Doppler effects of an oscillating line source in shear flow with a free surface

Wave Motion ◽  
2015 ◽  
Vol 52 ◽  
pp. 103-119 ◽  
Author(s):  
Peder A. Tyvand ◽  
Mikkel Elle Lepperød
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Line Source ◽  
Doppler Effects

scholarly journals Oscillating line source in a shear flow with a free surface: critical layer-like contributions

2016 ◽  
Vol 798 ◽  
pp. 201-231 ◽  
Author(s):  
Simen Å. Ellingsen ◽  
Peder A. Tyvand
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Line Source ◽  
Fluid Velocity ◽  
Wave Pattern ◽  
Critical Layer ◽  
Surface Velocity ◽  
Wave Radiation ◽  
Closed Curves ◽  
Surface Manifestation

The linearized water wave radiation problem for an oscillating submerged line source in an inviscid shear flow with a free surface is investigated analytically at finite, constant depth in the presence of a shear flow varying linearly with depth. The surface velocity is taken to be zero relative to the oscillating source, so that Doppler effects are absent. The radiated wave out from the source is calculated based on Euler’s equation of motion with the appropriate boundary and radiation conditions, and differs substantially from the solution obtained by assuming potential flow. To wit, an additional wave is found in the downstream direction in addition to the previously known dispersive wave solutions; this wave is non-dispersive and we show how it is the surface manifestation of a critical layer-like flow generated by the combination of shear and mass flux at the source, passively advected with the flow. As seen from a system moving at the fluid velocity at the source’s depth, streamlines form closed curves in a manner similar to Kelvin’s cat’s eye vortices. A resonant frequency exists at which the critical wave resonates with the downstream propagating wave, resulting in a downstream wave pattern diverging linearly in amplitude away from the source.

A note on the instabilities of a horizontal shear flow with a free surface

2000 ◽  
Vol 406 ◽  
pp. 337-346 ◽  
Author(s):  
L. ENGEVIK
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Shear Flow ◽  
Velocity Profile ◽  
Froude Number ◽  
The Other ◽  
Algebraic Equations ◽  
Horizontal Shear ◽  
Analytical Treatment ◽  
Unstable Region

The instabilities of a free surface shear flow are considered, with special emphasis on the shear flow with the velocity profile U* = U*0sech2 (by*). This velocity profile, which is found to model very well the shear flow in the wake of a hydrofoil, has been focused on in previous studies, for instance by Dimas & Triantyfallou who made a purely numerical investigation of this problem, and by Longuet-Higgins who simplified the problem by approximating the velocity profile with a piecewise-linear profile to make it amenable to an analytical treatment. However, none has so far recognized that this problem in fact has a very simple solution which can be found analytically; that is, the stability boundaries, i.e. the boundaries between the stable and the unstable regions in the wavenumber (k)–Froude number (F)-plane, are given by simple algebraic equations in k and F. This applies also when surface tension is included. With no surface tension present there exist two distinct regimes of unstable waves for all values of the Froude number F > 0. If 0 < F [Lt ] 1, then one of the regimes is given by 0 < k < (1 − F2/6), the other by F−2 < k < 9F−2, which is a very extended region on the k-axis. When F [Gt ] 1 there is one small unstable region close to k = 0, i.e. 0 < k < 9/(4F2), the other unstable region being (3/2)1/2F−1 < k < 2 + 27/(8F2). When surface tension is included there may be one, two or even three distinct regimes of unstable modes depending on the value of the Froude number. For small F there is only one instability region, for intermediate values of F there are two regimes of unstable modes, and when F is large enough there are three distinct instability regions.

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.

