On the equations of a thick axisymmetric turbulent boundary layer

(1)

OF A THICK AXISYMMETRIC

TURBULENT BOUNDARY LAYER

by

V. C. Pate!

et

_{czZ. 1968),}

nominally

two-dimensional turbulent boundary ìäyers developing on surfaces of a1Ï óurvature have beeh studied extensively, both experimentally and

theoretically. Although the axisymmetric case. is of considerable interest

in many engineering applications it does not appear to have received any special attention in the pasto The reason for ignoring this aspect almost certainly lies in the often quoted observation that the influence of transverse curvature can be neglected provide. tue boundary layer

thickness is small in comparison with the local radius. of the. body. This

of course begs the question: HOw all. d.oes the boundary layar thickness

need to be for this approximation to hold? If one considers the boundary

layer on a long thin cylinder df constant radius, with its axIs

oiiénted along the flow, it is clear that. this .aproximation may not be applicable sufficiently far from the. leading edge. Furthermore, on a body of revolution of finite length there is always a region close to the tail where the boundary layer thickness may become much larger than the

local radius of the 6ody. The puiosé of this paper is to explore the.

equations of such thick a.xisymmetric turbulent boundary layers and examine

the possibilities.of extending convehtional thin boundary-layer calcu-lation procedures to the solution of these equatIons,

II. DIFFERENTIAL EQUATIONS OF THICK AXISYNMETRIC BOUNDARY lAYERS

Let us consider axially symmetric flow on a body of revolution

(5)

respectively. We choose a curvilinear co-ordinate where x is the distance

measured along

a meridian,

measured

normal

to the surface of the body, and z

Then, from geometry (see Figure i) we have

r

=

r

+

y cos4,

dr o = sin4,

= -

hR

= -K,

dx Dr =

cos,

= (1 sine = (1-i-<y) sine,

where r is the distance from the axis of the body and is the angle bet-ween the axis and the tangent to the meridian. The metric coefficients, or linearizing factors, associated with the X, y, z, directions are

h11+Ky, h21, h3r,

(2)

respectively.

Following Nash and Patel (1972), the equations of conservation of mean-flow momentum, i.e. the Reynolds equations, for steady turbulent flow in the above co-ordinate system may then be written as follows:

D

,-K

_uv+

+

-tuv)

h1Dx

Dy

_;i

h1Bx

p Dy

+

)uv+

(2K

cost - sin

₂ sin

r r r

F 1 D2U D2U dx Sifl4))

i

DU

-' [ 2 +

-

ii1 2 dx r h1 Dx + (e-.

COS)

DU 2K DV

-

K2 + S1fl2)jJ

r

Dy2Dx

h12

(1

dK dx

sin24

+

Ata

dx

h13 dx

2r2

r

h1

y]

= o,

(3)

system (x,y,z),

y

is the distance

is the azimuthal angle.

(6)

hax

ay

sin

--I- uv-r +

u2 +

_h1-

-4

> + _3y ( +

v)

p

j:. av

a2v

2. + C0s4) ..YL r (C0 ₊ cs4( z 1 d , di

h2dx

h13dx

The equation of continuity in this co-ordináte

_systeecòmes

3Vsin4

_u+

_(COSe

h1ax

ay

r r h1

Here, U and V are the cponents of mean vélocity iñ dIrections x and. y, respectively. The velocity .fiuctuati'oñs in the x, y, z diect1öns are denoted by u, y, reSpéctive1y p is static pressure, p is density

and y is kinematic viscosity.

-In order to treat boundary layer flow on abody öf r&folution we

recognizé that, in general, the body has three dist-inct length scales,

namely the overall length L, the 1oigitudina1 radius of curvatuieR, and.

the transverse, radius r. in the usual thin boundary-layer _{theôr'y it is}

assumed that' the thickness of the layer, (5, is everywhere _{at least an}

order of magnitude _{a1ler than the three length scales, i.é. (5/L«1.}

6/R«1 and 5/r«l.

With this assumption, order of magnitude

onsidera-tions applied to equaonsidera-tions (3),

₍₄₎

_{and (5), lead t} _{the well known thn}

axisynimetrió boundary-ierer equations:

au au

a

-(6)

ay

ax

p ay

ay2

(7)

(1..

Si'fl) .;L_ìï

h12dx

r

2K U COS K2

r2

+ (4) sin2 sin ic h1 0. 2r2 r

v

o.

(5)

(7)

and

dr

i o

3x ay

U----O.

o

These are identical with the equations for plane surface flows except for the additional transverse curvature term in the equation of continuity. This additional term presents no difficulty and many of the existing

integral as well as differential calculation methods, constructed for plane surface boundary layers, can readily be extended to calculate the development of a thin boundary layer on a body of revolution. In what follows, however,

we shall consider the situation in which the boundary layer my be regarded as thin in comparison with the overall length L and the longitudinal radius of curvature R but not in comparison with the local transverse curvature of

the body. We shall therefore examine situations in which 6/L«l, tS/R«i

and 5/r0r 1. Two flows of practical interest where these conditions are closely realized are sketched in Figure 2. In both cases we have boundary layer behavior since, according to the rather broad definition, the direct influence of viscosity and turbulence is still confined to a "narrow" region

(in comparison with the infinite expanse of the overall flow field) close to

the boundaries. Although the two cases appear to have a number of features

in common we shall find later that they behave in quite different fashions and consequently have to be treated in quite different ways. The analysis of both, however, starts by examining the Reynolds and continuity equations with the longitudinal curvature terms neglected, viz

u + V au a + 2) + - - (ruy) +i a

-

six ax p

ray

r av a _{2) +}

-- (r)

+ COS (v2_w2) ax

ayay p

rax

r y[1 a

(r)+-- r

av i

a ay) sin2 (cos())2

- r ax ax r ay ay 2r2 r V] - O (io)

S19

1 0, (9) (Ur) + f(Vr) = 0. (8) r1 _a _au (Sifl)2 u

- vi- -

(r )

Lrax

ax +

ray

-- (r ay) r

(8)

5-We now proceed-to see what simplifications, _{if any, can be} _{ade'in these} equations for the two cases shown in Figuré 2.

We consider first the case of a thick boundary layer growing on a long slender cylinder of constant radius, _{so that} _{= O.}

Here., it

is reasonable to assume that, regardless of the relative thickness _of.the boundary layer, the mean-fio _{streamlines remain nearly parallel to the} surface so that the normal component of _{rneaíi vélocity is much: smaller than} the longitudinal component, i.e.

_V«U.

_{We further assume that the Reynolds} stresses will be similar in magnitude to those occurring in a flat-plate

boundary layer. _{With these assumptions, order of magnitude considerations}

appliedtÒ equations (9),(1O), and(ll) _{lead to}

au au

ia

-

au

U - + V r- + - {r (uy

_- _{= 0} (12) ax ay

ray

_ay and r + * (Vr) = 0.

