Normal Velocity Component Research Articles

Influence of rotations in energy transfer of a particle colliding on a flat surface

To understand the influence of rotations on the dissipation of energy in the interaction between a grain and its environment, we investigate the relaxation process of a single particle bouncing on a flat horizontal surface. For this purpose, faceted particles were used to promote the appearance of rotations in each bounce. The evolution of potential, translational and rotational kinetic energies was analyzed during the whole relaxation process, particularly focusing on the behavior just before and after each collision. The rebounding action of an individual grain results in energy dissipation commonly quantified by ϵ, the coefficient of restitution. This coefficient is defined as the ratio of the normal velocity component prior to impact (Vn) to the corresponding component immediately following the collision (Vn′), i.e. related to translational kinetic energy associated with motion in the normal direction. We identify a critical impact velocity below which, in certain collisions, ϵ>1. This phenomenon can be attributed to stored rotational kinetic energy which is transferred to translational kinetic energy during the collision, thereby increasing the normal velocity Vn′ and resulting in the observed high values of ϵ.

Global Regular Axially Symmetric Solutions to the Navier–Stokes Equations: Part 2

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder Ω⊂R3 with the axis of symmetry. S1 is the boundary of the cylinder parallel to the axis of symmetry, and S2 is perpendicular to it. We have two parts of S2. On S1 and S2, we impose vanishing of the normal component of velocity and the angular component of vorticity. Moreover, we assume that the angular component of velocity vanishes on S1 and the normal derivative of the angular component of velocity vanishes on S2. We prove the existence of global regular solutions. To prove this, the coordinate of velocity along the axis of symmetry must vanish on it. We have to emphasize that the technique of weighted spaces applied to the stream function plays a crucial role in the proof of global regular axially symmetric solutions. The paper is a generalization of Part 1, where the periodic boundary conditions are prescribed on S2. The transformation is not trivial because it needs to examine many additional boundary terms and derive new estimates.

Origin of High-velocity Ejecta, Excess Emission, and Redward Color Evolution in the Infant Type Ia Supernova 2021aefx

SN 2021aefx is a normal Type Ia supernova (SN) showing excess emission and redward color evolution over the first ∼ 2 days. We present analyses of this SN using our high-cadence KMTNet multiband photometry, spectroscopy, and publicly available data, including first measurements of its explosion epoch (MJD 59529.32 ± 0.16) and onset of power-law rise (t PL = MJD 59529.85 ± 0.55; often called first light) associated with the main ejecta 56Ni distribution. The first KMTNet detection of SN 2021aefx precedes t PL by ∼ 0.5 hr, indicating presence of additional power sources. Our peak-spectrum confirms its intermediate Type Ia subclassification between core-normal and broad-Line, and we estimate an ejecta mass of ∼ 1.34 M ⊙. The spectral evolution identifies material reaching >40,000 km s−1 (fastest ever observed in Type Ia SNe) and at least two split-velocity ejecta components expanding homologously: (1) a normal-velocity (∼ 12,400 km s−1) component consistent with typical photospheric evolution of near-Chandrasekhar-mass ejecta; and (2) a high-velocity (∼ 23,500 km s−1) secondary component visible during the first ∼ 3.6 days post-explosion, which locates the component within the outer <16% of the ejecta mass. Asymmetric subsonic explosion processes producing a nonspherical secondary photosphere provide an explanation for the simultaneous appearance of the two components, and may also explain the excess emission via a slight 56Ni enrichment in the outer ∼ 0.5% of the ejecta mass. Our 300 days post-peak nebular-phase spectrum advances constraints against nondegenerate companions and further supports a near-Chandrasekhar-mass explosion origin. Off-center ignited delayed-detonations are likely responsible for the observed features of SN 2021aefx in some normal Type Ia SNe.

Global Regular Axially Symmetric Solutions to the Navier–Stokes Equations: Part 1

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder Ω⊂R3 with the axis of symmetry. S1 is the boundary of the cylinder parallel to the axis of symmetry and S2 is perpendicular to it. We have two parts of S2. For simplicity, we assume the periodic boundary conditions on S2. On S1, we impose the vanishing of the normal component of velocity, the angular component of velocity, and the angular component of vorticity. We prove the existence of global regular solutions. To prove this, it is necessary that the coordinate of velocity along the axis of symmetry vanishes on it. We have to emphasize that the technique of weighted spaces applied to the stream function plays a crucial role in the proof of global regular axially symmetric solutions. The weighted spaces used are such that the stream function divided by the radius must vanish on the axis of symmetry. Currently, we do not know how to relax this restriction. In part 2 of this topic, the periodic boundary conditions on S2 are replaced by the conditions that both the normal component of velocity and the angular component of vorticity must vanish. Moreover, it is assumed that the normal derivative of the angular component of velocity also vanishes on S2. A transformation from part 1 to part 2 is not trivial because it needs new boundary value problems, so new estimates must be derived.

