Timothy W. Kao

Internal solitons on the pycnocline: generation, propagation, and shoaling and breaking over a slope

In Part 1 a study is made of the internal solitary wave on the pycnocline of a continuously stratified fluid. A Korteweg–de Vries (KdV) equation for the ‘interfacial’ displacement is developed following Benneys method for long nonlinear waves. Experiments were conducted in a long wave tank with the pycnocline at several different depths below the free surface, while keeping the total depth approximately constant. A step-like pool of light water, trapped behind a sliding gate, served as the initial disturbance condition. The number of solitons generated was verified to satisfy the prediction of inverse-scattering theory. The fully developed soliton was found to satisfy the KdV theory for all ratios of upper-layer thickness to total depth. In Part 2 of this study we investigate experimentally the evolution and breaking of an internal solitary wave as it shoals on a sloping bottom connecting the deeper region where the waves were generated to a shallower shelf region. It is found through quantitative measurements that the onset of wave-breaking was governed by shear instability, which was initiated when the local gradient Richardson number became less than ¼. The internal solitary wave of depression was found to steepen at the back of the wave before breaking, in contrast with waves of elevation. Two slopes were used, with ratios 1:16 and 1:9, and the fluid was a Boussinesq fluid with weak stratification using brine solutions.

A numerical investigation of circulation in the Arabian Gulf

A three-dimensional hydrodynamic model is developed to study the circulation in the Arabian Gulf. The model contains realistic basin geometries and bathymetries of the Arabian Gulf and a good portion of the Gulf of Oman and is driven by monthly climatological winds, evaporation, and net ocean heat gain in both gulfs and the Shatt-al-Arab discharge. It is found that the cyclonic circulation in the southern portion of the gulf is primarily driven by the evaporation-induced freshening from the Strait of Hormuz. In the northwestern corner of the gulf, the Shatt-al-Arab discharge maintains the cyclonic circulation, which would otherwise be anticyclonic. The northwestward intrusion of fresher water along the Iranian coast is weakened by northwesterly winds in winter but strengthens and extends almost to the head of the gulf in summer owing to the warming of the Gulf of Oman waters, the development of the summer thermocline in the Arabian Gulf, and diminishing winds. The southward coastal current along the Arabian coast is most prominent between the head of the gulf and Qatar. The model also predicts a strong southward coastal jet east of Qatar which is primarily wind driven. No similar coastal jet can be developed in the Gulf of Salwa, west of Qatar.

Vortex structure in the wake of a sphere

Results showing the three‐dimensional vortex shedding structure when a sphere is towed at a constant velocity through a stratified fluid are presented. It is found that for small Richardson numbers (weak stratification) and Reynolds numbers in the range from 4×103 to 2×104 the vortex is shed three‐dimensionally. However, stratification quickly and effectively inhibits the vertical motion and the initially turbulent wake collapses and reveals the vertically oriented portion of the vortex structure, reminiscent of a two‐dimensional vortex street behind a circular cylinder when viewed from above. The structure is, however, distinctly three‐dimensional. It is also found that the estimated vortex shedding frequency is in reasonable agreement with previously published results for a sphere in a homogeneous fluid. It is suggested that a weak stratification is an excellent means for revealing the vortex structure of a three‐dimensional body in a homogeneous fluid, and that the vortex tube in the wake of a sphere in a homogeneous fluid has a closed‐end double helical structure. Two branches of the double helix are continuously unwinding in an opposite sense from the formation region. Moreover, the present double helical model satisfies Thompson’s circulation theorem in contrast to previously proposed helical models.

Experimental investigations of the stability of channel flows. Part 2. Two-layered co-current flow in a rectangular channel

The stability of the laminar co-current flow of two fluids, oil and water, in a rectangular channel was investigated experimentally, with and without artificial excitation. For the ratio of viscosity explored, only the disturbances in water grew in the beginning stages of transition to turbulence. The critical water Reynolds number, based upon the hydraulic diameter of the channel and the superficial velocity defined by the ratio of flow rate of water to total cross-sectional area of the channel, was found to be 2300. The behaviour of damped and growing shear waves in water was examined in detail using artificial excitation and briefly compared with that observed in Part 1. Mean flow profiles, the amplitude distribution of disturbances in water, the amplification rate, wave speed and wavenumbers were obtained. A neutral stability boundary in the wave-number, water Reynolds number plane was also obtained experimentally. It was found that in natural transition the interfacial mode was not excited. The first appearance of interfacial waves was actually a manifestation of the shear waves in water. The role of the interface in the transition range from laminar to turbulent flow in water was to introduce and enhance spanwise oscillation in the water phase and to hasten the process of breakdown for growing disturbances.

