Hsien-Ping Pao

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.

Numerical Solution of the Navier‐Stokes Equations for Flows in the Disk‐Cylinder System

A numerical computation of a viscous incompressible fluid confined in a circular cylindrical chamber has been carried out, where the top disk is rotating with a constant angular velocity, and the bottom disk and side wall are held fixed. Using the full Navier‐Stokes equations, steady secondary flows were calculated for various Reynolds numbers (Re = Ωb2/ν) up to a maximum value of 400. It is found that, for the Reynolds numbers below 10, the governing equations are essentially linear and the numerical solution agrees very well with the analytic solution. For higher Reynolds numbers, the flow near the boundary is intensified. The detailed flow patterns are compared with previous work of a single disk and two infinite disks. The volume flow rate of the secondary flow due to the centrifugal action and the frictional moment of the disk were also calculated.

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...

Sources and Sinks at the Axis of a Viscous Rotating Fluid

Theoretical and experimental investigations have been made for the problem of a source or sink at the axis of a viscous, incompressible, steady, unbounded rotating fluid. For the sake of generality, the theoretical portion of this paper also includes a uniform axial velocity. It is found that the velocity distributions have similarity forms at the distant wake along the axis of rotation using the Fourier transform technique. An inverse coordinate expansion technique is then used. This procedure not only brings out the nature of the approximation very clearly but also allows higher‐order solutions to be calculated. The zeroth‐order solution indicates that there exists a withdrawal viscous core which grows in radius with the axial distance x* from the sink at the rate x*1/3. The first‐order solutions are also calculated. The experimental results agree very well with the theoretical predictions.

Note on the flow of a stratified fluid over a stationary obstacle in a channel

Abstract A new method is introduced to produce a uniform stratified flow over a stationary obstacle in an open channel. The flow is achieved by discharging the flow from the channel through a sink. The details of the sink are unimportant. The flow speed is limited only by the sink capacity. Selective withdrawal at lower densimetric Froude numbers is effectively eliminated through the use of a contraction. The standing, free-surface, long wave arising from the initiation of the flow is also eliminated by the contraction. Experiments are conducted for flow over a sphere for a range of Reynolds numbers from O(102) to O(103) and a range of Richardson numbers from O(10−1) to O(10). Dye and neutrally buoyant droplets are used for quantitative analysis of the wake structure. The wake is also probed by a hot-film anemometer. The frequency of vortex shedding is obtained. Comparison with data from towed experiments is also presented.

Stability of Swirling Flow of a Viscous Conducting Fluid in the Presence of a Circular Magnetic Field

A sufficient condition for the stability of a swirling flow in a circular magnetic field, for the case of small spacing between cylinders, is established. A new branch of solution which corresponds to negative critical Taylor number is calculated.