Numerical Heat Transfer -Paper1

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Numerical Simulation of the JetImpingement Cooling of a CircularCylinderDushyant Singh a , B. Premachandran a & Sangeeta Kohli aa Research Scholar, Department of Mechanical Engineering , IndianInstitute of Technology Delhi , New Delhi , IndiaPublished online: 02 May 2013.

To cite this article: Dushyant Singh , B. Premachandran & Sangeeta Kohli (2013): NumericalSimulation of the Jet Impingement Cooling of a Circular Cylinder, Numerical Heat Transfer, Part A:Applications: An International Journal of Computation and Methodology, 64:2, 153-185

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NUMERICAL SIMULATION OF THE JET IMPINGEMENTCOOLING OF A CIRCULAR CYLINDER

Dushyant Singh, B. Premachandran, and Sangeeta KohliResearch Scholar, Department of Mechanical Engineering, Indian Institute ofTechnology Delhi, New Delhi, India

A numerical investigation was carried out on circular jet impingement heat transfer from

a constant temperature circular cylinder to understand the major parameters which

influence the fluid flow and heat transfer characteristics. In this study, air was considered

as the working fluid. The flow was considered to be three-dimensional, incompressible,

and turbulent. To select a suitable turbulence model for the parametric study, numerical

simulations were carried out with standard k-e, standard k-x, RNG k-e, Realizable k-e,and SST k-x turbulence models for modeling Reynolds stress terms. Simulations were

also carried out using four low Reynolds number models. The results obtained using these

models were compared with the available experimental results of jet impingement heat

transfer from circular cylinder. It was identified that the RNG k-e model predicts heat

transfer characteristics better compared to all other turbulence models considered in this

study. Using this turbulence model, a parametric study was carried out for the Reynolds

number (Red), defined based on the diameter of the nozzle ranging from 10,000 to 50,000.

The ratio of distance between the nozzle exit and the cylinder surface to the diameter of

the jet (h/d) was varied from 4 to 16 and the ratio of nozzle diameter to cylinder diameter

(d/D) varied from 0.11 to 0.25. For a fixed Red and d/D, the stagnation point Nusselt

number increases as h/d decreases. The stagnation point Nusselt number decreases as

d/D increases for a fixed value of Red and h/d. The effects of change in h=d and d=D

are significant only near the stagnation region.

1. INTRODUCTION

Jet impingement heat transfer is one of the widely used techniques for heating,cooling, and drying of a surface. Impinging jet flows are employed in a wide variety ofapplications such as surface coating and cleaning, cooling of electronic components,metal cutting and forming, cooling of turbine blades and outer wall of combustionchamber, drying of textiles, aircraft wing leading-edge heating for anti-icing applica-tions, continuous casting of circular metal billets, surface cooling of pipes carryinghot air in aircraft cooling systems, etc. Martin [1], Livingood and Hrycak [2], Hrycak[3], Jambunathan et al. [4], Viskanta [5], and Zuckerman and Lion [6] presentedreviews on jet impingement cooling of flat surfaces. However, literature on jetimpingement cooling of curved surfaces, in particular, convex surfaces and circular

Received 12 June 2012; accepted 12 January 2013.

Address correspondence to B. Premachandran, Department of Mechanical Engineering, Indian

Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India. E-mail: [email protected]

Numerical Heat Transfer, Part A, 64: 153–185, 2013

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2013.772869

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cylinders, are scarce even though there are many applications related to materialprocessing and food processing in which the jet impingement cooling=heating of thesesurfaces are important.

Some researchers have studied jet impinging heat transfer from hemisphericalconvex surfaces. Cornaro et al. [7] studied experimentally the effects of high relativecurvature, i.e., the ratio of the diameter of the jet to the diameter of the convexsurface (d=D), on surface heat transfer for a round jet impinging perpendicularlyon a semi-cylindrical convex surface. The experiments were conducted for Reynoldsnumber Red¼ 6,000, 10,000, and 16,000 and the nondimensional distance betweenthe jet exit to the cylinder surface, h=d in the range 1–4 and the relative curvature,d=D¼ 0.18, 0.28, and 0.38. Lee et al. [8] carried out experimental investigation oncircular jet impingement heat transfer from a hemispherical convex surface forRed¼ 11,000–50,000, h=d¼ 2–10, and d=D¼ 0.034–0.089. The results show thatthe stagnation point Nusselt number increases with curvature and the curvatureeffect is less in the wall jet region compared to the stagnation region.

Gau and Chung [9] investigated the surface curvature effect on two-dimensional slot impinging jet heat transfer from concave and convex semi-cylindrical surfaces. Chan et al. [10] observed that the rate of decay of the averagecircumferential Nusselt number around the semi-circular convex surface was muchfaster than that which occurs laterally along the flat surface

Most of the researchers who worked on the jet impingement cooling ofcylinders considered only slot jets. Gori and Bossi [11] carried out experimentalinvestigation on single slot jet impingement heat transfer from a circular cylinderfor ReD¼ 4,000–20,000, h=S¼ 2–10, and D=S¼ 2 where S is the slot spacing. Based

NOMENCLATURE

d nozzle diameter, m

D cylinder diameter, m

Dh hydraulic diameter, m

h distance between the nozzle exit and

target, m

I turbulence intensity

kf fluid thermal conductivity, w=m k

k turbulence kinetic energy, m2=s2

L cylinder length, m

l pipe length, m

Nu nusselt number

Nuh circumferential nusselt number

Nuz axial nusselt number

p mean pressure, pa

Pr prandtl number

q00 heat flux, w

Red reynolds number based nozzle diameter

ReD reynolds number based heated cylinder

diameter

Res reynolds number based nozzle slot width

S slot width, m

Sij strain rate tensor, s�1

T temperture, k

Tjet jet exit temperature, k

Tw impingement wall temperature, k

ui mean velocity component in xidirection, m=s

u0i fluctuating velocity component in xidirection, m=s

us frictional velocity,ffiffiffiffiswq

q, m=s.

yþ nondimensional distance from.wall, usyn

z axial direction in cylinder coordinates, m

e dissipation rate, m2=s3

h angle

q density, kg=m3

m dynamic viscosity, kg=m-s

n kinematic viscosity, m2=s

nt turbulent (or eddy) viscosity, m2=s

s shear stress, n=m2

x dissipation rate, s�2

Subscripts

j jet

t turbulent

w wall

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on their experimental results, correlations for the average Nusselt number werepresented for 2� h=S� 6 and 6� h=S� 10. They also carried out an experimentalstudy on slot jet impingement on a circular cylinder [12] for the Reynolds numberReD¼ 4,000–22,000, h=D¼ 6, 8, 10, 12, 14, and 20, and D=S¼ 4. From the para-metric study they found that the heat transfer rate is maximum at h=D¼ 8. Basedon their studies on single slot jet impingement on circular cylinders, defining Rebased on the slot spacing (S), they reported that heat transfer rate is maximum atD=S¼ 2. McDaniel and Webb [13] conducted an experimental study on slot jetimpingement cooling of a cylindrical surface which was maintained at constant heatflux. They investigated the effect of a sharp-edged and contoured orifice on heattransfer rate for Reynolds number ReD¼ 600–8000, the cylinder to nozzle widthratio D=S¼ 0.66, 1.0, and 2.0, and cylinder forward stagnation point h=S¼ 1–11.They also observed that there is an optimum nondimensional cylinder spacing h=Swhich gives maximum heat transfer. Nada [14] studied experimentally the effect ofheat transfer rate for both single slot jet aligned with cylinder axis and multiple slotjets orthogonal to cylinder axis for Reynolds number ReD ranging from 1,000 to10,000. Considering vapor deposition is an important application, Kang and Greif[15] carried out a numerical study on vertical upward, laminar slot jet impingementcooling of a cylindrical surface. In their study, a Reynolds number defined based onthe radius of the cylinder varied from 100 to 1000 and the Richardson number variedfrom 0 to 1.0. Experimental and numerical results on similar configuration were alsoreported by Amiri et al. [16] for ReD¼ 120–1,210 and Richardson numbers up to 10.

A numerical study of heat transfer from a slot air jet impinging on a circularcylinder placed on a solid surface in a partially confined domain was carried outby Olsson et al. [17] for Res¼ 23,000–1,00,000, jet–cylinder distance, h=D¼ 2–8,and cylinder curvature S=D¼ 0.29–1.14. Based on the parametric study it was con-cluded that the effect of jet-cylinder distance on stagnation point Nusselt number islittle compared to that of Reynolds number and curvature. A similar study was alsoconducted numerically by Singh and Singh [18] to understand flow and heat transfercharacteristics. They also carried out flow field measurement using PIV to validatetheir numerical results of fluid flow. Dirita et al. [19] carried out numerical investi-gation on transient conjugate slot jet impingement cooling of a solid cylindrical fooditem placed on a flat surface for fixed flow, thermal, and geometric parameters. Nitinet al. [20] performed a numerical investigation of hot air impingent on cylindricalobject. Imraan and Sharma [21] studied the flow and heat transfer around a cylinderwith and without a confinement wall. They observed that a circular cylinder in a con-fined space yields heat transfer rates that are between those for the correspondinguniform cross flow (lower limit) and slot jet impingement on a non-confined jet(upper limit).

