An integral prediction method for three-dimensional turbulent boundary layers on ships

DTNSRDC.79/006

PREDICTION METHOD FOR THREE-DIMENSIONAL TURBULENT BOUNDARY —

nos
hy. F9/%6

DAVID W. TAYLOR NAVAL SHIP
RESEARCH AND DEVELOPMENT CENTER

Bethesda, Md. 20084 a

= a |
/ WHO’ NG

DOCUMENT
\ COLLECTION

AN INTEGRAL PREDICTION METHOD FOR THREE-DIMENSIONAL
TURBULENT BOUNDARY LAYERS ON SHIPS

by

Christian von Kerczek
Thomas J. Langan

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

SHIP PERFORMANCE DEPARTMENT
RESEARCH AND DEVELOPMENT REPORT

July 1979 DTNSRDC-79/006

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DTNSRDC-79/006

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

AN INTEGRAL PREDICTION METHOD FOR THREE-
DIMENSIONAL TURBULENT BOUNDARY LAYERS
ON SHIPS

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6. PERFORMING ORG. REPORT NUMBER

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7. AUTHOR(s)

| Christian von Kerczek
Thomas J. Langan

9. PERFORMING ORGANIZATION NAME AND ADDRESS
David W. Taylor Naval Ship Research
and Development Center
Bethesda, Maryland 20084
11. CONTROLLING OFFICE NAME AND ADDRESS
Naval Sea Systems Command (SEA-035)
Washington, D.C. 20362

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July 1979
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78

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19. KEY WORDS (Continue on reverse side if necessary and identify by block number)

Boundary layers, turbulent, three-dimensional, momentum integral

20. ABSTRACT (Continue on reverse side if necessary and identify by block number)

This report presents refinements of a previous momentum-integral
method for calculating three-dimensional turbulent boundary layers on

ship hulls. In particular the following refinements are made: the small
crossflow assumption is removed; numerical calculation of the double model
potential flow replaces the slender body potential flow; a more general

(Continued on reverse side)

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Program Element 61153 N 23
Project Number SR 02301
Task Area SR 0023-01-01
Work Unit 1552-070

(Block 20 continued)

and versatile orthogonal coordinate system is used in place of the
streamline surface coordinate system; and finally, an improved
numerical method is used for solving the momentum-integral boundary-
layer equations. It is shown that the boundary layer calculation
method, developed here, can be used to calculate certain boundary
layer parameters, such as boundary layer thickness or skin friction,
with fair accuracy over a large portion of hulls that maintain un-
separated flow. The surface coordinate system can also be used in
other methods for calculating the boundary layer.

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TABLE OF CONTENTS

LIST OF FIGURES . .

ABSTRACT

ADMINISTRATIVE INFORMATION

INTRODUCTION

THE BOUNDARY LAYER MOMENTUM INTEGRAL EQUATIONS
THE SHIP SURFACE COORDINATE SYSTEM

NUMERICAL ANALYSIS ..

COMPUTATIONAL RESULTS AND DISCUSSION
CONCLUDING REMARKS

APPENDIX - MATHEMATICAL DETAILS .

REBERENGES!) 2 2 3 6 ©

LIST OF FIGURES

1 - The Bottom and Side Views of the Coordinate Net
Consisting ot the Cross-Sections and Their
Orthogonal Trajectories, as Recommended by

Miloh and PaeeIne on the SSPA-720 Model ..

2 - The Bottom and Side Views of the ($,0) Coordinate

Net on the SSPA-720 Model

3 - Streamlines on the Lucy Ashton Model .

4 - Streamline Momentum Thickness versus Axial Distance

for Flow Along Streamlines of the Lucy Ashton

5 - Streamline Skin Friction Coefficient versus Axial

Distance for Flow Along Streamlines of the
inney ASIMEOD 6 6 56 0 6 0 6

6 - Crossflow Angle versus Axial Distance for Flow
Along Streamlines of the Lucy Ashton .

iii

27

29

SAL

33

37

41

10

11

12

13

Streamlines on the SSPA-720 Model Along Which
Larsson Measured the Boundary Layer ..... .

Starting Values of H, 8 and 914 at Station S =- 0.5
on SSPA-720 Model .

Boundary
for Flow

Boundary
for Flow

Boundary
for Flow

Boundary
for Flow

Boundary
for Flow

Layer
Along

Layer
Along

Layer
Along

Layer
Along

Layer
Along

Characteristics
Streamline 1 on

Characteristics
Streamline 3 on

Characteristics
Streamline 5 on

Characteristics
Streamline 7 on

Characteristics
Streamline 8 on

versus Axial Distance
Model SSPA-720 .

versus Axial Distance
Model SSPA-720 .

versus Axial Distance
Model SSPA-720 ..

versus Axial Distance
Model SSPA-720 .

versus Axial Distance
Model SSPA-720 .

iv

Page

43

44

45

48

51

54

57

ABSTRACT

This report presents refinements of a previous momentum-
integral method for calculating three-dimensional turbulent
boundary layers on ship hulls. In particular the following refine-
ments are made: the small crossflow assumption is removed;
numerical calculation of the double model potential flow replaces
the slender body potential flow; a more general and versatile or-
thogonal coordinate system is used in place of the streamline
surface coordinate system; and finally, an improved numerical
method is used for solving the momentum-integral boundary-layer
equations. It is shown that the boundary layer calculation method,
developed here, can be used to calculate certain boundary layer
parameters, such as boundary layer thickness or skin friction,
with fair accuracy over a large portion of hulls that maintain un-
separated flow. The surface coordinate system can also be used in
other methods for calculating the boundary layer.

ADMINISTRATIVE INFORMATION
The work reported herein was supported by the General Hydromechanics

Research Program under Task Area SR 023-01-01 and Work Unit 1552-070.

INTRODUCTION

This report presents some refinements of the momentum-integral method
for calculating three-dimensional turbulent boundary-layers as developed by
von Kerczelans for ship hulls. These refinements include: (1) the removal
of the boundary-layer small crossflow approximation; (2) the incorporation
of an exact numerical calculation of the double model potential flow in-
stead of using the slender-body theory potential flow; (3) the abandonment
of the streamline surface coordinate system in favor of a more general and
versatile orthogonal surface coordinate system; and (4) an improved
numerical method for solving the momentum-integral boundary layer equations.

The use of momentum-integral methods for calculating three-dimensional
boundary layers has come under severe criticism recently by the advocates
of the differential boundary layer equations (see, for example, Landweber
and Dafeeils” Cebeci et aie” and Spaildlsiays)): The main objection to the inte-
gral methods seems to center on the unavailability of a suitable crossflow

velocity profile function that adaquately approximates a variety of

*A complete listing of references is given on page 6/7.

crossflow velocity profiles. It is most often claimed by these critics
that an accurate representation of the boundary layer crossflow profile is
required for the prediction of longitudinal bilge vortices. Large-scale
longitudinal bilge vortices arise due to a complicated form of three-
dimensional separation; the vortex flow itself being the separated flow.
Thus, one cannot expect to be able to compute the bilge vortex flow, even
by the most sophisticated boundary layer methods, whether integral or
differential. Presently, boundary layer theory can be used only to calcu-
late the flow up to separation. Recently developed momentum-integral
methods for two-dimensional and axisymmetric boundary layers (see
Green et us”) are as accurate as, yet considerably more economical than,
differential methods. There seems to be no reason to believe that similar
improvements in three-dimensional momentum-integral methods cannot be found.

