Orthogonal Curvilinear Coordinate System Research Articles

Low Speed Preconditioning Algorithm Used in the Process of Calculating Euler Equation

This paper describes a numerical method for solving the unsteady Euler equation at any speed. In the process of calculating Euler's equation, the control equation in orthogonal curvilinear coordinate system is discretized by the finite -volume scheme based on the center-difference method, and convection flux used Jameson central deference scheme was solved at every pseudo time step, and the Runge-Kutta method, dual-time algorithm and the implicit LU-SGS add preconditioning algorithm are used for time-marching. For obtaining the numerical solution of two-dimensional unsteady flow around a cylinder and the flow of the Convex Hull, finding that the calculated results agree well with literature values and theoretical analytical solution.

Extension of Material Derivative Concept in Fluid Mechanics

The original definition of material derivative is changing rate relative to the time of a physical quantity which belongs to a fluid particle. But when studying fluid mechanics problems, need in certain cases to calculate the material derivative of some quantities which don’t belong to a fluid particle, such as calculating the material derivative of base vectors of orthogonal curvilinear coordinate system. The original definition of material derivative isn’t suitable for these quantities. Calculating material derivative of these quantities is not clear in concept. To solve this problem, thorough analysis about material derivative was made according to the original definition and calculation formula of material derivative. Results found that the concept of material derivative can be extended and so can apply to any quantities which don’t belong to a fluid particle and are defined in flow fields, including vector or scalar, physical quantities or non physical quantities.

STUDY ON INFLUENCE OF BAIMAO SANDBANK CONTROL PROJECT IN SOUTHERN BRANCH OF THE YANGTZE RIVER ESTUARY, CHINA

The author establishes the Yangtze estuary 2D tidal and sediment mathematical model based on the orthogonal curvilinear coordinate system. The erosion and siltation of engineering section under the hydrological conditions of flood season, dry season and the great flood after the Baimao Sand control engineering is calculated. Preliminarily expounds the correlation of Baimao Sand and Biandan Sand and the impact of Biandan Sand after the Baimao Sand control engineering is implemented.

X-FEM Applied to Three-Dimensional Curvilinear Crack Front

A three-dimensional extended finite element algorithm coupled with the level set method is developed for curvilinear crack front problems. This method does not need to force the discontinuities to be in conformity with the mesh. In order to compute the pointwise stress intensity factors along the curvilinear crack front, an orthogonal curvilinear coordinate system is constructed at the intersection point between the crack front and a surface of the crack front element. As the crack surface is expressed by the level set method, no explicit meshing of the crack geometry emerges on the crack front. This makes the domain integral more difficult. An independent grid of hexahedral cells around the crack front needs to be built as the virtual extension domain for accomplishing the domain integral. To assess the accuracy of the approach, an example with semi-elliptic crack surface is considered. The results show excellent agreement with the results calculated by the standard finite element method and those obtained from the literature.

A rigorous full-wave analysis of mode coupling in rectangular waveguide bends

From Maxwell's equations in an orthogonal curvilinear coordinate system, a rigorous full-wave analysis of mode coupling in rectangular waveguide bends is performed, obtaining the general and explicit formulas of coupling coefficients for any modes in rectangular waveguides. This full-wave analysis turns out to be necessary for analysing mode conversion in microwave devices involving waveguide bends. To validate the theory calculation, some miter-joint mode converters, which are composed of several miter-joint waveguides, are proposed and studied. It shows that, in some instances, miter-joint mode converters and bend converters have similar conversion characteristics if they have equal bend angles and equivalent axial lengths. The excellent behaviour of our results enables engineers to design these types of waveguide bends quickly.

Swirling turbulent flow between rotating profiled discs

Modeling of aerodynamics in an air-centrifugal classifier, the separation zone of which consists of rotating profiled disc elements, between which a turbulent swirling flow is observed in the direction to the rotation axis, is carried out. The problem was solved numerically in the acquired orthogonal curvilinear coordinate system using the known turbulence model by Wilcox. The influence of mode parameters and the additional supply of the carrying medium on the aerodynamics in the separation zone of the centrifugal apparatus is shown. The reliability of the results is substantiated by test investigations and by a comparison with reference data.

