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Numerical Computations of Film Cooling In Gas Turbine

Blades

Ahmed S Al-Adawy1, Samy M Morcos

2 and Essam E Khalil

2

1Teaching assistant

2Professor of Mechanical Engineering

Cairo University, Faculty of Engineering, Cairo, Egypt

Film cooling is vital to gas turbine blades to protect them from high temperatures and hence high

thermal stresses. Improvements have been made to the shaping of the cooling hole to provide higher

heat transfer effectiveness. Forward-diffused (laidback) holes and laterally-diffused (fan-shaped) holes

proved to give better effectiveness than cylindrical round ones. However, combining both effects

achieves better lateral distribution and less penetration into the mainstream. As a preliminary study of

the effect of varying lateral diffusion angle (γ) and forward diffusion angle (δ) on effectiveness

distribution on an adiabatic flat plate, a numerical model is built with GAMBIT commercial

preprocessor and flow field is solved with spatially-averaged film cooling effectiveness values in the

streamwise direction using FLUENT commercial solver. Numerical results are verified and compared

to corresponding experimental measurements.

I. Introduction

To increase the gas turbine power and efficiency, it was necessary that the inlet pressure and temperature be

increased to meet the high continuous demand over about the last two decades. However, maintaining such high

temperatures has been difficult due to material limits as creep and thermal stresses. That made the development of

gas turbine cooling technology an inevitable task to prevent failure of the turbine components. Many cooling

techniques were developed as (1) internal convective cooling, (2) impingement cooling and (3) external film

cooling. Film cooling has met a series of improvements to the shape of cooling hole to optimize the film cooling

effectiveness distribution. Effects such as orientation angles, length-to diameter ratio, hole spacing-to-diameter ratio,

blowing ratio, density ratio and temperature ratio have been studied, influencing local and spatially-averaged heat

transfer coefficients and film cooling effectiveness.

This literature studies the effect of lateral and forward diffusion angles on local and spatially-averaged cooling

effectiveness at the same blowing ratio. The blowing ratio (M) is defined by the equation

1 … (1)

Three sets of results are analyzed at three blowing ratios 0.5, 1 and 2. For the sake of improving film-cooling

effectiveness, holes are oriented to inject cooling fluid at a specified angle with streamwise direction (α); here taken

to be 35°. These holes are either simple- or compound-angle holes. For simple-angle holes, as shown in Figure 1, the

injection angle with span wise direction (β) is maintained at 90°. For compound-angle holes, as shown in Figure 2,

this angle is varied to give some lateral penetration in the span wise direction. In the presented literature, only

simple-angle holes are studied1-3

.

46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit25 - 28 July 2010, Nashville, TN

AIAA 2010-7088

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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Figure 1: Simple-Angle (SA) Holes Figure 2: Compound-Angle (CA) Holes

Simple-angle holes are cylindrical round simple-angle (CYSA), laterally-diffused simple angle (LDSA), forward-

diffused simple-angle (FDSA) or laterally- and forward diffused simple angle (LFDSA). CYSA holes are simple

cylindrical holes with no lateral or forward diffusion. This type is shown in Figure 3.

Figure 3: Cylindrical Round Simple-Angle Holes

LDSA and FDSA holes are both simple cylindrical holes for some metering length L1 and diffused for a length L2.

In LDSA holes, the diffusion is span wise with a lateral diffusion angle (γ). This type is also known as fan-shaped

holes. They provide better lateral distribution and hence improve the effectiveness through the pitch between holes

and away from centerlines of cooling holes. This type is illustrated in Figure 4. On the other hand, in FDSA, the

diffusion is streamwise with a forward diffusion angle (δ). This type is also known as laidback holes. They provide

less penetration to the mainstream at the hole exit area in FDSA than in CYSA and hence they provide less cooling

velocities and less viscous losses. This type is illustrated in Figure 5. However, both diffusion techniques can be

combined to produce laterally and forward diffusion FLDSA holes that are also known as laidback fan-shaped holes.

