A comparative study of heat transfer characteristics of wall-bounded jets using different turbulence models

International Journal of Thermal Sciences 89 (2015) 337e356

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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

A comparative study of heat transfer characteristics of wall-bounded jets using different turbulence models Sushil Kumar Rathore, Manab Kumar Das* Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, West Bengal, 721302, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 July 2013 Received in revised form 19 November 2014 Accepted 22 November 2014 Available online

A comparative study of heat transfer characteristics of turbulent offset jet and wall jet flows has been carried out using two different low-Reynolds number keε models, standard keε model and shear stress transport (SST) model. The heat transfer study has been carried out for wall-bounded jets in a quiescent medium and also in the presence of an external moving stream. The computational results obtained from different turbulence models have been compared with the experimental results available in the literature to judge the suitability of these models. The capability of low-Reynolds number keε models and SST model to resolve the entire boundary layer have enabled to capture the temperature profile in the thermal sublayer. The linear law Tþ ¼ Pr yþ has been observed up to yþ z 4.0 in the thermal sublayer for the wall jet flow. The variation of temperature profile in the thermal boundary layer, decay of the maximum streamwise temperature and streamwise variation of the local heat transfer coefficient have been computed for all models considered and compared with the experimental results. © 2014 Elsevier Masson SAS. All rights reserved.

Keywords: Wall jet Offset jet Heat transfer Standard keε model SST model

1. Introduction A jet is produced when a fluid is injected with the addition of some momentum from a nozzle into the surrounding fluid. The pressure drop across the nozzle results in the momentum gain of the fluid. The surrounding fluid may be moving or stagnant depending upon the application. Turbulent jets are encountered in many engineering devices for cooling and mixing applications, some of which include: cooling of turbine blades in gas turbine, boiler combustion chamber and electronic components, boundary layer separation control, and thrust augmentation in aircraft during vertical take-off. Some more applications of jet which may be cited are: fuel injection systems, heating and air conditioning applications, automobile defroster, environmental dischargers, heat exchangers, etc. The jet flow can broadly be divided as free and wall-bounded jets. If the jet spreads freely in the surrounding fluid in the absence of any wall, it is known as a free jet. The presence of a wall makes the flow more complicated. The formation of the viscous sublayer very close to the wall is one of the distinguished characteristics of the wall-bounded jets. The * Corresponding author. Tel.: þ91 3222 282924; fax: þ91 3222 282278. E-mail addresses: [email protected] (S.K. Rathore), [email protected]. ernet.in (M.K. Das). http://dx.doi.org/10.1016/j.ijthermalsci.2014.11.019 1290-0729/© 2014 Elsevier Masson SAS. All rights reserved.

momentum and heat transport in the viscous sublayer are mainly due to the viscous mechanism. In wall-bounded turbulent flows, the inner shear layer exhibits some of the characteristics of the wall boundary layer. The velocity increases from zero at the wall and attains a local maximum value while the structure of outer shear layer is similar to that of the free shear layer. The schematic diagram of an offset jet and a wall jet are shown in Fig. 1(a and b) [25] respectively. The flow field of an offset jet can be classified into three regions: the recirculation region, the impingement region, and the wall jet region (as shown in Fig. 1(a)). The entrainment of the surrounding fluid above and below an offset jet is unequal due to the presence of the impingement wall. As a result, a low pressure (sub-atmospheric) region forms between the jet and the impingement wall. The jet deflects towards the wall and finally attaches at the reattachment point due to the presence of a low pressure region just downstream of the nozzle exit. This is known as the Coanda effect [27]. This low pressure region is known as the recirculation region. The reattachment point is a location where the wall shear stress changes its direction (tw ¼ 0). The flow field of an offset jet attains the characteristics of a generic wall jet in the wall jet region. The study of flow and heat transfer characteristics of turbulent offset and wall jets was carried out by several authors. The relevant experimental and numerical works are discussed in the following paragraphs.

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Nomenclature Cfx cp d G h h(x) k p0 p P Pr Prt q Re Ret Rey T Tþ Tt Tin u; v U, V uþ U0 ut Umax umax

skin friction coefficient, tw =12 rU02 specific heat at constant pressure, J kg1 K1 non-dimensional distance from the nearest wall production by shear width of the nozzle, m local convective heat transfer coefficient, W m2 K1 turbulent kinetic energy, m2 s2 ambient pressure, Pa static pressure, Pa non-dimensional static pressure Prandtl number turbulent Prandtl number wall heat flux, W m2 Reynolds number, U0h/y turbulent Reynolds number,pkffiffiffi2 =n~ε or k/nu non-dimensional distance, ky=y dimensional temperature, K dimensionless temperature (TwT)/Tt friction temperature, q/rcput, K jet inlet temperature, K dimensional mean velocities in x,y-directions respectively, m s1 non-dimensional velocities in X,Y-directions respectively non-dimensional velocity, u=ut 1 average inlet jet velocity, pffiffiffiffiffiffiffiffiffiffiffi m s1 friction velocity, tw =r, m s non-dimensional local maximum streamwise velocity dimensional local maximum streamwise velocity, m s1

1.1. Literature related to turbulent jet flow in the quiescent medium Pelfrey and Liburdy [23] experimentally studied the mean flow and turbulence characteristics of a turbulent offset jet. The measurement was done using Laser Doppler Anemometry (LDA) for offset ratio and Reynolds number 7 and 15,000 respectively. The offset ratio (OR) is defined as the ratio of the jet center line height (H) to the jet width (h) and the Reynolds number (Re) is based on the jet inlet mean velocity (U0) and the jet width (h). They measured the mean velocity profile at different streamwise locations, ratio of curvature to shear strain rate, and the entrainment parameter in the recirculation and impingement regions. They mentioned that the flow cannot be modeled as thin shear flow as the ratio of curvature to shear strain rate, and ratio of jet width to radius of curvature were considerably higher in the recirculation region. In conclusion, the flow is subjected to additional large strain rate due to jet curvature in the recirculation and impingement regions that makes it challenging test case for testing of turbulence models. Holland and Liburdy [10] later on experimentally investigated the heat transfer characteristics of the heated offset and wall jet flows for a flow geometry similar to that of Pelfrey and Liburdy [23]. The objective was to provide detailed experimental investigation of heat transfer characteristics of turbulent offset jet as literature dealing with heat transfer characteristics of turbulent offset jet was very scarce. The offset ratios considered were 3, 7 and 11. The temperature measurements were done with a very fine thermocouple wire probe (0.0254 mm chromel and alumel wire) in the flow field. They measured the temperature profile at different streamwise locations, decay of the maximum streamwise

X, Y x, y xþ yþ Y0.5

Ymax Yq,0.5

Greek a,at ε ~ε

n, nt u r tw q qmax

non-dimensional coordinates dimensional coordinates, m non-dimensional distance, xut/n non-dimensional distance, yut/n non-dimensional distance in the cross-streamwise direction at which (UU∞) ¼ (UmaxU∞)/2, (U∞ ¼ 0 in case of the quiescent medium) non-dimensional distance in the cross-streamwise direction at which U ¼ Umax non-dimensional distance in the cross-streamwise direction at which q¼qmax/2

laminar and turbulent thermal diffusivities, respectively, m2 s1 rate of dissipation of turbulent kinetic energy, m2 s3 modified rate of dissipation of turbulent kinetic energy (εD), m2 s3 laminar and turbulent kinematic viscosities, respectively, m2 s1 rate of specific dissipation, s1 density, kg m3 wall shear stress, Pa non-dimensional temperature non-dimensional local maximum streamwise temperature

