t?
SEP.ARCH1E
Lab.
y.
.epbouik4t
Technische Hogeschool "r
S ¿-Jyj
Deift
On sculling propulsion by two elastically coupled profiles by J.A. Sparenberg and J. de Vries.
Summary
The sculling propulsion of two profiles which are coupled by an elastic hinge is considered. The leading profile carries out a
prescribed motion. The motion of the second profile is induced by the hinge and by the fluid pressures. This profile regains part of the kinetic energy left behind by the first one.
j Introduction
With respect to the propulsion of some species of fishes, inter-action can be expected of a protruding dorsal fin with the caudal fin.
The question arises to which extent such an interaction can be
favourable. In this paper we look at the problem more from the tech-nical point of view. We investigate if by means of two interacting wings a high efficiency propulsion device can be constructed. An ana-lytical discussion of such a device, for the two dimensional case
is given in [il.
In the following we revise the two dimensional problem and
for-mulate it in a different way. We use from the beginning the fact that
behind the second profile, in the linearized theory, no free vorticity is present when the configuration is optimum. The advantage of this
formulation is that no integrals over infinite regions have to be calculated. It is found that the numerical results obtained here
are in agreement with those following from the theory of [i].
In this paper we do not consider the stability of the motion of the second profile, we do not check if flutter can occur.
Several types of joints can be thought of. Here we confine our-selves to the case of two flat rigid profiles coupled by an elastic
hinge. The leading profile carries out a prescribed motion.
The motion of the second profile is passive, it is induced by the hinge and by pressures in the fluid. When the hinge is "infinitely stiff" the system can be considered as one slotted profile. This can be investigated rather simply because the number of essential para-meters is only three. When we have an elastic hinge the number of essential parameters increases to seven. This means that only to some extent insight in the working of this configuration can be obtained. The amount of numerical work involved to give a complete description of the possible motions in this case is large even for a nowadays computer. So we have restricted ourselves by considering selected areas of the parameters.
The construction of good working profile systems of the type under consideration here, has some diffículties. It is favourable when the second profile has not a too small chord length and is not too far
behind the first one. Further with respect to the stability of the flow it is desirable that the suction forces at the leading
edges of the profiles are small.
r
2It goes without saying that the efficiency has to be as large as
possible. By numerical means it is shown however,that for the region of parameters considered, some of these conditions are conflicting.
Larger values of the chord length of the second profile and low values of the energy losses, are coupled to high suction forces.
Small energy losses can be obtained more easily for lower values of
the reduced frequency. However then the second profile becomes
situated further down stream of the first one and it becomes ques-tionable if the free vorticity shed by the first profile behaves
as calculated by our linearized theory. So compromises have to be
L)
the numbersT5o
b. J F.z
J
F.(x)dx, j, 1,2. a. J-3-Formulation of the problem
We have a Cartesian coordinate system (x,y) embedded in an inviscid and incompressible fluid which has an incoming velocity U in the positive x direction. The problem is
S,
_Ji
77>o
Fig. 2.1. Two flat, infinitely thin profiles. two dimensional. For a1<x<b1 and for a2<x<b2, (b1 <a2) we have two flat and infinitely thin profiles, which can move in
the neighbourhood of the x axis. The profiles are coupled by two rigid arms (b1,s1) and (s1,a2) which are connected to each
other by means of a linear elastic hinge at xs1. The theory
will be linearized so we assume the vorticity of the profiles and the free vorticity to be at the x axis.
To start with we assume the motion of the two profiles to be prescribed independently of each other
y = h. (x,t) = (A. + A.2x)cos ut + (A.3 + A.4x)sin wt,
a. <x<b. ,j = 1,2. (2,1)
We omit constants in (2,1) because it is proved in [i], that these have to be equal in the case of least energy losses, hence both can be taken equal to zero.
For the normal velocity at the two profiles we find from (2,1)
v.(x,t) =v.1(x)cos wt v.2(x)sin wt, a. <x<b., (2,2)
v.(x) w(A.3+ 32+A.4x), v.2(x) w(A.1 4+A.2x), a. <x<b..
(2,3)
The vorticity in the region a1 <x<b2 is divided into three parts
F. (x.t) = F. (x)cos uit- F2(x)sin uit,
j I ,2,3
with a. <x<b. for j = 1,2 and b1 <x<a2 for j =3. We introduce
(2,4)
-4--When behind the two profiles, hence for x>b2 no free vor-ticity is present, we can write the condition that the flow passes along the profiles
2 b a2
F()
dJ
(c-x)
g(x),
a.<x<b.
-d=-2u
v.(x)-(g-x) ,i= 1,2.nI a
(2,6) nThe assumption of no free vorticity for x>b2 is not correct for arbitrary values of = l,2;i= 1,2,3,4,U and , later on
however we will determine these quantities such that indeed (2,6)
is valid.
First we direct our attention to the vorticity F31(x) and r32(x) in between the two profiles. Because for b1 <x<a2 no
boundary is present, the vorticity shed by the first profile is simply transported by the fluid, we find
F3k(x)
=_(_J)k
(3-k) cas (x-b1) + (_I)kFlk sin (x-b1)},k= 1,2,
(2,7)
the constants F11 and F12 being defined in (2,5). Using
(2,7) we can write the right hand side of (2,6) as
g.1(x) =-2uw(A. + A. +A. x)-F Q1(x) +F12Q2(x), w j2 j4 11 g2(x) =2uw(A. - A. +A. x)-F 1Q2(x)-F12Q1(x) ji w j4 j2 I where (2,8) (2,9) a2 w a w sin (-b1) 2 cas (-b1)
Q1(x)r-
J
1(c-x)
d,
(2,10) U b1 (e-x) d, Q2(x) =-U b1 with a.<x<b., j = 1,2.The solution of integral equations of the type (2,6) is straight forward [2] and will not be discussed here. We obtain
2 b Vzk() (1 a2 sin (-b1)d Fik(x) =_ W(x){-2Tr E
f
u ' aW()(-x)
Fikb1W()(-x)
J
d} , a.<x<b.; j,k=1,2,
b1W()(x)
where W(x) -Ix-b1 (x-b2I Ix-a11 jx-a2In the derivation of the last two terms of (2,11) we have used the relation
2 b,
Q()
a2 sin Wi a
W()(-x)
d=-
JW()(-x)
dfl, a. <x<b., j 1,2, (2,13)and the analogous one for Q2(), where we have to replace the "sin" in (2,13) by "cos".
