Source-sink Flow Research Articles

A source-sink flow in a rotating fluid

The flow from a line source to a line sink in a multiply connected region is discussed. It is shown that the transport from source to sink passes through the interior region and the Stewartson layers as well as through the Ekman layers.

A numerical investigation of western boundary currents in subtropical and subpolar gyres with a barotropic model

Dynamics of western boundary currents in the subtropical and subpolar gyres are studied as a source-sink flow of barotropic fluid by means of numerical integration of the time-dependent non-linear vorticity equation. The bottom topography consists of a continental shelf of uniform slope (120 km wide) parallel to the straight western coast and a flat bottom of uniform depth.

On nonlinear Ekman and Stewartson layers in a rotating fluid

Boundary and shear layers parallel to the axis of rotation in a rapidly rotating fluid are modified by inertial effects when the Rossby number R is not small enough to be neglected. If E is the Ekman number, inertial modifications to Stewartson E ¼ layers become important when the Rossby number is comparable with, or larger than, E ¼ and changes in the thickness and the structure of the layer then occur. The nature of the change depends on whether the total component of vorticity parallel to the axis of rotation is increased or diminished by the contribution from the layer. When inertial effects are important in these layers, the Ekman layers are also modified by inertia and a nonlinear Ekman condition must be used. Such a condition is obtained and it is used to discuss the flow between rotating disks when there is a discontinuity in their angular velocity and source-sink flow in an annulus.

Nonlinear source-sink flow in a rotating pie-shaped basin

Source-sink flows in a rotating pie-shaped basin provide a laboratory analogue of wind-driven ocean circulation (Stommel, Arons & Faller 1958). Experiments and theory are presented here for flows which are mildly nonlinear. Theory and experiment show satisfactory agreement for the intense flow in the western boundary-layer region which contains the strongest nonlinear effects. The strengths of the sources and sinks were increased in the experiments in an attempt to induce an instability in the western boundary layer. However, the western boundary layer was always stable, even for relatively large Rossby numbers. Photographs from experiments with a basin of semicircular cross-section show the difference between eastern and western boundary layers in a striking manner.

The source-sink flow in a rotating system and its oceanic analogy

Laboratory analogues of theoretical models of wind-driven ocean circulation are based on ideas presented by Stommel (1957). A particularly simple demonstration of the applicability of these ideas is contained in a paper by Stommel, Arons & Faller (1958). The present work develops the source-sink laboratory analogue of ocean circulation models to a point where chosen parametric values allow one to simulate the theoretical models of Stommel (1948) and Munk (1950) exactly. The investigation of the flow in a rotating cylinder generated by a source of fluid near the outer wall leads to a detailed description of the roles of the various boundary layers which occur. This knowledge is used to analyse the more complex source-sink flow in a pie-shaped basin. The laboratory analogue to the Stommel circulation model is analyzed in detail. It is shown that the change in the flow pattern brought about by a radial variation of the position of the eastern boundary in the pie-shaped basin is confined to the interior flow and the boundary layer is largely unaffected. When the bottom of the pie-shaped container slopes, the circulation pattern is changed significantly. For the particular case treated, the depth of the basin along the western boundary is unchanged and the maximum depth occurs at the southeast corner. The circulation generated by a source introduced at the apex of the pie has a gyre whose centre is shifted more toward the southwest corner than the corresponding centre of the gyre for a flat-bottomed basin. Two experiments are reported showing that the western boundary may separate because of the effect of bottom topography or because of the pressure of a cyclonic and an anti-cyclonic gyre generated by suitably placed sources and sinks.

On source-sink flows in a rotating fluid

An incompressible fluid fills a container of fixed shape and size and of uniform cross-section in the (x, y)-plane, themrigid side walls and the two rigid end walls being in contact with the fluid. Here (x, y, z) are the Cartesian co-ordinates of a general point in the frame of reference in which the container is stationary. Fluid is withdrawn from the container atQcm3/sec via certain permeable parts of the side walls and replaced at the same steady rate via other permeable parts of the side walls. As, by hypothesis, the vorticity of the entering and leaving fluid relative to the container is zero, the concomitant fluid motion within the container, Eulerian velocityu= −∇ϕ − ∇ ×A, is irrotational when the container is stationary in an inertial frame. The present paper is concerned with the effects onuof uniform rotation of the whole system with angular velocity Ω about thez-axis when the normal component ofuon the side walls is independent ofz.In the simplest conceivable case,D≡zu−zlis infinite (butD/Qremains finite). End effects are then negligible anduis everywhere independent ofz.The solenoidal component ofu, − ∇ ×A, corresponds tojgyres, one for each of thejirreducible sets of circuits across which the net flow of fluid does not vanish that can be drawn within them-ply connected region bounded by the side walls. While ∇ϕ, which satisfies ∇2ϕ = 0, depends onQbut not on Ω,jandv(the coefficient of kinematic viscosity), ∇ ×Adepends on all these quantities but vanishes identically whenjΩ = 0. WhenjΩ ≠ 0 butv→ 0, ∇2A+ 2Ω, the absolute vorticity, tends to zero everywhere except in certain singular regions near the bounding surfaces, where boundary layers form.End effects cannot be ignored whenDis finite. WhenDis independent ofxandyand equal toD0(say) and Ω is sufficiently large for the boundary layers on the end walls to be of the Ekman type, 95% thickness δ = 3(v/Ω)½(δ [Lt ]D0), the end effects that then arise are only confined to these boundary layers whenj= 0. Whenj≠ 0 boundary-layer suction influences the flow everywhere; thus ∇2Aand ∇ϕ (but not ∇ ×A) are reduced to zero in the main body of the fluid, the regions of non-zero ∇ϕ and ∇2Abeing the Ekman boundary layers on the end walls and boundary layers of another type, 95% thickness δs(typically greater than δ), on the side walls. A theoretical analysis of the structure of these boundary layers shows that non-linear effects, though unimportant in the end-wall boundary layers, can be significant and even dominant in the side-wall boundary layers. The analysis of an axisymmetric system, whose side walls are two coaxial cylinders, suggests an approximate expression for Δs. WhenDis not everywhere independent ofxandy,non-viscous end effects arise which produce relative vorticity in the main body of the fluid even whenj= 0.Experiments using a variety of source-sink distribution generally confirm the results of the theory, show that instabilities of various kinds may occur under certain circumstances, and suggest several promising lines for future work.