Computing Streamfunction and Velocity Potential in a Limited Domain of Arbitrary Shape.Part I:Theory and Integral Formulae

The non-uniqueness of solution and compatibility between the coupled boundary conditions in computing velocity potential and streamfunction from horizontal velocity in a limited domain of arbitrary shape are revisited theoretically with rigorous mathematic treatments.Classic integral formulas and their variants are used to formulate solutions for the coupled problems.In the absence of data holes,the total solution is the sum of two integral solutions.One is the internally induced solution produced purely and uniquely by the domain internal divergence and vorticity,and its two components（velocity potential and streamfunction） can be constructed by applying Green＇s function for Poisson equation in unbounded domain to the divergence and vorticity inside the domain.The other is the externally induced solution produced purely but non-uniquely by the domain external divergence and vorticity,and the non-uniqueness is caused by the harmonic nature of the solution and the unknown divergence and vorticity distributions outside the domain.By setting either the velocity potential（or streamfunction） component to zero,the other component of the externally induced solution can be expressed by the imaginary（or real） part of the Cauchy integral constructed using the coupled boundary conditions and solvability conditions that exclude the internally induced solution.The streamfunction（or velocity potential） for the externally induced solution can also be expressed by the boundary integral of a double-layer（or singlelayer） density function.In the presence of data holes,the total solution includes a data-hole-induced solution in addition to the above internally and externally induced solutions.

CMIP5 Simulated Change in the Intensity of the Hadley and Walker Circulations from the Perspective of Velocity Potential

Based on the simulations of 31 global models in CMIP5, the performance of the models in simulating the Hadley and Walker circulations is evaluated. In addition, their change in intensity by the end of the 21st century （2080-2099） under the RCP4.5 and RCP8.5 scenarios, relative to 1986-2005, is analyzed from the perspective of 200 hPa velocity potential. Validation shows good performance of the individual CMIP5 models and the multi-model ensemble mean （MME） in re- producing the meridional （zonal） structure and magnitude of Hadley （Walker） circulation. The MME can also capture the observed strengthening tendency of the winter Hadley circulation and weakening tendency of the Walker circulation. Such secular trends can be simulated by 39% and 74% of the models, respectively. The MME projection indicates that the winter Hadley circulation and the Walker circulation will weaken under both scenarios by the end of the 21st century. The weak- ening amplitude is larger under RCP8.5 than RCP4.5, due to stronger external forcing. The majority of the CMIP5 models show the same projection as the MME. However, for the summer Hadley circulation, the MME shows little change under RCP4.5 and large intermodel spread is apparent. Around half of the models project an increase, and the other half project a decrease. Under the RCP8.5 scenario, the MME and 65% of the models project a weakening of the summer southern Hadley circulation.

Computing Streamfunction and Velocity Potential in a Limited Domain of Arbitrary Shape.Part II:Numerical Methods and Test Experiments

Built on the integral formulas in Part I,numerical methods are developed for computing velocity potential and streamfunction in a limited domain.When there is no inner boundary（around a data hole） inside the domain,the total solution is the sum of the internally and externally induced parts.For the internally induced part,three numerical schemes（grid-staggering,local-nesting and piecewise continuous integration） are designed to deal with the singularity of the Green＇s function encountered in numerical calculations.For the externally induced part,by setting the velocity potential（or streamfunction） component to zero,the other component of the solution can be computed in two ways：（1） Solve for the density function from its boundary integral equation and then construct the solution from the boundary integral of the density function.（2） Use the Cauchy integral to construct the solution directly.The boundary integral can be discretized on a uniform grid along the boundary.By using local-nesting（or piecewise continuous integration）,the scheme is refined to enhance the discretization accuracy of the boundary integral around each corner point（or along the entire boundary）.When the domain is not free of data holes,the total solution contains a data-hole-induced part,and the Cauchy integral method is extended to construct the externally induced solution with irregular external and internal boundaries.An automated algorithm is designed to facilitate the integrations along the irregular external and internal boundaries.Numerical experiments are performed to evaluate the accuracy and efficiency of each scheme relative to others.

