Wave drag due to generation of capillary-gravity surface waves
Teodor Burghelea and Victor Steinberg
Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot 76100, Israel
Received 25 November 2001; revised manuscript received 3 September 2002; published 20 November 2002
The onset of the wave resistance via the generation of capillary-gravity waves by a small object moving with
a velocity V is investigated experimentally. Due to the existence of a minimum phase velocity V
c
for surface
waves, the problem is similar to the generation of rotons in superfluid helium near their minimum. In both
cases, waves or rotons are produced at V V
c
due to Cherenkov radiation. We find that the transition to the
wave drag state is continuous: in the vicinity of the bifurcation the wave resistance force is proportional to
V -V
c
for various fluids. This observation contradicts the theory of Raphae ¨l and de Gennes. We also find that
the reduced wave drag force for different fluids and different ball size may be scaled in such a way that all the
data collapse on a single curve. The capillary-gravity wave pattern and the shape of the wave-generating region
are investigated both experimentally and theoretically. Good agreement between the theory and the experimen-
tal data is found in this case.
DOI: 10.1103/PhysRevE.66.051204 PACS numbers: 47.35.+i, 68.03.-g
I. INTRODUCTION
An object, partially submerged in a fluid and moving uni-
formly on a free fluid surface, is subjected to a drag force
that can be of different physical nature. The most common
one is a viscous drag, which at low Reynolds numbers, Re
1, is the Stokes drag, R
f
=3 dV , that is proportional to
the object velocity V. Here d is the sphere diameter and is
the viscosity. At higher Re, there is another contribution to
the drag, which originates from either laminar or turbulent
wakes, called the eddy resistance R
e
1. However, there ex-
ists another contribution to the drag force, R
w
, due to the
generation of surface capillary-gravity waves by the uni-
formly moving object. These waves remove momentum from
the object into infinity, and in this way produces the wave
resistance force, acting on the object 2.
The problem of the wave resistance for long wavelength
gravity waves is rather old one. It was an active subject of
research for many years from the end of 19th century in
relation to ship navigation and naval architecture. Since the
viscosity of water is small, R
f
and R
e
contributions associ-
ated with the viscosity are small compared with the wave
resistance exerted on a ship. Thus, in the case of a large
object generating the gravity waves, the total drag consists
mainly of the wave resistance.
Kelvin was the first to introduce a theoretical model to
calculate the wave resistance 4. Instead of solving a full
hydrodynamic problem of a flow past a body of arbitrary
shape, he suggested to consider a moving pressure point ap-
plied along the body course to a free fluid surface. Later,
Kelvin’s model was modified into a moving pressure area
5. So the wave resistance was calculated for a given pres-
sure distribution. Further development brought more sophis-
ticated linear and nonlinear models, which provide a numeri-
cal solution of the problem and describe main features of the
phenomenon 6. On the other hand, it seems to be surpris-
ing, but, hitherto, nobody noticed that the wave resistance
appears due to a bifurcation similar to many other nonequi-
librium threshold phenomena, such as Rayleigh-Benard con-
vection, Couette-Taylor flow, etc.
In the case of the gravity waves, Shliomis and Steinberg
7 showed recently that there is a critical velocity V
c
such
that for V V
c
one finds R
w
=0. This critical velocity is
defined by the characteristic size of the object, L e.g., by the
ship length, so V
c
= gL /2 . Above the threshold the wave
drag increases continuously with the velocity. Such continu-
ous imperfect smoothed bifurcation was indeed observed in
the careful experiments with specially designed ship models
by Taylor about a century ago 8. The critical velocity ob-
tained from the fit of the experimental data agrees rather well
with the theoretically predicted value 7.
The dispersion relation for the capillary-gravity waves, on
the other hand, exhibits a minimum in the phase velocity V
c
at the wave number, which is called the capillary wave num-
ber . Below V
c
the surface waves cannot be emitted, so no
wave resistance force acts on the object. Thus, in this case
the bifurcation from a no-wave to the wave-generating state
is an intrinsic property of the dispersion law and is not re-
lated to the size of the perturbing object. The dispersion re-
lation for the surface waves is 1
2
=gk + k
3
/ , 1
where is the circular frequency, k is the wave number, is
the fluid density, g is the gravity acceleration, and is the
surface tension. According to Eq. 1, a phase velocity of the
waves c
p
= / k has a minimum V
c
=(4 g / )
1/4
at the cap-
illary wave number =
g / see Fig. 1. The stationarity
condition of the wave pattern in the frame moving with the
object leads to 2
=kV cos , 2
where is the angle between V and k.
Eliminating from Eqs. 1 and 2, one finds
cos k =c k / V . 3
This equation has evidently no solutions for V V
c
and de-
scribes the opening of the Cherenkov radiation cone at V
V
c
9. Generation of surface waves is analogous to the
PHYSICAL REVIEW E 66, 051204 2002
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