Qinglin Niu, Biao Chen,Zhihong He, Jianfei Tong,2and Shikui Dong*

(1. Key Laboratory of Aerospace Thermophysics,Ministry of Industry and Information Technology,Harbin Institute ofTechnology,Harbin 150001,China;2. Dongguan Branch,Institute of High Energy Physics,Chinese Academy of Sciences,Dongguan 523803,Guangdong,China)

Abstract:Local heat transfer and flow characteristics in a round turbulent impinging jet forRe≈23 000 is predicted numerically with the RANS approach and ak-ε-futurbulence model. The heat transfer predictions and turbulence parameters are verified against the axis-symmetric free jet impingement measurements and compared with previous other turbulence models, and results show thek-ε-fumodel has a good performance in predictions of the local wall heat transfer coefficient, and in agreement with measurements in mean velocity profiles at different radial positions as well. The numerical model is further used to examine the effect of the fully confined impingement jet on the local Nusselt number. Local Nusselt profiles inxandy-centerlines for the target plate over three separation distances are predicted. Compared with the experimental data, the numerical results are accurate in the central domain around the stagnation region and present a consistent structure distribution.

Keywords:heat transfer; impingement flow;k-ε-futurbulence model; Nusselt number

1 Introduction

Jet impingement technique has been widely applied in many industrial applications because of its high heat transfer rate. During the last several decades, scores of the experiments and relative numerical methods were studied on the mechanism of the impingement jet flow and the validation of the turbulence models, including different impingement distances, Reynolds numbers, nozzle lips, impingement angles, confines of the stream path, geometries of the nozzle, etc. Especially, a representative test case, in which a fully developed low-Reynolds turbulent jet injecting from a round-long pipe and then impinging perpendicularly upon a flat surface, was of great interest, such as, Baughn and Shimizus[1-2],Cooper[3], Yan[4], Lytle and Webb[5], Brignoni and Garimella[6]and Katti et al.[7]In addition, the heat transfer rate of the confined impingement jet flow also attracted more and more attentions in recent years, which were reported in quite a few investigations[8-12].

A nonlineark-ε-fumodel by means of the realizability constraint and Cayley-Hamilton theorem, based on the linear model developed by Park and Sung[23],was developed to predict an axisymmetric impinging jet flow[24-25], which had a high performance through comparisons of thek-εmodel andν2-fmodel. At the same time, another turbulence model, the wall-distance-free low-Rek-ε-fumodel, was reported by Goldberg and Palaniswamy[26]. In that model,its transport equation was used for determining the damping function. It was confirmed that Goldberg’s three-equationk-ε-fumodel could improve the prediction of some flows involving back-flow regions.

In current work, the local Nusselt number and flow parameters of a jet impinging to an isothermal constant temperature plate are calculated using the RANS approach. A refined Goldberg’sk-ε-fumodel modified in the turbulence production and the realizable time scale is applied to resolve the turbulence. The axisymmertric, compressible Navier-Stokes (N-S) equations based a reynolds averaged method are solved with the finite volume technique. Thek-εtransport and thefudamping function are considered as well. The solutions are verified against a set of previous experimental data[1-2,4,7]and six excellent turbulence models ofRe≈23 000 in aforementioned numerical investigations[16, 18, 21-22, 24, 27]aiming to validate the performance of the present turbulence model.

In the second part of the investigation, the effect of the fully confined impingement round jet on the local Nusselt number for the jet plate and target plate is analyzed. Computations of the symmetry model with present turbulence model are compared with the measurements conducted by Caggese et al.[12]recently. The main purpose of the analysis is to show the level of the heat transfer at different jet-to-plate distances in confined space having a single exit of the jet flow, and also to validate the effectiveness of the refined Goldberg’sk-ε-fumodel.

