Surface wave resonance of a liquid (water) layer confined in a circular channel is studied both experimentally and numerically. For the experiment, eight unevenly distributed ultrasonic distance sensors measure the local height of the wave surface. The resonance curves show maxima only for odd multiples of the fundamental resonance frequency $$f_0$$ f 0 . We explained this behavior using a simple intuitive “ping-pong” like model. Collision of wave fronts can be observed for higher frequencies. Also, the wave reflection on the walls can be treated as wave collision with itself. The non-linearity seems to be weak in our study so the delay in the wave propagation before and after the collision is small. Time-space plots show localized propagating waves with high amplitudes for frequencies near resonance. Between the peaks low amplitude and harmonic patterns are observed. However, for higher frequencies, the frequency band for localized waves becomes wider. In the Fourier space-time plane, this can be observed as a point for the harmonic patterns or a superposition of two lines: one line parallel to wave-vector k axis corresponding to the excitation frequency $$f_0$$ f 0 and a second line with inclination given by wave propagation velocity $$\sqrt{gh}$$ gh . For planned future work, this result will help us to reconstruct the whole water surface elevation using time-series from only a few measurement points
Instabilities and pattern formation in viscous fluids have been a major topic of non-linear fluid dynamics for several decades. The study of pattern formation in viscoelastic thin films offers the opportunity to find new fascinating structures that cannot be observed in viscous fluids. Rayleigh–Taylor and Faraday instabilities, such as the resulting patterns in thin films of viscoelastic fluids, are investigated. We use the long-wave approximation and a Karman–Pohlhausen approach to simplify the mass and momentum equations. The viscoelastic stress tensor is calculated applying the linear Maxwell model. Conditions for the Faraday instability have been found using Floquet’s theorem. It is shown that viscoelastic films can exhibit harmonic resonance under external vibration. Moreover, a simulation of the non-linear problem in 2D and 3D is conducted with a finite difference method. Unstable oscillating Rayleigh–Taylor modes occur in the 2D numerical solution. Furthermore, we find that the wavenumber changes with the relaxation time of the fluid. Faraday patterns in viscous films emerge as regular structures of the surface, like squares or hexagons. Numerical simulations of the viscoelastic fluid also show regular structures. However, they collapse into a chaotic stripe-like pattern after a certain time.
<p>Tidal bores are natural phenomena observed in at least 450 river estuaries all around the world from Europe to America and Asia. Tidal bores manifest as a series of waves propagating over long distances upstream in the estuarine zone of a river. Bores can be studied experimentally using sloshing water tanks where sloshing itself is a process with many applications, not only relevant for environmental flows. In a remarkable paper, Cox and Mortell (1986) showed that for an oscillating water tank, the evolution of small-amplitude, long-wavelength, resonantly forced waves follow a forced Korteweg-de Vries (fKdV) equation. The solutions of this model agree well with experimental results by Chester and Bones (1968). At first glance this is surprising since their experimental setup is in conflict with a number of assumptions made for deriving the fKdV equation. It is hence worth to repeat the experiment by Chester and Bones but using a long narrow channel setup.</p> <p>We use a long circular channel and repeat the experiments by Chester and Bones. We compare the results with solutions from the fKdV equation but also with the one from a full nonlinear model solving the Navier-Stokes equations. Under resonance conditions, depending on the parameters, we find a range of nonlinear localized wave types from single and multiple solitons to undular bores. As shown by Cox and Mortell, when the fluid is considered to be inviscid a kind of Fermi-Pasta-Ulam recurrence is observed for the fKdV model. Stationarity is reached by including a weak damping to the fKdV equation.&#160;</p> <p>References<br />A.A. Cox, M.P. Mortell 1986. J. Fluid Mech. 162, pp. 99-116.<br />W. Chester and J.A. Bones 1968. Proc. Roy. Soc. A, 306, 23 (Part II).</p>
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