Subscripts ∞ corresponds to freestream/surrounding fluid condition n non-dimensional quantity p corresponds to the first near-wall grid point w corresponds to wall

temperature and temperature variation along the impingement wall in the recirculation, impingement and developing wall jet regions. They observed that temperature within the recirculation region was approximately uniform for the range of offset ratios investigated. The other important outcome from the experiment was that the flow experiences greater strain rate with higher offset ratio and it increases with increase in offset ratio. The experimental work conducted by AbdulNour et al. [1] provides the temperature profile in the thermal sublayer which is very scarce in the literature. The experimental data for temperature profile in the thermal sublayer are very crucial to judge the suitability of different low-Reynolds number turbulence models in the near-wall region. AbdulNour et al. [1] experimentally investigated the heat transfer characteristics of a wall jet (Re ¼ 7700) for isothermal and constant heat flux boundary conditions. The measurements were done using micro-thermocouple and Infrared (IR) imaging. The temperature measurement in case of the isothermal boundary condition was done using micro-thermocouple, whereas in case of the constant heat flux boundary condition both microthermocouple and IR imaging were used. They measured the temperature profile in the thermal boundary layer including the thermal sublayer at different streamwise locations. Furthermore, the streamwise variation of the local heat transfer coefficient for isothermal and constant heat flux boundary conditions was also measured. They observed that the streamwise variation of the local heat transfer coefficient was insensitive to the thermal boundary conditions for streamwise locations X⩾5. They suggested IR imaging for more accurate measurement of the local convective heat transfer coefficient in case of the constant heat flux boundary condition.

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339

Fig. 1. Schematic diagram of an offset jet and a wall jet.

El-Gabry and Kaminski [7] numerically and experimentally investigated the three-dimensional jet impingement with cross flow for Reynolds number in the range 14733e34878. The jet angle was varied between 30 , 60 and 90 as measured from the impingement surface. The objective was to examine the performance of Yang and Shih [31] (YS) low-Reynolds number keε model and the standard keε model for the prediction of heat transfer characteristics of jet in cross flow. They compared the numerical results obtained from the Yang and Shih [31] low-Reynolds number keε model and the standard keε model using Fluent with their experimental results. They observed that the standard keε model performed better in some cases while the YS model preformed better in others. Finally, they concluded that there is no single turbulence model that performs better in all cases. The errors in predicting the average Nusselt number were 2e30% for the YS model and 0e60% for the standard keε model as compared to the experimental results. The jet impingement on a surface having a constant heat flux over a limited area was numerically investigated by Shuja et al. [26] for Re ¼ 23,000 and 70,000. They considered the low-Reynolds number keε model proposed by Lamand Bremhorst [12]; standard keε model and two Reynolds stress models for turbulence closure. They observed that temperature profiles predicted by the low-Reynolds number keε model and Reynolds stress model were better as compared to the standard keε model.

theoretical study was based on the integral formulation of the basic conservation laws. They took into account both variation in pressure and radius of curvature in the recirculation region. They observed a good agreement between experimental and theoretical results. The same authors (Hoch and Jiji [8]) later on studied the heat transfer characteristics of turbulent offset jets for the same geometry. They provided the experimental and analytical solutions for decay of the maximum streamwise temperature. They concluded that the freestream velocity had very small effects on decay of the maximum streamwise temperature for the range of offset ratios and free stream velocities considered. Dakos et al. [4] experimentally studied the flow, turbulence and heat transfer characteristics of plane and curved wall jets in an external stream for Reynolds number of 30,000. The flow and wall heat-transfer measurements were done using stagnation tube and heat-flux meters (9.4 mm diameter) of the Schmidt-Boelter multiple-thermocouple type respectively. The measurement of mean temperature was done with a 12 mm diameter chromel-alumel thermocouple sensor welded to a standard miniature hot-wire probe. The turbulence measurements were made using 5 mm diameter hot wires. They observed that flow was not self-preserving in an external stream. The position of zero-shear-stress was shifted to a point closer to the wall due to the effect of curvature. The turbulence intensity and stress in the curved outer layer were increased due to the extra strain resulting from the curvature of the wall.

1.2. Literature related to turbulent jet flow in an external stream Hoch and Jiji [9] experimentally and theoretically studied the flow field of turbulent offset jets in an external moving stream for the Reynolds number of 16,000. They considered offset ratios up to 8.7 and non-dimensional free stream velocity U∞ < 0.3. The

1.3. Objective of the work and different turbulence models considered The heat transfer study of turbulent offset jet is more complex as it involves the recirculation, impingement and wall jet regions. The

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experimental and computational study of fluid flow characteristics of turbulent offset and wall jet flows have been carried out by many researchers. But, very few computational works related to the heat transfer study of turbulent offset and wall jet flows have been reported in the literature. The present investigation is an attempt to bridge this gap in the literature. The objective of present investigation is a comparative study of heat transfer characteristics of turbulent offset and wall jet flows to judge the suitability of lowReynolds number keε models, standard keε model and SST model. The low-Reynolds number keε models proposed by Launder and Sharma [14] (LS), and Yang and Shih [31] (YS) are considered for numerical simulation. The low-Reynolds number turbulence model is first proposed by Jones and Launder [11]. Launder and Sharma [14] and many other low-Reynolds number models are proposed thereafter with several modifications. LS model is one of the classical model which have been successfully applied for many flow situations. However, the model performance is not reported earlier for turbulent offset jet. Yang and Shih (YS) model is proposed with several modifications which avoids solving for pseudo-dissipation rate as in the case of LS model. The damping function in this model is designed in such a way that satisfies the correct near-wall asymptotic behavior for shear stress ðO ðy3 ÞÞ. The inbuilt mechanism that takes into account the Kolmogorov time scale near the wall and conventional time scale (k/ε) away from the wall is another added feature in this model. The Kolmogorov time scale is more appropriate near the wall as the viscous mechanism dominates in the near-wall region. The more details are given in Sec. 2.1. The performance of YS model is reported for the jet in cross flow by El-Gabry and Kaminski [7]; however the suitability of model for relatively more complex case of wall bounded turbulent offset jet comprising of recirculation, impingement and wall jet regions have not been reported earlier. In addition, the additional large strain rate arising due to jet curvature in case of offset jet makes the flow problem more severe test case for testing of turbulence models. Another class of turbulence model is the two-equation eddy viscosity based keu model. The keu based turbulence models are more common in aerodynamic applications due to its robust and accurate near-wall formulation. SST model is discussed in more detail in Sec. 2.2. The performance of SST model is reported in the literature for wall jet [2] and slot impinging jet [6]; however the model performance is not yet reported for turbulent offset jet. Many modifications are incorporated into the original SST model in the last 15 years [18,19]. For the present study, the SST model described in [18] is implemented. The computational results have been compared with the experimental results of [1,10,20,21,23,30,4]. 2. Mathematical formulation The Reynolds averaged NaviereStokes (RANS) equations are used for predicting the turbulent flow. The mean flow is assumed to be steady and two-dimensional. The body forces are neglected and the fluid properties are assumed to be constant. The Boussinesq hypothesis is used to link the Reynolds stresses to the mean velocity gradients and eddy viscosity. The eddy viscosity is modeled as a product of length scale and velocity scale. The length scale and velocity scale are determined form kn and ~εn (or un in case of SST model) in two-equation eddy viscosity models. Low-Reynolds number models resolve the entire boundary layer including the viscous sublayer region near the wall whereas high-Reynolds number model does not resolve the viscous sublayer. The wall functions are used to bridge the viscous sublayer. Low-Reynolds number keε models employ damping functions (fm, f1 and f2) and some other terms (Dn and En) so that the governing equations

become valid in the low turbulence Reynolds number (Ret) region near the wall. Different low-Reynolds number models use different expressions for model functions (fm, f1 and f2). The governing equations are solved for incompressible flow which are presented below. The dimensionless variables are defined as:

