1

Numerical Simulation of Boundary Layer Transition for Turbine Blade

Heat Transfer Prediction

A. HARIZI

Mechanical Engineering Department, Science and Technology Faculty, University of Batna, Algeria

a.harizi@univ-oeb.dz

A. GAHMOUSSE

Energetic and Turbomachinery Laboratory LET, University of Tebessa, Algeria

E-A. MAHFOUDI

Mechanical Engineering Department, Science and Technology Faculty, University of Constantine, Algeria

A. MAMERI

Mechanical Engineering Department, Science and Technology Faculty, University of Oum El Boughi, Algeria

ABSTRACT

This paper deals with an external heat transfer numerical simulation for a two dimensional

transonic turbine blade cascade. We focused on the prediction of the laminar-turbulent

boundary layer transition which can have an important effect on the distribution of the heat

transfer around the turbine blade surface. The Reynolds Averaged Navier-Stokes equations

(RANS) with the correlation-based transitional model developed by Menter and later

modified by Langtry are solved. Comparisons with measurements for a high loaded transonic

turbine blade, experimentally studied at the von Karman Institute (VKI) test facility, show

good agreement especially for the prediction of the transition onset for the all test cases

considered. One of the major contributions of this paper is the implementation and evaluation

of a set of new empirical correlations published recently in the literature. The results show

that all correlations tested predict correctly the boundary layer transition onset with a relative

difference for the heat flux computed in the fully turbulent region.

KEY WORDS : Turbine blade, Boundary layer, Heat transfer, Transition, Turbulence

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NOMENCLATURE

C Blade chord

h Convective heat transfer coefficient

k Turbulent kinetic energy

M Mach number

P Static pressure

eR θ Momentum thickness Reynolds number

e tR θ Transition onset momentum thickness Reynolds number

Re tθɶ Local transition onset momentum thickness Reynolds number

RT Viscosity ratio

Rev Vorticity Reynolds number

S Absolute value of strain rate, blade arc length

T Static temperature

Tu Turbulence intensity

U Local velocity

y Distance to nearest wall

y+ Distance in wall coordinates

Greek symbols

δ Boundary layer thickness

γ Intermittency

λθ Pressure gradient parameter

µ Molecular viscosity

µt Eddy viscosity

ω Specific turbulence dissipation rate

ρ Density

3

θ Momentum thickness

Subscripts

t Transition onset or total condition

w Wall values

0 Free-Stream values

1 Inlet values

2 Exit values

1. INTRODUCTION

External turbine blade heat transfer is very complex and is affected by several mechanisms.

Different regimes can be identified as follows: stagnation region, laminar, transitional and

turbulent boundary layer, shock boundary layer interaction and separation with or without

reattachment. In addition, heat transfer is influenced by flow parameters such as Mach

number, Reynolds number, free-stream turbulence, pressure gradient and gas properties. Heat

transfer is particularly very sensitive to boundary layer transition which has a major effect on

the rate of heat transfer (Lakshminarayana, 1999).

Numerical simulation of the flow influenced by the phenomenon of laminar/turbulent

boundary layer transition is very important, particularly for the gas turbine configurations

where the Reynolds number, based on the blade chord, can be relatively low (of the order

106) so a large part of the flow along the turbine blades is often laminar or transitional.

Evaluation of the gas turbine blade heat transfer distribution depends strongly on the correct

prediction of the transition phenomenon. Indeed, the exact prediction of the transition zone

makes it possible to consider a better turbine blade protection by optimization of the cooling

methods in the turbulent regions where wall heat flux reaches its highest values. Direct

numerical simulation (DNS) and large eddy simulation (LES) methods remain the best way

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for the prediction of transition in the boundary layers, but they are difficult to implement and

require great computing effort. In addition, high Reynolds number models (HRN) present

serious difficulties in the prediction of the transition, they tend to overestimate the heat flux

in the normally laminar zone. On the other hand, low Reynolds number models (LRN),

although they allow obtaining good performances when implemented with a suitable

damping function to allow them correctly predict boundary layer transition, they present a

difficulty to accurately estimate heat transfer in the stagnation regions and are generally

suitable for predicting only bypass transition.

In this work, we try to test the transitional SST model developed by Menter et al.(2006a)

and later modified by Langtry (2006). This model introduces the concept of the intermittency

(γ) described firstly by Emmons (1951). There exists in the literature several versions of

turbulence models which use the intermittency factor, but the basic of all of them follows

Abu Ghannam (1980) which was later refined by Mayle (1991).

