An estimation of the normal force and the pitching moment of TEAR DROP underwater vehicles

AN ESTIMATION OF THE NORMAL FORCE

AND THE PITCHING MOMENT OF

"TEAR-DROP" UNDERWATER-VEHICLES

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECH-NISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE

RECTOR MAGNIFICUS PROF. DR. IR. H. VAN BEKKUM, HOOGLERAAR IN DE AFDELING DER SCHEIKUNDIGE TECHNOLOGIE, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN TE VERDEDIGEN

OP WOEINISDAG 19 MEI 1976 TE 14.00 UUR

DOOR

EDUARD VAN DEN POL

NATUURKUNDIG 1NGENIEUR GEBOREN TE MIDDELHARNIS

1976

Aan: Anneke

mijn schoonvader

Reinoudt en Adrienne

de Koninklijke Marine

page no.

1

LIST OF USED SYMBOLS ₄

CHAPTER I GENERAL INTRODUCTION 9

1.1. The origin of this thesis 9

1.2. Aim of this thesis 10

1.3. Synopsis 11

CHAPTER II ANALYSIS OF THE INVISCID FLOW AT ANGLE OF ATTACK 12

II.!'. Introduction 12

11.2. The longitudinal flow 14

11.3. The transverse flow

19

11.4. The normal force and its distribution ₂₃

Chapter III THE NON KARMAN-METHODS AT WORK ₂₇

III.1.First experiences ₂₇

III.2.An optimal approximation of the given contour ₂₈

III.3.Computational results ₃₃

III.4.Some remarks ₃₄

CHAPTER IV VORTEX FLOW CONSIDERATIONS ₃₆

Introduction ₃₆

Experimental evidence with respect to the

vortex generation ₃₈

The separation philosophy of this thesis ₄₂

CHAPTER V _{ANALYSIS OF THE "VISCOUS" FLOW AT ANGLE OF ATTACK}

45

The "viscous" flow field ₄₅

How to solve for CI and

r

₄₇

The "viscous" normal or transverse force ₄₉ LIST OF FIGURES TABLE OF CONTENTS .... . .... -IV.1. IV 2. 1V.3. _-... V.E.

...

V.2.

...

V..3m

FIGURES 1 - 54

APPENDIX A THE STREAM FUNCTION OF STOKES Definitions

A parallel, uniform stream A point source

APPENDIX B THE CONTINUITY IN THE FIRST DERIVATIVE OF THE CON-TOUR APPROXIMATION 8.1. Analysis B.2. Numerical example 77 124 124 125 126 128 128 129 VI 2c The "separation curve" experiment 52

CHAPTER VII RESULTS WITH THE VORTEX FLOW MODEL ₅₄

1. Introduction ₅₄

VII.2. "AKRON" and the used model ₅₅

VII.3. The fuselages of Lange

₅₅

VII.4. The DTME-model 4198/series 58 ₅₆

VII.5. The trajectory of a vortex core ₅₇

CHAPTER VIII THE INFLUENCE OF A STERN-MOUNTED PROPELLER ₅₈

VIII.1.Introduction 58

VIII.2.The boundary layer ₅₉

VIII.3.The induced flow field ₆₃

VIII.4.The relation between ca and KT ₆₅

VIII.5.The wake fraction ₆₇

VIII.6.Some remarks ₆₈ VIII.7.Resultg ₆₉ LIST OF REFERENCES 71 ... A.2. A.3. . . .

APPENDIX C THE SOURCE-TERM IN EQUATION (V.1.2) 131

APPENDIX D THE FORCES ON THE VORTEX AND VORTEX-SHEET ₁₃₂

Dc l. General theory ₁₃₂

0.2. The vortex with sheet ₁₃₄

APPENDIX E COMPUTATIONAL DETAILS

APPENDIX F THE OFF-SETS OF THE USED "TEAR-DROP" BODY OF

REVOLU-TION, DEPICTED IN (FIG. 27) 138

SUMMARY (IN DUTCH) ₁₃₉

CURRICULUM VITAE ₁₄₁

...

.. .

LIST OF FIGURES

Figure page no.

Line source nomenclature 77

Body of revolution in an uniform stream 77

Used coordinates 77

Used coordinates 78

Transverse flow around cylinder 78

Condition for transverse flow around body of revolution 79

Force situation in potential transverse flow 79

The meridian curve of a "tear-drop" body 80

The "Von Kaman treatment" 80

The surroundings of the "pivotal" point {xi, yi} 81

The approximation of the second derivative 81

The pressure distribution of body no. 2 82

The pressure distribution of body no. 3 83

The pressure distribution of body no. 5 84

The pressure distribution of body no. 6 85

The pressure distribution of body no. 7 86

The pressure distribution of US airship "AKRON" 87

The pressure distribution of DIMS-model 4198 88

The normal force distribution at a = 15°: US airship

"AKRON" ₈₉

The dragcoefficient as a function of Re-/Sh number for long cylinders in steady transverse flow

The dragcoefficient for a circular cylinder in impulsively started laminar flow

NACA vortex model

The flow around a "tear-drop" body of revolution at

incidence

The vortex generation along a "tear-drop" body at incidence The two-dimensional transverse or cross-flow

The calculated pressure distribution for the model, depicted in fig. 27 90 91 92 93 94 95 96 2. 4. ... 5- ... ... IQ. 04. 05. 22. ... 26'..

34, Lift-force and pitching moment as function of angle of attack

Normal force distributions as calculated for model, depicted in fig. 27

Lift- and moment coefficients, body no. 3

105

106

Lift- and moment coefficients, body no.. 6.-...--..- 107

Lift- and moment coefficients, body no. 7 . 108

390 Lift- and moment coefficients, body no. 5 Lift- and moment coefficients, body no. 2

Normal force distributions DTMB-model 4198

Normal potential force distribution for DTMB-model 4198 at incidence

Normal force distributions DTMB-model 4198 113

Normal force distributions DTMB-model 4198 114

The calculated vortex core trajectories of US airship

"AKRON" at a = 15° D15

The boundary layer thickness US airship "AKRON"

The propeller sink-disc 1117

The propeller sink

The validity of the propeller-representations 1419_

50. Velocity profile in boundary layer at a tip-radius ahead of the propeller, ref. 511. Velocity profile in boundary layer at a

tip-radius ahead of the propeller, ref. Calculated values local wake fraction Flow diagram for the computation of the for a given stern-mounted propeller

distance twice the

(28) 119

distance equal to the

(28) 120

124

induced flow field

522

The "tear-drop"' body of revolution used in the towing tank experiment

The streamlines over the bow; bow down 250 98

The streamlines over the mid-part of the body; bow down 15°. 99

The streamlines over the tail; bow down WOO

37. The position of the separation curve

The normal force distribution of US "AKRON"

The normal force distribution of US "AKRON" 103

-2-15° 97 101 102 104 109 110 111 112 117

[A] - a matrix

a - a line piece of constant length or an arbitrary

distance

al, a2, a3,

4,

b3- coefficients or constants

CD - cross-flow drag coefficient = (dN/dx)/(pmW2r)

pitching moment coefficient = 2.moment about

re-Cm

LL

ference point/(pm.V2.volume) = ( _{5al(x)dx)/volume}

CL - liftcoefficient = 2L/(pm.V2.volume2/3)

CT - propeller thrust-loading coefficient = 2T/(pmU:Feff)

Ca - sink strength per unit area

(matrix) coefficient = 1 +

(pL -

pL)/a

cki

c, c

, c - total velocity components in x-, r-, a-direction

x r

in the angle of attack situation

diameter propeller

Feff - effective propeller disc area = 11-(R2prop - R2)

h, h' - length of an interval along x-axis

- impulse

dimensionless velocity

t(ukx/U)2 + (u` kr/11\21/

- lift force

4

-LIST OF USED SYMBOLS

IL - length of the body

1 / 2

-a

-1 - a line piece of arbitrary length

- doublet strength

pitching moment (positive nose up) = 2.moment about reference point/(pm.V2.LL3)

