A simple integral method for the calculation of thick axisymmetric turbulent boundary layers

A SIMPLE INTEGRAL

FOR

THE CALCULATION OF THICK AXISYMMETRIC

TURB LENT BOUNDARY LAYERS

by

V. C. Patel

Sponsored by

General Hydromechanics Research Program

of the Naval Ship Systems Command

Naval Ship Research and Development Center

Contract No. N0O014-68-A-01960002

uHR Report No. 150

Iowa Institute of Hydraulic Research

The University of Iowa

Iowa City, Iowa

September 1973

ABSTRACT

in a recent paper the author examined the equations of a

thick, incompressible, axisymmetric, turbulent boundary-layer and suggested,

in a preliminary way, two different methods for their olution. In this papei', òne of the methods, namely the integ'aJ. approacì, is considered in

greatér detail.. Lt is shown that the developent of the thick boundary

layer can be predicted with acceptable accuracy by using an approximate form

of the momentum-integral equation,

an

appropriat skin-frictiox relation, and a±i ent-aiÍiment equatiön obtained Íor axisythetric boundary layers. The method also requires the explicit use of a velocity-pro-file family in order to interre3.ate some of the intégral parameters. Available

experi-mental results have been used to demonstrate the general accuracy of the

method.

ACKNOWLED GENT

This paper is baSed upon research conducted under the General Hydromechanics Research Program of the Naval Ship Systems Cozmnand,

technical-i a&ninistered by the Naval Ship Research and Development Center, under Contract NOO011_68-A=Ol96-OOÖ2.

-1].

LST 0F CONTENTS

Page

±. INTRODUCTION i

II.

BASIC ASUTIONS AND EQUATIONS

i

III. CALCULATION PRO.CURE

IV. COMARISON OF THE THOD WITH

ERIITS

9

BOdies öf Revolution 10

Internal Flows 12

Cylinder of Constant Radius .13

V. CONCLUDING REMARKS

REFERENCES 16

LIST OF FIGURES

Page Figure 1. Thick Axisymnietric Boundary Layers--Definition

₁₈

Sketches and Typical Situations

Long, slender cylinder of constant radius Tail of a body of revolution

Internal flow (Conical Diffuser)

Figure 2. Experiments of Freeman

₍₁₉₃₂₎

on the Model of the

19

Airship Akron

(a) Freestreazu velocity distribution

(b) Boundary layer thickness

(c)

Mentum thickness

(d) Shape parameter

Figure

_3.

Experiments of Patel, Nakayama and Damian

₍₁₉₇₃₎

21 on th Tail of a Modified Spheroid

Actual and hypothetical Í'reestream velocities Boundary layer thickness

Momentum thickness

Cd) Shape parameter

(e) Skin-friction coefficient

Figure . Experiments of Fraser

(1956)

in a 10-degree Conical

24

Diffuser, Case A

Freestream velocity Boundary layer thickness Momentum thickness

Shape parameter

Skin-friction coefficient

Figure

_5.

Experiments of Yu

₍₁₉₅₈₎

on a Cylinder of Constant 27

Radius, Ur/v =

_15,250

Boundary layer thickness Momentum thickness

Cc) Shape parameter

(d) Skin-friction coefficient

A SILE INTEGRAL }.THOD FOR THE CALCULATION OF

THICK AXISYMIvETRIC TURBULENT BOUNDARY LAYERS

i. ' INTRODUCTION

Heré we còns'ider the. pioble of calcülati.ng'the development of thcompressible, axisymmetric., turbulent boundary-layers in situations

here the layer thickness is hot small 1h comparison with: the transverse radius of curvaturé of the surfaôe. In order 'to.is1ate the influence

of transverse curvature, it will be assumed that the boundary-layer

thicknessis. much smaller than the longitudinal radius of curvature of

the. surface. These conditions are closely realise in a humber of práctical flow configurations, some examples being shown in Figure. 1. Experiments in flows of this type have been madé b a. nurnbe' òf woikers

but the theoretical studies on the influence 'Of transverse curvature have been òonfined to the relatively simple case of the thick axi-symmetri boundary-layer developing, on a cylihder of constant Í'adius placed axially in a unifOrm stream (Figure la). As pointed out recently

by Páte]L

_(.1973),

this particular case differs fro the others in that there is little or no interaction between the' boundary layer and the

poténtlal flow outside it. In the second case shown in Figurel, however,

such an interaction is certainly present, and as a consequenòe of this, the static pressure does not remain constant across the boundary layer. Further discussion of the interaction phenomenon is given in the paper of Patel.

Here we shall dévelopa simple integral method förthecalòulation'of

thick, 1Syxmnetrc, turbulent boundary-layers Of the type shown in Figure

1. Such a method will be ueful in the atuy Of the i'hteraction between

the boundary layer and the potential flow.

BASIC ASSUMPTIONS AND. EQUATIONS

As shown by Patel

(1973),

the momentuminteral equation for

a thick, axisymmetric, turbulent boundary-layer, across which there is a significant variation of static pressure, may be written

The influence of turbulent velocity fluctuations

has

been neglected for the. present, but will be considered later on in the light of experiénta1

datai Here, and

2 are the displacement

and

momentum thicknesses

defined in,

forms

appropriate for axiymmetric flow, i.e.

-=

(1-

;)

dy; (2)

x and

y are distances measured along and normal to the surfacê while r is the. distance from the axis of

syetry;

Cf is the skin-friction

coeffi-dent defined by

1

2'

PU

r is

wall

shear stress, p is. density and p is pressure. Subscripts e

and o. are used to denote values at the. edge of the boundary

layer (y)

and on the surface (.y=o)., respectively.

From geometry, it is easily

shown that

r

r0 +

y cos,

.

()

where is

the angle between the axis of sy3mnetry and

the

tangent to

t.e

surface..

All

the equations we deduce

can

be made applicablé to both

external as well as internal boundary layers if we regard

cos

to be

positive for externa]. flogs and

negative, for

internal

flows

The terms ön the right-hand-side öf equation

(i) become

appre-dable only when the

boundary

layer thickness is

comparable with thé

transverse radius or

curvature

of the surface (i . e. when S ,, r.) and. when

there is a significant variatiOn of pressure across

the boundary layer.

Since thèse termsare not known a priori, some

assumption cóncerning their

magnitude is required.

Fri an examination of the experimental data of

2 dc5 dU dr

2(2+\ i

e 2

°

dx '

2 l'U

- dx r dx 2 f e

o-1

d

- U2 dx

_e p

i:

dy +

i:

('e)

_.