Methodology for modeling spray cooling of a cylindrical tube heated in the film boiling regime

This study examines spray cooling of horizontal, circular tubes when the surface temperature exceeds the Leidenfrost point, and the individual drop impacts are in the film boiling regime. Although film boiling is not an optimal condition for heat transfer, it is encountered in transient cooling processes or when high temperatures persist along the tube. The goal of the study is to apply existing models for drop impact and heat transfer to surfaces more complex than flat plates and to determine the overall steady state heat removal rate as an essential parameter for heat exchanger design. The spray characteristics are prescribed by a local number flux and drop size, assuming a random distribution in space and a uniform velocity of all drops. The angle of individual drop impingement on the cylindrical tube is accounted for through the maximum spreading diameter of the drop on the surface and the normal component of impact velocity. Furthermore, drop-drop interaction on the heated surface is accounted for at high local number flux values through an effective coverage area coefficient.The results are expressed in the form of a local heat transfer coefficient. These results confirm that the most influential parameters regarding heat transfer are the liquid mass flow rate in the nozzle and the drop diameter; however, they also indicate that the non-normal impact of drops over portions of the cylindrical tube leads to a highly reduced number flux; hence heat transfer. This oblique impact angle arises for one from the local spray angle, but more drastically from the curved, cylindrical surface; the latter leading to a dramatic sharp decrease in heat transfer. This is important information when considering the use of multiple spray nozzles, either circumferentially or longitudinally along the tube, since it dictates the optimal spacing between neighbouring nozzles.

Performance Optimization of Underwater Crushing Unit Based on AHP

An underwater crushing unit loaded on the underwater cleaning robot was intended to handle marine biofouling that adhered to the surface of the ship and the dam, and a prototype was initially built. A Computational Fluid Dynamics–Discrete Element Model (CFD-DEM) was created to boost the prototype’s crushing performance, and its rationale was validated by contrasting the simulation results with the results of experimental tests. Accordingly, the primary influences on crushing performance and the laws governing their influence were investigated. The Analytical Hierarchy Process (AHP) method was then used to establish a prediction model for the comprehensive evaluation indicator of crushing performance. The AHP was used, in this case, because of its ability to generate the weight of indicators. The prediction model was a quadratic polynomial function with the rotational speed, the normal velocity component at the outlet of the propeller, the mass flow rate of the particles at the inlet of the unit, and the thickness of the bushing as independent variables. The prediction model fitting effect met the requirements after the test. The primary elements influencing the underwater crushing unit’s performance were optimized using the prediction model. The average accumulation speed of particles in the crushing unit was reduced by 59.05%, and the mass flow rate of particles at the outlet was reduced by 11.93%. The maximum wear height of the bushing was reduced by 33.36%. The specific power was up 20.88%, and the overall crushing performance was up 9.87% when compared to before optimization.

Fractional Modeling of Non-Newtonian Casson Fluid between Two Parallel Plates

In this manuscript, fractional modeling of non-Newtonian Casson fluid squeezed between two parallel plates is performed under the influence of magneto-hydro-dynamic and Darcian effects. The Casson fluid model is fractionally transformed through mixed similarity transformations. As a result, partial differential equations (PDEs) are transformed to a fractional ordinary differential equation (FODE). In the current modeling, the continuity equation is satisfied while the momentum equation of the integral order Casson fluid is recovered when the fractional parameter is taken as α = 1 . A modified homotopy perturbation algorithm is used for the solution and analysis of highly nonlinear and fully fractional ordinary differential equations. Obtained solutions and errors are compared with existing integral order results from the literature. Graphical analysis is also performed at normal and radial velocity components for different fluid and fractional parameters. Analysis reveals that a few parameters are showing different behavior in a fractional environment as compared to existing integer-order cases from the literature. These findings affirm the importance of fractional calculus in terms of more generalized analysis of physical phenomena.

