James D. Heidmann

Glenn Research Center, Cleveland, Ohio

Scott D. Hunter

General Electric Aircraft Engines, Evendale, Ohio

Coarse Grid Modeling of Turbine Film Cooling

Flows Using Volumetric Source Terms

NASA/TM—2001-210817

May 2001

2001–GT–138
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NASA/TM—2001-210817

May 2001

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

2001 Turbo Expo

sponsored by the American Society of Mechanical Engineers

New Orleans, Louisiana, June 4–7, 2001

2001–GT–138

James D. Heidmann

Glenn Research Center, Cleveland, Ohio

Scott D. Hunter

General Electric Aircraft Engines, Evendale, Ohio

Coarse Grid Modeling of Turbine Film Cooling

Flows Using Volumetric Source Terms
Available from

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Available electronically at http://gltrs.grc.nasa.gov/GLTRS
COARSE GRID MODELING OF TURBINE FILM COOLING FLOWS

USING VOLUMETRIC SOURCE TERMS

James D. Heidmann

Glenn Research Center

Cleveland, Ohio 44135

Scott D. Hunter

General Electric Aircraft Engines

Evendale, Ohio 45215

ABSTRACT

The recent trend in numerical modeling of turbine film cooling

flows has been toward higher fidelity grids and more complex

geometries. This trend has been enabled by the rapid increase in

computing power available to researchers. However, the turbine design

community requires fast turnaround time in its design computations,

rendering these comprehensive simulations ineffective in the design

cycle. The present study describes a methodology for implementing a

volumetric source term distribution in a coarse grid calculation that can

model the small-scale and three-dimensional effects present in turbine

film cooling flows. This model could be implemented in turbine design

codes or in multistage turbomachinery codes such as APNASA, where

the computational grid size may be larger than the film hole size. 

Detailed computations of a single row of 35 degree round holes on a flat

plate have been obtained for blowing ratios of 0.5, 0.8, and 1.0, and a

density ratios of 1.0 and 2.0 using a multiblock grid system to resolve the

flows on both sides of the plate as well as inside the hole itself. These

detailed flow fields were spatially averaged to generate a field of

volumetric source terms for each conservative flow variable. Solutions

were also obtained using three coarse grids having streamwise and

spanwise grid spacings of 3d, 1d, and d/3. These coarse grid solutions

used the integrated hole exit mass, momentum, energy, and turbulence

quantities from the detailed solutions as volumetric source terms. It is

shown that a uniform source term addition over a distance from the wall

on the order of the hole diameter is able to predict adiabatic film

effectiveness better than a near-wall source term model, while strictly

enforcing correct values of integrated boundary layer quantities.

NOMENCLATURE

Ah hole cross-sectional area

cp specific heat

cfl Courant number

d film cooling hole diameter

DR density ratio = ρc/ρin

e specific internal energy

I momentum ratio = (ρV2

)c/(ρV2)in

k turbulence kinetic energy

M blowing ratio = (ρV)c/(ρV)in

Pr Prandtl number

Prt turbulent Prandtl number

qn updated conservative variable

qo old conservative variable

R gas constant for air

Rq residual

Re Reynolds number

s effective slot width

Sq source term

∆t local time step

T temperature

u x component of velocity

v y component of velocity

V absolute velocity

(vol)R residual volume

(vol)S source term volume

w z component of velocity

x streamwise distance from hole leading edge

y transverse coordinate from hole centerline

y+ dimensionless grid spacing

z normal coordinate from wall

η film effectiveness

ρ density

ω dissipation per unit turbulence kinetic energy

Subscripts

aw adiabatic wall conditions

c coolant conditions

in freestream inlet conditions

o stagnation conditions

National Aeronautics and Space Administration

NASA/TMÑ—2001-210817 1
INTRODUCTION

As gas turbine inlet temperatures increase in the drive toward higher

cycle efficiencies, film cooling of high pressure turbine blades is

increasingly needed to ensure their durability. However, film cooling

flows are complex three-dimensional flows involving streams of different

temperature, and are not easily described using low-order models. In

addition, the scale of film cooling flows is much smaller than the scale

over which the aerodynamic design of turbine systems is typically

performed. Thus film cooling systems have generally been designed

using databases of information from detailed studies. However, in order

to optimize the aerodynamics and heat transfer of a film-cooled turbine

blade, it would be highly desirable to have a model which would account

for the detailed effects of film cooling on a coarser grid such as the type

used for turbine aerodynamic design. Such an optimization would reduce

the amount of cooling flow required by placing the cooling air only

where it is required, and reducing overcooling.

Goldstein (1971) summarizes early studies in the area of film

cooling. These studies concentrated on slot flows, and film cooling

effectiveness values were found to correlate well with the parameter

x/Ms, where x is the downstream distance, M is the blowing ratio, and s

is the slot width. This parameter has also been used for discrete hole

cooling, with s defined as the effective slot width for the row of holes.

 However, the physics of discrete hole cooling is quite different from that

of a slot. A row of discrete holes typically has a much lower spanaveraged downstream film effectiveness distribution for the same x/Ms

due to the formation of vortices which allow hot gas to penetrate to the

wall. These vortices are of the scale of the hole size, so if a numerical

simulation has a spanwise grid spacing greater than the film hole

spanwise pitch, as is typical for turbine blade aerodynamic design, their

effect is lost. In essence, any such calculation is two-dimensional on the

scale of the film holes.