Integration of (2,11) for j=1, k=J,2 with respect to x over the interval a1 <x<b1 yields two linear equations for the unknown
constants F1 and F12, from which they can be determined.
3. The condition on the free vorticity
We now discuss the condition that with respect to our linear theory, no free vorticity is behind the second profile so that our integral equations (2,11) have physical meaning. This condition is simply that the total vorticity at the interval (a1,b2) is
constant. In this paper we do not consider the case that a mean lift is produced by the two profIles hence the value zero for this constant is appropriate. In formula
b1 a2 b2
J
F1(x,t) dx+ J
F3(x,t) dx+ J
F2(x,t) dtO
a1 b1 a2
Using (2,4),(2,5) and (2,7) this condition can be replaced by
(2, II)
(2, 12)
(3,1)
sin (a2_b1)+(_l)k
F1(3k) cos (a2_b1)+(_l)k F2(3_k) =o, k=1,2.
-6-In order to treat our problem system atically, we introduce
the basic vorticity distributions Yflk(x); n,j,= 1,2; k= ,2,3,4, by
1nk
(x)r. (x)
(3,3)fo r
A
=6
6k r= 1,2 ; m 1,2,3,4,
rm in m
where 6 is the Kronecker symbol, 6rn = 1, r = n, 6 = O, r * n.
rn
rn
It is not difficult to show that the Iflkzi(x) satisfy the
following symmetry relations
3i(3-k) =
1)1ik(
' Y4j(3_k)(x) = (_1)kY2jk(X); k= 1,2,(3,5)
hence of the 32 functions yk.Q(x) there are 16 independent ones. Also we introduce the basic vorticity integrals by
b. q
J
y (x) dx , q=O,1,2, 1nkj nkj2. a. Jwhich, with respect to the lower indices, posses the same
symmetry properties as the YnkJ2(X) in (3,5).
From (3,3) it follows that the functions 1nkj(x) also satisfy the
integral equations (2,6), for any choice of; n,= 1,2; k= 1,2,3,4, with Ars 6rn 6sk' r = 1,2; s = I ,2,3,4.Hence for any set of values
of A we can write nk
2 4
Z Z A
n=1 k=I
nkj(X)
; j,1L=1,2.The condition (3,2) then changes into
2 4 Z Z A {-(-1) y cos
-
(a b ) + - 1nk1(3-) sin (a2-b1) + nk nkl2. U 2 1n1 k=1
(3,4) (3,6) (3,7) - (-1)nk2° ,
i=l,2. (3,8)-7-4. Pressure distributions and the motion of the second profile.
The pressure jump across a profile can be written as
[p(x,t)]=p(x,t)-p(x,t) =-pU(u-u)-p
(-p) ,
(4,1)where the + and - sides are denoted in figure 2.1 and (p(x,t)
is the velocity potential of the disturbance velocity field (u,v). We introduce the basic pressure jumps Pnkj(x); j = 1,2, for
which the coefficients A follow from (3,4), by
[p(x,t)]pkl(x) c°s
+ pflki2(x) sin wt, a<x<b 1,2.(4,2)From our previous definitions we find by (4,1)
X .(x) = (_1)33P{U1nklj(x) + (-1)3w
J
1flkI(3)()d}al <x<b1,
nk I j a b7Pnk2j(X) =P{(_1)33Unk2j(x)
wf
Ynk2(3_j)()d}a2 <x <b2,
X 1 (4,3) n,j = 1,2; k= 1,2,3,4.In the general case for arbitrary A we find nk
[p(x,t)]=p1(x) cos wt + p2(x) sin
wt, a<x<b,
whe re
2 4
p.(x)
Z Z An = I k = ¡ nk nkfj
(x) ,
j
1,2.We now introduce the basic pressure integrals by
b (x) dx , q = 0,1. a q nk2j (4,4) (4,5) (4,6)
and
-8--These integrals are triple integrals, however by a partial integration with respect to the variable boundary x in (4,3) and (4,4) they can be written as double integrals. We find
q pw q+1 q+1 o
k1j =(_1)JPUYnklj
(q+1)nk1
(3.)-(b1) 1nkI (3_j)}pu q+1 q+1 o
nk2j =_(_1)3pU1k2j + (q+1) {1flk2(3_)_(a2) nk2(3-j)
We next consider the motion of the second profile which has to satisfy the geometrical relation h1(s1,t) =h2(s1,t) for
4/
'
N 7'-' ¿t4>o
4
> )c
a, ,Fig. 4.1. External force K1 and moment M1 needed at
internal force K2 and moment M2 at second profile,
x=s1.
all t. By (2.1) this becomes
A21+A22s1=A11+Al2s1 ,A23+A24s1=A13+A14s1.
(4,10)The dynamic properties of this profile are determined by its centre of gravity at xs, by its mass m and by its moment of inertia I
around the centre of gravity. It experiences by the hinge at x=s1 a force K2 and a moment M2, of which the positive directions are
denoted in figure 4.1. Because our problem is two dimensional,m, I, K2 and M2 are per unit of span.
o
(4,8)
and
-9-The moment M2 has the value
M2(t) =-v(t) -{(A22-Al2) cos wt + (A24-A14) sin wt},
where y is the strength of the hinge per unit of span and c
the angle between the first and the second profile. Then we write the two equations of motion of the second profile as
b m 2 h2(s,t) K2(t)- J[p(x,t)] dx a2 (4,11) (4,12) 3 b2 2 h2(s,t) =M2(t)_K2(t)(s_si)_
J [p(x,t)]
(xs)
dx. (4,13) 3t x a2In (4,12) and (4,13) we substitute (2,1), (4,2), the related
pressure integrals and (4,11). Next we compare in each of the equations the factors of cos wt and sin wt and eliminate the
unknown factors of the cosine and sine components of K2. Using the abbreviation
o
Snk2=Pnk2_S1Pnk2
; £= 1,2,we obtain the following two equations
(mw2 (s - ) +S2121)A21 +
(2{I
+ms + S2221 +y)A22 +S2321A23 +z I
+
S2421A24-S1121A11+(-S1221+v)Al2-S1321A13-S1421A14, (4, 15)
and
S2122A21 + S2222A22 + (-m2(s-s1) + S2322)A23 +
(2{I
ms (s-s1)} ++
S2422+y)A24=-S1122A11-S1222Al2-51322A13+(-51422+y)A14 (4,16)
The equations (4,15), (4,16) together with (4,10)
are four equations for the four unknowns A21,... ,A24.
