Non-orthogonal Coordinate System Research Articles

Mathematical modeling of three-dimensional heat and fluid flow in a moving gas metal arc weld pool

Mathematical models capable of accurate prediction of the weld bead and weld pool geometry in gas metal arc (GMA) welding processes would be valuable for rapid development of welding procedures and empirical equations for control algorithms in automated welding applications. This article introduces a three-dimensional (3-D) model for heat and fluid flow in a moving GMA weld pool. The model takes the mass, momentum, and heat transfer of filler metal droplets into consideration and quantitatively analyzes their effects on the weld bead shape and weld pool geometry. The algorithm for calculating the weld reinforcement and weld pool surface deformation has been proved to be effective. Difficulties associated with the irregular shape of the weld bead and weld pool surface have been successfully overcome by adopting a boundary-fitted nonorthogonal coordinate system. It is found that the size and profile of the weld pool are strongly influenced by the volume of molten wire, impact of droplets, and heat content of droplets. Good agreement is demonstrated between predicted weld dimensions and experimently measured ones for bead-on-plate GMA welds on mild steel plate.

Numerical modeling of the disk-edge field emitter triode

In this article, a new numerical algorithm, a finite-difference method in a nonorthogonal curvilinear coordinate system, is presented. This algorithm can be used for efficient simulation of a disk-edge field emitter. With the boundary-conforming coordinate lines, this method can deal with arbitrary microemitter shapes. Simulation results are presented for a unit cell, consisting of a disk-edge field emitter with “cocktail glass” structure. Dependence of the field enhancement factor, electron trajectories, and emission current upon geometrical factors are presented.

NUMERICAL SIMULATION OF TURBULENT FLOW AND HEAT TRANSFER IN THE WEDGE-SHAPED LIQUID METAL POOL OF A TWIN-ROLL CASTER

Controlled flow and heat transfer are important for the quality of a strip in a twin-roll continuous casting process. A numerical study was carried out to investigate the two-dimensional turbulent flow and heat transfer in the liquid stainless-steet-filted wedge-shaped cavity formed by the two counterrotating roils in a twin-roll continuous casting system. The turbulent characteristics of the flow were modeled using a low-Reynolds-number k-e turbulence model due to Launder and Sharma. The arbitrary nature of the computational domain was accounted for through the use of a nonorthogonal boundary-fitted coordinate system on a staggered grid. A control-volume-based finite difference scheme was used to solve the transformed transport equations. This study is primarily focused on elucidating the inlet superheat dissipation in the melt pool with the rolls being maintained at a constant liquidus temperature of the steel. A parametric study was carried out to ascertain the effect of the inlet superheat, the casting speed, and the roll gap at the nip of the rotating rolls on the flow and heat transfer characteristics. The velocity fields show two counterrotating recirculation zones in the upstream region. The local Nusselt number on the roll surface shows significant variations. The contours of temperature and turbulent viscosity show the complex nature of the turbulent transport phenomena to be expected in a twin-roll casting process

On the dynamic strength of a disk with a central cylinder-conic opening

The problem of nonstationary dynamic behavior of a disk weakened by a central cylinder-conic opening under the action of suddenly applied loading is considered. Determining relations are given in the form of a system of integral equations in an arbitrary oblique coordinate system. A numerical solution has been obtained by using a hybrid difference scheme constructed on the basis of the method of flow correction. Our generalized versions of the S.K. Godunov method for the case of a nonorthogonal system of coordinates were used as basic schemes. The locations of maximum stresses that precede the appearance of plasticity zones have been determined for a particular elastic body.

A new approach to the classical Stokes flow problem: Part I Methodology and first-order analytical results

The problem of determining the axisymmetric Stokes flow past an arbitrary body, the boundary shape of which can be represented by an analytic function, is examined by developing an exact method. An appropriate nonorthogonal coordinate system is introduced, and it is shown that the Hilbert space to which the stream function belongs is spanned by the set of Gegenbauer polynomials based on the physical argument that the drag on a body should be finite. The partial differential equation of the original problem is then reduced to two simultaneous vector differential equations. By the truncation of this infinite-dimensional system to the one-dimensional subspace, an explicit analytic solution to the Stokes equation valid for all bodies in question is obtained as a first approximation.

