Curvilinear System Research Articles

Stress-strain state of an open spherical shell of a medium thickness

We study the stress-strain state of a medium thickness spherical shell with a hole in the vicinity of the pole in a geometrically nonlinear statement using the classical Kirchhoff-Love model and refined Timoshenko-type model that takes into account the transverse shear strains. The study is based on the construction of an orthogonal curvilinear system of coordinates of a median surface that presents a doubly-connected domain and also on the reduction of the original nonlinear boundary problem to a sequence of linear two-dimensional problems and these latter to one-dimensional ones whose integration is performed by a robust numerical method. Using an open spherical shell as an example, the influence of the transverse shear strain on its stress-strain state is investigated under the action of uniform and nonuniform loads. As compared to the classical model, the use of the Timoshenko-type model is shown to result in a considerable refinement of displacements near the hole.

Vortex Line Representation for the Hydrodynamic Type Equations

In this paper we give a brief review of the recent results obtained by the author and his co-authors for description of three-dimensional vortical incompressible flows in the hydrodynamic type systems. For such flows we introduce a new mixed Lagrangian-Eulerian description - the so called vortex line representation (VLR), which corresponds to transfer to the curvilinear system of coordinates moving together with vortex lines. Introducing the VLR allows to establish the role of the Cauchy invariants from the point of view of the Hamiltonian description. In particular, these (Lagrangian) invariants, characterizing the property of frozenness of the generalized vorticity into fluids, are shown to represent the infinite (continuous) number of Casimirs for the so-called non-canonical Poisson brackets. The VLR allows to integrate partially the equations of motion, to exclude the infinite degeneracy due to frozenness of the primitive Poisson brackets and to establish in new variables the variational principle. It is shown that the original Euler equations for vortical flows coincides with the equations of motion of a charged compressible fluid moving due to a self-consistent electromagnetic field. Transition to the Lagrangian description in a new hydrodynamics is equivalent to the VLR. The VLR, as a mapping, turns out to be compressible that gives a new opportunity for collapse in fluid systems - breaking of vortex lines, resulting in infinite vorticity. It is shown that such process is possible for three-dimensional integrable hydrodynamics with the Hamiltonian where Ω is the vorticity. We also discuss some arguments in the favor of existence of such type of collapses for the Euler hydrodynamics, based on the results of some numerics.

Vibration response and harmonic wave propagation of ultrasonic arc drivers

A piezoelectric curvilinear arc driver designed for an ultrasonic curvilinear motor is evaluated in this study. A design of piezoelectric curvilinear arc driver is proposed and its governing equations, vibration behaviour and wave propagation are investigated. Then, analysis of forced vibration response or driving characteristics to harmonic excitations in the modal domain is conducted. Finite element modelling and analysis of the arc driver are also discussed. Analytical results of free vibration characteristics are compared favourably with the finite element results. Harmonic analyses of the three finite element models reveal changes of dynamic behaviours of three models and also imply operating frequencies with a significant travelling wave component. Parametric study of mathematical and finite element simulation results suggests that stable travelling waves can be generated to drive a rotor on the proposed curvilinear arc motor system.

Godunov-Based Model for Nonhydrostatic Wave Dynamics

A new numerical model based on the incompressible, nonhydrostatic Navier-Stokes equations for free surface flow is developed. The equations are transformed vertically to the σ coordinate system and laterally to an orthogonal curvilinear system and solved in a fractional step manner in which the pressure is split into hydrostatic and nonhydrostatic components. The model treats the nonhydrostatic term implicitly and uses a collocated grid and pressure interpolation to prevent checkerboard solutions that occur when the velocity and pressure become decoupled. Advection and hydrostatic pressure terms are integrated explicitly with a second-order accurate predictor-corrector scheme. The corrector utilizes fluxes that are computed in a Godunov-based manner by solving a Riemann problem at each cell face. Flow variables are reconstructed at each cell face to obtain second-order spatial accuracy. Numerical simulations of Stokes, cnoidal, and solitary waves with the proposed method and a reference method in which th...

Buoyancy-induced periodic flow and heat transfer in lid-driven cavities with different cross-sectional shapes

This report is concerned with transient behavior of a buoyancy-induced periodic flow in different lid-driven cavities with different cross-sectional shapes. Periodic flow patterns and heat transfer characteristics for various geometries are predicted. Governing equations are discretized based on the finite volume method with a curvilinear grid system generated numerically by the body-fitted coordinate transformation. Attentions are focused at nine cases of cross-sectional shape. For all the cases considered, Reynolds number and Grashof number are fixed at 100 and 5×10 5, respectively. Results show that among the nine cases, cases 1/5C, 1/4C, 1/4T, and 1/5R tend to produce the periodic flow pattern and, case 1/5T exhibits the highest heat transfer performance.

