Turbulence structure and heat transfer in a sudden expansion with a porous insert using linear and non-linear turbulence models

Highlights

• Introduction of porous inserts past sudden expansions substantially affects drag and heat transfer on channel walls.

• New analyses show impact of porous material characteristics on mean and turbulent flow structures past a sudden expansion.

• Results show complete damping of turbulence kinetic energy generation along the channel for thick porous inserts.

Abstract

Flow past a sudden expansion is found in a number of engineering equipment of practical relevance. This article presents numerical results for turbulence structure and heat transfer in flow past a two-dimensional backward-facing-step channel with a porous insert using linear and non-linear eddy viscosity macroscopic models. The expansion ratio is 1:3. The non-linear turbulence models are known to perform better than classical eddy-diffusivity models due to their ability to simulate important characteristics of the flow. Parameters such as porosity, permeability and thickness of the porous insert are varied in order to analyze their effects on the flow pattern, particularly on the damping of the recirculating bubble after the porous insertion. The numerical technique employed for discretizing the governing equations is the control-volume method. The SIMPLE algorithm is used to correct the pressure field. Wall functions for velocity and temperature are used in order to bypass fine computational close to the wall. Results showed that the recirculating bubbles simulated with the linear model were shorter than those calculated with non-linear theories. Thickness of the insert had a more pronounced effect in suppressing the recirculating bubble than permeability or porosity. Results for the statistical field indicate that using porous inserts dampens generation of turbulence along the channel and concentrate conversion of mean mechanical energy into turbulence inside the porous material. Inserting a porous substrate past the expansion seems to be a practical way to decrease the sudden variations on C_f and St.

Introduction

Recirculating bubbles appear in flows past a backward-facing step, over sinusoidal surfaces or inside diffusers . Sometimes the attenuation or even the suppression of the recirculating bubble is desired. Honeycombs or screens are frequently used in wind tunnels where a uniform flow with low turbulence intensity is required in the test section. Depending on certain parameters, such as insert thickness, porosity and directional permeability, such components may be treated as a porous medium positioned within the flow. In addition, many engineering flows of practical relevance invole interfaces between a porous medium and a clear region. The problem of boundary conditions at the porous medium/clear fluid interface has been dealt by several authors [[1], [2], [3], [4]].

A number of researchers have used porous materials for improving heat transfer in engineering equipment, such as heat exchangers. They have investigated the heat transfer performance of different partially filled systems. Shokouhmand and Salimpour [5] studied the effect of porous insert position in a fully developed laminar flow with convective heat transfer in the channel. The walls of the channel were kept to a uniform constant temperature. They carried out numerical simulation using Lattice Boltzmann Method. Two configurations were investigated in function of the position of the porous insert: attached the channel walls or in the channel core. In addition, the effects of several parameters, such as, Darcy number, porous medium thickness and the ratio of porous insert effective conductivity to fluid conductivity were analyzed. They concluded that the position of the porous material influences significantly on the pressure loss and the heat transfer process of the channel. Mahmoudi and Karimi [6] conducted various numerical investigations in a forced convection flow inside of a pipe partially filled with a porous material under local thermal non-equilibrium condition and constant wall temperature boundary conditions. Two models of thermal boundary conditions at the porous-fluid interface were used. Darcy-Brinkman-Forchheimer model was assumed to model the flow in the porous medium. They analyzed the effects of porous layer thickness, Darcy number, inertia parameter, porosity, particle diameter and solid-to-fluid thermal conductivity ratio on the validity of local thermal equilibrium. An optimum porous thickness in order to obtain a better thermal performance with reasonable pressure drop is presented. In another work, Mahmoudi et al. [7] studied analytically forced convective heat transfer in a channel partially filled with a porous medium under local thermal non-equilibrium condition and constant wall heat flux boundary conditions. The thermal boundary conditions at the interface between the porous material and the clean region are prescribed by two different approaches. They developed exact solutions for the temperature profile of fluid and solid phases in the porous region by using two different interface models, and for Nusselt number as a function of thickness of the porous medium, thermal conductivity, Biot number and Darcy number. Recently, Zheng et al. [8] numerically investigated the heat transfer enhancement by forced convection of laminar flow in a circular tube partially or fully filled with porous material. A new method coupling Genetic Algorithm (GA) and Computational Fluid Dynamics (CFD) was proposed in order to optimize the configurations of porous insert to obtain a better thermo-hydraulic performance, particularly using multiple layers of porous insert in the radial direction with different porosity, which varies from 0.5 to 1.0. The surface of the tube was maintained with constant heat flux. For a Re = 800, they found out that there is an appropriate number of layer of porous insert, and the optimum porosity varies from 0.95 to 1.0 to reach an optimal thermo-hydraulic performance. Here, optimal thermal-hydraulic performance is associated with maximum heat transfer for a given head loss through the flow system.

