Transp Porous Med https://doi.org/10.1007/s11242-017-0995-9

Numerical Simulation of Turbulent Flow and Heat Transfer in a Three-Dimensional Channel Coupled with Flow Through Porous Structures Guang Yang1,2 · Bernhard Weigand2 · Alexandros Terzis2 · Kilian Weishaupt3 · Rainer Helmig3

Received: 22 August 2017 / Accepted: 18 December 2017 © Springer Science+Business Media B.V., part of Springer Nature 2018

Abstract This study investigates numerically the turbulent flow and heat transfer characteristics of a T-junction mixing, where a porous media flow is vertically discharged in a 3D fully developed channel flow. The fluid equations for the porous medium are solved in a pore structure level using an Speziale, Sarkar and Gatski turbulence model and validated with open literature data. Overall, two types of porous structures, consisted of square pores, are investigated over a wide range of Reynolds numbers: an in-line and a staggered pore structure arrangement. The flow patterns, including the reattachment length in the channel, the velocity field inside the porous medium as well as the fluctuation velocity at the interface, are found to be strongly affected by the velocity ratio between the transversely interacting flow streams. In addition, the heat transfer examination of the flow domain reveals that the temperature distribution in the porous structure is more uniform for the staggered array. The local heat transfer distributions inside the porous structure are also studied, and the general heat transfer rates are correlated in terms of area-averaged Nusselt number accounting for the effects of Reynolds number, velocity ratio as well as the geometrical arrangement of the porous structures. Keywords Porous structure · Convective heat transfer · Turbulence · Numerical simulation · Pore scale

List of Symbols cp D

B

Specific heat capacity of the fluid [J/(kg K)] Size of the microcolumn (m)

Guang Yang [email protected]; [email protected]

1

Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University, 200240 Shanghai, China

2

Institut für Thermodynamik der Luft- und Raumfahrt, Universität Stuttgart, 70569 Stuttgart, Germany

3

Institut für Wasser- und Umweltsystemmodellierung, Universität Stuttgart, 70569 Stuttgart, Germany

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Ek H k L Lr p P Pr r Re Rec,L Rep,D t T V VR X, Y, Z

Turbulence kinetic energy (m2 /s2 ) Distance of neighbouring columns (m) Thermal conductivity [W/(m K)] Length of the channel cross section (m) Reattachment length (–) Pressure (Pa) Pressure (–) Prandtl number (–) Grid refinement ratio Reynolds number (–) See Eq. 4 See Eq. 6 Time (s) Temperature (K) Velocity (–) Velocity ratio (–) Dimensionless coordinate system (–)

Greek Symbols ρ τ μ θ ϕ

Density (kg/m3 ) Dimensionless time (–) Dynamic viscosity of the fluid (Pa s) Temperature Porosity (–)

Subscripts c p x, y, z

Channel Porous media Cartesian coordinates

Superscript 

Dimensional variable

1 Introduction The understanding of coupled free flow and flow through coarse porous structures is of great significance due to its direct applicability in many industrial processes (transpiration cooling, fuel cells, oil extraction, etc), environmental flows (water flows over seabed, melt migration) as well as biological applications (flow at the wall of organic tissues) (Davis 2002; Kollet and Maxwell 2006; Levenberg et al. 2005; Wu et al. 2017). For example, porous media are used in transpiration cooling due to their large specific surface area that enhances heat transfer. The coolant flows through the porous structure removing heat by convection and then forms a film layer near the surface which blocks the heat transfer from the main flow.

