A numerical study of the unsteady flow and heat transfer in a transitional confined slot jet impinging on an isothermal surface

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Abstract

A numerical finite-difference approach was used to compute the steady and unsteady flow and heat transfer due to a confined two-dimensional slot jet impinging on an isothermal plate. The jet Reynolds number was varied from Re=250 to 750 for a Prandtl number of 0.7 and a fixed jet-to-plate spacing of H/W=5. The flow was found to become unsteady at a Reynolds number between 585 and 610. In the steady regime, the stagnation Nusselt number increased monotonically with Reynolds number, and the distribution of heat transfer in the wall jet region was influenced by flow separation caused by re-entrainment of the spent flow back into the jet. At a supercritical Reynolds number of 750 the flow was unsteady and the net effect in the time mean was that the area-averaged heat transfer coefficient was higher compared to what it would have been in the absence of jet unsteady effects. The unsteady jet exhibited a dominant frequency that corresponded to the formation of shear layer vortices at the jet exit. Asymmetry in the formation of the vortex sheets caused deformation or buckling of the jet that induced a low-frequency lateral jet “flapping” instability. The heat transfer responds to both effects and leads to a broadening of the cooled area.

Introduction

Impinging jets are used in many industrial applications because they produce high transfer coefficients with relatively low pressure drops. Applications include drying of papers and films, tempering of glass and metal during processing, cooling of gas turbine surfaces, and cooling of electronic components. In many industrial applications, such as in cooling of electronics or internal cooling of turbine blade/vane surfaces, the jet outflow is often confined between the target surface and an opposing surface in which the jet orifice is located. The presence of a confining top surface in jet impingement results in a complicated flow in which the free jet behavior is coupled to the behavior of the fluid in the channel formed between the two surfaces. Most previous research has focused on the heat transfer and fluid mechanics in free impinging jets, primarily in the turbulent regime. In microelectronics cooling, air velocities are often limited by acoustic concerns and, hence, impinging jet heat transfer in the turbulent regime may be impractical. From both a fundamental and practical perspective then, both the laminar and turbulent regimes are relevant for investigation. The confined impinging jet provides a practical solution to the microelectronics cooling problem because of the concentration of intense cooling over small areas. It seems reasonable to assume that cooling technology will follow the same path as the electronics, toward miniaturization and integration. For this reason, the potential emergence of milli-scale impinging jets as cooling solutions requires an understanding of the detailed behavior so that it can be scaled downward in physical length scales.

Because of its importance, heat transfer between single or multiple impinging air or liquid jets and a flat surface has been the subject of numerous investigations over the last four decades, most of them experimental. Lytle and Webb [1] experimentally investigated the impingement of a circular air jet on a flat plate for nozzle-to-plate spacings of less than one nozzle diameter and Reynolds numbers varying from 3600 to 27 600. They found that, for H/d of less than 0.25, the maximum local heat transfer did not occur at the stagnation point but at a radial (r) location that depended on Reynolds number and nozzle-to-plate distance, d. A local minimum heat transfer developed at the stagnation (r=0) point and two distinct peaks were observed in the radial direction. The first peak in the Nusselt number occurred in an annular region around r/d=0.5, due to the non-uniform, mixing-induced turbulence in the developing jet. The second peak occurred in the region 1.5<r/d<2.5, as a result of the retransition of the wall jet from laminar to turbulent.

Huang and El-Genk [2] investigated the heat transfer between a uniformly heated flat plate and a circular impinging jet. In their experiments, Re was varied from 6000 to 60 000, the radial distance from the stagnation point from 0 to 10 cm, and the nozzle-to-plate spacing, H, from 1 to 12 cm. For all Reynolds numbers and H⩾2 cm, the lowest wall temperature always occurred at the stagnation point (r=0). At smaller spacings (H⩽1 cm) and Re>13 000, the maximum Nu still occurred at the stagnation point. However, for lower Reynolds numbers, its location shifted outward to r/d=1.8–2.0 as Re was increased.

The influence of vortices on the thermal field of the jets was investigated experimentally by Kurosaka et al. [3]. When a thermally insulated flat plate was inserted into an impinging jet, the wall temperature distribution was modified by the presence of a secondary layer of vortices imposed on the plate by the primary vortex ring of the jet. When the plate was near the jet nozzle, a region of lower wall temperatures was observed as a result of the additional vortices. When placed farther from the jet nozzle, the secondary vortices were not present in the flow and the region of lowered wall temperatures vanished.

Anderson et al. [4] experimentally investigated stagnation zone flow in impinging jets. Small particles of glass beads were injected into air impinging jets at flow Reynolds numbers of 21 000. Particle velocities near the wall deviated strongly from fluid velocities, resulting in rebound. The deceleration associated with the rebound caused a significant increase in particle density above the impinging plate in the stagnation region. The shear layer vortices induced spatial variations in the particle concentration in the free jet, causing time fluctuations in the particle number density near the plate. Collisions with the plate were nearly elastic, with particles rebounding strongly into the impinging flow. Particles gained radial velocity near the peaks of their rebound trajectories at locations where their residence times were longest. Furthermore, in order to capture the particle–vortex interactions, the jet was acoustically forced at a frequency of 180 Hz, corresponding to the naturally occurring harmonic for the flow. Cross-sectional images showed that the free jet vortices produced spatial variations in the particle concentrations, causing temporal fluctuations in the density of particles in the stagnation zone.

