An experimental study of the thick turbulent boundary layer near the tail of a body of revolution

OF THE THICK TURBULENT BOUNDARY

LAYER NEAR THE TAIL

OF A BODY OF REVOLUTION

by

V. C. Patel, A. Nakayama, and R. Damian

Sponsored by

General Hydromechanics Research Program of the Naval Ship Systems Command Naval Ship Research and Development Center

Contract No. N00014-68-A-0196-0002

uHR Report No. 142

Iowa Institute of Hydrahulic Research The University of Iowa

Iowa City, Iowa

January 1973

ABSTRACT

Petalled rneasurements of pressure distributiozs, _{mean velocity}

profiles and Reynolds stresses were made in the _{thick axisymmetric}

turbu-lent boundary layer near the tail öf a bàdy of revolutiOn. _{The results}

indicate a zuthber of' important difference _{between the behavior of a thick} and a thin boUndary layer. _{The thick boundary layer is charaöterized by}

significant variations of static pressure across it ànd an abnoialÏy low

level of turbulence. _{The static pressure variation is associatéd with}

a

strong iñteraction between the boundary layer and the ötential flow

out-side ît, while

the

_{ähanges in the turbulence structure appear to be a}

consequence öf the transverse surface curvaturi. _{tn order to predict the}

behavior of the flow in the tail region of a body of revolUtiOn it is not

theÑfore possible to _{use cQntrentional thin-boundary-layer calculation}

procedures.

ACKNOWLEDGEMENTS

The authors wish to thank Professor L _{Landweber for introducing}

them to the problem. _{The authors also aóknowlege the assistance offered}

by Professor J.R. Glover in the use of' hot-wire anemometers, andby Mr.

Dale Harris, and his workshop staff, _{in the construction, of' the}

experi-mental apparatus.

1his relort is based _{upon research conducted under thê General} Hydroinechanics Research Program of the Naval _{Ship Systems Command.,}

teäh-nicaLly adinistered by the Naval Ship Research and Development Center, under Contract NOO011-68-A-0196-OOo2.

?agé no.

INTRÖDtJCTION

_.1

EXPERThTAI. ÁRRANGEIv.T AND _INSTRUMENTS

. 3

A.

Wind TuineJL

. 3

B..

_{Model, Mountings and Alignment}

. 3

C.,: Traversing Medhanism

D.

_{Measuring Instruments}

5

SO OBSATIoNS FROM

PRELIMINARY EXPERIMENTS ₆

MEAN-FLOW MEAStJRENTS

. 9

V..

_{ANALYSIS OF THE MEAN-FLOW LEAStffiEMENTS}

.

.

U

Böunda±y Layer Growth

_li

Static PrëssureDjstrjbuti

.

11

C

Velocity Distributions

.

12

The Mean-Flow Stréaljnes

13

Integrai parameters from

_{Velocity Profiles}

_f4

Wall Shear Stress

17

VI.

_{ASUREIvflTS OF THE REYNOLDS STRESSES}

VI.

CONCIAISIONS

20 ERENCES

22

APPENDIX: TABLES OF EXPERIMEW2At, _DATA

23

FIGURES

32

Table 2.

LIST, OF TABIS

Table i. _{Pressure Vaiation at}

y = and y O, and

Integral Parameters

Profiles o _{Pressure Distribution, Meaz}

VelOcities and Reynolds Stresses

Page no. 2 (a) (b) (e:) Cd) X/L = X/L = X/L XIL

0.662

o.8o

0.85

0.90

25

26 27

28

(e) (í) X/L X/L = 0.93

0.96

₃₀

29 (g) 'X/L

0.99

31

Page no. Figure 1. _{Model and Traversing Mechanism}

32

Figure

2.

_{Measuring Stations and Notation}

33

Figure

_3.

_{Variation of Total and Static Pressure}

Coefficients ₃₄

Figure

4.

_{Variation of Boundary Layer Thickness} and

Local Radius of the Body ₃₅

Figure

5.

_{Variation of Wall Static Pressure}

and

Pressure and Velocity at the Edge _{of the}

Boundary Layer

Figure

_6.

_{Mean Velocity Profiles}

37

Figure

7.

_{Variation of Skin-Friction}

Coefficient

₃₉

Figure

8.

_{Mean-Flow Streamlines Computed from Velocity}

Profiles

39

Figure 9. _{Variation of Shape Factors}

_Using

Different

Definitions

Figure 10. _{Variation of Momentum Thickness}

_Using

Different Definitions

Figure 11. _{Longitudinal Velocity Fluctuations,}

Figure

12.

_{Normal Velocity Fluctuations,}

Figure 13. _{Transverse Velocity Fluctuations,}

Figure

14.

_{Variation of Reynolds Shear Stress,}

Figure

_15.

Comparison Between the Turbulence Measurements at

X/L =

0.662

and the Data of Klebanoff

₍₁₉₅₅₎

Figure

16.

_Mixing

_{Length Profiles}

Figure 17. _{Eddy Viscosity Profiles}

iv 36

40

40

4'

42 43 414 45 146

46

AN EXERINTAL STUDY OF THE _{THICK TURBULENT BOUNDARY}

LAYER NEAR THE _{AIL OF A BODY OF REVOLÙ'IOÑ}

I. _INTRODUCTION

A number of previous studies _{haVe shown that, when}

the thickness of the boi.u4ary layer on a body of revolut-ion becomes

of the same order as,

the local radius of the iody _{the influence of transverse (or}

lateral)

curvature, which is usually neglected in thii boundary-layer _theory,

be-comes appreciable.' Such a situation arises in the _{case of the boundary layér}

developing on a long slender _{cylinder placed axially in a strea.}

Th'is

particular co±ifiguratjo _{has been examined}

experimentally and' thèoreticaliy

in some detail in previous _{investigations.}

In his récent papeP Cebeci

(1970)

_{has reviewed the previous stûdjes in both lamirar}

and turbulent boundary layers, and also, presented, the results of his _{own calculations}

using finitejfference _techniques.

For laminar' flow the situation appears

to be quite satisfactory _{insofar as Cebeci's}

numerical results cover _{a wide}

range of Conditions and also _{show substantial}

agréément with the results of

other, not so extensive, _{analytical studies.}

In the case of the _turbulent

boundary l'ayer Cebeci _{employed an eddy_viscosity}

model 'with the additional

assumption that this model is

Îlot directly affected by transverse curvature.

Thus,, the

nf1uence of curvature is taken into account only _{in the mean-flow}

momentum and continuity _equations..

While the results of _{these calculations}

show a plausible effect of trns,verse

curvature, and. also agree with the

experimental results of Richmond

₍₁₉₅₇₎

_{and Yasuhara.}

(1959),

the assumption

that 'the same _{eddy-viscosity model applies}

to both thin 'and thick _boundary

layers remáins to be verified directly sincé it _{implies that the}

turbu-lent motion itsélf is not explicitly influenced _{by trnsverse}

curvature.

In order to demonstrate the validity of such _{an assumption it is'of}

course

necessary to make detailed

turbulence measuréments in _{thick axisyinmetric}

boundary layers. _{Such measurements have not been reported}

so far,. 'From

the recent study of Patel

₍₁₉₇₂₎

_{it appears that}

the use of an universal

distribution in the wall region of a thick axisymmetric boundary layer. More experimental data are, however, needed to elucidate the turbulent motion in the outer region.

The cylinder problem cited above is ideally suited for studying the influence of transverse curvature on the development of the boundary layer since the absence of pressure gradients in this case enables one to

isolate the curvature effect. In many practical situations, however, sig-nificant transverse curvature effects occur in conjunction with streamwise

pressure gradients, and possibly with longitudinal surface curvature. Well

known examples of this type of flows are the boundary layers in the

mid-section of conical and annular diffusers. Another, equally important, case

is the flow in the tail region of a body of revolution. In this case the

boundary layer over the fore part of the body is thin and can be treated by conventional techniques, but if the body is sufficiently long and slender,

the boundary layer in the tail region will grow to a thickness which is

comparable with, or even much larger than, the local radius of the body.

The present study deals with this latter case. It will be obvious that here

we have a case in which both transverse curvature and pressure gradients may

dominate the flow. As we shall see later, this flow exhibits several other

features of interest: amongst these being a strong interaction between the

boundary layer and the potential flow outside it.

This paper describes mean-flow and turbulence measurements in the

thick axisymmetric boundary layer on a body of revolution. A parallel

theo-retical study was undertaken to develop methods for calculating such a thick boundary layer and also to investigate the possibilities of calculating,

simultaneously, the potential and boundary-layer flows, allowip the two to

interact. The results of these sudies will be published separately.