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Due to the complex geometry of the porous structures, a macroscopic model is used in most of numerical studies, where general transport equations are integrated over a representative elementary volume (REV) that accommodates both the fluid and solid phases. This approach was derived from an empirical point of view by Darcy’s equation, with its Brinkman’s correction or Forchheimer’s equation in proper cases. Dahmen et al. (2015) conducted a numerical investigation on the boundary layer of transpiration-cooled turbulent channel flow. Separate solvers for the hot gas flow in the channel and the porous medium flow were used, which were coupled to each other by boundary conditions imposed at the interface. Liu et al. (2013) studied the flow and heat transfer characteristics for a coupled flow through sintered porous flat plates with hot air flow. In their simulation, the main stream flow was calculated by solving the Navier–Stokes equations using the standard k − ε, SST k − ω and RNG k − ε turbulence models, while the Brinkman–Forchheimer-extended Darcy equation was used to model the coolant flow through the porous media. As for the energy equation within the porous medium, there are mainly two different models: the local thermal equilibrium model and the local thermal non-equilibrium model (where two energy equations are solved). The local thermal equilibrium model assumes that the solid temperature is equal to the fluid temperature, and hence, the heat exchanges between solid and fluid phases in the porous medium are not separately considered and only one energy equation is solved. As the assumption of local thermal equilibrium between the fluid and the solid is inadequate for many applications, a local thermal non-equilibrium model has also been developed for the investigation of conduction and convection heat transfer processes in porous media (Nuske et al. 2014; Vortmeyer and Schaefer 1974). However, both the momentum and two-equation energy equations of macroscopic models need additional input parameters from experiments, pore-scale numerical simulations or empirical equations (Mayer 2014; Yang and Nakayama 2010). Contrary to the macroscopic model, pore-level numerical simulations have been also performed in porous structures in order to obtain the microscopic aerothermal features of the flow. By using an upscaling procedure, the microscopic description can be used to improve macroscopic models. The pore-scale geometry of the porous media can be reconstructed from scanning electron microscope (SEM) or microcomputed tomography (CT) images (Acosta and Camacho 2009; Ghanbarzadeh et al. 2015), and then, the transports in solid and fluid phases are solved separately. The structural units composing the porous structures were also often simplified as basic shapes such as cylinders, cubes, spheres (Gao and Xu 2015; Torabi et al. 2016; Uth et al. 2016; Xu and Jiang 2008), as the geometry models are easier to build with high precision while some macroscopic parameters, such as porosity, can be directly inferred. By considering the effects of microstructure parameters, such as particle shape, orientation, pore to throat size ratio, and unit cell distributions (Ozgumus and Mobedi 2016; Yazdchi et al. 2011; Liu et al. 2017), the features for a particular configuration of porous medium can be extended to more general cases. If the velocity in the porous medium is high, e.g. the Reynolds number based on particle diameter is higher than 100–300 (Pedras and de Lemos 2001; Prausnitz and Wilhelm 1957; Uth et al. 2016), a turbulence-like structure will occur within the pores. The flow and heat transfer in the porous structures, the boundary layers in the channel as well as at the interface of the two domains will become much more complex (Brillant et al. 2008a, b; Suga et al. 2017). This is because the parameters in the two domains strongly interact with each other. Fetzer et al. (2016, 2017) extended the concept, where compositional single-phase free flow was coupled to compositional two-phase porous medium flow under non-isothermal conditions, to turbulent free-flow conditions. Rosti et al. (2015) carried out a number of direct numerical simulations of turbulent channel flow over two porous walls. Turbulence

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statistics and instantaneous flow fields were used to study the main changes introduced by the porous material. Zampogna and Bottaro (2016) studied the interaction between a fluid flow and a transversely isotropic porous medium, and different interface conditions were evaluated in their study. L¯acis and Bagheri (2017) computed the effective boundary conditions at the interface between a porous surface and an overlying flow. They derived a tensorial generalized version of the empirical condition suggested by Beavers &Joseph and compared the homogenized model results with DNS. However, the coupled effect considering both the imposed turbulent flows in a channel and in a porous medium, as well as the heat transfer effect have not been fully understood yet. Therefore, a fully resolved pore-scale numerical simulation is necessary to obtain the microscopical parameters at the flow domain. The objective of the present work is to gain a deep insight into the flow and heat transfer characteristics in coupled turbulent flows consisting of a 3D channel flow and a flow through a porous medium, and to reveal the interaction effect of the two kinds of flows. The continuity, energy and Navier–Stokes equations, together with a Reynolds stress SSG turbulence model for the fluid inside the pore structures, are solved simultaneously with those in the channel flow. Two types of porous structures are considered: an in-line and a staggered microarray of square pillars are examined over a wide range of Reynolds numbers. The pore-scale velocity and temperature profiles inside the porous structure and at the interface, as well as the flow reversal in the channel are in detail investigated over a wide range of Reynolds numbers. The results are analysed by various post-processing procedures aiming to quantify and clarify the flow and heat transfer characteristics of the coupled flow domain at various conditions. In addition, the heat transfer rates inside the porous medium are successfully correlated in terms of area-averaged Nusselt numbers.

2 Physical Description and Governing Equations The geometry and the relevant dimensions are schematically shown in Fig. 1a. Nitrogen gases with a uniform free-stream velocity (Vc ) and a constant temperature (Tc ) enter a channel with a square cross section of L × L . Similarly, nitrogen gases with a uniform velocity (Vp ) and constant temperature (Tp , higher or lower than Vc ), are injected through the porous structure and mix with the channel flow in a T-junction arrangement. Similar to the channel, the branch with the porous medium has a cross section of L × L and a length of L/2. The surfaces of the solid skeleton of the porous structure are kept at constant temperature Tc , equal to the channel flow. All other walls in the model are considered adiabatic. The channel length extends 10L upstream of the porous zone in order to ensure fully developed conditions. The channel length downstream the porous zone is 20L, in order to obtain the full flow patterns resulted from the transverse mixing of the two flow streams. All solid surfaces are considered by using a no-slip condition for the fluid velocity. An in-line and a staggered pattern geometrical arrangements are considered for the porous medium, both of them consisting of cubic square microarrays, as shown in Fig. 1b, c. All square columns have a cross section of D × D, and the centrelines are in Z -direction. The distance between neighbouring columns is H . The porosity of the porous structures can be determined by ϕ = 1 − (D/H )2 . In this study, D = L/40 and H = 2D resulting a porosity ϕ value of 0.75. The flow is considered to be incompressible with constant fluid properties (ρ = 1.2kg/m3 , c p = 1006 J/(kg K), k = 2.54 × 10−2 W/(m K), and μ = 1.82 × 10−5 Pa s), and the effect of buoyancy is neglected. This is reasonable in many applications since the Mach number is