Ma - mach number

- number of line sources or body-coordinates, also normal force

number of propeller revolutions

fluid pressure

Undisturbed fluid pressure at infinity

source strength

'source strength per unit length = Q/a

- radius in propeller-plane

Re - Reynolds number

radius of propeller-hub

Re - Reynolds number based on body's length

LL

prop - radius of propeller

- a difference Sh _ Strouhal number piece of arc - propeller thrust Rh

- time

uniform stream velocity in x-direction

Ue - fluid velocity through the propeller-plane in

ab-sence of the propeller

fluid velocity in vicinity of wall or

body

the axial velocity component in the k-th "pivotal" or reference point in an axisymmetric uniform, parallel flow

ukr - the radial velocity component

in the k-th "pivotal" or reference point in an axisymmetric uniform, parallel flow

V u2 w2

-

propeller induced velocity

x-component of propeller induced velocity in point

uniform stream velocity in y-direction

velocity components in x-, y-, a-direction due to a transverse flow

Px

w , w , w

x y

x, r - radial and axial cylindrical coordinates,

respec-tively

y equals r for

a =

0, also distance to wall or body

Zt - vertical force (positive downward) = 2V(om.V2.LL2)

+

-i-th dimensionless source strength = Qi/(2ffila2),

also i-th dimensionless doublet strength = p./(4rWa2)

angle of attack

= normal force width =

(2dN/dx)/(pm(e +

W2))

= angular spherical coordinate

vortex strength

angle between tangent and x-velocity component in a point of the, contour curve., also thickness

boun-dary layer'

-.complex coordinate in) normal plane .= y + iz rei0

angle of radius vector with respect to y-axis in complex (normal-)plane

propeller apparent speed of advance coefficient =

Ue/(n.2 R )

prop

M7dC = doublet strength per unit length

.7 kinematic viscosity'

v,p radial and angular spherical coordinates, respec-'

tively.

infinitesimal element of a line source.

!= distance from left end of i-th line source to k-th "pivotal" point

Pk-

distance from right end of i-th line source to k-Eh

]: "pivotal" point a -= -A

-Pm

To

-8-fluid mass density

angular cylindrical coordinate

shearing stress

the complex velocity potential = + iT

the velocity potential

wake fraction

stream function of Stokes

property in a point P

property for r = rmax

normal property

property of the separation point on the

body in

the normal plane

( )1 - property of the center of the vortex in the normal

plane

)re - real part of variable

( )im - imaginary part of variable

(*) - time derivative of variable

()

- complex conjugate potential property viscous property Subscripts ( )p ( max

--

_max (

-K

-Chapter

I

GENERAL INTRODUCTION

The origin of this thesis.

By using computers it is nowadays possible and already a established practice to predict the submerged performance of

underwater-vehicles. After the disaster in 1963 with the nuclear submarine USSN "Thresher" for example, the United States Naval Authorities ordered that no new submarine should be put out to sea unless - by using simulation techniques - its stability, and consequently its complete submerged behaviour, are fully understood.

The higher the speed of a manned underwater-vehicle the more it is necessary to explore the vehicle's submerged performance before-hand, because the hydrodynamic forces - increasing with the square of the speed - are enormous.

As in general such a manned underwater-vehicle can only operate in a relatively thin layer of several times its own length, the time-interval, available to its crew to deal succesfully with undesirable disturbances in the vehicle's behaviour, is therefore severely li-mited. Only by a thorough, systematically executed investigation one is able to minimize those submerged incidents that may well lead to a complete disaster.

The tools for such studies are modern analogue- and those digital computers, accessible for a simulation language. To simulate the trajectories of a particular vehicle, e.g. a submarine, one must have at one's disposal a set of characteristic data, concerning the motion of that type of submarine in water. These data, in the form of dimensionless coefficients, are obtained by towing-tank

experiments, as described a.o. by Goodman (34)1), van den Brug (50) and Gertler (54), using models of the submarines concerned.

1)

10

-When an underwater-vehicle is travelling at constant depth and is well trimmed, the vector, representing the vehicle's speed, in general will be in the same direction as the longitudinal axis of

the vehicle.

When manoeuvring submerged, however, the direction of the vector will make an angle with the longitudinal axis which is called the angle of attack.

For small values of the angle of attack it is well-known that li-nearized differential equations, describing the motion, are suf-ficiently adequate.

For larger angles of attack non-linear terms should be added. The problem now arises around the coefficients involved with the non-linear terms.

One way to obtain the necessary extra information is extrapolation of the forementioned towing-tank experiments but this cannot be accomplished without additional costly equipment and instruments, among others a so called rotating arm facility.

Obviously a theoretical way to determine the non-linear coefficients would be preferable and in this thesis a modest attempt is made to establish expressions for some, non-linear coefficients as a function of the angle of attack.

1.2. Aim of this thesis.

The aim of this thesis is to establish a procedure of tentative nature, enabling the computation of the normal force and pitching moment for underwater-vehicles of "tear-drop" configuration in the range of angle of attack of 10 degrees up to 25 at any desired

speed.

The vehicle is assumed to proceed in a fluid of infinite dimensions in order to elimenate functional dependance on the Froude number or the need to consider "bottom"-effects.

The whole thesis is based on the following philosophy:

If, however, original experimental work has to be done and if applicable, aggravation of the mathematical model is preferable if this opens the way to a simpler experiment.

Synopsis.

In chapter II the Von Kgrmgn (3) axial line source and doublet methods will be discussed, respectively.

Combination of these two techniques leads to the angle of attack situation, in which the viscosity is not considered.

In spite of its mathematical simplicity the Von Kgrmin method was inherently never suitable for rotational symmetric bodies of given, arbitrary shape.

Chapter III deals with a very effective and simple approximation of the meridian contour of so called "tear-drop" bodies of re-volution, which does make the Von Kgrmgn techniques for the very.

first time applicable to those bodies.

The role of the fluid-viscosity is introduced in chapter IV in the form of boundary layer separation followed by a steady vortex over the leeward side, inducing a viscous force distribution on the body of revolution.

Chapter V analyses the vortex development as described by Bryson

(33).

In chapter VI the results are discussed of an attempt to obtain the seperation line of a representative body of revolution. Chapter VII deals with the computational results based on chapter V, using the experimental figures from chapter VI as input data. Finally in chapter VIII the influence of a propeller on the fore-going results is discussed.

12

-Chapter

ANALYSIS OF THE INVISCID FLOW AT ANGLE OF ATTACK

II. Introduction.

Since the advent of the airship there has been interest for

methods that give the flow over a body of revolution; for interest ting historical reviews in this respect see reference (19) and

(39).

The direct problem, i.e. to determine the flow over a given body of revolution, was tackled for the first time by Von Kaman (3). He was asked by the "Zeppelin-Luftschiffbau" at Friedrichshaven

in 1927 to predict the force-distribution of the Zeppelin LZ 126, the later "Los Angles", and obtained results which seemed well in accordance with the experiments of Klemperer (2) in 1924. In 1974 Oberkampf and Watson (56) wonder that, since the original work of Von Kdrman, very little enlightening information on his

method has been published, although most text-books on fluid-dy-namics do refer to this method.

From a practical engineering point of view this may seem at first glance, very remarkable as its mathematical simplicity makes the Von Kirman method rather inviting and seemingly very accessible. Therefore one would expect that - at least for a restricted

num-ber of applications perhaps - this method would be at one's dis-posal now in a more or less developed state.

The reason that this is not the case arises from the fundamental objection that only given bodies of revolution of exceptional

shapes can be represented by a distribution of singularities on the axis of symmetry.

According to Von Kaman, considering the axisymmetric or longitu-dinal flow over a given airship-body:

"The representation by an axial source-sink distribution is only possible in the exceptional case when the analytical continuation of the potential function, free from singularities in the space outside the body, can be extented to the axis of symmetry without encountering singular spots".

