Improved turbulence modeling of film cooling flow and heat ...

Improved turbulence modeling of film cooling flow and heat transfer

R. Jones,1,2 S. Acharya1 & A. Harvey2

1 Mechanical Engineering Department, Louisiana State University, USA 2 Dow Chemical Company, Plaquemine, LA 70765,USA

Abstract

Film-cooling flows are characterized by a row of jets injected at an angle from the blade surface or endwalls into the heated crossflow. The resulting flowfield is quite complex, and accurate predictions of the flow and heat transfer have been difficult to obtain using traditional two-equation turbulence models employed with Reynolds- Averaged Navier–Stokes (RANS) solvers. Recently, a four-equation model, the

εkfv −2 model, has been shown to yield improved predictions in a wide range of applications including gas turbine flows. However, this model retains some of the drawbacks inherent with ε-based models, and in many applications exhibits significant numerical stiffness. In the present chapter, a new model is presented (the

ωkfv −2 model), which is based on ω as the relevant length scale, and provides

predictive accuracy that is comparable to or better than the εkfv −2 , but with significantly improved numerical stability. This new model is tested on a film-cooling flow application, and computations are compared with detailed flow and cooling effectiveness measurements. It is shown that the ωkfv −2 model provides improved flow predictions over other turbulence models, but that a variable Prandtl number formulation is necessary for improved cooling effectiveness predictions. A Prandtl number expression is proposed and shown to yield improved predictions.

www.witpress.com, ISSN 1755-8336 (on-line) WIT Transactions on State of the Art in Science and Engineering, Vol 15, © 2005 WIT Press

doi:10.2495/978-1-85312-956-8/04

Nomenclature

k Turbulence kinetic energy RANS Reynolds-averaged Navier–Stokes Vjet Jet velocity V∞ Crossflow velocity

2v Reynolds stress acting normal to a solid wall CFD Computational fluid dynamics Ui Cartesian velocity component xi Cartesian direction, x,y,z ν, νt Kinematic viscosity, turbulent eddy viscosity ω, ε Turbulence length scales 1 Introduction External film cooling of turbine blades involves the injection of coolant jet over the blade surfaces or the end walls (hub and blade tip). These flows are difficult to predict accurately due to the inherent complexity of the jet-crossflow interaction. Figure 1 shows a cartoon from Fric and Roshko [1] illustrating the various structures generated when a jet is injected normally into an unbounded crossflow. Unlike a rigid cylinder in crossflow, the boundaries of the jet are compliant and entraining, causing the jet to bend over. Periodic shedding of wake vortices has been observed particularly when the jet-blowing ratio (Vjet/V) is greater than 1. Considerable effort has gone into determining the origin of the wake vortices, and there is now experimental [1-3] and computational evidence [4,5] that the wake vortices are initiated by the entrainment of the crossflow boundary layer into the wake, and the upward reorientation of the entrained flow into the wake structures. The jet structure itself is dominated by a pair of kidney-shaped counter-rotating vortex pair (CVP), and both the shearing between the jet and the crossflow and the vorticity issuing from the jet exit has been attributed as the source of the CVP [3]. There are, however, different mechanisms proposed on the reorientation of the jet-hole vorticity into the CVP structure. Upstream of the jet, due to the adverse pressure gradients, a horseshoe vortex system is formed, which wraps around the base of the jet travelling downstream with vorticity counter to the CVP [2,3]. Shear-layer vortices on the leeward and windward edges of the jet have also been observed, and have been attributed to Kelvin–Helmholtz instabilities [2]. These and other studies provide unambiguous evidence of the importance of the coherent structures and their dynamics in the near field of the injected jet. Clearly, any predictive model must embody the physics of the coherent structures to accurately predict the near-field jet behavior.

∞


Figure 1: Schematic of the flow field of a jet in crossflow (Fric and Roshko [1])

The present chapter focuses attention on the jet-in crossflow configuration representative of film cooling (see Goldstein [6] for a review of film cooling). The main aim of the chapter is the development of a turbulence model that improves the predictions of both the flow and heat transfer, and therefore the chapter primarily presents the results using a new four-equation ( ωkfv −2 )

turbulence model developed by the authors. It is shown that the ωkfv −2 model provides satisfactory results for film-cooling applications considered here.

2 Previous studies

The majority of the RANS simulations for a jet in a cross flow have employed a variant of the k–ε models (originally proposed by Launder and Spalding [7]) to obtain the distribution of eddy viscosity. Patankar et al. [8] were among the early researchers to use this model to perform a detailed study of the jet in a cross flow, and even with a relatively coarse (15×15×10) grid, obtained reasonable agreement with experimental data for the jet trajectory and streamwise velocity. Jones and McGuirk [9] used a grid containing 20×15×15 nodes and obtained only qualitative agreement with measured data due to the inadequate grid resolution. Grid-resolution requirements were investigated by Demuren [10] in his computations for a row of jets in a crossflow. Results on a 37×70×14 (stream wise, vertical and spanwise directions) were shown to be grid independent and captured experimental trends fairly well. Demuren [11] also published a detailed analysis on modeling turbulent jets in cross flow, and presented a systematic review of the various models reported till 1985.

