Three-dimensional Equations Of Motion Research Articles

Modeling Ocean Swell and Overtopping Waves: Understanding Wave Shoaling with Varying Seafloor Topographies

One risk posed by hurricanes and typhoons is local inundation as ocean swell and storm surge bring a tremendous amount of energy and water flux to the shore. Numerical wave tanks are developed to understand the dynamics computationally. The three-dimensional equations of motion are solved by the software ‘Open Field Operation And Manipulation’ v2206. The ‘Large Eddy Simulation’ scheme is adopted as the turbulence model. A fifth-order Stokes wave is taken as the inlet condition. Breaking, ‘run-up’, and overtopping waves are studied for concave, convex, and straight-line seafloors for a fixed ocean depth. For small angles of inclination (<10°), a convex seafloor displays wave breaking sooner than a straight-line one and thus actually delivers a smaller volume flux to the shore. Physically, a convex floor exhibits a greater rate of depth reduction (on first encounter with the sloping seafloor) than a straight-line one. Long waves with a speed proportional to the square root of the depth thus experience a larger deceleration. Nonlinear (or ‘piling up’) effects occur earlier than in the straight-line case. All these scenarios and reasoning are reversed for a concave seafloor. For large angles of inclination (>30°), impingement, reflection, and deflection are the relevant processes. Empirical dependence for the setup and swash values for a convex seafloor is established. The reflection coefficient for waves reflected from the seafloor is explored through Fourier analysis, and a set of empirical formulas is developed for various seafloor topographies. Understanding these dynamical factors will help facilitate the more efficient designing and construction of coastal defense mechanisms against severe weather.

Backspin in Ruellia ciliatiflora does not maximize seed dispersal range, but provides moderate dispersal range thatisrobust to launch conditions.

Ruellia ciliatiflora is a perennial herb whose fruits explosively dehisce, launching their thin disc-like seeds over 6 m with a backspin up to 1660 Hz. Whileithas been previously shown that the backspin launch orientation minimizes the aerodynamic drag experienced by the seeds, it is not immediately obvious whether backspin is also the range-maximizing launch orientation. Here the three-dimensional equation of motion of a thin, spinning disc flying through a fluid medium was derived and solved numerically to simulate the flight of seeds of R. ciliatiflora under different launch conditions. Simulations of seed flights reveal that the range-maximizing launch orientation lies between sidespin and topspin, far from the backspin that is observed in nature. While this range-maximizing orientation results in dispersal ranges of nearly 10 m, the precise orientation is highly sensitive to other launch parameters, chiefly spin rate and launch angle. By contrast, backspin, which yields moderate dispersal ranges about 60% of the range-maximizing orientation, is robust to perturbations in launch parameters that the plant cannot precisely control.

Three-dimensional nonlinear dynamics of imperfectly supported pipes conveying fluid

This paper examines the effects of support imperfections on the dynamics and stability of pipes conveying fluid. The three-dimensional equations of motion for an imperfectly supported pipe conveying fluid are obtained using the extended Hamilton’s principle. The support imperfection at the upstream end is modelled by cubic rotational springs. The partial differential equations governing the dynamics of the pipe are discretized in space via the Galerkin method. Numerical results including two-/three-dimensional motion maps and bifurcation diagrams show that the dynamics of the system is generally affected by the support imperfection. In particular, depending on the mass ratio and the level of imperfections, the critical flow velocity may be decreased or increased compared to the pipe with no support imperfections (i.e. cantilevered pipe). In addition, quasi-periodic oscillations may occur in the post-critical flow regime while they do not occur for cantilevered pipes. Also, the amplitude of oscillations are generally higher, and two- or three-dimensional motions may occur following a sequence different from that for cantilevered pipes.

Breaking Wave Simulations by a New k−l Turbulence Model

The three-dimensional motion equations are used to simulate the wave and velocity fields. These equations are written in integral contravariant form on a time-dependent curvilinear coordinate system. In this paper a new ?−? turbulence model in contravariant form is proposed for three-dimensional simulation of breaking waves. In this model the mixing length is defined as a function of the first and second spatial derivatives of the maximum water surface elevation.