(u)

Notice that, within the approximations made, thé boÚndary layer _{on a cylinder} of constant radius develops in _{a constant pressure field.}

Pressure varia-.

tions across the boundary layer do, however, occur not only _{as a result. of}

the. normal Reynolds stresses

_{but also dueto thé dis1acent}

effect of the boundary layer. _{In öther words, the êxterial potential flog behaves as if}

it were developing on a cylinder _{of ever-increasing radius.}

Détailed calculations of boundary layer growth, however, indicate that _{Jthough the}

boundary layer 'itself may be _{very thick in comparison with the radius of}

the cylinder the rate of inörease of the displacement thickness is aìl,

and cömparable with that _{occurring on a flat plate..}

This implies that the

sélf-induced pressure gradients, _{and consequently the curvatures}

of the

mean-flow streamlines, are negligible. _{In the present case we may thereföre} say that the ThTERACTION between the

boundary layer and. the external potential

flow is WAK and the equations

of the boundary layer, narnely equations (12)

and (13), can be Éolved _{oncé the vélöcity of the' external flow is specified.}

Equations (12) and (13) have indeed' _{been used by a number of}

workers to study the influence of transverse surface curvature _{on the} develop-ment of laminar as well _{as turbulent boundary layers.}

(9)

recently reviewed the previous studies and also presented results obtained by the solution of equations (12) and (13) using finite-difference techniques. For laminar flow the numerical results of Cebeci showed excellent agreement with the previous analytic studies of Seban and Bond (1951), Kelly (l951),

and Stewartson (1955). In the case of turbulent flow Cebeci employed an

eddy-viscosity model for the Reynolds shear stress, with the additional assumption that this model is not directly affected by transverse curvature, and obtained good agreement with the experimental data of Richmond (1957)

and Yasuhara (1959), collected from turbulent boundary layers on slender

cylinders of constant radius. The overall success of Cebeci's calculations

would appear to vindicate the assumptions made in deducing equations (12)

and (13).

We consider next the second case shown in Figure 2, namely the boundary layer near the conical tail of a body of revolution. Here,

considerations of the principle of conservation of mass, applied to the flow within the bouñdary layer, immediately lead to the conclusion that the diminishing radius of the body must be accompanied by a rapid thickening of

the boundary layer. This thickening is associated not so much with the

adverse longitudinal pressure-gradients which are undoubtedly present as with the changing geometry of the surface. In the absence of premature

separation prior to this thickening we have a situation in which the boundary layer thickness may be much larger than the local radius of the body. While

this suggests that transverse curvature effects will be present hère just as in the previous case, there is an important difference in that the rapid thickening of the boundary layer leads to appreciable divergence of the mean-flow streamlines in planes normal to the surface and consequently it is

no longer possible to assume V«U. In other words, we now have substantial variation of pressure across the boundary layer. This variation of pressure is of course accounted for by the y-momentum equation, equation (10). The

flow close to the tail of a body of revolution is thus characterized by a STRONG INTERACTIOI'T between the boundary layer and the external potential flow, with the result that neither can be determined independently of the

other. In particular, we can no longer use potential flow pressure distribu-tion on the wall, together with the usual constant pressure assumpdistribu-tion (for

(10)

-7-iterative techniques in which potential fiow and boundary layer

calcu-lations are performed simultaneously have to be devised,. We shall discuss these later on in the paper.

Returning to the generai Reynolds equations,i.e. equations (9),

(io) _{and (ii), we see that the only simplification that can be made in}

these is that we can neglect some of the viscosity terms in the momentum equations since they are small and important only in the sublayer. In the

absence of any prior knowledge of the importance of the tubulènce. ters we shall retain them in the analysis for the present. If equation (io), with the viscous terms neglected, is integrated with respect to y, maldng some use of the equation of continuity, there results

p-p

-

ry

- _{+ -e--} (UV+uv) d.y

p o +

i:

UV+ÚV dy + cos J '

'"

dy, (1h)

where p(x) is the pressure distribution on the wall, y=O. The recent

experi-ments of Patel, Na.kayama and Damian (1973) indicate that the Reynolds stresses

are much smaller than the corresponding products of mean velocity coppnents

when the boundary layer is thick,i.e. uv«tJV and v2«V2. This leads to the

conclusion that the static pressure variation _{across the thick acisyetric} boundary layer is associated primarily with the mean flow. _{Consequently,}

the d.-ifferential equations fÓr this case may be written

y

px r3y

3y

py

- (Ur) ±

f

(vr) = 0. ₍₁₇₎

These equations form an elliptic set since they differ from Euler's potential flow equations only in the presence of the visòous and _{Reynolds shear-stress} terms in the first equation.

(11)

III. CLOSURE OF TIlE DIFFERENTIAL EQUATIONS.

It will be noticed that equations (15) through (17) include the somewhat simpler equations of the boundary layer on a cylinder of constant

radius as a special case. We shall therefore confine the subsequent

dis-cussion to the general case. These equations contain four unknowns, namely

U, V, p and uy. In order to make them determinate it is therefore necessary

to furnish one additional equation. This usually implies some assumption

concerning the Reynolds stress uy.

Perhaps the simplest way to effect closure of the differential equations is to employ the classical phenomenological theories in which the Reynolds stress is related to the mean flow via mixing-length or

eddy-viscosity functions. This, however, raises an additional uncertainty concerning the particular variation of mixing length or eddy viscosity. through the boundary layer that has to be chosen. In the treatment of the thick boundary layer on a cylinder Cebeci (1970) made the assumption that the eddy-viscosity model found most suitable for thin, plane-surface boundary

layers applies equally well to thick axisynunetric boundary layers. This

implies that there is no direct influence of transverse curvature on the

eddy viscosity. Bradshaw (1969) and others, on the other hand, snggest that longitudinal surface curvature has a marked influence on the turbulence structure, and by implication, on the mixing-length and eddy-viscosity

distributions through the boundary layer. The recent experiments of Patel,

Nakayama and. Damian (1973) in the thick boundary layer near the tail of a body of revolution also showed that mixing length as well as eddy viscosity are influenced directly and significantly by transverse curvature. The

problem of determining this influence quantitavely is therefore an impor-tant one and worthy of further research.

The closure of the differential equations can also be effected by

a number of other methods. Here we shall single out for discussion the procedure associated with the names of Townsend, Rotta, Bradshaw and others, in which a rate equation for the Reynolds stress uy is obtained by postulating plausible models for the various terms in the turbulent kinetic-enerr

equation. For a thin boundary layer developing on a plane surface the turbulent kinetic-enerr equation may be written

(12)

+ V(2/2)

..

.;.(9!+

_{) +}

E = 0,

(18)

convection

where

_pq2

p

(2+2+2)

is turbulent kinetic-ener

_and

_E

_represents

dissipation into heat. Following Townsend

_and

Rotta, Bradshaw, Per±iss

_and

AtweJ..l (1967)

assume

that

-=a,

q2 1 2 p

production diffusion disipation

where a1 is a constant (o.l5), while the diffusion

fuctioh

_{and the}

dissipation length 1

are

functions of y/6

only.

_{Iñtroductjon of equations}

(19) equation (18) leads to the required rate equation _{for the Reynolds}

stresS, T , in the

_form

W

_}-+

V

-

₊

- {G

----+

32

0. (20)

It can be shown that, when the convection

_and

_difusion

_terms

_{in this} equa-tion

are

_{neglected,, the.resulting equation (Production} _{= Dissipation)}

reduces to the familiar mixing length

_formula

T =

pL2(.F)2

_,

the mixing-length

L being related to the dissipatiofi length

J by

L312L

For the thick axisynmietric boundary

_{1arer the turbulent}

kinetic-enerr

equation, correct to the approximations _{already introduced,} _{ay be}

written IJ

.(q2/2)

V

q2/2)

₊

Fu2

₊ U + CO8 + ) convectIon _production E

(T)3"2

('9)

di f fu.sion

dissipation

rax

2 _p

r3y

2 _p E = 0. (21) - r

-.'