lo
-Next a short survey is given of the way in which the motion of the second profile can be calculated. We assume to be prescribed some values p,U,AIk(k 1,2,3,4), a1,b1,a2,b2,v,
I,m,s1 and s. We now can calculate the double integrals
(k= I,..,8;Q= 1,..,6) presented in table 4.1. Each of the integrals in column I has to be combined with each of the functions in t behind the dashed line in the same block. These combinations have the index k1,..,8.
Analogously each of the integrals in column II has to be
com-bined with each of the functions in II behind the dashed line, in the same block. These combinations have the index £ 1,..,6.
Then each of the combinations of column I has to be combined with each of the combinations of block II. In this way we obtain 48 double integrals denoted by
Table 4.1. The double integrals
; k= I,..,8; £ 1,..,6.
From the values of the double integrals follow the numbers 'nkj3,6 which by (3,3),(3,4) and (2,11) are linear
combina-tions of the
ki The numbers then follow from (4,8) and
(4,9).
We now calculate the coefficients
A2k(k= 1,..,4)(2,1) of
the motion of the second profile by means of the equations
(4,10), (4,15) and (4,16). I k II b
-
i 1 b 1J
dx W(x) a1 I x 2J
d a1 2W()(-x)'
b2 J dx I 1 4 5 b2J
d I i 3 W(x) a 2 I x 2 x 6 a2W()(-x)
-
---=
a2 Jdx W(x) x 7 a2Jd
Icos x 5Next we have to use the vorticity conditions (3,8) in order to
improve the chosen values a2 and b2. Substituting the chosen and
the calculated values into the left hand sides (2. = 1,2) of (3,8)
we find in general some values r1 *0 and r2 *0. We now repeat
the foregoing procedure with some slightly different values of a2 and b2 and determine the change of r1 and r2. Then by an iteration procedure based on a gradient method the left hand sides of (3,8) are brought to zero.
In this way the position of the leading edge a2 of the se-cond profile, its chord length (b2-a2) and its motion can be calculated.
5. The thrust, external lateral force
and external moment.
The mean value with respect to time of the total thrust
delivered by the two profiles can be divided into parts as follows
2 2
T=
T=Z (T
4-T)
2.=1 £=1 p,2. s,2.
where T2. is the mean value of the thrust of profile
i(2.=1,2),T2.
the contribution to T2. by the pressure jump across the profiles and T2. the contribution to T2. by the suction forces
at the
leading edges. Using the basic
pressure integrals (4,7) we find 21T w b2. + ah2. T
J
J [p(x,t)]
(x,t)dx dt - ax o a2.24
E
A k2.2nk2.l +A2.4P°k2) - 2 nn1 k1
ir 2The suction forces are - pC2. (t); 2.= 1,2, where
2 4
C2.(t) =lim
{Ix-a
r'2.(x,t)}=Ak(C
k2.1 cos wt-C sin wt), nk 2.2a2.x
n1 k=1(53)
(5,1)
12
-Hence the mean value of the suction force at the profile 2. becomes
w 2 2 4
w It
S
C(t) dt=
( ( Ank Cflk2.j)}.o j=1 n1 k=1
Another way of calculating the mean value T of the total thrust follows from the fact that the lost kinetic energy per period of the motion is o(), where c is the order of the magnitude of the amplitudes A..(2,1), as will be discussed in
section 6. Because the useful work U T per period of the motion is 0(c2) it follows that the power supplied to the profiles equals U T. The amount of lost energy is outside the field of vision of the linear theory. The justness of the mentioned equality can be proved also analytically. Besides it was
con-firmed by our numerical calculations, where it was used as a check on the computer program.
For this method we calculate the amount of work needed to keep the profiles moving. Consider the profiles carrying out
the motion under consideration,however without mass and without a connection by a hinge to each other. For the fluid there is no difference between this situation and the original one, hence the mean value of the external work to move the profiles will be the same. The work over one period of motion amounts to
b
A=
ç
2 J[p(x,t)] 2. (x,t) dx dt = o 2.=1 a2. 2 2 4 = n Z ZAk{Aj3P°k2.l +A2.4pk2.l)_(A2.Ip°k2.2 +A2.2p'k2.2)}. (5,6)
Hence we find for the mean value of the total thrust
=I
fl=I K1
wA T = 2iitJ (5,5) (5,7) with Ck2.. 11m {/ x-a; Yflk2.j)L (5,4) a2. ± x13
-We next assume that the motion of the two profiles is maintained by an external lateral force K1(t) and an external
moment M1(t) acting at the leading edge of the first profile
(figure 4.1). We write
K1(t) =K1 cos wt+K15 sin wt ; M1(t) M1 cos wt+M1 sin uit. (5,8)
The values of KIc,KIs,MIc and M can be derived easily from
Is
first (4,6),(4,7),(4,12) and (4,13). When we assume that the
2 4 I o M = Z Z A k I a1p kIl Ic n n n1 k=1 kI 2 - mw (s1-a1)(A2 +A s )-v(A22-Al2). I
22z
2 4 o M=Z
Z A{p1
-ap
° +n=1 k=I nkI2 1 nkl2 (sJ-al)pk22}+
Is nk
2
- mui (s a )(A
+A
s )-v(A-A ).
I 1 23
24z
24 14Lateron we give numerical values of the amplitudes of K1 (t)
and M1(t) defined by
K1
= c
+K2)
, M = (MFrom the fact that the lost kinetic energy is O(e3) it follows again (5,7) that
-aI)p°k2I} +
(5,11)
(5,12)
(5, 1 3) profile has no mass, we find
2 4 K = Z Z A
knk11
Ic nn1
=I nk12 (A+A
s ) 21 22 z (5,9) 2 4 K = Z Z A k Is n n1 k=1 o onk12nk22
mw(A
+A
s ) 2324z
(5, 10)14
-2n/w
T=
J
{K1(t) (0,t) +M1(t) 2h1(O,t)dt
o ax2t
which yields a check for the calculated values of K1(t)
and M1(t).
