"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 WOENSDAG 19 MEI 1976 TE 14.00 UUR
'??
DOOREDUARD VAN DEN POL
NATUURKUNDIG INGENIEUR GEBOREN TE MIDDELHARNIS
;
1976
DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOREN
Dit proefschrift is tot stand gekomen als
onderdeel van het onderzoekprogramma van
het Koninklijk Instituut voor de Marine.
LIST OF FIGURES 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. 1. Introduction 12 II.2. The longitudinal flow 14
II. 3. The transverse flow 19 II. 4. The normal force and its distribution 23
Chapter III THE VON KARMXN-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
IV. 1. Introduction 36 IV.2. Experimental evidence with respect to the
vortex generation 38 IV.3. The separation philosophy of this thesis 42
CHAPTER V ANALYSIS OF THE "VISCOUS" FLOW AT ANGLE OF ATTACK... 45
V. 1. The "viscous" flow field 45
V.2. How to solve for C,i and T 47
CHAPTER VI THE SEPARATION CURVE 52 VI. 1. Introduction 52 VI. 2. The "separation curve" experiment 52
CHAPTER VII RESULTS WITH THE VORTEX FLOW MODEL 54
V I I . 1. Introduction 54 V I I . 2. "AKRON" and the used m o d e l 55
V I I . 3. T h e fuselages of Lange 55 V I I . 4 . T h e DTMB-model 4198/series 5 8 56
V I I . 5. T h e trajectory of a v o r t e x core 57
CHAPTER V I I I T H E INFLUENCE O F A STERN-MOUNTED PROPELLER 5g
V I I I . 1. Introduction 50 V I I I . 2.The boundary layer eg V I I I . 3 . T h e induced flow field ^3 V I I I . 4 . T h e relation between c and K_ ^5
V I I I . 5 . T h e w a k e fraction ^-j
V I I I . 6 . S o m e remarks co
V I I I . 7.Results gg
LIST OF REFERENCES ^ j
FIGURES 1 - 5 4 77
A P P E N D I X A T H E STREAM FUNCTION OF STOKES 124
A. 1. D e f i n i t i o n s 124 A . 2 . A p a r a l l e l , uniform stream 125 A. 3. A point source 1 26 A P P E N D I X B T H E CONTINUITY IN T H E F I R S T DERIVATIVE OF T H E C O N -TOUR APPROXIMATION 128 B. 1. A n a l y s i s 128 B . 2 . Numerical example 129
D.l. General theory 132 D.2. The vortex with sheet I34
APPENDIX E COMPUTATIONAL DETAILS 136
APPENDIX F THE OFF-SETS OF THE USED "TEAR-DROP" BODY OF
REVOLU-TION, DEPICTED IN (FIG. 27) 138
SUMMARY (IN DUTCH) 139
LIST OF FIGURES
Figure page no.
1. Line source nomenclature 77 2. Body of revolution in an uniform stream 77
3. Used coordinates 77 4. Used coordinates 78 5. Transverse flow around cylinder 78
6. Condition for transverse flow around body of revolution 79
7. Force situation in potential transverse flow 79 8. The meridian curve of a "tear-drop" body 80
9. The "Von Karman treatment" 80 10. The surroundings of the "pivotal" point {x. , y.} 81
11. The approximation of the second derivative 81 12. The pressure distribution of body no. 2 / 82 13. The pressure distribution of body no. 3 ., 83
14. The pressure distribution of body no. 5 84 15. The pressure distribution of body no. 6 85 16. The pressure distribution of body no. 7 86 17. The pressure distribution of US airship "AKRON" 87
18. The pressure distribution of DTMB-model 4198 88 19. The normal force distribution at a = 15 : US airship
"AKRON" 89 20. The dragcoefficient as a function of Re-/Sh number for long
cylinders in steady transverse flow 90 21. The dragcoefficient for a circular cylinder in impulsively
started laminar flow 91 22. NACA vortex model 92 23. The flow around a "tear-drop" body of revolution at
incidence 93 24. The vortex generation along a "tear-drop" body at incidence. 94
25. The two-dimensional transverse or cross-flow 95 26. The calculated pressure distribution for the model, depicted
27. The "tear-drop" body of revolution used in the towing tank
experiment 97 28. The streamlines over the bow; bow down 25 98
29. The streamlines over the mid-part of the body; bow down 15 . 99
30. The streamlines over the tail; bow down 15 100 31. The position of the separation curve 101 32. The normal force distribution of US "AKRON" 102
33. The normal force distribution of US "AKRON" 103 34. Lift-force and pitching moment as function of angle of
attack 104 35. Normal force distributions as calculated for model, depicted
in fig. 27 105 36. Lift- and moment coefficients, body no. 3 106
37. Lift- and moment coefficients, body no. 6 107 38. Lift- and moment coefficients, body no. 7 108 39. Lift- and moment coefficients, body no. 5 109 40. Lift- and moment coefficients, body no. 2 110 41. Normal force distributions DTMB-model 4198 Ill 42. Normal potential force distribution for DTMB-model 4198 at
