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Numerical investigation of effect of semi-circular confinement bottom opening angle for slot jet impingement cooling on heated cylinder
International Journal of Thermal Sciences 149 (2020) 106148
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International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts
Numerical investigation of effect of semi-circular confinement bottom opening angle for slot jet impingement cooling on heated cylinder Ketan Atulkumar Ganatra, Dushyant Singh * Mechanical Engineering Department, National Institute of Technology, Manipur, Imphal, Manipur, 795004, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Air slot jet impingement Turbulence Semi-circular confinement Heat transfer
A numerical analysis for the air slot jet impingement cooling of a heated circular cylinder with a semi-circular bottom opening confinement is carried out. The Reynolds number (ReD) which is defined based on the cylin der diameter ranges from 30,000 to 90,000. The non-dimensional distance between the nozzle exit and the heated circular cylinder ranges from 4 to 10 and the bottom opening angle (α) for the semi-circular confinement ranges from 0� – 120� . The numerical results of the four turbulence models SST k-ω, realizable k-ε, RNG k-ε and v2f are compared to the experimental results. Further, the numerical results of the original and the modified v2f model are compared to the experimental results. From the comparison of the numerical and the experimental results, the SST k-ω model is found better from the other models for evaluating the fluid flow and the heat transfer characteristics. Hence, the further parametric investigation has been carried out by the SST k-ω model for h/S, ReD and α. The mean Nusselt number (NumD) is found maximum at h/S ¼ 4, 6, 8 and 10 for the value of the semi-circular confinement bottom opening angle α ¼ 70� , 60� , 100� and 90� . The Reynolds number (ReD) does not have much effect on the semi-circular confinement bottom opening angle (α) for the maximum value of the mean Nusselt number (NumD).
1. Introduction Impinging jets are used in many processes such as cooling, heating and drying of a surface. They are also used in manufacturing processes such as annealing of metals and glass surfaces. It has the advantages such as simple in construction and higher heat transfer rates at the impingement region. It has better heat removal rate as compared to the uniform flow. A large number of research articles of jet impingement cooling for a flat plate with single and multiple jets are available [1–6]. The avail ability of the literature for a curved surface either cylindrical or convex is less as compared to the flat plate. Gori and Bossi [7] studied the impinging air slot jet heat transfer over a cylindrical surface experimentally. The non-dimensional distance be tween the slot jet exit and the cylinder (h/S) ranges from 2 to 10. For D/S ¼ 2 and Re 4000–22,000, the maximum value of the mean Nusselt number observed is at h/S ¼ 6. A correlation for the mean Nusselt number (NumD) is also formulated. The same authors [8] studied experimentally the heat transfer characteristics for the different values surface curvature (S/D). The ratio of the diameter of the circular
cylinder (D) to the Slot width (S) are taken as D/S ¼ 1, 2 and 4. For ReD ¼ 4000–22,000 and D/S ¼ 4, the mean Nusselt number (NumD) is maximum at h/S ¼ 8. The mean Nusselt number (NumS) is maximum at D/S ¼ 2 when the Reynolds number which is defined based on the slot width (ReS). From their study the optimal slot width is taken as D/S ¼ 2 from the consideration of the same Reynolds number (ReS) means the same mass flow rate for different slot jets. Gori and Petracci [9] studied experimentally the heat transfer characteristics for an air slot jet over a cylinder for two different slot widths, 2.5 and 5 mm. The Nusselt number at the stagnation point is higher for a smaller width of the slot jet. The jet with a width of 2.5 mm has a greater efficiency as compared to that for the slot width of 5 mm. The mean Nusselt number is also independent from the slot width (S). They have compared the two jets by slot effi ciency and found that the jet with the slot width of 2.5 mm has a greater efficiency. McDaniel and Webb [10] investigated the heat transfer characteris tics for the slot jet and the cylinder experimentally. The impinging slot jet is provided with two different configurations of sharp edged and contoured type. They have concluded that the sharp edge configuration provides better heat removal rate as compared to the contoured orifice.
* Corresponding author. E-mail addresses: [email protected], [email protected] (D. Singh). https://doi.org/10.1016/j.ijthermalsci.2019.106148 Received 10 June 2018; Received in revised form 16 March 2019; Accepted 17 October 2019 1290-0729/© 2019 Published by Elsevier Masson SAS.
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
shapes considered are circular, quadrilateral and hexagonal. The flow confinement is provided with the top and the bottom opening. They have observed the increase in the average heat transfer from the cylinder to be 20–42% with the quadrilateral and the hexagonal flow confinements. The impinging jet over a heated cylinder with a round jet has studied by some authors [19–24]. They have investigated the effect of the various parameters namely the non-dimensional distance between the cylinder and the exit of the jet, Reynolds number and the ratio of jet to cylinder diameter. The detailed study has also been carried out for the effect of various shapes of nozzle. They have used different measuring technique to predict the Nusselt number distribution over a cylindrical surface. The impinging jet heat transfer for a convex surface with a single and multiple slot jets has studied by different authors [25–28] both experimentally as well as numerically. For an air slot jet impingement over a heated cylinder, the heat transfer is maximum at the stagnation point. Then after, the heat transfer decreases continuously till the rear stagnation point. Hence the present study is focussed at investigating the heat transfer characteristics mainly at the rear part of the cylinder. A bottom opening semi-circular confinement is taken for the evaluation of the heat transfer character istics at the rear part of the cylinder. Pachpute and Premchandran [16] have carried out the similar study for the jet impingement cooling with a semi-circular bottom opening confinement. They have studied the effect of the various parameters on the heat transfer from the cylindrical surface. They have done the nu merical simulation by the modified v2f turbulence model. They have used the modified v2f turbulence model without comparing them with the other two equations turbulence models. Further they have taken the width of the bottom opening for which the maximum size of the angular opening is around 20� . In the present study, the numerical simulation is carried out with the four different turbulence models both the two equations and four equations. Further the numerical simulation is also carried out with the original and the modified v2f turbulence models. The size of the angular opening taken is up to 120� . The present study is mainly aimed at selecting a turbulence model for the prediction of the fluid flow pattern and the heat transfer distribution better from the other models. The further parametric study is done from the best available turbulence model. The different parameters are the non-dimensional distance be tween the cylinder and the slot jet exit (h/S), Reynolds number (ReD) and the semi-circular confinement bottom opening angle (α). The effect of the above parameters for the heat transfer characteristics is investi gated. The study also focuses on identifying the value of the bottom opening angle (α) at which the heat transfer from the cylinder is maximum.
Fig. 1. The slot jet and the cylinder configuration with a semi-circular bottom opening confinement.
Brahma et al. [11] have studied the flow characteristics experi mentally for an air slot jet impingement over a cylinder. The static pressure and the velocity of the air are measured with the pitot tube and the manometer for the angular direction of the cylinder. They have concluded that the stagnation pressure is higher for the lower value of surface curvature (S/D) and higher Reynolds number (ReD). Olsson et al. [12] investigated impinging air slot jet heat transfer over a cylinder placed on a flat plate using ANSYS - CFX 5.5 as a solver. Three turbulence models standard k-ω, SST k-ω and standard k-ε are used for evaluating the heat transfer characteristics. The experimental and the numerical results are compared with each other for the cylinder in a crossflow. From their study, they observed that SST k-ω turbulence model is better from the other models. Singh and Singh [13] have made experimental as well as numerical investigation for the air slot jet over a cylinder placed on a flat plate. The experimental study has been carried out by Particle Image Velocimetry (PIV) technique. They have concluded that the flow separation occurs at lower angles for the lower value of the surface curvature. Nada [14] studied experimentally impinging air slot jet heat transfer over a cylinder with two different configurations of single and multiple slot jets. They have concluded that the multiple slot jets configuration is better for the heat transfer characteristics of the cylinder. Zuckerman and Lior [15] studied numerically the heat transfer characteristics for multiple slot jets over a cylinder. They have used SST k-ω, standard k-ε, realizable k-ε, Reynolds stress model and v2f turbulence models to obtain the heat transfer characteristics. From their study, they have pointed out that the realizable k-ε and the SST k-ω models are better from the other models. Pachpute and Premchandran [16] have studied experimentally as well as numerically the impinging air slot jet over a cylinder with and without a semi-circular bottom opening confinement. The maximum difference in the numerical and the experimental stagnation point Nusselt number is around 14%. The bottom opening confinement has significant effect at h/S ¼ 2 for the heat transfer from the cylindrical surface. The same authors [17] have also done a numerical simulation for a confined and an unconfined jet. They have reported that the flow confinement effect is significant for Lconf/D � 6 and h/S � 4. They [18] have studied experimentally and numerically the different shape of the flow confinement on the heat transfer from the cylinder. The different
2. Problem statement Fig. 1 shows the slot jet and the cylinder configuration. To evaluate the heat transfer characteristics at the rear part of the cylinder, a semicircular bottom opening confinement is used. The cylinder has a diam eter (D) equal to 36 mm. The heated cylinder is maintained at a uniform heat flux. The slot width (S) and the confinement radius (Rc) are taken as 3.6 mm and 22.5 mm. The ratio for the nozzle length (L) to the slot width (S) is taken as 64 for ensuring the fully developed velocity profile at the slot jet exit [16]. The non-dimensional distance between the jet exit and the cylinder is varied as 4, 6, 8 and 10. The Reynolds number (ReS ¼ ρUjet S=μ) based on the slot width (S) is varied from 3000–9000 to ensure the fully turbulent flow at the exit of the slot jet [29]. The Reynolds number which is defined based on the cylinder diameter (ReD ¼ ρUjet D=μ) is varied from 30,000 to 90,000. The semi-circular confine ment bottom opening angle (α) is varied from 0� to 120� . The local Nusselt number (Nuθ) can be evaluated from the following relation, 2
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Table 1 Range of the different geometric and flow parameters for the jet and the cylinder configuration. Parametric Notation
Values/Range
D h/S S/D ReD DC
36 4–10 0.1 30,000–90,000 45 0o– 120�
α
Table 2 Model constants of the original and the modified v2f turbulence models. Model Constant
Durbin [35] (Original)
Lien and Kalitzin [36] (Modified)
Cμ
0.19 1 1.3
0.22 1 1.3
σk σε
C1ε
C2ε C1 C2 CL Cη
0:25 � � #4 CL d 2 1þ 2L
1:3 þ "
1:4 1 þ 0:05
1.9 1.4 0.3 0.3 70
1.9 1.4 0.3 0.23 70
rffiffiffiffiffi ! k v2
Fig. 3. Grid independence study with different number of quadrilateral cells at h/S ¼ 8, S/D ¼ 0.25 and ReD ¼ 4000.
Table 3 Model specifications of v2f turbulence models. v2f Turbulence Models
Originality
v2f Model – 1 v2f Model – 2
Original v2f turbulence model developed by Durbin [35] Modified v2f turbulence model developed by Lien and Kalitzin [36]
Table 4 Percentage relative errors in numerical results for different Grids with respect to Grid D. Grid
Nustag
NumD
Grid A Grid B Grid C
10.92 8.09 4.96
1.19 1.64 0.93
Fig. 2. The computational domain used for the numerical simulation in the present study (a) closed semi-circular confinement (b) Opened semi-circular confinement. 3
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Nuθ ¼
TW
q’’ D � Tjet kf
The mean Nusselt number (NumD) over the circular cylinder can be calculated as Z 1 π NumD ¼ Nuθ dθ
π
0
The different parameters with their range are as shown in Table 1. 3. Mathematical modelling The flow is assumed as steady, incompressible, turbulent and twodimensional. For the prediction of the flow pattern and the heat trans fer distribution over the heated cylinder, the Reynolds time Averaged Navier-Stokes (RANS) equations are used. The continuity, momentum and energy equations with negligible amount of external forces are used for the numerical simulation as, Fig. 4. Effect of Turbulent Intensity on the local Nusselt number distribution for h/S ¼ 8, S/D ¼ 0.25 and ReD ¼ 20,000.