The presence of a confining top plate facing the target plate in jet impingement results in a more complicated flow structure and only recently have studies focused on this configuration. Fitzgerald and Garimella [5] experimentally investigated the flow field of an axisymmetric, confined, submerged, turbulent jet impinging normally on a flat plate. Reynolds numbers varied from 8500 to 23 000. Flow field measurements proved that, in the potential core of the air jet, the confinement increased the length of the core, decreased turbulence levels in the jet, and reduced the heat transfer at the stagnation point by up to 10% in comparison to the free jet. A recirculation zone was observed moving radially outward from the stagnation zone, with an increase in both Reynolds number and nozzle-to-plate spacing.

Garimella and Rice [6] investigated the heat transfer from a small heat source to a normally impinging axisymmetric and submerged liquid jet in confined and unconfined configurations for electronics cooling situations. The nozzle diameter (0.79⩽d⩽6.35mm), Reynolds number (4000–23 000), and nozzle-to-source spacing (1⩽H/d⩽14) were all varied. For a given H/d and Re, the smaller nozzles generally produced the higher heat transfer coefficients, especially in the stagnation region where the heat transfer coefficient for the 0.79 mm nozzle was 3.5 times greater than that for the 6.35 mm nozzle. The stagnation and area-averaged heat transfer coefficients increased with increasing Reynolds number. Secondary peaks, resulting from the transition to turbulence in the wall jet region, were noted in the local distributions around r/d=2. In some cases, the heat transfer coefficients for the secondary peak were higher in magnitude than those at the stagnation point.

Morris and Garimella [7] used a commercial finite-volume code to investigate the local heat transfer coefficient distribution on a square heat source due to a normally impinging, axisymmetric, confined, and submerged turbulent liquid jet. Numerical predictions were made for nozzle diameters of 3.18 and 6.35 mm at several nozzle-to-heat source spacings and the Reynolds number varied from 8500 to 13 000. The results showed good agreement, 16–20% accuracy, with experiments.

A review of the literature has revealed that the behavior of the two-dimensional impinging jet in the laminar and transitional regime is not at all well known. Principally, this is because most studies have been motivated by the industrial use of impinging jets with macro-scale dimensions, of order 1 cm and greater. If we consider the potential use of impinging jets at increasingly smaller scales, from milli-scale to even micro-scale, practical limitations will clearly limit their operation to the laminar regime. Hence, the objectives of the present study are to characterize the behavior of the confined laminar impinging jet and the attendant heat transfer removal to the target wall. Since the stability of this fundamental flow is not well documented, particular attention is paid to identifying transitions to unsteady regimes and their effects on the heat transfer.

Section snippets

Problem description

The problem geometry and nomenclature are shown in Fig. 1. A two-dimensional jet with uniform velocity Vj enters from a nozzle of width W into a channel with height H and length L, for a fixed geometry of H/W=5 and L/W=25. The jet Reynolds number, based on a hydraulic diameter of 2W, is varied from 250 to 750. The Prandtl number is assumed to be 0.72, with air as the cooling fluid. The upper confining wall at y=0 is adiabatic while the lower target wall at y=H is isothermal at Tw. The

Numerical procedure

The numerical calculations were performed using the fluid and heat transfer solver (FAHTSO) code, a custom CFD/CHT solver developed for solving the momentum and energy equations in two- or three-dimensional steady or unsteady flows. A brief description of the numerical procedure is presented here. Complete information can be found in [8], [9].

The discretized equations are derived from the differential equations by using a finite-volume discretization procedure. , , can be represented by a

Steady laminar regime

Fig. 2 illustrates typical velocity fields for subcritical, steady Reynolds numbers of 250 and 500, and Fig. 3 gives the respective temperature contours. As is typical of impinging jets, the flow undergoes severe acceleration as it turns and then evolves into a plane wall jet. For a Reynolds number of 250 (Fig. 2(a)), the flow separates from the impinging plate. The jet momentum at this Reynolds number is not able to overcome the opposing frictional forces of the wall and the retarding effects

Conclusions

A numerical investigation of the confined impinging jet was performed for Reynolds numbers of 250, 500, and 750, based on hydraulic diameter. Visualization of the numerical simulations showed the complexity of the flow field, steady for Reynolds numbers of 250 and 500 and unsteady for a Reynolds number of 750. Wall pressure, friction, and heat transfer coefficients reached a peak value in the vicinity of the stagnation point for all cases under investigation, as expected. The physical

Acknowledgements

This work would not have been possible without the dedicated efforts of our colleagues, Dr. Jorge Luis Rosales and Prof. J.A.C. Humphrey, in developing the FAHTSO code. We are grateful to them for their many contributions to this study.

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