Turbulent Forced Convection of Radiative Gas Flow in a Duct with Separation

In the present work, a numerical solution is described for turbulent forced convection flow of an absorbing, emitting, scattering and gray fluid over a two-dimensional backward facing step in a horizontal duct. The AKN low-Reynolds-number model is employed to predict turbulent flows with separation and heat transfer, while the radiation part of the problem is modeled by the discrete ordinate method (DOM). Discretized forms of the governing equations for fluid flow are obtained by finite volume approach and solved using SIMPLE algorithm. Results are presented for the distributions of Nusselt numbers as a function of the controlling parameters like radiation-conduction parameter (RC) and optical thickness.


Introduction
Flow in ducts with combined convection, conduction, and radiation in participating media occurs in many engineering applications, such as solar collectors, combustion chambers, industrial furnaces, gas turbine blades and so on.An extensively known geometry is the backward facing step (BFS) flow that has the most features of separated flows.Although the geometry of BFS flow is very simple, many aspects of the heat transfer and fluid flow structure remain incompletely explained.
Several investigations like [1,2] have been done over BFS convection flow in a duct, both about laminar and turbulent regimes.Some important measurements in turbulent convection flow downstream of a BFS were done by Adams et al. [3] and Vogel and Eaton [4].Abe et al. [5,6] found a quite successfully numerical turbulent model, and tested their codes with these experimental results.The present research work was carried out to add radiation effect to this problem with considering a participating media.Similar research studies have been done for fluid flows with simple geometries, such as pipe flow and flow between parallel plates [7,8].To the best of author's knowledge, the forced convection turbulent flow over BFS has not been studied using AKN low Reynolds turbulent model in flow calculation with DOM in solving radiation problem.

Problem Statement
Two-dimensional turbulent forced convection flow in a rectangular duct with a BFS is numerically simulated.A schematic of the computational domain is shown in Figure 1.The channel height, H, is 0.19 m, and the step height, h, is 0.038 m, which is considered as the characteristic length in the computation.The upstream and downstream lengths of the step are 0.076 and 0.760 m, respectively, which is corresponds to 2 2 x h    0, in the computational domain (Figure 1).In the test case related to numerical validation, the fluid physical properties are treated as constants and evaluated for air at the inlet temperature of T 0 = 20˚C (i.e.density (ρ) is , specific heat (C p ) is 1005 J/(kg˚C) and Prandtl number (Pr) is 0.71).The channel expansion ratio is 1.25, with a Reynolds number of 28,000 based on the centerline velocity at the inlet section (u 0 = 10.86 m/s) and step height.

Basic Equations
For predicting turbulent flow and heat transfer in separating and reattaching flows, quite successfully AKN model that introduced by Abe et al. [5,6], was selected for this study.
The governing equations for BFS flow, which are considered to be 2-D, steady, incompressible and turbulent are the equation of continuity, the Reynolds averaged Navier-Stokes equation, the equations of the turbulent kinetic energy k for the velocity field and its dissipation rate ε, the energy equation, and the equations of the turbulent kinetic energy t 2 for the thermal field and its dissipation rate ε t that can be written as follows: Continuity: Two-equation model for velocity field: with 2 3 Two-equation model for thermal field (note that molecular viscosity is negligible): where 1 exp 1 exp 14 14 2 exp 200 In the above equations, i j u u   is the Reynolds stress component and j u t  is the turbulent heat flux.Also, the constants parameters in the governing equations are given in Table 1.
At the inlet duct section, the fluid flow consists a uniform temperature profile (T 0 = 600 K).Also, the walls considered isotherm with temperature of 750 K.

nts appearing in the governing equations. Table 1. Model consta
where, and are position and direction of the ra-

2
In presence of participating media besides the convective and conductive terms in the energy equation, the radiative term r   q is also exist that can be calculated as [9]: For example, non-dimensional form of Equation ( 6) is: To obtain the radiation intensity field and then the term r   q , we should solve the radiative transfer equation (R firstly, that for an absorbing, emitting and scattering gray medium can be written as: in which is incoming and is scattered directions and   ,  s s is the scattering ase function which is equa ty for isotropic scattering media.The numerical procedure in solving RTE (that is the DOM) was given in detail by the second author in his previous work [10].By this method, heat flux may also be determined from surface energy balance, as: The boundary conditions for the radiative problem are treated as diffusely walls with constant emissivity of 0.8 w   . In addition, the inlet and outlet sections are d as pseudo-black walls at their temperatures equal to fluid temperature in inlet and outlet sections, respectively [11].
The local total N considere usselt number along the duct walls is defined as where t q represents the sum of c e heat flu s such that onvective and radiativ xe   t c r r q q q T y q         .Therefore, the function Nu is the sum of local conve