A 2D Numerical Model for Simulation of Two-Dimensional Circular Dam-Break

The purpose of this paper is to present a 2D depth-averaged model under orthogonal curvilinear coordinates for simulating two-dimensional circular dam-break flows. The proposed model uses an orthogonal curvilinear coordinate system efficiently and accurately to simulate the flow field with irregular boundaries. As for the numerical solution procedure, The SIMPLEC solution procedure has been used for the transformed governing equations in the transformed domain. Practical application of the model is illustrated by an example, which demonstrates that the mathematical model can capture hydraulic discontinuities accurately such as steep fronts, hydraulic jump and drop, etc.

Three-dimensional multi-field equations of a functionally graded piezoelectric thick shell with variable thickness, curvature and arbitrary nonhomogeneity

The present research develops a three-dimensional multi-field formulation of a functionally graded piezoelectric thick shell of revolution by using tensor analysis. An orthogonal curvilinear coordinate system was employed, and basic geometric equations were derived for an arbitrary thick shell of revolution with variable thickness and curvature. Mechanical and electrical properties were assumed to vary along a three-dimensional orthogonal coordinate system with arbitrary functional distribution. The functional of the introduced shell was derived by using kinetic and potential energy of the structure based on three orthogonal displacement components, electric potential and material properties. The final differential equations were derived in general state for every arbitrary structure and material property distributions. The obtained equations were reduced for functionally graded and functionally graded piezoelectric cylindrical shells and the mentioned reduced equations were verified by comparison with the literature. Trueness and generality of the present results can be justified by capability of these equations for different geometries and material properties.

On orthogonal curvilinear coordinate systems in constant curvature spaces

We describe a method for constructing an n-orthogonal coordinate system in constant curvature spaces. The construction proposed is actually a modification of the Krichever method for producing an orthogonal coordinate system in the n-dimensional Euclidean space. To demonstrate how this method works, we construct some examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.

Numerical study on the shape oscillation of an encapsulated microbubble in ultrasound field

The shape oscillation of an encapsulated microbubble in an ultrasound field is numerically investigated. To predict the nonlinear process, the continuity equation and the Navier–Stokes equation are directly solved by means of a boundary-fitted finite-volume method on an orthogonal curvilinear coordinate system. The mechanics of neo-Hookean membrane is incorporated into the dynamic equilibrium at the bubble surface. The numerical results show that the membrane raises the natural frequency of an encapsulated bubble especially for small bubble, whereas this effect is attenuated as the initial bubble size grows. For a small encapsulated bubble of which the natural frequency is sufficiently higher than the driving frequency, the oscillation is stable, namely, the oscillatory amplitude is small; besides, the radial mode and shape modes are out of resonance so that no deformation emerges. As the bubble becomes larger, the natural frequencies of encapsulated and gas bubbles get closer, leading to the less apparent difference in oscillatory amplitude between them. Furthermore, shape modes of an encapsulated bubble are prone to be induced when twice of the higher-order natural frequency is approximately equal to the frequency of radial mode particularly when the bubble is at radial resonance for which the large-amplitude pulsation enhances the compressive stress developing in the membrane. In contrast, the shape oscillation is less likely to occur for a gas bubble with micrometer size since the surface tension suppresses the developments of nonspherical shape modes.

Characteristics of wave propagation in piezoelectric bent rods with arbitrary curvature

The wave propagation in the piezoelectric bend rods with arbitrary curvature is studied in this paper. Basic three-dimensional equations in an orthogonal curvilinear coordinate system ( r, θ, s) are established. The Bessel functions in radial co-ordinate and triangle series in the angular co-ordinate are used to describe the displacements and electrical potential. Characteristics of dispersion, distributions of displacements and electrical potential over the cross section are calculated, respectively. In the numerical examples, the effects of the ratio of the two ellipse axes on the dispersion relations of the first three modes are observed. The characteristics of the distribution of displacements and electric potential in the cross section, along the radial and s direction are investigated.

