Note on the non-stationary actuator disc

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Note on the non-stationary actuator disc.

by

J.A. Sparenberg.

Summary.

The l i n e a r theory of an actuator d i s c , of which the pressure Jump i s a f u n c t i o n o f time or which moves along a curved path, i s discussed. I t i s shown that here free sources and sinks are shed, i n combination with a l o c a l fXow which compensates the divergence of the sources and sinks. The concept of the s t a r t i n g source, instead o f the s t a r t i n g vortex, i s introduced.

Introduction.

The actuator d i s c i s a model which to a c e r t a i n extent can replace the more complicated s h i p screw o r aiiy other propulsion system.. To a

c e r t a i n extent i s used here i n the sense that the i n f l u e n c e of the p r o p e l l e r at points which are not too close to i t can be c a l c u l a t e d by means of i t s g l o b a l a c t i v i t y , nataely inducing a p r o p e l l i n g pressure jump over a c e r t a i n area. Even i n the case of the shrouded p r o p e l l e r [1], where i n general the distance between screw and duct i s r a t h e r small i t i s assumed t h a t the average e f f e c t of the srew can be described by the actuator d i s c .

Several types of actuator d i s c s are p o s s i b l e . The most simple one i s a c i r c u l a r region placed perpendicular to a uniform flow at which we have a constant pressure jump. This pressure jump can also be taten as a f u n c t i o n of the radius. Other refinements can be made. For instance non l i n e a r e f f e c t s and the r o t a t i o n a l v e l o c i t i e s behind the d i s c can be taken i n t o account [2].

In t h i s note we w i l l extent the theory to the non-stationary case, however we keep the theory l i n e a r . A somewhat remarkable r e s u l t w i l l appear, instead of the concept s t a r t i n g vortex which i s usual i n l i f t i n g surface theoiy, the concept s t a r t i n g source or sink w i l l occur. During the

non-stationary a c t i o n of the actuator d i s c sources and sinks w i l l be shed continuously. At the same time a s t r i c t l y l o c a l v e l o c i t y f i e l d i s created of which the divergence exactly compensates the divergence of the sinks.

In accordance with [ l ] i t w i l l be p o s s i b l e to use t h i s theory f o r an estimation of the time dependent pressures induced on the nozzle of a shrouded p r o p e l l e r dùr.ing manoeuvring..

2

-2. External force a c t i n g on the f l u i d .

Consider an incompressible and i n v i s c i d f l u i d which i s at r e s t , f p r t < t ^ , with respect to a Cartesian coordinate system (x,y,2). Through

t h i s f l u i d s t a r t s to move at t = t ^ an "external" f o r c e f i e l d F = F(x,y,z,t) per u n i t of volume. The f i e l e , i s switched o f f a,t t = t ^ .

In the l i n e a r i z e d theory we have the f o l l o w i n g equations of motion

l ^ - i g r a a p H - i i , (2.1)

where v = v(x>y,2,t) i s the v e l o c i t y of the f l u i d , p i t s density and p = p(x,y,z,t) i t s pressure. Taking divergence of both sides of (2.1) we obtain

N2 ^2 ^.2 ÔF

A p = (--% ^ + p . (-^ + + , (2.2) dx ây ôz ôx ôy ôz

where F , F and F are the components of F. Hence outside the region where x^ y z * o -the external forces are applied, -the pressure i s a harmonic function.

We now s p e c i a l i z e to the case that the f i e l d F i s reduced to a s i n g u l a r force f i e l d . The point of a p p l i c a t i o n Q (^,7),^) of t h i s f i e l d moves along a l i n e I ( f i g . 2.T) with the parameter representation

e=:S(t), ri Ti(t), Ç ^ ^ ( t ) , (2.3)

with the v e l o c i t y

v = ( i ( t ) , n ( t ) , iit)) , |v| . V ^ ( f + ^ + f ) ^ , (2.4)

where we assume that ^, r\ and t, are continuous functions of time and do not vanish simultaneously. The singular f o r c e f i e l d i s described by

î = ( f ^ ( t ) , f ( t ) , fjt))ô(x-^(t)) &(y-ii(t)) ô(z-^(t)) , (2.5)

the symbol 5 denotes the d e l t a f u n c t i o n of D i r a c

Substitut:-'.g (2.5) i n t o (2.2) and s o l v i n g the equation f o r p, y i e l d s the dipole f i e l d f o r the pressure.

p = •r(t){K<(t)) + f (t)(y-Ti(t)) + f ( t ) ( z - ^ ( t ) ) X .V Zl 3 E (2.6)

where R i s the distance between the point Q and the point (x^y^z) where we c a l c u l a t e the pressure. The s o l u t i o n (2.6) i s unique because we t a c i t l y assumed that p vanishes f o r E <» and that i t s s i n g u l a r i t y at E = 0 i s of the type we used.

