Proceedings of ASME Turbo Expo 2012

GT2012
June 11-15, 2012, Copenhagen, Denmark

GT2012-68819

CALCULATION OF 3-D TEMPERATURE DISTRIBUTION IN FILM-COOLED FLAT

PLATES USING 2-D EMPIRICAL CORRELATIONS FOR FILM-COOLING EFFECTIVENESS
AND HEAT TRANSFER AUGMENTATION

Peter T. Ingram Savas Yavuzkurt

Department of Mechanical and Nuclear Department of Mechanical and Nuclear
Engineering Engineering
The Pennsylvania State University The Pennsylvania State University
University Park, PA 16802 University Park, PA 16802

ABSTRACT for high temperature study. The study showed that the
In existing gas turbine heat transfer literature there are several difference between conjugate and non-conjugate solutions
correlations developed for the spanwise-averaged film-cooling increases as the temperature levels increase. These differences
effectiveness and heat transfer augmentation for inline injection are quite important and should be taken into account during
on flat plates. More accurate and detailed prediction of film- design of turbine blades.
cooling performance, particularly 3-D metal temperatures are
needed for design purposes. 2-D correlations where INTRODUCTION
effectiveness and heat transfer augmentation are functions of An increase in the efficiency of gas turbines can be obtained by
streamwise and spanwise directions would help to satisfy this increasing turbine inlet temperature. For engine power to
need. Based on this fact, the current study extends the double, the rotor inlet temperature should increase from 2500⁰F
spanwise-averaged correlations into 2-D correlations by using to 3500⁰F, as stated by Han et al. [1]. These temperatures far
a Gaussian distribution in the transverse direction. The exceed the melting point of present blade materials, thus
correlations are obtained using limited spanwise data and more decreasing its life. An increase in the lifetime of Gas Turbine
available spanwise-averaged data and existing spanwise- blades can be achieved by employing film-cooling, wherein air
averaged correlations for a single row of holes with inline from compressor is bled through holes on to the blade surface.
injection. These correlations presented in this paper are
functions of different flow parameters such as mass flow ratio Many researchers have studied the application of film-cooling
M, density ratio DR, , transverse pitch P/D, and inline injection to flat plates and gas turbine blades using experimental,
angle α, with ranges of M:0.2-2.5, DR: 1.2,1.5,1.8, P/D: 2, 3,5, theoretical and numerical approaches. Goldstein [2] provided a
α: 30, 60, 90 degrees. The developed correlations match review of early film-cooling studies. Yuen & Martinez [3-6]
existing spanwise-averaged correlations when averaged. These did an exhaustive study on film-cooling characteristics of round
correlations are used to calculate solid flat plate temperatures hole and presented film-cooling effectiveness and heat transfer
for two well-documented cases of film-cooled flat plates. coefficient data for various injection angles and orientations.
Spanwise variations in the metal temperature were calculated Similarly, Baldauf et al. [7-10] conducted a detailed study on to
to be between 5-6K for a temperature difference of 40K and obtain film-cooling effectiveness and heat transfer coefficient
between 20-30K for a temperature difference of 250K, data at mainstream temperature of 550K for various injection
significant for design purposes. The study also contains the angles and blowing ratios.
comparison of solid temperatures for conjugate and non-
conjugate heat transfer cases using a Reduced Order Film Most manufacturers use what is termed a conventional
Model (ROFM) which is implemented in a loosely coupled technique to determine blade temperature distribution using
conjugate heat transfer technique called Iterative Conjugate experimental and numerical results which do not take into
Heat Transfer (ICHT)).The differences between conjugate and account the effect of conjugate heat transfer on heat transfer
non conjugate simulations are about 6K or 2% of the local coefficients. Silieti et al. [11] reported that the full conjugate
temperature for low temperature study and about 20K or 5% heat transfer (CHT) model shows a significant difference in