In the literature there are several publications which report meas.

urements on boundary layers in conical diffusers and on bodies of revolution

but none of' these was made with a view to study the interaction phenomenon just méntioned. Moreover, there is little information concerning the turbulence in a thick axisymmetric boundary layer. In the case of the body of revolution,

the interaction is important since the pressure distribution on the surface

sepaatean,-where. _{The pressure recovery at tie tail(or,,}

the prèssure

variation through the boundary layer at _{the tail) are obviously needed}

to

obtain better estimates of the thtal drag _{experienced by the body. The}

present experiments were designed specifically _{to study these aspects of}

the flow.

II.

EXPERI1vNTAL ARflA1JGEIvtENT MID INSTRUMENTS

Wind Tunnel.. _{The experiments were conducted in the largest}

closed-circuit wind tunnel of the Iowa _{Institute, of Hydraulic Research.}

The working-section of the tunnel is 214 feet _{long with a crós-.section in the}

form of a 5-foot octagon pròvided

by throating' a 12-foot squarè approach section. _{Although the maximum speed attainable in the tunnel is of the}

order of 90 ft/sec, all the measurements repOrted here weremáde at _{a nomi-'}

nal speed of 4O ft/sec in order to minimize the influence of _{tunnel vibratIon.}

Model, Mountings and Alignment. _{A six-to-one prolate spheroid,}

5 feet long and lO inches in maximum _{diameter, used. previously by}

Chevray.

(1969) to study axisylnmetric _{akes, and by}

atÏja (1971) to make preliminary investigations of axisymmetrjc boundary layers, was used in the present

ex-perimen-ts. _{Two rows of cylindrical lucite. beads,}

1/14 inch long, were attached

with a spacing of _{1/14 inch ata distance of 3 inches}

from the nose to ensure

early transition of the boundary _{layer. In order to avoid separation}

ahead

of the tail, Satija modified the original sheroid used _{oy Chevray by}

trun-cating it at a seëtion 2 inches _{from the tail and glueing}

a conical tail piece

such that the slope of the _{surface remained continuous}

at the junction. With

this modification the overall _{length of the. model became}

62.114. inchés. The

model i _{shown schematically in Figure 1.}

The model was mounted in the wind tunnel by _{means of eight 0.0141 inch}

diameter steel wires .intension, four at each end of the model. _{Each wire}

was provided with a screw-coupling

such that its length could be _easily

ad-justed and the model located _{at any desired position.}

In order to Obtain axial symmetry of the boundary ]ayer _the

geo-metric axis of the model

about 2-1/2 inches from the tail. Small adjustments were then made in the

lengths of the supporting wires until, the total pressures recorded by the three tubes became equal. This procedure was repeated by placing three

simi-lar tubes at about the mid-section of the model. Finally, six Pitot tubes

were used, three at each of the two sections, the tubes downstream being

offset by 60 degrees relative to those upstream. Only minor adjustments

were required in this final stage to obtain agreement between the readings of

the three tubes at each section. Since the readings of the Pitot tubes may

be regarded as a measure of the wall shear stress, and consequently of the

velocity distribution in the wall region, this procedure ensured axial syinme-try of the flow.

As an additional check on the axial symmetry, the axial components of mean and fluctuating velocities were measured by means of a hot-wire

ane-mometer in the wake of the body 3 inches downstream from the tail. Although

the measurements were made in only one plane across the wake, the distribu-tions of mean velocity and the root-mean-square values of the fluctuating velocity were found to be closely symmetrical about the geometric axis of

the model. From these measurements it was concluded that the boundary layer on the model was axially symmetric to a sufficient degree of accuracy.

C. Traversing Mechanism. The essential features of the traverse

mechanism used. in the experiments are shown in Figure 1. The mechanism con-sisted of a rigid rod with three degrees of freedom in the horizontal plane

passing through the axis of the model: transverse motion along the length

of the rod, transverse motion along a slide situated outside the tunnel, and

rotation about a pivot on this slide. The mechanism was thus capable of

traversing measuring probes, such as Pitot tubes and hot-wire probes, mounted on the end of the rod by suitable brackets, in the direction normal to the

model surface at a number of discrete streainwise locations on the model. The

length of the slide was such that the most upstream point on the model which

could be investigated was about 66 percent of the model length ñom the nose.

the othér end, measurements could be made right up to the tail of the _ode1. The main rod of the träverse mechanism was provided with a screw drive, a

scale and .a vernjér s.o that the _{normal distance of the probes from}

the model

surface could be adjusted and measured _{from outside the tunnel with.a}

reso-lution of 0.001 foot. _{The rod éntered the tunnel through a narrow slit ôut} out. of the tunnel wall. _{The portion of the slit not}

occupied by the rod was sealed b _{a rubber sheet to prevent any leakage of}

air from the tunnel.

The traersing rod had tO be made rigid enough tò prevent it from

vibrating in the wind. _{In order to minimize the interference between the} traversing mechani.sm and the _{'low being measured the}

probes were attached

such. that the distance between _{the probe tip and the traversing}

'od

was as large as pòásible.

D. _{Measuring Instruments. AU total (Pitot)}

and static pressures

were measurea using probes of standard design, made from hypodermic _tubing

of outside diameters 0.050 inch _{and 0.065 inch respectively, in}

conjunction

with micro-manometers capable of resolving pressure differences of the order

of 0,001 inch alcohol _{The air temperature and}

pressure ii. th tunnel were

measured iiediately downstream of _{tue contraction.}

These, together w±th

the barometric pressure and _{drj- and wet-bulb teùiperátures}

in te labòratory,

were used to find the density and viscosity óf the air aswell _{as to correct}

the manometer readings in the manner suggested by Naudascher (l96).

Mean velocities and the Reynolds stresses within the boundary layer

on the model were measured by means of single- and cross-wire probes _using

the twO-channel, _{constant-temperature,. "Old Gold Model,.Type}

4-2H Hot-Wire

Anemometer" and "Type 2 _{Mean-Product Cbxnputer", designed}

and manufactured by

the Iowa Institute of Hydraulic _Research.

These instruents are built with

all solid-state, electronic _{componeñts and equipped with a linearizing}

cir-cuit for each hot-wire channel _{and an analog_to_frequency}

converter. Other features of the desigi. and 'prïnciples _{of ope'ation of this anemometer}

ystem

have.been. described by _{lover (1972).}

The single-wire and cross-wire probes

used in the measurement of _{mean, and fluctuating., velocity}

components were all

made from copper-plated. tungsten wires _{of nominal diameter 0.000113 inch}

Preliminary experiments conducted at the beginning of the present study, as well as those made earlier by Satija, indicated an unrealistically

low level of turbulence in the boundary layer on the spheroidal model. After

a considerable amount of experimentation, the origin of this anamoly was

traced to an inadequate frequency response of the hot-wire anemometer. Once

this was discovered, modifications were made in the amplifier circuits to im-prove the frequency response and at the same time to cut down the noise level.

The necessary modifications were relatively minor. In order to ensure that

the modified system performed adequately, a number of tests were conducted

in fully-developed turbulent pipe flow. A 2-inch diameter, 30 feet long

copper pipe was constructed for this purpose. Measurements with Pitot tubes,

and single-wire and cross-wire probes, were then made a few inches upstream

of the pipe exit, and the results compared with those of Lauf er (195)-i). Satisfactory agreement was obtained with the data of Lauf er as regards the

distributions across the pipe of mean velocity, the Reynolds shear stress and the turbulent normal stresses (i.e. the mean-square values of the three com-ponents of velocity fluctuation).

In addition to providing a check on the performance of the hot-wire anemometer, the pipe experiments also served to highlight the problems, such as thift in the calibration curves and the necessity to have identical cali-bration curves for both wires in cross-wire operation, associated with the

use of the instruments. These tests also suggested the techniques to be followed in subsequent experiments in order to obtain reliable and consistent

data.

III. SOME OBSERVATIONS FROM PRELIMINARY EXPERIMENTS

Exploratory measurements in the boundary layer on the spheroidal model were first made by traversing the total-head and static-pressure tubes

separately across the boundary layer at a number of streaniwise stations rang-ing from about 66 percent of model length to the tail. Both tubes were oriented

parallel to the model surface. These measurements indicated, amongst other things, that the boundary layer remained attached right up to the tail and that

òn, it

that thé

parallel T

large variations of static pressure occurred across the boundary layer _over

the last 10 percent of the model _{length This variation in static}

pressure

was accompanied by a dramatic increase in the thickness of the boundary

layer.