Non-linear dynamics of elastic curvilinear systems. A new coordinateless approach

Using the theory of Lie groups, a mechanical model is developed and the equations of motion are written in coordinateless form for an arbitrary curvilinear model of an elastic system with dimension no higher than three, starting from the unique hypothesis of a continuous distribution of rigid bodies, called sections. The proposed method is illustrated using examples of the automatic derivation of the corresponding scalar equations, by means of which one can describe all possible models of such systems as beams, cables, strings, etc.

Seepage analysis based on boundary-fitted coordinate transformation method

Finite difference method (FDM) based on boundary-fitted coordinate (BFC) transformation is presented in this paper. The curvilinear grid system, with computational boundary being coincident with the physical boundary, is numerically obtained by solving the Poisson equation. Seepage analysis can then be done by FDM in a uniform transformed orthogonal coordinate system. The method was applied to analyze the steady seepage in a foundation pit, a lock foundation, and an embankment dam with a free surface. It shows that BFC transformation method can make it easy to fit and update the changeable free surface as well as to closely simulate the physical domain with complex geometrical boundaries.

Numerical modelling of the propagation of fast landslides using the finite element method

AbstractThis paper proposes a two‐dimensional (2D) model for the analysis of the propagation of fast landslides involving a fluidized material such as debris and mud flows, flowslides and avalanching flows. The model is based on the Navier–Stokes depth‐integrated equations. To incorporate the effect of steep slopes and centrifugal forces due to the high velocities characterizing the flowslides and the bed curvature, a curvilinear system of reference is used. The corresponding equations of motion are complemented by depth‐averaged constitutive equations and bed friction laws.The resulting set of differential equations are solved using the two‐step Taylor–Galerkin algorithm. This algorithm has been used by the authors to solve hydraulic and dam‐break problems using the finite element method. Owing to the importance of the source term compared to the advection component, the proposed algorithm follows a splitting scheme using a fourth‐order Runge–Kutta method for integrating the friction and slope components.The performance of the overall approach has been checked in a number of examples. The analysis of the results provides insights into the key elements of the model and shows the adequacy of the method to solve real problems where merging and splitting of the flow occur. Copyright © 2004 John Wiley & Sons, Ltd.

移動一般座標系での水・深積分モデルを用いた水槽内流体振動解析

移動一般座標系での水・深積分モデルを用いた水槽内流体振動解析

General equations of curvilinear systems: A coordinate-free approach by using Lie group theory

Using Lie group theory, it is proposed to give the intrinsic most general equations of any curvilinear system ( Σ), the only hypothesis being that each section is rigid. After giving these equations, we shall illustrate the power of the method by proposing some elements of the automatic generation of the corresponding scalar equations.

Propagation of Nonstationary Curved and Stretched Premixed Flames with Multistep Reaction Mechanisms

The propagation speed of a thin premixed flame disturbed by an unsteady fluid flow of a larger scale is considered. The flame may also have a general shape but the reaction zone is assumed to be thin compared to the flame thickness. Unlike in preceding publications, the presented asymptotic analysis is performed for a general multistep reaction mechanism and, at the same time, the flame front is curved by the fluid flow. The resulting equations define the propagation speed of disturbed flames in terms of the properties of undisturbed planar flames and the flame stretch. Special attention is paid to the near-equidiffusion limit. In this case, the flame propagation speed is shown to depend on the effective Zeldovich number Z f , and the flame stretch. Unlike the conventional Zeldovich number, the effective Zeldovich number is not necessarily linked directly to the activation energies of the reactions. Several examples of determining the effective Zeldovich number for reduced combustion mechanisms are given while, for realistic reactions, the effective Zeldovich number is determined from experiments. Another feature of the present approach is represented by the relatively simple asymptotic technique based on the adaptive generalized curvilinear system of coordinates attached to the flame (i.e., intrinsic disturbed flame equations [IDFE]).

Crosstalk and reflection for curvilinear conductors by utilizing a non-uniform transmission line approach

A novel approach to analyze curvilinear multiconductor transmission line systems is presented. Non-uniform transmission line equations for the systems are obtained and solved by using chain matrix approach. Crosstalk and reflection due to discontinuities such as bend and changes of radii are investigated. Per-unit-length parameters for the lines are discussed as well. It is observed that the discontinuities will increase the crosstalk significantly and bending will substantially increase crosstalk at the driving end.

Flow simulation in the human upper respiratory tract.

Computer simulations of airflow patterns within the human upper respiratory tract (URT) are presented. The URT model includes airways of the head (nasal and oral), throat (pharyngeal and laryngeal), and lungs (trachea and main bronchi). The head and throat morphology was based on a cast of a medical school teaching model; tracheobronchial airways were defined mathematically. A body-fitted three-dimensional curvilinear grid system and a multiblock method were employed to graphically represent the surface geometries of the respective airways and to generate the corresponding mesh for computational fluid dynamics simulations. Our results suggest that for a prescribed phase of breath (i.e., inspiration or expiration), convective respiratory airflow patterns are highly dependent on flow rate values. Moreover, velocity profiles were quite different during inhalation and exhalation, both in terms of the sizes, strengths, and locations of localized features such as recirculation zones and air jets. Pressure losses during inhalation were 30-35% higher than for exhalation and were proportional to the square of the flow rate. Because particles are entrained and transported within airstreams, these results may have important applications to the targeted delivery of inhaled drugs.

