Efficacy of impinging air jets under various cross-flow conditions

D

orifice diameter (m)

D_c

cross-flow inlet hydraulic diameter (m)

h

coefficient of heat transfer [W/(m² · K)]

k

thermal conductivity of air [W/(m · K)]

Nu

Nusselt number

Nu₀

stagnation Nusselt number

Pr

Prandtl number

q

heat flux (W/m²)

r

radial distance from center of plate (m)

R_e

jet Reynolds number

R_ec

cross-flow Reynolds number

St

Stanton number

St₀

Stanton number at stagnation point

St_a

average Stanton number

St_p

Stanton number at location of peak heat transfer

T_j

air jet temperature (K)

T_w

target surface temperature (K)

u

radial velocity (m/s)

V_c

cross-flow velocity (m/s)

V_j

jet speed (m/s)

Z

orifice-to-target distance (m)

Greek

α

ratio of cross-flow to jet Reynolds numbers

ν

kinematic viscosity of air (m²/s)

Jet cooling is an attractive cooling method owing to its superior heat transfer capability. An impinging jet produces strong turbulence near the target surface, and the boundary layer becomes narrower. In single-jet cooling, the parameters affecting the heat transfer rate include among others, the Reynolds number Re, the fluid properties, the size and shape of the nozzle, the normalized orifice-to-target spacing Z/D, and the steadiness of the flow. The characteristics of jet cooling are also influenced by the flow from the surroundings. In jet cooling, heat transfer coefficients as high as 100 kW/(m² · K) are possible with single-phase fluid.¹ Lytle and Webb² studied the influence of Z/D on jet performance and found that secondary peaks occur at low Z/D and high Re, owing to increased turbulence. Correlations have been proposed to calculate the Nusselt number Nu in the three regions near the impinging wall.³ Martin⁴ presented an extensive overview of jet cooling, with a focus on applications and empirical formulas for predicting heat transfer coefficients. The important variables identified were the Reynolds number, the properties of the cooling medium, the size and shape of the nozzle, and Z/D. Empirical correlations have been proposed for predicting the heat transfer coefficients for a single round nozzle, a slot nozzle, and arrays of these nozzles. Zuckerman and Lior⁵ presented a comprehensive review of jet impingement cooling. They described the physics of the jet flow, the heat transfer mechanism, and the suitability of various turbulent models for predicting the flow and heat transfer. They listed the various correlations used for predicting heat transfer. Bouchez and Goldstein⁶ reported that under certain cross-flow conditions, the effectiveness at the stagnation point is as high as 0.6. Metzger and Korstad⁷ studied the effect of cross flow on multiple jets and found that the cooling is affected by both Z/D and the normalized spacing between jets. They proposed a correlation for the average Stanton number. Studies at Re > 35 000 revealed that at low Z/D, there is a slight improvement in the maximum Nu at low cross flow. However at high Z/D, peak Nu decreases with increase in cross flow.⁸ Studies of cross flow by Huang et al.⁹ indicated that changes in heat transfer depend on the cross-flow velocity and direction. Jet impingement studies on a cylinder showed that Nu is influenced by cross flow and Kelvin–Helmholtz instability.¹⁰ Wang et al.¹¹ studied the effects of cross flow on an inclined jet and reported that the cross flow decreases the peak Nu but has no influence on wall jet heat transfer. In multiple-jet cooling, a staggered configuration has superior characteristics compared with other arrangements.¹² Interaction with neighboring jets and the turbulence produced affect the cooling. Experimental investigations indicate that cross flow improves the peak Nu in comparison with a pure jet.¹³ In gas turbine blade cooling, diverters can minimize the effect of cross flow.¹⁴ Cooling studies have revealed that cross flow considerably affects heat transfer. Higher values of peak Nu are obtained at high Mach number, lower D, and lower Z/D.¹⁵ Nonuniformity in temperature due to cross flow can be minimized by increasing the impingement Re and reducing Z/D.¹⁶ Masip et al.¹⁷ studied heat transfer from a hot surface placed in a channel subjected to jet flow and lateral flow. Lateral flow with jet-to-cross-flow ratios of 1.0 and 1.5 produced maximum cooling. In multiple-jet cooling, the influence of cross flow from adjacent jets can be minimized by providing diverters.¹⁸ Cylindrical diverters give the maximum enhancement of heat transfer (5.2%). Choudhari et al.¹⁹ studied the combined effect of synthetic jets and cross flows on cooling of a channel and reported that the introduction of cross flow reduces heat transfer by 10%.

