (x,t)  ni (x,t) (40) The Bounce Back condition is also used on the Among the 5 moments, only temperature is adiabatic walls: conserved: upper wall: 4 g2,n = g2,n1 (49) n0  T   gi (41) i0 lower isolated portions:

The non-conserved moments are written in g4,n = g4,n1 (50) equilibrium as follows: (Lallemand et al. 2003) and in the isothermal walls, we applied in the: neq  v T ; neq  v T ; neq  T ; neq  0 (42) 1 x 2 y 3 4 left wall:

The effective thermal diffusivity αe is defined as:  1   g1  2  TC  g3 (51)  4   1   5 20 e  T   (43) 10  2  right wall:

The thermal transfer between the hot cavity wall  1   and the cold fluid is characterized by the local (Nu) g3  2  TC  g1 (52)  5 20 and the mean (Nuavg) Nusselt numbers: lower heated portion: T 1 Nu   and Nuavg  Nu dx (44) y y0 0  1   g2  2  TH  g4 (53)  5 20 4.3 Boundary Conditions In the LBM, conditions should be introduced 5. CODE VALIDATION AND MESH through the distribution function. SENSITIVITY ANALYSIS Boundary conditions for flow: The computational code has been successfully Bounce Back conditions are employed for approved with the numerical works of Liu et al. specifying boundary conditions on the solid walls. (2014), Nithiarasu et al. (1997) and Guo and Zhao The distribution functions at the solid are equal to (2005), for the case of differentially heated porous the distribution functions of the fluid (Rahmati and cavity with isolated horizontal walls. This cavity of Najjarnezami 2016). For the D2Q9 model: porosity equal to 0.4 is filled with a Newtonian upper wall: fluid. Table 1 indicates a good accordance between the average Nusselt numbers obtained from this f4,n = f2,n , f7,n = f5,n , f8,n = f6,n (45) code and those of the mentioned papers, for different Darcy and Rayleigh numbers, with a lower wall:

465 A. Bourada et al. / JAFM, Vol. 14, No. 2, pp. 459-472, 2021.

Table 2 Grid independence study for n = 0.7, 1 and 1.3. Ra = 104; Da = 10−3; R = 0.3. n = 0.7 n = 1 n = 1.3

Relative gap Rela-tivegap Relative gap Grid Nuavg Nuavg Nuavg (%) (%) (%) 8080 4.7666 - 4.1222 - 3.8385 - 100100 4.7880 0.448 4.1347 0.303 3.8499 0.297 120120 4.8048 0.351 4.1441 0.227 3.8560 0.158 140140 4.8115 0.139 4.1488 0.113 3.8575 0.039 160160 4.8119 0.008 4.1499 0.026 3.8582 0.018 180180 4.8125 0.012 4.1505 0.014 3.8586 0.010 200200 4.8126 0.002 4.1505 0.000 3.8587 0.003

maximum relative deviation of about 3.62 %. The with Ra = 104, Da = 10−3 and R = 0.3. The values were compared with the three works and the relative gap of the average Nusselt number of each maximum relative deviation is reported. mesh, compared to the result of the prior mesh, is presented. Based on the simulation runs, the The code is also validated with the numerical work uniform mesh size of 180180 nodes was selected of Turan et al. (2011) in the case of differentially for the rest of the simulations. heated cavity filled by a power law fluid. Figure 4 shows a good correspondence between isotherms obtained from this code and those of Turan et al. 6. OUTCOMES WITH EXPLANATIONS (2011), for several power law indices and Numbers of Rayleigh. This study is performed primarily to exhibit the applicability of the MRT-LBM method and its simplicity of implementation, compared to more 4 5 Ra = 10 Ra = 10 classical methods. The analysis was, therefore, n = 0.6 restricted to the treatment of the effect of certain parameters, namely: power law index, Rayleigh

et number, Darcy number and the radius ratio of the porous deposit, on hydrodynamic and heat transfer.