Ui ¼ kn ¼

ui ; U0 k

; 2

U0

Xi ¼

xi ; h

~εn ¼

~ε . ; 3 U0 h

q¼

T  T∞ DTref nt;n ¼

P¼

nt ; n

p  p0 rU02

un ¼

u U0 =h

(1)

The non-dimensional equations are: Continuity Equation:

vUi ¼0 vXi

(2)

Momentum equations:

  " !# v P þ 23 kn  vUi vUj vUi 1 v  Uj ¼ þ þ 1 þ nt;n Re vXj vXi vXj vXj vXi

(3)

Energy equation:

Uj

" #  vq vq 1 v  ¼ 1 þ at;n vXj RePr vXj vXj

(4)

Eddy diffusivity (at,n) is given as:

at;n ¼

nt;n Prt

(5)

The turbulent Prandtl number (Prt) is taken as equal to 0.9 for air [3]. 2.1. Turbulent transport equations for the low-Reynolds number keε model and high-Reynolds number keε model Launder and Sharma model: The low-Reynolds number version of two-equation keε model is first proposed by Jones and Launder [11]. Launder and Sharma [14] model is similar to Jones and Launder [11] model with some modifications. The extra term Dn appeared in the turbulent kinetic energy equation due to application of zero Neumann boundary condition ð~ε ¼ 0Þ at the wall in LS model. Jones and Launder [11] first reported the decisive computational advantages of zero dissipation ð~ε ¼ 0Þ boundary condition in solving the dissipation equation. Thus, the extra term Dn is exactly equal to the rate of dissipation at the wall. The term En is added in the dissipation equation to get the desired profile of k close to the wall [11]. Yang and Shih model [31]: model was proposed to accurately capture the near-wall flows in wall bounded turbulent boundary layer. The pffiffiffidamping function pffiffiffi in this model is proposed as function of Rey ¼ ð ky=yÞ, where k is chosen as the turbulent velocity scale. The formulation of damping function in terms of Rey instead of yþ ensures its applicability in separated flow region as well. The nondimensional distance yþ depends on friction velocity ut; hence it cannot be applied in the separation region. In addition, this model has got a turbulence time scale which is bounded from the below by Kolmogorov time scale. Turbulent time scale is the ratio of the turbulence length scale to the turbulence velocity scale. Yang and Shih [31] mentioned that the time scale near the wall must be the Kolmogorov time scale as the viscous dissipation dominates near the wall. So, in this model turbulence time scale is given by k/ε away from the wall and by the Kolmogorov time scale near the wall.

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layer and the standard keε model in the outer boundary layer. This model takes into account the robust near-wall formulation of the Wilcox keu model [29] and the freestream independence of the keε model. The second version of BSL model is known as the shear stress transport (SST) model as proposed by [16]. The SST model additionally takes into account the transport of principal turbulent shear stress in an adverse pressure gradient. The SST model is obtained by transforming the standard keε model into keu formulation and this model is then added to the original model multiplied with a switching function F1. The function F1 is designed in such a way that it assumes a value of unity near the viscous wall region and the keu model is activated, and assumes a value of zero for yþ > 70 [17] and the standard keε model is activated in the outer region of boundary layer. The model constants are also obtained by blending the corresponding constants of the keu and keε models. Turbulent kinetic energy (kn) equation:

This can be expressed as follows:

¼

Tt |{z} Turbulent time scale

ðk=εÞ |fflffl{zfflffl} Time scale away from the wall

Ck ðn=εÞ1=2 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

þ

Kolmogorov time scale

It is to be noted that k/ε is much larger than the Kolmogorov time scale away from the wall and k/ε vanishes near the wall. Thus the turbulent time scale is more appropriate in either region. The model damping function is defined in such a way that shear stress near the wall satisfies the correct near-wall asymptotic behavior that is O ðy3 Þ [31]. In addition, model constants are similar to that of the standard keε model to match the model performance in the outer region. The value of constant Ck is taken as 1 [31]. Turbulent kinetic energy (kn) equation:

vkn 1 v Uj ¼ Re vXj vXj

"

#  nt;n vkn 1þ þ Gn  ~εn  Dn sk vXj

(6)

Uj

where modified dissipation is defined as

~ε ¼ ε  D

" #  vkn vkn 1 v  ¼ 1 þ sk nt;n þ MinðGn ; 10b kn un Þ  b kn un vXj Re vXj vXj (11)

(7) Specific dissipation rate (un) equation:

Rate of dissipation (~εn ) equation:

v~εn 1 v Uj ¼ vXj Re vXj

"

#

 nt;n v~εn ~εn ~ε2 1þ þ C1ε f1 Gn  C2ε f2 n þ En sε vXj kn kn

Uj

(8)

" #  vun vun 1 v  aRe ¼ MinðGn ; 10b kn un Þ 1 þ su nt;n þ Re vXj nt;n vXj vXj  bu2n þ 2ð1  F1 Þsu2

Production by shear (Gn):

nt;n Gn ¼ Re

"

341

vUi vUj þ vXj vXi

!#

vUi vXj

(12)

(9) The blending function F1 given as

i h F1 ¼ tanh fMinðz; hÞg4

Eddy viscosity (nt,n):

nt;n ¼ Cm fm Re

1 vkn vun un vXj vXj

k2n

(10)

~εn

(13)

where

! pffiffiffiffiffi kn 500 ; z ¼ Max  b un d Re d2 un

The model functions fm, f1 and f2 and extra terms Dn and En are defined in Table 1 [25]. The model constants for YS, LS and standard keε models are given as: sk ¼ 1.0, sε ¼ 1.30, C1ε ¼ 1.44, C2ε ¼ 1.92, and Cm ¼ 0.09 [14,15,31].

4su2 dkn2

h¼

2.2. Turbulent transport equations for the SST model [18]

Max

In two-equation keu turbulence models, turbulent eddy viscosity is determined from kn and un. The analytical solution of u in the near-wall region is first given by Wilcox [29] that makes it more robust and accurate near the wall. Wilcox [29] first proposed the keu model in the year 1988, which became very popular for aerodynamic applications. However, the model was later found to be highly sensitive to freestream values. To avoid this problem Menter [16], first proposed the baseline (BSL) model which is identical to the Wilcox keu model [29] in the near-wall boundary

(14)

!

(15)

n vun 10 2su2 u1n vk vXj vXj ; 10

Eddy viscosity (nt;n ):

2 nt;n

3

6 6 6 kn ¼ Re Min6 6 un ; 6 4 F

2

7 7 7 a1 kn 7 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 7 u uvU vU vUj 7 5 i t i þ vXj

vXj

(16)

vXi

Table 1 Summary of model functions. Model LS

fm

3

2

exp4

3:4 1þ

YS

Re 2

5

t 50

   1 ffi 1  expðaRey  bRe3y  cRe5y Þ 1 þ pffiffiffiffiffi Re

f1

f2

1

1  0:3 expðRe2t Þ

Dn 2 Re

En v

pffiffiffiffi!2 kn vXj

! 2nt;n Re2

v2 U i vXj2 v2 U i vXj2

pffiffiffiffiffiffi Reffiffiffiffiffi t ffi p 1þ Ret

pffiffiffiffiffiffi Reffiffiffiffiffi t ffi p 1þ Ret

0

2nt;n Re2

1

1

0

0

t

Standard keε

where a ¼ 1.5  1004, b ¼ 5  1007 and c ¼ 1  1010 1

!

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where

"( F2 ¼ tanh

!)2 # pffiffiffiffiffi 2 kn 500 ; Max  b un d Re d2 un

(17)

[5] of the residual is taken as convergence criteria. The convergence criterion for mass residual is taken as 105 and convergence criterion for temperature is taken as 106.

