3 results
The blockage of gravity and capillary waves by longer waves and currents
- Jinn-Hwa Shyu, O. M. Phillips
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- Journal:
- Journal of Fluid Mechanics / Volume 217 / August 1990
- Published online by Cambridge University Press:
- 26 April 2006, pp. 115-141
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Surface waves superimposed upon a larger-scale flow are blocked at the points where the group velocities balance the convection by the larger-scale flow. Two types of blockage, capillary and gravity, are investigated by using a new multiple-scale technique, in which the short waves are treated linearly and the underlying larger-scale flows are assumed steady but can have a considerably curved surface and uniform vorticity. The technique first provides a uniformly valid second-order ordinary differential equation, from which a consistent uniform asymptotic solution can readily be obtained by using a treatment suggested by the result of Smith (1975) who described the phenomenon of gravity blockage in an unsteady current with finite depth.
The corresponding WKBJ solution is also derived as a consistent asymptotic expansion of the uniform solution, which is valid at points away from the blockage point. This solution is obviously represented by a linear combination of the incident and reflected waves, and their amplitudes take explicit forms so that it can be shown that even with a significantly varied effective gravity g’ and constant vorticity, wave action will remain conserved for each wave. Furthermore, from the relative amplitudes of the incident and reflected waves, we clearly demonstrate that the action fluxes carried by the two waves towards and away from the blockage point are equal within the present approximation.
The blockage of gravity–capillary waves can occur at the forward slopes of a finite-amplitude dominant wave as suggested by Phillips (1981). The results show that the blocked waves will be reflected as extremely short capillaries and then dissipated rapidly by viscosity. Therefore, for a fixed dominant wave, all wavelets shorter than a limiting wavelength will be suppressed by this process. The minimum wavelengths coexisting with the long waves of various wavelengths and slopes are estimated.
An experiment on boundary mixing: mean circulation and transport rates
- O. M. Phillips, Jinn-Hwa Shyu, Haydee Salmun
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- Journal:
- Journal of Fluid Mechanics / Volume 173 / December 1986
- Published online by Cambridge University Press:
- 21 April 2006, pp. 473-499
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An experiment is described in which a turbulent boundary layer was generated along a sloping wall of a laboratory tank containing salt-stratified fluid. Initially, the pycnocline separating the upper fresh-water layer from the lower saline layer was relatively thin; it thickened in response to the mixing as the experiment proceeded. Two types of mean circulation developed as a result of the boundary mixing. In the boundary layer, counterflowing mean streams were observed that augmented the diffusion of salt up- and downslope. This augmented dispersion in turn tended to spread the salt beyond the depth range of the pycnocline in the ambient fluid, producing a mean convergence in the boundary layer and intrusions from the layer into the ambient fluid.
The evolving density structure in the ambient fluid was measured by conductivity-probe traverses, from which the net buoyancy flux (or salt flux) and volume flux in the boundary layer were determined. The level of zero volume flux was found to coincide closely with the level of maximum stability frequency N, so that the buoyancy transport across this level was entirely the result of turbulent dispersion in the boundary layer. At other levels, the convective transports up and down contribute significantly. A simple theory provides scaling in terms of the laboratory parameters; in terms of the inferred overall turbulent viscosity νe, the buoyancy transport resulting from boundary-layer dispersion was found to be \[ F_{\rm B} = 0.60 \left\{\frac{\nu_{\rm e}N(z)}{\sin\theta}\right\}^{\frac{3}{2}}\cos^2\theta, \] and the intrusion velocity into the ambient fluid is \[ v_1 = 0.42\frac{\nu_{\rm e}^{\frac{3}{2}}}{N^{\frac{1}{2}}h^2}\frac{\cos^2\theta}{(\sin\theta)^{\frac{3}{2}}}, \] where θ is the angle of the sloping bed. These expressions must become invalid when θ becomes vanishingly small; their application to estuarine and continental slope flows is discussed briefly.
Reflection of oblique waves by currents: analytical solutions and their application to numerical computations
- JINN-HWA SHYU, CHI-CHAO TUNG
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- Journal:
- Journal of Fluid Mechanics / Volume 396 / 10 October 1999
- Published online by Cambridge University Press:
- 10 October 1999, pp. 143-182
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Surface waves superimposed upon a larger-scale flow are blocked and reflected at the points where the group velocities balance the convection by the larger-scale flow. In this study, we first extended the theory of Shyu & Phillips (1990) to the situation when short deep-water gravity waves propagate obliquely upon a steady unidirectional irrotational current and are reflected by it. In this case, the uniformly valid solution and the WKBJ solution of the short waves were derived from the Laplace equation and the kinematical and dynamical boundary conditions. These solutions in terms of some parameters (the expressions for which have also been deduced in this case) take the same forms as those derived by Shyu & Phillips, which by referring to Smith's (1975) theory can even be proved to be valid for gravity waves in an intermediate-depth region and near a curved moving caustic induced by an unsteady multidirectional irrotational current. In this general case, the expressions for certain parameters in these solutions cannot be obtained so that their values must be estimated in a numerical calculation. The algorithm for estimates of some of these parameters that are responsible for the amplitude of the reflected wave not being equal to that of the incident wave in the vicinity of the caustic and therefore are crucial for the computer calculation of the ray solution to be continued after reflection, was illustrated through numerical tests. This algorithm can avoid the error magnification phenomenon that occurred in the previous estimates of the reflected wave in the vicinity of the caustic using the action conservation principle directly. The forms of the solutions have also been utilized to clarify the wave profiles near caustics in a general situation, which indicate that in storm conditions freak waves characterized by a steeper forward face preceded by a deep trough will probably occur in the caustic regions.