It should be mentioned that the works in Refs. [[5], [6], [7]] were limited to laminar flows and used distinct equations sets when involving both regions, namely clear and porous. As such, methods that used one unique set of equations, regardless of the characteristic of the medium (clear or porous) and that consider turbulence in all computational domain, can simplify and benefit analyses and optimization of a number of important flows with relevant engineering applications. Here, such novel technique is applied as it resolves the entire domain with one unique set of equations and, in addition, handles turbulent flow in the clear regions as well as in the porous substrate [9].

More specifically, the problem of flow past a backward-facing step with porous inserts has been study by Rocamora and de Lemos [10] and Chan et al. [11]. Both works presented laminar and turbulent results with forced convective heat transfer. They used for modeling turbulent flow a two-equation linear k-ε model with wall function for both the fluid region and the porous medium. Rocamora and de Lemos [10] treat the interface between the porous medium and the clear fluid following the work in Ochoa-Tapia and Whitaker [4]. Chan et al. [11] considered the flow at the interface between the fluid and porous medium as being continuous. The presence of the Brinkman's extension model (Brinkman [12]) in the porous media equation eliminates the need for imposing an explicit interface condition, in accordance with Nield and Bejan [13].

In present work, numerical results for heat transfer and turbulent flow past a backward facing step in a channel with or without a porous insert are presented. Both linear and non-linear eddy viscosity macroscopic models are employed. Here, the boundary conditions at the porous medium/clear fluid interface are the same used by Rocamora and de Lemos [10].

It is well established in the literature that for clear flows Linear Eddy-Viscosity Turbulence Models (LEVM) do not cope well with strong streamline curvature . Yet, turbulence-driven secondary motion and directional effects due to buoyancy cannot be fully simulated with LEVM. In spite of that, they are often used for engineering computations due to the numerical robustness obtained via its linear stress-strain rate relationship (Jones and Launder [14]). On the other hand, Non-Linear Eddy Viscosity Models (NLEVM)follow the idea used in obtaining constitutive equations for laminar flow of non-Newtonian fluids (Rivlin [15]). Example is the work of Speziale [16]. Essentially, the observed relationship between laminar flow of viscoelastic fluids and turbulent flow of Newtonian substances has motivated developments of such Non-Linear Models (NLEVM, Lumley [17]). Another possibility to pursue is the solution of transport equations for the individual turbulent stresses. Such Reynolds Stress Models (RSMs) [18] are not based on a stress–strain rate relationship.

Isothermal flows with a porous insert in a channel were calculated in Assato et al. [19] using linear and non-linear theories. The benefits of using a higher degree of implicitness when numerically solving non-linear turbulence models were documented by Assato and de Lemos [20]. Further, in the short conference article [21], results for the effect of a porous insert were presented, but no simulation for the behavior of the turbulence kinetic energy was shown or discussed upon. Important effects of a porous insert on turbulence structure still lacks documentation, to the best of the authors' knowledge.