_(i)

(3)

3

Patel, Nâkayama and Daznian

₍₁₉₇₃₎

obtained in the thick boundary layer near the tail of a body of revolution, the author [Patel

_(1973)]

found tha.t these terms can be accounted for in an approximate manner if equation

(i) is used in the form

_/+

(262+1)t+_4cf=o,

(5)

where U is the "hypothetidal freestrea.m velocity" distribution implied by the largest longitudinal pressure gradient pr.esnt within the boundary

layer. In the càse of the boundary J.ayer on body of revolutin- (igure

lb), for example, the maximum longitudinal pressure variation is known to occur at the surface and is significantly different from the prêssure varia-tion at the edge of the boundary layer. The magnitude of the largest

pressure gradient in any particular stuati6n will of course e determined by geometry, and the nature of the interaction referred to'previously, but, for the présent discussion, it will be assumed that this pressure gradient is. known, so that 5e can be found. It will be clear that since boh

and U(x) are presumed known, we are taking into account, rather

indirectly, the static pressure variation acrøss the boundary layer. In

order to integrate equation (5) to èalcuiate the growth of the momentum thickness we shall need wo further relations, namely a skin-fricibn law

and an "auxiliary" or "shape-parameter" equation.

For thin, plane-surface, boundary layers he skin-friction

coefficient is usually expressed as a function of two integral parameters: a shape paraméter such as H E

l'2 and a

Ñeynolds number such as

Th better-known friction formulae of this type are deduced directly from the observation that the velocity profiles across the. boundary layer form a two-parameter family of shapes [see,-for example, Coles

(1956)

and

Thompson

_(1965)].

cperiments in axisymsietric boundary layérs, in bòth external [Patél, Nakayama and Damian (1973)] and internal [Fraser

(1956)],

flows, indicate that even when the boundary layer is thick, the velocity

profiles do nt deviate appreciably from the two-parameter families

cònstructed primarily for thin bop.ndary layers, provided the integral para-meters involved are evaluated according to the usual plane-surface boundary layer definitions. In view of this, the skin-friction law for a thick

aisymmetric boundary-layer nay be written

where the bars denote vài.ues obtained purely from the sIape of the \reloeity proffle, i.e.

dy,

where a = 0.019521 - 0.386768ò + 0.Ó28345c2 - O.000701c3., b = 0.191511 - 0.8314891c 0.062588e.2 - 0.001953c3,

and

c&nR0.

e2

o , (7)

The problem of relating, these "planar" definitions to the usual axisymietric definitions given in equations (2) wifl be considered later on. Here

we may. no-te, however, that the friction law of

(1965), presented

graphicaily in the original paper, may

be approximated adequately by the

formula [Head sand Patel (1969)],

Cf =

exp

(ai-i-b),

F . (6a)

For

the auxiliary equation we shall use a mod.ified form

of the

well known entrainment equation of

Head (19.58).

For an axisymmetric

boundary

layer the volume flux,

_Q,

within the boundary layer at any

streà.mwise löcation

is given by

Q' ₌

J0

2irrUdy = 2TÍUe

{r01)

+

. 2

cost), (8)

.ç.

and. the rate of entrairmelit of freestream

fluid into the boundary layer

is dQ/dx. The entrainment equation may then be written in the form

= CEUer0, . . . (9)

wher'e

Q t

_{{r (15i) +}

. (io)

and

CE is

an-thknown,

dimensionless, coefficient of entrainment. For thin

boundary

layers (i.e. 6"r) Head postulated that CE depends upon the

freestream velocity, a length scaie. of the flow in, the outer' region of the boundary layer and. the shape of the velocity profile in this region,

and

and

5

from dimensional considerations deduced that CE is a function only of the shape parameter H* _{(6_6l)/62. Following similar logic we may}

gener-alize Head's result to consider thick axisymmetric boundary layers by assuming that CE is the same function of H*, i.e.

CE =

CE(H),

where

6-6

=

_l

62

reflects only the shape of the velocity profile without regard to the fact

that the profile is wrapped around an axisymmetric body. In view of the observed similarity between the velocity profiles in thin and thick boundary layers it is further assumed that the shape parameter H* is related to the conventional shape parameter H as suggested by Head, i.e.

= H*(H). ₍₁₃₎

For computer calculations, relations (11) and (13), which were presented

graphically in the original paper of Head, may be approximated by the following explicit formulae:

CE = exp {-3.512 - 0.617 in(H* - 3)}

= 3.30 + 1.535

(ff

0.1)_2.715

The first of these is due to Dvorak (1969) while the latter was proposed

by Standen (196k).

From the definition of 11* we have

(12a)

so that the boundary-layer thickness

cari be

expressed in

terms of

2 and

the

shape parameters. Equation (lo) can therefore be written in the alternative form

r

{(if*Th

Ô2 - + COSc, (*+if)2 2 =

2 U o

e

(lOa)

(n)

(12)

To complete the method of calculation, we now need to establish relationships between the usual axisymmetriò thicknesses

appearing in the meiitum-integra1 equation; and the planar thicknesses

a.

2troced via the skin-friction and entrainment equations.

Such relationships are readily obtained from velocity-profile families commonly used in thin plane-surface boundary layer methods since, as re-marked earlier, they appear to describe the profiles in thick axisynmetric

boundary-Iayes quite adequately. In the interest of simplicity, and alsó consisteî'icy with the one-parameter family of profiles implied by equation i (13), wé shall make se of power-law profiles, i.e.

U

e

wherè either n or li servés as a. parameter. From the definitions of the various thicknesses, it is then easy to show that

where 6 .+ (H+1), (ii) i 6 i2(i+i)

cos4

_r

(ff-i)(+3)

(i6)

(18)

(20) H (Hi.)

₍₁₅₎

(ii-1)

HH+F+1)

(19)

It will be noticed hère that equations (15) and (16) follow directly from

the power-law pofi1es. We will not, however,.make. use of them here since we already have two similar relations in equatio±is (l2a) and. (13a).

Equation (l3a) is to be preferred since it was established by empirical correlations and has proven reliable over a wie, range of values of H for. thin plane-surface boundary layers. In what follows, we shall màke explicit use of equations (17) through (20) in order to solve, simul-taneously, the momentum-integral equation and the entrainment and

skin-friction relations developed earlier.

The system of equations given in this section are sufficient for the calculation of the development of a thick exisyrnmetric boundary-layer.

It òan readily be verified that the geeal method proposed. here reduöes

idntically to the original method of Head when the boundary layer develops

on a plane surface (i.e. r = ),andto the extension of the method of

Head proposed recently by Shanebrook and. Simner (1970) when the boundary

layer is axisyetric but thin (i.e.

«r). This, the present method can

be used to predict the development of thin as well as thiòk boundary layers. A procedure for the step-by-step solution of the equations is described i

the next section.

-III.