Present models of the boundary layer crossflow are primitive and
further experimental data and research can be expected to uncover a simple
crossflow velocity profile family that is adequate for calculation methods.
For this reason it was thought desirable to make the technical improvements
mentioned above in the von Kerczek method so as to accommodate easily
detailed improvements in the crossflow model that may come about later.
However, the modified surface coordinate system and the incorporation of
the exact double model potential flow calculations (in lieu of the slender
body theory potential flow calculation method) are of independent value and
can be used with any other boundary layer calculation method. The surface
coordinate system for the boundary layer calculations developed in this
report has some advantages over the coordinate systems recommended by
others (see, for example, Cebeci et ails” and Miloh and Paieeil), The surface
coordinate system used in this report is very similar to the one used by
Cebeci et An but it does not have the complication of being nonorthogonal.
The present surface coordinate system is superior to the Miloh and Patel
coordinate system because it provides a better coordinate net coverage of
the hull surface for uniform spacing of the coordinate parameters.

This report is divided into six sections including the introduction.

The second section describes the formulation of the boundary layer

calculation problem in terms of the momentum-integral entrainment method.
The third section describes the surface coordinate system and the potential
flow calculation method. The fourth section describes the numerical method
for integrating the momentum-integral equations. The fifth section
describes and discusses some sample computational results and the sixth
section gives some concluding remarks on further developments of this
three-dimensional ship boundary layer calculation method. An appendix at
the end of the report gives some detailed formulas that are used in the

computational algorithm.

THE BOUNDARY LAYER MOMENTUM INTEGRAL EQUATIONS

It is assumed that the ship surface is hydraulically smooth and has
no abrupt changes in principal curvature anywhere on the hull. Also assumed
is that the boundary layer thickness is small compared to the principal
curvatures everywhere on the hull. Only the turbulent boundary layer
development is considered. The length scale used is half the length, L,
between the perpendiculars of the hull. The velocity scale is the steady
ship speed U,. Henceforth, all physical quantities that are discussed will
be dimensionless with respect to these scales. There is an orthogonal
surface coordinate system on the hull, which has lines of constant @ running
generally lengthwise along the ship and lines of constant $ running nearly
parallel to the cross-sections of the ship. This system will be described
in detail in Section 3. The coordinate perpendicular to the hull surface
can be described in terms of its arc length parameter }.

At an arbitrary point on the hull surface the potential flow velocity

vector U is given by

= +
U US = Uy &5 (1)
where ER and €y are unit tangent vectors in the direction of the ¢ and 0
coordinates, respectively, and U = | |u| | is the magnitude of the velocity

vector U. The angle that the velocity vector U makes with the $ coordinate

line is denoted by

U
ees eine (z) (2)

In terms of this coordinate system the momentum-integral boundary-layer
equations are a special case of the equations given by Nereis! and repro-
duced in the article by Reynolds and Caneeias These equations are the

momentum integrals for the flow in the $-direction;

iba i162) lt 2 OU
—— + = = ZC - 0 = —— -K
OX Ov Diets iL} ef OR
$ 8 on!
A. ou
Dn Wes ee edie Oy ie Meas ee Se
|| We ) DD Ww os
) )
A dU
ba ea e.
= = Ee visas KH (3a)
and for the flow in the 0-direction
22) ee Se ee L ae |i ae *, |
a dhe 2 fy 6) aL Zak |) Obes 0)
A oU
2 dU 1 s)
22 i an -x, | -2la KAU
0 w)
NOU
2 s)
ty | a Wy (3b)

where me and Le are arclength parameters in the $ and 6 directions,

respectively; K, and K, are the geodesic curvatures of the @ and Iv)

¢ 9

aGnat 5 r
coordinates, respectively; Ou O54> O15» Oo9> A> and A, are momentum and
displacement thicknesses defined, respectively, by Equations (4a,4f); and
C and C
=a fp

directions, respectively.

are wall skin friction coefficient components in the $ and 6

The usual turbulent boundary layer assumptions
of the neglect of turbulent normal stresses and the neglect of mean dif-

fusion in directions parallel to the hull surface are incorporated in

Equations (3a, 3b).

The momentum and displacement thicknesses are defined by

5
6... = a G)dA
Te Leap Upete (4a)
0 U
6
Oe -| i (UG) dd (4b)
0 U
6
5, = = (Up-#) dd (4c)
0 U
5
v nw
O55 = | ~Z y=) ar (4d)
0 U
: U, - a
A, =I aa ah (4e)
0
é Ty =
NS | as Tae di (4£)
0

where Gi and % are the components of the

g and @ directions, respectively.

mean boundary layer velocity in the

In the case of streamline coordinates, in which the coordinate curves

g are parallel to the inviscid streamlines on the hull surface, U

6 =U. For this case, the momentum and displacement thicknesses of

= 0 and

definitions (4a-4f) then will be denoted by corresponding lower case Greek

letters and these definitions reduce to

6
z u
| U
0
6
2 wv.
a, | U
0
6
|
0
6
Ho)
0
)
0
fC)
noe||
0

The quantity 6 is some overall nominal boundary layer thickness.

(.- ) di

ale

di

(5a)

(5b)

Ge)

(5d)

(5e)

(Sf)

The

relationships between the boundary layer thicknesses defined by Equations

(4a-4f£) and those defined by Equations (5a-5f) will be needed later and

can readily be worked out. They are

. 2
O11 = O14 cos Q- (855+ 951) sin @ cos a + 955 sin a (6a)
O.. = © ) +0, 2s On. ellie” (6b)
12 aL 9,5 sin Qa cos a cos Qa 21 sin a
6, 2 @& ) + 05 A O.. BR” @ (Be)
21 Li O55 sin @ cos Qa cos Qa 12 sl
6. = O., aa ae © ) of Hae Z (64)
22 LiL Ss Qa 12t O04 sin Q@ cos a 22 cos Qa
Ay = rT cos a - 5, sin Q (6e)
A, = oT sin a + 6, cos a (6£)

The inverses of Equations (6a-6f) that relate the lower case Greek symbols
8 and 6 to their upper case counterparts easily can be derived.

The two momentum equations, Equations (3a,3b) are insufficient to

determine the eight unknowns 0 O45 © e) [No (A)

i? 912° 921° P09 Ay C and C hence,

2 fo ay

some other integral equations and empirical information are needed. The
calculation method is based on the Cumpsty and tend” momentum-integral

8 ‘ oe : . ; soley)
method which utilizes the three-dimensional entrainment equation

(U 6-UA,)

6 —¢

il ;)
U (gosta) =
$
3 i
+ Wy (U,5-UA, ) = Ky (o,8-UA,)| =F (7)

where F is the three-dimensional rate of entrainment function. It is

assumed, in this three-dimensional boundary layer calculation method, that

F is the same function evaluated with respect to the flow components in the
streamline direction as the one used for two-dimensional flows. Thus,

, 5
according to Green, et al.

