Numerical Simulation of a Turbulent Confined Slot Impinging Jet

May 24, 2012 - turbulent slot jet impinging normally on a flat plate. The standard high ... et al.2 have reviewed the numerical studies to predict the...
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Numerical Simulation of a Turbulent Confined Slot Impinging Jet Shantanu Pramanik, A. Madhusudana Achari, and Manab Kumar Das* Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India 721302 ABSTRACT: A detailed numerical study has been conducted for a two-dimensional, steady, incompressible, single confined turbulent slot jet impinging normally on a flat plate. The standard high Reynolds number two-equation k−ε eddy viscosity model has been used as the turbulence model. The separation between the jet-impingement surface and the confinement surface is varied from 2 to 20; however, detailed results are shown for 6.0. The turbulence intensity and the Reynolds number at the inlet are 2% and 11000 respectively. The mean flow and turbulent characteristics have been investigated. The distribution of streamwise mean velocity, static pressure, and turbulence have been presented for thorough understanding of the flow field. The integral of mean momentum, pressure, and turbulence fluctuation have been computed along the direction of flow. The locations of the reattachment point and center of the recirculating vortex have also been determined. A similarity profile was observed downstream of the reattachment point. A detailed discussion is provided on the pressure field, Reynolds stress, and kinetic energy and its dissipation rate.

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INTRODUCTION In the field of heat and mass transfer, one of the important problems is the investigation of confined impinging jets. Tempering operations, turbine blade cooling, drying of paper and textiles or other thin films, secondary cooling of continuous casting of steels, and cooling of high power density electronic components, etc. are some of the industrial applications of impinging jets due to their highly favorable heat and mass transfer characteristics. The numerical and experimental study of impinging jets has been done by several authors. Martin1 has provided a collection of equations for predicting the velocity in the free jet and decaying jet regions based on low Reynolds number flow. Polat et al.2 have reviewed the numerical studies to predict the transport processes in laminar and turbulent jet impingement in light of the experimental data available. Zuckerman and Lior3 have described numerical simulation techniques and results for impinging jet flow and heat transfer. They compared the relative strengths and drawbacks of different turbulence models. Weigand and Spring4 have summarized relevant experimental and numerical results on multijet impingement heat transfer in focus of gas turbine applications. Saad et al.5 used the vorticitystream function formulation with an upwind finite-difference representation to predict the effects of Reynolds number, nozzle-to-impingement surface spacing, shape of the velocity profile at the nozzle exit on the flow and local heat transfer characteristics of a laminar semiconfined round jet impinging normally on a stationary plane wall. Deshpande and Vaishnav6 studied submerged laminar jet impingement on a plane using computation for the axisymmetric case and found how wall shear stress depends on Reynolds number, nozzle-to-impingement surface distance and the velocity profile at the nozzle exit. Saad et al.7 developed criteria to make a distinction between an array of confined impinging slot jets and a single jet by comparing the turbulence, mean flow and heat transfer characteristics. Craft et al.8 compared the performance of four transport models of turbulence, using low-Reynolds number k−ε model across the near-wall semiviscous sublayer, in © 2012 American Chemical Society

predicting the dynamic and thermal characteristics of the near-impingement region of the turbulent impinging jet. Cooper et al.9 in an experimental study reported the mean velocity profile in the vicinity of the impingement surface and the three Reynolds stress components in the x−r plane for different Reynolds number and nozzle-to-impingement surface spacing. Lytle and Webb10 examined experimentally the local heat transfer characteristics of air jet impingement at nozzle-toimpingement surface spacings of less than one nozzle diameter. They found a substantial increase in local heat transfer with decrease in nozzle-to-impingement surface spacing. Seyedein et al.11 numerically investigated two-dimensional flow field and heat transfer of a turbulent slot jet by using low-Reynolds and high-Reynolds number versions of k−ε turbulence model. Three low-Reynolds number models, viz., Lam−Bremhorst, Launder−Sharma, and Chien models were tested. They found that models presented by Lam−Bremhorst and Launder− Sharma show good agreement with the available experimental data. In an experimental study, heat transfer between a uniformly heated flat plate and an impinging circular air jet was investigated by Huang and El-Genk12 to determine the values of the local and average Nusselt numbers, particularly for small values of Reynolds number and jet spacing. From the study, they provided useful information for potential industrial applications concerning the combinations of jet radius and jet-to-plate distance values for maximizing the average Nusselt number and the corresponding values of maximum Nusselt number. In an experimental study, Frost et al.13 provided flow field measurement to complement the heat transfer data that enabled an understanding of the effect of near-wall flow characteristics on the heat transfer mechanism. Fitzgerald and Garimella14 studied experimentally the flow field of an Received: Revised: Accepted: Published: 9153