CALCULATION PROCURE

It will be seen from the previous section that there are

basically two ordinary diff±'ential equations, namely the momentum-integral equation and the entrainment equatiön, which have to be solved for the two basic variables ô2 and Q,. .11 other eqiations are simply

inter-relation-ships between the various quantitiés occurring in these differential equations. Unfortunately, the complexity of these. additional relationships is such

that the differential equations cannot readily be written explicitly in forms which contain only the two basic variables entioiëd abçve. The method of solutidn proposed below evolved after several trials of

alterna-tive procedures..

When the hypothetical freestream-velocity distribution,

e'

as well as. the x-component Of the true freestream velocity distribution U(x), are known and. the geometry of the axisyxietric surface is prescribed, in the

form r(x) and. x), the equations of the previous section can be integrated with respect to x starting from some initial streamwise position x0 where the boundary layer charà.cteristics are k±iown. Thus, if

₆₂

aÏid

are knom

at x, as is most often the case, the properties of the boundary layei at.:,

x0 + x can bè found by performing the following calculations:

i. Find c from equation (20), and therefore' from (17), 62 from (18), H from (19), from (13a), CE froth (lIa), Q from (lOa) and Cf from (6a).

Integrate the momentum-integral equation, equation (5), over the step x to find

₆₂

at x + x.

liitegrate the entrainment relation, equation (9), over the step Lx to find Q at + bc

Knowing 62 and Q at X0 + ¿x, thè problem now is to find all the quantities listed in step '1 e.t + x

so that the solution can proceed further. In particular, we wish to find 62 and H at x + Lx so that the other.

quañtities follow from step 1. This may be àòcom14shed

lE the following manner:

Using the definition of CL, equation

(i8)

may be

writteii

+ rÔ - r

, where i I 2H (H+l,)

-!.

(u_1(T-t.i.'

.'

Next, using te definitiôn of CL and , and equation. (i7),

equation (lOa) may be written

2 2r

cos

(ff*ff)

+ .=2.

if*

-

2Q - (if+L) = 0, (23)

ô

2

e2

Since

if*

is a function oÍ

if

(equation 13a), we see that equations

and can be solvéd. simuItaneous1y.

In the calculations described, later on, equations (21)

and

(23)

were solved using an iterative procedure by regarding equation (21,) as a quadratic in and' finding

the roots of equation

₍₂₃₎

using the well knöwn "secant" method.

The steps outlined so far constitute a simple "predictor" method. For better accuracy the values of the boundary layer parameters so compúted were used as first approxima-tions to re-integrate the momentum and entrainment equaapproxima-tions to obtain better approximations. For the step-lengths

chosen in typical calöulations, the number of such

"corrçc-tors" required fòr adequate accuray were of the order of

to

6.

Knowing the converged values Ó2 and at x0+&, stéps

1 through 5 were retraced to advance the calculation further

by another step Lix.

The pred±ctor-corrector scheme described above was programmed for an IBM

_360/365

computer i Fortran IV. Since the computer times are at most of the order of 0.03 second per step, the method is ideally suited for studies in which a large number of boundary-layer calculations have' to be performed rapi1y.

IV. CO}ARISON OF THE IVETEOD WITH FERIIVETTS

As stated in Section II, the method described here reduces

identically to that of Head. (1958) when the boundary layer develops on a plane surface, and to the extension of Head's method due to Shanebrook and Sumner (1970) when the boundary layer is axisynnnetric but thin. Since the general accuracy of Head's original method has been demonstrated on a number of previous occasions (see, for example, Thompson

(1965)

aiìd ine

lo

et ai

_(1968)),

_{it remains only to assess the performance of the new}

method in predicting sxismetric turbulent boundary layers.

A. Bodies of Revolution: In order to verify their extension of

Head's methodto treat thin axisymmetriò boundery layers, Shanebrook

and i.iner compared their calculations with the results of one of the

experiments of Freeman (1932) on a model. of the airship Akron. The present method was also applied to this case and the results obtained using the measured surface pressure distribution are shown in Figure 2. The curves

labelled "thin boundary layer" in this figure represent the results obtained using the extension of Shanebrook and. Sumner. These calculations indicate

the following: (a) The present method reduáes to that of Shanebrook and Sumner when the boundary layer thickness is small compared with the radius of the body (x/L <

0.7,

6/r < 0.115) and gives acceptable agreement

with experiment. (b) Beyond X/L 0.7, the boundary layer can not be.

regarded as thin. This is evidenced by the rather large differences between the usual a.xisy1metric definitions and the planar definitions of the

in-tegral parameters, and also by significant differences between the predictions, particularly of the boundary layer thicknéss and the shape parameter,

of the axisymmetric version of Head's method and the present method.

(c). Unfortunately, the data of Freeman do iiot extend sufficiently close to the tail so that it is not possible to assess the real success of the present method in the prediction of the thick axisymmetric boundary layer over

the rear 10 to 15 percent of the body length. A noticeable improvement is, however, seen in the predictions of the boundary layer thickness in the range 0.7 < X/L < 0.87.

There are a number of other., older, experimental studies in the literature, sûóh as those 6f yon (1931), Cornish and Boatwright

(1960)

and Gertler (1950), on bodies of revolution. These were examined during

the course of this ork for their suitability as test cases for the present method. Lyon made boundary layer measurements on two bodies of

revolution but these. did not extend into the last 10 to 15 percent length of the body. The calculatIons for these cases indicated good agreément with experimental data over the range of the measurements but the lack of data from the tail region preclude their use to vérify the capability of the

presentmethod to prédict the behavior of the thick axisymmetric boundary

11

modified full-scale airship. _{Although boundary layer measurements were}

made up to X/L = 0.96, the auhors do not give _{sufficient, details of the}

velocity profiles to evaluate the integral parameters with the desired

ac-curacy. _{Additional complications vere introduced by the presence Òf control}

surfaces so that the flow could not be regarded as axisymmetric in the region of interest. _{Finally, only overall drag coefficients were measured} by Gertler. _{Due to the absence of informatin regarding the behavior} of the thick boundary layer in the tail region of a body of revolution experiments were performed at this Institute oñ a modified spheroidal model and the results have been reported by Pat el, _{Nakayama and Damian (1973).} To the author's knowledge, this _{appears to be the only set of experimental} data which can provide a severe test case for the proposed method of

calculating thick axisymmetric boundary layers.