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Fig. 1 a Schematics of flow in a 3D channel coupled with flow through porous structures; b porous structure with in-line array and c staggered array

less than 0.25 and the Richardson number (Yang et al. 2013) is far less than 0.1, if L = 0.1m and the temperature difference between Tp and Tc is less than 100 K. As the flow in the porous structure is modelled on the pore scale, fluids in both the channel and within the porous structures are governed by the continuity equation, the Navier–Stokes equations and the energy equation. The non-dimensional form of equations is given by: ∂ Vi =0 ∂ xi      ∂Vj ∂ Vi 1 ∂ Vi ∂  ∂ ∂P   − Ui U j + + + Vi V j = − ∂τ ∂Xj ∂ Xi ∂ X j Rec,L ∂ X j ∂ Xi    ∂θ 1 ∂θ ∂ ∂  Vjθ = − V j θ  + ∂τ ∂Xj ∂ X j Pr Rec,L ∂ X j

(1) (2) (3)

where i = 1, 2 or 3, j = 1, 2 or 3, X 1 = X , X 2 = Y , X 3 = Z , V1 = Vx , V2 = Vy and V3 = Vz . The non-dimensional variables are defined as: τ = T −T

t·Vc L ,

Vi =

Vi Vc ,

Xi =

X i L ,

P = ρVp 2 , and θ = T −Tpc . −Ui U j is the Reynolds stress, computed using a Reynolds stress c turbulence model (RSM) (Hanjalic and Launder 1972). The quantity Rec,L in the above equations is the Reynolds number characterizing the channel flow, based on the inlet velocity into the channel Vc and the characteristic length L, which is expressed as:

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Rec,L =

ρVc L μ

(4)

μcp k

(5)

The Prandtl number is defined as: Pr =

The value of the Prandtl number is kept constant, 0.72, in the present study. The dimensionless physical boundary conditions are given by: ∂θ Channel inlet (X = −10): Vx = 1, Vy = Vz = 0, θ = 1, and ∂Y = ∂∂θZ = 0. Inlet of the porous structures (Y = 0): Vy = Vp /Vc , Vx = Vz = 0, θ = 0, and ∂∂θX = ∂θ ∂ Z = 0. Channel outlet (X = 21): ∂∅ ∂ X = 0, ∅ is any variable. ∂θ Channel walls: Vx = Vy = Vz = 0 (no-slip) and ∂n = 0. n is the direction perpendicular to the wall. Surface of porous structures: Vx = Vy = Vz = 0, and θ = 1. The Reynolds number based on the macroscopic velocity in the porous structures and the size of the microcolumns D is also used in this study, which is defined as Rep,D =

ρVp D μ

(6)

where Vp is the macroscopic uniform velocity in the porous structure, defined as: Vp = Vp,D ϕ

(7)

where Vp,D is the microscopic average velocity in the pores of the porous structure and is calculated by the superficial velocity at the inlet of the porous structure Vp : Vp,D =

H Vp D

(8)

The velocity ratio (VR) of the two inlets is also considered, which is expressed as: VR = Vp /Vc

(9)

The VR also equals to the ratio of the mass flow rate through the porous structures to that in the channel, since the density is considered to be constant and the areas of the two inlets are the same in this study. It can be inferred from Eq. (4), (6), (7), (8) and (9) that Rec,L , Rep,D , and VR are not independent, but related by the following equation: VR =

Re p,D L Rec,L H ϕ

(10)

The heat transfer between the solid porous structure and the fluid is calculated by the Nusselt number:  hL ∂θ  (11) Nu = =− k ∂n surface The average Nusselt number in the porous structure is obtained by averaging the local Nusselt number over the complete surface of the porous medium.