∂ui ¼0 ∂xi � �
ρuj
∂ui ¼ ∂xj
∂P ∂ ∂ui ∂uj þ μ þ ∂xi ∂xj ∂xj ∂xi
ρuj
∂T ∂ μ ∂T ¼ ∂xj ∂xj Pr ∂xj
�
�
�
ρu’i u’j
�
ρT ’ u’j
From the Boussinesq’ hypothesis the turbulent shear stress and the
Fig. 5. Comparison of numerical local Nusselt number distribution with the experimental results for a fixed value of surface curvature (S/D). 4
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 6. Comparison of numerical local Nusselt number distribution with the experimental results for a fixed value of non dimensional distance between the cylinder and the jet exit (h/S). Table 5 Percentage errors at stagnation Nusselt number (Nustag) for different surface curvature (S/D). Author/Reference
h/ S
S/D
Realizable
RNG
SST
Modified v2 f
Gori & Bossi [7] Pachpute & Premchandran [17]
2 2
0.5 0.12
25.25 2.74
43.83 12.83
23.71 12.98
21.21 1.49
Table 6 Percentage errors in the mean Nusselt number (NumD) for different surface curvature (S/D). Author/Reference
h/ S
S/D
Realizable
RNG
SST
Modified v2 f
Gori & Bossi [7] Pachpute & Premchandran [17]
2 2
0.5 0.12
10.48 40.40
38.33 25.37
3.19 44.19
8.30 40.77
Fig. 7. Comparison of numerical pressure coefficient with the experimental results for h/S ¼ 8, S/D ¼ 0.1 and Res ¼ 19,300.
5
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 8. Fluid flow pattern obtained by four different turbulence models for h/S ¼ 8, S/D ¼ 0.1 and ReS ¼ 19,300 (a) Realizable k-ε model (b) RNG k-ε model (c) SST kω model (d) v2f Model - 2.
heat flux can be calculated from the following relations,
ρu’i u’j ¼ μT
ρT ’ u’j ¼
turbulence dissipation rate (ε) are as follows, � � �� ∂ ∂ μ ∂k ρ ðkui Þ ¼ μþ T ρε þ Sk ∂xi ∂xj σ k ∂xj
∂ui ∂xj
μT ∂ T PrT ∂xj
ρ
PrT is the turbulence Prandtl number whose value is taken as 0.85 for the numerical analysis.
∂ ∂ ðεu Þ ¼ ∂xi i ∂xj
��
μþ
�
�
μT ∂ε þ C1 ρjSjε σ k ∂xj
C2 ρ
ε2 k þ ðϑεÞ0:5
þ Sε
The turbulent viscosity is calculated from the following relation,
μT ¼ ρ C μ
3.1. Realizable k-ε turbulence model
k2
ε
The values of the model constants are C1Ɛ ¼ 1.44, Cμ ¼ 0.09 and C2 ¼ 1.9.
The realizable k-ε model is formulated by Shih et al. [30]. The model transport equations for the turbulence kinetic energy (k) and the 6
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 9. Comparison of numerical pressure coefficient obtained by four different turbulence models for h/S ¼ 8, S/D ¼ 0.1 and ReS ¼ 19,300 (a) Realizable k-ε model (b) RNG k-ε model (c) SST k-ω model (d) v2f Model - 2.
3.2. RNG k-ε turbulence model The RNG k-ε model is formulated with the help of a statistical technique from the Navier-stokes equations. The used statistical tech nique is known as the renormalisation group theory [31]. The turbu lence model has a differential formula for the evaluation of the turbulence viscosity (μT). The model transport equations for the different parameters are as follows,
ρ
∂ ∂ ðkui Þ ¼ ∂xi ∂xj
ρ
∂ ∂ ðεu Þ ¼ ∂xi i ∂xj
��
μþ ��
μþ
�
μT ∂k σ k ∂xj �
7
ρε þ Sk �
μT ∂ε ε þ C1ε Pk σ k ∂xj k
where C*2ε ¼ C2ε þ
�
� Cμ η3 1 1 þ β η3
� η η0
C*2ε ρ
ε2 k
þ Sε
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 10. Comparison of numerical local Nusselt number distribution with the experimental results for a semi-circular confinement with bottom opening for h/S ¼ 2 and S/D ¼ 0.12.
η¼
Sk
ε
The blending function F1 is defined as, ! #)4 ) (( " pffiffi k 500ν 4ρσω2 k ; min max F1 ¼ tanh ; β*ωy y2 ω CDkω y2
pffiffiffiffiffiffiffiffiffiffiffi 2Sij Sij
And S ¼
The values of the model constants are, Cμ 0.09
C1ε 1.42
C2ε 1.68
σk
σε
0.7194
0.7194
η0
4.38
β 0.012
where � 1 ∂k ∂ω CDkω ¼ max 2ρσ ω2 ; 10 ω ∂xi ∂xi
3.3. SST k-ω turbulence model
The turbulence viscosity can be calculated from the following rela tion,
Wilcox [32] developed a model called the standard k-ω model which is very much sensitive to the inlet free stream turbulence. Menter and Esch [33] transformed the standard k-ε model into the k-ω form using a blending function F1. Hence, the model combines the advantages of the k-ε and the k-ω models with the help of the blending function F1. Near the walls the value of F1 is equal to one and zero for the remaining part of the flow [34]. The model transport equations for the different parameters are given by, � � ∂k � ∂ ∂k ρui ¼ Pk β * ρkω þ ðμ þ σ k μT Þ ∂xi ∂xi ∂xi
ρu i
∂ω ¼ αρS2 ∂xi
�
βρω2 þ
�
∂ ∂ω ðμ þ σω μT Þ þ 2ð1 ∂xi ∂xi
F1 Þρσω2
� 10
νT ¼
a1 k maxða1 ω; SF2 Þ
F2 is another blending function which is defined as, !#2 # "" pffiffi 2 k 500ν F2 ¼ tanh max ; β*ωy y2 ω The formation of turbulence in the stagnation region is prevented by the production limiter which is defined as follows, � � ∂ui ∂ui ∂uj P k ¼ μT þ ∂xj ∂xj ∂xi
1 ∂k ∂ω ω ∂xi ∂xi 8
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
uj
∂v2 ¼ kf ∂xj
v2 þ k ∂xj
∂2 f ∂x2j
1 ðC1 Tt
L2t
ε
f¼
∂
��
�
μT ∂v2 σk ∂xj
μþ
1Þ
� 2 v k
� 2 3
C2
�
Pk k
where kf ¼ ϕ*
ε* þ v2
ε k
Φ* ¼ Pressure Strain
ε* ¼ Dissipation Rate At the walls (Where y is the wall normal coordinate) f ð0Þ ¼
20ν2 v2 εð0Þy4
The ε (0) term in the denominator is not properly defined for the laminar and the transition region as shown by Lien and Kalitzin [36]. Hence they have reformulated the source term (kf v2 kε) in the v2 equation with the suggestion from Mansour et al. [37] as follows, ϕ* ¼
ε
ε
2v2 And ε* ¼ 4v2 k k
Accordingly, the transport equation for f changes as follows, � � ∂2 f 1 v2 2 Pk L2t 2 f ¼ ðC1 1Þ C2 ðC1 6Þ Tt ∂xj k 3 k The above equation for f together with the three equations for k, ε and v2 are known as Lien and Kalitzin [36] modified v2f model. However the asymptotic behaviour of the v2 near the wall remains the same for both the models. The turbulence time scale (Tt ), � � � � k ν 1=2 Tt ¼ max ; 6
Fig. 11. Comparison of numerical local Nusselt number distribution obtained by v2f model – 1 and 2 with the experimental results for S/D ¼ 0.5 (a) h/S ¼ 2 (b) h/S ¼ 6.
The turbulence length scale (Lt), � 3 �1=4 ! k3=2 ν Lt ¼CL max ; Cη
�
Pk ¼ minðPk ; 10β* ρkωÞ
ε
The values of model constants are, β* 0.09
σk1
0.85
σω1
0.5
α1
5/9
σk2
β1 3/40
1
σw2
0.856
α2
0.44
Durbin [35] developed a turbulence model named as v2f which is identical for the attached or the mildly separated boundary layers. It uses a velocity scale v2 for calculating the turbulent viscosity. The anisotropic wall effects are modelled through the elliptic relaxation function ‘f’. The model has four different transport equations for the different parameters namely (i) the turbulence kinetic energy (k) (ii) the turbulence dissipation rate (ε) (iii) the wall normal component of the fluctuating velocity (v2) (iv) Elliptic relaxation function (f). The model transport equations for the different parameters are as follows, �� � � ∂k ∂ μ ∂k u j ¼ Pk ε þ μþ T ∂xj ∂xj σ k ∂xj
∂ε C1ε Pk C2ε ε ∂ ¼ þ ∂xj ∂xj Tt
��
μþ
�
μT ∂ε σ ε ∂xj
ε
The model constants for both the models are as shown in Table 2. It should be noted that the model constants of Durbin [35] are dependent on the wall distance and hence on the yþ and that for the Lien and Kalitzin [36] are independent of the wall distance. Davidson et al. [38] has suggested two modifications in the Lien and Kalitzin [36] model. In the first modification they have set a limit on the source term of v2 as follows, � � � � 1 2k ðC1 1Þ þ C2 Pk v2 source ¼ min kf ; ðC1 6Þv2 Tt 3
β2 0.0828
3.4. v2f turbulence model
uj
ε
ε
With this modification the v2 � 2k/3 remains in the entire flow re gion but in the region where v2 � 2k/3, the turbulence viscosity will become 0.22v2Tt . Hence, a limit is also set on the value of the turbulence viscosity as below, � � k2 νT ¼ min 0:09 ; 0:22v2 Tt
ε
In the second modification they have proposed a method for calcu lating the turbulence viscosity for the wall normal and the parallel di rection as follows,
νT;? ¼ 0:22v2 Tt
�
9
and νT;jj ¼ 0:09kTt :
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 12. Fluid flow pattern obtained by v2f model – 1 and 2 for S/D ¼ 0.5 and ReD ¼ 20,000 (a) v2f Model – 1 at h/S ¼ 2 (b) v2f Model – 2 at h/S ¼ 2 (c) v2f Model – 1 at h/S ¼ 6 (d) v2f Model – 2 at h/S ¼ 6.
4. Numerical method
turbulent intensity and the viscosity ratio are fixed as 5% and 5 for all the cases. The flux of ‘f’ is assumed to be zero with zero gauge pressure. The other terms calculated are as follows,
The numerical simulation is carried out by using OpenFOAM solver. Four different turbulence models SST k-ω, v2f, RNG k-ε and realizable kεare used for modelling the turbulent flow. The SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm is used for the velocity and the pressure coupling. The terms for the turbulence and the con vection are discretised using Gauss upwind scheme in OpenFOAM [41]. The convergence is assumed to be reached when the residuals for all the convective and turbulence terms are less than 10 6. The fluid properties are evaluated at the jet exit temperature. The density, viscosity and thermal conductivity of air are taken as 1.2 kg/m3, 1.7894 � 10 5 kg/ms and 0.0257 W/mK. The under relaxation factors for all the turbulence and the convective terms and the tolerance value of 0.3 and 10 8 are used for the present study. The computational domain used for the closed and open semi-circular confinement is shown in Fig. 2.
3 ρk2 �μt � 1 ρk �μt � 1 2 2 ;ω ¼ ;v ¼ k k ¼ ðUIÞ2 ; ε ¼ Cμ 2 3 μ μ μ μ � Nozzle wall - The nozzle wall having no-slip boundary condition is taken as an adiabatic wall. � Heated Cylinder - The heat flux of constant intensity of 1000 W/m2 is imposed at the target heated cylinder wall with a no-slip boundary condition. � Pressure boundary - The pressure boundary is assumed to be the entrainment boundary where the atmospheric air interacts with the inlet air. The total temperature at the boundary is assumed to be 300 K. � Confinement wall - The confinement wall having no-slip boundary condition is assumed to be adiabatic.
