Non-orthogonal Coordinate System Research Articles

Finite element formulation for shear deformable thin-walled beams

Starting with the principle of stationary potential energy, this paper develops the governing differential equations of equilibrium and boundary conditions for shear deformable thin-walled beams with open cross-section. Unlike conventional solutions, the formulation is based on a non-orthogonal coordinate system, in which the selected origin is generally offset from the section centroid. The exact solution of the resulting coupled differential equations of equilibrium is derived and used to develop exact shape functions. A finite element based on the exact shape functions is then formulated. Through a series of examples, the adoption of non-orthogonal coordinates is shown to enable the seamless modelling of structural members with eccentric boundary conditions and (or) stepwise cross-sectional variations.

A Two-Dimensional Numerical Simulation of Combustion and Heat Transfer in Radiant Porous Burners

Premixed combustion inside a porous medium is presently under investigation due to the characteristics of this technology, which provides controlled flames. The authors present numerical methods for the simulation of premixed methane/air flames in porous media using a 6-step reduced mechanism. The laminar 2-dimensional model is based on a macroscopic formulation of the heat and mass transport equations for laminar flow. The finite volume method is used for simulation with a boundary-fitted nonorthogonal coordinate system. The effects of the model on temperature and species are examined in 2 porous burners with different geometries and materials. The results are compared with the experiments available in the literature. In the first reactor analyzed, the CO mass fraction calculated is compared with the available experimental data. The good agreement with the experimental data suggests that the numerical model is an excellent tool to investigate pollutants formation. In the second reactor studied, the temperature field calculated is compared with experimental data. The effect of an injection plate at the entry of the burner is predicted, the 2D field of temperatures and mass fractions of species, giving an insight of the 3D profile of the flame. The strong advective effect caused by the acceleration of the flow due to the injection plate causes a conical flame shape. The numerical simulation of the conical flame predicted the temperature field and the flame position. A sensitivity study to the Rosseland mean attenuation coefficient, to the thermal conductivity of the solid and to the porosity is presented. It is shown that the properties of the preheating region do not significantly affect the CO emissions at the exit of the reactor, although the properties of the combustion region have a strong influence.

Generalization of ray tracing in a linear inhomogeneous anisotropic medium: a coordinate-free approach

The Hamiltonian of an optical medium is important in both the design and the description of optical devices in the geometrical optics limit. The results calculated in this article show in detail how ray tracing in anisotropic materials in arbitrary coordinate systems and curved spaces can be carried out. Writing Maxwell's equations in the most general form, we derive a coordinate-free form for the eikonal equation and hence the Hamiltonian of a general purpose medium. The expression works for both orthogonal and non-orthogonal coordinate systems, and we show how it can be simplified for biaxial and uniaxial media in orthogonal coordinate systems. In order to show the utility of the equations in a real case, we study both the isotropic and the uniaxially transmuted birefringent Eaton lens and derive the ray trajectories in spherical coordinates for each case.

Expression of Joint Moment in the Joint Coordinate System

The question of using the nonorthogonal joint coordinate system (JCS) to report joint moments has risen in the literature. However, the expression of joint moments in a nonorthogonal system is still confusing. The purpose of this paper is to present a method to express any 3D vector in a nonorthogonal coordinate system. The interpretation of these expressions in the JCS is clarified and an example for the 3D joint moment vector at the shoulder and the knee is given. A nonorthogonal projection method is proposed based on the mixed product. These nonorthogonal projections represent, for a 3D joint moment vector, the net mechanical action on the JCS axes. Considering the net mechanical action on each axis seems important in order to assess joint resistance in the JCS. The orthogonal projections of the same 3D joint moment vector on the JCS axes can be characterized as "motor torque." However, this interpretation is dependent on the chosen kinematic model. The nonorthogonal and orthogonal projections of shoulder joint moment during wheelchair propulsion and knee joint moment during walking were compared using root mean squares (rmss). rmss showed differences ranging from 6 N m to 22.3 N m between both projections at the shoulder, while differences ranged from 0.8 N m to 3.0 N m at the knee. Generally, orthogonal projections were of lower amplitudes than nonorthogonal projections at both joints. The orthogonal projection on the proximal or distal coordinates systems represents the net mechanical actions on each axis, which is not the case for the orthogonal projection (i.e., motor torque) on JCS axes. In order to represent the net action at the joint in a JCS, the nonorthogonal projection should be used.