To overcome this problem, several computational studies have

computed turbine blade geometries with accurate resolution of the film

holes, and in some cases, of the hole pipes and plena as well. Garg and

Gaugler (1997) showed the importance of film hole exit profiles. Garg

and Rigby (1999) resolved the plenum and hole pipes for a three-row

showerhead film cooling arrangement, and Heidmann, et al. (2000)

computed the heat transfer for a realistic turbine vane with 12 rows of

film cooling holes with shaped holes and plena resolved. Garg (2000)

presented results of a full rotating blade with 172 film holes, resolving

the film hole exits, but not the hole pipes and plena. These studies are

appropriate in the research environment, but are too computationally

intensive for turbine design. A simplified model is thus needed.

Many published studies have discussed the physics and presented

data for discrete hole film cooling in simplified geometries. The most

basic geometry consists of a row of round holes in a flat plate. There is

a relatively large body of experimental data for 35 deg. round holes with

a pitch of 3d (Pedersen, et al., 1977; Foster and Lampard, 1980; Pietrzyk,

et al., 1989; Pietrzyk, et al., 1990; Sinha, et al., 1991). This geometry

allows for a study of jet lift-off behavior at various blowing ratios and is

perhaps the most realistic simplified geometry for turbine film cooling.

 In addition, the computational study of Leylek and Zerkle (1994) uses

this geometry and gives an excellent description of the vortical flows

associated with this geometry. The present study will therefore use this

geometry in the development of the film cooling model.

A source term model for film cooling flows has been described by

Hunter (1998), and implemented in the APNASA code (Adamczyk,

1985). In this model, source terms of mass, momentum, and energy are

added at the blade surface to account for the flux of these quantities due

to film cooling or other types of injection. The source terms are added

at a single computational cell adjacent to the wall at the hole exit

location. The typical distance from the wall for the cell adjacent to the

wall is 0.015d for these wall function-type grids. The source terms are

inferred from the known blowing ratio, hole angle, and temperature ratio.

 However, the flow produced by these source terms behaves as slot flow

for large spanwise grid spacings, because they are applied in a constant

manner across the blade span. It is thus the goal of this research effort to

develop a model to account for the three-dimensional mixing of discrete

hole injection on the coarse grids typical for turbine aerodynamic design.

 The test of this model will be its ability to predict span-averaged film

effectiveness downstream of a row of discrete film holes in a coarse grid

calculation.

COMPUTATIONAL METHOD

The initial approach of the study was to determine the source terms

of mass, momentum, energy, and turbulence quantities which must be

added in a two-dimensional (slot) calculation to yield the same flow

solution as the span-average of a full three-dimensional calculation

including the plenum and hole pipe. The geometry of choice is shown

in Figure 1. The holes are aligned with the streamwise direction with an

inclination angle of 35 deg. The hole spanwise pitch is 3d. The

geometry is the same as that of Leylek and Zerkle (1994) and Sinha et al.

(1991).

x

y

z

freestream inlet: x/d=-19

exit: x/d=30

symmetry: y/d=10

wall: y/d=0, y/d=-2

symmetry: z/d=-1.5, z/d=0

coolant inlet: y/d=-6

symmetry: x/d=-13

symmetry: x/d=9

35o round hole, s/d=3

Figure 1: Geometry definition and boundary conditions.

 A multi-block computational grid was developed using the GridPro

multiblock grid generator with 15 blocks and approximately 162,500

computational cells as shown in Figures 2 and 3. Viscous clustering was

employed at all solid walls with a y+

 value less than 1.0 at all locations.

Stretching ratios less than 1.2 were used normal to the viscous walls. A

second 15 block grid was produced having approximately 1,300,000

computational cells – twice the number of cells in all three directions.

The grid spacing at solid walls was also half that of the first grid. This

was done to ascertain the grid dependence of the solutions. Each solution

on the initial grid required approximately 10 CRAY C-90 CPU hours to

converge. The finer grid required approximately an additional 50 hours

to converge, starting from the coarser grid solution. Convergence was

considered achieved when both of the following criteria had been met:

1) reduction in all residuals of three orders of magnitude, and 2) no

observable change in surface temperature prediction for an additional 200

iterations. It was found that the solution indeed required at least the

resolution of the finer grid, since the calculation on the coarser grid

underpredicted jet reattachment and film effectiveness. Because of

computational limitations, the grid was not refined further. Symmetry

NASA/TMÑ—2001-210817 2
boundary conditions were employed in the spanwise direction, on all

sides of the plenum, and at the y/d=10 plane. Fixed mass flow rate and

stagnation temperature inlet boundary conditions to the plenum and

freestream were used to ensure proper density and blowing ratios. The

inlet flows were normal to the inlet planes. Adiabatic no-slip conditions

were applied at all solid walls, including the inner surface of the film hole

and the plenum. A turbulence intensity of 0.2% and a turbulence length

scale of 10% of the inlet height were used. The Reynolds number based

on hole diameter and inlet conditions was 16100. An exit boundary

condition with fixed static pressure was employed at x/d=30. A

maximum Mach number not exceeding 0.3 was achieved in the flow field

while maintaining the desired Reynolds number by scaling the

experimental geometry down by a factor of 5. This resulted in a hole

diameter of 2.54 mm, and was done to allow more rapid convergence of

the solution using the density-based formulation of the computer code

while minimizing compressibility effects.

Figure 2: Computational grid at z=0.

Figure 3: Computational surface grid near hole.

