W. H. Hui

Directional Spectra of Wind-Generated Waves

From observations of wind and of water surface elevation at 14 wave staffs in an array in Lake Ontario and in a large laboratory tank, the directional spectrum of wind-generated waves on deep water is determined by using a modification of Barber’s (1963) method. Systematic investigations reveal the following: (a) the frequency spectrum in the rear face is inversely proportional to the fourth power of the frequency w, with the equilibrium range parameter and the peak enhancement factor clearly dependent on the ratio of wind speed to peak wave speed; (b) the angular spreading 0 of the wave energy is of the form sech2 (fid), where /? is a function of frequency relative to the peak; (c) depending on the gradient of the fetch, the direction of the waves at the spectral peak may differ from the mean wind direction by up to 50°, but this observed difference is predictable by a similarity analysis; (d) under conditions of strong wind forcing, significant effects on the phase velocity caused by amplitude dispersion and the presence of bound harmonics are clearly observed and are in accordance with the Stokes theory, whereas (e) the waves under natural wind conditions show amplitude dispersion, but bound harmonics are too weak to be detected among the background of free waves.

Turbulent airflow over water waves-a numerical study

Turbulent airflow over a Stokes water-wave train of small amplitude is studied numerically based on the two-equation closure model of Saffman & Wilcox (1974) together with appropriate boundary conditions on the wave surface. The model calculates, instead of assuming, the viscous sublayer flow, and it is found that the energy transfer between wind and waves depends significantly on the flow being hydraulically rough, transitional or smooth. Systematic computations have yielded a simple approximate formula for the fractional rate of growth per radian \[ \zeta = \delta_{\rm i}\frac{\rho}{\rho_{\rm w}}\left(\frac{U_{\lambda}}{c}-1 \right)^2, \] with δ i = 0.04 for transitional or smooth flow and δ i = 0.06 for rough flow, where ρ is density of air, ρ w that of water, U λ wind speed at one wavelength height and c the wave phase velocity. This formula is in good agreement with most existing data from field experiments and from wave-tank experiments. In the case of waves travelling against wind, the corresponding values are δ i = −0.024 for transitional and smooth flow, and δ i = −0.04 for rough flow.

SIMILARITY SOLUTIONS OF THE TWO-DIMENSIONAL UNSTEADY BOUNDARY-LAYER EQUATIONS

The method of Lie group transformations is used to derive all group-invariant similarity solutions of the unsteady two-dimensional laminar boundary-layer equations. A new method of nonlinear superposition is then used to generate further similarity solutions from a group-invariant solution. Our results are shown to include all the existing solutions as special cases. A detailed analysis is given to several classes of solutions which are also solutions to the full Navier–Stokes equations and which exhibit flow separation.

A new Lagrangian method for steady supersonic flow computation I. Godunov scheme

Abstract This paper studies the problem of steady two-dimensional supersonic flow of an inviscid compressible fluid using the new Lagrangian formulation of Hui and Van Roessel, in which the stream function and the Lagrangian time are used as independent variables. A shock capturing method is developed by applying the first-order Godunov scheme to the conservation form equations of this formulation. The method is fast and robust. Furthermore, extensive comparisons with exact solutions and with the second-order Godunov scheme of Glaz and Wardlaw based on the Eulerian formulation show that the first-order Lagrangian method generally attains the same level of accuracy as the second-order Eulerian method and is even better in resolving slip line discontinuities.

Mechanics of Ocean Surface Waves

For over a century ocean surface waves have held a sufficiently abundant source of mysteries to excite the imagination of generations of applied mathematicians and physicists, theoretians and experimentalists alike. The quickly developing field of satellite remote sensing and its intimate relationship with surface texture have added new impetus to this aspect of fluid mechanics. In this chapter we present the basic theoretical framework and observational evidence on which our current understanding of the mechanics of ocean surface waves rests. Our aim is to cover the ground as completely as the thrust of this book demands. Where the material is long established we content ourselves with a brief summary with references to more complete treatments. Where the material is new or directly related to remote sensing we attempt to provide a more complete picture.

