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.

scholarly journals Flow caused by a point sink in a fluid having a free surface

1990 ◽  
Vol 32 (2) ◽  
pp. 231-249 ◽  
Author(s):  
Lawrence K. Forbes ◽  
Graeme C. Hocking
Keyword(s):  
Free Surface ◽  
Numerical Solution ◽  
Froude Number ◽  
Stagnation Point ◽  
Nonlinear Problem ◽  
Stagnation Line ◽  
Fully Nonlinear ◽  
Maximum Value ◽  
Low Froude Number ◽  
Nonlinear Solutions

AbstractThe flow caused by a point sink immersed in an otherwise stationary fluid is investigated. Low Froude number solutions are sought, in which the flow is radially symmetric and possesses a stagnation point at the surface, directly above the sink. A small-Froude-number expansion is derived and compared with the results of a numerical solution to the fully nonlinear problem. It is found that solutions of this type exist for all Froude numbers less than some maximum value, at which a secondary circular stagnation line is formed at the surface. The nonlinear solutions are reasonably well predicted by the small-Froude-number expansion, except for Froude numbers close to this maximum.

A free-streamline solution for stratified flow into a line sink

1965 ◽  
Vol 21 (3) ◽  
pp. 535-543 ◽  
Author(s):  
Timothy W. Kao
Keyword(s):  
Froude Number ◽  
Inverse Method ◽  
Separated Flows ◽  
Two Dimensional ◽  
Mathematical Solution ◽  
Velocity Discontinuity ◽  
Natural Processes ◽  
A Value ◽  
Line Sink ◽  
Two Dimensional Flow

An analysis is made of the two-dimensional flow under gravity of an inviscid non-diffusive stratified fluid into a line sink, involving a velocity discontinuity in the flow field. The fluid above the discontinuity is stagnant and hence is not drawn into the sink. At sufficiently low values of the modified Froude number, this is the only physically possible mode of flow, and is the cause of flow separation in many industrial and natural processes. A proper mathematical solution for flows with a stagnant zone has so far been lacking. This paper presents such a solution, after posing the problem as one involving a free-streamline, which is the line of velocity discontinuity. The solution to be given here is obtained by an inverse method. It is also found herein that the modified Froude number has a value of 0·345 for all separated flows of the kind in question.

scholarly journals Two infinite-Froude-number cusped free-surface flows due to a submerged line source or sink

1986 ◽  
Vol 28 (2) ◽  
pp. 260-270 ◽  
Author(s):  
I. L. Collings
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Ideal Fluid ◽  
Line Source ◽  
Steady Motion ◽  
Free Surface Flow ◽  
Free Surface Flows ◽  
Surface Flow ◽  
Surface Flows ◽  
Flow Problems

AbstractSolutions are found to two cusp-like free-surface flow problems involving the steady motion of an ideal fluid under the infinite-Froude-number approximation. The flow in each case is due to a submerged line source or sink, in the presence of a solid horizontal base.

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 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.

Free-surface effects on spin-up

1988 ◽  
Vol 187 ◽  
pp. 395-407 ◽  
Author(s):  
Ulf CederlÖf
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Froude Number ◽  
Reasonable Agreement ◽  
Laboratory Experiments ◽  
Dynamic Effect ◽  
Delayed Response ◽  
Inertial Oscillations ◽  
Homogeneous Fluid ◽  
Singular Vortex

The effects of a free surface on the spin-up of a homogeneous fluid are studied, both analytically and experimentally. The analysis is carried out in cylindrical geometry and shows that the spin-up process is strongly modified as the rotational Froude number F = 4ω2L2/gH becomes large. The dynamic effect of the free surface causes delayed response outside a sidewall boundary layer of thickness LF−½. The timescale in the slowly decaying core is larger than the usual spin-up time by a factor of order F. A set of laboratory experiments using a cylinder with a parabolic bottom were carried out in order to test the theory. Reasonable agreement is found in all the experiments except close to the centre where an interesting deviation was observed, especially in cases corresponding to smaller Froude numbers. The deviation consisted of an anticyclonic vortex at the centre. It is shown that this phenomenon might be explained by Lagrangian mean motion resulting from inertial oscillations. In fact, the analysis shows that this motion produces a singular vortex at the centre.

Impact of a rising stream on a horizontal plate of finite extent

2009 ◽  
Vol 621 ◽  
pp. 243-258 ◽  
Author(s):  
P. CHRISTODOULIDES ◽  
F. DIAS
Keyword(s):  
Galerkin Method ◽  
Froude Number ◽  
Steady Flow ◽  
Stagnation Point ◽  
Critical Value ◽  
Horizontal Plate ◽  
Free Surfaces ◽  
Bernoulli’S Equation ◽  
Asymmetric Flows ◽  
Finite Extent

The steady flow of a stream emerging from a nozzle, hitting a horizontal plate and falling under gravity is considered. Depending on the length of the plate L and the Froude number F, the plate can either divert the stream or lead to its detachment. First, the problem is reformulated using conformal mappings. The resulting problem is then solved by a collocation Galerkin method; a particular form is assumed for the solution, and certain coefficients in that representation are then found numerically by satisfying Bernoulli's equation on the free surfaces at certain discrete points. The resulting equations are solved by Newton's method, yielding various configurations of the solution based on the values of F and L. The lift exerted on the plate is computed and discussed. If the plate is long enough, physically meaningful solutions are found to exist only for values of F greater than or equal to a certain critical value F0, which is to be determined. Results are presented, both for F > F0 where the detachment is horizontal and for F = F0 where the detachment point is a stagnation point at a 120° corner. Related asymmetric flows where the rising stream is inclined are also studied.