incidence 112 43. Normal force distributions DTMB-model 4198 113
44. Normal force distributions DTMB-model 4198 114 45. The calculated vortex core trajectories of US airship
"AKRON" at a = 15° 115 46. The boundary layer thickness US airship "AKRON" 116
47. The propeller sink-disc II7 48. The propeller sink 1 1 7
49. The validity of the propeller-representations 118 50. Velocity profile in boundary layer at a distance twice the
tip-radius ahead of the propeller, ref. (28) II9 51. Velocity profile in boundary layer at a distance equal to the
tip-radius ahead of the propeller, ref. (28) 120
52. Calculated values local wake fraction 121 53. Flow diagram for the computation of the induced flow field
3
-54. The pressure distribution for DTMB-model 4198 with propeller
LIST OF USED SYMBOLS
[A] - a matrix
a - a line piece of constant length or an arbitrary
distance
aj, 32, aj, a^, bj- coefficients or constants
C - cross-flow drag coefficient = (dN/dx)/(p W^r)
C„ - pitching moment coefficient = 2.moment about re-M LL f e r e n c e p o i n t / ( p .V^.volume) = ( f 31(x)dx)/volume
o C, - liftcoefficient = 2L/(p .V^.volume^Z^)
L ^^m
C - propeller thrust-loading coefficient = 2T/(p U^F ,,)
c - sink strength per unit area
c, . - (matrix) coefficient = 1 + (p'. - p".)/a Kl K,l Kl
c , c , c - total velocity components in x-, r-, G-direction in the angle of attack situation
D - diameter propeller
F ., - effective propeller disc area = •f(R - R, )
eff '^ '^ p r o p h
h, h' - length of an interval along x-axis
I - impulse
f 2 2-1 ^/2
k - dimensionless velocity = J (u, /U) + (u, /U) I
L - lift force
5
-a line piece of -arbitr-ary length
doublet strength
pitching moment (positive nose up) = 2.moment about reference point/(p .V^.LL')
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
Reynolds number
radius of propeller-hub
Reynolds number based on body's length
radius of propeller
a difference
Strouhal number
piece of arc
- 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 axisjmmetric uniform, parallel flow
- the radial velocity component in the k-th "pivotal" or reference point in an axisymmetric uniform, parallel flow
- U2 + W2
- propeller induced velocity
- x-component of propeller induced velocity in point P
- uniform stream velocity in y-direction
- velocity components in x-, y-, a-direction due to a transverse flow
- radial and axial cylindrical coordinates, respec-tively
- y equals r for a = 0, also distance to wall or body
7
-i-th dimensionless source strength = Q./(2TrUa ) , also i-th dimensionless doublet strength = M./(4iTWa2)
angle of attack
normal force width = (2dN/dx)/{p (U^ + W^)}
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 = re
angle of radius vector with respect to y-axis in complex (normal-)plane
propeller apparent speed of advance coefficient = U /(n.2 R )
e prop
M/d^ = doublet strength per unit length
kinematic viscosity
radial and angular spherical coordinates, respec-tively
infinitesimal element of a line source
distance from left and of i-th line source to k-th "pivotal" point
distance from right end of i-th line source to k-th "pivotal" point
fluid mass density
angular cylindrical coordinate
shearing stress
the complex velocity potential = (J) + ii"
the velocity potential
wake fraction
stream function of Stokes
property in a point P
property for r = r max normal property
property of the separation point on the body in the normal plane
property of the center of the vortex in the normal plane
real part of variable
imaginary part of variable
time derivative of variable
complex conjugate
potential property
9
-C h a p t e r I
GENERAL INTRODUCTION
I.l. 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 analoque- 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)^), van den Brug
(50) and Gertler (54), using models of the submarines concerned.
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.
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:
11
-2. Maximum use of existing and published theoretical and experi-mental work of others, provided it is in accordance with rule
1.
3. 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.
1.3. Synopsis.
In chapter II the Von Karman (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 Karman 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 Karman 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.
C h a p t e r II
ANALYSIS OF THE INVISCID FLOW AT ANGLE OF ATTACK
II.1. Introduction.