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Fig. 2 Numerical grid of the flow domain. The porous structure of the in-line array is displayed

3 Numerical Details 3.1 Solution Methodology The simulations are conducted with the commercial CFD package ANSYS CFX-17.1 on a cluster of 28 CPU nodes with dual processor Intel(R) Xeon(R) CPU E5-2690 v4 @ 2.60 GHz. The Reynolds stress model (RSM) of Speziale, Sarkar and Gatski (SSG) (Speziale et al. 1991) is used for modelling the turbulence, including a scalable wall function (ANSYS Inc 2016). This model has been proven to perform better than any two-equation turbulence model for the simulation of the pressure loss in complex metallic porous structures by comparison with experimental data (Mayer 2014). High-resolution schemes are used for both advection and turbulence (ANSYS Inc 2016). The calculations were run in stationary mode and they were considered to convergences when the residuals of all variables were less than 10−5 . The grid structure is shown in Fig. 2 (only the porous structure of an in-line arrangement is displayed). The grid cells of the whole computational domain were rectangular cuboids. The structural grid distribution employed was non-uniform, varied in all directions, providing a finer mesh near the walls as well as the internal surface of the porous skeleton.

3.2 Grid Independence Test A grid sensitivity study was performed in order to select the proper number of grid cells to be used for the calculations. The test was performed for Rec,L = 1 × 105 and Rep,D = 1 × 103 for an in-line array porous structure. Total element numbers of about 4.59 million (N1), 6.4 million (N2), and 9.7 million (N3) were tested, as shown in Table 1. The grid convergence index (GCI) (Celik et al. 2008), which is based on a Richardson extrapolation and has been successfully used in complex flow patterns (Caggese et al. 2013; Fechter et al. 2013), is calculated to quantify the discretization error of the solution between the coarse and fine

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G. Yang et al. Table 1 Grid independence test results for Rec,L = 1 × 105 , and Rep,D = 1 × 103 , for the porous structure of in-line array

N1

N2

N3

4,594,212

6,441,344

9,017,212

r21

r32

Nu1

Nu2

Nu3

1.4

1.4

91.3729

92.3425

92.8134

p

GC I 21

GCI32

2.146

1.24%

0.60%

grids. For the present study, the GCI was computed based on the average Nusselt number of the porous structures. Grids N1 to N3, with global refinement factors of r = 1.4 for each refinement step, were examined. p = 2.146 is the order of convergence. The discretization error of N2 is 0.6%, and thus it was used for the calculations. Similar tests were also performed for the staggered array, and a total grid number of about 6.4 million has been also proven to be sufficient.

3.3 Validation of the Numerical Method In order to validate the present numerical method, direct comparisons with experimental and numerical studies are conducted, and the validation is presented in Fig. 3a, b. Figure 3a shows the variation of Nusselt number with Reynolds number for the flow through an array of two-dimensional staggered square columns. All the column surfaces are isothermal and maintained at a constant temperature different from the temperature of the bulk flowing fluid. Periodic boundaries are used for the simulations. Over a large range of Reynolds numbers (characterized by the size of the square column), the here predicted Nusselt numbers agree well with those reported by Saito and Lemos (2006), Wakao et al. (1979) and Kuwahara et al. (2001). In addition, the present numerical scheme and code were used to calculate the flow pattern in a three-dimensional T-junction flow without porous medium, which had been experimentally studied by Hirota et al. (2006). The same boundary parameters as described in their experiments are used for the simulation. The contours of Vx /Vin at the middle plane of the channel for the case of AR (aspect ratio) = 2 and VR (velocity ratio) = 1, are presented in

Fig. 3 Comparison of a area-averaged Nusselt number and b streamwise velocity Vx downstream of the mixing junction with benchmark results

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Fig. 3(b), indicating excellent agreement with the experimental flow pattern as well as the length of the reversal zone downstream of the junction. The comparison of the distribution of the local streamwise velocity at X/B = 1 is presented on the right-hand side of the contours in Fig. 3b, which also shows a good accordance. Geometrically, the model in the present study can be regarded as a combination of the above two models. Based on the above two validation cases, it can be safely concluded that the numerical method used in this study is able to represent reasonably the complex flow and heat transfer of coupled turbulent flow with the porous medium.

4 Results and Discussion Numerical simulations of the turbulent flow and heat transfer in a three-dimensional channel coupled with flow through a porous structure are performed for the parameter range of 1 × 104 ≤ Rec,L ≤ 5 × 105 and 5 × 102 ≤ Rep,D ≤ 1 × 104 , considering two types of porous structures.