EH) =n08 025) Hem 0n O22 (8)
where
6
H = os (9)
11

Furthermore, it is assumed that

6 = 0), (G+H) (10)
where
Pai aca ila
G= G1) (11)

Thus, the entrainment parameters, the length scale 6 and the entrainment
function F, are completely specified in terms of integral thicknesses
defined by Equations (5a-5f). A justification for the foregoing empirical
expressions can be obtained from the paper of Cumpsty and Head? and its
antecedents,

The components Ce and Ce of the skin friction coefficient can be ob-

(0) 6
tained from the skin friction coefficient Cy in the local inviscid stream-
line direction by the formulas
C = C,(cos a - sin @ tan 8) (11a)
fo £

C = C.(sin a + cos a@ tan 8) (11b)
fy ify

where 8 is the angle between the direction of the wall friction vector and

the inviscid streamline. The angle 8 is precisely defined by

Ov
see LNT ee ReeOu
tan 68 = limit — = limit —— (12)
u du
A>0 A>0 an

where the boundary-layer velocity components u and v are in the direction
of the inviscid streamline and its normal, respectively. By assuming that
the component of the boundary layer flow in the inviscid streamline direc-
tion satisfies the two-dimensional velocity similarity laws, the skin

friction coefficient C, can be evaluated using a two-dimensional skin

f
friction formula. The following skin friction formula given by Head and
Paton? is
Ce = exp (aHtb) (13)

where a = 0.019521 - 0.386768 c + 0.028345 ea - 0.000701 e

b = 0.191511 - 0.83489 c + 0.062588 ee - 0.001953 ee

ec =i1nR

Pan

and where

UL
Ry = oy vos, (14)

is the local streamline momentum thickness Reynolds number. Recall that U

is the dimensionless (scaled by U,,) magnitude of the local inviscid velocity

at the edge of the boundary layer and 814 is the streamline component of
the momentum thickness made dimensionless by the half length L of the
ship.

The final empirical formulas that are used to reduce the number of
unknown quantities to three, in order to match the number of differential
Equations (3a,3b) and (7), are the formulas that relate the crossflow
momentum and displacement thicknesses 915° 854° 55> and oy) to the stream-
line momentum and displacement thicknesses O14 and 61: These relationships
constitute the critical approximations of three-dimensional boundary layer
theory in the momentum-integral framework. It is easiest to simply make
crossflow and streamline flow boundary-layer profile assumptions and derive
the corresponding relationships that result from the definitions (5a-5f).
However it is not really necessary to proceed in this way. Basically,
definitions (5a-5f) simply say that if the streamwise velocity profile u is

described parametrically by parameters la, ] and the crossflow pro-
LA GS 6 omit
file v is described parametrically by the parameters [B8. ] then the
j=1l,...,m

boundary layer thicknesses OF and Sos k and 2 = 1,2 are each functions of

W

the parameters [a], [6,5] such as O19 (a 9015 Byo+++s8)- One could

500
determine these functions MO sre thereby completely bypass any
velocity profile assumptions.

In fact, such a scheme has already been used for the streamwise flow
by using the Head entrainment method. From two-dimensional and axisymmetric

flow theory, the entrainment method gives
6, = 6, (H,9,,) (15a)

Cane C,(H,6,,) (15b)

directly without any profile assumptions. In the most highly developed
entrainment method of Green et Allyn? a third independent parameter is added

to O14 and H, namely the entrainment coefficient Che so that

10

oT = 6, (H, 8 (16)

11°°p)

replaces Equation (15a). If a velocity profile is needed, then the velocity
profiles of Coilesam or Thompsonua may be used for specified values of the
momentum thickness 944 and shape factor H. Sufficient experimental data

for three-dimensional boundary layers are not yet available for a similar
program to be conducted. It does, however, seem that the flow in the
streamwise direction is sufficiently similar to two-dimensional flow that
the two-dimensional data can be directly applied to this component of the
boundary layer flow. However, insufficient data exist for the crossflow to
make more than a crude estimate of profile shapes. There has not even been
a sufficiently large collection and analysis of crossflow profiles to make

a reasonable estimate of the proper parameters [B. ] that need to be
GJSlbs ooo
used to approximate the crossflow. Thus, as the simplest first approxi-

mation of the crossflow profile shape, it is common practice to assume that
(1) the crossflow profile scales on the same length scale 6 as the stream-
wise profile and (2) the crossflow profile shape depends on only one or two
independent parameters, one of which is the shear stress angle 8. The fact
that condition (12) must be satisfied at the wall introduces the angle 8
into the description of the crossflow profile and also dictates (as a matter

of convenience) the shape assumption in the form
Ws
v ie f(A) tan 6 (17a)

In order to satisfy condition (12) at the wall and the condition that
v = 0 at X = 6 (in streamline coordinates), the function f must satisfy
£(0) = 1 and £(6) = 0. The assumption by Macerue is a popular first ap-

proximation that gives

£(A) = (1- ay? (17b)

IIL

It should be emphasized that a profile assumption, such as Equation (17b),
need not be employed at all but, at this time, lack of experimental data
forces the use of such an approximation. More sophisticated profile
assumptions than Equation (17b) have been made. For instance Clawnee

assumed that

f(A) = [+0(4)] (1- 2 (18)

The parameter C in Equation (18) is governed by an additional crossflow
integral equation. Clemo” chose the crossflow moment-of-momentum integral
equation for calculating the development of the parameter C and obtained
considerably better results for crossflow profile predictions than with the
Mager model of Equation (17b). However, there still were certain areas on
the ship hull at which the predicted crossflow velocity profiles were in
serious disagreement with experiment. The areas of serious disagreement
between the crossflow profiles given by Equation (18) and the experimental
ones on the Okuno test model (a Series 60 block 0.70 double model) are very
close to the stern on streamlines that turn upwards from the keel towards
the load waterline. From Okuno's eranelenoe, it seems likely that
further experiments and research will eventually lead to a fairly accurate
crossflow velocity profile shape function f(A) that involves only one or at
most two extra parameters.

This report is mainly concerned with setting the proper framework for
the three-dimensional momentum-integral boundary layer computational method,
so it will presently be confined to the simplest of the crossflow models,
namely that of Mager, Equation (17b). It is hoped that future developments
and availability of sufficient experimental three-dimensional boundary layer
data will warrant modifications of this method to include a more complete
and accurate crossflow model along the lines of the Okuno model.

By examining definitions, Equations (5b), (5c), and (5f), it is easy

to see that 8155 ts) and 6, satisfy the relationship

Dai

eZ)

al On ae) © (19)

This equation can be used to eliminate one of the unknowns from the boundary
layer momentum-integral equations. The Mager crossflow model, Equation (17),

in conjunction with the approximation

ca en

and Equation (10) for the nominal thickness 6, can be used to express the

cle

crossflow boundary layer thicknesses in terms of the streamwise momentum
thickness 814° the shape parameters G and H, and the crossflow angle 8 by
the equations

2 05, (Gt+H) tan 8

Cy = TAGES) GE) Ca Gan BB G9 (Ze)
i Sa ekectn eRe a eet Daher aaa He 7
859 = > Spy (Ge) eae la fea ) PED Y eS | = 8), tan 8 £,(H) (21b)
i 2 Oe sie aL Ar romney sect DF
Sng F = By Cer) Gam 8| § = ST GE) ERS al = Sig Han © EE) (Ze)
6. 2S 6. (em) ton 6) Seo A eB oe 8 EG (214)
2 11 on eS eee EEE a a 4 \

Thus Equations (8) through (11), (13), (17), (19), and (21), together with
the transformation Equations (6a-6f), can be used to reduce the total
number of unknown quantities to three. A convenient set of unknown
quantities, that are integrated in the (¢,8) coordinate system by Equations
(3) and (7) are the streamwise momentum thickness C1 the shape factor H,
and the tangent of the wall crossflow angle t = tan 8. The details of the
final forms of Equations (3) and (7) in terms of these variables are given

in the appendix.

