Onset of Wave Drag due to Capillary- Gravity Surface Waves

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Onset of Wave Drag due to Capillary- Gravity
Surface Waves

• V. Steinberg
• in collaboration with T.Burghelea
• Department of Physics of Complex Systems

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Dispersive (gravity) waves

• Different nature of the drag-viscous, , eddy
  resistance (laminar and turbulent wakes), ,
  and wave resistance, . For ships ,
  ,ltlt
• Pattern of gravity waves generated by a ship,
• Waves continuously remove momentum from a moving
  object to infinity-wave drag (DR)
• Onset of drag at the threshold at
  , where
• and the Froude number
  
• is an analog of the Mach number.
• Wave drag appears as a continuous bifurcation
• that was actually observed in experiment by
  Taylor(1908)
• (M. Shliomis V. S., PRL 79,
  4178 (1997).)

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Schematic sketch of a ship wake
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Wave drag coefficient for a ship moving at speed V
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.

Wave Pattern
The wave pattern generated on the water surface
by a moving disturbance was first theoretically
explained by Lord Kelvin, hence the name Kelvin
wake. Without citing the details of the solution
of the classical problem, we give its basic
features for a point-like disturbance moving with
constant speed V in a straight line on the
surface of water of uniform depth D (Newman,
1977). For deep water, which in this context is
defined by
, where g is the acceleration of gravity, the
wave pattern is confined to a wedge-shaped region
behind the ship with a half angle of 19.5
(the wake angle). Note that this angle is
independent of the speed of the disturbance. The
wave pattern consists of diverging and transverse
waves, as sketched in Fig. 1. The propagation
direction of the waves in the wake with respect
to the heading of the disturbance is between 0
and 35.25 for the transverse waves, and between
35.25 and 90 for the divergent waves, while the
wavelength decreases with increasing angle. The
outward edge of the wedge, where transverse and
divergent waves are superimposed to form
so-called cusp waves, is usually the region of
the wake with highest wave amplitudes, called
cusp region or cusp lines.
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Onset of Wave Drag due to Gravity Capillary
Surface Waves. V. Steinberg Department of
Physics of Complex Systems, Weizmann Institute of
Science, Israel. In collaboration with Teodor
Burghelea.
The qualitative features of this pattern are
preserved, if the point like disturbance is
replaced by a disturbance of finite spatial
extent which is more suitable to describe a real
ship. However, the region around the ship and
behind the ship up to a distance of several ship
lengths (local wave disturbance region) shows a
complex combination of breaking waves, bow and
stern waves, very much depending on the speed,
the shape and the propulsion system of the ship.
The distribution of waves amplitudes behind the
local disturbance region (free wave pattern
region) depends on the ship as well. Thus, e.g.,
either the transverse or the diverging waves may
dominate, and the cusp line may be more or less
prominent. When the depth D approaches
, the wedge widens For
the wake angle is and for
,
eventually approaching 90 at the singularity
(Havelock, 1908). For shallow
water which in this context is defined by
, the wave pattern is very different
There are no transverse waves , and no cusp
lines. The divergent waves form a wedge whose
half angle depends on the water depth and on the
speed of the ship, . For a
swift ship with V10 m/s (20kn), 5.6
m, thus the deep water limit is valid when D
exceeds about 10 m, which is generally the case
for ship routes in the sea. The wavelength of the
longest waves in the Kelvin wake is 64 m for V10
m/s and 16 m for V5 m/s.
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Capillary-gravity surface waves
Dispersion relation
For water cm/s
cm
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Detecting surface waves. Surface waves form at
upstream and downstream sides of a fisher spider
Dolomedes Triton (0.7 g) leg segment moving
across the water surface at relative velocities
greater than 20cm/s. The ovals in the images
indicate the approximate locations of the shadow
of the leg segment. (R. Suter et al,
J. Exp. Biology, 200, 2523 (1997).
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A disturbance moving with a constant velocity, V,
creates surface waves only if
-critical velocity

• Examples of similar systems
• Superfluid helium
• Cherenkov radiation

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Superfluid helium
-Landau critical velocity for breaking of
superfluidity via generation of
rotons-onset of drag.
m/s
Nonlinear Schreodinger equation as a model
(PomeauRica(1993)) Roton drag-
Cherenkov radiation of rotons
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Cherenkov radiation
Cherenkov process
Non-dispersive waves
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At (light speed in a
medium) Cherenkov cone is
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Supersound at (sound wave speed) Mach
cone
Dispersive waves
From stationary conditions in moving frame one
gets
angle between and
There is no solution at and
the Cherenkov opening cone appears at
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Onset of wave drag in capillary-gravity surface
waves
E.Raphael P.G. deGennes(1996)
Kelvins model (suggested for gravity waves)
P(x,y)-fixed point-like distribution
For -distribution leads to
, i.e. a finite jump
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For finite width pressure distribution,
p-is total vertical applied force.
The wave resistance is
-Fourier transform of the pressure distribution
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1
2
3
1. b/ 1-10 2-17 3-25,4-inf
2. b/ 1-0.89 2-1.19 3-1.79 3. b/ 1
Results of numerical calculations for finite
depth distributions (type of transition remains
(by jump) but functional behavior contradicts to
experiment)
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Capillary contribution to the measured force in
the case of wetting, and
the ball is allowed to move.
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Ratio between capillary and viscous forces acting
on a ball