Six numerical simulations were performed for this geometry using

different density and blowing ratios. These conditions are shown in

Table I. The primary focus of the study is on cases 1-3, for which the

density ratio is 2.0 – typical of gas turbine engine operation. The density

ratio 1.0 cases were primarily used as a check on source term behavior.

Table I: Detailed Calculation Cases

Case Density ratio Blowing ratio Momentum ratio

1

2

3

4

5

6

2.0

2.0

2.0

1.0

1.0

1.0

0.5

0.8

1.0

0.5

0.8

1.0

0.125

0.320

0.500

0.250

0.640

1.000

The simulations in this study were performed using a multiblock

computer code called Glenn-HT, previously known as TRAF3D.MB

(Steinthorsson et al., 1993) which is based on a single block code

designed by Arnone et al. (1991). This code is a general-purpose flow

solver designed for simulations of flows in complicated geometries. The

code solves the full compressible Reynolds-averaged Navier-Stokes

equations using a multi-stage Runge-Kutta-based multigrid method. It

uses the finite volume method to discretize the equations. The code uses

central differencing together with artificial dissipation to discretize the

convective terms. The overall accuracy of the code is second order. The

present version of the code (Rigby, 1996, Rigby et al., 1997 and Ameri

et al. 1997) employs the k-ω turbulence model developed by Wilcox

(1994a, 1994b), with subsequent modifications by Menter (1993) as

implemented by Chima (1996). The k-ω turbulence model is desirable

because it does not require specification of distance to the wall. Such a

specification is difficult for complex geometries requiring multi-block

grids, such as is considered in the present study. Accurate heat transfer

predictions are possible with the code because the model integrates to the

walls and no wall functions are used. Rather, the computational grid is

generated to be sufficiently fine near walls to produce a y+

 value of less

than 1.0 at the first grid point away from the wall. For heat transfer a

constant value of 0.9 for turbulent Prandtl number, Prt , is used. A

constant value of Pr=0.72 is used. Laminar viscosity is a function of

temperature through a 0.7 power law (Schlichting, 1979) and cp is taken

to be a constant.

 In order to facilitate the spanwise-averaging of solutions and source

terms, the full three-dimensional solutions were interpolated to the cell

centers of a rectilinear grid with the same near-wall grid spacing and

constant grid spacing in the streamwise and spanwise directions. The

rectilinear grid discretized only the region above the plate, since this is

the region for which it is desired to match solutions with the model. The

quantities interpolated were the conservative variables: density, three

components of momentum, energy, turbulence kinetic energy, and

dissipation per unit turbulence kinetic energy. The conservative variables

were then span-averaged on a volumetric basis to yield a twodimensional flow field that represents the span-average of the full threedimensional flow field above the plate. This is the two-dimensional

solution that is desired for the coarse grid solution with the model. This

two-dimensional flow field was used as the starting solution for a single

Runge-Kutta time step in order to compute the time derivatives of all

conservative variables in space. A no-slip boundary condition was used

at the hole exit plane. The time derivatives of conservative variables are

the quantities that must be subtracted from the residuals at each time step

to maintain the desired two-dimensional flow field in time. The

negatives of the time derivatives of conservative variables are known as

the source terms of these variables. At each time step in the solution, the

source terms are added in each conservative equation in the following

manner:

qn = qo + (cfl)∆t[Rq/(vol)R + Sq/(vol)S] (1)

NASA/TMÑ—2001-210817 3
where qn is the updated conservative variable (ρ, ρu, ρv, ρw, ρe, ρk, ρω),

 qo is the conservative variable from the previous iteration, cfl is the

global Courant number, ∆t is the local time step, Rq is the local residual

for the conservative variable q, (vol)R is the volume of the computational

cell, Sq is the source term of the conservative quantity q, and (vol)S is the

volume of the space over which the source term is applied. In this case,

where the source term is defined at each computational cell in the

domain, (vol)S is the volume of each computational cell.

The source terms must be corrected for interpolation and

discretization error incurred in the process. To do this, source terms for

the three-dimensional interpolated flow field were computed, spanaveraged, and subtracted from the original source terms. This process

corrects for the nonconservative interpolation process and the different

local grid spacing between the two grids. As a check on the correction

process, the predicted source terms of mass were found to be zero within

machine accuracy. This must be true, because mass must be conserved

in the two-dimensional flow field since density was volumetrically spanaveraged. The source term of z-momentum is also zero because the zcomponent of momentum is zero in the two-dimensional flow field. This

leaves x- and y-momentum, energy, and the two turbulence quantities as

source terms that must be modeled to precisely account for the spanaveraging.

Second and fourth order artificial dissipation were not used in the

determination of source terms. This assured that only its direct neighbors

determined the source terms at any cell. This also means that the only

source terms affected by varying boundary conditions were those for cells

immediately adjacent to boundaries. This becomes important at cells

next to the film hole inlet. Because it is very difficult to span-average

boundary conditions exactly, a jump in the source terms was sometimes

noted at these cells. It was decided to ignore these cells, since we are

primarily interested in a model of the source terms which varies smoothly

as it approaches the boundary. In fact, the source terms are found to be

smooth except at the boundary cell. In addition, we are interested in the

source terms that arise due to span-averaging in the flow field. The

source term model of Hunter (1998) models the source terms that arise

due to addition of flow at the boundary.

