NASA Contractor Report 3101

- - NASA CR

Full-Coverage Film Cooling: . 3-Dimensional Measurements of Turbulence Structure and Prediction of Recovery Region Hydrodynamics

S. Yavuzkurt, R. J. Moffat, and W. M. Kays

CONTRACT NAS3-14336 MARCH 1979

https://ntrs.nasa.gov/search.jsp?R=19790014257 2020-01-17T22:37:21+00:00Z

TECH LIBRARY KAFB, NM

, ,__... --... --..- . OObL89b

NASA Contractor Report 3104

Full-Coverage Film Cooling: +Dimensional Measurements of Turbulence Structure and Prediction of Recovery Region Hydrodynamics

S. Yavuzkurt, R. J. Moffat, and W. M. Kays Stanford Uthersity

Stutlford, Calzforuia

Prepared for Lewis Research Center under Contract NAS3-14336

National Aeronautics and Space Administration

Scientific and Technical Information Office

1979

Table of Contents

Page

NOMENCLATURE.......................... V

SUMMARY............................ Xi

Chapter

1 INTRODUCTION ..................... 1

1.1 Background .............. '. ..... 1 1.2 Objectives .................... 5 1.3 Outline of the Work. ............... 5

2 APPARATUS AND INSTRUMENTATION ............. 7

2.1 The Test Surface .................. 7 2.2 Instrumentation ................. 8

3 REAL-TIME HOT-WIRE MEASUREMENT IN THREE-DIMENSIONAL FLOWS......................... 15

3.1 3.2 3.3 3.4 3.5

3.6 3.7 3.8

Background in Hot-Wire Measurements. ....... The Present Approach ............... The Three-Dimensional Turbulent Flow Analyzer. .. Processing of the Data .............. Problems Created by the Size of the Triaxial WireProbe .................... Qualification Tests. ............... Further Work on Triaxial Wire System ....... Conclusions and Capabilities ...........

15 20 22 25

26 26 31 32

4 PREDICTION OF RECOVERY REGION HYDRODYNAMICS WITH A ONE-EQUATION MODEL OF TURBULENCE 45 4.1 Equations to be Solved . . . . . . . . . . . . . . 45 4.2 Mixing-Length Model. . . . . . . . . . . . . . . . 48 4.3 Predictions. . . . . . . . . . . . . . . . . . . . 58 4.4 Refining the Model . . . . . . . . . . . . . . . . 59

5 MEASUREMENTS AND DISCUSSIONS OF THE DATA. . . . . . . . 71

5.1 Experimental Conditions. . . . . . 5.2 Measurement Locations. . . . . . . 5.3 Mean Velocity Profiles . . . . . . 5.4 Turbulent Kinetic Energy Profiles. 5.5 Streamise Shear Stress Profiles . 5.6 On Spanwise Averaging . . . . . . 5.7 On the Three-Dimensionality of the 5.8 Correlation Coefficients . . . . . 5.9 Mixing Length. . . . . . . . . . .

........

........

........

........

........

........ Flow Field. . . ........ ........

71 72 73 79 86 88 89 92 93

iii

Table of Contents (cont.)

Page

Chapter

6 SUMMARY AND RECOMMENDATIONS .............. 120

6.1 Summary ...................... 120 6.2 Recommendations. ................. 122

REFERENCES...........................

APPENDIXES A TABULATION OF EXPERIMENTAL DATA ............

A.1 Individual Profiles. ............... A.2 Spanwise-Averaged Profiles ............

B WIRING DIAGRAMS OF THE 3-D TURBULENT FLOW ANALYZER. . . 170

123

127 127 159

iv

A 4

A+

B '7

C

D

v

E eff

F

F'

gc

H

3

kl

k2

R

WQa

(mmax

Proportionality constant, Newton's Second Law.

Shape factor = 61fs2.

Diffusion of turbulent kinetic energy.

Pitch sensitivity coefficient for hot wire.

Yaw sensitivity coefficient for hot-wire.

Mixing length.

Augmentation in mixing-length normalized on outer hound-

ary layer thickness, nixir,g-length model.

Maximum value of augmentation in mixing-length normal-

ized on outer boundary Layer thickness, mixing-length model.

(Jmmax ,- i 1nitia.L value of maximum augmentation in mixing-length

normalized in outer boundary-layer thic!:iless, mixing-length

model.

NOMENCLATURE

Production constant for turbulent kinetic energy.

Effective thickness of laminar sublayer.

Dissipation constant for turbulent kinetic energy.

Division constant for the total velocity magnitude CR) at

the output of 3-D Turbulent Flow Analyzer = 0.576.

Hole diameter of injection tube.

Van Driest damping function, mixing-length model.

Dissipation of turbulent kinetic energy.

Output of linearizers.

Blowing fraction = 'jet"jet T D2

p,u, T p2'

P. U2 Blowing fraction based onmomentum = w z D.

0003 P:

V

M Blowing ratio, p j etUj et"mUW'

Inverse Jorgensen matrix, Chapter 3.

N

P

P

q2

f

K

ReX

SC 4

T

U

U jet

UC0 7 IJ

%

u eff

@eff)max

U,ff

u T

U’

%ff

uf2

r UV

Coordinate transformation matrix, Chapter 3.

Hole spacing, or pitch.

Pressure.

Production of turbulent kinetic energy. ---

2 2 2 Turbulent kinetic energy = u' + v' + w' .

Value of the turbulent kinetic energy at free stream.

Magnitude of mean velocity.

x-Reynolds number = xlJoo/\).

Schmidt number of turbulent kinetic energy = E /E M (1'

Temperature.

Instantaneous velocity component in x-direction.

Injectant mean velocity.

Freestream velocity.

Mean velocity component in x-direction.

Centerline mean velocity in 2-D channel, Chapter 3.

Instantaneous effective velocity for hot--wire.

Naximum value of effective velocity in one calibration.

Time-averaged effective velocity.

Friction velocity = gc~o/~o.

Fluctuating velocity component in x-direction.

Fluctuating effective velocity.

Reynolds normal stress in x-direction.

Longitudinal-normal velocity correlation, Reynolds shear stress.

-i--T uw Longitudinal-tangential velocity correlation, Reynolds shear

stress.

--7-- Ueff

Mean square of fluctuatin,~ effectiae velocity.

Vi

Instantaneous velocity component in y-direction.

V'

y,2

v’w’

W

w W’

Wf2

X

X’

X’ 0

X” 0

Y

Y+

(Yf 51d

(Y/U i

z

Hean velocity component in y-direction.

Fluctuating velocity component in y-direction.

Reynolds normal stress in y-direction.

Normal-tangential velocity correlation, Reunolds shear stress.

Instantaneous velocity component in z-direction.

Mean velocity component in z-direction.

Fluctuating velocity component in z-direction.

Reynolds normal stress in z-direction.

Longitudinal coordinate.

Distance along test surface measured from nozzle exit.

Distance along test surface measured from start of recovery

region.

Distance, start of recovery region to virtual origin of in-

ternal two-dimensional layer, Chapter 4.

Distance, start of recovery region

fective jet action.

Coordinate normal to test surface.

Distance normal to test surface.

Nondimensional distance = yu,/v.

to virtual origin of ef-

Departure point from KY line in mixing-length model, Chap-

ter 4.

Intersection point of new mixing-length model with R/6 =

x iine, Chapter 4.

Transverse coordinate.

Distance in transverse direction on test surface measured

from centerline.

Vii

Greek Symbols

Angle of rotation around probe axis for triaxial hot-wire

probe.

Injection angle measured from test surface.

Angle between v and w components of mean velocity.

Angle between ?! and 7 components of mean velocity.

Boundary layer thickness, also 6.99.

Thickness ofouterboundarylayerinmixing-lengthmodel, Ch. 4.

Displacement thickness.

Momentum thickness.

Thickness of internal two-dimensiona

Chapter 4.

Eddy diffusivity for momentum.

.1 layer, 3

Eddy diffusivity for turbulent kinetic energy.

Fmil-cooling effectiveness.

Roll angle for slant hot-wire probe.

Non-dimensional temperature = (T j et-Tco) / (To-To,).

Von Karman constant = 0.41.

Outer region mixing-length proportionality constant = 0.085.

Dynamic viscosity.

Kinematic viscosity.

Density. -s---7

Shear stress = II -2 u'v'~ + u'w'

---2 + v'w' .

Slant angle for slant hot-wire probe.

Pitch angle for triaxial hot-wire probe (angle between probe

axis and s-axis when probe body is in X-Y plane).

viii

Subscripts

1 Wire number one for triaxial-wire probe, Chapter 3.

2 Wire number two for triaxial-wire probe, Chapter 3.

3 Wire number three for triaxial-wire probe, Chapter 3.

a Augmented value, mixing length model.

eff Effective value.

ilet Injectant value.

0 Wall value.

co Freestream value

Superscripts

Time averaged.

ix

-

SUMMARY

Hydrodynamic measurements were made with a triaxial hot-wire in the

full-coverage region and the recovery region following an array of injec-

tion holes inclined downstream, at 30' to the surface. The data were

taken under isothermal conditions at ambient temperature and pressure for

two blowing ratios: M = 0.9 and M = 0.4. (M = ~~~~~~~~~~~~~ where U

is the mean velocity and p is the density. Subscripts "jet" and "m"

stand for injectant and free stream, respectively.) Profiles of the three

mean velocity components and the six Reynolds stresses were obtained at

several spanwise positions at each of five locations down the test plate.

High turbulent kinetic energy (TKE) levels were found for low blow-

ing, with low TKE levels for high blowing. This observation is signifi-

cant when coupled with the fact that the heat transfer coefficient is high

for high blowing, and low for low blowing. This apparent paradox can be

resolved by the hypothesis that entrainment of the main stream fluid in a

3-D manner along the lanes between wall jets must be more important than

the turbulent mixing in heat transfer for the high blowing ratios (close

to unity).

The flow in the recovery region can be described approximately in

terms of a two-layer model: an outer boundary layer and a two-dimensional

(2-D) inner boundary layer.

A one-equation model of turbulence (using TKE with an algebriac mix-

ing length) was used with a 2-D computer program to predict the mean vel-

ocity and TKE profiles in the recovery region. Mixing-length values

calculated from the data were used in inputs to the program. The mixing-

length distribution was assumed to be piecewise continuous, a heuristic

fit to the data consistent with the concept of the two quasi-independent

layers observed in the recovery region. This mixing length, along with

a set of otherwise normal constants (for 2-D boundary layer predictions),

successfully predicted the spanwise-averaged features of the flow.

A new hot-wire scheme was developed to make measurements in the three-

dimensional (3-D) turbulent boundary layer over the full-coverage surface.

The method uses a triaxial hot-wire and an analog device for real-time

Xi

data reduction. It does not use the low-fluctuation assumption and can

tolerate an unknown flow direction within limits (?30° for t'he mean

velocity). The method is very fast and quite practical for taking large

amounts of data in 3D, high-fluctuation turbulent flows.

Xii

Chapter 1

INTRODUCTION

1.1 Background

Modern gas turbine technology requires high turbine inlet tempera-

tures to increase thermodynamic efficiency, well above the usable temper-

ature of the materials from which the turbine blades are manufactured.

One way to protect the blades is by injecting a coolant through the blade

surface, thus forming a cooling film between the hot combustion gases

and the metal surface. There are several ways of injecting the coolant

onto the blade surface. A comparison of different methods is given by

Brunner (1969). Two of the more comma:: are transpiration cooling through --~-~~ a porous plate surface and full-covcllge film ccoling through an array

of closeiy spaced multiple rows of smail i-holes, as discussed by Esgar

(1971). Transpiration cooling achieves lcwer heai transfer to the sjr-

face and has better aerodynamic performance than full-coverage film cool-

ing but has problems with structural strength and is susceptible to

plugging (Goldstein, 1971). Full-coverage f!lm cooling seems more prac-

tical, but not enough is known about the heat trransfer and hydrodynamic

mechanisms to allow accurate prediction. Such information is necessary

before accurate designs can be developed.

There have been many studies of heat transfer with film cooling.

A general review is given by Goldstein (1971). A review of discrete-hole

film cooling is given by Choe et al. (1975). Most of the early experi-

mental research on Full-coverage Film cooling concentrated on measurement

of film-cooling effectiveness with th e heat transfer coefficients. The

hydrodynamics was treated as secondary ill importance. A review of the

experimental work on this topic is also given by Crawford et al. (1976).

Le Brocq, Launder & Priddin (197'1) studied the effects on effective-

ness, n, of hole-pattern arrangement, injection angle, ratio of densi-

ties of the coolant and mainstream, and blowing ratio. Some proEiles of

mean velocity were taken near the holes, with a pitot probe. They found

that the staggered hole pattern was more effective than the in-line

1

pattern. A critical blowing ratio was found to exist, which gave rnaxinnrm

effectiveness. Above this blowing ratio effectiveness decreased again.

Slant-angle injection was found to be more effective than the normal in-

jection.

Launder & York (1973) studied the effects of mainstream acceleration

and turbulence level on film-cooling effectiveness, using a staggered 45"

slant-hole test section. Again, only a small number of mean velocity

profiles were taken. Acceleration seemed to increase effectiveness, but

the free stream turbulence did not have a significant effect.

Metzger, Takeuchi & Kuenstler (1973) studied both effectiveness and

heat transfer on a full-coverage surface with normal holes spaced 4.8

diameters apart, arranged in both i-n-line and staggered patterns. They

did not make any measurement of hydrodynamics. Again, it was observed

that a staggered pattern :i~as more effective.

Mayle & Camarata (1975) examined the effects of hole spacing and

blowing ratio on heat transfer and film effectiveness for a staggered-

hole array with compound-angle injection. Lt was found that for P/D =

10 and 8, higher effectiveness was obtained than with P/D = 14. No

measurements were made of the :Iydrodyndmics oi the flow fjeld.

Choe et al. (1975) studied the efEe:ts on heat transfer of hole

spacing, blowing ratio, mainstream velocity and conditions upstream of

the discrete-hole array. Normal injection was used with a staggered

array of holes spread 5 alld 10 diameters apart. T4esn vcllocity profiles

were taken with pitot probe tu obtain spanwise-averaged val.ues. From

the spanwise-averaged profile, mixing length was obtained and used in

predictions of heat transfer data with a "zero-equation model" of turbu-

lence. They observed that the P/D = 5 case with a comparable F was

more effective than P/D = 10. Furthermore there was no pronounced

effect in the initial blowing row.

Crawford et al. (1976) also give a summary of analytical work in

the field of full-coverage film cooling. Goldstein et al. (1969) and

Ericksen, Eckert & Goldstein (1971) used superposition of film-effective-

ness data for individual jets to predict n by modeling injection as a

point heat source. Mayle & Camarata (1975) developed an improved super-

position method to predict their full-coverage data. Pai & Whitelaw

2

(1971) and Patankar, Rastogi & Whitelaw (1973) investigated the predic-

tion of wall temperature and effectiveness downstream of 2- and 3-D film-

cooling slots. For the 2-D siot injection, the boundary layer equations

were used along with an augmented mixing length model to represent the

effect of injection. For the 3-D injection, 3-D Navier-Stokes equations

were solved numerically by reducing them to elliptic form in the lateral

plane and to parabolic form in the streamwise direction.

Herring (1975) used a finite-difference method for predicting the

flow over a full-coverage film-cooled surface. Lateral averaging in the

full-coverage region was invoked to justify using 2-D boundary layer

equations. Terms arising from the spanwise averaging were obtained from

consideration of the jet-boundary layer interaction. Predicted velocity

profiles were reported but showed some problems in the initial regions

of injection near the wall..

Choe et al. (1975) developed a finite-difference method for predict-

ing heat transfer with full-coverage film cooling, solving the two-

dimensional boundary layer equations. (These equations have a form simi-

lar to those given by Herring, 1975.) Choe et al. (1975) used the concept

of local averaging. with a different model for the injection process.

the nonlinear terms, and augmented turbulent mixing. By using an augmen-

ted mixing length in a zero-equation model for turbulence, he success-

fully predicted most of their Stanton number data for low and moderate

blowing in the full-coverage region. The predictions in the recovery

region and for high blowing were less accurate.

Crawford et al. (1976) repeated the conditions covered by Choe et

al. (1975); however, they used a different full-coverage surface: 30"

slant-hole injection with a staggered array of holes, with two different

hole spacings (5 and 10 diameters). For each injection, Stanton number

data were taken for two values of injectant temperature, corresponding

to 6 equal to zero and one. Linear superposition was then used to ob-

tain Stanton numbers as a continuous function of the injectant tempera-

ture. Stanton numbers were measured for a range of injection mass flux

ratios (M from 0 to 1.3) and Reynolds numbers (Rex from 1.5 x lo5

to 5 x lob>.

3

With the injection temperature equal to the wall temperature, the

Stanton number decreased below the normal flat-plate value and reached a

minimum at M of 0.4. A higher M caused an increase in the Stanton

number. For the recovery region, downstream of the five-diameter hole

array, two distinct data trends were observed. For low M the Stanton

number began to recover immediately from the effects of blowing, while

for high M the Stanton number either remained constant or dropped

throughout the recovery region. This latter behavior suggested that

interrupted hole patterns may be an effective cooling scheme.

In Crawford's work (Crawford et al., 1976) both integral and dif-

ferential analyses were carried out on the data sets. A differential

prediction method was developed which used a two-dimensional boundary

layer program. The injection process was modeled as slot injection, and

the turbulence was modeled with a zero-equation model, using an augmented

mixing length obtained from the spanwise-averaged mean velocity profiles.

Mean velocity profiles were obtained with a pitot probe. Downstream of

the jets the profiles may be in error because of the static-pressure

islands created around the jets. In Crawford's data, most of the five-

hole diameter data were successfully predicted in the full-coverage re-

gion. The predictions in the recovery region and for high blowing ratios

were not very good, as was also the case with Choe et al. (1975).

Very little of the literature mentioned above gives any information

about flow field measurements and, in particular, measurements of turbu-

lence quantities. In the following papers, however, some attempts were

made to make measurements of hydrodynamics. Ramsey Sr Goldstein (1971)

measured the mean velocity and turbulence intensity profiles (not the --

turbulent kinetic energy) for normal injection downstream of a heated jet

at blowing ratios of 1.0 and 2.0. Metzger, Carper & Warren (1972)

measured the mean velocity profiles both upstream and downstream of a

two-dimensional injection slct. Hartnett, Birkebak & Eckert (1961) mea-

sured boundary-layer velocity profiles for a number of positions down-

stream of injection through a single slot. Foster & Haji-Sheikh (1974)

measured mean velocities downstream of flush, normal-injection slots.

4

In these references it was frequently observed that the literature

is quite weak from the point of view of hydrodynamic data, especially

for full-coverage surfaces.

1.2 Objectives

This research represents an extension of the work by Crawford et al.

(19761, whose experiments consisted mainly of heat-transfer coefficient

measurements. As explained, the predictions of Crawford were good for

low and moderate blowing ratios in the full-coverage region, but problem-

atic in the recovery region and for high blowing ratios. It was felt

that the key to understanding these result lay in the structure of the

turbulence.

A decision was made to adapt a one-equation model of turbulence for

prediction of the hydrodynamics within the recovery region, since the

zero-equation model did not give satisfactory results. It was thought

that a higher-order model might better represent the real process.

The objectives of this research were twofold:

0 to make 3-D measurements of the turbulence structure and mean

velocity field in the 3-D turbulent boundary layer over the full-

coverage film-cooled surface and in its recovery region,

0 to apply a one-equation model of turbulence for predictions of the

recovery region hydrodynamics with the empirical input supplied by

the experiments.

1.3 Outline of the Work

A new hot-wire anemometer method of measurement was devised to allow

measurements of the turbulence structure in the 3-D turbulent boundary

layer over the full-coverage surface. Due to the interaction between the

mainstream and the jets, a complex flow field is created in the full-

coverage region where the local flow is 3-D in the vicinity of the jets,

the local flow direction is unknown, and the fluctuations are high.

The new hot-wire method is believed to be an important contribution.

The present research could hardly have been done without it. The method

is described in Chapter 3 of this thesis.

5

The physics of the recovery region flow are described in Chapter 4,

as well as the predictions of the recovery region hydrodynamics using

the one-equation model of turbulence. The physical modeling is accom-

plished by modeling the length scale of turbulence. The length-scale

model introduced in this chapter was successful in predicting the flow

with the one-equation model.

Chapter 5 presents the data taken on the full-coverage surface and

discussions of the data. The volume of the data taken was so large that

only typical results are discussed. As far as the author knows, these

data comprise the first detailed study of this process. There will be

much to explore in the data beyond the scop2 of the present report.

Chapter 6 contains the summary and recommendations about the com-

plete research.

The building of the 3-D Turbulent Flow Analyzer was funded by the

Thermosciences Affiliates. The Scientific and Technical Research Coun-

cil of Turkey (Tiirkiye Bilimsel ve Taknik Arastirma Kurumu) provided all

the first author's living expenses and tuition during his four-year stay

at Stanford University.

6

Chapter 2

APPARATUS AND INSTRUMENTATIO1J

The Discrete Hole Rig is the heat transfer apparatus which carries

the full-coverage film-cooling test section. It is a closed loop wind

tunnel which delivers air at ambient pressure and constant temperature.

This system was previously used by Choe et al. (1975) and Crawfcrd et

al. (1976). It is described in detail in their work. Only those aspects

relevant to the present research will be given here. A flow diagram of

the systems which make up the rig is shown in Fig. 2.1; its photograph

is given in Fig. 2.2.

2.1 The Test Surface

The test surface of the Discrete Hole Rig is made from copper plates.

it has three parts: a preplate, a full-coverage region, and an after-

plate.

The preplate is made of 24 individual copper plates, each abo,ut

2.6 cm long in the flow direction and 46 cm wide.

The full-coverage section is composed of 12 copper plates, each 0.6

cm thick, 46 cm wide, and 5 cm long in the flow direction. The first

plate does not have any holes, but the remaining 11 downstream plates

plates have alternately nine and eight holes each. The holes are each

1.03 cm in diameter and are spaced on five-diameter centers to form a

staggered hole array. A photograph of the full-coverage section can be

seen in Fig. 2.3.

Delivery tubes for secondary air extend beneath the surface at a 30"

angle to the plate surface, giving a 30" slant-angle injection in the

direction of the main flow. A schematic of the full-coverage surface can

be seen in Fig. 2.4.

The afterplate is identical to the preplate. This is the recovery

region for the flow after the blown part of the test section. See Craw-

ford et al. (1976) for more detailed information about the test surface.