Claus and Vanka [12] used a refined grid (256×96×96) and the k–ε model and found that they could not capture the horseshoe vortex. Kim and Benson [13] employed a multiple-time-scale turbulence model to perform a detailed analysis of the flowfield of a row of jets in a confined cross flow. The horseshoe structure


was predicted correctly using a non-uniform 165×59×80 grid and the good agreement was attributed partly to the multiple-time-scale model used for this study. An analysis of cooling jets near the leading edge of turbine blades was performed by Benz et al. [14]. The RANS equations coupled with the standard k–ε model was solved. A multi-block grid was used to simulate an actual blade geometry along with the coolant supply hole. Good agreement with experimental results was obtained due to the inclusion of the coolant delivery tube along with the main flow.

Garg and co-workers have systematically studied the effects of turbulence models [15] and the hole physics [16–19]. In Garg [15], an ACE rotor with five rows containing 93 film cooling holes were simulated. Three different turbulence models were explored (Wilcox’s k–ω, Coakley’s q–ω and the Baldwin–Lomax model). Results were compared with the experimental data of Abhari [20]. Overall, the k–ω model appears to provide the best agreement with the measurements, and particularly on the pressure side. Garg and Rigby [16] used Wilcox’s k–ω turbulence model, and found that the coolant velocity and temperature profiles at the hole exit did not conform to the commonly used parabolic or 1/7-th power law distribution. The exit-velocity profile appeared to significantly impact the heat-transfer-coefficient distribution on the suction side, and to obtain reasonable predictions, it was shown that the flow development in the coolant delivery tube must be accounted for. In another application of Wilcox’s k–ω turbulence model, Garg [17] computed heat transfer coefficients on the blade, hub and shroud for a rotating high-pressure turbine blade with 172 film-cooling holes in eight rows.

Leylek and Zerkle [21] included the coolant-supply hole and the plenum in their calculations and used the standard k–ε model employing generalized wall functions prescribed by Launder and Spalding [7]. York and Leylek [22] presented predictions for mainstream pressure gradient effects in film cooling. A realizable k–ε model was used and the computations demonstrated the ability of the applied computational methodology to accurately model film cooling in the presence of mainstream pressure gradients. A detailed analysis of film-cooling physics, in a four-part series, has been presented by Walters and Leylek [23], McGovern and Leylek [24], Hyams and Leylek [25], and Brittingham and Leylek [26], each dealing with different aspects of the film cooling problem. The standard k–ε model employing wall functions and a two layer model was used.

An aerodynamic and heat-transfer analysis of film-cooled turbine airfoils was conducted by Edwards et al. [27]. Ajersch et al. [28] made detailed measurements of multiple square jets injected normally into a cross flow and carried out an accompanying numerical simulation using a multi-grid, segmented, k−ε CFD code. Predictions and measurements did not compare well for velocities and stresses on the jet centerline, while values off the centerline matched those of the experiments much more closely. A similar study for an inclined jet was performed by Findlay et al. [29]. A numerical study of discrete-hole film cooling was conducted by Berhe and Patankar [30] on a three dimensional film-cooling geometry that included the main flow, injection hole and the plenum. The effect of various variables like blowing ratio, density ratio,


hole length, plenum height, plenum-flow direction and turbulence level at the inlet were discussed in detail. Berhe and Patankar [31,32] extended their flat- plate studies and included the effect of curvature using a Richardson-type correction and a two-equation model. The standard k–ε and the two-layer k–ε turbulence models were used by Lakehal et al. [33] for investigating film-cooling effectiveness of a flat plate by a row of laterally injected jets. In order to match the measured lateral spreading, they employed an anisotropic correction for eddy viscosity proposed by Bergeles et al. [34]. Hoda and Acharya [35] compared seven different turbulence models for film-cooling flows and concluded that the Lam–Bremhorst k–ε formulation provided the best comparison with the measurements. More complex configurations have also been studied. A transonic film- cooling investigation of the effect of hole shapes and orientations was carried out by Wittig et al. [36]. Bohn et al. [37] made detailed 3-D conjugate flow and heat- transfer calculations of a film-cooled turbine guide vane at different operational conditions. More recently, Heidmann et al. [38] have reported fully coupled calculations of an Allied-Signal film cooled vane with shaped holes. Their calculations included both the internal cooling channels, the coolant-delivery tubes, and the external flow. In these cases, the model predictions were only in qualitative agreement with data, and a need for improved turbulence modeling is clearly present.