4.1. Numerical boundary condition � Nozzle Inlet - At the nozzle inlet the uniform velocity profile is assumed. The temperature of the air is assumed to be 300 K. The 10
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
taken as 5 percentages for the further numerical study. 5. Selection of the turbulence model As discussed earlier, the numerical analysis is carried out with the four different turbulence models SST k-ω, RNG k-ε, v2f and realizable k-ε. The turbulence models have different characteristics for predicting the fluid flow and the heat transfer characteristics. They perform differently with the same type of the separated flow, rotational flow and the pres sure gradient. Further from the numerical study of Olsson et al. [12] and Zeyi Jiang et al. [27], it is observed that the realizable k-ε and the SST k-ω models work better from the other turbulence models. Hence a turbulence model is selected which predicts the heat transfer charac teristics well among the other models. To study the effects of the non-dimensional distance between the jet exit and the cylinder (h/S) and the Reynolds number (ReD) on the heat transfer characteristics, the numerical and the experimental results of [7] are compared with each other for h/S ¼ 2 and 6, ReD ¼ 4000 and 20, 000 and S/D ¼ 0.5 in Fig. 5. The stagnation point Nusselt number (Nustag.) is over predicted by all the turbulence models. The difference in the stagnation Nusselt number (Nustag.) obtained numerically and experimentally is minimum for the SST k-ω turbulence model. The heat transfer distribution over the cylindrical surface has a maximum value at the stagnation point, then it decreases and after a certain point it in creases till the rear part of the cylinder. The same trend is followed by all the turbulence models. This shows that all the turbulence models cap ture the recirculation zone but with different locations and size as shown in Fig. 5(a)–(d). As shown in Fig. 5(b) and (d), the RNG k-ε model pre dicts the peak Nusselt number at around 40� at h/S ¼ 2 and 6 and ReD ¼ 20,000. The numerical local Nusselt number distribution is compared with the experimental results for S/D ¼ 0.25 and 0.12 and h/S ¼ 2 in Fig. 6. In Fig. 6 (a) and (b), for S/D ¼ 0.25, the Nusselt number at the stagnation point is over predicted for ReD ¼ 4500 and under predicted for ReD ¼ 20,000 by all the turbulence models. As shown in Fig. 6 (c) and (d), for S/D ¼ 0.12 the Nusselt number at the front and the rear stag nation point remains almost equal to the experimental value. Hence, the trend of variation of the local Nusselt number remains unchanged for S/ D ¼ 0.12 as shown in Fig. 6 (c) and (d). For S/D ¼ 0.12, the size as well as the location of the recirculation zone remains almost same for all the turbulence models. However the size and the location of the recircula tion zone is not the same as for S/D ¼ 0.5 as shown in Fig. 6 (a) and (b). Table 5 and 6 show the percentage error in the stagnation Nusselt number (Nustag) and the mean Nusselt number (NumD) for different values of surface curvature (S/D) for h/S ¼ 2. All the turbulence models have less error for stagnation Nusselt number (Nustag) at lower value of surface curvature (S/D). While all the turbulence models except RNG k-ε have less error for the mean Nusselt number (NumD) for the higher value of surface curvature (S/D). Fig. 7 shows the distribution of the non-dimensional pressure coef ficient around the circumferential direction of the cylinder. The exper imental data shows that it is maximum at the stagnation point. Then after it decreases up to around 40� and remains almost constant up to around 125� from the stagnation point. Then after a slight increase in it is observed which shows that a higher pressure exists at the rear part of the cylinder. This is due to the higher value of the dynamic pressure at the rear part of the cylinder. Hence, there exists a region where the velocity of the air is more as compared to the cylindrical flow region. This zone is referred to as the recirculation zone. The SST k-ω and the RNG k-ε turbulence model follow the similar trend while the v2f and realizable k-ε models also capture a secondary minimum at the rear part of the cylinder as shown in Fig. 7. As shown in Fig. 8, the irregular vortices are found which is in opposite direction to that of the velocity of the air in the recirculation zone which is responsible for the reduced pressure in the recirculation zone. From Fig. 8 (a) and (d), it is observed that the irregular vortices in the recirculation zone are captured by v2f
Fig. 13. Comparison of numerical local Nusselt number distribution obtained by v2f model – 1 and 2 with the experimental results for S/D ¼ 0.12 and ReD ¼ 20,000 (a) h/S ¼ 2 (b) h/S ¼ 8.
4.2. Grid independence test A grid independence study is done with the four different grid sizes having 5548, 8265, 12889 and 20578 quadrilateral cells in Grid A, Grid B, Grid C and Grid D. The grid independence test has done with the v2f turbulence model for h/S ¼ 8, ReD ¼ 4500 and S/D ¼ 0.25 as shown in Fig. 3. All the grids near the walls are made such that yþ value remains below unity. The v2f turbulence model is selected for predicting the local Nusselt number distribution over the cylindrical surface. The percent relative error in numerical results for different grids with respect to Grid D is shown in Table 4. From Table 4, it is clear that the maximum relative error is around 11% at stagnation point of Nusselt number (Nustag) and 2% for mean Nusselt number (NumD). Hence, the grid size does not much affect the heat transfer distribution from the cylindrical surface and the Grid B is chosen for the further numerical simulation. 4.3. Selection of the value of the inflow turbulence intensity Gori and Bossi [8] have specified the value of the turbulence in tensity as 5% for ReD ¼ 6000 and 8% for ReD ¼ 20,000. Hence to check the effect of turbulence intensity on the heat transfer distribution, the values of turbulence intensity is taken in different intervals. The tur bulence intensity for the inflow is varied as 0.5, 1, 3 and 5 percentages to study the effect on the local Nusselt number distribution over a cylinder as shown in Fig. 4. As the turbulent intensity increases from 0.5 to 5 percentages, the local Nusselt number distribution remains almost the same over the entire cylindrical surface. Hence, the turbulence intensity has not very much effect for the local Nusselt number distribution over the cylindrical surface. Hence the value of the turbulent intensity is 11
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Fig. 14. Fluid flow pattern obtained by v2f model – 1 and 2 for S/D ¼ 0.12 and ReD ¼ 20,000 (a) v2f Model – 1 at h/S ¼ 2 (b) v2f Model – 2 at h/S ¼ 2 (c) v2f Model – 1 at h/S ¼ 8 (d) v2f Model – 2 at h/S ¼ 8.
and realizable k-ε models. To better understand the fluid flow pattern causing the heat transfer from the cylinder, the numerical non dimen sional pressure coefficient is shown in Fig. 9. The fluid flows in the di rection where the difference in the non dimensional pressure coefficient (Cp) is positive. As shown in Fig. 9, the numerical value of the non dimensional pressure coefficient (Cp) is around 1 predicted by all the turbulence models. This shows that all the kinetic energy of the air issuing from the jet is converted to the pressure energy which is responsible for the higher value of the Nusselt number in the stagnation region. A pressure difference exists in the stagnation region and the cylindrical flow region which is responsible for the flow over the cyl inder. Hence in the cylindrical flow region as the flow proceeds the velocity of the air decreases which reduces the heat transfer in the cy lindrical flow region. In the rear part of the cylinder, the positive value of the non dimensional pressure coefficient (Cp) is predicted by all the turbulence models which causes the recirculation of the flow, though the size of the recirculation zone may vary. Hence the higher value of the Nusselt number is observed in the recirculation zone as compared to the
cylindrical flow region. Fig. 10 shows the distribution for the local Nusselt number obtained numerically for h/S ¼ 2, S/D ¼ 0.12, α ¼ 7� and 20� and ReD ¼ 6000 and 20,000. The experimental results of [16] and the numerical results are compared with each other. The stagnation Nusselt number (Nustag.) predicted by all the turbulence models is almost equal to the experi mental value as shown in Fig. 10. Hence all the turbulence models predict almost the similar local Nusselt number distribution. However the value of the local Nusselt number at the rear stagnation is under predicted as compared to that without confinement. 5.1. Comparison of original and modified v2f turbulence models The numerical results obtained by the original and the modified v2f turbulence models suggested by two authors (Table 3) are compared from the experimental results. The local Nusselt number distribution over the cylindrical surface obtained by the v2f model – 1 and 2, at S/ D ¼ 0.5 and ReD ¼ 20,000 is shown in Fig. 11. The numerical and the 12
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International Journal of Thermal Sciences 149 (2020) 106148
Fig. 15. Effect of semi-circular confinement bottom opening angle (α) on the local Nusselt number distribution for Reynolds number (ReD) 30,000 and 90,000 at different values of h/S ¼ 4, 6, 8 and 10.
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Fig. 16. Fluid flow pattern for the different values of the semi-circular confinement bottom opening angle (α) at h/S ¼ 6 and ReD ¼ 90,000 (a) α ¼ 0� (b) α ¼ 40� (c) α ¼ 80� (d) α ¼ 120.
values of h/S ¼ 2 and 8 for both the v2f model – 1 and 2. Hence the value of h/S does not have much effect on the fluid flow and the heat transfer characteristics of the cylinder at lower value of S/D irrespective of the choice of the turbulence model. The absence of the irregular vortices and the reduced size of the recirculation zone are observed as shown in Figs. 12 and 14 for S/D ¼ 0.12 and 0.5. Hence, the main effect of the surface curvature (S/D) observed is the attached flow or the separated flow. For the lower values of S/D the flow remains attached to the cy lindrical surface more as compared to the higher values regardless of the flow region. Hence, the v2f model – 2 works better from the v2f model – 1 for predicting the fluid flow pattern and the heat transfer distribution. The different authors have explained the error between the experi mental data and numerical results. Behnia et al. [39] has suggested that the over prediction of the Nusselt number at the stagnation point is due to the over prediction of the turbulence kinetic energy (k). They observed an increase in the value of the turbulence kinetic energy (k) about 80% from the experimental value and therefore the increase in the Nusselt number at the stagnation point to about 170%. Behnia et al. [40] have explained the reason for the secondary peak in the Nusselt number away from the stagnation point is due to the transition from laminar to turbulent flow. Durbin [35] has suggested that as the stagnation region
experimental results of [16] are compared with each other. As shown in Fig. 11 (a), the local Nusselt number distribution has two secondary peak for the v2f model – 1 at ReD ¼ 20,000 and h/S ¼ 2. However, for the v2f model – 2 only a single peak is observed for the local Nusselt number as shown in Fig. 11 (a). As shown in Fig. 11, the difference in the nu merical and the experimental value of the local Nusselt number at the stagnation point is less with the v2f model - 2 as compared to the v2f model – 1 at h/S ¼ 2 and 6. The flow separation also occurs earlier with the v2f model – 2 as compared to that with the v2f model – 1 as shown in Fig. 12. In Fig. 12 (a) and (b), at h/S ¼ 2, the size of the irregular vortices is more for both the v2f model – 1 and 2. From Fig. 12 (c) and (d), at h/S ¼ 6 the presence of the irregular vortices is not observed for the v2f model – 2 and a decrease in the size of the irregular vortices is observed for the v2f model – 1. The fluid flow pattern causing the heat transfer observed is the stagnation zone, cylindrical flow region and recircula tion zone as shown in Fig. 12. As shown in Fig. 13 for S/D ¼ 0.12 and h/S ¼ 2 and 8, with the decrease in the value of surface curvature (S/D), there is more attachment of the fluid flow to the cylindrical surface. At h/S ¼ 2 and 8, the flow separation from the cylindrical surface occurs at around 135� from the stagnation point. It is clearly seen in Fig. 14, the size of the recirculation zone almost remains the same for the different 14
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International Journal of Thermal Sciences 149 (2020) 106148
Fig. 17. Effect of semi-circular confinement bottom opening angle (α) on mean Nusselt number for Reynolds number (ReD) 30, 000 to 90,000 (a) h/S ¼ 4 (b) h/S ¼ 6 (c) h/S ¼ 8 and (d) h/S ¼ 10.
Table 7 The percantage increase in the mean Nusselt number, α ¼ 60� (reference value for the mean Nusselt number is taken for closed semi circular confinement case). h/S 4 6 8 10
Table 9 The percantage increase in the mean Nusselt number, α ¼ 80� (reference value for the mean Nusselt number is taken for closed semi circular confinement case).
Reynolds Number (ReD)
h/S
30,000
50,000
70,000
90,000
36.69 48.59 35.11 41.14
40.08 53.60 41.78 42.72
44.76 57.09 43.13 44.94
48.17 60.20 45.67 47.85
4 6 8 10
Table 8 The percantage increase in the mean Nusselt number, α ¼ 70� (reference value for the mean Nusselt number is taken for closed semi circular confinement case). h/S 4 6 8 10
h/S
50,000
70,000
90,000
37.13 43.80 35.11 45.49
39.11 46.89 41.33 45.33
43.60 49.43 42.66 46.29
46.73 51.63 45.31 49.22
30,000
50,000
70,000
90,000
35.69 45.85 35.11 40.17
38.97 49.62 42.56 41.81
43.86 52.27 43.79 44.01
46.85 54.48 46.45 46.67
Table 10 The percantage increase in the mean Nusselt number, α ¼ 90� (reference value for the mean Nusselt number is taken for closed semi circular confinement case).
