Cfd Prediction for Confined Impingement Jet Heat Transfer

7/22/2019 Cfd Prediction for Confined Impingement Jet Heat Transfer

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CFD PREDICTION FOR CONFINED IMPINGEMENT JET HEAT TRANSFER

USING DIFFERENT TURBULENT MODELS

Y.Q. Zu1*

, Y.Y. Yan1, J.D. Maltson

2

1.School of the Built Environment, the University of Nottingham NG7 2RD, UK

2.Siemens Industrial Turbomachinery, Lincoln LN5 7FD, UK

ABSTRACT

The flow and heat transfer characteristics of a confined circular air jet vertically impinging on a flat plate are

numerical analyzed base on the CFD commercial code FLUENT 6.1.18. The relative performance of seven

versions of turbulent models, including the standard k model, the renormalization group k model, therealizable k model, the standard k model, the Shear-Stress Transport k model, the Reynolds stressmodel and the Large Eddy Simulation, for the prediction of this type of flow and heat transfer is investigated by

comparing the numerical results with available benchmark experimental data. It is found that Shear-Stress

Transport k model and Large Eddy Simulation time-variant model can give the better predictions of fluid

flow and heat transfer properties; especially, SST k model is recommended as the best compromise between

the computational cost and accuracy. Using Shear-Stress Transport k model, the effects of jet Reynoldsnumber (Re), jet plate length-to-jet diameter ratio (L/d), target spacing-to-jet diameter ratio (H/d) and jet platewidth-to-jet diameter ratio (W/d) on local Nusselt number (Nu) of the target plate are examined. A correlation for

the stagnation Nu is presented.


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INTRODUCTION

Jet impingement is one of the most efficient solutions of cooling hot objects in industrial processes as it

produces a very high heat transfer rate of forced-convection. There is a large class of industrial processes in

which jet impingement cooling is applied such as the cooling of blades/vanes in a gas turbine, the quench of

products in the steel and glass industries and the enhancement of cooling efficiency in the electronic industry.

Over the past 30 years, experimental and numerical investigations of flow and heat transfer characteristics undersingle or multiple impinging jets remain a very dynamic research area. The effects of nozzle geometry, jet-to-

surface spacing, jet-to-jet spacing, cross flow, operating conditions, etc. on flow and heat transfer have been

experimentally studied (Donaldso and Snedeker, 1971; Hollworth and Cole, 1987; Vantreuren et al., 1994; San

et al., 1997; Dano et al., 2005; Wang and Mujumdar, 2005; San and Shiao, 2006). Martin (1977), Jambunathan

et al. (1992), Viskanta (1993), and Zuckerman and Lior (2005) referred to a large panel of fundamental studies

and presented complete reviews on the above important parameters influence on heat transfer.

Most industrial applications of impinging jets are concerned with turbulent flow in whole domain

downstream of a nozzle. Modelling of turbulent flow presents the greatest challenge for rapidly and accurately

predicting impingement heat transfer even under a single round jet. Over the past decades, although no single

model has been universally accepted to be superior to all classes of problems, various turbulent models have

been developed successfully to roughly predict impingement flow and heat transfer. However, there are only a

limited number of studies concerned with comparisons of the reliability, availability and capability of different

turbulent models for impingement flows. Thakre and Joshi (2000) evaluated twelve versions of low Reynoldsnumber k models and two low Reynolds number RSM models for heat transfer in turbulent pipe flows.

