Patrick Weidman

Reflection of a high-amplitude solitary wave at a vertical wall

The collision of a solitary wave, travelling over a horizontal bed, with a vertical wall is investigated using a boundary-integral method to compute the potential fluid flow described by the Euler equations. We concentrate on reporting new results for that part of the motion when the wave is near the wall. The wall residence time, i.e. the time the wave crest remains attached to the wall, is introduced. It is shown that the wall residence time provides an unambiguous characterization of the phase shift incurred during reflection for waves of both small and large amplitude. Numerically computed attachment and detachment times and amplitudes are compared with asymptotic formulae developed using the perturbation results of Su & Mirie (1980). Other features of the flow, including the maximum run-up and the instantaneous wall force, are also presented. The numerically determined residence times are in good agreement with measurements taken from a cine film of solitary wave reflection experiments conducted by Maxworthy (1976).

On the stability of circular Couette flow with radial heating

The stability of circular Couette flow with radial heating across a vertically oriented annulus with inner cylinder rotating and outer cylinder stationary is investigated using linear stability theory. Infinite aspect ratio and constant fluid properties are assumed and critical stability boundaries are calculated for a conduction-regime base flow. Buoyancy is included through the Boussinesq approximation and stability is tested with respect to both toroidal and helical disturbances of uniform wavenumber. Symmetries of the linearized disturbance equations based on the sense of radial heating and the sense of cylinder rotation and their effect on the kinematics and morphology of instability waveforms are presented. The numerical investigation is primarily restricted to radius ratios 0.6 and 0.959 at Prandtl numbers 4.35, 15 and 100. The results follow the development of critical stability from Taylor cells at zero heating through a number of asymmetric modes to axisymmetric cellular convection at zero rotation. Increasing the Prandtl number profoundly destabilizes the flow in both wide and narrow gaps and the number of contending critical modes increases with increasing radius ratio. Specific calculations made to compare with the stability measurements of Snyder & Karlsson (1964) and Sorour & Coney (1979) exhibit good agreement considering the idealizations built into the linear stability analysis.

On the spin-up and spin-down of a rotating fluid. Part 1. Extending the Wedemeyer model

The Wedemeyer model describing the spin-up of a fluid in a rotating cylinder is generalized to include the case of spin-down. Attention is focused on spin-up and spin-down at a finite (constant) acceleration for small Ekman numbers En. It is found that when the full nonlinear Ekman suction is included for spin-up from rest, the characteristics near the propagating wave front intersect, thus yielding an O(1) velocity discontinuity for the inviscid model. This anomalous behaviour is limited to only a small range of Rossby numbers and does not appear in any of the spin-down solutions. Transient spin-down velocity profiles in the O(E1/4Ω) shear layer at the cylindrical wall are calculated for the quasi-steady flow which occurs for sufficiently small decelerations. Results for impulsive spin-up and spin-down between infinite parallel plates are presented and compared with the asymptotic solutions given by Greenspan & Weinbaum and Benton. Finally, characteristic spin-up and spin-down times are computed for both the contained cylinder and the infinite-parallel-plates geometry.

Stokes drag on hollow cylinders and conglomerates

An experimental study of the drag on hollow cylinders and conglomerates falling in a viscous fluid under Stokes flow conditions is described. The experiments were carried out in a tank of square cross section using silicone oil as the Newtonian fluid. Settling velocities of the free falling objects were measured and corrected to conditions of zero Reynolds number flow in an unbounded fluid. The results reveal that all objects tested have Stokes settling velocities smaller than that of a sphere of equal mass and volume. Measurements are reported in terms of the settling speed ratio defined as the ratio of the Stokes settling speed to that of a sphere of equal mass and volume. For the hollow cylinders two parameters are varied: the aspect ratio (length to outside diameter) and the radius ratio (inner to outer radius). Measurements show that the settling speed ratio decreases markedly as the hollowness of the cylinder increases. Each fixed radius ratio data set exhibits a maximum settling speed ratio near an...

Stability of stationary endwall boundary layers during spin-down

Since Bodewadts (1940) seminal work on the boundary layer flow produced by a fluid in solid-body rotation over a stationary disk of infinite radius there has been much interest in determining the stability of such flows. To date, it appears that there is no theoretical study of the stability of Bodewadts self-similar solution to perturbations that are not self-similar. Experimental studies have been compromised due to the difficulty in establishing these steady flows in the laboratory. Savac (1983, 1987) has studied the endwall boundary layers of flow in a circular cylinder following impulsive spin-down. During the first few radians of rotation, the endwall boundary layers have a structure very similar to Bodewadt layers. For certain conditions, SavaC has observed a series of axisymmetric waves travelling radially inwards in the endwall boundary layers. The conjecture is that these waves represent a mode of instability of the Bodewadt layer. Within a few radians of rotation however, the centrifugal instability of the sidewall layer dominates the spin-down process and the endwall waves are difficult to examine further. Here, the impulsive spin-down problem is examined numerically for Savac’ (1983, 1987) conditions and good agreement with his experiments is achieved. New experimental results are also presented, which include quantitative space-time information regarding the axisymmetric waves. These agree well with both the numerics and the earlier experimental work. Further, a related problem is considered numerically. This flow is also initially in solid-body rotation, but only the endwalls are impulsively stopped, keeping the sidewall rotating. This results in a flow virtually identical to the usual spin-down flow for the first few radians of rotation, except in the immediate vicinity of the sidewall. The sidewall layer is no longer centrifugally unstable and the circular waves on the endwalls are observed without the influence of the sidewall instability.