3D MODELING OF BIOLOGICAL WASTEWATER TREATMENT IN AERATION TANK

Purpose. The main purpose of the article is to develop a 3D CFD model for modeling the process of biological wastewater treatment in an aeration tank. Methodology. For mathematical modeling of the process of biological wastewater treatment in the reactor, taking into account the flow hydrodynamics, geometric shape of the aeration tank, convective-diffusion transfer of the substrate and activated sludge, a 3D CFD model was built. The model is based on the three-dimensional equation of motion of an ideal liquid and the equation of mass conservation for the substrate, activated sludge. The field of sewage flow rate in the aeration tank is calculated based on the velocity potential equation. The process of biological transformation of the substrate is calculated on the basis of the Monod model. The splitting scheme was used for numerical integration of the equations of convective-diffusion transfer of activated sludge and substrate. The splitting is carried out in such a way to take into account the transfer of substrate (activated sludge) in only one direction at each step of splitting. The calculation of the unknown value of the substrate (activated sludge) concentration is carried out according to an explicit scheme. The Richardson method is used to numerically integrate the three-dimensional equation for the velocity potential, and the unknown value of the velocity potential is calculated by an explicit formula. Euler's method is used for numerical integration of equations describing the process of substrate transformation and change in activated sludge concentration (Monod model). Findings. The software implementation of the constructed 3D CFD model is carried out. A description of the structure of the developed software package is provided. The results of a computer experiment to study the process of wastewater treatment in an aeration tank with additional elements are presented. Originality. A new multifactor 3D CFD model has been developed, which allows quick assessing the efficiency of biological treatment in an aeration tank. Practical value. The constructed 3D CFD model can be used to analyze the efficiency of the aeration tank under different operating conditions at the stage of sketch design of wastewater treatment systems.

Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

Boundary Conditions for the Simulation of Wave Breaking

In this paper we propose a new numerical model for the simulation of the wave breaking. The three-dimensional equations of motion are expressed in integral contravariant form and are solved on a curvilinear boundary conforming grid that is able to represent the complex geometry of coastal regions. A time-dependent transformation of the vertical coordinate that is a function of the oscillation of the turbulent wave boundary layer is proposed. A new numerical scheme for the simulation of the resulting equations is proposed. New boundary conditions at the free surface and bottom for the equations of motion expressed in contravariant form are proposed. We present an analysis of the importance of the correct positioning, inside the oscillating turbulent boundary layer, of the centre of the calculation grid cell closest to the bottom, in order to correctly simulate the height of the breaking waves.

ЧИСЛЕННОЕ ИССЛЕДОВАНИЕ ТЕЧЕНИЙ В ЦИМЛЯНСКОМ ВОДОХРАНИЛИЩЕ ПРИ РАЗНЫХ УРОВНЯХ НАПОЛНЕНИЯ ВОДОХРАНИЛИЩА

A description of the mathematical model of the hydrodynamics of the Tsimlyansk reservoir is given. In shallow water regions, the flows are described by the equations of shallow water, and in deep-water regions by both the equations of shallow water and the three-dimensional equations of motion of an incompressible viscous fluid. The problem is solved by finite difference methods on a uniform difference grid. The calculations were carried out at different levels of reservoir filling. A comparison of the results obtained with the surface currents scheme constructed based on the results of field work performed in the spring of 2019 is carried out. The pictures of currents were obtained at different water edges. For the period of expeditionary field work in May 2019, hydrodynamics was calculated on the near-dam site using wind data obtained for this period and compared with the results of field measurements of currents. Comparison of the obtained results of numerical modeling with the scheme of surface currents measured from the research vessel reveals a high proportion of convergence of the calculated and field data. The obtained estimates of the modeling results allow us to proceed to the problem of predicting unfavorable hydrometeorological conditions in the Tsimlyansk reservoir, which carry the risk of disrupting the operation of the water intakes of Volgodonsk located in the southern part of the near-dam site