Before describing the detailed measurements which were made later

is convenient to discuss briefly _{the consequences of these early}

ob-sextrations. _{The rapid thickening of the}

boundary layer and the large variations in the static

face. of coursé indicated

layer did reain closely

pressure in the direction normal to the body sur-mean-flow streamlines withinthe boundary

to the surface. _{Alternatively, the} com-ponent of' velocity normal to the

waIl could no longer be assuned to be an

order of magnitude smaller than the cömonent parallel _{to the wall, as is}

the case in thin boundary layers. _{A number of inpòrtant conclusiòns}

per-taining to the measuring techniques _{follow iediatély:}

(i) Since the direction _{of the mean-flow streamlines}

change

Continuously across the boundary layer, a simple static-pressure _probe,

mounted parallel to the body surface, cannot be relied upòn to give an

accurate measure of the locàl static _{pressure, especially at larSe}

dis-tances from the surface.

If suòh aprobe is to be used with any success its orientation must be changed

continuously to coincide with the local

strèam.line direction, which is not _{known a priori.}

. (ii) A properly designed

total-head tube can be made _insènsitive

to yaw angles up to 30 degrees

so that it canbe used in the present situa

tion without incurring _{appreciable errors.}

Owing to the difficulty _{of measuring static}

pressure men-tioned above, the Pitot-static

combination cannot he uséd to measure the velocity distribution across the bundary layer.

In order to measure the velocity profiles in thetái]. _region

it is therefore necessary to use hot-wire techniques. _{A single hot-wire}

probe, traversed normal to _{the body surface will,}

however, give the

distri-bution of. the resultant _{velocity through the boundary}

layer but nòt. the angle which the velocity, _{vector makes with the}

surface of the body.

and.

can in principle be determined by taking the difference between the total-head measured by a Pitot tube and the dynamic pressure recorded by means of

a single wire. As we shall see later, this procedure is not altogether

satis-factory, although consistent results can be obtained when due care is taken. To specify the velocity field in the tail region of the body completely, it is necessary to measure the components of mean velocity along

and normal to the body surface. This can best be accomplished by traversing a cross-wire probe in the direction normal to the surface.

The use of cross-wire probes to measure mean velocities has the additional advantage that the Reynolds normal and shear stresses can also be measured at the same time.

The above conclusions contributed in large measure to the final

procedures adopted for making the measurements reported below. Traverses

of total-head and static-pressure tubes, and single-wire and cross-wire

probes, were made at seven streaniwise stations on the model, namely X/L =

0.662, 0.80, 0.85, 0.90, 0.93, 0.96

and 0.99, where X denotes the distance

from the nose of the model measured along the axis and L the overall length

(= 62.l1

inches) of the model. All measurements were made at a nominal unit

Reynolds number of x l0 per foot.

The relative positions of the seven measuring stations and the

no-tation adopted for the presenno-tation of data are shown in Figure

2.

The

com-plete model is described by the radius distribution

ri(

_x)4

x

-i:- =

-

.)j

O <

< 0.9333

= o.I333(l -

_),

0.9333 <

< 1.000

The angle between the tangent to the surface and the axis of the model is

_4.

x and y are curvilinear coordinates measured along and normal to the surface,

respectively. If r is the distance of a point from the model axis, it follows

that

r=r +ycOs

The componénts of mean vlocity

along x and y are' U and V, respêctjvely.

The resultant velocity at _{any point is denoted by Q,}

so that Q (u2 +

V2).

The. velocit3r fluctuations will _{be denoted by loer-case}

letters. _In addi-tiori to these, we have the

total pressure P, the static pressure p, density

p, and

_kinematic

_viscosity

Subscripts w and _{will be used to signify}

values at the wall (y=o) _arid

_.t

_{the edge of the boundary}

layer ('y=s),

respectively. .

IV. MEAI FLOW MEASUREIV1EIÇ

The total-head (Fitot) tube described earlier _{was first tested for}

insensitivity to yaw and then traversed through the boundary _{layer, keeping}

it

parallel to. the body surface,, at each of the sevén

streamwise

easuring

stations.

The total pressure

_{was referended to the static}

pressure (Pe)

in the tunflel just downstream

from the contraction and _{made dimensionless by}

dividing by the tunnel dynamic _{pressure (pU2f) which}

was held constant in all tests.. The measured variations of the tOtaL-pressure

coefficient,

=

_(P-Pf)/PU2

_{are shown in FiSure 3.}

Since, CF becomes öönstant

with y

_{outside the boundary, layer,}

these measurements were used to estimate

the boundary layer thickness.

_{The variation of}

_{and the ratio}

6/r, with

strea.mwjse distance i

shown in.Figúre i.

Similar measurements were then made using the

static-pressure tube

and again the data were redered dimensionless _{using the same reference}

con-ditions as before. _{The'varia-tions of the}

static-pressure coeffiòjent,

Cp

refef'

are also shown in Figure 3. _{As indiöated in the last}

section,, these static

pressure measurements may be: _{suspect in' 'the outer}

regions of the boundary layer over the last 10 _{percent of the ]ength of the}

model 'due to the yaw sensitivity of the probe. _{They are, however, included}

hère for comparison with

iesults obtained by _{an alternative method. ince}

the measurements close to the' wall are not _{in doubt even up to the tail (the}

flow being loöally _{parallel to the wall),}

the streamwise variation _{of static}

pressure on the model Surface, _{C, was determined by}

ctràpòlating the

measurements to the wall. _{A plot of' C}

from which it win be seen that the pressure gradient on the waÏÏ is adverse

all the way from the first measuring station to the tail. The variation of

static pressure at the edge of the boundary layer, indicated by the

static probe, aligned with the model surface, is also shown in Figure 5. Although the accuracy of the results for X/L greater than 0.93 may be questionable, the figure indicates clearly the large static pressure

var-iations across the boundary layer near the tail of the model. The other

results shown in Figure 5 are described later on.

From the total and static pressure distributions described above,

it was possible to calculate the profiles of total velocity, i.e., Q/Q6.

These are shown in Figure

6.

A single hot-wire probe was then traversed through the boundary

layer at each of the seven streainwise stations. The wire was held normal

to the flow and parallel to the model surface. In these, as well as other

hot-wire measurements, care was taken to ensure that the drift of the wire

calibration curve was small. Those runs in which the calibration drifted

by unduly large amounts were repeated. The profiles of total velocity

measured by means of the single wire are compared with those obtained from

the Pitot-static measurements in Figure 6. The values of

Q,

the velocity

at the edge of the boundary layer, recorded by the single-wire probe are

shown in Figure 5. Also shown in this figure is the pressure variation along the edge of the boundary layer, C, implied by the measured values

of and the constancy of total pressure.

As mentioned in the last section, the total-pressure profiles mea-sured. by the Pitot tube (shown in Figure 3) and the total velocity profiles

measured by the single-wire probe were used to infer the static pressure

dis-tribut.ions through the boundary layer. The results of this exercise were not

altogether satisfactory primarily due to the fact that the static pressure

came out as a small difference between two relatively large quantities,

neither of which could be measured with the required accuracy. Some

consis-tency was, however, obtained by smoothing the data before taking the

dif-ferences. The resulting distributions of static pressure are compared with

11

In order to determine the _{components of mean velocity, U and V,}

along and nOrmal to the model _{surface, respectively,}

a cross-wire probe

was traversed across the boundary layer _{at èach of the seven measuring}

stations. _{A probe with the propér geometic ànd calibration}

characteris-tics was built àfter several trials.' The wires _{were located in the plane}

normal to the surface and _{thé flow.}

The results of the cross-wir

measûre-ments were converted to profiles _{of U/Qe, V/Q ad Q/Q6..}

These profiles are

also showi in Figure

_6.

Finally, a preston tube of outside d±ameter 0.1224 inch _{was used,}

in conjunction with the calibration curve ofPatel

₍₁₉₆₅₎

_{to determine the}

wan shear stress,,1. _{A plot of the skin-frjct±on}

coefficient, C. =

is made in Figure 7.

V. _{ANALYSIs OF THE MEAN-FLOW}

MEASURE1tENTS

Boundary Layer Growth.

From the variation of' boundary layer

thickness shown in Figure 4 _{it is clear that up' to about 85 percent of the}

body length the boundary _{layer may be regaided.}

as thin ±nsbfar as _{is muòb}

smaller than r, the local _{radius of the bOdy.}

Over the last 15 peròent _of

the body length, however, _{thin 'bounda.ry-1ayer theory}

_will

cease to appir

and transverse curvature _{effects are èxpected to}

play a dominant role in the

behavior of the flow.

Static_pressure Distribution Although, as, Indicated _earlier,

the' static pressure

distributions Obtained by the two methods' may be su5ect

as far as accuracy in the tail _{region of tie body 'is}

conôer,ned., the results

shown in, Figures 3 and 5 confirm the observatiOn thade _{above. The static'}

pressure rèmains substantially

constant through the boundarJ _{layer right'}

up to X/L =0.90. _{From Figure 5 we see thai up to this poiiit thepressure}

at the edge 'of the boundary _{layer is somewhat larger}

than that at the wall.