Zuckerman and Lion [22] carried out a numerical study on the cooling of a cir-cular cylinder by a number of slot jets positioned circumferentially. In this study, thenumber of nozzles was varied from 2 to 8, jet Reynolds number ReD varied from 5,000to 80,000, and target diameter varied from 5 to 10 times the nozzle hydraulic diameter.They also presented the results obtained from a numerical study of jet impingementcooling of a circular steel cylinder of finite thickness with a set of four circumferen-tially positioned slot jets using air [23]. The main focus of this work was to understandthe effect of conduction in the finite thick solid wall on heat transfer.

JET IMPINGEMENT COOLING OF A CIRCULAR CYLINDER 155

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Literature on a circular jet impingement cooling of a cylinder is very limited,compared to slot jet impingement cooling of a circular cylinder. Sparrow et al.[24] carried out experimental studies on circular jet impingement mass transfer froma circular cylinder using naphthalene sublimation technique and presented heattransfer correlation using the analogy between heat and mass transfer forRed¼ 4,000 to 25,000. The ratio of the jet diameter to the diameter of the cylinderd=D ratio ranged from 0.189 to 0.424, while the ratio of distance between the nozzleand the cylinder h=d varied from 5 to 15.

Tawfek [25] reported experimental results of circular jet impingement heattransfer from a constant temperature cylinder for Red¼ 3,800–40,000, h=d¼ 7–30and d=D¼ 0.06–0.14. It was observed that the Nusselt number decreases monotoni-cally along the axial and circumferential directions. The same researcher also studiedon inclined circular air jet impingement cooling of a constant temperature circularcylinder [26]. It was observed that the point of maximum heat transfer along the axisshifts upstream and the local heat transfer distribution changes as a function ofjet inclination. Apart from the above three studies, to the best of our knowledge,heat transfer studies on circular jet impingement cooling of circular cylinder arenot available.

Numerical study will be very useful in understanding the flow and heat transfercharacteristics of the jet impingement over a circular cylinder. For numerical simula-tion of jet impingement heat transfer, it is important to select a turbulence modelwhich predicts the flow and heat transfer characteristics accurately. Even thoughSharif and Mothe [27] recently compared numerical results obtained by usingvarious two-equation turbulence models for 2-D slot jet impingement on a confinedconcave surface and a confined circular jet impinging on a flat surface to identifysuitable turbulence model, further study is required for 3-D circular jet impingementcooling of a circular cylinder=convex surfaces to identify a suitable turbulence modelwhich predicts fluid flow and heat transfer characteristics close to the experimentalresults for a wide range of flow and geometric parameters.

Hence, the objectives of the present work are to identify a two-equation tur-bulence model which performs better among the standard k-e, standard k-x, RNGk-e, Realizable k-e and SST k-x models, and four low Reynolds number k-e mod-els for a circular jet impingement cooling of a circular cylinder and carry outa parametric study with the selected turbulence model on the effects of Reynoldsnumber Red jet diameter to cylinder diameter d=D and spacing between the nozzleexit and the cylinder to cylinder diameter h=D on fluid flow and heat transfercharacteristics. Results obtained from a 3-D numerical study on circular jetimpingement cooling of a cylinder will provide valuable information on fluid flowdistribution over the cylinder and the Nusselt number distribution on the surfaceof the cylinder.

2. DESCRIPTION OF THE PROBLEM AND COMPUTATIONAL DOMAIN

In this work, the jet impingement cooling of a circular cylinder by a circularunconfined jet is studied. Air is considered to be the working fluid. Figure 1(a) showsthe geometric configuration of the problem along with computational domainconsidered in the present study. The diameter D and length L of the cylinder

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are 50.5mm and 600mm, respectively. The cylinder is maintained at a constanttemperature. In the model, the impinging jet is aligned with y-axis and the cylinderaxis is aligned with z-axis. In the present study, a long pipe is considered to bea nozzle and a fully developed condition is considered at the exit. The diametersof the nozzles considered in this study are 6mm, 10.5mm, and 12.65mm. Modelinga very long nozzle will increase the computational time enormously if the flow field isobtained inside the pipe computationally. Hence, the fluid domain is not consideredinside the nozzle and mesh was not generated within the nozzle. Instead, a fullydeveloped flow profile is imposed at the exit of the nozzle using a user defined func-tion (UDF) in order to reduce the computational time. A long pipe is modeled inorder to represent the actual geometry of the nozzle. In order to verify whetherany changes occur in the heat transfer and fluid flow characteristics with respectto the domain size above the cylinder, simulations were also carried out for longernozzles (for a larger computational domain above the cylinder). As there was noeffect of flow field and heat transfer results, the nozzle length of 100mm was fixedfor all numerical simulations. Figure 1b shows the axial and circumferential direc-tions along which the local Nusselt number distributions are presented.

Figure 1. Geometric configuration along with the computational domain considered in the numerical

study (color figure available online).


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3. MATHEMATICAL FORMULATION

In the present study, the flow is assumed to be three dimensional, steady,incompressible, and turbulent. The flow is modeled with time-averaged continuity,momentum, and energy equations as follows.

quiqxi

¼ 0 ð1Þ

qujquiqxj

¼ � qpqxi

þ qqxj

lquiqxj

þ qujqxi

� �� qu0iu

0j

� �ð2Þ

qujqTqxj

¼ qqxj

lpr

qTqxj

� qT 0u0j

� �ð3Þ

where ui, T, and p are the average velocity components, temperature, and pressure,respectively. ui

0and T 0 are the fluctuating velocity and temperature components,

respectively.As one of the objectives of the present study is to identify suitable two-equation

turbulencemodels for the closure of the problem, the standard k-e, standard k-x, RNGk-e, Realizable k-e and SST k-xmodels, and four low Reynolds turbulence models areconsidered for comparing the performance of each model for the present problem.

In order to close Eq. (2), i.e., calculation of Reynolds stresses, �qu0iu0j.

Boussinesq hypothesis is used.

u0iu0j ¼ vt

quiqxj

þ qujqxi

� �� 2

3kdij ð4Þ

Analogous to kinetic theory, the turbulence viscosity can be expressed as follow.

vt / nolo ð5Þ

where vo and lo are the velocity and length scales. In two equation turbulence models,the turbulent quantities k and e or k and x are widely used to obtain velocity and

length scales for calculating vT .ffiffiffik

pis taken as velocity scale in both k-e and k-x

models. The length scales considered in the k-e and k- x are k3=2=e and k1=2=x

respectively. Hence in the k-e model, vt / k2

e and in the k-x model, vt / kx.

The standard k-e model proposed by Launder and Spalding [28] is asemi-empirical model with the assumption of fully developed turbulent flows. Itpredicts flow features in the core region of turbulent flows and not near the wallregion. To capture flow features near the wall without using wall functions, varioustwo-equation low Reynolds number turbulence models were developed. In order topredict complex wall bounded re-circulating flows, rotating flows and jet impinge-ment flows, many variants of k-e turbulence models were also developed apart fromlow Reynolds number turbulence models. In these models, RNG k-e and realizablek-e models are widely used in engineering calculations. The other widely used two

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equation model is the shear stress transport (SST) k-x model, wherein, theadvantages of both k-e and k-x models are utilized. Brief descriptions of thesemodels are given below.

3.1. Standard k-e Model

In the derivation of the standard k-e model, the flow is assumed to be fullyturbulent and the effects of molecular viscosity are negligible. The standard k-emodel is therefore valid only for fully turbulent flows. The model transport equationfor k is derived from the exact equation by using Boussinesq hypothesis.

qqxi

kuið Þ ¼ 1

qqqxj

mþ mtrk

� �qkqxj

� �þ 1

qGk � e ð6Þ

However, the transportation equation for e is obtained using physical reasoning asmathematically exact equation is difficult to simplify as it is done for the k equation.

qqxi

euið Þ ¼ 1

qqqxj

mþ mtrk

� �qeqxj

� �þ 1

qC1e

ekGk � C2e

e2

kð7Þ

In Eqs. (6) and (7), Gk is the turbulence production term, �qu0iu0jqujqxi. Based on the

Boussinesq hypothesis, this term is written as Gk ¼ mtS2, where S is the modulus

of the mean rate-of-strain tensor S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2SijSij

p. The model constants were obtained

experimentally for fundamental shear flows and decaying isotropic grid turbulence.

