Flow induced on a salt waterbody due to the impingement of a freshwater drop or a water source

The particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) techniques are used to study the flow induced on the surface of a body of saltwater when a drop impinges on its surface or when a source is present on the surface. The measurements show that the impingement of a fresh water drop causes a strong axisymmetric solutocapillary flow about the vertical line passing through the center of impact. The fluid directly below the center of impact rises upward, and near the surface it moves away from the center of impact. The flow, which develops within a fraction of second after the impact, persists for several seconds. In comparison, when a freshwater drop falls on a body of freshwater, the flow induced on the surface is much weaker and persists for a relatively shorter duration of time and the volume of water circulated is two orders of magnitude smaller. Similarly, when a fresh water source is present on a body of saltwater there is a solutocapillary flow which on the surface is away from the source and below the surface is towards the source.     

A spectral boundary integral method for inviscid water waves in a finite domain

In this paper, we show how the spectral formulation of Baker, Meiron and Orszag can be used to solve for waves on water of infinite depth confined between two flat, vertical walls, and also how it can be modified to take into account water of finite depth with a spatially varying bottom. In each case, we use Chebyshev polynomials as the basis of our representation of the solution and filtering to remove spurious high‐frequency modes. We show that spectral accuracy can be achieved until wave breaking, plunging or wall impingment occurs in two model problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Variational formulation of three-dimensional viscous free-surface flows: Natural imposition of surface tension boundary conditions

We present a new surface-intrinsic linear form for the treatment of normal and tangential surface tension boundary conditions in C⁰-geometry variational discretizations of viscous incompressible free-surface flows in three space dimensions. The new approach is illustrated by a finite (spectral) element unsteady Navier-Stokes analysis of the stability of a falling liquid film.     

A linearized Riemann solver for the steady supersonic Euler equations

A very simple linearization of the solution to the Riemann problem for the steady supersonic Euler equations is presented. When used locally in conjunction with the Godunov method, computing savings by a factor of about four relative to the use of exact Riemann solvers can be achieved. For severe flow regimes, however, the linearization loses accuracy and robustness. We then propose the use of a Riemann solver adaptation procedure. This retains the accuracy and robustness of the exact Riemann solver and the computational efficiency of the cheap linearized Riemann solver. Numerical results for two- and three-dimensional test problems are presented.     

Energy-balance equation for turbulent pulsations in the theory of the free turbulent boundary layer

Semiempirical expressions are proposed for the coefficient of turbulent viscosity and for the scale of turbulence in the equations for the free turbulent boundary layer in an incompressible fluid, these equations consisting of the equation of continuity, the equations of motion, and the equation for the average energy balance in the turbulent pulsations. The advantage of the expressions over the existing ones is that the two empirical constants in the equations have nearly the same values for circular and plane turbulent streams and also for a turbulent boundary layer at the edge of a semiinfinite homogeneous flow with a stationary fluid. The mean-energy distribution and the mean energy of the turbulent pulsations computed in this paper agree well with the experimental values.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 75–79, November–December, 1970.     

Nonlinear theory of the propagation of pulses of electromagnetic waves through a plasma boundary

A solution is obtained for the problem of the propagation of electromagnetic waves of arbitrary form through a plasma boundary on condition that the length of the wave train is much greater than the wave length. A solution is found both for the case of a wide spectrum of width much greater than the plasma frequency ₀, as well as for a narrow spectrum. The results obtained enable us to draw conclusions about the time and space variation of the shape of electromagnetic pulses in a plasma.The passage of high frequency electromagnetic waves through a plasma is similar to that of a beam of charged particles [1, 2]. This is associated with the fact that decay processes are similar to Cerenkov radiation effects. The dynamics of the development of transverse wave instabilities in a uniform Isotropic plasma were studied in [2] assuming that the wave phase behaves stochastically. It was calculated here that instabilities develop quite differently in the case of a wide frequency spectrum than in the case of a narrow monochromatic spectrum. If we can speak of transverse quanta diffusion effects in the field of the generated longitudinal quanta in the first case, and if the resulting effects are closely similar to the nonlinear effects arising when beam instability develops [3, 4], then the development of instabilities in the case of a narrow spectrum leads to the appearance of red satellites in the transverse wave spectrum differing from the basic frequency by a quantity ₀ (=1, 2, 3,...). In this case the development of the instability corresponds to a tendency for a plateau over the satellites to appear.Attention should however be drawn to the fact that the dynamics of instability development in a semibounded plasma may be quite different. This is associated first with the different values of group velocities of transverse and longitudinal waves, and what is also important, with the effect of longitudinal wave accumulation in the boundary region if the length of the wave train is sufficiently large. The treatment of a similar problem for beam instabilities in paper [5] showed that a narrow transition layer may arise with a transverse wave energy density greatly in excess of the energy density of the injected beam. In what follows we examine the part played by boundary effects in the passage of pulses of electromagnetic waves through the boundary of the plasma. The cases of both narrow and wide spectra are considered. We note that in the case of narrow spectra the wave train must necessarily be greatly in excess of ^–1, and the effects of the accumulation of oscillations will be appreciable.The phases of both transverse waves, and also generated longitudinal waves are assumed to be stochastic quantities. The boundary effects which have been treated may be applied both in the generation of longitudinal waves necessary for the effective acceleration of particles in a plasma as well as in the modulation and alteration of the initial transverse wave spectrum. It should also be stressed that these effects which have been considered could be applied for turbulent plasma diagnostics, as has already been pointed out in [2].The authors are grateful to Ya. B. Fainberg, M. S. Rabinovich, I. S. Danilkin, and M. D. Raizer for their interest in the paper and for valuable criticisms.     

