theoretical methods to phredict the action of propellers began to develop in the latter part of the...

8/12/2019 Theoretical Methods to Phredict the Action of Propellers Began to Develop in the Latter Part of the Nineteenth Cen

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Theoretical methods to predict the action ofpropellersbegan to develop in the latter part of the nineteenth cen-tury. Perhaps the most notable of these early works wasthat of Rankine, with his momentum theory, whichwas closely followed by the blade element theoriesof Froude. The modern theories of propeller action,however, had to await the more fundamental works inaerodynamics of Lanchester, utta, !oukowski, "unkand Prandtl in the early years of the last century beforethey could commence their development.

Lanchester, an #nglish automobile engineer and self-styled aerodynamicist, was the $rst to relate the idea ofcirculation with lift and he presented his ideas to the%irmingham &atural 'istory and Philosophical (ocietyin )*+. 'e subseuently wrote a paper to the Physical(ociety, who declined to publish these ideas. &ever-theless, he published two books, Aerodynamics and

Aerodonetics, in )+/ and )+* respectively. 0n thesebooks, which were subseuently translated into 1er-man and French, we $nd the $rst mention of vorticesthat trail downstream of the wing tips and the propos-ition that these trailing vortices must be connected by avorte2 that crosses the wing3 the $rst indication of the4horse-shoe5 vorte2 model.

0t appears that uite independently of Lanchester5swork in the $eld of aerodynamics, utta developed theidea that lift and circulation were related6 however, he

did not give the uantitative relation between these twoparameters. 0t was left to !oukowski, working in Russiain )+7, to propose the relation

L=V 8 9/.):

This has since been known as the utta-!oukowski the-orem. 'istory shows that !oukowski was completelyunaware of utta5s note on the sub;ect, but in recog-nition of both their contributions the theorem hasgenerally been known by their ;oint names.

Prandtl, generally acclaimed as the father of modernaerodynamics, e2tended the work of aerodynamics into$nite wing theory by developing a classical lifting linetheory. This theory evolved to the concept of a liftingline comprising an in$nite number of horse-shoe vor-tices as sketched in Figure /.). "unk, a colleague ofPrandtl at 1ottingen, $rst introduced the term 4induceddrag5 and also developed the aerofoil theory which has

produced such e2ceptionally good results in a widevariety of subsonic applications.

From these beginnings the development ofpropellertheories started, slowly at $rst but then gathering

pace through the )+


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)

model which has inherent assumptions built into it.=onseuently, these mathematical models ofpropelleraction rely on the same theoretical basis as that of aero-dynamic wing design, and therefore appeal to the samefundamental theorems of sub-sonic aerodynamics orhydrodynamics. >lthough aerodynamics is perhaps thewider ranging sub;ect in terms of its dealing with a

more e2tensive range of ?ow speeds, for e2amplesubsonic, supersonic and hypersonic ?ows, both non-cavitating hydrodynamics and aerodynamics can beconsidered the same sub;ect provided the "ach numberdoes not e2ceed a value of round . to .erofoil Theory Lifting (urface "odelsPressure Bistribution 9"organ et al., van-1ent,

=alculations %reslin:

CB=

and

EAV E

M

9/.E:

&>=> Pressure >dvanced Lifting Line LiftingBistribution (urface 'ybrid "odels>ppro2imation Aote2 Lattice "odels

%oundary Layer 1rowth 9erwin:over >erofoil %oundary #lement "ethods

Finite ing and (pecial Propeller Types3Bownwash =ontrollable Pitch

'ydrodynamic "odels of Bucted PropellersPropeller >ction =ontra-rotating

Aorte2 and (ource Panel (upercavitating"ethods

C" = )EAlV E

in which A is the wing area, l is a reference length, Vis the free stream incident velocity, the density of the

?uid,L andD are the lift and drag forces,perpendicularand parallel respectively to the incident ?ow, and M isthe pitching moment de$ned about a convenient point.

These coef$cients relate to the whole wing sectionand as such relate to average values for a $nite wing section.