A Turbulent Impinging Jet on a Plate Covered with a Porous Layer

This work shows numerical results for a turbulent jet impinging against a flat plane covered with a layer of permeable material, which is kept at a higher temperature than that of the incoming fluid. Parameters such as porosity, permeability, thickness, and thermal conductivity of the porous layer are varied in order to analyze their effects on the local distribution of Nu. The macroscopic equations for mass, momentum, and energy are obtained based on volume-average concept. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted nonorthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure–velocity coupling. Results indicate that inclusion of a porous layer decreases the peak in Nu avoiding excessive heating or cooling at the stagnation point. Also found, was that the integral heat flux from the wall is enhanced for certain ranges of values of porosity, layer thickness, and thermal conductivity ratio.

Simulation of laminar impinging jet on a porous medium with a thermal non-equilibrium model

This work shows numerical simulations of an impinging jet on a flat plate covered with a layer of a porous material. Macroscopic equations for mass and momentum are obtained based on the volume-average concept. Two macroscopic models are employed for analyzing energy transport, namely the one-energy equation model, based on the Local Thermal Equilibrium assumption (LTE), and the two-energy equation closure, where distinct transport equations for the fluid and the porous matrix follow the Local Non-Thermal Equilibrium hypothesis (LNTE). The numerical technique employed for discretizing the governing equations was the finite volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure–velocity coupling. Parameters such as porosity, porous layer thickness, material permeability and thermal conductivity ratio were varied in order to analyze their effects on flow and heat transport. Results indicate that for low porosities, low permeabilities, thin porous layers and for high thermal conductivity ratios, a different distribution of local Nusselt number at the wall is calculated depending on the energy model applied. The use of the LNTE model indicates that it is advantageous to use a layer of highly conducting and highly porous material attached to the hot wall.

Tilted Gaussian beam propagation in inhomogeneous media

The present work is concerned with applying a ray-centered non-orthogonal coordinate system which is a priori matched to linearly-phased localized aperture field distributions. The resulting beam-waveobjects serve as the building blocks for beam-type spectral expansions of aperture fields in 2D inhomogeneous media that are characterized by a generic wave-velocity profile. By applying a rigorous paraxial-asymptotic analysis, a novel parabolic wave equation is obtained and termed "Non-orthogonal domain parabolic equation"--NoDope. Tilted Gaussian beams, which are exact solutions to this equation, match Gaussian aperture distributions over a plane that is tilted with respect to the beam-axes initial directions. A numerical example, which demonstrates the enhanced accuracy of the tilted Gaussian beams over the conventional ones, is presented as well.

Study of steady viscous flow through a wavy channel: non-orthogonal coordinates

In this paper we consider the steady flow of a viscous fluid through a channel bounded by two sinusoidally varying plates differing in phase by π and separated by a mean distance 2h. For the non-varying channel, the classical parabolic velocity profile for the fully developed flow is well known. An attempt here is made to analyze the flow in a generalized non-orthogonal coordinate system that renders the wavy channels as plane walls. Continuity equation and Navier-Stokes equations are presented in the generalized coordinate system and simplified through use of small perturbation under small Reynolds number approximation. Flow characteristics such as centerline velocity and drag force have been evaluated and discussed.

Simulation of triaxial induction measurements in dipping, invaded, and anisotropic formations using a Fourier series expansion in a nonorthogonal system of coordinates and a self-adaptive hp finite-element method

Borehole triaxial induction instruments were designed to diagnose and measure rock electrical conductivity parallel and perpendicular to the bedding plane. Experience has shown that the interpretation of triaxial induction measurements often requires numerical modeling for a proper diagnosis of rock electrical conductivity anisotropy in the presence of geometric effects such as dipping wells, layer boundaries, and invasion. We introduce a new algorithm to simulate triaxial induction measurements that combines a Fourier series expansion in a nonorthogonal system of coordinates with a 2D goal-oriented, self-adaptive, high-order [Formula: see text] finite-element method. This procedure enables the accurate and reliable simulation of triaxial induction measurements across reservoir rock formations with extreme contrasts of electrical conductivity while reducing the 3D computational complexity associated with deviated wells. Numerical results indicate that borehole dip effects on triaxial induction measurements are larger than on standard coaxial induction measurements. The sensitivity of triaxial induction measurements to transversely isotropic rock formations decreases with increasing dip angle.

Statistical study of radiation loss from planar optical waveguides: The curvilinear coordinate method and the small perturbation method

This article presents an original method for the theoretical analysis of the intensity radiated by a dielectric waveguide with rough walls. The method is based on Maxwell's equations under their covariant form written in nonorthogonal coordinate systems adapted to the geometry of the waveguide. The solutions are found by using a perturbation method starting from a guide with smooth walls. The statistical characteristics of the radiant intensity, the mean value, and the probability density function are analytically determined.

