Computational investigation of slot jet impinging on a heated flat plate using low-Reynolds number modeling

The present work reports the computational investigation of turbulent slot jet impinging on a heated plate subjected to isothermal and isoflux boundary conditions. The eddy viscosity based 2-equation low-Reynolds number k − ε model of Yang and Shih has been considered. Low-Reynolds number (LRN) modeling employ near-wall integration which solves the entire boundary layer using very find grid near the wall to handle the sharp gradient of variables prevailing in these regions. The Reynolds number of jet at the inlet is 9900 and the non-dimensional distance between the top and bottom plate i.e. H/w is 7.5. The variation of centerline axial velocity, variation of local maximum axial velocity, profile of non-dimensional wall shear stress along impingement and confinement wall, distribution of local Nusselt number along impingement wall and confinement wall, and distribution of Reynolds shear stress in the domain are discussed.


Introduction
The jet impingement heat transfer is very efficient technique for cooling or heating the target surface. The excellent heat transfer coefficient associated with impingement heat transfer makes it suitable for wide ranges of practical and engineering applications. Some of the important applications areas are: cooling of turbine blades, boiler combustion chambers, cooling of hot metal plates and glass, food processing industries, thrust augmentation during short take-off and landing of fighter aircraft, automobile defroster, etc. [1][2][3][4][5][6]. Figure 1 shows physical domain of slot impinging jet flow problem under consideration. There are three regions of flow: free jet region, stagnation/impingement region and wall jet region. As the name implies, the free jet region remains unaffected from the effects of wall. The free jet region further comprises of two regions namely potential core region and shear layer region. The potential core region remains unaffected from the effect of viscosity and the velocity in this region is approximately equal to the jet inlet velocity. The impingement region forms when the jet strikes to the bottom plate and comes to the stagnation with subsequent conversion of Y-component of velocity into X-component of velocity. Sufficiently away from the impinging point, flow field resembles to characteristics of that of a wall jet flow. The present numerical simulation reports the turbulent jet impingement flow and heat transfer using LRN − model of Yang and Shih [7] (YS). The YS model is reported to perform well in many complex flow and heat transfer problems [8,9,10]. Yang and Shih model takes into account Kolmogorov time scale in the near-wall region and conventional time scale (k/ ) in the off-wall region which makes the model more appropriate for complex flow situations. Apart from composite time scale feature, this model correctly mimics the variation of near-wall shear stress. The correct prediction of near-wall behavior is key for the success of low-Reynolds number model. Very limited works have been reported on flow and heat transfer behavior of turbulent impinging slot jet using LRN turbulence models. The present study is an attempt to fill this gap. The advantage of LRN turbulence model is that it does not require wall functions unlike the high-Reynolds number turbulence model.
The first grid is placed in viscous sublayer with + = y /υ value close to 1 where = �τ / is friction velocity, τ is wall shear stress, is density of fluid, υ is kinematic viscosity of fluid. The LRN turbulence model is more suitable for the flow situations e.g. flow involving separation, presence of transition and/or laminar regions of flow, natural and mixed convection flow, etc.

Governing Equations
For present computations, RANS equations are used. For turbulence closure, LRN − model of Yang and Shih (YS-model) [7] has been considered.
Here 0 is the velocity at the jet inlet, w is width of nozzle at the inlet, 0 is ambient pressure, is kinematic viscosity of fluid.
Non-dimensionalized governing equations Turbulent kinetic energy ( ) equation: Rate of dissipation ( ) equation: Production by shear ( ): Eddy viscosity ( , ): The model functions are defined as follows: Where, a = 1.5 × 10 -4 , b = 5 × 10 -7 , c = 1 × 10 -10 Where � = νε 2 k and � = ν y k Figure 2 shows the computational domain of present problem under consideration. It the present work, it would be sufficient to consider half portion of domain for computation due to symmetry. At the jet inlet V=1 and k n = 1.5I 2 . The turbulence intensity I is taken as 0.02 for the present study. The value of turbulence intensity is taken as 0.02 following the earlier works [11][12]. The turbulent kinetic energy k n is taken as zero on the solid walls. The dissipation rate at the jet inlet is taken as ε n = ( k n 3/2 . C μ 3/4 )/0.07. Whereas, the dissipation rate at the solid wall is taken as ε n =