The results of the present method are compared with the data of Patel et al in Figure 3. _{Figure 3(a) shows the variation of the total} freestream velocity

e and the x-component of the frees-bream velocity Ue actually measured at the edge of the thick boundary layer. _{The hypothetical} freestream velocity

e deduced, from the wall static pressure measurements

is also shown. _{Here then is the first case where there is an appreciable} variation of static pressure across the boundary layer. The calculations shown by solid lines in Figures 3(b) through 3(e) vere performed' using

in the momentum integral equation and tJe in all other equations. Also shown in these figures are two other sets of results obtained from the

axisyetric version (Shanebrook ând Sumner) _{or Head's method,using U}

only and thenU only. I- _{will be observed from the comparisons with the}

experimental data that the method of Head (with either U or U used

e e

everywhere) agrees closely with the predictions of the, present method as well as with the' experimental data only up to. X/L = 0.85, i.e. over the region where the boundary layer may be regarded as thin., Past this point, however, the present method gives much improved agreement with

experimental data. _{Notice that the method accurately predicts not only the}

development of theboundary layer thickness and the _{usual axisymmetric}

definitions of the momentum thickness and the shape parameter but also the planar definitions, and. , of the integral parameters. _{It shpuld perhaps} 'be mentioned here that the present method' in no way uses the experimental data for the purpose of generalizing Head's'original method tò treat thick

12

B. Internal Flow: It, was mentioned in Sectin II that the

proposed method. should apply eqiiaJ-ly well to external as well as internal

thick ax-isymmétric boundary layers. The two boundary layer developments

measured by Fraser

(1956)

in a 10-degree total angle conical diffuser fall into the latter category , since the ratio of the boundary layer thickness to the diffuser radius approached values of the order of 0.5 just before

separation was encountered. While the S/r0 ratios are not as large as

those found near the tail of a hdy of revolution we should nevertheless

expect some unusual features in these cases. It will be recalled- thát the two séts of '.ata collected by Fraser were considered as "optional" test cases for the evaluation of two-dimensional boundary-layer calculation methods at the Stanford Conference (Kline at al

1968).

Of the few in-vestigators who applied their calculation procedures to these cases, none

succeeded in predicting either of these flows with any accuracy. As we shall see later this failure can not really be attributed to the thick boundary layer influence.

Because of the confined nature of the internal flow the mean streamlines in the conical diffuser are'unhikely to deviate much from their radial behavior in potential flow. This suggests that we 'can not

expect the variation of static pressure across the boundary layer to be a signIficant factor in the flow development. The' calculations presented in Figure 14 wére therefore performed using the freestream velocity distrIbutiOn deduced from the wall static pressure measurements. Since the two flows measured by Fraser were found to be quitè similar only one of them (Case A) will be discussed here in detail. The solid lines in Figures-14(b) through 14(e) 'are the results of the calculations using the present method while the

curves labelled "T" correspond to the axisymetric version òf Head's method. The first observation we make is that the difference between the' two sets

of calculations is small. This small difference may be attributed to the thick boundary layer effect which the present method seeks to elucidate; the relatively small difference is not surprising since the differences were equally small on the bodies of revolution considered previously for values of cS/r0 of the order of 0.5 (see, for example, X/L

= o.87

in Figure

3). Why then are the experimental data so different from the

calculations of the methods considered here as well as at the Stanford Conference? In Figure 14(c) the curve labelled "M" shows the development of. the momentum

13

thickness implied by the momentum integral equation alone, i.e. the result of i±itegrating this equation using the measured values of and Cf. The disagreemant between this and the experimental data suggest that either tiie momentum integrai equation used here does not represent the true momentum balance or that the experiment was not truly axisyinmetric. _{While we have}

ruled out the possibility of significant statiò pressure variations across the boundary layer there remains the influence of the turbu1ence terms which were omitted from the momentum integral equation. These were not

measured by Fraser but judging by the rather large decrease in the turbulence intensities observed by Patèl et al in the external thick boundary layer we may expect a corresponding increasé in the turbulence leVel in this internai flow. _{These terms, together with any departures from axial}

symetry (flow convergence) along the line of measurements could explain

the observed discrepancy in the growth of the, momentum thickness. In view of this, another calculation was performed in which the experimental values

of the momentum thickness were used m'piace of the momentum integral equation. _{The results of this calculation are shown by the dotted lines} in Figure (b) through 4(e). _{Notice that such a calculation represents} a tes-t of all thé other assumptions made in the present method. The

general improvementin the prediction of the boundary layer thickness, _the

shape parameter and the skin-friction coefficient is quite noticeable. It

is, howevér, rather surprising to find that the amount of momentum imbalance shown in Figure 14(c) could have such a marked effect on the development of the Other ±ntegral parameters.

C. Cylinder of Constant Radiùs: No calculation method for thick axisyimetric boundary-layers äan be regarded as complete unless it predicts the behavior of the boundary layer on a cylinder of cönstant radius placed axially in a uni'orm stream. (Case

l

shown in Figure 1(a)). Theoretical investIgations of this case have been made by a number of

previous workers, the latest and the most comprehensive investgtion

being the recent one of Cebeci (1970). With the exception. of Cèbeci, how-ever, most workers have fodussed their attention on the prediction of the influence of transverse curvature on the skin-friction and heat-transfer coefficients, _{On the experimental side, mêasurements have beefl reported} by Richmond (1957), Yu (1958), Yasuhara (1959) and Keshavan (1969). A

detailed analysis of this data, however, revealed that the present method could not be tested conclusively againât _{them since there was a} marked lack of axial symmetry in these, rather difficult to perform, experiments. _{Comparisons with one set of data collected by Yu are} never-theless shown in Figure

5.

These are the _{measurements made at the lowest}

cylinder Reynolds number (tJr/v =

_15,250)

_{and therefore represent the}

thickest boundary layer. It will be seen that the shape parameter and the skin-friction coefficient are predicted with reasonable accuracy but

the calculated momentum thickness and the boundary layer thickness are

not in satisfactory agreement with experimental data. Since, in the absence of pressure gradients, the momentum integral equation reduces simply to

d6

= _{- Cf}

it is clear that the disagreement between the predicted and measured

momentum thickness signifies either a lack of axial symmetry or the presence of streainwise pressure gradients. The comparisons shown in Figure 5 are typical of all the cylinder data examined during the course of this work. _{The rather large discrepancy in the boundary layer} thick-ness shown in Figure 5(a) is, however, unlikely to be explained by the lack of axia.l. symmetry, and it is possible that the assumption of the velocity profile family implied by equation (l3a) may not be entirely satisfactory.

V. CONCLUDING REMARKS

It hasbeen shown that the proposed method for the calculation

of thick axisymnetric turbulent boundary-layers represents a marked

improvement ovez the straightforward use of the usual thin boundary-layer calculation procedures. A number of simplifying assumptions weré made in order to account for the variation of static pressure across the boundary layer, and axial symmetry or transverse curvature. In the one case where appreciable variation of static pressure was present, the simple momentum-integral equation which uses .a hypothetical freestream veloity derived from the observed wail static pressure distrIbution appears to perform quite satisfactorily. The assumptioïm coñcerning the rate of entrainment,

15

namely that. it is a function of the 'p1anar" shape parameter H* as given by Head for two-dimensional flows is rather difficult to verify

irectl since the data available so far are not sufficiently accurate.