Conical diffraction of a plane wave by an inclined parallel-plate grating

We present a rigorous differential method describing the conical diffraction of an electromagnetic plane wave by an inclined parallel-plate grating. Above and below the grating, the fields are written with the use of Rayleigh’s expansion. Inside the grating, Maxwell’s equations are used in covariant form, written in a nonorthogonal coordinate system. Therefore the expression of boundary conditions on the perfectly conducting walls as well as on the planes delimiting the grating becomes simplified. In the classical diffraction case the numerical results compared successfully with those obtained with a Wiener–Hopf method [IEEE Trans. Antennas Propag.36, 1424 (1988)].

Nonstandard beam propagation

A nonorthogonal coordinate system that extends the present range of angled configurations for which the beam propagation method (BPM) is accurate is introduced. This offers an extra degree of freedom within configuration and method when the BPM is applied in the design of general photonic integrated circuits incorporating angled waveguide sections. © 1996 John Wiley & Sons, Inc.

Nonstandard beam propagation

A nonorthogonal coordinate system that extends the present range of angled configurations for which the beam propagation method (BPM) is accurate is introduced. This offers an extra degree of freedom within configuration and method when the BPM is applied in the design of general photonic integrated circuits incorporating angled waveguide sections. © 1996 John Wiley & Sons, Inc.

PARAMETRIC ANALYSIS OF RADIATIVE-CONVECTIVE HEAT TRANSFER AROUND A CIRCULAR CYLINDER IN A CROSS FLOW USING THE FINITE VOLUME RADIATION SOLUTION METHOD

The radiative-cancective heat transfer around a circular cylinder in a cross flow of a radiating gas has been numerically analyzed using the finite volume radiation solution method in a nonorthogonal coordinate system. The cross-flow Reynolds number based on the cylinder diameter is 40, and the fluid Prandtl number is assumed to be 0.7. The radiative heat transfer coupled with convection is reasonably predicted by the finite volume radiation solution method. Distributions of the local Nusselt number are investigated according to the variation of radiation parameters such as conduction-to-radiation parameter, optical thickness, scattering albedo, and cylinder wall emissivity.

A new form of the magnetohydrodynamic potential energy

In this Letter a useful and instructive form of the magnetohydrodynamic potential energy is presented. In this form all but one term is positive definite, each term is self-adjoint, and it can be directly evaluated numerically in nonorthogonal coordinate systems. The product of the local shear and the current parallel to the magnetic field is identified as a significant parameter for current-driven instabilities.

Tapered beam propagation

A novel beam propagation algorithm is presented that eliminates errors caused by modelling tapered optical structures with staircase approximations. In consequence, highly accurate results are obtained with a substantial improvement in computational efficiency. The algorithm is developed from a nonorthogonal co-ordinate system that intrinsically describes the taper geometry and can be implemented using conventional beam propagation techniques.

Thermophoretic Particle Deposition Around a Circular Cylinder in a Uniform Laminar Air Dlow

Thermophoretic particle deposition on a circular cylinder in a uniform laminar air flow was numerically investigated using a control volume method based on the generalized non-orthogonal coordinate system. Variation of air properties due to the change of temperature was taken into account. Effects of variable property on the distribution of heat transfer and deposition rates of particle were discussed. A new correlation of thermophoretic particle deposition on a circular cylinder was proposed in the present study.

Numerical Calculation of Bank Erosion and Free Meandering

A numerical model is proposed to investigate free meandering of river. Flow field, bed deformation, bank erosion and channel migration are calculated numerically with a general non-orthogonal coordinate system which can be applied to moving boundary conditions. Flow field is calculated from two dimensional momentum equations and continuity equation. Bed deformation is calculated from continuity equation of bed load transport. Bank erosion and deposition is estimated from critical slope of bank material and emergence of bed due to the bed deposition. Time dependent change of channel geometry is calculated with an iteration process of these procedures. Calculated results are favorably compared with experimental results with movable bed and bank, and thus the accuracy of new model is verified.

The energy and helicity of knotted magnetic flux tubes

Magnetic relaxation of a magnetic field embedded in a perfectly conducting incompressible fluid to minimum energy magnetostatic equilibrium states is considered. It is supposed that the magnetic field is confined to a single flux tube which may be knotted. A local non-orthogonal coordinate system, zero-framed with respect to the knot, is introduced, and the field is decomposed into toroidal and poloidal ingredients with respect to this system. The helicity of the field is then determined; this vanishes for a field that is either purely toroidal or purely poloidal. The magnetic energy functional is calculated under the simplifying assumptions that the tube is axially uniform and of circular cross-section. The case of a tube with helical axis is first considered, and new results concerning kink mode instability and associated bifurcations are obtained. The case of flux tubes in the form of torus knots is then considered and the ‘ground-state’ energy function ͞m ( h ) (where h is an internal twist parameter) is obtained; as expected, ͞m ( h ), which is a topological invariant of the knot, increases with increasing knot complexity. The function ͞m ( h ) provides an upper bound on the corresponding function m ( h ) that applies when the above constraints on tube structure are removed. The technique is applicable to any knot admitting a parametric representation, on condition that points of vanishing curvature are excluded.