This type can be taken as the generalized form of the previously-mentioned types and is illustrated in Figure 6.

Figure 4: Laterally-Diffused Simple-Angle Holes Figure 5: Forward-Diffusion Simple-Angle Holes

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Figure 6: Laterally- and Forward-Diffusion Simple-Angle Holes

The selected numerical model for the present work is illustrated here in Figure 7.

Figure 7: Numerical Model

A simple control volume is extracted out of this numerical model. The diameter of the cooling hole is D, its length is

L and the pitch between holes is P. The control volume has a height of 10D and a thickness of P/2, and is extended

upstream to 10D and downstream to 25D. Symmetry boundary conditions are maintained at the vertical mid-pitch

and hole center planes. This control volume is illustrated in Figure 8.

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Figure 8: Control Volume Studied

The diameter of cooling holes is taken to 5 mm and length of cooling holes for CYSA is taken to be 3D,

corresponding to 15 mm. For LDSA, FDSA and LFDSA holes, both the metering length and diffusion lengths are

taken as 1.5D, corresponding to 7.5 mm. The streamwise injection angle is maintained at 35° and compound angle is

kept at 90°. Pitch between holes is chosen as 3D, corresponding to 15 mm.

II. Mathematical Modelling and Assumptions

The present numerical investigation was based on solving the governing equations that described airflow inside the

selected control volume by a CFD program FLUENT 6.34-5

(commercially available CFD program). This numerical

approach solves the partial differential equations governing the transport of mass, three momentum, energy and

species in a fully turbulent three dimensional domain under steady state conditions in addition to standard k – ε

model equations for turbulence closure6-7

.

Computational Fluid Dynamics Models

The different governing partial differential equations are typically expressed in a general form as:

ΦΦΦΦ +

∂

Φ∂Γ

∂

∂+

∂

Φ∂Γ

∂

∂+

∂

Φ∂Γ

∂

∂=Φ

∂

∂+Φ

∂

∂+Φ

∂

∂S

zzyyxxW

zV

yU

xeffeffeff ,,,ρρρ

… (2)

Where ρ is the air density and Φ is the dependent variable, SΦ = Source term of Φ, and U, V, W are the velocity

vectors, and ΓΦ,eff is the effective diffusion coefficient. The effective diffusion coefficients and source terms for the

various differential equations8 are listed in the table 1.

Table 1. Terms of Partial Differential Equations (PDE) in Equation 1.

Φ ΓΦ,eff SΦ

Continuity 1 0 0

X-momentum U µeff -∂P/∂x +ρg+ SU

Y-momentum V µeff -∂P/∂y+ρg (1+β∆t) + SV

Z-momentum W µeff -∂P/∂z+ρg+ SW

H-equation H µeff/σH SH

k-equation k µeff/σk G - ρ ε ε-equation ε µeff/σε C1 ε G/k – C2 ρ ε2

/k

µeff = µlam + µ t µ t = ρ Cµ k2 / ε

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G = µt [2{(∂U/∂x)2 +(∂V/∂y)

2 +(∂W/∂z)

2}+(∂U/∂y + ∂V/∂x)

2 +(∂V/∂z + ∂W/∂y)

2 +(∂U/∂z + ∂W/∂x)

2]

SU = ∂/∂x(µeff ∂Φ/∂x)+∂/∂y(µeff ∂Φ/∂x)+∂/∂z(µeff ∂Φ/∂x)

SV = ∂/∂x(µeff ∂Φ/∂y)+∂/∂y(µeff ∂Φ/∂y)+∂/∂z(µeff ∂Φ/∂y)

SW = ∂/∂x(µeff ∂Φ/∂z)+∂/∂y(µeff ∂Φ/∂z)+∂/∂z(µeff ∂Φ/∂z)