13

THE SHIP SURFACE COORDINATE SYSTEM

The hull surface coordinate system that is used in the boundary layer
calculation method described in the previous section stems directly from the
hull surface representation of von Kerczek and Tuer ue This hull surface
representation utilizes conformal mapping onto a unit circle of the cross
sections of the hull and polynomial interpolation along the length of the
hull of the individual mapping coefficients. Let s be the longitudinal
coordinate, x the lateral coordinate, and y the vertical coordinate of the
ship hull. These coordinates are made dimensionless by the half length L
so that the bow and stern perpendiculars lie at s = + 1, respectively. The
load water-line is located at y = 0 and the keel at midships is located at
vy = SDs

The hull surface representation of von Kerczek and Tele results in a

parametric surface equation of the form

N M
z=xtiy = y » Aue a GHZa)o ee (22)

n=l m=l

where the matrix (A of coefficients specifies the hull form and is

computed from a set of defining hull offsets by an algorithm given by

von Kerezek and nels

The surface coordinate system used in the boundary layer equations
consists of the 6 = constant lines, obtained from Equation (22) and their
orthogonal trajectories, here denoted by ¢ = constant lines. It is not
necessary to specify the variable » since the arclength along the > =
constant lines will be used directly.

Equation (22) can be written in real form as

N M
m-1
x = x(s,9) = y > A Ss cos (3-2n)6 (23a)
n=l m=l1

14

N M
y= y¥@oo) = > > A gn sin(3-2n)6@ (23b)

n=l m=1
and, in the vector form,

2 = 26,0) = 20(S)10)) a te y(s,9)j + sk (24)

where r is the vector from the origin of the (x,y,s) coordinate system to a
point on the hull surface, and (i,j,k) are unit tangent vectors to the
(x,y,Ss) coordinates, respectively.

The surface coordinate lines 9 = constant, run along the length of the
hull surface, and the coordinate lines » = constant, are nearly parallel to
the hull cross-sections. The arclength increments along the @$ and 6

coordinates are Or and dha, respectively, and are given by

LID
8=constant

dk

(dredr)

|(2)* +(22)’ a | ds (252)

and

yf
g=constant

9 9 1/2
ds Ox dx ds (3 |
= ae E> Se ap || dé (25b)
\(3 ds dé i , dé i

where (ds/d8| ,) denotes the evaluation of the derivative ds/d68 along the

Qu
=
iT

(dredr)

2
ay , dy ds
(B® do
@ = constant line.

15)

Let eae eg» and cx be unit tangent vectors to the $, 9, and the hull

surface normal coordinates, respectively. Then e, and e, are easily

S)
computed using Equation (24) by

OE
oe
Os
— x —
or or
TS og
ds cls)
where || * || denotes the length of the vector. The requirement that the >
and 8 coordinates be orthogonal imposes the condition that
Sa Sa Sa (28)
The increment of arc along the curve $ = constant, dr| > can be
written as
or or He )
ella] 4 = = |} ald
telly Siam” Oe, al (29)
and by virtue of orthogonality
e, ° dr = 0 30
Sy) Els is
The derivative (GSHCI) can be obtained from Equation (30) by
( dx , dy ay)
Gey 2 UN GO ss i ee Os (31)
dé

16

The geodesic curvature terms K, and K, are defined, respectively, by

v)

(32a)

ai itesa bog De (32b)

The derivatives in Equations (32a,32b) can be evaluated by converting them
to derivatives with respect to s and 0 using Equations (25a,25b),
respectively.

Miloh and Pareto recommended the use of the orthogonal surface
coordinate system that consists of the cross-section curves and their
orthogonal trajectories on the hull surface. in terms of the surface
Equation (23) the cross-section curves are given by s = constant. Thus,
if e, in this case, is the unit tangent vector to the cross-section profile,
the orthogonal trajectories of the cross-sections can be computed by inte-

grating the differential equation

Q 0 Gk = ©) (33)

which, in expanded form, and making use of Equation (29), reads

or

e e
GOR Un ees (34)

ds or
+3)

Examples of hull surface coordinate grids for the (d,0) system

described earlier and the cross-section system of Equation (34) are given

in the section on Computational Results and Discussion.

17

The potential flow velocity on the surface of the hull was obtained by
using the computer program of Gia.” This program solves the double-
model Neumann problem by distributing a layer of doublets on the hull sur-
face and numerically solving the resulting integral equation by a panel
method. The main advantage of using a doublet distribution, rather than a
source distribution as in the Hess-Smith meeHodeat is that the surface
potential is obtained directly as the solution of the integral equation.

The hull surface representation, Equation (23), is used to generate
the input for the potential flow program of Gheneo” A uniform rectangular
distribution of points in the (¢,9) plane determines a set of curved quadri-
lateral elements covering the hull surface. These curved quadrilateral
elements are approximated by plane quadrilateral elements and then used as
input into the Chang program. The results of the potential flow calcu-
lation are values of the surface velocity potential at the geometric mid-
points of the plane quadrilateral elements. These values of the surface
velocity potential are then assumed to be accurate values of the exact
surface velocity potential at the points of the hull surface that correspond
to the geometric centers of the rectangular elements in the (@,9) plane.
The value of the surface velocity at each point is obtained by numerical
differentiation of the surface velocity potential. The values of the
potential are first interpolated along > = constant curves by a periodic
cubic gelllnes This interpolation yields accurate values of the surface
velocity potential at arbitrary locations on the 9 = constant curves. Then
the values of the surface velocity potential are interpolated along
6 = constant curves by another cubic spline. The surface velocities and
the derivatives of the surface velocities that are required in Equations
(3) and (7) are obtained, respectively, by differentiating the cubic
spline, evaluating the result, which gives the velocities, and then
refitting the velocities with cubic splines and differentiating the second
set of splines. This "spline-on-spline" mocademe seems to be one of the

best ways of obtaining two derivatives of a numerically defined function.

18

NUMERICAL ANALYSIS
The two momentum integral Equations (3a,3b), together with the entrain-

ment Equation (7), can be written completely in the form

OW
D(w(p,0)) SH + Bcw(g,e)) Se = cow) (35)
a) Iv)
where the vector W is given by
Wi Ona
We W. = te (36)
W, H

From Equations (50) and (55) in the appendix, the coefficient matrices are

defined as follows:

; 0811 3 9814
S11 ih Oe il Oe
dg dg
ie re 21 21 “
DED =) Dyed) | Bon) hia He ia Ge)
, ; ahi : dh,
iLL Tee Tt “On
A 9819 5 9819
812 il, ae iil Ai
dg og
a é 22 22
BOW) = (Boa) | Bq Sig THE O51 0H (37b)
f : dh» ‘ dhy 5
22 TSMC TE nil SE

19

and

C(W) = (C,) HG

2
os
where
sat et ET ns ol awl oa
1 Die U OX U Oo’
) w) 6
Dialers Givens Kae. ate ado. rls) C}))
U obs U dk, Tay ME) ALD OBA al
+ O41 Ky (84178997 hy sin a)
1 2 3U 3U
Oy ea Ty Neon A, Y Sar Be
Q 0) ic]
1 du, dU,
ae A, m7 A, Oh, + O14 Ky (81 9781 *hy cos Q)
+ 814 Ky (8597844 7h} cos Q)
and

20

(37c)

(37d)

(37e)

(SE)

The variables in Equation (35) and auxiliary formulas of Equation (37) are
defined in terms of the principal unknown momentum and displacement thick-
lb? O15 O54 O59 A> and A, in the appendix. It is assumed that
the values of W are known at a given station 9 = 9 for all values of 0 and

nesses 0

that the boundary layer is to be computed between 6 and a final station ¢
downstream of 9: Because of symmetry, it is only necessary to sclve
Equation (35) between 6 = - 71/2 and 6 = 0. An (N+1) x (Mtl) grid with
spacing Ad and A98, respectively, is superimposed on the region [(,8) |5<

<,.-1/2<8<0]. For simplicity of notation, wt will denote W(o tiAd ys

FaO)o Bor = O, dooodgil aincl 9 S OS tloocc slo MESO pi, Bt, and ci will
denote the values of the matrices B, C, and D,: respectively, at the point
(pp tidd, jAe).