This model, also called γ-Reθ model, is a highly generalized turbulence model built strictly

on local variables and compatible with modern CFD codes where unstructured grids and

parallel computing are generally integrated (Langtry and Menter, 2009). The model

introduces two new transport equations, one for the intermittency (γ) whose role is to control

the transition process and a second equation for the local transition onset momentum

thickness Reynolds number (Re tθɶ ). We have to couple these two transport equations with the

traditional Menter’s k-ω SST model, thus we will have four transport equations to solve for

this turbulence model.

The experimental test validation considered in the present work is the highly loaded

transonic linear turbine guide vane. This 2D turbine blade cascade was extensively studied by

Arts et al.(1990) at the von Karman Institute (VKI) using the isentropic light piston

compression tube facility (CT-2). Detailed information about this experimental apparatus can

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be found in Consigny et al.(1982). Arts et al.(1990) were able to get through their

experimental study a rich database covering wide range of test conditions.

All simulations presented in this work are performed using the Ansys-Fluent code which

its latest versions allow the use of the γ-Reθ model.

2. TURBULENCE MODELING

The γ-Reθ turbulence model is based on two transport equations, one for the intermittency (γ)

introduced to control the transition process and a second equation for the local transition

Reynolds number based on the boundary layer momentum thickness (Re tθɶ ), the first equation

is given as:

( ) ( ) ( )j

tj j j

UP E

t x x xγ γ

ρ γργ γµ µ∂ ∂ ∂ ∂+ = − + + ∂ ∂ ∂ ∂

(1)

The production of the intermittency and the destruction/relaminarization term are defined

by:

[ ] ( )0.52 1length onsetP F S Fγ ρ γ γ= − (2)

( )0.06 50 1turbE Fγ ρ γ γ= Ω − (3)

(Flength) in Eq. (2) is an empirical correlation which controls the length of the transition

zone and (Reθc) is the critical Reynolds number which gives the location where the

intermittency first starts to increase in the boundary layer. These two parameters are functions

of the local transition Reynolds number (Re tθɶ ). The boundary condition at the wall for (γ) is

a zero flux, at the entry (γ) is set equal to 1.

(Fonset) in Eq. (2) triggers the intermittency production and thus the transition onset, it is a

function of the vorticity Reynolds number (Rev), the critical transition Reynolds number

(Reθc) and the viscosity ratio (RT) as described below:

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2

Rev

y Sρµ

= (4)

1

Re

2.193Rev

onsetc

Fθ

= (5)

( )42 1 1min max , ,2.0onset onset onsetF F F = (6)

T

kR

ρµω

= (7)

3

3 max 1 ,02.5

Tonset

RF

= −

(8)

( )2 3max ,0onset onset onsetF F F= − (9)

4

4TR

turbF e − = (10)

In Eq. (3), (Fturb) is used to disable the destruction/relaminarisation term outside of the

laminar boundary layer.

The transport equation for the local transition momentum thickness Reynolds number is

given as:

( ) ( ) ( )Re Re Re

2t j t t

t tj j j

UP

t x x x

θ θ θθ

ρ ρµ µ

∂ ∂ ∂∂+ = + + ∂ ∂ ∂ ∂

ɶ ɶ ɶ

(11)

The source term in this equation is defined by:

( )( )0.03 Re Re 1t t t tP Fθ θ θ θρτ

= − −ɶ (12)

With τ a time scale introduced for dimensional reasons and defined as:

2

500

U

µτρ

= (13)

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(Fθt) in Eq. (12) is a blending function introduced to turn off the source term in the

boundary layer allowing the diffusion of the transported scalar (Reθt) from the free-stream.

This function is equal to zero in the free-stream zone and to1 inside the boundary layer.