Reynolds Number (ReD) 30,000
Reynolds Number (ReD)
4 6 8 10
is approached, the turbulence time scale becomes very large by which the turbulence kinetic energy (k) is over predicted by any of the two equation turbulence model. The numerical results obtained from the various turbulence models SST k-ω, v2f, realizable k-ε and RNG k-ε are compared from the experi mental results. Also the numerical results of the v2f model – 1 and 2 are
Reynolds Number (ReD) 30,000
50,000
70,000
90,000
36.54 45.37 34.86 57.23
39.76 48.91 47.14 54.08
44.69 51.41 47.46 53.81
47.76 53.53 46.19 55.28
compared with the experimental results. From Figs. 5–14, it can be concluded that the SST k-ω model predicts the fluid flow pattern and the heat transfer distribution better for an impinging air slot jet over a cir cular cylinder. Hence, the further parametric investigation of the jet and the cylinder configuration with a semi-circular bottom opening confinement is done with the SST k-ω model. 15
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6.2. Mean Nusselt number
Table 11 The percantage increase in the mean Nusselt number, α ¼ 100� (reference value for the mean Nusselt number is taken for closed semi circular confinement case). h/S 4 6 8 10
Fig. 17 (a) shows the values for the mean Nusselt number (NumD) for different values of the bottom opening angle (α) at h/S ¼ 4 and ReD ¼ 30,000, 50,000, 70,000 and 90,000. The mean Nusselt number (NumD) is minimum for a closed semi-circular bottom opening confine ment. With the increase in the bottom opening angle (α), the mean Nusselt number (NumD) increases and after reaching the maximum value, it starts decreasing. A similar trend of variation is found at different values of the Reynolds number (ReD) considered. The value of the mean Nusselt number (NumD) is found maximum for α ¼ 70� at different values for the Reynolds number (ReD). For all the values of the bottom opening angle (α), the mean Nusselt number (NumD) increases with increasing the value of the Reynolds number (ReD). However with increasing Reynolds number (ReD), no change in the value of the bottom opening angle (α) is observed at which the mean Nusselt number is maximum. At ReD ¼ 30,000 and 90,000, around 25% and 30% increase for maximum value of the mean Nusselt number (NumD) is observed from the value for a closed semi-circular confinement. Fig. 17 (b)–(d) show mean Nusselt number (NumD) for different values of the bottom opening angle (α) at ReD ¼ 30,000, 50,000, 70,000 and 90,000 and at h/S ¼ 6, 8 and 10. The observed trend of variation for the mean value of the Nusselt number (NumD) is much similar for h/ S ¼ 6, 8 and 10 as for h/S ¼ 4. For the different values of h/S, the value of the mean Nusselt number (NumD) remains almost equal. Hence, the mean Nusselt number (NumD) is not much affected by the nondimensional distance between the slot jet exit and the cylinder (h/S). The mean Nusselt number (NumD) is maximum for α ¼ 60� , 100� and 90� at h/S ¼ 6, 8 and 10. Tables 7–11 shows the percentage increase in the mean Nusselt number as compared to the value for the closed confine ment at different values of h/S and ReD. The percentage increase for the mean Nusselt number (NumD) is maximum at h/S ¼ 6 and ReD ¼ 90,000 for the values of α ¼ 60� , 70� , 80� and 100� except α ¼ 90� . The per centage increase in the mean Nusselt number ranges from 35 to 57% as compared to that for the closed semi-circular confinement.
Reynolds Number (ReD) 30,000
50,000
70,000
90,000
35.36 43.26 44.26 43.85
37.95 47.01 44.95 44.66
42.80 49.85 46.18 46.33
45.91 52.08 49.14 49.31
6. Effect of the semi-circular confinement bottom opening angle (α) 6.1. Local Nusselt number For an air slot jet, the stagnation point Nusselt number (Nustag.) is of primary concern due to the higher heat removal rate. The local Nusselt number increases with the increase in the Reynolds number (ReD) and with the decrease in the non-dimensional distance between the slot jet exit and the cylinder (h/S). The stagnation point Nusselt number (Nus tag.) follows the similar trend of variation. The numerical results ob tained agree very well with the literature available for the cylindrical surface [7,8]. In Fig. 15 (a) for a closed confinement the separation point predicted is at around 75� from the stagnation point and the recirculation zone starts from there and extends up to the rear stagnation point. The same flow pattern is observed by Olsson et al. [12] for an impinging jet over a cylinder placed on a flat plate. For α ¼ 40� , the local Nusselt number is minimum nearly at 160� and then after it increases up to the rear part of the cylinder. The separation point predicted numerically is at around 160� and the recirculation zone spreads around 40� over the entire cy lindrical surface. The size of the recirculation zone is almost equal to the bottom opening angle (α). The same trend is observed for the value of α ¼ 80� and the flow separation observed is at around 150� . But with the more increase in the bottom opening angle α ¼ 120� , the separation point predicted is at around 1400 with the reduced size of the recircu lation zone. The size of the recirculation zone does not match with the value of the bottom opening angle (α) as observed for the values of α ¼ 40� and 80� . This is because of the higher incoming velocity of the air in the cylindrical flow region. Fig. 15 (b) shows the local Nusselt number distribution at higher value of the Reynolds number, ReD ¼ 90,000 for h/S ¼ 4 and S/D ¼ 0.1. For the values of α ¼ 0� and 40� , the separation point predicted coincides with the values as predicted with the low Reynolds number (ReD). For the value of α ¼ 80� and 120� , the location of the separation point is at around 150� . This shows that the location of the separation point at higher Reynolds number shifts towards the higher angular location from the stagnation point. This agrees very well with the experimental results of Gori and Bossi [7] for an impinging air slot jet over a circular cylinder. With the increase in the value of Reynolds number (ReD), the size of the recirculation zone decreases. The reason for this is the higher incoming velocity of the air in the cylindrical flow region at higher Reynolds number (ReD) as pointed out earlier. As shown in Fig.15 (c)–(h), when h/S changes from 6 to 8 and 10, the local Nusselt number distribution is not much affected for the values of α ¼ 0� , 40� , 80� and 120� . The only change observed is for h/S ¼ 8 and α ¼ 80� . The numerical local Nusselt number is found minimum at around θ ¼ 160� and 150� at ReD ¼ 30,000 and 90,000. Fig. 16 (a) to (d) show the fluid flow pattern for the h/S ¼ 6, ReD ¼ 90,000 and α ¼ 0� , 40� , 80� and 120� . For a closed confinement the separation point is at around 80� from the stagnation point and beyond that the recirculation starts and extends up to the rear part of the cylinder. With the increase in the bottom opening angle (α), the size of the recirculation zone also increases for semi-circular bottom opening confinement.
7. Conclusion Present study shows the numerical analysis carried out by four various turbulence models SST k-ω, modified v2f, realizable k-ε and RNG k-ε. The local Nusselt number distribution is obtained from the original and the modified v2f model over the cylindrical surface numerically. By comparing the numerical results from various turbulence model, a parametric investigation is also done by the SST k-ω turbulence model for the study of the effect of the non-dimensional distance between the cylinder and the jet exit (h/S), Reynolds number (ReD) and the semicircular confinement bottom opening angle (α) on the local as well as the mean Nusselt number. The conclusions made from the detailed study are as follows, 1. As compared to the other models, the SST k-ω and the RNG k-ε predict the better fluid flow characteristics. 2. As compared to the other models, the SST k-ω predicts the better heat transfer characteristics. 3. The modified v2f turbulence model works better from the original v2f turbulence model for predicting the heat transfer characteristics for a circular cylinder. 4. At the rear part of the cylinder, the size of the recirculation zone increases with the increase in the bottom opening angle (α) and the decrease in the Reynolds number (ReD). 5. The value of the h/S does not have much effect on the local as well as mean Nusselt number. 6. As compared to the value for the closed confinement, the maximum increase in the value of the mean Nusselt number (NumD) is at h/ S ¼ 6 for all the values of ReD.
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Acknowledgements
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This work is thankful to the Department of Science and Technology (DST), New Delhi of Project No. ECR/2016/001364 for providing computational facility to carry out this study in Department of Me chanical Engineering at NIT Manipur. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.ijthermalsci.2019.106148. References [1] J.N.B. Livingood, P. Hrycak, Impingement Heat Transfer from Turbulent Air Jets to Flat Plates – A Literature Survey, NASA Technical Memorandum, NASA TM X2778, Lewis Research Centre, Cleveland, 1973. [2] P. Hrycak, Heat Transfer from Impinging Jets: A Literature Review, AFWAL-TR 813054, 1981. New Jersey. [3] H. Martin, Heat and mass transfer between impinging gas Jets and solid surfaces, Adv. Heat Tran. 13 (1977) 1–60. [4] K. Jambunathan, E. Lai, M.A. Moss, B.L. Button, A review of heat transfer data for single circular jet impingement, Int. J. Heat Fluid Flow 13 (1992) 106–115. [5] R. Viskanta, Heat transfer to impinging isothermal gas and flame jets, Exp. Therm. Fluid Sci. 6 (1993) 111–134. [6] N. Zuckerman, N. Lior, Jet impingement heat transfer physics, correlations and numerical modelling, Adv. Heat Tran. 39 (2006) 565–632. [7] F. Gori, L. Bossi, On the cooling effect of an air jet along the surface of a cylinder, Int. Commun. Heat Mass Transf. 27 (5) (2000) 667–676. [8] F. Gori, L. Bossi, Optimal slot height in the jet cooling of a circular cylinder, Appl. Therm. Eng. 23 (2003) 859–870. [9] F. Gori, I. Petracci, On the effect of the slot height in the cooling of a circular cylinder with a rectangular jet, Int. Commun. Heat Mass Transf. 48 (2013) 8–14. [10] C.S. McDaniel, B.W. Webb, Slot jet impingement heat transfer from circular cylinders, Int. J. Heat Mass Transf. 43 (2000) 1975–1985. [11] R.K. Brahma, O. Faruque, R.C. Arora, Experimental Investigation of Mean Flow Characteristics of Slot Jet Impingement on a Cylinder 26, Springer-Verlag, 1991, pp. 257–263. [12] E.E.M. Olsson, L.M. Ahrne, A.C. Tragardh, Heat transfer from a slot air jet impinging on a circular cylinder, J. Food Eng. 63 (2004) 393–401. [13] S.K. Singh, R.P. Singh, Air impingement cooling of cylindrical objects using slot jets, Food Engineering: integrated Approaches, Food Eng. Ser. 5 (2008) 89–104. [14] S.A. Nada, Slot/slots air jet impinging cooling of a cylinder for different jets–cylinder configurations, Heat Mass Transf. 43 (2006) 135–148. [15] N. Zuckerman, N. Lior, Radial slot jet impingement flow and heat transfer on a cylindrical target, J. Thermophys. Heat Transf. 21 (3) (2007) 548–561. [16] S. Pachpute, B. Premachandran, Experimental and numerical investigations of slot jet impingement with and without a semi-circular bottom confinement, Int. J. Heat Mass Transf. 114 (2017) 866–890. [17] S. Pachpute, B. Premachandran, Slot air jet impingement cooling over a heated circular cylinder with and without a flow confinement, Appl. Therm. Eng. 132 (2018) 352–367. [18] S. Pachpute, B. Premachandran, Effect of the shape of flow confinement on turbulent slot jet impingement cooling of a heated circular cylinder, Int. J. Therm. Sci. 131 (2018) 114–131. [19] E.W. Sparrow, C.A.C. Altemani, A. Chaboki, Jet impingement heat transfer for a circular jet impinging in cross flow on a cylinder, J. Heat Transf. 106 (1984) 570–577. [20] A.A. Tawfek, Heat transfer due to a round jet impinging normal to a circular cylinder, Heat Mass Trans. Springer-Verlag 35 (1999) 327–333. [21] D. Singh, B. Permachandran, S. kohli, Experimental and numerical investigation of jet impingement cooling of a circular cylinder, Int. J. Heat Mass Transf. 60 (2013) 672–688. [22] X.L. Wang, D. Motala, T.J. Lu, S.J. Song, T. Kim, Heat transfer of a circular impinging jet on a circular cylinder in cross flow, Int. J. Therm. Sci. 78 (2014) 1–8. [23] D. Singh, B. Premachandran, S. Kohli, Effect of nozzle shape on jet impingement heat transfer from a circular cylinder, Int. J. Therm. Sci. 96 (2015) 45–69. [24] D. Singh, B. Premachandran, S. Kohli, Circular air jet impingent cooling of a circular cylinder with flow confinement, Int. J. Heat Mass Transf. 91 (2015) 969–989. [25] C. Gau, C.M. Chung, Surface curvature effect on slot air-jet impingement cooling flow and heat transfer process, J. Heat Transf. 113 (4) (1991) 858–864. [26] T.L. Chan, C.W. Leung, K. Jambunathan, S. Ashforth –Frost, Y. Zhou, M.H. Liu, Heat transfer characteristics of a slot jet impinging on a semi circular convex surface, Int. J. Heat Mass Transf. 45 (2002) 993–1006.