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Theoretical methods Cbasic concepts

M

+

Figure 7.2 #2perimental single aerofoil characteristics 9&>=> 7


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Figure 7.3 "oment and force definitions for aerofoils

The results shown in Figure /.E also show the Dero liftangle for the section which is the intersection of the liftcurve with the abscissa6 as such, it is the angle at whichthe aerofoil should be set relative to the incident ?ow inorder to give Dero lift. The propeller problem, however,rather than dealing with the single aerofoil in isolationis concerned with the performance of aerofoils in cas-cades. %y this we mean a series of aerofoils, thebladesin the case of the propeller, working in suf$cient pro2-imity to each other so that they mutually affect eachother5s hydrodynamic characteristics. The effect of cas-cades on single aerofoil performance characteristics is

shown in Figure /.. From the $gure it is seen thatboththe lift slope and the Dero lift angle are altered. 0n the

Figure 7.4 #ffect of cascade on single aerofoilproperties

case of the lift slope this is reduced from the singleaerofoil case, as is the magnitude of the Dero lift angle.>s might be e2pected, the section drag coef$cient isalso in?uenced by the pro2imity of the other blades6however, this results in an increase in drag.

7.2 Vortex filaments and

sheets

The concept of the vorte2 $lament and the vorte2 sheetis central to the understanding of many mathematicalmodels of propeller action. The idea of a vorte2 ?ow,Figure /. vorte2 $lament cannot end in a ?uid. >s a conse-uence the vorte2 must e2tend to the boundaries ofthe ?uid which could be at or, alternatively, thevorte2 $lament must form a closed path within the?uid.

These theorems are particularly important since theygovern the formation and structure of inviscid vorte2

ll d l


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Figure 7.5 Aorte2flows3 9a: two-dimensional vorte2 and 9b: line vorte2

Figure 7. Aorte2sheet

The idea of the line vorte2 or vorte2 $lament canbe e2tended to that of a vorte2 sheet. For simplicity atthis stage we will consider a vorte2 sheet comprisingan in$nite number of straight line vorte2 $laments side

by side as shown in Figure /.7. >lthough we are hereconsidering straight line vorte2 $laments the conceptis readily e2tended to curved vorte2 $laments such asmight form a helical surface, as shown in Figure /./.Returning, however, to Figure /.7, let us consider the

sheet 4end-on5 looking in the direction Oy. 0f we de$nethe strength of the vorte2 sheet, per unit length, over

Figure 7.7 'elical vorte2sheet

the sheet as I 9s: where s is the distance measuredalong the vorte2 sheet in the edge view, we can thenwrite for an in$nitesimal portion of the sheet, ds, the

strength as being eual to I ds. This small portion ofthe sheet can then be treated as a distinct vorte2strength which canbe used to calculate the velocity at

i i h


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=

neighbourhood of the sheet. For the point P9x, z:shown in Figure /.7 the elemental velocity dV ,

perpendicular to the direction r, is givenby

I ds

7.3 Field !oint "elocities

The $eld point velocities are those ?uid velocities thatmay be in either close pro2imity to or remote from the

dV

EJr9/.7: body of interest. 0n the case of a propeller the $eld

point velocities are those that surround the propeller

=onseuently, the total velocity at the point P is thesum- mation of the elemental velocities at that pointarising from all the in$nitesimal sections from a to b.

The circulation 8 around the vorte2 sheet is eual tothe sum of the strengths of all the elemental vorticeslocated between a and b, and is givenby b8= I ds 9/./:

both upstream and downstream of it.

The mathematical models of propeller action aretoday based on systems of vortices combined in avariety of ways in order to give the desired physicalrepresentation. >s a conseuence of this a principaltool for calculating $eld point velocities is the %iotC(avart law. This law is a general result of potentialtheory and describes both electromagnetic $elds andinviscid, incompressible ?ows. 0n general terms thelaw canbe

a stated 9see Figure /.+: as the velocitydV

induced at

0n the case of a vorte2 sheet there is a discontinuity in

the tangential component of velocity across the sheet.This change in velocity can readily be related to the

a point P of radius r from a segment dsof a

vorte2 $lament of strength 8 givenby 8 dl rlocal sheet strength such that if we denote upper andlower velocities immediately above and below the vor-te2 sheet, by ) and E respectively, then the local;umpin tangential velocity across the vorte2 sheet is eual tothe local sheet strength3

I = ) E

The concept of the vorte2 sheet is instrumental inanalysing the properties of aerofoil sections and $ndsmany applications in propeller theory. For e2ample,

one such theory of aerofoil action might be to replacethe aerofoil with a vorte2 sheet of variable strength, asshown in Figure /.*. The problem then becomes to cal-culate the distribution of I 9s: so as to make the aerofoilsurface become a streamline to the ?ow.

dV=

J

|r|G9/.*:

Figure 7.# (imulation of an aerofoil sectionby avorte2 sheet

These analytical philosophies were known at the timeof Prandtl in the early )+Es6 however, they had toawait the advent of high-speed digital computers someforty years later before solutions on a general basiscouldbe attempted.