3 Four-equation turbulence models

The 2v f kε− model: Durbin [39] developed a new eddy-viscosity four-

equation model, the εkfv −2 model, which can be integrated down to the wall

without the need for near-wall damping functions. In the εkfv −2 model,

additional equations for 2v and f have to be solved, and the eddy viscosity is made proportional to ε/2kv instead of k2/ε as in traditional two-equation models. By comparing with direct numerical simulation (DNS) data, Durbin has shown that 2v is the appropriate velocity scale near walls compared to k, which is used in traditional two-equation models.

The model equations for the εkfv −2 model are:

( )

∂∂

+∂∂

+−=j

tj x

kx

PDtDk υυε (1)

∂∂

+

∂∂

+−

=j

t

j xxTCPC

DtD ε

συ

υεε

ε

εε 21 (2)


( )

∂∂

+∂∂

+−=j

tj x

vxk

vkfDtvD 222

υυε (3)

kPC

kv

TCffL 2

212

32

+

−=−∇ , (4)

where 2 t ij ijP S Sυ= . The time and length scales are computed as:

=

SvCkkT

266.0,6,maxminµ

ευ

ε (5)

= 4/1

4/3

2

2/32/3

,6

,minmaxευ

ε η

µ

CSvC

kkCL L, (6)

where ijij SSS = and ( )jiijij xUxUS ∂∂+∂∂= //5.0 . The eddy viscosity

and the various model coefficients are defined as:

TvCt2

µυ = (7)

70 ,19.0 ,3.0 ,4.0 ,3.1 ,92.1 212 ====== ηµεε σ CCCCC .

The wall boundary conditions for each of the above equations are

4

222

2

20 ,0 ,2 ,0y

vfvy

kkw

ww ευυε −==== , (8)

where y represents the distance from the wall to the first grid point off the wall. The value of 1εC in the ε equation is computed using an algebraic expression. This expression has two different forms in different versions of the model, and will be discussed later when comparing the different versions of the εkfv −2 model. Recently, Medic and Durbin [40] applied the εkfv −2 model to film-cooling

flows. They found that the εkfv −2 model significantly improves results over the k–ε and k–ω models. They also found that the k–ε and k–ω model


predictions can be improved if the model equations are formulated in terms of a time scale, such as in the eddy-viscosity formula kTCt µυ = . If a suitable bound is placed on the time scale, the eddy-viscosity predictions are improved. For instance, the k–ε eddy viscosity becomes

×=

SCkkkCtµ

µα

ευ

6,min (9)

instead of ευ µ /2kCt = , where α=0.6 for the εkfv −2 model. This bound was derived from the condition that the eigenvalues of the Reynolds-stress tensor should be positive, which could be thought of as a realizability constraint. They found that with this bound, the k–ε and εkfv −2 models give good predictions of the overall heat-transfer coefficient levels, except for high blowing ratios where the k–ε model predictions are not satisfactory. They also found that the

εkfv −2 model predicts stronger jet lateral spreading compared with the other models. In addition to the original version of the model (called version 1 here), two additional versions of the εkfv −2 model (versions 2 and 3) have been developed, with the goal to eliminate the wall-distance function of the original version (version 2), and the other to improve the numerical behavior related to the stiffness of the wall boundary condition for the f equation (version 3). A summary of the three different versions was given by Kalitzin [41] who applied these models to flow over an airfoil. He found that there were significant differences between the models in terms of both numerical convergence and accuracy. A brief summary is given below to show the differences in the model formulation for each version.

Version 1: Equations (1) – (8) are the governing equations for the original version. The expression for 1εC is defined as:

( )[ ]421 2/1/25.03.1 LdCC L+==ε , (10)

where 3.0=LC and d represents the distance of each grid point to the nearest wall. The other coefficients are as stated in eqn. (7).

Version 2: A new version was developed by Parneix and Durbin [42] which eliminates the need to compute the wall distance d for the Cε1 expression, but instead the relationship 2vk is used. The new formulation for Cε1 and the new closure coefficients are:


( )21 1.4 1 0.045 /= +C k vε (11)

85,25.0,22.0 === ηµ CCC L .

Version 3: To create a more numerically friendly model, the f equation has been reformulated by Lien and Durbin [43] so that instead of solving for f, a similar

variable ~f is solved. By adding an additional source term to the f equation and

subtracting a similar term from the 2v equation the new 2v and ~f equations

become:

( )

∂∂

+∂∂

+−=j

tj x

vx

vk

fkDtvD 2

2~2

6 υυε (12)

kTv

kPC

kv

TCffL

2

2

21

~~22 5

32

++

−=−∇ (13)

19.0 ,23.0 ,/045.014.1 21 ==

+= µε CCvkC L . (14)

The boundary condition for 2v and ~f at the wall is: 02

~== vf w .

Although the εkfv −2 model shows improved predictions, it is difficult to obtain a converged solution. In fact, a k–ε model solution must be obtained first in order to establish good approximations for the turbulence quantities, k and ε, before the εkfv −2 model can be restarted to obtain better results.

