ON THE USE OF LIFTING SURFACE THEORY FOR MODERATELY AND HEAVILY LOADED
ON THE USE OF LIFTING SURFACE
THEORY FOR MODERATELY
AND HEAVILY LOADED
SHIP PROPELLERS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE
HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. IR. L. HUISMAN, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP
WOENSDAG 11 MEI 1977 TE 14.00 UUR DOOR
WILLEM VAN GENT
NATUURKUNDIG INGENIEUR /GEBOREN TE AMERONGEN
H. V E E N M A N EN Z O N E N B.V. - W A G E N I N G E N - 1977
Dit proefschrift is goedgekeurd
door de promotor
Prof. Dr. Ir. J. D. van Manen
i/
Aan mijn ouders
CONTENTS
GENERAL INTRODUCTION 1
1. PROPELLER GEOMETRY AND HYDRODYNAMICAL BOUNDARY CONDITION... 4
1.1. Introduction 4 1.2. Mathematical formulation for geometry of thin blades-• 4
1.3. Hydrodynamical boundary condition 9
2. VELOCITY FIELD INDUCED BY PROPELLER LOAD 16
2.1. Introduction 16 2.2. Fundamental equations 17
2.3. Steady load on actuator disk 21 2.3.1. First approximation of flow field due to
external forces 21 2.3.2. Approximations of the effect of secondary
forces 27 2.3.3. Remarks about flow through disk 35
2.4. Steady load on rotating lifting lines and surfaces... 39
3. APPLICATION OF LIFTING SURFACE THEORY 4 4
3.1. Introduction 44 3.2. Formulation of integral equation and solution method- 46
3.3. Slipstream rotation and contraction 51
3.4. Flow around blade sections 52 3.5. Corrections to blade load and onset flow 54
4. PROPELLER ANALYSIS AND CORRELATION WITH EXPERIMENTS 56
4.1. Introduction 56 4.2. Effective axial velocity at propeller blade 58
4.3. Effects of some model components 60
4.4. Blade load distributions 62
5. REVIEW AND CONCLUSIONS 70
APPENDIX A : Calculation of radial velocities at a
REFERENCES 84 LIST OF SYMBOLS 86 SUMMARY 89 SAMENVATTING 90 DANKWOORD 91 LEVENSBESCHRIJVING 92
GENERAL INTRODUCTION
The design of a screw propeller and the analysis of its performance are important subjects in the field of ship hydrodynamics. Therefore a lot of work has been carried out on formulating an adequate
mathematical model for it. The flow in which a ship's propeller operates is non-uniform, unsteady, turbulent and affected by gravity. The difficulties met in describing this flow are
limited by the apparent incompressibility of the liquid,
but they are complicated by the occurrence of gas and/or vapour filled cavities. A much simpler flow is obtained, when the propeller on its driving shaft is disengaged from the ship and placed in a uniform, steady, axially directed flow in
which viscous effects are confined to boundary layers and in which gravity and cavitation are disregarded.
Even for this simple flow the description of the action of a
screw propeller has not yet the appropriate level of specification to allow for exact calculations. Hence it is not surprising that the simple case rather than the complicated case has engaged many investigators. For the understanding of the complicated case
the knowledge of the simple case is fundamental. Hence also for that reason the latter still deserves attention. For the simple flow model the name "propeller in open water" is current.
The description of the flow of a propeller in open water is, apart from the propeller geometry, governed by only one parameter, viz. the ratio between the translation velocity in axial direction and the rotational velocity. Of all possible values
of this parameter, those corresponding to cases in which a thrust in the advance direction is developed are of main interest for the propulsion of ships. In this thrust producing domain of the propeller operation three main types of loading are distinguished . The division into resp. light, moderate and heavy loading is
based on the magnitude of the flow velocities induced by the propeller action.
For lightly loaded propellers the induced velocities are assumed to be negligible, as far as they react on the blades. This model of the flow has been useful for the ^tudy of optimum propellers.
For the model of moderately loaded propellers the induced velo-cities are derived from the velocity field of a number of
helicoidal vortex sheets^ each trailing behind one of the propeller blades. At a constant radius the vortex strength in the sheets and the pitch of the sheets are constant in downstream direction. The vortex strength is related to the load distribution on the blades, but the radial pitch distribution of the sheets is not determinate. An obvious estimate for this pitch is the hydro-dynamic pitch,i.e. the pitch of the relative streamlines in the neighbourhood of the propeller blades. This choice leads to a good correlation between theory and experiment. In the induction-factor method a simple, but highly accurate, computation scheme for this model of the propeller wake has been developed. In combination with a lifting-line representation of the propeller blades this model is widely used. Its application requires, however, knowledge about the lifting properties of the propeller blade sections,supplemented by influencing correction factors associated with the complicated propeller geometry. These correction factors have to be derived from a lifting-surface representation of the propeller blades. The lifting-surface representation can do without fore-knowledge about the lifting properties of the blade sections, which is an obvious advantage of this representation in comparison with the lifting-line representation. A disadvantage is the relatively long time re-quired to work out the computation schemes for lifting-surface methods.
It is assumed that in the flow around a heavily loaded propeller the foregoing picture of the wake flow is insufficient. It has been
observed, by theoretical considerations and by experiments ,that the wake vortex sheets exhibit a radial contraction,that the pitch of these sheets is not constant and that in the wake a reduced pressure level exists. Moreover the effect of the radial component
-of the induced velocity, which is neglected completely in the model of moderately loading, has to be considered. The study of
the deformation of the helicoidal vortex sheets and its effects 3 / on the induced velocities is a difficult one. The formulation of
the hydrodynamical equations of motion is complicated by the instability of the vortex sheets. The transition region in which
the vortex sheets roll up is located quite close behind the . , propeller. An established model for this type of wake flow, by '^ . definition associated with the heavily loaded propeller, is not
yet available. In this study a model, accounting for the main ' ^""'^/'"^n effects, will be investigated. i) ^l^' The helicoidal geometry of the propeller blades and the consequent''' shape of the wake are inherent difficulties in the formulation , of an exact metheraatical model for the flow around a propeller.
It is often tried to eliminate these difficulties by consideration of the circumferential mean values of the quantities which are periodic functions of the angular position. An approximate model for the mean values is obtained by the conception of a propeller with a large number of blades. In the limiting case of an infinite number of blades the circumferential variations disappear. This representation of a propeller is called "actuator disk". It has been proved only for the lightly loaded propeller that the flow behind an actuator disk is equivalent with the circumferential mean flow of a propeller with finite blade number. For moderately and heavily loaded propellers the exact equivalence is unlikely, but it might be a good approximation when the equivalence is assumed. In this study it is postulated at first that the characteristic effects of a heavily loaded propeller model can be derived from this approximation. Estimates for these effects, obtained in this way, are combined with a lifting-surface represen-tation of the propeller. This model is verified by correlation with experimental results of systematic propeller series.
\
1. PROPELLER GEOMETRY AND HYDRODYNAMICAL BOUNDARY CONDITION.
1.1. Introduction.
In this chapter a description of the geometry of thin propeller blades is given. The formulation incorporates rake, skew and
variable pitch. The kinematical boundary condition at the propeller blade in a flow field is formulated inclusive the radial velocity component. It is shown that, when camber and thickness of the blade sections are considered to be small perturbations of a reference plane, a first order approximation leads to two equations, of which one relates the flow field to the thickness distribution and the other relates the flow field to the camber distribution.
1.2. Mathematical formulation for geometry of thin blades
For the description of the geometry of the propeller blades we use a reference plane built up of helical lines and given by the mathematical function F ;
F = (}) - a(x-z ) + ut = 0, (1-1)
in which a and z can be functions of r .
( X, r, $) is a set of cylindrical coordinates; the positive X -direction is opposite to the axial propeller motion. Looking into the positive x-direction the positive <^ -direction is clockwise and 41=0 at a radial line directed vertically upward. tot is the angular rotation with reference to the position at time t=0. For positive angular velocity u (right-handed propeller) the direction of the rotation is opposite to the positive (J) -direction.