Parallel and perpendicular flows of a couple stress fluid past a solid cylinder in cell model: Slip condition

The axisymmetric steady flow of a couple stress fluid between two concentric cylinders with a slip effect is investigated with the help of the cell model technique. Here, the inner cylinder is rigid, and the outer cylinder is fictitious. The tangential slip, vanishing of normal velocity, and zero couple stress conditions are applied on the inner cylindrical surface. In addition, zero shear stress (Happel's model), continuity of normal velocity component, and zero couple stress conditions are used on the outer cylindrical surface. We consider two flow problems: the first is the parallel flow, and the second is the perpendicular flow to the cylinder in the cell model. Also, we have discussed the random case. For all the cases, the Kozeny constant is calculated. We described some special cases and compared them with well-known results. The effects of slip and couple stress parameters on the Kozeny constant with fixed value of couple stress viscosity parameter are presented graphically. The influence of the couple stress viscosity parameter on the Kozeny constant with fixed values of couple stress, and slip parameters for parallel flow are expressed graphically. The numerical values for the Kozeny constant for different values of fractional void volume are tabulated. We also obtained the results of the consistent couple stress theory as a special case.

Crosswise Stream of Cu-H2O Nanofluid with Micro Rotation Effects: Heat Transfer Analysis.

The present study focuses on a crosswise stream of liquid-holding nano-sized particles over an elongating (stretching) surface. Tiny particles of copper are added into base liquid (water). The influence of the micro rotation phenomenon is also considered. By means of appropriate transformations non-linear coupled ordinary differential equations are attained that govern the flow problem. The Runge-Kutta-Fehlberg scheme, together with the shooting method, is engaged to acquire results numerically. Micropolar coupling parameter, microelements concentration and nanoparticles volume fraction effects are examined over the profiles of velocity, temperature and micro-rotation. Moreover, heat flux and shear stress are computed against pertinent parameters and presented through bar graphs. Outcomes revealed that material constant has increasing effects on normal components of flow velocity; however, it decreasingly influences the tangential velocity, micro-rotation components and temperature profile. Temperature profile appeared to be higher for weak concentration of microelements. It is further noticed that normal velocity profile is higher in magnitude for the case of strong concentration (n = 0) of microelements, whereas tangential velocity profile is higher near the surface for the case of weak concentration (n = 0.5) of microelements. An increase of 3.74% in heat flux is observed when the volume fraction of nanoparticles is increased from 1 to 5%.

Prediction model for multidirectional vortex-induced vibrations of catenary riser in convex/concave and perpendicular flows

This study presents an advanced numerical prediction model based on nonlinear wake oscillators for the investigation of multidirectional vortex-induced vibration (VIV) responses of a long catenary riser depending on the structural curved configuration versus the incoming flow direction. By considering a uniform free-stream flow aligned with the initial curvature plane of the catenary riser in a convex or concave shape, the normal flow velocity component is nonlinearly sheared owing to the spanwise variation of inclination angles. This is different from the perpendicular flow case in which the normal flow velocity is spatially constant. To capture the influence of flow direction on the three-dimensionally coupled cross-flow and in-line VIV, distributed wake oscillators are introduced and applied to simulate the fluctuating hydrodynamic lift and drag forces that account for the important effects of cylinder inclination, variable vortex excitation frequencies, global–local​ displacement relationships, and relative flow-cylinder velocities. The amplified mean drag force and the axial force depending on the tangential flow velocity are also considered. These empirical hydrodynamic effects are nonlinearly coupled with the equations of riser three-dimensional motion, and the overall governing equations are numerically solved to assess the multimode, multi-frequency and multi-degree-of-freedom responses. Validation of the model is carried out for VIV of flexible straight and curved cylinders based on experimental results in the literature. Parametric investigations are performed by varying the incoming flow velocities in perpendicular, convex and concave configurations for a long catenary riser with a low mass-damping ratio. In-plane (horizontal and vertical) and out-of-plane VIV features are presented and discussed in terms of space–time varying amplitudes, drag-amplified shape reconfigurations, oscillation frequencies, lock-in occurrences, dual resonances, orbital motion trajectories, fluid–structure energy transfer, and multimodal contributions. Overall prediction results highlight the influence of the cylinder geometric configuration versus the incoming flow orientation on multidirectional VIV characteristics that should be recognised and incorporated into practical analysis tools.