Later, Gallupo and de Lemos [22] applied only non-linear models to solve flow and heat transfer past an abrupt expansion with an inlet developing entry length. Differently from the work herein, no comparison between different classes of turbulence models was documented in Ref. [22]. It is important to note that the use of linear models in Ref. [22] was limited to the sole purpose of comparing reattachment lengths. As such, no information was given therein on the behavior of the mean and statistical flow fields when using simpler linear models. Below, a broader comparison between these two classes of models is presented.

It is interesting to emphasize here that although no novelty is here introduced as far as turbulence modeling is of concern, it is the new discretization scheme of [19,20] that has not been yet evaluated for heat transfer analysis in the configuration here investigate. As such, the contribution in this article is to extend the work in Refs. [19,20] simulating now heated flow past an abrupt expansion. Both linear and non-linear k-ε turbulence models for heat transfer in a backward-facing step channel flow are employed. Some important parameters such as porosity, permeability and thickness of the porous insert are varied and their effects on the flow and heat transfer are assessed. It is worth noting that in Ref. [19] no heat transfer analysis was included but only the time-mean and statistical flow fields were there investigated. Here, heat transfer analysis is focused on the behavior of Stanton number along the bottom wall for different insert thicknesses and turbulence models. Notwithstanding the fact that here a different analysis is presented when compared with reference [19], the mathematical model for the flow field here and in Refs. [19,20] are similar, and, for the sake of completeness, flow modeling is included below.

Flow over the back-step is schematically shown in Fig. 1a, with and without the porous insert, was computed using the control-volume method applied to a boundary-fitted coordinate system. The SIMPLE algorithm was used to relax the algebraic equations. Classical wall functions for the velocity and temperature were employed. Results were obtained considering an inlet Reynolds number of Re = 132 000 based on the height of the step H. Inlet conditions for U, k and ε, were used according to values proposed by Heyerichs and Pollard [34]. All boundary conditions are illustrated in Fig. 1. The working fluid is air ($\rho =1.25\left[kg/m^3\right]$, $\mu =1.8x{10}^{-5}[Ns/m^2]$, $\mathrm{Pr}=0.72$, $c_{pf}=1006.0\left[J/Kg{}^{\circ }C\right]$) with a uniform inlet temperature of $T_{in}=50{}^{\circ }C$. The boundary conditions for the thermal field are: constant heat flux, $q_w=2000.0[W/m^2]$, on the step side wall after the step, i.e., at $x>0$ including the porous insert and $q_w=0$ on the other wall surfaces. The turbulent Prandtl number was adopted constant, ${\mathrm{Pr}}_t=0.9$. The non-linear model employed was the Shih et al. [32] closure. An orthogonal mesh of size 200 × 60 (Fig. 1b) was used for channel dimensions of x = 15H and y = 3H.

The mathematical development presented here is fully described in the open literature and is presented in detail in the worldwide available book by de Lemos [9]. For that, there is no need to repeat here detailed derivations and to the interested reader reference [9] is suggested for further studying..

Before presenting the mathematical model, it is worth mentioning three basic differences between the work herein and similar works in the literature on flow [23] and heat transfer [11] past an abrupt expansion with porous inserts. First, as mentioned, although the configuration here and in Ref. [23] are similar, or say, one investigates flow past an abrupt expansion with porous inserts, in Ref. [23] no heat transfer analysis is presented. Secondly, the turbulence model used in Refs. [11,23] for turbulence in porous media is based on the so-called volume-time sequence of integration of local transport equations, leading for the turbulence kinetic energy the expression $k_m=\overline{〈{\mathbf{u}}^{\prime }{〉}^i\cdot 〈{\mathbf{u}}^{\prime }{〉}^i}/2$, where $〈{\mathbf{u}}^{\prime }{〉}^i$ is the intrinsic volume-average of the velocity fluctuation ${\mathbf{u}}^{\prime }$. On the other hand, if the sequence time-volume integration is applied, the flow turbulent kinetic energy turns out to be $〈k{〉}^i=〈\overline{{\mathbf{u}}^{\prime }\cdot {\mathbf{u}}^{\prime }}{〉}^i/2$. The relationship between these two quantities is $〈k{〉}^i=k_m+〈\overline{{}^i{\mathbf{u}}^{\prime }\cdot {}^i{\mathbf{u}}^{\prime }}{〉}^i/2$, which seems to indicate that models based on k_m do not fully account for all of the turbulent kinetic energy associated with the flow. A detailed discussion of such a matter is beyond the scope of the present article and interested readers are referred to Ref. [9] where an in depth discussion on the subject of turbulence in porous media is presented. Finally, the work in the [23] deals with linear turbulence model (equation (17) in Ref. [23]) whereas in the present article a non-linear model is employed. To the best of the author's knowledge, heat transfer simulation of turbulent flow past an expansion with porous insert is a novel application of non-linear turbulence models, which are known to perform better in problems involving boundary layer reattachment. Having said that, transport equations are presented below.