(2011) The power law index ranges from 0.6 to 1.4, the

Turan al. Rayleigh number varies between 103 and 106, Darcy number takes its values between 10−5 and 0 0.9 .9 −2 0.8 0.8 10 and the radius ratio of the porous deposit is

0.6 0.6 chosen between 0.05 and 0.5. 0.5 0.5 In the simulations below, the necessary parameters

work 0.4 0.4 Present are fixed at these values: Pr = 10,  = 0.6, e / = 0.2 0.2 0 .1 0 1 ( is the fluid's thermal diffusivity), J = 1 and .1 

n = 1.8 To respect the incompressible flow approximation and the stability criterion on , the

et based on U (Ma = U / CS) should be less than 0.3. U  g  T L is the characteristic velocity in

(2011)

Turan

al. thermal convective flows. Figure 5 presents the influence of power law index

0

. 0 9 . on streamlines for multiple Rayleigh numbers with 9

0 0. . 8 8 −3 Da = 10 and R = 0.3. For the elevated numbers

0 . 0.6 6 6 of Rayleigh (Ra = 10 ), the structure of streamlines 0 .4 0.4

work shows that the natural convection transport becomes 0 . 2 Present 0 0 .2

. 1 stronger for the pseudoplastic fluid compared to the 0 . 1 dilatant one. However, the low buoyancy forces in Fig. 4. Comparison of isotherms with those of the lower Rayleigh numbers significantly slow the Turan et al. (2011). movement of the fluid and the flow pattern becomes almost independent of n. Therefore, the effects of shear-thinning and shear-thickening appear Before tackling the simulations results, the mesh insignificant except inside the porous medium, sensitivity analysis was performed to assure a grid where the flux of the pseudoplastic fluid is weaker independent solution. The Mesh influence on the as compared to the Newtonian and dilatant fluids. mean Nusselt number of the hot horizontal wall is This is due to the lightness of this fluid that is shown on Table 2 for various power law indices, deflected by the porous deposit.

466 A. Bourada et al. / JAFM, Vol. 14, No. 2, pp. 459-472, 2021.

Darcy number on flow field is also investigated. Ra n = 0.7 n = 1 n = 1.3 Results are illustrated in Fig. 7 for Ra = 105 and R = 0.3. At higher numbers of Darcy (Da = 10−3 and −2 103 10 ), the flow is able to penetrate more deeply inside the porous deposit without disturbance for pseudoplastic, Newtonian and dilatant fluids because of the high permeability. Nevertheless, the convective flow in porous matrix is weak for lower 104 Darcy numbers (Da = 10−5 and 10−4), given that the deposit becomes quasi-solid and causes deflection of streamlines. But even so, the extremely external streamlines, induced by the strong convective currents existing at the vertical 5 10 walls, are only able to penetrate the porous deposit.

Da n = 0.7 n = 1 n = 1.3

106 10−5

Fig. 5. Streamlines for various power law indices and Rayleigh numbers. Da = 10−3 ; R = 0.3.

−4 The effect of power law index on isotherms for 10 several Rayleigh numbers at Da = 10−3 and R = 0.3 is presented on Fig. 6. For the smallest Rayleigh numbers (Ra = 103), it can be observed that isotherms are similar for all three types of 10−3 power law fluid and the conduction is dominant, given the low intensity of convective currents.

However, for larger Rayleigh numbers, the thermal transfer becomes more convective and the decrease in power law index reduces the thermal boundary 10−2 layer thickness near the walls. This is due to the low apparent viscosity that improves thermal transfer, referring to the increase in the rate of parietal Fig. 7. Streamlines for various power law indices gradients. It is of interest to note that the cold fluid and Darcy numbers. Ra = 105; R = 0.3. follows the vertical walls in a downward movement, in the form of two concentric cells, Figure 8 displays isotherms for different power law given the high density, and then rises through the indices and Darcy numbers with Ra = 105 and R = centre forming a thermal plume. 0.3. Clearly, improving the Darcy number reduces significantly the thickness of the thermal plume and boundary layer for the pseudoplastic fluid. For Da Ra n = 0.7 n = 1 n = 1.3 = 10−5 and Da = 10−4, the conduction heat transfer is dominant within the porous deposit for the three 103 considered fluids. In particular, for the pseudoplastic fluid, the cold fluid descending along the vertical walls goes around the porous deposit.

This phenomenon is attenuated when dealing with the dilatant fluid given the magnitude of the 104 apparent viscosity. Therefore, in the case of pseudoplastic fluid, the hot fluid inside the deposit is prevented from invading the upper regions of the cavity and only a plume of hot fluid is observed. This is not the case for the dilatant fluid where the 105 cold fluid is less reassembled and thus allows the expansion of the hot fluid.