-DIRECTION OF PROPELLER TRANSLATION
DIRECTION OF PROPELLER ROTATION
Fig. 1. Coordinate system for description of propeller geometry.
This notation is illustrated in Fig. 1. The meaning of a and becomes clear, when the intersection of the reference plane F=0 with an axial cylinder of radius r is considered. The line of intersection is a helical line with a pitch angle defined by
tan
f> ' v ... •- ^ T = " 7 OF/3<t>)y(3F/3x) = 1 / a r . ' ( 1 - 2 •= I
I- -. tof - -:
The pitch i s simply r e l a t e d to a according t o
P = 2TTr tan<I> = 2TT/a .
2>7
: - e \ ,^
3f
(1-3)
This relation shows that when a = constant,a constant P is recovered.
The rake z describes the shape of the so-called generatorline, which is the intersection of the reference plane F=0 with the plane i|>=0 at time t=0 . In that case eq.(l-l) reduces to
2< = ZR • (1-4)
z is the distance between the generatorline and the plane x=0, which is called the propeller plane. A survey of the recommended nomenclature for the description of ship propellers is given by Cumming / I / .
For a propeller with z blades an equal number of reference planes can be defined by substituting in eq.(l-l) <s>. for $ , which are related by
-k2TT/Z , k=0(l)Z-l (1-5)
For the description of the propeller blades they are also inter-sected with the cylinder of radius r. When this cylinder is unrolled the picture given in Fig. 2, which is taken from Ref. / I / , is obtained. The helical reference line becomes a straight line; its pitch can be chosen equal to the pitch of the nose-tail line of the blade section. In this way the blade section is situated between two points on the helical line viz, the leading edge
(x , (() ) and the trailing edge (x , <sf ) . This coordinates are still functions of the radius r and describe the blade contour.
We will consider the propeller blades as thin lifting surfaces.
Consequently the blade section shapes reduce to cambered mean lines.'if»^»*-i The following derivation, however, also holds for blade sections o<»^ with thickness if it is applied to the upper side or to the
lower side. This feature will be used later on to find a correc-tion for thickness effects.
The camber is measured in the direction normal to the helical line. The mean line is described by adding to every point of the
helical line, between the leading and trailing edge,a point accor-ding to
.(3tJ.er
Li
BLADE ROOT S E C T I O N , REFERENCE POINT OF BLADE ROOT SECTION AND PROPELLER
REFERENCE X^O (PITCH ANGLE OF LINE ^ \ SECTION AT RADIUS r )
HELICAL LINE
INTERSECTION OF GENERATOR LINE AND PLANE AT RADIUS r
FORWARD STARBOARD-PLANE CONTAINING SHAFT AXIS AND PROPELLER REFERENCE LINE
/ •"»
INTERSECTION OF BLADE REFERENCE LINE (LOCUS O F
BLADE SECTION REFERENCE ^,(> 9s=SKEW ANGLE POINT) AND PLANE AT RADIUS r
VIEW OF UNROLLED CYLINDRICAL SECTIONS AT BLADE ROOT AND AT ANY RADIUS r OF A RIGHT-HANDED PROPELLER (LOOKING DOWN) SHOWING RECOMMENDED LOCATION OF PROPELLER PLANE
BLADE REFERENCE \ LINE PROJECTED BLADE OUTLINE GENERATOR LINE PROPELLER REFENCE LINE (PROPELLER PLANE) REFENCE POINT OF BLADE ROOT SECTION
PROPELLER ^ HUB
SHAFT
TRAILING EDGE
PROPELLER REFERENCE LINE AND GENERATOR
LINE BLADE REFENCE LINE (LOCUS OF .BLADE SECTION REFERENCE POINTS) AXIS FORWARD -DOWN LEADING EDGE RENCE POINT OF ROOT SECTION SHAFT AXIS STARBOARD-DOWN
Fig. 2. Diagrams showing recommended reference lines and terminology.
< - f COS <1
+ f/r sin P = r
: i - 6 )
where ( x,r, <t> ) are the coordinates of the helical line and (5 , P ,9) are the coordinates of the mean line.
The camber f is a function of the position between the leading edge and trailing edge, which is given by the parameter s :
s = (x-Zrj,) sin$ + r ((j ,) cos<i (1-7)
lies within the range from -c/2 to +c/2 , where c is the chord. z and (J) are the coordinates of the midchord point
T
on the helical line. They are related to each other via eq.(l-l)
a (z^-Zp) + üjt = 0
z is called the total rake. When a skew angle is introduced, 0 =a(Z -Z ) , we can write
S I K z,„ = z„ + e /a T R s' - (jjt z„ + p8 tani} R s (1-S)
The mathematical formulation of the mean line sections of the lifting surfaces is obtained by means of eliminating the coordinates x, r and 0 from eq.(l-l, 6 and 7'.The result is
F + ait - (5-Zj^)/(ptanO) f/(psin<I>) = 0 (1-9)
where
f = f (s,p) (1-10)
-and
s = (C-z-p) sin$ + p(e-4'rp) coS({, (1-11)
A useful relation, v;hich has an obvious geometrical interpretation, can be derived from eqs. (1-8, 9 and 11) and is given here for refe-rence .
C-z = s siniji - f cos<l) ( 1 - 1 2 )
The set of equations (1-8, 9, 10 and IDconstitute the complete formulation for the lifting surfaces, provided that the functions for rake z (p) , skew angle B (p) , pitch angle $(p),
K S
chord c(p) and camber f(s,p) are specified. As stated before this set of equations can also bo used for blades with finite thickness. In that case the funct.ron f(s,p) is a double-valued function of s, thus specifying the upper and lower surface of the blade.
At the leading edge and the trailing edge f is not necessarily zero. If ft ^ ff' however, the length of the projection of the nose-tail line on the helical line differs from the chord. In practice this difference is negligible.
1.3 Hydrodynamical boundary condition.
When a propeller is rotating in a moving fluid, the velocities of the blades and the fluid have been matched. The blades have to move continuously with the fluid and can be considered as a material surface which properties do not change in time. This means in mathematical terms that the value of the function F , describing the blades, does not change. The material derivative of this function should be zero. Let the vector u with
components U, V and W in the (x, r, 41 ) set of cylindrical
coordinates describe the velocity field, then the boundary condition is
3 F * — ^*
| F !
, u I I , V 3 ^ t w ^ = o
3 t 3£; 3P p 3 6 w h e r e : 3 F * ^ _ 1 3 f 3 s 3 t " p s i n $ 3 s 3 t ' 1 1 3 f 3 s 35 p l . a n $ p s i n $ 3 s 3C 3 F * ^ ^R r^ ^ ., P d $ , , f •^-^— = —^ ^r- {tan<I) + —^^T^— 3—} + ^ 5 5 ^ 3P p ^ t a n ^ * c o s % "^^ p ^ s i n ^ * d $ , 1 '^^R 1 ( i f , M i l } •{sin* + pcos$ ^ > + - £ ^ ^ 5 ^ - -^^177^^ 9p 3s 3p 3F* ^ l _ 1 if 3s r3e p ps.in$ 3s p39These derivatives can be evaluated further by using the equations (1-3, 8, 9,11 and 12). Substitution of these expressions into eq. (1^13) and multiplication of this equation with psin* finally results in
i \ + G^ Vj^ } - { V^ + G^ Vj^}|f = 0 (1-14)
where: V , V and V are newly defined velocity components according to
V = (cop +W) sin$ - U cos$ , (1-15)
V = (top +W) cos* + U sin$ , (1-16)
V„ = V , (1-17)
-while G and G are expressions,which depend only on the geometry of the propeller blades
dz„ 9 +-cos$ ,„ ^ -, „ ^ N dp 2TT dp p 3 p ' dz„ 9 ,_, de r R j^ s dP, . . r s p dp 2TT dp dp COS'S - cos $J P , fT . . cos$ dP, f + {2sin* - -^— -r—} — cos^ 2iT dp p (1-19)
The meaning of the velocity components V and V is illustrated in Fig. 3, which gives a vector diagram of the velocities in
the unrolled cylinder plane of radius p. The effect of the velocity component V is neglected in this picture. It can be seen that V and V are the components normal to the helical line resp. along
the helical line.