Hybrid Methods for Computing the Streamfunction and Velocity Potential for Complex Flow Fields over Mesoscale Domains

Three types of previously used numerical methods are revisited for computing the streamfunctionψand velocity potentialχfrom the horizontal velocity v in limited domains.The first type,called the SOR-based method,uses a classical successive over-relaxation(SOR)scheme to computeψ(orχ)first with an arbitrary boundary condition(BC)and thenχ(orψ)with the BC derived from v.The second type,called the spectral method,uses spectral formulations to construct the inner part of(ψ,χ)-the inversion of(vorticity,divergence)with a homogeneous BC,and then the remaining harmonic part of(ψ,χ)with BCs from v.The third type,called the integral method,uses integral formulas to compute the internally induced(ψ,χ)-the inversion of domain-internal(vorticity,divergence)using the free-space Greenꞌs function without BCs and then the remaining harmonicψ(orχ)with BCs from v minus the internally-induced part.Although these methods have previously been successfully applied to flows in large-scale and synoptic-scale domains,their accuracy is compromised when applied to complex flows over mesoscale domains,as shown in this paper.To resolve this problem,two hybrid approaches,the integral-SOR method and the integral-spectral method,are developed by combining the first step of the integral method with the second step adopted from the SOR-based and spectral methods,respectively.Upon testing these methods on real-case complex flows,the integral-SOR method is significantly more accurate than the integral-spectral method,noting that the latter is still generally more accurate than the three previously-used methods.The integral-SOR method is recommended for future applications and diagnostic studies of complex flows.

Iterative and adjusting method for computing stream function and velocity potential in limited domains and convergence analysis

The stream function and the velocity potential can be easily computed by solving the Poisson equations in a unique way for the global domain. Because of the var- ious assumptions for handling the boundary conditions, the solution is not unique when a limited domain is concerned. Therefore, it is very important to reduce or eliminate the effects caused by the uncertain boundary condition. In this paper, an iterative and ad- justing method based on the Endlich iteration method is presented to compute the stream function and the velocity potential in limited domains. This method does not need an explicitly specifying boundary condition when used to obtain the effective solution, and it is proved to be successful in decomposing and reconstructing the horizontal wind field with very small errors. The convergence of the method depends on the relative value for the distances of grids in two different directions and the value of the adjusting factor. It is shown that applying the method in Arakawa grids and irregular domains can obtain the accurate vorticity and divergence and accurately decompose and reconstruct the original wind field. Hence, the iterative and adjusting method is accurate and reliable.

ON VELOCITY POTENTIALS DUE TO PULSATING PRESSURE DISTRIBUTIONS ON THE FREE SURFACE OF INFINITE-DEPTH WATERS AND THE RADIATION CONDITIONS FOR SECOND-ORDER DIFFRACTION PROBLEMS

In the present paper,two-and three-dimensional velocity potentials generated by pulsating pressure distributions of infinite extent on the free surface of infinite-depth waters are strictly derived based on special cases of concentrated pulsating pressure.The far-field asymptotic behaviour of the potentials and the radiation conditions to be satisfied by them are discussed. It is proved in a general sense that the potentials should be composed of a forced wave component,a free wave component and a local disturbance component.The radiation condition of the forced wave component should correspond to the far-field asymptotic behaviour of the pressure distribution,Hence,the formulation of radiation conditions for the second-order diffraction potentials has theoretically become clear,The radiation conditions for two-and three-dimensional problems are explicitly given in the paper.

Error Analysis of A New Higher Order Boundary Element Method for A Uniform Flow Passing Cylinders

A higher order boundary element method(HOBEM)is presented for inviscid flow passing cylinders in bounded or unbounded domain.The traditional boundary integral equation is established with respect to the velocity potential and its normal derivative.In present work,a new integral equation is derived for the tangential velocity.The boundary is discretized into higher order elements to ensure the continuity of slope at the element nodes.The velocity potential is also expanded with higher order shape functions,in which the unknown coefficients involve the tangential velocity.The expansion then ensures the continuities of the velocity and the slope of the boundary at element nodes.Through extensive comparison of the results for the analytical solution of cylinders,it is shown that the present HOBEM is much more accurate than the conventional BEM.

Analytical Research on Wave Diffraction on Arc-Shaped Bottom-Mounted Perforated Breakwaters

An analytical method is developed to study wave diffraction on arc-shaped and bottom-mounted perforated breakwaters. The breakwater is assumed to be rigid, thin, vertical, immovable and located in water of constant depth. The fluid domain is divided into two regions by imaginary interface. The velocity potential in each region is expanded by eigenfunctions. By satisfying the continuity of pressure and normal velocity across the imaginary fluid interface, a set of linear algebraic equations can be obtained to determine the unknown coefficients of eigenfunctions. Numerical results, in the form of contour maps of the relative wave amplitude around the breakwater, are presented for a range of wave and breakwater parameters. Results show that the wave diffraction on the arc-shaped and bottom-mounted perforated breakwater is related to the incident wavelength and the porosity of the breakwater. The porosity of the perforated breakwater may have great effect on the diffracted field.