Numerical investigation of shear-flow free-surface turbulence and air entrainment at large Froude and Weber numbers

2019 ◽  
Vol 880 ◽  
pp. 209-238 ◽  
Author(s):  
Xiangming Yu ◽  
Kelli Hendrickson ◽  
Bryce K. Campbell ◽  
Dick K. P. Yue
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Shear Flow ◽  
Power Law ◽  
Outer Surface ◽  
Air Entrainment ◽  
Entrainment Rate ◽  
Surface Layers ◽  
Surface Turbulence ◽  
Free Surface Turbulence

We investigate two-phase free-surface turbulence (FST) associated with an underlying shear flow under the condition of strong turbulence (SFST) characterized by large Froude ($Fr$) and Weber ($We$) numbers. We perform direct numerical simulations of three-dimensional viscous flows with air and water phases. In contrast to weak FST (WFST) with small free-surface distortions and anisotropic underlying turbulence with distinct inner/outer surface layers, we find SFST to be characterized by large surface deformation and breaking accompanied by substantial air entrainment. The interface inner/outer surface layers disappear under SFST, resulting in nearly isotropic turbulence with ${\sim}k^{-5/3}$ scaling of turbulence kinetic energy near the interface (where $k$ is wavenumber). The SFST air entrainment is observed to occur over a range of scales following a power law of slope $-10/3$. We derive this using a simple energy argument. The bubble size spectrum in the volume follows this power law (and slope) initially, but deviates from this in time due to a combination of ongoing broad-scale entrainment and bubble fragmentation by turbulence. For varying $Fr$ and $We$, we find that air entrainment is suppressed below critical values $Fr_{cr}$ and $We_{cr}$. When $Fr^{2}>Fr_{cr}^{2}$ and $We>We_{cr}$, the entrainment rate scales as $Fr^{2}$ when gravity dominates surface tension in the bubble formation process, while the entrainment rate scales linearly with $We$ when surface tension dominates.

Transport of passive scalar in turbulent shear flow under a clean or surfactant-contaminated free surface

2011 ◽  
Vol 670 ◽  
pp. 527-557 ◽  
Author(s):  
HAMID R. KHAKPOUR ◽  
LIAN SHEN ◽  
DICK K. P. YUE
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Solid Wall ◽  
Surface Flux ◽  
Scalar Fields ◽  
Transport Processes ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Scalar Flux ◽  
Conditional Averaging

Direct numerical simulation is performed to study the turbulent transport of passive scalars near clean and surfactant-contaminated free surfaces. As a canonical problem, a turbulent shear flow interacting with a flat free surface is considered, with a focus on the effect of splats and anti-splats on the scalar transport processes. Using conditional averaging of strong surface flux events, it is shown that these are associated with coherent hairpin vortex structures emerging from the shear flow. The upwelling at the splat side of the oblique hairpin vortices greatly enhances the scalar surface flux. In the presence of surfactants, the splats at the surface are suppressed by the surface tension gradients caused by spatial variation of surfactant concentration; as a result, scalar flux is reduced. Conditional averaging of weak surface flux events shows that these are caused by anti-splats with which surface-connected vortices are often associated. When surfactants are present, the downdraught transport at the surface-connected vortices is weakened. Turbulence statistics of the velocity and scalar fields are performed in terms of mean and fluctuation profiles, scalar flux, turbulent diffusivity and scalar variance budget. Using surface layer quantification based on an analytical similarity solution of the mean shear flow, it is shown that the depth of the scalar statistics variation is scaled on the basis of the Schmidt number. In the presence of surfactants, the scalar statistics have the characteristics of those near a solid wall in contrast to those near a clean surface, which leads to thickened scalar boundary layer and reduced surface flux.

The generation of surface waves over a sloping beach by an oscillating line source

1976 ◽  
Vol 79 (3) ◽  
pp. 573-585 ◽  
Author(s):  
Clare A. N. Morris
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Line Source ◽  
Short Wave ◽  
Long Waves ◽  
Edge Waves ◽  
Wave Regime ◽  
Wave Solutions ◽  
Sloping Beach

AbstractA line source whose strength varies sinusoidally with time and also with the co-ordinate measured along its length is situated parallel to the shoreline of a beach of angle ¼π0. Both long-and short-wave solutions are found. It is shown that for certain positions of the source, long waves are not radiated to infinity, while in the short-wave regime, the solutions take the form of edge-waves, with resonances occurring at certain wavenumbers. Computations of the free-surface contours are presented for a range of wavenumbers.

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

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