The boundary condition at the wall for (Re tθɶ ) is a zero flux, at the inlet (Re tθ

ɶ ) should be

calculated from an empirical correlation based on the inlet turbulence intensity, this

correlation is defined and published by Menter et al.(2006a) and recently modified by

Langtry (2006) to improve the prediction of natural transition (Tu0<1), the new correlation is

defined by the following relations:

( )0 20

0.2196Re 1173.5 589.428t u

u

T FTθ θλ

= − +

For Tu0 ≤1.3 (14)

[ ] ( )0.671

0Re 331.5 0.5658t uT Fθ θλ−= − For Tu0 >1.3 (15)

( )1.5

0

2 3 1.51 12.986 123.66 405.689uT

F eθ θ θ θλ λ λ λ − = − − − − For λθ ≤0 (16)

( )0

35 0.51 0.275 1uT

F e eθλθλ

− − = + − For λθ >0 (17)

With (λθ ) a pressure gradient parameter defined as:

2 dU

dSθρθλµ

= (18)

To close the problem, the model requires the definition of empirical correlations

for (Flength) and (Reθc), these two correlations are published recently by Langtry and Menter

(2009) and are defined as:

1 4 6 2

2 5 2 8 3

398.189.10 119.27.10 132.567.10 400

263.404 123.939.10 194.548.10 101.695.10 400 596

0.5 3.10

t t t

t t t tlength

Re Re Re

Re Re Re ReF

θ θ θ

θ θ θ θ

− − −

− − −

−

− − <

− + − ≤ < =

−

ɶ ɶ ɶ

ɶ ɶ ɶ ɶ

( )[ ]

4 596 596 1200

0.3188

t tRe Re

θ θ − ≤ <

ɶ ɶ

1200 t Reθ

≤

ɶ

(19)

8

( ) ( )( ) ( )

( )( )

2 4 6 2

9 3 12 4

396.035.10 120.656.10 868.23.101870

696.506.10 174.105.10Re

593.11 0.482 1870

t t

t t

t tc

t t

Re ReRe Re

Re Re

Re Re

θ θθ θ

θ θθ

θ θ

− − −

− −

+ − + − ≤

+ − +=

− + −

ɶ ɶ

ɶ ɶ

ɶ ɶ

ɶ ɶ 1870t Re θ

>

ɶ

(20)

In the present work we propose to use and test other correlations suggested in the

literature. This is made possible by means of the User Defined Function (UDF) feature of the

code used.

2.1 Correlations of Malan

According to Malan et al.(2009) these correlations are defined as below :

( )Re min 0.625Re 62,Rec t tθ θ θ= +ɶ ɶ (21)

( )( )min 0.01exp 12 0.022Re 0.57,300length tF θ= − +ɶ (22)

Malan has implemented these two correlations in the commercial code STAR-CCM+ and

tested them successfully for a variety of external and internal flows.

2.2 Correlation of Sorensen

The correlations proposed by Sorensen (2009) are defined as below:

( )Re 12000 7 Re 100Re 1

25 10t t

cθ θ

θ β β + += + −

ɶ ɶ

(23)

With 4

Re 100tanh

400tθβ

− =

ɶ

(24)

1.2Re

min 150exp 0.1,30120

tlengthF θ

= − +

ɶ

(25)

Sorensen has applied these two correlations to external 3D flows and has achieved good

results in comparison with fully turbulent simulations.

This transitional model, thus defined, must be coupled with the classical k-ω SST model

by multiplying both the production and destruction terms in the k-equation of k-ω SST model

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with the effective intermittency, more details about this equation can be found in Menter et

al.(2006a). So finally, we have four transport equations to solve for this turbulence model.

3. TEST CASES AND NUMERICAL METHOD

The configuration considered in this study is the so called VKI LS-89 turbine guide vane, the

shape of this blade was optimized for an exit Mach number equal to 0.9. The choice of this

configuration is justified by the availability of the experimental results. It represents a good

test case and was the subject of several tests validations for numerical simulations results for

many research groups.

The most important geometrical characteristics of this cascade are the blade chord 67.647

[mm], the cascade pitch 57.5 [mm], the blade stagger angle 55° and an inflow angle of zero

degree. The blade coordinates are given in Arts et al. (1990).

In order to facilitate the calculation, the computational domain is composed only of one

blade with imposed periodic boundary conditions. The grid generated is an unstructured grid

made up of quadrilateral mesh around the blade (in the boundary layer) and quadri-triangular

in the rest of the domain. The total size of the grid generated is 93232 nodes with 50 cells

placed in the boundary layer with a wall normal expansion ratio of 1.12. Figure 1 shows the

grid generated in the entire domain.

We considered six test cases in this paper, two cases for the pressure field at two flow

regimes, subsonic (Mis,2=0.875) and transonic (Mis,2=1.02). For the wall heat flux distribution,

we examined four test cases corresponding to a transonic condition (Mis,2≈1.1) for two values

of the inlet free-stream turbulence intensity (1% and 6%) and an exit Reynolds number

ranging from 106 to 2.106. We summarized data for all cases in Table1.