C1e ¼ 1:44;C2e ¼ 1:92;Cm ¼ 0:09;rk ¼ 1:0;re ¼ 1:3

From Eqs. (5) and (6), the turbulent viscosity is calculated as follows.

mt ¼ qcmk2

eð8Þ

3.2. Low Reynolds Number k-e Models

The standard k-e model is valid only in the core turbulent region. For wallbounded flows, this model is not applicable near the wall re.ion. In order to simulatewall bounded flows, the standard k-e was modified with damping functions forcapturing the near wall effects. In this study the low Reynolds number models ofLaunder and Sharma [29], Yang and Shih [30], Abe et al. [31] and Chang et al. [32]are considered to investigate the performance of the low Reynolds number two-equation models. The equations of k and e in the low Reynolds number turbulencemodels are similar to the standard k-e model. Equations (9) and (10) are the general-ized equations for the low Reynolds number turbulence models. Instead of using thevariable e the generalized equation is written in terms of ~ee, which is equal to Dþe.

qqxi

kuið Þ ¼ 1

qqqxj

mþ mtrk

� �qkqxj

� �þ 1

qGk � ~eeþD ð9Þ


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qqxi

~eeuið Þ ¼ 1

qqqxj

mþ mtrk

� �q~eeqxj

� �þ 1

qf1C1e

~eekGk � f2C2e

~ee2

kþ E ð10Þ

The damping functions fm, f1, and f2 are used to damp the constant values used in theabove equations when they are solved in the near wall region. The model constantsand the expressions used for D and E are given in Table 1, and the expressions forthe damping functions are given in Table 2.

The f functions are mainly used to modify the constant terms to account thenear wall effects. In the low Reynolds number turbulence models, the damping func-tion fm has been multiplied with the original term used in the standard k-e model forturbulent viscosity as follows.

mt ¼ qCmfmk2

~eeð11Þ

The damping function fm is used to damp Cm from 0.09 in the turbulent core region to

zero at the wall; f1 is used to enhance the near wall dissipation. The dampingfunction f2 multiplied with the destruction term of e equation is used to take in toaccount the low Reynolds number effects on the decay of isotropic turbulence. Eand D are added in some models in order to better represent the near wall behavior.

3.3. RNG k-e Model

RNG k-e model was developed by Yakhot and Orszag [33] from the instan-taneous Navier-Stokes equation using a mathematical method called Renormaliza-tion group theory. The final expression obtained for k is similar to that of standardk-e equation. However, the constants are obtained through analytical expressionsin contrast to the empirical values used in the standard k-e model. Scale eliminationprocedure used in RNG theory results in the following differential equation for eddyviscosity.

dq2kffiffiffiffiffiem

p� �

¼ 1:72v̂vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V̂V3 � 1þ cp dV̂V ð12Þ

Table 1. Summary of model constants and expressions of D and E used in the low Reynolds number k-emodels

Model D E ew-B.C. Cm C1e C2e rk re

LS2n q

ffiffik

p

qy

� �2

2mnt q2Uqy2

� �2 0 0.09 1.44 1.92 1.0 1.3

YS 0mnt q2U

qy2

� �2

2n qffiffik

p

qy

� �2 0.09 1.44 1.92 1.0 1.3

AKN 0 0 n q2kqy2

� �0.09 1.44 1.92 1.0 1.3

CHC 0 0 n q2kqy2

� �0.09 1.44 1.92 1.0 1.3

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where

v̂v ¼meffm

ð13Þ

Cv � 100 ð14Þ

For higher Reynolds number flow, Eq. (12) results in Eq. (13).The value of Cm obtained using analytical expression is 0.085 compared to the

empirical value of 0.09 used in the standard k-e model. The other model constantsobtained from the analytical expressions are C1e¼ 1.42, C1e¼ 1.68.

In the RNG k-e model, more terms appear in the dissipation rate transportequation for the treatment of non-equilibrium effects and flows in rapid distortionlimit such as separated flows and stagnation flows.

3.4. Realizable k-e Model

Shih et al. [34], developed the Realizable k-e model with the following twomodifications in standard k-e turbulence model: (1) new turbulence viscosity model,and (2) a modified dissipation equation developed using the mean vorticity fluctu-ation equation.

In the standard and many variants of k-emodels, including the RNG k-emodel,the turbulence viscosity value is calculated from Eq. (8) with Cm¼ 0.09. This leads to aunrealizable negative turbulent normal shear stress value when the stain rate is large.In order to ensure the realizable constraints, ie., mathematical constrains on theReynolds stresses, the value of Cmmust not be a constant and must be related to meanstrain rate. Shih et al. [34] implemented a new eddy viscosity model originallyproposed by Reynolds [35] to sensitize the flow situation to obtain Cm. The turbulencedissipation equation is derived from the equation for the mean square vorticityfluctuation. The final expression of the dissipation equation is as follow.

qqxj

euj�

¼ qqxj

vþ vtre

� �qeqxj

� �þ C1Se� C2

e2

k þffiffiffiffiffive

p ð15Þ

where C1 ¼ max 0:43; ggþ5

h i; g ¼ S k

e ;S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2SijSij

p

Table 2. Summary of the damping functions used in the low Reynolds number k-e models

Model fm f1 f2

LS exp �3:4= 1þRet=50ð Þ2h i

1.0 1� 0:3exp �Re2T�

YS ½1� expð�1:5� 10�4Rey � 5� 10�7Re3y � 10�10Re5yÞ�1=2

1þ 1=ffiffiffiffiffiffiffiffiffiReT

pffiffiffiffiffiffiffiffiffiReT

p

1þffiffiffiffiffiffiffiffiffiReT

pffiffiffiffiffiffiffiffiffiReT

p

1þffiffiffiffiffiffiffiffiffiReT

p

AKN 1þ 5:0=Re3=4T exp � ReT

200

� 2h in o1� exp Ree=14ð Þ½ �2 1.0 1� 0:3 exp �ðReT=6:5Þ2

h in o1� ðRee=3:1Þ½ �2

CHC 1� expð�0:0215ReyÞ �2

1þ 31:66=Re5=4T

� �1.0 1� 0:01exp �Re2T

�� 1� exp �0:0631Rey

� �


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Equation (15) does not involve production of k. The denominator of the lastterm in the dissipation Eq. (15) never vanishes, even if k is zero. This is in contrastwith traditional k-e models which have a singularity due to the presence of k in thedenominator (Eq. (7)).

3.5. Standard k-x Model

This model was proposed by Wilcox [36] and takes into account the nearwall effects. In this model, for length scale k1=2=x is used. The turbulent viscosityis calculated as follows.

mt ¼ qk

xð16Þ

The k and x equations are written as follows.

qqxi

kuið Þ ¼ 1

qqqxj

mþ r� mtrk

� �qkqxj

� �þ 1

qGk � b�kx ð17Þ

qqxi

xuið Þ ¼ 1

qqqxj

mþ rmtrk

� �qxqxj

� �þ c

xk

1

qGk � bx2 ð18Þ

The values of coefficients are b� ¼ 0.09, b¼ 3=40, c¼ 5=9, r

� ¼ 1=2, and r¼ 12. More

details on model coefficients and near wall corrections are given in reference [36].With the near wall modifications (without any wall functions), the k-x model pre-dicts free shear flow spreading rates that are in close agreement with measurementsboundary layer flows accurately.

3.6. SST k-x Model

The standard k-x model predicts flow better without modification of xequation specific to near-wall region, but very sensitive to prescribed free streamconditions. On the other hand, the standard k-e model is insensitive to prescribedfree stream conditions, but needs modification for better prediction of near wall flowfeatures. Menter [37] developed the shear stress transport (SST) k-x model in whichboth k-e and k-x models were combined in such a way that would allow them to beused in the region where they show the best advantage. The SST k-x model uses thek-x model near the wall but switches to the k-e model using a function F1. Menteralso fixed the problem of over prediction of shear stress in the adverse pressure gradi-ent boundary layers by imposing a bound on the shear stress intensity ratio uvj j=k bymodifying the k-x model. After these modifications, it is believed that the SST k-xperforms better for wall bounded flows and flows with adverse pressure gradient.The modified governing equations, functions, and constants used in this model areconcisely given in reference [38].

Analogous to turbulent shear stresses, the turbulent heat flux qT 0u0j given in

Eq. (3) can be modeled as qT 0u0j ¼mtPrt

qTqxj

� �, where Prt is the turbulent Prandtl number.

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The local Nusselt number distribution in the axial and circumferential direc-tions obtained from the numerical simulations are compared with the available

experimental result. The local Nusselt number Nu is calculated as Nu ¼ q

Tw�Tjð ÞDkf

4. SOLUTION METODOLOGY

In the present work, numerical simulations were carried out using the commer-cial CFD package, FLUENT 6.3 [38] to solve continuity, momentum, and energyequations. SIMPLE algorithm of Patankar [39] was used for pressure–velocitycoupling. The effects of turbulence were modeled by using two-equation turbulencemodels. Simulations were carried out using the standard k-e, standard k-x, RNGk-e, Realizable k-e, and SST k-x models, and four low Reynolds number k-e modelsseparately to identify the most suitable model for a circular jet impingement coolingof a circular cylinder. Near the wall, enhanced wall function [40] was used for thestandard k-e, RNG k-e, and Realizable k-e models. The density, viscosity, thermalconductivity, and specific heat values used in the calculations are 1.225 kg=m3,1.789� 10�5 kg=ms, 0.0242W=mK, and 1006.43 J=kgK, respectively. The solutionwas considered to be converged when the normalized residuals were less than1� 10�5 for continuity, velocity components and turbulence quantities. For theenergy equation, the normalized residual was less than 1� 10�8 at convergence.