Equation (35) is hyperbolic if there is a nonzero crossflow. The
three characteristics at a point (6,8) lie between the angles a and a + 6;
the equation is parabolic at a point if 8 is zero, as the characteristics
have the same tangent or, equivalently, the same direction. Along a line
of flow symmetry, such as at the keel or at the waterline on a double
model, the crossflow is zero and the governing Equation (35) is parabolic.
Consequently, in the present case of the double hull models, Equation (35)
is a mixed equation, that is hyperbolic and parabolic at different points
of the region of integration unless the crossflow is everywhere zero. In
this latter case, Equation (35) reduces to parabolic form. A solution
method which is applicable to both parabolic and hyperbolic equations must,
therefore, be used to solve Equation (35) for double hull models or hulls
for which the crossflow is zero or very small everywhere.

The O'Brien et Bike 5 implicit finite difference scheme is used to
solve Equation (35). It is a stable scheme for any positive grid spacing
ratio r = A@/Ad and is applicable to both hyperbolic and parabolic
equations. It consists of a one-step forward difference in the $-direction
and a central difference at the i+ 1 step in the @-direction. In this

numerical integration scheme, Equation (35) is approximated by the equation

= See pi (ae cael ee

hy Ad 2A0 (38)

21

where the metrics Bs and hy (defined by nee and Lg=hp_d6) are evaluated
at the point ($5 tidd, jA8). Rearranging the terms of Equation (38)
yields the equation

_ pid witt j-l 4 pid witli + pu wert are

= C= Das Wels ns Ad cv (39a)
where
as h ia
ed (— (39b)
2rh
8
The values of W) for j = 0, 1,...,M are specified initial conditions;

they may be obtained from experimental data. Along the symmetry lines

j = 0 (6=0) and j = M (6 -- 5) , the crossflow angle B is zero, so that
W, = t = 0, the equation Wo = 0 replaces the second momentum Equation (36)
along the two lines of symmetry, the load waterline and the keel. Moreover,
dh,
a = 0 on these lines, so that B19° 891° 89° hs hoos S Ce ; Ay> and Ta

s)

are identically zero. Thus, along the symmetry lines, the momentum integral

Equation (3a) reduces to the equation

dg
1 WD ayes a x
Ly ve Chse Tae an, 2 oe we Chg, @ UR (Go)

els) oh
G peelals +6 dG oH +6 22 Oe = F(H) - one G Ge -«,) (41)

Ey) 11 dH 22 ib aoe Oe
) ) )

22

(See the appendix for details). The system of Equation (35) can be used
also to represent Equations (40) and (41) together with t = 0, if the

coefficient matrices B, C, and D are modified to the following:

et Oe ao
Dp ao i @ (42a)
dc
OL) 5 10) panics
dg
12
0 Meme v
B(w) =| 0 0 0 (42b)
en: deere ee
ii, BE

and

C(W) = 0 (42c)

The crossflow angle is asymmetric with respect to 9 = 0, so t is also
asymmetric with respect to this line. Accordingly, the central difference
approximation for dt/dk, on the 6 = O line is

,il io Ge ,il

(43)

23

The zero crossflow finite difference formula analogous to Equation (38) is

m0 etl LO age LO 0) aeol 2 2 fo is
hy Ao ma) 0

After rearranging the terms of Equation (44), one obtains the equation

pi? wir 0 a 710 witt 1 4 ci0

(45a)
where
: h .
Pe ey (45b)
rh
8
A similar argument at the symmetry plane 6 =- 1/2 yields the finite
difference equation
il piM weet M-1 A pim witt Lie ciM (6a
where
h
=i Mi iM
a ase B (46b)

The finite difference Equations (35), (45), and (46) form a linear
system of algebraic equations for the unknowns Tiny J where i = Oolong oo il
hal a} = Oswego a ciills

Let matrix D be defined by

24

D B o 0 CO RSENS Ee mes
Book Sapo GOEL AR OUMERO CURA StA aehaney A Rete tar
0 Bact an aC ROMO a
e - GDOUDODOOOOOUDOOODDOOODODUUOODUODOOOUDd ecoceceow ecco eee ee oe eee (47)
0 0 pil pt Bal ciONnanaenas ee
onght ee 0 Oe Ba Te Te ie a Our tn’ Oy

eceecee eee eee eee eee oo ee ee ee oe oe eee Oe eo ee wow ee oe oe oo eB Oo

DX aC (48a)
where X = (x) and C = cc) for uw = 1,...,3(M+1l), where the components
a of the vector X are defined by
itl k
Saye Mh (ED)

where K = 1,2,3 and the components Ey of the vector C are defined by

fei fit
Seneine Fhe (se)

Gaussian reduction is used to solve Equation (48). The 3 x 3 sub-
matrices of D have been inverted explicitly so that the Gaussian reduction
of Equation (48) is very fast on the computer. Back substitution is used

to obtain the vector X.
COMPUTATIONAL RESULTS AND DISCUSSION

Some sample computational results of the boundary layer on two double

ship models are presented in this section. The first sample. computational

25

result is the boundary layer on the Lucy Ashton model for which experi-
mental boundary layer data are given by Joubert and Meee cone and which
von Kerezeley computed using the small crossflow approximation. The second
sample computational result described below is the boundary layer develop-
ment on the Swedish SSPA Model 720 for which boundary layer experimental
data and calculations are provided by Larsson.

It is first necessary to describe some details of the calculation of
the surface coordinate system before embarking on a description of the
results of the boundary layer calculation. There are many different surface
coordinate systems that one can use for three-dimensional boundary layer
calculation methods. The most prevalent coordinate systems used for ship
boundary layers are the streamline coordinate system and the coordinate
system made up of the cross-sectional curves and their orthogonal tra-
jectories on the hull ainetaees © henceforth referred to simply as the cross-—
section system. The streamline coordinate syeten has the advantage of
yielding the simplest form of the boundary layer equations, but it may be
difficult and costly to generate this system when flows about a ship hull
at nonzero Froude number are considered. Thus it is worthwhile to consider
coordinate systems that only depend on the ship hull geometry and not on
the inviscid flow.

Figure 1 shows a sample of the cross-section coordinate system recom-
mended by Miloh and Papen” on the Swedish SSPA Model 720. The calculation
of the network of coordinate lines shown in Figure 1 is described in the
previous section of this report, Calculation of the Surface Coordinate
System and Potential Flow. The main feature of the cross-sectional co-
ordinate system that has been found to be objectionable is that the length-
wise running coordinate lines seem to diverge on certain portions of the
hull (at keel near the bow and stern) where the opposite, i.e., convergence
of these lines, is desirable. Another, minor, annoyance of this coordinate
system is that it is difficult to find the set of starting values at any
particular station for the coordinate lines along the length (the orthogonal
trajectories of the cross-sections) that will result in a suitable surface

coordinate grid. Such a grid should not have large grid intervals or

26

T2ePOW OZL-VdSS eu uO g era pue yoTIN Aq pepueumosey se ‘setzoq00feay, TeuoZoyqIO ATeYA

pue suot}0es-ssoij oy. JO BUTISTSUOD JON 9eUTPI00ND BY} JO SMAETA OPTS pue wWoIJOg SUL - T PaNnsTy

Covi veo 08'0 0c o 09°0 9s°0 Ov’. oe'o 02°0 o1'o 00°0 oi'o- 02" 0- o€*0- Ov O- 05° 0- 03° O- 0L'0- 96° O- 06° 0°

4
(\\
[

i

21)