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 temperature prediction when compared to adiabatic cases and begins at a single point it is important to have, if possible, 2-D
confirmed that the CHT model can take into account the mutual correlations for 𝜂 and ℎ𝑓 as a function of axial (streamwise) and
influence of heat transfer on fluid flow and vice versa. He transverse (spanwise) directions, not just spanwise-averaged
compared the centerline effectiveness for both adiabatic and correlations so that 3-D temperature distributions of film cooled
conjugate cases and showed that a significant improvement in solid can be calculated.
conjugate effectiveness of up to 3 times in the immediate
region (x/D <6) is observed. Bohn et al.’s [12] conjugate As flow geometries become more complicated, prediction of
studies of a film-cooled turbine blade predicted 8% difference the blade temperature becomes very complex and cannot be
in temperatures for conjugate and decoupled conventional accurately predicted by traditional computational methods. A
approach. Lu et al. [13] concluded that the application of the method to be able to accurately predict 3-D temperature
conjugate method includes the influence of heat transfer on the distributions in the blades is needed. The present CFD study
velocity field within the film. Kane & Yavuzkurt [14], who deals with this need using numerical techniques of ICHT-
performed numerical simulation on a non-film-cooled blade, ROFM while implementing 2-D correlations for film-cooling
reported 30% deviation from data of Hylton and Nirmalan [15] effectiveness and heat-transfer augmentation (hf/h0) This
in using the conventional constant wall temperature approach, technique simplifies the computational process and takes into
whereas full conjugate results were much closer to the data account the effect of the thermal resistance of blade metal on
with an overall deviation about a few percent. From the above the temperature distribution.
mentioned review of literature, it can be inferred that conjugate
heat transfer plays a significant role in correctly predicting
surface temperatures and heat transfer coefficients, as it takes OBJECTIVES
into account effects arising out of internal convection and blade The objectives of this study are to extend the available
metal conduction. Dhiman and Yavuzkurt [16] developed an spanwise-averaged correlations for film-cooling effectiveness
Iterative Conjugate Heat Transfer Technique (ICHT) where and heat transfer coefficient to 2-D form as a function of axial
flow over a film-cooled blade is not solved directly and instead and spanwise directions and to obtain 3-D conjugate and non-
convective heat transfer calculated on a similar blade without conjugate temperatures distributions in film-cooled flat plates
film-cooling and under the same flow conditions are corrected using ICHT-ROFM and compare them.
by use of experimental data to incorporate the effect of film-
cooling on the heat transfer coefficients. The effect of conjugate
heat transfer is taken into account by using this iterative THEORY
technique. This approach is later named ICHT-Reduced Order
Film Modeling (ROFM) technique. Using ICHT for uncooled Techniques for calculating blade temperatures
surfaces, the deviations were as high as 3.5% between
conjugate and conventional technique results for the wall Conventional Technique: Manufacturers of turbine
temperature. In terms of convective heat transfer coefficient, blades often rely upon heat transfer coefficients obtained
this deviation is around 20%. Using ICHT-ROFM for film- through results of constant wall temperature or constant wall
cooled flat plates with high temperature differences, a deviation heat flux experiments to calculate temperature distribution in
of up to 10% in wall temperature and 60% in heat transfer the blade material. Calculating the blade temperature using
coefficients are observed. However, for film-cooling flows with such experimental data results in inaccuracies, since in real
low temperature differences a nominal deviation of up to 3% is applications neither the wall heat flux nor the surface
wall temperature is observed indicating that the conjugate heat temperature distribution remain constant, as discussed by
transfer effect diminishes with decreasing temperature Dhiman and Yavuzkurt[16].
difference.
Full Conjugate Heat Transfer (CHT) Analysis: Full
Several researches have taken detailed and complete data on CHT analysis is an analysis in which the both the external flow
film-cooling effectiveness (𝜂 ) and film heat transfer coefficient and metal conduction are coupled together and provides more
(ℎ𝑓 ) which are needed for calculation of temperatures of film- realistic results. Also it is easier to specify boundary conditions
cooled solid surfaces. Yuen & Martinez’s [4-6] measurements in case of full CHT analysis as conditions at the gas and solid
were made for a single hole and Baldauf et al. [7-8] for a row of interface need not be known. Kane & Yavuzkurt [14], who
holes. This data and others like it have been used to generate performed numerical simulation on a non-film-cooled blade
many correlations for 𝜂 and ℎ𝑓 for many different film-cooling ,reported 30% deviation from data of Hylton and Nirmalan [15]
geometries and flow characteristics. Bunker [17] gives several in using conventional constant wall temperature approach.
forms for correlations commonly used by industry. Colban et al. Whereas with full CHT analysis, the results were much closer
[18] generated film-cooling correlation for shaped holes. to the data with an overall deviation about a few percent. Since
Baldauf et al. [8-10]] used data and developed correlations for the steady turbulence models employed in CHT simulation
film-cooling at engine-like conditions. All of these correlations suffer from known inadequacies, completely reliable prediction
take many data sets into account but all are for spanwise- of heat transfer coefficients in the near field of the jets cannot
averaged film-cooling properties. Since failure of a turbine

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 be obtained. This leads to inaccurate results in full CHT was compared to analytical solution and the full CHT solution.
analysis after a lengthy simulation, Dhiman and Yavuzkurt [16]. The solution was found to be in agreement with both the
analytic solution and full CHT solutions Dhiman and
Iterative Conjugate Heat Transfer (ICHT) and Yavuzkurt [16].
Reduced Order Film Modeling (ROFM) Technique: The
details of ICHT-ROFM method used here are given by Dhiman
and Yavuzkurt [16]. The ICHT technique is a practical 2-D FILM-COOLING EFFECTIVENESS AND HEAT-
compromise between full conjugate simulations and TRANSFER AUGMENTATION CORRELATION
conventional techniques to determine the solid temperature DEVELOPMENT
distribution with which one can limit the inaccuracies of full
CHT by providing experimental input to correctly predict the Effectiveness
temperature field. In this methodology, convection and The correlation developed is partly based on the assumption
conduction domains are loosely coupled, wherein external that the film-cooling effectiveness profile was Gaussian in the
convective heat transfer coefficient provides the boundary spanwise direction centered on the cooling hole exit for inline
condition for conduction in blade metal, corrected by use of injection. This was observed by Ramsey et al. [19] for single
experimental data to incorporate the effect of film-cooling on hole cooling. To examine how the assumed spanwise
the heat transfer coefficients. The effect of conjugate heat distribution can capture the film-cooling effectiveness profile,
transfer is taken into account by using this iterative technique raw data from Lawson [20] is compared with a summation of
known as the ICHT-ROFM process and can be seen in Figure 1 normal distributions. As seen in Figure 2, the Gaussian
taken from [16]. Consideration of conjugate heat transfer distribution can capture much of the behavior in the spanwise
results in different surface temperatures after each iteration. direction. Using the assumed Gaussian profile, existing
This changes the temperature profiles near the wall leading to spanwise-averaged correlations can be used to develop 2-D
change in heat transfer coefficients. New heat transfer correlations. The developed 2-D correlations are formulated to
coefficients give new wall temperature distribution. Therefore satisfy the spanwise-averaged correlations they are based on.
this process is iterated until convergence is achieved and both
wall temperatures and heat transfer coefficients do not change Gaussian distribution: The effectiveness at any
anymore. Iterations stop when convergence is obtained, location (x,z) is assumed to be given by:
indicated by the continuity of temperature and heat flux at the
fluid-metal interface. This process was run on commercial CFD (𝑧�−𝑧�ℎ )2
−
2𝜎(𝑥�)2 (1)
solver ANSYS FLUENT 12. 𝜂(𝑥�, 𝑧̃ ) = 𝜂𝐶𝐿 (𝑥�) 𝑒