Specific Features of Charged Particle Confinement in the T-15 Tokamak in the Presence of Magnetic Perturbations

Charged particle motion in the regions of magnetic islands and near-separatrix ergodicity of magnetic field lines in the Т-15 tokamak is studied numerically. The trajectories are calculated by integration of exact three-dimensional equations of motion for trapped and passing charged particles starting at different pitch angles. It is shown that the presence of resonant magnetic perturbations qualitatively affects the trajectories of passing particles. In the region of magnetic islands, the orbits of passing particles acquire an island structure, while in the region of the near-separatrix ergodicity of magnetic field lines, the orbits are stochastized. It is shown that magnetic perturbations insignificantly affect trapped particle motion. Even in the absence of magnetic surfaces, the trajectories of trapped particles are regular and their cross sections have the shape of a standard banana orbit. The possibility of charged particles to pass between regions with different magnetic topologies is analyzed.

Investigation of state vector computational solution on modeling of wave propagation through functionally graded nanocomposite doubly curved thick structures

In this paper, a solution strategy based on the state vector technique is proposed to study the vibroacoustic performance of carbon nanotube (CNT)-reinforced composite doubly curved thick shells employing three-dimensional theory. For this purpose, the properties of the nanocomposite media are investigated by the rule of mixture which includes some productivity parameters. This approach considers various distributions of CNT which can be identical or functionally graded through thickness direction. Since the state vector solution is used to solve the three-dimensional equations of motion, the nanocomposites are divided into s layers for stress and strain components according to approximate layer model along with local transfer matrix. This process is followed by the global transfer matrix for all of the layers of thick shell. After obtaining the sound transmission loss of structure, various configurations are presented to validate the results. Hence, not only the precision of the formulation is confirmed but also the importance of considering the three-dimensional theory is remarked due to very reliable outcomes at high-frequency domain for CNT-reinforced composite thick shells. Furthermore, it is shown that the CNTs are able to enhance the sound insulation property of doubly curved structures. Finally, a new approach is made to present the sound pressure level of CNT doubly curved shells with respect to various CNT materials in different distributions.

Analysis of axisymmetric fractional vibration of an isotropic thin disc in finite deformation

This study investigates axisymmetric fractional vibration of an isotropic hyperelastic semi-linear thin disc with a view to examine effects of finite deformation associated with the material of the disc and effects of fractional vibration associated with the motion of the disc. The generalized three-dimensional equation of motion is reduced to an equivalent time fraction one-dimensional vibration equation. Using the method of variable separable, the resulting equation is further decomposed into second-order ordinary differential equation in spatial variable and fractional differential equation in temporal variable. The obtained solution of the fractional vibration problem under consideration is described by product of one-parameter Mittag-Leffler and Bessel functions in temporal and spatial variables respectively. The obtained solution reduces to the solution of the free vibration problem in literature. Finally, and amongst other things, the Cauchy\'s stress distribution in thin disc under finite deformation exhibits nonlinearity with respect to the displacement fields whereas in infinitesimal deformation hypothesis, these stresses exhibit linear relation with the displacement field.

To mathematical modeling of deformation of micropolar thin bodies with two small sizes

We consider some problems of modeling the deformation of micropolar thin bodies with two small sizes. Using the three-dimensional equations of motion, the constitutive relations and the boundary conditions of the micropolar elasticity theory [1, 2], we got the equations of motion, the constitutive relations and the boundary conditions of the micropolar theory of thin bodies under the parametrization of the thin body domain with two small sizes [3]. The boundary conditions and various representations of the system of equations of motion and the constitutive relations of physical content in moments with respect to the Legendre polynomials are obtained. Note that using this method of construction the thin bodies theory with two small sizes, we get an infinite system of ordinary differential equations. This system contains quantities which depends on one variable, namely, depends on the parameter of the base line. Thus, decreasing the number of independent variables from three to one we increase the number of equations to infinity, which, of course, has its obvious practical inconveniences. In this regard, we reduce an infinite system to the finite system. The initial-boundary value problems are formulated. To satisfy the boundary conditions on the front surfaces we constructed correcting terms [4]. As a special case, we considered a prismatic body. We used the tensor calculus to do this research [5–8].