COMPUTATIONAL RESULTS

Figure 4 shows the computed hole centerline profiles of stagnation

temperature for cases 1-3: density ratio of 2.0 and blowing ratios of 0.5,

0.8, and 1.0. It can be seen that the colder jet fluid penetrates deeper into

the freestream as the blowing ratio increases. This is accompanied by an

increase in entrained hot freestream fluid under the coolant jet,

particularly in the first few hole diameters downstream of the hole

trailing edge. This is in contrast to the stagnation temperature profile for

a two-dimensional slot. In a two-dimensional calculation, the colder jet

fluid is pushed much closer to the wall by the freestream because of the

inability of the freestream to penetrate the slot flow. This difference in

behavior increases with blowing ratio. The difference in coolant flow

trajectory is responsible for a much higher span-averaged film

effectiveness in a two-dimensional slot calculation for the same global

blowing parameters, as described by Goldstein (1971) and many others.

 It may also be noted that, particularly for the higher blowing ratio cases,

the increase in stagnation temperature along the jet centerline is not

monotonic. That is, an island of relatively cool fluid appears downstream

of the jet exit. This is due to the vortical nature of the flow. The cool

region is actually the cool jet core, which has been swept away from the

hole centerline by a pair of counterrotating vortices, and is brought back

to the centerline further downstream. Figure 5 shows the appearance of

these vortices for cases 1-3. It can be seen that the jet lift-off behavior

increases with blowing ratio as the increased strength of the

counterrotating vortices enhances entrainment of the hot freestream under

the cool jet. In fact, the coolant jet core has actually been divided into

two at the highest blowing ratio of 1.0 as the vortices push hot gas up

through the center of the jet.

M=0.5

M=0.8

M=1.0

Figure 4: Computed centerline stagnation temperature

contours, DR=2.0.

Film effectiveness for a low Mach number flow may be defined as:

Figure 6 shows comparisons of the computed span-averaged film

effectiveness with the data of Sinha et al. (1991) for cases 1-3. The

computations capture the reduction in film effectiveness with increased

blowing ratio that results from the increased entrainment and jet lift-off.

 The computations predict a sharp minimum in effectiveness at the film

hole trailing edge (x/d=1.74), and a rapid increase until about x/d=5,

which corresponds to jet reattachment. This sharp minimum is not

reflected in the experimental data, but it may be missed due to the

sparseness of the data. The experimental peak effectiveness occurs

further downstream for the two higher blowing ratio cases. The crossover in experimental effectiveness between the two lower blowing ratio

cases at x/d=10 is hinted at in the computation, but also appears further

downstream. In general, however, the agreement with experimental data

is very good for computational film cooling, which is known to suffer

from the presence of anisotropy not modeled by the present turbulence

model.

As described in the computational method section, the detailed

three-dimensional solutions were span-averaged and the source term

distributions required to maintain the span-averaged solution in a twodimensional solution were computed. Figure 7 shows a vector plot of the

momentum source terms for case 1. The source terms of momentum are

)2( ,

,

cino

awino

TT

TT

−

− =η

NASA/TMÑ—2001-210817 4
M=0.5

M=0.8

M=1.0

Figure 5: Crossflow vectors and stagnation temperature

contours for DR=2.0, x/d=5.0.

0 5 10 15 20 25 30

distance downstream of hole leading edge, x/d

0.0

0.1

0.2

0.3

0.4

spanaveraged film effectiveness, eta

M=0.5, computation

M=0.8, computation

M=1.0, computation

M=0.5, data of Sinha et al.

M=0.8, data of Sinha et al.

M=1.0, data of Sinha et al.

Figure 6: Comparison of computed span-averaged

effectiveness with data of Sinha et al. (1991).

Figure 7: Momentum source term vectors computed from

span-average of full 3-D flow field, DR=2.0, M=0.5.

highest near the hole exit, and are generally positive in the x direction

and negative in the y direction. It should again be noted that these values

are at least two cells from the inlet, and thus unaffected by the boundary

condition at the wall. In fact, the direction of the momentum source

vector is different from that of the momentum vector itself at the

boundary. This source term makes intuitive sense, because a slot flow

disrupts the freestream much more than a row of discrete holes with the

same gross flow properties. While the discrete jets may themselves

penetrate deeper into the freestream, the span-average of this flow is

much more aligned with the freestream than the slot flow. The

momentum source terms are in the right direction to flatten the jet as it

is predicted on a coarse grid. Downstream of the hole, there is a

momentum source term away from the wall, which decays slowly in the

streamwise direction.

The source terms of energy are shown in Figure 8. Unlike the

momentum source terms, the maximum residuals occur away from the

NASA/TMÑ—2001-210817 5
wall in the upstream interaction zone of the jets with the freestream. This

is again due to the penetration of the discrete jets into the mainstream.

 Referring to Figure 4, the stagnation temperature is lower near the

forward edge of the jet due to its penetration than it would be for twodimensional slot flow. This results in a negative source term of energy

in this interaction zone. A positive source term of energy exists

immediately downstream of the hole exit, where in the three-dimensional

solution the stagnation temperature increases due to hot freestream fluid

entrainment.

Figure 8: Energy source term contours computed from

span-average of full 3-D flow field, DR=2.0, M=0.5.