In addition to the _{well known 'influence' of the}

iormal, Reynolds _tress ,

this increase in pressure with distance from the _{wall may partly bé}

attri-buted to the convex _{curvature, 'of the .méan.flow}

stréanij.júes associated with

¶he deórease in statiè

ressure from the wall towards' the edè

öf the boufldary layer in the tail region òf the body (x/L> 0.90), on the

other 'hand, appears tO be a cönsequênáe of the concave curvature of thé'

'ëan-flow' streamlineS associated prithàriiy

th the rapid thickening of the

boünd-a.ry layer .

The chäñe from thê thin t'o the thiòk boundary layer behavior

seems to take place in 'the region of thé X/L

0.90

Eation where cS/r0 i

approxiina'tely O.62

'

The rapid inci'se in the thcknes df the boundary layer near th

tail of the body may of course be regarded as a direct consequence of the ever

increasing influence of transverse curvature, but the large variátion in

static pressure across the boundary layer associated with his thickening

suggests that thre is a stro

iñtéractiòn between the boundary laye±'

flow and the potentiäl flow outside it.

Owing to the presence of the thicik

bourdary layer, potential flow theo±ies can no longer be expected to predict

the tres sure field in the tail region correctly.

At the same time., it is

unlike1y that usual thin boundary-layer theory, which asses cònstant

static pressure in the direction normal to the surfäce, can adequately

prè-diet the bou ndar

layer behavior even when the. experimentally determined wall

essure distribution is prescribed. Thus, it appears that any ratIonal

theory describing the flow in the tail region of a body of revolution with a

thick. boundary laye,r must attempt to solve the potential flow and the boundary

layer flow simultaneously.

C

Velocity Distributions

Referring to Figure 6, it will be seen

-that the profiles of the total velocity, Q/Q, measured by the three diffe±'ent

methods, namely Pitot and statiò tubes, single hòt-wire, and cross-wire probes,

are. in reasonable agteement at' all streamwisé stations.

ilthough ve had

exp.êcted to observe systematic' differences between the Pitöt-static and the

hot-wiré results. in the oiter parts of the boundary layer near the tail of

the:bor,oing tohe yàw sensitivity of the static-pressure tube, Figure

6 indicates thát any differenòe

which tháy é±ist are swamped by the general

scatter of the datä.

Detailed' calculatiöns 'iñdicated that the expected

differences were too small to be distinguished from experimental scatter

since, the érrors incurred in the .easurement of static pressure were

were calculated. _{The agreement between the results of three different}

instruments in a complex f10 _{situatiö± suh as this}

was cônsidered. very encouraging.

The profiles of the _{longitudinal and normál}

components of mean

velocity measured by means of the cross-wire probe

clearly shoi the

differ-ence between the thin and the thick bbundar layer. Up to X/L =

_0.90

thè

normal component of velocity _{is seen to be small}

compared with the longitudinai

component, as required in thin _{boundary-layer theory.}

By X/L

_0.99,

_however,

the normal component of velocity is almost 32 _{percent of the. longitudinal}

cornponent at the edge of the toundary layer. _{Apart from indicating}

the-breakdown of the usual thin

boundary-larer assumptions in the tail region,

these results confirm the observation made _{earlier regaiding the rapid}

divergence of the _{mean-flòw.streaines in planes}

normal to the surface. Indeed, the angle bétween- _{the surface and the}

streàthljne passing through

y & at XIL =

0.99

_{is tan1O.32 =}

_i8.o

_degrees.

Since the angle between

the tangent to the body surface at this point _{and the body axis is cos1}

0.9?

23°,

_{this indicates that the flow nearthe édge of}

thé bdundary layér

is more nea'ly parallel to the axis than _{the body surface.}

The general shape of the

vélocity profile at the last. measuring

station shows that the flow theie

is close to separation. The profilé at

the most upstream _{station, X/L =}

_0.662,

was found to òonfórm well with _the

two-parameter veloòity_profi1e family of Thompson

₍₁₉₆₅₎

indicating that

the boundary layr there has essentially _{the same characteristics}

as one developing. on a piane twO-dimensional surface.

D _{The Mean-Flow Streamlines}

The longitudinal _{components of}

velocity measured by means of the cross-wire

probe, namely U, were used _to

compute the distributions-of _the

trearn function., _{b, at each streamwise}

measurin _{station,, using the definition:}

fi.

ip(y)

= j

Urdy

j

U(r+y cos 4)dy.

o o

Figure 8 shows the meanf1ow

Streamlines within the bounda'y layer

deter-mined from these _{distributions of}

. It

will

be sen that the

st1eam1ines

are convex and nearly parallel t _{the surface in the}

and

where the boundary layer is thin, and concave and divergent over the last

lO percent of the body length. Figure 8 thus verifies directly the obser-vations made in section V-B simply on the basis of static pressure variations.

The angles between the streamlines and the body surface obtained from Figure 8

were found to be in good agreement with those deduced from the direct

measure-ments of the normal and longitudinal components of velocity. This may be

re-garded as a check on the axial symmetry of the boundary layer.

Figure 8 shows yet another interesting feature of the thick

boundary layer near the tail. From the near coincidence of the edge of the

boundary layer with a mean-flow streamline it may be concluded that the en-trainment of free-stream fluid into the thick boundary layer is small.

E. Integral Parameters from Velocity Profiles. The thickness of

the boundary layer at each measuring station was determined from the

total-pressure profiles measured by means of the Pitot tube. Thus, was defined

as the normal distance from the wall where the total pressure became

_0.99

times the constant value in the free-stream. This definition appears to be

the most appropriate one here for two reasons. First, it gives a unique

value öf even when the static pressure, and therefore the velocity, vary

with distance from the wall outside the boundary layer. Secondly, it reduces

to the usual definition of 5, as being the distance where the local velocity

is 0.995

times the free-stream velocity, when the boundary layer is thin and the static pressure substantially constant across it.

For an axisyinmetric boundary layer there appear to be a number of

différent ways of defining integral parameters such as the displacement and momentum thicknesses.

Perhaps the most meaningful definitions are the ysical definitions:

r +5*cos

r +-os

10

1 27rrpUth o 2irrp(U-U)dr (1) r cOS r 6cos4

Jo

2,rrptr2dr = J o r r o o (2)

and

and.

so that

where U is the velocity distribution if the flow _{were potential right up to.}

the wall, is the physical mass-flux deficit thickness, _{is the momentum-.}

flux deficit thickness and, as noted before, r r + .y cos . .The. shape

factor of the velocity profile and. the momentum-thickness _{Reynolds number of}

the bqundary layer may then be defined as

= _and

respectively. _{The displacement thickness defined in this 'manner gives the}

physical displacement of the exte±naì flow streamlines _{due to the presence}

of the boundary layer, while the momentum thickness _{s closely related to}

the drag experienced by the body. _{The evaluation of these thicknesses from}

measured velocity profiles is, however, made difficult by the fact that

nothing is known about t.he variation of the _{velocity distributiön U in the}

hypothetical potential flow over the distance _{occupied by the. boundary layer.}

if it is assumed that U _{remains constant., and equal to U, over this distähce,}

then equatiöns (i) and (2) siplify to yield

's

+ .;cos 4))

=

+

cos 4))'

=..

Jj(.J.

-

L.

dy

If the axisymmetric boundary-layer _{equations are integrated across}

the layer i the usual manner to obtain, the _{momentumintegral equation, it}

is found that the integrals _{on the right-hand-side of equations ()4) and ('5)}

arise quite naturally. In almost _{ll previous stúdiés of açisymmetric boundary}

layers, thereföre, the displacement and _{moenturn thicknesses have,been d.efined}

simply as.

-15

l-t

p-dy

o

H=

and R0

Q662

(8)

We shall refer to these as the usual axisyinmetric definitions. A major advantage of using these is that they enable the momentum-integral equation for axisynetric flow to be written in a form that is simple and very similar

to that for plane-surface boundary layers. As we shall see later, however,

these definitions lead to some anamolies when the boundary layer is thick in

comparison with the local radius of the body. From the expressions given

above it will be clear that the usual axisyinmetric definitions are related to

the physical definitions (using u u6) by the formulae

6=6(l+cos)

and 6

=6*

--cosc)

2 2 (1 r 1 2 (5* (lo) o

Finally, if one is interested only in describing the shape of the velocity profiles, without regard to the geometry of the surface, one can determine the thicknesses using the usual definitions:

1=J

(5

(1--)dy,

(11) o

2=J

6 ...L!_(1_ _{) dy} 6 o

Q62

=T'

and )

We shall refer to these as the planar definitions for obvious reasons. It will be clear that the physical definitions as well as the

usual axisyinmetric definitions reduce to the planar definitions given above

when the boundary layer is thin, i.e. when 6 « r. For thick boundary

layers, however, the numerical values of the various integral parameters

calculated using the three definitjons are considerably different.