0n addition to being a convenient mathematicaldevice for modelling aerofoil action, the idea ofreplacing the aerofoil surface with a vorte2 sheet alsohas a physical signi$cance. The thin boundary layerwhich is formed over the aerofoil surface is a highlyviscous region in which the large velocity gradients

produce substantial amounts of vorticity. =onseuently,there is a distribution of vorticity along the aerofoil sur-face due to viscosity and the philosophy of replacingthe aerofoil surface with a vorte2 sheet can be

an inviscid ?ow.


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Figure 7.$ >pplication of the %iotC(avart law toa general vorte2 filament

To illustrate the application of the %iotC(avart law,two common e2amples of direct application to

propeller theory are cited here3 the $rst is a semi-in$nite line vorte2 and the second is a semi-in$niteregular helical vorte2. %oth of these e2amplescommonly represent systems of free vorticesemanating from thepropeller.

First, the semi-in$nite line vorte2. =onsider the sys-tem shown in Figure /.), which shows a segment dsof a straight line vorte2 originating at O and e2tendingto in$nity in the positive x-direction. &ote that in

practice, according to 'elmholtD5 theorem, the vorte2could not end at the point Obut must be ;oined to someother system of vortices. 'owever, for our purposeshere it is suf$cient to consider this part of the system in

isolation. &ow the velocity induced at the point Pdistant r from dsis given by euation 9/.*: as

dV8 sin K

ds=

J rE


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Figure 7.1% >pplication of the %iotC(avart law to a semi-infinite line vorte2 filament

Figure 7.11 >pplication of %iotC(avart law to a semi-infinite regular helical vorte2 filament

from which the velocity atP is written as

8 K= sin K dsconcept is the same. =onsider the case where a helicalvorte2 $lament starts at the propeller disc and e2tends

VP =J K= r

Eto in$nity having a constant radius and pitch angle, asshown in Figure /.)). From euation 9/.*: the velocity

and sinces=h 9cot K cot : we

have

at the pointP due to the segment dsis givenby

8

8d 9ds a:

VP = J K=

sin K dK =

J|a|G

that is

8VP =

Jh9) cos : 9/.+:

The direction of VP is normal to the plane of thepaper,by the de$nition of a vector cross product.

0n the second case of a regular helical vorte2 the

and from the geometry of the problem we can derivefrom

a= ax i+ ay !+ az "

that

a= r sin9K+ M:i 9y+ y :!


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M::"


10/16

| |

dK

(imilarly,

s9K:= r sin 9K+ M:i+ rK tan Ni !+ r cos 9K

+ M:"

from which we can derive

8d=

J a G

the theoretical solution for an aerofoil in potential ?ow6however, nature selects ;ust one of these solutions.

0n )+E, utta made the observation that the ?owleaves the top and bottom surfaces of an aerofoilsmoothly at the trailing edge. This, in general terms,is the utta condition. "ore speci$cally, however, thiscondition can be e2pressed as follows3

i ! "). The value of the circulation 8 for a given aerofoil at a

r cos 9K+ M: rK tan Ni r sin 9K+

M: particular angle of attack is such that the ?ow leavesrsin9K+M: 9y+y : r rcos9K+

M:

where the scalar a is givenby

O9y+ y :E+ r

E+ r

E Er r cos 9K+ M:

GHE

'ence the component velocities x , y and z are givenby the relations

r8

x=

J tan Ni 9r cos 9 K + M :: 9y+ y : sin

9K + M:

O9y+ y :E+ rE+ rE Err cos 9K+

M:GHE

r8y =

J r r cos 9K+ M:

dK O9y+ y :

E+ rE+ rE Err cos 9K+

M:GHE

r8z =

J

r tan Ni sin 9 K + M : 9y+ y : cos 9 K

+ M : dK

the trailing edge smoothly.E. 0f the angle made by the upper and lower surfaces of

the aerofoil is $nite, that is non-Dero, then thetrailing edge is a stagnation point at which thevelocity is Dero.

G. 0f the trailing edge is 4cusped5, that is the anglebetween the surfaces is Dero, the velocities arenon-Dero and eual in magnitude and direction.