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Experimental and numerical methods have been applied to study these phenomena. Many researchers have applied several types of turbulence models to check the accuracy of the results with the experimental one. The main observations made are on the shear stress distribution, the mean velocity profile, turbulent fluctuations for the fluid flow problems and local Nusselt number, average Nusselt number for the heat transfer problem. In the present study, a detailed numerical simulation of a two-dimensional, steady, incompressible, single confined turbulent slot jet impinging normally on a flat plate has been conducted where the nozzle-to-impingement surface spacing is varied over a range of 2−20. A single nozzle-to-impingement surface spacing is presented to keep the range of the results limited. The standard high Reynolds number two-equation k−ε model has been used as the turbulence model. Mean flow field and the turbulent characteristics are investigated in details. Mean flow has been studied in terms of span-wise distribution of mean velocity in the X-direction and maximum velocity. The variation of location of maximum velocity along stream-wise direction and the distribution of ∂U/∂Y have also been studied for a thorough understanding of the mean flow. The characterization of the turbulence has been done by plotting u′u′, v′v′, and u′v′ contour distribution and also by span-wise distribution of u′u′ along the stream-wise direction. It is found that the complex flow field of the impinging jet can better be understood by examining the span-wise integral of mean momentum, pressure, and turbulence fluctuation for all the stream-wise locations. Similar characteristics of mean streamwise velocity and variation of span-wise pressure distribution along stream-wise locations have been studied with an aim to investigate the development of the flow field along the streamwise direction.

axisymmetric, confined, and submerged turbulent jet impinging normally on a flat plate using laser-Doppler velocimetry and mapped the toroidal recirculation pattern in the outflow region characteristic of confined jets for nozzle aspect ratio (length to diameter) of unity with nozzle diameters of 3.18 and 6.35 mm, nozzle-to-impingement surface spacings of up to four jet diameters, and Reynolds numbers in the range of 850 and 23000. Behnia et al.15 studied numerically the problem of cooling of a heated plate by an axisymmetric isothermal fully developed turbulent jet. They found that the normal-velocity relaxation turbulence model heat transfer prediction are in better agreement with the experimental data as compared to the k−ε model. The same authors in another study16 simulated the flow and heat transfer in circular confined and unconfined impinging jet configurations and determined the effects of confinement on the local heat transfer behavior by using an elliptic relaxation turbulence model. They found that the effect of confinement is only significant in very low nozzle-toimpingement surface spacings and the flow characteristics in the nozzle strongly affects the heat transfer rate, especially in the stagnation region. In an experimental study, Beitelmal et al.17 determined the effect of the inclination of an impinging two-dimensional air jet, nozzle-to-impingement surface spacing and Reynolds number on the heat transfer from a uniformly heated flat plate. Large eddy simulation (LES) was used by Cziesla et al.18 to numerically investigate the distribution of the mean velocities, turbulent stresses, the velocity fluctuations, the temperature field due to an impinging jet from a rectangular slot nozzle. Shi et al.19 discussed the effects of turbulence models, near wall functions, jet turbulence, jet Reynolds number, and type of thermal boundary condition at the target surface and compared favorably with the experimental data.11 The same authors20 examined the effect of thermophysical properties of fluid and heat transfer and gave a correlation of the effect of fluid Prandtl number in a semiconfined laminar slot jet. Sahoo and Sharif21 investigated numerically the flow and mixed convection heat transfer characteristics in the jet impingement cooling of a constant heat flux surface. They found that the average Nusselt number at the heat flux surface increases with increasing jet exit Reynolds number for a given domain aspect ratio and Richardson number. They also found that for a given aspect ratio and Reynolds number, the average Nusselt number does not change significantly with Richardson number indicating that the buoyancy effects are not very significant on the overall heat transfer process for the range of jet Reynolds number considered. In another study, Lee et al.22 found out the effect of Reynolds number and height ratio (ratio of nozzle-to-impingement surface spacing and nozzle diameter) on the flow and temperature field in the channel. Liping et al.23 found numerically the variation of Nusselt number at stagnation point with increase in nozzle-to-impingement surface spacing and the little change in temperature fields with the presence of vortex of a rectangular impinging jet. By conducting a thorough literature survey, it has been observed that fluid flow and the associated cooling by the slot jet impingement situation is governed by the following parameters: (1) nozzle to impingement surface spacing. This is the single-most important parameter for this kind of problem. Various researchers have taken different spacing from a very low of less than a jet diameter to large values over 20. (2) shape of the velocity profile at the nozzle outlet. (3) jet exit Reynolds number. (4) arrangement of jet as a single jet or as an array of jets.

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MATHEMATICAL FORMULATION For the present simulation, it has been assumed that the flow field is two-dimensional, turbulent and, in steady state. The fluid is assumed to be incompressible. To predict the turbulent flow, Reynolds averaged Navier−Stokes (RANS) equations are used. For modeling the Reynolds stresses, a standard high Reynolds number two-equation k−ε model has been used. The Reynolds stresses are linked to the velocity gradients by using the Boussinesq approximation. The nondimensional governing equations are presented by assuming the above conditions. The dimensionless variable are defined as y u v x , V= , X= , Y= , U= U0 U0 w w p − p0 ν ε k P= , kn = 2 , εn = 3 , νt , n = t 2 ν ρU0 U0 U0 /w (1)

The nondimensionalized equations are Continuity equation: ∂U ∂V =0 + ∂X ∂Y

(2)

X-momentum equation: ∂(U 2) ∂(UV ) ∂ ⎛ 2 ⎞ + = − ⎜P + k n ⎟ ∂X ∂Y ∂X ⎝ 3 ⎠ 1 ∂ ⎡ 1 ∂ ⎡ ∂U ⎤ ∂U ⎤ + ⎢(1 + νt , n) ⎥⎦ + ⎢(1 + νt , n) ⎥⎦ Re ∂X ⎣ Re ∂Y ⎣ ∂X ∂Y 9154

(3)

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Figure 1. Schematic diagram of a single confined turbulent slot impinging jet.

