scholarly journals A cusp-like free-surface flow due to a submerged source or sink

1984 ◽  
Vol 25 (4) ◽  
pp. 443-450 ◽  
Author(s):  
E. O. Tuck ◽  
J.-M. Vanden Broeck
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Line Source ◽  
Free Surface Flow ◽  
Surface Flow

AbstractA solution is found for a line source or sink beneath a free surface, at a unique squared Froude number of 12.622.

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 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 Flow from a source above a sloping base

2020 ◽  
Vol 61 ◽  
pp. C75-C88
Author(s):  
Shaymaa Mukhlif Shraida ◽  
Graeme Hocking
Keyword(s):  
Free Surface ◽  
Flow Rate ◽  
Water Waves ◽  
Line Source ◽  
Free Surface Flow ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Surface Flow ◽  
Theoretical Research ◽  
Line Sink

We consider the outflow of water from the peak of a triangular ridge into a channel of finite depth. Solutions are computed for different flow rates and bottom angles. A numerical method is used to compute the flow from the source for small values of flow rate and it is found that there is a maximum flow rate beyond which steady solutions do not seem to exist. Limiting flows are computed for each geometrical configuration. One application of this work is as a model of saline water being returned to the ocean after desalination. References Craya, A. ''Theoretical research on the flow of nonhomogeneous fluids''. La Houille Blanche, (1):22–55, 1949. doi:10.1051/lhb/1949017 Dun, C. R. and Hocking, G. C. ''Withdrawal of fluid through a line sink beneath a free surface above a sloping boundary''. J. Eng. Math. 29:1–10, 1995. doi:10.1007/bf00046379 Hocking, G. ''Cusp-like free-surface flows due to a submerged source or sink in the presence of a flat or sloping bottom''. ANZIAM J. 26:470–486, 1985. doi:10.1017/s0334270000004665 Hocking, G. C. and Forbes, L. K. ''Subcritical free-surface flow caused by a line source in a fluid of finite depth''. J. Eng. Math. 26:455-466, 1992. doi:10.1007/bf00042763 Hocking, G. C. ''Supercritical withdrawal from a two-layer fluid through a line sink", J. Fluid Mech. 297:37–47, 1995. doi:10.1017/s0022112095002990 Hocking, G. C., Nguyen, H. H. N., Forbes, L. K. and Stokes,T. E. ''The effect of surface tension on free surface flow induced by a point sink''. ANZIAM J., 57:417–428, 2016. doi:10.1017/S1446181116000018 Landrini, M. and Tyvand, P. A. ''Generation of water waves and bores by impulsive bottom flux'', J. Eng. Math. 39(1–4):131-170, 2001. doi:10.1023/A:1004857624937 Lustri, C. J., McCue, S. W. and Chapman, S. J. ''Exponential asymptotics of free surface flow due to a line source''. IMA J. Appl. Math., 78(4):697–713, 2013. doi:10.1093/imamat/hxt016 Stokes, T. E., Hocking, G. C. and Forbes, L.K. ''Unsteady free surface flow induced by a line sink in a fluid of finite depth'', Comp. Fluids, 37(3):236–249, 2008. doi:10.1016/j.compfluid.2007.06.002 Tuck, E. O. and Vanden-Broeck, J.-M. ''A cusp-like free-surface flow due to a submerged source or sink''. ANZIAM J. 25:443–450, 1984. doi:10.1017/s0334270000004197 Vanden-Broeck, J.-M., Schwartz, L. W. and Tuck, E. O. ''Divergent low-Froude-number series expansion of nonlinear free-surface flow problems". Proc. Roy. Soc. A., 361(1705):207–224, 1978. doi:10.1098/rspa.1978.0099 Vanden-Broeck, J.-M. and Keller, J. B. ''Free surface flow due to a sink'', J. Fluid Mech, 175:109–117, 1987. doi:10.1017/s0022112087000314 Yih, C.-S. Stratified flows. Academic Press, New York, 1980. doi:10.1016/B978-0-12-771050-1.X5001-3

Free Surface Flow over Obstacles at Subcritical Froude Numbers

1978 ◽  
Vol 5 (2) ◽  
pp. 89-94
Author(s):  
J.S.C. Tong ◽  
I.G. Currie
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Finite Length ◽  
Theoretical Approach ◽  
Wave Train ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Horizontal Surface ◽  
Lee Waves ◽  
The Mean