èS / \ VN'V-rt.n^c - %
J^''"
---u copFor the interpretation of eq. (1-14) as a boundary condition at the propeller blade, it has to be realized that it has to be fulfilled at both sides of this surface. The velocity components consist of two parts viz. the velocity components of the flow field without propeller and the velocity components induced by the action of the propeller.
When the propeller blades are approximated by lifting surfaces without thickness the induced velocities can have discontinuities across these surfaces and across vortex sheets in the flow field. The correct approximation will be derived from eq. (1-14) by star-ting from the case of finite thickness. This is done by substitustar-ting for f
f = f + ^ f^ (1-20) c - t
where f =f (s, p) is the camber and f^=f^(s,p) is the thickness. c c t t
The geometrical expressions G and G become
S = % " %c ± ^ % t ' <1-21)
G^ = g ^ + 5 ^ 0 i ^ ^ T t ' ' ^ - 2 2 ) w h e r e s d z 9 +—cos<J> g „ = {—^ + - ^ §^} c o s * , ( 1 - 2 3 ) N d p 2TT d p ' 3 f f c . 2 c g . , = — s i n ' * - ^ , ( 1 - 2 4 ) Nc p 3p ' ^ t 2 ^ ^ g ^ ^ = ^ s i n * - , — , ( 1 - 2 5 ) ^Nt p " " " " 3p R s d P , . ^ r S P s 2 . , q = - {^-— + —) s i n * - {- ^^-r - - c o s * } ^T dp 2TI d p d p c o s * P : i - 2 6 ) 1 2-^Tc=^2^^"*-ir'^^-=^ ' '1-2^)
g_,^= {2 sin*-|^ ^ - } — cos* . (1-28) ^Tt dp 2TT p
In this way G and G are split into terms which depend resp. on blade contour,, camber and thickness.
Application of eq. (1-14) to the upper resp. lower surface of the blade and addition or subtraction of these equations leads to
+ - ^^c + - + - + _ 3ft
+ kq (V +V )} - {(V TV )-Kg +q ) (V +V )+^q (V +V )]h =0 ^^Tt R R 3s T T ' ^ T ^ T C ' R R '^Tt R - R'^ ' 3s
In the description of the geometry the camber f and the thickness f are small parameters. A first order approximation
of the above equations ^s(g^^<<g^,g^^<<g^,g^^<<g^,g^^<<g^):
( ^ N ^ V R '
- 'V5T^R'-3f=° ' '^-25'
3f
'^^N^^N^V%t^'-<WR>-Ti=° ' '1-3°'
where V = 's (V +V ) and AV=V -V for the three components and it is assumed that AV << V and AV„ <<V„.
T T R R ~ This approximation is not valid at the blade edges when the section V
shape is rounded there. ' It is consistent with the first order approximation to treat eqs. S'^'i"'P
•13-ó
, (1-29 and 30) as boundary conditions to be fulfilled at the helical reference plane instead of at the blade surface; the velocities V , V and V are average velocities at the reference plane, the
N R T
velocity jumps AV and AV are discontinuities across this plane.
Eqs. (1-29 and 30) are two conditions in which the camber effect and the thickness effect are separated. Only the velocity
components occur in both equations. For eq. (1-30) it is easy to find a satisfying hydrodynamical model. (V„+q„V„) and (AV,+q, AV„)
N N R N N R
are resp. the velocity and the velocity jump in the direction normal to the reference plane. Eq. (1-30) specifies the velocity jump in relation to the thickness distribution and the average velocities V and V according to:
" % + % ^ \ = ^ ^ " ^ ( ^ T ^ - ^Nt' '1-31'
It is well-known that the flow field fulfilling this condition is given by a source distribution over the reference plane
with a strength equal to the velocity jump. So the right-hand side of the above equation is the source distribution. In general, however, such a distribution also has a continuous normal
velocity component. This component is one of the velocities, induced by the propeller action, which has to be taken into account in eq. (1-29), and can be considered as a thickness correction.
In eq. (1-29) only continuous velocity components occur in combination with camber effects. To satisfy this equation, together with a flow condition at the trailing edge, a vortex distribution is required. This contribution
is related to the flow field in which the propeller is placed. The velocity components in eq. (1-29) have to be split in velocity components of the flow field without propeller, velocity components induced by the thickness of the blades and velocity components induced by the lifting action of the blades. When these components are given the indices o, t and i eq. (1-29) becomes:
-14-f ( ^ N O + ^ N t + \ i ' + % < V " ^ R t ^ ^ R i ' ^
^f'V+^Ti'^^T'V^^Ri'^ t^'^^ = °
3f Q where tan S = — P T — . C dSIn the foregoing derivations the radial velocity component is taken into account irrespective of its magnitude. The coefficients g and g of the radial velocity can be of order one for propellers with rake and/or variable pitch. It has already been noted by Brockett / 2 / that also in the case of no rake and constant pitch, however, the latter coefficient is not zero. In /2/ a strict deriviation of the boundary condition is given in terms of the velocity
potential. Here we have used the velocity components, which are required when we develop calculation schemes.
(1-32)
2. VELOCITY FIELD INDUCED BY PROPELLER LOAD.
2.1. Introduction
In this chapter we will derive the relations which exist between the three vector components of the velocity field and the lifting forces on the propeller blades. Two methods are in use.
The first method is based on the knowledge that there is a direct relation between the lifting forces and the production of vorticity. This vorticity is distributed over a narrow region downstream of the lifting surfaces, which is called the wake. From a specified vorticity distribution the asso-ciated velocity field can be calculated by means of a vector relation, which is a generalisation of the law of Biot-Savart for a line vortex.
In the second method integration of the differential equations for the fluid motion is pursued. To handle the non-linearity of these equations the non-linear terms are partly combined with the pressure and partly considered as induced forces. The
combination with the pressure is called the acceleration potential. The solution can be obtained by means of successive approximations. The principal scheme for this method was already given by Von Karman and Burgers / 3 / . To investigate how this method can be used for the propeller induced velocities first an idealized case will be treated in this chapter. This case is an actuator disk for which the associated velocity field is steady. This model is a representation of a propeller with infinitely many narrow blades. To allow for the effect of a finite number of blades, the results of the steady actuator disk are modified. The load is no longer constant in circumferential direction, but is concentrated in narrow regions on the disk. The positions of these regions rotate with constant angular velocity. In this way a lifting line representation of the propeller blades is obtained. The wake of these lines are trailing helicoidal sheets. In a lifting surface representation of the propeller blades the chord-wise distribution of the load is accounted for, but the wake is not essentially different.
-2.2. Fundamental equations
We start with the equation of motion for a frictionless fluid under influence of external forces:
1^ + (U.grad) u = - ^ grad p + ^ f (2-1)
^^ Pw Pw
where p = pressure, f = force vector,
p = fluid specific mass, t = time,
U = velocity vector.
Let the velocity of the fluid be composed of a steady and constant component U in axial direction and an additional component
u = (u , u , u ) , so the total field is described by the vector U = (U+u , u , u . ) . For an incompressible fluid this
X r (J) "^ field satisfies:
div U = div u = 0. (2-2)
Using also the vector relation
(u.grad) u = grad (^u ) - u x rot u,
the equation of motion can be transformed into:
I T + " 1 ^ = - ^ 9 " ^ (P+'sP^u^) + ^ f + (u X rot u) dt 3X p ^ W p ^
It is convenient to define the following new variables:
q = P + ^%^'^ ' '2-3)
k = f + g , (2-5)
with which the equation of motion becomes
3u ^ „ 3u 1 , ^ 1 r
•r-r + U TT— = - — grad q + — k. /^ ^\
3t 3x p ^ ^ p • (2-6) W W
In this equation we see on the left-hand side the linear acceleration terms, while on the right-hand side two terms representing forces occur:
grad q is a rotation free force and
k is a generalized force composed of ? the external force and
g the secondary or induced force.