Constrained Trajectory Optimization for Planetary Entry via Sequential Convex Programming

In this paper, the highly nonlinear planetary-entry optimal control problem is formulated as a sequence of convex problems to facilitate rapid solution. The nonconvex control constraint is avoided by introducing a new state variable to the original three-dimensional equations of motion. The nonconvex objective function and path constraints are convexified by first-order Taylor-series expansions, and the nonconvex terms in the dynamics are approximated by successive linearizations. A successive solution procedure is developed to find an approximated solution to the original problem, and its convergence is discussed. In each iteration, a convex optimization problem is solved by the state-of-the-art interior-point method with deterministic convergence properties. Finally, the proposed method is verified and compared to a general-purpose optimal control solver by numerical solutions of minimum terminal-velocity and minimum heat-load entry problems. The sequential method converges to accurate solutions with faster speed than the general-purpose solver using MATLAB on a desktop computer with a 64-bit operating system and an Intel Xeon E3-1225 V2 3.2 GHz processor, which demonstrates its potential real-time application for computational guidance.

Dynamic analysis of functionally graded piezoelectric cylindrical panels by a three-dimensional mesh-free model

In this work, dynamic analysis of functionally graded piezoelectric cylindrical panels is carried out under different mechanical and electrical loads and boundary conditions by a three-dimensional mesh-free model. Moving least squares approximation is used in the weak form of governing equations including three-dimensional equations of motion and Maxwell’s equation. Transformation method is applied to impose the essential boundary conditions. A power-law distribution is used to determine the effective material properties in the panel. The resulting system of differential equations is solved using Newmark time integration method. After validation of the proposed model, parametric study is carried out to investigate the effects of boundary conditions, geometry of panel, and distribution of constituent materials on natural frequencies and dynamic response of functionally graded piezoelectric panel.

A rational derivation of dynamic higher order equations for functionally graded micropolar plates

The dynamics of functionally graded micropolar plates is considered. The derivation process is based on power series expansions in the thickness coordinate. Using the three-dimensional equations of motion for micropolar continuum, variationally consistent equations of motion and end boundary conditions are derived in a systematic fashion up to arbitrary order. Numerical results are presented for simply supported plates using different material distributions for both low and high order truncation orders. These results illustrate that the present approach renders benchmark solutions provided higher order truncations are used, and act as engineering plate equations using low order truncation.

Nonlinear dynamics of straight fluid-conveying pipes with general boundary conditions and additional springs and masses

In this paper, nonlinear equations of three-dimensional motion are established for straight fluid-conveying pipes with general boundary conditions. Firstly ten springs at the two ends of the pipes are used to represent the general boundary conditions, and the displacements are expressed as the superposition of a Fourier cosine series and four supplementary functions to satisfy the boundary conditions. Then in the Lagrangian framework, three coupled equations of motion in compact matrix form are derived by using the extended Hamilton's principle, and the effect of the extra linear springs and lumped masses at arbitrary positions along the pipe are also considered in these equations. The natural frequencies of pipes with several different boundary conditions are computed by the linearized dynamic equations in order to validate the method through comparisons with results from other reliable sources. Nonlinear dynamic behavior of the pipes with different boundary conditions is analyzed by solving the nonlinear equations using the Runge–Kutta scheme at last. It is found that the method presented in the paper could conveniently and effectively solve the nonlinear vibration problem of fluid-conveying pipes with general boundary conditions and additional springs and masses.