The assumption that the velocity profiles in a thick axisyetric boundary

layer can be represented by a simple one-parameter fathily of shapes is per-haps the most important one since it attempts to take account of most of the thick boundary layer effects. An Obvious impovement. One can maie

here is to incorporate the more, refined two-praméter models such, as those proposed by Coles (1956) and Thompson (1965). While this may lean to some gain in the accuracy, particularlr in thé prediction of the boundary layer thickness, the additional complexities will be considerable.,

Overall, the proposed method appears to perfrm best for the

boundary layer on bodies of revolution, including the region near the tail where the boundary layer 'grows to a thickness many times the local radIus of the body. Proper teatment 'of 'the boundary layer in the tail region i.s important for the estimation of'the viscous resistance of such bodies. When the resistance is calòulated either by a straightforward

application 'of the wé11 known Squire-Young formula. or 'by 'a continuation of the boundary layer calculation into the near and far wake of the body, the f i:flal result depends upon the characteristics of the boundary layer

at the tail or just ahead of separation. In a recent study on the calcu-lation of the drag of bodies of. revolution performed by Nakayama and SPatel

(1973) it was fouiid that the use of the proposed method, in conjunction with a squire-Young type formula, gave accurate prediction of the drag

-16

REFERENCES

Cebeci, T.

_1970,

"Laminar and Turbulent Incompressible Boundary Layers on Slender Bodies of Revolution in Axial Flow",

J Basic Eng.,

Trans. ASME, Ser. D,

,

515.

Coles, D.E.

₁₉₅₆

"The Law of the Wake in the Turbtilént Boundary Layer",

J.

Fluid Meché, 1, 191.

Cornish, J.J. III and Boatwright, D.W.

₁₉₆₀

"Application of Full-Scale Bourid.ary Layer Measurements to Drag Reduction of Airships", Miss

State Uni., Aerophysics Dept., Rept.. No.

28.

Dvoràk, F.A.

₁₉₆₉

"CalculatiOn of Turbulent Boundary Layers on Rough Surfaces in Pressure Gradient ",

AIAA Journal, T, 1752.

Fraser, H.R. 1958 "Study of an Incompressible Turbulent Boundary Layer in

a Conical Diffuser", Ph.D. Thesis, Uni. of Illinois; also,

J.

Hyd. Div., Trans. ASCE, 3)4, 16814/1.

Freeman, H.B.

₁₉₃₂

"Measurements of Flow in the Boundary Layer of a 1/40-Scale Model of the U.S. Airship Akron", NACA Rept. No 1430.

Gerbier, M. 1950 "Resistance Eperiments on a Systematic Series of

Stream-lined Bodies of RevolutiOn--for Application to the Design of High-Speed Submarines", David Taylor Model Basin, Rept. No.

C-297.

Hed, M.R. 1958 "Entrainment in the ThrbulentBoundary Layer", British Aeron. Res. Council R & M

_3152.

Head, M.R. and Patel, V.C. 19:9 "Improved Entrainment Method for Calcu-lating Turbulent Boundary Layer Development", British Aeron. Res. CouncIl R & M 36143.

Keshavan,N.R.'

₁₉₆₉

"Axisyminetric Incompressible Turbulent Boundary Layers in Zero Pressure Gradient Flow", M.Sc. Thesis, Indian Inst. of Science, Bangalore, India.

Kline, S.J., Morkovin, M.V., Sovran, G. and Cockrell, D.J. (Editors)

₁₉₆₈

Proceedings of "Computation of Thrbulent Boundary

Layers--AFOSR-IFP-Stanford COnference".

Lyon, H.M. 19314 "A Study of the Flow in the Boundary Layer of Streamline Bodies", British Aeron. Res. Council R & M

1622.

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

1973

"Calculation of the Viscous Resistance

of Bodies of Revolution", Iowa Inst. of Hydraulic Res., Rept. No.

151.

Patel, V.C. 1973 "On the Equations of a ThIck Axisetric Turbulent

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

₁₉₇₃

"An Experimental Study of the Thick Turbulént Boundary Layer Near tie Tail of a. Body of RevolutIon",. Iowa Inst. of Hydraulic Res., Rept.. No. 142. To be published in the J.

fluid Mechanics.

Ribmond, R.L.

₁₉₅₇

"Experimefltal Investigation of Thick

Axially

Smetric

Boundary Layers on Cr1inders at Subsonic and HypersOnic Speeds", Ph.D. Thesis, Cal±fornia Inst. of Tech., Pasadena.

Shanébrook., J. R. and Sumner,

W.J. 1970

"Entraiument Theory for Axisinmetric,

Turbulent, Incompressible Boundary ers", AIM,

J

Hydronauti.cs,

Standen, N.M.

_l96

"A Concept ofMass' EtrainmehtApp1ied to Compressible

Turbulent Boundary. Layers -in Adverse Pressure Gradients",

AIM

POEper No.

_6-58]4.

Thompson, B.G.J.

_{1965 "A}

New Two-Parameter Family of Mean Velocity Profiles for Incompressible Boundary Layers on Smooth Wails", British

Aeron. Res. Council R & M

3463.

Yasuhara, M. 1959 "Experiments of Axi'synñnetric Boundary Layers Along a

Cylindér inIncompressib1eF1ow",

_Tran.s

Japan Soc. Aerospace

Sci., 2, 33. .. .

Yu, Y.S. 1958 "Effect 0±' Transverse Cuiature on Turbulént Boundary

(a) Long, Slender Cylinder of Cçnstant Radius

r+ y

8

(b) Tail of a Body of Revolution r = r0 + ycos4

(e) Internal Flow (Conical Diffuser)

r=r-ycos4

Figure 1. THICK AXISYIvflvIETRIC BOtJDPRY LAYS Definition Sketches and Ty-pical Situations

0.14

0.2

0I

o Experiment.

T, Thin Boundary Layer

- Present Methöd

- I-19 Figure 2 (continued) 0 0.2 0.14 .0.6 0.8 i, o X/L

(a) Freestream Veiòcity Distribution

0.2

o.6

0.8 _X/L,

io

(b) Boundary Layer Thickness 1.1 1.0 0.9 o.6 feet 20 15 10 5

feet 0.06 o 014 0.02 2.0 1.2 o 1.0 o O '2' Experiment 2' Experiment

Present Method

---T, Thin Boundary Layer

I- I

20

o H, Experiment

V , Experiment

T, Thin Boundary Layer Present Method (c) Momentum Thickness

(d) Shape Pareter

0.8 X/L

10

2:. 0 1..₅ 1.0

0.5

o cm.

igure 2. Experiments of Freeman (1932) on the Model of the Airship Akron

'0.2 0.14

0.6

0.8

X/L

10

O 14 0.6

0.6 0.5 feet 0.11.. 0. 3 -0.2 0.1 0.7 0.8 21

0.7'

0.8.