When we take the divergence of eq. (2-6) and apply also eq. (2-2) we obtain a relation between q and ïc
2 —
div grad q = V q = div k. (2-7)
A solution of this Poisson equation is
'S = - / <^« f f f ^ = 77 / <^« '^^•g^^'^ 5'- '2-8)
G G The integration extends throughout the region G where
the force vector k is non-zero. The equality of both integrals can be derived from the vector relation:
ƒ dG (A.grad *) = ƒ dS $ (A.n) - ƒ dG * div A G S G when S is the bounding surface of the region G and n is the
local normal to this surface, pointing outwards. Substitution of A = k, with E=0 on S, and $= - proves eq. (2-8).
In eq. (2-8) R is the distance between the point (x, r, $),where g is calculated,and the variable point (C,P,9) of the integration
-R^ = (x-5)^ + r^ -I- p2 - 2rpcos {<i>-6) . (2-9)
For R the notation R(x;f;) will be used, where x resp. E, indicate the positions of the two points involved in the expression.
In the regions where the force field k is zero, we see from 2
eq. (2-7) that q satisfies the Laplace equation V q = 0. In this case the acceleration terms in eq. (2-6) are derived from grad q and therefore q is called the acceleration potential.
To find a solution for the system of eqs. (2-6) and (2-7) it is convenient to split the velocity field u in a rotation free component v and a component w.
Writing
V = grad * (2-10)
and
u = V + w (2-11)
and substituting these relations into eq. (2-6) leads to an equation for which a solution has been suggested in / 3 / , by separate solution of
|w + U 1^ = i- k
>
(2-12)
3t 3x p w | | . u | | = - ^ q - (2-13) w 3w 3* — In steady cases, when -rr: and -r-r- are zero, w and $ can be0 t o t derived from integral expressions
w = - ^ ƒ dx' k {x-} , • (2-14) w -'"
It is assumed that * = 0 for x'->--°°.
To find the velocity field u = w + grad $»when the external force field f is given,successive approximations can be used. In the first approximation the secondary forces g are neglected. We then have k = f,according to eq. (2-5), while the integra-tions in eq. (2-8) and eqs. (2-14, 15) can be performed, resulting in the first approximations w., q and * .
For the second approximation eq. (2-4) yields the secondary forces q., = p (u, x rot u, ) and with k_ = f + g_ the
inte-2 w 1 1 inte-2 =inte-2 grations can be repeated. An example of a steady case is treated in section 2.3. where it is observed that, besides a useful first approximation, also general relations can be derived.
In unsteady cases, integral expressions similar to eq. (2-14) and (2-15) can be used.
t w = (2-16) — ƒ dt' k {x-U(t-t'),t'} = W -" ^ } d x - k { x - , t - ^ } , w -<» 1 t $ = ƒ dt' q {x-U{t-t'),t'} w -<» ^ ƒ dx' q { x ' , t - ^ ^ } . (2-17) x p W -o
In section 2.4 an unsteady case will be considered.
2.3. Steady load on actuator disk.
In the conception of a steady actuator disk the external force field is confined to the region inside a circular cylinder with outer radius r^ and with a small length A. tet the axis of the cylinder be in C-direction. The axial, radial and tangential components
of the force vector can vary in radial direction, but are constant in circumferential direction. The distribution of the force components in axial direction is such that in the limiting case, where the length A becomes zero, finite values for the integrated force components remain. At first it is left out of consideration in which way the forces are generated. Later on the relation to infinitely many lifting blades is introduced.
2.3.1. First approximation of flow field due to external forces.
For the first approximation the generalised force vector is equal to
^ = h'l'^ ^ ^ P ^ S ^ ^e«i^e- '2-18)
The force components are written as products of the factors fr , f. and f , which depend on the radial position only, and
s P 6
the factor 6j,giving the axial distribution. This distribution is such that always
+A/2 +A/2
|f6^d5 = f |ó^dC = f, (2-19)
-A/2 -A/2
even in the limit A^O.
From eq. (2-14) we obtain as a first approximation of the vector field w:
for the region inside the disk
^ = ^ f ƒ S^? . '2-21)
w -A/2
for p<r^,C>A/2 w^ = - ^ f. (2-22) w
The region downstream of the disk is called the wake. It can be concluded that only inside the disk and the wake the velocity field w, is affected by the external force field f. In the limit A->0 this velocity field is discontinuous at the disk. As div U=0, the irrotational field v must have a discontinuity of opposite sign.
Hereafter we will refer to the components of w separately and use the notation:
w, = w^, i^+w ,i +WQ,iQ •
1 Cl 5 pi p 91 9
The velocity potential * from which v is derived will be considered for each vector component of the force f separately. The symbols q and $ are given the corresponding indices. According to eq.
(2-8) we obtain for q: div{f^6 i^)
^^a
=
' ^ P i '-/ 4 T T R ( X ; 5 ) / 4 T I R ( X ; C ) ^D (2-23) (2-24) div(f„6 i ) dG = 0, (2-25) '1 / 4TrR(x;5) ^D 22-where G is the volume of the disk.
The third expression is zero, because the force field does not vary in circumferential direction. Consequently there is no corresponding velocity potential according to eq. (2-15):
*Q^ = 0. (2-26)
For a point outside the disk(R f 0) , eq. (2-23) can be further evaluated: 2Tr r +A/2
q^^(x,r,(t>) = - ^ |d((t.-e) ƒ dp P f ^ | d 5
d6j/d5 R(x;S) A/2By partial integration and by taking the limit A -> 0 we obtain:
q^,(x,r,<t.)
= -7~ f dS
f [ ^ 1
R(x;C)^5^ 0
where S is the circular disk surface.
The corresponding velocity potential field is obtained from eq. (2-15) :
w S^
using
3? R{x;C) 3x R(x;C)
This result can be interpreted as the potential of a sink dis-tribution of strength f /p U over the disk surface S . The axial
C ^ D velocity component at the disk is:
ïïï * u | = lim • 1 „ / dS f i - l — i — _ } = i — i
' n ry 4TIp U J göx R(x;0) p U ^D
The jump in this velocity component is equal to the discontinuity found for the axial component of w and consequently the complete field
(u) = w i + grad $ (2-28)
is continuous at the disk.
It is useful to show that this field can also be described by
a vorticity distribution in the wake. The strength of the vorticity - I df^ _
in the wake is: rot{u)-, = rot (w^^i,) = 77 -r-— IQ . c, i t, 1 c, P U uP U
W
The associated velocity field can be derived from a general vector relation
00 T" —
O _, d f , , ipXR dC
'^*'a = h'l
0 0 ^"" " ' C 4 . R 3 ( X ; C )^' 'ft
^'/'
where the integral over C is an integration along a circle with radius p at the axial position E, = constant. For further use we recall that R/R^ = grad (1/R).
According to a variant of Stokes' theorem the following relation holds:
/dC{igXA}= /dS{(nxV)xA }.
Substitution of this relation with A = grad — and use of a rule K
for vector triple products
(nxV)xV-^ = V{n.vi}-n{v2 i} , K K K yields ^o df — .. I f f ü*)n=:; hr I'^^ f dp (--r-^) /dsfgrad(-^ — ) -n (V^ — ^ — )1 0 0 s (2-29) where we have taken for S a circular disk surface with radius p.
-Partial integration over p is possible and when it is assumed that f, = 0 for p=r the result is:
? o
oo
° ^D
The first part of this integral can be evaluated by performing first the integration over C• Further the operation grad —. -.
