SERIES PERTURBATIONS APPROXIMATE SOLUTIONS TO N-S EQUATIONS AND MODIFICATION TO ASYMPTOTIC EXPANSION MATCHED METHOD

A method that series perturbations approximate solutions to N-S equations with boundary conditions was discussed and adopted. Then the method was proved in which the asymptotic solutions of viscous fluid flow past a sphere were deducted. By the ameliorative asymptotic expansion matched method, the matched functions, are determined easily and the ameliorative curve of drag coefficient is coincident well with measured data in the case that Reynolds number is less than or equal to 40 000.

Oblique wave scattering by a semi-infinite elastic plate with finite draft floating on a step topography

On the basis of a potential theory and Euler-Bernoulli beam theory, an analytical solution for oblique wave scattering by a semi-infinite elastic plate with finite draft floating on a step topography is developed using matched eigenfunction expansions. Different from previous studies, the effects of a wave incident angle, a plate draft, three different plate edge conditions （free, simply supported and built-in） and a sea-bottom topography are all taken into account. Moreover, the plate edge conditions are directly incorporated into linear algebraic equations for determining unknown expansion coefficients in velocity potentials, which leads to a simple and efficient solving procedure. Numerical results show that the convergence of the present solution is good, and an energy conservation relation is well satisfied. Also, the present predictions are in good agreement with known results for special cases. The effects of the wave incident angle, the plate draft, the plate edge conditions and the sea-bottom topography on various hydrodynamic quantities are analyzed. Some useful results are presented for engineering designs.

Dynamics of Vapor Bubble in a Variable Pressure Field

This paper presents analytical and numerical results of vapor bubble dynamics and acoustics in a variable pressure field.First,a classical model problem of bubble collapse due to sudden pressure increase is introduced.In this problem,the Rayleigh–Plesset equation is treated considering gas content,surface tension,and viscosity,displaying possible multiple expansion–compression cycles.Second,a similar investigation is conducted for the case when the bubble originates near the rounded leading edge of a thin and slightly curved foil at a small angle of attack.Mathematically the flow field around the foil is constructed using the method of matched asymptotic expansions.The outer flow past the hydrofoil is described by linear(small perturbations)theory,which furnishes closed-form solutions for any analytical foil.By stretching local coordinates inversely proportionally to the radius of curvature of the rounded leading edge,the inner flow problem is derived as that past a semi-infinite osculating parabola for any analytical foil with a rounded leading edge.Assuming that the pressure outside the bubble at any moment of time is equal to that at the corresponding point of the streamline,the dynamics problem of a vapor bubble is reduced to solving the Rayleigh-Plesset equation for the spherical bubble evolution in a time-dependent pressure field.For the case of bubble collapse in an adverse pressure field,the spectral parameters of the induced acoustic pressure impulses are determined similarly to equivalent triangular ones.The present analysis can be extended to 3D flows around wings and screw propellers.In this case,the outer expansion of the solution corresponds to a linear lifting surface theory,and the local inner flow remains quasi-2D in the planes normal to the planform contour of the leading edge of the wing(or screw propeller blade).Note that a typical bubble contraction time,ending up with its collapse,is very small compared to typical time of any variation in the flow.Therefore,the approach can also be applied to unsteady flow problems.

Hydroelastic interaction between water waves and thin elastic plate floating on three-layer fluid

The wave-induced hydroelastic responses of a thin elastic plate floating on a three-layer fluid, under the assumption of linear potential flow, are investigated for two-dimensional cases. The effect of the lateral stretching or compressive stress is taken into account for plates of either semi-infinite or finite length. An explicit expression for the dispersion relation of the flexural-gravity wave in a three-layer fluid is analytically deduced. The equations for the velocity potential and the wave elevations are solved with the method of matched eigenfunction expansions. To simplify the calculation on the unknown expansion coefficients, a new inner product with orthogonality is proposed for the three-layer fluid, in which the vertical eigenfunctions in the open-water region are involved. The accuracy of the numerical results is checked with an energy conservation equation, representing the energy flux relation among three incident wave modes and the elastic plate. The effects of the lateral stresses on the hydroelastic responses are discussed in detail.