Nomenclature D: Diameter of the heated circular cylinder (m) DC: Diameter of the semi-circular confinement (m) f: Elliptic relaxation function (1/s) h: Distance between the nozzle exit and the cylinder (m) k: Turbulent kinetic energy (m2/s2) kf: Thermal conductivity of the air (W/mK) � � Nuθ: Local Nusselt number, Nuθ ¼ q00 D� kf ðTW Tjet Þ � � Z π Nuθ dθ NumD: Mean Nusselt number NumD ¼ 1π 0
Pr: Prandtl number PrT: Turbulence Prandtl number q0 0 : Heat flux (W/m2) RC: Radius of the semi-circular confinement (m) ReD: Reynolds number based heated cylinder diameter, (ReD ¼ ρUjet D=μ ) ReS: Reynolds number based nozzle slot width, (ReS ¼ ρUjet S=μ ) S: Slot width of the jet (m) TW: Cylinder wall temperature (K) Tjet: Jet temperature or nozzle exit temperature (K) Ujet: Magnitude of the velocity of the air at the nozzle exit (m/s) U: Local velocity of air (m/s) v2 : Velocity fluctuations normal to the streamlines (m2/s2) X: – Cartesian co-ordinate in x direction (m) Y: Cartesian co-ordinate in y direction (m) yþ: Non-dimensional wall distanceGreek ε: Turbulent dissipation rate (m2/s3) ω: Turbulence specific dissipation rate (1/s) μ: Dynamic viscosity of the air (kg/ms) ρ: Density of the air (kg/m3) ν: Kinematic viscosity of the air (m2/s) α: Semi-circular confinement bottom opening angle (o)Suffix j: Jet t: Turbulence w: Wall C: Semi-circular confinement
17
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts
Numerical investigation of effect of semi-circular confinement bottom opening angle for slot jet impingement cooling on heated cylinder Ketan Atulkumar Ganatra, Dushyant Singh * Mechanical Engineering Department, National Institute of Technology, Manipur, Imphal, Manipur, 795004, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Air slot jet impingement Turbulence Semi-circular confinement Heat transfer
A numerical analysis for the air slot jet impingement cooling of a heated circular cylinder with a semi-circular bottom opening confinement is carried out. The Reynolds number (ReD) which is defined based on the cylin der diameter ranges from 30,000 to 90,000. The non-dimensional distance between the nozzle exit and the heated circular cylinder ranges from 4 to 10 and the bottom opening angle (α) for the semi-circular confinement ranges from 0� – 120� . The numerical results of the four turbulence models SST k-ω, realizable k-ε, RNG k-ε and v2f are compared to the experimental results. Further, the numerical results of the original and the modified v2f model are compared to the experimental results. From the comparison of the numerical and the experimental results, the SST k-ω model is found better from the other models for evaluating the fluid flow and the heat transfer characteristics. Hence, the further parametric investigation has been carried out by the SST k-ω model for h/S, ReD and α. The mean Nusselt number (NumD) is found maximum at h/S ¼ 4, 6, 8 and 10 for the value of the semi-circular confinement bottom opening angle α ¼ 70� , 60� , 100� and 90� . The Reynolds number (ReD) does not have much effect on the semi-circular confinement bottom opening angle (α) for the maximum value of the mean Nusselt number (NumD).
1. Introduction Impinging jets are used in many processes such as cooling, heating and drying of a surface. They are also used in manufacturing processes such as annealing of metals and glass surfaces. It has the advantages such as simple in construction and higher heat transfer rates at the impingement region. It has better heat removal rate as compared to the uniform flow. A large number of research articles of jet impingement cooling for a flat plate with single and multiple jets are available [1–6]. The avail ability of the literature for a curved surface either cylindrical or convex is less as compared to the flat plate. Gori and Bossi [7] studied the impinging air slot jet heat transfer over a cylindrical surface experimentally. The non-dimensional distance be tween the slot jet exit and the cylinder (h/S) ranges from 2 to 10. For D/S ¼ 2 and Re 4000–22,000, the maximum value of the mean Nusselt number observed is at h/S ¼ 6. A correlation for the mean Nusselt number (NumD) is also formulated. The same authors [8] studied experimentally the heat transfer characteristics for the different values surface curvature (S/D). The ratio of the diameter of the circular
cylinder (D) to the Slot width (S) are taken as D/S ¼ 1, 2 and 4. For ReD ¼ 4000–22,000 and D/S ¼ 4, the mean Nusselt number (NumD) is maximum at h/S ¼ 8. The mean Nusselt number (NumS) is maximum at D/S ¼ 2 when the Reynolds number which is defined based on the slot width (ReS). From their study the optimal slot width is taken as D/S ¼ 2 from the consideration of the same Reynolds number (ReS) means the same mass flow rate for different slot jets. Gori and Petracci [9] studied experimentally the heat transfer characteristics for an air slot jet over a cylinder for two different slot widths, 2.5 and 5 mm. The Nusselt number at the stagnation point is higher for a smaller width of the slot jet. The jet with a width of 2.5 mm has a greater efficiency as compared to that for the slot width of 5 mm. The mean Nusselt number is also independent from the slot width (S). They have compared the two jets by slot effi ciency and found that the jet with the slot width of 2.5 mm has a greater efficiency. McDaniel and Webb [10] investigated the heat transfer characteris tics for the slot jet and the cylinder experimentally. The impinging slot jet is provided with two different configurations of sharp edged and contoured type. They have concluded that the sharp edge configuration provides better heat removal rate as compared to the contoured orifice.
* Corresponding author. E-mail addresses: [email protected], [email protected] (D. Singh). https://doi.org/10.1016/j.ijthermalsci.2019.106148 Received 10 June 2018; Received in revised form 16 March 2019; Accepted 17 October 2019 1290-0729/© 2019 Published by Elsevier Masson SAS.
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
shapes considered are circular, quadrilateral and hexagonal. The flow confinement is provided with the top and the bottom opening. They have observed the increase in the average heat transfer from the cylinder to be 20–42% with the quadrilateral and the hexagonal flow confinements. The impinging jet over a heated cylinder with a round jet has studied by some authors [19–24]. They have investigated the effect of the various parameters namely the non-dimensional distance between the cylinder and the exit of the jet, Reynolds number and the ratio of jet to cylinder diameter. The detailed study has also been carried out for the effect of various shapes of nozzle. They have used different measuring technique to predict the Nusselt number distribution over a cylindrical surface. The impinging jet heat transfer for a convex surface with a single and multiple slot jets has studied by different authors [25–28] both experimentally as well as numerically. For an air slot jet impingement over a heated cylinder, the heat transfer is maximum at the stagnation point. Then after, the heat transfer decreases continuously till the rear stagnation point. Hence the present study is focussed at investigating the heat transfer characteristics mainly at the rear part of the cylinder. A bottom opening semi-circular confinement is taken for the evaluation of the heat transfer character istics at the rear part of the cylinder. Pachpute and Premchandran [16] have carried out the similar study for the jet impingement cooling with a semi-circular bottom opening confinement. They have studied the effect of the various parameters on the heat transfer from the cylindrical surface. They have done the nu merical simulation by the modified v2f turbulence model. They have used the modified v2f turbulence model without comparing them with the other two equations turbulence models. Further they have taken the width of the bottom opening for which the maximum size of the angular opening is around 20� . In the present study, the numerical simulation is carried out with the four different turbulence models both the two equations and four equations. Further the numerical simulation is also carried out with the original and the modified v2f turbulence models. The size of the angular opening taken is up to 120� . The present study is mainly aimed at selecting a turbulence model for the prediction of the fluid flow pattern and the heat transfer distribution better from the other models. The further parametric study is done from the best available turbulence model. The different parameters are the non-dimensional distance be tween the cylinder and the slot jet exit (h/S), Reynolds number (ReD) and the semi-circular confinement bottom opening angle (α). The effect of the above parameters for the heat transfer characteristics is investi gated. The study also focuses on identifying the value of the bottom opening angle (α) at which the heat transfer from the cylinder is maximum.
Fig. 1. The slot jet and the cylinder configuration with a semi-circular bottom opening confinement.
Brahma et al. [11] have studied the flow characteristics experi mentally for an air slot jet impingement over a cylinder. The static pressure and the velocity of the air are measured with the pitot tube and the manometer for the angular direction of the cylinder. They have concluded that the stagnation pressure is higher for the lower value of surface curvature (S/D) and higher Reynolds number (ReD). Olsson et al. [12] investigated impinging air slot jet heat transfer over a cylinder placed on a flat plate using ANSYS - CFX 5.5 as a solver. Three turbulence models standard k-ω, SST k-ω and standard k-ε are used for evaluating the heat transfer characteristics. The experimental and the numerical results are compared with each other for the cylinder in a crossflow. From their study, they observed that SST k-ω turbulence model is better from the other models. Singh and Singh [13] have made experimental as well as numerical investigation for the air slot jet over a cylinder placed on a flat plate. The experimental study has been carried out by Particle Image Velocimetry (PIV) technique. They have concluded that the flow separation occurs at lower angles for the lower value of the surface curvature. Nada [14] studied experimentally impinging air slot jet heat transfer over a cylinder with two different configurations of single and multiple slot jets. They have concluded that the multiple slot jets configuration is better for the heat transfer characteristics of the cylinder. Zuckerman and Lior [15] studied numerically the heat transfer characteristics for multiple slot jets over a cylinder. They have used SST k-ω, standard k-ε, realizable k-ε, Reynolds stress model and v2f turbulence models to obtain the heat transfer characteristics. From their study, they have pointed out that the realizable k-ε and the SST k-ω models are better from the other models. Pachpute and Premchandran [16] have studied experimentally as well as numerically the impinging air slot jet over a cylinder with and without a semi-circular bottom opening confinement. The maximum difference in the numerical and the experimental stagnation point Nusselt number is around 14%. The bottom opening confinement has significant effect at h/S ¼ 2 for the heat transfer from the cylindrical surface. The same authors [17] have also done a numerical simulation for a confined and an unconfined jet. They have reported that the flow confinement effect is significant for Lconf/D � 6 and h/S � 4. They [18] have studied experimentally and numerically the different shape of the flow confinement on the heat transfer from the cylinder. The different
2. Problem statement Fig. 1 shows the slot jet and the cylinder configuration. To evaluate the heat transfer characteristics at the rear part of the cylinder, a semicircular bottom opening confinement is used. The cylinder has a diam eter (D) equal to 36 mm. The heated cylinder is maintained at a uniform heat flux. The slot width (S) and the confinement radius (Rc) are taken as 3.6 mm and 22.5 mm. The ratio for the nozzle length (L) to the slot width (S) is taken as 64 for ensuring the fully developed velocity profile at the slot jet exit [16]. The non-dimensional distance between the jet exit and the cylinder is varied as 4, 6, 8 and 10. The Reynolds number (ReS ¼ ρUjet S=μ) based on the slot width (S) is varied from 3000–9000 to ensure the fully turbulent flow at the exit of the slot jet [29]. The Reynolds number which is defined based on the cylinder diameter (ReD ¼ ρUjet D=μ) is varied from 30,000 to 90,000. The semi-circular confine ment bottom opening angle (α) is varied from 0� to 120� . The local Nusselt number (Nuθ) can be evaluated from the following relation, 2
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Table 1 Range of the different geometric and flow parameters for the jet and the cylinder configuration. Parametric Notation
Values/Range
D h/S S/D ReD DC
36 4–10 0.1 30,000–90,000 45 0o– 120�
α
Table 2 Model constants of the original and the modified v2f turbulence models. Model Constant
Durbin [35] (Original)
Lien and Kalitzin [36] (Modified)
Cμ
0.19 1 1.3
0.22 1 1.3
σk σε
C1ε
C2ε C1 C2 CL Cη
0:25 � � #4 CL d 2 1þ 2L
1:3 þ "
1:4 1 þ 0:05
1.9 1.4 0.3 0.3 70
1.9 1.4 0.3 0.23 70
rffiffiffiffiffi ! k v2
Fig. 3. Grid independence study with different number of quadrilateral cells at h/S ¼ 8, S/D ¼ 0.25 and ReD ¼ 4000.