Numerical Scheme and Method of Solution. A schematic diagram of the single confined turbulent slot jet impinging normally on a flat plate is shown in Figure 1. Because of symmetry, simulation of only a half-domain is sufficient for complete characterization of the flow. The choice of grid layouts and the density in the simulation domain plays a very important role in numerical simulations. Because of the presence of large gradients at streamline curvature, the impingement region is the most difficult domain to simulate accurately. The grid layout in this region was specified to be much finer than that in the downstream region. A nonuniform grid was used where the grid density in region B decreased along the X direction, and the grid density in region A decreased toward the plane of symmetry. The finest grid layout in the Y direction was set near the wall in both regions A and B. The dimensionless governing equations are discretized using the control volume approach following Patankar.25 The convective terms are discretized by the power-law upwinding scheme and the diffusive terms are discretized by the central difference scheme to ensure the stability of the solution. The wall function method by Launder and Spalding,24 appropriate for high Reynolds number flows, has been used to avoid the fine mesh required for resolving the viscous sublayer near the wall. To couple the velocity and pressure equations, the SIMPLE algorithm of Patankar25 is followed . The pseudotransient approach as described in Versteeg and Malalasekera26 is used to under-relax the momentum and the turbulent equations. An under-relaxation of 0.1 is used for the pressure correction equation. The detailed discretization of the turbulent kinetic energy equation and the rate of dissipation equation has been done following Biswas and Eswaran.27 The size of the domain considered is 60 × 6. The code has been written in C+ + language. All the computations have been carried out on an intel Core 2 Duo 2.8 GHz machine on Linux platform.

Y-momentum equation: ∂(UV ) ∂(V 2) ∂ ⎛ 2 ⎞ + = − ⎜P + k n ⎟ ∂X ∂Y ∂Y ⎝ 3 ⎠ 1 ∂ ⎡ 1 ∂ ⎡ ∂V ⎤ ∂V ⎤ + ⎢(1 + νt , n) ⎥⎦ + ⎢(1 + νt , n) ⎥⎦ Re ∂X ⎣ Re ∂Y ⎣ ∂X ∂Y

(4)

Turbulent kinetic energy (kn) equation is ν ⎞ ∂k ⎤ ∂(Ukn) ∂(Vkn) 1 ∂ ⎡⎛ ⎢⎜1 + t , n ⎟ n ⎥ + = . ∂X ∂Y σk ⎠ ∂X ⎥⎦ Re ∂X ⎢⎣⎝ +

ν ⎞ ∂k ⎤ 1 ∂ ⎡⎛ ⎢⎜1 + t , n ⎟ n ⎥ + Gn − εn . σk ⎠ ∂Y ⎥⎦ Re ∂Y ⎢⎣⎝

(5)

Rate of dissipation (εn) equation is ν ⎞ ∂ε ⎤ ∂(Uεn) ∂(Vεn) 1 ∂ ⎡⎛ ⎢⎜1 + t , n ⎟ n ⎥ + = . ∂X ∂Y Re ∂X ⎢⎣⎝ σε ⎠ ∂X ⎥⎦ +

ν ⎞ ∂ε ⎤ ε ε2 1 ∂ ⎡⎛ ⎢⎜1 + t , n ⎟ n ⎥ + C1ε n Gn − C2ε n . Re ∂Y ⎢⎣⎝ kn kn σε ⎠ ∂Y ⎥⎦

(6)

Production (Gn): Gn =

2 νt , n ⎡ ⎛ ∂U ⎞2 ⎛ ∂V ⎞2 ⎛ ∂U ∂V ⎞⎟ ⎤ ⎢2⎜ ⎟ + 2⎜ ⎟ + ⎜ ⎥ + ⎝ ∂Y ⎠ ⎝ ∂Y ∂X ⎠ ⎦ Re ⎣ ⎝ ∂X ⎠

(7)

Eddy viscosity (νt,n):

νt , n = CμRe

kn2 εn

(8)

The model constants are given as: σk = 1.0, σε = 1.30, C1ε = 1.44, C2ε = 1.92, and Cμ = 0.09 (Launder and Spalding24). 9155

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Figure 2. Validation of the present code.

