2 results
On Lagrangian drift in shallow-water waves on moderate shear
- W. R. C. PHILLIPS, A. DAI, K. K. TJAN
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- Journal:
- Journal of Fluid Mechanics / Volume 660 / 10 October 2010
- Published online by Cambridge University Press:
- 16 July 2010, pp. 221-239
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The Lagrangian drift in an O(ϵ) monochromatic wave field on a shear flow, whose characteristic velocity is O(ϵ) smaller than the phase velocity of the waves, is considered. It is found that although shear has only a minor influence on drift in deep-water waves, its influence becomes increasingly important as the depth decreases, to the point that it plays a significant role in shallow-water waves. Details of the shear flow likewise affect the drift. Because of this, two temporal cases common in coastal waters are studied, viz. stress-induced shear, as would arise were the boundary layer wind-driven, and a current-driven shear, as would arise from coastal currents. In the former, the magnitude of the drift (maximum minus minimum) in shallow-water waves is increased significantly above its counterpart, viz. the Stokes drift, in like waves in otherwise quiescent surroundings. In the latter, on the other hand, the magnitude decreases. However, while the drift at the free surface is always oriented in the direction of wave propagation in stress-driven shear, this is not always the case in current-driven shear, especially in long waves as the boundary layer grows to fill the layer. This latter finding is of particular interest vis-à-vis Langmuir circulations, which arise through an instability that requires differential drift and shear of the same sign. This means that while Langmuir circulations form near the surface and grow downwards (top down), perhaps to fill the layer, in stress-driven shear, their counterparts in current-driven flows grow from the sea floor upwards (bottom up) but can never fill the layer.
On impulsively generated inviscid axisymmetric surface jets, waves and drops
- K. K. TJAN, W. R. C. PHILLIPS
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- Journal:
- Journal of Fluid Mechanics / Volume 576 / 10 April 2007
- Published online by Cambridge University Press:
- 28 March 2007, pp. 377-403
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The evolution of an unbounded inviscid free surface subjected to a velocity potential of Gaussian form and also to the influence of inertial, interfacial and gravitational forces is considered. This construct was motivated by the occurrence of lung haemorrhage resulting from ultrasonic imaging and pursues the notion that bursts of ultrasound act to expel droplets that puncture the soft air-filled sacs in the lung plural surface, allowing them to fill with blood. The tissue adjacent to the sacs is modelled as a liquid and the air–tissue interface in the sacs as a free surface. The evolution of the free surface is described by a boundary-integral formulation and, since the free surface evolves slowly relative to the bursts of ultrasound, they are realized as an impulse at the free surface, represented by the velocity potential. As the free surface evolves, it is seen to form axisymmetric surface jets, waves or droplets, depending upon the levels of gravity and surface tension. Moreover the droplets may be spherical and ejected away from the surface or an inverted tear shape and fall back to the surface. These conclusions are expressed in a phase diagram of inverse Froude number Fr−1 versus inverse Weber number We−1. Specifically, while axisymmetric surface jets form in the absence of surface tension and gravity, gravity acts to bound their height, rendering them waves, although instability overrides the calculation prior to its reaching that bound. Surface tension acts to suppress the instability (provided that We−1 > 0.045) and to form drops; if sufficiently strong it can also damp the evolving wave, causing it to collapse. The pinchoff which effects spherical drops is of power-law type with exponent 2/3, and the universal constant that relates the necking radius to the time from pinchoff, thereby realizing a finite-time singularity, has the value . Finally, drops can occur once the mechanical index, a dimensional measure used in ultrasonography, exceeds 0.5.