2 Mathematical Formulation

2.1 Governing Equations

In Cartesian tensor notation, the two-dimensional governing equations can be expressed as:

(1)

(2)

In accordance with the Boussinesq model, Reynolds stresses are related to the mean strain in an eddy viscosity model, hence the expression of Reynolds stresses can be given as follows:

(3)

The eddy-viscosity, expressed by the damping functionfμ, densityρ, turbulence kinetic energykand dissipation rateε, is shown as:

(4)

It is noteworthy that this equation has been modified from the old expressionμt=fμρk2/εin Ref.[16], where the damping coefficient was over-predicted. Thus the transport equations fork,εandfμcan be modeled as:

(5)

(6)

To get a refinement enhancing the model’s as the modify formula of thek-lmodel in Ref.[22], the realizable time scale is:

(7)

whereRt=ρk2/μεis the turbulence Reynolds number with the fluid viscosityμ.

In equilibrium conditions, the advection and diffusion terms are absent. The damping function only varies in the wall-normal direction under the condition of two-dimensional flow, which depends on its own transport equation. Thus, the following expression is derived from above analyses as:

(8)

The model constants above are given respectively as:

Cμ=0.09,σε=1.3,σk=1.0,σf=50.0

Cε1=1.44,Cε2=1.92,Aμ=0.001

In the present implementation, the heat flux can be expressed as the following formula:

q=-Cp(μ/Pr+μt/Prt)T

(9)

2.2 Numerical Procedure

The solution of the steady-state at lager times was obtained when the absolute residual errors over all the nodes was less than 10-6. The second-order scheme with out-of-face viscous polynomials was used to discretize the governing equations. The finite volume technique[28]was applied to solved the N-S equation. The fourth-order central-differencing format is selected for the treatment of the viscous term. In the turbulence transport equations, the source terms is evaluated using the same treatment as the two-equation turbulence model reported by Merci[29]. A hybrid-scheme time integration was applied, in which the four-order Runge-Kutta method was used for the explicitly treated terms and the Crank-Nicolson method for the implicitly treated terms, respectively. Details about this hybrid scheme is presented by You[30]and also found in Park’s work[25].

3 Validation Test-long Circular Pipe

3.1 Test Case Description

A typical test case of a low-Reynolds circular turbulent jet, impinging onto a perpendicular flat surface, has been measured by many researchers. To obtain a fully developed jet at the exit plane, the fluid need to pass along a smooth and sufficiently long round pipe at Reynolds number of 23 000 and then impinges on a heated large plate at a 2 or 6 diameter distance. In fact, the diameter of the pipe is not fixed value but changeable (see Table 1), since the Reynolds number has been a function of characteristic parameters containing the diameterD, the bulk velocityUband the properties of the fluid (ρa(bǔ)ndμ), consequently expressed asRe=(ρUbD)/μ.

Table 1 List of experimental conditions at Re≈23 000 in published investigations

The unheated air is selected as the fluid medium generally. The elaborate set of data is available for the jet-to-plate distance of two-pipe diameters in most tests. The length of the straight long pipe ranges from 56 to 81 diameters and its wall-thickness could be about 0.13 diameters, illustrated the geometry sketch in Fig.1. Besides, the surface of the heated plate impinged by air jet covers a special thin paint to measure the properties of the flow near the wall, where the temperature difference between jet and plate keeps 20 K so that both can perform the convective heat transfer each other.

3.2 Computational Grid

The specific parameters of simulation are given as: the diameter of the pipeD=25 mm, the lengthL=72D, the wall thicknessδw=0.031 3D[16], the radius of the plateRmax=10Dand its thicknessδp=0 due to that cannot affect the structure of the flow, and the outlet-plate distanceZ=2D. The entire computational domain for 2Daxisymmetrical model is consisted of the inner part of the whole pipe and the part between the plate and the line overtop the outlet 1/2D, shown as the closure regiona-b-c-d-e-O-ain Fig.1.