6. The efficiency
It follows from (3,1) that no free vorticity is taken into account for x>b2. This is correct with respect to a linearized
theory. However more realisticly for x>b2, there are two vortex sheets at a distance of 0(c), which are of opposite strength (c is the order of the magnitude of A..(2,1)).
The velocity induced by these sheets at the two profiles, is 0(c2) and can therefore be neglected.
In order to calculate the kinetic energy belonging to these sheets we have to find their
strength as well as their shape. Their shape can be calculated upto and including an accuracy of O(e) by the linear theory. This is sufficient for our purpose.
First we calculate the disturbance velocity in the y direc-tion in between the two profiles, by (2,4),(2,7) and (2,10)
2 b -v(x,o,t) {
()
d-f12Q2(x) +F11Q1(x)} cos wt+ a 2 bF2()
def 2fl (c-x) d-F12Q1 (x)-F1 1Q2(x)} sin wt y1 (x) cos wt i=1 a + 2(x) sin wt, (6,1)for b1 <x<a2. The integrals in (6,1) are by (2,11)
double
inte-grals. By enlarging the domain of
x in the denominator of the integrals from the real axis to the complex domain it is
not dif-ficult to show that these integrals
can be written as 2 b
Fk()
d W(x) 2 b_2k()_FlkQl()_(_I)krI(3k)Q2()
J f
(e-x)
U =1 aW()(-x)
(6,2)for k=1,2 and b
<x<a2.
r
Next we consider the integrals at the right hand side of (6,2) over the functions Q,()(2,lO). For instance
2 b a2 sin ) 2 b
dd
d = b1 (e-x) (c-x)- (_)}
w()
=1 a2 (n-b1)wit1
U -- U b1 (r-x)W()
}d , b1 <x<a2.This result differs from (2,13) because here x is in between the two profiles.
In this way we find the following expression for
k(x),k= 1,2, in (6,1) for b1 <x<a2, 2 b
vk()
a2 {(1)(3-k) . w Flksln -(rrb1)+ W(x)[ uW() (e-x)
d + W() (n-x) £=1 a cos (-b1)}d]
(6,3) (6,4)where the integrals are one dimensional.
The vorticity shed by the traíling edge of the first profile is transported by the fluid with a velocity U in the x direction and with a velocity v(x,o,t)(6,1) in the y direction. Then it reaches the lea-ding edge of the second profile. For a2<x<b2 the velocity in the
y direction equals the velocity (2,2)(j=2) of the second profile with an accuracy of 0(c). Hence theoretically the vorticity does not change its distance to the second profile when floating along it.
Having passed the second profile it leaves its trailing edge at x=b2,
where the second vortex sheet starts. The two sheets being at a
small distance of 0(c), will keep because of our linear theory, their distance upto 0(c2).
The distance Ay(x,t)y1(x,t)-y2(x,t), x>b2, where y(x,t)(.z.=1,2)
is the y coordinate of the vortex sheet shed by the corresponding profile, is found to be
16 -b a a2 (a2,t
+----)
+-¿y(x,t)=h1(b1,t-
+_!)_1
x 2 IJv(
ot
+) d
U U 2 U U U''
U U def 1yIcos w
+ '2 sin w(t-, b2<X.
(6,5)
The numbers y1 and y2 follow from a straightforward substitution
of (2,1), (6,1) and (6,4) into (6,5).
The strength of the vorticity shed by the first profile can be
written as
wb wb
y(x,t)
=r31(x) cos wt-r32(x) sin wt= {F12 cos sin _!}cos w(t- ) +wb wb
xdef
x+ {-F12
sin
jj--+ F11 cossin w(t-
)-y cos w(t- +
(6,6)
X+ sin
w(t-for x>b1.
Next we calculate the mean value of the kinetic energy E lost per unit of time. This energy is, at large distances downstream of the
two profiles, theoretically confined to the region in between the two vortex sheets. It reads
w
E=
p J
ï2(x,O)ly(x,O)dx. 2n
o
Using (6,5) and (6,6) we carry out the integration and find
1 2 2
+y2y2)2
E= (Ay +
y)
2{ +2 311
+
The efficiency rì becomes
=UT(UT+E)1=1-O().
(6,9)
(6,7)
- 17
-o
h1 (x, t) = cos wt + B1 x cos (wt +
hence we have the following 8 parameters which can be chosen
F =
, B, s1, m, I, y, sr 2rr z
The order relation follows from the fact that TO(2)
and E=O(3),
as can be seen by (5,7) and (6,8) respectively.In calculating E we have by using (6,4) to cope only with double integrals.
7. Presentation of numerical results
Our numerical results are based on the following fixed values
p=I,U=1,a1=O,b1=I.
(7,1)The motion of the first profile will be given here by
A shortcoming of (7,2) is that it cannot
represent a purely pitching motion around the leading edge, this has to be considered
separate-Ly. The problem described above will be called problem I. The computer program gives with respect to problem I a number of results which are
scaled to a total thrust T1.
The results are the following. The position a2 of the leading edge
of the second profile and its chord length (b2-a2). The motion of the two profiles are described by
h(x,t) =A cos(wt+a)
+B
x cos(wt+) , £1,2.
(7,4)
where A,B,cz2 and
2 are calculated. The values a1 =0 and are already known from (7,2). It is clear that when the quantities
used in (7,4) are known, also the previously used A.
(2,1) can be
easily calculated.
Further the program gives the contribution T1 of the first
pro-file to the total thrust T = I, the quotient Z. of the magnitude
of the suction force T .(j = 1,2) and thrust T., hence Z.