Non-Dimensional Forms of the Governing
In t ution of governing equations, the Equations he numerical sol following dimensionless parameters are used to obtain the non-dimensional forms of the equations: Two physical quantities of interest in heat transfer study are the mean bulk temperature and the convective and radiative Nusselt numbers which are defined by:

Numerical Procedure lved numerically by the velocity and temperature s of 430(x) × 28
The governing equations are so CFD techniques to obtain the fields.Discrete procedure utilizes the method of line-byline in conjunction with finite volumes that coded into a computer program in FORTRAN and solved by SIMPLE algorithm of Patankar and Spalding [12].
Based on the grid-independent study, several grid distributions were performed and the grid 0(y) downstream of the step were selected for the numerical analysis, while using denser mesh of 470(x) × 330(y) resulted in less than 2% difference in the value of maximum total Nusselt number on the bottom wall (Table 2).Non-uniformly structured with highly concen-trated close to the wall surfaces and near the step corners and the reattachment zone, were used in order to ensure the accuracy of numerical solution.
Since, in the DOM, different numbers of discrete directions can be chosen during S N approximation, the results ob accuracy of convective heat benchmark problem was selec-tained by the S 4 , S 8 and S 12 approximations were compared and there was a small difference, less than 2% error, between S 8 and S 12 approximations.Therefore, S 8 approximation has been used in subsequent calculations.

Code Validation
In order to validate the transfer computations, a ted.It deals to a turbulent convection flow over a BFS in a duct in which the bottom wall downstream of the step is supplied with a uniform heat flux   2 270 W m w q  , while other walls are treated as adiabatic surface.So predicted Stanton number profile on th tained by two-equation turbulence model compared with experimental data [4] and a numerical data [13] with assumption of constant turbulent Prandtl number, where exhibited in Figure 2. It can be seen that the two-equation turbulence model prediction is in better agreement with experiment.
It should be noted that as the radiating effect of the gas flow is neglected i e bottom wall ob n that test case, the gas flow is consider ed non-participating media in the computation of Fig- ure 2, where the validation of combined conductiveradiative heat transfer results was given by the second author in his previous work [10].The numerical results are presented for a turbulent sepalow of a radiating gas rated and reattached convection f over a 2-D BFS in a horizontal duct.The results represent how well the energy transfer from the wall to the gas as the fluid flow passes through the channel.

Results and Discussions
In order to show the variations of Nusselt numbers (Nu c,r,t ) along the bottom wall, Figure 3 is plotted with considering the effect of RC parameter, which shows the relative importance of the radiation mechanism compared with its conduction counterpart. .It should be noted that similar results have been reported by Tsai and ozisic [14].

Conclusion
Numerical simulation of 2-D turbulent forced convection S has been studied, including thermal in a duct with a BF radiation.The effects of RC parameter and optical thickness on the Nusselt numbers distribution along with the bottom wall downstream of the channel step were presented.Numerical results show that by increasing in RC parameter, the Nu c decreases whereas the Nu t increases along the bottom wall.Also, numerical results revealed that by increasing the optical thickness, the Nu c decreases

t
ctive Nusselt number, Nu c , and local radiative Nusselt number, Nu r .

Figure 2 .
Figure 2. Comparison of the Stanton number with the experimental and theoretical results.

Figure 4 .
Figure 4. Effect of optical thickness on the Nusselt number distribution along the bottom wall, RC = 25,  = 0.5: (a) Convective Nusselt number; Total Nusselt number.

4 s
: mean temperature and temperature fluctuation t : time u i : general notation for mean velocity components molecular and eddy diffusivity  : extinction coefficient ij  : Kronecker delta  : dissipation rate of turbulent kinetic energy, molecular kinematic and eddy viscosities  : Stefan Boltsman's constant, 5.67 ×10 -8 W/m 2 K