The external force (2.5), shorthand f o r external singular force f i e l d , caji be decomposed i n t o two parts o f which one i s perpendicular to the

v e l o c i t y V and the other one i s i n the d5.rcction of V. The influence of the t o t a l force can be found by super imposing the influences o f i t s both component

F i r s t tho component of F perpendicula-^ to V y i e l d s the usual pressi-u-e d i p o l e of the theory of l i f t i n g surfaces. I t i s w e l l known that the v e l o c i t i e s induced by t h i s pressure dipole can be represented by elementary horseshoe vortex configurations

However, the second component o f F gives d i f f e r e n t r e s u l t s and we w i l l consider i n the f o l l o w i n g only forces i n the d i r e c t i o n of t h e i r motion ( f i g . 2.1).

X

Fig-. 2.1 . An external force F i n the d i r e c t i o n of i t s v e l o c i t y V.

We now investigate the v e l o c i t y f i e l d induced by t h i s f o r c e . F i r s t we describe t h i s flow and then prove that i t s a t i s f i e s our equations. I t w i l l consist of three d i f f e r e n t parts which have to be superimposed.

J+

-F ( t ) / p V ( t ) , t ^ < t < t ^ . (2-7)

F = ( f -!- f + f )^ i s the magnitude o f the f o r c e reckoned posn;ive when X y z ' ^

F points i n the d i r e c t i o n of V.

The second part i s induced by sources d i s t r i b u t e d along the path 1 which have the strength per u n i t of length

'TX-éiM)

,

Mhen F ( t ^ ) ^ 0 and V(t^) ^ 0 then wo have a s t a r t i n g source of strength

- F ( t ^ ) / p V ( t ) , (2.9)

when at t = t , the time where the force i s switched o f f , F ( t ) ^ 0 and

e e V(t^) 0, wc have an "ending" source of strength

F ( t ^ ) / p V ( t ^ ) . (2.10)

The t h i r d part consists of a flow i n the d i r e c t "neighbourhood" of the l i n e I. Everywhere i n space the v e l o c i t y of t h i s part, i s zero except at I where i t i s i n f i n i t e and tangent to I. VJhen we consider a plane which cuts I perpendicularly the transport of f l u i d per u n i t of time, has the value

F(-r)/pV(T) , t ^ < T < t ^ (2.11)

i n the d i r e c t i o n of 7(1") at the moment t = T. Hence t h i s v e l o c i t y f i e l d has the character of a d e l t a f u n c t i o n o f Direac.

V?e now show that t h i s flow s a t i s f i e s the conservation o f mass p r i n c i p l e . Consider a closed surface S which cuts the l i n e I at two places, corresponding to t = and t = , l'l < Tg- When we c a l c u l a t e tho amount of f l u i d entci-in^-^ and leaving S we need only to consider the c o n t r i b u t i o n by the part of I i n s i d e S. At the places TI and we have the t o t a l i n f l u x (2.11)

The sources on I f o r i"i < x < however d e l i v e r an amount of f l u i d which amounts to (2.8) 1

^ ( I f l l ) v ( . ) , . . - l c

which passes through S. However because (2.13) equals (2.12) thore i s no net outflow through S.

Next we i n v e s t i g a t e i f the equations of motion are s a t i s f i e d . To t h i s end

\JG compare the pressin-e f i e l d which i s caused by the f o r c e F given by (2.6) and

the pressure f i e l d induced by the three parts which describe the f l u i d motion. We use the l i n e a r i z e d v e r s i o n of B e r n o u l l i ' s equation which has the form

P = -P 1^ -^(xjyjZjt) , (2.11;)

v/here 0 represents the v e l o c i t y p o t e n t i a l o f the flow,. Hence wo have to c a l c u l a t e the time d e r i v a t i v e ôo/ôt. In order to s i m p l i f y these calculations" we consider a new coordinate system 'x,yj'z at r e s t with respect to tho f l u i d ^ of which the o r i g i n 0 coincides with Q, at the moment t , while the y axis i s along F(t}, ( f i g . 2.2).

Consider the increment of our c o n f i g u r a t i o n when time increases from t towards t + At. Then we have the f o l l o w i n g increase per second of the p o t e n t i a l at the point x , y j Z ,

-/} (x,y,z)

6 -ÔO ,. 1 T— = l i m TT At-.0 Fft+At) J+«p V(t+At) R(t+Aty 4ïtp V(t) E ( t ) t+At 1 V ( T ) dT ^ V ( T ) ^ R ( T ) 1 _JL 1 F.E 4np ' (2.15)

where R i s the veetor p o i n t i n g from Q towards x,,yjZ.