Twenty-four instrumented strips in the afterplate permit measurement of

the Stanton number distribution in the recovery region.

7

2.2 Instrumentation

This section will give information on the instruments used for mea-

surements in the turbulent flow field. The amount of detail given about

a certain instrument depends on the importance of its use in the measure-

ments. Detailed information is given about the calibration of the tri-

axial hot-wire probe.

2.2.1 Temperature

Since the experiments were performed under isothermal conditions and

the purpose was to make measurements of hydrodynamics, there was no need

for extensive temperature measurements. Only the free-stream temperature

was measured. This measurement was made with an iron-constantan thermo-

couple whose junction was parallel to the fl.ow. The thermocouple probe

did not require any special shielding or precautions because the operat-

ing temperatures were about ambient and the velocities were low. An in-

tegrating digital voltmeter (HP 24OlC) was used to read out the thermo-

couple output.

2.2.2 Pressure

Tunnel static pressure was measured with an inclined Meriam manom-

eter using a 0.824 specific gravity fluid. The manometer had a range of

0.5 in with a smallest division of 0.005 inches of water. Static pres-

sure was measured from the taps located in one of the tunnel sidewalls.

Their locations and properties are given by Choe et al. (1976). The main-

stream dynamic pressure was measured with a kiel probe corrected to a

Combist micromanometer whose smallest division was 0.0005 inches of manom-

eter fluid of specific gravity 0.82. The tunnel geometry was adjusted to

achieve a uniform static pressure throughout the test section.

2.2.3 Secondary Air Flow Rate

Flow rates through the seconda.ry air delivery tubes were measured

with hot-wire flow meters permanently built into the rig. More informa-

tion about the flow meters is given by Choe et al. (1975). The digital

voltmeter mentioned in section 2.2.1 was used to read out the flow meter

outputs. Calibration of the meters was done individually, using reference

flow meters.

8

2.2.4 Turbulent Flow Field

a. Instrumentation

The main instrumentation for the experiments was the hot-wire system

used in the measurements of the turbulent flow field. It consists of a

DISA claw-type triaxial hot-wire probe, three constant temperature ane-

mometers (TSI lOSO>, three polynomial linearizers (TSI 1052), and the 3-D

Turbulent Flow Analyzer.

An HP-2100 digital computer was used for data acquisition and reduc-

tion.

An integrating digital voltmeter (IDVM) (HP-2401C) and an RMS meter

(TSI 1076) were used occasionally to reduce and read the outputs of the

3-D Turbulent Flow Analyzer instead of the computer.

More detailed information about the hot-wire instrumentation is

given in Chapter 3.

b. Calibration of the triaxial hot-wire probe

The triaxial hot-wire probe was calibrated on the calibrator de-

scribed by Pimenta et al. (1975). This calibrator produced a uniform flow

of air at constant but adjustable temperature, with moderate-to-low turbu-

lence levels (about 0.8%).

The probe was calibrated in the free jet at the exit of the calibra-

tor. After the flow in the calibrator was adjusted to the desired tem-

perature and the anemometers were set, the following procedure was applied

for the process of calibration.

The axis of the probe was aligned with the flow direction by trial

and error. To check the alignment, the output of each anemometer was re-

corded at a certain orientation of the probe. Then the probe tip was

rotated 180" (the probe holder was designed so that the probe tip could

be rotated to 12 positions, each 30" apart) and the anemometer outputs

were recorded again. If the probe was aligned properly, the outputs of

the same anemometer at 0" and 180" positions of the probe tip should have

read the same (for 10 seconds integration of the output signal, the ac-

ceptable value is x.xxx _+ 0.002 volts for a velocity about 10 m/set).

If the numbers were not the same, the probe position was readjusted until

the alignment criterion was satisfied. It usually took about five trials

9

to obtain the correct alignment. The final alignment position was within

l-2' of the first visual alignment. During this procedure it was very

important to make sure that any change in readings was not caused by a

change in either the temperature or the flow rate.

After the probe was aligned with the flow direction the calibration

started. First the flow speed was adjusted to the desired value by ad-

justing the valve of the calibrator. The flow speed was measured

through a static pressure tap located on the plenum chamber where the

pressure is equal to the total pressure of the jet at the nozzle exit.

The pressure was read with the Combist micromanometer mentioned in sec-

tion 2.2.2. Afterwards, the output of the three anemometers was read

with the IDVM, described before. Ten seconds of integration of the out-

put was enough for the calibration.

It was necessary to check the temperature during calibration. This

was done by checking the cold resistance of one of the wires from time to

time.

Finally, the obtained values of static pressure and anemometer out-

puts aiong with the temperature, ambient pressure, and humidity were fed

to a computer program prepared by Coleman (Coleman et al., 1975) to ob-

tain the linearized calibration equation.

The typical values used for some of th e variables during the cali-

bration were the following:

Cold wire resistances: = 9.5 R

Overheat ratio: = 1.5

Maximum change in the cold resistance

during one calibration: 52 + .02 R

Temperature range (for different cali-

brations): 18"C-26°C

Velocity range during one calibration: 4-22 mlsec.

io

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Chapter 3

REAL-TIME HOT-WIRE MEASUREMENTS IN THREE-DIMENSIONAL FLOWS

Hot-wire anemometers are the most widely used devices for making

measurements in turbulent flow fields. Satisfactory results are obtain-

able in one- and two-dimensional flows whose directions are known and

whose turbulence structures are stationary and of low amplitude (.typi-

tally on the order of 10% of the mean velocity). Three-dimensional flows

of unknown flow direction, with high fluctuations or large intermittent

structures, cannot be measured accurately with classical techniques, and

even the stationary, low-fluctuation cases are very tedious.

The first step in most classical methods involves finding the time-

averaged value of the fluctuating component of the anemometer output.

Detail.s of the turbulence structure are deduced by solving sets of simul-

taneous equations using measurements at different angular positions.

These equations are formed by expanding the expression for effective

velocity in a Taylor's series around the mean and truncating at second

order (usually). When the equations are weakly convergent (as when many

angular positions are used), the uncertainty in the calculated values of

the turbllience quantities is high, especially for the higher-ordered

terms.

The object of this work was to develop a real-time method for hot-

wire anemometry which would be applicable to three-dimensional flows of

unknown flow direction (within limits) and with high fluctuation levels.

3.1 Background in Hot-Wire Measurements -_.___

3.1.1 Classical time-averaged reductions --____ -__ -- Most of the existing hot-wire methods are good in 2-D flows with low

fluctuations and known flow direction. Almost all use either single

horizontal wi.res, single rotatable slant wires or cross wires. The usual

practice in data-taking is to align the probe with the known flow direc-

tion, to read the mean output of the anemometer or linearizer with a

digit-al voltmeter, and the average fluctuation value with an RMS meter.

15

The desired quantities, mean velocities and Reynolds stresses are deduced

from the truncated Taylor series expansions of these effective quantities.

The truncation and solution process requires some simplifying assumptions,

in particular, that the fluctuations are small compared to the mean values.

When the flow direction is unknown, the data-reduction process is

quite complicated. To explain this point better, the relation between

the effective velocity and the velocity components is given in Eqn. (3.1)

taken from Jorgensen(1971), with the nomenclature shown in the following

figure.

WIRE

Ufff = X2 + k;Y2 + k;Z2 (3.1)

Here, X, I', Z are the velocity components in the directions of the wire

coordinates x', y', and z', respectively, and kl and k2 are pitch

and yaw sensitivities of the wire. The following figure shows a single,

rotatable slant wire and the wire orientation with respect to the labora-

tory coordinates x, y, and z.

-

16

In this figure 9 is the slant angle, in the plane of prongs, between

the wire and the y' direction. The angle 4 is zero when the wire is

perpendicular to the axis of the prongs. The angle 8 is the roll angle

in the (~2) plane, and is zero when the prongs are in the (xy) plane.

Velocity components U, V, and W are in the laboratory coordinates, in

the x, Y, and z directions, respectively. With this nomenclature,

Eqn. (3.1) can be written in terms of velocities in laboratory coordinates,

as follows:

Ufff = A(U2) + B(V2) + C(W2) + D(W) + E(VW) + F(UW) (3.2)

where

A = cos2$+k:sin2$,

B = (sin2 I$ + k: cos2 Q)cos2 0 + ki sin2 8,

C = (sin2 4 + kt cos2 $)sin2 0 + k; cos2 0,

D = (1-k:)sin 2$ cos 8, (3.3)

E = (sin2 4 + k: cos2 $I - kc)sin 20,

F = (1-k:)sin 21$ sin 8.

In Eqn. (3.2), each velocity can be written as the sum of the mean and

fluctuating parts:

u = u + u'

v = iT+v' (3.4)

w = W+w'

and

(3.5)

In the classical time-averaged method, the quantities recorded are

ceff and uLEf for each of several angular positions. If the first-

order fluctuation descriptions are substituted into Eqn. (3.2), a very

complicated equations results which can be brought to a theoretically

solvable form by a Taylor series expansion of both sides. This expansion

17

is valid only when the flow has a strongly preferred direction and low

fluctuations (i.e., 7, W, and u', v', and w' are each at least one

order of magnitude smaller than 5) . The resulting equations are given

below:

feff = A--- r 2fiYL

(2Ac2 +

+ (Au' 2

+ Bv' 2

,2 Ueff

=

-- -- -- By' + Ci2) + (DLJV + EVW + FUW)

+ Cw'?) + (Du'v' + Ev'w' + Fm) 1 -

(3.6)

+ O(3)

v w -- - -+ 4AD-+ 4AF-+ 2DFVW 2

u u u2 -- -

E2 -- w2 7 k w - vw 2

u2 + 4BD + 2DE + 4BE

ii u u2 + F2 + E2 $+ 4C2 5

' 2

-- - + 2EF-+ 4CF-++CEy w' 2

(3.7) u i u )

+ v2 w2 -

4.AD + 4BD - + 2EF - + (8AB + 2D2) v + (4-4E + 2DF) -- w

v2 jJ2 v u

+ (2DE + 4BF) 7) u'v' ( + 4-W + 2DE !- + 4CF W- + (4% + 2DF) - ,: ,: ;

+ (SAC + 2F2) ; + (2EF + 4CD) $)m + (2DF+4BE $ + 4CE 5

v ii -- + (2DE + 4BF) - + (4CD + 2EF) - + (8BC + 2E2) vw 1

1 v'w'

u + O(3)

u t 21

Even with these assumptions the resulting equations are quite ccm-

plicated. In two-dimensional flows with low fluctuations, where the flow

direction is known, aligning the probe with the flow direction sets v

and w to zero, allowing the simplified equation to be used. In cases

where there is symmetry (u'w 7 = 0 and v'w' = 0), the equations reduce

to the following form, which is generally used for rotatable slant wires:

'eff = fi v + O(2) (3.8)

--- -- 2 --- -- U’ 2 2

eff = Au' D t2 +xew~’ 7-7 + 4A v

4A + D u v + O(3) (3.9)

18

One measures ceff and u:f, in each of four different positions, thus

obtaining enough equations to solve for these unknowns.

There are several ways to make measurements in 3-D turbulent flows

using the classical time-averaged equations. The most important problem

in 3-D flows arises when the mean flow direction is unknown. This direc-

tion can be found with one of the following ways. Johnston (1970) mea-

sured the local pitch and yaw angles for the mean velocity with a Conrad

probe and then measured the Reynolds stresses with a horizontal wire and

a rotatable cross wire aligned with the flow. The low-fluctuation assump-

t-ion was invoked. Moussa & Eskinazi (1975) tried to measure the mean

flow direction by using a rotatable slant wire. Making use of the direc-

tional properties of hot wires, they calibrated the probe for all possible

angles and prepared detailed charts which included the flow angles as

functions of four mean voltages obtained at different rotations of the

probe. Delleur (1966) used a cross wire to measure the flow direction,

arguing that the cross-wire technique was twice as accurate as the single-

wire technique. Both methods require the use of a hot-wire calibration

curve for flow-direction measurement and consequently require frequent

calibration of the hot wire to renew the calibration charts.

Some other 3-D hot-wire methods were developed which do not require

the flow direction to be known. Mojola (1974) gives a hot-wire method

(rotatable slant wire or cross wire) for measuring the three mean compo-

nents of the velocity and six Reynolds stresses without knowing the flow

direction. His equations are valid when the probe is approximately

aligned with the flow (i.e., a strongly preferred mean flow direction)

and has only low fluctuations. Hoffmeister (1972) describes a scheme

which employs a single rotatable slant wire to obtain three mean veloci-

ties and six Reynolds stresses. In this scheme the interpretation of

the anemometer voltages is based on calibrations of the probe over the

entire range of angles between the wire and the flow which may exist dur-

ing measurements.

None of the preceding methods is practical for taking large amounts

of data. Further, the accuracy with which the higher-ordered terms can

be measured is seriously limited. As the number of terms retained in

Eqn. (3.7) increases, the number of independent realizations required

19

increases; thus the number of probe rotational positions increases and

the strength with which the equations converge diminishes.

The problem lies, basically, in the time-averaged approach to tur-

bulence measurement. With one or two wires, one does not have enough

information to solve for the instantaneous velocities, and, hence, time- averaging is required. With three wires, however, the whole problem is

resolved and one can deal directly with the instantaneous velocities.

3.1.2 Real-time reduction

A real-time data-reduction scheme was used by Zimmerman & Abbott

(1975). Using a triaxial hot-wire probe and an analog device, they

solved for the three unknowns (three instantaneous velocities) in the

effective velocity expression, in real time. The outputs of their analog

device were the three mean velocities, the three instantaneous fluctua-

tions, and the means of three correlations (off-diagonal components of

Reynolds stress tensor). This scheme was used to measure the Reynolds

stresses in a 3-D turbulent boundary layer created by a skewed-leading-

edged flat plate in zero pressure gradient air flow, which yields high

cross-stream gradients of the turbulent quantities.

3.2 The Present Approach

The present approach also uses the real-time data-reduction idea.

In fact, this work was finished at the same time as Abbott & Zimmerman's

(1975) work, but the publishing of results has been delayed due to un-

foreseen circumstances.

Equation (3.1) introduced the effective velocity expression. The

three unknowns are the three instantaneous velocity components in wire

coordinates. To solve for these velocities in real time, one needs three

simultaneous and independent values of the Ueff at one point in the

flow field. Probes are available which have an orthogonal array of wires,

such that the x' axis for one is the y' and z' axis for the others.

The probe used in the present studies was a DISA triaxial-wire probe.

The DISA triaxial wire probe has three wires each of which has two

leads and each of which is driven by a separate anemometer. The wires

are mutually orthogonal, forming a right-angled coordinate system. The

20

sensors provided by DISA are 3.2 mm long, with 5 micron Pt-plated tungsten

wires having copper- and gold-plated ends leaving a sensitive length of

1.25 mm. In the present work, bare Pt-plated tungsten wires were used

without gold plating, giving 3.2 mm active length. The three wires form

a cone of apex angle 70.6" around the axis of the probe stem. The wire

coordinate system can be seen in Fig. 3.1. With the special prong struc-

ture of the probe, the effective velocity indicated by each wire is rela-

ted to the velocity components in the wire coordinates in the following

manner.

Ufff

Cffl

= k;lX2 + Y2 + k;lZ2

= k;2X2 + ki2Y2 .+ Z2

u:,,' = x2 + kz3Y2 + k;3Z2 3

(3.10)

and the linearized effective voltages (linearizer outputs) are related to

the effective velocities as follows:

U eff 1

= Al + BlEeff1

U eff, = A2 + B2Eeff, (3.11) L L

U eff 3

= A3 + B3Eeff3

In Eqns. (3.11), the A's and B's are constants obtained from calibra-

tions of the wires.

Equations (3.10) have three unknowns -- the instantaneous velocities

in the wire coordinates -- which can be obtained from the equations shown

below: x2- I- Y2 Z2

=

k211 I 2

k22

I 1 L

1

kC2

2 k23

-1 -

U:ff 1

Ufff 2 2

L'eff 3

(3.12)

21

or

(3.13)

Once the instantaneous velocities in the wire coordinates are obtained,

the instantaneous velocities in the laboratory coordinates can be ob-

tained easily with a transformation of coordinates.

IJ-

L

v

i W

= N

-x-

Y

2 --

(3.14)

where N is the coordinate transformation matrix from wire coordinates

to laboratory coordinates. With the three instantaneous velocity compo-

nents available in laboratory coordinates, the mean values and the Rey-

nolds stresses can be obtained by using digital voltmeters, RMS meters

and turbulence correlators, or with a digital computer having a fast

enough A/D (analog-to-digital) converter.

3.3 The Three-Dimensional Turbulent Flow Analyzer _---- An analog device was built to solve the equations (3.13) and (3.14)

using high-speed analog components. All quantities were magnitude-scaled

using OJeff)max9 the value which resulted in the maximum output of the

linearizer. (For the present case (Ueff)max can be cbtained by substi-

tuting 10 volts for Eeff, i.e., (Ueff)max = A + B x 10.) Also, in

calibration the problem is scaled such that the output in the real. experi-

ment will be between 4 and 8 volts, never reaching the limits of 0 volts

and 10 volts. Under these normalization conditions it can be shown that

none of the quantities in the equations to be solved goes over 10 volts

during an experiment. After normalization, the equations to be solved

22

by the analog system are (3.13) and (3.14), with all the quantities nor-

malized on (Ueff)max.

A and B values can be set to be the same for all wires during

calibration, since they are functions only of the minimum and maximum

velocities of the calibration range and the voltages desired. If the

stem of the three-wire probe is aligned with the calibration flow, all

the wires are exposed to the same minimum and maximum velocities.

The coefficient matrix M can easily be obtained using the known

values of kl and k2 for each wire. For example, for kl = 0.15 and

k2 = 1.02, the matrix is:

-0.5058 r

0.5155 0.4750

M = 1 0.4750 -0.5058 0.5155 1 (3.15)

0.5155 0.4750 -0.5058

In this matrix the. values used for kl and k2 are the same for all

three wires. Uniform values of k1 and k2 were used in the present

study, but the analog system is designed such that it can easily handle

different values for kl and k2 for each wire. An uncertainty analy-

sis showed that a 1% change in kl causes a change of only 0.02% in the

indicated velocity, while a 1% change in k2 yields a change of l-04%,

regardless of the flow angle. Thus the final results depend only slightly

kl but nearly linearly on the value of k2. As was shown by Jorgerl-

sen (1971), the change in k2 with flow angle is very small.

0 0 JsI3 G/37

cos a sin a -G/6 G/3 (3.16)

-sin a cos a &I2 0,

where the angles a and w are shown in the following figure: a is

the angle of rotation around the probe axis and w is the angle between

the probe axis and the x axis when the probe body is in the X-Y plane.

The angles are measured in the direction indicated by the arrows. Angle

a is taken to be zero when the third wire is in the x-y plane and the

longer prong is below the shorter one.

23

The 3-D Turbulent Flow Analyzer has been designed and built to take

the linearizer OlJtpUts E,ffl' Eeff2. and Eeff3 as inputs and give

instantaneous II, v, and W (the normalized velocities in laboratory

coordinates) as outputs. Fig. 3.2 shows a photograph of the two control

panels of the 3-D Turbulent Flow Analyzer. On the lower panel, there is

one potentiometer for setting the A coefficient, and three potenti-

ometers for setting the B coefficients. Three potentiometers are needed

for B values because of circuit requirements. The nine coefficient po-

tentiometers for the inverse Jorgensen Matrix CM) are shown, each with

its own pair of test points for checking values. The meter provides a

continuous display of E/C, the time-averaged value of the magnitude of

the normalized velocity vector. The outputs are the normalized velocity

components in wire coordinates X, Y, Z and a value of R/C, the root

sum square of the components, divided by three. The upper panel accepts

the X, Y, Z as inputs and contains nine potentiometers whose values can

be set according to the wire position to calculate the values of II, v,

and W, the normalized velocity components in laboratory coordinates.

A flow diagram of the complete system is shown in Fig. 3.3.

The electrical performance of the 3-D Turbulent Flow Analyzer was

checked up to 20 kHz for phase shift and attenuation of magnitude.

24

Maximum phase shift was between 1 and 2", and maximum attenuation of

signal magnitude was approximately 0.1%.

3.4 Processing of the Data - There are several options which may be exercised once the instanta-

neous values of the three components of velocity are available as analog

electrical signals. Analog multipliers could be used to form instanta-

neous products, with integrating voltmeters for time averages and RN'S

meters for fluctuation measurements. In the work reported here, the three

signals were digitized and processed through an HP-2100 system. The A/D

converter used had a capability of 45,000 points per second with which to

process all three channels. This would have !.imitcd the acceptance to

about 7500 Hz, using two points per cycle. Peripheral limitations on the

current system in fact limited the response to 5882 Hz. Thus the high-

frequency events cannot be seen through the present system, but the re-

striction is in the A/D system, not the analog system. Values of u' 2

measured by analog and digital methods agreed to within 2-3%; thus the

frequency limitation on the digital system seems not to have been impor-

tant. Data were recorded onto digital tape and processes in computer-

idle time, overnight. Twenty-two seconds of data-taking per point re-

quired 11 minutes of post-acquisition processing on the HP-2100 to produce

the values of the three mean components and t-he six Reynolds stresses.

With all-analog processing, the nine outputs could be generated in what-

ever time was felt necessary to achieve stationary values, based on the

flow.

The functions of the Flow Analyzer could all be performed digitally,

by passing the three linearizer outputs through A/D converters by exe-

cuting the matrix-inversion processes digitaiiy. Using the same HP-2100

system, with the same A/D conversion unit, the total time required for

ail-digital processing was 75 minutes to process 22.6 seconds of data-

taking. The use of analog operations for the two matrix inversions re-

duced the computing time by a factor of 8.5. Different computer systems,

with different A/D conversion and higher speed would surely make the di-

rect digital system look more attractive than these results. Whatever

digital system is available, however, the two analog stages presently

used will continue to be attractive from a cost-of-running standpoint.

25

3.5 Problems Created by the Size of the Triaxial-Wire Probe

The triaxial wire probe creates some problems in the measurement

because of its relatively large size compared to the single- and cross-

wire probes. The probe has three wires, with their centers located on a

sphere of diameter 3 mm to prevent interference due to the wakes of wires.