- -

(13)

-satisfactorily if it is assied that (- + is equal to (. +

so hat the same diffusion function can be used to model both. terms.

Experi-ments of Patel

et al.

further indicate that the ratio of -uy to i is nearly

constant and equal to 0.15, the value found for thin boundary layers.. If the above observations are intröduced. in equation. (2].) we obtain

.

/T

2a1 ax ay ay r.ax a13/2 p

-lo-In order to develöp a rate equation similar to equation (20),retaining a1i the esséntial elements of the method of Bradshaw

et

al.,

it is necessary not only to assess the effect of the extra production and diffusion terms

0±1 the ôverall ènerr balance but also to say somethi±1g about the direct influence of transverse curvature on the three empirical fuflctions a, :

d I. (or L). Of the. production terms, the dominant one is uv, as

id

the case in thin boundary layers. Since V is no longr small, however, he

Other terms, although smaller than the dominant one, cannot be neglected

a priori Fortunately the turbi.±lence measurements of Patel, Nakayama an4

Damiàn cited earlier iïidicate that in a thick boundary layer all the Reyrols stresses are much smaller than those expected jn a corespon.ing boundari

layer (haviflg the same mean-velocity profile, say) developing on a plane surface. this implies that retention òf only the major production term will not involve any appreciable error. The extra diffusion térm in equation (21), whiãh is expected to be generally smaller than the usual one, can be handled

____ +

____ = o. -(22)

Heré it. has beeh assumed thatthe diÍfusioh and issipation-length functïon,

G*arld T. L*/a13/2, respectively, may be different from their thin

boundary layer counterparts owing to the direct i±1fJ.uence of t±à.nsverse

curvaturé on the turbulence.. As suggested earlier, L* can be identified. with the conventional mixing length if convection and diffusion are neglected..

The variation of mixing length measured by Ñtei

et ai.

in their thick _H

boundary layer experiments is compared i4th the L function. cpòsed by

Bradahaw, Ferriss and Atwell (l96T) in Figure 3 From this it is clear thajt

(14)

-11-the mixing length decreases markedly as -11-the boundary layer thickness in-creases in relation tO the local radiu ¿f curvature. _{The increase in}

dissipation implied by this, and the _{observed decrease in production} mentioned earlier, suggest that the convection and diffusion of turbulent kinet1c-enerr, which are relatively unimportant in a thin boundary layer, become apprejab1e in _{a thick boundary layer.}

This, in turn, implies that it is no longer possible to associate the dissipation length with the conven-tion&1 mixing length. _{Previous experience with the use of equation (20)}

in the calculation of thin boundary layers has shown that the dissipation

length L, _{is the most important one of the three empirical functions (as}

might be expected from its _{association with mixing length).}

The observations

made above, however, appear to suggest that in the treatment of t1ick

-boundary layers the diffusion function _{and, to a lesser extent, the} _convective

constant will play a greater role in _{the performance of equation (22)}

as a

closure relation. _{F\rther work is obviously required to find the quantitative}

behavior of G* and L* _{across the boundary layer and their dependence}

on transverse curvature.

IV. _{ON THE SOLUTION OF THE DIFFERENTIAL EQUATIONS}

Regardless of the method used to close _{the diffexential equations} of the mean flow, equations (15) _through

₍₁₇₎

remain elliptic. .j contrast, the usual thin boundary-layer equatiöns are either parabolic _{or hyperbolic}

-depending onhow the Reynolds stress is related to the mean flow. _The ellipticity of the equations of the thick boundary layer means that it is

not strictly possible to use conventional forward-marohlng numerical _techniques. These equations have to be solved as a boundary válue problem and herein

lies a major difficully since not all the boundary cönditions _{are well defined.} The primary difÍiculty _{concerns the specification of the pötential} flow outside the boundary layer and the pressure distribution _{onthe surface.}

For à. thin boundary layer the _{constancy of static pressure}

açross it simplifies

the problem considerably since _{then the wall pressure dIstribution}

is sImply

related to the velocity in _{the freestream.}

For a thick boundary layer, however, the pressure remains _{an unknown quantity which we seek to}

determine. The

y-mornentum equation, which is _{usually ignored, now serves to relate the} pressure fiei'd to the velocity field, _{but in order to solve it we need to}

(15)

prescribe either the pressure at the wall _{or that at the edge of the boundary} layer It is clear that neither is known _{a priori owing to the stiong}

interaction between the boundary layer and the external potential flow In oHer to pröceed at all we therefore _{need sorne sort of an iterative}

scheme iíi which successive approximations are inadê for the external and

boun-dary Larer flows. One such procedure my invo1e the following _steps: _A

potential flow solution ay be obtained for the given _{istmmetr1c body ignoring} the boundary layer altogether., The _{pressure distribution onthewafl so}

obtned can then be usedto solve the x-momenti _{equation, the equation or}

continuity and. the closure equation, ignoring the variation _{of static pressure,} to obtain a first approximation for the bound.ary layer behavior. _{Notice that}

this destror, artificially, the ellipticity of the mean flow equations.

Since the first bounarr layer calculation leads to thé velocity distxibutions through the boundary layer, we can find the value of the stream function at

the edge of the bou.ndary layer. _{A second potential flow calculation} _{can then} be performed. in' which the condition of tangency of the _{flow on the body surface}

is replaced by the stream function at y = 6 and some suitable extension o?

this stream ?unction to infinity to account for the wake of the body. Thìs,

tbe second'potential flow solution involves the stream function boundary cbndition extending to infinity along the edge of the boundary layer and the

wake, and. not the shape of the body as such. The pressure distribution Ialòng the edge of the' boundary layer determined in this _{manner can then be used,}

together with the velocity field within the bcundáry layer obtained ear1.e-,

to fInd the static pressure variation acröss the boundary layer implied by

the y-monientum equation. 'Note that this leads to a first approximation foi'

the hitherto unknown press _{e distribution on the wali.} _{A secònd boux4ar} layer calculation can then be performed using the pressure variation acros the boundary layer thus obtained. Again, this involves theolution of thi

.x-momèntum equation, the equation of continutiy and. 'the closure _elation,

that the eflipticity of the complete set of equations is avoided A number of iterations of this type will eventually lead not only to the predicti9n

of the boundary layer development but. also 'the 'pressure field asociated with it _{Perhaps the weakest link in this approach is the necessity to make some}

asstptions conceining the behavior Of the wake, but it is éxpécted that tie po'ential flow at the edge. of the boundary layer on thebody will nöt be undu1r sensitive to the precise assumptions that are made.

(16)

-13-The amount of numerical computation involved in a procedure oi

the type described above is not to6 large when one considers thé fact that methods fOr the solution of potential flow equations as well as the differen-tial equations of thé boundary layer are aJ.ready in existence. The author

and his colleagues have recently extended the method. of Bradshaw, Ferriss and. Atweil along the lines suggested in the previous section in order to

calculate the development of a thick boundary layer when the statiò _pressure variätion across the boundary layer is prescribed. It is hoped to cobine this method with a suitable potential flow calculation _{procedure to test the} iteration scheme described above.