Since the advent of the airship there has been interest for methods that give the flow over a body of revolution; for
interes-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 Karman (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 Karman, 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 Karman 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 Karman, considering the axisymmetric or longitu-dinal flow over a given airship-body:
13
-"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 Karman 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 te very convenient for first, tentative in-vestigations of the flow around "tear-drop" bodies of
revolu-tion if the Von Karman techniques could be suitable developed. If so, useful results can be obtained in a fairly "inexpensive" way.
In the following paragraph's, therefore, the Von Karman methods will be discussed in detail.
II.2. The longitudinal flow.
In appendix A the following expression is derived for the stream-function fp in a point P, situated in the flow of a simple point source:
iip = - ^ {\ ^ cos V) (II.2.1)
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. 1) and d^ is an infinitesimal element of the line source then the stream function in a point P, due to this element is:
# p = - ^ (1 + cos V) (II.2.2)
For the whole line source this amounts to: a
*P ^ ~ fiF -^ (1 + cos V) d? (II.2.3) o
Consider (fig. 1) again:
cos V = - -TI (II.2.4)
The negative sign since an increase in ^ is coupled with a de-crease in p.
Expression (II.2.4) substituted in (II.2.3) gives: P'
f
P'
h^ ' h \ ^ ^^ - ^ '^P ^ (II.2.5)
- - ^ (a + p' - p") (II.2.6)
The total discharge of the line source is:
15
So, using (II.2.7), for (II.2.6) can be written:
0 D' - p"
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 fo- Pj _ P- 1
In order to create rotational-symmetric stream surfaces the stream-function of a parallel, uniform stream (see appendix A) is super-imposed on (II.2.9):
Ur^ i:« fo: ,, Pi - P i .
*P= — - .f,
\t^
^' ^ - T - n ("-^-lO)
Recalling the definition and the use of Stokes stream function, expression (II.2.10) is investigated,
When point P is on the negative x-axis then p. - p- = -a for every line source, while r = 0, so from (II.2.10):
iJp = 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
(II.2.11) 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
\ii = 0.
If therefore, point P (x,r) is situated (fig. 2) on the dividing streamline expression (II.2.10) will be:
,,2 i=N f _. pi p'.' 1
2 after multiplication with T T T :
-_ rL,2 _ 'T I _Q^
„ , ^
Suppose there is a whole series of points P and referring to the k-th point, the following notation will be used:
P 'ki - P"ki
1 + _!ii ^ = c, . (II.2.14)
a kl
Likewise when dealing with the i-th source in dimensionless form:
Zi = 2 ^ (II.2.15)
Using ( I I . 2 . 1 4 ) and ( I I . 2 . 1 5 ) transforms ( I I . 2 . 1 3 ) i n t o :
i=N
l c, . z. = (—)2 (II.2.16)
i=l
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 (II.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
Z z. = 0 (II.2.17) i=l ^
This causes a linear system of equations for the z^, N-1 of which are described by (II.2.16) and one by (II.2.17), leading to the solution of N sources (sinks included).
17
-In this case k = I, 2 N-1 in (II.2.16).
Following the Von Karman methods - as will be done in this thesis - condition (II.2.17) is not applicable and consequently k runs from k = 1 up to N in (II.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 P can be found through the expressions:
^kx -kr _1_ ^k 9r 9x (II.2.18)
Using relation (II.2.10):
''kx \ r i=N V + l i=l i=N - £ i=l _Qi^ 4irarij 4iTarij
r M i
-dr 3x 3r 'M± - Ml
9x ( I I . 2 . 1 9 )From (fig. 1) it can be seen that:
^r^ + {x - (C + 1)}^' , diff
erentiation gives:9p r -r^ = — = sin V 9r p d£ ^ X - (g -t- 1) 8x p cos V (II.2.20)
i=N u, = U + E J 7^^^— (sin v.". - sin v' ) kx - = 1 1 4TTar ^ '-' '— ki ki' (II.2.21) ^=^
r
0-u, = - E ) -^ (cos V" - cos v' ) kr ._. I 4iTari^^ ^ ki ki
Using now (II.2. 15) : y^— = z^ -^— and substituting this result in (II.2.21) :
\x = " f ' *
kr^
i!i {'^ ("" v;;. - 5in V'.)
•7;— I J zi (cos v.". - cos V,'.)L 2rk i=i 1 kl kl J (II.2.22) Suppose P Q is the not disturbed fluid pressure e.g. at infinity, then according to Bernouilli's line theorem:P ^ ^ <-kx ^ "kr> = PC - 1^ "^ introduce: k^ = {-rr-j + (-p-) p + -^ (kU)2
Po
- f
U^
Ap = P - Po = f^ (1 - k^) U^ (II.2.23) (II.2.24)While k^ can be found by use of (II.2.22), it is possible to cal-culate for every "pivotal" point of the given contour-curve:
P - Pc M u^ 2
= 1 - k" (II.2.25)
Expression (II.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.