%y returning to the concept discussed in (ection /.E,in which the aerofoil surface was replaced with asystem of vorte2 sheets and where it was noted that thestrength of the vorte2 sheet I 9s: was variable alongits length, then according to the utta condition thevelocities on the upper and lower surfaces of theaerofoil are eual at the trailing edge. Then fromeuation 9/./: we have

I9T#: = ) E

which implies in order to satisfy the utta condition

I9T#: = 9/.)):

O9y+ y :E+ rE + rE Err cos 9K+ M:

GHE

9/.):

These two e2amples are suf$cient to illustrate the

proce-dure behind the calculation of the $eld point velocities

in inviscid ?ow. =learly these principles can bee2tended to include horse-shoe vorte2 systems,irregular helical vortices, that is ones where the pitchand radius vary, and other more comple2 systems asreuired by the modelling techniues employed.

0t is, however, important to keep in mind, whenapply- ing these vorte2 $lament techniues tocalculate the velocities at various $eld points, that theyare simply conceptual hydrodynamic tools forsynthesiDing more comple2 ?ows of an inviscidnature. >s such they are a convenient means ofsolving Laplace5s euation, the euation governingthese types of ?ow, and are not by themselves of any

great signi$cance. 'owever, when a number ofvorte2 $laments are used in con;unction with a freestream ?ow function it becomes possible to

application.

7.4 The &utta condition

For potential ?ow over a cylinder we know that,

depend- ing on the strength of the circulation, a numberofpossi-ble solutions are attainable. > similar situationapplies to


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7.5 The starting

"ortex

elvin5s circulation theorem states that the rate ofchange of circulation with time around a closed curvecomprising the same ?uid element is Dero. 0n math-ematical form this is e2pressed as

D8Dt= 9/.)E:

This theorem is important since it helps e2plain thegen- eration of circulation about an aerofoil.=onsider an aerofoil at rest as shown by Figure/.)E9a:6 clearly in this case the circulation 8 about theaerofoil is Dero. &ow as the aerofoil beings to movethe streamline pattern in this initial transient state lookssimilar to that shown in Figure /.)E9b:. From the $gurewe observe that high- velocity gradients are formed atthe trailing edge and these will lead to high levels ofvorticity. This high vorticity is attached to a set of

?uid elements which will then move downstream asthey move away from the trailing edge. >s they moveaway this thin sheet of intense vorticity is unstableand conseuently tends to roll up to give a pointvorte2 which is called the starting vorte2 9Figure/.)E9c::. >fter a short period of time the ?ow stabiliDesaround the aerofoil, the ?ow leaves the trailing edgesmoothly and the vorticity tends to decrease anddisappear as the utta condition establishes itself.


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Figure 7.12 #stablishment of the starting vorte23 9a: aerofoil at rest6 9b: streamlines on starting prior to uttaconditionbeing established and 9c: conditions at some time after starting

The starting vorte2 has, however, been formed duringthe starting process, and then continues to movesteadily downstream away from the aerofoil.

0f we consider for a moment the same contour com-prising the same ?uid elements both when the aerofoilis at rest and also after some time interval when theaerofoil is in steady motion, elvin5s theorem tells usthat the circulation remains constant. 0n Figure /.)E9a:and 9c: this implies that

8)= 8E=

for the curves C) and CE which embrace the same ?uidelements at different times, since 8) = when the

aerofoil was at rest. Let us now consider CE splitinto tworegions, CG enclosing the starting vorte2 and C theaerofoil. Then the circulation around these contours 8Gand 8 is givenby

8G+ 8,= 8E

but since 8E = ,

then

8,= 8G 9/.)G:

which implies that the circulation around the aerofoil iseual and opposite to that of the starting vorte2.