Measurements made on this sphere must be attributed to the center of the

sphere. When making measurements in high shear flows, there may be some

uncertainty about where the effective measurement center is, since it

will change depending on the orientation of the wires with respect.to the

gradient. Another problem is the overall size of the probe, about 0.8 cm

in diameter. It is not possible to obtain data closer than 0.4 cm to the

wall.

3.6 Qualification Tests

The Thermosciences Laboratory in the Mechanical Engineering Depart-

ment of Stanford University has a two-dimensional channel which gives

fully developed mean-velocity and turbulence profiles at least to the

second-order turbulence quantities. The performance characteristics of

this channel have been fully explored; it. has been used by several re-

cent experimenters to calibrate their hot-wire technique, their probes,

and their systems. The Flow Analyzer PTas tested in this channel to qual-

ify its performance and to explore its limitations for turbulence mcasure-

ments.

For the qualification tests, the probe was mounted in a two-axis

probe holder so that it could be rotated around the streamwise axis (roll

angle aj and also -tilted (pitch angle bj> against the approaching flow,

as well as traversed to several different distances from the wall. By

measuring at several distances from the wall, the system performance was

recorded both for high shear regions (near the wall) and zero shear re-

gions (at the centerline of the channel). The outputs were compared with

the outputs of the other acceptable methods of measurement in the channel

(single horizontal wire, pitot tube measurements, and linear shear stress

distribution calculated frcm the pressure gradient along the channel).

26

The two-dimensional channel is 6.35 cm wide and 117 cm high. The

experiments were made with air flow at ambient conditions and with a

centerline speed of 11.2 m/.sec.

In Fig. 3.4 the values of c (the streamwise mean velocity) .obtained

from a pitot probe and from the triaxial probe are compared. The pitot

probe was modified to create the same stem blockage effect in the tunnel

as did the triaxial probe, as seen in Fig. 3.5. In this test the probe

was set to zero roll (a = 0") and the axis was aligned with the flow

(w = 0"). The readings of the pitot probe were corrected for shear dis-

placement effect and for turbulence level. The maximum difference between

the pitot probe and the triple wire probe occurs near the wall -- about

2.4%. The difference diminishes rapidly as the distance from the wall

increases. The difference near the wall may be due to the finite size of

the triaxial probe, as was explained earlier, interacting with the veloc-

ity gradient.

The effect of rotation of the probe around its axis was investigated.

When the probe axis is aligned with the flow direction (w = 0"), rota-

tion around its streamwise axis should not affect the result if the vel-

ocity is uniform, but may affect the result in a shear flow. To investi-

gate this, the probe was aligned with the flow direction (LL~ = O"), and

for each transverse position across the tunnel, the probe was rotated

around its axis to the values of a = 0", 90", 180", and 270". This

angle range covers the extreme positions for the wires and exposes dif-

ferent configurations of wires to the shear at different angles.

Figure 3.6 shows the three mean velocity components 6, 6 and w>

as a function of the distance from the wall for several values of the

roll angle a. Roll around the probe axis does not affect the v values.

The effect on 7 and w is small, but not negligible. In this figure

7 and 7 should be zero, but due to the probe size, some deviation from

zero is observed within the shear region. The.most meaningful comparison

for error in v and w is to compare them to the c at the same loca-

tion. The largest deviations occur at the point near the wall for a = 0, -- -- V/U = l%, and for a = 270", W/U = 4.5%. These deviations from zero

become smaller as the distance from the wall increases. In the zero gra-

dient region at the centerline there is no deviation.

27

Figure 3.7 shows the turbulent kinetic energy and shear stress dis-

tributions as a function of the distance from the wall for several values

of the roll angle. The shear stress (-u'v') measurements are compared

with the linear shear stress distribution obtained from the pressure gra-

dient along the channel (dP/dx). As is seen, all the experimental data

lies inside the *lo% error band, but at angles a = 90" and a = 270"

the deviations are much smaller; therefore, one would like to measure -i-i -u v at these angles. The measured turbulent kinetic energy is not

much affected by the roll. The largest difference between results occurs

at the high shear region near the wall; it is about 3.5%. In the zero

shear region there is no effect of roll. 2

Figure 3.8 shows the diagonal Reynolds stress components (u' ,

7, w12) as a function of the distance from the walLnormalized with

centerline velocity. The streamwise normal stress (u'~) does not seem

to be affected much by roll around the probe axis Leven in the high shear

regions near the wall. On the same figure the u' 2

distribution obtained

with a conventional single horizontal wire is also given, and its agree-

ment with the triaxial wire data is good. The other normal Reynolds

stresses (v' 2 and wf2) are affected by the roll angle, especially in

the high shear region, but the data collapse on each other quickly as the

shear decreases. In the zero shear region on the centerline there are no

deviations. One important point to observe is that, at a certain a 2

value, if v' reads high compared to the value at a = 0", then w' 2

reads low, or vice versa, while u' 2 does not change much with a.

This combination leads to qL values which are quite insensitive to the

changes in a, a fortuitous result for the measurement of q 2 -- the

main interest of the general research.

The data discussed above are enough to qualify this system for mea-

surements when the probe axis is aligned with the flow direction. But

one of the most important objectives of this research was to find a method

which would work in a flow of unknown direction. To investigate this,

the probe axis was tilted against the approaching flow direction (w),

again in the 2-D channel. Some rotations around the probe axis (a)

were also tested to see the combined effect of bcth a and w. The re-

sults are discussed in the following paragraphs.

28

Figure 3.9 shows the three mean velocity components as a functicn of

distance from the wall for several values of the angle between the flow

and the probe axis (u). There the value of a was held constant, be-

cause it was seen above that the mean velocities were not much affected

by the roll engle. Up to w = 20" the data for i collapse on top of

each other and the deviation for w = 30" is not very large. The larg-

est deviation at this angle is about 3.5% in the high shear region near

the wall, and about 2% in the zero shear region. Deviation is calculated

as the difference between two extremes, not from the pitot probe data.

This result means that mean velocity can be measured with good accuracy

if the approaching flow direction is within f 30" of the probe axis;

i.e., one does not have to know the flow direction better than within a

cone of half apex angle 30" around the probe axis to measure the mean

velocity with acceptable accuracy. (If 7 = 0.15 this will give an

angle of about ?6%.) As the angle between the flow direction and the

probe axis increases, the errors in v and w also increase. Some of

this may be the effect of the probe size, as was explained earlier, and

some may be due to the probe blockage effect in the channel.

In Fig. 3.10 the turbulent kinetic energy (TKE) and shear stress

are plotted for several values of roll angle (a) and pitch angle (w).

In the TKE plot the line at the center is faired through the data at

w = O", a = o" (this measurement should be the one closest to reality).

The other lines denote the + 10% and 515% error bands around the refer-

ence. Again the data points converge rapidly as the distance from the

wall increases. Deviations are much smaller in the zero shear region.

In the same figure also, the q* distribution for a = 90", o = 20" is

shown to demonstrate the increase in deviation as w increases. Another

important point to observe from this figure is that for a = 90" and

w = 10" the data lie very close to the center profile. This shows that,

depending on the quantity being measured, there are angle combinations CI

and w for which the measurement cone can be enlarged. For example, for

a = 90" and w = 10" and 20", it appears that even in the highest

shear region q 2 can be measured within 12% inside a cone of 15" half

apex angle around the probe axis. The deviations in q 2 are not like

uncertainty scatters, but rather have a preferred direction. It may be

29

possible to d.evise a scheme to correct the data based on the first

estimate of the flow direction, to improve the accuracy. In the shear

stress part of Fig. 3.10, the straight line in the middle of the figure

is the shear stress distribution obtained from the pressure gradient in

the streamwise direction. The other straight lines are the boundaries

for + 10% and+15% error. Most of the data up to the angle w = 10" lie

within 10% error band, except a few points near the wall for angles a =

270' and a = 90". Almost all the data, including 1~ = 15' and a = O",

lie within the +15% error band. In conclusion, it can be said that the

shear stress 1 can be measured within 10% within a cone of half -u v

apex angle 10" and within 15% inside a cone of half apex angle 15", ex-

cept very near the wall. -- Figure 3.11 shows the normal Reynolds stress u'

2 as a function

of the distance from the wall for several pitch angles. It was shown be-

fore that uq2 does not change much with roll angle CL. For this reason,

different a and w combinations are not included in the figure, and

attention is given only to the w values. up to w = 15" the measure-

ments collapse on top of each other except at the point nearest the wall,

where the data for w = 15O deviate from others about 5%; but these data

converge rapidly to the others with increasing distance from the wall.

The data for w = 20" deviate about 15% from the data for smaller w's

near the wall, but again conver=fast with increasing distance frcn the

wall. The conclusion is that u' 2

can be measured with good accuracy

(maximum error being 5% near the wall) up to the angle w = 15", i.e.,

within a cone of half apex angle 15". The deviation in the zero shear

region is about 1% or 2% up to w = 30".

Figure 3.12 shows the normal Reynolds stresses v' 2

and w' 2 for

different roll and pitch angles. The data for v' 2 collapse on top of

each other for w = 0" and w = 5". For ((II = lo", CL = 0") and (QJ =

10", a = 180"), the data generally lie in the 10% error band, except the

next-to-wall point for (w = lo", u = O"), for which the deviation is

about 15%. Generally, it can be said that within a cone of half apex

angle 5", v' 2 can be measured with very good accuracy (less than 2%

deviation). The measurement cone angle can be enlarged to 15" in a zero

30

shear region. 2 In the high shear region one can measure v' within 5%

for the case of w = lo", a = 0", and within l&15% for w = lo", a =

180".

Considering now the wVL data, in the zero shear region the data

converge very well, and good accuracies can be obtained up to w = 15".

up to W'5O the data lie within 10% of each other in the high shear

region. For the larger values of w, certain combinations of a and w

are necessary to get the data within lo-15% error bands in a high shear

region. For example, all of the w = 10" data is out of the 15% error

band for all a values, but the data for w = 20" and a = 90" fall

within the 15% error band.

The qualification data discussed above show that the size of the

measurement cone changes, dsnding on the quantity measured. -- Some quan-

tities, such as v' 2 2 and w' , are affected by rotation around the

probe axis, especially in high-velocity gradient regions. The reason for

these changes may be the large probe size.

An important point shown by the qualification data is that the major

mean velocity component f can be measured quite accurately within a

cone of 30" half apex angle around the probe axis. In three-dimensional

flows where the probe axis makes a large angle with the unknown flow di-

rection, the errors in the measured turbulence quantities may be unaccep-

table. In critical cases, one might use a two-step process, first finding

the flow direction approximately (within 3-4"), and then aligning the

probe with this direction. In this position all of the turbulence quan-

tities could be measured quite accurately.

3.7 Further Work on Triaxial Wire System

As has been seen, the size of the probe creates some problems in

high shear flows, by causing artificial velocity components to be indi-

cated which depend on the arrangement of the wires in the velocity gra-

dient. Work is in progress currently to predict the magnitude of the

artificial velocity components indicated, as a function of the orienta-

tion of the probe in flow and the value of the gradient of mean velocity.

It is hoped that this work will lead to a method of correcting the turbu-

lence quantities.

31

Secondly, to eliminate the probe size effects, a new, smaller probe

design is being considered for possible future extensions of this work.

3.8 Conclusions and Capabilities

A hot-wire method and system have been developed to make measure-

ments in 3-D turbulent flows with high fluctuations and partially unknown

flow direction. The results are summarized below. The error figures are

for high-velocity gradient (about 1600 l/set\ regions; for zero or low

shear regions the errors are smaller and consequently the measurement

cones are larger. With this system:

l The mean velocity can be measured with good accuracy within a cone

of 30" half apex angle around the probe axis.

a The turbulence kinetic energy can b? measured with lo-15% accuracy

within a cone of half apex angle 10-12".

l The shear stress (-u'v') can be measured within 210% accuracy

in a cone of half apex angle of 10" and "15% accuracy within a cone

of 15" half apex angle.

l The streamwise normal Reynolds stress component can be measured

with a maximum error of 5% within a cone of half apex angle 15".

The errors in the quantities mentioned above are not random but

rather have a preferred direction. Some of these effects can be predic-

ted from the known effects of probe size and position in the vel.ocity

field. Work in this area is continuing.

The triple wire method is very fast, with the data-acquisition time

being governed by the time required to establish a stationary value of

the process being studied. Thus, large amounts of data can be taken.

With the addition of RM and correlating circuits, quantities such

as TKE and mean velocity could be displayed directly with analog instru-

ments. This would eliminate the need for complicated data reduction --

usually the most tedious part of an experiment.

32

-

Fig. 3.1. Triaxial wire probe tip and wire coordinate system.

33

W

P

Fig.

3.

2.

Phot

ogra

ph

of

the

3-D

Tu

rbul

ent

Flow

An

alyz

er.

r -

3-D

TU

RBU

LEN

T FL

OW

AN

ALYZ

ER-l

,

‘3

c\1

9 I

‘c”

.E

PRO

BE

q J

POTS

11

I L

---

----

v-

--I

Fig.

3.

3.

Flow

di

agra

m

of

the

com

plet

e sy

stem

.

r

3-c

y (cm

2.c

0 TRIAXIAL WIRE a=O” w=O”

V PITOT PROBE 8

&?

-

8

U( m/set)

Fig. 3.4. Comparison of triaxial probe and pitot probe measurements of if in the 2-D channel.

36

Fig.

3.

5.

Phot

ogra

ph

of

the

triax

ial

prob

e an

d m

odifi

ed

pito

t pr

obe.

4.0-

Y (cm)-

3.0-

2.0-

l.O-

u=OO Q

0 o” 0 180°

1

TRIAXIAL A 90° WIRE V 270'

PITOT PROBE ----p--+

f mm W(m/sec) 0.0 , 5.0 L I I I I I I

0.0 I

50 V (m/set) O@OO.O

I I I I I I I I I I I I

5.0 10.0 Cl (m /set)

Fig. 3.6. Effect of roll angle about the probe axis on the mean velocity components.

(cm) k

m 3.0

-2

t

ff

L

2.0-

l.O-

0 Q = o” 0 Q =180° V a = 90° A a =. 270°

“r dx

0.2 -27 (m*/sec*)

q*( m * /set*)

Fig. 3.7. Effect of roll angle about the probe axis on shear stress and turbulent kinetic energy.

39

P w=OO

a

m a m cl o” 0 180°

y (cm) v 9o” Q m a A 270°

3.0

ilP ff O&?a

1.0 0-

ml

OAOV OAOV VOQh VOQh oar oar I I I I C?Aay C?Aay VW VW a9 a9

0.05 0.05 I I I I I I I I I I I I l/7& l/7&

0.05 0.05

O-Oo.0 O-Oo.0

I I I I I I I I I I I I I I GWJ~ GWJ~

0.05 0.05 4-t 4-t u’2 UE u’2 UE

Fig. 3.8. Effect of roll angle about the probe axis on the normal Reynolds stresses.

40

4.0 t-

Y m-o-

3.0 -

2.0 -

l.O-

w Q 0 o” o” A 5” 0” v 10” o” 0 15O o” 0 2o” o” l 3o” o”

PROBE

1

W(m/sec) J(m/sec)

U(m/sec)

Fig. 3.9. Effect of pitch angle on the mean veloc- ity components,

41

--

3.0

I\

\ y

(cm

) -7

7 \

w

a

0 o”

o”

A

5’

0’

10%

ER

RO

R B

AND

v

10°

O”

15%

ER

RO

R B

AND

.

2o”

9o”

v lo

o o”

A

10”

90”

. 10

’ 18

0°

n 10

’ 27

0°

0.01

I

I I

I, I,

I !

, I

I I,,

,

, ,

, ,

r 0.

0 0.

5 1.

0 1.

5 2.

0 q*

( m

*/sec

*)

Fig.

3.

10.

Effe

ct

of

com

bine

d ro

ll an

d pi

tch

angl

es

on

shea

r st

ress

an

d tu

rbul

ent

kine

tic

ener

gy.

Y (cm)

1.c

0.c

)--

‘0.c

l lm3

w 0 o” A 5O v loo 0 15O 0 2o" a 3o"

Q

O0 O0 O0 O0 O0 O0

anax

I I I I I I ,* I I

3 0.5 u’2( m2/sec2)

Fig. 3.11. Effect of pitch angle on the u' 2

Reynolds stress.

43

3.0-

A 5O 0’ v loo o”

Y (cm> -

O*Oo.o I 1 I , 1 1 I I I I 0.5 0.0 0.5 I I 1

c d( m*/sec*) v12( m2/sec2)

Fig. 3.12. Effect of combined roll and pitch angles 2 on the v' and w'

2 Reynolds stresses.

44

Chapter 4

PREDICTION OF RECOVERY REGION HYDRODYNAMICS

WITH A ONE-EQUATION MODEL OF TURBULENCE

One of the main objectives of this research was to see whether or

not the recovery region hydrodynamics could be predicted using a one-

equation model of turbulence in a two-dimensional boundary layer compu-

tation program, STAN5 (Crawford & Kays, 1975).

An algebraic equation for the mixing length was needed for the

model. The actual mixing length profiles were first obtained from the

experimental profiles (u'v' and E) and purely empirical curve fits were

used to model the behavior. The piecewise model was used in STAN5, as

a check, and the predictions of the data were very successful. Each

region in the mixing-length profile was then interpreted physically and

alternative, plausible equations identified. Thi,; procedure showed that

the recovery region hydrodynamics can be satisfactorily explained by

postulating a two-dimensional boundary layer growing inside the thicker

initial boundary layer. The following sections present the details of

these steps.

4.1 Equations to be Solved

The following equations must be solved to obtain mean velocity and

turbulent kinetic energy (TKE) profiles. For a two-dimensional, turbu-

lent boundary layer under isothermal conditions, at ambient temperatures

and low speeds, at the constant free stream velocity and without any ex-

ternal body forces:

Continuity:

x-Momentum:

7 -Y ?L+” = 0

ay

- - pyy 5 + pv av = -a aG - ay ay ( !Jay-PU 'V' 1

45

(4.1)

2 TKE: pTi&+ pv -?gf = -7T@-

ay -puv ay v

+ ay a (8 (4.3)

pX dission diffusion

Boundary conditions:

Momentum eqn. y = 0

(4.4)

The equation for TKE is not solved all the way to the wall in STAN5,

but only to y+ = 2A+ + where A is a measure of sublayer thickness. It

is assumed that the flow is in local equilibrium (experiments confirm th

assumption) below y + = 2A+ and Prandtl's mixing length can be used in

this region. + The TKE at this point is calculated such that at y = 2A+

the eddy viscosity obtained from the mixing length model is equal to the

one obtained from a one-equation model of turbulence. This condition

gives the following boundary conditions on TKE:

- F II E) at yf = 2A+

4

2 lim .$ = qm

2 V- ,

(4.5)

The following terms must be modeled in equations (4.2) and (4.3) in

order to obtain 2 soluble set of equations: -7-i IlV , D, J. The term --

for u'v' will be modeled after Boussinesq (1877) with eddy viscosity

model:

(4.6)

The term ~~ will be modeled after Prandtl (1945) and Kolmogorov (1942):

2 Aq 2 3- _--

EM = K J 2 (4.7)

is

The term "D" dissipation of 'IKE will be modeled 2s given in Launder &

Spalding (1972):

46

(4.8)

and diffusion of TKE "J" is also to be modeled as in Launder & Spalding

(1972).

J = (v + Eq) a (q2/2) ay (4.9)

where SC = E /E 9 M q'

The production of TKE, "P", can be expressed as

7 au p = --pu’v - = p ay

(>) 2&2Ei’ (4.10)

To complete the model, the following constants or functions need to

be specified: R, A B SC A q' 9' 4' q

is the production constant and can

be obtained from the value of the correlation coefficient - u'v'/q2

near the wall. It is generally about 0.22 (Wolfstein, 1969). B is 4

called the dissipation constant and can be evaluated under the condition

that the production of TKE is equal to its dissipation in the region near

the wall. This condition gives the following relation:

(4.11)

K = 0.41 (4.12)

From this equation, B = 0.377 is obtained. q

"scq" y the Schmidt number of TKE, is expressed as is shown in the

following figure. It has a value of 1.75 near the wall and 0.5 near the

free stream. Launder & Spalding (1972) suggest that.for film-cooling

applications SC should follow a linear distribution from 1.75 near the q

wall down to 0.5 near free stream, but during the present predictions,

the distribution shown in the preceding figure was found to work better.

The extended region of SC 4

= 0.5 near free stream played an important

role in modeling the correct diffusion of TKE near the edge of the momen-

tum boundary layer. The value near the wall did not seem to be as impor-

tant because of the dominant role of production and dissipation in this

region.

The last quantity to be modeled to complete the set is the mixing

length, "R". Predictions and data show that all other constants men-

tioned above have their usual values (i.e., the values used for 2-D flat-

plate boundary predictions). The success of predictions depended very

strongly on the correct modeling of "1" t as will be shown and discussed

in the following sections.

4.2 Mixing-Length Model - A mixing-length model was developed specifically for the recovery

region. This same model can later be applied to the full-coverage re-

gion, however, because the boundary layer should behave almost like a

recovery region between the injection rows.

The following requirements should be met by the mixing-length model

for it to have at least some limited universality:

a It should be possible to relate the deviations from the 2-D mixing

length to physical events taking place in the flow fieid.

l The dynamics of the model shouldall@wit to relax back to a 2-D

flat-plate mixing length.

The general approach taken in modeling the mixing iength was ex-

plained at the beginning of this chapter. More specifically, the model-

ing process will be explained in Sections 4.2.1-4.2.9 in the following

order: First the flow structure in the recovery region will be explained

briefly (Fig. 4.1). Then each region in the mixing length profiles will

be discussed and empirical equations will be given with supporting physi-

cal arguments. Figures 5.27, 5.28, 4.1, and 4.2 are used for these

discussions, and they will not be mentioned separately in the discussions

4s

of each region. Figures 5.27 and 5.28 show the mixing-length distribu-

tions obtained from experiments by

(4.13)

Three streamwise positions were used: the stari: of the recovery region

(x = 188 cm) and tvc do;lnstream recovery region stations (x = 214 cm,

256 cm, or 27 hole diameters and 67 hole diameters downstream of the last

row of injection. The mixing length at the start of recovery region was

obtained from the spanwise-averaged profiles at this location. Fig. 4.2

shows the modeled mixing length superposed on Fig. 4.1 to show the rela-

tion between the regions of mixing length and the flow structure of the

recovery region. The new information is on 1 vs. y coordinates.

