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This study proposes a multi-ship collision avoidance and route generating algorithm based on the general requirements of the International Regulations for Preventing Collisions at Sea (COLREGs) and the artificial potential field (APF) method. The velocity potential field is used as the field function of APF method. The algorithm consists of two modes, the course-changing and the track-keeping modes based on the velocity potential of vortex and dipole, respectively. The course-changing mode guides the ship to turn away from the obstacles according to the vector field of vortex potential. The track-keeping mode steers the ship back to and along on a pre-designed track in accordance with the vector field of dipole potential. The data of distance to the closest point of approach (DCPA), time to the closest point of approach (TCPA) and bearing angle evaluated from a maneuvering simulation are the acquired parameters of the proposed collision avoidance algorithm. The algorithm is straightforward and very simple to implement, and is suitable for the real time and distributed intelligent collision avoidance system. Simulation results indicate that the anti-collision formulation can avoid collision safely with the desired distance and indicate that the algorithm proposed can work effectively.

Water storage tanks are critical infrastructure that requires maintenance to ensure they function efficiently. Earthquake‐induced sloshing and hydrodynamic pressures can cause significant stress and damage in liquid storage tanks. Various methods have been used to reduce sloshing effects in these structures. This study proposes a new method to reduce the sloshing of partially filled rectangular water storage tanks during earthquake ground motions. A pendulum steel wall system is used to create a sloshing control system. The hydrodynamic equation of motion for these systems is derived analytically using the velocity potential function. Eight different ground motions, categorized as near‐fault and far‐fault, were chosen to analyze the dynamic responses of liquid storage tanks equipped with this system. The study also examines the effect of different wall connection placements relative to the water’s free surface on the system’s performance. The results, in terms of wave elevation, base shear, and base moment, show that the submerged pendulum wall system is more successful in reducing base shear and base moment levels than wave elevation. The study also recommends the optimal connection location in the tanks.

In this paper, the east–west tropical ‘Walker circulation’ and its linear association with sea surface temperature (SST) of the Nino 3 region and Indian summer monsoon rainfall (ISMR) have been investigated. ‘Walker circulation’/‘reverse Walker circulation’ is primarily forced by SSTs of the equatorial Pacific Ocean. In this study, velocity potential field of 0.21 sigma level over the tropics was considered as proxy of the zonal tropical circulation (‘Walker circulation’/‘reverse Walker circulation’). Principal component analysis of the monsoon season tropical velocity potential data of 0.21 sigma level for the two periods 1951–1980 and 1981–2010, was done separately. We find that earlier, two different patterns of the velocity potential field, forced by probably two distinct modes of El Nino episodes, were associated with the ISMR. These two El Nino episodes, respectively, correspond to the strong El Nino events, where in warming was extended up to the date line (primarily zonal) and the moderate El Nino events in which, warming having north south extension, was limited to the eastern Pacific Ocean only. However, in recent years, only the first pattern of the velocity potential field, induced by the strong El Nino events (warming extending up to the date line), was correlated with the ISMR. Further, in the later period (1981–2010), velocity potential field at 0.21 sigma level over the tropical Pacific and Indian Oceans, which appeared to be primarily driven by SST anomalies of the equatorial Pacific Ocean in the first period, was found to be significantly correlated with the extra tropical circulation anomalies also. Therefore, modulation of the ISMR through velocity potential field over the tropical Indian and Pacific Oceans, in the later period, may have additional significant impact of the extra-tropical circulation anomalies. This might have led weak correlation between the ISMR and SSTs of the Nino 3 region, which is actually being observed in recent years.

Three choices of control variables for meteorological variational analysis (3DVAR or 4DVAR) are associated with horizontal wind: (1) streamfunction and velocity potential, (2) eastward and northward velocity, and (3) vorticity and divergence. This study shows theoretical and numerical differences of these variables in practical 3DVAR data assimilation through statistical analysis and numerical experiments. This paper demonstrates that (a) streamfunction and velocity potential could potentially introduce analysis errors; (b) A 3DVAR using velocity or vorticity and divergence provides a natural scale dependent influence radius in addition to the covariance; (c) for a regional analysis, streamfunction and velocity potential are retrieved from the background velocity field with Neumann boundary condition. Improper boundary conditions could result in further analysis errors; (d) a variational data assimilation or an inverse problem using derivatives as control variables yields smoother analyses, for example, a 3DVAR using vorticity and divergence as controls yields smoother wind analyses than those analyses obtained by a 3DVAR using either velocity or streamfunction/velocity potential as control variables; and (e) statistical errors of higher order derivatives of variables are more independent, e.g., the statistical correlation between U and V is smaller than the one between streamfunction and velocity potential, and thus the variables in higher derivatives are more appropriate for a variational system when a cross-correlation between variables is neglected for efficiency or other reasons. In summary, eastward and northward velocity, or vorticity and divergence are preferable control variables for variational systems and the former is more attractive because of its numerical efficiency. Numerical experiments are presented using analytic functions and real atmospheric observations.