V. MOIvTUM INTEGRAL EQUATION

In view of the difficulties associated with the _{solution of the}

differential, equations of a thick axisymmetric boundary layer 'it may be moré

profitable to examine the possibilities of extending _{one or moré of the} well known approximate methods of calculation which _{are based on the}

integrated forms of the differential equations. _{If integral methods are to be} considered it Is necessary to obtain the momentum integral equation which includes the variation of static pressure across the boundary layer. This

equation is readily öbtained using standard _procedures.

We write equation (15) in the form

U

+ V-+ ---+

₊ _.{r(-' _{)} = O,}

(23)

where the subscript e denotes the value at the edge _{of the boundary layer,} i.e. at y . Then, using the equation of continuity, this òan be

re-arranged' to obtain _(U2r)

f(u J

d.p Ur]dy)

+L_

+

-{r(-v

= O.

Integration of this with respect toy from the wall to the edge of the bOun-dary layer gives

Ju2rdy

- U

J0rdY

+ ₊

i:

- 0, (24)

(17)

equation (214) can be written dO

le

dU O dr 2 o

-+ (2o2+Ol).

e o o _o I r 1

ra

PP

U 2

dx _p 2 e j dy + U 2

_J

r dx e) (27) e

00

_e

_oo

This is the basic form of the momentum integral equation for a thick

axisymmetric boundary layer across which there is appreciable static pressure

variation. _{The first term on the right hand side of this equation can be}

expressed in terms of the normal

_component

_{of velocity at the edge of the}

boundary layer and the layer thickness by making use of the Bernoulli equation,

p +

p(U 2 + y 2) =

constant,

e 2 e e

which applies with sifficient accuracy at y = O, and the fact that

d.y = O {l +

_cos4}.

2r

o

The second term on the right hand side of equation (27) represents simply the rate of change of the integrated pressure force _{across the boundary layer,} and can be written in a number of different ways using the integrated

y-momentum equation _y = e +

J

(u L

_v)dy

w

-

Jcu

+ V ) dy. (30) Io

r

r o _o (3U where T U

w is the wall shear stress.

If we now define the displacement _{thickness 6, and the momentum} thickness 02 in forms appropriate to axisyrnmetric _{flow, viz}

f = - )

t

dy O = 2

JU

e o (25) e

and introduce the skin-friction _coefficient

T

(18)

-15-It will be noticed that, when the ìou.ndary l8yer Is thin (i.e. _when

and V«tJ), both terms _{on the right hand side of èquation (27)}

can

be neglected, and the _{equation reduces to the well known form}

Obtainable

directly from the equations ôf thin axisyetrje boundary layers, _nathely

equations

(6)

_{and (8).}

Yl. _{CLOSURE 0F THE INTEGRAL EQUATIONS}

For a thin boundary layer the momentum integral _{equation còntains} tiree unknown,.dimensjoess, _{integral quantities: R, H}

and. Cf where

R0 _{Ue62/v is the moxentum-thjckness}

Reynolds number and H _61/62 _is the shape parameter of the _{velocity profile.}

When the velocity &istributjön

in the external flow,

or the pressure distribution on the wall, is specified this equation can be _{solved for R0 or.ly by providing}

two additional

relation-ships. _{.A large number of}

suggestions have been thade for this _{purpose and}

most of these were examined at the Stanford Conference (laine

_{et al.1968).}

Here, it suffices to note that closure of the _{momentum integral equation is} usually effected by the introduction of a skin-friction _{formula of the form} Cf = Cf(H, R0) and an auxiliary, or shape-parameter, _{equation which relates} either directly or indirectly, the rate of chánge _{of H with x to the other} variables in the momentum _equation.

The skin-friction formula _{and the} shape-parameter equation often nvo1ve the explicit use of _{a velocity profile family.}

In what follows _{we shall explore the possibilitje}

of extending some of

these ideas to the _{treatment of thick axisysmetric}

bouñdary layers.

Examination òf the momentum integral equation for.the thick boundary layer obtained inthe previous section shows _{that, even whén the} velocities and the _{pressure in the ex-bernal flow}

are known, we have two additional unknowns, namely the boundary layer thickness _{and the integral} involving the static _{pressure variation across.the boundary}

layer. These

unknowns are contained in the

two terms on the right hand sidé of equation

(27). _{The relative importance öf these terms can best be}

judged by

referring to experimental _data.

The recent measurements of Patel, Nakayama

and Damian. suggest that _{both terms are much}

larger than the other terms _in the equation but the first is negative while the _{seàond is positive.}

The

sum of the two, which is

poSitive,however, is of the same order of magnitude as the other terms in the _equation.

In estimating the magnitude of the right hand side of equation

₍₂₇₎

_{it is therefore advisable to regard it as}

(19)

a single term. In Order to Obtain sorné idicatio±i of the imprtance of

these terms we have used the data of Patel et al..rnentioned above. In

?igure the experimental values of are compared with those

lela

from equation (2'r), with the Mght hand sde omitted, usin the measuredi

values of ti, , and. Cf. The disagreement between experiment and

calcu-lätion indicates the importance of the terms whidh were omitted. Notice

that these terms become appreciable only over the last ten percent of the bodj length where the böundary layer is thiek and there is significant,

variation of static pressure across it. Va.rious attempts were made to rèlate

thesé terms with the other quantities in equation (27) but noné of these

proved very successful. An attempt was .so made to evaluate these terms

from the measured data but it was abandoned owing to the uncertainties involved in taking small differences between two large terms which

them-selves involved differentiation of ifl-defined. quantities Such as the boundary

layer thickness. During the course of thesé calculations,however, it was

observed that the momentum thickness could be predicted accurátely withotit ilncluding the right hand side of equation (27) if the second term on the eft

hand side of this equation was inreased artificially. This could be accomplished quite simply by using the hypothetical freestream velocity

Ue implied by the measured wafl presste d.istributi.on and BernOulli's eqution

in place of the real measured variation. of tJe Thus, the momentum integrál equation was approximated by

dr 2 o 1,.. dx

2¼f_V

o where + dU dp

-

e w -(31

The results Of integratnghis equatiôn using the measured values of p,1

and f are also shown in Figure 4. From this we see that the usual thin

boundary-layer momentum integral equation can be used. to calcula.te the thomentpm

thickness development with acceptable accuracy provided the usual presue

dU

gradiènt term is identifiéd with the largest pressure gradient

e.

experienced by the boundary layer, namely the pressure gradient at the wall

This then represents an approximate but simple way in which the. 'terms on the right hand side of equation (27) can. be taken into account.

(20)

-17-To use equation (31)in a calcúlation procedure we still neèd

two additional relations. As remarked earlier, the skin-friction coefficient

in a thin boundáry layer is usually taken to be a function of H andR0.