0.9

_X/L i

(a) Actu.i & Hypothetical Freesti-eam Ve1o±ties

o Experiment

T, Thin Boundary

-- T, Th±ÍI Boundary

Present Method

(b.) Boundary Layer Thickness

Figure 3 (continued).

Layer (Using

Layer (Using _tie)!!

6/

Q9

_x/L

o

10

15 cm 10 1.05 1.00

Q/U

0.95

\

U/U

-y-

--q

0.90

U/u

_e

0.85

I

--0.2 Ô2 feet 0.1 o _{62, Experiment} v Experiment

T, Thin Boundary Layer (Using

---T, Thin Boundary. Layer (Using U)

Present Method

22

(e) Momentum Thickness

Figure 3 (continued)

-H, Cf 2.5 2.0 1.5 1.0 001L .003 .002 .001 23 O _{H, Experiment} V H, Experinient

T, Thin Boundary Layer (Using

T, Thin Boundary Layer (Using U)

Present Method

(d) Shape Parameter

0.9

X/L

T, Thin Boundary Layer T, Thin Boundary Layer Present Method

10

(e) Skin-Friction Coefficient

Figure 3. Experiments of Patel, Nakayama & Damian (1973) on the Tail of a Modified Spheroid

o

0.7 0.8 0.9 _X/L

10

180 160 ft/see 110 120 100 , feet 0.3 0.2 0.1 21 0.25

x, meters050

0.2 Experim.nt

Present Methöd

---Present Methöd. (expt.

---T, Thin Boundary

Layer..

igure; lt (cóntinued.) 0.5 50 rn/sec lt5 35 io 5 cm 1.0 2.0 ( ) Freestream Velocity x, meters 2.0

1.0

_{X., feét}

0.25 meters

M Momentum equation only

Figure 1 (continued) 2.0 cm 0.50 0.25 1.0 x, feet (e) Momentum Thickness

H cf o .004 0.25. P.50 003 .002 .001 O Present Method O 26 o 0.25 meters _0.50 x, feet (d) Shape Paraméter

X,

meters 2.0 2.0

Figure ii. Experiments of Fraser (1956) in a i0-d.gree Conical Diffuser (Case A)

1.0

x, feet (e) Skin-Friction Coefficient

0.02 2' 62 feet 0.01 Fiùre 5 (continued) O _Experiment - Present Method

T, Thin Boundary Layer

2 6

x,feet

8

(a) Boundary Layer Thickness

o 62, Eieriment

2' Eperiinent

Present Method

- - - T., Thin Bound.ary Layer

2.5 cm .0.50 0.25 o _1.0

X,

meters 2.0 1.0

x, meters20

2 _{x, feet} 6 (b) Momentum Thickness 0.2 feet 0.1 5.0 cm

Cf 003 002 .001 28 O H, Experiment y

H,Eçpriment

- Present Method

T; Thin Boundary Layer

(c) Shape Parameter

X,

meters O 1.O 2.0 .Q05 o Experiment -. Présent Method

-- - T ,

Thin BoindarY Layer

2 feet

(a) Skin-Friction Coefficient

Figure 5. ExperimentS of Yu

(1958)

on a Cyiiner

öf Constant Radius,

Ui=

15,250

Unclassified

NOV

1473

D D

FORM I (PAGE 1) Uncias sified

.1Uur1, V V.. LtlSi 11V..,

DOCUMENT CONTROL DATA. R & D

S ,'r,,rjty ,l0s,jfje1atjii1 cit title, h,civ if .11,,itr,,« t irnd iticicXjtijl il)flOtiitt(Pfl nr,,,.t be e,pter,cIu Ile?, tipe ov,raII rif,.,?? I,. ç1.,i..ss(,,'ii)

oNr(;INaNG AC1IVIIY (Cor, oteouflcor)

Institute of Hydraulic Research

The University of Iowa

Iowa City, Iowa 52212

¿L'.REP'OHT SECUP.'TV CLATOIt CA TI)TJ

Unclassified

2h. GrL0UP

-I. RCPOIPT ti TLPC .

"A Simple Integral Method for the Calculation of Thick Axisymmetric Turbulent Boundary Layers"

4. DESCRIPTIVF NOTES (Type of report andinclusive dates)

uHR Report No. 150

, AU THOR(S) (First r,,,me. m,ddie initial, last nome)

V.C. Patel

6. REPORT DATE

September 1973

78. TOTAL NO. OF PAGES

28

7b. NO. OF REF5

21

88. CON TRACT OR GRANT NO.

N0O014-68-A-0l96-0002

b. PROJECT NO.

C.

d.

9e. ORIGINATORS REPORT NUMBER(S)

lIER Report No. 150

Sb. OTHER REPORT NO(S) (Any other numbers that may be asslned this report)

10. DISTRIBUTiON STATEMENT

Approved for public release; distribution unlimited

il. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Naval Ship Research & Development Center Bethesda, Maryland

13. ABSTRACT

In a recent paper the author examined the equations of a thick, incompressibi-axisymmetric, turbulent boundary-layer and suggested, in a preliminary way, two differ-ent methods for their solution. In this paper, one of the methods, namely the integral approach, is considered in greater detail. It is shown that the development of the thick boundary layer can be predicted with acceptable accuracy by using an approximate foim of the momentum-integral equation, an appropriate skin-friction relation, and an entrair3ment equation obtained for axisymmetric boundary layers. The method also re-quires the explicit use of a velocity-profile family in order to interrelate some of the integral parameters. Available experimental results have been used to demonstrate the general accuracy of the method.

urn

ssif99

Sccurity ClaKsiflcation

Thrbulent Boimdary Layer

Thick, Adsyétric Flow

Calcul at ions Integral Method Entrainment Equation KEY WORDS R O L E LINK A wr LINK *3 ROL E Wr RO LI N Wi.

D051473 (BACK)

_Unclassified

NSRDC LIST

December

5, 1913

Docs/Repts/Trans Section

Scripps Inst. öf Oceanögraphy Library University of Calif., San Diego

P.O. Box

36T

La Jolla, Calif.

₉₂₀₃₇

University of Calif. at San Diego

La Jolla, Calif.

92038

Attn: Dr. A.T. Ellis Dept. of Applied Math

McDonnell Douglas Aircraft Co. .Attn: A.M.0. Smith, J. Hess

3855

Lakewood Blvd o

Long Beach, Calif.

90801

Stanford Research Institute Menlo Park

California

1502

Naval Postgraduate School Monterey, Calif.

₉₃₉₄₀

Attn: Library, Code

2124

Attn: Dr. T. Sarpkaya

Attn: Prof. J. Miller

Nielsen Engineering & Research, Inc.