4.2.1 Flow structure in the recovery region - -~- Figure 4.1 shows a sketch of the flow structure in the recovery re-

gion. The basic structure is an internal two-dimensional (2-D) boundary

layer growing inside the thick boundary layer. The thickness of the

inner 2-D boundary layer is 6' and the thickness of the overall bound-

ary layer thickened with the injection process in the full-coverage re-

gion is 6. In the middle regions the two boundary layers blend (the

cross-hatched area) with the help of cumulative jet spread. The region

next to the wall is not cross-hatched, however, because the processes in

this region are ccmpletely controlled by the wall. The 2-D internal

boundary layer has an initial thickness at the start of the recovery re-

gion which depends on the upstream conditions (e.g., the blowing ratio,).

This internal boundary layer growth has been observed by several other

experimenters whenever there is a sudden change of the surface conditions.-

For example, Antonia 60 Luxton (1972) observed such an internal boundary

layer in their experiments on the response of a turbulent boundary layer

to a step change in surface roughness.

When the mixing-length profiles are examined in Figs. 5.27 and 5.28,

five distinguishable mixing length regions can be identified, numbered in

Fig. 4.2. In the same figure, three distinguishable flow regions are

49

shown, numbered I, II, and III. Region I is the near-wall region of the

2-D internal boundary layer, II is the blend region dominated by cumula-

tive jet effects, and region III is the outer region, dominated by the

outer region of the thick boundary layer.

4.2.2 Region 1 of the mixing length - A region very close to the wall, where L = KY, is termed region 1.

Experiments show that this region extends up to y/6 = 0.14 for M = 0.4,

but only up to y/d = 0.055 for M = 0.9.

This is the inner region of the 2-D internal boundary layer where

the length scale is based on the distance from the wall 91 ,I y . For pro-

gramming convenience, the mixing length and the distance from the wall

in all the regions were normalized on the total boundary layer of thick-

ness 6. In this region, however, the wall effects are dominant and de-

termine the heat transfer regardless of the blowing ratio. The effect

of blowing is mainly to change the initial thic!mess of the internal 2-D

boundary layer. The proposed model for this region is:

D for 0 < (8 2 ($)d (4.14)

where D is the damping function (Van Driest-) and (yb)d is the depar-

ture point from "KY" line. This point corresponds to y/6' = X/K for

the inner 2-D boundary layer of thickness 6'.

The following empirical equation is given for (Ymd:

+ ClM[(f)- C2] (4.15)

where Cl = 0.0045 and C2 = 37. It is seen that as M gets large the

initial value of (Y/B)d = l/K - ClC2M gets smaller, indicating that a

higher blowing ratio destroys the near wall layer, causing a smaller ini-

tial thickness for the 2-D internal boundary layer. This fits the physi-

cal situation very well. C2 is the number of boundary layer thicknesses

at which (yh)d reaches the A/K point for the outer boundary layer.

In fact, then, the recovery is completed. (Antonia*also supports C,

being 37.) Cl controls the rate of recovery of (y/&Id point.

* Personal communication. 50

4.2.3 Region 2 of the mixing length

Experiments show that in this region the mixing length is constant

and low, compared to the outer region of a 2-D flat-plate boundary layer

of the same thickness. This region is barely visible fcr M = 0.4 but

is obvious for M = 0.9. It extends up to about (y/6) = 0.2 for all

stations and corresponds to the outer region o f the inner 2-D boundary

layer, where R = Xb'. Here the eddy size does not depend on the dis-

tance from the wall, nor does it depend on the total thickness of the

outer boundary layer.

The distance of this region from the outer edge means that the outer

length scale does not affect it. This region does not extend all the way

up to y=6' for the reasons stated below.

The following empirical equation is proposed:

& = K (4.16)

This region does.not extend all the way up to 6'. As soon as the edge

of the inner layer of the outer boundary layer (y/b = X/K) is encoun-

tered, the length scale begins to be affected by the outer length scale

of the outer boundary layer. The branch point is given as X/K.

4.2.4 Region 3 of the mixing length -- Experiments show that in this region the mixing length is still below

a 2-D flat-plate value. This is barely visible for M = 0.4 but is ob-

vious for M = 0.9. In this region, the mixing length rises from the

value in region 2 to 0.085. The region starts around y/6 = X/K " 0.2

and extends up to y/6 = 0.3 for M = 0.9 and y/6 2 0.25 for M = 0.4

for the first station. It moves away from the wall in the recovery re-

gion.

This is the first section of the blend region of inner and outer

boundary layers (region II in Fig. 4.2). Here the length scale changes

from the 2-D inner layer value to the value in the outer layer in a me-

dium dominated by the jet flow regime.

The following empirical equation is given for this region:

51

R ( 1 6 =a ( ) $ + b for f < (4.17)

The linear combination is all that is justified and fits the experiments.

Here, (p/6) i is the intersection point of the new mixing length line

and the (a/s> = X line. The (y/6) i point is the effective center-

line of rising jets and moves out in the recovery region as the jets

spread outwards. This will later be shown quantitatively in section

4.4.2. The following emPirica equation is given for this point:

+ C3M [($, + C4] (4.18)

It has an initial value of ii/K + C3C4M and an advance rate of C3M(x'/G)

in the downstream direction. The initial value increases with M, indi-

cating a deeper penetration for high blowing, which fits the physics.

The values of the constants are:

c3 = 0.0275 , C4 = 4.0 (4.19)

and the equations for coefficients a and b in equation (4.6) are:

A - K(Y/s>d a =

(Y/a>i - ~

2

b = - $ + K(Y/6)d (Y/s&

cymi - ;

4.2.5 Region 4 of the mixing length

Experiments show that a region of

between 0.24 ( (y/6) < 0.37 for M =

augmented mixing length is located

0.4 and between 0.35 r_ (y/6) <

(4.20)

(4.21)

0.55 for M = 0.9 at the start of the recovery region. This region is

wider and higher for M = 0.9 than for M = 0.4. The augmented region

vanishes in the recovery regions.

The augmentation in the mixing length occurs due to high shear be-

tween the cumulative jet spread and the outer layer fluid. The maximum

in the augmentation occurs somewhere between the outer edge of the jet

52

spread and the effective jet centerline. In the case of higher blowing,

this region is moved outwards because of the greater jet penetration.

The augmented region also moves outwards in the downstream direction in

the recovery region, as jets rise; but at the same time it vanishes, as

there is a decrease in shear between the cumulative jet spread and the

outer layer. This is discussed further in section 5.9.

This region was modeled as a parabola, the simplest curve which fit

the data:

c (zf + d ($)+ e (4.22)

and

R ( 1 6 = h+(i)a for (f)i < (5) L(:)+' (4.23)

Here, c, d, and e are constant coefficients. 'Their values depend on

end points and will be given after the discussion of the end points.

W is the width of the augmented region at the (R/6) = A level. Based

on the experimental evidence, it was made a function of blowing ratio M.

Since no better formulation was possible, it was expressed as a linear

function.

W = C7M (4.24)

In Figure 4.2, ('lmmar is the maximum value of the augmentation. Its . decay in the downstream direction was expressed as an exponential func-

tion.

R

( 1

R

6 max = 6 max jm ( 1

, exp(- 'x~~s') (4.25)

(us> max,i is the initial value of augmentation. It is given from the

experimental evidence, as follows:

II ( 1 6 = C5M (4.26)

max,i

53

The coefficients for the parabola are given as:

4(R/6) max C =

W2

d =

4(U6) e = - --f= (f)i kq + w]

The values of the constants are:

c5 = 0.496 , C6 = 0.435 , c7 = 0.333 (4.30)

(4.27)

(4.28)

(4.29)

4.2.6 Region 5 of the mixing length

Experiments show that in this region the length scale is constant

and it is (i/6) = X a 0.085. The region begins after the augmented re-

gion and continues up to the free stream for both blowing ratios.

This region is the outer layer of the outer boundary layer, where

the mixing length scaies on the total thickness 6. In this region, the

following empirical equation is given:

R ( ) 6 = X for g +W < s

( ) i ( 1

4.2.7 Summary of the model equations

Region 1 (4.14): = K ($)D , 0 c (5) 5 (5)

d

Region 2 (4.16): R

( ) Y ' =K6d' ( 1

(4.31)

Region 3 (4.17) i

54

Region 4 (4.23) $ ( 1 = x + ($) a 9 (5) i

< ($1 5 (5) +w i

Region 5 (4.31): i

Unknowns in the above equations are:

Region 1 (4.15):

Region 3 (4.20):

(4.21):

(4.18):

Region 4 (4.22)

(4.27)

(4.28)

(4.29)

(4.25)

(4.26)

; + ClM C

(~'/a) - C2 1 D = 1.0 - exp(-y+/A+)

x - dY/G)d a =

(ymi - X/K

2

b - $ + K(Y/6jd (Y/f$

= (Y/q - 5

: + C3M [($) + C4]

II ( 1 s a = c (z)2 + d(z) + e

4(1/6) max C = -

W2

8(11/g) d =

W2max[(X)i +!-I

d = -

R ( )

R 2. max = 2. max i ( 1

, exp(- v)

II ( ) ;j- =

max,i C5M

55

(4.32)

(4.24): w = c7M

Constants:

c1 = 0.0045 (controls the rate of recovery of (y/6)$ -

c2 = 37.0 (number of boundary layer thicknesses at which (~/6)~

approaches (X/K) or the 2-D internal layer coalesces with

outer layer).

c3 = 0.0275 (controls the rate at which the (v/S)~ point advan-

ces, i.e., the rising of the effective jet centerline).

c4 = 4.0 (controls the initial value of the (Y/S)~ point at the

start of the recovery region).

c5 = 0.496 (controls the magnitude of the initial augmentation).

c6 = 0.435 (decay rate of augmentation).

c7 = 0.333 (controls the width of the augmented region).

Figure 4.3 shows the experimental and modeled mixing lengths for

M = 0.4 and M = 0.9 at the three different stations in the recovery

region.

4.2.8 Comments on the number of constants --- Except for the usual universal values of K = 0.41 and A = 0.085,

seven constants were used to model the mixing length. The number of con-

stants looks very large at first glance, but when examined more closely

one can see that this number is reasonable. Three different flow struc-

tures are being modeled: inner and outer boundary layers and blend re-

gion in between, as well as their dynamics in the streamwise direction.

During the prediction process it was observed that the most important

constants in the mixing length model were Cl' C2’ C3’ and c4' The con-

stants used to specify the augmented region (C 5' '6" and C7) did not

prove significant. The augmented region did not halre much effect on the

predictions because it lies in the outer region of the boundary layer.

It also decays rapidly. Durir,g computer experiments with W and

56

WQmax, changing the values of C5 and c7 did not affect the results

at all. It is possible that the augmented region can be completely elimi-

nated for the recovery region predictions, thus reducing the number of

constants to four. The constants were obtained from the empirical data

rather than from computer experiments. Kacker & Whitelaw (1970) used five

constants to model the mixing length for prediction of wall jet and wall-

wake flows, which is similar to, but not more complicated than, the re-

covery region of the present full-coverage film-cooled surface. In the

present model the augmented region is kept for convenience and for easy

adaptation of this mixing length model to the full-coverage region where

the peak of augmentation moves closer to the wall and is important for

the prediction of heat transfer.

In the recovery region predictions, especially for M = 0.9, the

most important region of the mixing length turned out to be the reduced

mixing length region near the wall. Without ccrrect modeling of this

region predictions always failed. The first four constants are the im-

portant ones in the recovery region predictions.

4.2.9 General remarks about the mixing length

It was observed that only regions 2, 3, and 4 show deviations from

a usual 2-D mixing length model. Region 2 proved especially interesting,

because it had been thought that the jet mainstream interaction would

increase the mixing length above the 2-D value (Choe et al., 1975; Craw-

ford et al., 1976). Choe et al. (1975) and Crawford et al. (1976) cal-

culated the mixing length for low blowing and in the fuli-coverage region.

They observed that the augmented region (Region 4) moved closer to the

wall and did not observe Regions 2 and 3, which are more easily observable

for high blowing. In reality, Regions 2 and 3 are very important in pre-

dictions, as is explained in Section 4.3.1.

4.2.10 Physical explanation of recovery to 2-D state in the mixii -~ length model

The relaxation to the 2-D state takes place in the following manner.

As the cumulative effects of jets move downstream and diffuse out. they

lose their strength and the augmentation dies out (cb3 or 46), but the

57

effective centerline of the jets (the (Y/'>i point) continues to rise.

At the same time the inner 2-D boundary layer thickness 6' grows faster

than 6 (6 almost stops growing after the injection stops), and fi-

nally, when they merge (in approximately 406), the recovery process is

complete.

4.3 Predictions

The mixing length model, along with the IKE equation, was used as a

one-equation model of turbulence in the 2-D boundary layer program STAN5

(Crawford & Kays, 1975). Starting with the spanwise-averaged mean veloc-

ity and TEE profiles at the beginning of the recovery region (x = 188 cm),

very successful predictions of TEE and the mean velocity profiles for

M = 0.4 and M = 0.9 were obtained at two downstream stations in the

recovery region (x = 214 cm, 256 cm).

Figure 4.4 shows a comparison of the predicted mean velocity and TEE

profiles with the experimental data for M = 0.4 on the two stations in

the recovery region, starting with the spanwise-averaged initial profiles.

The prediction of mean velocity on plate 11 (x = 214 cm) is somewhat

high in the middle region of the boundary layer, but the difference les-

sens by the 27th plate (x = 256 cm). For IKE profiles the predictions

are somewhat low. These figures show that the model predicts a slightly

faster recovery to the 2-D state than the physical process, but the dif-

ference is not that great. The reason for lower T!LE predictions might

be the mixing length. In Fig. 4.3 the mixing-length model for M = 0.4

is slightly lower compared to measurements in the region 0.15 < (y/6) <

0.3. This means that the model results in a smaller production and a

higher dissipation than the reality, yielding lower 'IKE values compared

to the experiment.

Figure 4.5 shows a comparison of the mean velocity and TKE profiles

with the experimental data for M = 0.9 on the two stations in the re-

covery region, starting with the spanwise-averaged initial profiles. The

suggested model predicts the M = 0.4 case well. The predictions for

the M = 0.9 case are excellent. It is important to remember that the

constants do not change with the blowing ratio. Both cases were predic-

ted with the same set of constants for the mixing length and the other

58

constants in the one-equation model are the usual ones employed for 2-D

boundary layer predictions. The author believes that the model will suc-

cessfully predict blowing ratios up to M = 1.0. Further extrapolation

of the model might be dangerous due to the changing hydrodynamic charac-

ter of the flows.

4.3.1 The effect of reduced mixing on predictions

As previously mentioned, it is important to consider the effect of

the reduced mixing near the wall in making predictions. This is demon-

strated by Fig. 4.6. This figure shows the comparisons of the experimen-

tal data with the predictions of TKE (in the recovery region on the 27th

plate for M = 0.9). The predictions were made with two different mixing

length models. One of them is the normal 2-D flat-plate mixing length,

and the other is the model developed in this study, which has a smaller

value near the wall relative to the former. The new model predicts per-

fectly, whereas the usual 2-D mixing length predicts much higher TKEs

almost up to the first half of the boundary layer. The,reason lies in

the relationship between the mixing length and the production and dissi-

pation of TKE. Higher mixing lengths increase the production and reduce

the dissipation, giving rise to higher TKEs.

The reduced mixing region did not have a significant effect cn near-

velocity profiles; however, it is important in heat transfer behavior - because of its influence on the turbulence level.

4.4 Refining the Model

The empirical curve fits described in section 4.2.7 do in fact match

the boundary layer behavior. The physical arguments are plausible and

suggest that the empirical relations could be replaced (if desired) by

more conventional forms. In this section some possible alternative equa-

tions are examined -- without. changing the physical arguments -- both for

the purpose of testing the physical argument for reasonableness of the

magnitudes and to take advantage of established forms where possible.

59

4.4.1 The outer edge of the inner boundary layer

(4.15) $+ 0.0045 M [($) - 371

Using a conventional equation to describe the growth of the internal

boundary layer growth (Schlichting, i968) yields:

pu<x + x;, -l/5

(x ::,, = 0.37

( u 1

6’ and (y;/Sjd are related as follows:

At(f) , ($,)=+ d

Therefore,

(Y/ad (Y/Qd

(Y/ 6’) = (A/K)

or

6’ (Y/Qd

T-= (h/K)

(4.33)

(4.34)

(4.35)

(4.36)

Consequently, if one knows the initial value of 6 and (Y/S)~ at

X l- - 0, then the initial 6' can be calculated and X' o determined

from the formula. After x' the rest of 6' 0

is found, and (~1s)~

can be obtained from the boundary layer growth formula.

In Eqn. (4.33) the value of Urn can be used for the value of -

since U at "lgg is never far from U co- For example, for M = 0 g .

the velocity profile is so flat that near 6' .99

For M = 0.4 the initial thickness cf this boundary layer is large (be-

cause there is less disturbance), and again around Y fh 6'9 iJ TJ urn.

One might question the use cf Eqn. (4.33) to predict the growth of the

inner layer, since it neglects the effect of the turbulence of the outer

regions. It seems justified, however, because around 8' the turbulence

levels are quite small for both cases studied here (3%-5%).

60

By using t = U, = 16.7 m/set, p = 1.2 kg/m3, u = 0.9 x 10 -5

kg/msec, the virtual origin of the internal 2-D boundary layer can be

calculated for both cases from the initial value of (yb)d- The follow-

ing values were obtained:

For M = 0.4 , x: = 1.29 m

For M = 0.9 , x: = 0.41 m

(x;)M=0.4 is greater than (x~>~=~.~, which is normal because for low

blowing, the undisturbed region near the wall is larger. After calcula-

tion of virtual origins, the following comparisons were obtained between

the piecewise model and the internal boundary layer growth formula.

M = 0.4

Piecewise Model 0.028 0.036 0.045 __-- .__._.^_ ~_____

Boundary Layer Formula 0.028 0.033 0.040

M = 0.9

Piecewise Model 0.011

Boundary Layer Formula 0.011

As is seen from the comparison, the present linear model and the boundary

layer growth results compare quite well. In fact, when one looks at the

data, the boundary layer formula seems to be closer.

The following changes can be suggested in the present model. Instead

of calculating (~/6)~ from the piecewise linear formula (Eqn. (4.15)),

(Y/6) d,initial can be calculated, which involves only one empirical con-

stant. Then, using

61

*initial (Y/6) = d,initial

6 (A/K)

'initial can be found, and

(4.37)

(4.38)

Other 6' can be calculated as follows:

-l/5

6’ = (x’+xA) 0.37 ( pv (x’ +x;>

IJ ) (4.39)

and

6 ), = (3 (3 (4.40)

Here, even though the equation for boundary layer growth is still empiri-

cal, it is more universal than the constant supplied in the formula for

(Y/Ud. So the (y/&Id formula can be reduced to one empirical constant

supplied by the present experiments and a more universally accepted em-

pirical equation.

4.4.2 Comparison of the development of the (Y/&)~ point with jet

spreading theory

It is argued that the dynamics of the (Y/Ui point corresponds to

the rising of the effective jet centerline. In this section, the output

of the empirical equation for (Y/&>~ (Eqn. (4.18)) is compared with

the results of an-equation describing the rising of the centerline of a

jet in crossflow (Abramovich, 1960):

(3) = (j)“’ ($i’ + ($) cot a’ (4.41)

Here x'o' is the virtual origin for the effective jet action, D is the

diameter of a single jet, and cl' is the injection angle. The solution

of this equation depends on four parameters: 1: ;9 D, M and c('. The last

three are supplied by the problem physj.cs, and the only unknown is the

virtual origin, which can be calculateci from :he empirical input, i.e.,

62

from the initial value of (~/6)~, assuming that effective jet spread

occurs according to the equation (4.41).

The results of the virtuai origin calculation are:

For M = 0.4 , x: = 0.13 m

For M = 0.9 , x: = 0.055 m

Then, with these results and from Eqn. (4.41), the spreading of the jets

can be calculated and compared with the present model values of (Ymi’

The results:

M = 0.4

w - 0 %5 %12 i

Piecewise Model 0.244 0.3 0.37

Jet Spread For- mula 0.244 0.3 0.333

M = 0.9

\JI VI. 1

Piecewise Model 0.3

Jet Spread For- mula

As is seen again, the comparison is not bad. This supports the argument

that (y/6) i point.moves out with distance similar to the jet centerline.

The following changes can be made in the model to replace the con-

stant c3 with the more accepted empirical equation; or with this physi-

cally supporting agreement, the old model for (Y/6) i can be left as it

is. The change is as follows. Cal.culate

initial = K x + C3C4M

, (4.42)

63

Then

Y initial = 4:) i,initial

(4.43)

Calculate xi from

f = ($)1'? (Yinfttial)3 + 'iniDtial cot u1 (4.44)

Then calculate new y.'s from

(4.41) "::" = (2g3 (5,' + g cot a'

4.4.3 Other changes suggested

To include the effect of P/D (pitch-to-diameter ratio, the blowing

ratio M in the formulas can be replaced by

F

or by

F'

which reflects the relative effects of the jet and free stream momentum.

It is thought that it will be better to replace them by F'; by inclu-

sion of an empirical jet spread equation (4.10), the variation in the

injection angle (c1') can be represented in the model.

64

GR

OW

ING

2-

D

/-IN

TER

NAL

I

CU

MU

LATI

VE

/ !%

!!!!A

RI

t r.J

FT

/ LA

YtH

- O

UTE

R

$IAR

Y

NEW

W

AI

I I

AYFR

FULL

- tX

iX&y

GE

--I---

R

ECO

VER

Y R

EGIO

N

I 1

Fig.

4.

1.

Flow

st

ruct

ure

in

the

reco

very

re

gion

.

X’

+-e-o- 2-D MIXING LENGTH

MODELED MIXING LENGTH

c OUTER B.L. 8 l

I CUMULATIVE JET SPREAD-q EFFECTIVE JET “-7

I’ -IN TE_R!AL,?-DA ;

i tl. L.

NEAR

i!!Ati!L~d

/S)i F99 1

Fig. 4.2. Mixing-length model and flow structure in the recovery region.

66

0.;

w99

0.1

0 8

n .

L

M=O.4 - FLAT PLATE

MODEL DATA x(cm) l 0 188

A A 214

0 0 256

0 0 1.69 97

0 l

0 0

M=0.9 - FLAT PLATE

MODEL DATA x(cm) l 0 188

A A 214

0 0 256

Fig. 4.3. Comparisons of the piecewise mixing-length model and the data in the recovery region.