Their comparative analysis between the k models and RSM models for the Nusselt number prediction is in

favour of the applicability of the k models even though the RSM model overcomes the assumption ofisotropy and the constancy of turbulent Prandtl number. Shi et al. (2002) presented simulation results for a single

semi-confined turbulent slot jet impinging normally on a flat plate. The effects of turbulence models, near wall

treatments, turbulent intensity, jet Reynolds number, as well as the type of thermal boundary condition on the

heat transfer were studied using the standard k and RSM models. Their results indicate that both standard

k and Reynolds stress model (RSM) models predict the heat transfer rates inadequately, especially for lownozzle-to-target spacing. For wall-bounded flows, large gradients of velocity, temperature and turbulent scalar

quantities exist in the near wall region and thus to incorporate the viscous effects it is necessary to integrate

equations through the viscous sublayer using finer grids with the aid of turbulence models. In the study of Wang

and Mujumdar (2005), five versions of low Reynolds number

k models for the prediction of the heattransfer under a two-dimensional turbulent slot jet were analyzed by comparison with the available experimental

data. Effects of the magnitudes of the turbulence model constants were also carried out.

In this paper, a numerical study of a confined circular air jet vertically impinging on a flat plate is performed.

The jet flow after impingement is constrained to exit in two opposite directions. The paper aims to recommend

the most suitable model(s) in predicting this type of flow and heat transfer through an investigation of the

relative performance of different turbulent models. Numerical calculations base on the CFD code FLUENT

6.1.18 are conducted, the justification of the models are carried out by comparing the numerical results with

available benchmark experimental data. Then, using the most suitable model, the effects of jet Reynolds number ,

jet plate length-to-jet diameter ratio, target spacing-to-jet diameter ratio and jet plate width-to-jet diameter ratio

on local Nusselt number of the target plate are examined. Also, a correlation for the stagnation Nu is presented.

PROBLEM DESCRIPTION

Geometry and Boundary Condition

Fig. 1 shows the physical domain and boundary conditions of the modelling. Air flow at high velocity

passes through a round jet with both length and diameter d=6mm, vertically impinging on the confined target

plate with two side walls positioned at spanwise distances of y=W/2. The jet after impingement was restricted

to discharge in two opposite directions parallel to the x-axis, and two channel outlets are placed at x=L/2. The

target plate was kept at constant heat flux of 1000W/m2; all other walls are adiabatic.

For Re=10000, H/d=2.0, L/d=41.7, W/d=10.42, the local Nusselt number distribution along the positive x-axis is numerical simulated in different turbulent models to investigate the relative performance of these models

by comparing the numerical results with available benchmark experimental data (San and Shiao, 2006). Then,

using the most suitable model, the effects of jet Reynolds number, jet plate length-to-jet diameter ratio, target

spacing-to-jet diameter ratio and jet plate width-to-jet diameter ratio on local Nusselt number of the target plateare examined. A correlation for the stagnation Nu is presented.


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Fig. 1: The physical domain and boundary conditions

Turbulent Models

In this study, the standard k model, the renormalization group (RNG) k model, the realizable k

model, the standard k model, the Shear-Stress Transport (SST) k model, the Reynolds stress model(RSM) and the Large Eddy Simulation (LES) time-variant model (FLUENT_6.1_Documentation, 2003) wereused.

All the k , k and RSM turbulence models belong to the Reynolds-Averaged approach, in which theReynolds-averaged Navier-Stokes (RANS) equations were used as the transport equations for the mean flowquantity, all the scales of the turbulence are modelled based on certain assumptions. The RANS equation in a

Cartesian tensor form can be written as:

0

i

i

uxt

(1)

jijl

lij

i

j

j

i

ji

ji

j

i uuxx

u

x

u

x

u

xx

puu

xu

t

3

2(2)

For steady state flow, the time derivative terms drop out. Comparing the RANS momentum equation Eq. (2) tothe to the Navier-Stokes momentum equation, additional terms appear that represent the effect of turbulence. The

Reynolds stresses, jiuu , must be modelled in order to close Eq. (2).