Fig.1 The geometry sketch of the jet impingement

In the region near the wall, theReis lower as the effect of viscosity. Thus boundary layers with a propery+value were built on the region near the wall. In this work, a refined grid near the pipe wall withy+=0.5 is used in computations. The instructed meshes are given in Fig.2(a). A detail description for the limiting number of constrained wall layers was also given in Wang’s work[31]. A non-uniform structured grid of 120×170 cells is used andy+≈0.5 is selected for a high resolution near wall boundaries. Fig.2(b) shows the grid-independence of a finer doubled-grid solution. The number of cells along the plate is 120 with a geometric distribution. Radially, there are 45 cells within the pipe, where the grid is also re-fined near the axis (Oa) to capture the flow structure at the stagnation point. Axially, the cells of the pipe are uniform and consistent with the downstream grids. There are five equal spacing cells in the thickness of the pipe so that the influence of the lip structure in the jet outlet can be considered.

Fig.2 Computational grid and Grid-independence

3.3 Boundary Conditions

3.4 Validation

The different turbulence models of the jet impingement with numerical methods, performed under the same conditions as the experiment in Ref.[1], are listed in Table 2, where all turbulence models have been validated through experimental data or verified by other models. Within thek-εmodel is over-predictive in the stagnation region, while LES and DNS are accurate but expensive due to the dependence of three-dimensional sophisticated grids, so these models are not illustrated in Fig.3. From Fig.3, the result obtained from the present refined Goldberg’sk-ε-fumodel has good agreement with the experimental data reported.

Besides, the prediction of the Nusselt number in the stagnation region is excellent. There are two turbulent models,k-land nonlineark-ε-fumodels, having excellent results analogous to the present model in the whole region. However, the former is too over-predicted and the latter is little under-predicted in the first peak region. The predictions using the current turbulence model are slightly higher than early experimental data at the second peak, but they are within the range of Katti and Brabhu’s recent data[7]. In other radial regions, the profile of the Nusselt number is excellently consistent with the published data, especially with Baughn and Shimizu’s.

Table 2 Overview of turbulence models at Z/D=2 with Re≈23 000

Fig.3 Profiles of heat transfer coefficient for different turbulent models

Furthermore, two important parameters to determine the accuracy of the prediction, namely the mean velocity and turbulent shear stress, were used for comparison with the data collected from Refs.[1-5] by Merci and Dick[18]. In their works, it was confirmed that the cubick-εturbulence model had good agreement with measurements in Ref.[18]. Thus distributions of the mean velocity with present model compared with data described above are shown in Fig.4. From Fig.4, profiles of mean velocity are consistent with the experimental data at different radial positions for jet-to-plate distanceZ/D=2.

4 Effect of Fully Confined Impinging Domain

4.1 Experimental Case

This section investigates the confinement effects of the fully confined impingement jet on the heat transfer coefficient for the impingement plate and the target plate. Results of simulations using the SST turbulence model, withRe=23 350 at different separation distances varying fromZ/D=0.5 and 1.5, were validated against the experimental data by Caggese et al.[12]

Fig.4 Comparison of mean velocity at R/D=0.5,1.0,2.5,3.0,for Z/D=2

In Caggese’s work, the confined geometry consists of a single exit for spent air of the jet, a 5D×5Dsquare target plate for the jet impingement, a jet-plate drilled a hole connected with a single round straight nozzle. The schematic representation of the confined impingement box is shown in Fig.5, where the arrow indicates the inlet and the outlet of the jet in the box. In experiments, the heated air through the heater mesh fixed on the pipe injects from the hole on the jet-plate and impinges onto the target plate, whose surfaces cover the liquid crystal coat to acquire the local Nusselt number around the whole flow path indirectly. During the impinging, as reported in Ref.[8], the target plate almost keeps a constant temperature.