Wave Defending Effects of V-Type Bottom-Mounted Breakwaters

An analytical method is developed for the study of the wave defending effects of the V-type bottom-mounted breakwater. The breakwater is assumed to be rigid, thin, impermeable and vertically located in water of constant depth. The fluid domain is divided into three sub-regions by an imaginary interface. The velocity potential in each region is expanded by eigenfunctions. By satisfying the corresponding boundary conditions and matching conditions in and between sub-regions, a set of linear algebraic equations can be obtained to determine the unknown coefficients for the eigenfunction expansions for each sub-region. The accuracy of the present model is verified by a comparison with existing results for the case of an isolated breakwater. Numerical results, in the form of contour maps of the relative wave amplitude around the breakwater, are presented for a range of wave and breakwater parameters. The results show that the V-type bottom-mounted breakwater is generally effective in defending against waves. In general, the wave height in the protected area is about 20～50 percent of the incident wave height.

THE SINGULAR BEHAVIOUR IN THE VICINITY OF INTERSECTION BETWEEN THE BODY AND FREE SURFACE

The singular behaviour in [he vicinity of intersection between the body and free surface is presented. It is shown that in the linear regime the singularity of velocity potential for transient problem is in d2/nd. The singular behaviour for harmonic problem is the same as the result for the transient problem. In particular, the singularity for the harmonic problem with infinite frequency is in d2/nd for velocity potential (d is the distance between field point and intersection).

Theoretical investigation on the cavitation bubble dynamics near three spherical particles based on Weiss theorem

To research the dynamics of the cavitation bubble under the interaction of particle clusters,the bubble morphological evolutionary characteristics near three equal-sized spherical particles are theoretically explored in the present study based on the Weiss theorem and the velocity potential superposition theory.The three particles are arranged symmetrically,and the fluid velocity field near the three particles and the cavitation bubble is obtained.Moreover,the effects of the bubble-particle distance and the maximum radius of the cavitation bubble on the fluid velocity are investigated,and the contribution mechanisms of the fluid velocity field constituents are compared.The analysis has found that:(1)The fluid velocity between the bubble and the particle is lower than that at the other locations in both the growth and collapse phases,thus the bubble cannot always maintain a standard spherical shape.(2)The bubble-particle distance and the maximum radius of the cavitation bubble are the key parameters affecting the circumferential inhomogeneity of the radial velocity of the fluid around the bubble.The larger the maximum radius or the smaller the bubble-particle distance is,the more visible the non-circularity of the bubble morphology.(3)The image bubbles and the linear sinks contribute oppositely to the fluid velocity field,and the presence of the image bubble reduces the fluid velocity.In the low velocity region,the image bubble is the main mechanism contributing to the effect of the particle on the fluid velocity.

WAVE DIFFRACTION ON ARC-SHAPED BOTTOM-MOUNTED BREAKWATERS

An analytical method was developed to study the wave diffraction effects on arc-shaped bottom-mounted breakwaters. The breakwater was assumed to be rigid, thin, impermeable and vertically located in water of constant depth. The fluid domain was divided into two regions by imaginary cylindrical interface. The velocity potential in each region was expanded with cigcnfunctions. By satisfying continuity of pressure and normal velocity across the imaginary fluid interface, a set of linear algebraic equations could be obtained to determine the unknown coefficients for eigenfunction expansions. The accuracy of the present model was verified by a comparison with existing results for the case of an isolated straight-line breakwater. Numerical results, in the form of contour maps of the non-dimensional wave amplitude around the breakwater and diffracted wave amplitude at three typical sections, were presented for a range of wave parameters. Results show the arc-shaped bottom-mounted breakwater is generally effective in defending against waves. The wave amplitudes at most sheltered areas are commonly 10%-50% of incident wave amplitudes under most wave conditions.

ANALYTICAL RESEARCH ON THE WAVE DIFFRACTION OF ARC-SHAPED FLOATING BREAKWATERS

An analytical method was developed to study the wave diffraction on are-shaped floating breakwaters. The floating breakwater was assumed to be rigid, thin, vertical, immovable and located in water of constant depth. The fluid domain was divided into two regions by imaginary interface, The velocity potential in each region is expanded by eigenfunctions. By satisfying continuity of pressure and normal velocity across the imaginary fluid interface, a set of linear algebraic equations could be obtained to determine the unknown coefficients for eigenfunctions. The accuracy of present model and the computer program were verified by a comparison with ex isting results for the case of arc-shaped bottom-mounted breakwaters. Numerical results, in the form of contour maps of the non-dimension wave amplitude around the breakwater, were presented for a range of wave and breakwater parame ters. Results show the wave diffraction on the arc-shaped floating breakwater is related to the incident wavelength and the draft of the breakwater.

SECOND ORDER TRANSIENT WAVES AROUND A VERTICAL CYLINDER IN A TANK

The free surface problem bound by two cylinders is analysed based on the velocity potential theory. An analytical solution in the take domain is obtained up to the second order in the perturbation expansion. The results are compared with those obtainal from the fully nonlinear theory based on a finite element formulation.It is found that the second order solutiongives a fsr better agreement with the fully nonlinear solution.