R (X; t,)
applied at the coordinates(C,P,9) is the opposite of this operation applied at the coordinates (x,r,<j)). Consequently the first part of the integral becomes:
4TIP U '^'^^^ f / dS — ^ ,1 = grad *.., . (2-30) w I ^ R(x;o)J "^ 51
^D
2 1 The second part of the integral has an integrand in which V :r-; F ^ R(x;5) is only non-zero when the points (5,P,9) and (x,r,*) coincide. Therefore the value of the integral reduces to
ü^l '^ h' ^K
^duV h
j ^ l ^^ h[
v 2 1 4TTP U 7 '^•^ j ^^ - 5 [ ' R ( x ; 5 ) l C 4TTP U 7 5 1 R ( x ; C )O S „ G D W
f^ i^ inside the wake P.u 5 C
outside the wake
" ^ - (w^^ i^). (2-31)
Eqs. (2-30) and (2-31) prove the equality of the vector fields (u )r, and (u)j-,, given in eq. (2-29) resp. eq.(2-28).
Now we have to proceed with the evaluation of eq. (2-24) to find the velocity potential field associated with the radial force in the disk. For a point outside the disk we have:
2iT r -l-A/2
Si'-'^'*' = -iv/ d(*-e)/°dpf^(pfp) / d 5 ^ ) _
For the limiting case A-^0 the result is i ^ f n f ) 'ïpl<-'-'*'= - 4 7 / ' ^ ^ ^ R(x;0) ^D and 1 d , ^ . 1 d , _ , (pf„) , r 7 :r:(pf_)
$ (^,r,^) = L. / dx' / ds
^-^
^ = — i — - / dG L^Ê £
^^ 4.p^U J / R(x;0) ^^^w"
I
R(x;a
D Wwhere G is the volume of the w a k e . If w e define a f u n c t i o n F ,
w pi by: ^ ( F ,) = f (2-32) d p pi p with F „ , = 0 for p = r w e o b t a i n : PI o V^F
% i ' - ' - ' * ' = 4 7 ^ ü - / ^ ^ R ü 7 i y • '2-33)
'^WTo this expression the Green's second theorem can be applied with the final result
%i"^'^'*' -j^\^''\. [A'^)'l
+C (2-34) where C = - — F (r) inside the wake,w
•-^ = 0 outside the wake.
The complete velocity field associated with a radial force in the disk is, according to the definitions in eqs. (2-16, 11)
(u) , = grad * , + w i . pl pi pi p
Combination of the eqs. (2-20, 21, 22) and (2-34) shows that
-'-'pl = ï^fü
'^'-^ I ^^
%1 [A ^ ( ^ 1 = ^^"^*P1 '^-3^'
S^ 5=0
The integral represents the potential of a doublet distribution over the disk S„. It has to be noted that the field (u) , is
D P 1 irrotational.
The vector field (u) , can also be described by an equivalent pl
distribution of ring vortices with strength f /p Ui„, over the disk. p W 6
Using the general vector relation
^0 f , i„x R '^•'pl = / ^' ' ^ ' / < ^ ^
0 w ^^ 4^ R
—# — and using again eqs. (2-28) and (2-29) the equality (u ) = (u) can be proved in a similar way as has been done for (u ) . =(u)j. .
2.3.2. Approximations of the effect of secondary forces.
Now we have derived expressions for the complete first approximation of the velocity field associated with a general steady force
distribution inside an axisymmetric actuator disk. We recapitulate that this field consists in three parts
'^'ci = ^ 1 'h
^
^^^"^ '51
'
'^'pl ='"pl ' \ * 5^^^^ %i=g^ad **^, (2-36)
'^'ei = "ei ^9 •
In Fig. 4 the streamlines of these velocity fields are illustrated. At the disk the axial velocity is continuous and equal to
3 I -^
u =w + — ( * +$* )=\ vi + — * *
CD SI 3c^ SI * p r ^ ^ s i 3 s ' p i
where
(D)p,»GRAD(Pp^
Fig. 4. First approximation of velocity field of actuator disk.
-28-The radial and tangential velocities have discontinuities at the disk
< i = ^ / ' P w " ' '
^91 =
^e/'^w"'-The mean values are
%D = ^pi + l^'^i^%i' = h\i'
"9D = ^^e/'Pw"' = H i = K i
-On the base of this velocity field we can derive approximations for the secondary forces and their influence on the flow.
At the disk the secondary forces are surface forces instead of volume forces, similar to the external forces, see eq. (2-19). An estimate for the rotation of the velocity at the disk is
+ + , , , '*91 ^''pl -'^°t - ' D = - ^ - ^ P ^ ^ ^9
A suitable definition for the secondary forces at the disk, accor-ing to eq. (2-4) , is
+A/2
g^ = lim ƒ p (u x(rot u)j^}dC
*^° -*/2 ,2-37)
= Pw[^-pD ^ l ^ ^ ' " 9 l ' ^ ^ ^S-^'^SD^pl'i
-^"sD^gi'^e] •
The secondary forces in the wake can be evaluated starting from the definition in eq. (2-4)
— — — — # — • — * g-, = p (uxrotu) = p (v + w ) x rot w (2-38)
with
V* = grad(*^,+**,) (2-39) 51 pl
and w = w i + w i (2-40)
It is useful to consider two separate parts of g_, viz.
g- = p (v X rot w ) , (2-41)
— —* —»
g_ = p (w X rot w ) . (2-42) 2w w
I t w i l l be observed t h a t the force g^ i s c o n s t a n t in downstream
^2w
direction, but the force g_ disappears far downstream as the
— . I f
velocity component v goes to zero. Therefore the latter force can be related with the contraction of the wake in the region behind the disk. To find the effects of g^ on the flow through the disk, we have to investigate the associated velocity potential. From eqs. (2-8, 15) it follows
X div g^
/"dx' ƒ dG „,. • .^y (2-43) v2 4TIP U •' Z R(x' ;S)
w -Of G „ W
where G„ is the volume of the wake. W
div q_ = p div (v*x rot w* ^2v w
—4 —# — 4 —# 1
(rot V ) . (rot w )-V . (rot rot w ) » f 2 —* —* 1
= P V -|V (w ) - grad (div w )l
with eqs. (2-43,44) the velocity potential * _ is related to the first approximation of the flow field.
Evaluation of the expression for g^ , eq.(2-42), reveals that it is a radially directed force
-30-r d ^*'91 d , ,1
^2w = Pw[^5i dF " 5 1 ^ ^ dïï'p^ei')
1 PThis equation can also be written as
'2w " dp'^w'^p
where
go,.. = f x ( F j i . (2-45)
r
^w'p' = pw^^i^^r I Ki*^^^ "-'''
The function F„ depends on p only.
w
It will be proved that the effect of the force g^ on the flow can be described by a secondary force at the disk, in addition to the force g in eq.(2-37).
The proof can be given in two ways.
One way is based on the investigation of the associated velocity potential: X 1 r f °"^^ '^•y f -, = A -TT / d x ' / dG ^ , ,-7! , ( 2 - 4 7 ) w2 4TTP U y / R ( x ' ; 5 ) W dF d i v 5 , = 1 f- { p - - W } = V 2 F „ ( 2 - 4 8 ) ^2w p dp dp W
The volume integral in eq.(2-47) can be transformed into surface integrals by application of Green's second theorem.
Without going into the details of the manipulations involved, we give the result
. = _ ^ r . . : ' - ' ' '
w2
, , -f^II^P'
= 7 „ / dS „, -, + C (2-49) 4TTP U / R(x;0)
where C = -xF (r)/(p U) inside the wake, w w
C = 0 outside the wake.
It has been assumed that F,, and dF,,/dp are zero at p = r .
W W o
Comparison of the two terms on the right-hand side of eq.(2-49) with eq.(2-27) resp. (2-22) reveals that * - is the potential of a velocity field completely similar to (u)j-, associated with the
the external force f, on the disk.
Consequently -F i.can be considered as an axial secondary force at the disk.