Boundary Conditions. The turbulent flow of an impinging jet emanating into the quiescent fluid is considered and is shown in Figure 1. A uniform velocity U0 is used at the nozzle inlet. No-slip and no-penetration conditions are considered for the impingement and confinement surfaces. The flow at the outlet has been considered as fully developed. Neumann boundary condition is provided for the exit boundary, a developed condition of ∂φ/∂n = 0 is considered where φ = U, V, kn, εn. The symmetry is taken at symmetry axis. U = 0 and ∂φ/∂n = 0, where φ=V, kn, εn have been considered along the symmetry axis. The Reynolds number for the impinging jet flow is defined as (U0w)/ν where w, the jet width, is the characteristic length. The Reynolds number considered for all the present computations is 11000 at which the flow becomes completely turbulent and independent of Reynolds number. For the turbulent kinetic energy equation, the boundary condition at the inlet is kn = 1.5I2, where I is the turbulence intensity. Turbulence intensity is equal to 0.02. For the dissipation equation, the boundary condition is ε n = (kn3/2Cμ3/4)/l, where l = 0.07w is considered where w is the jet width.19 The first grid point near the wall has been ensured to fall in the logarithmic region, that is, 30 < Y+ < 100 where Y+ = yuτ/ν, uτ being the friction velocity. Validation of the Computer Code. To validate the code developed, results obtained from the present computation are compared with the results given by Seyedein et al.11 In Figure 2a, the decay of the nondimensional centerline axial velocity from unity at the nozzle exit, to zero at the impingement surface for Re = 9900 and H/w = 7.5 has been compared with the results of Seyedein et al.11 as predicted by the Lam− Bremhorst (LB) and Launder−Sharma (LS) versions of the low-Reynolds number k−ε model. From the comparison, it is clear that the centerline velocity drops to 95% of the nozzle exit value at distances of 5.2, 5.9, and 5.8 × the nozzle width from the nozzle exit for the LB, LS, and the present computation, respectively. This distance expresses the length of the potential core region. The rapid decrease of the centerline axial velocity within the last 1.5−2.0 slot widths is due to the exchange of Ydirection momentum into X-direction momentum in the stagnation region. The present computation matches well with both the results of LB and LS models except in the region of 0.2 to 0.4, where the computational result is closer to the result of the LS model. To further validate the numerical work, comparison of the variation of normalized static pressure along the impingement surface with the results of Gardon and Akfirat,28 Cadek29 and Seyedin et al.11 has been presented in Figure 2b. The static pressure along the impinging surface varies from a maximum at

the stagnation point to zero at the end of the stagnation region in a confined slot impinging jet. The end of the stagnation region has a minimum value for a confined impingement system. Along the wall jet region, the static pressure increases slightly with the downstream distance to reach the atmospheric pressure at the outlet. The normalized form of the static pressure can be defined as (P − Pmin)/(Pstagnation − Pmin). The present computational results (H/w = 2.35, 6.0) show good agreement with the results obtained in ref 28 (H/w = 2) and ref 11 (H/w = 7.5). Downstream of X = 1.5, the computational result for H/w = 2.35 matches closely with that of Gardon and Akfirat28 while the computational result for H/w = 6.0 matches with the result of ref 11. The computational results show significant deviation from that of ref 29 along the length. Grid Independency Test. Three sets of grid densities have been tested and the results are presented in Figure 3. It can be

Figure 3. Grid independence test.

noted that almost the same stream-wise mean velocities are obtained for grid densities of 201 × 121, 251 × 151, and 301 × 181. A grid density of 251 × 151 has been considered for the rest of the simulations. The domain (in the Y direction) has been divided into four subdomains of sizes 0.5, 2.5, 2.5, and 0.5. These four subdomains are provided with 31, 45, 45, and 30 numbers of grids and expansion ratios of 1.000025, 1.048, 0.9542, and 0.9999, respectively. The domain (in the X direction) has been divided into three subdomains of sizes 0.5, 14.5, and 45. These three subdomains are provided with 32, 9156

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173, and 46 numbers of grids and expansion ratios of 0.9995, 1.0154, 1.055, respectively. Geometric expansion scheme Σ = (a(1 − rn)/(1 − r)) is used for the nonuniform grids where a is the size of the first grid, r is the expansion ratio, Σ is the domain size, and n is the number of grids in the domain. A typical grid distribution (zoomed view) near the stagnation region is shown in Figure 4.

Figure 5. Streamline and velocity vector distribution.

Figure 4. Typical grid distribution zoomed near the stagnation region.

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RESULTS AND DISCUSSION The fully developed turbulent two-dimensional, steady, incompressible, single confined impinging jet flow is numerically simulated. The impingement surface and confinement surface separation (H) is varied from 2 to 20. The turbulence intensity and the Reynolds number at the inlet are 2% and 11000, respectively. The mean flow characteristics, the pressure distribution, the momentum flux and the similarity velocity profiles are presented. However, the results are plotted and discussed for the case H = 6 with a domain width of L = 60 to keep the results within limit. Flow Characteristics. In this section, the characteristics of the flow in the X−Y plane are presented. The streamlines and the mean velocity vector plots are shown in Figure 5. Inside the recirculation zone, a counter-rotating vortex is observed with the vortex center at (10.04, 2.87). The position of the reattachment point is located at X = 27.73. Opposite to the reattachment region, on the impingement wall, the velocity gradient becomes very small; however, it recovers further downstream and maintains a constant positive value which is equal to that on the confinement wall. From Figure 6, it is clear that in the impingement region, the flow over the impingement surface resembles a wall jet like flow;30 however, due to the presence of recirculation zone, the flow is in the reverse direction away from the wall. It is also observed that the velocity gradient on the impingement wall is very large at X = 5 which gradually decreases in the flow direction. The velocity gradient is always positive on the impingement surface and downstream of the reattachment region, the flow gradually becomes developed. The vortex center is the point where both U and V are zero, that is, the point of intersection of U = 0 and V = 0 contour lines. The U−velocity gradient is computed and the contour plot is shown in Figure 7. The positive velocity gradients are shown by

