Onset of Wave Drag due to Capillary- Gravity Surface Waves V. Steinberg in collaboration with T.Burghelea Department of Physics of Complex Systems

Dispersive (gravity) waves Different nature of the drag-viscous, , eddy resistance (laminar and turbulent wakes), , and wave resistance, . For ships: , ,

.

Schematic sketch of a ship wake

Wave drag coefficient for a ship moving at speed V

.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 isconfined 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.Wave Pattern

Onset of Wave Drag due to Gravity Capillary Surface Waves. V. SteinbergDepartment of Physics of ComplexSystems,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 V=10 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 V=10 m/s and 16 m for V=5 m/s.

Capillary-gravity surface wavesDispersion relation: For water: cm/s; cm

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).

A disturbance moving with a constant velocity, V, creates surface waves only if-critical velocityExamples of similar systems: Superfluid helium

Cherenkov radiation

Superfluid helium-Landau critical velocity for breaking of superfluidity via generation of rotons-onset of drag.m/sNonlinear Schreodinger equation as a model (Pomeau&Rica(1993)):

Roton drag-Cherenkov radiation of rotons

Cherenkov radiation12Cherenkov processAt (light speed in a medium)Cherenkov cone is Supersound at (sound wave speed)Mach cone Non-dispersive wavesDispersive wavesFrom stationary conditions in moving frame one getsThere is no solution at and the Cherenkov opening cone appears at angle between and

Onset of wave drag in capillary-gravity surface waves E.Raphael & P.G. deGennes(1996)Kelvins model (suggested for gravity waves)P(x,y)-fixedpoint-likedistributionFor -distribution leads to , i.e. a finite jump

For finite width pressure distribution,The wave resistance isp-is total vertical applied force.-Fourier transform of the pressure distribution

1231. 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)

Capillary contribution to the measured force in the case of wetting, and the ball is allowed to move.

Ratio between capillary and viscous forces acting on a ball

Vc=23 cm/s, R=0.157 cm, =1cSt =73 dyn/cm

for water

for oil

For a wire: d=0.7 mm, l=1mm this ratio is about 30 for oil and above 300 for water Balls of 1.57, 2.35, and 3.14 mm were used

Experimental setup: (a) wave visualization; (b) force measurements

(Channels with R/gap: 12.4/3.6 and 17.5/ 15 cm)

Physical properties of fluids used in the experiments (at 25 C)

The forces acting on the pendulum: is the magnetic force, is the restoring force of the thin fiber F.

Calibration data for 3.14 mm ball in air (full circles) and the fit (full line)

Expression for calibration to find and

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.(from fit 8.5 cSt)

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

- DC200/50 cSt - glycerin-water-30cSt- glycerin-water-46cSt3.14 mm ballFit by the Ginzburg-Landau equation:The inset: water

-water -silicone oil DC200/50 cSt-silicone oil DC200/50 cSt2.35 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))

The reduced scaled wave drag force vs. the reduced velocity for four fluids and 1.57 mm ball.-silicone oil DC200/50 cSt*-glycerin-water-10 cSt -glycerin-water-30 cSt -glycerin-water-46 cSt 1.57 mm ball (more general scaling: was also checked and discussed further)

The reduced wave drag force vs. the velocity for a silicone oil DC200/50 cSt and 3.14 mm ball.-with feedback -without feedback silicone oil DC200/50 cSt3.14 mm ball

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.

The reduced water-silicone drag force vs. the reduced velocity for oil DC200/50 cSt interface and 3.14 mm ballExperimental value of and that found from independent surface tension measurements agree well

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 )

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.

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

Europhys.Lett. 53, 209 (2001)J. Browaeys, J. Bacri, R. Perzynski, M. Shliomis

Wire of 0.3 mm in water; squares-increasing velocity, circles-decreasing

Fiber of 0.7 mm in silicone oil DC200/10 cSt; circles-increasing velocity, squares-decreasing

Comparison with theoryRaphael-de Gennes (1996): Kelvins model, 2D-singular solution at , 3D-discontinuous transition, for finite width 1/b-wrong functional dependenceRichard-Raphael (1999): 2D + viscosity-wave resistance increases continuously but steeply and remained boundedSun-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-Shliomis&V.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.

V=25.33 cm/s

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.(1)(2)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!

Geometry of the wave crest

Data on the wave crests generated in water by 3.14 mm ball (squares). Calculated wave crests presented by full lines

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).