Axisymmetric stagnation-point flow impinging on a transversely oscillating plate with suction

The viscous fluid motion generated by axisymmetric stagnation-point flow of strain rate a impinging on a flat plate oscillating in its own plane with velocity amplitude U0 and frequency ω, including uniform suction of strength W0 is considered. A coordinate decomposition transforms the full Navier-Stokes equations into a primary equation describing the steady flow and a secondary equation describing the unsteady motion coupled to the primary solution. The solution to the boundary-value problem is governed by two dimensionless groups: the suction parameter S = W0 √aν and the frequency parameter Ω = ω/a, where ν is the kinematic viscosity. Numerical integrations performed with a Runge-Kutta routine provide an exact solution to the Navier-Stokes equations. Values of the steady shear stress are found to agree with asymptotic results for large values of |S|, with S>0 representing suction and S<0 representing blowing. The magnitude and phase of the unsteady shear stress are given over a range of frequencies sufficient to recover analytical asymptotic results at large values of Ω. The unsteady shear stress lags the wall motion by π radians for Ω → 0 and by 5π/4 radians when Ω → ∞. Velocity profiles at selected parameter values during a period of plate oscillation are presented and discussed.

Boundary layer similarity flow driven by power-law shear

SummarySimilarity solution of the Prandtl boundary layer equations describing wallbounded flows and symmetric free-shear flows driven by rotational velocitiesU(y)=βyα are determined for a range of exponents α and amplitudes β. Asymptotic analysis of the equations shows that for α<−1 no similarity solutions with proper algebraic decay exist. For wall-bounded flow, exact solutions found at α=−1/2 and α=1 correspond to an Airy function wall jet and uniform planar Couette flow. Numerical integration of the governing similarity equation reveals singular behaviour for wall-bounded flows as α→α0 = −2/3, and no solutions are found in the range −1−2/3 the shear stressf″(0) parameter is determined as a function of α and β. Symmetric free-shear flow solutions become singular as α→α0 = −1/2 and no solutions are found in the range −1−1/2 the centerline velocityf′(0) is determined as a function of α and β. An asymptotic analysis of the singular behavior of these two problems as α→α0, given in a separate Appendix, shows excellent comparison with the numerical results. Similarity solutions at the critical values α0 have exponential decay in the far field and correspond to the Glauert wall jet for wall-bounded flow and to the Schlichting/Bickley planar jet for symmetric free-shear flow.

A phase space analysis of baroclinic flow

Abstract The qualitative dynamics of a baroclinic flow experiment are studied by constructing phase space coordinates from a single time series. As the stress on the flow is increased we observe steady, periodic, quasiperiodic, and chaotic flow. The chaotic attractor we observe near the transition has the appearance of a thickened torus.

On the inverse Magnus effect in free molecular flow

A Newton-inspired particle interaction model is introduced to compute the sideways force on spinning projectiles translating through a rarefied gas. The simple model reproduces the inverse Magnus force on a sphere reported by Borg, Soderholm and Essen [Phys. Fluids 15, 736 (2003)] using probability theory. Further analyses given for cylinders and parallelepipeds of rectangular and regular polygon section point to a universal law for this class of geometric shapes: when the inverse Magnus force is steady, it is proportional to one-half the mass M of gas displaced by the body.

Onset of convection in a vertical slab of saturated porous media between two impermeable conducting blocks

Abstract The onset of natural convection in a vertically oriented, finite thin slab of saturated porous material is considered. The slab is embedded between two impermeable conducting blocks of finite dimension. A vertical temperature difference is imposed between the upper and lower horizontal surfaces of the slab and blocks, and a linear temperature distribution is imposed on the outer vertical surfaces of the blocks. This configuration is used to model convection in a saturated, fractured rock zone like that associated with faulting. A linear stability analysis is developed for both convection in the slab and conduction in the block. The objective of the study is to obtain the critical Rayleigh number and mode of convection in the slab. When the block and the slab widths are both small compared to the other two dimensions one finds a large number of tall, narrow, three-dimensional cells. In contrast, a block of relatively large width promotes the formation of longer wavelength, weakly three-dimensional cells in the slab at a much lower Rayleigh number. The difference is related to the character of the temperature distribution in the solid block.