In many physical situations, the oscillations of the free surface of a fluid are a random process in space and time. This is equally correct for ripples in a tea cup as well as for large ocean waves. In both cases, the situation must be described by the averaged equations imposed on a certain set of correlation functions. The derivation of such equations is not a simple problem even on a `physical' level of rigor. It is especially important to determine correctly the conditions of applicability for a given statistical description. For some physical reasons they might happen to be narrow. In this context, the statistical description of potential surface waves on the surface of an ideal fluid of finite depth is described.
Three proposed local criteria for the identification of vortices are analyzed and discussed; they are based on the analysis of invariants of the velocity gradient tensor ▿u or invariants of the tensor +Ω , where and are the symmetric and antisymmetric parts of ▿u. Moreover, a tentative non-local procedure is proposed, which takes advantage of the observation that vortices tend to be made up of the same fluid particles; this leads to the definition of a Galilean invariant quantity, which can be computed and used to identify vortical structures. Three analytical flow fields are used for a comparative evaluation of both local and non-local criteria, which allows a deeper understanding of the physical meaning of the considered techniques.
We consider two problems of nonlinear flow in porous media: 1) a derivation of a cubic weak inertia correction of Darcy's law which is valid for any matrix anisotropy, and 2) a description of flow by the weak inertia equation for low Reynolds numbers and a Forchheimer equation for high Reynolds number laminar flow. Recent homogenization studies show that the weak inertia correction to Darcy's law is not a square term in velocity, as it is in the Forchheimer equation, but instead a cubic term in velocity. By imposing that the pressure loss is invariant under flow reversion, it has been shown that the weak inertia equation is valid even for anisotropic media. We show, by using the homogenization technique, that the weak inertia equation is valid for any anisotropic matrix symmetry without imposing a reversed flow symmetry. For the second problem, we reexamine published data. We find that the description 2) applies well. A spline may be applied in the crossover regime. (C) Elsevier, Paris.
In this paper we consider the circulation induced by waves breaking near a coast. We show that the vertical nonuniformity of the wave-averaged horizontal velocities leads to mixing-like terms for the horizontal velocity in the depth-integrated equations of momentum. The mechanism is analogous to Taylor's (1953, 1954) shear-dispersion mechanism for solutes in a shear flow. The results presented here are an extension of the results found by Svendsen & Putrevu (1994) to the general case of unsteady flow over an arbitrary bottom topography.
A special form of the Boltzmann collision operator for the hard spheres model is introduced. The possibilities of fast numerical computation of the collision operator based on this form and the Fast Fourier Transform are discussed. A new difference scheme for the Boltzmann equation for the hard spheres model is developed. The results of some numerical tests and accuracy comparisons with the Direct Simulation Monte Carlo (DSMC) method are presented.
The paper addresses a new approach for investigating and evaluating the basic properties of distributive laminar mixing in two-dimensional creeping flows by analyzing a periodic Stokes flow in an annular wedge cavity. Flow is induced by the repetitive motion of the curved top and bottom walls, with prescribed velocities. An analytical solution for the velocity field in the cavity is presented, along with the algorithm for line tracking, which conserves the topological properties of any closed contour. A technique for finding all periodic points in the flow, and quantitative measures for the estimation of the mixing quality at any instant, are derived and applied to the flow in the wedge cavity.
The stability of the interface separating two immiscible incompressible fluids of different densities and viscosities is considered in the case of fluids filling a cavity which performs horizontal harmonic oscillations. There exists a simple basic state which corresponds to the unperturbed interface and plane-parallel unsteady counter flows; the properties of this state are examined. A linear stability problem for the interface is formulated and solved for both (a) inviscid and (b) viscous fluids. A transformation is found which reduces the linear stability problem under the inviscid approximation to the Mathieu equation. The parametric resonant regions of instability associated with the intensification of capillary-gravity waves at the interface are examined and the results are compared to those found in the viscous case in a fully numerical investigation.
Analytical and experimental investigations were conducted on short-crested wave fields generated by a sea-wall reflection of an incident plane wave. A perturbation method was used to compute analytically the solution of the basic equations up to the sixth order for capillary-gravity waves in finite depth, and up to the ninth order for gravity waves in deep water. For the experiments, we developed a new video-optical tool to measure the full three dimensional wave field η(x, y, t). A good agreement was found between theory and experiments. The spatio-temporal bi-orthogonal decomposition technique was used to exhibit the periodic and progressive properties of the short-crested wave field.
The velocity field and mixing behaviour in the so-called partitioned pipe mixer were studied. Starting with the same physical model as in previous studies, an exact analytical solution was developed which yields a more accurate description of the flow than the previously used approximate solution. Also, the results are in better accordance with the reported experimental data.
It is often quoted that Gortler vortices cannot be described by a local eigenvalue analysis. In this work, by using the inverse of the Gortler number as a small expansion parameter, we derive an asymptotic sequence continuable to all orders which is similar, in principle, to the one that justifies the application of the Orr-Sommerfeld equation to two-dimensional boundary-layer instabilities. Existing local theories from the literature can be framed within the leading term of this expansion; however, none of the heuristically proposed non-parallel corrections fully captures the next higher term. We show that, when this term is included, locally computed growth rates quickly collapse onto those obtained from numerical simulations of the parabolic linear stability equations, with initial conditions applied at the leading edge. The Gortler number (or, equivalently, the downstream distance) beyond which this non-parallel local theory is found out to be accurate encloses the commonly recognized experimental range. The small Gortler number (short distance) effect of initial conditions is described in a companion paper. (C) Elsevier, Paris.