Low Reynolds Number Flow in Spiral Microchannels

We study the creeping flow of an incompressible fluid in spiral microchannels such as that used in DNA identifying “lab-on-a-chip” installations. The equations of motion for incompressible, time-independent flow are developed in a three-dimensional orthogonal curvilinear spiral coordinate system where two of the dimensions are orthogonal spirals. The small size of the channels results in a low Reynolds number flow in the system, which reduces the Navier–Stokes set of equations to the Stokes equations for creeping flow. We obtain analytical solutions of the Stokes equations that calculate velocity profiles and pressure drop in several practical configurations of channels. Both pressure and velocity have exponential dependence on the expansion/contraction parameter and on the streamwise position along the channel. In both expanding and converging channels, the pressure drop is increased when the expansion/contraction parameter k and/or the curvature is increased.

The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized H -function using integral transformation methods. The fractional heat conduction equation in the case 0 < α ≤ 1 interpolates the standard heat conduction equation ( α = 1 ) and the Localized heat conduction equation ( α → 0 ) . Finally, numerical results are presented graphically for various values of order of fractional derivative.

Debye series for light scattering by a nonspherical particle

The Debye series is developed for scattering of light by a homogeneous nonspherical particle to interpret the angular dependence of the scattered intensity in terms of various physical processes. In contrast to the previously developed Debye series for several regularly shaped particles that mirror the orthogonal curvilinear coordinate system where the variable-separation method can be applied, we develop and verify the Debye series in a coordinate-independent way using the extended boundary condition method. Verification computations are made for an oblate spheroidal water droplet of equivalent-volume sphere radius 10 $\ensuremath{\mu}$m.

On two models of arbitrarily curved three-dimensional thin interphases in elasticity

In boundary value problems involving thin interphases, it is often desirable to have a model of an interphase which makes possible to solve for the fields in the adjacent media without having to solve for the fields in it. This is usually achieved in the literature by replacing the interphase by a geometrical surface with appropriately designed “imperfect interface” conditions on it. In the present study, carried out in the setting of elasticity, another option is explored: the geometry of the interphase is left intact, and conditions are devised for the displacements and tractions pertaining to the media adjacent to the interphase and evaluated at both sides of it such that they will simulate the presence of the interphase. Those conditions do not involve the fields within the interphase, yet they depend on its material properties and on those of the adjacent media as well, and make possible to solve for the fields in the adjacent media without having to solve for the fields in the interphase. The formulation is given in a parallel orthogonal curvilinear coordinate system suitable for the modeling arbitrarily curved three-dimensional interphases of constant thickness. Both types of the above described interphase models are tested in the setting of a coated infinite fiber embedded in a matrix which is subjected to an anti-plane shear loading and an in-plane transverse shear loading at infinity, and their predictions are compared with the exact solutions for the fields in the three-phase configuration consisting of the interphase and its adjacent media. The model in which the interphase geometry is left intact is observed to perform generally better than the one in which the interphase is replaced by an interface.

Deposition and erosion in the fluctuating backwater reach of the Three Gorges Project after upstream reservoir adjustment

The construction of hydropower projects, such as Xiluodu and Xiangjiaba Reservoirs, in the upper Yangtze River will lead to variations of the incoming water and sediment conditions and in turn changes in the deposition and erosion processes in the fluctuating backwater reach of the Three Gorges Project (TGP). In this paper, based on the water and sediment characteristics, a 2D mathematical model in the boundary-fitting orthogonal curvilinear coordinate system has been employed to predict the space-time changes of sedimentation in the Chongqing reach, part of the fluctuating backwater reach of TGP, in a period of 100 years with repetition of the 1961–1970 hydrological series with and without the construction of upstream reservoirs. The results show that in the case of natural hydrological series (NHS) without upstream reservoirs, severe deposition occurs in bays with concave bank lines, recirculation flow areas and wide-shallow reaches. In the case with the adjustment of upstream reservoirs, only a little deposition exists in bays with concave bank lines, and the amounts of deposition in the Yangtze River and Jialing River reaches are about 10–20% of those in the case of NHS. Therefore, this is helpful to maintain the effective capacity of the TGP Reservoir and utilize the bank lines of the Chongqing reach.