Numerical Scheme
The present CFD simulation has been done using computer code written in C++. Finite volume method and the staggered grid arrangement are used. The discretization of convection and diffusion terms has been carried out using Power law upwind scheme (due to stability of the scheme) and central difference scheme, respectively. The SIMPLE algorithm [13] has been used for velocity pressure coupling. In present work, TDMA and line-by-line solver are used to find the variables of interest. The pseudo-transient approach [14] has been implemented for under-relaxation of momentum equations and turbulence equations. The domain size chosen for the present study is 66×7.5 (nondimensional). The Reynolds number at the jet inlet is 9900. Figure 3 (a) shows comparison of the centerline axial velocity (V c ) with the published results of Seyedein et al. [5]. In order to further validate, the profile of local Nusselt number is compared with the work of Cadec [15] and Gardon and Akfirat [16] as shown in Figure 3(b). Overall, good agreements have been noticed. The grid sizes of 351×171, 381×191 and 401×201 are considered for grid independence study. Figure (4) presents the grid independence study where non-dimensional variation of wall shear stress is plotted for grid sizes 351×171, 381×191 and 401×201. The variation of non-dimensional wall shear stress with three grid densities is very small however to be on safer side, grid size of 381×191 is chosen for the present computation. The grid layout for the present problem is shown in Figure 5.

Results and Discussion
The flow pattern of the present problem is shown with the help of streamlines plot as shown in Figure  6. A recirculation region is evident below the top confinement wall with the vortex center (12, 3.4). The variation of centerline velocity (V c ) for H/w=7.5 is demonstrated in Figure 7. It can be observed from Figure 7 that the maximum value of 1 is observed at the jet inlet and the minimum value of 0 at the stagnation point. The value of centerline axial velocity (V c ) is equal to jet inlet velocity equal to 1 (one) up to approximately y/H = 0.4 due to the presence of potential core region. Figure 8 shows the axial variation of local maximum axial velocity U max . At the jet inlet, U max is zero as the jet is directed to the impingement surface at 90 0 . It reaches a maximum value of 0.95 at axial location X = 3 and then decreases very rapidly in the downstream due to interaction with recirculation region and finally decays steadily in the wall jet region.
The distribution of local Nusselt number along the heated impingement plate is illustrated in Figure 9. The heated impingement plate is at higher temperature than the cold impinging jet. The temperature of the jet at inlet is same as that of the ambient. The local Nusselt number distribution for isothermal boundary condition is shown by solid lines whereas dashed lines are used to show the local Nusselt number distribution for isoflux boundary condition. The peak value (maxima) of local Nusselt number is observed in the stagnation region, as the normal velocity gradient in the X-direction is high resulting in better heat transfer. The off-stagnation minima of local Nusselt number occurs due to interconversion of normal component of velocity into axial component of velocity. The zone of high heat transfer is area of interest for the heat transfer problem. The qualitative variation of local Nusselt number for isothermal and isoflux boundary condition is almost similar. Figure 10 illustrates the non-dimensional temperature profile at various axial locations for isothermal and isoflux boundary conditions on the impingement wall. The higher temperature is observed in the thermal boundary layer close to the wall. The temperature gradient along Y direction is very small away from the impingement wall. The temperature decreases in the downstream locations for both isothermal and isoflux wall as evident from Figure 10. Figure 11 shows the distribution of Reynolds shear stress in the domain. The Reynolds shear stresses (− ′ ′ ������ ) are higher in the surroundings region of stagnation point due to turning of jet and the interaction of jet with the recirculating fluid. The Reynolds shear stresses outside the recirculation region are moderate.

Conclusion
The computational study of two-dimensional turbulent slot jet impinging on a heated plate whose temperature is higher that jet inlet temperature is presented. The Reynolds number of the flow considered is 9900 and non-dimensional spacing between the top and bottom wall i.e. H/w is 7.5. Low-Reynolds number − model proposed by Yang and Shih [7] has been chosen for turbulence closure. The capability of low-Reynolds number modelling has enabled to capture the various parameters like velocity, wall shear stress, temperature, local Nusselt number profile, etc. in the viscous sublayer region also which otherwise cannot be captured by high-Reynolds number modeling. The higher value of Local Nusselt number is observed in the impingement region indicating the zone of higher heat transfer. Wall shear stress is zero at stagnation point and the maximum at X = 1 as observed in case of impingement surface. The Reynolds shear stresses (− ′ ′ ������ ) are higher in the impingement region.