Table 3 Model specifications of v2f turbulence models. v2f Turbulence Models
Originality
v2f Model – 1 v2f Model – 2
Original v2f turbulence model developed by Durbin [35] Modified v2f turbulence model developed by Lien and Kalitzin [36]
Table 4 Percentage relative errors in numerical results for different Grids with respect to Grid D. Grid
Nustag
NumD
Grid A Grid B Grid C
10.92 8.09 4.96
1.19 1.64 0.93
Fig. 2. The computational domain used for the numerical simulation in the present study (a) closed semi-circular confinement (b) Opened semi-circular confinement. 3
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Nuθ ¼
TW
q’’ D � Tjet kf
The mean Nusselt number (NumD) over the circular cylinder can be calculated as Z 1 π NumD ¼ Nuθ dθ
π
0
The different parameters with their range are as shown in Table 1. 3. Mathematical modelling The flow is assumed as steady, incompressible, turbulent and twodimensional. For the prediction of the flow pattern and the heat trans fer distribution over the heated cylinder, the Reynolds time Averaged Navier-Stokes (RANS) equations are used. The continuity, momentum and energy equations with negligible amount of external forces are used for the numerical simulation as, Fig. 4. Effect of Turbulent Intensity on the local Nusselt number distribution for h/S ¼ 8, S/D ¼ 0.25 and ReD ¼ 20,000.
∂ui ¼0 ∂xi � �
ρuj
∂ui ¼ ∂xj
∂P ∂ ∂ui ∂uj þ μ þ ∂xi ∂xj ∂xj ∂xi
ρuj
∂T ∂ μ ∂T ¼ ∂xj ∂xj Pr ∂xj
�
�
�
ρu’i u’j
�
ρT ’ u’j
From the Boussinesq’ hypothesis the turbulent shear stress and the
Fig. 5. Comparison of numerical local Nusselt number distribution with the experimental results for a fixed value of surface curvature (S/D). 4
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 6. Comparison of numerical local Nusselt number distribution with the experimental results for a fixed value of non dimensional distance between the cylinder and the jet exit (h/S). Table 5 Percentage errors at stagnation Nusselt number (Nustag) for different surface curvature (S/D). Author/Reference
h/ S
S/D
Realizable
RNG
SST
Modified v2 f
Gori & Bossi [7] Pachpute & Premchandran [17]
2 2
0.5 0.12
25.25 2.74
43.83 12.83
23.71 12.98
21.21 1.49
Table 6 Percentage errors in the mean Nusselt number (NumD) for different surface curvature (S/D). Author/Reference
h/ S
S/D
Realizable
RNG
SST
Modified v2 f
Gori & Bossi [7] Pachpute & Premchandran [17]
2 2
0.5 0.12
10.48 40.40
38.33 25.37
3.19 44.19
8.30 40.77
Fig. 7. Comparison of numerical pressure coefficient with the experimental results for h/S ¼ 8, S/D ¼ 0.1 and Res ¼ 19,300.
5
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 8. Fluid flow pattern obtained by four different turbulence models for h/S ¼ 8, S/D ¼ 0.1 and ReS ¼ 19,300 (a) Realizable k-ε model (b) RNG k-ε model (c) SST kω model (d) v2f Model - 2.
heat flux can be calculated from the following relations,
ρu’i u’j ¼ μT
ρT ’ u’j ¼
turbulence dissipation rate (ε) are as follows, � � �� ∂ ∂ μ ∂k ρ ðkui Þ ¼ μþ T ρε þ Sk ∂xi ∂xj σ k ∂xj
∂ui ∂xj
μT ∂ T PrT ∂xj
ρ
PrT is the turbulence Prandtl number whose value is taken as 0.85 for the numerical analysis.
∂ ∂ ðεu Þ ¼ ∂xi i ∂xj
��
μþ
�
�
μT ∂ε þ C1 ρjSjε σ k ∂xj
C2 ρ
ε2 k þ ðϑεÞ0:5
þ Sε
The turbulent viscosity is calculated from the following relation,
μT ¼ ρ C μ
3.1. Realizable k-ε turbulence model
k2
ε
The values of the model constants are C1Ɛ ¼ 1.44, Cμ ¼ 0.09 and C2 ¼ 1.9.
The realizable k-ε model is formulated by Shih et al. [30]. The model transport equations for the turbulence kinetic energy (k) and the 6
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 9. Comparison of numerical pressure coefficient obtained by four different turbulence models for h/S ¼ 8, S/D ¼ 0.1 and ReS ¼ 19,300 (a) Realizable k-ε model (b) RNG k-ε model (c) SST k-ω model (d) v2f Model - 2.
3.2. RNG k-ε turbulence model The RNG k-ε model is formulated with the help of a statistical technique from the Navier-stokes equations. The used statistical tech nique is known as the renormalisation group theory [31]. The turbu lence model has a differential formula for the evaluation of the turbulence viscosity (μT). The model transport equations for the different parameters are as follows,
ρ
∂ ∂ ðkui Þ ¼ ∂xi ∂xj
ρ
∂ ∂ ðεu Þ ¼ ∂xi i ∂xj
��
μþ ��
μþ
�
μT ∂k σ k ∂xj �
7
ρε þ Sk �
μT ∂ε ε þ C1ε Pk σ k ∂xj k
where C*2ε ¼ C2ε þ
�
� Cμ η3 1 1 þ β η3
� η η0
C*2ε ρ
ε2 k
þ Sε
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 10. Comparison of numerical local Nusselt number distribution with the experimental results for a semi-circular confinement with bottom opening for h/S ¼ 2 and S/D ¼ 0.12.
η¼
Sk
ε
The blending function F1 is defined as, ! #)4 ) (( " pffiffi k 500ν 4ρσω2 k ; min max F1 ¼ tanh ; β*ωy y2 ω CDkω y2
pffiffiffiffiffiffiffiffiffiffiffi 2Sij Sij
And S ¼
The values of the model constants are, Cμ 0.09
C1ε 1.42
C2ε 1.68
σk
σε
0.7194
0.7194
η0
4.38
β 0.012
where � 1 ∂k ∂ω CDkω ¼ max 2ρσ ω2 ; 10 ω ∂xi ∂xi
3.3. SST k-ω turbulence model
The turbulence viscosity can be calculated from the following rela tion,
Wilcox [32] developed a model called the standard k-ω model which is very much sensitive to the inlet free stream turbulence. Menter and Esch [33] transformed the standard k-ε model into the k-ω form using a blending function F1. Hence, the model combines the advantages of the k-ε and the k-ω models with the help of the blending function F1. Near the walls the value of F1 is equal to one and zero for the remaining part of the flow [34]. The model transport equations for the different parameters are given by, � � ∂k � ∂ ∂k ρui ¼ Pk β * ρkω þ ðμ þ σ k μT Þ ∂xi ∂xi ∂xi
ρu i
∂ω ¼ αρS2 ∂xi
�
βρω2 þ
�
∂ ∂ω ðμ þ σω μT Þ þ 2ð1 ∂xi ∂xi
F1 Þρσω2
� 10
νT ¼
a1 k maxða1 ω; SF2 Þ
F2 is another blending function which is defined as, !#2 # "" pffiffi 2 k 500ν F2 ¼ tanh max ; β*ωy y2 ω The formation of turbulence in the stagnation region is prevented by the production limiter which is defined as follows, � � ∂ui ∂ui ∂uj P k ¼ μT þ ∂xj ∂xj ∂xi
1 ∂k ∂ω ω ∂xi ∂xi 8
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
uj
∂v2 ¼ kf ∂xj
v2 þ k ∂xj
∂2 f ∂x2j
1 ðC1 Tt
L2t
ε
f¼
∂
��
�
μT ∂v2 σk ∂xj
μþ
1Þ
� 2 v k
� 2 3
C2
�
Pk k
where kf ¼ ϕ*
ε* þ v2
ε k
Φ* ¼ Pressure Strain
ε* ¼ Dissipation Rate At the walls (Where y is the wall normal coordinate) f ð0Þ ¼
20ν2 v2 εð0Þy4
The ε (0) term in the denominator is not properly defined for the laminar and the transition region as shown by Lien and Kalitzin [36]. Hence they have reformulated the source term (kf v2 kε) in the v2 equation with the suggestion from Mansour et al. [37] as follows, ϕ* ¼
ε
ε
2v2 And ε* ¼ 4v2 k k
Accordingly, the transport equation for f changes as follows, � � ∂2 f 1 v2 2 Pk L2t 2 f ¼ ðC1 1Þ C2 ðC1 6Þ Tt ∂xj k 3 k The above equation for f together with the three equations for k, ε and v2 are known as Lien and Kalitzin [36] modified v2f model. However the asymptotic behaviour of the v2 near the wall remains the same for both the models. The turbulence time scale (Tt ), � � � � k ν 1=2 Tt ¼ max ; 6
Fig. 11. Comparison of numerical local Nusselt number distribution obtained by v2f model – 1 and 2 with the experimental results for S/D ¼ 0.5 (a) h/S ¼ 2 (b) h/S ¼ 6.
The turbulence length scale (Lt), � 3 �1=4 ! k3=2 ν Lt ¼CL max ; Cη
�
Pk ¼ minðPk ; 10β* ρkωÞ
ε
The values of model constants are, β* 0.09
σk1
0.85
σω1
0.5
α1
5/9
σk2
β1 3/40
1
σw2
0.856
α2
0.44
Durbin [35] developed a turbulence model named as v2f which is identical for the attached or the mildly separated boundary layers. It uses a velocity scale v2 for calculating the turbulent viscosity. The anisotropic wall effects are modelled through the elliptic relaxation function ‘f’. The model has four different transport equations for the different parameters namely (i) the turbulence kinetic energy (k) (ii) the turbulence dissipation rate (ε) (iii) the wall normal component of the fluctuating velocity (v2) (iv) Elliptic relaxation function (f). The model transport equations for the different parameters are as follows, �� � � ∂k ∂ μ ∂k u j ¼ Pk ε þ μþ T ∂xj ∂xj σ k ∂xj
∂ε C1ε Pk C2ε ε ∂ ¼ þ ∂xj ∂xj Tt
��
μþ
�
μT ∂ε σ ε ∂xj
ε
The model constants for both the models are as shown in Table 2. It should be noted that the model constants of Durbin [35] are dependent on the wall distance and hence on the yþ and that for the Lien and Kalitzin [36] are independent of the wall distance. Davidson et al. [38] has suggested two modifications in the Lien and Kalitzin [36] model. In the first modification they have set a limit on the source term of v2 as follows, � � � � 1 2k ðC1 1Þ þ C2 Pk v2 source ¼ min kf ; ðC1 6Þv2 Tt 3
β2 0.0828
3.4. v2f turbulence model
uj
ε
ε
With this modification the v2 � 2k/3 remains in the entire flow re gion but in the region where v2 � 2k/3, the turbulence viscosity will become 0.22v2Tt . Hence, a limit is also set on the value of the turbulence viscosity as below, � � k2 νT ¼ min 0:09 ; 0:22v2 Tt
ε
In the second modification they have proposed a method for calcu lating the turbulence viscosity for the wall normal and the parallel di rection as follows,
νT;? ¼ 0:22v2 Tt
�
9
and νT;jj ¼ 0:09kTt :
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 12. Fluid flow pattern obtained by v2f model – 1 and 2 for S/D ¼ 0.5 and ReD ¼ 20,000 (a) v2f Model – 1 at h/S ¼ 2 (b) v2f Model – 2 at h/S ¼ 2 (c) v2f Model – 1 at h/S ¼ 6 (d) v2f Model – 2 at h/S ¼ 6.