It is also interesting to point out the advantages of using the model below when simulating hybrid media, involving a porous medium and a clear flow region. In the model to follow, the benefits are twofold. First, the use on non-linear turbulence models is known to be advantageous to more accurately simulate the size of a recirculating region. Second, such hybrid media is here modeled by one unique set of transport equations, being used both inside the porous insert as well as in the clear flow region. Combining these two benefits in a unique numerical tool might be useful in designing and optimizing thermal systems that can be model by the schematic shown in Fig. 1a.

$$\nabla \cdot {\overline{\mathbf{u}}}_D=0$$

where, ${\overline{\mathbf{u}}}_D$ is the average surface velocity (‘seepage’ or Darcy velocity). Equation (1) represents the macroscopic continuity equation for an incompressible fluid.

$$\left[\nabla \cdot \left(\rho \frac{{\overline{\mathbf{u}}}_D{\overline{\mathbf{u}}}_D}{\varphi }\right)\right]=-\nabla \left(\varphi ⟨\overline{p}{⟩}^i\right)+\mu {\nabla }^2{\overline{\mathbf{u}}}_D+\nabla \cdot \left(-\rho \varphi ⟨\overline{{\mathbf{u}}^{\text{'}}{\mathbf{u}}^{\text{'}}}{⟩}^i\right)-\left[\frac{\mu \varphi }{K}{\overline{\mathbf{u}}}_D+\frac{c_F\varphi \rho \left|{\overline{\mathbf{u}}}_D\right|{\overline{\mathbf{u}}}_D}{\sqrt{K}}\right]$$

where the last two terms in equation (2), represent the Darcy-Forchheimer contribution. The symbol K is the porous medium permeability, c_F is the form drag coefficient (Forchheimer coefficient), $〈\overline{p}{〉}^i$ is the intrinsic average pressure of the fluid, ρ is the fluid density, μ represents the fluid viscosity and ϕ is the porosity of the porous medium. It is interesting to point out that equation (2) differs from the so-called “standard Darcy-Brinkman-Forchheimer model”, which does not consider the advection term on the left neither the macroscopic Reynods stress term on the right-hand-side of equation (2). The macroscopic Reynolds stress $-\rho \varphi ⟨\overline{{\mathbf{u}}^{\text{'}}{\mathbf{u}}^{\text{'}}}{⟩}^i$, in either linear and non-linear forms, are given below.

$$\nabla \cdot \left(\rho {\overline{\mathbf{u}}}_D〈\overline{T}{〉}^i-\frac{k_{eff}}{c_{pf}}\nabla 〈\overline{T}{〉}^i\right)=s_T$$

where $k_{eff}$ is the effective conductivity of the saturated porous medium, $c_{pf}$ is the fluid specific heat and $〈\overline{T}{〉}^i$ is time-volume average temperature of the porous medium considering thermal equilibrium between the porous matrix and the working fluid.