The aim of the present study is to investigate the changes in flow field and cooling of a single jet due to lateral flow. A correlation is proposed to calculate Nu without cross flow. This study provides new insight into the cooling performance of jets with cross flow.

Details of the domain considered is shown in Fig. 1. The target plate (stainless steel) has dimensions 160 × 80 × 0.06 mm³.³ The face abfe is the target surface to be cooled, and the heat inflow is from the bottom surface of the target plate. Th face abcd is the cross-flow inlet. All other sides are open (outlets), and pressure boundary conditions are applied. The nozzle diameter is 7.35 mm, and the length-to-diameter ratio is 7. For meshing the computational domain, an unstructured grid system with an inflation layer near the impinging surface is employed (Fig. 2). An appropriate y⁺ value was obtained by cell refinements in the near-wall regions and is close to 1. This is done to precisely capture the process of heat transfer in the area near the target surface.

The turbulent scheme used is the renormalization group (RNG) k-ε model. This turbulence model is very successful in predicting the thermohydraulic behavior of impinging jets.^20,21 For pressure–velocity coupling, the SIMPLE algorithm is used. A second-order upwind method is used for momentum and energy, and first-order equations are employed for the turbulence parameters. The convergence criterion used is <10⁻⁶ for residuals of the continuity, momentum, and energy. The temperature of the inlet air is 300 K. At open sides other than inlets, pressure boundary conditions are applied. The flow is assumed to be steady, turbulent, and incompressible. The effects of temperature on the thermophysical properties of air are taken into account.

The equations used are as follows:

The influence of Re, Z/D, and α on the heat transfer performance of a turbulent jet impinging on a flat surface is studied with the help of the flow field.

Figure 3 shows the results of the mesh-independence study. The grid is improved by raising inflation and lowering edge sizing. At both values of Re, the optimum number of elements is found to be 3.32 × 10⁶.

Figure 4 depicts the stream-wise variation of Nu. The Reynolds numbers considered are 5100 and 23 000. The experimental results obtained in some other studies^2,3,21 are also shown for comparison. The results from the present numerical study are in fair agreement with experiment at both Reynolds numbers. Under these flow conditions, Nu is maximum at the stagnation point and decreases radially.

The Nusselt number distributions at different Reynolds numbers and Z/D values are displayed in Fig. 5. For Z/D = 6 at all Reynolds numbers, the curves are qualitatively similar. For Z/D = 2, at higher Reynolds numbers, there are secondary peaks in the transition region due to the high turbulent kinetic energy.

Figure 6 shows the effect of Reynolds number on stagnation Nusselt number. Nu₀ increases with increasing Reynolds number. For Z/D = 6, when the Reynolds number is raised from 5100 to 23 000, the Nu₀ increases by 150%. The corresponding increase for Z/D = 2 is 156%. As Re increases, the jet centerline velocity increases and so does the turbulence close to the wall, and both help to increase Nu₀.

Figure 8 shows the dependence of the stagnation heat transfer (based on the correlation proposed) on Z/D. Data from other studies^3,22,23 are also shown. Equation (7) is successful in predicting stagnation heat transfer (within 6%).

Figure 9 shows the variation of the normalized gauge pressure (close to the plate) in the radial direction. Cross flow causes the profile to become asymmetrical, and the peak pressure location shifts radially. Mixing of the cross-flow stream coming inward and the jet stream results in recirculation, which causes creation of a low-pressure region (see Fig. 11).

Figure 10(a) shows the radial velocity profiles without external flow. At the jet axis, i.e., r/D = 0, the radial velocity is close to zero, since the jet is perpendicular to the plate.

Near the surface, the flow is accelerated radially and attains its peak velocity at a normalized radial distance r/D = 1. Farther downstream, the velocity decreases because of the thickening of the boundary layer. Higher radial velocities are observed for smaller Z/D. For example, for Z/D = 6, the peak value of the normalized velocity is 0.64, and the corresponding value for Z/D = 2 is 0.91. At very low Z/D, the radial velocity can be more than the jet velocity.

Figure 10(b) illustrates the radial velocity profile upstream (r/D > 0) of jet in the presence of cross flow. For Z/D = 6, the peak radial velocity is obtained at r/D = 0 and it diminishes radially up to r/D = 1. With cross flow, the jet becomes as an inclined jet and the stagnation condition is not attained. In the region r/D > 1, the flow is radially inward. For Z/D = 2, the cross flow has a considerable influence, and in all regions (r/D ≥ 0), the flow is in the cross-flow direction.

