ARCHIEF
THE WAVE RESISTANCE OF
A SURFACE PRESSURE DISTRIBUTION
IN UNSTEADY MOTION
V. K. Djachenko
Translated by:
M. Aleksandrov
1. J. Doctors TP o, 1811TIlE
DEP4RTMi4i
CTHE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Lab.
v.
ScheepsbouwkundèTechnische Hogeschool
No. 044
Deift
January 1970
0
M/IRINE ENGINEERINGPROCEEDINGS OF THE
LENINGRAD SHIPBUILDING INSTITUTE
Volume 52
HYDROCHANICS AND THEORY OF SHIPS
1966
The Wave Resistance of a
Surface Pressure Distribution in Unsteady Notion
by.:
V.K. Djachenko
Translated by:
M. Aleksaridrov
L.J. Doctors
The University of Michigan
Department of Naval Architecture and Marine Engineering
TRANSLATORS'
NOTE:-In this paper the symbols used for the hyperbolic
functions have the following meanings:
sh sinh
ch cosh
1
We consider the problem of non-steady motion of a surface pressure distribution Of arbitrary form p(, y) over an area .
A coordinate system (0, x, y, z) is fixed on the undisturbed surface, where is in the direction of motion and z is vert-ically upwards.
The disturbance potential which is caused by this pressure distribution is defined by the. soution of Laplace'sEquation,
= 0 with the followthg boundary conditions: On the free surface:
dD'Oq ôtz
5
ox2 2z .dt Ox Ot Ox .atz 0. (1) the bottOm: Liz at z = -H. (2)In expressions (1) and (2)
all
the variables arenon-dimensionalized with respect to 1, the half-length of the pressure distribution (see Fig. 1), po, the characteristic
pressure, and g, the acceleration of gravity. The disturbance potential can be represented by the following integral form:
2
and
Making the substitutions
o=gcos8,
flk
Stn9 we get f(x,y, Z) , SrPJJkdk(de _!fI
àx
J
J
4r2JJ Ox.0
1/' ./
/ / /
//
/
// /
/ / / / / Z/ / /
Fig. 1Substituting (3.) and (4) in (1) and (2), and equating
the integrands (because the
k and
e),
then expressing B and substituting in (1),. weI .
-vz i(xcose+sinBJ k
?T.[wdwfde fA(f)e
B(t)e
]e
ejh'f(x-.z0)cos9 (W-D)5"idx0cJy,
(3)
I
equality must hold for all
in terms of A from equation (2) obtain
2
We consider the case when Let us assume
-f--t-F't/7kH t- AL
ikco56)2224
(iKco5B)A -J'jg co 6)
2kH..1(f np,.,
-iK(;cos8t0sins)
=
(1e
)JJdPii-y--e
dx0d4q0Now let us introduce the function
-tkCO56f7dt
p
To determine A1 we note that it satisfies the differential equation
(7)
where
_ei
oc0se+o5Lfle?i10 (8)and
5JL7clt
(8a)Solving (7) with zero initial conditions,
d41
dt
(equivalent to
/
b and a ), by the method of variation of the integration constant, we getA4=(jrfçcoi 8tdt) sin
8t(-f(strz
Itdt)
co,$t
+(- -f(cotdt)sin$t+(ff.5i7iddt)co58t
(9)t
where
Lr=
Therefore the disturbance potential is
Y(x,(,2,t)=
ifledN[cJe.A1.e"
x
[e_e2K]e
iiqrcoiepin
8)The wave resistance is obtained from
R=-7(X,y)-- dxdi,
assuming that R is equal to the horizontal component of the pressure acting on the area . In formula (11) all variables
are dimensional. In non-dimensional form it would be
ffPu x. )
::i dxdy;
Q
8 aPu LT0 82!f
-
fxPu ft dxd
y
-i;
dxcly-a'
dxd, =In the last equation the first term is the horizontal component of the pressure, the second is equal to the
resist-ance during steady motion, and the third term occurs only
because of the unsteadiness. We can obtain
Y?JJ3fdwJde.A. (,
etMM)(jgcogof 8) eINC0S8Si,qxco$+ jsin8)
&(k,e)=Jfie
dxdy (12),
(13)
(14) 4(10)
R
p(Z
j-dxd4,-
d'iwcoJ9)1(M,8)Ofr,e) '(g,e,t)
de,;(23)
and = if kdA'f do (i t.p
)
(icoso)Ofro)t 2
(15)
The wave resistance can be evaluated from
fJXPu
dxdi-
e2)
(16)
13('k', 8)ii'8
On the basis of (9)
= c,0 8coi8t -c20 8s,78t
cos8tffcoi8tdt sin8tJ(sin6tdt.
(17)And from (8)
fccoi8tcit=fxe
LW C0g0S
8t(i+
e2K)ff&L ei0&*JL
'dy0dt ==x(11-e2NN -) C (*'o)Jve
cos8tdt;
(18)where fr
9)ff.
dxd
(19)Identically
8tdt
z
(i+e2')'
(A,8)JOe ' 0gB S.8tdt
(20)If the motion started from rest, i.e. v-O, tO, then,
using (8) , (9) , (18) and (20)
= x
(ii.e2VH)t L*(A,,9) e,t),
(21)where
r
-iico8S
-i,cos8Sand
1Ncos8S LI.%cOS8
8, t)= e (cosoltfire to,8tdt +
+sLndJ
-j$coS8.