3.1. Boundary conditions The model constants are given as [18]: a1 ¼ 0.31, b* ¼ 0.09, a1 ¼ 5/9, a2 ¼ 0.44, b1 ¼ 0.075, b1 ¼ 0.0828, sk1 ¼ 0.85, sk2 ¼ 1.0, su1 ¼ 0.5, su2 ¼ 0.856, a ¼ a1F1 þ a2(1F1), b ¼ b1F1 þ b2(1F1), sk ¼ sk1F1 þ sk2(1F1), au ¼ su1F1 þ su2(1F1). 3. Solution methodology The governing differential equations are solved in the nondimensional form using the in-house code. The governing differential equations are discretized using the finite volume approach on a staggered grid. The power-law upwind and second-order central-difference schemes are used to discretize the convective and diffusive terms respectively. The velocity-pressure coupling is done using semi-implicit method for pressure-linked equation (SIMPLE) [22]. The under-relaxation factor of 0.2e0.4 is used in the pressure-correction equation. The algebraic equations are solved using Tri-diagonal matrix algorithm (TDMA) and line-by-line solver. The TDMA traverse is employed in the cross-streamwise direction and a sweep from upstream to downstream is preferred to obtain a faster convergence. The under-relaxation of momentum, turbulent and energy equations have been done using the pseudotransient approach as given in [28]. The source terms appeared in the momentum and turbulent transport equations are suitably linearized to increase the diagonal dominance. The Euclidean norm

The no-slip (U ¼ 0) and no-penetration boundary conditions (V ¼ 0) are imposed at the solid walls. The value of turbulent kinetic energy at the jet inlet is kn ¼ 1.5I2. In the present study, the turbulence intensity at the jet inlet, I, is taken equal to 0.05. Likewise, the rate of dissipation ð~εn Þ and rate of specific dissipation ðun Þ at the jet 3=2 3=4 1=2 1=4 inlet are ~εn ¼ ðkn Cm Þ=0:07 and un ¼ kn =ðCm 0:07Þ for keε models and SST model respectively. The value of turbulent kinetic energy at the solid wall is taken as kn ¼ 0. The LS model imposes zero rate of dissipation i.e. ~εn ¼ 0 ð~εn ¼ εn  Dn Þ at the solid walls and an additional term Dn is added in the transport equation for the turbulent kinetic energy (Eq. (6)). The decisive computational advantages of zero dissipation ð~ε ¼ 0Þ boundary condition are first reported by Jones and Launder [11]. Thus, the extra term Dn is exactly equal to the rate of dissipation at the wall. The rate of dissipation at pffiffiffiffiffi the solid walls for the YS model is ~εn ¼ ð2=ReÞðv kn =vnÞ2 and the additional term Dn ¼ 0 (i.e. ~εn ¼ εn ), where n is the non-dimensional coordinate perpendicular to the wall. In case of the SST model, the rate of specific dissipation at the solid wall is prescribed as un ¼ 60=ðRe b1 DYp2 Þ following the recommendation of Menter [17]; where DYp is the non-dimensional distance to the next near-wall grid point. The rate of dissipation at near-wall grid points are prescribed based on the wall functions given by Launder and Spalding [15] in case of the standard keε model.

Fig. 2. Grid independence study.

S.K. Rathore, M.K. Das / International Journal of Thermal Sciences 89 (2015) 337e356

Fig. 3. Grid distribution for offset ratio 3.

The thermal boundary conditions for low-Reynolds number keε and SST models are q ¼ 1 and vq=vY ¼ 1 for isothermal and constant heat flux surfaces respectively. The corresponding thermal boundary conditions for the standard keε model are prescribed based on the wall functions as given in Biswas and Eswaran [3]. The Neumann boundary condition vf=vY ¼ 0 is considered for the top boundary (i.e. entrainment side) in case of keε models where f ¼ U, V, kn and ~εn . Likewise, the entrainment boundary conditions in case of the SST model are vf=vY ¼ 0, kn ¼ 106 and un ¼ lV∞/L where f ¼ U, V. L is the approximate non-dimensional length of the computational domain and l is a constant. The value of l is taken as 40 for mixing layer [2]. The developed condition of vf=vn ¼ 0 ðf ¼ U; V; kn ; ~εn ; un and qÞ is imposed at the exit boundary for all models considered. 3.2. Grid independence study The grid independence study has been done using three grid sizes viz. 241  241 (¼58081), 313  313 (¼97969) and 443  443 (¼196249). Non-uniform grids are considered with a staggered arrangement. More number of grid points are laid in the near-wall and the jet entrance regions. It is ensured that the first near-wall þ grid points lie in the viscous sublayer (yþ p < 1.0 and xp < 1.0) for low-Reynolds number keε models. On the other hand, the first near-wall grid points are placed in the logarithmic region þ (30 < yþ p < 100 and 30 < xp < 100) for the standard keε model. The

343

grid distribution in case of the SST model is similar to that of the low-Reynolds number keε models. Mentor [17] has mentioned that the specific dissipation rate (un) at a solid surface un ¼ 60/ (Re b1DYp2) simulates the smooth wall boundary as given by Wilcox   [29] ðun /6= Re b1 DY 2 Þ as (DY / 0) as long as DYpþ < 3. However, to be on the safer side, the first near-wall grid points are laid in the þ viscous sublayer which satisfies yþ p z1:5 and xp z1:5. Fig. 2 shows the grid independence study carried out for the offset jet (OR ¼ 3 and Re ¼ 15,000) at three grid sizes viz. 241  241, 313  313 and 443  443 for all models considered. The thermal boundary conditions considered for the grid independence study are heated jet at the inlet, adiabatic impingement surface and isothermal vertical surface (temperature same as that of the quiescent medium T∞). The non-dimensional temperature profiles in the recirculation region (X ¼ 2.49) have almost merged on each other for the three grid sizes (241  241, 313  313 and 443  443) considered as shown in Fig. 2. Thus, the grid size of 241  241 is considered for simulation. Fig. 3 shows the grid distribution for offset ratio 3. The near-wall region and jet inlet region are clustered with more number of grid points as shown in Fig. 3. The non-dimensional domain size considered is 75  60. This choice is based on the study of effect of domain size on the important flow quantities like reattachment length, wall shear stress and velocity profile. The more details about the selection of domain size are given in [25]. 4. Results and discussion 4.1. Study of wall-bounded jets in the quiescent medium 4.1.1. Fluid flow and thermal characteristics of heated offset and wall jets over adiabatic surface The computations of flow and thermal characteristics of heated offset and wall jets have been carried out in the stagnant surrounding (quiescent medium) at the Reynolds number of 15,000. The flow geometry, thermal boundary conditions and Reynolds number of flow considered are similar to the experimental study of Holland and Liburdy [10]. The boundary conditions considered are heated jet at the inlet, adiabatic impingement surface and isothermal vertical surface (temperature same as that of the quiescent medium T∞). The offset ratios considered are 3, 5, 7, 9 and 11. The coordinate axes of Holland and Liburdy [10] are properly rescaled to match the present non-dimensional coordinates. Fig. 4 shows the domain extent (non-dimensional distance) and nondimensional boundary conditions for low-Reynolds number models at various boundaries. Standard wall function is implemented in case of the standard keε model. 4.1.2. Skin friction coefficient and self-similarity profile Fig. 5 [25] shows the streamwise variation of the normalized skin friction coefficient (Cfx/Cfx,max) for the offset ratio 7. This is

Fig. 4. Block diagram showing domain extent and boundary conditions at various boundaries.