The r e s u l t (2.15) i s i n agreement with (2.6). When however at each time the pressures are equal, the f l u i d motions, which s t a r t from rest at t = t ^ , have to be equal too.

The flow f i e l d we have found i n t h i s way can be considered as a

3- The non-stationary actuator d i s c .

We s t a r t with an actuator d i s c , c o n s i s t i n g o f a c i r c u l a r region

2 2 2

y -V -L < a perpendicular to the x a x i s , which moves with constant v e l o c i t y V i n the p o s i t i v e x d i r e c t i o n . Over t h i s area we assume a pressure jump q = q ( t ) , independent o f y and z, which i s the d i f f e r e n c e between the pressure j u s t i n f r o n t of the d i s c and j u s t behind the d i s c . Then wo can consider each elementary area dO o f the d i s c as a force of strength q(t)dO which acts i n the d i r e c t i o n of i t s motion. Hence the theory o f section 2 i s a p p l i c a b l e , and we can determine the t o t a l induced v e l o c i t y f i e l d qf t h i s actuator d i s c by i n t e g r a t i o n of the Green f u n c t i o n of the previous s e c t i o n over the d i s c . We remark that when'the d i s c , gives a thrust i n the d i r e c t i o n o f i t s motion, the e x t e m a l force on the f l u i d i s i n the opposite d i r e c t i o n .

Consider f o r instance the f o l l o w i n g case. Th_e pressure jump i s switched on at t = t at x = 0 and has the value q . A f t e r t h i s i t remains constant up to

O "^0

t = t and i s switched o f f at that time. The v e l o c i t y f i e l d at a time t < t < t

e 0 c can be described as follows. A t t h e place x = 0 we have a layer of sources of

strength q /pV per u n i t of area and at x = Vt we have a l a y e r of sinks of

" , 2 2 2

strength q^/pV, both layers f o r y -i- z ^ a . Between these two layers and within the c y l i n d e r determined by them, we have a p a r a l l e l flow o f i n t e n s i t y q^/pV i n the negative x d i r e c t i o n . The superposition of the v e l o c i t y f i e l d s induced by the source and sink layers and the j u s t mentioned p a r a l l e l f l o w ' i s f r e e of divergence as i s e a s i l y seen. For values o f t > t ^ , hence when the

y 4V

X

= v(t J

8

-actuator disc i s switched o f f , the v e l o c i t y f i e l d i s independent of time and i s drawn i n f i g u r e 2.1.

Next we discuss the case of an actuator d i s c c o n s i s t i n g again of a c i r c u l a r region of radius a, the centre of which moves along a c i r c l e C with radius b. This c i r c l e C l i e s i n the x,z plane and i t s centre coincides with the o r i g i n 0 of the coordinate system. During i t s motion the d i s c i s perpendicular to C. The p o s i t i o n of the d i s c i s described by the angle a and we assume that a = cot.

The pressure jump q i s again assumed to be constant ovei.' the area, independent of time and i s switched on at a = 0, t = t ^ . Then the f o l l o w i n g

happens. F i r s t we have a l a y e r of s t a r t i n g sources of strength q/'l^^-"

2 2 Â

(r = (x + z )^) per u n i t of area, over the c i r c u l a r region at the place a = 0. When the d i s c i s moving i t c a r r i e s with i t a layer of sinks of strength q/p'o^r. The d i s c leaves behind a flow tangent to the c i r c u l a r paths of the points of the d i s c i n the negative a d i r e c t i o n of iragnitiide v := q/p:i)r.

Now consider the f o l l o w i n g experiment. Ifcen the d i s c a r r i v e s at i t s

s t a r t i n g point a - 2n, i t i s switched o f f . Then i t sheds a layer o f sinks which exactly compensates the l a y e r of s t a r t i n g sources. Hence a l l that remains i s a c i r c u l a r flow of strength v, w i t h i n the torus passed through by the

X

a cx)t

F i g . 3.2. The torus passed through by the actuator d i s c

actuator d i s c .

It i s c l e a r that when an actuator d i s c changes i t s strength more

continuously i n s t e a d of switching on and off,, the shed sources and sinks are d i s t r i b u t e d continuously i n space. A l s o the l o c a l v e l o c i t y f i e l d then becomes a function of the p o s i t i o n of the d i s c .

References.

1. Morgan, B.W. Some r e s u l t s from the inverse problem of the annular

a i r f o i l and ducted p r o p e l l e r , Journal of Ship Research, March 1969*

2. Wu, T.Y. Flow through a h e a v i l y loaded actuator d i s c ,

S c h i f f s technik 9, Heft k7•

3. Sparenberg, J.A. A p p l i c a t i o n of l i f t i n g surface theory to ship screws.

K o n i n k l i j k e Nederlandse Aliademie van Wetenschappen. Series B,