4.1 General Flow Patterns Figure 4a, b shows the dimensionless streamwise velocity Vx at the middle plane of the channel and the porous zone (Z = 0.5) for the two types of porous structures at Rec,L = 1 × 105 and Rep,D = 1 × 103 , that result in a velocity ratio of VR = 0.267. First, it can

L

Vx 26 24 22 20 18 16 14 12 10 8 6 4 2 0 -2 -4 -6 -8 -10

L/2

D

(a)

(b)

Fig. 4 Distribution of dimensionless streamwise velocity Vx in the middle plane of the channel (Z = 0.5) for Rec,L = 1 × 105 and Rep,D = 1 × 103 for a in-line and b staggered array arrangements of porous structures

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L/2

D

(a)

(b)

Fig. 5 Distribution of dimensionless crosswise velocity Vy in the middle plane of the channel (Z = 0.5) for Rec,L = 1 × 105 and Rep,D = 1 × 103 for a in-line and b staggered arrangements of porous structures

be seen that the velocity distributions in the channel are similar for both pore structures arrangements. A flow reversal zone in the channel can be found downstream of the porous zone, which is similar to the those observed in clear T-junction flows (Hirota et al. 2006). In addition, a close-up look at the interface in the middle of the channel shows that the overall macroscopic velocity profiles also do not significantly differ. However, on the pore scale of the porous structures, the inter-pore velocities behave strongly different for the two types of porous structures. For Rec,L = 1 × 105 and Rep,D = 1 × 103 , a larger region of negative velocity can be found in the pores of the staggered array and the magnitudes of both negative and positive velocities are higher than those in the in-line arrangement. The dimensionless crosswise velocity Vy is presented in Fig. 5. Similarly to Fig. 4, the velocity contours in the channel are also similar for the two column arrangements, while they are quite different inside the pores of the porous medium. In the channel, the maximum Vy is located above the porous zone while the velocity at the trailing edge of the porous structure and near the interface is significantly larger for the in-line array. This is due to a clearer flow path where the flow through the porous medium encounters less solid obstacles towards the free flow channel, and hence, lower pressure drop (The dimensionless pressure differences (P/(1/2ρVp2 )) between the inlet and outlet of the porous medium region are 21.78 and 51.40, respectively, for in-line and staggered porous structures at the same boundary conditions in Fig. 5.) With other words, the pillars of the staggered arrangement somehow eliminate the development of the crosswise velocity component due to the systematic blockage of the vertical flow.

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Fig. 6 Distribution of dimensionless velocities Vx and Vy near the interface between the channel and the porous structure for VR = 0.267. a Vx for in-line array, b Vy for in-line array, c Vx for staggered array, d Vy for staggered array

4.2 Velocity Near the Interface A quantitative comparison of the velocity distributions at the interface between the channel and the porous medium is shown in Fig. 6, which plots the normalized Vx and Vy velocity components along the X -coordinate of the porous structure. The locations are chosen 0.5D/L above and below the interface Y = 0.5. As expected, both Vx and Vy fluctuate according to the locations of the pores of the porous structure. The trends of the velocities at the two locations are similar for the in-line array but differ for the staggered array. In Fig. 6a, different inlet velocities at both the channel and porous structure are chosen, but the velocity ratios are always kept constant at VR = 0.267. It can be seen that the velocities at the same location are coincident for the different combinations of inlet velocities but the same velocity ratio, which means that the overall flow pattern is only affected by the velocity ratio. As qualitatively shown in Fig. 5, a higher Vy velocity component at X = 1 for the in-line array can also be found when comparing Fig. 6b, d. Although the velocity level is similar at X = 0, it can be clearly observed that Vy is about 40–45% higher for the in-line array compared to the staggered arrangement. The variations of dimensionless streamwise velocity Vx with Y-coordinate at different locations of the middle plane (Z = 0.5) for in-line porous structure at Rec,L = 1 × 105 and Rep,D = 1 × 103 are presented in Fig. 7. Though microscale oscillations exist in the porous

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Vx 1.5

0

0.8

Vx 1.6

0

0.8

1.6

Y

1.0

0.5

0

0

0.8

1.6

Vx

-0.4

0.4

1.2

2.0

Vx

Fig. 7 Variation of Vx with Y-coordinate at different locations

structures, a slip streamwise velocity can be found at the interface between the channel and the porous media. It has been proved that slip velocity may have a significant physical effect on the characteristics of the overlying fluid. As the flow develops downstream, the thickness of the velocity boundary layer in the channel is clearly increased, which is affected by the upward flow from the porous structures.

4.3 Flow Reversal in the Channel As observed in Figs. 4 and 7, there is a flow reversal zone in the channel downstream of the porous structure. The dimensionless lengths of the flow reversal zone for the two types of porous structures and different velocity ratios are shown in Fig. 8. The length of the reversal zone (reattachment length, L r ), for both the in-line and the staggered arrays, increases monotonously with increasing VR. This is expected since the boundary layer in the channel is affected by the flow through the porous structures. Over the complete range of velocity ratios, the reattachment length of the in-line array is about 20% longer than the staggered one, as a direct consequence of the higher Vy velocity components at trailing edge of the porous medium (X = 1), shown in Fig. 6b. Furthermore, at constant VR, the reattachment length L r is almost independent from the Reynolds number for both porous structure arrangements, indicating the incompressible character of the flow. This further proves that for a fixed type of porous structure, the flow pattern is mainly affected by the velocity ratio.