The second way to prove this relationship is based on the considera tion that radially directed forces can be associated with vorticity rings. In section 2.3.1. it has been shown that a radial force
f i concentrated at the disk induces a vortex ring with strength P P _ 3 ^ f /(p U)i.. It has also been shown that a vorticity distribution
p w 9 _ ^
in the wake with strength -(df^/dp)/(p U)i„ is induced by an axial _ c, w 0
force frif concentrated at the disk. Replacing f, by -F then, in view of eq. (2-4 5) , it can be concluded that the radial force g, in the wake is equivalent with the axial force -F^^irat the disk. Having obtained an approximation for the secondary forces at the disk, we can consider the second approximation for the generalised force according to eq.(2-5). At the disk we have
k2 = f + g^
^w^C
In view of eq.(2-l4) the components of this force are directly connected with the discontinuities in the velocity w at the disk:
r
. '^2c = ^w" ^^2 = fe^''w[-pD^r'^'^5i'^V p'"9i''dp]'
'2p p U w ^ = f - p u ^ „ w , , w p2 p ^w SD pl '
Pw""92 ^g-Pw^SD'^gr
(It has to be noted that w_,= w^, and w„ = w . , ) . A review of the Si SI 91 91
steps which are made to obtain this result reveals that similar relations will apply to higher-order approximations. We write
+ V 2 Pw" "sn+1 = ^5 " Pw^'^Sn' ^ ^Sn' p U w , , = f - p u,„w , '^w pn+1 p "^w SD pn p U W Q j^, w 9n+l P u_„w. , ^w SD en' (2-50)
-32-where
r
gr = P |u r, w"^ + f -(Wo )^dp1 (2-51) ^5n '^w[ pD pn } p en J
P
When we assume that for a high value of n the successive approximations are convergent, the following relations can be derived
w^ = k^/(p^U) = (f^+g'n)/ {P^^(U+^w^) }, (2-52)
"p = % / ' P w " ' = ^ / ^ P W ' " ^ ^ D ' > ' '2-53)
^J = '^e/'Pw"' = fe/^Pw'"^-SD>>' '2-54)
It has been assumed tacitly that at each approximation step thetotal velocities at the disk, u,_^ and u , are also improved. It has to be realized, however, that these velocities also depend on the velocity potential * , for which an approximation is given in eq. (2-43). Successive approximations of * seem not very easy to handle. Neglection of * means that the effects of the contraction of the wake are disregarded. An estimate for these effects can be made by investigating some general features of flow in the wake.
We start with the general formulation of the secondary forces in the wake, according to eqs. (2-4, 10, 1 1 ) .
g^ = p^(u X rot u) = p^{v+w)x rot w (2-55)
3w 3w 3w = P„{(v^+w^) (-^?- - ~^) + w _ ^ } ï , w p p 3^ 3p 6 3S 5 w 3w 3w
^ ^ f T 3^ "'^9'-'V"5"^-^'^^P
9 w^Pw^-'-s""5' - ^ - ' ^ % '
iè'P'e'^^6-In the wake the generalized force k is equal to g and, accordinc to eq.(2-12), we have for a steady wake
-33-g^ = p^U3w/3S (2-56)
From the eqs.(2-55, 56) three relations can be derived, viz.
3w 3w
(U+u^) - ^ + U p ^ = 0, (2-57)
3w_ 3w 3w
^ - y f * "p 35^ ^ ^ 9 ^ S = °' '2-58) 3(PW„) 3(pw )
The first and the third expression reveal that along a streamline w^ and (pw_) have constant values.
S 9
This feature of the flow in the wake can be used in the inter-pretation of an equation which is derived in the general momentum theory for an actuator disk. Suffice it to say that it is obtained by application of Bernoulli's relation to the flow on both sides of the disk. For the details is referred to Glauert/4/. In our notation this relation is (inclusive the effect of radial forces)
{f5(p)+^P„u2 (p) + p^w; Up}^^Q= (2-60)
p^{Uu^(r)+%
ulir)^Hul(r)}^_ -ip^-pj .
This relation holds along a streamline of which the radius at the disk (x=0) is p and for downstream (x=<») is r.
We can substitute:
{u (r) = w (r)} = w+(p),
c, c, X — c,
{rug(r) = rwg(r)}^^^ = pWg(p) ,
'Po-PJ = ^w
/"^"e'^"^'^-The third relation follows directly from the equation of motion
-34-(2-1), when applied far downstream where the outer radius of the wake is r,,. The result of substitution in eq. (2-50) is:
W
K
= ^—r (fr+gf) (2-61)
P„(U+W^)
where r . o< = ^KDV^'<''^i-'f''^-/ ^^;''^'f''
g^H-pWhen comparing eqs. (2-51, 52) with eqs. (2-61, 6 2 ) , we see-that we have obtained a slightly different expression for w^. . Instead of gf I which was derived from successive approximations, we now have g? , which has been derived from general considerations. The difference between the two expressions is connected with the radial contraction of the streamlines through the disk. Eq.(2-62) can be considered as the correct expression, incorporating the effect of contraction of the wake on w,. Its evaluation, however, requires the knowledge about the radial contraction of the streamlines.
The considerations in this section lead to the conclusion that the velocity vector w can be derived from a generalised force at the disk. For the axial and tangential components of w the courses along
a streamline have been found. For the complete flow through the disk the magnitude of the irrotational velocity vector v has to be known. Its relation to the secondary forces in the wake is less simple. In the next section some remarks about the effects of these secondary forces will be made.
2.3.3. Remarks about flow through disk.
For the total axial velocity at the disk we write
\ B
= K ^[IT *vj5=0 (2-63)
We recall that in this notation, it is incorporated that the dis-continuity w. at the disk always corresponds to a continuous velo-city h w.. For the other term we write
^5D ik M^=o ' ^'"'''
where * is the velocity potential associated with the secondary forces in the wake which can not be included in the generalized force at the disk.
The velocity potential * connected with the secondary forces in the wake has to be found from a solution of the equations of motion for the flow in the wake, eq. (2-57,58,59) . This solution will also yield the radial velocity u , which appears in eq.
(2-62) . A relation between the radial contraction of the stream-lines and the axial flow through the disk is given by the condition of continuity of the flow inside an annular streamtube:
(U+w^)rdr = (U+u )PdP (2-65)
For the solution of the equations of motion for the flow in the wake, we refer to the work of Wu /s/ and Greenberg /è/ .They
use a formulation in terms of the vorticity and the stream function, which is more convenient. Their results indicate that v is rather small. For the present purpose we write
, +
u^„ = ïw^ + V, 5D '' S 5D
\^= e ^w^ (2-66;
and assume that e is has a small value, which is known, either from theory or from experiment. Its actual value will depend on the radial position, on the load and on the load distribution.
Another conclusion, which can be drawn from /6/,concerns the radial velocity component at the disk. It appears that, except for the region close to the outer radius
•36-of the disk, the effect •36-of the contraction •36-of the wake on the radial velocity component is small. Consequently a good estimate of this velocity component u can be obtained from the first approximation of the flow field, treated in section 2.3.1. If there is no radial force at the disk, the velocity vector field
(u) , see eq. (2-36), has to be investigated. This is done in Appendix A, where special attention is given to singular behaviour encountered in the evaluation of the integral expressions.
A computational scheme for the distribution of the radial velocity component u results. An example is presented. These results supplement the calculations of Hough and Ordway / 7 / , who also calculated the radial velocities from such a model, but not at the disk itself.
Hitherto the treatment of the actuator disk has been rather general. From the requirement that the disk has to represent a propeller with infinitely many blades, rotating with angular velocity -OJ, an additional equation can be derived. To the
flow relative to the propeller blades Bernoulli's relation can be applied. See Glauert / 4 / . The result can be written as
f^= P X ' ^ ^ P - K ' - P W V P D . " - " '
Combination of this equation with eqs. (2-53, 54) yields'"^^D'^S
^ ^ D % + (K-'^P) fe= 0
o r i n v e c t o r n o t a t i o n
(Ul^ - 'JPÏg+ u p ) . f = 0. ( 2 - 6 8 )
This result shows that the resulting velocity vector at the disk relative to the blades is always normal to the force vector.