T

at each streawise station using all three definitions given above.

_{In te}

calculation of the thicknesses given by tite physical definitions, howevei, the

assumption of U was notj used. Instead., use wasmadeof the distribution

of U implied by thé constancy öf total pressure and the observed variation

çf static pressure, since this would appear to represent the _{true variation}

of U morerealistically. _{The difference between thevalues calculated in}

this manner and those obtained using equations (9) and (io) was found to be

of the order of a few percent. Since the profiles of U/U _{and Q/Q were not}

substantiálly different at the first three measuring stations, _nämely

X/L = 0.662, 0.80 and 0.85, it was decided to use for these the values

of _{indicated by thé Pitot and static tubes so as to minimize the}

influence of the scatter in the hot-wire data. _{For the last four measuring}

stations, however, the profiles of U/Us recorded _{by the cross-wire probe}

were used. _{The results of these calculations aré presented in Figures 9} and. lQ.

From Figure 9 we see that the nearness to separation at the tail

of the body is indicated only by the large value _{of the shape parameter,}

f, which is based solely on the shape of the velocity profile. _{The other}

two definitions do not convey this important _{infOrmation. Examination òf}

Figure 10 indicates the large differences in the numerical values of the

momentum thickness resulting from the three alternative _{definitions. It}

is interesting to note that, when the _{boundary layer is muôh thicker than}

the local radius of the body (e.g. at X/L = 0.99), the usuai. axisyetric

definitions lead to the rathe± incongruous _{stuatión where the momentum and}

displacement thicknessés become larger than the physical thickness of the

boundary layer. _{(This is of course due to the ever-increasing factor hr0}

appearing in the definitions.) _{Considerable care iiiay therefore be required}

in choosing the most meäninfiil and appropriate definitions of the integral parameters when attempts are made to extend some of the more successful

integrai calculation methods to treat thick _{axisy7imletric boundary layers.}

F. Wall Shear-Stress. _{The values of the wall shêar-stress,}

measured directly b _{eans of Preston tubes, are compared. in Figure 7}

with thOse Obtained by applying the method of Clauser (1956) to the

pro-files of longitudinal velocity, U/Us. _{Also shown in the figure is the}

values of the planar parameters and. The disagreement between this formula and the Preston-tube and Clauser-plot results appears to imply that the well known two-parameter representation of velocity profiles, upon which the skin-friction formula of Thompson is based, may not adequately

describe the velocity profiles in thick axisyimnetric boundary layers. This observation was indeed confirmed by detailed comparisons of the measured

profiles with Thompson's profile family. It may be remarked here that the

use of integral parameters other than those obtained from the planar

defini-tions will not lead to improved correlation between experiment and

skin-friction and velocity profile relations commonly used in thin boundary-layer

analysis.

VI. 4EASUREMENTS OF THE REYNOLDS STRESSES

In the present experiments the Reynolds stresses and were

assumed to be identically zero on account of the axial symmetry. The

re-maining components of the Reynolds stress tensor, namely , , and

uy, were measured by means of cross-wire probes. The results were made

di-mensionless using the velocity at the edge of the boundary layer and are

shown in Figures 11 through 111.

The well known turbulence measurements of Kiebanoff (1955) in a flat-plate boundary layer are compared with the present measurements at the

most upstream station, X/L =

0.662,

in Figure 15. The small disagreement

between the two sets of data may be attributed largely to the small adverse

pressure gradient which exists at this statiòn and the uncertainties

asso-ciated with the determination of boundary layer thickness. Nevertheless, the

trends shown in Figure 15 indicate that the boundary layer at the most up-stream measuring station has the properties of a fully-developed, thin,

tur-bulent boundary layer.

Perhaps the most striking characteristic of the data shown in Figures

11 through i1 is the generally low level of turbulence in the thick boundary

9

towards separation the. velocity

fluctuations and the shear stresses are much

larger than those observed

_Ire.

_{From ti'}

measurements of shear stress and

mean velocities the distributions of mixing length and eddy kinematic

vis-cosity were determined using the usual rélations

- = L (.!L)2

= C _(i1)

These, are shown in Figures. 16 _{and IT. It. may be}

emarked here that the

variations of mixing length were also föund using an axisyetriò

defini-tion in the form

-

_{= Ij {- (Ur)}.}

(15)

The values of LA determined in this manner _{were found to be substantially}

lower than those shown _{in Figure}

_i6,

especially near the tail. _{In Figure}

16 a comparison is hade between the experimental

distributions of L and the

universal distribution used by Bradshaw, Ferriss and _Atwell

₍₁₉₆₇₎

in the

calculation of thin boundary layers. _Fro

Figures.16 and 17 it is clear that there is a systematic

and dramatic decrease in the mixing length and

eddy viscosity as the boundary layer thickness increases _{in relatjoü to the}

local radius of the srface.

In recent discussions of _{ênergy trá.nsport}

processes in thin

böund-.ry layers the mixing length is

often associated with a dissipation length

on the. assumption that the. _{production and dissipation}

of turbulent kinetic

energy are much larger than either

diffision or convection in _{the wafl region,}

and néarly balance _{each other. If such an interpretation}

is accepted for

the present case, the reduction in mixing length _{observed here implies that}

the rate o,f dissipation in

a thick boundary layer is larger _{than that in a} thin böundary layer.

this, coupled with the lower rates of _prothiction

re-sulting from the reduced Reyholds stresses, _{would appear to suggest that}

the

near equï1ibri _{between production and dissipation is no longer}

maintained

iti the thick boündary _{layer, and that increasedrate}

of dissipation must be

accompanied by increased rates _{of convection and}

diffusion. _{More detailed}

turbulence measurements

are obviously, needed in ord.ér to verify these observations.

VII. CONCLUSIONS

Perhaps the most useful purpose served by the present study is the collection of a complete set of experimental data in a hitherto unexplored

situation. This data can form the basis for further theoretical studies on

a number of aspects of turbulent boundary layer behavior. In view of this,

all the experimental results are reproduced in the form of tables in the

Appendix.

The major conclusions of this study may be suimnarized as follows:

The turbulent boundary layer on the conical tail of a body of

revolution thickens very rapidly. This thickening is accompanied by (a)

significant variations in static pressure across the boundary layer such that fluid elements further away from the surface experience less adverse pressure gradients than those nearer the surface; (b) a strong divergence of the mean-flow streamlines in planes normal to the surface, so that the normal velocity component cannot be neglected in comparison with the longitudinal component; and (c) a dramatic decrease in the Reynolds stresses, so that empirical laws established for turbulence behavior in thin boundary layers cannot be used, unmodified, for the prediction of thick boundary layers.

The static pressure variation across the boundary layer implies

an interaction between the turbulent rotational flow within the boundary layer

and the potential flow outside, with the result that neither can be calcula-ted independently of the other.

In order to calculate the development of the thick boundary layer, it will be necessary to include not only the direct effects of pressure

varia-tion but also the indirect effect of transverse curvature on the turbulence

as reflected in the decrease of mixing length and eddy viscosity.

The boundary layer calculation is made all the more difficult

by the fact that potential flow theory, which ignores the presence of the

boundary layer, can no longer be relied upon to predict the pressure field

21

flow in the tail region of a body of revolution

n'Íust therefore, oe accôna-pushed by aÍ iteative

proeeiire in ïhii potential flow azìd'boundar lyer

calculations are performed simultaneously _{Further discusio of the}

differ-ential and integrai

equations of thick, ax-isynetrjc boundár _{láyers, and the}

problems associated with their Solution, is _{given in a recent paper by Patel}

REFREN CES

Bradshaw, P., Ferriss, D.H., and Atwell, H.P.,

_1967,

"Calculation of Boundary

Layer Development Using the Turbulent Energy Equation," J. Fluid Mech.,

28, 593.

Cebeci, T., 1970, "Laminar and Turbulent Incompressible Boundary Layers on Slender Bodies of Revolution in Axial Flow," J. Basic Eng., Trans. ASME,

Ser. D, 92, 5)45.

Chevray, R.,

_1968,

"The Turbulent Wake of a Body of Revolution," J. Basic

Eng.,

Trans. ASME, Ser. D, 90, 275; Also Ph.D. Thesis, University of

Iowa, Iowa City, Iowa,

_1967.

