scholarly journals A note on withdrawal through a point sink in fluid of finite depth

1996 ◽  
Vol 37 (3) ◽  
pp. 406-416 ◽  
Author(s):  
Lawrence K. Forbes ◽  
Graeme C. Hocking ◽  
Graeme A. Chandler
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Stagnation Point ◽  
Numerical Solutions ◽  
Finite Depth ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Flow Through

AbstractWithdrawal flow through a point sink on the bottom of a fluid of finite depth is considered. The fluid is at rest at infinity, and a stagnation point is present at the free surface, directly above the point sink. Numerical solutions are computed by means of the method of fundamental solutions, and it is observed that flows of this type are apparently possible for Froude number less than about 1.5. Relationships to previous work are discussed.

scholarly journals A note on the free surface induced by a submerged source at infinite Froude number

1988 ◽  
Vol 30 (2) ◽  
pp. 147-156 ◽  
Author(s):  
A. C. King ◽  
M. I. G. Bloor
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Conformal Transformation ◽  
Numerical Solutions ◽  
Finite Depth ◽  
Closed Form Expression ◽  
Free Surface Elevation ◽  
Form Expression ◽  
Transformation Technique ◽  
Surface Angle

AbstractThe free surface due to a submerged source in a fluid of finite depth at infinite Froude number is reconsidered. A conformal transformation technique is used to formulate this problem as an integral equation for the free-surface angle. An elementary solution is found for the equation, which results in a closed form expression for the free-surface elevation. Comparison is made with previous numerical solutions.

The bath-plug vortex

1995 ◽  
Vol 284 ◽  
pp. 43-62 ◽  
Author(s):  
Lawrence K. Forbes ◽  
Graeme C. Hocking
Keyword(s):  
Free Surface ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Fluid Flows ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Fluid Circulation ◽  
Vertical Drain ◽  
Drain Hole ◽  
Outflow Rate

Steady flow with constant circulation into a vertical drain is considered. The precise details of the outflow are simplified by assuming that the drain is equivalent to a distributed volume sink, into which the fluid flows with uniform downward speed. It is shown that a maximum outflow rate exists, corresponding to no fluid circulation and vertical entry into the drain hole. Numerical solutions to the full nonlinear problem are computed, using the method of fundamental solutions. An approximate analysis, based on the use of the shallow-water equations, is presented for flows in which the free surface enters the drain. There is, in addition, a second type of solution, having a stagnation point at the free surface and no fluid circulation. These flows are also computed numerically, and results are presented.

scholarly journals A model for the free-surface flow due to a submerged source in water of infinite depth

1998 ◽  
Vol 39 (4) ◽  
pp. 528-538 ◽  
Author(s):  
J.-M. Vanden-Broeck
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Stagnation Point ◽  
Surface Condition ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Nonuniform Distribution ◽  
Infinite Depth ◽  
Collocation Points ◽  
Series Truncation Method

AbstractWe consider a free-surface flow due to a source submerged in a fluid of infinite depth. It is assumed that there is a stagnation point on the free surface just above the source. The free-surface condition is linearized around the rigid-lid solution, and the resulting equations are solved numerically by a series truncation method with a nonuniform distribution of collocation points. Solutions are presented for various values of the Froude number. It is shown that for sufficiently large values of the Froude number, there is a train of waves on the free surface. The wavelength of these waves decreases as the distance from the source increases.

scholarly journals Free surface flow past topography: A beyond-all-orders approach

2012 ◽  
Vol 23 (4) ◽  
pp. 441-467 ◽  
Author(s):  
CHRISTOPHER J. LUSTRI ◽  
SCOTT W. MCCUE ◽  
BENJAMIN J. BINDER
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Numerical Solutions ◽  
Power Series Expansion ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Asymptotic Results ◽  
Stagnation Points ◽  
Flow Past ◽  
Stokes Lines

The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck, (2006) Exponential asymptotics and gravity waves. J. Fluid Mech.567, 299–326, the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of inclination. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.

scholarly journals A note on the flow induced by a line sink beneath a free surface

1991 ◽  
Vol 32 (3) ◽  
pp. 251-260 ◽  
Author(s):  
G. C. Hocking ◽  
L. K. Forbes
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Stagnation Point ◽  
Ideal Fluid ◽  
Critical Value ◽  
Homogeneous Fluid ◽  
Substitution Method ◽  
A Value ◽  
Line Sink ◽  
Low Froude Number

AbstractThe problem of withdrawing water through a line sink from a region containing an homogeneous fluid beneath a free surface is considered. Assuming steady, irrotational flow of an ideal fluid, solutions with low Froude number containing a stagnation point on the free surface above the sink are sought using a series substitution method. The solutions are shown to exist for a value of the Froude number up to a critical value of about 1.4. No solutions of this type are found for Froude numbers greater than this value.