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4.4 Pore-Scale Flow Pattern in the Porous Structures This subsection focuses on the pore-scale flow pattern inside the porous structures. Figure 8 shows the velocity vectors in the porous structures of the in-line array close to the interface area with representative regions of 0.4 < X < 0.65 and 0.375 < Y < 0.575. The interpore vortex structure is also marked with red arrows. In Fig. 9a, for VR = 1, the fluid flows vertically from the bottom to the free flow channel. However, as the velocity ratio decreases (or the momentum of the porous medium flow decreases), the clockwise flow between vertical porous elements vanishes, and anticlockwise flow occurs beside the elements. For VR = 0.0267, the momentum of the flow in the porous structure is significantly lower compared to the free channel flow and the overall flow field becomes more complex since the rear flow of the porous structure is quickly occupied at the interface by the channel flow which flows almost parallel to the streamwise direction, see Fig. 9d. Figure 10 shows the velocity vectors in the porous structure for the porous structure with a staggered arrangement. For VR = 1, Fig. 10a shows that the effect of the channel flow on the flow pattern inside the porous medium is small, and symmetrical vortices occur behind the porous elements. This is similar to the flow past a single rectangular column (Yang and Wu 2013). As VR decreases, the effect of channel flow increases, which makes the vortex inclined towards the direction of the free flow. Interestingly, it can be also observed that when VR decreases to 0.0267, e.g. Fig. 10d, the flow field around the porous elements is similar to that for an in-line array, as shown in Fig. 9d. This indicates that the influence of the type of the porous structure on the flow pattern is significantly attenuated for small values of low velocity ratios, or with other words, that the momentum of the flow through the porous medium is much lower compared to the free flow channel which dominates the overall flow field close to the interface. In order to quantitatively study the effects of velocity ratio and the type of the porous structure, averaged velocity angles in the porous structures for the various cases are presented in Fig. 11. The velocity angle is calculated using the volume averaged velocity Vx and the volume averaged velocity Vy in the porous structure. Thus, it reflects the relative inclination to the Y -axis. The larger the value of the velocity angle, the stronger the effect of the channel flow on the porous medium flow. Theoretically, the value should be zero, if VR is infinitely large, in which case the porous structure flow has no influence on the channel flow. It can be seen that as VR increases from 0.0267 to 1, the velocity angle significantly decreases from

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Fig. 9 Velocity vectors in the porous structure and at the interface for a VR = 1, b VR = 0.267, c VR = 0.133 and d VR = 0.0267 for in-line array. The representative regions are chosen as: 0.4 < X < 0.65, 0.375 < Y < 0.575, Z = 0.5

about 29.2◦ to 2.2◦ for the in-line, and from 28.2◦ to 1.1◦ for the staggered arrangement. The difference in the velocity angle for the two types of the porous structure is as small as 3.36% for VR = 0.0267, which further proves the effect of the type of the porous structure is negligible for small VR. Note also that the reduction of the velocity angle is more pronounced for the staggered porous structure for small velocity ratios, e.g. VR < 0.267, which indicates that flow field generated with a staggered arrangement is more sensitive to free flow channel velocity variations.

4.5 Turbulence Figure 12a, b shows the dimensionless turbulence kinetic energy (E k /Vin2 ) at the middle plane of the channel and the porous zone (Z = 0.5) for the two types of porous structures at Rec,L = 1 × 105 and Rep,D = 1 × 103 , that result in a velocity ratio of VR = 0.267. In the channel downstream the porous medium, the turbulence kinetic energy is large in the mixing layer and around the shear layer formed outside the flow reversal region. After the fluids are fully mixed, the turbulence kinetic energy is found to decrease along the flow direction.

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Fig. 10 Velocity vectors in the porous structure and at the interface for a VR = 1, b VR = 0.267, c VR = 0.133 and d VR = 0.0267 for the staggered array. The representative regions are chosen as: 0.4 < X < 0.65, 0.375 < Y < 0.575, Z = 0.5 Fig. 11 Average velocity angle in the porous structure for various velocity ratios

In the pores of the porous structures, the turbulence kinetic energy is larger for staggered arrangement than in-line arrangement, which indicates the former structure produces more velocity fluctuations. As the average velocity angle in the porous structures with Y -axis is

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Ek/Vin 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

L/2

D

(a)

(b)

2 ) in the middle plane of the channel Fig. 12 Distribution of dimensionless turbulence kinetic energy (E k /Vin 5 3 (Z = 0.5) for Rec,L = 1 × 10 and Rep,D = 1 × 10 for a in-line and b staggered arrangements of porous

structures

larger for in-line arrangement, the turbulence kinetic energy increases from lower left to upper right, as the flow develops.