Non-Orthogonal Domain Parabolic Equation and Its Tilted Gaussian Beam Solutions

A non-orthogonal coordinate system which is <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> matched to localized initial field distributions for time-harmonic wave propagation is presented. Applying, in addition, a rigorous paraxial-asymptotic approximation, results in a novel parabolic wave equation for beam-type field propagation in 3D homogeneous media. Localized solutions to this equation that exactly match linearly-phased Gaussian aperture distributions are termed tilted Gaussian beams. These beams serve as the building blocks for various beam-type expansion schemes. Application of the scalar waveobjects to electromagnetic field beam-type expansion, as well as reflection and transmission of these waveobjects by planar velocity (dielectric) discontinuity are presented. A numerical example which demonstrates the enhanced accuracy of the tilted Gaussian beams over the conventional ones concludes the paper.

PARAMETERIZATION OF THE TILTED GAUSSIAN BEAM WAVEOBJECTS

Novel time-harmonic beam fields have been recently obtained by utilizing a non-orthogonal coordinate system which is a priori matched to the field’s planar linearly-phased Gaussian aperture distribution. These waveobjects were termed tilted Gaussian beams. The present investigation is concerned with parameterization of these time-harmonic tilted Gaussian beams and of the wave phenomena associated with them. Specific types of tilted Gaussian beams that are characterized by their aperture complex curvature matrices, are parameterized in term of beam-widths, waist-locations, collimationlengths, radii of curvature, and other features. Emphasis is placed on the difference in the parameterization between the conventional (orthogonal coordinates) beams and the tilted ones.

Validation of atmospheric boundary layer turbulence model by on-site measurements

Modeling atmosperic boundary layer with standard linear models does not sufficiently reproduce wind conditions in complex terrain, especially on leeward sides of terrain slopes. More complex models, based on Reynolds averaged Navier-Stokes equations and two-equation k-? turbulence models for neutral conditions in atmospheric boundary layer, written in general curvilinear non-orthogonal co-ordinate system, have been evaluated. In order to quantify the differences and level of accuracy of different turbulence models, investigation has been performed using standard k-? model without additional production terms and k-? turbulence models with modified set of model coefficients. The sets of full conservation equations are numerically solved by computational fluid dynamics technique. Numerical calculations of turbulence models are compared to the reference experimental data of Askervein hill measurements.

Turbulent Flow in Wavy Channels Simulated with Nonlinear Models and a New Implicit Formulation

This work examines the performance of linear and nonlinear eddy-viscosity models when used to predict the turbulent flow in periodically sinusoidal-wave channels. Two geometries are investigated, namely a converging-diverging channel and a channel with concave-convex walls. The numerical method employed for the discretization of the equations is the control-volume method in a boundary-fitted nonorthogonal coordinate system. The SIMPLE algorithm is used for correcting the pressure field. The classical wall function and a low Reynolds model are used to describe the flow near the wall. Comparisons between those two approaches using linear and nonlinear turbulence models are done. Here, a new implicit numerical treatment is proposed for the nonlinear diffusion terms of the momentum equations in order to increase the robustness. Results show that by decomposing and treating terms as presented, solutions using nonlinear models and the high Reynolds wall treatment, which combine accuracy and economy, are more stable and easier to be obtained.

Diffusion Experiments with a Global Discontinuous Galerkin Shallow-Water Model

Abstract A second-order diffusion scheme is developed for the discontinuous Galerkin (DG) global shallow-water model. The shallow-water equations are discretized on the cubed sphere tiled with quadrilateral elements relying on a nonorthogonal curvilinear coordinate system. In the viscous shallow-water model the diffusion terms (viscous fluxes) are approximated with two different approaches: 1) the element-wise localized discretization without considering the interelement contributions and 2) the discretization based on the local discontinuous Galerkin (LDG) method. In the LDG formulation the advection–diffusion equation is solved as a first-order system. All of the curvature terms resulting from the cubed-sphere geometry are incorporated into the first-order system. The effectiveness of each diffusion scheme is studied using the standard shallow-water test cases. The approach of element-wise localized discretization of the diffusion term is easy to implement but found to be less effective, and with relatively high diffusion coefficients, it can adversely affect the solution. The shallow-water tests show that the LDG scheme converges monotonically and that the rate of convergence is dependent on the coefficient of diffusion. Also the LDG scheme successfully eliminates small-scale noise, and the simulated results are smooth and comparable to the reference solution.

Laminar heat transfer in a porous channel simulated with a two-energy equation model

Laminar heat transfer in a porous channel is numerically simulated with a two-energy equation model for conduction and convection. Macroscopic equations for continuity, momentum and energy transport for the fluid and solid phases are presented. The numerical methodology employed is based on the control volume approach with a boundary-fitted non-orthogonal coordinate system. Fully developed forced convection in a porous channel bounded by parallel plates is considered. Solutions for Nusselt numbers along the channel are presented for laminar flows. Results simulate the effects Reynolds number Re, porosity, particle size and solid-to-fluid thermal conductivity ratio on Nusselt sumber, Nu, which is defined for both the solid and fluid phases. High Re, low porosities, low particle diameters and low thermal conductivity ratios promote thermal equilibrium between phases leading to higher values of Nu.