The new v2f–kω model: In view of the poor numerical convergence behavior of the εkfv −2 model, Jones [44] developed a new model, the ωkfv −2 model,

which does not suffer from the numerical stiffness of the εkfv −2 model, but

shows improved predictions similar to that of the εkfv −2 model. The basic philosophy in developing this new model lies in introducing a new ε–ω relationship (instead of the kωβε *= that is used with the standard k–ω model) that is based not only on dimensional arguments but also on the need for preserving the v2k/ε scaling in the eddy-viscosity expression. The following equation is proposed for relating ε and ω


( ) 2

12* 21y

kfuncvkfuncnn υωβε −+

=

−, (15)

where

[ ]υ

ykfunc yy =−−= Re ; )Re02.0exp(1 2 . (16)

The relationship between ε and ω proposed in eqn. (15) has been shown by Jones [44] to provide the best fit to eddy viscosity computed based on DNS data. The second part of eqn. (15) is introduced to provide the right asymptotic behavior near the wall. A value of n = 0.7 was found to give the best predictions of eddy viscosity for flow in a channel, and also shows good agreement with experiments for flow in a coaxial jet, and flow over a heated cavity [44]. The resulting governing equations for the ωkfv −2 model are:

( )

∂∂

+∂∂

+−=j

tj x

kx

PDtDk υυε (17)

∂∂

+

∂∂

+

−=

−

j

t

j

n

xxkvP

kDtD ω

συ

υβωωαω

ω

12

2 (18)

( )

∂∂

+∂∂

+−=j

tj x

vxk

vkfDtvD 222

υυε (19)

kPC

kv

TCffL 2

212

32

+

−=−∇ . (20)

The time and length scales are computed by eqns. (5) and (6) and the eddy viscosity and other coefficients are defined as:

TvCt2

µυ = (21)


−=

=======

*2*

21**

1

,25.0 ,23.0 ,5.1 ,3.0 ,4.0 ,54 ,09.0

βκσβ

βα

σβββ

ω

µω CCCC L

The complete derivation of the coefficients can be found in Jones [44] and have shown to produce good agreement with experimental data for several flows. The wall boundary conditions for each of the above equations are

4

222

1

22*

20 ,0 ,2 ,0y

vfvvk

yk

ww

n

w ευ

βυω −==

==

−

, (22)

where k and 2v are those values at the first grid point from the wall. One of the advantages of using a ω-type equation instead of a ε-type equation is that a ω-type equation does not contain singularities as the wall is approached. This is a problem encountered when solving the ε equation. To eliminate this problem, typically the ε equation is formulated in terms of a time scale with appropriate bounds. This should mathematically fix the problem but since iterative methods are used to solve the equations, the solution is usually unphysical until convergence is achieved. Only at this point can it be expected that these remedies will be effective. The singularity problem typically causes k–ε models to be numerically stiff and as expected, causes instabilities in the

εkfv −2 model as well. By using a ω-type equation the ωkfv −2 model does

not suffer from these instabilities. The εkfv −2 model also suffers from numerical instabilities originating from eqn. (22), which is the wall boundary condition of the f equation. Jones [44] showed that for highly stretched grids the boundary condition, wf , can take on large negative values. Since the f equation is a source-term-dominated equation, large values of wf can create difficulties for conventional solvers. Jones [44] also showed that in certain flow situations, the value of wε can take on very small values also causing wf to become large. This can become an even bigger problem since wε is changing as the solution develops causing large fluctuations in the value of f. By using an expression for ε similar to the k–ω model expression, kωβε *= , the value of ε does not become excessively small eliminating the large fluctuations in f as the solution develops. The new ωkfv −2 model has been shown to produce better results for several different flows [44].


Figure 2: Convergence history for the εkfv −2 (version 1) and ωkfv −2 models.

The convergence history for flow over a backward-facing step using the ωkfv −2 model and version 1 of the εkfv −2 model is shown in Fig. 2. Both

models were started from an arbitrary initial condition for the velocity and turbulence quantities. As Fig. 2 shows the εkfv −2 model becomes unstable

after 40 or 50 iterations, while the ωkfv −2 model converges to a reasonable level. Similar convergence for both models has been observed for other flows such as the film-cooling problem described next.

4 Numerical details

The steady-state incompressible Reynolds-averaged Navier–Stokes equations were solved in a generalized curvilinear coordinate system using a pseudo-compressibility formulation to couple the pressure and velocity fields. This is done using a low Mach number preconditioning method. The convective terms are computed using a second-order, low-diffusion flux-splitting scheme of Edwards [45], and second-order central differences are used for the viscous terms.