19
-II.3. The tranverse flow.
For the potential in a point P in the flow of a doublet of strength M, Von Karman (3) uses - after a detailled derivation the expression (fig. 3 ) :
(j)p M
4TTP cos Y (II.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 \i 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:
d(j)p pdC
47rp^ cos Y
From (fig. 4) it can be seen that: cos Y - sin V cos a
and consequently: Md5 d<J>P *^P 4Trp2 J sin V cos a furthermore: X - (5 + 1) = r cotg V r so that - dj; and
r^-
dv sin V r = p sin VSubstitution of (II.3.5) in (II.3.4) gives: d(j)p = - —i=— sin V cos adv
For the whole line doublet (II.3.6) becomes:
(Jjp =
-4iTr / sin V cos adv
(II.3.2) (II.3.3) (II.3.4) (II.3.5) (II.3.6) (II.3.7) (j)p
Integration of interpretation
X = + oo
*P
(II.3.7) from v = -- 0 till V of a line doublet extending for such a doublet the
- v
4TTr (11) cos o =
-The velocity component in the r
w^ 8*p
" 9
r~ ''= 2l!r2 ^ ° ^
°
potential TT^— cos a 27rr -direction = IT is the mathematical from X = - oo to in P amounts to: can be(II.3
found by:(II.3
9) 10)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 (II.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:
w - W cos a = 0 or using (II.3.10):
„ -I cos o - W cos a = 0 iTrr'^
The radius from the resulting cylinder is:
— (II.3 1 1)
2ITW (.ix.j.ii;
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 (II.3.4) such a system is expressed by:
(|) = - cos a 2 (-^ p ^ ) (II.3. 12)
(|) = cos a [({)] ^ ^ (II.3.13) Apparently the flow-solution for every angle a is found by solving
21
-Starting then by using (II.3.8) and bearing in mind that for o = 0 : y = r (see fig. 4):
i=N
4TTy ^ \ >^i (cos VV - cos v p [
i=l '• •' (II.3.14)
The velocity components in the k-th "pivotal" point P, can be found through the expressions:
i=N 9^
"kx I9x k ^•^Yk i=i • V ] u i ( s i n v : | ^ - s i n v v | ^ )
"ky i=N k 4Try2 . k 1=1 (cos V- - cos v^) J/ . , 3vi • " 8Vi\ (II.3.15)
In general from (fig. 4 ) :
V = arctg X - (5 + 1) 9V ^ -y 9x y2 + {x - U + 1)}' — = X - (g + 1) 9y y^ + Ix - (C + l)i^ (II.3.16) and: sin V cos V
Vy' + {x - (1 + oy
X - (1 + g)Vy' + {x - (1 + oy
(II.3.17) S u b s t i t u t i n g ( I I . 3 . 1 6 ) and ( I I . 3 . 1 7 ) in ( I I . 3 . 1 5 ) : w, = 7 - V 2 \]i. (sin'v'.' - s i n ' v l ) kx 47ry2 .^1 1^1 1 i ' ( I I . 3 . 1 8 )1
'f
ky 47ryj^ .^, VI. ] 2 (cos v! - cos v'.') - (cos vl - cos v'.')
(II.3.18)
Introduce the abbrevations f and g, then for the whole doublet system: i=N 1
", , , E u. (fV - fl)
kx 4TTy^ i^j 1^1 1 1'1
^ = ^ r
"^ = -r-^ E p . (g.' - g V ) ky 4TTyg i^j ^1 *i ^1' (II.3.19)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 Wjj, Wy of the doublet-system coincides with the tangent.
y from (fig. 6 ) : tg 6 = -^—,
w - W
tgSw W (II.3.20)
Substitute in (II.3.20) the velocity components of (II.3.19)
1 4Try^ i=N E i=l Mi {(g! - gV) + t g 6 ( f : - f V ) | Define a n a l o g o u s t o ( I I . 2 . 1 5 ) : V^i i itfia'-^V W ( I I . 3 . 2 1 ) ( I I . 3 . 2 2 )
23
-then ( I I . 3 . 2 1 ) transforms inl
i=N r
= i {(g; - gV) + tg 6(f: - fp}
E
i=l
= rZk^2
(^)'= ( I I . 3 . 2 3 )
To solve this system for N line doublets N reference points on the given body-contour are necessary.