From the experiments of Patel et al. it appears that even in the thick boundary

layer the longitudinal.velocjty profiles conform wellwith the

families, such as those of Coles,

_(1956),

Thompson

_(1965),

and otïiers, from

which the skin-friction formulae are deduced, provided the integral

para-meters are evaluated according to the thin plane-surface bondary _layer

definItions. _{Thus, the skin-friction law may be represented by}

Cf =

Cf(L Ç

₍₃₂₎

where the bars denote. "planár" definitions, i.e.

-

Ç= j

(1

tJe2,

H = (33)

It can be shown that the axisyetric thicknesses defined _{by equation (25)} can readily be reited to the planar thicknesses above and the ratio 6/r when expliit use is mà.de of a particular profile family, _{so that the} skin-frictiOn coeffjclent for the thick boundary layer _{can be expressed as a} function of H, R0 an& /r.

Considering the shape parameter equation next, it is clear that

we can not possibly discuss thé use of all the different équatIons proposed

to date. _{Here we shall consider the well known entri±ment method of Head}

(1958)

since its extension to treat thick axiSythmetriè _{boundary layers can be} _de

rather simply. _{For an axisymznetric boundary layer the volume fluì,}

Q,, at any streamwisê location is'

=

i: Q irr U dy = 2irU{r

+ cos

o}

(314)

The rate of change of t.is with _{x is the rate of entrainment of freestream}

fluid into the bouidary layer. _{Introdticing a non-dimensional coefficient Of} entrainment, 0E' we have '

(21)

en tie boundary.iayer is thin (i.e _{when (S«r ).this expression reduce}

to that given by head. _{In order to take use of this equation in} _{a calcu1a}

tion procedure we. now need to rnake _{some assumption concerning C. Fo tii}

boundary layrs Head postulated that CE depends upon the freestream veio:cir, a length scale of the flow in the outer region of the boundary layer an the

shape of the veloöity profile in this region, and from dimensional conSidea-tions deduced that CE is a function only of the shape parameter H*

-Following similar logic we xay generalize Head's result to consider thick axiSmetic böundary layers by asuming that C is the same ftnction oÍ he

'H

shape parameter H* 1 _{which reflects only the shape of the vocity}

profile without regard to the local radius of the surface. Whether this assumption is adequate, or we ieed to introduce an explicit dependence of

C _{on a curvature parameter such as (Sir, can bést be judged by detailèd} comparisons with experiment.

qÙat.-ions (31), (32) and

_(35.),

_{togethe±- with the velocity prbf±le}

family chosen to relate the planar definitions of the integral _{parameters to}

the conventional .axisymmetric définitions, foi a closed set in the thre

unknowns Cf H. and R0. _{Although the. boundary layer thickness} _(S _appears

explicitly in equation

(35)

_{and iplicitIy in the friction formula. it is} clear that it can be related

th

2 and. the other integral. parameters via

the velocity profile family.

The method outlined above has been employed bytheauthor to .prdict thé devé1opment of thick axisymetric turbu1et bot dary layers. _{The det.ils} of the method and its experimental verification are given in _{a sparate}

paper(Patei 1973.).

VII ON THE SOLUTION OF THE INTEGRAL EQUATIONS

To.calcu:late the development of. the boundary layer using. the,

integral method of the previous section it is of course necessary to pecribe

the: préssure distribution: on the wall. _{This, however, is not known a pr.ori.}

öwing to the strong interaction between the thick boundary layer and the external potential flow It is necessary therefore to resort once again to an iterative procedure in whih successive aprbximations are made fo' the boundary layer and the potential flow. This aspect of the problem has;

(22)

-19-already been discussed in detail in section V in _{conjunòtiön with the}

solution Of thé. differential equations. ,When _{Integral equations are used}

for the boundary layer ca1culatjoi the iterative procedure is identical

but the _{oliition of'the y-momentum equátion required to find the static}

pressure varlatiön across the boundary layer will _{now 'involve theuse of}

the velocity, profile family.

VIII. CONCLUDING RARKS

Here we have considered the problem _{of the thick axisymmetrïc} boundary layer near the tail of a body of revolution in some detail not only because of its practical importance _{in detérmining the form drag Of} such bodies but also because this is _{one of the few situations for which} experimental data is available to guide _{our discussion.} _{The case of a} thick boundary layer developing on a cylinder of constant radius mentioned briefly in Section II has received rather more attention in the past. _This

problem is somewhat simpler, insofar _{as the interaction between the boundary}

layer and the external potential flow _{may. be assumed to be negligible ¿n}

account of the simple geometry. _{Even in this case, however, it is likely}

that there is some direct influence of transverse curvature on the turbulence sò that.it may not be realistic _{to adopt cloSure relations estä.blished}

for thin boundary layers as has been done in the literature.

The problem of. the interaction _{between the boundary layer and}

the external potential flow is of _{courSe not new.} _{Such an interaction occurs} in many situations and, for low-speed'flows, has been Studied in _connection with the determination of the _{influence of thin boundary layers. on preassure} distributions on airfoils and _{other shapes. . The calculation ôf the pressure} distribution by potential flow theory after adding the displacement thickness of the boundary layer to the body surface is a well known example of the

procedures used. _{The work described her} _{differs from these conventional}

'procedures in two respects. _{First, an attempt is made to show how} the

caculation methods used for thin boundary layers may be extended to

cOn-sider thick boundary layers

across which there is an appreciable' variation of

static pressure. _{Secondly, a procedure has been}

suggested for the òalculation:

f the interaction between such a thick boundary layer and the external flow using the more realistic matching condition at the edge of the boundary layer.

(23)

In addition

to

the problem of

obtaining

_{better estimates for the drag} o. bodies of revôlution the proosêd Ieth6d _{may also. find}

_{app1iation i}

the ca1ciato

of the flòw

in

long conicJ. ánd _{nu1ai difTusers.}

(24)

REFERENCES:

Bradahaw, P.

₁₉₆₉

_{"The Analor Between Streathiine Curvature and Buoyancy}

in Thrbulent Shear Flow", J. Fluid Mech., 36,

_177.

Bradshaw, P., Ferriss, D .H. and Atwéll, N..P.

₁₉₆₇

_{"Calculation of} Boundary-layer. Development Using the Turbulent Enerr Eq.uation", J. Fluid

Mech., 28,

_593.

Cebeci, T.

₁₉₇₀

_{"Laminar and Turbulent Incompressible Bouñdary Layers on} Slender Bodies of Rerò1ütion in Axial Flow".., _{J. Basic Eg., Trans.}

Ser. D, 92, 5145.

Coles, D.

₁₉₅₆

_{"The Law of the Wake in the Turbulent}

Boundary Liayer",

J. Fluid Mech., i, 191.

Head, M.R.

_.1958

_{"Entrainment ïn the Turbulent Boundary Layer",}

British

Aeron. Res. Counc., R &M

_3152.

Kelly, H.R. ₁₉₅₁₄ _{"A Note on the Laminar Boundary Layer}

on a CIrcular

Cylinder in Axial Incompressible Flow", J. Aeron. Sci., 21, 6314.

Kline, S.J., Morkovin, M.V., Sovran, _{G. and Cockrêll, D.J.}

1968

"Proceedings:

Computation of Turbulent Boundary Layers- 1968 _{AFOSR-IFP-STAI'TFORD}

CONFERENCE", _{Stanford University, California.}

Nash, J.F. and Patel, V.C.