The capture zone equations of a multi-well system in bounded confined and unconfined aquifers are derived. The aquifer is rectangular-shaped in plan view and bounded along all four sides. The boundaries could be inflow (constant head) or no-flow (barrier) or a combination of both, and hence, six boundary configurations are formed. Using the method of images, the flow field in bounded aquifers is first transformed to its equivalent in extensive aquifers, and then, the complex velocity potential theory is applied for the generation of stream function delineating the capture envelope. We show that the derived solution is general, and it may be easily reformulated for some existing solutions of well capture zone. Our solution is flexible in terms of well number, well location, well type, extraction/injection rate, uniform regional flow rate and direction, and number of boundaries. The derived equations are presented in the form of capture type curves that may be used for the remediation of contaminated groundwater project design, containment of contaminant plumes, the evaluation of surface-subsurface water interaction, and the verification of numerical models. Drawdown equations for the above six boundary configurations are also derived.

In this paper an extension of the theoretical model of Molin (J. Fluid Mech., vol. 430, 2001, pp. 27–50) is proposed, where the assumptions of infinite depth and infinite horizontal extent of the support are released. The fluid domain is decomposed into two subdomains: the moonpool (or the gap) and a lower subdomain bounded by the seafloor and by an outer cylinder where the linearized velocity potential is assumed to be nil. Eigenfunction expansions are used to describe the velocity potential in both subdomains. Garrett’s method is then applied to match the velocity potentials at the common boundary and an eigenvalue problem is formulated and solved, yielding the natural frequencies and associated modal shapes of the free surface. Applications are made, first in the case of a circular moonpool, then in the rectangular gap and moonpool cases. Based on so-called single-mode approximations, simple formulas are proposed that give the resonant frequencies.

The hydroelastic waves in a channel covered by an ice sheet, without or with crack and subject to various edge constraints at channel banks, are investigated based on the linearized velocity potential theory for the fluid domain and the thin-plate elastic theory for the ice sheet. An effective analytical solution procedure is developed through expanding the velocity potential and the fourth derivative of the ice deflection to a series of cosine functions with unknown coefficients. The latter are integrated to obtain the expression for the deflection, which involves four constants. The procedure is then extended to the case with a longitudinal crack in the ice sheet by using the Dirac delta function and its derivatives at the crack in the dynamic equation, with unknown jumps of deflection and slope at the crack. Conditions at the edges and crack are then imposed, from which a system of linear equations for the unknowns is established. From this, the dispersion relation between the wave frequency and wavenumber is found, as well as the natural frequency of the channel. Extensive results are then provided for wave celerity, wave profiles and strain in the ice sheet. In-depth discussions are made on the effects of the edge condition, and the crack.

The so-called traditional approximation, wherein the component of the Coriolis force proportional to the cosine of latitude is ignored, is frequently made in order to simplify the equations of atmospheric circulation. For velocity fields whose vertical component is comparable to their horizontal component (such as convective circulations), and in the tropics where the sine of latitude vanishes, the traditional approximation is not justified. We introduce a framework for studying the effect of diabatic heating on circulations in the presence of both traditional and nontraditional terms in the Coriolis force. The framework is intended to describe steady convective circulations on an f plane in the presence of radiation and momentum damping. We derive a single elliptic equation for the horizontal velocity potential, which is a generalization of the weak temperature gradient (WTG) approximation. The elliptic operator depends on latitude, radiative damping, and momentum damping coefficients. We show how all other dynamical fields can be diagnosed from this velocity potential; the horizontal velocity induced by the Coriolis force has a particularly simple expression in terms of the velocity potential. Limiting examples occur at the equator, where only the nontraditional terms are present, at the poles, where only the traditional terms appear, and in the absence of radiative damping where the WTG approximation is recovered. We discuss how the framework will be used to construct dynamical, nonlinear convective models, in order to diagnose their consequent upscale momentum and temperature fluxes.

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.

The interaction between the flow in a channel with an obstruction on the bottom and an elastic sheet representing the ice covering the liquid is considered for the case of steady flow. The mathematical model based on the velocity potential theory and the theory of thin elastic shells fully accounts for the nonlinear boundary conditions at the elastic sheet/liquid interface and on the bottom of the channel. The integral hodograph method is employed to derive the complex velocity potential of the flow, which contains the velocity magnitude at the interface in explicit form. This allows one to formulate the coupled ice/liquid interaction problem and reduce it to a system of nonlinear equations in the unknown magnitude of the velocity at the interface. Case studies are carried out for a semi-circular obstruction on the bottom of the channel. Three flow regimes are studied: a subcritical regime, for which the interface deflection decays upstream and downstream; an ice supercritical and channel subcritical regime, for which two waves of different lengths may exist; and a channel supercritical regime, for which the elastic wave is found to extend downstream to infinity. All these regimes are in full agreement with the dispersion equation. The obtained results demonstrate a strongly nonlinear interaction between the elastic and the gravity wave near the first critical Froude number where their lengths approach each other. The interface shape, the bending moment and the pressure along the interface are presented for wide ranges of the Froude number and the obstruction height.