The steady compressible RANS equations are solved using the density-based solver of the

Ansys-Fluent code. The convective fluxes are evaluated with a standard Roe flux-difference

splitting method with an implicit pseudo-time integration scheme. The new steering option of

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the code used, allows automatic variation of the CFL number during the calculation process

for better numerical stability and convergence of the solution. Second order upwind scheme

with a differentiable limiter is used for the spatial discretization of both viscous and

turbulence terms. Simulations are performed with a double precision number, and

convergence criterion associated with the iterative solver is satisfied when values of the

unscaled residuals of all equations become under 10-5. For all cases considered the

simulations started from a preliminary initial solution interpolated on the current grid.

Boundary conditions are applied to satisfy the experimental conditions so at the outlet, a

static pressure is applied. The inlet boundary conditions reproduce the experimental test

conditions with a particular effort for the estimation of inlet turbulence parameters. In all

cases, blade surface is treated as an isothermal wall.

To be able to correctly capture the laminar-turbulent transition which occurs in the

boundary layer, particularly on the blade suction side, the grid used must have an y+ of about

1. If y+ is too large (y+> 5), then the transition onset location moves upstream with

increasing y+ (Langtry, 2006). Figure 2 shows the evolution of y+ around the turbine blade

surface for the extreme case with Re2=2.106 and Tu0=6%. We note that for the entire blade

surface, y+ remain under the value of 1, allowing the viscous sub-layer to be correctly

treated. Note that negative values of the reduced coordinate(S/C) describe the pressure side

surface of the blade while positive values are for the blade suction side.

4. RESULTS AND DISCUSSION

4.1 Pressure Distribution

In this section we try to discuss the capacity of the numerical approach adopted to reproduce

the distribution of the wall pressure on both blade suction and pressure sides, for the two test

cases chosen. This task is necessary before performing heat transfer calculation.

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Figure 3 shows the distribution of the wall static pressure around the blade according to

the reduced coordinate S/C. We can particularly observe a good agreement of the numerical

results with the measurement on the pressure side where it is noted that the pressure

continues to decrease from the leading edge to the trailing edge without any abrupt variation

except at the trailing edge. On the suction side, a light variation in the results is observed on a

small plateau (S/C= 0.5 to 0.95), in this region the simulation overestimates slightly the wall

pressure.

For the case at high exit Mach number Mis,2=1.02 ( see Figure 4), the flow on the suction

side is accelerated until (S/C=0.95), which results in a fall of the wall pressure, then we

observe the appearance of a small shock wave which causes a deceleration of the flow

resulting in an increase in the pressure. In addition, the pressure on the pressure side

decreases gradually from the leading edge until the vicinity of the trailing edge. The position

of the shock wave is captured correctly by the model and coincides well with that

experimental (S/C≈ 1.05) as illustrated in Figure 4.

For the two cases examined, the agreement between the RANS simulations and the

experiments is excellent over most of the blade surface. The small discrepancy between

numerical results and measurement observed on the rear part of the suction side surface could

be attributed to the uncertainty in the measurement (Bhaskaran, 2010). Arts et al.(1990)

reported the uncertainty on the measurements of pressure ( ± 0.5 %), temperature (± 1.5%),

heat transfer coefficient (± 5 %), loss coefficient (± 0.2 %) and exit flow angle (± 0.5 %) but

they did not indicate any information about the uncertainty on the flow angle of attack. This

last parameter may have an important effect on the results (Michelassi et al, 2002). We can

also mention the effects of end walls flow and three dimensional character of the flow which

cannot be considered with a two dimensional simulation.

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4.2 Heat Transfer Calculation

The convective heat transfer coefficient (h) studied in the present work, is defined as the ratio

of the calculated wall heat flux and the difference between the total free-stream and the local

wall temperatures:

1

w

t w

qh

T T=

− (26)

Since there are cases at low free-stream turbulence intensity Tu0=1%, it is logical to

consider in such cases the flow as laminar. The distribution of the convective heat transfer

coefficient obtained with a laminar simulation is plotted in Figure 5. On the pressure side

there is no notable difference between numerical and experimental results indicating clearly

the laminar state of the boundary layer. On the suction side the numerical results agree well

with measurement until (S/C = 0.6), then results from laminar simulation become much

smaller than those experimental due to the boundary layer transition which occurs in this

zone. It is evident from this figure that the laminar part of the boundary layer includes the

stagnation region, the entire pressure side and the favorable pressure gradient zone on the

suction side, indicating clearly that the boundary layer may be laminar, transitional or

turbulent on different parts of the turbine blade surface making the prediction of the wall heat

transfer by RANS methods an interesting challenge.