Fig.5 Schematic representation of the confined impingement box

4.2 Results and Analysis

For the numerical simulation, the turbulence model described in section two is used, and other conditions are the same to these given in Ref.[12], such as types of boundary conditions, the quality of mesh grids, the distance of the first grid node and the residuals of the calculated parameters. Given the overall symmetry of the geometry, half of the flow domain is selected for simulation. Structured grids are generated using the grid-generation software ANSYS ICEM and is given in Fig.6. About 0.73 × 106hexahedral elements are used for an accurate spatial discretization. Dimensionless distances from centroid to wall of first layer grid, namelyy+, are used for near wall regions with different Re numbers. In this work,y+=0.5 andy+=3 are used for the lowest and the highestRe, respectively.

Fig.6 Numerical grid of the confined impingement jet

Calculated and experimental local Nusselt numbers at all tested separation distances inxandycenterlines for target plates are given in Fig.7 and Fig.8, respectively. For the target plate, all tested separation distances show similar shapes of the local Nusselt number inydirection: two distinct peaks around the ring of which radii arex/D=0.5 and 1.5, and the minimum value is located at the center of the stagnation region. From the left of Fig.7, the tendency of the experimental data in this confined geometry, referring to the results ofRe=23 000, is also coincident with that of the free-impingement, whereas the secondary peak of the confined box is outward than free-impingement by 0.5D, and the primary peaks of both cases decrease slightly and the secondary peaks have a distinct decrease as the increase of the separation distance. In the right of Fig.7, comparisons between CFD and experiments show the donut shape distribution and the level of the local Nusselt number have agreement except for the region near the secondary peak. As the same to the description of the validation part, the use of the present turbulence model to capture the heat transfer actually, due to a common over-prediction of 6% in the secondary peak, is acceptable in numerical simulation.

In Fig.8, a degradation of the primary peak in the stagnation appears because of the asymmetric geometry: a confined wall in the upstream direction and a single exit in the downstream. As theZ/Dincreases, the primary peaks in the wall side keep the same shape to these in Fig.7 but slightly below the value of the symmetry in the exit side. Additionally, all the secondary peaks at three separation distances are almost disappear but relatively strong forZ/D=0.5. Besides, the declining trend of the profiles is more abrupt, but relatively weak, in the upstream direction than the downstream direction. Note that the CFD results show a similar distribution with experiments within the region fenced by the secondary peak ring inxdirection, but the rest region, especially the position and intensity of the secondary peak, is not able to be predicted accurately. This could be attributed to the overestimation of the turbulence model itself and the increase values of GCI discussed in Ref.[12]. In some literatures on numerical solutions of the confined impingement, the secondary peak was common and its intensity was obvious in the results reported as Ref.[33].

Fig.7 Local NuD distributions on the target plate at Re=23 350:y-direction,x=0

Fig.8 Local NuD distributions on the target plate at Re=23 350:x-direction,y=0

5 Conclusions

Results have been presented for the local heat transfer in turbulent impingement jets, impinging onto a flat plate and a fully confined space for the Reynolds number of about 23 000, with a non-lineark-ε-futurbulence model. Comparison of the numerical results with experimental data shows that the RANS approach with thek-ε-fumodel is relatively accurate to be applied for low-Reynolds engineering predictions of local heat transfer. A series of validation tests considering the jet impingement, which is fully developed through a long pipe, show that the local heat transfer prediction is highly affected by the turbulence profile.

The second part of this study aims to investigate the effect of the fully confined impingement jet on the local Nusselt number. The numerical results indicate that the increasing of separation distance has no obvious effect on the local heat transfer in the center region around the first stagnation zone but has a dramatic decline in the second peak inside the confined box. The overestimated predictions of the heat transfer level around the second peak show this model still need to be further improved for the fully confined impingement case.

Acknowledgment

Authors thank Dr. Alexandros, professor of Group of Thermal Turbomachinery (GTT), Swiss federal Institute of Technology in Lausanne, Switzerland, for his help in experimental data and numerical initial parameters.