=T ./T. for the
SJ
j
j
sjj
first rofi1e (
= ) for th second. one ( 2)
(7,2)
18
-Also it gives the lost kinetic energy E per unit of time. At last it calculates the amplitude of the external lateral force
K1 and of the external moment M1 at the nose of the first
profile, (5,9),...,(5,13), needed to maintain the
motion.-Now suppose we want to calculate the performance of a system of two profiles of which, with respect to an arbitrary system of units, the following parameters are given,
p, U, a1 = Q, b1 = , s1, m, I, y, s
while the motion of the first profile is prescribed by
í1(x,t) cos t+''1x cos(t+''1).
This is called problem II.
The question is how can we find the defining quantities of problem I which correspond to those of problem II and what are
the factors by which the results of problem I have to be multi-plied in order to get the results of problem II.
It is not difficult to obtain by dimensional considerations the following scheme. The defining parameters of the correspon-ding problem I are
B , s1 m s I y z
pZ
pZU
£The quantities (with ) which are asked in problem II
can be ex-pressed in the results of problem I by
a2=a2, 2=b2, A2=
A1 A2, B2=B2, a2a2,
22'
A 2( )=T1,
. =Z. (j = 1,2) 2. A J J (7,5) (7,6) (7,7)19 -K1
-2'
M1K1=pUA1-,M1=p9.0 AIA-,
( ) __;_, fl{1+
E-1
={1 + 9. ATU
9. IIn the relations (7,8) we have used the fact that when we multiply the motion of the first profile (7,6) by a dimension-less quantity À, the force and the moment are multiplied
by X, the thrust and its parts
'
T.
by X2and the ener-by by An example of the use of the formules (7,7) and (7,8)is given at the end of section 10.
Finally we make a remark about equivalent sets of parameters. Suppose we have found a motion (7,4)(scaled for total thrust 1=1) of the profile system, which belongs to 8 chosen parameters (7,3). From this motion we calculate the coefficients A.
11
(2,1).
Now consider an other set of 8 parameters F=w/2îr,B',
,
s1, m*, I, v, s* where the last four ones have been
altered. Then it follows from (4,10), (4,15) and (4,16) that this
set of parameters yield the same motion if the 2 expressions.
- Al2*
A21w2m*(s -s1) +A22{_2(I*+s* m(s*+*},
and
-
A14v* _A23w2m*(s* -s ) +A {_w2(I*+s*m (s* _ ) +*},z 1 24 z
have the same value for the altered parameters as they have for the original ones m, I, y and s. This means that from one system
of profiles, we obtain a two parameter family of systems which have equivalent dynamic parameters and which carry out the same motion. Hence these systems have the same hydrodynamic properties, T= I,
ir, K1, M1, etc.
(7,8)
(7,9)
8. Numerical results, rigid connection
In this section we give some numerical results with respect to the performance of two rigidly connected profiles, which
to-gether form one "slotted" profile. We discuss the heaving motion and the pitching motion around the leading edge of the first profile.
First the heaving motion, represented in our graphs by drawn lines , dashed lines are used for the pitching motion. The
cal-culations are based on (7,1) and v=. The motions (7,4)( 1)
(scaled to total thrust T = 1) are in this case restricted to
A1 =A2, a1
a20 ,
B1B20.
In figure 8. 1 we give the position of the second profile. The first
profile is represented by the interval a1=0,b1 = 1, the second one
by (a2, b2).
20
-2. 3
.,.
Fig. 8. 1 Leading edge a2 and trailing edge b2 of
second profile,
leaving, - - - pitching.
Such a figure is also presented in [i], there however, due to a computational error, an incorrect dent occurred in the graph. We remark that the given slots (b1, a2) are the smallest possible
ones. There exists for each frequency a series of slotted profiles
of which the slots are about one, two or more wavelengthswider than.
the slots given here and which also satisfy the free vorticity
con-dition (3.1).
21
-In figure 8.2 we have drawn the values of A1 (7,4) (T= 1) and
the thrust contribution T1 of the first profile. It follows from
our calculations that T1 nearly equals the thrust produced by
a single profile of length 1 under the same conditions, hence
there is only a slight upstream influence of the second profile on the first one. Because we consider the heaving motion we have
T
=T
=O,T
p,i p,2 s,1 ' A, -B,
T, 0.2-B,
/
A,
/
2. Q _-_---O.2Ç0.$0
¿ç
i.zS-J.0
Fig. 8.2 Amplitude A1; thrust first profile T1;
heaving.
Amplitude B1; ,, ,, ,, T1; - - - pitching.
Figure 8.3 gives the ratio of the total length L={1 + (b2-a2)}
of the two parts of the slotted profile and the length L* of a non-slotted full profile which delivers the same thrust T = I by the same amplitude A1 and the same frequency as the slotted profile.
=T1andT52=T2=I-T1,Z1=Z2=1.
o. ô.S0 o7ç /.ao '.2S
Fig. 8.3 Ratio of total length L of slotted profile and L* of corresponding full profile, heaving,
hence the slotted profile has a smaller wetted area, possibly due to the occurrence of a second suction force at the leading edge of the second profile.
The kinetic energy E lost per unit of time by the slotted profile and the kinetic energy E* lost by the full profile
mentioned before, are drawn in figure 8.4. When A1 is multiplied by a parameter A it follows from the theory that T is multiplied by A2 both for the slotted and for the full profile, however E is multiplied by A3 and E* by A2. This means that for A suffi-ciently small the lost energy A3E of the slotted profile becomes smaller than A2E* of the full profile and hence its efficiency becomes better. When A3E>A2E*, it is likely that higher order terms are no longer negligible, hence the linearized theory is not valid. This happens in figure 8.4 over the whole range of frequencies(for A = 1). So the given results can most probably be applied only for values A1(fig.8.2) which are "sufficiently" smaller than those which belong to T = 1. In order to give some
information with regard to this we have drawn in figure 8.5 a relation between A1 and Fr such that A3E=A2E*. The values of
A are denoted along the curve, by which the corresponding values
of A2T=A2, A3E=A2E* can be calculated.
-o,LS Q,50
o;S
1,oOì,25
1,O
Fig. 8.4 Energy loss E,of slotted profile, E* of corresponding full profile, heaving, pitching.
i;,
= i.. s-
23-o,25
0.S0
û.7S
i-00
/.25
/.50
Fig. 8.5. Graph of A3E=À2E*, A1 heaving, B1 pitching.