It is important before performing turbulent simulation to estimate correctly the turbulence

parameters at the entry particularly the turbulence intensity, because its decay from the inlet

can have an important effect on the heat transfer calculation since it affects directly the value

of the viscosity ratio to be defined at the inlet. If this value is very high the skin friction and

heat transfer coefficient can deviate significantly from their laminar values. Langtry

suggested, since it is possible, to use small values for the viscosity ratio close to 10.

The results obtained for moderate Reynolds number Re2=106 are shown in Figure 6. For

the two levels of the free-stream turbulence we can clearly observe in the stagnation region a

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high level of heat transfer due essentially to the fact that, in this region, boundary layer is

very thin with small velocity values. On the suction side for the case of Tu0=1% and as the

laminar boundary layer grows, heat transfer decreases reaching its lowest values near the

trailing edge where a rapid increase in the heat transfer rate is observed due to the existence

of an exit shock wave. For this case numerical results are in good agreement with the

measurement except near the trailing edge. On the pressure side, no transition is observed

denoting the laminar nature of the boundary layer. In this region numerical results agree well

with those experimental. For the second case, at high free-stream turbulence intensity, we

observe for the suction side part that numerical simulation under predicts slightly the heat

transfer in the laminar region, the difference is about 10% maximum. Transition occurs at

about S/C=0.6 and is captured correctly by the model. In the fully turbulent zone, numerical

results deviate from those of experiment.

Figure 7 illustrates the distribution of the convective heat transfer coefficient for the

extreme cases with high exit Reynolds number Re2 =2.106. For the two values of the inlet

turbulence intensity, the model correctly predicts the transition point on the suction side

S/C=0.6 with very good agreement of the results in the laminar zone. In case of high

turbulence intensity Tu0=6% and due to the strong acceleration on the suction side, a shock

wave appears followed by a deceleration which causes a boundary layer transition. This

phenomenon is not observed for the case with low turbulence intensity Tu0=1%, where the

boundary layer crosses the deceleration section and is maintained in its laminar state. For the

case with low turbulence intensity, we observe that on the pressure side the model accurately

reproduces the distribution of the heat transfer coefficient but for the case with high free-

stream turbulence intensity we observe on the pressure side a significant variation compared

to the experimental results on a large part of the pressure side, while approaching the trailing

edge the results are in good agreement. This difference can be explained by the high value of

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the viscosity ratio which we have to specify at the entry to take into account the inlet

turbulence intensity decay. Langtry and Menter (2009) studied the cases with highest exit

Reynolds number using the same turbulent model implemented in the CFX code. From its

results we found some discrepancy in comparison with our results, this fact can be attributed

to the choice of the inlet turbulence parameters.

In addition, we notice for cases with Tu0=6%, in the fully turbulent zone, that the

simulation results deviate from those experimental, at this stage of the study we can be

tempted to explain this difference by the effect of the local Mach number Ma on the transition

and on the heat transfer distribution in this region. This is not, unfortunately, taken into

account in the original version of the transitional SST model (Langtry, 2006). Some authors

suggested to introduce the local Mach number into the correlation of Reθt in the following

way:

( ) ( )Re Ret t aoriginalg Mθ θ= (27)

According to this idea, Zhang and Gao. (2010, 2011) introduced a modification in the Reθt

correlation using the relation defined as:

( ) 3 20.00987 0.144407 0.75109 1a a a ag M M M M= − + + For Ma ∈ [0,12] (28)

Figure 8 shows the heat transfer distribution on the suction side for the case of Re2=2.106

at Tu0=6% obtained by original and modified versions of Reθt correlation. It is evident from

this figure that this modification has not any effect in the laminar and fully turbulent regions.

The only difference in the results is observed in the transition zone where the modified Reθt

correlation gives a slight improvement in the results. This modification can particularly be

interesting for flows at important Mach number (supersonic regime for example).