Role of viscosity stratification in the stability of two-layer flow down an incline

The stability of the flow of two layers of viscous liquids down an incline is investigated. The problem is governed by two competing long-wavelength modes associated with the surfaces. One mode, calling it the first mode, is faster than the second one. When the two layers have the same coefficient of dynamic viscosity, the second mode was found earlier by the author to be the governing mode for most cases. The situation is greatly altered when viscosity is not the same for the two layers. The second mode is dramatically stabilized for the range of viscosity ratio, m , less than unity, and the first mode is now generally the governing mode in that range. The overall effect is stabilizing compared with m = 1. A relative stability index is also introduced to compare the result with that of the homogeneous case. It is found that the presence of the upper layer is generally destabilizing compared with that of a homogeneous fluid of the same total depth.

Experimental investigations of the stability of channel flows. Part 1. Flow of a single liquid in a rectangular channel

The stability of the laminar flow in a rectangular channel with aspect ratio 1:8 was investigated experimentally, with and without artificial excitation. The critical Reynolds number based on the hydraulic diameter and the average velocity was found to be 2600. Behaviour of damped and growing waves, using artificial excitation, was examined in detail. In particular the progress of growing disturbances was followed. Breaking was found to be the ultimate fate of a growing wave. Spectra of growing and damped waves were also obtained. Measurements were made for wavelengths, wave speeds and amplification or damping rates. The neutral stability boundary in the α r , R plane was determined. In the damped region, comparison of several aspects of the behaviour of the measured disturbances with the plane Poiseuille theory for spatial decay yielded good agreement. Three-dimensionality and non-linear subcritical instability were briefly examined. Neutral subcritical waves at low Reynolds numbers appeared possible when the exciter amplitude was quadrupled. The possible bearings of the present study on the stability of plane Poiseuille flow are suggested.

Role of the Interface in the Stability of Stratified Flow down an Inclined Plane

For the case in which the density of the upper layer is somewhat less than that of the lower layer, a second mode essentially associated with the interface is found to govern the stability of the flow. However, when the upper density becomes much smaller, the free surface mode and the second mode compete for governing the stability. A clarification of the mechanism involved is given. The Reynolds stress transfer of energy at the interface is estimated and is found to be insignificant for long waves.

Numerical study of the breaking of an internal soliton and its interaction with a slope

Abstract The full Navier-Stokes and diffusion equations are applied to study the breaking of an internal soliton on the continuously stratified pycnocline in a two-layer system and its interaction with a slope. First, these equations are solved numerically to study the limiting height and breaking of the soliton in the case of constant total depth. Breaking occurs when the particle velocity in a region of flow field exceeds the wave celerity. This results in a gravitational instability with a patch of dense water entraining into the upper layer in the lee of the wave. The numerically determined breaking criterion is supported by an estimate using the first-order Korteweg-de Vries (KdV) theory. Then, the model is used to examine the interaction of the soliton with a slope-shelf topography and a uniform slope. In both cases, the relative depths of the layers change at the turning point along the slope. Mechanisms of the wave breaking and wave propagation processes for both cases are described. Scaled bottom stresses and total wave run-up on the slope are also presented.

Experimental study of upstream influence in the two‐dimensional flow of a stratified fluid over an obstacle†

An experimental investigation was made of the upstream influence in front of two‐dimensional obstacles when they were towed in a linearly stratified fluid. The experiments were performed in a plexiglas channel 30.5 feet long, 2 feet high and 14 inches wide filled with a linearly stratified salt solution. Velocity measurements and flow visualization were obtained by neutrally buoyant liquid droplets and dye lines. Density measurements were made by a salinity probe. The existence of unattenuated upstream influence in front of an obstacle was quantitatively documented for the first time. It occurred in the form of multiple unattenuated horizontal jets when there was a separated open wake behind the obstacle. These jets were identified to be the super‐position of “columnar disturbance modes”. The total number of columnar modes was determined solely by the Froude number of the flow and was equal to the number of lee‐wave modes excited. The drag due to upstream columnar modes was estimated and found to be lower...

Inflows, density currents, and fronts

The inflow of a fluid of different density into an ambient fluid of uniform density and finite depth is studied numerically using the full Navier–Stokes and diffusion equations. The problem is posed as an initial‐boundary‐value problem. In both the overflow (lighter buoyant inflow) and underflow (heavier inflow) cases, a gravity current is established with a pronounced headwave. A significant feature of the flows is the presence of strong downwelling (overflow case) and upwelling (underflow case) activity near the front. The overflow case has been applied by Kao, Park, and Pao to the study of small‐scale oceanic fronts formed by the river discharge into the coastal region. The present paper deals with the fluid mechanical and numerical aspects of the overall problem in addition to presenting the results of the underflow case. It is found that the characteristics of the flow in the near field of the inflow are mainly governed by an inflow densimetric Froude number, Fe. In particular, the flow regime in the...