-37-For further reference we rewrite the equations for w- and w„ , eqs. (2-52,54).(The effect of a radial force component is disregarded and therefore w is irrelevant.)
^ P w^ = w^^ + w ^ , (2-69) 5 fS gS < = ^f9 - % ' '2-70' with and ^fS= ^ 5 / " ^ w V ' '2-71' ^fe= ^ g / ' P w V ' '2-72)
^S= </'^wV = h'<''^l-'^''> ƒ ^<''^'|'' g 4 " m '
(2-73) % = < / ' P w V = [ ' " 5 D - K ' / ' " ^ ^ S D ' ] f 9 / ' P w V = ^ S D " > m ' 2 - V 4 ) U = U+h'wt . (2-75) m SThe components w and w can be considered as the components tS 19
of a vector:
w , = f/(P, U ). (2-76) f w m
This composition of the velocities can be interpreted tentatively. Four contributions can be recognized.
The vector w- is parallel to and simply related to the external force vector f. The relation (2-76) is similar to eq. (2-22), which is derived for the first approximation of w. In the nominator
the velocity U is replaced by the velocity U . We conclude that for w a quasi-linear actuator disk model holds.
The vector (w r^r) is an axial velocity component induced by the secondary force (g i ) . This force is related to the pressure decrease in the wake, due to the rotational velocity. A balance
^-38-for the axial momentum of the flow through a propeller shows that this pressure decrease means a loss of effective thrust of the propeller. Obviously the expression for w is the proper des-cription of this connection.
The vector (w i ) is proportional to the effect of the secondary
ge
e
forces in the wake on the disk, viz. proportional to v . i t has been observed that, for an actuator disk representation of a propeller, v^ is small. If such a representation is extended, to include e.g. the velocities induced b,y a propeller duct, V and w may become of importance.
Just like for vi^', the expressions for (w i ) and (w i ) correspond f ^ g? 5 ge 9
to a quasi-linear model of an actuator disk with U as reference m
velocity in the nominator. Consequently the velocities at the disk are the halves of these vectors.
In Fig. 5 these velocities at the disk are combined in a vector diagram. The velocity component Vj_ , for which we have
an approximation in eq. (2-66), describes the deviation from the quasi-linear model. In the diagram of Fig.5, the condition of normality, eq. (2-68), for the resulting velocity vector and the external force vector is satisfied.
2.4. Steady load on rotating lifting lines and surfaces.
Various distinctions have to be made between the representation of the propeller as a finite number of thin blades and the re-presentation of the propeller as a thin circular actuator disk. Formally the former representation can be obtained by a modification of the latter. Let the actuator disk rotate about the axis and allow its load to be not constant in circumferential direction. The associated flow field becomes unsteady in the original reference frame, but is steady in a reference frame that rotates with the disk. When the regions on the disk where the forces are non-zero
Fig. 5. Velocity and force vectors at actuator disk.
r"
\ t (jjpV >
/A
-'
Vw,
r y.Wg^ u Um UbFig. 6. Velocity and force vectors at propeller blade,
-40-shrink to a finite number of identical and equidistant lines, the lifting line representation is obtained. Lifting lines are,however, idealisations of material lifting surfaces, which stretch in
streamwise direction. The flow field of such an actuator disk with a rotating load distribution can also be split in a irrotati-onal and remaining part, as defined in the foregoing sections. The first approximation for the wake, that is the region where the vector w is non zero, is a number of helicoidal sheets. This can be seen from eq. (2-16), where we put the generalised force k equal to the external force f at x'=0. Let the distribution of f on the disk be given by a number of Dirac functions:
f = f6{(l)+ü)t+k^} (2-77)
where Z = number of blades and k=0(l) Z-1.
Evaluation of eq. (2-16) yields
w = - ^ f6{9+wt- ^x + I 27T } . (2-78) w
The argument of 6 describes helicoidal sheets with a constant pitch 2TTU/ÜJ. These sheets are wakes of zero thickness. Just like we have done for the actuator disk a description can be given of these wakes starting with finite thickness. Studying the limiting properties for vanishing thickness reveals that the velocity vector vanishes too, but not the rotation of this vector. Thus the welUcnown picture of the wake as a vortex sheet is obtained. The vector product of the vortex strength vector and the velocity vector is a measure for the concentrated secondary forces in the wake. So the main part of the flow through the actuator disk is
irrotational. To distinguish this flow from the wake of an actuator disk with constant circumferential load distribution it is called slipstream.
The flow in the slipstream can be described by a velocity potential and has to be matched with the flow induced by the secondary
the relation between the time averaged flow in the slipstream and the flow in the wake of an actuator disk with a
circum-ferentially averaged load equal to the load of the lifting lines. It has been shown by Hough and Ordway 111 that these flows are equivalent in the limiting case of light loading. For heavy
loading such an equivalence is unlikely as shown by Greenberg /6/.
Nevertheless it can be assumed that some effects of heavy loading on the flow through the propeller disk are described sufficiently accurate by the model of an actuator disk with the same time-averaged load. (For convenience we use the terms propeller disk and actuator disk to distinguish between the disk with load on lifting lines or surfaces and the disk with circumferentially constant load). In the foregoing section the flow through an actuator disk is separated in locally induced velocities and wake induced velociti,e'= .For propeller disk flow it is assumed that the velocity induced by the rotation of the slipstream is equal to the corresponding velocity of the actuator disk, so circumferential variations are disregarded.
For the locally induced velocities holds that they are directly related to the external forces (see eq. (2-76)). The relation is quasi-linear as the effective m.ean flowvelocity U depends on the
m
induced velocities and can vary with the radius. Application
of this quasi-linearity also to the propeller disk flow means that in eq. (2-16) k has to be replaced by f and U by U , which results in X f p . w m m 1 f6{e+u)t-;^ X + ^ 2n} (2-79) p U U Z w m m
Comparison of this equation with eq. (2-78) shows that the pitch of the helicoidal wakes is changed to 2ii
this pitch is adapted to the local flow.
The total velocity in the neighbourhood of the propeller disk has to be derived from the velocity potential. According to eq.
(2-17) we have X
'
= r ^
h-'
qf(x,t-^-} (2-80)
^w m •' m —<» w h e r e q , f o l l o w s from e q . ( 2 - 8 ) q^ = - r i / dS ( f . g r a d i ) ( 2 - 8 1 ) ^ f 4TT y RThe integration area S extends over all the lifting surfaces. B
For the velocities induced by the contraction of the slipstream it can be expected that they differ considerably from the veloci-ties induced by the contraction of the wake of the actuator disk. In the slipstream the secondary forces, which describe the contraction, are located in the trailing vortex sheets close behind the blades. Therefore it seems reasonable to assume that these forces induce a velocity which is directed tangentially to the pitch line of the lifting surface. See the velocity v* in Fig. 6. This velocity is larger in magnitude than the circumfe-rentially constant values of v and ^w 9 in the actuator disk model. A consequence of this modification is that also the secon-dary forces concentrated on the lifting surface have to be correc-ted. An estimate for these forces is based on the following consideration of eqs.(2-22) and (2-76). The former gives a first approximation of the locally induced velocity at the disk, without the effect of secondary forces. The latter includes the effect of secondary forces. The apparent difference is the replacement of the main flow velocity U by the local axial velocity U . In Fig.6 we see that U is proportional to the resultant velocity, which
m
is tangential to the lifting surface. We conclude that the secondary forces at the lifting surface associated with the increase of this tangential velocity can be accounted for by replacing U by U .
3. APPLICATION OF LIFTING SURFACE THEORY.
3.1. Introduction
In chapter 1 and 2 the basic principles are given, with which it is possible to find the geometry of the propeller blades and their load. The formulation of such a relation is the subject of this chapter.