What this means in practice may be illustrated with the example that difficulties can be encountered, even in the relatively simple case of a body of revolution consisting of a cylindrical mainbody and a hemispherical nose-piece, since the transition nose-piece/mainbody brings about discontinuities in the body's

curvature.

Lotz (5), dealing with the inviscid transverse flow of airship-bodies, mentioned the appearance of "certain difficulties" in her calculations. She stated that it is not predictable when the Von Kgrman method will behave badly; her statement is supported by the experiences of Watson (53).

Because of this irrational behaviour she developed a new method, in which the sources and sinks are placed upon the surface of the rotational-symmetric body.

Her method is finally brought to peak development by the work of Hess, Pierce and Smith (47), (32).

Although they have succeeded to produce superior methods one should be aware of the fact that these require the evaluation of very involved simultaneous integral equations, which even on fast computers demands relatively long computation time.

Therefore, it would be very convenient for first, tentative in-vestigations of the flow around "tear-drop" bodies of revolu-tion if the Von Kgrmgn techniques could be suitable developed. If so, useful results can be obtained in a fairly "inexpensive" way.

A4

-In the following paragraph's, therefore the Von Kaman methods will be discussed in detail.,

11.2. The longitudinal flow.

In appendix A the following expression is derived for the

stream-function LP in a point P, situated in the flow of a simple point

source:

V

-- (E +. cos

V)

P' 47

If instead of a point source a line source, the length of which is a, is placed on the x-axis with a constant discharge q per unit length (see fig. I) and dE is an infinitesimal element of the line source then the stream function in a point P. due to this element is:

q dE

ti*P 47 tos v) tiI.2.2)

.

For

the whole line source this amounts tent a

LI) = - 1-

f

(ill +

cos v) dE

47

Consider (fig.

0

again:

cos V

-dE

(11.2.1)1,

p11.2.4),

The negative sign since an increase, in is coupled with a de-crease in p.

Expression (11.2.4) substituted in (11.2.3) gives.:

a _ID"

V

_P 3-₄₇

{1

dc _dP (11.2.5)

Pt

- (a + Pi - p")

47

The total discharge of the line source

M.2.6)

Q q . a (11.2..7) = - (1 + = -= is: (11.2.1)

So, using (11.2.7), for (11.2.6) can be written:

PI P"

th

'P - 4n (1 a )

For N in number line sources on the x-axis (sinks are negative sources) the streamfunction in point P will be given by:

i=N in, _Pi a pi; } 1 ipp = - E =II (1 + (11.2.9) i= 1.47T

In order to create rotational-symmetric stream surfaces the stream-function of a parallel, uniform stream (see appendix A) is super-imposed on (11.2.9): *p = 2 - E =(1- (1 + _ Pi i=N n. 47 a Pi Ur2 i=1

Recalling the definition and the use of Stokes stream function, expression (11.2.10) is investigated.

When point P is on the negative x-axis then p! -

4 =

1

for every line source, while r = 0, so from (11.2.10):

= 0 (II.2.11)

Besides that (II.2.11) is in full agreement with all that has been said in appendix A about the zero streamline, expression (11.2. II) is also valid if P coincides with the stagnation point S (see fig. 2), here the zero streamline becomes a dividing streamline.

In other words: the dividing streamline is also characterized by

= 0.

If therefore, point P (x,r) is situated (fig. 2) on the dividing streamline expression (II.2.10) will be:

i=N I Ur-7 E

a

(1 + P! -= 0 -= a 1) _(11.2.12) 2 47 i=1 2

after multiplication with

(11.2.8)

(11.2.10) ---)

i=N

(11.2.13)

0 = ) 2 - 274-7J'a- (I + Pi -a Pi)}

i=1

Suppose there is a whole series of points P and referring to the k-th point, the following notation will be used:

I +

C)'ki -pu

ki

_{-c .}

_(11.2.14)

a ki

Likewise when dealing with the i-th source in dimensionless

form:

Qi

z.

-2

1 21-11a

Using (11.2.14) and (11.2.15) transforms (11.2.13) into:

i=N _rk

E c . z. =

ki 1 a

i=1

To apply the foregoing in order to solve the flow-problem of given bodies of revolution at angle of attack the dividing

stream-line must be forced to pass through a given set of body coordi-nates, belonging to a number of points of the meridian or contour

curve of the body. These points

will

be referred to as "pivotal" points and have to satisfy (11.2.16).

If one intends - contrary to Von Karman, who is cutting the given body of revolution in two halves and deals with them separately -to treat the body as a whole, another requirement should be added, because the sources (and sinks) must form a flow-system closed

in itself.

The so called "condition of closure" is:

i=N

Ez.=-0

₁ i=1 16 -(II:. 2 .,M5) (II. 2.16) (11.2.17)

This causes a linear system of equations for the zi, N-1 of which are described by (11.2.16) and one by (11.2.17), leading to the solution of N sources (sinks included).

In this case k = 1, 2, , N-1 in (11.2.16).

Following the Von Karman methods - as will be done in this thesis - condition (11.2.17) is not applicable and consequently k runs

from k = 1 up to N in (11.2.16).

It should be noted in this respect that this is also true when dealing with given bodies of revolution which are symmetric fore and aft as has been demonstrated by Watson (53).

The velocity components in the k-th "pivotal" point _{Pk can be} found through the expressions:

u = . kx rk

ar

lk (11.2.18) 1 ukr = rk

From '(fig. can be seen that:

p V1.2 1)12 , differentiation gives:

ac

r = = sin

v

Dr p 1) - cos v ax p

Substituting (11.2.20) in (11.2.19) changes the latter:

(11.2.20) Using relation (11.2.10): i=N ukx 1 1=

1r4

ukr = E .=1 1 (api (II. 2-19) 47ark

_a__

471-ark ar

(aAki

ar ax

4'1'6)1

ax I [

x

= + E +

E

I=N

ukx=..U. + E Qi (sin vki". - sin v'.) ki i=1

(11.2.22)

Suppose pc) is the not disturbed fluid pressure e.g. at infinity, then according to Bernouilli's line theorem:

pr 4.

im

u)

v= oi 4. 2m u2 2 '-kx kr 2 ,UkX,2 Ukr 2 -introduce:

k2 = m

U e 4Th) p + Jim Kk0.2" po + U2

AP = P

Pd =

(r

0)

U2

While k2 can be found by use of (11.2.22), it is possible to cal= culate for every "pivotal" point of the given contour-curve:

P Po _kz

(11.2.25)

Expression (11.2-25) gives the pressure distribution in dimension-less form of a given body of revolution at zero angle of attack:.

the case of longitudinal or axisymmetric flow. Pm u2 2 XII.2.23) (II.2-24) ukr i=N E (cos v".

cos

v1,31

kJ. ki (11.2.2) (mar k i=V Using now (11.2.15) : Qi - ---Ua this

4fark

z.

2tft" and substituting result in (11.2.2401: i=N ukx U a E i=1 i=N (stn ki - sin

vt )1

ki ' ! + 2rk ukr Ua E 1=1,

.(z.

(cos V. - COS V

k -2tk 18 -= = -

-47ark

-De uitspraak van Lessing (51)1) dat de methode van bron-put belegging op de rotatie-as slechts t.a.v. omwentelingslichamen met een lengte/ diameter verhouding groter dan 10 acceptabele resultaten geeft is niet alleen in tegenspraak met de ervaringen van Watson. (53) maar wordt

tevens in dit proefschrift ale onjuist ervaren,

Ten aanzien van de distributie van de normaalkracht over een "druppel-vormig" omwentelingslichaam is de ligging van de zg. "loslaat-kromme" een kritische parameter, die geldt vooral voor het voorste, min of

meer horizontale gedeelte.

3'

De schroef, die een "druppelvormig" onderwater-vaartuig voortstuwt, heeft geem invloed op de grootte en de distributie van de normaalkracht.

