Variational Principle for Velocity-Pressure Formulation of Navier-Stokes Equations

The work described here shows that the known variational principle for the Navier-Stokes equations and the adjoint system can be modified to produce a set of Euler-Lagrange variational equations which have the same order and same solution as the Navier-Stokes equations provided the adjoint system has a unique solution, and provided in the steady state case, that the Reynolds number remains finite.Comment: 10 page

An Asymptotic Analysis for Generation of Unsteady Surface Waves on Deep Water by Turbulence

The detailed mathematical study of the recent paper by Sajjadi, Hunt and Drullion (2014) is presented. The mathematical developement considered by them, for unsteady growing monochromatic waves is also extended to Stokes waves. The present contribution also demonstrates agreement with the pioneering work of Belcher and Hunt (1993) which is valid in the limit of the complex part of the wave phase speed ci ↓ 0. It is further shown that the energy-transfer parameter and the surface shear stress for a Stokes wave reverts to a monochromatic wave when the second harmonic is excluded. Furthermore, the present theory can be used to estimate the amount of energy transferred to each component of nonlinear surface waves on deep water from a turbulent shear flow blowing over it. Finally, it is demonstrated that in the presence of turbulent eddy viscosity the Miles (1957) critical layer does not play an important role. Thus, it is concluded that in the limit of zero growth rate the effect of the wave growth arises from the elevated critical layer by finite turbulent diffusivity, so that the perturbed flow and the drag force is determined by the asymmetric and sheltering flow in the surface shear layer and its matched interaction with the upper region

Growth of Stokes Waves Induced by Wind on a Viscous Liquid of Infinite Depth

The original investigation of Lamb (1932, x349) for the effect of viscosity on monochromatic surface waves is extended to account for second-order Stokes surface waves on deep water in the presence of surface tension. This extension is used to evaluate interfacial impedance for Stokes waves under the assumption that the waves are growing and hence the surface waves are unsteady. Thus, the previous investigation of Sajjadi et al. (2014) is further explored in that (i) the surface wave is unsteady and nonlinear, and (ii) the effect of the water viscosity, which affects surface stresses, is taken into account. The determination of energy-transfer parameter, from wind to waves, are calculated through a turbulence closure model but it is shown the contribution due to turbulent shear flow is some 20% lower than that obtained previously. A derivation leading to an expression for the closed streamlines (Kelvin cat-eyes), which arise in the vicinity of the critical height, is found for unsteady surface waves. From this expression it is deduced that as waves grow or decay, the cats-eye are no longer symmetrical. Also investigated is the energy transfer from wind to short Stokes waves through the viscous Reynolds stresses in the immediate neighborhood of the water surface. It is shown that the resonance between the Tollmien-Schlichting waves for a given turbulent wind velocity profile and the free-surface Stokes waves give rise to an additional contribution to the growth of nonlinear surface waves

The Time Periodic Solution of the Burgers Equation on the Half-Line and an Application to Steady Streaming

The evolution of large amplitude Tollmien-Schlichting waves in boundary layer flows over wavy surfaces is considered for two-dimensional disturbances which are locally periodic in time and space. Consideration is given to both large Re 1 and Re ∼ O(1) Reynolds numbers using asymptotic methods. The large Reynolds number analysis is valid for oscillatory two-dimensional turbulent boundary layer. In both cases the phase equation approach shows that the wavenumber and frequency will develop shocks or other discontinuities as the disturbance evolves. It is shown that the evolution of constant frequency/wavenumber disturbances and their modulational instability is controlled by Burgers equation at finite Reynolds number and by a new integro-differential evolution equation at large Reynolds numbers. The Burgers equation is formulated on the half line, using Fokas’ method, which provides a simple model of the above phenomenon. The physical situation corresponds to the solution of the Dirichlet problem on the half-line, which decays as x → ∞ and which is time periodic. It is shown that the Dirichlet problem, where the usual prescription of the initial condition is now replaced by the requirement of the time periodicity, yields a well posed problem. Furthermore, it is also shown that the solution of this problem tends to the “inner” and “outer” solutions obtained by the perturbation expansions. For the large Reynolds number case the evolution equation points to the development of a spatially localized singularity at a finite time. The three-dimensional generalizations of the evolution equations is also given for the case of weak spanwise modulations

Growth of Groups of Wind Generated Waves

In this paper we demonstrate numerical computations of turbulent wind blowing over group of waves that are growing in time. The numerical model adopted for the turbulence model is based on differential second-moment model that was adopted for growing idealized waves by Drullion & Sajjadi (2014). The results obtained here demonstrate the formation of cat\u27s-eye which appear asymmetrically over the waves within a group

Numerical Studies of Particle Laden Flow in Dispersed Phase

To better understand the hydrodynamic flow behavior in turbulence, Particle-Fluid flow have been studied using our Direct Numerical(DNS) based software DSM on MUSCL-QUICK and finite volume algorithm. The particle flow was studied using Eulerian-Eulerian Quasi Brownian Motion(QBM) based approach. The dynamics is shown for various particle sizes which are very relevant to spray mechanism for Industrial applications and Bio medical applications

A Study of Energy Transfer of Wind and Ocean Waves

To develop a better understanding of energy transfer between wind and different types of waves a model was created to determine growth factors and energy transfers on breaking waves and the resulting velocity vectors. This model was used to build on the research of Sajjadi et al (1996) on the growth of waves by sheared flow and takes models of wave velocities developed by Weber and Melsom (1993) and end energy transfer by Sajjadi, Hunt and Drullion (2012)