Figure 9 reveals the effect of power law index and Darcy number on the average Nusselt number, 106 calculated on the hot portion of the bottom wall for Ra = 105 and R = 0.3. Note that the heat transfer rate decreases with the increase of power law index. Fig. 6. Isotherms for various power law indices The previous behaviour is owed to the fact that the −3 and Rayleigh numbers. Da = 10 ; R = 0.3. augmentation of n leads to an enhancement in the effective viscosity which reduces convective The impact of varying the power law index and currents in the cavity. Moreover, the heat transfer

467 A. Bourada et al. / JAFM, Vol. 14, No. 2, pp. 459-472, 2021. rate rises sharply with the increase of Da, in of Da = 10−3. It can be explained by the particular in the case of pseudoplastic fluid. This improvement of the convective transport. derives from the reduction in the hydraulic resistance of the porous matrix. In this manner, the 3 fluid circulation and the heat transfer are thereby 24 n = 1.3 Da = 10 increased. When the Darcy number is equal to 10−5 n = 1 Da = 103 and 10−4, it is noticed that n influences slightly 21 n = 0.7 Da = 103 n = 1.3 Da = 105 the heat transfer rate. On the other hand, as Da 18 5 −4 −3 n = 1 Da = 10 increases from 10 to 10 , the impact of n 5 15 n = 0.7 Da = 10

becomes more prominent. For Da = 10−2 the heat avg

transfer rate decreases more sharply with the Nu 12 increase of n. 9

6

Da n = 0.7 n = 1 n = 1.3 3 3 4 5 6 log (Ra) Fig. 10. Effect of Rayleigh number for various −5 10 power law indices on average Nusselt number. Da = 10−5 and Da = 10−3 ; R = 0.3.

The influence of radius ratio, of the semi-cylindrical porous deposit, and Darcy number on the mean 10−4 Nusselt number with n = 0.7 and 1.3 is shown in Fig. 11 and Fig. 12, respectively. Since the hot

partition is in a larger contact with the fluid for the lowest radius ratios, the average Nusselt number 10−3 becomes higher. The rate of heat transfer is less important for lower values of Darcy number, because the porous deposit behaves like a solid and therefor presents an important hydrodynamic resistance. Then, the reduction of its radius reduces 10−2 this resistance, which is less important in the high Darcy numbers.

Fig. 8. Isotherms for various power law indices and Darcy numbers. Ra = 105 ; R = 0.3.

16

20 2 14 Da = 10 18 Da = 103 4 12 16 Da = 10 5

Da = 10 avg

10

14 Nu

12 8

avg 2

Da = 10 3 Nu 10 6 Da = 10 Da = 104 8 5 4 Da = 10 6 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 4 R Fig. 11. Effect of radius ratio for various Darcy 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 n numbers on average Nusselt number. n = 0.7 ; Fig. 9. Effect of power law index for various Ra = 105. Darcy numbers on average Nusselt number. Ra = 105; R = 0.3. Figure 13 illustrates the effect of radius ratio for different power law indices on the mean Nusselt Figure 10 demonstrates the effect of Rayleigh number with Da = 10−4 and Ra = 104. The mean number for different power law indices on the heat number of Nusselt decreases sharply with the transfer rate when Da = 10−5 and 10−3 and R = increase in radius ratio for the pseudoplastic fluid, 0.3. Obviously, the mean Nusselt number rises with while it decreases slightly for the Newtonian one, the rise in the number of Rayleigh. For 103  Ra  and in particular for the dilatant fluid, where an 104, n has a small impact on the rateof heat independence of the Nusselt number according to transfer. This may be attributed to the slow R is noted. By observing those curves closely, an movement of the fluid caused by the low buoyancy unusual behaviour is noticed for the pseudoplastic forces and thus the effects of shear-thinning and and Newtonian fluids, which offer the same transfer shear-thickening did not appear significantly. Above rate as the dilatant fluid for radius ratio greater than Ra = 104, the heat transfer rate is enhanced 0.4. A more careful observation of the isotherms significantly by increasing n, especially in the case corresponding to these cases shows that for the

468 A. Bourada et al. / JAFM, Vol. 14, No. 2, pp. 459-472, 2021. pseudoplastic case, the descending currents along viscosity. For Ra = 104, the local temperature of the cold vertical walls come to bypass the porous Newtonian fluid becomes superior to that of the deposit (less permeable: Da = 10−4) and confine pseudoplastic. This is attributed to the strong the hot fluid to the lower (hot) wall. Then, the heat currents of cold fluid flowing down along the is only evacuated by a central restricted region. vertical walls that reduce the temperature in the Furthermore, for the dilatant fluid, the heat is only case of pseudoplastic fluid. When Rayleigh prevented in its upward ascent by the value of the increases to 105 and 106, the temperature profiles apparent viscosity. In the case of pseudoplastic and are reversed in the part away from the cold walls Newtonian fluids, this upward ascent is hindered by and the dilatant fluid offers the highest temperature this cold fluid cover, which settles above the porous in this part. This can be explained by the fact that deposit and hence the hot fluid cannot cross or when buoyancy forces rise through the increase in bypass. Rayleigh number, the descending currents along the cold vertical walls confine the hot fluid to the lower wall. Therefore, the fluid with the highest apparent viscosity displays the highest temperature values at 6.5 the cavity centre. On the other hand, the parietal