The energy source terms are consistent with the vortical structure of

the film cooling flow as shown in Figure 5. These results agree with

previous experimental data of Pietrzyk (1989) and calculations by Leylek

and Zerkle (1994), as all three show the counterrotating vortex structure

downstream of the hole. As the freestream encounters the film cooling

jet, it is entrained in the counterrotating vortex structure and moved away

from the centerline of the jet. This allows colder jet fluid to displace the

hotter freestream fluid. However, the span-averaged y-momentum does

not reflect this penetration due to the motion of the opposite edge of the

vortex, which is downward. Thus a negative source term of energy arises

to model the jet penetration in the absence of convective motion in the

span-average. Since the goal of this study is to model the adiabatic wall

temperature downstream of the film cooling holes, it is very important to

model the hot fluid entrainment which gives rise to the energy source

terms.

A consequence of the preceding explanation is that the energy

source terms will depend strongly on the temperature ratio (or,

equivalently for single constituent flows, on the density ratio). To test

this hypothesis, cases 4-6 was computed, having the same blowing ratios

as cases 1-3, but with coolant temperature equal to the freestream

(density ratio of 1.0). It was found that the momentum source term

distributions are very similar to the corresponding cold jet cases, while

the energy source terms are nearly zero. This data indicates that the

energy source terms are caused by the temperature difference between the

coolant and the freestream.

MODELING

When the precise source term fields were applied in their entirety to

the two-dimensional solution, the exact span-average of the full threedimensional solution was recovered. This is of course expected, because

they are functionally defined in this manner. However, this model is too

complex since it requires detailed descriptions of all 7 unknowns:

density, three components of momentum, energy, and two turbulence

model unknowns. Even when the span-averaging is done conservatively,

which drops the mass source term, and z-momentum is dropped because

the span-averaged z-momentum is identically zero over a hole pitch,

there are still 5 source term fields to model. A first attempt to reduce the

complexity of the model was to ignore turbulence variable source terms.

 But even with exact representations of the momentum and energy source

terms, the adiabatic film effectiveness downstream of the hole was

completely different in the two-dimensional case. Several other attempts

at simplified models indicated that anything less than full source term

modeling resulted in poor representation of the three-dimensional effect.

Because of these difficulties, a different tack was taken. The turbine

heat transfer community is primarily concerned with the wall temperature

downstream of the film hole. The problem with the existing source term

model is that for spanwise grid spacings larger than the film hole pitch,

the model represents a slot and overestimates film effectiveness. This is

because the source terms are added only at the cell adjacent to the wall.

 For a design calculation using wall functions, this would correspond to

a distance of about 0.015d from the wall. Clearly, this will underestimate

the jet penetration and mixing. Even for coarse grids fine enough to

resolve the spacing between holes (spacings on the order of hole

diameter), the streamwise vorticity will be underpredicted in the

calculation, leading to overprediction of film effectiveness.

A simple way to overcome these difficulties is to distribute the

source terms of mass, momentum, energy, and turbulence quantities over

a thicker layer - on the order of the hole diameter - rather than on the

order of one wall function grid cell (0.015d). This would have the

benefit of accounting for the jet penetration and mixing that the coarse

grid calculation cannot resolve due to its inability to capture the vorticity

which produces it. This approach is akin to the penetration length

approach of Dahlander (1998) and Holmer et al. (2000). While

Dahlander (1998) recommended a penetration length of Md, Holmer et

al. (2000) found their best agreement for realistic conditions with a

penetration length of 1.0d. For the present study, a distributed source

term layer of 1.0d is used. It is expected that the problems with the

existing source term model would become more pronounced at larger

grid spacings, especially as the spanwise grid spacing exceeds the hole

pitch. For this reason, a series of calculations were performed with

Glenn-HT for the flat plate with 35 degree hole with three constant

spanwise and streamwise grid spacings: 3d, 1d, and d/3. Only the flow

domain above the plate was modeled. The grid spacing normal to the

wall of y+ < 1.0 was maintained from the detailed grid. The integrated

source terms of mass, momentum, energy, and turbulence quantities were

found from integration of the hole exits fluxes of these quantities from

the detailed calculations, and are shown in Table II. Referring to

Equation (1), (vol)S is now the volume over which the source terms are

applied per hole. For example, for the 3d grid with the 1d source term

layer, the volume per hole is 3d x 3d x 1d, or 9d3

. The quantities in

Table II have been normalized by the appropriate average inlet flow

quantities and hole length scales to facilitate application to new cases.

 It can be seen that the mass and energy fluxes scale linearly with

blowing ratio, as expected. The other quantities follow a higher order

relation.

Figures 9-11 show the effect of adding source terms uniformly over

a distance from the wall of 1d (32 cells from the wall on the current grid)

at the hole centerline grid cell location versus the near-wall model of one

computational cell on a wall function grid (0.015d, or 10 cells from the

wall on the current grid) for coarse grids having streamwise and

transverse cell sizes of 3d, 1d, and d/3, respectively. All three figures are

for case 3, DR=2.0, M=1.0. This case was chosen because it has the

highest level of vorticity, entrainment, and jet lift-off. The results are

representative of cases 1 and 2 as well. For the two finer grids (1d and

d/3), the source terms were only added in cells which lay above the actual

hole exit, so these cases did have some spanwise nonuniformity, unlike

NASA/TMÑ—2001-210817 6
Table II: Integrated Source Terms (Sq) Per Hole for Cases 1-3, DR=2.0