4.6 Temperature Profiles The dimensionless temperature profiles in the middle plane of the channel (Z = 0.5) and in the porous zone for Rec,L = 1 × 105 and Rep,D = 1 × 103 are shown in Fig. 13a, b for in-line and staggered porous elements, respectively. It can be observed that, under the same boundary conditions, the temperature in the channel is generally higher for the staggered array, while the temperature distribution is more homogenized in the streamwise direction. The temperature distributions along the X -coordinate near the interface of the porous structure (in-line array) and the channel (Y = 0.5 ± 0.5D/L) for various velocity ratios are presented in Fig. 14. The Reynolds number in the channel is fixed to Rec,L = 1 × 105 and the corresponding Rep,D is varied to meet the required velocity ratios. For all velocity ratios, the temperatures at the two lines fluctuate with X , due to the alternately presence of solid and fluid phases of the porous structure. As VR increases, the average temperature and the deviation of the temperature between the two lines decrease. This is because the flow rate in the porous structure is increased with increasing Rep,D , which also increases the heat transfer efficiency. Note that at VR = 1, the temperature variations between the two lines are almost identical. Figure 15 shows temperature distribution along the X -coordinate near the interface for the porous structure of staggered arrangement. Contrary to the in-line array, the overall trends in

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L/2

(a)

(b)

Fig. 13 Dimensionless temperature distribution in the middle plane of the channel (Z = 0.5) for Rec,L = 1 × 105 and Rep,D = 1 × 103 a in-line and b staggered pore structure arrangement

Fig. 14 Temperature distributions near the interface of the in-line array for various velocity ratios of a VR = 0.133, b VR = 0.2, c VR = 0.267 and d VR = 1, at Rec,L = 1 × 105

the temperatures fluctuations for the two locations are opposite. As VR increases, the average temperature is found to decrease, due to the increasing velocity in the porous structure. As previously shown in Fig. 13, the streamwise temperature distribution is more homogeneous for the staggered array. This can be clearly observed by comparing Fig. 15 with Fig. 14 under the same VR.

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Fig. 15 Temperature distributions near the interface for velocity ratios of a VR = 0.133, b VR = 0.2, c VR = 0.267 and d VR = 1 of the staggered array at RecL = 1 × 105

4.7 Heat Transfer in the Porous Structure The local Nusselt number distributions at the solid surface of the porous structure for Rep,D = 5 × 102 and VR = 0.267, Rep,D = 5 × 103 and VR = 0.267, and Rep,D = 5 × 102 and VR = 0.0267, for both in-line and staggered arrays are presented in Fig. 16. The velocity ratios are the same for Fig. 16a, b, but the velocities at the inlets are larger for the later case. Therefore, the Nusselt number is generally increased in Fig. 16b. As the flow patterns, which are determined by the velocity ratio, are the same for Fig. 16a, b, the distributions of the Nusselt number on the surfaces are also similar for the two cases. In Fig. 16c, Rep,D is the same as in Fig. 16a, but Rec,L is as high as that in Fig. 16b. It can be seen that in the case of Fig. 16c, the Nusselt number distribution is not as uniform as that for VR = 0.267 in Fig. 16a, b. The distributions of the local Nusselt number for in-line and staggered arrays are also different, as the maximum values are located at the bottom edges of the porous elements for the in-line array but at the whole bottom surface for the staggered array. The area-averaged Nusselt number in the porous structures for various Rec,L at a fixed Rep,D = 1 × 103 is presented in Fig. 17a for both porous structures. The Nusselt number for the in-line array is always smaller than that of the staggered array, as the fluid mixes better with a staggered arrangement. As the flow rate in the channel increases, the Nusselt number for in-line array increases drastically, but the increase in the Nusselt number for the

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z x

Nu

Nu

Nu 1000

120 0

0

(a)

270 0

(b)

(c)

Fig. 16 Local Nusselt number distributions on the surface of the porous structures for a Rep,D = 5 × 102 , VR = 0.267, b Rep,D = 5 × 103 , VR = 0.267, c Rep,D = 5 × 102 , VR = 0.0267 (First row: in-line array; second row: staggered array)

Fig. 17 Variation of area-averaged Nusselt number in the porous structure a with Rec,L for Rep,D = 1 × 103 and b with Rep,D for Rec,L = 1 × 105 .

staggered array is relatively small. For example, when the Rec,L increases from 2.67 ×104 to 5 × 105 , the Nusselt number for in-line array increases about 44% and only 6.2% for the staggered configuration. Therefore, as Rec,L increases the Nusselt number for the in-line array approaches that of the staggered array. This is expected since the velocity profile in the in-line array changed a lot with the variation of velocity, becoming similar to the one of the staggered arrays if the velocity ratio is as small as 0.0267, see Fig. 9. When Rec,L is kept constant at 1 × 105 , the average Nusselt number for various Rep,D is presented in Fig. 17b. The logarithmically plotted Nusselt numbers increase linearly for both the in-line and the staggered arrays.