4.1. Boundary Conditions

. Nozzle exit. As mentioned in the problem description, flow inside the nozzle wasnot considered. Instead, a fully developed velocity distribution of a turbulentcircular pipe flow was imposed at the nozzle exit through a user defined function(UDF) for validation of the computation model for heat transfer and parametricstudies. Only for the validation of flow field predictions, a uniform velocity wasused at the nozzle exit. For all the cases considered in the parametric study, theturbulence intensity (I) specified at the exit of the nozzle was 5%. The temperatureof the air exit from the nozzle was taken to be 30�C.

. Target wall. For velocity, no-slip condition was used on the circular cylinder.A constant temperature condition was specified over the cylinder surface. Thetemperature difference between the inlet round jet and circular cylinder surfacewas considered to be 7�C.

. Surface of the nozzle. The nozzle outer surface is specified as an adiabatic wall.

. Outlet. All other surfaces were considered as outflow boundaries, and the gaugepressure was set as zero.

4.2. Grid Independence Study

In the present study, structured hexahedral grids were used for numerical simu-lations. These grids were generated using ANSYS ICEM-CFD, a commercial gridgeneration software. Figure 2a shows a representative computational grid patternused on the surface of the target cylinder. In order to capture the wall effects, the


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mesh was refined near the surfaces of the heated cylinder and the surface of thenozzle. Hence, at the exit of the nozzle and near the target cylinder, a large numberof cells were used. Grid independence studies were carried out to obtain gridindependent solutions with optimum grid sizes. The RNG k-e turbulence modelwas selected for the grid independence study. For selecting an optimum grid, thelocal Nusselt number variation along the axial and circumferential direction of thecylinder for various grid sizes was compared. Figures 2b and c show the representativeresults of a grid independence study carried out for the case of Red¼ 15,980, h=d¼ 15,and d=D¼ 0.14. From the results it is clear that there is no change in local Nusseltnumber variations in both axial and circumferential directions when the grid sizechanged from 5.2� 106 to 6.2� 106. Hence, for the above mentioned case 5.2� 106

cells were considered. The grid was sufficiently refined near the wall to ensure theyþ value near unity in order to capture flow and heat transfer characteristics in the

Figure 2. (a) Typical grid used in the numerical study, (b) variation of local Nusselt number along the axial

direction obtained from the grid independence study, and (c) variation of local Nusselt number along the

circumferential direction obtained from the grid independence study.

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viscous sublayer. Similar grid independence studies were carried out for variousgeometric and flow parameters as the same grid cannot be used for these cases.

5. RESULTS AND DISCUSSION

5.1. Comparison of Present Numerical Results with theExperimental Results

In this study, nine different two-equation turbulence models are considered forsimulating turbulent jet impingement heat transfer from a circular cylinder. As noneof the existing models perform satisfactorily for all types of flow and heat transferproblems, it is important to identify suitable turbulence model for the flow and heattransfer problem under consideration. Literature on systematic study on selection ofa two-equation model for the circular jet impingement cooling of convex surfaces orcylinders is scarce. Hence, it is necessary to carry out a detailed study on the jetimpingement cooling of cylindrical surfaces with various two-equation turbulencemodels and identify the best model for the parametric study by comparing thenumerical results obtained from these models with the available experimental data.In the present study, the standard k-e, standard k-x, RNG k-e, Realizable k-e, SSTk-x, and four low Reynolds number turbulence models were selected to compare theperformance for the circular jet impingement cooling of a cylinder.

5.1.1. Fluid flow. For comparison of flow field, the experimental results pre-sented by Esirgemez et al. [41] were considered. They measured the velocity distri-bution of a circular jet at various distances from the nozzle exit to just above thesurface of the cylinder using laser doppler velocimetry for Red¼ 25,000, d=D¼ 0.252,0.252, and h=D¼ 4. Velocity measurements were done in both parallel and perpen-dicular directions to the cylinder axis. In their experimental study, air exits froma plenum with a uniform velocity profile. Hence, for the validation purpose,simulation was carried out with uniform velocity profile at the exit of the nozzlefor the above mentioned parameters. They have not presented velocity distributionover the cylinder.

In Figure 3 the experimental velocity distribution presented by Esirgemez et al.[41] and velocity distributions obtained from the present numerical study carried outusing the two equation models, standard k-e, standard k-x, RNG k-e, Realizable k-e,and SST k-x are compared. Velocity distributions obtained along the z-axis at thenondimensional distances from the nozzle exit to the cylinder surface; h1=d¼ 2,h2=d¼ 3, h3=d¼ 3.25, h4=d¼ 3.75, and h5=d¼ 3.95. The predictions of velocity distri-butions obtained using the turbulence models at h1=d¼ 2 and h2=d¼ 3 matches wellwith the experimental results. All turbulence models fail to predict velocity profilesaccurately at h3=d¼ 3.25, h4=d¼ 3.75, and h5=d¼ 3.95. However, the shape of thevelocity profiles predicted within the core region by all turbulence model consideredin the numerical study resembles the experimental results even though the magni-tudes are slightly different, except the results of the RNG k-e model.

Esirgemez et al. [41] have also presented the velocity distributions of the jet inthe direction perpendicular to the cylinder axis (variation in the x-axis) at the nondi-mensional distances from the nozzle exit: h1=d¼ 2, h2=d¼ 3, h3=d¼ 3.25, h4=d¼ 3.75,and h5=d¼ 3.95. Figure 4 shows the comparison of the velocity distributions obtained


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from the present numerical study and those of reference [41]. The results obtainedusing the standard k-e standard k-x Realizable k-e, and the SST k-x turbulence mod-els match well with the experimental results excluding h5=d¼ 3.95. These two modelspredict the jet spread accurately. On the other hand, the numerical results obtainedusing the RNG k-e model does not match exactly with the experimental results in

Figure 3. Comparison of velocity distribution of the impinging jet along the nondimensional distance z=d

at various distances from the nozzle exit. (a) Standard k-e model, (b) RNG k-e model, (c) Realizable k-emodel, (d) standard k-x model, and (e) SST k-x model (color figure available online).

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terms of the jet spread and the magnitude of velocity. Overall, the predictions ofvelocity distributions obtained using all five turbulence models deviate much formthe experimental results at h5=d¼ 3.95. As the low Reynolds number k-emodels differonly in the damping functions which were introduced only to capture the wall effectcompared to the standard k-e model, the modifications in the low Reynolds numberk-emodels do not affect the flow predictions significantly away from the wall. Due tothis reason, the flow field obtained using the low Reynolds number models are the

Figure 4. Comparison of velocity distribution of the impinging jet along the nondimensional distance x=d

at various distances from the nozzle exit. (a) Standard k-e model, (b) RNG k-e model, (c) Realizable k-emodel, (d) standard k-x model, and (e) SST k-x model (color figure available online).


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same as that of standard k-e model. Hence the results obtained from these modelsare not shown here.

5.1.2. Heat transfer. To evaluate the performance of the standard k-e,standard k-x, RNG k-e, the Realizable k-e, SST k-x models, and four low Reynoldsmodels on heat transfer predictions, results presented by Tawfek [25] for variousRed, h=d, and d=D were compared. The local Nusselt number variation along theaxial and circumferential directions (Figure 1) obtained from the numerical simula-tions using the standard k-e, standard k-x, RNG k-e, the Realizable k-e, and SSTk-x models were compared with the experimental results in Figure 5.

Figure 5a shows the variation of local Nusselt number from the stagnationpoint to the end of the cylinder along the axial direction predicted by using thestandard k-e, standard k-x, RNG k-e, Realizable k-e, and SST k-x models and theexperimental data of Tawfek [25] for Red¼ 15,980, h=d¼ 15, and d=D¼ 0.14. Allthe abovementioned turbulence models overpredict the local Nusselt number distri-bution in the region 0< z=D< 1. The difference between the experimental resultsand present numerical results obtained using the RNG k-e model is minimum com-pared to the results obtained from the other turbulence models. Compared to theexperimental value, the stagnation point Nusselt number is over predicted by RNGk-e model and the over prediction is around 22%. In the region z=D> 1, the localNusselt number variation obtained from the numerical simulations carried out usingthe standard k-e, standard k-x, RNG k-e, Realizable k-e, and SST k-x models agreewell with the experimental results. In particular, agreement between the numericalresults obtained using the RNG k-e model and the experimental results is excellent.Experimental results on the variation of local Nusselt number with respect to angle(h) are not available for this case. As the predictions of the local Nusselt number dis-tributions by the low Reynolds number k-e models are much higher than the experi-mental results of Tawfek [25], the local Nusselt number distributions obtained usingthe low Reynolds number k-e models are not presented in this paper. However, thestagnation point Nusselt number values obtained using these models are presentedin Table 3 along with the other results obtained using other turbulence models forcomparison.

Keeping h=d and d=D values the same as those for the last case, simulation wascarried out for Red¼ 38,800. Both the axial and circumferential variations of localNusselt numbers obtained from the numerical results are compared with the experi-mental results in Figures 5b and 5c. In Figure 5b, the axial variation of local Nusseltnumber is shown. At the stagnation point, the difference between the experimentalresults and the results obtained from the numerical simulation using the RNG k-eturbulence model is around 38%. On the other hand, the standard k-e, Realizablek-e, and the SST k-e models predict the Nusselt number more than twice theexperimental results. The predictions of stagnation point Nusselt numbers obtainedby using the standard k-x model and the low Reynolds number k-e models areextremely poor. The local Nusselt number distributions obtained using low Reynoldsnumber k-e models are not presented in this figure. The local Nusselt numbervariation along the circumferential direction was compared with the experimentalresults in Figure 5c. Near the stagnation region, the RNG k-e model performs bettercompared to the other turbulence models. Beyond 90�, i.e., in the recirculation

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region, the results obtained from all turbulence models deviate from the experi-mental results. However, compared to the stagnation region, percentage differencebetween the experimental and numerical results obtained from the other turbulencemodels are not very high in this region.