₁₉₇₂

_{"Three-Dimensional Turbulent Boundary}

Layers", SBC Technical Book, Atlanta.

Patel, V.C.

₁₉₇₃

_{"A Simple Integral Method for the Calculation of Thick}

Axisymetric Turbulent Boundary Layers1t _{To be published.}

Patel, V..C., Nakayàma, A. and bamian, R.

₁₉₇₃

_{"Añ Experimental Study of}

the Thick Turbulent Boundary Layer Near the Tail of a Body of Revolution", Iowa Institute of _{Hydraulic Research, Report No.} _1142.

Richmond, R.L.

₁₉₅₇

_{"Experimental Investigation òf Thick}

_Axially

_Symmetric

Boundary Layers on _{y1inders at Subsonic and Htpersonic Speeds'!,} _Ph.D. Thesis, California Inst. of Techno1or, Pasadena.

Seban R.A. and Bond, R. _1951. _{"Skin Friction and Heat Trasfer}

Char.cte'istjcs

of a Laminar Boundary Layer on a Cylinder in Axial Incompressible Flow", J. Aeron. Sci., 18,

_671.

Stewartson, K.

₁₉₅₅

_{"The Asymptotic Boundary Layer}

on a Ciräular Cylinder", Quart. Appi. Math.., 13, 113.

.Thomspon, B.G.J.

₁₉₆₅

_{"A New Two-Parameter Family of Mean Velocity Profiles}

for Incompressible Turbulent Boundary Layers pn Smooth

_Walls",

British Aeron. Res. Counc., R & M 31463.

Yasuhara, M.

₁₉₅₉

_{"Experiments of' Axiymmetric Boundary Layers Along a}

Cylinder in incompressible Flow", _{Trwis. Jan Soc. Aerospace Sci.,} 2,

33.

(25)

-21-U,u

(26)

-23-.

(a) Thick Adsymmettic Boundary Layer on

Long

Slender cylinder of Constant Radius.

(b) ThikAxisyrnmeric Boundary Layer Near.the

Conical Tail of a Body of Revolution.

FIGURE.2.

TWÖ.EXAMPLSQF FLOWS. IN WHICH

SIGNIFICANT TRANSVERSE CURVATURE EFFECTS

ARE PRESENT.

(27)

0.10 LI 0.08 0.06 0.04 0.02 O 0.2 o D

Bradshaw, Ferriss & Atwell

(1967), thin boundary layers

o D o

0 Xe

G

X

FIGURE 3. THE IIFLUENCE 0F TRANSVERSE CURVATURE ON

MIXING LERGTE

+

0.8

iR

1.0 o + 1.2 X/L ô/r o 0.662 0.152

+

X

0.80 0.85 0.90 0.93 0.261 0.381 0.619 1.09 Data of Patel, Nakayamä and Damián (1973) e _0.96 _2.56 O _0.99 13.00

(28)

-25--Resultant velocity at

_y

----x-component velocity atyb

.___fHypothetical velocity from \waII-pressure distribution o Cf

I

O _{Cf Preston Tube} / / x _{Cf Clauser Plots}

/

5, Experimental

/

o»,

-s-

3%

-0.9 0.7 o _Experiment

- - - Using

_{0e in place of Ele}

-- - - Using Ue in

_{place of} Using 0e o 0.7 _0.8

s-

.4-U X-. t I I ¡ 0.9 X/L 1.0

FIGURE 4. _{CALCULATION OF MOMENTUM}

THICKNESS FROM

EQUATION _{1) USING MEASURED}

_{VALUES OF Cf AND 6.}

EXPERIMENTS OF

_{Patel.Nakayama and Damian}

(1973). 4-w

Q3 z

_X L)

I

I-02 -

_z

w w

0.1v

-J Q .00

Fo

z

'n .001

(29)

130 _Commander

-Naval Ship Research and

Development Center

Bethesday, Maryland

_200313:

Attn:

Code. 1505

CÓd

_{56113 (39 cys)}

Officer-in-Charge

Annapölis Laboratory

Nava]. Ship Research and

Development Center

Annapolis, Maryland 21402

.Attn: Còde 561e2(L±brary)

Commander

Naval Ship Systems Coxrunand

Washington, D.C. 20360

Attn:

_{SHIPS 2052 (3 cys)}

SHIPS 0313123

SHIPS 0372

SHIPS 0342.

12 _Director

Defense Poçumentatjon Center

5010 Duke Street

.Alexanth'ia, Virginia

₂₂₃₁₁₃

Office of. Nava].. Research

800 N. Quincy Street

Arlington, Virginia

₂₂₂₁₇

Attn: Mr. R.D. Cóoper (Code 438)

3.

Office of Nava]. Research

Branch Office

1392 Summer Streét

Boston, Mass.

02210

Office of Naval Research

Branch OffIce (493)

536 S. Clark Street

Chicago

_{Illinois 60605}

Chief Scientist

Office of Naval ReSearch

Branch Office

1030 E.Green Street.

Pasadena, CA 91106

Officé of NávSI Réseárch

Resident Representative

2Ó7 Wèt 213th St.

New York, New York 10011

March 19, 1973

Office of Naval Research

Resident Representative

50 Fell Street

San Francisco3 Ca

913102.

Direc tör

Naval Research Laboratory

Washington, D.C.

₂₀₃₉₀

Attn:

Code. 2027

Code 2629 (0NRL)

COmmander

Naval Facilities Engineering

Command (Code 032c)

.Washingtoù, D.C. 20390

1 Library of Congress

Science & Techno1or Division

Washington,DC. 205130

Commander

Naval Ordnance SystémS

Coiand (ORD 035)

Washington, D.C. 20360

Commander

Naval Electronïcs Labòratory

Center (Library)

_:

San Diègo, Ca 92152

Commander

Naval Ship Engineering Center

Center Building

Prince Georges Center

Hyattsville, Maryland 20782

Attn:

SEC 6034B

.

SEC 6110

SEC 6114H

sEC 6120

SEC 6136

SEC 611313G SEC 6113ÓB

sc 61138

Naval S1ip Engineering

Center

Norfolk Division

m1l Craft Engr. Dept.

NorfOlk, Virginia 23511

(30)

i Library (COde i614o)

Naval Oceanographic Office Washington, D.C. 20390 i TechnIcal Library

Naval Pròving Ground Dehigren, Virginia 221448

J. Commander (ADL)

Naval Air Development Center

Warininster, Penna _1897le

i Commanding Óffier (I.a3l)

Nava]. Civil Engineering LaboratOry

Port Hueneme, CA 930143

Conunander

Naval .Undérsea Cénter

11400 Wilson Blvd.

Ai'1iriton., Virginia 222Q9

-2-1 .AFFOL/FYS (J. Olsen)

Wright Patterson AFB Dayton, Ohio

4433

1 Dept. of Transportation Library TAD-491.'i'

- 7th Street, 'S.W..

Washington., D.C. 2O59

Boston Nava]. Shipyard

Planning Dept. Bldg 39

Technical Library, Code 202.2 BoSton, Mass.

Oì2

Charleston Naval Shipyard Technical Libraxy Naval Base Charleston, S.C. 291408 Code 202.3 Vallejo, CA

_959?