Free-surface flows past a surface-piercing object of finite length

1994 ◽  
Vol 273 ◽  
pp. 109-124 ◽  
Author(s):  
J. Asavanant ◽  
J.-M. Vanden-Broeck
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Surface Condition ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Boundary Integral ◽  
Three Solutions ◽  
Surface Flows ◽  
Sharp Crest ◽  
Supercritical Flows

Steady two-dimensional flows past a parabolic obstacle lying on the free surface in water of finite depth are considered. The fluid is treated as inviscid and incompressible and the flow is assumed to be irrotational. Gravity is included in the free-surface condition. The problem is solved numerically by using boundary integral equation techniques. It is shown that there are solutions for which the flow is supercritical both upstream and downstream and others for which the flow is subcritical both upstream and downstream. These flows have continuous tangents at both ends of the obstacle at which separation occurs. For supercritical flows, there are up to three solutions corresponding to the same value of the Froude number when the obstacle is concave and up to two solutions when the obstacle is convex. For subcritical flows, there are solutions with waves behind the obstacle. As the Froude number decreases, these waves become steeper and the numerical calculations suggest that they, ultimately, reach limiting configurations with a sharp crest forming a 120° angle.

scholarly journals A note on withdrawal from a fluid of finite depth through a point sink

2002 ◽  
Vol 44 (2) ◽  
pp. 181-191 ◽  
Author(s):  
G. C. Hocking ◽  
J.-M. Vanden-Broeck ◽  
L. K. Forbes
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Finite Depth ◽  
Experimental Results ◽  
Problem Comparison

The problem of withdrawal through a point sink of water from a fluid of finite depth with a free surface is considered. Assuming the flow to be axisymmetric, it is found that there is a maximum Froude number at which such flows can exist. This maximum corresponds to the formation of a secondary stagnation ring on the free surface. This result extends earlier work on this problem. Comparison is made with a small Froude number solution and past experimental results.

scholarly journals Steady free surface flows induced by a submerged ring source or sink

2012 ◽  
Vol 694 ◽  
pp. 352-370 ◽  
Author(s):  
T. E. Stokes ◽  
G. C. Hocking ◽  
L. K. Forbes
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Numerical Solutions ◽  
Surface Deformation ◽  
Axisymmetric Flow ◽  
Free Surface Flows ◽  
Inviscid Fluid ◽  
Physical Reason ◽  
Small Surface ◽  
Source Radius

AbstractThe steady axisymmetric flow induced by a ring sink (or source) submerged in an unbounded inviscid fluid is computed and the resulting deformation of the free surface is obtained. Solutions are obtained analytically in the limit of small Froude number (and hence small surface deformation) and numerically for the full nonlinear problem. The small Froude number solutions are found to have the property that if the non-dimensional radius of the ring sink is less than $\rho = \sqrt{2} $, there is a central stagnation point on the surface surrounded by a dip which rises to the stagnation level in the far distance. However, as the radius of the ring sink increases beyond $\rho = \sqrt{2} $, a surface stagnation ring forms and moves outward as the ring sink radius increases. It is also shown that as the radius of the sink increases, the solutions in the vicinity of the ring sink/source change continuously from those due to a point sink/source ($\rho = 0$) to those due to a line sink/source ($\rho \ensuremath{\rightarrow} \infty $). These properties are confirmed by the numerical solutions to the full nonlinear equations for finite Froude numbers. At small values of the Froude number and sink or source radius, the nonlinear solutions look like the approximate solutions, but as the flow rate increases a limiting maximum Froude number solution with a secondary stagnation ring is obtained. At large values of sink or source radius, however, this ring does not form and there is no obvious physical reason for the limit on solutions. The maximum Froude numbers at which steady solutions exist for each radius are computed.

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