Comparative investigation on macroscopic and microscopic characteristics of impingement spray of gasoline and ethanol from a GDI injector under injection pressure up to 50 MPa

Particulate Matter (PM) emissions from passenger vehicles have attracted considerable interest over the last decade. In order to reduce PM emissions, improving maximum injection pressure has been a developing trend for new generation GDI engines. However, comparing gasoline and ethanol impingement spray characteristics from a GDI injector under high injection pressure is still unclear. In this paper, a comparative investigation on both the macroscopic and microscopic characteristics of impingement spray from a GDI injector fuelled with gasoline and ethanol was performed under injection pressure up to 50 MPa, providing new findings to promote a more homogeneous air–fuel mixture and reduce PM emissions. The experimental results show that under the same PI (injection pressure), rebound height of gasoline impingement spray is a bit higher than ethanol. AS (spray area) of gasoline is slightly higher than ethanol under PI=10MPa. However, under PI=30MPa and PI=50MPa, AS of gasoline is gradually exceeded by that of ethanol as time progresses. By increasing PI to 50 MPa, the difference in DN (diffusion distance of the near side) between gasoline and ethanol is greatly reduced, meantime DF (diffusion distance of the far side) becomes weaker than ethanol. For both gasoline and ethanol, with the increase PI from 10 MPa to 50 MPa, VN (average normal component of droplet velocity) and VT (average tangential component of droplet velocity) of incident droplets increase by around 1 m/s. Meantime, there is a slight decrease in the absolute value of VN and VT of reflected droplets. DSMD (Sauter mean diameter of droplets) presents a significant decreasing trend with the increase of PI. Besides, a smaller DSMD can be seen for the gasoline impingement spray compared to ethanol under the same PI.

Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks

A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich's paper [44] in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed. In this paper, we investigate the existence of weak solutions to this system by relying on {\it a priori} bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a $L^p$ integrability in space, with $p $ in $[1,+\infty]$, and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where $p > \frac43$, in the renormalized sense in the case where $p > 1$, and in a symmetrized sense in the case where $p =1$.

Transient Characteristics of Three-Dimensional Flow in a Centrifugal Impeller Perturbed by Simple Pre-Swirl Inflow

The pre-swirl inflow generated by guide vanes could improve the hydrodynamic performances of centrifugal pumps as long as the inflow matches the patterns of internal flow of the impeller. In this work, we present a numerical investigation on the internal flow in a centrifugal impeller subjected to inflow artificially constructed with simple pre-swirling; unsteady Reynolds-Averaged Navier-Stokes (URANS) simulations are performed at the designed flow rate with five values of rotating velocity of the inflow, i.e., Urot/Uref = −0.5, −0.3, 0.0, 0.3 and 0.5, where Urot and Uref denote the rotating and normal velocity component at the entrance of the inflow tube, respectively. The primary objective of this work is to reveal the three-dimensional characteristics of internal flow of the impeller as influenced by the superimposed pre-swirl inflow, and to identify the propagation of inflow within the impeller. The numerical data are presented and analyzed in terms of the streamline fields, the distributions of various velocity components along the circumferential and axial directions, the pressure distribution and limiting streamlines on the surfaces of a blade. Numerical results reveal that separation occurs around the leading edge of the blades and occasionally at the trailing edge, and the internal flow is more uniform in the central region of the channels. A noticeable fluctuation of both radial and circumferential velocities is observed at the outlet of the impeller as it is subjected to counter-rotating inflow, and the greatest fluctuation is close to the hub instead of the middle channel and shroud as for the co-rotating inflow. The boundary layer flow of suction surface is more sensitive to the inflow; occasional small-scale separation bubble occurs on the suction surface around the leading edge for some blades, and reattachment of separated flow is reduced for the counter-rotating inflow.

Magnetic Switchbacks Heat the Solar Corona

Magnetic switchbacks are short magnetic field reversals ubiquitously observed in the solar wind. The origin of switchbacks remains an important open science question, because of switchbacks’ possible role in the heating and acceleration of the solar wind. Here, we report observations of 501 robust switchbacks, using magnetic and plasma measurements from the first eight encounters by the Parker Solar Probe. More than 46% (6%) of switchbacks are rotational (tangential; TD) discontinuities (RD), defined as magnetic discontinuities with large (small) relative normal components of magnetic field and proton velocity. Magnetic reconnection in the solar atmosphere can be a source of the observed RD-type switchbacks. It is discovered that: (1) the RD-to-TD ratio exponentially decays with increasing heliocentric distance at rate 0.06 [R S −1], and (2) TD-type switchbacks contain 64% less magnetic energy than RD-type switchbacks, suggesting that RD-type switchbacks may relax into TD-type switchbacks. It is estimated that relaxing switchbacks generated via magnetic reconnection in the solar atmosphere can transfer an additional 16% of the total reconnected magnetic energy into heating and/or accelerating the solar corona, within 11.6 [R S] of the reconnection site, below the critical Alfvén surface.