This 2-D correlation contains two main parameters: 𝜂𝐶𝐿 (𝑥�): the
centerline effectiveness and 𝜎(𝑥�) the transverse width of
effectiveness. It is spanwise-averaged and the result is
compared with spanwise-averaged correlations given by
Baldauf et al. [8-10] to determine the parameters 𝜂𝐶𝐿 (𝑥�) and
𝜎(𝑥�).

Assuming that for a row of holes the transverse distribution of

effectiveness can be approximated by a sum of Gaussian
distributions, then for H holes:
𝜂𝑡𝑜𝑡𝑎𝑙 (𝑥�, 𝑧̃ ) = 𝜂1 (𝑥�, 𝑧̃ ) + 𝜂2 (𝑥�, 𝑧̃ ) + ⋯ + 𝜂𝐻 (𝑥�, 𝑧̃ )
𝐻

= � 𝜂ℎ (𝑥�, 𝑧̃ )
ℎ=1 (2)
𝐻 (𝑧�−𝑧�ℎ )2
−
= � 𝜂𝐶𝐿 ℎ (𝑥�) 𝑒 2𝜎ℎ(𝑥�)2
ℎ=1

Figure 1. GRAPHICAL REPRESENTAION OF THE ICHT-

ROFM METHOD, DHIMAN AND YAVUZKURT [16]. Spanwise-average of Equation (1) is:

𝐻 (𝑧�−𝑧�ℎ ) 2
Validation of ICHT Approach: Validation of ICHT- 1 ∆𝑧/2 −
ROFM approach has been carried out using spanwise-averaged 𝜂̅ (𝑥�) = � � 𝜂𝐶𝐿 ℎ (𝑥�) 𝑒 2𝜎ℎ(𝑥�)2 𝑑𝑧̃ (3)
∆𝑧 ∆𝑧/2
film-cooling effectiveness and heat-transfer augmentation by ℎ=1
Dhiman and Yavuzkurt [16]. In order to validate the ICHT
approach, the numerical solution obtained by the ICHT scheme

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 1.2
1
Film-cooling effectivness

0.8
0.6
0.4
η

0.2
0
-3 -2 -1 0 1
Correlation Z/P
Data (Lawson [20])
Figure 3. ILLUSTRATION OF DISTANCE FROM HOLE
Figure 2. SPANWISE PROFILE OF FILM-COOLING CENTER TO REGION OF WHERE DATA IS GIVEN
EFFECTIVENESS AT DOWNSTREAM LOCATION
�=0.00981) 𝒙
(𝒙 � = 𝒙/𝑫, 𝒛� = 𝒛/𝑫 Assuming a Gaussian distribution in the lateral direction, then
the spanwise-averaged effectiveness is obtained as follows:

If ∆𝑧 is an order of magnitude larger then the lateral spreading √2𝜋

𝜂̅ = 𝜂𝑝∗ 𝜂 ∗ 𝑓 = 𝜂 (𝑥�) 𝜎(𝑥�) (7)
of the effectiveness, then the spanwise-average simplifies to: ∆𝑧̃ 𝐶𝐿
𝐻
1 Where, 𝜂𝐶𝐿 (𝑥�) is the centerline effectiveness and σ(𝑥�) is the
𝜂̅ (𝑥�)~ � √2𝜋 𝜂𝐶𝐿 ℎ (𝑥�) 𝜎ℎ (𝑥�) (4) standard deviation of the lateral spreading given by the
∆𝑧
ℎ=1 following equations:
𝐶1
Baldauf et al. [8-10] has presented a correlation for a single 𝜂𝐶𝐿 (𝑥�) = (8)
𝜉 (𝑎∗+𝑏∗)𝑐 ∗ 𝑚
hole in cross-flow. The streamwise scaling for this correlation is [1 + ( ) ]
𝜉0
a function of the lateral pitch, which is undefined for a single
hole. The correlation presented appears to begin before the 𝜉 ∗
hole, since the value of the spanwise-averaged effectiveness 𝐶2 ( )𝑎
𝜉0
begins at zero and increases. Baldauf et al. [8] claim this is due 𝜎(𝑥�) = (9)
𝜉 ∗ ∗ ∗
to high levels of entrainment. The recovery region begins some [1 + ( )(𝑎 +𝑏 )𝑐 ]𝑛
𝜉0
distance behind the hole. The placement of the streamwise
coordinate axis is shown in Figure 3. Where
𝜂𝑝∗ 𝜂0 𝑓∆𝑧̃
If 𝜉𝑅 is the distance between the hole center and the location 𝐶1 𝐶2 = (10)
where research data begins, then 𝜉𝑅 could be several hole √2𝜋
diameters(~2-3 D) downstream depending on where 𝜉 is
defined. If 𝑎∗ , 𝑏 ∗ , 𝑐 ∗ match those given in Baldauf et al. [8-10], then the
spanwise-averaged correlation developed by Baldauf et al. [8-
According to Baldauf et al. [8], the spanwise-averaged 10] will automatically be satisfied by the distribution assumed.
effectiveness for a single hole should be given by the following: In the far-field, the spanwise spreading grows asymptotically,
such that the spanwise variations become negligible compared
0.9 to the axial decay of effectiveness. This assumption appears to
�𝑠
𝐷𝑅 𝐷 (5) be true for a row of holes in which the local effectiveness
𝜂̅ = 𝜂𝑝∗ 𝜂 ∗ becomes uniform by 20-30D downstream. From the work done
𝑆𝑖𝑛(𝛼)
on Yuen [3-5] data on single holes, this assumption also seems
to be true for single holes. This assumption gives values of ‘n’
Where 𝜂𝑝∗ & 𝜂∗ are as given in Bauldauf et al. [8] . If
and ‘m’ as follows:
0.9
𝑎∗
�𝑠 𝑛= ∗ (11)
𝐷𝑅 𝐷 (6) (𝑎 + 𝑏 ∗ )𝑐 ∗
𝑓= 𝑏∗
𝑆𝑖𝑛(𝛼) 𝑚= ∗ (12)
(𝑎 + 𝑏 ∗ )𝑐 ∗

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 Assuming that at the hole the value of the local effectiveness is
approximately “1” then C1 and C2, will be:

𝐶1 = 1
𝜂𝑝∗ 𝜂0 𝑓∆𝑧̃ (13)
𝐶2 =
√2𝜋

Where ∆z is the lateral distance over which the effectiveness

was integrated. For the geometry used by Baldauf et al. [8-10],
the width of the test section as 44D which is 0.22 meters. Based
on these, the final distribution for local effectiveness is given
as: Figure 4. ASSUMED TRANSVERSE DISTRIBUTION OF hf/h0
FOR A FILM-COOLING HOLE.
(𝑧�−𝑧�ℎ )2
− (14)
𝜂(𝑥�, 𝑧̃ ) = 𝜂𝐶𝐿 (𝑥�) 𝑒 2𝜎(𝑥�)2
fluid is entrained beneath the jet, causing an increase in heat
transfer. This is consistent with the findings of Yuen et al. [5,6]
Resulting in centerline effectiveness and lateral spreading as �,
on their study with a single hole. Therefore, for a single hole 𝑍’
follows: would be a strong function of M and would effectively go to
1 zero for high blowing ratios (M>1). This is shown in Figure 5.
𝜂𝐶𝐿 (𝑥�) = 𝑏∗
𝜉 ∗ ∗ ∗ (15)
[1 + ( )(𝑎 +𝑏 )𝑐 ](𝑎∗+𝑏∗)𝑐 ∗
𝜉0
1.8
1.7 M=0.33
𝜂𝑝∗ 𝜂0 𝑓∆𝑧̃ 𝜉 𝑎∗
( ) 1.6
√2𝜋 𝜉0 M=0.5
hf/h0