De singulariteiten-methoden van Von Kgrmgh zijn vooral bijzonder ge-voelig voor de continuiteit in de tweede afgeleide van de

contourbe-schrijving. lie literatuurlijst, blz. 71L I) 1 2 4

Literatuurdocumentatie is een onmisbaar stuk gereedschap voor een vetenschappelijk onderzoek; de kwaliteit van dit gereedschap wordt

echter uitsluitend bepaald door het vertrouwen in de mate van volle-digheid van de door de documentalist geleverde bibliografie.

Het bestaansrecht van het Koninkrijk instituut voor de marine t.a.cr. de opleiding voor marine-officier, is alleen dan te verdedigen als naast het onderwijs de overige facetten, die bijdragen tot devorming,

van de toekomstige officier, dezelfde aandacht kriigen.

Het komt als onjuist voor de energie-opwekking en de voortstuwing van een oorlogsschip op hoog niveau te automatiseren als dit alleen geschiedt om een reduktie van (ter zake kundig) personeel te kunnen

effektueren..

8

Door de zg. nvermaatschappelijking" van de krijgsmacht is de over-levingskans in vredestijd van de Nederlandse dienstplichtige militair vrijwel op. Len te stellen.

9

Gegeven dat een proefschrift een proeve van o.4. logisch denken is, impliceert dit nog geenszins dat in het leven van alle dag gepromo= veerden tot de meest logisch denkenden behoren.

5

Het schriSven van een wetenschappe/ijke verhandeling in een andere taal is als de plotselinge conversatie met een onbekende, beeldschone vrouw: in beide gevallen weet men niet altiid wat men zegt.

pdE

dctip = -

---7

470

From (fig. 4) it can be seen that: cos y = sin v cos a

and consequently: pdE &pp = 4702 sin v cos a

furthermore:

x - (E +

1) = r cotg v so that - dE -

_sin2

v dv and r = p sin v Substitution of (11.3.5) in (11.3.4) gives:

cl(Pp = - sin v cos adv (11.3.6)

torr

For the whole line doublet (11.3.6) becomes: VI!

q)P =

-v'

sin v cos adv

471.

OP=

_4irr (cos v" - cos v') cos a

cos y

- 19

-11.3. The tranverse flow.

For the potential in a point P in the flow of a doublet of strength M, Von

Kaman

(3) uses after a detailled derivation -the expression (fig. 3):

Op =

74-i7

COST (11.3.1)

Turning to (fig. 4) assume on the x-axis on a arbitrary distance 1 form the origin a line doublet, of length a and strength p per unit length, is placed.

If d is an infinitesimal element of this line doublet then the potential in a point P, due to this element, is:

(II .3.2)

(11.3.5)

Integration of (11.3.7) from v = 0 till

v =

if is the mathematical interpretation of a line doublet extending from x

= - co

to

ic

= + 03,

for such a doublet the potential in P amounts to

Op = (1h1) cos a =

-7r 27r (11.3.9)

(11.3.10)

Apparently the flow-solution for every angle a is found by solving the problem for

a

= Oland then multiplicating with cos

a.

The velocity component in the r-direction can be found by:

r ar

P 1.1

27r2 cos

Superposition of a uniform, parallel flow in the negative y-di-rection (with velocity - W), and the flow-field due to a doublet, as expressed by (11.3.9), creates the potential flow around an infinite long cylinder.

When using (fig. 5) it is obvious that the condition for flow around such a cylinder demands:

wr - W cos CY = 0

or using (II.3.10):

27r2 cos a - W cos

a = 0

The radius from the resulting cylinder is:

%/ P

r = (II.3.11)

27W

If in the above superposition a doublet system, of N in number line doublets, of which the intensity is a function of x, is used the transverse flow of a body of revolution can be obtained. Using (11.3.4) such a system is expressed by:

i=N ,Mi sin

vi

(1) =

-.cos

a E (11.3.12) 47 Pi' i=1 =

cos a [4)]

_{a = 0}

(11.3.13) cos =

(cos

Vi

- cos

Vi)}]

In general from (fig.

4):

V = arctg

x -( + 1)

Dv -y so

ax

y2 + tx

- (C +

1)/2

x - (C +

1) - 2 "Y Y

LX -

+ 1)}7 and: sin

v

%/y2 {x - (1

u}2

21

-Starting then by using (11.3.8) and bearing in mind that for

a = 0 : y = r (see fig.

4):

i=N

(P=I

iE1 {11.(cos

v

-

cos

v!)/

47y (11.3.14)

The velocity components in the k-th "pivotal" point Pk can be

found through the expressions:

+

Substituting (11.3.16) and (11.3.17) in (11.3.15):

wkx tiny? i=1E (sin3ve: - sin3v1)}

(11.3.15) (11.3.16) (11.3.17) (11.3.18) = f ao1 = i=N I Dv. 3vr:

wkx L3xjk 47 ill pi (sin V! --I -Dx sin

ax

= w,Y

Kbib.

30 [ - 1 i=N i=1 ,

av;

klii.(Yk " - yk sin

Vi

_i

avY 47172 sin v: Dy ay

COS V -

X -

(1 + VY2 + {x - (1 + -+ + v? i=N

i

(11.3.18)

Introduce the abbrevations f and g,, then for the whole doublet system: i=N 1

w

= E

Ip. (r!

-kx

4ryi / i=N

-, 1

4zyi

I

-1 Define analogous to (11.2.15): Pi

4i

-43714

1( .1 9)

To study the transverse flow of a given body of revolution,, of which the contour-curve and consequently its slope are known it is imperative that the contour coincides with a streamline. Referring to (fig. 6) it is to be said that this condition is satisfied in a point P of the 'contour if the normal velocity com-ponent equals zero.

This is the case when in P the resultant of the velocities of the uniform, parallel flow - W and wx, wy of the doublet-system coincides with the tangent.

w

-w

from (fig. 6)Z tg

-or r

tgowx <11.1.20)

Substitute in (II.3.10)1 the velocity components of 01.3.197

i=N A E

[p.

f(gl! -

e) +

tg

oar -

w

4731

i=1

411.3.22) .,;ky ..1 ----2-

i'll I

E I i=1 p.A 2 (cos V! cos vP) -1 -A cosi3v, P) - nos3v 1 A A irtryk i=1 w

-f)}]=

(11.3.21)

or by use of (11.3.22)

I r' 2

Z.

_2a

-

23

-then (11.3.21) transforms in:

i=N

zi

t(gi -

+ tg 6(fi _ f)j =

(.5?)2

i=1

(11.3.23)

To solve this system for N line doublets N reference points on the given body-contour are necessary.

A fast approximation of (11.3.23) can be obtained as follows: Each annular part of the given body of revolution, situated around each of the points of reference is replaced by a small cylinder with a radius equal to the radius ri of the annular part at the reference point.

Then from (II.3.11): p. = 27r.2W

1 1

(11.3.24)

Using (11.3.22) substitution of either the zi's found from (11.3.23) or (11.3.24) (together with the proper addition of the uniform, parallel flow velocity - W) in the system (11.3.19) will produce the transverse velocity components.