AE I [ =i i Cs === =
5 Z Zz Se 15 Sec 1 T S = = =
Z E Et rt [eal i a
Io ee iat je =a
MIIA SOIS
90° 06's oe'o oL°O 09°9 9S°0 Ov'O O€°O 02°0 o1°o fii Ol0= 02°0- O€*0- Ov’ O- | 0s‘0- 0S°0- 0L°0- 98°0- 06°0- (ofa)o

M3SIA WOLLOS

excessive grid crowding on some portion of the hull (for instance note the
trajectories on the bilge and and those at the load waterline near the
stern). For these reasons the coordinate system shown in Figure 2 for
SSPA Model 720, which consists of the lines of constant 98 and their
orthogonal trajectories of constant ~, was chosen in preference to the
cross-section system. Recall from the section on Calculation of the
Surface Coordinate System that the lines of constant 9 are defined by the
surface representation Equations (23a,23b). Thus, it is necessary to
always first represent the ship hull by a surface equation of the type of
Equation (23) in order to use the coordinate system of Figure 2, whereas
the cross-section coordinate system does not require a prior analytical
representation of the ship nl However, the calculation of the ship
surface Equation (23) for a typical ship hull such as the Swedish SSPA
Model 720 requires only about one to one and a half minutes of CDC 6/700
computer execution time. This calculation of the surface Equation (23)
(i.e., the matrix (An need only be done once and then it is available for
several other uses. Furthermore, the computer method used to calculate
the matrix (An is an old one and several modifications of this method
are presently under development that are expected to reduce the compu-
tationai time by a factor of about 100. Thus, the need to calculate the
surface representation of the Equation (23) type is not seen as a dis-
advantage of the (9,)-coordinate system.

The Lucy Ashton double model boundary layer was computed using the
earlier slender body theory potential flow meenodn because the first test
of the present calculation method was to check the complete crossflow
formulation. It was shown previously by von Rerenek that the slender
body theory gives fairly accurate values of the double-body pressure
distribution on the Lucy Ashton.

The boundary layer calculation method of this report is implemented
in terms of the (6,¢)-coordinate system but the computed boundary layer
results are given in terms of the streamline momentum thickness 81> dis-
placement thickness O45 shape factor H, and the stream coefficient of skin

friction Ce. This is done to facilitate the comparison of the present

28

Me
MLL
ML
1M

BOTTOM VIE

SIDE VIEW

29

Figure 2 - The Bottom and Side Views of the (¢,9) Coordinate Net on the SSPA-720 Model

results with previous boundary layer calculation methods and experi-

mentee

Earlier calculations of the Lucy Ashton double body boundary
layer by von Reena showed that crossflow is small almost everywhere on
the hull.

Figure 3 shows several streamlines computed by slender body potential
flow theory. In bottom, elevation, front, and rear views on the Lucy
Ashton double model. Computed boundary layer results will be shown along
these streamlines. Figures 4a-4d show a comparison of the distribution of

the streamline momentum thickness 0 along the streamlines 1, 6, 10, and

ila.
13 shown in Figure 3b, as computed by the present complete crossflow
method and the small crossflow method of von Reoneles Figures 5a-5d show

the distribution of streamline skin friction coefficient C, and Figures 6a-

6c show the distribution of the crossflow angle in i ee these same
streamlines. Note from Figures 3 through 6 that there is little difference
in the boundary layer characteristics that are predicted by the present
complete crossflow method and the small crossflow method. This is not an
unexpected result because the Lucy Ashton is a fairly slender hull with
very slowly changing cross-section shape along the length of the ship.
Hence the values of the coefficient Ky are small everywhere along the hull
and it is reasonable to expect fairly small boundary layer crossflow
effects. The differences in the two sets of results shown in Figures 3
through 6 are due mainly to the differences in the numerical integration
method used by von Kerezeky and the present method. This is indicated by
the differences in the results on the keel, shown in Figures 4d and 5d,
where the two methods solve identical equations.

The second test calculation is of the boundary layer on the Swedish
SSPA Model 720 double body. Figure 7, taken from Larsson's ee~ORE, shows
front and rear views of the streamlines along which measured and computed
boundary layer properties were given. Larsson's calculation method starts
from a momentum-integral-entrainment method closely related to the one
described in this report. The main difference between these two boundary

layer calculation methods is in the auxiliary data used for the crossflow

velocity profiles and the numerical implementation of the methods.

Dal
Larsson's method computes the boundary layer in the streamline surface

30

SoUuTTWe2TIS JO SMOTA ePTS pue wojJOg - ee oANSTYy

M3IA WOLLOd SANITINVAYLS

Tepon uoqysy Aon] oy UO SaUTTWeezqS - ¢€ sANSTY

0s0°0—

000°0—

000°0

0s0°0

OOL'O

31

SOUTTWPSIIS FO SMATA UIAqIS pue Mog — qE vANSTY
x

OcL'0 O0L0 0800 0900 oro00 0200 0000 0200 Ooro0o0 0900 0800

OOLO O2LO
080°0—

M3IA NYALs MalA MO€@

(penutquoj) ¢€ ean3Ty

090°0—-

0v0'0—

020°0—

000°0

32

Figure 4 - Streamline Momentum Thickness versus Axial Distance for Flow
Along Streamlines of the Lucy Ashton

x 10°

LEGEND

COMPLETE 3-D MOMENTUM
INTEGRAL METHOD

SMALL CROSSFLOW
APPROXIMATION

Figure 4a - Streamline l

33

Figure 4 (Continued)

x 103

LEGEND

PRESENT THEORY
VON KERCZEK

Figure 4b - Streamline 6

34

Figure 4 (Continued)

x 10°

LEGEND

PRESENT THEORY
VON KERCZEK

Figure 4c - Streamline 10

35

Figure 4 (Continued)

x 10°

LEGEND

PRESENT THEORY
VON KERCZEK

Figure 4d - Streamline 13

36

Figure 5 - Streamline Skin Friction Coefficient versus Axial
Distance for Flow Along Streamlines of the Lucy Ashton

x 102
6
Ss LEGEND
‘ PRESENT THEORY
VON KERCZEK ——-—-——
5
4

Figure 5a - Streamline 1

37

1.0

Figure 5 (Continued)

LEGEND

PRESENT THEORY
VON KERCZEK

Figure 5b - Streamline 6

38

Figure 5 (Continued)

x 10°

LEGEND

PRESENT THEORY
VON KERCZEK

Figure 5c - Streamline 10

39

Figure 5 (Continued)

x 103

LEGEND

PRESENT THEORY
VON KERCZEK

Figure 5d - Streamline 13

40

Figure 6 - Crossflow Angle versus Axial Distance for Flow Along
Streamlines of the Lucy Ashton

0.1

0.0
SSS?
LEGEND
mo PRESENT THEORY
VON KERCZEK
-0.2 |
-1.0
s
Figure 6a - Streamline 1

Ss
=

LEGEND

PRESENT THEORY
VON KERCZEK

LEGEND
PRESENT THEORY
VON KERCZEK

Figure 6c - Streamline 10

Al

coordinate system and uses a numerical integration method that includes
the crossflow terms by an explicit method. The present calculation
scheme may improve on Larsson's results only in detail but not in a funda-
mental way.

The surface coordinate grid on which the present calculations were
made is similar to but slightly coarser than the one shown in Figure 2.
The grid spacing is A® = 6 deg and As = 0.025 (As = Ao). The Cheae”
potential flow method was used to compute the values of the potential at
the center of rectangular elements in the (8,6) plane. These potential
flow rectangular elements (A0=10 deg, Ad=0.045) were much larger than the
grid elements of the boundary layer calculation so that the interpolation
technique with the cubic spline-on-spline described in the section on The
Ship Surface Coordinate System was used.