0n summary, therefore, we see that when the aerofoilis started large velocity gradients at the trailing edge areformed leading to intense vorticity in this region which

ll d t f th f il t f th t ti

vorte2. (ince this vorte2 has associated with it ananticlockwise circulation it induces a clockwise circula-tion around the aerofoil. This system of vorticesbuilds

up during the starting process until the vorte2 aroundthe aerofoil gains the correct strength to satisfy the

utta condition, at which point the shed vorticityceases and steady conditions prevail around theaerofoil. The starting vorte2 then trails away

downstream of the aerofoil. These conditions havebeen veri$ed e2perimentally

by ?ow visualiDation studies on many occasions6 theclassic pictures taken by Prandtl and Tiet;ens9Reference


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vorticity along the camber line of an aerofoil. For the


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Figure 7.13 Thin aerofoil representation of an aerofoil

camber line to be a streamline in the ?ow $eld the com-ponent of velocity normal to the camber line must beDero along its entire length. This implies that

Vn + Qn 9s:= 9/.):

where Vn is the component of free stream velocity nor-mal to the camber line, see inset in Figure /.)G6 andQn 9s: is the normal velocity induced by the vorte2sheet at some distance s around the camber line fromthe leading edge.

0f we now consider the components of euation9/.): separately. From Figure /.)G it is apparent,again from the inset, that for any point # along thecamber line,

assume that normal velocity at the chord line will beappro2imately that at the corresponding point on thecamber line and to consider the distribution of vorticityalong the camber line to be represented by an identical

distribution along the chord without incurring any sig-ni$cant error. Furthermore, implicit in this assumptionis that the distance s around the camber line appro2i-mates the distance x along the section chord. &ow todevelop an e2pression for Qn 9s: consider Figure /.),which incorporates these assumptions.

From euation 9/.7: we can write the followinge2pression for the component of velocity dQn 9x:normal to the chord line resulting from the vorticityelement dR whose strength is I 9R:3

Vn = Vsin

+

tan)

dzdx

dQn 9x: I 9R:dR= EJ9x :

For small values of and dzHdx, which are condi-tions of thin aerofoil theory and are almost always metin steady propeller theory, the general condition thatsin K tan K K holds and, conseuently, we maywrite for the above euation

'ence the total velocity Qn 9x: resulting from all thecontributions of vorticity along the chord of the aerofoilis givenby

c I 9R:dRQn 9x:=

dz EJ9x :

Vn = V dx

9/.)


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V

=

J

Figure 7.14 =alculation of induced velocity at the chord line

incidence angle and camber pro$le. 0n this euation, as in all of the previous discussion, is simply adummy variable along the Ox a2is or chord line.

0n order to $nd a solution to the general problem

of a cambered aerofoil, and the one of most practicalimportance to the propeller analyst, it is necessary touse the substitutions

c =

E9) cos K:

which implies d = 9cHE: sin K dK

and

cx =

E9) cos K :

which then transforms euation 9/.)7: into

a camber line of a given shape and at a particularincidence angle so as to obey the utta condition atthe trailing edge. The restrictions to this theoreticaltreatment are that3

). the aerofoils are two-dimensional and operating asisolated aerofoils,

E. the thickness and camber chord ratios are small,G. the incidence angle is also small.

=onditions 9E: and 9G: are normally met in propellertechnology, certainly in the outer blade sections. 'ow-ever, because the aspect ratio of a propeller blade issmall and all propeller blades operate in a cascade,=on- dition 9): is never satis$ed and corrections havetobe

)J

I 9 K: sin KdK dz

9/.)/:

introduced for this type of analysis, as will be seen later.

ith these reservations in mind, euation 9/.)*: can

EJ cos K cos K :=

dx be developed further, so as to obtain relationships for

0n this euation the limits of integration K = Jcorres-ponds to =c and K = to =, as can

be deduced from the above substitutions.&ow the solution of euation 9/.)/:, which obeys the

utta condition at the trailing edge, that is I 9J: =, and make the camber line a streamline to the ?ow, isfound tobe

the normal aerodynamic properties of an aerofoil.From euation 9/./: the circulation around the cam-

ber line is givenbyc

8= I 9R:dR

which, by using the earlier substitution of =

9cHE: 9)

cos K:, takes the form )+ cos K c cI 9K:= EVA

sin K+ An sin

9nK:

n=)

9/.)*: 8 I 9K: sin K dK 9/.)+:E

in which the Fourier coef$cients A and An can be from which euation 9/.)*: can be written asshown, as stated below, to relate to the shape of thecamber line and the angle of the incidence ?ow by thesubstitution of euation 9/.)*: into 9/.)/: followed bysome algebraic manipulation3

8= cV

J

A

9)+ cos K:dK

J

) J

dz

+

An

n=)

sin K sin9nK:dK

A= dx

dK 9/.)*a:which, by reference to any table of standard integrals,

E J reduces to


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theoretical methods to phredict the action of propellers began to develop in the latter part of the...

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