A fast approximation of (II.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 r^ of the annular part at the reference point.
Then from (II.3.11):
\i. = 2iTr.^W ^ 1 1
or by use of (II.3.22)
Zi = i (fi)^ (II.3.24)
Using (II.3.22) substitution of either the z^'s found from (II.3.23) or (II.3.24) (together with the proper addition of the uniform, parallel flow velocity - W) in the system (II.3.19) will produce the transverse velocity components.
II'4. The normal force and its distribution.
The following total velocity components at angle of attack are defined:
(II.4.1) Cr = Uj. + Wy cos a J
Ujj and Up can be found from (II.2.21) while Wj^ and Wy follow from one of the two above mentioned substitutions of z^ in (II.3.19) adding the term - W in the expression for w w .
The used potential flow theory justifies the application of Ber-nouilli theorem:
Po + f® (U' + W^) = p + fDl (c^ + cr^ + co^ ) (II.4.3)
Po = undisturbed fluid pressure
p - p„ = f^ (U^ . W^ - c,^ - c,^ - Co^ )
From (II.4.1) and (II.4.2):
p - P Q = l^^l U^ + W^ - U^j^ _ ^^2 _ ^^2 ^Qg2^ _ ,^^2 ^^^2^ _
v^ sin^a - 2 (ux w^j + u,- Wy) cos a\ (II. 4.4)
Focussing the attention on (fig. 7):
P|Tro Po = I ^ i " ' + W' "X u# Wx^ co^o w^cos^a
-w^sin^O + 2 ( u w + u w ) cos a I 0 ^ r y X x' J PI - Pa = ^{^^ * ^^ - '^x - ^J - vj cos^g w^cos^a -1 Q ^ Z k r X y w^sin^a - 2 ( u w + u w ) cos a \ a ^ r y X X J
Api = PI - P I = 2p (u^ Wy + u„ w ) cos a (II.4.5) la lir-a la ^ -^ A
The net contribution on an annular part of the given body of re-volution of length dx will therefore be:
Tr/2
dNp = J Ap| cos a dx r da (II.4.6)
-TT/2 I ^
S u b s t i t u t e ( I I . 4 . 5 ) i n ( I I . 4 . 6 ) :
TT/2
dNp = 2pjjjr (Up Wy + Ux w^^) dx J cos^da ( I I . 4 . 7 )
-rr/2
25 -^ / 2 '^^P " ^Pm'^ ^ " r Wy + Ux wx) d x dNp = Pmirr ( U j w^ + Ux w^) d x a 1 J * -^ sin a cos a - T/2 (II.4.8)
The normal force per unit of body-length is therefore given by: dN
j ^ = Pn,TTr (u^ Wy + u,, Wjj) (II.4.9)
In order to compare the calculations, based on (II.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 !2 V^ ^ (U^ + W^) Combined w i t h ( I I . 4 . 9 ) 27rr U2 + \}2 - T - y UW ( " r " y + " x ^ x ) = 2Trr U^ + W W " r Wy + U x Wx 2 UW and w h i l e t g a = 77 : B = 2lTr / " r ^ ^ " x c o s a s i n a U ' W U ' W
—)
( I I . 4 . 1 0 ) s i n a cos a o o^ s i n 2 g Ur w^ ^ " x Wx 3 = 2^r — ^ — ( _ . J l + _ . _ ) " r w u w B = Trr s i n 2a ( / . / . - . ^ ) ( I I . 4 . 1 1 ) For f u r t h e r comparisons i t i s n e c e s s a r y t o e s t a b l i s h t h e r e l a t i o n between 3 and t h e l i f t - c o e f f i c i e n t C L : L •'L 1 2 ^/3 -^ p V . v o l u m e 2 m ( I I . 4 . 1 2 )From (II.4.10) and realizing N = L cos a: 1 2 LL
^ P V . J 3dx
2 m - ' L = (II.4.13) cos aSubstitution of (II.4.13) in (II.4.12) gives: LL
/ 6dx o
L volume^ ' . cos a
(II.4.14)
Nowadays it is an established practice to define the normal force:
V. _ N \ PmV^ • LL^ (II.4.15) With (II.4.10): LL
J 3dx
Z' = -^"Yp (II.4.16)
In a similar way: , _ volume M = - ^ L ? — % (II.4. 17)11
-C h a p t e r I I I
THE VON KARMAN-METHODS AT WORK
III.l. First experiences .
Bearing in mind the words of Von Karman (cited in paragraph II. 1.) 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 (II.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 Karman 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 Karman 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.
III.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.l., this has been done by satisfying (II.2.17) along with.