67

1 : 0 I- c I

- PR

EDIC

TIO

N

1 - S

TAR

T O

F R

ECO

VER

Y (IN

ITIA

L PR

OFI

LES)

2

- R

ECO

VER

Y R

EGlO

N

11 th

PLA

TE

3-

REC

OVE

RY

REG

ION

37

th

PLAT

E

7

O’

015

I I

I I

’ I

I ’

’ I

’ I

I I

I I

I I

I I

I I

I I

1 1

1 0

0.1

0.1

0.1

wu*

J-

1 92 U

C0

J Fi

g.

4.4.

Pr

edic

tions

of

st

ream

wis

e m

ean

velo

city

co

mpo

nent

(3

an

d tu

rbul

ent

kine

tic

ener

gy

(q2)

fo

r M

=

0.4.

1 0 0

Y (c

m)

Y (c

m) 7 7

1 6 6

t t

0

5.

5.

4 4 I I

0 l a

3 3 0

2, 1 1

0 2,

:

2

I ! .o

- PR

EDIC

TIO

N

l D

ATA

1 - S

TAR

T O

F R

ECO

VER

Y (IN

ITIA

L PR

OFI

LES)

2

- REC

OVE

RY

REG

ION

11

th P

LATE

3-

REC

OVE

RY

REG

ION

27

th

PLAT

E M

= 0.9

J I

1 I

I !

, ,

I I

I I

! 0.

1 '

0.1

' 0.

1

\q2

02

T/U

Fig.

4.

5.

Pred

ictio

ns

of

stre

amw

ise

mea

n ve

loci

ty

com

pone

nt

(3

and

turb

ulen

t

kine

tic

ener

gy

(q*)

fo

r M

=

0.9.

y(cm) ’

R 0 DATA

7 - PRESENT MODEL - - 2-D FLAT PLATE PREDICTIONS

MIXING LENGTH G

M = 0.9 RECOVERY REGION 27 th PLATE

4-

3-

2-

l-

I OO

1 I I I I I I I I I I __ ~- 0.1 0.2-

x/- q*wo

Fig. 4.6. Effect of mixing-length on predic- tions of turbulent kinetic energy.

70

Chapter 5

MEASUREMENTS AND DISCUSSIONS OF THE DATA

Measurements were made to supply the necessary empirical input for

development of turbulence models for the recovery region following a

full-coverage, film-cooled region. The final objective of the turbulence

modeling was to obtain correct predictions of the heat transfer coeffi-

cient. This chapter will discuss the experimental conditions, measure-

ment locations, and the data. Due to the large volume of the data, only

the parts which directly concern the prediction process will be discussed

in detail.

5.1 Experimental Conditions

The experiments were made under isothermal conditions, with air tem-

peratures around 22-25°C. Since the heat transfer experiments were made

with small temperature differences (16"), it was believed that the hydro-

dynamics of the isothermal flow field would be the same as that of non-

isothermal heat transfer experiments. This question was investigated by

Harnett, Birkebak & Eckert (1961). The pressure levels were about atmos-

pheric (s 760 mm of mercury), and the experiments were made at a uniform

free stream velocity of approximately 16 m/set. The velocity of the free

stream was kept uniform along the test section by adjusting the top wall

(as explained in Crawford et al., 1976), so that the static pressure

change along the tunnel was less than to.82 of freestream dynamic head

(? 0.005 inches of water).

Experiments were made at two different blowing ratios: M = 0.4 and

M = 0.9 (M = I! j et'Um) * The M = 0.4 blowing ratio was chosen because,

during heat transfer experiments, minimum Stanton numbers on the test

section were cbtained here, as demonstrated by Crawford et al. (1976).

The second blowing ratio, M = 0.9, was chosen because a definite change

in the heat transfer behavior was seen at this high blowing ratio.

71

5.2 Measurement Locations

The schematic of the test section is given in Fig. 2.4. The jets

are injected into the main stream at a 30" angle with the horizontal

plane, and the injection is in line with the main flow direction. Due

to the interaction of the jets and the main flow, a 3-D high-fluctuation

flow field is created, at least locally, around the secondary air jets.

All the measurements of the velocity field were made with the 3-D Turbu-

lent Flow Analyzer system introduced in Chapter 3.

The measurement locations can be seen in Fig. 5.1. Five streamwise

positions were selected for measurements to observe the development of

the flow field on the full-coverage surface and its relaxation in the re-

covery region. The first is located at the center of the 5th plate (x =

148 cm) after three rows of blowing, and the second is at the center of

the 9th plate (x = 168 cm) after seven rows of blowing in the full-

coverage region. The third one is at the center cf the first plate (x =

188 cm) in the recovery region after 11 rows of blowing. The last two

are at the centers of the 11th (x = 214 cm) and 27th plates (x= 256cm)

and are, respectively, 27 and 67 hole diameters downstream from the last

row of injection holes. In order tc see spanwise variations in the flow,

several measurements were made at each streamwise position. At the first

and third stations, five spanwlse locations were used, situated at z/P =

+ 05, + 0.3, 0.0, -0.3, and -0.5. Since the flow field is periodic, it

repeats itself after z/P = + 0.5 and - 0.5. At the second streamwise

location three spanwise locations were selected at z/P = + 0.5, 0.0,

and - 0.5. In the recovery region for M = 0.9, at each streamwise

location three spanwise locations uere used. After the experiments it

was seen that the profiles in the recovery region station were the same

in all three spanwise locations, as illustrated by Figures 5.4 and 5.9.

So for M = 0.4 these spanwise locations were eliminated in the recovery

region. The profiles were used to obtain the necessary spanwise-averaged

input to the 2-D boundary layer program STAPI5. Measurements were made at

each location of the three mean velocities and the six Reynolds stresses.

Each quantity was averaged over 22 seconds to cnclude the effects of low-

frequency fluctuations.

72

5.3 Mean Velocity Profiles

In this section attention will be given to the streamwise component

of the mean velocity f, rather than discussing the three-dimensionality

of the flow around the holes. Two sets of profiles for different blowing

ratios will be discussed and compared with each other.

5.3.1 Mean velocity profiles for M = 0.4

Figure 5.2 shows the mean velocity profiles at each spanwise loca-

tion, at each of the five streamwise stations, for the blowing ratio

M = 0.4. The location of each of the traversing stations is marked on

each figure with respect to the injection locations.

The first observation on this figure is the remarkable symmetry of

the profiles from positions which are symmetric with respect to the cen-

terline. This proves that there is periodicity in the spanwise direction

and that the coalescing of the upstream jets is regular. Periodicity is

very helpful in spanwise averaging.

In the full-coverage section, two distinct regions appear in each of

the profiles. In the outer region all spanwise profiles are the same;

however, in the near-wall region, differences are obser'ved in the span-

wise profiles because of injection. The same observation was made by

LeBrocq, Launder & Priddin (1971) for a 45" injection. The spanwise pro-

files also show some common behavior. The central profiles just down-

stream of a jet show the greatest velocity deEect, because they are

affected directly by injection. The profiles at the outer edges show

less defect than the central profiles but more defect than the intermedi-

ate positions (between two holes). This is expected, because the outer

profiles are affected by the injection one row upstream and are still

recovering from the effect of that injection, while the profiles between

two holes have no injection in line with them and are affected only by

the lateral spreading of jets. This is clearly seen in Fig. 5.2, where,

after three rows of blowing, the intermediate profiles show almost no

defect, thus indicating that the effect of lateral spreading has not

reached them with its full force yet. After 11 rcws of blowing, however,

the momentum defect in the intermediate profiles can easily be observed.

73

The size of the near-wall momentum defect region increases down-

stream. The velocity defect region after li rows of blowing is almost

twice as large as after three rows of blowing. This indicates the cumu-

lative effect of upstream jets which keep rising as they spread down-

stream. The jets penetrate only about 2 cm, immediately after injection,

as can be seen from the figure, the point where there is an abrupt change

in the slope of the profiles. The immediate penetration distance is about

two jet diameters for this blowing ratio. This means that the mass shed

into the boundary layer by each injection row penetrates to an almost

constant distance for a certain blowing ratio at the injection location

before jets are knocked down by the main stream. The immediate penetra-

tion distance increases slightly in a downstream direction as the bound-

ary layer loses momentum due to the effect of previous injections. This

constant penetration distance was also observed by Le Brocq, Launder &

Priddin (1971), Crawford et al. (1976), Collzday & Russel (1975), and

Ramsey & Goldstein (1971).

There is no indication of a reverse flow and separation of the jet

from the surface in any of the full-coverage region profiles. 'This might

be due to the fact that the profiles were measured at locations where

jets have already reattached themselves to the surface. The measurement

locations were 2.5 hole diameters downstream from the edge of the closest

hole. The separation of jet from the surface was observed by Bergeles,

Gosman & Launder (1975) for normal blowing at a blowing ratio of M = 0.24,

and extended only to 1.5 diameters downstream of the hole. It is gener-

ally agreed in the literature (Le Brocq, Launder & Priddin, 1971; Coli.a-

day & Russell, 1972) that slant-injection jets stay attached to the sur-

face at much higher blowing ratios compared to a normal injection.

Launder & York (1973) state that for a 45" in-line injection with P/D =

8, jets attach to the surface at 3-4 hole diameters downstream of the

last injection for M = 0.6. These observations help one arrive at the

conclusion that the jets for the 30" slant injection for M = 0.4 re-

attach to the surface in a very short distance downstream or perhaps

don't separate at all.

The points which are closest to the sdall in all the mean vel.ocitv

profiles have almost the same value OF i'/Um for the same spanwise

74

locations at all streamwise stations. For the central profiles the value

is u/urn = 0.35-0.4. This number can be obtained by multiplying the -

value of the blowing ratio (U jet/Um = 0.4) with the cosine of the in-

jection angle (30"). This is a definite indication of the average jet

velocity in the near-wall region. It was observed by Bergeles, Gosman &

Launder (1975) that the velocity in the exit plane of jets is not uniform

at all; however, the present work shows that even though the velocity is

not uniform, the effect is governed by the average jet velocity given by

M x U,.

In Fig. 5.2, curves representing (l/7) th power profiles are shown

at the start of the recovery region (after the last row of injection) and

at the two measurement stations in the recovery region as reference lines

to make the cross-comparisons easier. It was expected that, finally, in

the recovery region, the boundary layer would return to a normal 2-D

flat-plate boundary layer. A (1/7)th power profile was chosen to rep-

resent the flat-plate mean velocity profile, although any curve which is

close to a 2-D turbulent boundary layer profile could have been chosen

as a reference line. A spanwise-averaged profile is also shown (for the

profiles at the start of the recovery region only). When compared to

the (1/7)th power profiles, it can be seen that the momentum defect de-

creases as the boundary layer recovers from the blowing. However, there

is a considerable momentum deficit even at the last recovery region sta-

tion due to the large initial deficit. The reason for the long recovery

distance is the "momentum-sink" action of the wall. The diffusion of

momentum from the free stream is the only way for these profiles to re-

cover to 2-D flat-plate profiles, and diffusion is a slow process. The

long recovery distance indicates that the effects of jets near the wall

prevail over long distances for low blowing.

Velocity gradients at the wall are smaller than for a 2-D flat-plate

profile; this will give lower friction factors. There is also no indica-

tion of boundary layer separation except locally at the injection points.

Thus, the aerodynamic behavior of the 30" slant-hole in-line injection

with M = 0.4 is quite good. Since this blowing also gives the lowest

heat transfer coefficients (Crawford et al., 1976), it is a very important

75

blowing ratio in full-coverage film-cooling applications. Le.Brocq,

Launder & Priddin (1971) agree with this conclusion.

Even though the jets initially penetrate to a constant distance (two

hole diameters), comparisons with the (l/7)th power profiles show that

their cumulative effect on the momentum deficit can be observed at much

larger distances from the surface dlle to the rising of the effective jet

centerlines in the downstream direction. The jets continue to rise even

at large distances from the last row of holes. This observation was also

made by Keffer and Baines (1963).

Figure 5.3 shows the spanwise-averaged mean velocity for M = 0.4

in (y/6) and u/U= coordinates. The profile at x = 168 cm is not

to be trusted quantitatively near the wall. Only three spanwise profiles

were taken at this location, and the spanwise averaging of three profiles

does not represent a physically correct average. This point is further

explained in section 5.6. It is plotted here, however, for comparisons

in the outer layers where the spanwise profiles are not very different

from each other,and the resulting average is correct. The profile after

three rows of blowing (x = 148 cm) has the largest momentum defect, but

the difference from the others is not great. This means that the region

near the wall is governed by the latest injection and stays almost at the

same level of U/UW. The effects of previous rcws can be seen in the

outer layers. For example, in the outer regions, the profile at x=

148 cm is very close to the profile at the last recovery station (x =

256 cm). This means that the jets cannot strongly affect the outer

layers at this distance. The outer region profiles are very similar after

7 and 11 rows of blowing (x = 168, 188 cm, respectively) and in the re-

covery region profile at 27 hole diameters downstream cf the last row of

injection (x = 214 cm). The result of these observations can be summa-

rized as follows: after four or five rows of blowing it appears that the

boundary layer re:iches a state of equilibrium in a large part of the

boundary layer (almost above y/d ~0.1) for low blowing. This may be

very helpful in the modeling o f the injection process in the full-

coverage region.

76

5.3.2 Mean velocity profiles for M = 0.9

Figure 5.4 shows the spanwise mean velocity profiles at five stream-

wise stations for the blowing ratio of M = 0.9. Spanwise symmetry can

be observed, as in M = 0.4. This indicates uniform lateral jet spread.

Nina 6 Whitelaw (1971) cautioned against non-uniform coalescing of jets in

full-coverage film-cooling applications; such a phenomenon is not observed

here.

Again, two distinct regions can be observed in the profiles, an outer

and an inner one. In the outer region all the spanwise profiles are the

same at a certain streamwise station. In the inner region, however, there

are differences in the profiles, depending on the spanwise location. The

size of the inner region is very large compared to the M = 0.4 case be-

cause of the higher rate of blowing and deeper penetration of the jets.

The jets penetrate up to three hole diameters at M = 0.9, and this dis-

tance is almost the same for all the stations.

Another difference between the cases is the excess momentum near the

wall in M = 0.9 because of the high blowing ratio. This canbeobserved

by comparing the spanwise-averaged profile to the (1/7>th po;Jer reference

curve at the third streamwise location.

In the central profiles just downstream of a jet, a local maximum in

velocity occurs around y=lcm with v/UC0 = 0.85-0.9 (this was not

observed for M = 0.4). This point is the centerline of the new injected

jet, about one hole diameter above the surface. This can be observed at

the same location in every station o f the full-coverage region. This

maximuminthe streamwise mean velocity was also observed by Le Brocq,

Launder & Priddin (1971) for M = 0.5 and P/D = 8 for 45" injection.

This is another indication of the fact that the depth of immediate pene-

tration is constant.

It is important to observe that at the point closest to the wall in

the central profiles, there is a sharp decrease in the velocity. This

indicates that the jet has separated from the surface and has not reat-

tached yet at the measurement location (about 2.5 diameters downstream

of the hole). However, the same phenomenon is not observed in the side

profiles, which are 7.5 hole diameters away from the last injection lo-

cation. Therefore, it can be concluded that. for the case of M = 0.9,

77

jets reattach to the surface somewhere between 2.5 and 7.5 hole diameters

downstream of their injection location. This was also observed by Colla-

day & Russell (1975) and by Launder & York (1973). The separation of the

protective jet from the surface explains the poor heat transfer beha#vior

(high heat transfer coefficients) of the M = 0.9 case observed by Craw-

ford et al. (1976).

In Fig. 5.4, W7Ph power profiles are shown at the start of the

recovery region and at the two recovery region stations as reference

lines for comparison. A spanwise average is also shown for the profiles

at the beginning of the recovery region.

The spanwise-averaged mean velocity profile has a region of negative

slope (as can be seen in section 5) which gives rise to a locally nega-

tive shear stress.

There is a rapid relaxation back to the 2-D state in the recovery

region, as can be seen from the comparison of the profiles in this re-

gion with (1/7)th power profiles. In fact, the profile at the last re-

covery region station is almost the same as the (1/7)th profiles, except

for a small momentum deficit in the outer layer. Relaxation back to the

2-D state is much faster for M = 0.9 than for M = 0.4 because of the

excess momentum near the wall. The wall acts as a momentum sink and fa-

cilitates the relaxation process. In the outer layer, where there is

some momentum deficit, the relaxation process is very slow. The momentum

supplied by the free stream entrainment and by the transport is not large

enough for rapid relaxation.

Higher velocity gradients near the wall (when compared with the

(1/7Ph profile and the M = 0.4 profiles) lead to larger skin friction

coefficients, thus increasing the aerodynamic drag. Increase in the

skin friction coefficitn with blowing was observed by Kacker & Whitelaw

(1970) for a 2-D wall jet.

One observes generally the lack of velocity gradient all over the

M = 0.9 mean profiles except at the wall. This yields low shear

stresses and low TKEs (low turbulent mixing) in a large part of the

boundary layer. Samuel & Joubert (1965) also observed that the closer

the M is to unity, the lower the turbulent mixing is for a 2-D

78

injection. Seban C Back (1362) made the same observation in the case of

a 2-D tangential injection.

In Fig. 5.5, spanwise-averaged profiles for five streamwise loca-

tions are plotted on (y/6) and t/U, coordinates for M = 0.9. A very

clear outer layer similarity is observed in these coordinates; the pro-

files lie much closer to each other than inthesame case for M = 0.4.

The similarity is not observable below (y/6) = 0.25. The high rate of

blowing causes non-equilibrium in the region near the wall. Another in-

dication of this non-equilibrium can be seen in Fig. 5.4. The interme,-

diate profiles in each streamwise location get fuller downstream as the

preceding jets expand laterally; they do not reach an equilibrium even

after 11 rows of blowing.

Figure 5.6 shows the spanwise-averaged mean velocity components in

semi-logarithmic coordinates (V/U, vs. Iln y) for M = 0.4 and M = 0.9.

No logarithmic region can be seen in the profiles for either case, in the

full-coverage region. The signs of the curvatures of the profiles in the

full-coverage region are opposite for different blowing ratios; this is

an indication of the momentum deficit in one case and of the momentum

excess in the other. In the recovery region, however, there is a defi-

nite indication of a logarithmic region near the wall. The two profiles

at x = 214 cm and x = 256 cm exhibit linear regions for both blowing

ratios; this proves that as soon as the blowing stops, wall effects are

dominant and the "law of the wall" seems to apply.

5.4 Turbulent Kinetic Energy Profiles

Turbulent kinetic energy will be discussed, like the mean velocity

profiles, in the form of two sets -- first for M = 0.4, second for

M = 0.9. Comparisons between the two cases will be deferred until the

section discussing the M = 0.9 data.

5.4.1 Turbulent kinetic energy profiles for M = 0.4 ~- Figure 5.7 shows the TKE profiles at several spanwise locations at

each of five streamwise stations for M = 0.4. Spanwise symmetry can be

observed in these profiles, indicating an orderly behavior of the turbu-

lent flow field.

79

The position of the 6.9g thickness is marked on the profiles in

the full-coverage region. As seen, the TKE boundary layer extends up to

Y = 1.3 6.gg. This was also observed by several other workers -- for

example, Klebanoff (1955).

In the full-coverage region, there is a strong convergence above

y=2cm in the TKE profiles at all spanwise locations, but large dif-

ferences exist between the profiles below y = 2 cm. This relates to

the 2 cm penetration distance and indicates that the region near the

wall is locally affected by the jets.

Near the wall, the highest TKE is observed in the central profiles;

the second-highest one is observed in the side profiles. The intermedi-

ate profiles have the lowest TKE, as expected (no jets are in line with

them). There is a dip in the center profiles around y = 1 cm. This

can be explained as follows. There is a high velocity gradient near the

wall which causes high shear, resulting in higher TKEs. However, there

exists a very low velocity gradient region (see Fig. 5.2) just above it,

because of injection. In this region the local TKE production is de-

creased but local dissipation is high, leading to low TKE values. The

low velocity gradient region ends at about y = 2 cm (penetration dis-

tance). Above this level, higher velocity gradients lead to higher TKE.

As a result, a double maximum is observed in the central TKE profiles.

The side TKE profiles have only one maximum. At some distance from the

injection location, the TKE near the wall dissipates and the maximum near

the wall vanishes. TKE diffuses from the second maximum to;lrards the wall.

The result is a smooth profile with one maximum.

In Fig. 5.7, 2-D flat-plate TKEs are also shown both at the start of

the recovery region and at the two recovery region stations as references

for comparisons. The 2-D flat-plate TKE profiles were obtained with the

STAN5 computer program (Crawford & Kays, 1975) for the same free stream

turbulence level as the experimental profiles. A spanwise-averaged TKE

profile is shown for the start of the recovery region. Comparisons with

the 2-D profile show that there is excess TKE throughout the region, but

most of the excess energy lies in the middle section of the boundary

layer. There is more violent turbulent mixing in the outer layers rela-

tive to layers next to the wall. Even though the jets do not penetrate

80

very far into the boundary layer, the TKE diffuses towards the free

stream, causing large TKE values far from the wall.

In the recovery region, there is still high TKE compared to the 2-D

value. The reversion of the turbulence quantities to the 2-D state is

very slow. This interesting process is shown in the following sketch.

- FLAT PLATE

( RECOVERY

OO I I I I I I I l I,

0.1

I/- q2/uco

Again, there are two distinct regions in.the relaxation process, an outer

region and an inner region. The relaxation to the 2-D state is in the

expected direction in the outer region and is caused by the diffusion of

TKE. However, an opposite "relaxation" is seen in the inner layer.

This means that the rate of diffusion of the TKE toward the wall

exceeds the rate of dissipation near the wall. After the profile

81

becomes flat in the last recovery region station (67 diameters downstream),

the diffusion will stop and the dissipation will dominate, causing the re-

version of the profile to the 2-D state.

The high TKE (turbulent mixing) levels in the recovery region must

be the reason for the slowly rising Stanton numbers here for M = 0.4.

Crawford et al. (1976) observed that Stanton numbers for M = 0.4 de-

crease in the full-coverage region and rise slowly in the recovery re-

gion. This process can be explained by the TKE levels, as follows. The

TKE levels are higher than the 2-D boundary layer values. This higher

turbulence level should cause high Stanton numbers; however, at the same

time, an energy sink is created in the full-coverage region by the cold

injection. The combination of these two opposing phenomena results in

the low and decreasing Stanton numbers in this region. There is no in-

jection in the recovery region (i.e., the energy sink is lost), but the

turbulence level (turbulent mixing) is still high. The mixing process,

unopposed, increases the Stanton numbers and they rise gradually in the

recovery region.