Fig. 5. Streamwise variation of normalized skin friction coefficient for the offset ratio 7.

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Fig. 6. Velocity similarity profile for wall jet and offset jet with various offset ratios in the similarity wall jet region (X ¼ 60).

discussed earlier in [25] (Fig. 7) but Fig. 5 includes the prediction of SST model in addition to other models. The normalized skin friction coefficient increases very rapidly in the impingement region (after the reattachment point) due to the conversion of static pressure into velocity. It then gradually decreases in the wall jet region due to retardation of the flow. The streamwise variation of skin friction coefficient obtained from the SST model is in closer agreement with the experimental results of [23] with some deviations in the

impingement region (X  Xrp (3) near the reattachment location (after the reattachment point). LS model shows very good agreement for low values of XXrp (X  Xrp (3) however it deviates from experimental results in the wall jet region. The flow in the impingement region is very complex as it undergoes stagnation followed by recovery and the formation of wall boundary layer. All low-Reynolds number models show better agreement with the experimental results as compared to the standard keε model. The

Fig. 7. Non-dimensional velocity profile (uþ  yþ) for wall jet and offset jet with various offset ratios in the similarity wall jet region (X ¼ 60).

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Fig. 8. Streamwise variation of the jet half-width (Y0.5) for wall jet.

poor performance of standard keε model may be due to the application of wall functions in the recirculation region where logarithmic profile assumption is not valid. The experimental uncertainty data for measurement of skin friction coefficient is not available and hence not included here. The flow field of an offset jet is similar to that of a wall jet flow after the impingement region and thus it is called the wall jet region. Rajaratnam and Subramanyam [24] also mentioned that the flow field of an offset jet attains the characteristics of a wall jet flow beyond the impingement region for any offset ratio. Therefore, offset jet is expected to attain a similarity behavior in the wall jet region. The scaling laws U/Umax and Y/Y0.5 are utilized to plot the similarity profile. Umax is the non-dimensional local maximum streamwise velocity and the jet half-width Y0.5 is the non-dimensional distance measured from the wall in the cross-streamwise direction at which U¼Umax/2. Fig. 6 illustrates the velocity similarity profile for the offset and wall jets in the similarity wall jet region (X ¼ 60) for all models. The experimental results of Wygnanski et al. [30] for the wall jet flow (Re ¼ 19,000) are taken for comparison as the offset jet behaves similar to that of a generic wall jet flow in the similarity region (far downstream in the wall jet region). The velocity similarity profiles for wall jet as well as offset jet with various offset ratios (OR ¼ 3, 5, 7, 9 and 11) follow the similar trend as shown in Fig. 6 which demonstrates that offset jet (various offset ratio) flow resembles to that of wall jet flow in the self-similarity region. The present computational results show good agreement with the experimental results of Wygnanski et al. [30]; however, some deviations have been observed in the free mixing region (outer shear layer). The deviations observed in the outer shear layer may be due to the possible presence of room draught as mentioned by

Wygnanski et al. [30]. The experimental measurements of jet flow in the quiescent medium require a very controlled surrounding so that the surrounding fluid is stagnant during the experiment. The experimental data taken in the outer part of the free shear layer (Y/ Y0.5 > 1.3) are not very reliable due to the possible movement of the surrounding fluid as reported by Wygnanski et al. [30]. 4.1.3. Velocity profile (uþyþ) and jet spread in the streamwise direction Fig. 7 shows the non-dimensional velocity profile (uþyþ) for offset and wall jets in the similarity wall jet region (X ¼ 60), where uþ ¼ u=ut and yþ ¼ yut =n. The experimental results of Nizou and Tida [21] for the wall jet flow at Re ¼ 16,000 is taken for comparison which is closer to the present Reynolds number of 15,000. There are some important observations from Fig. 7. The first observation is that velocity profiles for wall jet and offset jets have almost merged on the same profile with some deviations in the logarithmic region. It is expected as the offset jet attains the characteristics of a generic wall jet flow in the similarity wall jet region. The second observation is that both low-Reynolds number keε models and SST model have accurately mimicked the law of the wall (uþ ¼ yþ) in the viscous sublayer as per the experimental results of Nizou and Tida [21]. The universal log law (uþ ¼ 2.44 ln(yþ) þ 5.0) which is known to be valid in the near-wall region (outside the region of direct viscous effects) is also shown in Fig. 7. The computational results obtained from the SST model are in excellent agreement with the experimental results both near and away from the wall as shown in Fig. 7(d). This is expected as the SST model predicts the wall shear stress accurately (see Fig. 5) which in turn utilized to calculate the pffiffiffiffiffiffiffiffiffiffi ffi friction velocity ðut ¼ tw =rÞ.

Fig. 9. Streamwise decay of local maximum velocity for different offset ratios.

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Table 2 Length of potential core, recirculation and impingement regions. Model

Offset ratio

Length of potential core region

Length of recirculation region

Length of impingement region

YS LS Standard keε SST YS LS Standard keε SST YS LS Standard keε SST

3 3 3 3 7 7 7 7 11 11 11 11

0.00e1.91 0.00e1.54 0.00e2.16 0.00e2.82 0.00e3.29 0.00e2.86 0.00e3.48 0.00e4.66 0.00e4.61 0.00e4.61 0.00e4.98 0.00e6.15

0.00e6.84 0.00e6.93 0.00e6.87 0.00e7.29 0.00e13.71 0.00e13.76 0.00e13.66 0.00e13.00 0.00e19.51 0.00e19.55 0.00e19.16 0.00e17.79

6.84e9.45 6.93e9.74 6.87e10.13 7.29e10.51 13.71e20.72 13.76e20.82 13.66e20.80 13.00e20.63 19.51e29.97 19.55e29.50 19.16e29.10 17.79e27.18

The other important parameter which characterizes the flow involving the mixing layer is the rate of growth of outer shear layer (free mixing region). Launder and Rodi [13] have done a comprehensive review of available experimental data for two-dimensional turbulent wall jet flows and suggested the following expression for the rate of growth of outer shear layer: dY0:5 =dX ¼ 0:073±0:002. Fig. 8(a) demonstrates the growth of jet half-width as predicted by different models for the wall jet flow. The jet half-width in the similarity region can be expressed by a linear relation as shown in Fig. 8(b) for the SST model. The rate of growth of outer shear layer ðdY0:5 =dXÞ are 0.094, 0.093, 0.091 and 0.069 for YS, LS, standard keε and SST models respectively. The rate of growth of outer shear layer obtained from the SST model closely agrees with the expression suggested by Launder and Rodi [13]; whereas other models overpredict the rate of growth of outer shear layer. 4.1.4. Decay of local maximum streamwise velocity The streamwise decay of local maximum velocity is illustrated in Fig. 9 for offset ratios 3, 7 and 11. The velocity just downstream of nozzle exit remains constant in the potential core region as shown in Fig. 9. The potential core region remains unaffected from the viscous effects which is also illustrated in Fig. 1. The viscous effects penetrate up to the jet centerline once the potential core region is consumed. Umax decays very sharply just beyond the potential core