8tdr]
.
ire Sw
Let us now use this solution to calculate the wave re-sistance for the case of a constant pressure distribution
acting on a rectangular area with b/i moving at a constant
speed over deep water. Let us assume that the speed changes according to Fig. 2 below.
Fig. 2
For t + , the required functions become:
k L7&iNC0S8)
(
)- o"-
cog2 8 O4LPIin(*'COS8)Sin(A?1sLfl8) /"siao
l?('A',8)
=
sia(*'P sin 0)sin (*' cog8)'stn 9
s cog8t i 32 U rz(Au sin 9) Sin'fr cosO)
N2s1n19.*'cosO
R
n Pt
--
1Ndk1A'cos8). 32p5ir,A'u31r79)Sifl'(A' c0S8)4,yJ
J
k"SLrfO NcoI9iY'(-icoS6)
9'f!ktco329. Ll
Substituting czl'=Acos&, we obtain 8PX5I
tisjnh&2)sjnA
&dd -(tl)ft)
V&'-cx'The integrand has four poles on the real. axis, at. a and a = ±v'T/v (Fig. 3, below). To calculate the inner in-tegral we use the theorem of residues. For the contour of integration we take the semicircle Cr of infinitely large
radius in the upper half of the complex plane a, going under-neath the poles as shown. Substituting A = /v2 in (26),
we obtain finally, as a particular case of (23), the
estab-lished formula of B.P. Bolshakov.
Fig. 3
7
1öp C
f
R=Jfin
where 1(M,t)= 11-!fpuksin kudu; XL LofsJ.0
dx
J 8u 0 -2 eJA'tVF':7dt
T2 2(tz_!).j
co'q'u-x)du =
=fffr
t)cof kxdn'f (ii',t)5r
A'.x di1',?LfpNcoiN udu.
8
(27)
After substituting (28) into (2) we obtain the following
system of equations:
where A' = 2g1
TWO - DIMENSIONAL
PROBLEM:-In this case the velocity potential can be expressed as
Y(JC,2,t)f[(ACOIMX+ CSLfl
x)ch
*'z(Rco *'x + Dsin *'x)s/i it'Z] dw (28)
where A, B, C, rare unknown functions of t and
k.
For the solution we employ (1) and (2). Assuming ap/at = 0, thewhere
AShkW=BthNH
Csh NH = Dch A'H
B =Ath NH D=Cth I(HJ
Sibstituting (28) in (1) and equating the coefficients
of cos kx and sin kx we obtain the following system.:
4_Ns_gbo2A_2v.K-!_w C =
!4
gD_g272C2k k*A
(N,tjEliminating B and D from (31)
and (30),
we have--at
dt
Multiplying the second equation by I and adding to the
first, and introducing
itS
= ce
jg5
f(M,t)=e [(Mt)L/2(Nt)]. 5=fUdt
we obtain the following differential equation for W1 which is represented as
+I
+ =.
is
10
The solution of this equation with zero initial conditions
8tdt -
COS8tf(stntdt
--+
(f(sin8tdt)coJ6t.+(( f(coi8tdt)sin
St(33)
If the motion starts from rest, the two last terms
be-come zero. Then
fi co, 8tdt
8t dt The functions A, C areARe(e
-INS);
and C
The velocity potential is then
Y(x,z,t)=J{Re (%e5) fchkz+th1H.shM2J
COf MX +tJrn(e')(chIthkH. .hw] S'iaNx)dk,
where W i is given by (34) and a = Iv dt.
The wave resistance is
R I g 2
Pu
dZzfPu
Pu dx+ fPu
dx--f
U dxöt
dx
at
where
Substituting (37) in (38) and reversing the order of integration, we obtain po'
dx
!JAi2(.c)dN_(k
(Jm()]dN,
(39) 1,0 0,5 -INS I -jI(XG(N)Pe
dx.
a0
-- q = a = 11 (40)Using these solutions for the case of constant speed,
we get Lamb's formula for the resistance:
I
L7.2(41)
The result.s of the investigation of the wave resistance for the case of a constant acceleration, using (39) and the
approximate method of Stationary Phase, are presented in
Fig. 4 below. The hump speed for this motion is
consider-ably less than for the case of steady motion.
I!
4p2i;
15
0,5
FormulaS (23) and (39) are recommended for the de-termination of wave resistance for non-steady motion.
REFRENCES :
-Bolshakov, B.P. "The Wave Resistance of a Surface Pressure Distribution." Proceedings of the
Scien-tific Society of Shipbuilding, Vol. 49, 1963.
Sretensky, L.N. "On the Motion of a Cylinder under the Free Surface." Proceedings of the Central Aero-Hydrodynamics Institute, Vol. 346, 1938.