Formulation of three-dimensional equations of motion for train–slab track–bridge interaction system and its application to random vibration analysis

This study presents the formulation of three-dimensional equations of motion for a train–slab track–bridge interaction system and its application to random vibration analysis using the finite element and pseudo-excitation methods. In this study, a train, slab track, and bridge are regarded as an integrated system, each vehicle is modeled as a four-wheelset mass-spring-damper system with a two-layer suspension system at 23 degrees of freedom, and the rail, slab, girder, and pier are modeled as elastic Bernoulli–Euler beams connected with each other by discrete or continuous spring and damper elements. Three-dimensional equations of motion for the entire system are derived using the energy principle. Dynamic contact forces between moving vehicles and rails are considered as internal forces, and thus, the excitation vectors of load between a wheel and rail, induced by a vehicle's weight and random track irregularities, are easily formulated using the pseudo-excitation method. These equations can be solved by a step-by-step integration method to simultaneously obtain the random dynamic responses of the system. The three-dimensional random vibration characteristics of the system are investigated using an example of a nine-span simply supported beam bridge on which a train consisting of 8 cars travels.

Retraction Note to: Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating disk flow

The main interest of the present paper is to generate exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow motion due to a disk rotating with a constant angular speed. In place of the traditional von Karman’s axisymmetric evolution of the flow, the rotational non-axisymmetric stationary conducting flow is taken into consideration here. As a consequence, for an external uniform magnetic field applied perpendicular to the plane of the disk, the governing equations allow an exact solution to develop, which is influenced by a fixed point on the disk and also is bounded everywhere in the normal direction to the wall. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions, which differ from those of corresponding to the classical von Karman’s conducting flow. Making use of this solution, analytical formulas for the angular velocity components as well as for the wall shear stresses are extracted. The critical peripheral locations at which extrema of the local skin friction occur are also determined. It is proved from the analytical results that for the specific flow the properly defined thicknesses decay as the magnetic field strength increases in magnitude. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation and the Joule heating. According to the Fourier’s heat law, a constant heat transfer from the disk to the fluid occurs, though decreases for small magnetic fields because of the dominance of Joule heating, it eventually increases for growing magnetic field parameters.

Elasticity-based free vibration of anisotropic thin-walled beams

The free-vibration of anisotropic open and closed thin-walled beams is studied using approximate solutions to the three-dimensional equations of motion of a linear elastic solid. The three displacement components are approximated using grouped polynomials over both the cross-section coordinates and independent polynomial or trigonometric functions over the axial coordinate of several beams with representative boundary conditions. This method is applied to several sections with varying degrees of anisotropy and shape with both isotropic and orthotropic material properties. Results are compared with those computing using beam or shell theories, and several new results are presented for these geometries.

Long waves in a straight channel with non-uniform cross-section

A cross-sectionally averaged one-dimensional long-wave model is developed. Three-dimensional equations of motion for inviscid and incompressible fluid are first integrated over a channel cross-section. To express the resulting one-dimensional equations in terms of the cross-sectional-averaged longitudinal velocity and spanwise-averaged free-surface elevation, the characteristic depth and width of the channel cross-section are assumed to be smaller than the typical wavelength, resulting in Boussinesq-type equations. Viscous effects are also considered. The new model is, therefore, adequate for describing weakly nonlinear and weakly dispersive wave propagation along a non-uniform channel with arbitrary cross-section. More specifically, the new model has the following new properties: (i) the arbitrary channel cross-section can be asymmetric with respect to the direction of wave propagation, (ii) the channel cross-section can change appreciably within a wavelength, (iii) the effects of viscosity inside the bottom boundary layer can be considered, and (iv) the three-dimensional flow features can be recovered from the perturbation solutions. Analytical and numerical examples for uniform channels, channels where the cross-sectional geometry changes slowly and channels where the depth and width variation is appreciable within the wavelength scale are discussed to illustrate the validity and capability of the present model. With the consideration of viscous boundary layer effects, the present theory agrees reasonably well with experimental results presented by Chang et al. (J. Fluid Mech., vol. 95, 1979, pp. 401–414) for converging/diverging channels and those of Liu et al. (Coast. Engng, vol. 53, 2006, pp. 181–190) for a uniform channel with a sloping beach. The numerical results for a solitary wave propagating in a channel where the width variation is appreciable within a wavelength are discussed.