Clauser, F.H.,

_1956,

"The Turbulent Boundary Layer," Advances

in

Applied Mechanics, 4, 1, Academic Press.

Glover, J.R., 1972, "Old Gold. Model, Type 4-2H Hot-Wire Anemometer and Type 2 Mean-Product Computer,'! Iowa Institute of Hydraulic Research, Report

No.

136.

Klebanof f, P.S., 1955, "Characteristics of Turbulence in a Boundary Layer

with Zero Pressure Gradient," HACA Tech. Report No. 12)47.

Laufer, J., 1954, "The Structure of Turbulence in Fully Developed Pipe Flow,"

HACA Tech. Report No. _117)4.

Naudascher, E., _196)4, "Effect of Density on Air-Tunnel Measurements," J.

Royal Aeron. Soc.,

68, )4l9.

Patel, V.C., 1965, "Calibration of the Preston Tube and Limitations on Its

Use in Pressure Gradients," J. Fluid Mech., 23,

_185.

Patèl, V.C., 1972, "A Unified View of the Law of the Wall Using Mixing

Length Theory," Iowa Institute of Hydraulic Research, Report No.

_137.

Patel, V.C., 1973, "On the Equations of a Thick Axisymmetric Turbulent Boundary Layer," Iowa Institute of Hydraulic Research, Report No. 1)43.

Richmond, R.L., 1957, "Experimental Investigation of Thick Axially Symmetric Boundary Layers on Cylinders at Subsonic and Hypersonic Speeds," Ph.D. Thesis, California Institute of Technology, Pasadena, California.

Satija, K.S., 1971, "On the Thick Boundary Layer Near the Tail of a Body of Revolution," Ph.D. Thesis, University of Iowa, Iowa City, Iowa.

Thonpson, B.G.J., 1965, "A New Two-Parameter Family of Mean Velocity Profiles for Incompressible Turbulent Boundary Layers on Smooth Walls," British

Aero. Res. Council, R & M 3)463.

Yasuhara, M., 1959, "Experinents of Axisynmitric Boundary Layers Along a

APPDI)

-' Implied by Q6 measured by ei ngle vire probe and Bernoulli equation assumed to bold along y

6

() based on Pitot Static

Table 1.

Pressure Variation at y = S and. y = O, and Integral

Parameters

Sta- ¡IL tion

r0

_itt)

6 H

Q/Ur

C6

itt)

(it) (Ct) -0.040

(it)

2 (rt)

(it)

6? _(it) H Pr.,- ton Pube Clau-

er

Plot 1 0.662 0.387 0.059 1.030 -0.0461 -.0.035' 0.0100 0.0071 1.4009 0.0103 0.0074 1.1.099 0.0110 0.0078 1.3961. 3.1124 -0.030" (0.0096) (0.0068) (1.4055) (o.0099) (0.0071) (1.3978) 0.0082 2 0.800 0.31). 0.082 1.0085 -0.0132 0.011' 0.0142 0.0101 1.4111 0.0152 0.0108 1.3978 0.0151 0.0108 1.1.009 3.030 0.029" (0.0142) (0.0098) (1.41.5) (0.0152) (0.0106) (1.4287) 0.0505 3 0.850 0.270 0.103 0.996 0.0395 0.031.' o.o167 0.0117 1.4299 0.0183 0.0130 1.1.093 0.0179 0.0128 1.1.006 2.726 0.051" (o.oi81.) (0.0124) (1.4878) (0.0202) (0.0138) (1.4623) 0.0952 I. 0.900 0.210 0.130 0.971 0.0943 0.066' 0.0238 0.0156 1.5262 0.0213 0.0181. 1.4852 0.0256 0.0176 1.1.585 2.185 0.089" 0.102 5 0.930 0.157 0.171 0.965 0.1510 0.095' 0.0386 0.0232 1.6630 0.01.86 0.0307 1.5837 0.0401. 0.0270 1.5005 L321e 0.116" 0.109 6 0.960 0.090 0.230 0.963 0.2130 0.101' 0.0658 0.0338 1.91.91 0.1067 0.0617 1.7281 0.0713 0.01.73 1.5098 0.51.4 0. 130*1 0.103 7 0.990 0.022 0.286 0.967 0.2310 0.091' 0.0971 0.01.36 2.2256 0.3776 0.2169 1.71.12 0.1137 0.081.3 1.31.95 0.111." Note:

'

Crôes Wire Probe Sin1e Wire Probe

PjtÔt-Statjc

Table. 2(a),

Profiles of Pressure

DistrIbution, Mean Velocities

and 'Reynolds Stresses

at X/L =

0.662

1. ft.

cp Q':Ó LJQ6 V/Q6

//Q6

'':'6

'!'6

-20 0.003 0.377 0.:586 0.645 0.005 0.391 -0.037 0.634 0.008 0.682 0.010. 0.012 _'0.015 0.018 0.498 O.577

-o.00

-o. o4o o.ii 0.761 0.783 0.686

_0.10

0.7143 0.666 0.710 0.743 0.001 0.009 0.007 0.0771e 0.0756 _0.Ô731 0.01465 0.014514 0.014145 0.01455 0.0521 0.0529 0.0320 0.0300 '0.02914 0.020 0.023 0.6143 .-o.o14o 0.801 0.822 Q.189 0.789 0.006 0.0690 0.01428 0.0500 0.0270 0.025 _0.038 0.715 -0.0140 0.8142 0.8514 0.830 0.830 0.007 0.o67 0.0397 0.01482 0.0240 0.030 0.033 0.772 -0.037 0.866 0.887

086i

0.861 0.005 0.0595 0.0369 0.0452 0.0208 0.035 0.038 0.830 -0.037 0.902 0.911 0.891 0.891 0.0014 0.05142 0.03144 oo14i14 0.0170

0.00

0.043 0.891 -0.035 0.932 0.9143 0.922 0.922 0.012 0.01495 '0.0313 0.0357 0.0128' o 0145 0 050 0.053 0"055 0 9141 0 983 -0 035 -0 035 0 958 0 978 0.989 0 955 0 978 0 955 0 978 0 018 0 021 0 0422 0 0330 0 0270 0 0228 0 0312 0 02514 0 0098 _{0 0060} 0 060 0.063 0 017 J. 036 -0 033 -0 033 0 993 1 002 1.007 0 992 1 002 0 992 1 002 0 016 0 020 0 02514 0 0170 0 0187 0 01414 0 0195 0 0138 0 0028 0 0016 0 065

0.00

0.073 1 040 1.045 -0 031 -0.031 . 1 003 1.005' L011 1 007 ioi4 1 006 1.003 0 026 0.028 0 0109 0.0073 0 0106 0.0082 0 0100 0.0067 0 00014 0.0002 ' 0.075 o.o8o 0 090 0.093 1.047 -0.029 1.005 1.012 1.018 1.011 1 009 0.998 1.007 1.010 1 008 0.998 0.025 _{.0.022 0 030} 0.0314 0.00514 0.0037 0 0034 0.0025 0.0058 0.00146 0 0035 0.0032 0.0057 0';0052 0 0039 0.0045 0.0000 '0.0000 0 0002 0.0000

Cross Wire Probe Single Wire Probe

Pitot-Static

Table 2(b).

Profiles of Pressure Distribution, Mean Velocities and Reynolds Stresses at X/L = 0.80

y, ft.

C C P Q/q6 e. Q/Q6 U/Q6

.

V/Q6 e

//Q6

//Q

v"/Q6

0.003 0.272 0.0014 0.291e -0.009 0.005 0.501 0.008 o.6a1. 0.010 O.1i23 -o.00le 0.645 0.6141 o.oi4 0.677 0.671 0.020 0.0751e 0.01e33 0.0535 0.015 0.e93 -0.002 0.6914 0.688 0.017 0.687 0.681 0.018 0.07143 0.01436 0.0533 0.020 0.556 -0.002 0.738 0.730 0.712 0.712 0.019 0.0727 0.0142]. 0.05214 0.025 0.6114 -0.002 0.7514 0.751 0.750 0.019 0.0692 0.01e07 0.05114 0.030 0.658 -0.002 0.801 0.781 0.786 0.785 0.028 0.06614 0.0370 0.0502 0.035 0.712 -0.000 0.832 0.823 0.820 0.819 0.032 0.06314 0.0369 0.01490 0.0140 0.759 0.000 0.859 0.8149 0.8148 0.847 0.033 0.0605 0.0367 0.0472 0.045 0.805 0.0014 0.883 0.811 0.878 0.877 0.037 0.0582 0.0350 0.0443 0.050 0.848 0.007 0.905 0.890 0.900 0.899 0.0143 0.05145 0.0330 0.01419 0.055 0.923 0.922 0.047 o.0014 0.0307 0.0402 0.060 0.928 0.009 0.9145 0.933 0.91411 0.9142 0.056 0.01472 0.0281 0.0370 0.065 0.951 0.9149 0.055 0.01418 0.02148 0.0329 0.070 0.992 0.011 0.976 0.965 0.967 0.965 0.061 0.0372 0.02214 0.02814 0.080 1.037 0.013 0.998 0.998 0.997 0.9914 0.071 0.0253 0.0173 0.0188 0.090 1.058 0.015 1.006 1.008 1.012 1.009 0.078 0.0131 0.0118 0.0125 0.100 1.060 o.o18 1.006 1.006 1,015 1.012 0.082 0.0067 0.0073 0.0068 0.110 1.060 0.018 1.006 0.120 1.060 0.020 1.005 i.o08 1.015 1.010 0.093 0.0032 0.0038 0.00148 0.130 0.022 1.006 0.1140 0.0211 i.006 i.004 1.013 1.008 0.098 0.0030 0.00149 0.0045

Cross Wire Probe Single Wire Probe

Pitot-Statjc

Table 2(c).