In Fi . 5.8, spanwise-averaged TKE profiles are plotted in (y/6)

and I? q21um coordinates for each streamwise station. There is no defi-

nite similarity in any region, even though all the profiles seem very

similar (again, for the x = 168 cm profile, the points near the wall

are to be neglected, as discussed in Sec. 5.6). There is an indication

of invariance in the TKE profiles in the full-coverage region after four

or five fows of blowing (over (y/6) = 0.3). This was also the case with

the streamwise mean velocity profiles.

5.4.2 Turbulent kinetic energy profiles for M = 0.9

Figure 5.9 shows the TKE profiles for several spanwise locations at

each of five streamwise stations for M = 0.9.

Spanwise symmetry can be observed in the profiles within the experi-

mental uncertainty limits.

The behavior of the TKE is qualitatively similar to the M = 0.4

case -- contrary to the mean velocity profiles which exhibit different

behavior. Once more, the highest TKE is seen intie central profiles,

the next highest in the side profiles, and the lowest in the intermediate

a2

profiles. However, quantitatively, there are big differences between the

TKE profiles for the M = 0.4 case and the M = 0.9 case.

The double maximum observed in the M = 0.4 TKE profiles is not

found here. The highest TKE occurs next to the wall rather than in the

outer region, owing to the high velocity gradient there. Above y =

2.0 cm there is a sharp drop in TKE. Also, the TKE levels for M = 0.9

are much lower than in the case of M = 0.4 within a large region of

the boundary layer outboard of y/6 6 0.15 (compare the spanwise-averaged

profile at the start of the recovery region with the 2-D TKE profile).

This is clear evidence of a decrease in turbulence level as the blowing

ratio approaches 1. The TKE levels are lower because of the lack of

velocity gradients for the M = 0.9 case, as shown in Fig. 5.11. This

figure shows the central profiles of mean velocity and TKE at the start

of the recovery region for M = 0.4 and M = 0.9. The mean velocity

gradient is smaller and the TKE is lower for the M = 0.9 case than for

the M = 0.4 case everywhere above (y/6) 2-z 0.2. This behavior may be

due primarily to the dependence of the 'IKE production on the mean velocity

gradient -- the larger the mean velocity gradient, the larger the TKE pro-

duction and the TKE itselF.

The lower TKE levels for the higher blowing case seen?, at first

glance, to be inconsistent with the heat transfer data obtained by Craw-

ford et al. (1976), but when examined closely it can be seen that there

may be an explanation: the governing process for the heat transfer may

be different. Crawford et al. (1976) observed that the Stanton number

for the II = 9.9 case is below the 2-D flat-plate value 2nd above the

value for the M = 0.4 case. The Stanton number values decreased in the

downstream direction in the full-coverage region and also in the recovery

region. If turbulent mixing were the governing process in the heat trans-

fer , one woul.d expect the Stanton numbers for the II = 0.9 case to be

lower than for the f.1 = 0.4 case, for two reasons: first, more cold

fluid is injected into the boundary layer for high blowing and, second,

there is less turbulent mixing. Instead, higher Stanton numbers were ob-

served. The reason could be that, for high blowing, the jets separate

from the wall and penetrate dee!)1.y into the boundary layer, entraining the

83

hot free stream fluid towards the wall in the middle lanes. This process

is schematically explained in the following figure.

This figure was also used by Le Brocq, Launder & Priddin (1971) to de-

scribe the in-line blowing process. This same process seems to occur in

a staggered array, at high blowing. Thie hypothesis is also supported

by the v component of the mean velocity, as can be seen in Figs. 5.16

and 5.17.

In addition to the entrainment process, the cold fluid (energy sink)

is also deposited in the outer layer due to the separation of the jets

from the surface. Thus, two opposing mechanisms result in the decreas-

ing Stanton numbers in the full-coverage region. (They are, nevertheless,

higher than the low-blowing values.)

In summary, the two pairs of opposing processes which control the

heat transfer are: (i) turbulent mixing versus energy sink near the wall

for low blowing, and (ii) outer-layer fluid entrainment versus energy

sink in the outer layer for high blowing.

In the recovery region, the Stanton numbers continue to decrease

because there is rapid decay of entrainment of the outer-layer

fluid by the jets and the energy sink is no longer being injected

at the wall. In the absence of these mechanisms, the process continues

84

in its natural direction until the turbulent mixing is strong enough to

reverse the trend. Since the three-dimensionality is important in heat

transfer for high blowing, any 2-D computer programwillhave difficulty

in predicting the heat transfer in the full-coverage region for these

cases.

In Fig. 5.9, the recovery region TKE profiles are compared with 2-D

flat-plate TKE profiles. The behavior of the TKE is very interesting.

At the start of the recovery region, the spanwise-averaged TKE is higher

than the 2-D value; however, by 27 hole diameters into the recovery re-

gion, TKE values are lower than the 2-D value below y -N 3 cm. In fact,

the deviation from the 2-D value increases in the downstream direction.

Normally one would expect the difference to decrease. This can be ex-

plained as follows. In Fig. 5.4, the spanwise-averaged mean velocity

profile at the start of the recovery region has negative velocity gradi-

ents close to the wall. There are also negative shear stresses in the

same area as seen in Fig. 5.13. In fact, since the flow is three-

dimensional, there are y locations where the mean velocity gradient is

positive and the shear stress negative, and vice versa. This results in

the apparent negative production of TKE. The high TKE near the wall is

dissipated rapidly as soon as the injection stops and thus cannot feed

the low TKE regions by diffusion. As a result of the "negative produc-

tion", the dissipation, and the decreased diffusion, the TKE falls below

the 2-D flat-plate value. Even though the negative velocity gradient

region diminishes rapidly after the last row of injection, its effect tin

TKE continues because the turbulence profiles respcnd slowly to changes.

The production first becomes zero and then positive. A considerable dis-

tance is required before the production can adjust itself to increase the

TKE level.

The relaxation of TKE in the recovery region occurs in the expected

direction in the outer layer. This rate is again slow, as was observed

in the M = 0.4 case. In the near-wall region, TKE responds rapidly.

At the point nearest the wall, TKE drops from d- q2/Um = 0.134 to 0.1

quickly, after the last row of in:jection, and remains at that level

throughout the recovery region. This is also true for M = 0.4, in the

85

recovery region (see Fig. 5.7) and for a 2-D flat-plate boundary layer.

These observations illustrate the rapid recovery to the 2-D state in the

near-wall layer (y/S 5 O.l>, which occurs as soon as the injection

stops.

Figure 5.10 shows the spanwise-averaged TKE profiles for M = 0,9

at the five streamwise stations (discard the x = 168 cm, as discussed

in Section 5.6). There is a fair similarity above (y/6) = 0.8. This is

due to the undisturbed TKE region here. Below (y/6) = 0.8, there is

similarity between the full-coverage region profiles and the recovery

region profiles. This may mean that a state of invariance for TKE is

reached in the full-coverage region. This phenomenon may simplify pre-

dictions of the full-coverage region and might lead to better insight

into the injection process. The similarity observed in the recovery re-

gion may be due to the slow rate of relaxation to the 2-D state.

5.5 Streamwise Shear Stress Profiles

Figures 5.12 and 5.13 show the streamwise shear stress profiles at

the start of the recovery region (just after the last row of injections)

for M = 0.4 and M = 0.9, respectively. The 2-D flat-plate shear

stress profiles (Klebanoff, 1955) are also shown in these figures as

reference lines for comparison. In both cases there is some symmetry ob-

served for the profiles which are in symmetrical positions with respect

to the centerline. Compared to the 2-D shear stress, excesses are ob-

served for the M = 0.4 case, above y/6 = 0.2, for all of the span-

wise profiles. In general, for the M = 0.9 case, however, defects can

be seen when compared to the reference profiles below y/g = 0.6. Above

y/6 = 0.6, there are some excesses, but the magnitudes of these are less

than for the M = 0.4 case owing to the higher shear action of the ef-

fective jet spread for the low blowing. Thus, the shear stress levels

are much higher for M = 0.4 than for M = 0.9, 4s was expected from

the mean velocity gradients. This is another indicaticn of high turbu-

lence mixing for the low blowing. The extremely high shear stress next

to the wall in the central profile for the M = 0.9 case must be due to

the separated jet, which creates a high-shear and vortex region there.

86

Some negative shear stresses can be seen for M = 0.9 in the cen-

tral (No. 3) and side profiles (Nos. 1 and 5) due to the separation of

jets. These negative shear stress regions occur close to and at the

points of negative streamwise mean velocity gradients. It appears that

-u'v' is more or less following the direction of 2V/ay. Shear stress

regions close to zero can be seen in the intermediate profiles (Nos. 2

and 4) next to the wall. These regions correspond to the aE/ay = 0

region of the mean velocity profiles.

Another indication of the jet separation from the surface for M =

0.9 is the highly disturbed region of shear stress in the central pro-

file extending up to (y/B) = 0.5 (about y N 3 cm or three hole diam-

eters). This is the penetration distance for the last row of injections.

In fact, the same disturbed region can also be observed in the side pro-

files because of the jets injected one row before. However, the profiles

in the middle lanes are very smooth and do not show any sign of distur-

bance. They also indicate the low level of turbulence for the M = 0.9

case.

Figure 5.14 shows spanwise-averaged streamwise shear stress profiles

at five streamwise positions for M = 0.4. The 2-D flat-plate shear

stress profiles are also shown for comparison. The development of the

shear stress in the full-coverage region and its relaxation in the recov-

ery region can easily be observed. In the full-coverage region the pro-

files grow fuller in the downstream direction as the injection effects

diffuse outwards. In the recovery region the relaxation process is quite

slow. The shear stress levels are much higher in comparison to the 2-D

values at each station, indicating high turbulent mixing.

Figure 5.15 shows spanwise-averaged streamwise shear stress profiles

at the start of the recovery region, as well as at the two recovery re-

gion stations for M = 0.9. As can be seen, the relaxation process in

the recovery region is also slow for this case. When compared to the

2-D values, the shear stresses are quite small, indicating low turbulent

mixing below (y/6) = 0.65. In the outer layers (above (y/6) = 0.65),

there is some excess shear, but it is not significant. The low shear

level is consistent with the mean velocity gradients at the start of the

87

recovery region. The spanwise-averaged mean velocity profile is quite

flat, resulting in small gradients and small shear (see Fig. 5.4). The

mean velocity reverts back to the 2-D flat-plate value rapidly in the

recovery region. However, because of the slow relaxation speed of the

turbulence field, the shear stress remains low for long distances.

5.6 On Spanwise Averaging

Most of the data above was discussed on spanwise-averaged bases.

This section will explain the hows and whys of spanwise averaging.

In section 5.7 it is shown that the three dimensionality of the flow

field was local and restricted to the vicinity of the jets. The flow

field was expected to be periodic due to the fact that the jets are in a

regular array. The spanwice uniformity and the symmetry of the flow

field were shown in the discussions of the mean velocity, TECE, and the

shear stress profiles.

The values of the heat transfer coefficients sought here are not the

local values, butratherthe average behavior of the Stanton numbers in the

streamwise direction. 'This requirement plus the properties from the pre-

ceding paragraph allow for a simpler two-dimensional analysis instead of

a 3-D one.

The following method was used tc spanwise average the flow proper-

ties. A spline fit in the lateral (z) direction was made to the pro-

files at each y level within a pitch. Each point of the spanwise aver-

aged profiles was obtained from the following formulas. To be consistent

with the physics, mean velocities and TKES were mas5 averaged.

(5.1)

where G stands for f or q2, as the case may be. Shear stresses

were area averaged.

+p/2

J (-u'v')dz

(-u'v') -P/2 = - average P (5.2)

88

As was seen in section 5.2, five profiles were taken at the first

and last full-coverage stations and three profiles at the second station.

Results of spanwise averaging showed that three profiles were not enough to

obtainagood spanwise average, since this process gavemore weight than was

proper to the central profile. Therefore, for the profiles at x = 168 cm

(second full-coverage region station), little importance is attached to

the results of spanwise averaging near the wall, blhen the spanwise pro-

files differ significantly from each other due to the injection. How-

ever, spanwise averages obtained from five profiles appear to represent

the physics adequately. The spanwise-averaged profiles of mean velocity

and TKE at the start of the recovery region bears a strong resemblance

(at least from the standpoint of qualitative tendencies) to the profiles

in the recovery region, where no spanwise averaging was done because all

the spanwise profiles are the same. From this it can be concluded that

the physical averaging process taking place in the flow must coincide

with the spanwise averaging of five profiles here.

5.7 On the Three-Dimensionality of the Flow Field

As noted, the three-dimensionality of the flow field was limited to

the close vicinity of jets; the flow had a strongly preferred direction.

Elaboration follows in sections 5.7.1 and 5.7.2. Space and time did not

permit plotting all the data (such as the components of the mean velocity

and Reynolds stress tensor) at every measurement location. Some interest-

ing representative data of this nature are given, however, as an aid to

for heat .ications understanding the three-dimensionality and its imp1

transfer.

5.7.1 i and w components of the mean velocity -- Figure 5.16 shows the v component of the mean velocity after the

last row of injection for several spanwise locations for the case of

M = 0.4. Together with the 7, the flow angle y is also shown. Only

the profiles on one side of the centerline are plotted because of sym-

metry. As expected,because of the inline jet, the highest v occurs in

the centerline profile at the point next to the wall. The smallest

7 values occur in the intermediate profile, which has no jet in l.ine

89

with it. It is significant that the direction of fT depends on the pro-

file position. At the central and side positions, V is in the positive

direction (towards the free stream), whereas for the intermediate profile

it is in the negative direction. The fluid is pushed upwards by jets at

the position in line with them and is entrained towards the wall in be-

tween. Maximum values of 7 occur near the wall, reaching approximately

8% of the free stream velocity; whereas in most of the boundary layer

they are only from O-4% of the free stream velocity. The largest flow

angle is about 10" and occurs next to the wall. The flow angles get

smaller rapidly as the distance from the wall increases.

Figure 5.17 shows the v component of mean velocity after the last

row of injection at several spanwise locations, for M = 0.9. The obser-

vations for H = 0.4 also apply here qualitatively. Quantitatively, the

values of v are higher because of the higher rate of blowing -- there

is a larger drift towards the wall in the middle lanes. This is evidence

of the strong role which entrainment must play in the heat transfer for

high blowing ratios.

The largest value of 7 is around 16% of the free stream. The

largest flow angle is about 12'.

Figure 5.18 shows the w component for M = 0.4 at the same stream-

wise location as the 7 components. An additional spanwiae profile on

the other side of the centerline is also shown. w is biased, not symmet-

ric. Except for a few points near the wall, the w values are insignifi-

cantly small (l-3% of the U,). The flow angles 6 are also plotted on

the same figure. Generally, 7 and w values for the M = 0.4 case are

small enough SO that locating the spanwise-averaged flow as a 2-D boundary

layer is a valid approximation.

Figure 5.19 shows the w component for M = 0.9. The profile loca-

tions are the same as on Figure 5.18. There is no preferred direction of

the w, as seen in the case of M = 0.4; indeed, the W values are

smaller than the values for M = 0.4. This is because of the low span-

wise gradients in the streamwise mean velocity as compared to the low

blowing case. The lower w values support both the spanwise averaging

and the weak three-dimensionality argument, except perhaps very close to

the wall, where w values are about + 2% of the ?-I,.

90

The large and highly disturbed values of 7 and w near the wall

for M = 0.9 in the central profile are indications of vortices created

by the separated jet.

In both cases of blowing, the angle range of the flow field is en-

tirely suitable for measurements, within a good accuracy, of most of the

turbulence quantities with the triaxial wire system, except perhaps at

the point next to the wall just downstream of a jet for the M = 0.9

case.

5.7.2 Reynolds stresses

Figure 5.20 shows the Reynolds shear stresses for the first station

in the recovery region (27 hole diameters downstream of the last row of

injection) for M = 0.4 and 0.9. These prcfiles are given to show a

sample of the data; the rest are found in Appendix A. These profiles are

not discussed in detail because of their secondary importance in the pres-

ent work.

The (u'v') component of shear stress for M = 0.4 is signifi-

cantly higher than for the others. The (u'w') and (v'w') components

are almost one order of magnitude smaller than (u'v'), except for a

few points near the wall. This indicates a return to two-dimensionality

for the stress field.

All of the shear stress values for M = 0.9 are low in comparison -.

to M = 0.4 as expected, with the exception of (u'v') near the wall,

which is high (still remembering its upstream history).

Figure 5.21 shows the normal Reynolds stresses at the same station

as in Fig. 5.19 for M = 0.4 and M = 0.9. Again, the general observa-

tion is of the expected low turbulence level of M = 0.9 in the outer

region. The two cases also differ in the near isotropy of the M = 0.9

case. All the normal stress components are very close in magnitude,

which is not the case for M = 0.4. This distribution of the normal

stress may be attributed to the flat mean velocity profiles and the

small spanwise gradients in the M = 0.9 case.

For M = 0.4 the u' 2

profile looks very much like the u $2 pro-

file obtained by Vilson & Goldstein (1975) in a turbulent plane wall jet,

91

except that here the levels are lower. The reason is that the plane wall

jet was injected to 'a still environment, thus causing higher shear rate.

5.8 Correlation Coefficients

Figure 5.22 shows the spanwise-averaged stress-energy ratio

(-u'v'/q2> for M = 0.4 at five streamwise stations. The 2-D flat-plate

value (0.15) is also given in the same figure for comparison (taken from

Hinze, 1975). The values near the wall are smaller than the 2-D values

because of the disturbance caused by the injection there. The values

reach the 2-D level above (y/6) = 0.5, where the jet effects are weak.

Recovery to the 2-D values near the wall can be observed in the recovery

region stations.

The first profile after three rows of blowing does not show the large

area of low stress-energy ratio shown in the other full-coverage region

profiles. This must be because the effects of i.njection have not yet pen-

etrated that far.

Figure 5.23 shows the same profiles for M = 0.9. A much larger area

of reduced and disturbed stress-energy ratio is observed, attributed to

the deeper jet penetration; otherwise the same observations can be made

for this case as in M = 0.4.

Figures 5.24 and 5.25 show the spanwise-averaged 3-D stress-energy

ratio, (l?l/Pmi2, for M = 0.4 and M = 0.9, respectively. Since

there is some three-dimensionality in the flow field, it was thought that

it would be interesting to see the effect of three-dimensionality. Fig.

5.24 shows that the total stress-energy ratio I.ies closer to the 2-D value

than -uV/q2, except for a few points near the wall. This suggests

that the value of 0.15 may not be restricted to 2-D boundary layers, but

may be useful in weakly 3-D flow such as this one. The same observations

can be made for M = 0.9, also, except that the effect of high blowing

still shows itself in large disturbances.

Figure 5.26 shows the correlation coefficient, -m,dFdp,

for both blowing ratios at the two recovery region stations. The flat

plate value (0.45) of the correlation coefficient is also shown for

92

comparison (Schlichting, 1968). Again, for low blowing, the values are

much closer to the flat-piate value than for high blowing. The effect of

the injection can be observed near the wall in the form of decreased cor-

relation coefficients for both cases. Reversion to the 2-D flat-plate

state can be seen in the downstream direction for both blowing ratios.

5.9 Mixing Length

The mixing length can be obtained using Prandtl's (1925) definition

(4.13) L llz = -- av

( 1 ay

Figures 5.27 and 5.28 show the mixing length obtained from the shear

stress and the mean velocity data using the above formula in the reco-very

region for M = 0.4 and 0.9. The same figures also show the mixing-

length profile for a 2-D flat plate boundary layer (from Escudier, 1966).

An augmented mixing-length region is observed at the station just after

the last row of blowing (x = 188 cm) for both cases. This augmentation

is a result of the injection process, which creates shear stresses as

well as regions of low ailGay. Evidence in support of this hypothesis is

found in the large region of augmentation for the M = 0.9 case, where

the velocity profiles are quite flat. Having a high mixing length does

not always imply a high rate of mixing; it may just be a result of defi-

nition. For example, even though low shear stress and TKE levels are ob-

served for M = 0.9 (Figs. 5.9 and 5.13), the height and width of the

augmented region are much larger than for M = 0.4. This is an indica-

tion of the inadequacy of the mixing-length models in regions with a zero

mean velocity gradient. Fortunately, the augmented region moved out at

the start of recovery region so that its effect on the near wall behavior

is negligible. Furthermore, the peak in the mixing length dies out quite

fast, and is not observable in the recovery region stations.

Another interesting point is the existence of regions of low mixing

length near the wall, as compared to the 2-D flat-plate case. This region

cannot be seen clearly for M = 0.4, but is quite clear for M = 0.9.

The details of this region were discussed in Chapter 4, along wit11 the

93

predictions. Except for these two regions, the mixing length behaves

like a 2-D mixing length.

Figure 5.29 shows the mixing-length distributions obtained from the

spanwise-averaged profiles in the full-coverage region for both blowing

ratios, as compared to a 2-D flat-plate boundary layer. Some very gen-

eral remarks can be made on these trends, even though the whole picture

is quite complex. In the outer layers, the mixing length is similar to

that for a 2-D layer. In the inner layer, the profile for M = 0.9 is

reduced below the reference. Ho.qever, a diffe-rent behavior can be ob-

served for M = 0.4. It appears that the profiles at the first and sec-

ond recovery region stations have higher values than the 2-D below

(Y/&l = ,0.15, at which point their values drop below the reference be-

txeen (y/6) = 0.1.5-0.4. This leads one to the conclusion that for

low blowing ratios, the peak in the mixing length occurs near the wall

and moves outwards in the downstream direction in the full-coverage re-

gion, whereas for high blowing, it occurs in the outer layers to start

with, due to deeper jet penetration. The points which are outside the

figure limits are shown on top of the figure, with arrows indicating

their magnitude.

Figure 5.30 shows the mixing length obtained from the TKE using the

following definitions:

-pp- = au EMF ' EM = (5-3)

It was interesting to observe its behavior, since a one-equation model of

turbulence was used in predictions. Again, there is some scatter in the

data; the same trends discussed before apply here. Therefore it was de-

cided to use the mixing length obtained from Eqn. (4.13) -- since it was

already programmed in the computer program by Crawford & Kays (1975).