5 Film-cooling jet in a crossflow

The present work is focused on simulating a film-cooling jet injected into a crossflow. Configuration and parameters selected correspond to the measurements of Lavrich and Chiapetta [46]. Results with the ωkfv −2 model are compared to the high Reynolds number k–ω model, the two-layer k–ε model, version 1 of the εkfv −2 model, and experimental measurements [46]. In these simulations, the jet enters the crossflow at a 35 degree angle and the blowing ratio (mean jet-exit velocity/mean crossflow velocity) is one. The jet inlet

iterations

L∞norm

100 101 102 10310-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

v2f-kω modelv2f-kε model

Convergence History( k-equation )

iterations

L∞norm

100 101 102 10310-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


Convergence History( v2-equation )

iterations

L∞norm

100 101 102 10310-5

10-4

10-3

10-2

10-1

100

101

102


Convergence History( f22-equation )


temperature is 310.78 K and the crossflow temperature is 288 K. The Reynolds number based on the jet velocity and jet diameter is 22 200. The velocity and temperature measurements were taken using a hot wire, while the adiabatic effectiveness was measured using heat-sensitive paint and an infrared camera. The computational domain extended 5D upstream of the hole, 13D downstream of the hole, and 3.5D in the spanwise direction. The height of the domain extended to 5D above the hole. The computational grid is created using the GRIDPRO software and consists of 1.3 million grid points (Fig. 3). Each block on the left (Fig. 3) represents a different zone. The grid in Fig. 3 only shows the grid zones that outline the solid surfaces. For grid independence, the simulations were also computed using 0.7 million grid points. The predictions differed by less than 0.1 per cent, hence the solutions presented here are assumed to be grid independent. All results presented in this section were performed using the 1.3 million grid-point domain.

Each simulation was carried out in parallel across 15 PCs with 1.1-GHz Athalon processors. The domain decomposition can also be seen in Fig. 3 on the right-hand side and shows the outline of each zone in the domain. The zones have been shaded such that zones of the same grayscale reside on the same processor. Since the domain is symmetric about the jet centerline, z=0, only one half of the domain in Fig. 3 was simulated. The other half of the grid is shown only to represent the complete configuration.

X

Y

Z

X

Y

Z

Figure 3: Computational grid and domain decomposition

Figure 4 shows contours of the ωkfv −2 model predictions of streamwise velocity compared with the experiments. The left-hand side, z < 0, represent the computations and the right-hand side, z > 0, represents the experiments. The experiments do not extend down to the surface but instead start at D/10 above the surface. This is why the boundary-layer development seen on the computational side is not seen on the experimental side. The predictions show, like the experiments, that the jet exhausts into the crossflow and forms a plume that


spreads as it moves downstream. It also shows that the ωkfv −2 model compares quite well with the experiments. Figure 5 shows a closer view of the streamwise contours at X/D=5 for each of the model predictions. The k–ω model predictions in Fig. 5 show that the core of the jet is located at a higher position off the surface compared with the experiments. The k–ε model predictions show the core of the jet to be at a position slightly lower than experiments. The predictions of the εkfv −2 and

ωkfv −2 models are in good agreement with the experiments with very little mismatch between the contours. A careful examination of the contours clearly indicates that the ωkfv −2 model exhibits better agreement with the data than any of the other models.

Figure 4: Streamwise velocity predicted by the ωkfv −2 model.

Figure 6 shows the streamwise velocity predictions of each model at the individual locations labeled a–c in Fig. 4. The locations a, b, and c correspond to the spanwise centerplane (z/D=0) and streamwise locations of x/D=0, 5 and 10. Each of the locations shown in Fig. 6 has been scaled for clarity. The scaling was done by adding a constant value of 10 to the velocity at each successive x/D location. Therefore, the scale along the abscissa does not reflect the actual value of the velocity. At z/D = 0, the k–ω model overpredicts the jet penetration (relative to the other models) at both x/D = 5 and x/D = 10. This was also observed in Fig. 5. The other model predictions are relatively close to each other with the ωkfv −2 model showing the best agreement with experiments at x/D =5.


Figure 7 shows the streamwise velocity predictions of each model at the individual locations labeled d–f in Fig. 4. The locations d, e, and f correspond to a spanwise plane at the edge of the jet hole (z/D=0.44) and streamwise locations of x/D=0, 5 and 10. The velocities have been scaled as stated above for Fig. 6. All of the models produce similar and satisfactory predictions at this spanwise location of the flow.

Figure 5: Streamwise velocity at X/D=5.


u

y

10 20 30 400

0.02

0.04

0.06

0.08

0.1Streamwise Component of Velocity

z/D = 0x/D = 0 x/D = 5 x/D = 10

epx.v2f-kωk-ωk-εv2f-kε

a b c

Figure 6: Streamwise-velocity predictions at z/D=0.

u

y

10 20 30 400

0.02

0.04

0.06

0.08


z/D = 0.44x/D = 0 x/D = 5 x/D = 10


fd e

Figure 7: Streamwise-velocity predictions at z/D = 0.44.


u

y

10 20 30 400

0.02

0.04

0.06

0.08


z/D = 0.88x/D = 0 x/D = 5 x/D = 10


g h i

Figure 8: Streamwise-velocity predictions at z/D = 0.88.