The k models and k models employ the Boussinesq hypothesis (Hinze, 1975) to relate theReynolds stresses to the mean flow velocity gradients:

ij

i

it

i

j

j

itji

x

uk

x

u

x

uuu

3

2(3)

The standard, RNG and realizable k models have similar forms with transport equations for turbulence

kinetic energy k and its dissipation rate . The main issue is how the turbulent viscosity t is computed. In the

standard k model (Launder and Spalding, 1972) the transport equations for the turbulence kinetic energy kand its dissipation rate is solved and t is computed as:

2k

Ct (4)

where C is a constant. In the RNG k model (Yakhot and Orszag, 1986), the scale elimination results in a

differential equation for turbulent viscosity which gives Eq. (4) in the high-Reynolds-number limit and also

enables us to include low-Reynolds-number effects by using the original differential relation. While in the

realizable k model (Shih et al., 1995), the coefficient C is a function of the mean strain and rotation rates,

the angular velocity of the system rotation, and the turbulence fields.

In the standard k model (Wilcox, 1998), the transport equations for turbulence kinetic energy k and the

specific dissipation rate are solved, and t is computed as:


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kt * (5)

Where, * is a function of Reynolds number and is one for high Re.

In the SST k model (Menter, 1994), definition of the turbulent viscosity is modified to account for the

transport of the principal turbulent shear stress. Here, t is given as:

]/*,/1max[

1

12

F

kt

(6)

Where, is a function of the mean rate-of-rotation tensor, 2F is the blending function, and 1 is a constant.

This feature gives the SST k model an advantage in term of performance over both the standard k

model and the standard k model.The RSM model (Launder et al., 1975) solves exact transport equations for the transport of the Reynolds

stresses, jiuu . It also includes an additional scale determining equation for . Then the turbulent viscosity is

calculated similarly to the k models. The RSM is superior for situations in which anisotropy of turbulencehas a dominant effect on the mean flow. Such cases include highly swirling flows and stress-driven secondary

flows.

Unlike the k , k and RSM models which are all based on the RANS equations, LES provides andalternative approach in which the large eddies are computed in a time dependent simulation that uses a set of

filtered Navier-Stokes (NS) equations (Galperin and Orszag, 1993). The attraction of LES is that, by modelling

less of the turbulence (and solving more), the error induced by the turbulence model will be reduced. However,

the application of LES to industrial fluid simulations is in its infancy. So far, the model has mainly been used to

simulate the fluid flows in simple geometries because of the large computer resources required to resolve the

energy-containing turbulent eddies.

Numerical Procedure

The numerical simulations were carried out using the commercial flow solver FLUENT 6.1.18 based on the

finite volume method. The momentum and energy equations were discretized using the second-order upwindscheme and other transport equations were discretized using the power law scheme. The discretized equations

were solved using the SIMPLEC algorithm. Default values in FLUENT for all the parameters in the turbulent

models adopted. The commercial package Gambit 2.0.4 was used to generate the geometry and mesh for the

computational domain.

Fig. 2: Computational domain

,The computational domain is a quarter of the real physical geometry due to the symmetry of the problem as

shown in Fig. 2. Hexahedral elements are used for meshing the geometry. Different types of boundary conditions

were used for different zones of the flow domain. For the jet discharge, a mass flow inlet type of boundary

conditions is used. The temperature of the inlet is at 300 K. At the outlet of the computational domain (rightboundary in Fig. 1), the outflow type of boundary condition is used. At two symmetric sections (x=0 and y=0),

the symmetry type of boundary is specified. The no slip wall condition is used for all the other boundaries.

Hereinto, a constant heat flux of 1000W/m2

at the bottom wall, and a zero heat flux on the other walls. In the

computation, the mean velocity and temperature were normalized with the velocity and temperature of the jet,respectively. The material of the inlet air is described by the following parameters, 225.1 kg/m3,


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n=0.0242W/mK, 7894.1 10-5kg/ms. The implicit and segregated solver is used for the solution of the

system of governing equations. During the simulations, the value of y+

for the wall-adjacent cells was fixed

approximately at 1. The solution is assumed to be converged when the normalized residual of the energy

equation is lower than 106

and the normalized residuals of continuity and other variables are less than 103

.

RESULTS AND DISCUSSION

Comparison of Turbulent Models

For Re=10000, H/d=2.0, L/d=41.7, W/d=10.42, the local Nusselt number distribution along the positive x-

axis is numerically simulated. Seven different turbulent models are used in the simulation to investigate the

relative performance of these models by comparing the numerical results with available benchmark experimental

data (San and Shiao, 2006).