Blowing Ratio (M) 0.5 0.8 1.0

mass flux 0.494(ρV)inAh 0.794(ρV)inAh 0.997(ρV)inAh

x-momentum flux 0.162(ρV2

)inAh 0.364(ρV2)inAh 0.544(ρV2)inAh

y-momentum flux 0.082(ρV2

)inAh 0.200(ρV2)inAh 0.312(ρV2)inAh

z-momentum flux 0 0 0

energy flux 0.640(ρV)inRTinAh 1.023(ρV)inRTinAh 1.281(ρV)inRTinAh

turbulence kinetic

energy flux

0.000540(ρV3

)inAh 0.000915(ρV3)inAh 0.001741(ρV3)inAh

dissipation per unit

turbulence kinetic

energy flux

1.87(ρV2

)ind 4.49(ρV2)ind 6.91(ρV2)ind

−20 −10 0102030

distance downstream of hole leading edge, x/d

−0.5

0.0

0.5

1.0

1.5

2.0

span−averaged film effectiveness, eta 

data of Sinha et al. 	

detailed computation 	

coarse grid computation, near wall source terms 


coarse grid computation, thick layer source terms 


Figure 9: Comparison of span-averaged effectiveness for 2

source term models, grid spacing=3d, DR=2.0, M=1.0.

−20 −10 0
10
20
30


distance downstream of hole leading edge, x/d

−0.5

0.0

0.5

1.0

1.5

2.0

span−averaged film effectiveness, eta

data of Sinha et al. 

detailed computation 

coarse grid computation, near wall source terms 

coarse grid computation, thick layer source terms 

Figure 10: Comparison of span-averaged effectiveness for

2 source term models, grid spacing=1d, DR=2.0, M=1.0.

−20 −10 0102030

distance downstream of hole leading edge, x/d 

−0.5

0.0

0.5

1.0

1.5

2.0

span−averaged film effectiveness, eta 

data of Sinha et al. 

detailed computation 

coarse grid computation, near wall source terms 

coarse grid computation, thick layer source terms 

Figure 11: Comparison of span-averaged effectiveness for

2 source term models, grid spacing=d/3, DR=2.0, M=1.0.

the coarsest grid (3d). The standard no-slip wall boundary condition was

applied at all wall locations, including over the hole exit. It can be seen

that for all three grids the near-wall source term model grossly

overpredicts film effectiveness downstream of the hole due to the

unrealistic introduction of the coolant source terms so close to the wall.

 This makes intuitive sense, because the near-wall model squeezes all the

coolant into a very thin region at the wall. The coarse grids are not fine

enough to capture the vorticity that allows the three-dimensional flow to

advect from the wall, so the coolant remains too close to the wall and

produces film effectiveness predictions which are too high. The nearwall model does slightly better on the finest coarse grid (d/3, Figure 11)

because some of the three-dimensional effects are beginning to be

predicted, but the distributed source term model is still needed to better

model the entrainment effect. The near-wall solutions also exhibit a large

overshoot in the temperature at the wall just upstream of the hole. The

distributed source term solutions exhibit this overshoot as well, but to a

much lesser degree. This is due to Gibbs phenomenon (Foster and

Richards, 1991), where an overshoot results from the discontinuous

application of source terms. The problem is much worse in the near-wall

model due to the much higher source terms per unit volume.

NASA/TMÑ—2001-210817 7
Figures 12-14 show the effect of grid spacing on the thick source

term model implementation for cases 1-3, respectively. The finest grid

(d/3) better matches the downstream (x/d > 5) behavior of the data and

detailed calculation, but overpredicts the effectiveness near the hole (x/d

< 5). This is due to the inability of the model to capture the jet

detachment that causes the lower effectiveness values in the data and

detailed calculation. This might be remedied through a more realistic

distribution of source terms, instead of the uniform distribution of the

present model. However, such an improvement would be at the expense

of the simplicity of the present model, since it would require knowledge

of the distribution of the source terms for various blowing ratios. The

coarser grids (3d and 1d) actually better predict the near-hole film

effectiveness. This is because their source term additions are not as large

per unit volume, resulting in a coolant profile that is more evenly

distributed. This indicates that the distributed source term model loses

some of its advantages as the grid becomes finer. It is expected that as

the grid approaches the resolution of the detailed three-dimensional grid,

the near-wall source term model would surpass the distributed source

term model in predictive capability due to the increased ability to resolve

the vorticity. However, it appears that for grids having streamwise and

transverse cell sizes on the order of the film cooling hole diameter or

larger, the distributed source term model greatly improves upon the nearwall model. The inability of the distributed source term model to match

the highest blowing ratio case is a result of the absence of downstream

vortex modeling. The finest grid (d/3) does capture this effect far

downstream, but all grids fail near the hole, due to the strong jet

separation for the highest blowing ratio. Thus it appears that

volumetrically-uniform source term models are better suited for lower

blowing ratio cases where jet separation is not the dominant feature.

SUMMARY AND CONCLUSIONS

Detailed calculations have been performed for a flat plate with a row

of 35 deg. round holes at blowing ratios of 0.5, 0.8, and 1.0, and density

ratios of 1.0 and 2.0. The results were compared with experimental data

of Sinha et al. (1991). Some of the local phenomena are not

corroborated, but in general the predictions are in very good agreement

with the data, especially considering the known difficulty in predicting

film effectiveness values for film cooling. The detailed results were then

used to compute the source term distributions of mass, momentum,

energy, and turbulence quantities resulting from the span-averaging of

this highly three-dimensional flow field. It was shown that exact

implementation of these source term fields indeed produces the desired

result: a two-dimensional calculation of the film-cooled plate on a coarse

grid that matches the span-averaged three-dimensional solution. 