(II.2.16), as is outlined in paragraph II.2. Consequently the= 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 Karman tackled the problem^) 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 III.l. 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.
^) to which in the following text will be referred to as the "Von Karman treatment"
29
-Consider (Fig. 10) the "pivotal" point | x . ,y.l, which is surounded by the "pivotal" points;
{\-2' ^i-l} '
{Vl'
Vl}
' {Vl'
Vl}
• {V2'
V2}
The part of the meridian curve between Ix. „, y. ,|and
I x . ^ . , y-^21 •'•^ '*°*' ^^Pl^ced, in the sense of the least squares
by a parabola of the form:
r(x) = ai + a2X + asx^ (III.2.1)
The least square requirement i s : J = i+2
S = Z W ' ~ ^^1 * a2X. + a 3 X . ^ ) l ^ = m i n i m u m
j=i-2 '• ^ J ^ ' (III.2.2.)
•g—- = -2 E X y. - (ai + a2X + ajx 2) = Q
'^\ j = i-2 J t J J J J (III.2.3.)
(k = 1,2,3)
Equation (III.2.3.) can be written, respectively:
iy^,
k-i
.
j=;^2
.
^_,
.
Z (x. y.) - 2 <^ X. (ai + a2X. + a^ii.)\ = 0
j=i-2 J J j=i-2 ^ ^ J J J
E | x •""' (ai + a2X + asx 2)1 » i (x.''"'y.) j = i-2 I- J J J J j=i_2 J J
This result leads to the following system of normal equations: a, 5 + a2 E X. •*• aa E x? = E y. 1 J J J aj E X. + aj E x? + as E x. = E x. y J J ai E X . + 32 E X? + aj E x : J J J E X? y. J J
This system written as a matrix product:
E X. E X? J J E X? J J :? E X? J J J
(all summing from j = i - 2 t o j = i + 2 ) E X. ^ 2 E X. E X? J E x-; ai 32 as
Ey.
E X. y. J J E X? y.J J
(III.2.4)From (III. 2.4) the coefficients: aj , a 2 , a-^ can be solved and by differentiating (III.2.1) twice a representation for the second derivative in XJ 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.
This 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 b s , while in the same way
r" (xi+|) = 2 as (fig. 11). If h = x^+i - x^ then the second de-rivative of the contour curve on the interval x^, xj^+] will now be defined by:
31
-this gives for the first derivative:
r' (X) = B + 2 |b3 (h Y i ) - ^3Xi|
(III.2.6) and for the contour itself on the considered interval:
as — bs 2 X + —^ x^ h r(x) = C + Bx Jbs(h ^ xi) - asxi .,2 , as ~ h3 3 "" 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 (III.2.7). And a higher degree means automatically oscillations in the contour-approximation, because of the occurrence of inflexion-points.
As in (III.2.5) the second derivative is already conditioned be-forehand, resulting in the lowest possible degree in (III.2.7), this phenomenon is eliminated.
The constants B and C in (III.2.7) are determined by the bounda-ry conditions, using (III.2.1) for r(x.) and r(x. . ) , 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 (III.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:
(bs + 2 aa) h '^i + i - '^i ^ - (2 as + a p h ' ri^2 - '^i+l 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) = \jaix + a2X^ + ajx' + a^x'*" (III.2.9) This type of representation guarantees the nose of a "tear-drop"
body:
r (0) = 0 r'(0) = «>
The coefficients a. (i = 1, 2, 3, 4) are here determined by the boundary conditions:
r (X2) = y2 ("pivotal" point) "] I (III.2.11) r (xa) = ra; r' (X3) = r\ ; r" (X3) = r^" J
ra from (III.2.1) while r' and r" are found by substituting Xa in the appropriate derivative of (III.2.7).
As is stated before (paragraph III.2.), in the "Von Karman treat-ment" the original tail-end is used as nose-piece for a semi-in-finite halfbody (fig. 9).
Recalling also (paragraph III.l.) that in the first attempts to apply the Von Karman line source technique to "tear-drop" bodies it was here that difficulties were encountered.
In order to prevent this, expression (III.2.7) determined for the interval [xi,-«C5] is used beyond this interval for smaller x-values till equation (III.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.
33
-III.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 Karman techniques.
This may be demonstrated by calculating the pressure distribution, based on equation (II.2.25), for a number of "tear-drop" bodies coming from Lange (16). The results represented in (fig. 12 to fig. 16) 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 fa 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 (II.4.10).
In addition to the value of 3. computed with the full doublet-system and expressed by equation (II.3.23), the results with the approximate formula (II.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 (II.3.23) over the fore-part and (II.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.