Figure 6. Spanwise distribution of velocity.

Figure 7. Contour plot of ∂U/∂Y in the domain (solid lines positive; dashed lines, negative).

solid lines and the negative gradients are shown by dashed lines. Based on the discussion on the velocity profiles, it is found that the positive velocity gradient contour ∂U/∂Y = 0.01 lies very close to the impingement surface up to around X = 27. It means that the boundary layer effect is predominant up to that distance and beyond that, its behavior is diminished. On the 9157

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confinement surface, it is observed that the positive contour line finally merges at the reattachment point at X = 27.73, and after that, there is no negative U-velocity. The maximum Uvelocity lies on the ∂U/∂Y = 0 line. The ∂U/∂Y = 0.01 line behaves like a point of inflection. The Reynolds stresses −u′u′, −v′v′, −u′v′ are computed and plotted as shown in Figure 8a, Figure 8b, and Figure 8c,

Figure 9. Turbulence stress u′u′ distribution at different axial distances.

Figure 10. U-contour distribution in the domain.

up to the reattachment point where the negative U-velocity ceases to exist. The positive U-contour lines merge near the impingement region where Y-momentum is converted into Xmomentum. High values of U-velocity are visible in the vicinity of the stagnation point. It can be asserted from the figure that there always exists positive U-velocity along the impingement plate and except for in the recirculation bubble, the velocity near the confinement plate is positive. The V-velocity contour is shown in Figure 11. The negative bubble of the V-contour adjacent to the axis of the jet depicts the downward flow of the recirculating fluid which gets entrained along the outer edge of the vertical jet. Attached to the negative region, there is a large region of positive V-contour lines which starts near X = 3 and extends up to X = 30 resembling a wall jet like flow after impingement that spreads

Figure 8. Contour plots of the normalized Reynolds stresses for the impinging-jet flow (solid lines positive, dashed lines negative).

respectively. The jet with 2% turbulence intensity and 11000 Reynolds number is impinging on the surface thus creating a vigorous turbulence which is reflected in the plots. In the corner zone near the stagnation point, the stresses are large which gradually decreases in the downstream direction. The spanwise distribution of the turbulence is given in Figure 9. It is observed from the figure that the magnitude of u′u′ is more near the impingement surface in the upstream region and maximum value is observed at a streamwise location X = 8 which remains constant up to X = 12, then decreases further downstream. Downstream of the reattachment region, the spanwise variation of u′u′ diminishes and at X = 40, it becomes almost zero, reaffirming a developed flow where ∂U/∂X = 0. Figure 10 shows the contour lines for the axial velocity component. A large negative velocity bubble can be seen attached to the confinement wall near the nozzle which extends

Figure 11. V-contour distribution in the domain. 9158

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The normalized kinetic energy contour plot shown in Figure 14 is an indication of the level of turbulence present in the flow

rapidly toward the confinement wall at about reattachment point. A region of negative V-contour lines of small magnitudes downstream of the reattachment shows a negative slope of the streamlines as can be confirmed from Figure 5a. Figure 12 shows the mean static pressure contour. The jet, possessing a high amount of kinetic energy, directly impinges

Figure 14. Kinetic energy contour distribution.

at different locations in the flow domain. The turbulent kinetic energy kn has a maximum value in the stagnation region and decreases in the stream-wise and the cross-wise direction. In the downstream locations after X = 30, it gradually dies down. Near the stagnation region, it increases to 0.03 indicating the presence of high shearing and turbulence which gradually subsides in the downstream direction. The kinetic energy dissipation contour is shown in Figure 15. Maximum value of

Figure 12. Pressure contour distribution in the domain.

on the solid wall and the kinetic energy is converted into pressure energy. The contour plot gives a qualitative information of the complex phenomena of conversion of kinetic energy into pressure energy and vice versa. Pressure is very high in the stagnation region up to X = 1, then there is a negative pressure gradient which contributes to the local flow acceleration along the impingement surface up to X = 10. Further downstream, the pressure increases and as a consequence flow decelerates. The lowest pressure occurs at the center of the recirculation bubble. The reattachment region on the confinement surface is shown by a relatively high pressure zone. Because of the accelerating reverse flow, pressure decreases along the confinement surface from the reattachment point toward the nozzle exit. Downstream of the reattachment point, pressure does not change along the flow direction and at about X = 40, ∂P/∂X becomes ≈ zero suggesting a developed flow. Spanwise distribution of mean static pressure (Figure 13) shows that except near the stagnation region, there is no lateral variation of pressure, and from X = 40 onward, ∂P/∂X is ∼0 as shown in the contour plot.