4. Numerical method
turbulent intensity and the viscosity ratio are fixed as 5% and 5 for all the cases. The flux of ‘f’ is assumed to be zero with zero gauge pressure. The other terms calculated are as follows,
The numerical simulation is carried out by using OpenFOAM solver. Four different turbulence models SST k-ω, v2f, RNG k-ε and realizable kεare used for modelling the turbulent flow. The SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm is used for the velocity and the pressure coupling. The terms for the turbulence and the con vection are discretised using Gauss upwind scheme in OpenFOAM [41]. The convergence is assumed to be reached when the residuals for all the convective and turbulence terms are less than 10 6. The fluid properties are evaluated at the jet exit temperature. The density, viscosity and thermal conductivity of air are taken as 1.2 kg/m3, 1.7894 � 10 5 kg/ms and 0.0257 W/mK. The under relaxation factors for all the turbulence and the convective terms and the tolerance value of 0.3 and 10 8 are used for the present study. The computational domain used for the closed and open semi-circular confinement is shown in Fig. 2.
3 ρk2 �μt � 1 ρk �μt � 1 2 2 ;ω ¼ ;v ¼ k k ¼ ðUIÞ2 ; ε ¼ Cμ 2 3 μ μ μ μ � Nozzle wall - The nozzle wall having no-slip boundary condition is taken as an adiabatic wall. � Heated Cylinder - The heat flux of constant intensity of 1000 W/m2 is imposed at the target heated cylinder wall with a no-slip boundary condition. � Pressure boundary - The pressure boundary is assumed to be the entrainment boundary where the atmospheric air interacts with the inlet air. The total temperature at the boundary is assumed to be 300 K. � Confinement wall - The confinement wall having no-slip boundary condition is assumed to be adiabatic.
4.1. Numerical boundary condition � Nozzle Inlet - At the nozzle inlet the uniform velocity profile is assumed. The temperature of the air is assumed to be 300 K. The 10
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
taken as 5 percentages for the further numerical study. 5. Selection of the turbulence model As discussed earlier, the numerical analysis is carried out with the four different turbulence models SST k-ω, RNG k-ε, v2f and realizable k-ε. The turbulence models have different characteristics for predicting the fluid flow and the heat transfer characteristics. They perform differently with the same type of the separated flow, rotational flow and the pres sure gradient. Further from the numerical study of Olsson et al. [12] and Zeyi Jiang et al. [27], it is observed that the realizable k-ε and the SST k-ω models work better from the other turbulence models. Hence a turbulence model is selected which predicts the heat transfer charac teristics well among the other models. To study the effects of the non-dimensional distance between the jet exit and the cylinder (h/S) and the Reynolds number (ReD) on the heat transfer characteristics, the numerical and the experimental results of [7] are compared with each other for h/S ¼ 2 and 6, ReD ¼ 4000 and 20, 000 and S/D ¼ 0.5 in Fig. 5. The stagnation point Nusselt number (Nustag.) is over predicted by all the turbulence models. The difference in the stagnation Nusselt number (Nustag.) obtained numerically and experimentally is minimum for the SST k-ω turbulence model. The heat transfer distribution over the cylindrical surface has a maximum value at the stagnation point, then it decreases and after a certain point it in creases till the rear part of the cylinder. The same trend is followed by all the turbulence models. This shows that all the turbulence models cap ture the recirculation zone but with different locations and size as shown in Fig. 5(a)–(d). As shown in Fig. 5(b) and (d), the RNG k-ε model pre dicts the peak Nusselt number at around 40� at h/S ¼ 2 and 6 and ReD ¼ 20,000. The numerical local Nusselt number distribution is compared with the experimental results for S/D ¼ 0.25 and 0.12 and h/S ¼ 2 in Fig. 6. In Fig. 6 (a) and (b), for S/D ¼ 0.25, the Nusselt number at the stagnation point is over predicted for ReD ¼ 4500 and under predicted for ReD ¼ 20,000 by all the turbulence models. As shown in Fig. 6 (c) and (d), for S/D ¼ 0.12 the Nusselt number at the front and the rear stag nation point remains almost equal to the experimental value. Hence, the trend of variation of the local Nusselt number remains unchanged for S/ D ¼ 0.12 as shown in Fig. 6 (c) and (d). For S/D ¼ 0.12, the size as well as the location of the recirculation zone remains almost same for all the turbulence models. However the size and the location of the recircula tion zone is not the same as for S/D ¼ 0.5 as shown in Fig. 6 (a) and (b). Table 5 and 6 show the percentage error in the stagnation Nusselt number (Nustag) and the mean Nusselt number (NumD) for different values of surface curvature (S/D) for h/S ¼ 2. All the turbulence models have less error for stagnation Nusselt number (Nustag) at lower value of surface curvature (S/D). While all the turbulence models except RNG k-ε have less error for the mean Nusselt number (NumD) for the higher value of surface curvature (S/D). Fig. 7 shows the distribution of the non-dimensional pressure coef ficient around the circumferential direction of the cylinder. The exper imental data shows that it is maximum at the stagnation point. Then after it decreases up to around 40� and remains almost constant up to around 125� from the stagnation point. Then after a slight increase in it is observed which shows that a higher pressure exists at the rear part of the cylinder. This is due to the higher value of the dynamic pressure at the rear part of the cylinder. Hence, there exists a region where the velocity of the air is more as compared to the cylindrical flow region. This zone is referred to as the recirculation zone. The SST k-ω and the RNG k-ε turbulence model follow the similar trend while the v2f and realizable k-ε models also capture a secondary minimum at the rear part of the cylinder as shown in Fig. 7. As shown in Fig. 8, the irregular vortices are found which is in opposite direction to that of the velocity of the air in the recirculation zone which is responsible for the reduced pressure in the recirculation zone. From Fig. 8 (a) and (d), it is observed that the irregular vortices in the recirculation zone are captured by v2f
Fig. 13. Comparison of numerical local Nusselt number distribution obtained by v2f model – 1 and 2 with the experimental results for S/D ¼ 0.12 and ReD ¼ 20,000 (a) h/S ¼ 2 (b) h/S ¼ 8.
4.2. Grid independence test A grid independence study is done with the four different grid sizes having 5548, 8265, 12889 and 20578 quadrilateral cells in Grid A, Grid B, Grid C and Grid D. The grid independence test has done with the v2f turbulence model for h/S ¼ 8, ReD ¼ 4500 and S/D ¼ 0.25 as shown in Fig. 3. All the grids near the walls are made such that yþ value remains below unity. The v2f turbulence model is selected for predicting the local Nusselt number distribution over the cylindrical surface. The percent relative error in numerical results for different grids with respect to Grid D is shown in Table 4. From Table 4, it is clear that the maximum relative error is around 11% at stagnation point of Nusselt number (Nustag) and 2% for mean Nusselt number (NumD). Hence, the grid size does not much affect the heat transfer distribution from the cylindrical surface and the Grid B is chosen for the further numerical simulation. 4.3. Selection of the value of the inflow turbulence intensity Gori and Bossi [8] have specified the value of the turbulence in tensity as 5% for ReD ¼ 6000 and 8% for ReD ¼ 20,000. Hence to check the effect of turbulence intensity on the heat transfer distribution, the values of turbulence intensity is taken in different intervals. The tur bulence intensity for the inflow is varied as 0.5, 1, 3 and 5 percentages to study the effect on the local Nusselt number distribution over a cylinder as shown in Fig. 4. As the turbulent intensity increases from 0.5 to 5 percentages, the local Nusselt number distribution remains almost the same over the entire cylindrical surface. Hence, the turbulence intensity has not very much effect for the local Nusselt number distribution over the cylindrical surface. Hence the value of the turbulent intensity is 11
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 14. Fluid flow pattern obtained by v2f model – 1 and 2 for S/D ¼ 0.12 and ReD ¼ 20,000 (a) v2f Model – 1 at h/S ¼ 2 (b) v2f Model – 2 at h/S ¼ 2 (c) v2f Model – 1 at h/S ¼ 8 (d) v2f Model – 2 at h/S ¼ 8.
and realizable k-ε models. To better understand the fluid flow pattern causing the heat transfer from the cylinder, the numerical non dimen sional pressure coefficient is shown in Fig. 9. The fluid flows in the di rection where the difference in the non dimensional pressure coefficient (Cp) is positive. As shown in Fig. 9, the numerical value of the non dimensional pressure coefficient (Cp) is around 1 predicted by all the turbulence models. This shows that all the kinetic energy of the air issuing from the jet is converted to the pressure energy which is responsible for the higher value of the Nusselt number in the stagnation region. A pressure difference exists in the stagnation region and the cylindrical flow region which is responsible for the flow over the cyl inder. Hence in the cylindrical flow region as the flow proceeds the velocity of the air decreases which reduces the heat transfer in the cy lindrical flow region. In the rear part of the cylinder, the positive value of the non dimensional pressure coefficient (Cp) is predicted by all the turbulence models which causes the recirculation of the flow, though the size of the recirculation zone may vary. Hence the higher value of the Nusselt number is observed in the recirculation zone as compared to the
cylindrical flow region. Fig. 10 shows the distribution for the local Nusselt number obtained numerically for h/S ¼ 2, S/D ¼ 0.12, α ¼ 7� and 20� and ReD ¼ 6000 and 20,000. The experimental results of [16] and the numerical results are compared with each other. The stagnation Nusselt number (Nustag.) predicted by all the turbulence models is almost equal to the experi mental value as shown in Fig. 10. Hence all the turbulence models predict almost the similar local Nusselt number distribution. However the value of the local Nusselt number at the rear stagnation is under predicted as compared to that without confinement. 5.1. Comparison of original and modified v2f turbulence models The numerical results obtained by the original and the modified v2f turbulence models suggested by two authors (Table 3) are compared from the experimental results. The local Nusselt number distribution over the cylindrical surface obtained by the v2f model – 1 and 2, at S/ D ¼ 0.5 and ReD ¼ 20,000 is shown in Fig. 11. The numerical and the 12
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 15. Effect of semi-circular confinement bottom opening angle (α) on the local Nusselt number distribution for Reynolds number (ReD) 30,000 and 90,000 at different values of h/S ¼ 4, 6, 8 and 10.
13
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 16. Fluid flow pattern for the different values of the semi-circular confinement bottom opening angle (α) at h/S ¼ 6 and ReD ¼ 90,000 (a) α ¼ 0� (b) α ¼ 40� (c) α ¼ 80� (d) α ¼ 120.
values of h/S ¼ 2 and 8 for both the v2f model – 1 and 2. Hence the value of h/S does not have much effect on the fluid flow and the heat transfer characteristics of the cylinder at lower value of S/D irrespective of the choice of the turbulence model. The absence of the irregular vortices and the reduced size of the recirculation zone are observed as shown in Figs. 12 and 14 for S/D ¼ 0.12 and 0.5. Hence, the main effect of the surface curvature (S/D) observed is the attached flow or the separated flow. For the lower values of S/D the flow remains attached to the cy lindrical surface more as compared to the higher values regardless of the flow region. Hence, the v2f model – 2 works better from the v2f model – 1 for predicting the fluid flow pattern and the heat transfer distribution. The different authors have explained the error between the experi mental data and numerical results. Behnia et al. [39] has suggested that the over prediction of the Nusselt number at the stagnation point is due to the over prediction of the turbulence kinetic energy (k). They observed an increase in the value of the turbulence kinetic energy (k) about 80% from the experimental value and therefore the increase in the Nusselt number at the stagnation point to about 170%. Behnia et al. [40] have explained the reason for the secondary peak in the Nusselt number away from the stagnation point is due to the transition from laminar to turbulent flow. Durbin [35] has suggested that as the stagnation region
experimental results of [16] are compared with each other. As shown in Fig. 11 (a), the local Nusselt number distribution has two secondary peak for the v2f model – 1 at ReD ¼ 20,000 and h/S ¼ 2. However, for the v2f model – 2 only a single peak is observed for the local Nusselt number as shown in Fig. 11 (a). As shown in Fig. 11, the difference in the nu merical and the experimental value of the local Nusselt number at the stagnation point is less with the v2f model - 2 as compared to the v2f model – 1 at h/S ¼ 2 and 6. The flow separation also occurs earlier with the v2f model – 2 as compared to that with the v2f model – 1 as shown in Fig. 12. In Fig. 12 (a) and (b), at h/S ¼ 2, the size of the irregular vortices is more for both the v2f model – 1 and 2. From Fig. 12 (c) and (d), at h/S ¼ 6 the presence of the irregular vortices is not observed for the v2f model – 2 and a decrease in the size of the irregular vortices is observed for the v2f model – 1. The fluid flow pattern causing the heat transfer observed is the stagnation zone, cylindrical flow region and recircula tion zone as shown in Fig. 12. As shown in Fig. 13 for S/D ¼ 0.12 and h/S ¼ 2 and 8, with the decrease in the value of surface curvature (S/D), there is more attachment of the fluid flow to the cylindrical surface. At h/S ¼ 2 and 8, the flow separation from the cylindrical surface occurs at around 135� from the stagnation point. It is clearly seen in Fig. 14, the size of the recirculation zone almost remains the same for the different 14
K.A. Ganatra and D. Singh
International Journal of Thermal Sciences 149 (2020) 106148
Fig. 17. Effect of semi-circular confinement bottom opening angle (α) on mean Nusselt number for Reynolds number (ReD) 30, 000 to 90,000 (a) h/S ¼ 4 (b) h/S ¼ 6 (c) h/S ¼ 8 and (d) h/S ¼ 10.