III.4. Some remarks•
In order to decide on the optimal number of sources to use in the Von Karman 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 Karman methods one should be aware of the fact that Von Karman 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 Karman 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 which describes that particular body in the best way even when
35
-this means that there is still a substantial difference between the created dividing (zero-) streamline and the given contour.
C h a p t e r 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 100 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 Karman 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 (11) 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 10° ^ a < 15°.
[Grosche (52) found already at a = 7° a steady symmetric vor-tex system, his experiments were carried out at Re,, = 7,5.10^
and Mg « 0,12 with a long body of cylindrical configuration, length/diameter ratio = 15].
2) A steady asymmetric configuration was present in the angle of attack range of 15° ^ a < 28°.
3) The blunter the nose the greater the angle of attack at which the vortex flow became unsteady.
4) 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 (otjuax *** ^^°^ '
5) 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:
6) 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.
7) 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.
8) 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 (11) 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°; 15°; 21,5°; 28°) while the Reynolds number was about 5,72 . 10^.
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.
ll)There exists a well defined separation curve along the body.
12)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 6 and a Reynolds number of 2,8 . 10 , 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 6° 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 (11).
IV.3. The separation philosophy of this thesis.
1) In the angle of attack range 0° - 10° an acceptable flow mo-del is not feasible.
Between 0 and 5 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 5° and 10° the boundary layer will start breaking up and separation will increase, but no visible vortex formation yet.
2) For the angle of attack range 10° ^ a < 15° 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 10^, while the "cross-flow" Reynolds number (based on maximum boat-diameter) will be about 10^ to 10^.
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).
3) 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 > 25°.
4) 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 10° £ a < 15° and is minimum for a = 25°.
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.
Both investigators mention 1,2 for a complete non-stationa-ry, irregular wake, this situation is supposed to correspond with a ^ 25° 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 25° proportionate values will be ap-plied, for instance 0,95 for a = 15° and 0,75 for a = 20°.
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.1., can only be obtained by solving the Navier-Stokes equation for the given boundary conditions.
45
-C h a p t e r V
ANALYSIS OF THE "VISCOUS" FLOW AT ANGLE OF ATTACK
V.l. 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) which gives:
$ = -iV sin ex (C - f-) - i ^ In ^ ~ ^l
'
v-%
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:
$ = -i V sin o (5 - — ) - i ^ In ['^ ~ ^\ . 4i- 1+ rr In C 5 2TT \ ^ _ r £ - - I
V
"f.
The complex, conjugated velocity W is defined by:
— dij>
W = ^ (V.1.3)
Differentiation of (V.1.2) and substitution of the result in (V.1.3) gives:
r^ . r
W = - 1 V s m a (1 + -2") - 1 -^r2' ^ 2TT \ C - ?i , + r2
^ - ? :
C + Cl
(v.l.4)
The complex, conjugated velocity at the center (fig. 25) of the right (external) vortex can be obtained by substituting C = ?i in (V.l.4) realizing that a vortex does not induce a velocity in its own center:
- . . r^ . r / 1 1 . Wj = -1 V sin a (1 + —2-) - 1 TT —
Ci^ 2TT r2 , r2
+ P-
(V.l.5)
Cl + Cl / Cl
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 rest of the flow field:
ti + (Cl - Co) f = Wi (V.l.6)
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 ? = Cg equation (V.i.4) reduces to:
wL _ =7^ (V.l.7)
'C - Co Co
47 -or:
r-^ r
i V sin a (1 + ——) - i i;o2' 271 Co - Cl r2 ^ _ r2 Co + Cl Co Co (V.l.8)Using (V.l.5), (V.l.6) and (V.1.8) a non-linear first order differ-ential-equation can be constructed from which Ci and V can be
solved as functions of time if r and Co (also time dependent) are known.
V.2. How to solve for Cl and I.
Equation (V.1.6) is divided into a real and imaginary part:
yi + (yi - yo) f = Wi^^a (v.2.1)
- ^ f X I = Wi ,-„ (V.2.2)
zi + (zi - Z Q ) V '^^
Likewise (V.l.8) after a considerable amount of algebra:
r ^ ( ( y i - vn)^ + (zi - zn)^} { ( y i + vn)^ + (zi - zn)^} 2iT V s i n o 2 (yi2 + zi2 - r2) y i
( V . 2 . 3 )
r
•s—T;—: i s a p p a r e n t l y r e a l . 2ir V sin a '^'^ •'
Differentiating (V.2.3) in respect of time and dividing then by
r
-^—T.—: gives: 2ir V sin a ^
i n which:
AA =-, V-.-^^
( y i - y o ) ! + ( z i - Z Q )„ + n_LZii
2 ( y i + y o ) ^ + ( z i - z o ) 2_n.