.850

Maude Avenue

Attn: Mr. S.B. Spangler Mountain VIew, Calif. 94040

0ffi cer-In-Charge

Naval Undersea Research and Development Center

Attn: Dr. J. Hoty

₍₂₅₀₁₎

Library

Pasadena, Calif.

91107

Cörnmanding Officer

(131)

Naval Civil Engineering Lab

Port Hueneme, Calif.

₉₃₀₄₃

Commander

Naval Undersea Research and Development Center

Attn.: Dr. A. Fabula

₍₆₀₀₅₎

San Diego, Calif.

₉₂₁₃₂

Commánder

Electronics Laboratory Center

(Library)

San Diego, Çlif..

92125

Director

Office of Naval Research

Branch Office

50

Fell Street

San Franciso, Califonia

94102

Hunters Poizit Nával Shipyard

Technical Library (Code

202.3)

San Francisco, Calif.

94135

Stanford University

Attn: Engineering Librarr

Dr. R. Street

Stanford, Calif.

94305

Lockheed. Missiles & Space Cò. P.Ö. Box

o4

Attn. Mr. R.L. Waid, Dept.

_57-74

Bldg.

150,

Facility 1.

Sunnyvale, CalIf.

94088

Mare Island Naval Shipyard Shipyard Technical Library

Code

_202.3

Vallejo, Calif.

,9592

Assistant Chief Design Engineer For Naval Architecture (Code

₂₅₀₎

Mare Island Naval Shipyard

Vallejo, California

₉₄₅₉₂

Colorado State University

Föothills Campus

Fort Collins, Colorado

₈₀₅₂₁

Attn: Reading Room

Engr. Res. Center

University f Bridgeport

Attn. Prof. Earl Uram

Mechanical Engineering Dept.

Electric Boat Division

Attn: Mr. V. Boatwright., Jr. Generai Dynamics Corporation Groton, Connecticut 03610 University of Connecticùt Box U-37

Storrs, Conn. 06268

Attn: Dr. V. Scottron 'Hydraulics Res. Lab

2 Fi rida Atlantic. University Ocean Engineering Dept.

Attn: Technical Library Dr. S. Dunne

Boca Raton, Fiàrida 33132

Director

Naval Research Lab

Underwater Souñd Ref. Division

P.O. Box 8337

Orlando, Florida 32806 University of Hawaii

Dept. öf Ocean Engineering 2565 The Mall Attn: Dr. C. Bretschneider Honolulu, Hawaii 96822 University of Illinois Urbana, Illinois 61801 Attn: Dr. J. Robertson 2 College of EngIneering University of-Notre Darne Notre Dame, ndiana 46556

Attn: Engineering Library Attn: Dr. A. Strandhagen University of Kansas

Chm., Civil Engr. 'Dept. Library

Lawrence, Kansa.s 66o Kansas- State University

Engineering Experiment Station Séaton Hall

'Manhattan, Kansas 66502

.Attn: Prof. D. Nesmith

-2-Officer-in charge

Naval Ship Research and Development Laboratory

Annapolis, Maryland 21402 Attn: Library

3 'U.S. Naval Academy

Annapolis, Md. 214O2 Attn: Technical Library

Attn: Dr.. Bruce Johnston Attn: Profe. P. Van Mater, Jr.

The John Hopkins University

Baltimore, Md. 21218 Attn: Prof. O. Phillips Mech. Depts.

40 Commander

Naval Ship Res. and Devel. Center Attn: _{Code 1505}

Code 5611 (39 cys)

Bethesda, Maryland 2003b

8 Commander

Naval Ship Engineering Center Dept. of the Navy Center Bldg. Prince Georges Center

Hyattsville, Maì-yland 20782 Sec 603LB, 6110, 6114H,. 6120, 6136 61!iJ.G, 6lloB, 618 Hydronautics, Incorporated -Attn: Library Attn: Mr. Gertler Pindell School Ròad Howard County

Laurel, Maryland 20810 Bethlehem Stee1 Corporation

Attn: Mr. A. Haft, Technical Mgt. Central Technical Division

Sparrows Point Yard

Sparrows Point, Maryland. 21219 Bolt, Berañek and Newman

Atth: Library 50 Moulton Street

Cambridge, Mass. 02138 Bolt, Beranek and Newman 1501 Wilson Boyd

Arl±n'ton, Virginia 22209 Attn: Dr. F. Jackson

Cambridge Acoustical Associates, Inc. 1033 Mass. Avenue Cambridge, Mass. 02138 Attn: Dr. M. Junger 2 Harvard University Pierce Hall Cambridge, Mass. 02138 Prof. G. Carrier

Gordon McKay Library

5 Massachusetts Inst. of Tecbnölogy

Dept. Ocean Engineering

Attn: Library, Prof. P. Mandel, Dr. Leehey

Attn: Prof. M. Abkowitz, Dr., J. Newman

Cambridge, Massachusetts 02159

Parsons Lab., Prof. A. Ippen M.I.T.

Cambridge, Mass. 02319

Attn: Heferénce Room

Woods Hole Oceanographic Inst. Woods Hole, Mass. O2543

Worcester Polytechnic Inst. Alden Research Lab

Attn: Technical Library Worcester, Mass. 06109

-3-University of Michigan

Attn: Dr. T.F. Ogilvie, Prof. F. Hazmit Dept. of Naval Architecture

and Marine Engineering

Ann Arbor,- Michigan 481Q5

St. Anthony Falls Hydraulic Lab. University of Minnesota

Mississippi River at 3rd Ave., S.E., Minneapolis, Minn. 55414

Attn: Director

Attn: Mr. J. Wetzel, Mr. F. Schiebe

Attn: Mr'. J.' Killen, Dr. C. Song

Research Center Library Waterways Experiment Station

Corps of Engineers P.O. Box

63].

Vickaburg, Mississippi

₃₉₁₈₀

3 Davidson Lab

Stevens Institute of Technology 71]. JÏu.son Streèt

Hoboken, New Jersey 07030

Mr. J. Breslin, Mr. S. Tsakonas

Attn: Library

Depart. cf Aerospace and Mechanical Sciences Princeton University

Princeton, New Jersey 085140 Attn: Prof. G.' Me110

New York University, W Pierson, Jr. University Heights

Bronx, New York' 101453

Cornell, Aeronautical Lab

Aerodynamics Research Dept.

P.O. Box 235

Buffalo, New York 112'21

Attri: Dr. A. Ritter

Long Island University'

Graduate Dept. of Marine Science 140 Merrick Avenue

East Meadow, New York 115514.