11.4. The normal force and its distribution.

The following total velocity components at angle of attack are defined:

cx = ux + wx cos

a

(II.4.1)

Cr = ur Wy cos

a

LI, and ur can be found from (11.2.21) while wx and wy follow from one of the two above mentioned substitutions of zi in (11.3.19) adding the term - W in the expression for wky.

ca = wa

sin 0 (11.4.2)

E

= 1

The used potential flow theory justifies the application of Ber-nouilli theorem:

pc, p. (U2 4. w2) lit +

4

4. c102) Po = undisturbed fluid pressure

p - po = (U2 +w2

-

C2 -c

co? )_a

From (II.4.1) and (11.4.2):

2 22

P Po = .2m{ U2 + W2 - ux2 u1

-

w cos2a - W COS2

-X

2

Wa sin a - 2 (uxw + ur wy) cos a

Focussing the attention on (fig. 7): 0

= 2

{T.32 w2 ux2 2 _{wx coia -}2

_w2cos2a

-PI 7T-0

r0

2 sin2_{a + 2 (u} w + u w ) cos al a r y x x -- + W2 - 1.1)1

-

U2

w 2 COS2G w2cos2a -Pia -° 2 r x w2sinzo - 2 (u w + u w ) cos a r y x x

41

_{= PI7}

-131 = 2m (ur wy + ux wx) cos -a a

The net contribution on an annular part of the given body of re-volution of length dx will therefore be:

7/2 dNp =

f

Api cos o dx r do -7/2 la Substitute (11.4.5) in (11.4.6): 7/2 dNp = 2pmr (ur wy + ux wx) dx

f

cos2do -1r/2 (11.4.3) (11.4.4) (11.4.5) (11.4.6) (11.4.7)

-a

In order to compare the calculations, based on (11.4.9), with the experimental pressure difference results of the old airships in-vestigations it appeared necessary to define, following Klemperer (9), the normal force width:

dN dN dx dx 13 = 1(II . 4 0) v2 -11 (U2 4' i012) 2 Combined with (11.4.9): 27r -2 .4. W2 (Ur WY Ux WX) or: 25 -7/2

a

I

dNp = 2pmr (ur wy + ux wx) dx [ y + 'y sin a cos a

- 7/2

dNp = pm7r (ur wy + ux wx) dx (11.4.8)

The normal force per unit of body-length is therefore given by:

dN --a = pm7r (ur wy + ux wx) dx UW ur + ux wx 6 = 27r U2 + W2 UW and while tg a = : 1 ux wx 8 = 2nr . + cos a sin

a

'11 U

'

sin

_a

_{cos a} sin 2

a111

ux wx 6 = 27r 2 U U ' W ur 1/y ux wx

6 =

7r sin 2a . w _(11.4.11) C

L 12/3

-2 pmV2. volume (11.4.9)

For further comparisons it is necessary to establish the relation between 6 and the lift-coefficient CL:

(11.4.12)

+

From (II.4.110) and realizing N t cos. at LL Pm v2.

I

adx dos a Substitution of

(/1:4.43)

An (11.4-V2) gives: LL

f

adx CL -

0

(11.4.44) volume213

cos a

Nowadays it is an established -practice to define the normal force:

1 2 pmV . LL2 With (II.4.40)t LL adx Z' G

/12

In a .similar way; volume , M" (II.4.

r

(II.4.145) (11.4-16) = - V . L -. LL3 (11.4.17)

27

-Chapter

III

THE VON KARMAN-METHODS AT WORK

111.1. First experiences.

Bearing in mind the words of Von Kaman (cited in paragraph in connection with the flow over a given body of revolution it seems imperative that the given body's meridian or contour curve is free of discontinuities, while it is to coincide completely with a streamline.

Therefore it appeared perfectly logical to approximate the con-tour curve by finding the best least squares fit, which for so called "tear-drop" bodies of revolution (see fig. 8) results in polynomials of the sixth or seventh degree.

The derived pressure distributions on basis of (11.2.25) showed as a function of length along the body fair correlation with experimental results except at the extreme tail-end..

This was not so much due to the d'Alembert paradox as well to the least squares approximation of the body's meridian or

con-tour curve, which - as it turned out - caused unrealistic bumbs in the pressure distribution, especially aft.

Therefore an improved attempt was made using the more smooth contour-approximation, as suggested by Williams (36), which is a further development of the method of Landweber and Gertler (17), however, the results were hardly any better. Apparently the Von Kirman method is very sensitive to the small, ever present oscil-lations in a polynomial description of the meridian curve.

Improvement was then sought in a 2nd-degree approximation of the meridian contour per interval in such a way that the approxima-tion itself and the first derivative were continuous over the whole lenght of the body.

Nevertheless this procedure in combination with the Von Kirmin axial line source method failed again to produce a sound and acceptable pressure distribution.

Evidently a continuous representation of the second derivative of the contour curve is also an imperative condition.

111.2. An optimal approximation of the given contour,

Firstly the ways in which the condition of closure could be met will looked at. In the first instance, however, the considera-tions have be restricted to longitudinal flow.

In the first attempts, described in the preceding paragraph III.!., this has been done by satisfying (11.2.17) along with

(11.2.16), as is outlined in paragraph 11.2. Consequently thet complete source/sink distribution to represent the whole body is found at one time. This thesis will be conducted more or less in the same way as Von 1Cirmgn tackled the probleml) i.e. by cutting the given "tear-drop" body of revolution at those place(s) where the radius is just no longer equal to the maximum radius of the body.

Each converging part of the cut, original body is lengthened by a cilindrical main piece, so that the total length is twice that of the original body (Fig. 9).

In other words: the given, original body is replaced by two semi-infinite half-bodies, each having either the nose-or the tail-piece of the original body as frontpart.

The pressure distributions, resulting from the calculated line source distributions are summed in the end in the right way to give the pressure distribution along the given body.

As outlined in paragraph 111.1. the smoothness of the contour-approximation is a highly important condition for a succesful application of the Von Karman methods.

This gives rise to another problem as the approximation should not deviate too much from the original contour.

1) to which in the following text will be referred to as the

-

29

-Consider (Fig. 10) the "pivotal" point ixi,yi}, which is surounded by the "pivotal" points:

{xi 2'Yi-2} '

xi-1

Yi-1/

'

lxi+I' Y1+1/ '

{xi+2' Yi+2}

The part of the meridian curve between txi_2, y1_21and

yi+21

is now replaced, in the sense of the least squares, by a parabola of the form:

r(x) = al + ax +

a3x2

(III.2.1)

The least square requirement is:

j=i+2

=E{y4-(al+a2x.+a3x.2)}2

= minimum

j=i-2

(111.2.2.)

or:

j=i+2

as k-I iy. (al 4.

I. a_x. ,,x.2)1 aak I

.

0

= -2

x.

3 3

j=i-2

(111.2.3.)

(k = 1,2,3)

Equation

(111.2.3.)

can be written, respectively:

j=i+2

_(k1)

k-1 , E (x. .) - E x, = 0

j=i-2

3'3

j=i-2

j=i+2

k-I

j=1+2

k-Iy

{x.

(all'a2x.+a3x.2)1=

(x. .)

j=i-2

j=i-2

+ j-i+2

This result leads to the following system of normal equations:

a15

+ az E x. a3 E ]( = E . 3

Yj

E + az E 3( + a3 E x3. = E

This system written as a matrix product:

5

Ex.

E

(all summing from j = i - 2 to j i + 2)

From (111.2.4) the coefficients: al, a2, a3 can be solved and by differentiating (III.2.1) twice a representation for the second derivative in xi can be obtained. The whole procedure can be re-peated for all the "pivotal" points, except the first and the last two, because those do not have two adjoining points on both

sides.

Ibis makes it necessary to look for different but matching solu-tions for the extreme nose- and tail-end of the given body, but that will be postponed for a while.

Assume now, using the preceding procedure, for the second derivative in xi has been found r"(xi) = 2 b3, while in the same way

r"

(xi.")

= 2 a3 (fig. II). If h = x1+1 - xi then the second

de-rivative of the contour curve on the interval [x.[xi, xi +i] will now'11

be defined by: r"(x) = 2 b3 (h + xi - x) + 2 a3(x - x. (111.2.5)

Li.

Ex

E )c

Ex

3 J J

Ex

Ex

J J J J J a2 a3 Z yj E x.. Yj )c. Yj (111.2.4) + a2 + a3 E x4. = E x2. y. 3 3 J x. y.

33

E 3

31

-this gives for the first derivative:

r,(x) B 2 {b3 (h + xi) - a3xil

X + 53 b3 ₂

h

(111.2.6)

and for the contour itself on the considered interval:

r(x) = C + Bx +fb3(h + xi) - a3xilX 2 al -' b3 x3

3h

(111.2.7)

At this point the question may arise why the first derivative is not treated in the same way as was done with the second

deriva-tive?

The answer is that this would lead to a polynomial with a higher degree than is given in (111.2.7). And a higher degree means automatically oscillations in the contour-approximation, because of the occurrence of inflexion-points.