Eigenvalue system for the scattering from rough surfaces – Saving in computation time by a physical approach

The curvilinear coordinate method is an efficient theoretical tool for analysing rough surfaces. It consists on solving Maxwell’s equations written in a nonorthogonal coordinate system. The C method leads to eigenvalue systems and the scattered fields can be expanded as a linear combination of eigensolutions. The boundary conditions allow the combination coefficients to be determined. The dominant computational cost for the C method is the eigenvalue problem solution which is of order of N 3 where N is the size of eigenvalue systems. In this paper, we propose a new approach based on the association of the C method with the beam simulation method (BSM) in order to reduce the computational time. The BSM is based on decomposing a large incident beam into narrower subbeams and then synthesizing the large beam by coherent superposition. The adopted procedure consists of two stages. First, the surface fields are obtained by the C method associated with each elementary beam illuminating smaller surfaces. Second, the total surface field is deduced from a coherent superposition of elementary surface current densities. The far-field and the scattering coefficients are derived from the Huygens principle applied to the total surface fields. We confirm the efficiency and the validity of the approach and show that the BSM applied with the C method allows a significant saving in computation time.

Simulation of DC dual-laterolog measurements in complex formations: A Fourier-series approach with nonorthogonal coordinates and self-adapting finite elements

Dual laterolog (DLL) makes use of a galvanic conduction principle to focus electrical currents into rock formations, thereby minimizing shoulder and borehole effects in the measurement of formation resistivity. The tool includes two separate focusing systems: deep-sensing (LLd) and shallow-sensing modes (LLs). Laterolog current-focusing systems were designed for operation primarily in vertical boreholes penetrating horizontal layers; only recently their design has been revised for operation in deviated wells in the presence of electrical anisotropy. We simulated three-dimensional (3D) DLL measurements in dipping, invaded, and electrically anisotropic formations and appraised the corresponding effects on apparent resistivity logs. Simulations were performed by combining the use of a Fourier-series expansion in a nonorthogonal system of coordinates with an existing 2D goal-oriented, higher-order, and self-adaptive finite-element method. This numerical algorithm yields accurate solutions in limited CPU time because only a few Fourier modes are needed to simulate practical applications. For the calculation of focused currents, we introduced an embedded postprocessing method that incorporates a synthetic focusing principle to compute current intensities at each iterative step of optimal mesh refinements. Our numerical method accurately simulates 3D DLL measurements in rock formations that exhibit extreme contrasts of electrical resistivity. Simulations indicate that LLs resistivity logs are more sensitive to both invaded and anisotropic layers than LLd resistivity logs. In deviated wells, shoulder-bed effects on apparent resistivity logs increase with an increase of dip angle, and are emphasized across thin conductive layers. Electrical anisotropy effects on apparent resistivity logs increase substantially with dip angle.

Numerical simulations of ice accumulation under ice cover along a river bend

In this paper, the k-έ two-equation turbulence model has been used to simulate ice accumulation under ice cover along a river bend. A 2D depth-averaged numerical model has been developed in a nonorthogonal coordinate system with nonstaggered curvilinear grids. In this model, the contravariant velocity has been treated as an independent variable. To avoid the pressure oscillation in the nonstaggered grids, the momentum interpolation has been introduced to interpolate variables at the interface. The discretization equations have been solved by using pressure correction algorithms. An equation has been developed for describing the deformation of ice jam bottom. The thickness distribution of ice accumulation (ice jam) along the bend has been simulated. The developed model has been applied to the experimental studies under different conditions carried out at the Hefei University of Technology. Results indicate that all simulated thickness of ice accumulation agrees reasonably well with the measured thickness of ice accumulation in laboratory.

Thermal Analysis of an Impinging Jet on a Plate With and Without a Porous Layer

This work shows numerical results for a jet impinging against a flat plane covered with a layer of a porous material, which is maintained at a higher temperature than the incoming fluid. Parameters such as permeability and thickness of the porous layer and thermal conductivity ration are varied in order to analyze their effects on the local distribution of Nu. The macroscopic equations for mass, momentum, and energy are obtained based on a volume-average concept. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted nonorthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Results indicate that inclusion of a porous layer decreases the peak in Nu avoiding excessive heating or cooling at the stagnation point. Also found was that the integral heat flux from the wall is enhanced for certain range of values of layer thickness, porosity, and thermal conductivity ratio.