Finally, as a point of interest, the shape factor H is shown in

Fig. 5.31 for both blowing ratios. It was obtained from the spanwise-

averaged profiles. A representative value of H for a 2-D boundary layer

(by Hinze, 1975) is also shown for comparison. The shape factor lies very

close to the 2-D value for both blowing ratios. This is an indication of

spanwise-averaged aerodynamic behavior which is not violently damaged.

94

It is over the 2-D value for M = 0.4 and below it for M = 0.9. The

shape factor converges towards the 2-D value downstream in the recovery

region for both blowing ratios.

95

FULL

-CO

VER

AGE

--REC

OVE

RY

REG

ION

Fig.

5.

1.

Mea

sure

men

t po

sitio

ns.

-- %

y (c

m) 8

! p

FULL

-CO

VER

AGZ

---+-

--REC

OVE

RY

--

/ 5

-- w

-1

/ .

. 7-

VI=

0.4

VI=

0.4

6 + U

CO

= 16.

5 m

/set

7 r

UC

O= 1

6.5

m/s

et

-- -

SPAN

WIS

E --

- SP

ANW

ISE

AVER

AGE

AVER

AGE

5 5-

- (;/

7)th

PR

OFI

LE

- (;/

7)th

PR

OFI

LE

t

Fig.

5.

2.

Prof

iles

of

the

stre

amw

ise

mea

n ve

loci

ty

com

pone

nt

(3

for

M =

0.

4.

1.c

Y 699

0’ .b i-

o-o- 0.0

I I I I I I I I I I

M= 0.4 Ua= 16.5 m/set

x(cm) 0 148 A 168 0 188

: 214 256

A A

00 a;l,

A 08 0 OO A vv A0

5?

A ‘00 A

f 00 OVO

$a0

0.5 U/U*

1 .o

Fig. 5.3. Spanwise-averaged profiles of the streamwise mean velocity component (U) for M = 0.4.

98

Y (c

r n) 8-

7 --

6-

5-

4-

3-

2-

l- O- 0

p FU

LL-C

OVE

RAG

E --+

----R

ECO

VER

Y-

I I

M =

0.9

Uio

= 16

.7 m

/set

- -

- SP

ANW

ISE

AvER

AGE

- (1

/7)th

PR

OFI

LE

I

I 0

0

Fig.

5.

4.

Prof

iles

of

the

stre

amw

ise

mea

n ve

loci

ty

com

pone

nt

(5)

for

M =

0.

9.

1.t

Y’? 99

O.!

- 1)

M = 0.9 Uao= 16.7 m/set

x b-4

0 148 A 168 n 188 v 219 0 256

Fig. 5.5. Spanwise-averaged profiles of the streamwise mean velocity component (V) for M = 0.9.

100

WJC

O

1.C

0.5

I

M

x(cm

) 0 A 0 v l : m

v

0.9

168

0:9

:;

188

214

256

0.4

148

0.4

168

0.4

188

00::

z:z

0.01

“““

“’ 1

11

1 II,

I

I, I,

-:.0

-0.5

0.

0 0.

5 1.

0 In

y

(cm

)

1

Fig.

5.

6.

Span

wis

e-av

erag

ed

prof

iles

of

the

stre

amw

ise

mea

n ve

loci

ty

I'om

pone

nt

<!)

plot

ted

in

sem

i-log

co

ordi

nate

s.

FULL

-cov

mG

E +

REC

OVE

RY

---

n =

0.4

Jco =

16.

5 m

hec

.--

SPAN

WIS

E .&

‘Ei?

AGE

- 2-

D

FLAT

PL

ATE

TKE

7

Y (c

m)

6

L

0

\

0 00

00

0 0 0 0

24-i-

G 0.

1

Fig.

5.

7.

Turb

ulen

t ki

netic

en

ergy

pr

ofile

s fo

r M

=

0.4.

0.c

M=0.4 Ua= 16.5 m/set

x(cm) o 148

f: ;;g v 214 o 256

0 0%

A O ova

00 O $

A

SOA

0.0

Fig. 5.8. Spanwise-averaged turbulent kinetic energy profiles for M = 0.4.

103

------

- FU

LL-C

OVE

RAG

E __

_ci_

r___

R

ECO

VER

Y.

1

- 2-

D

“:.:T

Pi

ATE

TKE

01

’ ”

” ”

“1

” ”

” ”

’ ”

” ’

” B

0 0.

1 0.

1 0.

1 0.

1 0.

1

&/ua

J

Fig.

5.

9.

Turb

ulen

t ki

netic

en

ergy

pr

ofile

s fo

r M

=

0.9.

1" .\

Yh9I

1.c

5-

! 1

3

I-

0.: F-

M = 0.9 UCXI = 16.7 m/set

x (cm) l 148 A 168 l 188 v 214 l 256

l_. I I I I I I I I I

‘0.1 0.2

d-1 q2 ual

Fig. 5.10. Spanwise-averaged turbulent kinetic energy profiles for M = 0.9.

105

y’8*99

1.c

0.5

C

1.0

I I I \ \ \

FULL-COVERAGE

- M=

I I I I I I I I I I 0.5 1.0

U/U,

-

.-

I!- 3

x- - %-

12

I I I I I I I I I I I I I I I 0.1

A/- 0.2 0.3

92mo

Fig. 5.11. Relation between mean velocity gradient and turbu- lent kinetic energy level.

106

FLAT

PLA

TE

AD S

TRES

S

t /\"

')""I

\ 0.

5 1.

0 I

’ ’

1 ’

1 ’

’ ’

’ 1

0 0.

5 1.

0 -7

7 (m

2/se

c2)

Fig.

5.

12.

Stre

amw

ise

turb

ulen

t sh

ear

stre

ss

com

pone

nt

(-u'

v')

at

the

star

t of

th

e re

cove

ry

regi

on

for

M =

0.4

.

. .

CO

VER

AGE

1

- C

UR

VES

THR

OU

GH

L l-w

--

2-D

FL

AT P

LATE

SH

EAR

ST

RES

S

III1

I I

I1

I I

I V.

V I.V

I ,

, ,

I I

I,,

I

5 0

0.5 ;

In I."

II=.

I.J

-27

(m2/

sec2

)

Fig.

5.

13.

Stre

amw

ise

turb

ulen

t sh

ear

stre

ss

com

pone

nt

(-u'

v')

at

the

star

t of

th

e re

cove

ry

regi

on

for

M =

0.9

.

-

- FU

LL-C

OVE

RAG

E ---

--+--R

ECO

VER

Y -

- ;M

AVES

TH

RO

UG

H

---

2-D

FL

AT P

LATE

SH

EAR

ST

RES

S

0.5

I

. %

F t- l--

\

z \ \ \ O

o l,l1,

1’1

’ 0.

5 III

II 0.

5 I‘I

IIIIII

IIIII

0.5

0.5

0.5

-77

(m2/

sec2

)

Fig.

5.

14.

Spa

nwis

e-av

erag

ed

prof

iles

of

the

stre

amw

ise

turb

ulen

t sh

ear

stre

ss

com

pone

nt

for

M =

0.4

.

FI

T-

JLL-

CO

VER

AGE-

-+---

-REC

OVE

RY

JLL-

CO

VER

AGE-

-+---

-REC

OVE

RY

-

- FL

Ob

-4

f&VE

S TH

RO

UG

t

2-D

FL

AT P

LATE

SH

EAR

ST

RES

S M

= 0.

9 U

,=

16.7

m/s

et

-27

(m%

ec2)

Fig.

5.

15.

Spa

nwis

e-av

erag

ed

prof

iles

of

the

stre

amw

ise

turb

ulen

t sh

ear

stre

ss

com

pone

nt

for

M =

0.9

.

SY

M,= 0.4

I I II I II I I II 1 0

Vii 0.2

I I I I I I I I I I I ’ 1 - 0 5 10

y(degrees)

Fig. 5.16. 7 component of the mean velocity at the start of the recovery region for M = 0.4.

M=0.9 Uoo= 16.7 mlsec

I I I I I I I I I II 1 I 0 0.1 0.2

V/U, I I I I I I I I I I I 1 1 1

0 5 10 y(degrees)

Fig. 5.17. v component of the mean velocity at the start of the recovery region for M = 0.9.

111

I FULL- YJS COVERAGE t

.99 tnl- I

u11424-4-J

E4 M=0.4 (I

c- U,= 16.5 m/set - LA I-

0 0 0

I,,,, 1, , , , -0.2 -0.1 _ 0

WI ‘JaJ I I I I I I , I , , l ,

-10 -5 0 B(degrees).

Fig, 5.18. w component of the mean velocity at the start of the recovery re- gion for M = 0.4.

Y/~QQ FULL- COVERAGE 1

M=0.9 U, = 16.7 m/set

A P Fi u

0 0

0

I I I I I I I I , , I , , , -0.2- -0.1

P&U, 0

I I I I I I I I I I I I I. .~ -10 5 :,

B(degrees) -

Fig. 5.19, w component of the mean velocity at the start of the recovery region for M = 0.9.

112

l.O-

n m &-I IJU3 (m2/sec) (m2/sec) (m2/sec) (mhec)

0 16.5 l 16.7

STRESSES

Fig. 5.20. Reynolds shear stresses at the first recovery region station.

x=214 cm

-420 M ~b-b&%J, o/U, U&m/sec)

A0 0 0.4 0 A 0 16.5

STRESSES

Fig. 5.21. Reynolds normal stresses at the first recovery region station.

113

cl 0 188 214 g 256

- - FLAT PLATE VALUE

Fig. 5.22. Spanwise-averaged stress-energy ratio for M = 0.4.

----- --- -A----

A -5

8

vv AA A v ~~Yi~.7 m/set

x(cm) v’r A

-*r 168

8 188 1. 214

8 : 256 -- FLAT PLATE VALUE

A l

OO I I I I I I I I I I

0.5 1.0 Y’! 99

Fig. 5.23. Spanwise-averaged stress-energy ratio for M = 0.9.

114

Ua,= 16.5 m/set x(cm)

V 148

ii 168 188 li 256 214

-- FLAT PLAT

0 I I I I I I I I I

0 0.5 1.0 YF99

Fig. 5.24. Spanwise-averaged 3-D stress-energy ratio for M = 0.4.

I - I

0

----k-w-- 4 A---

1 OB 000 Cl s q A M 0.9

@ oA 0 Oo08, e 0 O”o 0 0 = 0.1

I A U,= 16.7 m/set I- -AA x(cm)

0 168 - P (uv)* + (“‘w’)* + (Ad)*

A 188 0 214 0 256

-- FLAT PLATE

I.. 2 I I I I I I

0.5 1.0 Y/F99

Fig. 5.25. Spanwise-averaged 3-D stress-energy ratio for M = 0.9.

115

x (cm) 214 256

M=O.4 M= 0.9

-- FLAT PLATE

1 I I I I I .I 1 I I

0.5 1.t

e99

Fig. 5.26. Correlation coefficient for M = 0.4, 0.9 at the two recovery region stations.

116

9 M - = 0.4 FLAT PLATE 2/S~~ _ 0.212 x(cm)

ii 214 188

0 256

q A

I I I I I I I I I

0.5 1.0 Y’a. 99

Fig. 5.27. Mixing-length distribution in the recovery region for M = 0.4.

0 P 0 M = 0.9

2’8QQ - 1 -FLAT PLATE

1.69 0.7 x(tm)

z $84 0 0 0 256

0.1 -

+QQ

Fig. 5.28. Mixing-length distribution in the recovery region for M = 0.9.

117

m99

0.:

0.1

M=0.4 2- - xm1

0 a 148 0 168 n .- 188 - FLAT PLATE

0 0

0

/ l @ ‘m 0

/ m

0 I I I I I I I I I I 0 0.5 1.0

M = 0.9 x(cm)

0 168 0 188

yk99

Fig. 5.29. Spanwise-averaged mixing-length distribution in the full-coverage region for M = 0.4 and M = 0.9.

118

0.087

"6,Q

Mz0.4 M=0.9 x(cm) 0 188

: 214 256 - FLAT PLATE

I I I 0 0.5 1.0

y”.QQ Fig. 5.30. Mixing-length obtained from the turbu-

lent kinetic energy in the recovery region.

LO 0 0

0 -----_____~_~~~

H 0 0 0 0

l.O- 0 cY4 0 019

- - FLAT PLATE

START OF RECOVERY

0 I I I I I I I L--IL- -

150 200 250 x (cm)

Fig. 5.31. Shape factor.

119

Chapter 6

SUMMARY AND RECOMMENDATIONS

6.1 Summary

The work reported here is a part of continuing study of the heat

transfer and hydrodynamics associated with full-coverage film cooling.

In the present work, hydrodynamic measurements were made with a triaxial

hot-wire in the full-coverage region and the recovery region of a 30"

slant-angle injection pattern. These measurements were made in isothermal

flows at ambient temperature and pressure (about 20°C and 760 mm Hg) for

two blowing ratios: M = 0.9 and M = 0.4. Profiles of the three mean

velocity components and the six Reynolds stresses were obtained at sev-

eral spanwise positions at each of five locations down the test plate: two

within the full-coverage region, two within the recovery region, and one

at the dividing line between the two regions.

One of the most important observations from these experiments is the

peculiar dependence of TKE on the blowing ratio: high TKEs were observed

for low blowing (M = 0.4), low TKEs for high blowing (M = 0.9). This

phenomenon results from the low mean velocity gradients for high blowing

(close to unity). The difference in the TKE levels has a significant ef-

fect on heat transfer behavior.

As a result, it was concluded that two different pairs of processes

compete to govern the heat transfer, depending on the level of blowLng:

for low blowing, the energy sink near the wall competes with high turbu-

lent mixing; for high blowing, the energy sink in the outer layer competes

with the entrainment and convection of mainstream fluid toward the wall in

the lanes between the jets. The present 2-D spanwise-averaged solution

methods are not expected to predict the heat transfer weli for high blow-

ing ratios -- except in a purely formal way -- because they cannot handle

the entrainment process and by definition neglect the spanwise 3-D flows.

The flow in the recovery region can be described in terms of a two--

layer model: an outer boundary layer where the length scale scales on

the total thickness of the layer, and an inner layer where the mixing

120

length is the usual KY and X6. The two layers blend into each other

with the spreading of jets. Recovery to a 2-D boundary layer is completed

when the inner and outer layers finally merge.

A one-equation model of turbulence was used in the 2-D finite-

difference boundary layer computer program (STAN5) to predict the mean

velocity and TKE profiles in the recovery region. The one-equation model

employs the TKE conservation equation with an algebraic relationship for

the mixing length. Mixing-length values calculated from the data were

input to the program using a piecewise continuous, heuristic fit consistent

with the concept of the two quasi-independent layers observed in the re-

covery region. This mixing length pattern, used with a set of otherwise

normal constants (for 2-D boundary layer predictions),successfully pre-

dicted all the spanwise-averaged features of the flow. This strongly sug-

gests that the principal mechanisms were modeled adequately by the mixing-

length formulation. It was also shown that the piecewise continuous heur-

istic model can be replaced by a set of relations taken from the literature

describing the spreading of jets and the growth of boundary layers, thus

supporting the physical arguments behind the mixing-length model.

Large amounts of data were required by the present program, and a

new hot-wire scheme was developed for this work: a triaxial hot-wire with

an analog device for real-time data reduction. The method is basically

simple; i.e., with three orthogonal wires, the components of the instan-

taneous velocity vector can be deduced without recourse to any time aver-

aging and without invoking the low-fluctuation assumption. The present

method can tolerate an unknown flow direction within the following limits.

The mean velocity was measured within a cone of ? 30" half apex angle

around the probe axis with a maximum error of 4%, even in a high-velocity

gradient (about 1600 l/set). The turbulent kinetic energy was measured

within a cone of 12" half apex angle with a maximum error of 12% up to

the same velocity gradient point and within 5.5% in a region of zero vel-

ocity gradient. The method is very fast and practical for taking large

amounts of data in three-dimensional, high-fluctuation turbulent flows.

121

6.2 Recommendations

a The present model for the length scale, developed for the recovery

region, can probably be used also for the full-coverage region provided

that the injection process is modeled correctly.

l Either 3-D solution methods should be used or new 2-D solution pro-

cedures should be devised to simulate the entrainment process for high

blowing ratios, in order to better predict the heat transfer.

l For the 3-D turbulent flow measurement scheme, a smaller probe

should be manufactured to decrease the errors arising from t'ne probe

size and to permit a closer approach to the wall.

RMS and correlator circuits can be added to the 3-D Turbulent Flow

Analyzer which will increase the speed of data-taking, as well as allow-

ing for reading important quantities such as mean velocity and TKE di-

rectly from the dials. This will greatly increase the range of applica-

bility.

122

REFERENCES

Abramovich, G. N., 1960, The Theory of Turbulent Jets, MIT Press, ---- p. 544.

Antonia, R. A., & Luxton, R. E., 1972, "The Response of a Turbulent Bound- ary Layer to a Step Change in Surface Roughness. Part 2. Rough to Smooth," J. Fluid Mechanics, 53, Part 4, 737-757.

Bergeles, G., Gosman, A. D., & Launder, B. E., 1975, "The Prediction of Three-Dimensional Discrete-Hole Cooling Processes: I - Laminar Flow," ASME Paper 75-WA/HT-109.

Boussinesq, J., 1877, "Theorie de l'ecoulement Tourbillant," Mem. Pre. Par. Div. Sav. 23, Paris.

Brunner, M. S., 1969, "Active Cooling Heat Protection," J. Space Craft, June 1969, Vol. 6, No. 6.

Choe, H., et al., 1975, "Turbulent Boundary Layer on a Full-Coverage Film-Cooled Surface -- An Experimental Heat Transfer Study with Nor- mal Injection," NASA Rep. CR-2642 (also Stanford Univ., Mech. Engrg. Dept. Rep. HMT-22).

Colladay, R. S., & Russell, L. M., 1975, "Flow Visualization of Discrete- Hole Film Cooling for Gas Turbine Applications," NASA Kep. TM X-71766.

Crawford, M. E., & Kays, W. M., 1975, "STAN5 -- A Program for Numerical Computation of Two-Dimensional Internal/External Boundary Layer Flows," Stanford Univ., Mech. Engrg. Dept. Rep. HMT-23.

Crawford, M. E., et al., 1976, "Heat Transfer to a .Full-Coverage Film- Cooled Surface with 3%Deg. Slant-Hole Injection," Stanford Univ., Mech. Engrg. Dept. Rep. HMT-25.

Delleur, J. W., 1966, "Flow Direction Measurement by Hot-Wire Anemometry," Journal A.S.C.E., Engineering Mechanics Division, Aug. 1966, Vol. 92, p. 45.

Eriksen, V. L., Eckert, E. R. G., & Goldstein, R. J., 1971, "A Model for Analysis of the Temperature Field Downstream of a Heated Jet Injected into an Isothermal Crossflow at an Angle of go"," NASA Rep. CR-72990.

Escudier, M. P., 1966, "The Distribution of Mixing-Length in Turbulent Flows Near Walls," Imperial College, Heat Transfer Section, Rep. TwFITN/l.

Esgar, 3. B., 1971, "Turbine Cooling -- Its Limitations and Its Future,v High Temperature Turbines, AGARD Conf. Proc. No. 73: 14.1-14.24.

123

Foster, R. C., Haji-Sheikh, A., 1974, "An Experimental Investigation of Boundary Layer Heat Transfer in the Region of Separated Flow Down- stream of Normal Injection Slots," ASME Paper No. 74-HT-i2.

Goldstein, R. J., 19.71, "Film Cooling," Advances in Heat Transfer L, 321-279.

Goldstein, R. J., et al., 1969, "Film Cooling Following Injection through Inclined Circular Tubes," NASA Rep. CR-73612.

Hartnett, J. P., Birkebak, R. C., & Eckert, E. R. G., 1961, "Velocity Distributions, Temperature Distributions, Effectiveness and Heat Transfer for Air Injected through a Tangential Slot into a Turbu- lent Boundary Layer," J. Heat Transfer, Aug. 1961, 293-306.

Herring, H. J., 1975, "A Method of Predicting the Behavior of a Turbulent Boundary Layer with Discrete Transpiration Jets," J. Eng. Power, 97, 214-224.

Hinze, J. O., 1975, Turbulence, McGraw-Hill Book Co., Second Edition, p. 643.

Hoffmeister, M., 1972, "Using a Single Hot-Wire Probe in Three-Dimensional Turbulent Flow Fields," DISA Information, May 1972, No. 13, 26-28.

Johnston, J. P., 1970, "Measurements in a Three-Dimensional Turbulent Boundary Layer Induced by a Swept Forward-Facing Step," J. Fluid Mechanics, 42, Part 4, 823-844.

Jorgensen, F. E., 1971, "Directional Sensitivity of Wire and Fiber Film Probes, An Experiemntal Study," DISA Information, May 1971, No. 11, 31-37.

Kacker, S. C., & Whitelaw, J. H., 1970, "Prediction of Wall-Jet and Wall- Wake Flows," J. Mech. Engrg. Science, 1.2, No. 6.

Keffer, J. F., & Bains, W. D., 1963, "The Round Turbulent Jet in a Cross- wind," J. Fluid Mechanics, 15, 481-496.

Klebanoff, P. S., 1955, "Characteristics of Turbulence in a Boundary Layer with Zero Pressure Gradient," NACA TN 1247.

Kolmogorov, A. N., 1942, "Equations of Turbulent Motion of an Incompress- w ible Turbulent Fluid," Izv. Akad. Nauk SSSR, ser Phys. VI, No. 1-2,

p. 56.

Launder, B. E., & Spalding, D. B., 1972, Lectures in Mathematical Models of Turbulence, Academic Press, London and New York.

LeBrocq, P. V., Launder, B. E., & Priddin, C. H., 1971, "Discrete Hole Injection as a Means of Transpiration Cooling -- An Experimental Study," Imp. Coil. Rep. HTS/71/37.

124

Mayle, R. E., & Camarata, F. J., 1975, "Multihole Cooling Film Effective- ness and Heat Transfer," J. Heat Transfer, 97, 534-538.

Metzger, D. E., Carper, H. J., & Warren, J. M., 1972, "Predicted Film Cooling near Flush Slots -- Comparison with Experiment," J. Air- craft, Dec. 1972, 2, No. 12, 857-863.

Metzger, D. E., Takeuchi, D. I., & Kuenstler, P. A., 1973, "Effectiveness and Heat Transfer with Full-Coverage Film-Cooling," J. Eng. Power, 9J, 180-184.