Fig. 3: Local Nusselt number distribution on the positive x-axis(Re=10000, H/d=2.0, L/d=41.7, W/d=10.42)

The comparison of numerical results with those of experiments is presented in Fig. 3 and shows that SST kmodel and LES time-variant model can give the better predictions of fluid flow and heat transfer properties,

while the simulations by using standard, RNG and realizable k models, standard k model, and RSMmodel all give large errors compared to available experiment data even with high resolution grids. Comparing

Nusg obtained by different turbulent models using the same grids with that obtained by experiment of San and

Shiao (2006), the k models, standard k model and RSM model have the errors as large as 28-116%.Moreover, they all give the bad predictions of the global distribution of the Nusselt number.

Table 1: Comparison of different turbulence models for confined impingement heat transfer

Turbulence Models Computational Cost Accuracy Error ofNusg(comparison with experiments)

Standard k Low Poor 101%

RNG k Low Poor 28%

Realizable k Low Poor 116%

Standard k Moderate Poor 67%

SST k Moderate Good 7%

RSM Moderate Poor 107%

LES High Excellent 4%

The computational cost and accuracy of the different turbulence models on predicting the flow and heat transfercharacteristics of this case are compared in Table 1. With the implementation of the LES time-variant model, the

cost in terms of the computational resource is very high although it gives excellent results.


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Fig. 4: Local Nusselt number distribution on the positive x-axis

(Re=10000, d=6.0mm, H/d=2.0, L/d=41.7, W/d=10.42)

As a further check on the consistency of the experimental and numerical results obtained by SST kmodel, the local Nusselt number distribution for Re=30000, H/d=2.0, L/d=41.7, W/d=10.42 are compared. Fig. 4

shows the distribution of Nu along the positive x-axis. Some representative points taken from the experimental

study (San and Shiao, 2006) are additionally shown to illustrate the degree of the quantitative comparison.

Therefore, SST k model is recommended as the best compromise between the computational cost andaccuracy and is used for all of the following simulations.

Effects ofW/don Nusg

Fig. 5 (a), (b) show the variation with W/dof Nusg when H/d=3.0, L/d=50 and H/d=5.0, L/d=50, respectively.

From the figures, it can be seen that with the increase of W/d, Nusg descreases and meanwile the corresponding

decreasing rate grows. Moreover, the figures indicate that the increasing Reynolds number results in a rise of

Nusg.

(a) H/d=3.0, L/d=50 (b) H/d=5.0, L/d=50

Fig. 5: Effects ofW/don Nusg.

Effects ofL /don Nusg

For L/d increasing from 10 to 150, the Nusselt numbers on stagnation point are calculated when H/d=3.0,

W/d=10 and H/d=3.0, W/d=20 at different Reynolds numbers as shown in Fig.6. It is noted that, the effects ofL/d are quite similar with those of W/d, i.e. the increasing L/d results in the decrease ofNusg and the augment of

decreasing rate. However, it should be pointed out that the effects of L/d are relatively weaker than W/d. The


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explanation for this phenomenon was presented by San, J. Y., and Shiao (2006). They believed that for an

increase of W/d, the decrease ofNusg is caused by the enhancement of mixing of the hotter recirculation air from

downstream with the air near the jet orifice; while for increasing L/d, the decrease ofNusg is mainly caused by an

enhancement of the flow-mixing near the stagnation point which has less effect than the enhancement of flow-

mixing near the jet orifice on the impingement heat transfer.

(a) H/d=3.0, W/d=10 (b) H/d=3.0, W/d=20Fig. 6: Effects ofL/don Nusg.

Effects ofH/don Nusg

As shown in Fig. 7, Nusg decreases with the increasing H/dbecause the larger jet-to-target spacting can allow

a longer time of mixing and heat transfer between the hotter jet and the surrounding cooler air and therefore

decrease the heat transfer near the stagnation point.