𝜎(𝑥�) = 𝑎∗
(16) 1.5
𝜉 ∗ ∗ ∗
[1 + ( )(𝑎 +𝑏 )𝑐 ](𝑎∗+𝑏∗)𝑐 ∗ 1.4 M=1.0
𝜉0
1.3
These functions are valid in the recovery region where 𝜉 > 𝜉𝑅 . M=2.0
1.2
For large values of x or ξ, it can be shown that the centerline
1 0 1 2
effectiveness approximately behaves like 𝜂𝐶𝐿 ~ 𝑏∗ and the
𝜉 Z/D
1
spanwise-average also behaves like 𝜂̅ ~ ∗ .. This agrees with Figure 5. SPANWISE DISTRIBUTION OF HEAT TRANSFER
𝜉𝑏
the idea that in the far-field the effectiveness is no longer AUGMENTATION AT x/D=2 FOR A SINGLE HOLE IN CROSS-
FLOW WITH Α=30⁰
strongly affected by the lateral spreading, but is dominated by
the decay of the centerline effectiveness. This is consistent with
For a single row of holes, Yu et al. [22] discuss the transverse
the asymptotic growth in the lateral spreading and that in the
distribution shown in Figure 5 at two transverse locations. The
far-field the effectiveness is more uniform.
centerline is at the same transverse location as the center of one
of the cooling holes, and the mid-span is the axial line midway
Heat Transfer Augmentation
between centers of two cooling holes. They discuss two
Figure 4 shows an assumed transverse (spanwise) distribution
competing factors that influence the magnitude of the heat
of the heat transfer augmentation (hf/h0) around a film-cooling
transfer coefficient. The first is the increasing boundary layer
hole. There are three functions that describe this assumed
thickness due to the injection; this causes an increase in
Gaussian distribution. The first is 𝑍�’ = 𝑧 ′ /𝐷 which describes
convective resistance. The second is the enhanced flow shear
the dimensionless distance from the hole to the location of the
induced by the jet interaction with the mainstream flow
max heat transfer coefficient enhancement. The second is the
resulting in an increase in heat transfer. The centerline is
value of the max heat transfer augmentation ((hf/h0) max). The
expected to be more affected by the first factor. Whereas
third is the dimensionless standard deviation around the
between the holes, the second factor would greatly dominate,
maximum heat transfer augmentation locations. All three are
causing an increased heat transfer. So the maximum heat-
functions of the flow parameters and the streamwise location.
transfer augmentation should occur at mid-span between the
Ammari et al. [21] found high heat transfer coefficients at the � of approximately P/2D.This is shown in Figure 7,
edge of the jet for low blowing ratios (M<0.5) due to high shear holes or at 𝑍’
between the main flow and the jet. Large values were on the taken from Baldauf et al. [8].
centerline in the vicinity of the hole for intermediate (M=0.5-
1.0) and large blowing ratios (M>1). This occurs because when
the jet detaches, the hot mainstream

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 ℎ�𝑓
For a single hole assumed � � distribution is:
ℎ0

∆𝑧
���
ℎ𝑓 1 2 ℎ𝑓 −
(𝑧�−𝑧� ′ )2
� �−1= � (� � − 1) 𝑒 2𝜎2 𝑑𝑧 (17)
ℎ0 ∆𝑧 −∆𝑧 ℎ0 𝑚𝑎𝑥
2

For H similar holes:

∆𝑧
���
ℎ𝑓 𝐻 2 ℎ𝑓 −
(𝑧�−𝑧� ′ )2
� �−1= � (� � − 1) 𝑒 2𝜎2 𝑑𝑧 (18)
ℎ0 ∆𝑧 −∆𝑧 ℎ0 𝑚𝑎𝑥
2
The spanwise-average is developed by a similar approach used
in Equations (3) and (4) which yields:

���
ℎ𝑓 𝐻 ℎ𝑓
� �−1= (� � − 1)𝜎 (19)
ℎ0 ∆𝑧√2𝜋 ℎ 0 𝑚𝑎𝑥

Assuming that the same mechanism that governs the spreading

of the film effectiveness also governs the spreading of the heat-
transfer augmentation, then the same correlation for the
spreading can be assumed and is given in Equation (16). With
this assumption the following equation is obtained.

ℎ𝑓 ���
ℎ𝑓 ∆𝑧√2𝜋
� � = �� � − 1� +1 (20)
ℎ0 𝑚𝑎𝑥 ℎ0 𝐻𝜎

ℎ�𝑓
Where� � is given by Baldauf et al. [7] or any other
ℎ0
appropriate spanwise-averaged correlation.

NUMERICAL SIMULATION
Spanwise-averaged-simulations were carried out and given in
Figure 6. CENTERLINE AND MID-SPAN STREAMWISE
by Dhiman and Yavuzkurt [16]. Here new 3-D simulations will
DISTRIBUTION OF HEAT TRANSFER COEFFICIENT FOR A
SINGLE ROW OF HOLES, P/D=3 FOR 3 DIFFERENT
be discussed. The conditions chosen match those used by
SHAPED HOLES AS PRESENTED BY Yu et al. [22]. Dhiman and Yavuzkurt [16] and the spanwised-average of the
developed correlations in this paper agree with those used in
the previous study. This allows comparison between the
spanwise-averaged results from this study and the results from
the previous study by Dhiman and Yavuzkurt [16].

Data Sets Used

3-D Flat Plate Film-Cooling Simulation at Low

Temperature Difference using ICHT-ROFM: Flat plate film-
cooling was simulated using a 3-D computational domain with
experimental input from Yuen et al. [4] which has an extensive
data set on various film-cooling configurations. Table 1 gives
experimental conditions used by Dhiman and Yavuzkurt [16]
for spanwise-averaged simulations. A constant heat flux of 410
W/m2 was used on the bottom surface of the metal plate. A slab
of 163mm thickness was used to investigate the effect of
Figure 7. PHYSICAL PHENOMENON THAT CAUSE HEAT- composite conduction. Numerical simulation employed a three-
TRANSFER AUGMENTATION (hf/h0) ACCORDING TO dimensional grid of size 150 x 250 x 110 was employed for
BALDAUF et al. [8].
external convection and 200x50x100 for metal conduction. The
convection side grid was selected after performing a mesh