An intermediate wavelength, weakly nonlinear theory for the evolution of capillary gravity waves

A new nonlinear evolution equation is derived for surface solitary waves propagating on a liquid-air interface where the wave motion is induced by a harmonic forcing. Instead of the traditional approach involving a base state of the long wave limit, a base state of harmonic waves is assumed for the perturbation analysis. This approach is considered to be more appropriate for channels of length just a few multiples of the depth. The dispersion relation found approaches the classical long wave limit. The weakly nonlinear dispersive waves satisfy a KdV-like nonlinear evolution equation with steeper nonlinearity.     

Finite element implementation of boundary conditions for the pressure Poisson equation of incompressible flow

In this paper we address the problem of the implementation of boundary conditions for the derived pressure Poisson equation of incompressible flow. It is shown that the direct Galerkin finite element formulation of the pressure Poisson equation automatically satisfies the inhomogeneous Neumann boundary conditions, thus avoiding the difficulty in specifying boundary conditions for pressure. This ensures that only physically meaningful pressure boundary conditions consistent with the Navier-Stokes equations are imposed. Since second derivatives appear in this formulation, the conforming finite element method requires C¹ continuity. However, for many problems of practical interest (i.e. high Reynolds numbers) the second derivatives need not be included, thus allowing the use of more conventional C⁰ elements. Numerical results using this approach for a wall-driven contained flow within a square cavity verify the validity of the approach. Although the results were obtained for a two-dimensional problem using the p-version of the finite element method, the approach presented here is general and remains valid for the conventional h-version as well as three-dimensional problems.     

A method for analysing nesting techniques for the linearized shallow water equations

A method for analysing different nesting techniques for the linearized shallow water equations is presented. The problem is formulated as an eigenvector–eigenvalue problem. A necessary condition for stability is that the spectral radius of the propagation matrix is less than or equal to one. Two test cases are presented. The first test case is analysed, and effects of enforcing volume conservation and nudging in time are studied. A nesting technique is found that causes no growth of any eigenvectors for reasonable time steps. This nesting technique is then used on both test cases, and results are compared to an everywhere refined model and a coarse grid model. Copyright © 2002 John Wiley & Sons, Ltd.     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

Wronskian,Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves

Application of the shallow water waves in environmental engineering and hydraulic engineering is seen. In this paper, a (3+1)-dimensional generalized nonlinear evolution equation (gNLEE) for the shallow water waves is investigated. The Nth-order Wronskian, Gramian and Pfaffian solutions are proved, where N is a positive integer. Soliton solutions are constructed from the Nth-order Wronskian, Gramian and Pfaffian solutions. Moreover, we analyze the second-order solitons with the influence of the coefficients in the equation and illustrate them with graphs. Through the Hirota-Riemann method, one-periodic-wave solutions are derived. Relationship between the one-periodic-wave solutions and one-soliton solutions is investigated, which shows that the one-periodic-wave solutions can approach to the one-soliton solutions under certain conditions. We reduce the (3+1)-dimensional gNLEE to a two-dimensional planar dynamic system. Based on the qualitative analysis, we give the phase portraits of the dynamic system.

     

An Analytical Model of the Dynamics of the Liquefied Vitreous Induced by Saccadic Eye Movements

An analytical model of the dynamics of the vitreous humour induced by saccadic movements within the eye globe is presented. The vitreous is treated as a weakly viscous Newtonian incompressible fluid, an assumption which is appropriate when the vitreous is liquefied or when it is replaced by aqueous humour after surgery. The thin viscous boundary layer generated during a saccadic movement on the side wall is neglected and the flow field is assumed to be irrotational. The vitreous chamber is described as a weakly deformed sphere and this assumption allows a linear treatment of the problem. An analytical solution is found in the form of an expansion of spherical harmonics. Results show that the non-spherical shape of the container generates a flow field characterised by significant velocities and strong three-dimensionality. The model allows the computation of the dynamic pressure on the wall, which may play a role in the generation of retinal detachments. Moreover, results suggest that the irregular shape of the globe may significantly modify tangential stresses on the boundary with respect to the case of motion within a sphere. A simplified analytical solution, for the case of two-dimensional flow within an impulsively rotated container, shows that boundary layer detachment is expected to occur for angles of rotation larger than a threshold value of 15° circa.     

Wave kinematic fields from the boundary integral method

The predictive potential of interior domain solutions from the boundary integral method for 2D extreme wave kinematics is explored. Comparisons with analytical solutions for near‐limit waves confirms the susceptibility of the boundary integral method to poor precision at near‐boundary locations. Additionally, these comparisons identify a domain‐wide precision challenge that is associated with the relatively rapid changes in water surface geometry and kinematics that are typical of extreme waves. A numerical evaluation of Green's integral around the boundary addresses these precision issues through formulation of the integration as a simultaneous system of ordinary differential equations at a cubic level of approximation. Careful attention is given to consistent interpolation of all contributions to the Green's integral. Copyright © 2005 John Wiley & Sons, Ltd.