Initial conditions for the quantities 6 H, and t = tan) 6 at sitatdon

s = -0.5 (which very nearly coincides with Re of the 6-coordinate curves
on the surface of the hull) were obtained from hereason e- experimental
results. These initial data are shown in Figure 8. Intermediate values of
the data shown in Figure 8 were obtained by linear interpolation because

it was felt that the sparsity and quality of the experimental data did not
justify a more accurate interpolation. Figures 9 through 13 show the
distributions of the streamline momentum thickness C1 crossflow angle £,
and skin friction coefficient C, along the streamlines 1 through 8 of

f£
Figure 7. In each case, the results of the present calculation are
compared to the corresponding results of ieirecomntg experiment and his
calculation that includes the complete crossflow but not the modification
of the hull offsets by the values of the local displacement thickness. It
can be seen in Figures 9 through 13 that the present predictions seem to
correlate with the experimental data, on the average, about as well, or
possibly slightly better than, Larsson's computational results. In
particular, the present boundary layer predictions on streamline 5 extend
nearly to the stern of the model, in fairly good agreement with the experi-
mental data, whereas Larsson's results on this streamline seem to terminate

somewhat earlier, apparently because of some breakdown in his calculation

method.

42

zokey] Arepunog 24}

LEZ ANY NS \ Sa

ROS Pall
TRE TIAN
HT TAS
ee LTS

AINIAVWSAYLS

SNMINWIYLS

0.05

0.04

0.03

0.02

0.01
0.00
1.50 -—-—0.01
1.45 --—0.02
1.40 --—0.03
1.35 --—0.04
1.30 ——90.05
7)
Figure 8 - Starting Values of H, 6 and @ at Station S =- 0.5 on

SSPA-720 Model

44

Figure 9 - Boundary Layer Characteristics versus Axial Distance

x 103
12

10

—1.0

for Flow Along Streamline 1 on Model SSPA-720

LEGEND
EXPERIMENT oO
PRESENT THEORY ——©@——

LARSSON

—0.8 -06 -0.4 —0.2 0.0 0.2 0.4 0.6

Figure 9a - Streamline Momentum Thickness

45

Figure 9 (Continued)

a)

a} Oo a)

LEGEND
EXPERIMENT O

THEORETICAL RESULTS
ARE IDENTICALLY ZERO

Figure 9b - Crossflow Angle 6

46

Figure 9 (Continued)

x 103
6
LEGEND

EXPERIMENT oO

PRESENT THEORY ——©——
5 LARSSON SaaS SS
4
3
2
1
0
—1.0 -0.8 —0.6 —0.4 —0.2 0.0 0.2 0.4 0.6

S
Figure 9c - Streamline Skin Friction Coefficient

47

1.0

Figure 10 - Boundary Layer Characteristics versus Axial Distance for Flow
Along Streamline 3 on Model SSPA-720

16.4
x 10% i

LEGEND
EXPERIMENT O

PRESENT THEORY ——-GQ——
LARSSON ee

Figure 10a - Streamline Momentum Thickness

48

Figure 10 (Continued)

LEGEND

EXPERIMENT O

PRESENT THEORY ——G——

LARSSON

Figure 10b - Crossflow Angle 6

49

Figure 10 (Continued)

x 10°

LEGEND

EXPERIMENT Oo

PRESENT THEORY ——@Q——

LARSSON Se

Figure 10c - Streamline Skin Friction Coefficient

50

Figure 1l - Boundary Layer Characteristics versus Axial Distance for Flow
Along Streamline 5 on Model SSPA-720

x 10°
12

LEGEND

EXPERIMENT Oo

10 PRESENT THEORY ——O——
LARSSON os

-—10 -08 -—0.6 —0.4 —0.2 0.0 0.2 0.4 0.6 0.8 1.0

Figure lla - Streamline Momentum Thickness

51

Figure 11 (Continued)

LEGEND

EXPERIMENT oO

PRESENT THEORY ——©o-——

LARSSON

Figure 1llb - Crossflow Angle 6

52

Figure 11 (Continued)

x 10%

LEGEND

EXPERIMENT oO

PRESENT THEORY -——Q——

LARSSON

Figure lle - Streamline Skin Friction Coefficient

53

Figure 12 - Boundary Layer Characteristics versus Axial Distance for Flow
Along Streamline 7 on Model SSPA-720

x 103 {

LEGEND

EXPERIMENT Oo
PRESENT THEORY ——©O——

LARSSON

Figure 12a - Streamline Momentum Thickness

54

Figure 12 (Continued)

LEGEND

EXPERIMENT Oo

PRESENT THEORY ——©@——

LARSSON ee

Figure 12b - Crossflow Angle 8

a)

Figure 12 (Continued)

x 103

LEGEND
EXPERIMENT

PRESENT THEORY a

LARSSON

Figure 12c - Streamline Skin Friction Coefficient

56

Figure 13 - Boundary Layer Characteristics versus Axial Distance for Flow
Along Streamline 8 on Model SSPA-720

x 103
12

LEGEND

EXPERIMENT O

Bs PRESENT THEORY —<——
LARSSON

—1.0

Figure 13a - Streamline Momentum Thickness

Si

Figure 13 (Continued)

LEGEND

EXPERIMENT O

PRESENT THEORY ——G——

LARSSON SSS

Figure 13b - Crossflow Angle 8

58

Figure 13 (Continued)

X 103

LEGEND

EXPERIMENT 0

PRESENT THEORY ——G——

LARSSON oa ee

Figure 13c - Streamline Skin Friction Coefficient

59

It is to be noted that the experimental values of the crossflow angle
8 shown in Figures 9b, 10b, 11b, 12b, and 13b all exhibit a change in sign
(hence crossflow reversal) between the stations s = 0.6 and s = 0.8. The
results of the boundary layer calculations shown in these figures consist—
ently miss this flow reversal. This indicates that the crossflow model may
need considerable improvement in order to reliably predict crossflow
reversal as the boundary layer approaches separation. However, the cross-
flow discrepancy at the stern of the SSPM Model 720 may also be due, in
large part, to the discrepancy between the potential flow pressure
distribution and the actual pressure distribution at the stern. The
experimental and computational results indicate that the boundary layer
is very thick at the stern of the model; hence, the pressure distribution
must be different from the potential flow values there. Note also that the
degree of accuracy in predicting the primary quantities of interest, the
boundary layer momentum thickness 814 and skin friction coefficient Ces is
considerably better than the prediction of the relatively small values of
the crossflow angle 8.

On the basis of the comparisons with experimental boundary layer data
shown in Figures 9 through 13, the overall assessment of the present
boundary layer calculation method is that it can predict boundary layers on
relatively fine double ship models with fair accuracy to within a distance
of the stern of about 10 percent of the ship's length. In this area, the
boundary layer thickens very rapidly and approaches separation. Calcu-
lation of this near-separated boundary layer region must await further

developments of boundary layer theory.

CONCLUDING REMARKS
This report presents a momentum integral method for computing three-
dimensional boundary layers for ships. Most of the technical details for
carrying out the computational problem of solving the momentum-integral
boundary layer equations are worked out here and have been implemented in a

set of computer programs. The basic method can be used to calculate

60

certain boundary layer parameters, such as boundary layer thickness or

skin friction, with fair accuracy over a large portion of hulls where
unseparated flow is maintained. The computer programs are now ready to be
modified so as to improve the crossflow modeling, with whatever new
experimental data becomes available. Alternatively, portions of the
developments described in this report, such as the surface coordinate
system and inviscid flow calculation, can be used in other methods for cal-

culating the boundary layer.