Figure 10(c) depicts the radial velocity profile downstream of the jet (r/D < 0) in the presence of an external flow. For both Z/D values, in a small region (−0.5 < r/D < 0) downstream, the flow near the target surface is opposite to the cross-flow, and beyond r/D = −0.5, the flow direction changes. The anomalous flow in the zone −0.5 < r/D < 0.5 is due to mixing of the main stream and the external flow (Fig. 11). The maximum velocity occurs at r/D = −1, and it diminishes in the downstream direction.

Figure 11 illustrates the velocity vectors in a midplane coinciding with the axis of the jet. The jet is deflected owing to cross flow, and there in mixing of the jet flow and the lateral flow. At the midpoint of the target, the stagnation condition is not achieved.

Figure 12 depicts the influence of α on the local Nu. Cross flow can occur from neighboring jets or other cooling equipment. The cross flow deflects the jet downstream, and the curves are not symmetrical about the centerline. For both values of Z/D, at smaller α, the peak Nu does not vary significantly. In fact, for Z/D = 2, there is a slight increase in the peak Nusselt number, and this is in line with the findings of other investigators. A significant reduction in heat transfer occurs at higher values of α.

The cooling effect upstream of the jet axis is decrease, while that downstream increases. From Fig. 10, it is clear that owing to cross flow, the fluid upstream decelerates and that downstream accelerates. For example, for Z/D = 2, in the absence of cross flow, the normalized radial velocity at a distance r/D = 1 is 0.9, and with cross flow, it decreases to 0.12. Similarly, without cross flow, the normalized velocity at r/D = −1 is 0.9, and there is acceleration in the presence of cross flow.

Figure 13 shows the effect of cross flow on cooling in terms of the Stanton number at the stagnation point (St₀) and at the location of peak heat transfer (St_p). For low Z/D, cross flow with low α helps to improve cooling. At higher values of α, the jet is deflected, and the jet fluid impinges on the plate at an acute angle, and there is also mixing between the two fluid streams.

When the jet impinges orthogonally on the surface, extreme cooling occurs at the stagnation point/center of the plate, and when the jet is inclined owing to cross flow, the peak heat transfer occurs downstream. At higher α, there is a considerable reduction in St_o and St_p. Cross flow has a greater influence at lower Z/D.

Figure 14 shows the ratio of the average Stanton number St_a in the upstream and downstream zones with and without cross flow. Cross flow reduces St_a in the upstream zone irrespective of α. St_a in the downstream zone increases with α to a certain value and then decreases with further increase in α. As α increases, the angle at which the jet impinges decreases and there is more circulation owing to the mixing of the two streams. Improved cooling downstream and reduced cooling upstream depend strongly on Z/D and α. The augmentation in cooling downstream and the reduction in cooling upstream are greater at lower Z/D. For Z/D = 2, at α = 1, the value of St_a downstream increases by 15%, and the corresponding decrease upstream is 35%.

The cooling characteristics of an air jet exposed to cross flow have been studied numerically under a constant wall heat flux condition. Heat transfer coefficients have been discussed in terms of the Nusselt number and Stanton number. The Stanton number in the presence of cross flow has been normalized by dividing it by the corresponding value for lateral flow. The major findings are as follows:

1. As reported by other investigators, the optimum Z/D is 6. At this Z/D, when the Reynolds number is raised from 5100 to 23 000, the stagnation Nusselt number increases by 150%.

2. The proposed correlation is successful in calculating Nu₀ in the absence of cross flow.

3. At higher values of α, cross flow results in a strong interaction between the fluid streams and causes recirculation.

4. Cross flow significantly affects the heat transfer and flow pattern of fluid near the target surface. For Z/D = 6, the stagnation Nu decreases by 84% when α is increased from 0 to 2. The corresponding decrease for Z/D = 2 is 75%.

5. With cross flow, anomalous flow behavior is observed in the zone −0.5 < r/D < 0.5, owing to recirculation near the wall resulting from the mixing of the two flows.

6. Without cross flow, the peak radial velocity occurs at r/D = 1, and cross flow causes the position of this peak velocity to shifts.

7. At low values of α and Z/D, cross flow results in a marginal improvement in peak Nu.

8. The augmentation in cooling in the downstream region and the reduction in cooling in the upstream region depend strongly on Z/D and α.

9. Cross flow has a stronger negative impact at lower Z/D and higher α.

The authors have no conflicts to disclose.

Vivek Mathew Jose: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Writing – original draft (lead); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.