6.0 (horizontal) temperature gradients are more important for the pseudoplastic fluid compared to 6 5.5 the dilatant one, except in the case of Ra = 10 and Da = 10−5, where the gradients are essentially

avg 5.0 similar.

Nu 4.5 2 ONCLUSION Da = 10 7. C Da = 103 4.0 Da = 104 Through this paper, numerical simulation of natural Da = 105 3.5 convection of power law fluid within a partially 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 heated square cavity, with a porous deposit, were R performed by employing a home made code. The Fig. 12. Effect of radius ratio for various Darcy code is based on MRT-LBM. Thereby, the effects numbers on average Nusselt number.n = 1.3 ; of the pertinent parameters, including power-law 5 Ra = 10 . index, number of Darcy, number of Rayleigh and the radius of the semi-cylindrical porous deposit, on hydrodynamic and heat transfer were extensively examined. The resulting predictions produced the 9 n = 0.7 conclusions as follows: n = 1 - The good consistency with previous predictions 8 n = 1.3 indicates the suitability of the used MRT-LBM 7 model with this kind of industrial problems. - The increase in the numbers of Rayleigh and

avg 6 Darcy and the decrease in the power law index, Nu increase convective flow in the cavity and 5 enhance the heat transfer rate.

4 - For lower Darcy numbers, the descending currents of pseudoplastic fluid along the cold vertical walls 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 come to bypass the porous deposit and confine the R hot fluid on the lower wall. The heat is then only Fig. 13. Effect of radius ratio for various power evacuated by a central restricted region which is law indices on average Nusselt number. Da = as low as n is small. 10−4 ; Ra = 104. - The average number of Nusselt is higher for the lowest radius ratios of the semi-cylindrical porous deposit, and the improvement in the Darcy Figure 14 exposes the local temperature at Y = 0.5 number enhances the thermal transfer rate for various power law indices and Rayleigh especially for pseudoplastic fluid. numbers with R = 0.3, Da = 10−5 and Da = 10−2. For each Rayleigh number, the increase in Darcy - For a porous deposit radius ratio greater than 0.4, number from 10−5 to 10−2 shows a noticeable the pseudoplastic and Newtonian fluids have impact on the temperature profile. This is attributed similar heat transfer rates to that of the dilatant to the increased role of convection on the heat fluid in the case of low permeability. transport inside the cavity as the permeability improves. For Ra = 103, Similar temperature - An unusual phenomenon is noticed for high profiles are obtained with all three types of fluid for Rayleigh numbers, where better heat evacuation Da = 10−5. However, for Da = 10−2, the local from the porous deposit is noticed for the dilatant temperature increases by decreasing the power law fluid compared to the pseudoplastic one. index. This is attributed to the decrease in apparent

469 A. Bourada et al. / JAFM, Vol. x, No. x, pp. x-x, 200x.

Ra Da = 10−5 Da = 10−2 0.0 0.0 n = 1.3 n = 1.3 n = 1 n = 1 -0.1 n = 0.7 -0.1 n = 0.7

-0.2 -0.2

  3 10 -0.3 -0.3

-0.4 -0.4

-0.5 -0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X X 0.0 0.0 n = 1.3 n = 1.3 n = 1 n = 1 n = 0.7 -0.1 -0.1 n = 0.7

-0.2 -0.2 

4 10 -0.3 -0.3

-0.4 -0.4

-0.5 -0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X X 0.0 0.0 n = 1.3 n = 1.3 n = 1 n = 1 -0.1 n = 0.7 -0.1 n = 0.7

-0.2 -0.2

5 10 -0.3 -0.3

-0.4 -0.4

-0.5 -0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X X 0.0 0.0 n = 1.3 n = 1.3 n = 1 n = 1 -0.1 n = 0.7 -0.1 n = 0.7

-0.2 -0.2

 

6 10 -0.3 -0.3

-0.4 -0.4

-0.5 -0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X X Fig. 14. Temperature distribution at Y = 0.5 for different power law indices and Rayleigh numbers. Da = 10−5 and Da = 10−2 ; R = 0.3.

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