As in (111.2.5) the second derivative is already conditioned be-forehand, resulting in the lowest possible degree in (111.2.7), this phenomenon is eliminated.

The constants B and C in (111.2.7) are determined

by

the bounda-ry conditions, using (III.2.1) for r(xi) and r(xi4.1), see also appendix B.

The above mentioned procedure can be repeated along the whole given contour curve and the continuity in r(x) and r"(x) is as-sured, this is, however, not quite true for the first derivative. Based on (111.2.5) it is shown in appendix B that for two adja-cent intervals (fig. 10) the requirement for continuity in the first derivative is:

(b3 + 2 a3) h

ri+1 -

ri - (2 a3 + al)111+ r1+2 - ri+1

3 h 3 h'

(111.2.8)

For "tear-drop" bodies of revolution this condition is not exactly satisfied in that part of the meridian contour where the change in curvature is relatively large.

In this respect it is advantageous to have here as many "pivotal" points as possible.

With regard to the first two "pivotal" points, the nose-piece of the body is represented by:

r(x)

= Vaix

_a2x2

_{a3x3 a (111.2.9)4x4}

This type of representation guarantees the nose of a "tear-drop"

body:

(0) = 0

(III.2.10)

re(0) = co

Thecoefficientsa.(i =

1, 2, 3, 4) are here determined by the boundary conditions:

(x2)

= y2

("pivotal" point)

(x3) = r3; r' (x3) = r; ; r" (x3) = r3"

r3 from (111.2. 1) while r' and r" are found by substituting x3 in the appropriate derivative of (111.2.7).

As is stated before (paragraph 111.2.), in the "Von 'Carman treat-ment" the original tail-end is used as nose-piece for a semi-in-finite halfbody (fig. 9).

Recalling also

(paragraph,

that

in the first attempts to apply the Von Kaman line source technique to "tear-drop" bodies it was here that difficulties were encountered.

In order to prevent this, expression (111.2.7) determined for the interval

[x44x5]

is used beyond this interval for smaller x-values till equation (111.2.7) equals zero.

Generally this will lengthen the original tail-end to some

ex-tent.

That, however, is fairly advantageous as it will oppose the d'Alembert paradox at the real tail-ending.

1

(III.2.11)

+ + +

r

33

-111.3. Computional results.

Although the meridian-contour approximation of a given "tear' drop" body of revolution, as described in the preceding paragraph does show very small discontinuities in the first derivative, it

gives extremely satisfactory results in connection with the Von Karmgh techniques.

This may be demonstrated by calculating the pressure distribution, based on equation (11.2.25), for a number of "tear-drop" bodies

coming from Lange (16). The results represented in (fig. 12 to.

fig. 46) were obtained with 100 sources (sinks included) per semi-infinite halfbody, comparing with 50 sources for the

origi-nal body.

Another application is the pressure change (fig. 17) along the airship "AKRON", on which Freeman (6), (7), (8) has done a lot of experimental work.

An interesting feature is that Freeman was not quite at ease, with the hump in the pressure curve as measured at x/LL gs 0,075 and that the curve as calculated reveals a more or less similar abberation.

To explore the line-source technique somewhat further, together with the developed contour approximation, the pressure distribu-tion along DTMB-model 4198 was calculated.

This "tear-drop" body of revolution, the offsets 'of- which were'

taken from Beveridge (44), is of slender configuration with a length/diameter ratio of 10:1.

The results of the calculation are represented in (fig. 18)..

In (fig. 19) the potential normal force distribution for the air-ship "AKRON"' at angle of attack is pictured (a = 15°) on basis of (11.4.10).

In addition to the value of

a,

computed with the full doublet-system and expressed by equation (11.3.23), the results with the approximate formula (11.3.24) are also mentioned.

In connection with the studied "tear-drop" bodies it is the ex-perience of the author of this thesis that only over the nose part the two approaches show some difference as for example "AKRON" shows very clearly.

For this thesis, therefore, the general rule has been adopted to apply (11.3.23) over the fore-part and (11.3.24) over the aft-part of the given body of revolution.

This is of great advantage as doing so the problem, mentioned by Lotz (5), is evaded.

Lotz showed in fact that a doublet-system has great difficulty to deal with the extreme, pointed tail-end of the studied "tear-drop" bodies.

111.4. Some remarks.

In order to decide on the optimal number of sources to use in the Von Kgruldn method, computer-runs were made with respectively 50,

100, 150 and 200 sources per infinite half-body.

In connection with the studied "tear-drop" bodies no substantial difference was observed in the pressure distributions, so it was decided to standardize on 100 sources per semi-infinite

half-body.

Considering the accuracy of the Von Kgrmin methods one should be aware of the fact that Von Kgrmgn only requires that the zero streamline passes through a discrete set of coordinates, no

mat-ter how.

Maybe here is a possibility to increase the accuracy when not only the desired coordinate is prescribed but also the matching

tangent.

If in the "Von Kgrmgn treatment" the number of control-coordi-nates equals M this will lead to 2 M sources.

On the other hand it should be realized that for each body of re-volution there exists one distribution of line sources or doublets

35

-this

means that there is still a substantial difference between the created dividing (zero-) streamline and the given contour.

Ch apter

IV

VORTEX FLOW CONSIDERATIONS

IV. 1. Introduction.

The calculation by potential flow theory of the total normal or transverse force on a "tear-drop" body of revolution immersed com-pletely in a uniform, parallel flow and at incidence, will yield a zero net force.

This is due to the fact that the potential flow theory has been developed disregarding the viscosity of the fluid.

Therefore application of this theory is only acceptable when the so called "inertia"-forces are much larger than the "viscous"-forces and consequently the equation of Navier-Stokes reduces to Euler's equation of inviscid motion.

In case this condition is not satisfied, because the "inertia" forces are of the same order as the "viscous" forces (as for in-stance in the boundary layer of a body of revolution) the only solution left - in principle - is to solve the equation of Navier - Stokes for the given boundary conditions.

Even with the help of the computers presently available this is only possible for a restricted number of flow configurations, in some cases not without far reaching simplifications.

The viscous flow along a body of revolution is one of the flow problems still in existence, for which no Navier-Stokes solution is yet available.

In such a case a flow model is required which provides as good as possible the influence of the viscosity.

In 1950 Allen and Perkins (18) presented a method attempting to account for the influence of the fluid's viscosity on the normal force and moment on bodies of revolution at incidence.

In their publication a viscous cross-flow term is added to the conventional, inviscid flow approach.

37

-Allen and Perkins assume that every, annular part of the given body can be replaced by a cylindrical part of the same length and experiences a equal cross-force.

This cross-force is derived from empirical data on long cylin-ders placed in a uniform, steady cross-flow (fig. 20).

This method is of adequate accuracy to predict the total normal force upon long cylindrical missile bodies, for the distribution of this force, however, it is not sufficiently reliable. This de-ficiency is caused by the fact that the local, viscous cross-flow drag coefficient is taken constant over the whole body. In 1954 Kelly (23) not only takes the value of the cross-flow drag coefficient as a function of the length along the body but is also inspired by the transient behaviour of the drag coeffi-cient of a circular cylinder impulsively set in transverse motion from rest (fig. 21).

He claims very good results for cylindrical missile-bodies with small length/diameter ratio.

Bryson (33) observes also the analogy between the circular cylin-der accelerated impulsively from rest to a speed V and bodies of revolution at angle of attack.

Following Bryson's ideas, imagine a fixed plane in the fluid (fig. 22) perpendicular to the axis of the body, then as the body pier-ces this plane its trace moves laterally in this plane with velo-city W and time is related to distance x along the body by

t = x/U.

With the fluid flow in the plane approximated as two-dimensional the picture is almost identical to the cylinder in impulsive

mo-tion.

The main difference is the "expanding circle" as the nose pierces the fixed plane.

It should be emphasized that Bryson developed his method only for application to cylindrical missiles and cones.

In this thesis, although Bryson's analysis is followed, it will be nevertheless applied to "tear-drop" bodies of revolution.