Most of the turbulence models fail to predict heat transfer rate accurately forlow h=d. Hence, to know the capabilities of selected turbulence models in predictingthe heat transfer rate for the present case, simulations were carried out for h=d¼ 7.5and d=D¼ 0.14 and compared the results with the experimental results of Tawfek[25]. Figure 5d shows the circumferential variation of local Nusselt number for

Figure 5. Comparison between the variation of local Nusselt number obtained from the present numerical

simulation and the experimental results of Tawfek [25] (color figure available online).


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Red¼ 16,000. As experimental results are available only for the variation of Nusseltnumber in the circumferential direction for this case, no comparison was made forthe local Nusselt number variation along the axial direction. At the stagnationregion, the Nusselt number predicted by the RNG k-e model is around 140% higherthan the experimental results and the other turbulence models predict the stagnationNusselt number higher than the RNG k-e model. However, agreement between thenumerical results and experimental results are good around h¼ 90�. In the recircula-tion zone, even though the numerical simulations could not predict heat transfer rateaccurately, deviation from the experimental results is not significant.

Figures 5e and f show the comparison of experimental and numerical resultsfor the case of h=d¼ 7.5, d=D¼ 0.14, and Red¼ 38,800. The variation of localNusselt number along the axial direction obtained from the numerical study withvarious turbulence models are compared with the experimental results of [25] inFigure 5e. As can be seen in Figures 5a, 5b, and 5e, the results obtained from theRNG k-e and the Realizable k-e models match the experimental results well in theregion z=D> 1 However, the error in the stagnation point region is around 150%.Figure 5f shows the comparison between the experimental and numerical resultsof local Nusselt number variation along the circumferential direction. The matchbetween the experimental and numerical results is good only for h� 90�. As theh=d value decreases, the local Nusselt number obtained in the stagnation region fromthe numerical simulations deviates much from the experimental results.

In Table 3, the percentage deviation of present numerical results compared to theexperimental results presented in reference [25] are summarized. It is observed that thenumerical models over predict the heat transfer rate at the stagnation point significantlyat low h=d ratio. Even though the values of stagnation point Nusselt number deviatemuch from the experimental results, the results agree reasonably well for z=D> 1along the axial direction and h> 90� along the circumferential direction. From thecomparison of numerical and experimental results, it is concluded that the RNG k-emodel performs better compared to the other two-equation turbulences models.

The computational time taken for simulations using various turbulence modelswas monitored. The standard k-e model has taken less time for the simulations

Table 3. Percentage deviations in the stagnation point Nusselt number obtained from the present numeri-

cal simulations and the experimental results of reference [25]

Model

% error in Nustag compared to

experimental results at h=d¼ 15

% error in Nustag compared to

experimental results at h=d¼ 7.5

Red¼ 15,980 Red¼ 38,500 Red¼ 16,000 Red¼ 26,000 Red¼ 38,800

STD k-e þ91 þ104 þ272 þ265

STD k-x þ363 þ405 þ215 þ203

Low Re LS k-e þ479 þ582 þ737 þ740

Low Re YS k-e þ408 þ500 þ218 þ332

Low Re AKN k-e þ268 þ372 þ443 þ538

Low Re CHC k-e þ300 þ420 þ508 þ609

RNG k-e þ23 þ38 þ135 þ140 þ146

Realizable k-e þ110 þ128 þ200 þ220 þ230

SST k-x þ150 þ157 þ234 þ249 þ255

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compared to other turbulence models for all the cases considered. Computationaltime taken for the simulations using various turbulence models for h=d¼ 15 andRed¼ 15,980 and 38,800with respect to that of standard k-emodel are shown inTable 4.

It is important to analyze why the two-equation models perform poorly inpredicting the stagnation region Nusselt number distribution. To quote Craft et al.[42], the fluid motion in the vicinity of the stagnation point comprises a nearlyirrotational normal straining, rather than simple shearing, while that near the edgeof the impinging jet combines strong rotationality and streamline curvature. Dueto complex features, jet impingement heat transfer is considered to be an importanttest case to evaluate the performance of any turbulence model.

It is known that the standard wall function cannot be used for jet impingementand recirculating flows. As there is no need of any wall function requirement tosimulate wall bounded flow if a low Reynolds number k-e turbulence model is used,one may tempt to use it for various wall bounded flows. When these low Reynoldsnumber models were developed, the main focus was on avoiding singularity aroundthe region of flow separation and ensuring the limiting values turbulent quantitieson the wall. It has to be kept in mind that these models were validated only forsimple boundary layer flows, channel flows, and flows with separation. None ofthe developers of these models considered the jet impingement problem as a test case.

When the k-e and k-x based two-equation turbulence models are used forthe jet impingement studies, the stagnation point Nusselt number is over predicted.Various researchers viewed this problem in various perspectives. Yap [43] reasonedthat the poor prediction of stagnation Nusselt number was due to an increase in thelength scale near the wall in the stagnation region. He introduced a correction termin the e equation to correct the length scale which is high near the wall in the none-quilibrium flows such as jet impingement flows and recirculating flows. The sourcetermed is written as follows.

se ¼ 0:83 e2=k�

max l=leð Þ l=leð Þ2; 0h i

ð19Þ

Table 4. Computational time taken by various turbulence models

Model

CPU time taken for a simulation

with a turbulence model=CPU

time taken for a simulation with

the standard k-e model (h=d¼ 15)

Red¼ 15,980 Red¼ 38,800

STD. k-e 1 1

STD. k-x 1.07 1.04

LS low Re k-e 3.41 1.49

YS low Re k-e 1.17 1.32

AKN low Re k-e 2.71 2.2

CHC low Re k-e 3.71 2.38

RNG k-e 4.82 3.87

Realizable k-e 10.29 7.81

SST k-x 26.72 20.2


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where l ¼ k3=2=e, is a turbulent length scale and lE¼Cly is the near wall equilibriumlength scale. Cl is typically around 2.5 and y is the normal distance from the wall.Finding y may be difficult in complex geometries. Hosseinalipour and Mujumdar[44] simulated jet impingement flows for h=S¼ 1.5 and Re¼ 8,000 with various lowReynolds number turbulence models with Yap correction. They found no improve-ment in the prediction of stagnation point Nusselt Number. Craft et al. [42] assesseda low Reynolds number model and three second-moment closure models with Yapcorrection. They found that the implementation of Yap correction in the low Rey-nolds number k-e model had a minor effect on the prediction of turbulent velocityand no effect on mean velocity. The prediction of stagnation point Nusselt numberwas also unsatisfactory, and its value is almost double even after implementation ofYap correction. They also pointed out that without Yap rrection, the stagnation Nus-selt number is nearly twice as high as when it is included. However, in contrast to theabove two numerical investigations, the recent investigation of jet impingement flowusing a low Reynolds number k-e model with Yap function carried out by Wang andMujumdar [45] for h=S¼ 2.6 and 6 for Re¼ 10,400 and 5,400, respectively, showedsignificant improvement in the stagnation point Nusselt number. As the Yap correc-tion, term uses normal distance from the wall, it is difficult to implement for complexcomputational domain. Due to the complexity in the implementation of Yap correc-tion in this study, no effort has been taken to include this correction. In the presentstudy, the standard k-e model performs relatively better than all low Reynoldsnumber models considered. The underperformance of low Reynolds number modelsmay be attributed to the arbitrary damping functions used in these models.

Durbin [46] explained that as the stagnation region was approached, the turbu-lence time scale becomes very large which resulted in over-prediction of k using anytwo-equation model. He derived a bound for the turbulent time scale in reference[46] from the condition that the eigenvalues of the Reynolds stress tensor shouldbe non-negative. The time scale bound for both k-e and k-x models are prescribedby Medic and Durbin [47], as follows.

to ¼ mink

e;

affiffiffi6

pCms

" #ð20Þ

and

to ¼ min1

cmx;

affiffiffi6

pCms

" #ð21Þ

where to is the time scale. Here, a is taken as 0.6. S is the modulus of the mean

rate-of-strain tensor, S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2SijSij

p.

Sofialides [48] tested Wilcox’s k-x model in an impinging jet flow. It was men-tioned that stagnation heat transfer was overpredicted, but the very sharp peak oftenreturned by e-based schemes without Yap correction was not present. However, thepresent numerical simulations with the k-x model showed the same trend observedby the e-based length scales. If the time bound present [47] is introduced in the k-e

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and k-x family turbulence models, the problem of overprediction of k may beavoided, which will result in a reasonable accurate stagnation Nusselt number.