Analytic investigation of the compatibility condition and the initial evolution of a smooth velocity field for the Navier–Stokes equation in a channel configuration

A partial differential equation has usually a regular solution at the initial time if the initial condition is smooth in space, fulfills the governing equations and is compatible with the boundary condition. In the case of the Navier–Stokes equation, the initial velocity field must also be divergence–free. It is common belief that the initial condition is compatible with the boundary condition if the initial condition fulfills the boundary condition but this is not sufficient. Such a field does not necessarily fulfill the full compatibility condition of the Navier–Stokes equation. If the condition is violated, the solution is not regular at the initial time (). This issue has been known for a while but not in the full breadth of the engineering fluid dynamics community. In this paper, a practical calculation method is presented for checking the compatibility condition. Furthermore, a smooth initial condition is presented in a periodic channel flow that violates the compatibility condition and has therefore no smooth solution at the initial instant. The calculations were performed in an analytical framework. The results for a channel configuration show that in the absence of wall–normal velocity the condition is always fulfilled and the problem has a regular solution. If the wall–normal velocity component is non-zero, the condition is usually not fulfilled but exceptional cases, fulfilling the compatibility condition can be generated with optimization methods. The generation procedure of such a field is useful to provide correct initial conditions for such time-dependent numerical flow simulations, where the very first instances are important from an engineering point of view. The presented methods can provide insight about the applicability of the chosen initial conditions.

Free convection of temperature-dependent thermal conductivity based ethylene glycol-Al2O3 nanofluid in an open cavity with wall heat flux

The natural or free convection flow in an open cavity filled with ethylene glycol-Al2O3 nanofluid has been integrated numerically using the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) with the Graphics Process Unit (GPU) acceleration. The applications of this type of problem frequently appear in open-door ovens and refrigerators and in the cavities of central solar receptors. The ceiling and bottom walls of the cavity are thermally adiabatic. The right side of the cavity is fully opened while heat flux condition is imposed on its opposite wall. The thermal conductivity of the nanofluid varies with fluid temperature. Numerical simulations have been conducted for a fixed Prandtl number, Pr=16.6, with varying other controlling parameters, i.e., the Rayleigh number from Ra=105 to 107, and nanoparticle volume fraction from 0 to 5%. The streamlines, isotherms, mid-line temperature and velocity distributions, and the rate of heat transfer regarding the Nusselt number have been presented in this article. The isolines and tabular form give the local and total entropy generation due to the fluid friction and temperature gradients. The tangential and normal velocity components grow dramatically as Ra increases; however, the temperature diminishes with Ra. Also, the locally distributed Nusselt number increases in magnitude with nanofluids’ volume fraction and Rayleigh number. The total entropy enhances significantly for the augmentation of the nanoparticles volume fraction, and the entropy due to fluid friction dominates over the entropy generated by the temperature gradients.

Hawking radiation from acoustic black holes in hydrodynamic flow of electrons

Acoustic black holes are formed when a fluid flowing with subsonic velocities accelerates and becomes supersonic. When the flow is directed from the subsonic to supersonic region, the surface on which the normal component of fluid velocity equals the local speed of sound acts as an acoustic horizon. This is because no acoustic perturbation from the supersonic region can cross it to reach the subsonic part of the fluid. One can show that if the fluid velocity is locally irrotational, the field equations for acoustic perturbations of the velocity potential are identical to that of a massless scalar field propagating in a black hole background. One, therefore, expects Hawking radiation in the form of a thermal spectrum of phonons. There have been numerous investigations of this possibility, theoretically, as well as experimentally, in systems ranging from cold atom systems to quark-gluon plasma formed in relativistic heavy-ion collisions. Here we investigate this possibility in the hydrodynamic flow of electrons. The resulting Hawking radiation in this case should be observable in terms of current fluctuations. Further, current fluctuations on both sides of the acoustic horizon should show correlations expected for pairs of Hawking particles.