Figure 8 shows the streamwise-velocity predictions of each model at the individual locations labeled g–i in Fig. 4. The locations g, h, and i correspond to a spanwise plane away from the jet hole (z/D=0.88) and streamwise locations of x/D=0, 5 and 10. This spanwise plane is far away from the hole such that there is no influence from the jet and the velocity profiles are representative of a turbulent flat-plate flow. All of the models are in good agreement with the experiments.

Figure 9 shows contours of the ωkfv −2 model predictions of the vertical velocity compared with the experiments. The left-hand side, z < 0, represent the computations and the right-hand side, z > 0, represents the experiments. As stated for the streamwise-velocity, the experiments do not extend down to the surface, but instead start at D/10 above the surface. Figure 9 shows that the

ωkfv −2 model predictions are qualitatively in good agreement with the experimental data. Figure 10 shows the details of the vertical velocity predictions from all of the models at x/D = 5. The k–ω model predictions show that the vertical velocity is appreciably overpredicted. This explains why the core of the jet seen in Fig. 5 is predicted at a higher location off the surface than in the experiments. The k–ε model also overpredicts the vertical velocity, but the magnitude of the overprediction is smaller, and a careful comparison of the mismatch between the contours indicate that the k–ε predictions are in closer agreement with the measurements relative to the k–ω model predictions. Both the εkfv −2 and the

ωkfv −2 models predict lower vertical velocities, and the model predictions


are in much better agreement with the data. The counter-rotating vortex pair is characterized by the circular contours on the right- (measured) and left- (predicted) hand sides of the centerline. The k–ε model shows more concentrated contours in this region with a greater spanwise extent than that of the k–ω model, indicating a larger overprediction of the circulation rate or the strength of the CVP. The εkfv −2 model shows better predictions of vertical velocity along the centerline compared to the k–ω and k–ε models but also shows an overprediction of the circulation rate of the counter-rotating vortex pair. The

ωkfv −2 model produces the best results since the vertical velocity along the centerline and the circulation rate of the counter rotating pair are in better agreement with the experiments.

Figure 9: Vertical component of velocity predicted by the ωkfv −2 model.

Figure 11 shows the vertical velocity predictions of each model at z/D=0 at the individual locations labeled a–c in Fig. 9. As was shown in Fig. 10, the k–ω model produces the largest overprediction of vertical velocity compared to the other models. The k–ε model also overpredicts the peak vertical velocity at all x/D locations. As in Fig. 10, the best agreement with the data comes from the

ωkfv −2 model. Figure 12 shows the vertical velocity at z/D=0.44, at the individual locations labeled d–f in Fig. 9. All of the models agree well with the experimental data except the k–ω model that exhibits significant differences with the measured velocity at x/D = 5. Figure 13 shows the vertical-velocity predictions at the locations labeled g–i in Fig. 9. At x/D = 5 the k–ε model overpredicts the peak vertical velocity significantly. The εkfv −2 model shows improvements over the k–ε model but still shows a slight overprediction at


x/D = 5. The ωkfv −2 model shows the best agreement with experiments at all of the locations compared with the other models.

Figure 10: Vertical-velocity predictions at x/D = 5.


v

y

0 4 8 12 16 20 240

0.02

0.04

0.06

0.08

0.1Vertical Component of Velocity

z/D = 0x/D = 0 x/D = 5 x/D = 10


a b c

Figure 11: Vertical-velocity predictions at z/D = 0.

v

y

0 4 8 120

0.02

0.04

0.06

0.08


z/D = 0.44x/D = 0 x/D = 5 x/D = 10


d e f

Figure 12: Vertical velocity predictions at z/D = 0.44.


v

y

0 4 8 120

0.02

0.04

0.06

0.08


z/D = 0.88x/D = 0 x/D = 5 x/D = 10


g h i

Figure 13: Vertical velocity predictions at z/D = 0.88.

Figure 14: Temperature contours predicted by the ωkfv −2 model.