Zu et al. [49] investigated the jet impingement cooling of flat surface with stan-dard k-e, RNG k-e, Realizable k-e, standard k-x, and SST k-x models apart fromconsidering time-consuming RSM and LES. Compared to the Reynolds stress models(RSM) and large eddy simulations (LES), two-equation turbulence models take lesscomputational time. Even though the predictions at the stagnation region are notaccurate, they preferred two-equation models over RSM and LES. Among the sevenmodels, they selected the best performing SST k-x model as the best compromisebetween the computational cost and accuracy. Using this model they studied theeffects of Reynolds number and various nondimensional geometric parameters.

The numerical study of jet impingement cooling of flat plate and concavesurfaces by Sharif and Mothe [27] revealed that the performance of the Reynoldsstress model (RSM) is also similar to that of the two-equation turbulence modeland not much benefit can be expected by using the RSM. However, Jaramillo et al.[50] highlighted the lack of generality of models. They found that the models withgood performance in the round jet case show poor results in the plane jet configu-ration. Hence, one has to investigate which turbulence model perform better forthe problem under consideration. Studies similar to Uddin et al. [51] and Jefferson-Loveday and Tucker [52], are not viable for any parametric study in the near future.

In spite of the limitations of two-equation turbulence models in predicting heattransfer rate accurately, one can still use numerical simulations of jet impingementflows to understand the effects of the variation of various flow and geometric parameteron heat transfer rate at least quantitatively. With the same line of argument of Zu et al.[49], based on the conclusions arrived at from the comparative study in the presentwork, the RNG k-e model has been selected to investigate flow and heat transfercharacteristics for a wide range of nondimensional flow and geometric parameters.

In the present numerical study, the following ranges of parameters areconsidered: Reynolds number, Red¼ 10,000, 15,000, 25,000 and 50,000, the nozzleto cylinder spacing ratio h=d¼ 4, 8, 12, and 16, and the nozzle diameter to cylinderdiameter ratio d=D¼ 0.11, 0.2, and 0.25. The effects of these parameters on heattransfer and fluid flow characteristics are discussed in detail in the following sections.

5.2. Effect of Reynolds Number (Red)

As mentioned in the previous subsection, to understand the flow and heat trans-fer characteristics of circular air jet impingement on a circular heated cylinder, a para-metric study was carried out using the RNG k-e model which predicts heat transfercharacteristics better than the other two-equation models considered in this study.

The effects of Reynolds number on flow and heat transfer characteristics werestudied by varying Reynolds number from 10,000 to 50,000 while maintaining h=dand d=D at 4 and 0.11, respectively. After jet impingement, the fluid spreads overthe surface of the cylinder and forms velocity boundary layers on both the leftand right sides of the jet impingement region. The overall flow pattern over thecylinder is shown in Figure 6a for Red¼ 10,000. As shown in Figure 6b, the sizeof the recirculation formed just behind the cylinder is much smaller compared tothe size of the cylinder. The overall flow pattern does not change much as the


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Reynolds number increases from 10,000 to 50,000. As the Reynolds numberincreases from 10,000 to 50,000, an increase in the size of recirculation along thecircumference of the cylinder and the main flow direction is only marginal as shownin Figures 6c and 6d. As the distance, z=D increases along the axial direction of thecylinder from the rear stagnation point, the size of the recirculation behind thecylinder decreases. Beyond a certain distance, no recirculation is observed behindthe cylinder. The variation in the color of the stream tubes signifies the localtemperature of the fluid.

Figure 7 shows the fluid temperature distribution over the cylinder at two cutsections in the computational domain. Figures 7a and 7b show the temperaturedistribution in the x-y plane which passes through z=D¼ 0, and that in the z-y plane

Figure 6. Flow characteristics for h=d¼ 4 and d=D¼ 0.11, (a) Flow over the cylinder for Red¼ 10,000, (b)

three- dimensional view of flow pattern behind the cylinder for Red¼ 10,000, (c) flow pattern in the cut

plane x-y which passes through the stagnation point for Red¼ 10,000, (d) flow patter in the cut plane

x-y which passes through the stagnation point for Red¼ 50,000 (color figure available online).

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which passes through h¼ 0 along the cylinder axis, respectively, for Red¼ 10,000.Similar plots are also shown for Red¼ 50,000 in Figures 7c and 7d. As the Redincreases, the thermal boundary layer thickness over the cylinder decreases signifi-cantly. In Figure 8, Nusselt number distributions are shown along with the flow pat-tern for various Red. As the Reynolds number varies from 10,000 to 50,000, thechange in the stagnation Nusselt number is around 130%. As the Reynolds numberincreases, the heat transfer rate increases significantly in the jet impingement region.In the region of rear stagnation point of the cylinder, the percentage change in theNusselt number is only marginal when the Reynolds number is varied from 10,000

Figure 7. Temperature distribution over a cylinder for h=d¼ 4 and d=D¼ 0.11. (a) in the x-y plane which

passes through z=d¼ 0 for Red¼ 10,000, (b) in the z-y plane which passes through h¼ 0 for Red¼ 10,000

(c) in the x-y plane which passes through z=d¼ 0 for Red¼ 50,000, and (d) in the z-y plane which passes

through h¼ 0 for Red¼ 50,000 (color figure available online).


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to 50,000. The variations of local Nusselt number along the axial and the circum-ferential directions are shown in Figure 9 for Red¼ 10,000 and 50,000. For the axialvariation, the values of the local Nusselt number are taken at h¼ 0. The local Nusseltnumber continuously decreases from the stagnation point region due to increase in thethermal boundary layer thickness. Significant reduction in local Nusselt number isobserved within 1.0D in the axial direction for all the Reynolds number consideredin the parametric study. Due to thermal boundary layer growth in the circumferentialdirection, the variation of the local Nusselt number is drastic up to 90�. The variationof the local Nusselt number is gradual thereafter due to recirculation. The trend in thevariation of the local Nusselt number shows that most of the fluid flow takes placearound the center portion of the cylinder where the jet impinges over the cylinder.

5.3. Effect of h/d

To study the effect of the distance between the nozzle exit and the stagnationpoint of the cylinder surface on fluid flow and heat transfer rate characteristics,

Figure 8. Effect of Reynolds number on flow pattern and local Nusselt number distribution. (a) Red¼10,000, (b) Red¼ 15,000, (c) Red¼ 25,000, and (d) Red¼ 50,000 (color figure available online).

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a parametric study was carried out for the nondimensional distances h=d¼ 4,8,12,and 16. Effects of h=d were studied for Reynolds number 10,000 and d=D¼ 0.11.Figures 10a–10d show the flow pattern in the x-y plane which passes through thestagnation point. As the value of h=d increases, the size of the recirculation increasesin both circumferential and along the jet flow directions. The size of the recirculationzone is highly influenced by h=d compared to Red.

The effect of h=d on local Nusselt number variation was studied forRed¼ 10,000 and 50,000. Figures 11a and 11b show the variation of local Nusseltnumber in the axial and circumferential directions for Red¼ 10,000 for varioush=d values. Along the longitudinal direction, the effect of change in h=d on thevariation of local Nusselt number is noticeable only for z=D< 1. Beyond this region,there is no effect of h=d on the variation of the local Nusselt number. Along the cir-cumferential direction, the effect of h=d on the local Nusselt number is significant, upto 90�. However, beyond 90� the effect of h=d on the local Nusselt number variationis insignificant (Figure 11b). At h=d¼ 4, the maximum Nusselt number obtained atthe stagnation point is 843; but for the same Red, as h=d changed from 4 to 16 thestagnation point Nusselt number decreases to around 220. The main reason forthe higher Nusselt number at lower h=d is the interaction of potential core withthe cylinder wall. At higher h=d, the length of the potential core region is shorterthan the distance between the nozzle exit to the cylinder surface, and the jet spreadsbecause of surrounding air interaction. Due to these reasons, the heat transfer rateat higher h=d is lesser than that of lower h=d. A similar trend is also observed onthe variation local Nusselt number in the circumferential and axial directions atRed¼ 50,000. Figure 11c shows the axial variation of local Nusselt number, andFigure 11d shows the circumferential variation of the Nusselt number at Red¼ 50,000.

Figure 11e shows the variation of Nusselt number at the stagnation point for allh=d and Red considered in the parametric study for a nozzle diameter to cylinderdiameter (d=D) of 0.11. For the range of Reynolds numbers considered in this study,the stagnation Nusselt number decreases nonlinearly as h=d increases. There is nooptimum nondimensional distance h=d at which the heat transfer rate is maximum.This is consistent with the results presented by Sparrow et al. [24] and Tawfek [25].

Figure 9. Effect of Reynolds number on local Nusselt number distribution. (a) Along the axial direction

z=d, and (b) effect of local Nusselt number along the circumferential direction h.


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However, Lee et al. [8] reported an optimum distance from circular nozzle exit tothe surface of the convex hemispherical surface for maximum heat transfer.