Profiles of Pressure Distribution,

Mean Velocities

and Reynolds Stresses at X/L

= 0.. 85 y,. C Q/Q6 Q/Q6 Q/Q6 U/Q6 V/Q6

//Q6

J!/Q6

/'/Q6

-20 0.003 0.263 0.530 0.001e 0.263 0.02I 0.1886 0.005 0.008 0.554 0.010 0.015 0.020 0.025 0.030 0.035 o.04o

0.05

0.050 0.055 0.060 0.065 0.070 0.080 0.090 0.100 0.110 0.3814 0.1458 0.500 0.5148 0.596 0.649 0.691 0.776 o.818 _0.912 0.967 1.013 1.0184 1.055 0.0214 0.0218 0.0218 0.0214 0.024 0.024 0.026 0.021e 0.029 0.029 0.031 0.033 0.035 .0.035 0.596 _{0.655 '0.685} 0.719 0.751 0.786 0.810 0.862 0.896 .0.9314 0.961 0.983 0.999 1.00Ie 0.6114 0.660 0.699 0.734 0.761 0.785 0.807 _{0.839 0.882} 0.9214 0.949 0.975 0.988 0.998 0.6018 0.621 0.6To 0.707 0.738 0.771 0.799 0.825 0.844 0.869 0.883 0.901 0.927 0.944 0.957 O:.992 0.999 0.6014 . 0.621 0.67.0 0.707

0738

0.771 0.798 0.825 0.8418 0.869 0.883 0.901 0.926 0.943 0.956 0.991 0.998 -0.001e 0.001 0.003 0.005 0.007 0.013 0.017 0.019 0.020 0.018 0.025 0.030 0.032 0.036 0.0187 0.052 .0.060 .

0.0753 0.0732 0.0699 0.0671 0.06184 0.0638 0.0602 Ô.058o

.055I

0.0532 0.01899 o.ô4614 0.0439 o.o4io 0.0325 0.0222

o.oii4

0.018314 0.01825 0.01816 . 0.O140Ii 0.0380 0.0380 0.0368 0.0353 0.0339 0.0321 0.0306 0.0285 .0.0267 0.02185 0.0196 0.0146 0.0099 0.0515 0.051? 001890 00le73 0.0471 0.0463 0.0455 0.0441 001827 0.0410 0.0383 0.0367 0.0340 0.0303 0.0237 0.0172 0.0100 0.0308 0.02918 0.0276 0.0252 0.0222 0.0216 0.02018 0.0178 o.oio 0.0146 ' 00130 0.0110 0.0090 0.0078 0.0042 0.00i18 0.0004 0.120 0.130 1.059 1.061 0.037 ' 0.o4o 1.005 _1.005 1.011 1.0018 1.002 0:.072 '0.0035 0.0040 00O56 '0.0000 0.140 0.150 0.160 1.064 1.0o4 0.0182 0.042 1.005 1.005 1.002 ' 1.000 . 0.997 ' '0.075 0.0028 0.0038 .0.0035 0.0000 1.005 0.994 0.991 0.0818 0.0031 0.0030 0.0044 0.0000

*

Cross Wire Probe

"

Single Wire Probe

Pitot-Static

Table 2(d).

Profiles of Pressure Distribution, Mean Velocities and Reynolds Stresses at X/L = 0.90

yft. C C, Q/Q Q/Q Q/Q L1/Q6 V/Q6

//Q6

//Q

¡/Q

0.003 0.0014 0.21*6 0.081 0.1*12 0.1*35 0.005 0.1*68 0.010 0.3144 0.081 0.522 0.540 0.012 0.537 0.537 0.015 0.0728 0.0399 0.01*67 0.015 0.395 0.081 0.570 0.580 0.578 0.578 0.011 0.0705 0.03914 0.01*57 o.orr 0.600 0.600 0.016 0.0686 0.0389 0.0447 0.020 0.141*3 0.079 0.614 0.603 0.628 0.628 0.021 0.0678 0.0382 0.041*2 0.025 0.187 0.079 0.650 0.620 0.660 0.659 0.024 0.0663 0.0375 0.0442 0.030 0.522 0.079 0.677 0.662 0.692 0.691 0.034 0.0646 0.0369 0.0440 0.035 0.691* 0.720 0.719 0.01*1 0.061*1 0.0361 0.01*31 0.01*0 0.594 0.081 0.729 0.706 0.71*6 0.744 0.01*9 0.0616 0.0359 0.01*23 0.045 0.723 0.171 0.769 0.054 0.0600 0.031*6 0.01419 0.050 0.675 0.083 0.783 0.752 0.792 0.789 0.062 0.05914 0.0338 0.01*13 o.o 0.816 0.813 0.071 0.0579 0.0319 0.0398 0.060 0.739 o.o86 0.823 0.7814 0.831* 0.830 0.075 0.d557 0.0318 0.0387 0.070 0.792 0.083 0.857 0.853 0.871 0.866 0.093 0.0528 0.0296 0.0365 0.080 0.853 0.083 0.893 0.88? 0.897 0.891 0.103 0.0472 0.0268 0.0331 0.090 0.908 0.083 0.9214 0.926 0.928 0.921 0.115 0.0127 0.0236 0.0285 0.100 0.961 0.083 0.951* 0.951 0.951 0.9149 0.121* 0.0371 0.0195 0.021*0 0.110 1.002 0.083 0.976 0.972 0.120 1.033 0.083 0.992 0.990 0.993 0.981 0.150 0.0203 0.01214 0.0136 0.130 1.I*8 0.083 1.000 o.i1*0 1.055 0.081 1.005 1.010 1.007 0.991* 0.165 0.0067 0.0068 0.0070 0.150 1.055 0.019 1.006 0.160 1.055 0.079 1.006 1.016 1.010 0.9914 0.176 0.0039 0.001,1 0.001*1 0.170 1.053 0.077 1.006 o.i80 1.O.3 0.077 1.006 1.018 0.190 1.053 0.077 1.006 0.200 1.006 1.021 1.007 0.988 0.195 0.0022 0.0057 0.00141

Crcss WIre. Probe Single Wire Probe

Pitot-Statjc

Table 2(e)'. Profiles of Pressure Distribution, 'Méan Velocities and eyno1ds Stresses at X/L = 0.93 y,

'ft.