Hence for values of A1 in some region below this graph it can be hoped that a linearized theory will have validity.
Next we consider the pitching motion. The calculations are again based on (7,1). The motions (7,4)(= I) (scaled to total thrust T= 1) are now restricted to
A1=A2=O,B=B2,1=2=O
In figure 8. I (dashed linies) we denote the values
of a2 and
b2 of the second profile for this case. In figure 8.2 are given the values of B1 and of the contribution T1 of the first profile to the total thrust T 1.
In figure 8.6 we give the ratio's Z1 and Z2 which show which part
of the thrust is generated by the suction forces. When Z.
> I (j = 1 2),
it means that the pressure forces at the corresponding part of the slotted profile act in the "wrong" x direction and have to be overcome by the suction forces, which is rather unfavourable.
In figures 8.3 and 8.4 we again compare quantities of the slotted profile with those of the corresponding full profile.
(8,2) 0.1/.
'I
0.3 Ào.1\
O.L-)o.$
O.'24
-The latter one carries out a pitching motion around its leading edge with the same value of B1 and of Fr as the slotted
profile. Its length L* is determined such that it also generates a thrust T= 1. At last we have drawn in figure 8.5 for the case
of the pitching motion the line X3E=À2E*, for which an analogous
discussion holds as has been given for the heaving motion, when A1 is replaced by B1.
A good working propulsion system of two profiles coupled rigidly or by an elastic hinge has to have, more or less, the
following properties:
Small energy loss E.
Small suction forces, hence small values of Z1 and Z2.
Well balanced distribution of thrust over the two profiles, hence not too small values of ( b2a2) and T1.
A slot which is not too wide, hence (a2b1) not too large.
It is clear from our results that a system of two profiles rigidly coupled to each other, will not be very effective as a practical propulsion device for the two types of motion we considered. The chord length of the second profile is small (figure 8.1) with respect to the chord length of the first one. Even in the case of pitching around the leading edge of the first profile, the thrust is delivered mostly by the suction force of the second profile (figures 8.2 and
(8.6)). From our rather extensive calculations it follows that also for more general motions of the slotted profile, either the chord length of the second profile or its Suction force or both are unfavourable.
In the following sections we direct our attention to profiles coupled by an elastic hinge.
25
-9. Elastic hinge, miscellaneous numerical results.
The calculations in the case of the elastic hinge are based on (7,1) and the eight parameters denoted in (7,3). From two of these parameters namely s1 and s it is difficult to have
a feeling in advance about their meaning with respect to the second profile. Before a calculation has been carried out the values of a2 and b2 are still unknown, hence it is not known
if a chosen x coordinate s1 of the hinge or s of the centre
of gravity will be at the second profile or outside of it.
To this end we will introduce the relative position Rs1 of the hinge and Rs of the centre of gravity.
For instance
Rs1 is
denoted by a number a.b. The digita can have the values 0 or I or 2, it indicates that the hinge is at
the first profile, or in between the two profiles or at the
second profile, respectively. The number b determines the fraction
s1/(b1-a1)=s1=O.bfora=0,(s1-b1)/(a2-b1)--
0,bforal
and (s1-a2)/(b2-a2) =0.b for a=2. Analogously the relative position of the centre of gravity is given by a number RsSo Rs
=2.6 means that the centre of gravity is at the secondprofile while (s-a2)/(b2-a2) =0.6.
Besides the remark at the end of section 7 about equivalent sets of parameters in which we used the still unknown A. ., we
13
can have a more explicite, be it less general, reduction of the
number of parameters. From equations (7,9) and (7,10) it follows that the motion of the second profile does not depend
on the three parameters m,I and s separately, but for instance only on their combinations
m(s_s1) , I +s m(s -s ).
z z I
By this
only two of them need to be independent and we take a fixed choice for s, or in connection with the first
paragraph of this section for Rs , namely Rs =2.6.
z z
U <m<U.3,
26
-Hence we reduce the number of parameters from eight to
seven. On the basis of preliminary calculations we have chosen for the parameters the following intervals
O.5<F<U.9, 1<B?<6, 15U0<6 <3600, O.5<Rs1<2.I,
0<1<0.1, 0<v<O.7
Rs =2.6.z
In these calculations it turned out that m(s_si) is larger than
or equal to zero for reasonable profile combinations, this is in agree-ment with the intervals for Rs and m and with the value of Rs
in (9.2).
z Because of the large number of parameters it is impossible to use a sufficiently dense grid on the intervals given in (9.2), for which all systems belonging to the grid points can be calcu-lated. Therefore we have made a stochastic approach.
A random choice is made of the parameters from the regions (9.2) For this choice it is tried to find values of a2 and b2 so that
the "no free vorticity condition" (3,1) is satisfied.
If this is possible we calculate the values of T1,ZJ,Z7,E,A1,B1, A2,c2,B2,2. Then we check if the obtained wing combination is
not too unfavourable, by demanding
1.5<a2<2.3 , U.4<(b2-a2)<1, 0.4<T1<I, 0<Z1<0.6,
0<Z,<0.6, 0<E<4.
(9,3)
When the calculated values satisfy (9,3) the system is accepted, otherwise rejected. Then a new random choice is made. The advan-tage of this program is that by simply putting it again and again
in the computer, additional profile
systems are obtained without the need of choosing new grids in the parameter space.
We made about 2.10 trials and obtained 57
results which satisfied (9,3).
o.
r
I5O orLT
I>
o.3-
27-Tuìe parameter values of the 57 accepted results are separately drawn in figure 9.1. The intervals of the accepted values are
i.78<a2<2.30, O.4<b2-a2<O.67, O.4<T1<O.58,
O<Z1 <0.59, 0.46<Z2<0.60, 2.06<E<4.
6J O 0.936d'
0.5 '-Ti,J
_r
Fig. 9.1. Number (n) of accepted results for each twenty fifth part of considered ranges.
Hence in the selection most probably the lower bounds of (b2-a2),
T1 and Z1 were active and the upper bounds of a2,Z1,Z2 and E.
Another program which has been useful is an optimization procedure. Starting from a calculated wing system one of the values of
F,B,1,m,I,Rs1 ,,a2,b2 -a2,T1 ,ZJ,Z2,E,
can be lowered or raised while the other ones are allowed to vary in prescribed intervals.
