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.

scholarly journals Waves from an oscillating point source with a free surface in the presence of a shear current

2016 ◽  
Vol 798 ◽  
pp. 232-255 ◽  
Author(s):  
Simen Å. Ellingsen ◽  
Peder A. Tyvand
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Point Source ◽  
Wave Field ◽  
Froude Number ◽  
Critical Layer ◽  
Surface Velocity ◽  
Wave Radiation ◽  
Gravitational Acceleration ◽  
Regular Waves

We investigate analytically the linearised water wave radiation problem for an oscillating submerged point source in an inviscid shear flow with a free surface. A constant depth is taken into account and the shear flow increases linearly with depth. The surface velocity relative to the source is taken to be zero, so that Doppler effects are absent. We solve the linearised Euler equations to calculate the resulting wave field as well as its far-field asymptotics. For values of the Froude number $F^{2}={\it\omega}^{2}D/g$ (where ${\it\omega}$ is the oscillation frequency, $D$ is the submergence depth and $g$ is the gravitational acceleration) below a resonant value $F_{res}^{2}$, the wave field splits cleanly into separate contributions from regular dispersive propagating waves and non-dispersive ‘critical waves’ resulting from a critical layer-like street of flow structures directly downstream of the source. In the subresonant regime, the regular waves behave like sheared ring waves, while the critical layer wave forms a street with a constant width of order $D\sqrt{S/{\it\omega}}$ (where $S$ is the shear flow vorticity) and is convected downstream at the fluid velocity at the depth of the source. When the Froude number approaches its resonant value, the downstream critical and regular waves resonate, producing a train of waves of linearly increasing amplitude contained within a downstream wedge.

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

Development of the boundary layer at a free surface from a uniform shear flow

1966 ◽  
Vol 25 (1) ◽  
pp. 87-95 ◽  
Author(s):  
Simon L. Goren
Keyword(s):  
Boundary Layer ◽  
Mass Transfer ◽  
Free Surface ◽  
Shear Flow ◽  
Surface Velocity ◽  
Cube Root ◽  
Two Dimensional ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
Capillary Jets

The development of the boundary layer accompanying the formation of a free surface at y′ = 0, from the two-dimensional uniform shear flow u′ = ωyω, is discussed. The analysis shows that the surface velocity and surface position vary as the cube root of the distance downstream, while the mass-transfer coefficient varies inversely as the cube root of this distance. It is shown how these may be applied to the formation of capillary jets.

scholarly journals Wave Emission From Bottom Vibrations in Subsurface Open-channel Shear Flow

Water Waves ◽  
2020 ◽  
Vol 2 (2) ◽  
pp. 415-432
Author(s):  
Peder A. Tyvand ◽  
Eivind B. Sveen
Keyword(s):  
Shear Flow ◽  
Open Channel ◽  
Surface Velocity ◽  
Wave Number ◽  
Wave Radiation ◽  
Radiation Problem ◽  
Upstream Direction ◽  
Wave Emission ◽  
Hydrostatic Limit ◽  
Radiation Conditions

Abstract The linearized water-wave radiation problem for a 2D oscillating bottom source in an inviscid shear flow with a free surface is investigated analytically. The fluid depth is constant. The velocity of the basic flow varies linearly with depth (uniform vorticity), with zero surface velocity. The far-field surface waves radiated out from the 2D source are calculated, based on Euler’s equation of motion with the application of radiation conditions. There are always two waves, one emitted in the upstream direction and the other in the downstream direction. The energy fluxes of these two waves are calculated. The hydrostatic limit of zero wave number is related to the theory of undular bores.

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 Mathematical modelling of the overflowing cylinder experiment

2003 ◽  
Vol 474 ◽  
pp. 275-298 ◽  
Author(s):  
P. D. HOWELL ◽  
C. J. W. BREWARD
Keyword(s):  
Free Surface ◽  
Surfactant Concentration ◽  
Dynamic Properties ◽  
Vertical Cylinder ◽  
Surfactant Solution ◽  
Surface Concentration ◽  
Experimental Results ◽  
Surface Velocity ◽  
Experimental Apparatus ◽  
Finite Distance

The overflowing cylinder (OFC) is an experimental apparatus designed to generate a controlled straining flow at a free surface, whose dynamic properties may then be investigated. Surfactant solution is pumped up slowly through a vertical cylinder. On reaching the top, the liquid forms a flat free surface which expands radially before over flowing down the side of the cylinder. The velocity, surface tension and surfactant concentration on the expanding free surface are measured using a variety of non-invasive techniques.A mathematical model for the OFC has been previously derived by Breward et al. (2001) and shown to give satisfactory agreement with experimental results. However, a puzzling indeterminacy in the model renders it unable to predict one scalar parameter (e.g. the surfactant concentration at the centre of the cylinder), which must be therefore be taken from the experiments.In this paper we analyse the OFC model asymptotically and numerically. We show that solutions typically develop one of two possible singularities. In the first, the surface concentration of surfactant reaches zero a finite distance from the cylinder axis, while the surface velocity tends to infinity there. In the second, the surfactant concentration is exponentially large and a stagnation point forms just inside the rim of the cylinder. We propose a criterion for selecting the free parameter, based on the elimination of both singularities, and show that it leads to good agreement with experimental results.

scholarly journals Experiments on generation of surface waves by an underwater moving bottom

2015 ◽  
Vol 471 (2178) ◽  
pp. 20150069 ◽  
Author(s):  
Timothée Jamin ◽  
Leonardo Gordillo ◽  
Gerardo Ruiz-Chavarría ◽  
Michael Berhanu ◽  
Eric Falcon
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Surface Deformation ◽  
Fluid Layer ◽  
Fluid Velocity ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass ◽  
Spatio Temporal ◽  
Experimental Findings

We report laboratory experiments on surface waves generated in a uniform fluid layer whose bottom undergoes an upward motion. Simultaneous measurements of the free-surface deformation and the fluid velocity field are focused on the role of the bottom kinematics (i.e. its spatio-temporal features) in wave generation. We observe that the fluid layer transfers bottom motion to the free surface as a temporal high-pass filter coupled with a spatial low-pass filter. Both filter effects are often neglected in tsunami warning systems, particularly in real-time forecast. Our results display good agreement with a prevailing linear theory without any parameter fitting. Based on our experimental findings, we provide a simple theoretical approach for modelling the rapid kinematics limit that is applicable even for initially non-flat bottoms: this may be a key step for more realistic varying bathymetry in tsunami scenarios.

scholarly journals Identification of Ellis rheological law from free surface velocity

2019 ◽  
Vol 263 ◽  
pp. 15-23 ◽  
Author(s):  
Abdulrahman Al-Behadili ◽  
Mathieu Sellier ◽  
James N. Hewett ◽  
Roger I. Nokes ◽  
Miguel Moyers-Gonzalez
Keyword(s):  
Free Surface ◽  
Surface Velocity ◽  
Free Surface Velocity
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