SH is the source of Energy at nodal points

Turbulence model constants C1 = 1.44, C2 = 1.92, Cµ = 0.09

σH = 0.9, σRH = 0.9, στ = 0.9, σk = 0.9, σε = 1.225

Boundary Conditions

The solution of the governing equations can be realized through the specifications of appropriate boundary

conditions. The values of velocity, temperature, kinetic energy, and its dissipation rate are specified at all

boundaries. The mainstream gases are approximated as air, with a temperature of 333 K. The inlet mainstream flow

is considered to have a uniform velocity of 10 m/s as measured by Nassir et al1, the turbulence level is 11% and

turbulent length scale is 5 mm. The coolant inlet is taken to be air, with a temperature of 293 K. The inlet coolant

flow is considered to have a uniform velocity. Due to the temperature ratio of 333/293, density ratio is maintained at

1.14. All computations in the present paper, which is a preliminary study to the effect of diffusion angles and

blowing ratios, are performed at a blowing ratio of 0.5 and hence cooling velocity inlet of 4.39 m/s. As measured by

Nassir et al1, the turbulence level is 1% and turbulent length scale is 5 mm. Symmetry boundary condition is applied

to hole-centre and mid-pitch planes. The flat plate surfaces upstream and downstream of the hole injection and the

internal walls of the coolant pipe are considered adiabatic (zero heat flux). Symmetry boundary condition is applied

to the upper boundary of the control volume, as the gradients of flow field properties are assumed to disappear at

ten-times the hole metering section diameter. A non-slip condition at all solid wall is applied to the velocities. The

logarithmic law of the wall (wall function) of Launder and Spalding6 was used here, for the near wall boundary

layer. At inlets, the air velocity was assumed to have a uniform distribution; inlet values of the temperature were

assumed to be of a constant value and uniform distribution. All velocity components were set as zeros initially, and

temperatures were assumed to be initially equal to the average of cooling and mainstream temperatures.

Numerical Procedure

The Computer Program, FLUENT5 was used to solve the time-independent (steady state) conservation

equations together with the standard k-ε model as Launder and Spalding6

and the corresponding boundary

conditions. The numerical solution grid divided the space of the control volume into discretized computational cells

of the order of 1,000,000 tetrahedral cells. The discrete finite difference equations were solved with the SIMPLE

algorithm, Khalil9. Solution convergence criteria, was applied at each iteration and ensured the summations of

normalized residuals were less than 0.001 for flow, 0.001 for k andε, and 10-6

for energy. The predictions of flow

and turbulence characteristics are in general qualitative agreement with the corresponding experiments and

numerical simulations published by others, Neilsen4. Nevertheless discrepancies exist and particularly in the vicinity

of recirculation zone boundaries. More discrepancies were also observed in situation with heating flows than those

of cooling.

Convergence and Stability

The simultaneous and non-linear characteristics of the finite difference equations necessitate that special measures

are employed to procure numerical stability (convergence); these include under relaxation of the solution of the

momentum and turbulence equations by under relaxation factors which relate the old and the new values of Φ as

follows,

( ) oldnew 1 Φγ−+Φγ=Φ}… (3)

Where γ is the under-relaxation factor. It was varied between 0.2 and 0.3 for the three velocity components as the

number of iteration increases. For the turbulence quantities, γ was taken between 0.2 and 0.4 and for other variables

between 0.2 and 0.6. The required iterations for convergence are based on the nature of the problem and the

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numerical conditions (grid nodes, under-relaxation factor, initial guess, etc.). So the time (on the computer

processor) required to obtain the results is based on many factors. The computational number of iterative steps is

selected according space cell (spatial difference) to yield converged solutions9. The validity of the present

computational technique was assessed previously in the open literature, for example Khalil 4, 7 where reference

should be made for more detailed readings.

III. Results and Discussions

The control volume is meshed for a 0.5 mm spacing of tetrahedral-hybrid grid on the tested flat plate and the cooling

hole. A size function is generated with a growth rate of 1.02 and maximum size of 2 mm. The number of cells is in

the order of a million cells. A snapshot of the meshed control volume is shown in Figure 9.

