Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-07-07T03:58:33.502Z Has data issue: false hasContentIssue false
  • English
  • Français

On parasitic capillary waves generated by steep gravity waves: an experimental investigation with spatial and temporal measurements

Published online by Cambridge University Press:  26 April 2006

Marc Perlin ,
Huanjay Lin  and
Chao-Lung Ting
Marc Perlin
Affiliation:
Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA
Huanjay Lin
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109, USA
Chao-Lung Ting
Affiliation:
Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

An experimental investigation of steep, high-frequency gravity waves (∼ 4 to 5 Hz) and the parasitic capillary waves they generate is reported. Spatial, as well as temporal, non-intrusive surface measurements are made using a new technique. This technique employs cylindrical lenses to magnify the vertical dimension in conjunction with an intensified, high-speed imaging system, facilitating the measurement of the disparate scales with a vertical surface-elevation resolution on the order of 10 μm. Thus, high-frequency parasitic capillary waves and the underlying gravity wave are measured simultaneously and accurately in space and time. Time series of spatial surface-elevation measurements are presented. It is shown that the location of the capillary waves is quasi-stationary in a coordinate system moving with the phase speed of the underlying gravity wave. Amplitudes and wavenumbers of the capillaries are modulated in space; however, they do not propagate with respect to the gravity wave. As capillary amplitudes are seen to decrease significantly and then increase again in a recurrence-like phenomenon, it is conjectured that resonance mechanisms are present. Measured surface profiles are compared to the theories of Longuet-Higgins (1963) and Crapper (1970) and the exact, two-dimensional numerical formulation of Schwartz & Vanden-Broeck (1979). Significant discrepancies are found between experimental and theoretical wavetrains in both amplitude and wavenumber. The theoretical predictions of the capillary wave amplitudes are much smaller than the measured amplitudes when the measured phase speed, amplitude, and wavelength of the gravity wave are used in the Longuet-Higgins model. In addition, this theory predicts larger wavenumbers of the capillaries as compared to experiments. The Crapper model predicts the correct order-of-magnitude capillary wave amplitude on the forward face of the gravity wave, but predicts larger amplitudes on the leeward face in comparison to the experiments. Also, it predicts larger capillary wavenumbers than are experimentally determined. Comparison of the measured profiles to multiple solutions of the stationary, symmetric, periodic solutions determined using the Schwartz & Vanden-Broeck numerical formulation show similar discrepancies. In particular, the assumed symmetry of the waveform about crest and trough in the numerical model precludes a positive comparison with the experiments, whose underlying waves exhibit significantly larger capillaries on their forward face than on their leeward face. Also, the a priori unknown multiplicity of numerical solutions for the same dimensionless surface tension and steepness parameters complicates comparison. Finally, using the temporal periodicity of the wave field, composite images of several successive wavelengths are constructed from which potential energy and surface energy are calculated as a function of distance downstream.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 255 , October 1993 , pp. 597 - 620
Copyright
© 1993 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chang, J. H., Wagner, R. N. & Yuen, H. C. 1978 Measurement of high frequency capillary waves on steep gravity waves. J. Fluid Mech. 86, 401–413.Google Scholar
Chen, B. & Saffman, P. G. 1979 Steady gravity–capillary waves on deep water–I. Weakly nonlinear waves. Stud. Appl. Maths. 60, 183–210.Google Scholar
Chen, B. & Saffman, P. G. 1980 Steady gravity–capillary waves on deep water–II. Numerical results for finite amplitude. Stud. Appl. Maths. 62, 95–111.Google Scholar
Chen, B. & Saffman, P. G. 1985 Three-dimensional stability and bifurcation of capillary and capillary waves on deep water. Stud. Appl. Maths. 72, 125–147.Google Scholar
Cox, C. S. 1958 Measurement of slopes of high-frequency wind waves. J. Mar. Res. 16, 199–225.Google Scholar
Crapper, G. D. 1957 An exact solution for progressive capillary waves. J. Fluid Mech. 2, 532–540.Google Scholar
Crapper, G. D. 1970 Non-linear capillary waves generated by steep gravity waves. J. Fluid Mech. 40, 149–159.Google Scholar
Davies, T. V. 1951 The theory of symmetrical gravity waves of finite amplitude. I. Proc. R. Soc. Lond.. A 208, 475–486.Google Scholar
Gonzalez, R. C. & Wintz, P. 1987 Digital Image Processing. Addison-Wesley.
Huh, J. 1991 A numerical study of capillary–gravity waves. PhD dissertation, Applied Mechanics, University of Michigan.
Longuet-Higgins, M. S. 1963 The generation of capillary waves by steep gravity waves. J. Fluid Mech. 16, 138–159.Google Scholar
Longuet-Higgins, M. S. 1992 Capillary rollers and bores. J. Fluid Mech. 240, 659–679.Google Scholar
Miller, R. L. 1972 The role of surface tension in breaking waves. Tech. Rep. Univ. Chicago, no. 13, 1 October.
Perlin, M. & Ting, C.-L. 1992 Steep gravity–capillary waves within the internal resonance. Phys. Fluids A 4 (11), 2466–2478.Google Scholar
Ruvinsky, K. D., Feldstein, F. I. & Freidman, G. I. 1991 Numerical simulation of the quasistationary stage of ripple excitation by steep gravity–capillary waves. J. Fluid Mech. 230, 339–353.Google Scholar
Ruvinsky, K. D. & Freidman, G. I. 1981 The generation of capillary–gravity waves by steep gravity waves. Izv. Atmos. Ocean Phys. 17, 548–553.Google Scholar
Ruvsinky, K. D. & Freidman, G. I. 1985 Improvement of first Stokes method for the investigation of finite-amplitude potential gravity–capillary waves. In IX All-Union Symp. on Diffraction and Propagation Waves, Tbilisi: Theses of Reports, vol. 2, pp. 22–25.
Schwartz, L. W. & Vanden-Broeck, J.-M. 1979 Numerical solutions of the exact equations for capillary–gravity waves. J. Fluid Mech. 95, 119–139.Google Scholar
Yermakov, S. A., Ruvinsky, K. D., Salashin, S. G. & Freidman, G. I. 1986 Experimental investigations of the generation of capillary–gravity ripples by strongly nonlinear waves on the surface of a deep fluid. Izv. Atmos. Ocean Phys. 22, 835–842.Google Scholar