However, this exact implementation is impractical since the source term

distribution can only be found by first performing the detailed solution.

 In addition, implementation of simplified source term distributions failed

to adequately match the span-average of the detailed solution.

The near-wall source term model of Hunter (1998) is shown to

accurately model the global addition of mass, momentum, energy, and

turbulence quantities, but to overpredict film effectiveness due to the

underprediction of vortical flows and hot freestream gas entrainment on

grids having streamwise and transverse grid spacings on the order of the

hole diameter or larger. A source term model is proposed which better

predicts the adiabatic film effectiveness downstream of the film holes by

uniformly distributing the source term addition over a thicker layer – on

051015202530

distance downstream of hole leading edge, x/d 

0.0

0.1

0.2

0.3

0.4

span−averaged film effectiveness, eta

data of Sinha et al. 

detailed computation 

coarse grid computation, grid=d/3 

coarse grid computation, grid=1d 

coarse grid computation, grid=3d 

Figure 12: Comparison of span-averaged effectiveness for

3 grid spacings, thick source term model, DR=2.0, M=0.5.

051015202530

distance downstream of hole leading edge, x/d

0.0

0.1

0.2

0.3

0.4

0.5

span−averaged film effectiveness, eta

data of Sinha et al. 

detailed computation 

coarse grid computation, grid=d/3 

coarse grid computation, grid=1d 

coarse grid computation, grid=3d 

Figure 13: Comparison of span-averaged effectiveness for

3 grid spacings, thick source tem model, DR=2.0, M=0.8.

0 5!10 15!20 25!30 

distance downstream of hole leading edge, x/d

0.0

0.1

0.2

0.3

0.4

0.5

span−averaged film effectiveness, eta

data of Sinha et al. "

detailed computation "

coarse grid computation, grid=d/3 #

coarse grid computation, grid=1d #

coarse grid computation, grid=3d #

Figure 14: Comparison of span-averaged effectiveness for

3 grid spacings, thick source term model, DR=2.0, M=1.0.

NASA/TMÑ—2001-210817 8
the order of the hole diameter. It is found that the new model provides

a much better prediction of film effectiveness downstream of the film

hole, while maintaining global conservation of all quantities. The model

performs best for lower blowing ratios (attached jets), since the jet

detachment behavior is not explicitly modeled. It is also shown that the

distributed source term addition performs best on grids having

streamwise and transverse grid spacings on the order of the hole diameter

or larger, such as those used for engineering design. Finer grids are

expected to progressively favor the near-wall source term implementation

of Hunter (1998) due to their ability to resolve the vortical flows that

produce hot gas entrainment.

Future work in this area will focus on more realistic descriptions of

the coolant boundary layer profile in conjunction with downstream

entrainment models to model the effect of the downstream vortices which

coarse grids cannot resolve. However a more realistic description of the

coolant boundary layer profile at the hole exit - in the absence of a

downstream mixing model - may actually worsen film effectiveness

predictions compared to the current model due to a lower minimum jet

temperature and lack of entrainment.

REFERENCES

Adamczyk, J. J., 1985, “Model Equation for Simulating Flows in

Multistage Turbomachines”, ASME Paper 85-GT-226.

Ameri, A. A., Steinthorsson, E., and Rigby, D. L., 1997, “Effect of

Squealer Tip on Rotor Heat Transfer and Efficiency”, ASME Paper 97-

GT-128.

Arnone, A., Liou, M.-S., and Povinelli, L. A., 1991, “Multigrid

Calculation of Three-Dimensional Viscous Cascade Flows”, AIAA Paper

91-3238.

Chima, R. V., 1996, “A k-ω Turbulence Model for Quasi-ThreeDimensional Turbomachinery Flows”, NASA TM-107051.

Dahlander, P., 1998, “An Engineering Tool for Film Cooling

Simulations”, Thesis for the Degree Licentiate of Engineering 98/10,

Department of Thermo and Fluid Dynamics, Chalmers University of

Technology, Goteborg, Sweden.

Foster, J. And Richards, F. B., 1991, “The Gibbs Phenomenon for

Piecewise-Linear Approximation”, American Mathematics Monthly, Vol.

98, pp. 47-49.

Foster, N. W. and Lampard, D., 1980, “The Flow and Film Cooling

Effectiveness Following Injection Through a Row of Holes”, Journal of

Engineering for Power, Vol. 102, pp. 584-588.

Garg, V. K. and Gaugler, R. E., 1997, "Effect of Velocity and

Temperature Distribution at the Hole Exit on Film Cooling of Turbine

Blades", Journal of Turbomachinery, Vol. 119, pp. 343-351.

Garg, V. K. and Rigby, D. L., 1999, "Heat Transfer on a FilmCooled Blade - Effect of Hole Physics", Int. J. Heat and Fluid Flow, Vol.

20, pp.10-25.

Garg, V. K., 2000, "Heat Transfer on a Film-Cooled Rotating

Blade", Int. J. Heat and Fluid Flow, Vol. 21, pp. 134-145.

Goldstein, R. J., 1971, Film Cooling, Advances in Heat Transfer,

Vol. 7, pp. 321-379.

Heidmann, J. D., Rigby, D. L., and Ameri, A. A., 2000, "A ThreeDimensional Coupled Internal/External Simulation of a Film-Cooled

Turbine Vane", Journal of Turbomachinery, Vol. 122, pp. 348-359.