Figure 9: Snapshot of the Meshed Control Volume

A common definition for the local adiabatic film effectiveness on the surface is

… (4)

The mainstream (hot) gas recovery temperature is a local value, but is in some cases taken to be a constant, or

assumed equal to the gas total temperature. The coolant temperature is typically taken to be equal to the coolant exit

bulk temperature at the injection point into the mainstream, though in compressible flows it may be more

appropriate to use the coolant recovery or total temperature.

The adiabatic wall temperature is the local mixed fluid temperature that drives heat transfer to or from the wall. It is

this adiabatic wall temperature that is primarily dependent on many parameters of the fluid streams, both local and

developing, as well as the surfaces and geometry.

This tetrahedral-hybrid mesh has proven to be effective when compared to corresponding experimental results.

When applying the same boundary conditions of the study of each of Nassir et al1, Ai et al

2 and Jung and Lee

3, the

laterally- (spatially-) averaged cooling effectiveness distribution meets the experimental results presented in each of

those studies as presented in Figure 10, corresponding only to the case of cylindrically-round simple-angle holes.

The predicted spatially averaged film cooling effectiveness is in reasonable agreement with the previous

measurements of Nassir et al1, Ai et al

2 and Jung and Lee

3. The present predictions are well within the various

experimental values that are relatively scattered. However the comparisons indicated good trend wise agreement. It

is to be noted that z/D represent the streamwise distance-to-metering diameter ratio and hence a dimensionless form

of the distance in the streamwise direction.

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Figure 10: Numerical results in comparison to corresponding experimental results

Contours of temperature distribution on the tested plate and on the hole-center planes are shown in Figure 11.

Figure 11: Temperature Contours on tested plate and hole-center plane (°C) for CYSA holes (M = 0.5)

IV. Concluding Remarks

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This paper analyzes verified numerical results of adiabatic wall film cooling effectiveness on a flat plate solved

using FLUENT commercial solver and modeled by GAMBIT preprocessor, as a preliminary step of studying the

effect of varying lateral diffusion angle (γ) and forward diffusion angle (δ) on effectiveness distribution on an

adiabatic flat plate. Numerical computations can well represent the measured data as highlighted in the case shown

in the paper. Further investigations are to follow to clearly identify the parameters affecting film cooling

effectiveness.

References

1 Nassir, H., Ekkad, S.V., Acharya, S., 2000, Effect of compound angle injection on flat surface film cooling with

large streamwise injection angle, Proceeding of the 8th

International Symposium on Transport Phenomena and

Dynamics of Rotating Machinery 2, 763- 768. 2 Ai D., Ding, P.P., Chen, P.H., 2001, The selection criterion of injection temperature pair for transient liquid crystal,

Int. J. Heat Mass Transfer 44, 1389- 1399. 3 Jung, I.S., Lee, J.S., 2000, Effects of Orientation Angles on Film Cooling over a Flat Plate: Boundary Layer

Temperature Distributions and Adiabatic Film Cooling Effectiveness, ASME J. Turbomach., 122, 153- 160. 4

Khalil, E.E., 2010, Modelling Of Flow Regimes and Thermal Patterns Interactions in Complex Applications,

AIAA-2010-1554, Orlando, January 2010. 5 FLUENT 6.3 Documentation, © Fluent Inc. 2005.

6 Launder, B. E., and Spalding D. B., 1974, The Numerical Computation of Turbulent Flows, Computer Methods

App. Mech., pp. 269-275. 7

Khalil, E.E. 2009, Computational Analyses and Design of Industrial Furnaces and Combustion Chambers,

Proceedings of ICGSI, MC04, Thailand, December 2009. 8

Spalding, D. B., and Patankar, S. V., 1974, A Calculation Procedure for Heat, Mass and Momentum Transfer in

Three Dimensional Parabolic Flows, Int. J. Heat & Mass Transfer, 15, pp. 1787. 9 Khalil, E.E., 1983, Modelling of Furnaces and Combustors, Abacus Press, UK.