5.4. Effect of d/D

As a part of the parametric study, simulations were also carried out to under-stand the effect of d=D on the heat transfer rate for Red¼ 10,000 and 50,000 ath=d¼ 4.0. In the present study, the diameter of the heater is fixed and the diameterof the nozzle is varied to change the d=D ratio. The d=D values considered are 0.11,0.2, and 0.25. As the d=D ratio is increased from 0.11 to 0.25, there is no significantchange in the circulation behind the cylinder. The axial and circumferential variationof local Nusselt number is shown in Figures 12a and 12b for Red¼ 10,000 andh=d¼ 4.0. This figure reveals that the effect of d=D is significant only up to z=D¼ 1.01.0 in the axial direction. In the circumferential direction, the variation of localNusselt number is significant only up to h¼ 90�. The same trend is also observed

Figure 10. Effect of nondimensional distance from the nozzle exit to the circular cylinder h=d on flow

pattern behind the cylinder for Red¼ 10,000. (a) h=d¼ 4, (b) h=d¼ 8, (c) h=d¼ 12, and (d) h=d¼ 16.

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for Red¼ 50,000, as shown in Figures 12c and 12d. For a fixed Reynolds number, inthe present simulations d=D is increased by increasing the nozzle diameter d. Thisresults in a decrease in the jet velocity, but the mass flow rate increases due to an

Figure 11. Effect of the nondimensional distance, h=d on heat transfer. (a) Local Nusselt number variation

along the axial direction for Red¼ 10,000, (b) local Nusselt number variation along the circumferential

direction for Red¼ 10,000, (c) local Nusselt number variation along the axial direction for Red¼ 50,000,

(d) local Nusselt number variation along the circumferential direction for Red¼ 50,000, and (e) the

stagnation point Nusselt number at various Red and h=d.


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Figure 12. Effect of d=D on heat transfer. (a) Local Nusselt number variation along the axial direction

for Red¼ 10,000, (b) local Nusselt number variation along the circumferential direction for Red¼ 10,000,

(c) local Nusselt number variation along the axial direction for Red¼ 50,000, (d) local Nusselt number

variation along the circumferential direction for Red¼ 50,000, and (e) the stagnation point Nusselt number

at various Red.

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increase in the cross-sectional area of the nozzle. Due to a decrease in the jet velocity,the stagnation Nusselt number is found to decrease. Similar, if d=D is decreased bydecreasing d, the jet velocity increases for the same Reynolds number which leads toan increase in the stagnation Nusselt number. However, the mass flow rate decreasesin this case. This leads to a sharp reduction in the local Nusselt numbers away fromthe stagnation region in both axial and circumferential directions. At the same time,it is noteworthy that due to an increase in mass flow rate with an increase in d, morefluid spreads over the cylinder. Hence, the drop in the local Nusselt number at highd=D in the axial and circumferential directions is not as sharp as observed in the lowd=D cases.

For the range of parameters considered in this study, the maximum localNusselt number is slightly away from the stagnation point for the following twocases: (1) at Red¼ 50,000, h=d¼ 4, and d=D¼ 0.2; and (2) at Red¼ 50,000, h=d¼ 4,and d=D¼ 0.25. For these two cases, a peak Nusselt number in the axial directionis observed at around z=D¼ 0.2. This trend is observed only for the low d=D values.As the range of Reynolds number considered in the experimental studies of Sparrowet al. [24] and Tawfek [25] is much smaller than the present study, this trend is notobserved in the variation of local Nusselt number distribution along the longitudinaland circumferential directions.

It is important to note that in the present case, the Nusselt number has beendefined with cylinder diameter D as the characteristics length, while some researchers[7, 8] have defined Nu based on jet diameter d. In the present case, an increase in jetdiameter is found to result in a decrease in stagnation Nu, while the trend is reversedfor those researchers who have defined Nu based on d. It must be understood thatthis difference is only due to the choice of the characteristic length, and that the trendin heat transfer coefficient is the same in both cases. It is also interesting to note thatwith the use of Nu defined based on the diameter of the target surface, this leads to adecrease in Nu with a decrease in jet velocity, which is physically more acceptablethan the case of Nud which increases when the jet velocity decreases.

The stagnation point Nusselt number varies linearly at lower Reynolds num-ber, as the d=D value varies from 0.11 to 0.25. However, at high Reynolds numberthe variation is steeper for d=D ratio varied from 0.11 to 0.20 compared to that from0.2 to 0.25 (Figure 12e).

6. CONCLUSION

In this article, the results obtained from a numerical study on turbulent air jetimpingement cooling of a circular cylinder with a constant wall temperature havebeen presented. To identify a suitable two-equation turbulence model, the numericalresults obtained from the present study using the turbulence models standard k-e,standard k-x, RNG k-e, Realizable k-e, SST k-e, and four low Reynolds numbermodels were compared with the experimental results available for flow and heattransfer. With the selected two-equation turbulence model, a detailed parametricstudy was carried out for a wide range of geometric and flow parameters. Basedon the numerical study, the following important conclusions were made.


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. The Realizable k-e and SST k-x models predict velocity distribution of the jetimpinging on the cylinder reasonably well compared to the RNG k-e model.

. All turbulence models considered in this study overpredict the stagnation Nusseltnumber. The overprediction of the stagnation Nusselt number is much higherwhen the low Reynolds number models are used. However, for the range of para-meters considered in the study, the RNG k-e model predicts the Nusselt numberdistribution better than the other two turbulence models. At Re¼ 38,500 andh=d¼ 15.0, the RNG k-e model predicts the stagnation Nusslet number 38%higher than the experimental value. At Red¼ 38,800 and h=d¼ 7.5, the RNG k-emodel overpredicts the stagnation point Nusselt number around 146% comparedto the experimental results. However, in the region z=D> 1, the local Nusseltnumber distribution along the axis of the cylinder matches well with the experi-mental results.

. The heat transfer rate increases significy near the stagnation region as theReynolds number of the jet increases. The overall flow feature does not changewith an increase in Reynolds number.

. The local Nusselt number decreases monotonically along both axial and circum-ferential directions. However, the peak Nusselt number was slightly away from thestagnation point for the following two cases: (1) Red¼ 50,000, h=d¼ 4, andd=D¼ 0.2; and (2) Red¼ 50,000, h=d¼ 4, and d=D¼ 0.25.

. As h=d increases, the local Nusselt number decreases along the axial directionconsiderably in the region z=D< 0.6 for fixed d=D and Red values. Beyond thisregion, there is no effect on Nusselt number distribution along the axial directionis noticed. Along the circumferential direction, the effect of h=d is considerable upto 90�. As h=d increases, the size of the recirculation behind the cylinder increases.There is no considerable change in the local Nusselt number in the rear stagnationregion as h=d changes. The stagnation Nusselt number decreases monotonically asthe nondimensional distance h=d increases.

. The stagnation Nusselt number decreases as d=D increases.

. The effects of changes in Red, h=d, and d=D on the local Nusselt number aresignificant in the circumferential direction compared to the axial direction.

REFERENCES

1. H. Martin, Heat andMass Transfer between Impinging Gas Jet and Solid Surfaces, Adv. inHeat Transfer, vol. 13, pp. 1–60, 1977.

2. J. N. B. Livingood, and P. Hrycak, Impingement Heat Transfer from Turbulent air Jets toFlat Plates—A Literature Survey, NASA Technical Memorandum, NASA TM X-2778,Lewis Research Center, Cleveland, 1973.

3. P. Hrycak, Heat Transfer from Impinging Jets: A Literature Review, AFWAL-TR-81–3054, New Jersey Institute of Technology, New Jersey, 1981.

4. K. Jambunathan, E. Lai, M. A. Moss, and B. L. Button, A Review of Het Transfer Datafor Single Circular Jet Impingement, Int. J. Heat Fluid Flow, vol. 13, pp. 106–115, 1992.

5. R. Viskanta, Heat Transfer to Impinging Isothermal Gas and Flame Jets, Exp. Thermaland Fluid Sci., vol. 6, pp. 111–134, 1993.

6. N. Zuckerman, and N. Lior, Jet Impingement Heat Transfer Physics, Correlations andNumerical Modeling, Adv. in Heat Transfer, vol. 39, pp. 565–632, 2006.

182 D. SINGH ET AL.

Dow

nloa

ded

by [

Indi

an I

nstit

ute

of T

echn

olog

y -

Del

hi]

at 1

9:53

02

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201

3

7. C. Cornaro, A. S. Fleischer, M. Rounds, and R. J. Goldstein, Jet Impingement Cooling ofConvex Semi-Cylindrical Surface, Int. J. Thermal Sci., vol. 40, pp. 890–898, 2001.

8. D. H. Lee, Y. S. Chungand, and D. S. Kim, Turbulent Flow and Heat Transfer Measure-ments on a Curved Surface with a Fully Developed Round Impinging Jet, Int. J. Heat andFluid Flow, vol. 18, pp. 160–169, 1997.

9. C. Gau and C. M. Chung, Surface Curvature Effect on Slot Air Jet Impingement CoolingFlow and Heat Transfer Process, J. Heat Transfer, vol. 113, pp. 858–864, 1991.

10. T. L. Chan, C. W. Leung, K. Jambunathan, S. Ashforth–Frost, Y. Zhou, and M. H. Liu,Heat Transfer Characteristics of Slot Jet Impingement on Semicircular Convex Surface,Int. J. Heat Mass Transfer, vol. 45, pp. 993–1006, 2002.