A comparison of the results for the heat transfer coefficient obtained by the three empirical

correlations is shown in Figure 9. We observe that on the blade pressure side, the two new

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correlations tested give about the same results as that original of Menter et al. (2006a). The

numerical results remain slightly higher in comparison with measurement for the same reason

developed in the precedent section. On the other hand, on the suction side we observe that the

three correlations predict the wall heat transfer differently. Thus, in the laminar zone (until

S/C=0.4) the results are almost identical and are in good agreement with the experiment.

Then, we observe on a small plateau (S/C=0.4 to S/C=0.6) that the original correlation

predicts best the convective heat transfer coefficient compared to the two others which give

lower values. It is necessary to note that in all cases the transition onset is correctly captured

and coincides with that given in experiment. In the fully turbulent zone, the values of the

coefficient (h) closest to those given by measurement are obtained by the correlation of

Malan on a small region between (S/C =0.7 to S/C=1.0), in this zone, Malan’s correlation

gives a relative improvement of the numerical results.

In order to investigate how the three correlations act on the intermittency γ responsible on

the activation of the kinetic energy production inside the boundary layer, we present in Figure

10 the distribution of the turbulent kinetic energy at different positions on the blade suction

side. At S/C=0.6 the difference between results obtained by the three correlations is

appreciable, at this position corresponding to the start of transition, Menter’s correlation gives

a much higher values for the turbulent kinetic energy in comparison with the two other

correlations, denoting the rapid increase of the intermittency from its lowest value close to

zero to its highest value close to 1 resulting in a short transition length. This difference in the

results decreases significantly when moving towards the trailing edge in the fully turbulent

region (S/C=1) which explains the convergence of the heat transfer coefficient values

predicted by the three correlations.

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SUMMURY AND CONCLUSIONS

To investigate heat transfer over a transonic turbine blade, a correlation-based turbulence

model is used and results are compared with available experimental data. The numerical

results are in general in good agreement with measurement except in the fully turbulent

region where we found that the present simulation overestimates the heat flux in this region.

The comparison of the results obtained with two other empirical correlations published in the

literature shows a difference in the heat flux computed especially in the transition region.

This fact proves that this kind of models is intimately related to the empirical correlations. By

introducing a modification in the correlation of (Reθt) we could obtain a relative improvement

of the results in a small region in the suction side transition zone, demonstrating that further

improvement of this modification can give more accurate results particularly for important

Mach number regimes. This paper demonstrates that the γ-Reθ model is very suitable for

predicting flows over high pressure turbine blades where transition occurs, particularly, when

associated with a robust transition criterion.

ACKNOWLEDGMENTS

The authors express their thanks to Professor K. Talbi from the University of Constantine,

who gave them the opportunity to conduct this research using his laboratory’s computing

resources.

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19

TABLE. 1: Test Cases Data

Cases Pt [N/m2]

PPPP1 [N/m2]

Tt

[°k] P2

[N/m2] Re2 Tu0

[%] Mis,2 Tw

[°k]

1 147500 145200 416 89583 106 0.8 0.875 300

2 160500 158000 416 82820 106 0.8 1.02 300

3 168900 166300 414.6 81500 106 1 1.07 297.35

4 167300 164600 413.2 82200 106 6 1.06 298.45

5 326900 321300 418.9 155000 2.106 1 1.09 297.55

6 325700 320700 416.4 154700 2.106 6 1.08 299.75

20

Figures captions

Fig. 1. Computational grid

Fig. 2. Evolution of y+ around the blade, case Re2=2.106

Fig. 3. Pressure distribution, case Mis,2=0.875

Fig. 4. Pressure distribution, case Mis,2=1.02

Fig. 5. Convective heat transfer coefficient, laminar simulation, Re2=2.106

Fig. 6. Convective heat transfer coefficient, Re2=106

Fig. 7. Convective heat transfer coefficient, Re2=2.106

Fig. 8. Convective heat transfer coefficient obtained by original and modified correlations,

Re2=2.106

Fig. 9. Convective heat transfer coefficient obtained by three different correlations, Re2=2.106

and Tu0=6%

Fig. 10. Turbulent kinetic energy at different positions, Re2=2.106 and Tu0=6%

21

FIGURE 1.

FIGURE 2.

22

FIGURE 3.

FIGURE 4.

FIGURE 5.

23

FIGURE 6.

FIGURE 7.

FIGURE 8.

24

FIGURE 9.

FIGURE 10.