San Diego, CA 92132

Attn: - Dr. A. Fabula (6005) 1 Nor folk Nava]. Shipyard Technical Library..

2 Officer-in-Charge _{Portsmouth, Virginia} ₂₃₇₀₉

Naval Undersea Center

Pasadena, CA

₉₁₁₀₇

J. Philadelphia Navä.1

Shipyard

Attn:. Dr. J. Hoyt (2501) _{Philadelphia, Penna 19112}

Library (13111) Attn: Code. 240

Director 1 Portsmouth Naval Shipyard

Naval Research Laboratory Technical Library

Underwater Sound Reference Division Portsmouth, N.H. 03801

P.O. Box 8337

Orlando, Flordia 32806 1 Puget Sound Nava]. Shipyard

Engineering Library

i Library .Bremei'ton, Wash. ₉₈₃₁₄

Nava]. Underwater

Systems

Center

Newport, R.I. 028140

_i

Long Beach Naval Shipyard Technical Library (246L)

Eesearch Center Library Long Beach,

CA 90801

Waterways Experiment Station

Corps of Engineers 1 Hunters Point . Nava]. Shipyard

P.O. Box 631

Vicksburg, Mississippi 39180

Technical Library (Code 202.3) an Franciáco, CA .94135

2 National Bureau of Standards 2. Pearl Harbor Naval Shipyard

Washingion, D.C. 20234 Code 202.32

Attn: P. Klebanoff (FM 105) Box

4öo,

PPO

Fluid Mechanics San FrancIsco, CA 96610

Hydraulic Section

3. Mare Island Nava]. Shipyard

(31)

Assistant ChiefDesign

_Engineer

for Naval Architecture (Code

₂₅₀₎

'Mare Island 'Naval Shipyárd

Vallejo, CA

₉₄₅₉₂

3

U.S. Naval Academy

Annapolis, Maryland

214Q2

Attn:

_{Technical Library}

Dr. Bruce Johnson

Prof. P. Van Mater, Jr.

Naval Postgraduate

School

Monterey, CA

₉₃₉₄₀

Attn:

_{Library, Code 2124'}

Dr. T. Sarpkaya

Prof. J. Miller

Capt.

L. S. McCready, TJSMS

Director, National

_Maritime

Research Center

U.S. Merchant Marine

Acadeni'y

Kings Point, L. I.,

_{New York 11204}

Attn:

_{Acadenr Library}

U. S. Merchant Marine Äcadezr

Kings Point, L.I.,

_{New York 11204}

Attn:

Academy' Library

Library

The Pennsylvania State University

Ordnance Research Laboratory

P.O. Box 30

State College, Penna

i68oì

Bolt, Béranek' & Newman

1501 Wilson Blvd.

Arlington,, Vir&.nia

₂₂₂₀₉

Attn:

_{Dr. F. Jackson}

Bolt, .Ber.anek & Newman

50 Moultòn Street

Cambridge., Mass.

₀₂₁₃₈

Attn:

Library

L

Bethlehem Steel Corporation

Center Technical Division

Sparrows Point Yard

Sparrows Point, Maryland

₂₁₂₁₉

L

Bethlehem Steel' Corporation

25 Broadway

New York, New York 10004

Attn:

_{Library (Sl2ipbulicijng)}

Cambridge icousticaj.

_{Asociatee, Inc.}

.033 Mass Avenue

Cambridge, Mass.

02138

Attn:

Dr. M. Junger

Cornell Aeronautical Laboratory

Aerodynamics Rêsearch 'Dept.

P.O. Box 235'

Buffalo, New York 14221

Attn:

Dr. A. Ritter

Eastern Research: Group

P.Ö. Box 222

church Street Station

New York, New York 10008

Esso International

Design Division, Tanker Dept.

15 West 51st Strebt

New York, Néw York 10019

J.

Mr. V'. Boatvright, Jr.

R & D Manager

Electric Boat Division

General Dynamics Corporation

Groton, Conn.

06340

Gibbs & Cox, Inc

21 Wést Street

New YOrk, New York 10006

Attn:

_{Technical Info. 'Control}

Hydronautics, Inc.

Pindell School Road

Howard County

Laurel, Maryland

20810

Attn:

Libráry

2

_{McDonnell Douglas Aircraft}

_Co.

3855 Lakewood. Blvd.

Long Beach, CA

_906b3.

Attn:

J. Hess

A.M.O. Smith

Lockheed Mis8i1e

& Space Co.

P.O. Box 504

'

Sunnyvale, CA

₉₄₀₈₈

Attn:

_{Mr. R.L. Wald, Dept.}

_57-74

Bldg. 150, Facility'

_i

Newport News

_hipbuildj

_{& Dry}

Dock Company

4ioi Wühington

Avenué

Newport News, Virpini

₂₃₆₀₇

(32)

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

122114 Lakewood Blvd.

Downey, CA 902141

Attn: Mr. Ben Ujihara (SL-20)

Nielsen Engineering & Research, Iñe.

850

Maude Avenue

Mountain View, CA .940140 Áttn:.Mr.. S.B. Spangler

1 Oceanica, Inc.

Technical Industria]. Park

Plainveiv, L.I., New York 11803 Socièty of Naval Architects

and Marine Engineers 74 Trinity Place'

Néw York, New York 10006

Attn:, Technical Library.

Sun Shipbuilding & Dry Dock Co'.

Chester, Penna 19000'

Attn: Chief Naval Architect

Sperry $ystems Management Division Sperry Rand Corporation

Great Neck,' New York 11020

Attn: Technical: Library

Stanford Research Institute Menlo Park, CA' 94025

Attn: Library G-021

2 ' SouthweSt Research Institute

P.O. Drawer 28510

San Antonio 'Texas 782814,

Attn: Applied Mechanics Review

Pr. H. Abramson Tracor, Inc. 6500 Tracor Láne AÚstin, 'Texas 78721 1 ' Mr.' Robert Taggart 393Ó Walnut Street Fairfax, Virginia ' 22030

'Ocean Engr. Departient

Wóods Hole Oceanographic Inst. Woods Hole, Mass. 02543

Worcester Polytechnic Inst.

Alden Reaearc!h Laboratories

Wòrceateî', Møs t1

Attn

Ff44

Applied Physics tboratory University of Wühington

'1013 N. 'E. 40th Street.'

Seattle, Washington '8105

Attn: Technical, Library

]. University' of Bridgeport

Bridgeport, Conn. 06602

Attn: Dr. E. Urani

J. Cornell University

Graduate School of Aerospace Engrf Ithaca, New'York '].4850

Attn: Prof. W.R. 'Sears

14 University of California

Naval' Architecture Departmzit Ço].lege of Engineering Berkeley, CA ₉₁₄₇₂₀ Attn: Library Prof. W. Webster Prof. J. Paullirig Prof. J'. Wehausen

California Institute o'f Techno1or Pasadena, CA 91109

Attn: Aeronautics Library

Dr. T.Y. Wu ',

Dr. A.J. Acosta Docs/Repts/Trans Section

Scripps Institution of Oceanography Library

University of California, Sn Diego P.O. Box 2367

La Jolla, CA 92037

Catholic University of AmerIca Washington, D.C. 20017

Attn: Dr. S. Heller, Dept. of

of Civil & Mech. Engr. 'Colorado State University

Foothills Campus

Fort Collins, Colorado 80521

Attn: Reading Room, Engr. Res. Cente:

University of Calif. at San Diego 'La Jolla, CA 92038

Attn: Dr. A.T. ,Ellis

(33)

2. Flordia Atlantic University Ocean Engineering Department Boca Raton, Fia

_3343?