region due to exchange of momentum with the primary vortex present in the recirculation region. The, vortex sustains its motion by extracting the mean kinetic energy of jet in the recirculation region. Umax attains the lower value near the reattachment point. It then increases in the impingement region due to the conversion of static pressure into velocity. The wall jet region is characterized by the gradual decrease of Umax as demonstrated in Fig. 9. The length of potential core, recirculation and impingement regions are given in Table 2. The length of potential core region is calculated according to the criterion U ¼ 0.999Umax and the recirculation region is taken from the nozzle exit to the reattachment point. The length of impingement region is taken from the reattachment point to the point up to which Umax increases in the impingement region due to conversion of static pressure into velocity. 4.1.5. Contour plots and non-dimensional temperature profile at various streamwise locations Fig. 10 shows the contour plot of temperature and various streamwise locations where computational results are compared with the experimental results for wall and offset jets which are discussed in the subsequent paragraph. The temperature distribution (q) in the domain is depicted in Fig. 11 for OR ¼ 7. The temperature inside the recirculation region is almost uniform due to better mixing caused by the primary vortex. However, the average temperature inside the recirculation region decreases as the offset ratio increases as shown in Fig. 10. This is due to enhanced heat transfer brought in by an increase in the entrainment with offset ratio [10]. The standard keε model predicts higher temperature in the recirculation region as compared to other models considered. Fig. 12 shows the non-dimensional temperature profile (q) at different streamwise locations viz. at X ¼ 10.4, 15.4, 20.3 and 25.2 for wall jet flow and comparison with the experimental results. The comparison with experimental results have been done at selected streamwise locations where experimental data are available. The temperature at any streamwise location is maximum at the bottom wall due to the adiabatic boundary condition; it then decays in the cross-streamwise direction and finally assumes the freestream temperature. The standard keε model underpredicts the temperature as compared to the experimental results, whereas lowReynolds number keε models and SST model slightly overpredict the temperature as shown in Fig. 12. All low-Reynolds number

Fig. 10. Contour plot of temperature showing various streamwise locations chosen for comparison with experimental results.

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Fig. 11. Temperature distribution (q) in the domain for offset ratio 7.

models predict the similar variation of temperature as shown in Fig. 12. The maximum discrepancies between the experimental and numerical predictions are approximately 16%, 13%, 22% and 21% for YS, LS, standard keε and SST models respectively. Figs. 13e15 show non-dimensional temperature profile (q) for offset ratios 3, 7 and 11 at different streamwise locations. The

streamwise locations X ¼ 2.49 (recirculation region), X ¼ 4.99 (recirculation region), X ¼ 8.06 (impingement region) and X ¼ 16.68 (wall jet region) have been considered for the offset ratio 3. For offset ratio 7, streamwise locations considered are X ¼ 3.35 (recirculation region), X ¼ 6.69 (recirculation region), X ¼ 15.75 (impingement region) and X ¼ 38.2 (wall jet region). Likewise for

Fig. 12. Non-dimensional temperature profile for wall jet at different axial locations.

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Fig. 13. Non-dimensional temperature profile at different axial locations for offset ratio 3.

offset ratio 11, comparisons have been done at five streamwise locations viz. X ¼ 5.04 (recirculation region), X ¼ 10.06 (recirculation region), X ¼ 18.12 (impingement region), X ¼ 29.06 (wall jet region) and X ¼ 39.5 (wall jet region). The various streamwise locations have been chosen for comparison in such a way that experimental results are available for comparison and it covers all regions (recirculation, impingement and wall jet regions). It is observed from Figs. 13e15 that temperature remains constant up to a certain distance in the cross-streamwise direction inside the recirculation region. It then attains the maximum value inside the jet and finally assumes the freestream temperature. The temperature inside the recirculation region is approximately uniform due to the presence of the primary vortex. The primary vortex causes better mixing in the recirculation region. The computational results obtained from all models slightly overpredict the temperature in the near-wall region as compared to the experimental results of Holland and Liburdy [10] for OR ¼ 3; this deviation increases in the recirculation and impingement regions as the offset ratio increases. This may be due to the additional large strain rate due to curvature of flow as mentioned by Holland and Liburdy [10]. The flow experiences much higher curvature strain rate with higher offset ratio. In the two-equation eddy viscosity keε models, Reynolds stress terms are modeled using Boussinesq hypothesis which results in the isotropic eddy (turbulent) viscosity. As a result, it is unable to take into account the large anisotropy and large strain rate resulting from the jet curvature in the recirculation region. Furthermore, the numerical diffusion increases with increase in the oblique angle of flow with grid lines. Therefore, very fine grids are used in the recirculation region to minimize the numerical diffusion. The maximum deviations between the experimental and numerical results are

observed for the offset ratio 11. These are approximately 30%, 30%, 39% and 32% for YS, LS, standard keε and SST models respectively in the recirculation region. Likewise, the maximum deviations between the experimental data and numerical results obtained from different models are approximately 79%, 79%, 93% and 41% for YS, LS, standard keε and SST models respectively in the impingement region. The maximum deviations are observed in the impingement region as the flow is highly complex, anisotropic and subjected to stagnation in this region. Low-Reynolds number keε models specially YS model performs better in the recirculation region among the models considered. The accuracy of predictions obtained from SST model lies within the low-Reynolds number keε models and the standard keε model. However, SST model shows minimum deviation with respect to experimental results in the impingement region. Overall, YS model performs better followed by the SST model. 4.1.6. Streamwise decay of local maximum temperature Fig. 16 shows the streamwise decay of local maximum temperature (qmax) for wall jet and offset ratios 3, 7 and 11. The thermal energy of heated jet is mainly lost to the surrounding entrained fluid as the impingement surface is adiabatic. The streamwise decay of the maximum temperature is higher for offset jet with higher offset ratio as shown in Fig 16. Holland and Liburdy [10] have mentioned that the entrainment of the surrounding fluid increases with the offset ratio. Thus, entrainment cooling is higher for offset jet with higher offset ratio. The present computational results show good agreement with the experimental results of [10] near the jet inlet region but overpredict in the downstream direction. The reason for this deviation may be due to heat loss from the

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Fig. 14. Non-dimensional temperature profile at different axial locations for offset ratio 7.

impingement wall during the experiment as it is very difficult to achieve perfect insulation (i.e. adiabatic condition). Thus, computational results predict higher temperature as compared to the experimental results. The maximum discrepancies between the experimental and numerical results are approximately 19%, 21%, 27% and 23% for YS, LS, standard keε and SST models respectively for the range of experimental data available (Fig. 16). 4.1.7. Thermal characteristics of wall jet flow under isothermal and constant heat flux boundary conditions The numerical investigation of thermal characteristics of wall jet flow has been done in the quiescent medium for isothermal and isoflux conditions on the bottom wall. The flow geometry, thermal boundary conditions and Reynolds number of flow considered are similar to the experimental study of AbdulNour et al. [1]. The Reynolds number of the flow considered is 7700. The unheated air jet supplied by the nozzle (temperature at the inlet same as that of the quiescent medium) flows tangentially along the bottom wall which is either maintained at constant temperature or constant heat flux condition. It is be noted that some portion of the bottom wall (1.5 times the nozzle width) just downstream of nozzle exit is thermally inactive (adiabatic) in the experimental study of AbdulNour et al. [1]. Hence, only the remaining portion of the bottom wall is maintained at constant temperature or constant heat flux condition. The experimental work of AbdulNour et al. [1] is chosen for study due to availability of experimental results for temperature distribution in the thermal sublayer which is very scarce in the literature. The comparison of temperature profile has been done with the experimental results to illustrate the performance of low-Reynolds number models near the wall.