Iterative analytical solution for wave reflection by a multichamber partially perforated caisson breakwater

This study examines wave reflection by a multi-chamber partially perforated caisson breakwater based on potential theory.A quadratic pressure drop boundary condition at perforated walls is adopted,which can well consider the effect of wave height on the wave dissipation by perforated walls.The matched eigenfunction expansions with iterative calculations are applied to develop an analytical solution for the present problem.The convergences of both the iterative calculations and the series solution itself are confirmed to be satisfactory.The calculation results of the present analytical solution are in excellent agreement with the numerical results of a multi-domain boundary element solution.Also,the predictions by the present solution are in reasonable agreement with experimental data in literature.Major factors that affect the reflection coefficient of the perforated caisson breakwater are examined by calculation examples.The analysis results show that the multi-chamber perforated caisson breakwater has a better wave energy dissipation function(lower reflection coefficient)than the single-chamber type over a broad range of wave frequency and may perform better if the perforated walls have larger porosities.When the porosities of the perforated walls decrease along the incident wave direction,the perforated caisson breakwater can achieve a lower reflection coefficient.The present analytical solution is simple and reliable,and it can be used as an efficient tool for analyzing the hydrodynamic performance of perforated breakwaters in preliminary engineering design.

LIQUID-GAS COEXISTENCE EQUILIBRIUM IN A RELAXATION MODEL

Stability of liquid-gas coexistence equilibrium in a relaxation model for isothermal phase transition in a sealed one-dimensional tube was discussed. With matched asymptotic expansion, a linear system for first order perturbations was derived formally. By solving this system analytically, it is shown that small initial perturbations are damped out in general; yet they may maintain at certain level for special cases. Numerical evidence is presented. This manifests the regularization effects of relaxation.

Thermoelastic stresses in SiC single crystals grown by the physical yapor transport method

A finite element-based thermoelastic anisotropic stress model for hexagonal silicon carbide polytype is developed for the calculation of thermal stresses in SiC crystals grown by the physical vapor transport method. The composite structure of the growing SiC crystal and graphite lid is considered in the model. The thermal expansion match between the crucible lid and SiC crystal is studied for the first time. The influence of thermal stress on the dislocation density and crystal quality is discussed.

Matched Asymptotic Expansions of the Eigenvalues of a 3-D Boundary-Value Problem Relative to Two Cavities Linked by a Hole of Small Size

In this article,we consider a domain consisting of two cavities linked by a hole of small size.We derive a numerical method to compute an approximation of the eigenvalues of an elliptic operator without refining in the neighborhood of the hole.Several convergence rates are obtained and illustrated by numerical simulations.

A NOTE ON THE NARROW OPEN CHANNEL RESONANCE

A narrow open channel resonant phenomenon, newly found by the authors in corresponding numerical calculations , was proved to exist based on the method of matching asymptotic expansions for three different channel configurations. It is shown that the resonant wave numbe rk emerges around kL=nπ, n=1,2,3,…∞ with a corresponding frequency s hift, where L is the length of the channel. It is also clear that the resona nce in a narrow open channel is an essential property of a channel as long as it is uniformly narrow.

Hydroelastic interaction between water waves and a thin elastic plate of arbitrary geometry

An analytical method is developed for the hydroelastic interaction between surface incident waves and a thin elastic plate of arbitrary geometry floating on an inviscid fluid of finite depth in the framework of linear potential flow.Three kinds of edge conditions are considered and the corresponding analytical representations are derived in the polar coordinate system.According to the surface boundary conditions,the fluid domain is divided into two regions,namely,an open water region and a plate-covered region.With the assumption that all the motion is time-harmonic,the series solutions for the spatial velocity potentials are derived by the method of eigenfunction expansion.The matching conditions for the continuities of the velocity and pressure are transformed by taking the inner products successively with respect to the vertical eigenfunction for the free surface and the angular eigenfunction.A system of simultaneous equations,including two edge conditions and two matching conditions,is set up for deriving the expansion coefficients.As an example,numerical computation for the expansion coefficients of truncated series is performed for an elliptic plate.The results show that the method suggested here is useful to revealing the physical features of the gravity wave scattering in the open water and the hydroelastic response in the plate.

Waves propagating over a two-layer porous barrier on a seabed

A research of wave propagation over a two-layer porous barrier, each layer of which is with different values of porosity and friction, is conducted with a theoretical model in the frame of linear potential flow theory. The model is more appropriate when the seabed consists of two different properties, such as rocks and breakwaters. It is assumed that the fluid is inviscid and incompressible and the motion is irrotational. The wave numbers in the porous region are complex ones, which are related to the decaying and propagating behaviors of wave modes. With the aid of the eigenfunction expansions, a new inner product of the eigenfunctions in the two-layer porous region is proposed to simplify the calculation. The eigenfunctions, under this new definition, possess the orthogonality from which the expansion coefficients can be easily deduced. Selecting the optimum truncation of the series, we derive a closed system of simultaneous linear equations for the same number of the unknown reflection and transmission coefficients. The effects of several physical parameters, including the porosity, friction, width, and depth of the porous barrier, on the dispersion relation, reflection and transmission coefficients are discussed in detail through the graphical representations of the solutions. It is concluded that these parameters have certain impacts on the reflection and transmission energy.