In the following we will discuss some of the properties 1f.. ,4
listed in the last part of the previous section, which are demanded for a good working propulsion system.
1
[L
r
z., (9,4) (9,5)the second profile in vacuum is
F*=
r 2u
28
-This has been done only to a certain extent because again the
large number of parameters prevents a complete investigation. First we consider the energy loss E by using the optimization procedure with respect to this quantity.
It turnes out that when E becomes really small the hinge moves to the place of the centre of gravity of the second profile,
Rs1 -p2.6 (table 9.1). This means that the quantity m(s-s1)
(9.1) tends to zero. Then the mass m has no longer influence and can be taken equal to zero. Another phenomenon is that the values of Z1 and Z2 increase extremely. The calculations were
carried Out up to values of Z1 and Z2 of the order of magnitude of 40. From this it follows that the pressure forces at the
profiles generate large negative thrust which is overcome by still larger suction forces. Of course this is not at all a practical situation. It is seen that the amplitude of the second profile is relatively large with respect to the amplitude
of the first one. Hence it can be expected that some resonance like motion occurs. When m=0 the resonance frequency F* of
Table 9.1 Small values of E.
The value of this quantity for F =0.5 or F =0.9 (first and second
row of table 9.1, respectively) becomes F*=O.5 or F*=0.93 which
confirms our conjecture.
(9,6) F B?
Rsm
I Rsa2b2_a2
0.50 1.22 203.0 2.60 2.60 0 0 0.045 2.6 2.6 0.4401.00
2.10 1.65 0.529 0.534 0.90 1.28 192.0 0.032 T1 Z1 Z2 E A1 B1 A2 B2 2 0.179 43.6 31.5 0.119 2.318 2.835 33.80 274.4 13.4210L7
0.149 46.6 46.2 0.214 1.139 1.456 18.59 260.3 9.03 85.3.,.
.29
-Because the motion of the hinge, when the hinge is at s has no
influence on the excitation of the second profile, its motion is caused by the action of the fluid pressures only.
Next we discuss the case of small values of the suction
force of the second profile, hence small values of Z2. Using the
optimization procedure with respect to Z2 we found a system with
Z2 0.12. In a small neighbourhood of the parameters
Frv
of this system we used the stochastic program in order toin-vestigate the stability of its working with respect to small
deviations of the parameters. From the obtained results we mention for F =0.5 the two systems given in table 9.2.
Table 9.2. Small values of Z2, two neighbouring systems.
It is seen that these systems have several drawbacks. The chord-length (b2-a2) of the second profile is small, the contribution T1
of the first profile to the total thrust T = I is small and the suction force at the first profile is relatively large. Although the two sets of parameters F,... ,v in table 9.2 are close to each other there are large differences in T1,E,A2,B2,K and M.
It follows that these configurations with low values of Z2 are not interesting from the practical point of view. Analogous behaviour was found for F =0.7 and F = 0.9.
r r
At last we discuss a type of resonance of the oscillation of the second profile around the elastic hinge which differs from the type which occurred in the case of small values of E.
Fr B Rs m I - Rs1 a2 b2-a2 T1 0.50 1.81 213.6 --2.6 --0.005 0.001 2.493 0.005 2.053 0.295 0.31
O.H
0.50 1.80 213.7 2.6 0.006 0.001 2.413 0.007 1.955 0.263 Z1 Z2 E A1 B1 A2 2 B2 2 Kn M 1.00 0.13 8.31 0.75 1.35 15.22 295.8 6.67 124.8 2.54 2.64 1.85 ¡.04 0.17 ¡.83 0.46 0.83 7.68 292.9 3.53 122.8 1.7530
-The existence of this can be established easily by taking for
instance large values of in and . Then the forces exerted by the
fluid on the second profile can be neglected with respect to the inertia and elastic forces. We consider a purely heaving motion of the first profile. The parameters we used are
FO.5, B=0, Rs1=l.5, m=100, 10, v1000, Rs =2.6.
Then we lowered and resonance was found at about v242. The
results in this case are given in the first row of table 9.3.
Considering the mass hinge system as a separate harmonic oscil-lator in vacuum, we find for the resonance frequency
'
r 21T(ss1) m
(9,7)
(9,8)
Table 9.3. Two resonance motions; m= 100, =242 first
row; m=0.29, v= 1.26 second row.
1hen we use Rs =2..6 and Rs = 1.5 we find from the first row of z
table 9.3, 5=1.9704 and s1 = 1.481. Taking '=242, we obtain
'r0509 which is rather close to the used frequency
F =0.5
Because we decreased y before arriving at resonance it is clear that our results belong to an excitation
below
the eigen frequency,hence the angular motion of the second profile has to be in phase with the motion of the first one. This is in agreement with the first row of table 9.3 where 360°.
It is shown numerically that it is not possible to con-Linue the calculations when in remains fixed and y is past its critical value, say somewhat below v=240. The reason is that then
a2(b2a2)
T1 Z1 Z2 E A1 B1 A2 B212
1.961 0.010 0.039 1 1.44 0.418 0.096 0 7.078 179.9 4.85 359.9
31
-the mass hinge system will be excited
above
its "mechanical" eigen frequency, hence a phase shift of2 of about
1800 occurs,
by which the free vorticity of the first profile cannot be
com-pensated anymore by the vorticity shed by the second one, unless the whole system is drastically changed.
In order to find other resonance situations we now fix y
and decrease m to some lower value by which we leave the reso-nance area. Then again we fix in and decrease till another
resonance is found. After repeating this procedure we arrived at the values in = 0.29 and y = 1 .26 for which the motion is given
in the second row of table 9.3. It is seen that here the fluid
pressures have some influence because 2360. It was not possible
to decrease in and any further without changing entirely the configuration by which resonance did not occurred any more.
Although E becomes small in table 9.3, these systems have
no practical importance because (b2-a2) becomes very small, as
can be expected because of the relatively large amplitude of
the resonating second profile. Opposite to the case of resonance which occurs in table 9.1, here the resonance is caused in
essence by the motion of the hinge which does not coincides with
the centre of gravity of the second profile (Rs1 * Rs).