In chapter 1 the hydrodynamical boundary condition is given as an equation,(1-32),for the flow velocity, which is locally decomposed in a radial component V , a component V tangential to the pitch
R T
line and a component V„ normal to the pitch line and normal to the N
radial direction. The pitch line is the intersection of the blade reference plane with a cylinder of constant radius and coincides with the nose-tail line of the blade sections. It is assumed that eq. (1-32) has to be satisfied on the blade reference plane instead of on the blade surface.
For the hydrodynamical load a similar approximation is used. It is described by a distribution over the blade reference plane instead of over the blade itself. The blade force normal to the reference plane is given by a pressure jump across the reference plane. The blade force tangential to the reference plane, connected with camber, friction and leading edge suction, is much smaller than the normal force and its effect will be dealt with as corrections to the model with a normal force only.
The analysis of the flow through an actuator disk, in
chapter 2, has revealed that this flow can be decomposed in locally induced velocities and wake-induced velocities. The locally in-duced velocities are directly related to the external forces
(f-forces, see eqs. (2-71, 72)). The wake-induced velocities at the disk can partly be described by secondary forces at the disk
(g*-forces, see eqs. (2-73,74)), which account for the effects of the rotation of the wake. The other part of the wake-induced velocities at the disk is connected with the contraction of the
-44-wake, which takes place in the region close behind the disk. For the effects of the f-forces and the g*-forces quasi-linear actuator disk relations ceui be used, but the effects of the contraction require a solution of the equation of motion for the flow in the wake.
The considerations about the flow through a propeller disk, which are also presented in chapter 2, have to be matched with the formulation of the boundary condition. This is accomplished by the following prescriptions.
- The velocities induced by the rotation of the slipstream are derived from an actuator disk model with the same radial load distribution as the propeller.
- The locally induced velocities are derived from a velocity potential, which is obtained by integration of the distribution of normal forces over the blade reference plane.
- The velocities induced by the contraction of the helicoidal wake sheets behind the blades are assumed to be tangential to the blade reference plane (see Fig.6).
The magnitude of the velocity U is still unknown. It will be found by correlation of the theoretical model with experimental results. Also the velocity factor e as defined in eq. (2-66) has to be fixed to complete the model.
3.2. Formulation of integral equation and solution method.
To introduce the properties of the load-induced velocity field, as discussed in the foregoing section, into the hydrodynamical boundary condition, we use a modification of eq.(l-32):
<^No"%^VRg'^'^To+V"VRi'"""'^"^' = -'ll^^ll'- '3-1'
To obtain this equation from eq.(l-32) we have used the following relations and assumptions:
- The velocity components induced by the thickness of the blades are expressed in a correction angle 6 , which has to be subtracted from the camber angle 6 .
c
In the flow field without propeller, there is no radial velocity component,hence V =0.
The velocity components induced by the blade load are written as
V = V + V , Ni Ng Nf' V = V + V , Ri Rg Rf' V = V + V , Ti Tg Tf where 3* 3* \ f = 3N ' ''Rf = 3R ' \ f = neglected, V,, = - M w ^cos B +w sing ) , Ng "^ gS P g9 P
^g = +^(V^'" V"ge^°^^'*^*D'
withpitch angle; the notation 6 is introduced to avoid confusion with the velocity potential
3/3N and 3/3R are differentiations in the directions of v and N V , see section 3.1.
H
-46-V . resp.v are related to the radial distribution of w^ resp. Ri Rg S ^ w ^ according to the relations between radial and axial velocity
g5
components given in Appendix A.
On the left-hand side of eq. (3-1) the terms are collected which are either known from the flow without propeller, viz. V and V , or estimated from the actuator disk model, viz. V , V , g V and g V„.. On the right-hand side of eq. (3-1) the terms ^N Rg ^T Ri ^
are collected which are derived from the velocity potential of the lifting surfaces.
The evaluation of the expression on the right-hand side is laborious. The integral expressions from which * can be derived, see eqs.
(2-80, 8 1 ) , involve integration of the load distribution over the lifting surfaces which represent the propeller. Further the integrand has singular points and requires a detailed analysis to make calculations possible. Finally the more difficult case for which eq. (3-1) is used is the analysis of the load
distribution on a propeller of given geometry. To obtain a solution of eq. (3-1) for that case, it is necessary to approximate the integral by a linear combination of a number of load function parameters with coefficients depending only on the geometry.
The formulation of the velocity potential * on the base of an external load distribution on the blades of a ship propeller has been studied by Sparenberg /8/, Hanaoka /9/ , Pien and
Strora-Tejsen /lO/ and Tsakonas et al. /ll/. Most of these studies are not restricted to the case of steady blade load but include also
unsteady blade loads which are periodic with the shaft rotation rate or a multiple of it as basic frequency. The case of steady load corresponds to zero frequency.
Calculation procedures for the evaluation of the right-hand side of eq.(3-l) have restrictions concerning the blade geometry, i.e. the pitch is constant and mostly there is no rake. To keep the computation time short it is appropriate to evaluate the expression
3$ , 3* 3N ^N 3R
as a whole and not in separate terms. For this study one of the two calculation procedures developed at the Netherlands Ship Model Basin is used. The first one, which applies to propellers with constant pitch and without rake, is prepared by Verbrugh /12/ and applied by Van Gent /l3/. Recently the second one, which applies to propellers with constant pitch and constant rake angle, has become available and has been prepared by Vis /14/.
For constant pitch and no rake g^, = 0 and the right-hand side of eq. (3-1) reduces to
3* • 3N •
To allow for the application of the corresponding calculation procedure also in cases of small radial pitch variations and small rake angles, we do not neglect the term
3* % 3R '
but transfer it to the left-hand side of eq.(3-l), and approximate it by g,,V^,. This term can be combined with the term q V as
N Rf ^N Rg
g V = g (V +V ) ^N Ri ^N^ Rg Rf'
and V„. is derived from the actuator disk. Hence the equation Ri
which will be investigated is
3*
(V +V +g V )+(V +V +g V .)tan(6 -B ) = - ^^ • (3-2) ^ No Ng ^N Ri To Tg ^T Ri' ^^c t' 3N
For the details of the calculation procedures we refer to the forementioned literature. Some main points are reviewed here. The distribution of the normal forces over the blade reference plane is represented by a truncated series of elementary chordwise functions
fj^(p,s) = I C (p) H (s) (3-3)
-48-where , ^ ^ . , , , % ^ _ COS(p()) ) +cos(p+l)(fi
H (s) = H ((t> ) = - ? - , P=0(1)P. (3-4) p P s TT Sin *
* is a chordwise parameter defined by
cos (l)g = 1-2 (-^"^^^2) ^ (3_5)
thus * =0 at the leading edge and * =TT at the trailing edge. C (p) are the weight factors of the functions H (s) and they are
P P functions of the radial position. It has to be noted that the
func-tions H (s) are zero at the trailing edge, thus satisfying the Kutta
condition for the flow. At the leading edge all the functions H (s) ij [^,^. have a singular behaviour. In the section 3.4 some properties of , « uji*» these functions will be further investigated. ^'*, A-i\ . By eq. (3-3) a separation of variables is introduced, which can
be further exploited by using an interpolation formula for radial distributions and functions. The final result of the manipulations involved, see /13/ , is that the calculation procedure for 3*/3N can be formally written as
„. P N
V^_(r, ,s.) = -f^ = sin B^ S I C (p^)M (r ,s.) (3-6) Nf V ] 3N a p^Q ^^^ p n pn V :
where
r and s . specify the radial and chordwise position of the considered point on the reference plane,
C (P ) is the weight factor of the function H at the radial
p n ^ P position p ,
M (r ,s.) are the coefficients of complicated structure, but pn V 3
depending only on geometrical properties of the reference plane,
6 is the pitch angle at the radial position r of the a
helicoidal plane at which the load is situated. It should be equal to 6 > the pitch angle of the reference plane, but the distinct notation is appropriate for approximations in cases of pitch deviations.