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 sensitivity test. The depth of the mesh is 22D. 3-D mesh used is Table 2. FLOW CONDITIONS FOR HIGH TEMPERATURE
shown in Figure 8.The correlations were further modified to DIFFERENCE.
satisfy the spanwise-averaged data presented by Yuen et al. [4] BOUNDARY CONDITIONS
Injection angle α=30⁰
Tabel 1. FLOW CONDITIONS FOR LOW TEMPERATURE
Diameter of hole D 5mm
DIFFERENCE.
Mainstream gas velocity, VG 60m/s
BOUNDARY CONDITIONS
Mainstream temperature, TG 550K
Injection angle α=30⁰
Free stream turbulence, Tu % 1.5 %
Diameter of hole D 10mm
Coolant Temperature Tc 300K
Mainstream gas velocity, VG 13m/s
Blowing Ratio M 1
Mainstream temperature, TG 300K
Free stream turbulence, Tu % Material Properties
2.7 %
Coolant Temperature Tc 280K Material of Plate Corning Macor,
Blowing Ratio M 0.5 Conductivity Corning Macor k 2.0 W/mK
Material Properties
Material of Plate Composite
Conductivity Composite k 1.5 W/mK

Figure 9. 3-D MESH USED FOR SIMULATING THE FLAT

PLATE FILM-COOLING EXPERIMENT FOR HIGH
TEMPERATURE DIFFERENCE.

Figure 8. 3-D MESH USED FOR SIMULATING THE FLAT

PLATE FILM-COOLING EXPERIMENT LOW TEMPERATURE
RESULTS AND DISCUSSION
DIFFERENCE.
Flat Plate Film-Cooling Simulation at Low Temperature
Difference using ICHT-ROFM
Flat Plate Film-Cooling Simulation at High
The distribution of effectiveness and heat-transfer augmentation
Temperature Difference using ICHT-ROFM: Flat plate
obtained from the correlations developed here can be seen in
film-cooling was simulated with a 3-D computational domain
Figures 10 and 11, respectively. The effectiveness shown in
using the geometry of the Baldauf et al. [10] experiment as was
Figure 10 begins at a value around 0.5 and decays downstream.
done in 2-D by Dhiman and Yavuzkurt [16] The geometry has
Around 30D the effectiveness is less than 0.1 and becomes
a single row of seven film-cooling holes with flow parameters
more uniform across the span. In Figure 11, the heat-transfer
are given in Table 2.
augmentation begins at 1.4 and quickly decays to close to 1
around 25D downstream. The two peaks occur at the edge of
The grid shown in Figure 9 was used for the Baldauf et al. [10].
the hole, which agrees with observations by Yuen et al. [5-6]
A mesh size of 150 x 250 x 110 was employed for external
and Ammari et al. [21]. The contours of effectiveness and heat-
convection and 200x50 x100 for metal conduction. This mesh
transfer augmentation is compared with data of Yuen et al.
was selected after performing a mesh sensitivity test and found
[5,6]. This is shown in Figures 12 and 13 for M=0.5 and α=30⁰.
to be sufficient. The Depth of the mesh is 44D to match the
The peak value of heat-transfer augmentation is located at the
geometry in Baldauf et al. [7-10]
hole edge to agree with the theory. The correlation starts with
an effectiveness value of about 0.4 at 2D distance from the hole
and decays to a value of 0.1 at a distance of 18D. This agrees
with the Yuen et al. [5] data, which begins at a value between
0.35-0.4 at a distance of 2D from the hole and decays to a value
of 0.1 around 15D. This is well within the resolution given by

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 Yuen et al. [5]. The contours in Figure 13 show the similarities
between the correlation results and the data given by Yuen et al.
[6]. At around 15D the spanwise variation of the heat-transfer
augmentation has dropped below the resolution of Yuen et al.
[6] data.

a)

Figure 10. EFFECTIVENESS DISTRIBUTION FOR A SINGLE

HOLE WITH M=0.5, α=30⁰ FROM CORRELATIONS.
b)
Figure 12. CONTOURS OF EFFECTIVENESS DATA FOR A
SINGLE HOLE WITH M=0.5, α=30⁰ a) AS GIVEN BY YUEN et
al. [5] b) AS PREDICTED BY CORRELATIONS.

Figure 11. HEAT-TRANSFER AUGMENTATION (hf/h0)

DISTRIBUTION FOR A SINGLE HOLE WITH M=0.5, α=30⁰
FROM CORRELATIONS. a)

Figure 14 shows the results of Dhiman and Yavuzkurt [16] for

the spanwise-averaged geometry. The difference between the
conjugate (Iteration 7) and non conjugate temperature (iteration
1) results is about 5 K as can be seen in Figure 14. The solution
quickly converges as can be seen in Figure 14, as iterations 3-7
give very similar results and lie on top of each other in the
figure.

Figures 12b and 13b show the 2-D contours of the effectiveness
b)
and heat-transfer augmentation from Yuen et al.[5,6] The
conjugate and non-conjugate results were obtained using the 2- Figure 13 CONTOURS OF HEAT-TRANSFER
D correlations developed here for film-cooling on the geometry AUGMENTATION (hf/h0) DATA FOR A SINGLE HOLE WITH
used by Yuen et al. [4]. Boundary conditions employed were M=0.5, α=30⁰ a) AS GIVEN BY YUEN et al. [6] and b) AS
the same as the spanwise-averaged study by Dhiman and PREDICTED BY CORRELATIONS.
Yavuzkurt [16] , the inlet mainstream gas temperature was kept
at 300K and secondary gas coolant temperature at 280K.
Results for surface temperature contours are shown in Figures
15-17.

