A stratified liquid with two layers separated by a fast oscillating interface in the case of Raleigh--Taylor instability is considered. The averaged equations are derived, and it is shown that a mushy region of a certain density appears after averaging. The similarity between this fact and the case of unstable jump decay is discussed.
The bottom pressure distribution under solitonic waves, travelling or fully reflected at a wall is analysed here. Results given by two kind of numerical models are compared. One of the models is based on the Green–Naghdi equations, while the other one is based on the fully nonlinear potential equations. The two models differ through the way in which wave dispersion is taken into account. This approach allows us to emphasize the influence of dispersion, in the case of travelling or fully reflected waves. The Green–Naghdi model is found to predict well the bottom pressure distribution, even when the quantitative representation of the runup height is not satisfactorily described.
An asymptotic solution with double-deck boundary layer structure is constructed for the problem of an incompressible fluid flow along a semi-infinite plate with small localized or periodic (fast-oscillating) irregularities on the surface whose shape depends on time. Numerical simulation of flow in the near-plate region is presented for two types of the shape change: oscillations of the amplitude of irregularities and motion of the irregularity in the upstream and downstream directions. The influence of the time-dependent shape on the flow behavior (including the vortex formation process) is shown in detail.
We consider the problem of flow of a viscous compressible subsonic fluid along a flat plate with small localized (hump-type) irregularities on the surface for large Reynolds numbers. We obtain a formal asymptotic solution with double-deck structure of the boundary layer. We present the results of numerical simulation of the flow in the thin boundary layer (i.e., in the near-boundary region).
We consider the problem of a viscous compressible subsonic fluid flow along a flat plate with small periodic irregularities on the surface for large Reynolds numbers. We obtain a formal asymptotic solution with double-deck structure of the boundary layer. We present the results of numerical simulation of flow in the thin boundary layer (i.e., in the near-boundary region).
Direct numerical simulations of irregular unidirectional nonlinear wave evolution are performed within the framework of the Korteweg–de Vries equation for bimodal wave spectra model cases. The additional wave system co-existence effect on the evolution of the wave statistical characteristics and spectral shapes, and also on the attained equilibrium state is studied. The concerned problem describes, for example, the interaction between wind waves and swell in shallow seas. It is next demonstrated that a low-frequency spectral component yields more asymmetric waves with more extreme statistical properties.
The bottom pressure distribution beneath large amplitude waves is studied within linear theory in time and space domain, weakly dispersive Serre–Green–Naghdi system and fully nonlinear potential equations. These approaches are used to compare pressure fields induced by solitary waves, but also by transient wave groups. It is shown that linear analysis in time domain is in good agreement with Serre– Green–Naghdi predictions for solitary waves with highest amplitude A = 0.7h, h being water depth. In the meantime, when comparing results to fully nonlinear potential equations, neither linear theory in time domain, nor in space domain, provide a good description of the pressure peak. The linear theory in time domain underestimates the peak by an amount similar to the overestimation by linear theory in space domain. For transient wave groups (up to A = 0.52h), linear analysis in time domain provides results very similar to those based on the Serre–Green–Naghdi system. In the meantime, linear theory in space domain provides a surprisingly good comparison with prediction of fully nonlinear theory. In all cases, it has to be emphasized that a discrepancy between linear theory in space domain and in time domain was always found, and presented an averaged value of 20%. Since linear theory is often used by coastal engineers to reconstruct water elevation from bottom mounted sensors, the so-called inverse problem, an important result of this work is that special caution should be given when doing so. The method might surprisingly work with strongly nonlinear waves, but is highly sensitive to the imbalance between nonlinearity and dispersion. In most cases, linear theory, in both time and space domain, will lead to important errors when solving this inverse problem.
The paper presents results of numerical simulations of freely rising solid spheres in a viscous fluid. The diameter of spheres was 5 mm, 7 mm, 10 mm, and 20 mm, and the corresponding Reynolds numbers varies in the interval 1400 < Re < 10100. It has been found that the free rise path varies, as the Galileo number increases. The paper describes the principal zigzag path generating mechanism, which is determined by regularly generated and freed, oppositely oriented vortex structures, which carry away the momentum and mass of fluid from a rising sphere. Quantitative characteristics are given with respect to oscillation periods of a sphere path, transverse velocity components, and accumulation of fluid mass involved in the attached vorticity. It has been found that the vortex detachment period is coincident with the half-period of oscillations. A critical mass of a fluid in the attached vorticity has been calculated, and a mechanism of changeover from zigzagging to spiraling has been suggested
A weakly-nonlinear potential theory is developed for the description of deep penetrating pressure fields caused by single and colliding wave groups of collinear waves due to the second-order nonlinear interactions. The result is applied to the representative case of groups with the sech-shape of envelope solitons in deep water. When solitary groups experience a head-on collision, the induced due to nonlinearity dynamic pressure may have magnitude comparable with the magnitude of the linear solution. It attenuates with depth with characteristic length of the group, which may greatly exceed the individual wave length. In general the picture of the dynamic pressure beneath intense wave groups looks complicated. The qualitative difference in the structure of the induced pressure field for unidirectional and opposite wave trains is emphasized.