4.1.8. Temperature profile in the thermal sublayer and entire thermal boundary layer Fig. 17 shows non-dimensional temperature profile in the thermal sublayer for isothermal boundary condition on the bottom wall at different streamwise locations. It is well-known that temperature varies linearly in the thermal sublayer. YS, LS and SST models also show the linear decay of temperature but the rate of decay of temperature is different as shown in Fig. 17. The temperature profiles obtained from SST model are in closer agreement with the experimental results of AbdulNour et al. [1] at X ¼ 2.22 and 3.18, whereas at other streamwise locations, YS model shows better agreement as compared to other models. Thus, there in no single turbulence model which performs better in all situations which is peculiarity of turbulence modeling. The standard keε model could not capture the near-wall temperature variation as the first nearwall grid points are placed in the logarithmic region. The maximum discrepancies between the experimental and numerical results are approximately 29%, 52% and 34% for YS, LS and SST models respectively. The non-dimensional temperature profiles in the entire thermal boundary layer are shown in Fig. 18 for isothermal boundary condition at different streamwise locations. Results obtained from all models show good agreement with the experimental results near the nozzle exit at X ¼ 2.22 and X ¼ 3.18. On the other hand, computational results at other streamwise locations match well near the wall (Y z 0e0.04) and far away from wall (Y z 0.15e0.25) but departs considerably from the experimental result in the middle part (Y z 0.04e0.15). Overall, the temperature profiles obtained from YS model show better agreement with the experimental results followed by SST model. The good predictions obtained from YS model as compared to other

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Fig. 15. Non-dimensional temperature profile at different axial locations for offset ratio 11.

models may be due to the composite time scale which takes into account Kolmogorov time scale near the wall and conventional time scale (k/ε) away from the wall. Yang and Shih [31] mentioned that Kolmogorov time scale is more appropriate near the wall as viscous effects dominate in the viscous sublayer and conventional time scale (k/ε) away from the wall. The Kolmogorov time scale is much smaller than the conventional time scale (k/ε) away from the wall and thus Kolmogorov time scale vanishes away from the wall. Similarly, k/ε vanishes near the wall and as a result Kolmogorov time scale works in this region. The maximum deviations with respect to experimental data are approximately 86%, 99%, 80% and 99% for YS, LS, standard keε and SST models respectively for the range of experimental data available in Fig. 18.

4.1.9. Streamwise variation of local heat transfer coefficient The estimation of heat transfer coefficient is very important from heat transfer perspective. Fig. 19(a and b) illustrate the streamwise variation of local heat transfer coefficient (h(x)) for isothermal and isoflux boundary conditions respectively. The calculation of local heat transfer coefficient is based on the slot height of jet which is taken as 2 cm similar to the experimental study of [1]. The streamwise variation of heat transfer coefficient is qualitatively and quantitatively similar for isothermal as well as isoflux boundary conditions on the wall. However, heat transfer coefficient in case of isoflux boundary condition is slightly higher as compared to the isothermal boundary condition as shown in Fig. 19(c) for the YS model. Heat transfer coefficient is very high

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Fig. 16. The streamwise decay of maximum temperature (qmax) for wall jet and offset jet with various offset ratios.

near the nozzle exit due to higher jet momentum. The jet momentum decreases in the downstream due to exchange of momentum with the surrounding fluid. The computational results obtained from YS and SST models show a closer agreement with the experimental observation as shown in Fig. 19(a and b). The maximum deviations with respect to experimental data are approximately 32%, 63%, 36% and 39% for YS, LS, standard keε and SST models respectively for the range of experimental data available in Fig. 19(a and b). It is be noted that AbdulNour et al. [1] have mentioned the estimated uncertainty in h(x) measurement as 11% and 6% for the isothermal and isoflux conditions (thermocouple measurement) respectively. 4.1.10. Temperature profile in wall coordinates (Tþyþ) The numerical simulation of wall jet flow has also been done at Reynolds number of 14,400 similar to the experimental study of Nizou [20]. The non-dimensional temperature (Tþyþ) profile for isoflux and isothermal boundary conditions is shown in Fig. 20 at X ¼ 60 where T þ ¼ ðTw  TÞ=Tt and Tt is the friction temperature. It is observed from Fig. 20 that the low-Reynolds number keε models and the standard keε model show good agreement in the thermal sublayer and the logarithmic region respectively as compared to the experimental results of Nizou [20]. On the other hand, SST model shows good agreement near the wall but slightly departs from the experimental results away from the wall as shown in Fig. 20(a). Overall, SST model performs reasonably well in the thermal sublayer as well as logarithmic region. Fig. 21 shows the non-dimensional temperature profile in wall coordinates (Tþyþ) near the wall at X ¼ 60. The linear relationship Tþ ¼ Pr yþ which is valid near the wall is also shown in Fig. 21. It is concluded from Fig. 21 that all models (YS, LS and SST models) follow the linear relation (Tþ ¼ Pr yþ) strictly only up to yþ z 4.0. Experimental results of Nizou [20]

(experimental data not available very close to the wall) also depart from the linear relationship Tþ ¼ Pr yþ before yþ ¼ 7 is reached. 4.2. Study of wall-bounded jet in the presence of an external moving stream The surrounding fluid may be in motion in some practical applications. Therefore, the study of wall jet flow has also been done in the presence of an external moving stream. Fig. 22 shows the schematic diagram of a wall jet in the presence of an external moving stream. The flow geometry and other details are similar to that of Dakos et al. [4]. The Reynolds number of the flow considered is 30,000. The non-dimensional velocity (U∞) of moving stream is 0.3. The turbulence intensities in the freestream and at the nozzle exit are taken as 0.4% and 0.5% respectively, similar to the experimental study of Dakos et al. [4]. 4.2.1. Velocity similarity profile and velocity profile (uþyþ) Fig. 23 shows the velocity similarity profile for all models at streamwise locations X ¼ 41.82 and 69.09. The non-dimensional variables ðU  U∞ Þ=ðUmax  U∞ Þ and Y/Y0.5 are utilized to plot the similarity profile. The jet half-width Y0.5 is the non-dimensional distance measured from the wall in the cross-streamwise direction at which ðU  U∞ Þ ¼ ðUmax  U∞ Þ=2. A good agreement has been obtained between experimental and numerical results in the inner as well as outer shear layers as shown in Fig. 23. On the contrary, the velocity similarity profile in the absence of an external moving stream (quiescent medium) deviates from the experimental results of Wygnanski et al. [30] in the outer shear layer (Fig. 6). The comparison of velocity similarity profile in the stagnant surrounding (quiescent medium) and in the presence of an moving stream with the experimental results of Wygnanski et al. [30] and

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Fig. 17. Non-dimensional temperature profile in the thermal sublayer at different axial locations for isothermal surface.

Dakos et al. [4] respectively has confirmed the observation of Wygnanski et al. [30] that experimental data taken in the outer shear layer are prone to errors due to room draught. The non-dimensional velocity profile in wall coordinates (uþyþ) is illustrated in Fig. 24 at streamwise locations X ¼ 41.82 and 69.09. A good agreement has been observed between the experimental results and the numerical results obtained from YS and standard keε models. The universal log law (uþ ¼ 2.44 ln(yþ) þ 5.0) is observed to fit the experimental results and the numerical results obtained from YS and standard keε models over the region 40(yþ (600 as shown in Fig. 24. The SST and LS models show good agreement in the viscous sublayer but some deviations are observed in the logarithmic region as compared to the experimental results. The maximum discrepancies between the experimental and numerical predictions are approximately 3%, 10%, 6% and 13% for YS, LS, standard keε and SST models respectively. Table 3 shows the comparison of the jet spread and the maximum

streamwise velocity at streamwise locations X ¼ 41.82 and 69.09. It is observed that the maximum streamwise velocity obtained from the standard keε model and YS model are in closer agreement with the experimental results. However, SST model predicts the jet spread more accurately among the models considered. 4.2.2. Non-dimensional temperature profile and variation of local heat transfer coefficient Fig. 25 shows the non-dimensional temperature profile q for isothermal boundary condition at streamwise locations X ¼ 41.82 and 69.09. It is observed from Fig. 25 that temperature gradient is very sharp near the wall. It then decays gradually and finally assumes the freestream temperature. The maximum deviations with respect to experimental data are approximately 9%, 24%, 23% and 18% for YS, LS, standard keε and SST models respectively for the range of experimental data available. The streamwise variation of local heat transfer coefficient (h(x)) for isothermal and isoflux