Vc23 cm/s, R0.157
cm, ?1cSt
s73 dyn/cm for water for oil For a
wire d0.7 mm, l1mm
this ratio is about 30 for oil and above 300 for
water Balls of 1.57, 2.35, and 3.14 mm
were used
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Experimental setup (a) wave visualization (b)
force measurements
(Channels with R/gap 12.4/3.6 and 17.5/ 15 cm)
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Physical properties of fluids used in the
experiments (at 25 C)
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The forces acting on the pendulum is the
magnetic force, is the restoring force
of the thin fiber F.
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Calibration data for 3.14 mm ball in air (full
circles) and the fit (full line)
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Expression for calibration to find and
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(from fit 8.5 cSt)
The drag force vs velocity for silicone oil
DC200/10 cSt and 3.14 mm ball. The upper inset
the same data but for reduced force. The lower
inset the viscous drag force below the
transition vs object velocity. Solid line is the
second-order polynomial fit.
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Re numbers at the transition for different fluids
and ball size vary from about 2 till 700. It
means that below transition the drag consists of
a viscous Stokes drag that changes linearly with
and drag due to the eddy viscosity that is
proportional to at
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- DC200/50 cSt ?- glycerin-water-30cS
t ?- glycerin-water-46cSt 3.14 mm ball
The inset water
Fit by the Ginzburg-Landau equation
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-water ?-silicone
oil DC200/50 cSt ?-silicone oil DC200/50 cSt 2.35
mm ball
The reduced scaled wave drag force vs the reduced
velocity for two fluids and 2.35 mm ball
(scaling with the factor )
(Dimple is an example of meniscus position
instability in oil (non-wetting fluid))
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?-silicone oil DC200/50 cSt -glycerin-water-10
cSt ?-glycerin-water-30 cSt
?-glycerin-water-46 cSt 1.57 mm ball

The reduced scaled wave drag force vs. the
reduced velocity for four fluids
and 1.57 mm ball.
(more general scaling was also
checked and discussed further)
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-with feedback ?-without feedback
silicone oil DC200/50 cSt 3.14 mm ball

The reduced wave drag force vs. the velocity
for a silicone oil DC200/50 cSt and 3.14 mm ball.
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The reduced drag force vs. the reduced velocity
for water-silicone oil DC200/10 cSt
interface and 3.14 mm ball.
Bump is due to the interface instability
sometimes exchange of a bump to a dimple was
observed, particularly with one wetting and
another non-wetting fluids.
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The reduced water-silicone drag force vs. the
reduced velocity for oil DC200/50 cSt interface
and 3.14 mm ball Experimental value of and
that found from independent surface tension
measurements agree well
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Scaling relations
1. Scaling for the critical wave drag force
2. Together with another scaling factor found
experimentally the scaling relation can be
written
is viscosity independent and is caused only by
wave emission!
3. Finally, the scaling compatible with
dimensionless analysis has the form
(Buckingham -theorem and relevant parameters
)
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3.14 mm ball -water
?-silicone oil DC200/50
cSt ?-glycerin-water-30 cSt
?-glycerin-water-46 cSt 2.35 mm ball
?-silicone oil DC200/50 cSt
The reduced wave drag force divided by
non-dimensional parameter A vs. the reduced
velocity for five different fluids with 3.14 and
2.35 mm ball. Solid line is the fit by the
stationary Ginzburg-Landau equation.
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Europhys.Lett. 53, 209 (2001) J. Browaeys, J.
Bacri, R. Perzynski, M. Shliomis
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Fiber of 0.7 mm in silicone oil DC200/10 cSt
circles-increasing velocity, squares-decreasing
Wire of 0.3 mm in water squares- increasing
velocity, circles-decreasing
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Comparison with theory

• Raphael-de Gennes (1996) Kelvins model,
  2D-singular solution at ,
• 3D-discontinuous transition, for finite width
  1/b-wrong functional dependence
• Richard-Raphael (1999) 2D viscosity-wave
  resistance increases continuously but steeply and
  remained bounded
• Sun-Keller (2001) wave drag far away from
  transition (Kelvins model)
• Chevy-Raphael (2003) constant force-discontinuous
  transition, constant depth-continuous one but
  very sharp (not of a second order) with different
  functional dependence behavior of the wave drag
  above the transition also disagrees with
  experiment. It does not resolve contradiction
  with French experiment since it was not
  conducted at constant force.
• Pham-Nore-Brachet (2005) first step in right
  direction-taking into account wetting and
  capillary force. However, the problem is
  2D-shallow water problem, dispersion law without
  minimum, appearance of wave drag is similar to
  phonon excitation and onset of dissipation in BE
  condensate ( or generation of gravity waves by
  finite length ship-ShliomisV.S.(1997)). Wetting
  is taking into account via boundary conditions.
  At subcritical speed unstable dynamics lead to
  dewetting instability. At critical speed
  continuous transition to the state with no
  stationary solution is observed.

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V25.33 cm/s
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(1)
(2)
of the Cherenkov cone as a function of the
velocity for the capillary ripples on a
water in front of the ball. The solid line is
calculation based on Eqs. (1) and (2) using the
data on k(V), presented in the inset.
At there are two values of k , at
which wave, for
which , and , for
which . Thus, short ripples are
found upstream (in front of object) and long
waves downstream (behind object)-look picture!
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Geometry of the wave crest
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Data on the wave crests generated in water by
3.14 mm ball (squares). Calculated wave
crests presented by full lines
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Conclusions.

• Transition to the wave drag state in gravity-
  capillary surface waves is continuous.
• Ginzburg-Landau type model with a field describes
  well the data for various fluids and various
  sizes of a moving object.
• Scaling of the parameters of G-L equation with
  physical parameters is found experimentally.
• References PRL, 86, 2557 (2001)
• and Phys. Rev. E 66, 051204 (2002).