Numerical solution for tidal flow in opening channel using combined MacCormack - finite analysis method

A new numerical scheme for solving the tidal flow in an opening channel using the advective-diffusion shallow-water equations as the governing equations is proposed based on the combination of the MacCormack and the finite analysis methods. In the present scheme, the finite analysis method is used to discretize the momentum equation and the MacCormack technique is used to discretize the continuity equation in a single grid system. The matrix of the discretized momentum equation is characterized by predominantly main diagonal elements, which ensures favorable convergence and stability for the numerical simulation by the combined method. To verify the present method, hydraulics simulation is carried out for a section down mainstream of the Huangpu River. The computational results agree with the measured data. By use of orthogonal curvilinear coordinate system, the methods can be easily extended to the numerical simulation of the tidal flow in a tortuous channel.

Two-dimensional central-upwind schemes for curvilinear grids and application to gas dynamics with angular momentum

In this work we present new second order semi-discrete central schemes for systems of hyperbolic conservation laws on curvilinear grids. Our methods generalise the two-dimensional central-upwind schemes developed by Kurganov and Tadmor [A. Kurganov, E. Tadmor, Numer. Methods Partial Differential Equations 18 (2002) 584. [1]]. In these schemes we account for area and volume changes in the numerical flux functions due to the non-cartesian geometries. In case of vectorial conservation laws we introduce a general prescription of the geometrical source terms valid for various orthogonal curvilinear coordinate systems. The methods are applied to the two-dimensional Euler equations of inviscid gas dynamics with and without angular momentum transport. In the latter case we introduce a new test problem to examine the detailed conservation of specific angular momentum.

Vibration characteristics and snapping behavior of hygro-thermo-elastic composite doubly curved shells

Abstract Due to the change in environmental conditions, hygrothermal residual stresses may induce buckling and dynamic instability in composite shell structures. This paper investigates the vibration characteristics of pre- and post-buckled hygro-thermo-elastic laminated composite doubly curved shells. The geometrically nonlinear finite element method is used for the analysis. Both deep and shallow shell surfaces are modeled using an orthogonal curvilinear coordinate system. The mechanical linear, nonlinear stiffnesses, the hygrothermal nonlinear geometric stiffness and the hygrothermal load vector are presented. The arc-length method is implemented to capture the snapping behavior of the structures under hygrothermal environmental conditions. Numerical results of nonlinear deflections and fundamental frequencies along with the deflection shapes and the mode shapes are presented and discussed for spherical and cylindrical shell panels. The fundamental frequency decreases in the pre-buckling region and increases in the post-buckling domain of the snap-through process. It is also noted that as the thickness of the panel decreases, the limit point load for buckling decreases; if the radius of curvature increases, the fundamental frequency decreases under a uniform hygral condition. The “curve veering phenomena” is observed for spherical panel under thermal load. The geometrically nonlinear deflection behavior affects the vibration characteristics in a hygrothermal environment.

Two-Dimensional Nonequilibrium Noncohesive and Cohesive Sediment Transport Model

The purpose of this paper is to develop an unsteady 2D depth-averaged model for nonuniform sediment transport in alluvial channels. In this model, the orthogonal curvilinear coordinate system is adopted; the transport mechanisms of cohesive and noncohesive sediment are both embedded; the suspended load and bed load are treated separately. In addition, the processes of hydraulic sorting, armoring, and bed consolidation are also included in the model. The implicit two-step split-operator approach is used to solve the flow governing equations and the coupling approach with iterative method are used to solve the mass-conservation equation of suspended sediment, mass-conservation equation of active-layer sediment, and global mass-conservation equation for bed sediment simultaneously. Three sets of data, including suspension transport, degradation and aggradation cases for noncohesive sediment, and aggradation, degra- dation, and consolidation cases for cohesive sediment, have been demonstrated to show the rationality and accuracy of the model. Finally, the model is applied to evaluate the desilting efficiency for Ah Gong Diann Reservoir located in Taiwan to show its applicability.