Experiments were carried out on free-surface flow over obstacles of finite length. The obstacles were located on the otherwise horizontal surface which contained the free-surface flow. The Froude number in each case was subcritical and resulted in a train of lee waves on the surface, downstream of the obstacles. The results confirm the predicted phenomenon of ‘upstream influence’ – that the mean upstream depth and the mean downstream depth should differ. Serious discrepancies between the observed results and the results from existing theories are noted, however. Not only is the amplitude of the lee waves at variance with the theory, but the phasing of the wave train, relative to the obstacle, is different. An alternative theoretical approach is proposed, the results from which are in much better agreement with the observed results.

scholarly journals The Effects of the Upstream Froude Number on the Free Surface Flow over the Side Weirs

2012 ◽  
Vol 28 ◽  
pp. 644-647 ◽  
Author(s):  
Sharareh Mahmodinia ◽  
Mitra Javan ◽  
Afshin Eghbalzadeh
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Side Weirs

Non-uniqueness of steady free-surface flow at critical Froude number

2014 ◽  
Vol 105 (4) ◽  
pp. 44003 ◽  
Author(s):  
Benjamin J. Binder ◽  
Mark G. Blyth ◽  
Sanjeeva Balasuriya
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Free Surface Flow ◽  
Surface Flow

scholarly journals Exponential asymptotics of free surface flow due to a line source

2013 ◽  
Vol 78 (4) ◽  
pp. 697-713 ◽  
Author(s):  
C. J. Lustri ◽  
S. W. McCue ◽  
S. J. Chapman
Keyword(s):  
Free Surface ◽  
Line Source ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Exponential Asymptotics

Spilling breakers in shallow water: applications to Favre waves and to the shoaling and breaking of solitary waves

2016 ◽  
Vol 808 ◽  
pp. 441-468 ◽  
Author(s):  
S. L. Gavrilyuk ◽  
V. Yu. Liapidevskii ◽  
A. A. Chesnokov
Keyword(s):  
Experimental Data ◽  
Free Surface ◽  
Shallow Water ◽  
Froude Number ◽  
Fluid Velocity ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Undular Bore ◽  
Characteristic Wavelength ◽  
Good Agreement

A two-layer long-wave approximation of the homogeneous Euler equations for a free-surface flow evolving over mild slopes is derived. The upper layer is turbulent and is described by depth-averaged equations for the layer thickness, average fluid velocity and fluid turbulent energy. The lower layer is almost potential and can be described by Serre–Su–Gardner–Green–Naghdi equations (a second-order shallow water approximation with respect to the parameter $H/L$, where $H$ is a characteristic water depth and $L$ is a characteristic wavelength). A simple model for vertical turbulent mixing is proposed governing the interaction between these layers. Stationary supercritical solutions to this model are first constructed, containing, in particular, a local turbulent subcritical zone at the forward slope of the wave. The non-stationary model was then numerically solved and compared with experimental data for the following two problems. The first one is the study of surface waves resulting from the interaction of a uniform free-surface flow with an immobile wall (the water hammer problem with a free surface). These waves are sometimes called ‘Favre waves’ in homage to Henry Favre and his contribution to the study of this phenomenon. When the Froude number is between 1 and approximately 1.3, an undular bore appears. The characteristics of the leading wave in an undular bore are in good agreement with experimental data by Favre (Ondes de Translation dans les Canaux Découverts, 1935, Dunod) and Treske (J. Hydraul Res., vol. 32 (3), 1994, pp. 355–370). When the Froude number is between 1.3 and 1.4, the transition from an undular bore to a breaking (monotone) bore occurs. The shoaling and breaking of a solitary wave propagating in a long channel (300 m) of mild slope (1/60) was then studied. Good agreement with experimental data by Hsiao et al. (Coast. Engng, vol. 55, 2008, pp. 975–988) for the wave profile evolution was found.

Divergent low-Froude-number series expansion of nonlinear free-surface flow problems

1978 ◽  
Vol 361 (1705) ◽  
pp. 207-224 ◽  
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Continuous Wave ◽  
Free Surface Flow ◽  
Formal Series ◽  
Surface Flow ◽  
Exponential Integral ◽  
Flow Problems ◽  
Low Froude Number ◽  
Nonlinear Free Surface

The low-Froude-number approximation in free-surface hydrodynamics is singular, and leads to formal series in powers of the Froude number with zero radius of convergence. The properties of these divergent series are discussed for several types of two-dimensional flows. It is shown that the divergence is of ‘n!' or exponential-integral character. A potential or actual lack of uniqueness is discovered and discussed. The series are summed by use of suitable nonlinear iterative transformations, giving good accuracy even for moderately large Froude number. Converged ' solutions ’ are obtained in this way, which possess jump discontinuities on the free surface. These jumps can be explained and, in principle, removed, by consideration of appropriate choices for the branch cut of the limiting exponential-integral solution. For example, we provide here a solution for a continuous wave-like flow, behind a semi-infinite moving body.

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