Supersonic and hypersonic flow with attached shock waves over delta wings

A unified theory is developed for supersonic and hypersonic flow with attached shock waves over the lower surface of a delta wing at an angle of attack. The flow field on the lower surface of a delta wing consists of uniform flow regions near the leading edges, where the cross flow is supersonic and a nonuniform flow region near the central part, where the cross flow is subsonic. In the nonuniform flow region, the theory is based on the assumption that the flow differs slightly from the corresponding two-dimensional flow over a flat plate. Thus a linearized perturbation on a nonlinear flow field is first calculated and then strained and corrected so that the flow is matched continuously to the uniform flow which is obtained exactly. When compared with available exact numerical solutions the theory gives, in all cases, almost identical results, except near the crossflow sonic line where existing numerical methods fail to produce a discontinuous slope in the pressure curve, whereas the present theory predicts such a discontinuity and shows that the slope has a square root singularity at the crossflow sonic line similar to that in the supersonic linear theory.

Exact solutions of a three-dimensional nonlinear Schrödinger equation applied to gravity waves

The three-dimensional evolution of packets of gravity waves is studied using a nonlinear Schrodinger equation (the Davey–Stewartson equation). It is shown that permanent wave groups of the elliptic en and dn functions and their common limiting solitary sech forms exist and propagate along directions making an angle less than ψ c = tan −1 (1/√2) = 35° with the underlying wave field, whilst, along directions making an angle greater than ψ c , there exist permanent wave groups of elliptic sn and negative solitary tanh form. Furthermore, exact general solutions are given showing wave groups travelling along the two characteristic directions at ψ c or − ψ c . These latter solutions tend to form regions of large wave slope and are used to discuss the waves produced by a ship, in particular the nonlinear evolution of the leading edge of the pattern.

A new Lagrangian method for steady supersonic flow computation part II. Slip-line resolution

Abstract It is well known that high-order accurate shock-capturing schemes, e.g., second-order TVD and ENO schemes, based on Eulerian formulation are capable of resolving a shock discontinuity in two grid points, but they smear a slip-line (contact-line) discontinuity over several grid points. In this paper we show theoretically and numerically that the first-order Godunov scheme based on the new Lagrangian formulation of Hui and Van Roessel for steady supersonic flow always resolves an isolated slip-line discontinuity crisply, provided it is initially aligned with a grid line. Moreover, a simple extension of the second-order scalar TVD scheme of Sweby to the system of Euler equations based on the new Lagrangian formulation, with no special procedure for slip-line detection, resolves slip-line discontinuities in at most two grid points. Many examples are given, showing excellent agreement with known exact solutions.

Some classes of exact solutions of the nonlinear Boltzmann equation

We derive and discuss some exact solutions of the full (nonlinear) Boltzmann equation. One of these is the similarity solution recently found by Krook and Wu for the velocity relaxation problem. Other similarity solutions do exist, and we point out their usefulness in the search for exact solutions of the spatially inhomogeneous Boltzmann equation.

Computation of the Shallow Water Equations Using the Unified Coordinates

Two general coordinate systems have been used extensively in computational fluid dynamics: the Eulerian and the Lagrangian. The Eulerian coordinates cause excessive numerical diffusion across flow discontinuities, slip lines in particular. The Lagrangian coordinates, on the other hand, can resolve slip lines sharply but cause severe grid deformation, resulting in large errors and even breakdown of the computation. Recently, in the spirit of the arbitrary Lagrangian--Eulerian (ALE) approach, W. H. Hui, P. Y. Li, and Z. W. Li, [J. Comput. Phys., 153 (1999), pp. 596--637] have introduced a unified coordinate system which moves with velocity hq, q being the velocity of the fluid particle. It includes the Eulerian system as a special case when h=0, and the Lagrangian when h=1, and was shown for the two-dimensional Euler equations of gas dynamics to be superior to both Eulerian and Lagrangian systems. The main purpose of this paper is to adopt this unified coordinate system to solve the shallow water equations. It will be shown that computational results using the unified system are superior to existing results based on either the Eulerian system or Lagrangian system in that it (a) resolves slip lines sharply, especially for steady flow, (b) avoids grid deformation and computation breakdown in Lagrangian coordinates, and (c) avoids spurious flow produced by Lagrangian coordinates.