(a) L/d=50, W/d=10 (b) L/d=50, W/d=20Fig. 7: Effects ofH/don Nusg.

Correlation for Nusg

In the experimental study of San and Shiao (2006) for a confined circular air jet impinging on a flat surface,

Nusg was found to be a function of)]/(011.0)/(044.0[3.0)/( dLdWedH

, which has also been validated by the present

simulations. Therefore, it is assumed that Nusg can be expressed by the following correlation,

)]/(011.0)/(044.0[3.0)/( dLdWBsg edHAReNu (7)


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By analyzing the results obtained by the present simulation, it is found that the exponent for the jet Reynolds

number, B=0.642; while the constant A=0.423. Thus, the complete correction for the Nusselt number on the

stagnation point can be written as,

.3000010000;61;15010;405for

)/(423.0 )]/(011.0)/(044.0[3.0642.0

ReH/dL/dW/d

edHReNu dLdWsg

(8)

To show the quantitatively agreement between the numerical results and the correlation, Fig. 8 show the

variation of })//{( )]/(011.0)/(044.0[3.0 dLdWsg edHNu with Reynolds number under the different geometry

conditions. It is noted that the values of })//{( )]/(011.0)/(044.0[3.0 dLdWsg edHNu can be approximated well by

fitting curve, 642.0423.0 Re .

(a) H/d=3.0, L/d=50; (b) H/d=3.0, W/d=20; (c) L/d=50, W/d=20

Fig. 8: Variation of })//{( )]/(011.0)/(044.0[3.0 dLdWsg edHNu with Re

CONCLUSIONS

The relative performance of seven versions of turbulent models, including the standard k model, the

renormalization group k model, the realizable k model, the standard k model, the Shear-Stress

Transport k model, the Reynolds stress model and the Large Eddy Simulation, for the prediction of this typeof flow and heat transfer is investigated by comparing the numerical results with available benchmark

experimental data. It is found that the Shear-Stress Transport k model and the Large Eddy Simulation time-

variant model can give better predictions of fluid flow and heat transfer. The SST k model is recommended

as the best compromise between the computational cost and accuracy. Using the Shear-Stress Transport kmodel, the effects of jet Reynolds number, jet plate length-to-jet diameter ratio, target spacing-to-jet diameter

ratio and jet plate width-to-jet diameter ratio on local Nusselt number of the target plate are examined. It is found

that Nusg increases with the rise of the jet Reynolds number; while the increasing W/d, H/dand H/d can all result

in the decrease of Nusg and the augment of decreasing rate. Moreover, for 3000010000 Re , 405 W/d ,15010 L/d and 61 H/d , a correlation )]/(011.0)/(044.0[3.0642.0 )/(423.0 dLdWsg edHReNu

, is presented.

NOMENCLATURE

English Symbols

A, B constants

d jet diameter (m)h convective heat transfer coefficient (W/m

2K)

H jet plate-to-impingement plate spacing (m)

k turbulence kinetic energy

L jet plate length (m)

n thermal conductivity of air (W/m K)


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Nu local Nusselt number, hd/n

Pr Prandtl number, /

q surface heat flux (W/m2)

Q volumetric flow rate (m3/s)

Re jet Reynolds number, dQ /4

aw

T adiabatic wall temperature (K)

jT jet total temperature (K)

wT local wall temperature (K)

U jet mean velocity (m/s)

iu , ju component of velocity

W jet plate width or heated surface width (m)

x x-coordinate (m)

y y-coordinate (m)y dimensionless distance, /yuy

z z-coordinate (m)

Greek Symbols

thermal diffusivity (m2/s)

dynamic viscosity (kg/ms)

t turbulent viscosity

density (kg/m3)

dissipation rate of turbulence kinetic energy k

specific dissipation rate of turbulence kinetic energy k

Subscripts and Superscripts

aw adiabatic wall

sg stagnation point

w wall

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Cfd Prediction for Confined Impingement Jet Heat Transfer

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