Figure 14 shows the temperature contours of the ωkfv −2 model compared

with experiments. At x/D = 5 the ωkfv −2 model overpredicts the temperature in the core of the jet. However, there is general agreement in the extent of the vertical and lateral spreading of the jet. Figure 15 shows the temperature contours for all the model predictions at x/D = 5. The k–ω model predictions show that the temperature is significantly overpredicted in the core of the jet. The other models also show an overprediction of temperature in the core region but the extent of overprediction is considerably lower, particularly along the centerline. All of these models utilized a constant Prandtl number of 0.9. This is the typical value used in turbulence models to compute the temperature field. The value of 0.9 was found experimentally to correspond to the logarithmic region of the boundary layer for flow in a channel. It should not, however, be considered to be a universal value since the turbulent Prandtl number is a property of the flow. Since Figs. 4–13 show that each velocity component predicted by the ωkfv −2 model is in good agreement with experiments, and Figs. 14 and 15 show that there are relatively large discrepancies between the temperature predictions and experiments, this indicates that the constant Prandtl number assumption may not be appropriate in such a flow. In the rest of the chapter, attention is focused on developing a suitable expression for the turbulent Prandtl number. He et al. [47] studied the effect of turbulent Schmidt number on scalar mixing in a jet in crossflow. In their study the governing equation for species concentration was identical to the equation for enthalpy, therefore their turbulent Schmidt number should be considered to be the same as the turbulent Prandtl number. They compared a number of simulations for turbulent Schmidt number ranging from 0.2 to 1.5 and a jet-to-crossflow velocity ratio of 2.3. They concluded that the turbulent Schmidt number in the jet should be 0.2 for best results. This conclusion was made because the decay of the thermal plume as the jet moves downstream is in good agreement with experiments for a turbulent Schmidt number of 0.2. A semi-empirical analysis was carried out to suggest a variable turbulent Schmidt number throughout the flow field. Kamotani and Greber [48] developed correlations for the turbulent Schmidt number for a jet in crossflow and show that the turbulent Schmidt number increases with increasing momentum flux ratio and density ratio, and with increasing x/D. This suggests that the turbulent Schmidt number, or turbulent Prandtl number, should be a variable instead of a constant. As shown by Moffatt and Kays [49] the turbulent Prandtl number in a channel flow takes on a value of about two near the wall and decreases to about 0.85 in the center of the channel. Chamber et al [50] found that the turbulent Prandtl number for a turbulent plane jet is about 0.4 in regions of the flow where the Reynolds stresses and heat fluxes are large. Since the jet in a crossflow is a combination of both a jet flow and a wall-bounded flow the turbulent Prandtl number should vary between about two or three near the wall and decrease to about 0.2 in the jet region.


Figure 15: Temperature contours at x/D = 5.

The following Prandtl number formulation of Moffatt and Kays [49] was initially used in the present study. It was formulated to match the turbulent Prandtl number variation in a channel flow, which ranges between 1.7 near the wall, to 0.85 in the fully turbulent region of the boundary layer.


( ) κν t

tttt

t PePePePe

=−−++

= 1)/5exp(08.04.01

7.1Pr 2. (23)

No noticeable difference in the solution shown in Figs. 15 was observed when this expression was used. This observation indicated that the lower limit of 0.85 in the jet region was inappropriate and that the value should be much lower as pointed out by He [47] and Kamotani and Gerber [48]. In view of the need to define a Prt expression that spanned the appropriate range of values observed experimentally, the following expression for the turbulent Prandtl number was developed in this study for the jet in a crossflow.

µρ yk

yt =

+= yRe ;

Re08.0110Pr . (24)

Figure 16 shows contour plots for the turbulent Prandtl number computed by eqn. (24) at x/D = 5 and 10. In the core of the jet the turbulent Prandtl number is approximately 0.2 and approximately 4 near the wall. These limits are more representative of the measured values [47, 48]. Figure 17 shows the vertical variation of the turbulent Prandtl number at x/D=5, z/D=0, which corresponds to the centerline of the jet. Near the wall eqn. (24) takes on values of three or four, while in the core of the jet the turbulent Prandlt number is approximately 0.2, which was suggested by He et al. [47].

3.853.392.942.482.021.571.110.660.20

Turbulent Prandtl Numberx/D = 5

Prt = 10/(1 + 0.008Rey)

3.853.392.942.482.021.571.110.660.20

Turbulent Prandtl Numberx/D = 10

Prt = 10/(1 + 0.008Rey)

Figure 16: Turbulent Prandtl number.


y/D

TurbulentPrandtlnumber

0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Recommended value of PrtPresent study

Kays and Moffatt

Turbulent Prandtl number at x/D=5, z/D=0

Figure 17: Turbulent Prandtl number compared with the suggested value of He [47].

Figure 18 shows the temperature contours predicted by the ωkfv −2 model at x/D = 5 for both a constant Prandtl number and with the variable Prandtl number calculation of eqn. (24). The contours show that the spreading rate for the predictions with the variable Prandtl number are in much better agreement with experiments compared to the predictions with a constant Prandtl number. Although the temperature is still overpredicted in the core of the jet the size of the overpredicted region has been reduced compared to the constant Prandtl number simulations. This indicates that the variable Prandtl number expression proposed here moves the predictions in the right direction, but further model developments are needed. Figure 19 shows the temperature predictions for each model at the individual locations labeled a–b in Fig. 14. Results are shown at two streamwise locations, x/D = 5, 10, and for a spanwise station, z/D = 0. When a constant value of 0.9 is used all of the models overpredict the temperature in the core of the jet, but the predictions with the variable Prandtl number (Fig. 19) show better agreement with the experiments. Figure 20 shows the temperature predictions for each model at the individual locations labeled c–d in Fig. 14. Results are shown at two streamwise locations, x/D = 5, 10, and for a spanwise station, z/D = 0.44. This spanwise location corresponds to the edge of the jet. At both locations all of the models

overpredict the temperature but the ωkfv −2 model does show some improvement. Despite this improvement, further model developments are needed. The overprediction indicates that the temperature-spreading rate in the core of jet is underpredicted.


