Vortical source-sink flow against a wall: The initial value problem and exact steady states

Johnson, E. R.; McDonald, N. Robb

doi:10.1063/1.2221353

Cited by 13 publications

(24 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The injected momentum flux is negative and the current turns upstream as in the vorticity-dominated solutions of Johnson & McDonald (2006). Southwick et al (2017) discuss these effects in greater detail showing explicitly for steady hydraulic solutions that the flux of momentum from the source exactly balances the increase of downstream momentum in the current and how this result and its extension to quasi-geostrophic flows helps resolve the momentum paradox of Pichevin & Nof (1997) and Nof (2005).…”

Section: Hydraulic Solutionsmentioning

confidence: 99%

“…For negative PVa outflows the terminal speed of the downstream leading edge is the speed of the rarefaction at the wall and so follows from substituting Z c = 1 into (4.11) yielding, once again, u e = 1/a, precisely the speed, downstream from the source, of √ 2, as in the vorticity-dominated limit (Johnson & McDonald 2006). The solid line in figure 10(b) shows this upstream terminal speed, which is small compared to the downstream KW flow speed when vortex effects are weak (a 2) but exceeds the KW flow speed for strong vortex effects (a 2).…”

Section: Terminal Speeds Of the Leading Edges Of The Anomaliesmentioning

confidence: 99%

“…It is the important parameter in determining the form of the outflow evolution, measuring the relative strength of the flow driven by image vorticity compared to the strength of the linear turning flow set up by the initial Kelvin wave (KW), with large a corresponding to vortically dominated evolutions as in Johnson & McDonald (2006) and small a corresponding to KW-advected zero PVa evolutions. It is noted in § 2.1, and in greater detail in § 4.5, that the parameter a gives the ratio of the scale for the speed of the image-vorticity-driven flow to that of the KW-driven flow.…”

Section: Rotating Buoyant Outflowsmentioning

confidence: 99%

“…Thus consider a 1(1/2)-layer quasi-geostrophic free-surface flow driven when a source of total volume flux Q 0 D (giving area flux Q 0 ) is switched on impulsively at time t = 0 and maintained for t > 0 (although the analysis extends immediately to unsteady sources). A natural scaling that retains the vorticity-dominated limit, for which analytical solutions are known (Johnson & McDonald 2006), is to scale horizontal lengths on the vortex-source length…”

Section: Rotating Buoyant Outflowsmentioning

confidence: 99%

“…McCreary et al (1997) integrated a full-physics version of the 1(1/2)-layer model obtaining results that supported those of Kubokawa (1991) in the small Rossby number limit. Johnson & McDonald (2006) discuss vortical outflows for one-layer flows, presenting CD integrations and an exact closed-form solution for the asymptotic steady state solution which proves useful below in giving the behaviour in the vorticity-dominated limit for the flows discussed here.…”

mentioning

confidence: 99%

See 4 more Smart Citations

The long-wave vorticity dynamics of rotating buoyant outflows

2017

Self Cite

View full text Add to dashboard Cite

This paper discusses the evolution of coastal currents by considering, relative to a rotating frame, the flow development when buoyant fluid is injected into a quiescent fluid bounded by a solid wall. The initial rapid response is determined by the Coriolis force-pressure gradient balance with a Kelvin wave propagating rapidly, at the long-wave speed, with the bounding wall to its right (for positive rotation). However fluid columns can stretch or squash on ejection from coastal outflows so that the ejected fluid gains positive or negative relative vorticity. Depending on its sign, the image in the solid wall of this vorticity can reinforce or oppose the zero potential-vorticity-anomaly (PVa) current set up by the Kelvin wave (KW). This paper presents a simple, fully nonlinear, dispersive, quasi-geostrophic model to discuss the form of coastal outflows as the relative strength of vortex to KW driving is varied. The model retains sufficient physics to capture both effects at finite amplitude and thus the essential nonlinearity of the flow, but is sufficiently simple so as to allow highly accurate numerical integration of the full problem and also explicit, fully nonlinear solutions for the evolution of a uniform PVa outflow in the hydraulic limit. Outflow evolutions are shown to depend strongly on the sign of the PVa of the expelled fluid, which determines whether the vortex and KW driving are reinforcing or opposing, and on the ratio of the internal Rossby radius to the vortex-source scale,, of the flow (where D measures the outflow depth, Π 0 the PVa of the outflow and V 0 the volume flux of the outflow), which measures the relative strengths of the two drivers. Comparison of the explicit hydraulic solutions with the numerical integrations shows that the analytical solutions predict the flow development well with differences ascribable to dispersive Rossby waves on the current boundary and changes in the source region captured by the full equations but not present in the hydraulic solutions.

show abstract

Section: Hydraulic Solutionsmentioning

confidence: 99%

Section: Terminal Speeds Of the Leading Edges Of The Anomaliesmentioning

confidence: 99%

Section: Rotating Buoyant Outflowsmentioning

confidence: 99%

Section: Rotating Buoyant Outflowsmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

The long-wave vorticity dynamics of rotating buoyant outflows

2017

Self Cite

View full text Add to dashboard Cite

show abstract

Analytical formulae for source and sink flows in multiply connected domains

Crowdy

2012

Theor. Comput. Fluid Dyn.

View full text Add to dashboard Cite

Analytical expressions for the complex potentials associated with irrotational incompressible flows due to a finite collection of point sources and sinks in a circular fluid region, of arbitrary finite connectivity, are derived. By the conformal invariance of the boundary value problem, the flows due to point sources and sinks in arbitrary multiply connected fluid domains are then determined up to conformal mapping. As an application of the theory, it is combined with recent results on conformal slit mappings to derive explicit formulae for the effective size of an array of holes in a screen, or grating, following considerations presented by Tuck (Adv Appl Mech 15:89-158, 1975).

show abstract

Necking in coating flow over periodic substrates

Johnson

McDonald

2009

J Eng Math

Self Cite

View full text Add to dashboard Cite

The free-boundary problem determining the shape of a layer of viscous fluid coating a substrate while draining steadily under gravity is solved analytically for substrates taking the form of a periodic array of long plates of arbitrary width and spacing. The mathematical problem involves solving Poisson's equation with constant forcing term in the fluid layer subject to vanishing Neumann and Dirichlet conditions on the free boundary. By considering the problem in a potential plane and using conformal mapping, a two-parameter family of solutions is obtained in the form of an infinite series. Explicit, closed-form solutions are derived in the limiting cases of a single gap perforating an infinitely wide plate, and for an array of evenly spaced point plates. In these cases explicit expressions are obtained for the thinning, or necking, of the fluid layer in the gap regions between plates.

show abstract

Vortical source-sink flow against a wall: The initial value problem and exact steady states

Cited by 13 publications

References 17 publications

The long-wave vorticity dynamics of rotating buoyant outflows

The long-wave vorticity dynamics of rotating buoyant outflows

Analytical formulae for source and sink flows in multiply connected domains

Necking in coating flow over periodic substrates

Contact Info

Product

Resources

About