Table 7 The percantage increase in the mean Nusselt number, α ¼ 60� (reference value for the mean Nusselt number is taken for closed semi circular confinement case). h/S 4 6 8 10
Table 9 The percantage increase in the mean Nusselt number, α ¼ 80� (reference value for the mean Nusselt number is taken for closed semi circular confinement case).
Reynolds Number (ReD)
h/S
30,000
50,000
70,000
90,000
36.69 48.59 35.11 41.14
40.08 53.60 41.78 42.72
44.76 57.09 43.13 44.94
48.17 60.20 45.67 47.85
4 6 8 10
Table 8 The percantage increase in the mean Nusselt number, α ¼ 70� (reference value for the mean Nusselt number is taken for closed semi circular confinement case). h/S 4 6 8 10
h/S
50,000
70,000
90,000
37.13 43.80 35.11 45.49
39.11 46.89 41.33 45.33
43.60 49.43 42.66 46.29
46.73 51.63 45.31 49.22
30,000
50,000
70,000
90,000
35.69 45.85 35.11 40.17
38.97 49.62 42.56 41.81
43.86 52.27 43.79 44.01
46.85 54.48 46.45 46.67
Table 10 The percantage increase in the mean Nusselt number, α ¼ 90� (reference value for the mean Nusselt number is taken for closed semi circular confinement case).
Reynolds Number (ReD) 30,000
Reynolds Number (ReD)
4 6 8 10
is approached, the turbulence time scale becomes very large by which the turbulence kinetic energy (k) is over predicted by any of the two equation turbulence model. The numerical results obtained from the various turbulence models SST k-ω, v2f, realizable k-ε and RNG k-ε are compared from the experi mental results. Also the numerical results of the v2f model – 1 and 2 are
Reynolds Number (ReD) 30,000
50,000
70,000
90,000
36.54 45.37 34.86 57.23
39.76 48.91 47.14 54.08
44.69 51.41 47.46 53.81
47.76 53.53 46.19 55.28
compared with the experimental results. From Figs. 5–14, it can be concluded that the SST k-ω model predicts the fluid flow pattern and the heat transfer distribution better for an impinging air slot jet over a cir cular cylinder. Hence, the further parametric investigation of the jet and the cylinder configuration with a semi-circular bottom opening confinement is done with the SST k-ω model. 15
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International Journal of Thermal Sciences 149 (2020) 106148
6.2. Mean Nusselt number
Table 11 The percantage increase in the mean Nusselt number, α ¼ 100� (reference value for the mean Nusselt number is taken for closed semi circular confinement case). h/S 4 6 8 10
Fig. 17 (a) shows the values for the mean Nusselt number (NumD) for different values of the bottom opening angle (α) at h/S ¼ 4 and ReD ¼ 30,000, 50,000, 70,000 and 90,000. The mean Nusselt number (NumD) is minimum for a closed semi-circular bottom opening confine ment. With the increase in the bottom opening angle (α), the mean Nusselt number (NumD) increases and after reaching the maximum value, it starts decreasing. A similar trend of variation is found at different values of the Reynolds number (ReD) considered. The value of the mean Nusselt number (NumD) is found maximum for α ¼ 70� at different values for the Reynolds number (ReD). For all the values of the bottom opening angle (α), the mean Nusselt number (NumD) increases with increasing the value of the Reynolds number (ReD). However with increasing Reynolds number (ReD), no change in the value of the bottom opening angle (α) is observed at which the mean Nusselt number is maximum. At ReD ¼ 30,000 and 90,000, around 25% and 30% increase for maximum value of the mean Nusselt number (NumD) is observed from the value for a closed semi-circular confinement. Fig. 17 (b)–(d) show mean Nusselt number (NumD) for different values of the bottom opening angle (α) at ReD ¼ 30,000, 50,000, 70,000 and 90,000 and at h/S ¼ 6, 8 and 10. The observed trend of variation for the mean value of the Nusselt number (NumD) is much similar for h/ S ¼ 6, 8 and 10 as for h/S ¼ 4. For the different values of h/S, the value of the mean Nusselt number (NumD) remains almost equal. Hence, the mean Nusselt number (NumD) is not much affected by the nondimensional distance between the slot jet exit and the cylinder (h/S). The mean Nusselt number (NumD) is maximum for α ¼ 60� , 100� and 90� at h/S ¼ 6, 8 and 10. Tables 7–11 shows the percentage increase in the mean Nusselt number as compared to the value for the closed confine ment at different values of h/S and ReD. The percentage increase for the mean Nusselt number (NumD) is maximum at h/S ¼ 6 and ReD ¼ 90,000 for the values of α ¼ 60� , 70� , 80� and 100� except α ¼ 90� . The per centage increase in the mean Nusselt number ranges from 35 to 57% as compared to that for the closed semi-circular confinement.
Reynolds Number (ReD) 30,000
50,000
70,000
90,000
35.36 43.26 44.26 43.85
37.95 47.01 44.95 44.66
42.80 49.85 46.18 46.33
45.91 52.08 49.14 49.31
6. Effect of the semi-circular confinement bottom opening angle (α) 6.1. Local Nusselt number For an air slot jet, the stagnation point Nusselt number (Nustag.) is of primary concern due to the higher heat removal rate. The local Nusselt number increases with the increase in the Reynolds number (ReD) and with the decrease in the non-dimensional distance between the slot jet exit and the cylinder (h/S). The stagnation point Nusselt number (Nus tag.) follows the similar trend of variation. The numerical results ob tained agree very well with the literature available for the cylindrical surface [7,8]. In Fig. 15 (a) for a closed confinement the separation point predicted is at around 75� from the stagnation point and the recirculation zone starts from there and extends up to the rear stagnation point. The same flow pattern is observed by Olsson et al. [12] for an impinging jet over a cylinder placed on a flat plate. For α ¼ 40� , the local Nusselt number is minimum nearly at 160� and then after it increases up to the rear part of the cylinder. The separation point predicted numerically is at around 160� and the recirculation zone spreads around 40� over the entire cy lindrical surface. The size of the recirculation zone is almost equal to the bottom opening angle (α). The same trend is observed for the value of α ¼ 80� and the flow separation observed is at around 150� . But with the more increase in the bottom opening angle α ¼ 120� , the separation point predicted is at around 1400 with the reduced size of the recircu lation zone. The size of the recirculation zone does not match with the value of the bottom opening angle (α) as observed for the values of α ¼ 40� and 80� . This is because of the higher incoming velocity of the air in the cylindrical flow region. Fig. 15 (b) shows the local Nusselt number distribution at higher value of the Reynolds number, ReD ¼ 90,000 for h/S ¼ 4 and S/D ¼ 0.1. For the values of α ¼ 0� and 40� , the separation point predicted coincides with the values as predicted with the low Reynolds number (ReD). For the value of α ¼ 80� and 120� , the location of the separation point is at around 150� . This shows that the location of the separation point at higher Reynolds number shifts towards the higher angular location from the stagnation point. This agrees very well with the experimental results of Gori and Bossi [7] for an impinging air slot jet over a circular cylinder. With the increase in the value of Reynolds number (ReD), the size of the recirculation zone decreases. The reason for this is the higher incoming velocity of the air in the cylindrical flow region at higher Reynolds number (ReD) as pointed out earlier. As shown in Fig.15 (c)–(h), when h/S changes from 6 to 8 and 10, the local Nusselt number distribution is not much affected for the values of α ¼ 0� , 40� , 80� and 120� . The only change observed is for h/S ¼ 8 and α ¼ 80� . The numerical local Nusselt number is found minimum at around θ ¼ 160� and 150� at ReD ¼ 30,000 and 90,000. Fig. 16 (a) to (d) show the fluid flow pattern for the h/S ¼ 6, ReD ¼ 90,000 and α ¼ 0� , 40� , 80� and 120� . For a closed confinement the separation point is at around 80� from the stagnation point and beyond that the recirculation starts and extends up to the rear part of the cylinder. With the increase in the bottom opening angle (α), the size of the recirculation zone also increases for semi-circular bottom opening confinement.
7. Conclusion Present study shows the numerical analysis carried out by four various turbulence models SST k-ω, modified v2f, realizable k-ε and RNG k-ε. The local Nusselt number distribution is obtained from the original and the modified v2f model over the cylindrical surface numerically. By comparing the numerical results from various turbulence model, a parametric investigation is also done by the SST k-ω turbulence model for the study of the effect of the non-dimensional distance between the cylinder and the jet exit (h/S), Reynolds number (ReD) and the semicircular confinement bottom opening angle (α) on the local as well as the mean Nusselt number. The conclusions made from the detailed study are as follows, 1. As compared to the other models, the SST k-ω and the RNG k-ε predict the better fluid flow characteristics. 2. As compared to the other models, the SST k-ω predicts the better heat transfer characteristics. 3. The modified v2f turbulence model works better from the original v2f turbulence model for predicting the heat transfer characteristics for a circular cylinder. 4. At the rear part of the cylinder, the size of the recirculation zone increases with the increase in the bottom opening angle (α) and the decrease in the Reynolds number (ReD). 5. The value of the h/S does not have much effect on the local as well as mean Nusselt number. 6. As compared to the value for the closed confinement, the maximum increase in the value of the mean Nusselt number (NumD) is at h/ S ¼ 6 for all the values of ReD.
16
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International Journal of Thermal Sciences 149 (2020) 106148
Acknowledgements
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Nomenclature D: Diameter of the heated circular cylinder (m) DC: Diameter of the semi-circular confinement (m) f: Elliptic relaxation function (1/s) h: Distance between the nozzle exit and the cylinder (m) k: Turbulent kinetic energy (m2/s2) kf: Thermal conductivity of the air (W/mK) � � Nuθ: Local Nusselt number, Nuθ ¼ q00 D� kf ðTW Tjet Þ � � Z π Nuθ dθ NumD: Mean Nusselt number NumD ¼ 1π 0
Pr: Prandtl number PrT: Turbulence Prandtl number q0 0 : Heat flux (W/m2) RC: Radius of the semi-circular confinement (m) ReD: Reynolds number based heated cylinder diameter, (ReD ¼ ρUjet D=μ ) ReS: Reynolds number based nozzle slot width, (ReS ¼ ρUjet S=μ ) S: Slot width of the jet (m) TW: Cylinder wall temperature (K) Tjet: Jet temperature or nozzle exit temperature (K) Ujet: Magnitude of the velocity of the air at the nozzle exit (m/s) U: Local velocity of air (m/s) v2 : Velocity fluctuations normal to the streamlines (m2/s2) X: – Cartesian co-ordinate in x direction (m) Y: Cartesian co-ordinate in y direction (m) yþ: Non-dimensional wall distanceGreek ε: Turbulent dissipation rate (m2/s3) ω: Turbulence specific dissipation rate (1/s) μ: Dynamic viscosity of the air (kg/ms) ρ: Density of the air (kg/m3) ν: Kinematic viscosity of the air (m2/s) α: Semi-circular confinement bottom opening angle (o)Suffix j: Jet t: Turbulence w: Wall C: Semi-circular confinement
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