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Fig. 18. Variation of temperature in the thermal boundary layer at different axial locations for isothermal boundary condition.

boundary conditions is shown in Fig. 26(a and b) respectively. The calculation of local heat transfer coefficient is based on the slot height of jet which is taken as 11 mm similar to the experimental study of Dakos et al. [4]. The qualitative and quantitative variation of local heat transfer coefficient is similar for isothermal and isoflux boundary conditions however it is slightly higher in case of isoflux boundary condition as shown in Fig. 26(c) for keu SST model. The streamwise variation of the local heat transfer coefficient is insensitive to the thermal boundary conditions for streamwise locations Xa4 as observed in Fig. 26(c). 5. Conclusion A comparative study of the thermal characteristics of offset and wall jet flows has been carried out where the bottom wall is subjected to different thermal boundary conditions. The standard keε model, SST model and two different low-Reynolds number keε

models (YS and LS models) have been considered for numerical simulation. The streamwise variation of normalized skin friction coefficient obtained from SST model has shown very good agreement with the experimental results of Pelfrey and Liburdy [23] for the offset ratio 7. Subsequently, the comparison of velocity profile (uþyþ) with the experimental results of Nizou and Tida [21] has demonstrated the capability of SST model to accurately predict the velocity profile in the viscous sublayer as well as logarithmic region. The low-Reynolds number keε models have accurately mimicked the velocity profile in the viscous sublayer but slightly deviate in the logarithmic region. The rate of growth of outer shear layer obtained from the SST model has shown closer agreement with the expression suggested by Launder and Rodi [13]ðdY0:5 =dX ¼ 0:073±0:002Þ as observed for the wall jet flow at Re ¼ 15,000. When the temperature distribution is compared for various offset ratios, all the models overpredict near the wall as compared to the experimental results; the discrepancy decreases with

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Fig. 19. Streamwise variation of heat transfer coefficient h(x) for different boundary conditions.

Fig. 20. Dimensionless temperature profile (Tþ  yþ) for different boundary conditions in the similarity region (X ¼ 60).

Fig. 21. Non-dimensional temperature profile (Tþ  yþ) near the wall for different boundary conditions in the similarity region (X ¼ 60).

Fig. 22. Schematic diagram of a wall jet in the presence of an external moving stream.

decrease in OR. The YS model performs a shade better than others. Similar trend is observed when the decay of the maximum streamwise temperature is considered. The near-wall temperature variations obtained from YS and SST models are in better agreement with the experimental results of AbdulNour et al. [1] as compared to other models. The streamwise variation of heat transfer coefficient obtained from YS and SST models agrees both qualitatively and quantitatively with the experimental results. The variation of non-dimensional temperature profile (Tþyþ) in the similarity region suggests that the low-Reynolds number keε models and standard keε model show good agreement with the experimental results of Nizou [20] in the near-wall and off-wall regions respectively. In contrast, the SST model performs reasonably good in the near-wall as well as off-wall region. The analysis of numerical and experimental results for velocity similarity profile in

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355

Fig. 23. Similarity solution in the presence of an external stream at X ¼ 41.82 and 69.09.

Fig. 24. Non-dimensional velocity profile (uþ  yþ) in wall coordinates in the presence of an external stream at X ¼ 41.82 and 69.09.

Table 3 Comparison of the maximum streamwise velocity and jet spread with the experimental result at X ¼ 41.82 and 69.09. X

Flow parameter

Exp. (Dakos et al.)

YS model

%error

LS model

%error

Standard keε model

%error

SST model

%error

41.82 69.09 41.82 69.09

Umax Umax Y0.5 Y0.5

0.863 0.774 1.390 1.845

0.883 0.788 1.548 2.167

2.31 1.80 11.36 17.45

0.908 0.803 1.456 2.067

5.21 3.74 4.74 12.03

0.882 0.788 1.511 2.103

2.20 1.80 8.70 13.98

0.916 0.812 1.440 2.013

6.14 4.90 3.59 9.10

the absence as well as presence of an external moving stream has confirmed the observation of Wygnanski et al. [30] that experimental data taken in the outer shear layer are prone to errors due to room draught. The comparisons of thermal characteristics for a range of offset ratios for offset jet, and wall jet with different boundary conditions

and Reynolds number have shown that the YS and SST model performs better. However, the fluid flow characteristics obtained from the SST model have shown better agreement with the experimental results among all models considered. To sum up, the reasonably good thermal predictions are obtained from YS and SST models for wall-attached and mildly separated flows. However,

Fig. 25. Non-dimensional temperature profile in the presence of an external stream at X ¼ 41.82 and 69.09.

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Fig. 26. Streamwise variation of heat transfer coefficient h(x) for different boundary conditions in the presence of an external moving stream.

more sophisticated models are required which take into account the anisotropy and large strain rate in the flow due to the jet curvature as in the case of higher offset ratio. Acknowledgment The helpful comments and suggestions of the reviewers are gratefully acknowledged by the authors. References [1] R.S. AbdulNour, K. Willenborg, J.J. McGrath, J.F. Foss, B.S. AbdulNour, Measurements of the convection heat transfer coefficient for a planar wall jet: uniform temperature and uniform heat flux boundary conditions, Exp. Therm. Fluid Sci. 22 (2000) 123e131. [2] J.E. Bardina, P.G. Huang, T.J. Coakley, Turbulence Modeling Validation, Testing and Development, Tech. rep., Nasa Technical Memorandum 110446, Ames Research Center, 1997. [3] G. Biswas, V. Eswaran, Turbulent Flows: Fundamentals, Experiments and Modeling, IIT Kanpur Series of Advanced Texts, Narosa Publishing House, New Delhi, India, 2002. [4] T. Dakos, C.A. Verriopoulos, M.M. Gibson, Turbulent flow with heat transfer in plane and curved wall jets, J. Fluid Mech. 145 (1984) 339e360. [5] J.P.V. Doormaal, G. Raithby, Enhancements of the simple method for predicting incompressible fluid flows, Numer. Heat. Transf. 7 (1984) 147e163. [6] R. Dutta, A. Dewan, B. Srinivasan, Comparison of various integration to wall (ITW) RANS models for predicting turbulent slot jet impingement heat transfer, Int. J. Heat Mass Transf. 65 (2013) 750e764. [7] L.A. El-Gabry, D.A. Kaminski, Numerical investigation of jet impingement with cross flow-comparison of Yang-Shih and standard keε turbulence models, Numer. Heat. Transf. Part A 47 (2006) 441e469. [8] J. Hoch, L.M. Jiji, Theoretical and experimental temperature distribution in two-dimensional turbulent jet-boundary interaction, Trans. ASME J. Heat Transf. 103 (1981) 331e335. [9] J. Hoch, L.M. Jiji, Two-dimensional turbulent offset jet-boundary interaction, Trans. ASME J. Fluid Eng. 103 (1981) 154e161. [10] J.T. Holland, J.A. Liburdy, Measurements of the thermal characteristics of a turbulent offset jets, Int. J. Heat Mass Transf. 33 (1) (1990) 69e78. [11] W.P. Jones, B.E. Launder, The calculation of low-Reynolds-number phenomena with a two-equation model of turbulence, Int. J. Heat Mass Transf. 16 (1972) 1119e1130. [12] C.K.G. Lam, K.A. Bremhorst, Modified form of the keε model for predicting wall turbulence, Trans. ASME J. Fluids Eng. 103 (1981) 456e460. [13] B.E. Launder, W. Rodi, The turbulent wall jet, Prog. Aerosp. Sci. 19 (1981) 81e128.

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