Mojola, 0. O., 1974, "A Hot-Wire Method for Three-Dimensional Shear Flows," DISA Information, July 1974, 16 ---p 11-14.

Moussa, 2. M., & Eskinazi, S., 1975, "Directional Mean Flow Measurements Using a Single Inclined Hot-Wire," Physics of Fluids, March 1975, ltJ, No. 3, 298-305.

Nina, M. N. R., & Whitelaw, J. H., 1971, "The Effectiveness of Film Cool- ing with Three-Dimensional Slot Geometry," Gas Turbine Conference and Products Show, Houston, Texas, March 1971, ASME Paper No. 71-GT-11.

Pai, B. R., & Whitelaw, J. H., 1971, "The Prediction of Wall Temperature in the Presence of Film-Cooling," Int. J. Heat Mass Transfer 14, 409-426.

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Pimenta, M. M., 1975, "The Turbulent Boundary Layer: An Experimental Study of the Transport of Momentum and Heat with the Effect of Roughness," Stanford Univ. Mech. Engrg. Dept. Rep. HMT-21.

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Samuel, A. E., & Joubert, P. N., 1965, "Film Cooling of an Adiabatic Flat Plate in Zero Pressure Gradient in the Presence of a Ibt Main- stream and Cold Tangential Secondary Injection," J. Heat Transfer, 87, 409-419.

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125

Seban, R. A., & Back, L. H., 1962, "Velocity and Temperature Profiles in Turbulent.Boundary Layers with Tangential Injection," J. Heat Trans- fer, 84, 45-54.

Wilson, D. J., & Goldstein, R. J., 1976, "Turbulent Wall Jets with Cylin- drical Streamwise Surface Curvature," J. Fluids Engineering, Sept. 1976, 550-556.

Wolfshtein, M., 1969, "The Velocity and Temperature Distribution in One- Dimensional Flow with Turbulence Augmentation and Pressure Gradient," J. Heat Mass Transfer, 12, 301-318.

Zimmerman, D. R., & Abbott, D. E., 1975, "An Experimental Investigation of a Three-Dimensional Turbulent Boundary Layer," Tech. Rep. CFMTR-75-1, May 1975.

126

APPENDIX A

TABULATION OF EXPERIMENTAL DATA

A.l. Individual Profiles

This section contains the dimensional and non-dimensional profiles

of three mean velocity components and six Reynolds stresses at each mea-

surement location for blowing ratios M = 0.4 and 0.9. Profiles of tur-

bulent kinetic energy, flow angles, mixing-length, and correlation coeffi-

cients are also given. The numberofdiameters downstream at the top of

appropriate pages representsthedistanceofthe measurement location from

the last row of injection in terms of jet diameter D. The nomenclature

given in this section also applies in Section A-2.

Nomenclature

BETA

CM

DELM

DELMl

DELM2

GAMA

L

M

M/S

M2/S2

P

PR

Q

42

R

- 8, angle between U and W components of mean velocity in degrees.

cm.

6, boundary layer thickness.

%, displacement thickness.

62' momentum thickness. - -

Y, angle between U and V components of mean velocity in degrees.

Q.3 mixing-length.

P jetUjetiP2~' blowing ratio.

m/set.

2 2 m /set .

p, hole spacing or pitch.

p, pressure

I/ 2= ---

2 2 q u I2 + v' + w'

--- q2 = u'

2 2 2 + v' + w' , turbulent kinetic energy.

if = lf u2 + y2 + 72, magnitude of mean velocity.

127

RF

RF2

TF

U

UDERV

UF

UF2

UINF

UP

UP2

uv

uw

V

VP

VP2

VW

W

WP

WP2

X

Y

Z

U co' value of X at freestream.

iT2 at freestream.

T 43' freestream temperature ("C).

C component of mean velocity in x-direction.

may, gradient of 5 component of mean velocity.

U

Ui:

freestream velocity.

freestream velocity squared.

U al' freestream velocity.

u', component of fluctuating velocity in x-direction.

2 u' , Reynolds normal stress in x-direction.

u'v' , longitudinal-normal velocity correlation, Reynolds shear

stress.

u'w' , longitudinal-tangential velocity correlation, Reynolds

shear stress.

v, component of mean velocity in y-direction.

v’ , component of fluctuating velocity in y-direction.

2 v' , Reynolds normal stress in y-direction.

7 v'w , normal-tangential velocity correlation, Reynolds shear

stress.

K component of mean velocity in z-direction.

, w , component of fluctuating velocity in z-direction.

2 w' , Reynolds normal stress in z-direction.

x, distance along test surface measured from nozzle exit.

YY distance normal to test surface.

Z, distance in transverse direction on test surface measured

from centerline.

128

-

H10

.4

BLO

I~IN

G

REG

ION

AF

TER

3

RO

HS

Of

HO

LES,

X=

148

CM

AN

D

L/P=

+O.S

RU

N

1215

7602

TF

= 20

.00

C

PR=

760.

00

HH

-HG

D

ELM

= 2.

952

CH

D

ELH

l=

0.78

2 C

M

DEL

HL=

0.

455

CM

OlM

ENSI

ON

AL

PRO

FILE

S

(CL

iH:S

, LH

k,

LM7S

)

0.41

9 8.

00

0.29

8.

01

0.50

0 8.

17

0.27

:;*

g 8.

18

0.60

0

9:6&

l K

i

0.30

-0

:21

8.39

00.;;

; 0.

29

1:20

0 -0

.37

-0.6

9 !*

77':

-0.7

9 11

:33

1.50

0 -0

.47

13.0

7 1.

800

2.10

0 iv

i:

KG

-0.1

4

3:oo

o

16.0

2

:i:::

16.0

2 15

:53

16.3

2 4.

000

::*ts

' . 4.

09

0.12

16

.46

NO

N-D

IHEN

SIO

NAL

PR

OFI

LES

Y/D

ELM

0.14

2

O.Y

l5

:*Sfg

.

Y/O

ELM

0.14

2 0.

169

0.20

3 0.

254

0.32

2

UP2

(Y

Z/S2

)

0.79

3 0.

795

P'E

r:e

53

2.23

6 1.

356

0.83

3

VP2

1H2/

52)

:*;g

1:31

0 :*;

g

1135

0 0.

790

0.51

5 0.

370

8%;

8:

p;

.

HP2

(H

2/S2

1

1.36

2 L.

38U

1.

458

:*E%

11

357

O.tl

69

92

tM2

52)

3.36

3 0.

260

O.L

l3

U/R

F V/

RF

H/R

F G

AMA

BETA

U

P/R

F VP

/RF

WP/

RF

Q/R

F U

V/R

FZ

UW

/RFZ

Vd

/RFZ

L/O

EM

UV/

Q2

uv/L

uP*v

P)

0.09

9

i- 8:

: r",

*:;;

-0:1

10

:g=j

:: .

-o:j7

5 -0

.451

-0

.418

0.11

1 0.

112

0.11

6 0.

124

8-

:z

0:10

5

oo%

l 0:

058

oo= 0

04:

0:oo

a

-0.1

3D-0

2 -0

.130

-02

-0.1

50-3

2 -0

.220

-02

-0.3

00-0

2 -0

.300

-02

Zoo'

;g;

-g;

-0:7

50-0

3 -0

.480

-03

-0.2

5D-0

3 '0

0.

w-M

Y0

.

H=0

.4

BLO

WIN

G

REG

ION

AF

TER

3

RO

WS

OF

HO

LES,

X=

148

CM

AN

D

L/P=

+O.3

RU

N

1215

7601

TF

= 20

.00

C

PR=

760.

00

HH

-HG

D

ELH

I 2.

783

CH

D

ELM

l=

0.52

3 C

M

OEL

ML=

0.

314

CH

DIM

ENSI

ON

AL

PRO

FILE

S

KnY1

(H

YSI

dS,

(ML,

(H

!SI

UP2

IY

Z/S2

I

1.03

6

ii*Ei

s 1:

001

VP2

(M2/

S21

WP2

(M

L/S2

1

0.16

3 0.

058

0.01

1

0.41

9 0.

500

k%

0:95

0

~=25

0000

1:

eoo

f l t88

<h

o”;

4:oo

o

10.9

6 -0

.21

-0.3

5 -0

.28

2x .

10.9

7 11

.15

::%

12:5

8 13

.21

:a-

2 15

:26

15.0

3 16

.23

16.4

3 16

.47

0.82

0 0.

(120

E-

g;

0:so

o 0.

710

0.58

8 0.

464

0.90

1 0.

8b6

t. :g

0:71

1 fj=

Sp:t

0142

6 0.

309

0.19

3

8.

EoO

0:

003

2.26

6

: 'E

1:

147

0.74

3 0.

390

Oo-

ii:;;

.

-0.2

95

:;*;‘U

: -g

::W&

-0:3

32

$.f$

I$;::

: -0

mO

bl

-0.0

21

0.00

0

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53

-0m

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.i7S

-0.1

75

-0.1

13

1;:;;

; -0

.0Ll

O

mO

U7

8-z!

0:

001

:tmt3

16

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I,“-‘,

: -0

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r,"=#

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1;:::

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3 -0

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-0.0

9 -0

.04

0.04

0.

13

NO

N-D

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SIO

NAL

PR

OFI

LES

Y/D

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U

/RF

V/R

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GAM

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TA

0.15

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321:

0:

431

0.53

9

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P-E

1:43

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: -1

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-1 m

l 3

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08

I;:$

-8%

0:

ooo

4 -0

.6

-0.6

:g=;

7.

:: -0

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6

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ELY

L/O

ELM

U

V/Q

Z uv

/ (u

P*vP

J

0.15

1 0.

180

oom

%06

0:

341

0.43

1 0.

53Y

g.g

0:86

2

Pm90

T0

1:43

7

0.07

5

m:

0:06

8 0.

080

k?t%

0:

otIo

0.

077

00%

0:

121

0.00

0

I;':;:

-0

1156

-0

.153

-0

.128

0.

000

0.00

0

UP/

RF

VP/R

F W

P/R

F O

/RF

UV/

RFZ

U

W/R

Fi!

VII/H

FZ

t-l

w 0

0.06

2 0.

055

0.05

8 0.

101

-0.1

10-0

2 -0

.200

-0~

-0.7

03-0

3 0.

061

0.05

5 0.

057

0.10

0 -0

.110

-02

-0.3

40-0

3 0.

061

0.05

5 0.

055

-U.fi

tJU

-J3

0.09

8 -O

.llD

-02

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LO-0

3 -0

.420

-u3

0.06

1 0.

055

0.05

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.OY8

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120-

02

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5i.I-

03

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062

0.05

4 0.

051

0.09

7 -0

.130

-02

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50-0

3 -O

.r!c)

U-3

3 0.

058

0.05

1 0.

048

0.09

1 -0

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-02

-0.4

20-O

;, 0.

054

0.04

7 0.

044

-O.L

9d-3

3 0.

084

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i)3

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041

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034

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REG

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RO

WS

OF

HO

LES,

X=

148

CM

AN

D

2/P=

0.

0

RU

N

1214

7604

TF

= 21

.00

C

PR=

760.

00

MM

-HG

D

EL.L

\=

2.Y3

4 C

M

DIM

ENSI

ON

AL

PRO

FILE

S

DEL

MA=

0.

898

CM

D

ELM

Z=

Om

44Y

CH

CM

%,

ItA,

dS,

it&,

UP2

tM

2/S2

J

8%

0.59

0:

600

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:: p;

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:* ;g

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0.

160-

03

0.24

0-04

0.

140-

03

uv/(u

P*vP

)

-0.2

63

m53

-0

:202

A.2. Spanwise-Averaged Profiles

'This section contains the spanwise-averaged profiles of streamwise

mean velocity component (I!>, turbulent kinetic energy and streamwise

turbulent shear stress component (-u'v') for blowing ratios M = 0.4

and 0.9. There is a pair of pages for each location. On the first page

are the run numbers, spanwise positions, and profiles for which the span-

wise averages were obtained. The dimensional and non-dimensional spanwise-

averaged profiles are given on the second page.

159

.4

BLO

kiIY

G

REG

ION

AF

TER

3

RO

WS

OF

HO

LES,

X=

14&

CM

H

=O

E Y

0.60

0

00-;5

5:

1:zo

o

:%z

2:10

0

I* ;1

:: 3:

ooo

4.00

0

0.41

9 0.

500

0.60

0

oo- z

;: 1:

200

3.00

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000

k 2:

0:

60

0 g.

;;;

1:zo

o 1.

500

1.80

0 2.

100

2.40

0 7.

700

3.00

0 4.

000

UV(

l 1

uv(2

) uv

(3)

bQls

2)

uv(4

) uv

(5)

(K/S

21

W/S

2)

oQ/s

2)

bQ/S

2)

-0.3

54

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95

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55

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71

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3-E

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(h

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) 3.

363

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0 3.

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54%

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015

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k?::;

3:

019

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2)

w/s

2 1

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266

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743

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02;

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3)

(m/s

2)

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5.

5.

6b7

5.82

5 4.

804

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3 p

kg;

2:80

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bb

a 0.

961

0.51

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42(4

) (h

Q./S

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0.43

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5)

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1576

03

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5J:

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R

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t5):

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WIS

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PRO

FILE

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3

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F H

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M

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M=O

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REG

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TER

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HO

LES,

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168

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1)

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1476

01

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1376

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300

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90

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40

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00

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4.80

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EPAG

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0.4

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EGIO

N A

FTER

7

RO

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F H

OLE

S,

x=16

8 C

M

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AGE

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5 H

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ENSI

ON

AL

PRO

FILE

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x:;

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342

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2.39

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uv(3

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FILE

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Y AFT

ER 1

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FILE

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WIN

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REG

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TER

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SPAN

WIS

E AV

ERAG

ED

PRO

FILE

S W

0.9

BLO

WIN

G R

EGIO

N A

FTER

7

RO

WS O

F H

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S, X

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C

M

AVER

AGE

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F =

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7 H

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ON

AL

PRO

FILE

S

Y(C

HI

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l/S)

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000

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100

10.8

44

12.6

24

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ART

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11

RO

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OF

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LES,

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188

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:- E

i 1:

500

1.75

0 2.

300

6.10

0

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0 2.

300

E%

3:50

0

E%~

6:

100

i? ;

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1: 0

00

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ty1

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: 31

7760

2 -0.5

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(S)

: 31

7760

3

u(4)

1(

5)

(M/S

) (M

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15

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15

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r; l ‘2

:: -0

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(M

/S)

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13

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13

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) w

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) W

P)

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1 0.

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0.01

5 0.

004

-5.0

19

1;.

fg

-0:2

79

-0.4

91

-0.2

08

0.21

4 -0

.096

w/s

21

3.35

0 3.

651

44%

8 3:

897

:*%

1:31

2 1.

419

1.34

0 0.

797

0.06

6 0.

019

WR

IS2)

0.

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’$g

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1..2

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2)

15.7

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/w

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) W

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i- z!

2:

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8 0.

016

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UIS

E AV

ERAG

E0

PRO

FILE

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=0.9

ST

ART

OF

REC

OVE

RY A

FTER

11

RO

WS O

F H

OLE

S,

X=18

8 C

M

AVER

AGE

UIN

F =

16.5

3 M

/S

DIM

ENSI

ON

AL

PRO

FILE

S

Y(C

M)

U(H

/S)

UO

ERVt

l/S

J

0.41

9 0.

575

1.00

0 1.

250

34-~

%

5:O

DD

6.

100

12.8

62

:z-

%

1319

47

14*1

0&t

i4;ii

9 14

.200

14

.288

14

.858

K-

3 16

1532

16

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0.78

5D

0. S

LOO

-C

.l5bD

-8

-E," .

-0.6

05~

0.34

50

DE

LM=

4.52

6 C

M

OEL

HI

= 0.

584

CU

D

ELU

Z=

0.47

9 C

M

QZ(

H2/

521

UV[

MZ/

SZl

4.88

9 4.

891

3.29

1

0.01

9

-0,5

67

-0.5

36

0.05

5

-0.1

06

-rpp .

NO

bDIH

ENSI

ON

AL

PRO

FILE

S

Y/D

ELM

U

/UF

Q/U

F U

V/U

FZ

L/D

EL

M

UV/

PZ

APPENDIX B

WIRING DIAGRAMS OF THE 3-D TURBULENT FLOW ANALYZER

Nomenclature

Linearizer outputs.

uxJJyJJz Normalized effective velocities, same as Ueff , Ueff , 1 2

U eff ' 3

Dx,Dy,DZ,Ex,Ey,EZ,Fx,Fy,F Coefficients in the inverted Jorgensen's Z

matrix M.

x,y, z Normalized velocities in wire coordinates.

xu,Yu,zu,xv,Yv,zv,xw,yw,z Coefficients of the coordinate transfor- W

mation matrix N.

u,v,w Normalized velocities in laboratory coordinates.

FLO

W

CO

W~J

TER

FR

ON

T PA

NFL

SC

HEY

hTIC

D

~~KG

RAM

R

EFI(

RD

A R

EFA

+1ov

+1

ov

SIO

V 5-

7 l

OV

REF

B

t

l-7

P.SA

bL

-I-

3-Y

ccw

es.

q

I I-X

v,

B”G

+&+

+I0

V R

EF 8

t1

5v

LEL

s-3

. ST

S I P

N!

CAR

L s.

...r”P

F%

--I- 3-

3,

ccw

p.

s.Fz

3

FLO

W C

OM

PUTE

R F

RO

NT

END

SU

MM

ERS

ANb

\NVE

RTF

RS

(ChR

b 1)

SFLE

hO

’lE

I

NO

TE

2 -- AL

L FI

XED

R

ESIS

TQR

S AR

E 0.

1%

25~p

y/oc

i/k

%’

DAL

E TY

PE

WF

U~J

LESS

OTH

ERW

ISE

SPEc

\FIE

D

*:518

K PI

NS

+15V

PI

N7

-15

v PM

4

FLO

W

CO

MPU

TER

M

UL,

TIPL

\ER

S AN

D

INVE

RTE

RS

. . 2

$q

+4&3

2&a-

~ @

OV

s 80

13 A

C

PIN

S +1

5v

PIN

2 1

@IV

Ov

PIN

8 -Y

N

OTE

\

wn-

E 2

Y -It

i\/

PIN

S +

3~51

8 K

PIN

S C

ARV

PIN

S I,%

,Y

AND

3

SHAL

L B

E W

ED T

OG

ETH

ER

ALLF

IXED

R

ESl=

OftS

SISV

PI

N7

AND

CO

NS\

OER

ED

A C

RD

Wb

ARE

0.1%

25

wn

/“c

-)5V

PIN

S BU

S TO

W H

1Cb-

i AU

cl

%U

\T

‘/bW

DAL

E TY

PE M

FF

()v

f?lN

TS

ARE

INbl

VID

I)NLy

U

NLE

SS

OTH

ERW

ISE

TIED

SP

ECIF

LED

FLO

W

CO

MPU

TER

SU

MM

ING

AM

PLIF

IER

S (C

ARD

1)

2OK

2%

J ’

*ov

LOK

2OK

CAU

D P

INS

L%Y,

Z SH

ALL

BE

TIED

TO

GET

HER

AN

D

CO

NSI

DER

ED

A G

RO

UN

D

BUS

TO W

H\C

H

ALI-

CIR

CU

IT

OV

POIN

TS

ARE

NM

VI D

UAL

LY

TIED

*.-

Be

n

FLO

W C

OM

PUTE

R

SQU

ARE

RO

OTS

(C

ARb4

)

*%I1

3 AC

f’lN

S SI

SV

PIN

2

O\l

PIN

%

-15’

4 PI

NS

FLO

W

CO

MPU

TER

PR

ECIS

ION

+

IOV

, R

EFER

ENC

F SC

t4EM

RTl

C

DtA

GKR

Iq

(ChR

D5)

-r-P2

+mJ

@

I

ov

@---

I *

II +I

OVf

IH

V

ov

G+

l8M

A AD

270

0~

ov

@-

7 4

ov

@

II +k

Jv A

lHV

AD27

00L

7

-

BNC

0 0

1 I

I I

I

I - b 0 P

OT S

E..

T.P.

&

PO; p

”ET

. .

b Pi

n SE

T T.

P.

2 i

2OK.

l%

S , W

C

+15v

o\

l -IS

V

bbd

1. Report No.

4. Title and Subtitle

FULL-COVERAGE FILM COOLING: S-DIMENSIONAL MEASUREMENTS OF TURBULENCE STRUCTURE AND PREDIC’TION OF RECOVERY REGION HYDRODYNAMICS

7. Author(s)

S. Yavuzkurt, R. J. Moffat, and W. M. Kays

9. Performing Organization Name and Address

Stanford University Stanford, California 94305

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, D. C. 20546

15. Supplementary Notes

..__ ----. -_-- 3. Recipient’s Catalog No.

~ -- 5. Report Date

March 1979 6. Performing Organization Code

8. Performing Organization Report No.

HMT-27 10. Work Unit No.

11. Contract or Grant No.

NAS3-14336 13. Type of Report and Period Covered

Contractor Report 14. Sponsoring Agency Code

___

Final report. Project Manager, Raymond E. Gaugler, Fluid System Components Division,

NASA Lewis Research Center, Cleveland, Ohio 44135.

16. Abstract

Hydrodynamic measurements were made with a triaxial hot-wire in the full-coverage region and the recovery region following an array of injection holes inclined downstream, at 30’ to the surface. The data were taken under isothermal conditions at ambient temperature and pressure for two blowing ratios: M = 0.9 and M = 0.4. (M = ojetUjet/o,Um where U is the mean velocity and p is the density. Subscripts “jet” and “a” stand for injectant and

free stream, respectively. ) Profiles of the three main velocity components and the six Rey- nolds stresses were obtained at several spanwise positions at each of the five locations down the test plate. A one-equation model of turbulence (using turbulent kinetic energy with an algebraic mixing length) was used in a two-dimensional computer program to predict the mean velocity and turbulent kinetic energy profiles in the recovery region. A new real-time hot- wire scheme was developed to make measurements in the three-dimensional turbulent boundary layer over the full-coverage surface.

22. Price-

A09

NASA-Langley, 1979

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Boundary layer; Heat transfer; Film cooling; Unclassified - unlimited

Computer program; Turbulence; Numerical STAR Category 34

19. Security Classif. (of this report1

Unclassified

I 20. Security Classif. (of this page)

Unclassified 21. No. of Pages

187

* For sale by the National Technical InformatIon Service, Sprinefleld. Virginia 22161