Figure 21 shows the temperature predictions for each model at the individual locations labeled e–f in Fig. 14. At these locations the ωkfv −2 model with variable Prandtl number shows better agreement with experiments. The other models underpredict the temperature indicating that the jet-spreading rate is underpredicted.

Figure 18: Temperature contours with constant Prandtl number (top) and with the variable Prandtl number of eqn. (24), (bottom).

Figure 22 shows the adiabatic effectiveness contours, while Fig. 23 shows the centerline adiabatic effectiveness. All of the models overpredict the adiabatic effectiveness when a constant Prandtl number is used while the ωkfv −2 model with a variable Prandtl number shows good agreement with experiments. When using eqn. (24) the Prandtl number takes on values of three to four near the wall reducing the temperature diffusion in this region. This considerably improves the predictions. Figure 24 shows the spanwise variation of adiabatic effectiveness computed by each of the turbulence models using a constant turbulent Prandtl number of 0.9. The variable Prandtl number, as given in eqn. (24) was also employed with the ωkfv −2 model. All of the models overpredict the adiabatic effectiveness when using a constant turbulent Prandtl number. When eqn. (24) is used the


adiabatic effectiveness is in good agreement with experiments. This clearly points to the importance of using a variable Prandtl number in film-cooling computations.

T

y

280 290 300 310 3200

0.02

0.04

0.06

0.08

0.1Temperaturez/D = 0

T T + 20

exp.v2f-kωk-ωv2f-kω, Prt = 10/(1 + 0.008Rey)k-εv2f-kε

a b

Figure 19: Temperature predictions by each model at z/D = 0.

T

y

280 290 300 310 3200

0.02

0.04

0.06

0.08

0.1Temperaturez/D = 0.44

T T + 20


c d

Figure 20: Temperature predictions by each model at z/D = 0.44.


T

y

280 290 300 310 3200

0.02

0.04

0.06

0.08

0.1Temperaturez/D = 0.88

T T + 20


e f

Figure 21: Temperature predictions by each model at z/D = 0.88.

Figure 22: Adiabatic effectiveness for each model; k–ε (first), k–ω (second), εkfv −2 (third), ωkfv −2 (fourth; Prt=0.9), ωkfv −2 (fifth; Prt from Eqn. (24))


x/D

η

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

v2f-kω model Prt = 10/(1 + 0.008Rey)v2f-kω model Prt = 0.9

v2f-kε model Prt = 0.9

k-ε model Prt = 0.9k-ω model Prt = 0.9

exp.

Centerline Adiabatic EffectivenessBlowing Ratio = 1.0

Figure 23: Centerline adiabatic effectiveness.

z/D

η

-2 -1 0 1 2

0

0.1

0.2

0.3

0.4Spanwise adiabatic effectiveness

x/D = 4

exp.

k-εk-ω model

v2f-kε modelv2f-kω model

v2f-kω modelPrt = 10/(1+0.008Rey)

z/D

η

-2 -1 0 1 2

0

0.1

0.2

0.3


x/D = 5

exp.k-ε

k-ωmodel

v2f-kεmodel

v2f-kωmodelv2f-kωmodel

Prt = 10/(1+0.008Rey)

z/D

η

-2 -1 0 1 2

0

0.1

0.2

0.3


x/D = 6

exp.k-εk-ω modelv2f-kε modelv2f-kω modelv2f-kω model

Prt = 10/(1+0.008Rey)

Figure 24: Spanwise variation of adiabatic effectiveness.


Concluding remarks

The present study has examined the predictive capability of existing turbulence models for film-cooling flow applications, and the corresponding limitations have been identified. A new model called the ωkfv −2 has been proposed, and the benefits of the model from robustness and accuracy perspectives have been identified. While the εkfv −2 model of Durbin [39] has been shown to improve predictions for a variety of flows, the same version of the model does not always show the best results. The ωkfv −2 model was shown in the present study to produce good results for each of the problems presented and is numerically robust. The εkfv −2 and ωkfv −2 model predictions for the film-cooling flow show good agreement with experiments for the velocity but grossly overpredict the temperature and adiabatic effectiveness when a constant turbulent Prandtl number is used. This demonstrates that the constant Prandtl number assumption does not hold for this flow. An empirical formulation for the turbulent Prandtl number was developed and shown to improve the predictions for the adiabatic effectiveness.

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