Figure 15. Dissipation contour distribution.

dissipation occurs in the stagnation region. The dissipation decreases along the stream-wise direction as well as in the cross-wise direction. It is observed that the energy mostly gets dissipated near the stagnation region where turbulent kinetic energy is very high. Because of the complex nature of the flow field of impingement jet, it is necessary to investigate the conservation of momentum flux along the flow direction. The momentum flux is calculated by integrating the X-momentum equation over a control volume that defines the computational domain. It results in the integral constant: ∫ YYmax (UU + P̅ + 2/3kn − (2/ min Ret)(∂U̅ /∂X)) dY = constant, evaluated at the planes of a particular axial length X. Here, Ret = Re/νt,n is called the turbulent Reynolds number. The Reynolds stress term is computed using the Boussinesq approximation, u′u′ = 2/3kn − (2/Ret)(∂U̅ /∂X). The individual components, Ju = ∫ UU dY, Jp = ∫ P dY, Jk = ∫ 2/3kn dY and Jr = ∫ (2/Ret)(∂U/∂X) dY are computed. Figure 16 shows the distribution of Jp, Ju, Jk, Jr, and J (= Jp + Ju + Jk + Jr) along the flow direction. From this figure, the integral of normal Reynolds stress terms Jk and Jr show nearly constant value (are of lesser significance), while the integral Ju decreases with X and the pressure integral Jp shows an inverse tendency. The result of the summation J is almost constant along X because the decrease in the amount of momentum is compensated by the recovery of pressure from lower to higher one. After the reattachment point location, the

Figure 13. Span-wise distribution of static pressure (×103). 9159

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Figure 16. Momentum flux distribution along the axial direction.

Figure 18. Decay of Vmax with downstream distance.

contribution from the Reynolds stress terms becomes insignificant, and remaining terms remain constant and contribute to the total momentum flux. Similarity Solution. Like an offset jet flow problem, the flow is expected to attain a behavior like a wall jet in the region after the reattachment point. The traditional outer-scaling method Umax and Y/0.5H, as the flow is similar to a channel flow downstream the reattachment point, is used to check whether the flow has approached a self-similar property or not. The Umax profiles at X = 40, 50, and 60 are shown in Figure 17. It is observed that all the profiles merge on the same line indicating a similarity solution has been obtained.

Figure 19. Locus of the variation of Vmax with downstream distance.

Ymax suddenly shoots up to a value of 0.79 indicating a flow toward the reattachment point. It then decreases gradually to a value of 0.5 and remains constant at 0.5 in the downstream region reaffirming the fully developed flow. Figure 20 shows the variation of wall pressure with X. From the plot of variation of wall pressure along the impingement surface shown by a continuous line, it can be found that the pressure at the stagnation point (X = 0) has the maximum value of 0.745 as expected and then goes down to a minimum value of 0.235 and remains almost constant up to X ≈ 10. The wall pressure recovers at the reattachment point and attains a constant value of 0.31. The other plot represented by a dotted line in the same figure depicts the variation of wall pressure along the confinement surface and is similar to that along the impingement surface with the exception of the stagnation region. The variation of shear stress as discussed earlier is an extremely important parameter from the point of view of heat transfer by impingement jet flow mechanism. The variation of (CfX × Re) along the wall with X is given in Figure 21 where CfX is the skin friction coefficient. Along the impingement surface, CfX is having the minimum value of zero at the stagnation point. It shoots up to a maximum value of 88.0 very sharply and then decreases gradually up to the reattachment point. It then attains a minimum constant value of 2.2 at the reattachment point and

Figure 17. Similarity solution.

The maximum velocity |V⃗ |max decay with downstream distance X is shown in Figure 18. It is observed that for the flow from the edge of the jet region, |V⃗ |max has a value of 0.62 at X = 0.61, attains a maximum value of 0.95 at X = 2.35, and then drops to a minimum value of 0.1 at X = 31.75 beyond which it remains constant for the rest of the downstream flow. Figure 19 shows the variation of Y-coordinate with X where the velocity is maximum, that is, |V⃗ |max . From the figure it can be seen that the locations where the velocity is maximum are very much adjacent to the impingement surface in the recirculation region. But near the recirculation point, that is, X = 27.73, the value of 9160

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Figure 20. Variation of wall pressure with downstream distance. Figure 22. Variation of reattachment length with nozzle-to-impingement surface spacing.