10. Elastic hinge, systematic calculations.
In this section we consider again systems for which (7.1) holds and Rs =2.6. We discuss some results of calculations carried out for a systematic variation of the parameters F,...,v. We have
chosen the intervals as given in table 10.1, which are in the crowded area's of figure 9.1. All possible combinations have been considered.
Each of these combinations is characterised by an index N of 7 digits, followed by a letter a or b, denoting of which row y has been taken. For instance N = 3485322b belongs to
Fr = 0.5333, B = 1.7,
=275, m=O.1, 1=0.0233, Rs1 = 1.4333 and y=O.5733. Although the number of possible combinations is large the grid is still with
32
-accepted only those of which the calculated values satisfy
(b2-a2)>0.5, T1 >04, Z <0.26, Z2<0.45, E<2.5. (10,1)
By this only 6 profile systems out of the 2.106 cases, are leftover, which are given in table 10.2.
Table 10. 1 Parameter values, systematic calculations.
Table 10.2 Profile systems which satisfy (10.1)
In table 10.2 we have also left out of consideration a small
num-ber of profiles which although they satisfy (10.1), were worse in
every respect than an other one. Hence those of which (b2-a2) and T1
1 2 3 4 5 6 7 8 Fr 0.5 0.5167 0.5333 0.55 0.5667 0.5833 0.6 B 1 1.2333 1.4667 1.7 1.9333 2.1667 2.4 215 223.57 232.14 240.71 249.29 257.86 266.43 275 m 0 0.025 0.05 0.075 0.1 0.125 0.15 I 0 0.01167 0.02333 0.035 0.04667 0.05833 0.07 Rs1 1.3 1.4333 1.5667 1.7 1.8333 1.9667 2.1 a j 0.28 0.3033 0.3267 0.35 0.3733 0.3967 0.42 b 0.550 0.5733 0.5967 0.62 0.6433 0.6667 0.69 a2 b2-a2 T1 Z1
Z2E
A1 K1 M1 N 2.257 0.512 0.43 0.20 0.45 2.18 0.464 0.620 1.082 1333341 a 2.276 0.526 0.45 0.20 0.45 2.37 0.477 0.600 1.159 1333444 a 2.232 0.528 0.42 0.14 0.45 2.40 0.324 1.103 1.024 1353454 a 2.254 0.520 0.43 0.14 0.44 2.39 0.382 0.657 1.824 1341754 b 2.220 0.523 0.42 0.16 0.45 2.46 0.285 1.126 1.448 1362652 b 2.243 0.503 0.46 0.21. 0.44 2.41 0.457 0.376 1.913 2331746 b33
-were smaller and Z1, Z2 and E -were larger than the corresponding values of another one of the accepted profiles.
The six systems of table 10.2 were first optimized with
res-pect to large values of (b2-a2). As constraints we used the last four conditions of (10.1). It turned out that in all six cases the optimization procedure converged to one and the same system given in the first row of table 10.3. The same holds mutatis
mutandis for the optimization of the six systems of table 10.2 with respect to T1, Z1, Z2 and E respectively. In this way we obtain the five systems given in table 10.3, which by the fore-going, could also have been derived from any one of the systems of table 10.2. For these systems we also give the motion of the
second profile so that we can calculate the coefficients A. . in
13
(7.9)
and (7.10). Hence again, to each system of table (10.2) we have,as discussed at the end of section 7,a two parameterfamily of equivalent profiles with different values of m, I, y and s
z
At last we discuss shortly a calculation of a more or less realistic propulsion device. We assume the following system of units; time, I second; length, 1 meter; mass, 1 kilogram mass;
force, 0. 1 kilogram force.
Table 10.3 Profiles with optimized values of (b-a2) T, Z1,
Z2 'd r:
'veiy
z a2 b2-a2 T1 Z1 Z2 E A1 B1 A2 a2 B2 2.337 0.539 0.514 0.260 0.450 2.500 0.473 0.629 3.328 276.5 1.195 2.359 0.503 0.523 0.260 0.450 2.500 0.486 0.652 2.778 274.8 1.000 2.213 0.500 0.400 0.049 0.450 2.500 0.344 0.597 2.931 276.1 1.109 2.345 0.500 0.504 0.260 0.403 2.500 0.466 0.617 3.869 279.5 1.L04 2.266 0.500 0.422 0.260 0.450 1.921 0.420 0.551 3.521 282.1 1.308 K1 M1 B s1 m I s 104.7 0.665 1.602 1.329 236.1 1.969 0.028 0.068 0.607 2.658 103.5 0.725 0.848 1.342 235.3 1.836 0.093 0.015 0.280 2.661 105.0 0.903 1.878 1.734 239.4 1.839 0.012 0.069 0.665 2.513 107.6 0.805 0.886 1.325 236.LL 2.060 0.062 0.028 0.280 2.645 110.3 0.703 1.013 1.312 237.3 2.037 0.035 0.026 0.280 2.566Suppose we want to construct a propulsion device that moves with a velocity
U=
1.5, delivers a mean value of thrust 1=20 (per unit span) and of which the first profile has achord length '0.3. Here and in the following a
"'
denotes a quantity with dimension. For the density of water we take7'= io3.
We choose from table 10.3 for instance the system optimized with respect to the energy, hence the one for which E= 1.921. From the expression for i in (7,8) we obtainA
r
__!=0.0217.
p U
The other quantities follow from (7,7) or (7,8), we find A
=F
=2.50;=2n
=15.71;=B0_=0.095;
r r r I o r.jr.j') n.ir.s4 n.jí31=1237.3;m=pi'm=3.i5; IpiIO.211;a2=a2j=Q.680;
= (b2-a2)=0.15;
= (Rs1-1)(2-i) += 1.910;= (Rs_2)(2-2)
= 2.566; A1 -I 2 K1 189 rì=(1+ ---)O.751;i='
-81.72;
i 1 (10,2) (10,3) Lt.r
3435
-References.
J.A. Sparenberg and A.K. Wiersma. On the efficíency increasing
interaction of thrust producing lifting surfaces, Swimming and Flying in Nature, Vol.2, Plenum Publishing Corporation, New York,
1975.
N.J. Muskhelishvily Singular Integral Equations,