Water wave breaking is of considerable importance in the transfer of momentum, and in other transfers, between the atmosphere and oceans. Typically breaking occurs on deep water as events that have finite duration and finite spatial extent. Near shore lines most of the water motions are dominated by breaking waves. Recent work on the generation of vorticity by breaking waves and bores in the surf zone on beaches is considered and typical vortical structures are briefly discussed. Consideration of deep water breaking leads to the proposal that the end result of a breaking event in deep water may be a coherent structure within the resulting current field. Such a structure is topologically equivalent to half a vortex ring.
Numerical simulation of dynamical equations for capillary waves excited by long-scale forcing shows the presence of both Kolmogorov spectrum at high wavenumbers (with the index predicted by weak-turbulent theory) and non-monotonic spectrum at low wavenumbers. The value of the Kolmogorov constant measured in numerical experiments happens to be different from the theoretical one. We explain the difference by the coexistence of Kolmogorov and `frozen' turbulence with the help of maps of quasi-resonances. Observed results are believed to be generic for different physical dispersive systems and are confirmed by laboratory experiments.
Spectral methods are very efficient and powerful for solving periodic problems. A new spectral method is developed for problems with no spatial periodicity, and demonstrated for water waves. The method splits the potential into the sum of a prescribed non-periodic component and an unknown periodic component. Computed results are compared with experiments by Shemer et al (1998).
An accurate numerical solution of the momentum and the heat transfer through a rarefied gas confined between two cylinders rotating with different angular velocities and having different temperatures has been obtained over a wide range of the Knudsen number on the basis of the Bhatnagar, Gross, Krook model equation. The viscous stress tensor, heat flux, and the fields of density, temperature and velocity are found. An analysis of the influence of the angular velocities and the temperature ratio on these quantities is given. (C) Elsevier, Paris.
Steady two-dimensional laminar flow through an infinite array of parallel circular cylinders is computed numerically for values of the Reynolds number , based on oncoming velocity and cylinder diameter, up to 40, and for values of the spacing parameter, (the ratio of the distance between cylinder axes to the cylinder radius), ranging from 2.3 to 10. The method used is that of . Results are presented for the dimensionless drag on a cylinder, (or, equivalently, the dimensionless permeability of the array, β = W/D) and are compared with previous analytical results for very small and either wide or very narrow gaps. Results are also presented for the efficiency with which a filter consisting of such an array would capture spherical particles of radius by direct interception, assuming that particle centres follow streamlines and that a particle is captured whenever it touches a cylinder. Such results are applicable to the study of filter feeding by small aquatic organisms.
We report the first systematic laboratory observations of 3-D features of wind waves at early stages of wave field development. The experiments performed in the large IRPHE-Luminy wind-wave tank provided instantaneous reconstruction of the decimeter-scale water surface motions based on simultaneous imaging of the wave slopes in two perpendicular directions. Five essentially distinct regimes in the 3-D evolution of the dominant waves have been identified. Each regime is characterized by different types of 3-D wave patterns associated with specific ranges of wave scale and wave steepness. The likely scenario of the evolution and the possible physical mechanisms of the pattern formation are discussed.
Atmospheric boundary layer flow over surface water waves of small slope is analyzed with a new heuristic method that clearly shows the underlying physical mechanisms. In this method we consider how the wave displaces mean streamlines in the air flow. Turbulence in the air flow is found to affect the flow over the wave only in a thin inner region that lies close to the interface. The streamline displacement at the top of this inner region has three contributions: displacement over the undulating wave surface; a Bernoulli contribution associated with pressure variations over the wave (which is associated with higher wind speeds at the waves crests and lower wind speeds in the troughs); and a displacement caused by the turbulent stresses in the air flow. The displacement caused by turbulent stresses is a factor (u*/U ) smaller than the other two contributions (u* is the friction velocity and U is the wind speed at the top of the inner region), but is important because it leads to the winds being slightly faster on the upwind side of the wave crest compared to in the lee and to the streamline-displacement pattern being shifted slightly downwind of the wave crest. This then leads to a small surface pressure difference across the wave crest and thence wave growth. This is the non-separated sheltering mechanism. The solutions obtained here using physically-based heuristic arguments are in full agreement with those calculated using formal asymptotic methods by Belcher & Hunt (1993) and Cohen & Belcher (1999). The understanding gained from the new method suggests a nonlinear correction to the formula for wave growth that tends to reduce the wave growth rate for steeper waves, in agreement with computations.
The one-dimensional steady heat flow in a dense hard sphere gas is studied solving the Enskog equation numerically by a recently proposed DSMC-like particle scheme. The accuracy of the solutions is assessed through a comparison with solutions obtained from a semi-regular method which combines finite difference discretization with Monte Carlo quadrature techniques. It is shown that excellent agreement is found between the two numerical methods. The solutions obtained from the Enskog equation have also been found in good agreement with the results of molecular dynamics simulations. (C) Elsevier, Paris.