11. F. Gori and L. Bossi, On the Cooling Effect of an Air Jong the Surface of a Cylinder,Int. Comm. Heat Mass Transfer, vol. 27, pp. 667–676, 2000.

12. F. Gori and L. Bossi, Optimal Slot Height in Jet Cooling of a Circular Cylinder,Appl. Thermal Eng., vol. 23, pp. 859–870, 2003.

13. C. S. McDaniel and B. W. Webb, Slot Jet Impingement Heat Transfer from CircularCylinders, Int. J. Heat Mass Transfer, vol. 43, pp. 1975–1985, 2000.

14. S. A. Nada, Slot=Slots Air Jet Impinging Cooling of a Cylinder for Different Jets-CylinderConfigurations, Heat Mass Transfer, vol. 43, pp. 135–148, 2006.

15. S. H. Kang and R. Greif, Flow and Heat Transfer to a Circular Cylinder with a HotImpinging Air Jet, Int. J. Heat Mass Transfer, vol. 35, pp. 2173–2183, 1992.

16. S. Amiri, K. Habibi, E. Faghani, and M. Ashjaee, Mixed Convection Cooling of a HeatedCircular by Laminar Upward–directed Slot Jet Impingement,Heat Mass Transfer, vol. 46,pp. 225–236, 2009.

17. E. E. M. Olsson, L. M. Ahrne, and A. C. Tragardh, Heat Transfer from a Slot Air JetImpinging on a Circular Cylinder, J. Food Eng., vol. 63, pp. 393–401, 2004.

18. S. K. Singh and R. P. Singh, Air Impingement Cooling of Cylindrical Objects UsingSlot Jets, In G. F. Gutierr-Lopez, J. Waltti-Chanes, and E. Parada-Arias, (eds.), FoodEngineering: Integrated Approaches, chap. 5, Springer, New York, 2008.

19. C. Dirita, M. V. D. Bonis, and G. Ruocco, Analysis of Food Cooling by Jet Impingement,Including Inherent Conduction, J. Food Eng., vol. 81, pp. 12–20, 2006.

20. N. Nitin, R. P. Gadiraju, and M. V. Karwe, Conjugate Heat Transfer Associatedwith a Turbulent Hot Air Jet Impinging on a Cylindrical Object, J. Food Process Eng.,vol. 29, pp. 386–399, 2006.

21. M. Imraan and R. N. Sharma, Jet Impingement Heat Transfer in a Frost–Free Refriger-ator: The Influence of Confinement, Int. J. Refrigeration, vol. 32, pp. 515–523, 2009.

22. N. Zuckerman and N. Lior, Radial Slot Jet Impingement Flow and Heat Transfer ona Cylindrical Target, J. Thermophysics and Heat Transfer, vol. 21, pp. 548–561, 2007.

23. N. Zuckerman and N. Lior, The Relation between the Distributions of Slot Jet Impinge-ment Convective Heat Transfer and the Temperature in the Cooled Solid Cylinder,Numer. Heat Transfer, vol. 53, pp. 1271–1293, 2008.

24. E. M. Sparrow, C. A. C. Altemani, and A. Chaboki, Jet–Impingement Heat Transferfor a Circular Jet Impinging in Crossflow on a Cylinder, J. Heat Transfer, vol. 106,pp. 570–577, 1984.

25. A. A. Tawfek, Heat Transfer due to Round Jet Impinging Normal to a Circular Cylinder,Heat and Mass Transfer, vol. 35, pp. 327–333, 1999.

26. A. A. Tawfek, Heat Transfer Studies of Oblique Impingement of Round Jets on CurvedSurface, Heat and Mass Transfer, vol. 38, pp. 467–475, 2002.

27. M. A. R. Sharif and K. K. Mothe, Evaluation of Turbulence Models in the Prediction ofHeat Transfer due to Slot Jet Impingement on Plane and Concave Surfaces, Numer. HeatTransfer, vol. 55, pp. 273–294, 2009.


Dow

nloa

ded

by [

Indi

an I

nstit

ute

of T

echn

olog

y -

Del

hi]

at 1

9:53

02

May

201

3

28. B. E. Launder and D. B. Spalding, Lectures in Mathematical Models of Turbulence,Academic Press, London, England, 1972.

29. B. E. Launder and B. I. Sharma, Application of the Energy–Dissipation Model ofTurbulence to the Calculation of Flow Near a Spinning Disc, Lett. Heat and MassTransfer, vol. 1, pp. 131–138, 1974.

30. Z. Yang and T. H. Shih, A New Time Scale Based k-e Model for Near Wall Turbulence,AIAA J., vol. 31, pp. 1191–1198, 1993.

31. K. Abe, T. Kondoh, and Y. Nagano, A New Turbulence Model for Predicting Fluid Flowand Heat Transfer in Separating and Attaching Flows—1. Flow Field Calculations, Int. J.Heat Mass Transfer, vol. 37, pp. 139–151, 1994.

32. K. C. Chang, W. D. Hsieh, and C. S. Chen, A Modified Low-Reynolds NumberTurbulence Model Applicable to Recirculating Flow in Pipe Expansion, J. Fluid Eng.,vol. 117, pp. 417–423, 1995.

33. V. Yakhot and S. A. Orszag, Renormalization Group Analysis of Turbulence, J. Scienceand Computing, vol. 1, pp. 3–51, 1986.

34. T. Shih, W. W. Liou, A. Shabbir, Z. Yang, and J. Zhu, A New k � e Eddy ViscosityModel for High Reynolds Number Turbulent Flows, Computers & Fluids, vol. 24,pp. 227–238, 1995.

35. W. C. Reynolds, Fundamentals of Turbulence for Turbulence Modeling and Simulation,Lecture Notes for Von Karman Institute, AGARD Report No. 755, 1987.

36. D. C. Wilcox, Turbulence Modeling for CFD, 3rd ed., pp. 124–128, DCW Industries Inc.,

La Canada, California, 1998.37. F. R. Menter, Two-Equation Eddy-Viscosity Turbulence Models for Engineering

Applications, AIAA-J., vol. 32, pp. 269–289, 1994.38. Fluent 6.3 User’s Guide, Fluent, Inc., 2006.39. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, pp. 126–129, Hemisphere,

Washington, D.C., 1990.40. M. Wolfshtein, The Velocity and Temperature Distribution in One-Dimensional

Flow with Turbulence Augmentation and Pressure Gradient, Int. J. Heat Mass Transfer,vol. 12, pp. 301–318, 1969.

41. E. Esirgemez, J. W. Newby, N. Cameron, S. M. Olcmen, and V. Otugen, ExperimentalStudy of a Round Jet Impinging on a Convex Cylinder, Measurement Sci. and Tech.,vol. 18, pp. 1800–1810, 2007.

42. T. J. Craft, J. W. Graham, B. E. Launder, Impinging Jet Studies for Turbulence ModelAssessment—II. An Examination of the Performance of Four Turbulence Models, Int.J. Heat Mass Transfer, vol. 36, pp. 2685–2697, 1993.

43. C. R. Yap, Turbulent Heat and Momentum Transfer in Recirculating and ImpingingFlows, Ph. D. thesis, University of Manchester, Manchester, 1987.

44. S. M. Hossenalipour and A. S. Mujumdar, Comparative Evaluation of DifferentTurbulenceModels for Confined Impinging and Opposing Jet Flows,Numer. Heat Transfer,vol. 28, pp. 647–666, 1995.

45. S. J. Wang and A. S. Mujumdar, A Comparative Study of Five Low Reynolds Numberk-e Models for Impingement Heat Transfer, Appl. Thermal Eng., vol. 25, pp. 31–44,2005.

46. P. A. Durbin, On the k-e Stagnation Point Anomaly, Int. J. Heat and Fluid Flow, vol. 17,pp. 89–90, 1996.

47. G. Medic and P. A. Durbin, Towards Improved Prediction of Heat Transfer on TurbineBlades, J. Turbomachinery, vol. 124, pp. 187–192.

48. D. Sofialides, The Strain-Dependent Non-Linear k-x Model of Turbulence and itsApplication, M.Sc. Dissertation, University of Manchester, 1993.

184 D. SINGH ET AL.

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49. Y. Q. Zu, Y. Y. Yan, and J. Maltson, Numerical Study on Stagnation Point Heat Transferby Jet Impingement in a Confined Narrow Gap, J. Heat Transfer, vol. 131, pp. 094504.1–094504.4, 2009.

50. J. E. Jaramillo, C. D. Perez-Segarra, I. Rodriguez, and A. Oliva, Numerical Study ofPlane and Round Impinging Jets Using RANS Models, Numer. Heat Transfer, vol. 54,pp. 213–237, 2008 .

51. N. Uddin, S. O. Neumann, B. Weigand, and B. A. Younis, Large-Eddy Simulationsand Heat-flux Modeling in a Turbulent Impinging Jet, Numer. Heat Transfer, vol. 55,pp. 906–930, 2009.

52. R. J. Jefferson-Loveday and P. G. Tucker, Wall Resolved LES and Zonal LES ofRound Jet Impingement Heat Transfer on a Flat Plate, Numer. Heat Transfer, vol. 59,pp. 190–208, 2011.


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Numerical Heat Transfer -Paper1

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