'C. Q/Q6 Q/Q6 Q/Q6, U/Q6 V'/Q6

//Q5

//Q6

//Q6

-20 0.003 0.0014 0.008 0.2l3 0.1514 0.307 0.010 0.013 0.300 0.151 0.395 0.376 0.015 0.020. 0.025 0.030 0.035 0.0140 0.0145 0.050 o.o6o 0.070 0 080 0 090 o.ioo _0.110 0.375 0 14142 0.507 0.563 0.616 0.671 0 726 0 772 0.822 0.873

0Ï5j

. 0.151 . 0.149 0.1149 0.1147 0.1143 0 138 0 134 0.129 0.125 .0.1483 0.552 0.610 0.658 '0.700 0.7143. 0 783 0 816 0.850 0.884 0.1431 0.1486 _{0.513 0.556 0.581}

o.6ïi

_{0.659 0.702 0.735 0 766 0 808 0839} 0.1435 0.1451 0.1487 0.5214 0555 0.5814 0.6Ï2 0.635 :0.663 0.708 0.7147 0 785 o 819 0.851 . 0.1435 0.1451 0.1486 0.523 0.552 0.580 0.608 0.629 _{0.656 0.700} 0.735 0 771 0 802 0.833 0.012 '0.020 _'0.031 o.o4i. 0.055 0.067, 0.074 0.082 0.092 0.110 0.131 0 150 0 165 . 0.177 0.0753 0.0727 00706 0.0692 0.0672 0.065]. 0.06146 0.063ß 0.0636 0.0620 0.0595 0 0579 0 0553 _.0.0515 0.01417 0.01403 0.0387 0.0378 0.0369 0.0367 0.0357 0.0353 0:03148 0.03142 0.0332 0 0312 0 0302 0.0277 0.01465 0.01468 0.014514 0.01457 0.o144 0.014147 0.04146 0.0442 0.01436 0.0429 0014ì9 0 01404 0 0388 0.0363 0;0242 0.0230' 0.0204 0.0180 0.0166 0.0152 0.0142 0.0126 0.0120 0;:0102 0.0086 0 0072 0 0056 0.00142 0.120 '0.130 0.140 0.914 0.956 0.121 0.liIe 0.910 0.938 0.902 0.913 0.887 0.216. .0.01447 0.0255 0.0295 0.0020

0.ì6o 0.18o 0 200 0.220 0.:21e0 0.260

.0.991 1.0146 1.0614 1 068 1.061e 1.059 0.110 0101 0.092 o o88 0.081 0.077 ' 0.959 0.9914 1.006 1 010 1.011 1.011 0.954 0.987 1.001 1 001 0.963 0.996 1.0014 1 000 1.002 0.93]. 0.959 0.964 0 958 0.957 , 0.2145 0.267 0.279 0 288 0.298 0.0325 0.0162 0.0053 0 0036 0'.0031 0.0171 00i30 0.0051 . 0 004]. 0.0035 00206 0.0101 0.00149 0 0039 0.0038 0.00014 0,0004 0.0000 0 0002 0.0000 1.053 0.070 1.011 1.007 0.957. 0.313 0.0033 0.00147 0.0036 0.0002

*

Cr088 Wire Probe

**

Single Wire Probe

** Pitot-Statle

Table 2(f).

Profiles of Pressure Distribution, Mean Velocities and Reynolds Stresses at XIL =

0.96

y, ft.

Q/Q6 cee Q/Q6 *0 Q/Q * U/Q6 O V/Q6 O

f!/Q6

e TT/Q e

V''/Q6

* 0.006 0.008 0.191 0.215 0.010 0.263 0.219 0.213 0.233 0.015 0.262 0.020 0.296 0.217 0.288 0.293 0.025 0.333 0.356 0.356 0.017 0.0638 0.03145 0.0399 0.027 0.367 0.367 0.017 0.0650 0.03514 0.01410 0.030 0.336 0.217 0.352 0.376 0.3814 0.383 0.026 0.06149 0.03514 0.01430 0.035 0.14014 0.14114 0.1412 0.036 0.0656 0.0357 0.01429 0.0140 0.377 0.215 0.1413 0.14143 0.14148 0.14146 0.0148 0.06147 0.03514 0.01426 0.0145 0.1465 0.050 0.1421 0.213 0.1468 0.1493 0.1496 0.1492 0.065 0.0628 0.03141 0.01420 0.055 0.5214 0.060 0.1467 0.211 0.521 0.5145 0.5142 0.536 o.oßi 0.0611 0.0338 0.0141? 0.065 0.567 0.010 0.511 0.206 0.566 0.583 0.588 0.579 0.103 0.0593 0.0330 0.01411 0.075 0.6014 0.080 0.550 0.202 0.606 0.630 0.626 0.611e o.iiß o.o87 0.0319 0.01403 0.090 0.599 0.195 0.653 0.668 0.662 0.6148 0.136 0.0566 0.0311 0.0397 0.100 0.6147 0.193 0.693 0.710 0.695 0.678 0.153 o.o6o 0.0298 0.0391 0.110 0.686 0.1814 0.129 0.7149 0.120 0.728 0.178 0.761 0.178 0.766 0.7143 0.1814 0.0538 0.0282 0.0356 0.130 0.800 0.1140 0.809 0.160 0.828 0831 0.838 0.812 0.210 0.01491 0.02614 0.0321 0.160 0.893 0.151 o.885 0.881e 0.902 0.870 0.237 O.01e27 0.0215 0.0270 o.i8o 0.965 0.136 0.935 0.935 0.933 0.897 0.256 0.0333 0.0165 0.01914 0.200 1.026 0.127 0.9714 0.970 0.968 0.926 0.282 0.0200 0.0118 0.0120 0.220 1.059 0.118 0.996 0.996 0.996 0.952 0.295 0.0086 0.0080 0.0058 0.2140 1.061 0.107 1.003 1.0014 i.0014 0.956 0.3014 0.0038 0.00141 0.00143 0.260 1.0614 0.101 1.008 1.006 1.008 0.957 0.317 0.0025 0.00143 0.00314 0.280 1.012 0.300 1.017 0.961e 0.326 0.00149 0.0030 0.0030

y,ft:.

C

C.

U/Q6 VQ6

/Y/Q

/Y/Q

/'/Q

-20 0.005. o.o0î 0.160 0.010 0.268 . 0.239 0.170 0.182 0.015 . 0.198 0.020 0.2714 0.237 0.196 0.215 0.025 _0.027 0.030 0.035 0.0140 0.0145 0.050 0.060 0.070 0.080 0.090 0.100 0.120 _{'0.140 0.160 0.180 0.200 0.220} 0. 2110 0.287 0.301e 0.322 0.344 0.373 0.1101 0.1165 0.535 0.601 _{0.678 0.752} 0.818 0.899 0.232 0.228 _0.226 0.2214 _.Ó.219 0.217 .0.208 0.197 0.1814 _0.175 0.1611 0.154 0.11i3 0.239 0.2814 0.319 0.357 0.1404 0.14113 0.523 0.599 0.666 _{0.731 0.793} 0.8142 0.899. 0.233 0.2119 0.3014 0.355 0.387 _.0.1432 0.473 0.559 0.628 0.693' 0.7514 0.813 0876 0.928. 0.212 0.223 0.241 0.269 0.288 0.309 0.333 0.388 0.1136 0.1476 0.518 . 0.539 0.611 0.667 0.739 _0.806 0.862 0.212 0.223 0.2110 0.269 0.286 0.307 0.330 0.383. 0.430 0.468 0.507 0.526. 0.593 0.645 0.716 0.774 0.825 0.012 .0.013 0.015, 0.021 0.032 0.035 0.0145 0.058 0.073 0.087 0.1011 0.120 0.149 0,170 0.181 _.0.222 0.249 . 0.0376 0.01400 0.04314 0.01482 0.0517 0.05146. 0.0577 0.0619 0.0621 0.06Ï9 0.0595 0.0586 _0.0577 0.0561 0.05142 0.0507 0.01469 0.0198 0.02114 0.0231 0.0256 0.0271 0.0291 0.0309 0.0335 0.0335 0.0330 0.0323 .0.0319 0.0307 0.0297 0.028Ê .0.0266 0.0238 0.Ó2Ï8 -0.0231 0.02148 0.0270 0.0296 0.0321 0.0333 0.0365 0.0391 00394 0.0392 0.0387 0.0380 0.0371 0.0362 0.0338 0.0298 0.0072 0.00814 0.0098 0.0118 0.0132 0.01148 0.0160 0.0182 0.0170 0.0150 ' 0.01140 0.01311 0. 0120 0.0088 0.0086 0.0072 0.00118 0.260 0.280 0.961 1.015 .0.129 .0.123 0.9112 0.977 0.951 0.987 0.957 '0.91'? 0.289 0.0323 0.0153-0.0185 0.0006 0.300: _'0.320 1.046 1.050 0.112 0.105 0.999 ' 1.005. Ó.993 1.002 0.999 0.949 '0.313 0.00811 0.0064 0.00511 0.0000 0.3110 1.050 1.050 0.099 0.0914 1.007 1.0Ï1 1.001 1.004 0951 0.321 0;00311 '0.0036 0.0038 0.00Ö2 0.360 1.050 0.094 1.013. ' 1.007 0.953 0.325 0.0038 0.0035 0.00314 0.0000 *

Cross Wire Probe

0*

Single Wire Probe

*0*'

Pitot-Static.

Table 2(g).

Profiles of Pressure

DIstribution, 'Mean Veloetjes

and -Reynolds Stresses

at X/L

62. 114 in.

Boundary .Lyer Trip

/./.ff/fff/f///f/f/ff f f/f/f/I/f//i

Figure I.

Model and Traversing Mechanism

Còiica1 Tall Piece

-X,

U, u

Figure 2.

Measuring Stations and Notation'

y T

0.6

0.662

0.7

0.85 0. 9: Q.93

96. 0.99

X/L