Thermal Energy Transfer between Helium Gas and Graphene Surface According to Molecular Dynamics Simulations and the Monte Carlo Method.

The scattering of gases on solid surfaces plays a vital role in many advanced technologies. In this study, the scattering behavior of helium on graphene surfaces was investigated, including the thermal accommodation coefficient (TAC), outgoing zenith angle of helium, bounce number, and interaction time. First, we performed molecular dynamics simulations to describe the incident angle-resolved behaviors, and showed that the scattering is highly dependent on the zenith angle of incident helium but insensitive to the azimuthal angle. The contribution of the normal velocity component of the incident helium dominated the energy transfer. The nonlinear relationship of the parameters to the zenith angle of the incident helium could be suppressed by increasing the graphene temperature or decreasing the speed of the incident helium. Subsequently, the scattering performance considering all gas molecules in the hemispherical space was evaluated using the Monte Carlo method with angle-resolved results. The result showed that the TAC, its nominal components, and the zenith angle of the scattered helium increased with higher speeds of incident helium and lower temperatures of graphene. This study should provide a fundamental understanding of energy transfer between gas and two-dimensional materials and guidelines to tune the scattering behavior between them.

Three-Dimensional Simulations of Anisotropic Slip Microflows Using the Discrete Unified Gas Kinetic Scheme.

The specific objective of the present work study is to propose an anisotropic slip boundary condition for three-dimensional (3D) simulations with adjustable streamwise and spanwise slip length by the discrete unified gas kinetic scheme (DUGKS). The present boundary condition is proposed based on the assumption of nonlinear velocity profiles near the wall instead of linear velocity profiles in a unidirectional steady flow. Moreover, a 3D corner boundary condition is introduced to the DUGKS to reduce the singularities. Numerical tests validate the effectiveness of the present method, which is more accurate than the bounce-back and specular reflection slip boundary condition in the lattice Boltzmann method. It is of significance to study the lid-driven cavity flow due to its applications and its capability in exhibiting important phenomena. Then, the present work explores, for the first time, the effects of anisotropic slip on the two-sided orthogonal oscillating micro-lid-driven cavity flow by adopting the present method. This work will generate fresh insight into the effects of anisotropic slip on the 3D flow in a two-sided orthogonal oscillating micro-lid-driven cavity. Some findings are obtained: The oscillating velocity of the wall has a weaker influence on the normal velocity component than on the tangential velocity component. In most cases, large slip length has a more significant influence on velocity profiles than small slip length. Compared with pure slip in both top and bottom walls, anisotropic slip on the top wall has a greater influence on flow, increasing the 3D mixing of flow. In short, the influence of slip on the flow field depends not only on slip length but also on the relative direction of the wall motion and the slip velocity. The findings can help in better understanding the anisotropic slip effect on the unsteady microflow and the design of microdevices.

Three alternative boundary integral flow representations for wave diffraction-radiation by offshore structures

Three alternatives to the classical boundary integral representation of diffraction-radiation of regular waves by large stationary bodies, such as offshore structures and moored ships, that underlies existing panel methods are defined. These three alternative flow representations are associated with two alternative linear flow models, called ‘free waterplane flow model’ and ‘rigid waterplane flow model’, of potential flow around an offshore structure in regular waves. As was shown previously, these two alternative linear flow models of diffraction-radiation of regular waves by stationary bodies yield identical boundary integral flow representations and are then consistent, although the rigid waterplane flow model precludes irregular frequencies. Moreover, this flow model – and mathematical transformations – yield two other boundary integral flow representations, so that three alternatives to the classical boundary integral flow representation used in existing panel methods are defined. These three alternative flow representations are weakly singular. The first of these flow representations, given previously, involves a surface integral over the waterplane inside the body, while the second flow representation involves a line integral around the waterline of the body, i.e. the intersection curve between the body and the undisturbed free surface. The third flow representation is of particular interest because it involves neither a surface integral over the waterplane nor a line integral around the waterline. Moreover, this boundary integral flow representation does not involve a distribution of dipoles over the body surface; indeed, it expresses the flow potential in terms of the normal and tangential components of the flow velocity at the surface of the body. It is also notable that this boundary integral flow representation is identical to the boundary integral representation previously obtained for flows around ships advancing at a constant speed in calm water or in regular waves, and thus provides a common basis for the analysis of three major classes of flows in ship and offshore hydrodynamics.