IV.2. Experimental evidence with respect to the vortex generation. The cross-flow drag coefficient for long circular cylinders in steady uniform cross-flow is pictured in (fig. 20) and borrowed from Goldstein (42).

For values of the Reynold's number between 20 and BOO the lee side shows two symmetrical standing vortices.

With increasing Reynolds number these vortices stretch farther and farther downstream from the cylinder. Eventually the standing vortices are drawn out to a considerable length, become distorted

and break down.

Then the characteristic state of flow is developed in which vor-tices are shed alternately and at regular intervals from the sides of the cylinder (Von Karmen vortex street).

When a circular cylinder is given an impulsive motion which is started from rest the flow pattern, initially the potential flow pattern, ultimately changes into the pattern corresponding to the final flow regime at the (steady) end-velocity V.

Assume this final flow regime corresponds with a point on the steady cross-flow curve of (fig.20), marked by a Reynolds num-ber based on the end-velocity V.

Also imaginable is that during an impulsive motion the part of the curve of (fig. 20), preceding the final flow regime, is passed through in an accelerated rate.

The photo-series of impulsive motions, depicted in (41), (29), (43) and (55) demonstrate indeed that the successive flow-phe-nomena appearing at different values of the Reynolds number in the

steady and uniform flow, will come more or less after each other in an accelerated rate.

This apparent similarity between the steady and transient beha-viour of cylinders is necessary if one wants to apply a dynamic

39

-method like Kelly's or Bryson's on one hand and for comperative purposes the experimental wind-tunnel results of several investi-gators on the other.

Bryson as well as Kelly observes the vortex generation along the body from the point of view of an observer, fixed in the

fluid-space, doing so the vortex formation transforms into a dyna-mic affair.

Such an observer will not be familar with steady phenomena i.e. a steady vortex pair on the body.

Results in wind-tunnel experiments are obtained from the point of view of an observer linked with the model and, therefore, he is

able to witness the occurrence of steady flow phenomena on the body's surface.

Concluding our short review of cylinders in motion a particular property, concerning the impulsive motion and necessary in the

analogy:

cylinder in impulsive motion/body of revolution at incidence, should be mentioned.

During the time-interval preceding the attainment of the final, steady cylinder speed the drag coefficient, according to Sarp-kaya (43), will rise to 1,6 while in the constant speed condition

its value will settle at around 1,2 eventually.

Focussing now on bodies of revolution at incidence the following experimental facts can be derived from the investigations of

Al-len & Perkins (18), Gowen & Perkins (21), Jorgensen & Perkins (31), Perkins & Jorgensen (26), Harrington (Ill and Grosche (52):

1) For the greater part of the models investigated a steady

sym-metric pair of vortices at the lee side was found in the angle of attack range of approximately 100 5- a < 150.

[Grosche (52) found already at a = 70 a steady symmetric vor-tex system, his experiments were carried out at Ren = 7,5.106

and MarJ 0,12 with

a

long body of cylindrical configuration, length/diameter ratio = 15].

A steady asymmetric configuration was present in the angle of attack range of 150 a < 28°.

The blunter the nose the greater the angle of attack at which the vortex flow became unsteady.

For those blunt noses that resemble the fore part of a "tear-drop" it was shown that the wake flow was both symmetric and

steady throughout the available angle of attack and Reynold number ranges (amax _320).

It appeared that for the more blunt-nosed models the effect of the body shape overshadowed any effect of the Reynolds

num-ber.

Perkins & Jorgensen (26) make a very explicit statement of the following:

Flow separation, which occurs at all angles of attack greater than approximately 5° is the principal cause of the failure of potential flow theory to predict normal force coefficients. In other words the influence of viscosity will manifest itself by flow separation.

The effects of Reynolds number on the pressure distributions result principally from the changes in the boundary layer-separation characteristics and consequently depend in the first place on the boundary layer being laminar or turbulent.

Concerning the distribution of the viscous cross-forces upon the body the experimental data show that if transition of the boundary layer occurs, the cross-flow cannot be considered to

41

-be independent of the axial flow for cross-flow Mach num-bers less than about 0,6.

For this thesis the findings of Harrington (Pr) are of extreme importance. In his investigation measurements were carried out in the wake of a ellipsoid of revolution (length/diameter ratio: 5,92) at different angles of attack (10°; I5°; 21,5°; 28°) while the Reynolds number was about 5,72 . 105.

The most salient features of this report are:

9) At the lee side of the body there are two distinct vortex

-cores.

10)The vortex generation at the lee side can be used to account for a net total lift force on a "tear-drop" body at

inciden-ce.

11)There exists a well defined separation curve along the body.

I2)A very clear insight is given in the extent the potential flow deviates from the real situation (fig. 23).

In fact the outline of the real streamlines does correspond with the ideas of Maskell (24), who states that flow separation

along a finite body is inevitable.

To conclude the work of Rodgers (39) should be mentioned, also dealing with experimental investigations of the flow over "tear-drop" bodies, this time an ellipsoid (length/diameter ratio : 8)

at a constant angle of attack of 60 and a Reynolds number of

2,8 . 106, while the Mach number was about 0,1.

Although in this publication is referred to work of others with similar ideas about the vortex formation - for instance Gowen & Perkins (21) - the idea of a distinct vortex pair at the lee side

is condemned on the ground of the results of Rodgers' experiments. This is odd because the same results constitute evidence of _two

vortex-cores in the extreme aft plane of observation.

Rodgers' observations were made at an angle of attack of 60 only, and apparently the fact was overlooked that, although in that case a separation curve can be observed, the incidence is still too small to cause vortex generation with distinct cores. Had the

angle of attack been increased this should have become quite clear as is shown by the work of Harrington (II).

IV.3. The separation philosophy of this thesis.

I) In the angle of attack range 00 - 10° an acceptable flow mo-del is not feasible.

Between 00 and 50 the boundary layer will become thinner on on the windward side and thicker and broader on the leeward side. It is this change in the boundary layer that is the cause of a net normal force.

Between 50 and 100 the boundary layer will start breaking up and separation will increase, but no visible vortex formation

yet.

2) For the angle of attack range 100 a < 150 it is assumed that a symmetrical, steady vortex pair exists at the lee side of

the body (fig. 24 a).

For a modern, conventional submarine for example, with a sub-merged top speed of about 20 knots, the Reynolds number (based

on boat-length) will be in the order of

ion,

while the "cross-flow" Reynolds number (based on maximum boat-diameter) will be about 105 to 106.

Apparently the vortex configuration at this "cross-flow" Rey-nolds number appearing at the lee side of a "tear-drop" body of revolution is comparable with a cylinder in steady trans-verse flow at a "cross-flow" Reynolds number between about 50 and 100 (fig. 20).

43

-The difference in Reynolds "cross-flow" number can only be ex-plained by assuming that the longitudinal velocity component

in the case of the body of revolution possesses a stabilizing influence (in fact this is what Kelly (23) in contrast with Allen & Perkins (18) already assumed: between cross-flow and

longitudinal flow is some interdependence).

For the angle of attack range 15° a 25° it is supposed that the vortex pair is still steady but its asymmetry is in-creasing (fig. 24 b) but loses its steady character for a > 250.

In this thesis it is further assumed that an increase in asym-metry is accompanied by a reduction of the normal force, the latter as computed by use of the method of Bryson (33). This method, however, is only suitable for a symmetric vortex pair; to account for the asymmetry a reduction factor is ap-plied in this thesis.

It may seem rather crude when the vortex formation, depicted in (fig. 24 b), is replaced by that of (fig. 24 a) multiplied by a reduction factor.

One should be aware of the fact that the alternative, a vortex model based on (fig. 24 b), needs information about the

fre-quency of vortex shedding i.e. the Strouhal number, which is not determined so very easily.

This reduction factor is set to equal one for 100

a

< 150

and is minimum for a = 250.

Sarpkaya (43) shows (fig. 21) that for a circular cylinder the cross-flow dragcoefficient equals 1,6 when the first asymmetry appears (which corresponds with a = 15° for bodies of revolu-tion at incidence).