Attn: _{Technical Library}

Dr. S. Durzne

Harv8rd University

Pierce Ha].].'

Cambridge, Mass

₀₂₁₃₈

Attn': _{Prof. G. Carrier}

Gordon McKay Library University of Hawaii

Department of Ocean Engineering

2565

The Mall

Honolulu, Hawaii.

₉₆₈₂₂

Attn: Dr. C. Bretschnejder

i University of Illinois Urbana, Illinois

6i80i

Attn: Dr. J. Robertson

institute Of Hydraulic Research The University of Iowa

Iowa City, Iowa

₅₂₂₄₁

Attn: Library

Dr. L. Landweber Dr.,J. Kennedy

The John opki.ns University

Baltimore, Md.

21218

Attn: Prof. 0. Phillips

Mechanjôs Depts. Kansas State University

Engineering Experiment Station Seaton Hall

Manhattan, .Kañsas _665Ò2

Attn: PrOf. D. Nesñith

2. University of Kansas

Chm. Civil -Engr.. Dept. Library

Lawrence, Kañsas

66044

Fritz Engr. Laboratory Library

'Depart. of Civil Engr.

Lehigh University

Bethlehem, Penna

₁₈₀₁₅

5 Department f Ocean Engineering Massachusetts Inst. 'of Téchno1or

Cambridge, Mass.-

₀₂₁₃₉

Attn: _{Department Library}

Prof:. P. Leéhey,

Profs. M. Abkowitz , P. Mandel

Dr. J. Newman

Parsons Laboratory

Massachusetts Institute of. Techno1Or Cambridge, Máss

₀₂₁₃₉

Attn.: ' Prof. A. Ippen

St. 'Anthony .Fá11s.Hydr. Laboratory University of -Minnesota

Mississippi .River at '3rd Ave.', ß..

Minneapolis, Minn.

₅₅₄₁₄

Attri: Diréctor Mr.. J. Wetze]. Mr, F. Schiebe. Mr. J. Kj].lçn Dr. C. Söng

Department of Naval Architecture and Marine Engineering

University of Michigan Ann Arbor, Michigan

₄₈₁₀₄

Attn: Library

Dr. T.F. Ogilvie

Prof. F. Hanunitt

2 College -of Engineering

University ö Notre Dame Notre Dame, Indiana

₄₆₅₅₆

'Attn: Engineering Library

Dr. A Strandhagen

2 New York University.

Courant Inst. of Math. Sciences

251

'Mercier Street .

New York', New York ₁₀₀₁₂ Attn: Prof.' A. Péters

Prof. J. Stoker. New York University University Heights Bronx, New York

₁₀₄₅₃

Attn: Prof. W. Pierson, Jr.

Department of Aerospace and Mechanical Sciences

Princeton University

Princeton, New Jersey

₀₈₅₄₀

Attn: Prof. G. 'Me2ior.

3 DavidSon Lab'

Stevens Institute of Techrio10

711 Hudson Street

Hoboken, New Jersey 07030

Attn: Library

Mr. J. Breslin Mr. S. Taakonas

(34)

Department of Mathematics St. John's University Jamaica, New YORK _U1&32

Attn: Prof. J. Lurye

Applied Research Lab Library University of Texas

P.O. Box

₈₀₂₉

Austin, Texas

₇₈₇₁₂

College of Engineering Utah state University Logan, Utah

₈₄₃₂₁

Attn: Dr. R. Jeppson

2

Stanford University Stanford, CA

₉₄₃₀₅

Attn: Engineering Library

Dr. R. Street

3 Webb Institute of Naval Architecture Crescent Beach Road

Glen Cover, L.I., New York

₁₁₅₄₂

Attn: Library

Prof. E.V. Lewis Prof. L.W. Ward

National Science Foundation Engineering Division Library

1800

G Street N.W. Washington, D.C.

₂₀₅₅₀

University of Connecticut Box

_U-37

Storrs, Conn.

06268

Attn: Dr. V. Scottron

Hydraulic Res. Lab Long Island University Graduate Department of

Marine Science

40 Merrick Avenue

East Meadow, L.I., N.Y.

₁₁₅₅₄

Attn: Prof. David Price

i Dr. Douglas E. Humphreys (Code 712)

Nava]. Coastal Systems Lab

Panama City, Florida

32401

(35)

-6-DOCUMENT CONTROL DATA - R & D

.urity r his.Iffrat ¡on ot tUSo, hodj, of abslparf and ¡ndc*in _{nnno?ntj(,fl muet be c,,tcrcd whrn rl,,, ovoi'nfl lepo,, I rin.. 'uSed}

Za. REPORT SECURITY CLASSIFI(ATION

Unclajfjed

2b. GROUP

I. O*IiNa TINO AC T'VI TY (Corporate r3iIthor)

Institute, of Hydraulic Research

The University of löwa

Iowa Ciy Ia.

.1. REPORT TITLE

"On the Equations of a Thick Axisymmetric Turbulent Bouñdary _Laye±"

OESCRiPj.I NOTES (Typ. of report_{and.jncfuejve dateS)}

II _{Report No. 143}

AU THOR(S) (Firat name, middle Stdsiái. laitname)

V.C. Pate3.

6. REPORT OATE

January 1973

Sa. CONTRAY OR GRANT NO.

N000l4-68-A_0196_0002,

b. PROJECT NO.

II SU PPL EMENTA PV NO TES

la. TOTAL NO. OF PAGES

28

9A. ORIGINA TORS REPORT ÑUMBER(S)

II Report No. .43

lb. NO. Or NEFS

15

Sb. OTHER REPORT NO(S) (Any àfhetnua,bera _{thal.máy b. .asig,,ed}

this mport)

d.

IO. DISTRIBUTION STATEMENT

Approved for public release.; _{distribution unlimited}

Il..SPONSORING MILITARY ACTIV,ry

Naval Ship Research & Deveiopziènt Center

I). ABSTRACT

--From an examination of the Reynolds equations for _{axisymmetric.}

turbulent flow in, situations where

the thickness of the boundAry _{layer is of the} same order as the transverse _{radius of curvature, of thé}

surface, it has been shown that, in general, neither the boundary layer _nor _{he poténtial flow outid.e} it can be calculated

independently of the other, owIng _{to significant, interaction} between th two flow regimes.

Following a discussion óf the various procedures for extending conventional thin _{boundary-layer calculatioñ methods}

to treat thick axi-symmetric turbulent boundary-layers, taking intQ account the _{influénce of 'transverse} curvature either at the differential

or the integral level, a method is proposed

for the simultaneous _{solution of the boundary layer}

and. the' potential flow _equations,

allowing the twà flow _{regimes to interact.}

(36)

TJnc1assifie.

a a asas. - a

a

-securty ..IHSMIII9.I1VH

4 _{KEY WORDS}

Turbulent Boundary Layers Transverse Curvature Interactipfl Calculation methods Differential equations Integral equations Reynolds stresses