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G. Yang et al. Fig. 18 Comparison between the simulated Nusselt number and the correlation

As the averaged Nusselt number in the porous structure is affected by the Reynolds number in the porous structure, the velocity ratio, and the type of the porous structure, it can be correlated according to the semi-empirical expression: Nuavg = nRem p,D

(12)

where, m = 0.816 − 0.069δVR2 n = 0.426 − 0.021VR − 0.384VRδ + 0.456δVR2 1, for in-line array δ= 0, for staggered array In Eq. (12), m is in the order of 0.8 or more, indicating the turbulent character of the flow domain. Note that for the in-line array, m slightly decreases with the velocity ratio. For a staggered array, m is always 0.816. The value of m is close to 0.84, as reported by Zukauskas (1972) using experimental data for the flow past arrays of tubes and is also very similar to the values reported at the direct wake of surface mounted obstacles (Terzis et al. 2015). Equation (12) has an average deviation of 0.16% and a maximum deviation of 4.12%, respectively, for the 41 sets of numerical data in the range of 1 × 104 ≤ Rec,L ≤ 5 × 105 and 5 × 102 ≤ Rep,D ≤ 1 × 104 . A comparison of the results of the proposed correlation and the numerical data is depicted in Fig. 18, which shows a good agreement.

5 Conclusion In this study, the turbulent flow and heat transfer characteristics in a three-dimensional channel coupled with flow through porous structures is investigated numerically, for the ranges of 1 × 104 ≤ Rec,L ≤ 5 × 105 and 5 × 102 ≤ Rep,D ≤ 1 × 104 . The present calculations are validated by comparison with benchmark numerical and experimental results from open literature data. For Rec,L = 1 × 105 and Rep,D = 1 × 104 (VR = 0.267), the velocity distributions for the two types of porous structures, in-line and staggered arrays, show significant differences in the porous structure on the pore scale, while the differences are relative small for the averaged velocity and for that in the channel. A larger region of negative Vx is found in the pores for

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the staggered array, and magnitudes of both negative and positive Vx are higher than for the in-line array. Vy at the interface is more uniform for the staggered array as for the in-line array, where locally large values are found in the vicinity of X = 1. For a fixed geometry, the flow patterns in both the channel and the porous structures are determined by the velocity ratio between the transversely interacting flow streams. A flow reversal zone (negative Vx ) is found in the channel downstream of the porous structure. The length of this flow reversal zone increases with increasing velocity ratios, and it is always larger for the in-line arranged porous structure. The pore-scale flow field in the porous structures is also strongly affected by the velocity ratio. However, when the velocity ratio decreases to values as low as 0.0267, the velocities behave similar for the in-line and staggered arrays. Under the same inlet boundary conditions, the temperature in the channel is generally higher for the staggered array, and the streamwise distribution of the local temperature is also more uniform for this case. The temperature at the interface fluctuates along the flow axial coordinate due to the alternately presence of fluid and solid phases of the porous structure, but the average value decreases with increasing Rep,D for a fixed Rec,L . In contrast to the flow pattern, the heat transfer in the porous structure is not only affected by the velocity ratio, but also directly by the absolute velocity magnitude at the inlets. The average Nusselt number in the porous structure increases drastically with increasing Rec,L (or decreasing VR) for a constant Rep,D for the porous structure of an in-line array, while for the staggered array it is insensitive. A correlation for Nu is also proposed considering the effects of Reynolds number, velocity ratio and the type of porous structures, which has an average deviation of 0.16% and a maximum deviation of 4.12%, respectively, compared to the numerical data. The problem of turbulent flow and heat transfer in a three-dimensional channel coupled with flow through porous structures is of particular importance. This study, for the first time, presents fully resolved numerical results for such a model, without any filtering errors. Even though idealized porous structures are used for the porous medium, our future studies will further investigate the effect of different microscale structural parameters to extend the present results to general porous structures. Furthermore, the present pore-scale results can be used to validate and improve REV and macroscopic models for flow and heat transports in porous media. Acknowledgements Guang Yang kindly acknowledges the support of the Sino-German (CSC-DAAD) Postdoc Scholarship, the National Postdoctoral Program for Innovative Talents (No. BX201600102) and the China Postdoctoral Science Foundation (No. 2016M601591). In addition, Alexandros Terzis is grateful to Alexander von Humboldt Foundation for the funding support.

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