Schwabe (12), who was the first to investigate the impulsive motion of cylinders, records in this respect a dragcoefficient

of 2,1.

3)1

Both investigators mention 1,2 for a complete non-stationa-ry, irregular wake, this situation is supposed to correspond with a f s 250 for a body of revolution at incidence

In this thesis for a = 25° reduction factor values 1,2/1,6

a 1,2/2,1 will be used, yielding a mean value = 0,6. For angles of attack between 15° and 250 proportionate values will be ap-plied, for instance 0,95 fora = 15° and 0,75 for a = 200.

5) Finally it should be emphasized that the so called "viscid" flow, to be analyzed in chapter V, is in essence a potential flow model containing a certain representation of the influen-ce of the fluid's viscosity.

Real viscous flow, as explained in paragraph IV.!., can only be obtained by solving the Navier-Stokes equation for the given boundary conditions.

ANALYSIS OF THE "VISCOUS" FLOW AT ANGLE OF ATTACK

V.I. The "viscous" flow field,

The complex velocity potential for the two-dimensional transverse flow, represented in (fig. 25), around a circular cylinder

(radius r) accompanied by a symmetrical vortex pair can be derived easily by using the first circle theorem of Milne-Thomson (45)

If it concerns a rotational symmetric body at incidence, attention should be given to the variation of the body-radius as function of the length along the body.

This adds another term, explained in appendix C, to expression (V.1.1) and the complex velocity potential for the transverse or cross-flow velocity component of a body of revolution at incidence will become:

r2

(1) = -i V sin a (C - - i .L_In CI

cl

27 r2

c + 74

Z1

45

-Chapter

\ C + ri In C (V.1.2)

The complex, conjugated velocity isT is defined by:

7. dgt

Differentiation of (V.1.2) and substitution of the result in (V.1.3)

gives: which gives: r2 r2 = -iV sin a

(c -

- i

-r 2n in Cl +

ci

r2 + V _ 71 (v.4

(

I 1 ri-+ --r2 C + 71 c

--7.1 ri = Co CO Cr + Cr ) ? + ri 07.11.4)

The complex, conjugated velocity at the center (fig. 25)) of the right (external) vortex can be obtained by substituting c

cl

in (V.11.4)1 realizing that a vortex does not induce a velocity in

its awn center;

= v sin a + TIT/ - 1 (

1

r2 r2

r2 ,

c/ + j-c

si -

ri-In appendix D a relation is established between on the one hand the motion, the strength and the position of a vortex at the lee

side and on the other hand the velocity induced at its center by the ream of the flow field:

CI + kcl

co), f-

01.1.01

Furthermore another relation exists between the strength of the vortex and its position. The underlying condition for this relation is that no flow should exist in the normal plane along the surface of the body through the vortex-sheet at the separation point. In fact this separation point is a two-dimensional stagnation point, this means that for

c = co,

equation (V.1.4) reduces to:

(V.1:7).

=

-i (1

(V.1.5)

Or:

47

-- i V sin a (1 + ) -2 i

--co:

r

7 1

- CI

co +

CO ra Cl r2 gl r2 ri + = ri co CO,

Using (V.1.5), (V.1.6) and (V.1.8) a non-linear first order differ-ential-equation can be constructed from which i and

r

can be

solved as functions of time if r and j (also time dependent) are known.

V.2. How to solve for CI and F.

Equation (V.1.6) is divided into a real and imaginary part:

,

r

5,1 (5,1 -

yo) T

Wire

+ (z1

_zo)

_F=

wl'i'

Likewise (V.1.8) after a considerable amount of algebra:

y0)2 (z1 zn)21 f(yi yo)2 (z1 z0)21

27 V sin a _{2 (y12} z12 r2) yi

(V.2.3)

is apparently real. 27 V sin a

Differentiating (V.2.3) in respect of time and dividing then by

gives: 27 V sin a . AAyi + BBil + CC (V.2.1) (V.2.2) (V.2.4) / + (17.1.8)

r

f(yi

Y12 + zi2 - r2

cc

_ Y11) in(zi zn). 57o(Y1 Yn)

in(z1 -

zn).

(yl

Yo)2 (z1

20)2

(Y1

Yo)2

(z1

zo)2

yi2 + zi2

-Dividing (V.1.5) into a real and an imaginary part gives finally:

, )

21

(

1

- --2

r2

r2

r1

w1,re= -V sine 2 ylzi

ri4

LT

(

r2

)2 2

(.

r2 )2212

1

r12

+

ri2

,/ 21 1 _try],

_(V.2.5)

r12

-

r2

r12 2 1 Vii r

,im = V sin a 1 +

TT kyl2

z12)1

{ 1

r

+

I

(1 + 12. .) (1 _rr212 ) y12 (1

---2

r/2 /

_r2

2 1 1

1)

212 ( Y

r12 -

r2 2371 +

----)}

+

rizi

(V.2.6)

2 in which: Y1 AA - - Yn +_Yo

(yi

yo)2

(21 - 20)2

(Y1 Yo)

+ (z1

_ZO)2

YI

Y12 Z12

r2

2y1

2)

-

zo zi - zo

BB -

, +

(Yl Yo)/ (zl

z0;2

(y1 YD)2

(zl

zO)t

-

+

-+

+ +

+

49

-In equation (V.2.5) and (V.2.6) is tacitly assumed:

ri2 = yI2

+ z21

Substitution of '

from (V.2.3), in resp. (V.2.5) and (V.2.6)

27

results in:

Substitute finally -f-, , from (V.2.4) in resp. (V.2.1) and (V.2.2) and

solve the result for Y1 and

BB {W1,1m- 2 (zi - zo) CC} 1 + CC W re - 2 KY/ - Yo) 1 + 2 (z1 - zo) BB ,91 ' -2 BB

_(z1

_zo) 1 + 2 AA (yi - Y ' o) _{+ 2 (z1 - zo) BB}

(V.2.9)

W1,im - 2 (z1 - zo)[AA

{W

1,re - 2 (Y1 1 + 2 (y/ -

yo)

AA - yo) CC1 + CC

ZI

-t

2 AA (Y1

-

_Yo) 1 + 2 BB (zi - zo) 1 1 _{+ 2 (yi - yo) AA} (V.2.10)

With the help of (V.2.7) and (V.2.8) the equations (V.2.9), (V.2.10) can be solved for yr and zi if yo, zo, Yo and io are known.

V.3. The "viscous" normal or transverse force.

A symmetrical vortex-pair (45) possesses an impulse:

= pm F. (distance between the vortices (V.3.1)

= F (yi, Yo, zl, zo)

(V.2.7)

1,re

W1,im = f (Yr, Yo,

zl, 20)

(V.2.8)

Applied to the flow field, depicted in (fig. 25) and described by (V.1.4), it results in:

r 2 r2

I =

pm

r.

(1

- - +

- -

(V.3.2)

The force active upon the flow field outside the body being zero -appendix D refers - clearly the force upon the total flow field is the force active upon the body.

The "viscous" normal force then is due to the rate of change of the impulse of the vortex-pair at the lee side of the body.

dNv

{.0

(

r2 r2\}

(V.3.3)

dx dt dt 1 mi '1 t77 41 )

When (V.3.3) is separated in a real and imaginary part it appears, that the local normal force is completely imaginary:

dNv {i 2pr( yr, r2Y1 ) dx dt Y12 4" z12 dNv Ax = 2oyir 2ri i + 2 (-1-2-)2ziii

r2

ri (V.3.4)

After differentiating expression (V.3.4) and using r12= y12+

zi2

r2

I -)

(17

1 2

)2

7 (I

-

7-1-2) t Y1

(

+ 2 7-1 r -i Yil (V.3.5)

At each value of time in the computation the value of each of the variables on the right-hand side is known and consequently - time and place being interchangable - the "viscous" normal force per unit

length can be determined as a function of x.

Expression (V.3.5) can be transformed into a dimensionless number CL or a non-dimensionless number like 3 in (11.4.10)

d

v

-\