Mathematical Modelling of a 2.25 MWtSwirling Natural Gas Flame. Part 2: Conserved Scalar Approach for Turbulent Combustion

ABSTRACT A mathematical model is presented for the swirling unstaged natural gas flame. The performance of the model is critically assessed by comparison with a detailed set of experimental measurements. The model makes use of a conserved scalar approach. Turbulence is modelled using k-epsilon model with turbulence chemistry interaction accounted for using a two parameter presumed p.d.f. Radiative heat transfer is treated by means of the discrete transfer method and chemistry models making use of the “mixed is burnt” and partial equilibrium assumptions are evaluated. Numerical solution is by means of a collocated finite volume approach in a general non-orthogonal curvilinear coordinate system. It is seen that predictions to reasonable engineering accuracy can be obtained, with the partial equilibrium model offering some advantage over fast irreversible chemistry. Recommendations for future development are given.

Comment on 'Application of wall functions to generalized nonorthogonal curvilinear coordinate systems'

Comment on 'Application of wall functions to generalized nonorthogonal curvilinear coordinate systems'

Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system

The author extends the rigorous Chandezon formalism (1982) to accommodate three-dimensional modulation profiles. The basic feature lies in the use of a coordinate system that maps the interface onto a plane. As in the two-dimensional case, one is led to a linear system of differential equations whose solution is obtained through the calculation of the eigenvalues and eigenvectors of a matrix dependent on the geometry and on the index of the medium. Results are compared with those obtained by the methods of others.

Application of a three dimensional model for the prediction of pollutant dispersion in Cyprus coastal waters

A recently developed numerical model for environmental flows has been applied to simulate the water circulation and pollutant dispersion in a confined coastal system in Cyprus. The model is based on the numerical solution of the time averaged Navier-Stokes equations, written in their contravariant strong conservation form, in a generalised non-orthogonal coordinate system. The model predictions are verified compared with experimental data incorporating radioactive tracer injection at the surface of the sea, then the model is applied to predict the dispersion of waste from the outfall pipe of a sewage treatment plant. The predicted results agree favourably with the measurements, demonstrating the code's validity, flexibility and economy for predicting accurately flows in three-dimensional complex sea bed terrain basins. The utilisation of such a model can be an advantageous alternative to the expensive and time consuming experimental field work, enabling the study of the plume dispersion under different discharge and hydrological prevailing conditions.

Prediction of Surface Depression of a Tungsten Inert Gas Weld Pool in the Full-Penetration Condition

A three-dimensional model is set up to predict the surface depression of a tungsten inert gas (TIG) weld pool in a full-penetration condition in order to find out the relation between pool depression and weld penetration. It solves pool surface depression, fluid flow and heat transfer simultaneously and determines the configuration of a weld pool surface based on the dynamic balance among arc pressure, pool gravity and surface tension at the deformed weld pool surface. In the numerical simulation, difficulties associated with the irregular shape of the deformed weld pool surface and the liquid/solid interface have been overcome by adopting a boundary-fitted non-orthogonal curvilinear coordinate system. A series of data about pool surface depression under different TIG welding conditions are obtained. The validity of the model is verified through TIG welding experiments.

New perturbation theory of diffraction gratings and its application to the study of ghosts

A new perturbation method for the diffraction of a plane wave by a grating with periodic imperfections is presented. The originality of the method lies in the fact that the perturbation occurs on a reference profile that is not a plane but a grating. First, the diffraction by a reference grating is treated. At this stage Maxwell’s equations are used in covariant form written in a nonorthogonal coordinate system fitted to the surface geometry. Second, the periodic errors are taken into account. The tensorial formalism permits the elaboration of this two-roughness-level model. The grating profile appears only through two fundamental functions. The variations of these functions under the effect of variation in the profile are expanded in power series of the perturbation parameter v1. v1 represents the maximum of the derivative of the function describing the perturbation. By using this formalism, we determine the efficiencies.