It is interesting to point out that the so-called Two-Energy Equation Model, when distinct energy balances for the permeating fluid and the solid matrix are employed, is more appropriate to situations where the size of the porous structure is sufficient large when compared with that the clear region, if there is one. In addition, most works in the literature endorse that the thermal non-equilibrium hypothesis is more relevant to situations where large temperature difference are expected, such as in a thermally developing flow [24], in parallel [25] or countercurrent moving beds [26] or when analyzing combustion processes in porous materials [27]. In the present article, the porous material accounts for no more than 5% of the total channel volume, which is distinct from a number of work in the literature that consider either a fully fitted channel [[24], [25], [26]], or else, a partially filled duct with porous material [28] or an impiging jet [29]. For the reasons above, the thermal equilibrium assumption expressed by Eqn. (3) was here applied.

$$-\rho \varphi ⟨\overline{{\mathbf{u}}^{\text{'}}{\mathbf{u}}^{\text{'}}}{⟩}^i={\mu }_{t_{\varphi }}⟨\overline{\mathbf{D}}{⟩}^{\mathbf{v}}-\frac{2}{3}\varphi \rho ⟨k{⟩}^i\mathbf{I},$$

where,

$$〈\overline{\mathbf{D}}{〉}^v=\left[{\nabla {\overline{\mathbf{u}}}_D+[\nabla {\overline{\mathbf{u}}}_D]}^T\right],$$

represents the mean deformation tensor and I is the unity tensor.

$${\mu }_{t_{\varphi }}=\rho c_{\mu }\frac{⟨k{{⟩}^i}^2}{⟨\epsilon {⟩}^i}$$

where $c_{\mu }=0.09$ and $〈k{〉}^i$ and $〈\epsilon {〉}^i$ are the intrinsic averages of the turbulent kinetic energy and its dissipation rate, respectively.

$$\rho \nabla \cdot ({\overline{\mathbf{u}}}_D〈k{〉}^i)=\nabla \cdot \left[(\mu +\frac{{\mu }_{t_{\phi }}}{{\sigma }_k})\nabla (\phi 〈k{〉}^i)\right]\underset{P_k}{\underbrace{-\rho 〈\overline{{\mathbf{u}}^{\prime }{\mathbf{u}}^{\prime }}{〉}^i:\nabla {\overline{\mathbf{u}}}_D}}+\underset{G_k}{\underbrace{c_k\rho \frac{\phi 〈k{〉}^i\left|{\overline{\mathbf{u}}}_D\right|}{\sqrt{K}}}}-\rho \phi 〈\epsilon {〉}^i$$

where $-\rho 〈\overline{{\mathbf{u}}^{\prime }{\mathbf{u}}^{\prime }}{〉}^i$ is defined by equation (4).

Terms $P_k=-\rho ⟨\overline{{\mathbf{u}}^{\text{'}}{\mathbf{u}}^{\text{'}}}{⟩}^i:\nabla {\overline{\mathbf{u}}}_D$ and $G_k=c_k\rho \frac{\varphi ⟨k{⟩}^i\left|{\overline{\mathbf{u}}}_D\right|}{\sqrt{K}}$ represent production rates of k due to shear and porous structure, respectively. The parameters σk and $c_k$ are constants with values 1.0 and 0.28, respectively.

$$\begin{array}{l}\rho \nabla \cdot \left({\overline{\mathbf{u}}}_D⟨\epsilon {⟩}^i\right)=\\ \nabla \cdot \left[\left(\mu +\frac{{\mu }_{t_{\varphi }}}{{\sigma }_{\epsilon }}\right)\nabla \left(\varphi ⟨\epsilon {⟩}^i\right)\right]+c_{1\epsilon }\left(-\rho ⟨\overline{{\mathbf{u}}^{\text{'}}{\mathbf{u}}^{\text{'}}}{⟩}^i:\nabla {\overline{\mathbf{u}}}_D\right)\frac{⟨\epsilon {⟩}^i}{⟨k{⟩}^i}+c_{2\epsilon }{c}_k\rho \frac{\varphi {\epsilon }_{\varphi }\left|{\overline{\mathbf{u}}}_D\right|}{\sqrt{K}}-c_{2\epsilon }\rho \varphi \frac{⟨\epsilon {{⟩}^i}^2}{⟨k{⟩}^i}\end{array}$$