A New Approach To Derive and Analyze Nonuniform Transmission Line Equations for Curvilinear Multiconductor Systems

A novel approach to derive nonuniform transmission line equations for curvilinear multiconductor systems is presented by directly using the scalar and vector potentials associated with boundary conditions. The per-unit-length parameters of the line can be obtained simultaneously in derivation and are explicitly given for straight conductor systems as an example. Results for a bent conductor of different radii over ground plane also verify the approach.

Molecular Fluids and Liquid Crystals in Convex-Body Coordinate Systems

The article discusses in a unified and lucid manner the prerequisites for the study of statistical physics of such convex body (CB) fluids that undergo transition from isotropic to liquid crystalline phases. The geometry of anisotropic rigid convex molecules is reviewed in an up-to-date and coherent manner limiting the discussions to the uniaxial and biaxial ellipsoidal molecules. Some old results, available from the use of the spherical polar coordinates or through the use of the CB coordinates, pertaining to the geometrical parameters needed in the study of thermodynamic, structural, and transport properties are rederived using our own approach and some new ones are reported. The transformation properties of the differential operator ▿→ in the CB coordinates are shown to be typically akin to the Cartesian coordinates. The suitability and elegancy inherent in the use of the CB system in the context of the study of aspherical polyatomic fluids are amply demonstrated. Furthermore, approach and methodology for the advancement based on the application of this orthogonal curvilinear system in the investigation of fluids composed of ellipsoidal molecules in isotropic as well as liquid crystalline phases are suggested.

On Edge Stresses Control in Strengthened RC Beams with FRP Strips: Adhesive Layer Profile Effect

A high-order analytical model for the analysis of reinforced concrete (RC) beams strengthened with fiber-reinforced plastic (FRP) strips bonded with an adhesive layer of variable thickness is presented. The model is based on the closed-form high-order approach, and it provides the means for the analysis of beams retrofitted with generally curved FRP strips and adhesive layers of arbitrary profile. The analysis is comprehensive and includes the local and overall response of the structure. An emphasis is put on the stress concentration that occurs at the edge of the FRP strip and in many cases leads to brittle and sudden failure of the strengthened member. The field equations and the boundary and continuity conditions are derived using the variational principle of virtual work along with the kinematic relations of small deformations. The governing equations of the generally curved FRP strip include large curvatures and are introduced via coordinate transformation from its local curvilinear system into the global Cartesian one. The derived model is used for the investigation of various adhesive profiles and their influence on the shear and vertical normal stresses at the edges of the FRP strip. The results focus on the stress concentrations involved and reveal that proper design and application of the adhesive profile can significantly reduce the edge stresses, thus preventing the brittle mode of failure. The paper is concluded with a summary and recommendations for the analysis, design, and application of the strengthening process.

Inverse optical magnus effect in the Majorana representation

An equation describing the effect of additional bending during photon motion along a twisted path is obtained from Maxwell’s equations in the Majorana representation. The generalized Rytov law in an arbitrary curvilinear system of coordinates forms the basis for the effect. The effect considered is inverse to the optical Magnus effect.

On Premixed Flames as Gasdynamic Discontinuities: A Simple Approach to Derive their Propagation Speed

We consider the propagation speed of thin premixed flames disturbed by the fluid flow and suggest a new approach based on using the generalized curvilinear system of coordinates and tensor calculus. The suggested technique is shown to be more simple than the traditional technique used by Sivashinsky (1976), Matalon and Matkowsky (1982) and also in other publications. The result of Matalon and Matkowsky (1982) is generalized for an arbitrary temperature dependent diffusion coefficient.

Polarization effects in the Majorana representation

Maxwell’s equations in the Majorana form are extended to the case of a orthogonal curvilinear system of coordinates. The optical Magnus effect is considered as an example. The results obtained are extended to massless spin particles with half-integer spins.

Tutorial / Article didactique Derivatives of curvilinear coordinate unit vectors: A concise matrix-vector approach

The time evolution of a curvilinear coordinate system can be identifiedwith an angular velocity matrix or vector, which contains the complete differentiationrule for all unit coordinate vectors of the curvilinear system, providinga concise method for their computations. When generalized the technique provides a powerful tool fordifferentiation, with respect to any parameter, of not only unit coordinate vectors but any vector that remains fixed in a general orthogonal curvilinear coordinate system. Several curvilinear systems, as well as the normal and tangentialcoordinate system are examined.PACS No.: 46.00