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 325

320

315 initial guess

iteration_1
310
iteration_2
305
Tw (K)

iteration_3
300 iteration_4
iteration_5
295
iteration_6
290
iteration_7
285
Figure 15. SURFACE TEMPERATURE CONTOURS FOR THE
280 LOW TEMPERATURE DIFFERENCE STUDY FOR NON-
0 0.1 0.2 0.3 0.4 CONJUGATE SOLUTION (FIRST ITERATION).
X(m)
Figure 14. SPANWISED-AVERAGE SURFACE
TEMPERATURE DURING ICHT PROCESS FROM DHIMAN
AND YAVUZKURT [16] FOR LOW TEMPERATURE
DIFFERENCE STUDY.

The resulting temperature distribution of the solid surface, after

the first iteration using the ICHT-ROFM technique (non-
conjugate), is shown in Figure 15. The conjugate results found
in the final iteration of ICHT-ROFM technique can be seen in
Figure 16. The difference of the two iterations can be seen in
Figure 16. SURFACE TEMPERATURE CONTOURS FOR THE
Figure 17. This figure shows a maximum temperature LOW TEMPERATURE DIFFERENCE STUDY FOR
difference of about 6K, or about 10% the maximum CONJUGATE SOLUTION (FINAL ITERATION).
temperature difference or 2% of the local temperature between
conjugate and nonconjugate cases. The conjugate case for this
geometry is cooler overall than the non-conjugate case. This is
because the plate is heated and is being cooled by the
mainstream air and cooling air. This is the reverse of what will
occur in turbines, as the mainstream air is hotter than the
turbine blade. This agrees with the spanwise-averaged run done
by Dhiman and Yavuzkurt [16] shown in Figure 14. The
temperature gradients in the conjugate case are higher than that
in the non-conjugate case.

The variations in the spanwise temperature distribution for the Figure 17. SURFACE CONTOURS OF THE TEMPERATURE
conjugate case are around 6 degrees, as can be seen in Figure DIFFERENCE BETWEEN THE CONJUGATE AND NON-
16. This is on the same order as the temperature difference CONJUGATE HEAT-TRANSFER FOR THE LOW
between the conjugate and non-conjugate cases as can be seen TEMPERATURE DIFFERENCE STUDY.
in Figure 17. This shows that both conjugate and 3-D
simulations are needed to accurately predict the surface Figures 19 and 21 show the contours of the film-cooling
temperature. effectiveness and heat-transfer augmentation obtained from
developed correlations in this study used for the 3-D simulation
Flat Plate Film-Cooling Simulation of High Temperature of Baldauf et al. [10] study. Figures 20 and 22 are more
Difference using ICHT-ROFM detailed contours in the near hole region. Due to the high level
The simulation was performed to investigate the workings of of entrainment near the hole, the location for maximum heat-
the ICHT-ROFM method using the developed 2-D film-cooling transfer augmentation was chosen to occur near the mid-pitch.
correlations for effectiveness and heat-transfer augmentation. This agrees with the observations of Yu et al. [22]. In Figure
Baldauf et al. [7,8] experiment was chosen due to its simple 20, the effectiveness begins with a value of 0.5 at the hole. At
geometry and completeness of experimental conditions. This x/D of 20, the effectiveness has become fairly uniform. This
data was also used by Dhiman and Yavuzkurt [16] for agrees with theory applied when developing these correlations.
spanwise-averaged simulations shown in Figure 18. It shows The heat-transfer augmentation begins at 1.6 and decays to a
that the ICHT process for this case converges quickly in about value below 1.1 around x/D=20 as shown in Figure 22.
5 iterations for the 2-D geometry.

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 The temperature profile along a centerline cut through the plate
is shown in Figure 23. As expected near the hole the plate
temperature is significantly cooler than downstream
temperatures.

The resulting temperature distribution of the solid surface after

the first iteration using the ICHT-ROFM technique (non-
conjugate) is shown in Figure 24 with the near hole region
shown in Figure 25. The conjugate results found in the final
iteration of ICHT-ROFM technique can be seen in Figure 26
and the near hole region in Figure 27. High temperatures can be
seen along the leading edge of the plate and at the mid-pitch Figure 19. EFFECTIVENESS CONTOURS USED IN 3-D
between the holes. The difference of the two iterations can be SIMULATION FOR THE HIGH TEMPERATURE DIFFERENCE
STUDY.
seen Figure 28, with a maximum temperature difference of
about 20K or about 5% of the total temperature or 8% of the
temperature difference.