where ${\sigma }_{\epsilon }=1.33$, $c_{1\epsilon }=1.44$, $c_{2\epsilon }=1.92$ and $c_k$ assumes a value equal to 0.28 found in the book by de Lemos [9]. Further, it is worthwhile to mention that volume average quantities are related to intrinsic average quantities through the porosity ϕ as:

$$⟨\phi {⟩}^v=\varphi ⟨\phi {⟩}^i$$

Also, the equations given above are valid for the clear medium as well setting ϕ = 1 (K→∞) and discarding the last two terms in equation (2).

In this work, results produced by non-linear eddy-viscosity models (NLEVM) are investigated. However, only in the 80's such closures had greater progress, particularly due to the works of [16], [30], [31], [32], among others. In these works, quadratic products were introduced involving the strain and vorticity tensors with different derivations and calibrations for each model. These quadratic forms produce a certain anisotropy degree among the normal tensions, which make possible to predict, among other processes, the presence of secondary motion in non-circular ducts.

The macroscopic non-linear turbulence model, here proposed, is constituted by the same system of equations (1), (2), (3), (4), (5), (6), (7), (8) fully described in de Lemos [9]. The difference between both macroscopic models (Boussinesq and Non-Linear) lies in the expression for the Reynolds stress, kept to second order, here rewritten using indexed notation in the form:

$$\begin{array}{ll}-\rho \varphi ⟨\overline{{u^{\text{'}}}_i{u^{\text{'}}}_j}{⟩}^i& ={\left({\mu }_{t_{\varphi }}⟨{\overline{D}}_{ij}{⟩}^v\right)}^B-{\left(c_{1NL}{\mu }_{t_{\varphi }}\frac{⟨k{⟩}^i}{⟨\epsilon {⟩}^i}\left[⟨{\overline{D}}_{ik}{⟩}^v⟨{\overline{D}}_{kj}{⟩}^v-\frac{1}{3}⟨{\overline{D}}_{kl}{⟩}^v⟨{\overline{D}}_{kl}{⟩}^v{\delta }_{ij}\right]\right)}^{NL1}\\ & -{\left(c_{2NL}{\mu }_{t_{\varphi }}\frac{⟨k{⟩}^i}{⟨\epsilon {⟩}^i}\left[⟨{\overline{\Omega }}_{ik}{⟩}^v⟨{\overline{S}}_{kj}{⟩}^v+⟨{\overline{\Omega }}_{jk}{⟩}^v⟨{\overline{S}}_{ki}{⟩}^v\right]\right)}^{NL2}\\ & -{\left(c_{3NL}{\mu }_{t_{\varphi }}\frac{⟨k{⟩}^i}{⟨\epsilon {⟩}^i}\left[⟨{\overline{\Omega }}_{ik}{⟩}^v⟨{\overline{\Omega }}_{jk}{⟩}^v-\frac{1}{3}⟨{\overline{\Omega }}_{lk}{⟩}^v⟨{\overline{\Omega }}_{lk}{⟩}^v{\delta }_{ij}\right]\right)}^{NL3}\\ & -\frac{2}{3}\varphi \rho ⟨k{⟩}^i{\delta }_{ij}\end{array}$$

where ${\delta }_{ij}$ is the Kronecker delta; the superscript NL in Eqn. (10) indicates Non-Linear contributions, ${\mu }_{t_{\phi }}$ represents the same macroscopic turbulent viscosity given by equation (6), $〈{\overline{D}}_{ij}{〉}^v$and $〈{\overline{\Omega }}_{ij}{〉}^v$ are the deformation and vorticity tensors, written in the indexed form, respectively, as:

$$〈{\overline{D}}_{ij}{〉}^v=\left(\frac{\partial {\overline{u}}_{i_D}}{\partial x_j}+\frac{\partial {\overline{u}}_{j_D}}{\partial x_i}\right),〈{\overline{\Omega }}_{ij}{〉}^v=\left(\frac{\partial {\overline{u}}_{i_D}}{\partial x_j}-\frac{\partial {\overline{u}}_{j_D}}{\partial x_i}\right)$$

