The Proceedings Fifth International Conference on Numerical Ship Hydrodynamics (1990) / Chapter Skim
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Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves
Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves
Pages 455-468
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From page 455... ...
= P = ~ = q* = 455 = fluid velocity vector = vorticity vector = density surface tension wave number group velocity kinematic viscosity acceleration of gravity pressure wave resistance velocity potential perturbation potential simple layer density p ql IP - q *
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From page 456... ...
for the radiation condition, proposed some kinds of iterative procedures to solve the nonlinear problem [6,11,153. The present paper is a first approach to our future aim to develope a full nonlinear method, implementing the exact boundary conditions taking into account viscosity and surface tension effects at the free surface.
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From page 457... ...
We are going to c kinds of approximation previously described, boundary conditions obtained from.(3) and for computational precisely four models every model we extra boundary on must be imposed at the botton omain is supposed Therefore, in this case, the linear model describing the motion of a submerged body is constituted by (2)
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From page 458... ...
The introduction of a boundary layer at the free surface can be considered as a first step of a zonal splitting of the global problem into several interacting ones: - the irrotational subdomain; - the boundary layer at the free surface; - the boundary layer about the body km 1q| = U - the wake; 1 a= Ho Furthermore the new dynamic boundary condition improves some convergence properties of the numerical procedure t153. To deduce the above mentioned condition we start from the consideration that the irrotational motion does not fulfil the condition of zero tangential stresses at the free surface (if the free boundary has a non zero curvature)
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From page 459... ...
V) Furthermore we observe velocity distribution in domain can be decomposed as: ~ vT qua where q is required to satisfy: or V'9`r=0 Vx9~~t of the thickness that the the fluid In particular the velocity at the free boundary can be thought as the sum of [V\' and the jump ^|9~| of the rotational component across the boundary layer.
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From page 460... ...
~ = ~ (I ~ ) 4kT - Us The positive sign must be gravity waves, and the negat capillary waves: it is recognize that the positive for 460 chosen for ive one foleasy to .= group velocity is the farmers, and
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From page 461... ...
n that gravity waves and ripples propagate in d 2.7 A simplified formulation for the linear gravity-capillarity problem Now we reconsider the linear formulation described in 2.2. Let us assume: ~ c Y= ~ k!
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From page 462... ...
The number of panels per wave length has been chosen equal to 40, and about 80 free surface elements hare beeen L? lacecl upstream the boa: leading edge.
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From page 463... ...
The model with surface tension can be a very deep tool for describing nonlinear energy transfer phenomena; the results obtained with the simplified linear model are preliminary steps for studying a radiation condition suitable also for the nonlinear model. Acknowledgements We wish to acknowledge Prof.
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From page 464... ...
31, ., Wang,H.,"A Nume_ r Solving Nonlinear rem", ITTO, Kobe, os, P D., Nakoc ity Analysis of E or Free-Surface ~ Speed", 17th Syn ydro., The Hague, H
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From page 465... ...
,e submerged hydrofoil, ~che arrows indicate the posi~cion of the trailing cdge. Dawson method full nonlinear method *
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From page 466... ...
SO 20 .00 2~ . SO 25 .00 7 SO 10.00 Fig.4: U/V~ = .704 / /^ / ~ / / full nonlinear method full nonlinear with viscous correction / / ~ ~ w __ _ U/ 466 Figs: Wave resistance versus U/V~ experimental [3]
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From page 467... ...
~ ~ 1 0.48 0.64 0.~0 Fig. 6: Gravity-capillarity steady free surface flow past a submerged circular cyi inder: - gravity and capillary waves — superposition o,— _ :~ — 0 _ ~ At_ ~ _ ~ m__ 1 ~ ~ 00 ~ 6 00 r ~ I I I I r 32.00 &8 .
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From page 468... ...
Anyway, since the attenuation increases with the curvature of the free surface, as it has been pointed out dealing with viscosity effects at the free boundary, the damping of capillary ripples must be much stronger with respect to the gravity waves. For the 2nd part of the question, the effect of surface tension on the wave behind the body is easily evaluated.
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