Attn: Prof. David Price

3 Webb Ïnst. of Naval Äròh.

Attn. PrOf. E.V. Lewis, Library Prof. L.W. Ward

Crescent Beach Road

Glen Cove, L.I., New York 115142

Cornell University

Graduate School of Aerospace Engr. Ithaca, New York

i148o

Attn: Prof. W.R., Sears U.S. Merchant Marine Academy

Attn: Academy Library

Kings Points, L.I.., New York 112014

U.S. Merchant Marine Academy

Attn. Cap. L.S.. McCready, Head :Department of

Ezgineering

Dept. of Matheatics

St. Johns University Jarnaca, ew York llI.32

Attn: Prof. J. Lurye

Sperry Systems Management Div. Sperry Gyroscope Co.

Great Neck

Long I1and, New York 11020

Bethlehem Steel Company 25 Bröadwäy

New York, New York iOOO1

Attn. Library (hipbui1ding)

Eastern Research Group P.O. Box 222

Church Street Station New York, New York 10008 ESSO Intérnational

Attn. Mr. R. J. Taylor

Const. ard Develop. Div. Tanker Department

15 West 51 Street

New York, New York lÓOl9

New York University

Coura.nt Inst. of Math.. Sciences

251 Mercier St.

New York, New ?ok 10012

Technical mf. Control Section

Gibbs and Cox, inc 21 West Street

Ìew York, New-York 10006 FFOL/FYS (J. Olsen) Wright Patterson AFB Dayton, Ohio 145133

Fritz Engr. Lab Library Dept. of Civil Engr. Legigh University Bethlehem, Pa. 18015

Sun ShIpbuilding and bD Company Attn. Chief Naval Architect

Chester, Pa. 19013

Commander

Philadelphia Naval Shipyard Attn: Code ?40

Philadelphia., Pennsylvania ₁₉₁₁₂ Library

The Pa. State University Ordnance Research Lab

P.O.. Box 30

State College, Pennsylvania 16801 Director

Attn: Dr. G. Wislicenus

Ordnance Research Lab.

Pennsylvania State University

University Park, Pennsylvania

Commanding Officer

U.S. Naval Air Development Center Johnsville

Waririi±ìster, Penisylvania ₁₈₉₁₁₄

Library

Naval Underwater Systems Center Newport, Rhode Island 028240

Applied Research Lab Library

University of Texas P.O. Box 8029 Austin, Texas 78712 Tracor Incorporated 6500 Tracor Lane Austin, Texas 78721

Southwest Research Institute 8500 Colegra Road

Applied Mechaniós REview Attn: Dr. H. Abranison

Sañ Antonio, Texas .78206

College oÍ' Engineeiing

Utah State University Logan, Utah 814321

Office of Naval Research

800 N. Quincy Street

Arlington, Virginia '22217

4ttn:

Mr. R.D. Cooper (Code

AFOSR/NAM

11400 Wilson Blvd.

A±'lington, Virginia 22209

Technical Library

Naval Proving Ground

Dehigren,

Virginia

2214148

RObert Taggart1 Inc.

Attn. Mr. Robert Taggart

.3930 Walnut Street

Fairfax, Virginia 22030

Newport News Shipbuilding

Attn:

Technical Library Dept.

14101 Washington .Mnue

Newport News, Virginia 23607

Naval Ship Engineering Center

Norfolk Division

Small Craft Engr. Dept.

Attn:

D. Blount (6660.03)

Norfolk, Virginia 23511

Engiíieering Library

Puget Sound Na'al Shipyard

Bremerton, Washington 983114

Applied Physics Lab

University of Washington

1013 N.E. 140th Streêt

Attn:

Technical Library

Seattle, Washington 98105

Catholic University of America

Attn:

Dr. S. Heller.

Dept. of' Clvi]. & Mech. Ear.

Washington, D.C. '20017

Commander

Naval Oceanographic Officé

Li'orary

1438)

Depa±tment of the Navy

Washington, D.C. 20390

Commander

Naval Ordance Systems

Command (ORB 035)

Washington; D.C.

20360

C ominan de r

Naval Shi.p Systés Command

Washington., D .C... 20360

Ships 2052 (3

ys), Ships 032412B

Ships 0372,

Dept. of Transportation

Libráry TAD-1491.1

1400 7thStreet., S..W.

Washington ,D .C. 20590

Dept.. o.f Transportation Library

Acquisitions Section, TAD 1491.1

lj.Q0 7th Street, S.W.

Washington, D.C. 20590

2

Director

Attn.:

Library

Attn.:

Code 2027, 2629 (ONRL)

U.S. Naval Research Lab.

Washington, D.C. 20390

2

NatiOnal Bureau of Standà.rds

Washington, D.C. 202314

Attn:.

P. ebanoff (FM 105)

Fluid Mechànics

Hydraulics Section

National Science Foundation

Eng.neering Division Library

1800 G. Street, Ñ.W.

Washington, D.C. 205.50

12

Director

COmmander

Defense Doewnentation Center

Nava]. Facilities Engineering Command

5010 Duke Street

Code 032C

California Inst. of Tech. Attn: Aeronautics Library

Attn: Dr. T.Y. Wu

Attn: Dr. A.J. Acosta Pasadena, Calif. 91109

Chief Scientist

Office of Naval Research

Branch Office. L

1030 East Green Street

Pasadena, Calif. 91101

Tchnical Library

Charleston Naval Shipyard U.S. Naval Base

Charleston, South Carolina 291108

Technical Library

Nòrfolk Naval Shipyard

Port sinouth, Virginia 23709

Commander

Pearl Harbor Naval Shipyard

OX. 1400

Attn: Code 202.32

F'PO San Francisco, Calif. 96610

Technical Library

Portsmouth Naval Shipyard

Portsmouth, New Hampshire 038014

Commanding Officer

Office of Naval Research Branch Office

1495 Summer Street

Boston, Mass. 02210

Commanding

Officer. Office of Naval Research

Branc,h Office Chicago 536 South Clark Street

ChicagO, Illinois 60605

Library of Congress

Science and Technology Div. Washington, D.C. 205140

Oceanic s, Incorporated

Dr. Paul Kaplan

Technical Inudstria3. Park

Plainview,L.I., New York

11803

-b-Office of Naval Research

Aréa Office - Scientific Section 207 West 214 Street

New York, New YorkiO011

Planning

Dept. Bldg.. 39

Technical Library, Code 202.2 Bostoi Naval Shipyard

Boston, Mast. 02129

Society of Naval Architects

and Marine Engineers 714 Trinity Place

New York, New York lOÓO4

Technical Library (2146L) Long Beach Naval .Shipyard

Long Beach, Calif. 90801

14 _{University of California}

Naval Architecture Department

Attn: Prof. J.V. Wehausen

Attn: Librè.ry

Attn: J. Paulling, W. Webster Berkeley, Calif. 914720

North American Aviation, Inc. Space & Information Systems Div.

Attn: Mr. Ben Ujihara (SL-20)

122114 Lakevood Blvd. Downey, Calif. 902141