It is interesting to note that Eqn. (10) degenerates to Eqn. (4) when the Stress – Strain-rate relationship becomes linear and the NL terms vanish. Table 1 shows the different values of $c_{\mu }$, $c_{1NL}$, $c_{2NL}$ and $c_{3NL}$ proposed in the literature.

The interface between the porous medium and the clear fluid is treated by following expressions:

$${{\overline{\mathbf{u}}}_D|}_{\varphi \ne 1}={{\overline{\mathbf{u}}}_D|}_{\varphi =1}$$

$${⟨\overline{p}{⟩}^i|}_{\varphi \ne 1}={⟨\overline{p}{⟩}^i|}_{\varphi =1}$$

$${{\varphi }^{-1}\frac{\partial {\overline{u}}_{D_1}}{\partial n}|}_{\varphi \ne 1}-{\frac{\partial {\overline{u}}_{D_1}}{\partial n}|}_{\varphi =1}=\frac{\beta }{\sqrt{K}}{{\overline{u}}_{D_1}|}_{\text{interface}}$$

$${⟨k{⟩}^v|}_{\varphi \ne 1}={⟨k{⟩}^v|}_{\varphi =1}$$

$${\left(\mu +\frac{{\mu }_{t_{\varphi }}}{{\sigma }_k}\right)\frac{\partial ⟨k{⟩}^v}{\partial n}|}_{\varphi \ne 1}={\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial ⟨k{⟩}^v}{\partial n}|}_{\varphi =1}$$

$${⟨\epsilon {⟩}^v|}_{\varphi \ne 1}={⟨\epsilon {⟩}^v|}_{\varphi =1}$$

$${\left(\mu +\frac{{\mu }_{t\varphi }}{{\sigma }_{\epsilon }}\right)\frac{\partial ⟨\epsilon {⟩}^v}{\partial n}|}_{\varphi \ne 1}={\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial ⟨\epsilon {⟩}^v}{\partial n}|}_{\varphi =1}$$

$${⟨\overline{T}{⟩}^i|}_{\varphi \ne 1}={⟨\overline{T}{⟩}^i|}_{\varphi =1}$$

$${{\mathbf{e}}_2\cdot \left({\underset{¯}{\underset{¯}{K}}}_{eff}\cdot \nabla ⟨\overline{T}{⟩}^i\right)|}_{\varphi \ne 1}={k_f\frac{\partial ⟨\overline{T}{⟩}^i}{\partial n}|}_{\varphi =1}$$

where ${\overline{u}}_{D_1}$ is the component of the average surface velocity parallel to the interface, n is the coordinate normal to the interface from the porous medium to the clear medium, and β is a coefficient which expresses the stress jump condition at the interface. For all the cases treated in this article the coefficient β was assumed to be null, i.e., β = 0. The interface conditions from Equation (12) to Equation (14) were proposed by Ochoa-Tapia and Whitaker [4] using the concept of the excess functions. Lee & Howell [33] proposed Equations (15)–(18) assuming the continuity of the $k$-$\epsilon$ and their diffusive fluxes at interface.

Integral parameters discussed below are the friction coefficient, C_f and for the Stanton number, St, along the bottom heated wall. These coefficients are given by:

$$C_f=\frac{{\tau }_w}{\rho U_0^2/2},St=\frac{q_w}{\rho c_{pf}{U}_0\left(T_w-T_{in}\right)}$$

where τ_w is the local wall shear stress calculated on the lower wall for x > 0.