Holmer, M.-L., Eriksson, L.-E., and Sunden, B., 2000, “Heat

Transfer on a Film Cooled Inlet Guide Vane”, Proceedings of the ASME

Heat Transfer Division, HTD-Vol. 366-3, pp. 43-50.

Hunter, S. D., 1998, "Source Term Modeling of Endwall Cavity

Flow Effects on Gaspath Aerodynamics in an Axial Flow Turbine", Ph.D.

Thesis, U. of Cincinnati.

Leylek, J. H., and Zerkle, R. D., 1994, “Discrete-Jet Film Cooling:

A Comparison of Computational Results With Experiments”, Journal of

Turbomachinery, Vol. 116, pp. 358-368.

Menter, F. R., 1993, “Zonal Two-Equation k-ω Turbulence Models

for Aerodynamic Flows”, AIAA Paper 93-2906.

Pedersen, D. R., Eckert, E. R. G., and Goldstein, R. J., 1977, “Film

Cooling With Large Density Differences Between the Mainstream and

the Secondary Fluid Measured by the Heat-Mass Transfer Analogy”,

Journal of Heat Transfer, Vol. 99, pp. 620-627.

Pietrzyk, J. R., 1989, “Experimental Study of the Interaction of

Dense Jets With a Crossflow for Gas Turbine Applications”, Ph. D.

Dissertation, University of Texas at Austin.

Pietrzyk, J. R., Bogard, D. G., and Crawford, M. E., 1989,

“Hydrodynamic Measurements of Jets in Crossflow for Gas Turbine Film

Cooling Applications”, Journal of Turbomachinery, Vol. 111, pp. 139-

145.

Pietrzyk, J. R., Bogard, D. G., and Crawford, M. E., 1990, “Effects

of Density Ratio on the Hydrodynamics of Film Cooling”, Journal of

Turbomachinery, Vol. 112, pp. 437-443.

Rigby, D. L., 1996, “Method of Weakest Descent for Automatic

Block Merging”, 15th International Conference on Numerical Methods

in Fluid Dynamics, Monterey, CA, June 1996.

Rigby, D. L., Ameri, A. A., and Steinthorsson, E., 1997, “Numerical

Prediction of Heat Transfer in a Channel with Ribs and Bleed”, ASME

Paper 97-GT-431.

Schlichting, H., 1979, Boundary Layer Theory, McGraw-Hill, New

York, 7th edition, pp. 312-313.

Sinha, A. K., Bogard, D. G., and Crawford, M. E., 1991, “FilmCooling Effectiveness Downstream of a Single Row of Holes With

Variable Density Ratio”, Journal of Turbomachinery, Vol. 113, pp. 442-

449.

Steinthorsson, E., Liou, M.-S., and Povinelli, L. A., 1993,

“Development of an Explicit Multiblock/Multigrid Flow Solver for

Viscous Flows in Complex Geometries”, AIAA Paper 93-2380.

Wilcox, D. C., 1994a, Turbulence Modeling for CFD, DCW

Industries, Inc., LaCanada, CA.

Wilcox, D. C., 1994b, “Simulation of Transition With a TwoEquation Turbulence Model”, AIAA Journal, Vol. 32, No. 2, pp. 247-

255.

NASA/TMÑ—2001-210817 9
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National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135–3191

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NASA TM—2001-210817

2001–GT–138

E–12732

WU–708–28–13–00

15

James D. Heidmann and Scott D. Hunter

Turbine; Film cooling

Unclassified -Unlimited

Subject Categories: 02, 07 and 34 Distribution: Nonstandard

Prepared for the 2001 Turbo Expo sponsored by the American Society of Mechanical Engineers, New Orleans, Louisiana,

June 4–7, 2001. James D. Heidmann, NASA Glenn Research Center, and Scott D. Hunter, General Electric Aircraft

Engines, Evendale, Ohio 45215. Responsible person, James D. Heidmann, organization code 5820, 216–433–3604.

Coarse Grid Modeling of Turbine Film Cooling Flows Using Volumetric

Source Terms

The recent trend in numerical modeling of turbine film cooling flows has been toward higher fidelity grids and more complex geometries. This trend has been enabled by the rapid increase in computing power available to researchers. However, the turbine design

community requires fast turnaround time in its design computations, rendering these comprehensive simulations ineffective in the

design cycle. The present study describes a methodology for implementing a volumetric source term distribution in a coarse grid

calculation that can model the small-scale and three-dimensional effects present in turbine film cooling flows. This model could be

implemented in turbine design codes or in multistage turbomachinery codes such as APNASA, where the computational grid size may

be larger than the film hole size. Detailed computations of a single row of 35° round holes on a flat plate have been obtained for

blowing ratios of 0.5, 0.8, and 1.0, and density ratios of 1.0 and 2.0 using a multiblock grid system to resolve the flows on both sides of

the plate as well as inside the hole itself. These detailed flow fields were spatially averaged to generate a field of volumetric source

terms for each conservative flow variable. Solutions were also obtained using three coarse grids having streamwise and spanwise grid

spacings of 3d, 1d, and d/3. These coarse grid solutions used the integrated hole exit mass, momentum, energy, and turbulence

quantities from the detailed solutions as volumetric source terms. It is shown that a uniform source term addition over a distance from

the wall on the order of the hole diameter is able to predict adiabatic film effectiveness better than a near-wall source term model, while

strictly enforcing correct values of integrated boundary layer quantities.