y i ^ + Zj^ - r2 2yi BB = Zi - zo '1 - zn ( y i - y o ) * + ( z i - z o ) 2 ( y i + y o ) * + ( z i - Z Q ) * Z i y j S + z i 2 - r 2CC = ~yn(yi ~ yp) - zn(zi - zn) ^ yn(yi + vn) - z o ( z i - zp) ^ (yi - yo)2 * (zj - Z Q ) 2 (yi + yo)2 + (zi - zo)2
+ ., hi yi2 + Zl2 -
r'-Dividing (V.1.5) into a real and an imaginary part gives finally:
2TT
r^ r
Wi,re = -V sine — 2 yizi - —+ llii
ri2Zl
V
- - 2
ri? - r2(-f7=r-^*('-Sir-^
(V.2.5)Wl,im = V sin a ^1 + f ^ (yi^ -r' z^-)\ + |^ ri"
J:I- 1
rrzi
ri2 - r2 2yi
49
In equation (V.2.5) and (V.2.6) is tacitly assumed: 2 2 2
ri = yi + Zl
Substitution of Y' ' ^^°^ (V.2.3), in resp. (V.2.5) and (V.2.6) results in:
W = F (yi, yo. Zl, Zo)
1 .re (V.2.7)
W . = f (yi, yo. Zl, Zo)
1 ,im (V.2.8) Substitute finally -p , from (V.2.4) in resp. (V.2.1) and (V.2.2) and solve the result for yi and zi:
2 (yi - yo) yi BB { W j ^ ^ ^ - 2 ( z ^ - Z o ) C c } 1 + 2 (zi - Zo) BB + CC 1 + 2 AA (yi - yo) 1 2 BB (Zl - Z Q )
}
Zl = ^ , i m - 2 (Zl - zo) 1 + 2 (zi - Zo) BB (V.2.9)AA {"i^re " ^ (yi - yo) Cc} 1 + 2 (yi - yo) AA + CC
(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 yi and zi if yo, Z Q , yo and zo are known.
V.3. The "viscous" normal or transverse force.
A symmetrical vortex-pair (45) possesses an impulse:
I = i p r. (distance between the vortices
Applied to the flow field, depicted in (fig. 25) and described by (V.l.4), it results in:
I = 1 p_
' • ( - - l ^ - . - f ^ )
(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.
dN V dx dl dt d_ dt m
.-|?-.-0
(V.3.3)When (V.3.3) is separated in a real and imaginary part it appears, that the local normal force is completely imaginary:
'^'Tt \i^-'{n-^^^)
After differentiating expression (V.3.4) and using rj = yj + zj (V.3.4) 2 . dN V dx = 2pyir 2rf ri=^
^(.-T^zi^n
+ 2m
zizi^ - ( 7 7 = - N '
(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 C or a non-dimensionless number like g in (II.4.10)
51
-With the local force parameters for the inviscid and viscous flow, expressed in a similar way, added the total force distribution along the body can be obtained.
Integration over the length of the body gives the total normal force, caused by the vortex formation.
The integral of the product of the local force per unit length and the axial distance to a defined reference point equals the total pitching moment.
C h a p t e r VI
THE SEPARATION CURVE
VI.1. Introduction.
The literature search for this thesis made it clear that no cohe-rent separation data concerning "tear-drop" bodies of revolution, necessary to solve the equations (V.2.9) and (V.2.10), seem to be available.
Therefore an experiment was performed to establish the position of the separation curve over a "tear-drop" body of a - from a na-val-constructors point of view - realistic geometry.
By chance such a model, the off-sets of which are given in appen-dix F, was on hand. Although possessing a longer cylindrical mid-part, this polyester model has a great resemblance to the
US-air-ship "AKRON", a fortunate coincidence, as from this airUS-air-ship a wealth of experimental data is available (6), (7) and (8). The hydrodynamic similarity of the two bodies is demonstrated adequately by comparing the curves of the pressure distribution during longitudinal flow (zero angle of attack), depicted in (fig.
17) and (fig. 26). [It should be noted that (fig. 26) gives the pressure curve computed by means of the Von Karman axial line source method together with the contour approximation of para-graph III.2.]
Particularly conspicuous is the similarity of the slope-increase along the tail-part of both bodies.
Although of course not true for one hunderd percent it is assumed in this thesis that the separation curves of the two bodies will be the same in actual practice.
VI.2. The "separation curve" experiment.
The experiments were carried out in the towing-tank facility of the Shipbuilding Laboratory of the Delft University.