Nonlinear Ship Waves at Low Froude Number

1989 ◽  
Vol 33 (03) ◽  
pp. 176-193
Author(s):  
A. J. Hermans ◽  
F. J. Brandsma
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Continuous Dependence ◽  
Wave Pattern ◽  
Ray Method ◽  
Ship Waves ◽  
Asymptotic Evaluation ◽  
Low Froude Number ◽  
Field Wave

The wave pattern of a thick ship-like object with finite bow and stern angles 0 <β<π/2 is studied. The completely blunt form p = π/2 is excluded. It turns out that the wave pattern is strongly influenced by the nonlinear terms at the free surface. The wave pattern is determined by means of the ray method. The rays are generated mainly at the bow and the stern. A crucial step is the determination of the so-called excitation coefficients. They are constructed by means of an asymptotic evaluation of a distribution of "local"sources at the free surface. It is shown that for small angles β<< 1 the excitation coefficients are the same as the ones obtained by means of an asymptotic expansion for small values of the Froude number of the results of Michell's thin-ship theory. For increasing values of β, the excitation coefficients change asymptotically. The theory herein shows a continuous dependence, nevertheless. Similar changes are observed in the far-field wave pattern.

scholarly journals The Froude number for solitary water waves with vorticity

2015 ◽  
Vol 768 ◽  
pp. 91-112 ◽  
Author(s):  
Miles H. Wheeler
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Froude Number ◽  
Simple Proof ◽  
Upper Bound ◽  
Water Waves ◽  
Wave Speed ◽  
Critical Value ◽  
Arbitrary Distribution ◽  
Two Dimensional

We consider two-dimensional solitary water waves on a shear flow with an arbitrary distribution of vorticity. Assuming that the horizontal velocity in the fluid never exceeds the wave speed and that the free surface lies everywhere above its asymptotic level, we give a very simple proof that a suitably defined Froude number $F$ must be strictly greater than the critical value $F=1$. We also prove a related upper bound on $F$, and hence on the amplitude, under more restrictive assumptions on the vorticity.

Unstable jets generated by a sphere descending in a very strongly stratified fluid

2019 ◽  
Vol 867 ◽  
pp. 26-44 ◽  
Author(s):  
Shinsaku Akiyama ◽  
Yusuke Waki ◽  
Shinya Okino ◽  
Hideshi Hanazaki
Keyword(s):  
Froude Number ◽  
Particle Image ◽  
Stratified Fluid ◽  
Turbulent Jet ◽  
Homogeneous Fluid ◽  
Image Velocimetry ◽  
Low Froude Number ◽  
Long Time ◽  
Short Time ◽  
Unstable States

The flow around a sphere descending at constant speed in a very strongly stratified fluid ($Fr\lesssim 0.2$) is investigated by the shadowgraph method and particle image velocimetry. Unlike the flow under moderately strong stratification ($Fr\gtrsim 0.2$), which supports a thin cylindrical jet, the flow generates an unstable jet, which often develops into turbulence. The transition from a stable jet to an unstable jet occurs for a sufficiently low Froude number $Fr$ that satisfies $Fr/Re<1.57\times 10^{-3}$. The Froude number $Fr$ here is in the range of $0.0157<Fr<0.157$ or lower, while the Reynolds number $Re$ is in the range of $10\lesssim Re\lesssim 100$ for which the homogeneous fluid shows steady and axisymmetric flows. Since the radius of the jet can be estimated by the primitive length scale of the stratified fluid, i.e. $l_{\unicode[STIX]{x1D708}}^{\ast }=\sqrt{\unicode[STIX]{x1D708}^{\ast }/N^{\ast }}$ or $l_{\unicode[STIX]{x1D708}}^{\ast }/2a^{\ast }=\sqrt{Fr/2Re}$, this predicts that the jet becomes unstable when it becomes thinner than approximately $l_{\unicode[STIX]{x1D708}}^{\ast }/2a^{\ast }=0.028$, where $N^{\ast }$ is the Brunt–Väisälä frequency, $a^{\ast }$ the radius of the sphere and $\unicode[STIX]{x1D708}^{\ast }$ the kinematic viscosity of the fluid. The instability begins when the boundary-layer thickness becomes thin, and the disturbances generated by shear instabilities would be transferred into the jet. When the flow is marginally unstable, two unstable states, i.e. a meandering jet and a turbulent jet, can appear. The meandering jet is thin with a high vertical velocity, while the turbulent jet is broad with a much smaller velocity. The meandering jet may persist for a long time, or develop into a turbulent jet in a short time. When the instability is sufficiently strong, only the turbulent jet could be observed.

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