The potential core length of the impinging jet is defined as the distance between the nozzle exit and the point where the jet velocity attains a value of 99.9% of the nozzle exit velocity. The effect of the nozzle-to-impingement surface spacing on the potential core length has been shown in Figure 23. It can be

Figure 21. Variation of (CfX × Re) with downstream distance.

remains constant thereafter in the downstream region. At the edge of the confinement surface, (CfX × Re) is just above zero and attains a minimum value of −19.0 at X = 12 before attaining a value just above zero at X = 30. It can be seen that (CfX × Re) has equal values along both impingement and confinement surfaces that indicate that the flow is fully developed beyond X = 35. Effect of Nozzle-to-Impingement Surface Spacing on Flow Characteristics. The effect of the variation of the nozzle-to-impingement surface spacing H on the fluid flow characteristics thus becomes an important controlling factor. Figure 22 shows the variation of reattachment length with the nozzle-to-impingement surface spacing H. It can be seen that the reattachment length increases with the increase in the nozzle-to-impingement surface spacing. The rate of increase of reattachment length with nozzle-to-impingement surface spacing is relatively high up to H = 10. For higher values of H, the reattachment length increases at a lesser rate. The variation of reattachment length with nozzle-to-impingement surface spacing can be correlated by the correlation Xr = 14H0.57 − 0.35H0.9 − 10. The reattachment length thus will also affect the shear stress distribution (τw) in the same way as has been shown in Figure 21.

Figure 23. Variation of potential core length with nozzle-toimpingement surface spacing.

seen that the potential core length increases monotonically with an increase in nozzle-to-impingement surface spacing. The variation of the potential core length with the nozzle-toimpingement surface spacing follows the empirical correlation, lpc = 0.05H1.58.

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CONCLUSION A detailed numerical investigation on the fluid flow of a confined impinging slot jet has been carried out. The standard high-Reynolds turbulence k−ε eddy viscosity model has been used as the flow becomes fully turbulent at high Reynolds number. The separation between the jet-impingement surface and the confinement surface separation (H) is varied from 2 to 20; however detailed results are shown for H = 6.0. The turbulence intensity and the Reynolds number at the inlet are 9161

dx.doi.org/10.1021/ie300321f | Ind. Eng. Chem. Res. 2012, 51, 9153−9163

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2% and 11 000, respectively. The recirculation flow, the vortex center, the reattachment point have been identified by the numerical computation. The effect of nozzle-to-impingement surface spacing on different flow parameters have been discussed. The variation of static pressure is from 0.745 at the stagnation point which decreases to 0.235 and then recovers to a constant value of 0.31. The pattern is similar to the pressure distribution observed in a developed channel flow. The velocity profile attains a developed behavior at a downstream location X ≈ 40. The position of the reattachment point is at X = 27.73. The momentum fluxes are calculated viz. Ju, Jp, Jk and Jr where J = Ju + Jp + Jk + Jr. It is found that J remains a constant after the reattachment point indicating that the flux remains the same; the flow variations are at a minimum due to a developed profile. Within the reattachment point, Ju and Jp have an opposite behavior. After the reattachment point, Jk and Jr are insignificant, implying that the turbulence levels are small. The turbulent kinetic energy and its rate of dissipation show that the mixing is predominant only near to the impingement region and within the recirculation region. The developed velocity profile resembles a nice shape observed in developed turbulent plane channel flow type. The shear stress variation at the wall is important only within the recirculation region. Outside the region, it is insignificantly small and thus can be neglected. So the important region of concern is the recirculation region only. Empirical correlations have been developed to relate the potential core length and the reattachment length of the impinging jet with the nozzle-to-impingement surface spacing H.

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w = width of the jet x, y = dimensional coordinates J = total monmentum flux Greek Symbols

εn κn

dissipation turbulent kinetic energy, = 1/2( u′2 + v′2)/U02 ν, νt laminar and turbulent kinematic viscosity σk, σε turbulence model constants