As can be seen, the differences in temperatures in the spanwise

direction are quite significant, showing a need for 3-D
simulations. Along the spanwise direction, temperatures near
the hole vary between 450K near the mid-pitch and 390K near
the hole. These temperature variations are of the same order of
magnitude as the difference between the conjugate and non-
conjugate solutions.
Figure 20. SIMULATED EFFECTIVENESS CONTOURS OF
Figure 29 shows the spanwise-averaged temperature obtained NEAR HOLES FOR CENTER HOLES FOR THE HIGH
from the 3-D simulation. This is comparable to the final TEMPERATURE DIFFERENCE STUDY OBTAINED FROM
iteration of Dhiman and Yavuzkurt [16] shown in Figure 18. DEVELOPED CORRELATIONS.
The two are similar except the near hole region. The spanwise-
average over estimates the cooling near the hole compared to
the 3-D simulation. This occurs because near the hole there is
significant entrainment of the mainstream gas. The entrainment
causes high values of heat-transfer augmentation near the mid-
pitch while the effectiveness is high near the centerline of the
holes. This phenomenon cannot be captured with only a 2-D
simulation.
550

500
Figure 21. HEAT-TRANSFER AUGMENTATION (hf/h0)
450 CONTOURS USED FOR THE HIGH TEMPERATURE
DIFFERENCE STUDY OBTAINED FROM DEVELOPED
400 Iteration1 (Initial guess)
CORRELATIONS.
Tw (K)

Iteration 2
350
Iteration 3

300 Iteration 4
Iteration 5
250

200
0 0.1 0.2 0.3

X (m)

Figure 18. VARIATION OF SURFACE TEMPERATURE

DURING ICHT PROCESS ON A FILM-COOLED FLAT PLATE- Figure 22. SIMULATED HEAT-TRANSFER AUGMENTATION
SPANWISE-AVERAGED RESULT DHIMAN AND (hf/h0) CONTOURS NEAR CENTER HOLES FOR THE HIGH
YAVUZKURT [16] FOR THE HIGH TEMPERATURE TEMPERATURE DIFFERENCE STUDY OBTAINED FROM
DIFFERENCE STUDY. DEVELOPED CORRELATIONS.

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 Figure 23. CALCULATED TEMPERATURE DISTRIBUTION Figure 27. TEMPERATURE CONTOURS FOR THE HIGH
ALONG CENTERLINE CUT OF SOLID FOR THE HIGH TEMPERATURE DIFFERENCE STUDY, CONJUGATE
TEMPERATURE DIFFERENCE STUDY. SOLUTION (FINAL ITERATION), NEAR- HOLE REGION.

Figure 24. SIMULATED SURFACE TEMPERATURE Figure 28. RESULTS FROM THE CURRENT SIMULATION
CONTOURS FOR THE HIGH TEMPERATURE DIFFERENCE FOR SURFACE TEMPERATURE DIFFERENCE BETWEEN
STUDY, NON-CONJUGATE SOLUTION (FIRST ITERATION). THE CONJUGATE AND NON-CONJUGATE HEAT-
TRANSFER FOR THE HIGH TEMPERATURE DIFFERENCE
STUDY.

Figure 25. SIMULATED SURFACE TEMPERATURE

CONTOURS FOR THE HIGH TEMPERATURE DIFFERENCE
STUDY, NON-CONJUGATE SOLUTION (FIRST ITERATION),
NEAR-HOLE REGION.
Figure 29. SPANWISED-AVERAGED TEMPERATURE OF
FINAL ITERATION FOR THE 3-D SIMULATION OF FOR THE
HIGH TEMPERATURE DIFFERENCE STUDY.

CONCLUSIONS
The current study extends the spanwise-averaged correlations
for film-cooling effectiveness and film heat transfer coefficients
into 2-D correlations using a Gaussian distribution in the
transverse direction. These correlations are obtained using
limited spanwise data and more available spanwise-averaged
data and existing spanwise-averaged correlations for single
Figure 26. SIMULATED SURFACE TEMPERATURE row of holes with inline injection.. These correlations are
CONTOURS FOR THE HIGH TEMPERATURE DIFFERENCE functions of different flow parameters such as mass flow ratio
STUDY, CONJUGATE SOLUTION (FINAL ITERATION). M, density ratio DR, , transverse pitch P/D, and inline injection
angle α, with ranges of M:0.2-2.5, DR: 1.2,1.5,1.8, P/D: 2, 3,5,
α: 30, 60, 90 degrees. These correlations are used to calculate

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 solid flat plate temperatures using ICHT-ROFM techniques 0 = No film-cooling
developed earlier for two well-documented cases of film-cooled CL = Centerline
flat plates. Spanwise variations in the metal temperature are h = relating to a particular hole
temperature are between 5-6K for a temperature difference of P or max=maximum value
40K and 20-30K for a temperature difference of 250K and are
quite significant, showing the need for 3-D simulations. The Superscript
comparisons of solid temperatures for conjugate and non- *= baseline case Baldauf et al. [8]
conjugate heat transfer cases show about 6K or 2% of the local
temperature for low temperature study and about 20K or 5% for Accent
high temperature study. The study showed the difference 𝑋� = laterally averaged quantity X like 𝜂�
between conjugate and non-conjugate solutions increases as the 𝑥� = normalized on the hole diameter D
temperature levels increase. Therefore it can be said that the
calculations of 3-D temperature distributions using conjugate
heat transfer are very important for design purposes. REFERENCES
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� � =dimensionless streamwise coordinate Baldauf et al. [8]
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C = Secondary gas or coolant side [10] Baldauf, S., Schulz, A., and Wittig, S., 1999, ‘‘High
G = mainstream gas side Resolution Measurements of Local Heat Transfer Coefficients

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 From Discrete Hole Film Cooling,’’ASME J. Turbomach., 123, [22] Yu, Y., Yen, C.H., Shih, T. I.P., Chyu, M. K., and
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