61

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APPENDIX
MATHEMATICAL DETAILS

Substituting directly from Equation (21) into Equation (6) yields

2} 2S aw , 2= pay iss
O14 = O14 [cos a - t(£,+£,) sina cosatt f. Sain O4)| = 0, 184, (tH) (49a)
6) = 6 n@eae ) sin a cos at tf moe) = er sino] = 6. (ep is) (495)
12 11 3} 2 Al fie el OTe es ee
@,. € O.. (eee) ef ee 49 = CE, sim el] € OB (t,H) (49c)
D1 Lit 3) Sin a cos a 1 CoS O y sina] = 6,,85,(ts Cc
C) = 6 feinee + t(f,+f.) sin a cos a + oR ROSE = ¢ (e518) (49d)
22 ii be 2 3 = “sneD9 S=2
A, = 814 (H cos a - tf) Sati ©) = 65, h, (t.H) (49e)
A, = O14 (H sin’ @ + tf, COS ©) = 6, hp (t>H) (49£)
The momentum integral Equations (3) can then be written
p eerie ee 0811 93t pn 9811 9H fo Wa 5)
iL aye ii 68e |} OM ii, Our SO WZ Os
w) w) w) 9
dg dg
12 at 1D BE
On ae Bn 11 on ae G5 Cage) (si)
and
Senna Oe a 2m ey, eer
S91 92 i Se. Bs il Fe Of, ° 222 0
w) w) w) 9
og og
22 dt D2 Ae
Oi Be ly 7 in En Oe, Cann te (Sb)
where

63

u u
i Ee CLO a)
Ci an oes On a
Hence,
20 du du
aval 4 dk OU UM \ eal me) pL)
Sy Wagotold) G. om (na BO” Sue 90 U (, wm, ° oD
) ) v) w) 6
+ oul (844 78997hy sin a) + Ko91 1 (81 9+851 thy sina) (5la)
Similarly,
20 du du
a ee Moral au SU) a\p bl 0 0
Co Wein Bok) oy G, v (s 32, 1 822 92 U (, ay, 7 OD BE
) ) 0 0) 9
+ Ky 944 (85578, 7hy cos Qa) + aia (854 +8, 9th, cos a) (51b)

Let T and S be defined by

6-UA, ) (52a)

c|rR

and

Sa y (ugs-UA,) (52b)

64

The entrainment Equation (7) can then be written in the form:

oT oS a ae ) oy

Se ea RH) eh ee eK eS

at,” Re U, , U; af, ‘

Moreover, using Equations (10), (49e), and (49f)

T= 81,66 cos a+ tf, sin a) = 051) (to)
S = 85,66 sin a - tf, Cosa) = 8 Boo (tS)
The final form of the entrainment equation is
Shin ’ ese. Vy. ‘ SSG AT lta 055,
11 a2 iL. Oe ” Os I OH OL Ov
$ p o )
oh dh
DO, Oe 22 OH
7 On Ge Oe ab a a Ceol)
where
dU dU
Ls He do SNS a PSS)
Cys F(H) U, av) U, apt TK, + SK

65

(53)

(54a)

(54b)

(55)

(56)

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aS i se

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| ale + MSE Beet
EO eye al)

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82 fe TP Se #07 > ies
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ae tk ’

cae

REFERENCES
1. von Kerezek, C., "Calculation of the Turbulent Boundary Layer on

a Ship Hull,'' Journal of Ship Research, Vol. 17, pp. 106-120 (1973).

2. Landweber, L. and V.C. Patel, "Ship Boundary Layers,'' to be
published in Annual Reviews of Fluid Mechanics, Vol. 12 (1979).

3. Cebeci, T. et al., "A General Method for Calculating Three-

"W

Dimensional Laminar and Turbulent Boundary Layers on Ship Hulls," presented

at the 12th ONR Symposium on Naval Hydrodynamics, Washington, D.C. (1978).

4, Spalding, D.B., "Theories of the Turbulent Boundary Layer,"
Applied Mechanics Reviews, Vol. 20 (1967).

5. Green, J.E. et al., "Prediction of Turbulent Boundary Layers and
Wakes in Compressible Flow by a Lag-Entrainment Method," Aeronautical

Research Council of Great Britain, R and M 3791 (1977).

6. Miloh, T. and V.C. Patel, "Orthogonal Coordinate Systems for
Three-Dimensional Boundary Layers with Particular Reference to Ship

HonMshae Journal or Ship) Research, Voll 175 Now 1 1973)

7. Myring, D.F., "An Integral Prediction Method for Three-
Dimensional Turbulent Boundary Layers in Incompressible Flow,'' Royal

Aircraft Establishment, Technical Report 70147 (1970).

8. Reynolds, W.C. and T. Cebeci, "Calculation of Turbulent Flow,"
in Topics in Applied Physics, Vol. 12, P. Bradshaw, editor, Springer-
Verlag (1976).

9. Cumpsty, N.A. and M.R. Head, "The Calculation of Three-Dimensional
Turbulent Boundary Layers, Part 1: Flow Over the Rear of an Infinite

Swept Wing," The Aeronautical Quarterly, Vol. 18, pp. 55-84 (1965).

10. Head, M.R. and V.C. Patel, “Improved Entrainment Method for
Calculating Turbulent Boundary Layer Development," Aeronautical Research

Council (Great Britain), R and M 3643 (1968).

67

11. Coles, D.E., "The Young Person's Guide to the Data,"' Proceedings
Computation of Turbulent Boundary Layers - 1968, Air Force Office of
Scientific Research IFP-Stanford Conference, Vol. II, D.E. Coles and E.A.
Hirst, editors (1968).

12. Thompson, J.G., "A New Two-Parameter Family of Mean Velocity
Profiles for Incompressible Turbulent Boundary Layers on Smooth Walls,"

Aeronautical Research Council (Great Britain), R and M 3463 (1976).

13. Mager, A., "Generalization of Boundary-Layer Momentum-Integral
Equations to Three-Dimensional Flows, Including Those of Rotating Systems,"

National Advisory Committee for Aeronuatics Report 1067 (1952).

14. Okuno, T., "Distribution of Wall Shear Stress and Cross Flow in
Three-Dimensional Turbulent Layer on Ship Hull," Journal Society Naval

Architects of Japan, Vol. 139, pp. 10-22 (1976).

15. von Kerczek, C. and E.O. Tuck, "The Representation of Ship Hulls
by Conformal Mapping Functions,'' Journal of Ship Research, Vol. 13,
pp. 284-298 (1969).

16. Chang, M.S. and P.C. Pien, "Hydrodynamic Forces on a Body Moving
Beneath a Free Surface," Proceedings of the First International Conference
on Numerical Ship Hydrodynamics, J. Schot and N. Salvesen, editors,

Gaithersburg (1975).

Ive) SHesSsdiliaand ASMo Ose smith. mGalculatdiongot. NonilattinemePotentelall
Flow About Abritrary Three-Dimensional Bodies," Journal of Ship Research,

WOlLS 85 Moo 2 (leo).

18. Alberg, J.H. et al., "The Theory of Splines and Their
Applications,"’ Academic Press, New York (1967).

19. O'Brien, G.G. et al., "A Study of the Numerical Solution of
Partial Differential Equations," Journal of Mathematics and Physics,

Wil, ZO, pao DAIS CEI).

68

20. Joubert, P.N. and N. Matheson, "Wind Tunnel Tests of Two Lucy
Ashton Reflex Geosims," Journal of Ship Research, Vol. 14 (1970).

21. Larsson, L., "Boundary Layers on Ships, Part IV: Calculations
of the Turbulent Boundary Layer on a Ship Model,'' The Swedish State
Shipbuilding Experimental Tank, Goteborg, Sweden, Report 47 (1974).

69

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