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REFERENCES

(1) Martin, H. Heat and mass transfer between impinging gas jets and solid surfaces. Adv. Heat Transfer 1977, 13, 1−60. (2) Polat, S.; Huang, B.; Mujumdar, A.; Douglas, W. Numerical flow and heat transfer under impinging jet. Annu. Rev. Numer. Fluid Mech. Heat Transfer 1989, 2, 157−197. (3) Zuckerman, N.; Lior, N. Impingement Heat Transfer: Correlations and Numerical Modeling. J. Heat Transfer, Trans. ASME 2005, 127, 544−552. (4) Weigand, B.; Spring, S. Multiple Jet ImpingementA Review. Heat Transfer Res. 2011, 42, 101−142. (5) Saad, N. R.; Douglas, W. J. M.; Mujumdar, A. S. Prediction of heat transfer under an axisymmetric laminar impinging jet. Ind. Eng. Chem. Fundam. 1977, 16, 148−154. (6) Deshpande, M. D.; Vaishnav, R. N. Submerged laminar jet impingement on a plane. J. Fluid Mech. 1982, 114, 213−236. (7) Saad, N. R.; Polat, S.; Douglas, W. J. M. Confined multiple impinging slot jets without crossflow effects. Int. J. Heat Fluid Flow 1992, 13, 2−14. (8) Craft, T. J.; Graham, L. J. W.; Launder, B. E. Impinging jet studies for turbulence model assessment. II. An examination of the performance of four turbulence models. Int. J. Heat Mass Transfer 1993, 36, 2685−2697. (9) Cooper, D.; Jackson, D.; Launder, B.; Liao, G. Impinging jet studies for turbulence model assessment. I. Flow-field experiments. Int. J. Heat Mass Transfer 1993, 36, 2675−2684. (10) Lytle, D.; Webb, B. Air jet impingement heat transfer at low nozzle-plate spacings. Int. J. Heat Mass Transfer 1994, 37, 1687−1697. (11) Seyedein, S. H.; Hasan, M.; Mujumdar, A. S. Modelling of a single confined turbulent slot jet impingement using various k-ε turbulence models. Appl. Math. Modell. 1994, 18, 526−537. (12) Lianmin Huang, L.; El-Genk, M. S. Heat transfer of an impinging jet on a flat surface. Int. J. Heat Mass Transfer 1994, 37, 1915−1923. (13) Frost, S. A.; Jambunathan, K.; Whitney, C. F. Velocity and turbulence characteristics of a semiconfined orthogonally impinging slot jet. Exp. Therm. Fluid Sci. 1997, 14, 60−67. (14) Fitzgerald, J. A.; Garimella, S. V. A study of the flow field of a confined and submerged impinging jet. Int. J. Heat Mass Transfer 1998, 41, 1025−1034. (15) Behnia, M.; Parneix, S.; Durbin, P. A. Prediction of heat transfer in an axisymmetric turbulent jet impinging on a flat plate. Int. J. Heat Mass Transfer 1998, 41, 1845−1855. (16) Behnia, M.; Parneix, S.; Durbin, P. Numerical study of turbulent heat transfer in confined and unconfined impinging jets. Int. J. Heat Fluid Flow 1998, 20, 1−9. (17) Beitelmal, A. H.; Saad, M. A.; Patel, C. D. The effect of inclination on the heat transfer between a flat surface and an impinging two-dimensional air jet. Int. J. Heat Fluid Flow 2000, 21, 156−163. (18) Cziesla, T.; Biswas, G.; Chattopadhyay, H.; Mitra, N. K. Largeeddy simulation of flow and heat transfer in an impinging slot jet. Int. J. Heat Fluid Flow 2001, 22, 500−508. (19) Shi, Y.; Ray, M. B.; Mujumdar, A. S. Computational study of impingement heat transfer under a turbulent slot jet. Ind. Eng. Chem. Res. 2002, 41, 4643−4651. (20) Shi, Y. L.; Ray, M. B.; Mujumdar, A. S. Effects of Prandtl number on impinging jet heat transfer under a semi-confined laminar slot jet. Int. Commun. Heat Mass Transfer 2003, 30, 455−464.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: 0091-3222-282924. Fax: 0091-3222-282278. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the helpful comments and suggestions of the reviewers and the Editor. A.M.A. sincerely acknowledges the support from the research grant obtained from the Department of Science and Technology, New Delhi, India.

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NOMENCLATURE C1ε, C2ε, Cμ = turbulence model constants w = width of the jet G = production by shear, eq 7 H = distance between the impingement wall and confinement wall p = static pressure p0 = ambient pressure P = nondimensional static pressure Re = Reynolds number, U0w/ν Uo = average inlet jet velocity u, v = dimensional mean velocities in x, y-directions, respectively U, V = nondimensional velocities in X, Y-directions, respectively 9162

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(21) Sahoo, D.; Sharif, M. A. R. Numerical modeling of slot-jet impingement cooling of a constant heat flux surface confined by a parallel wall. Int. J. Therm. Sci. 2004, 43, 877−887. (22) Lee, H. G.; Yoon, H. S.; Ha, M. Y. A numerical investigation on the fluid flow and heat transfer in the confined impinging slot jet in the low Reynolds number region for different channel heights. Int. J. Heat Mass Transfer 2008, 51, 4055−4068. (23) Liping, G.; Jingwei, Z.; Feifei, H.; Mo, Y. Numerical modeling of slot-jet impingement cooling of a constant heat flux surface confined by a parallel wall. Power Energy Eng. Conf. (APPEEC) 2011, 1−4. (24) Launder, B.; Spalding, D. The numerical computation of turbulent flows. Comput. Methods Appl. Mech. Eng. 1974, 3, 269−289. (25) Patankar, S. V. Numerical heat transfer and fluid flow; Hemisphere: New York, 1980. (26) Versteeg, H. K.; Malalasekera, W. An Introduction to computational fluid dynamics. The Finite Volume Method; Longman: London, 1996. (27) Turbulent Flows: Fundamentals, Experiments and Modelling; Biswas, G., Eswaran, V., Eds.; IIT Kanpur Series of Advanced Texts; Narosa Publishing House: New Delhi, India, 2002; pp 319−392. (28) Gardon, R.; Akfirat, J. C. Heat transfer characteristics of impinging two-dimensional air jets. J. Heat Transfer 1966, 88 (1), 101−108. (29) Cadek, F. F. Ph.D. Thesis, University of Cincinnati, 1968. (30) Vishnuvardhanarao, E.; Das, M. K. Computational study of heat transfer in a conjugate turbulent wall jet flow at high Reynolds number. J. Heat Transfer 2008, 130, 072201-1.

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