Orthogonal Curvilinear Coordinate System Research Articles

More considerations about the symmetry of the stress tensor of fluids.

Regarding a recent dispute about the symmetry of the stress tensor of fluids, more considerations are presented. The usual proofs of this symmetry are reviewed, and contradictions between this symmetry and the mechanism of gas viscosity are analyzed for simple gas flows. It is emphasized that these proofs depend on the theorem of angular momentum that assumes that internal forces between any two fluid particles are always along the line connecting them. Without this assumption, from Newton's laws of motion one can only obtain the theorem of angular momentum with an additional term. It is proved within classical continuum mechanics that this additional term represents the total moment of internal forces, and its volume density is similar to the body couple introduced in generalized continuum mechanics. When discrete structure of matter is considered, this term corresponds to the total moment of the forces exerted on the nuclei by the internal electrons. This moment of internal forces may lead to nonsymmetry of the stress tensor and is, in general, nonzero as long as shear stress exists. A nonsymmetrical stress tensor suggested in the literature is discussed in terms of its effects in eliminating the contradictions and simplifying the Navier-Stokes equation. The derivation of this stress tensor for ideal gases based on the kinetic theory of gas molecules is presented, and its form in a general orthogonal curvilinear coordinate system is given. Finally, a possible experimental verification of this nonsymmetrical stress tensor is discussed.

Global well-posedness of the 3D incompressible Navier-Stokes and Euler equations in orthogonal curvilinear coordinate systems

Global well-posedness of the 3D incompressible Navier-Stokes and Euler equations in orthogonal curvilinear coordinate systems

Orthogonal Curvilinear Coordinate Systems and Torsion-Free Sheaves over Reducible Spectral Curves

Orthogonal Curvilinear Coordinate Systems and Torsion-Free Sheaves over Reducible Spectral Curves

Principal Stress Trajectories in Plasticity under Plane Strain and Axial Symmetry

The two families of principal stress trajectories can be regarded as an orthogonal curvilinear coordinate system under plane strain and axial symmetry. Under plane strain, the equilibrium equations in conjunction with a yield criterion comprise a statically determinate system. Under axial symmetry, a statically determinate system results from the above equations supplemented with the hypothesis of Haar von Karman. In both cases, the compatibility equations for mapping the principal line coordinate system to a given coordinate system show that the scale factors of the former satisfy a simple algebraic or transcendental equation for many yield criteria. Using this equation, one can develop a method for reducing boundary value problems in plasticity to purely geometric problems. The method is independent of any flow rule that can be chosen to calculate displacement or velocity fields, as well as independent whether elastic strains are included. The present paper summarizes available results related to using principal stress trajectories in plasticity and emphasizes the advantages of the method above.

INFINITESIMAL AND FINITE DEFORMATIONS IN THE POLAR COORDINATE SYSTEM

The deformation problem of elasticity theory with regard to nonlinear deformations is examined. The expressions of deformations through displacements in the orthogonal curvilinear coordinate system are recorded. The relations for finite deformations in cylindrical and polar coordinate systems are derived. Physical relations for finite deformations and corresponding generalized stresses are recorded. 

Lagrangian 3-form structure for the Darboux system and the KP hierarchy

A Lagrangian multiform structure is established for a generalisation of the Darboux system describing orthogonal curvilinear coordinate systems. It has been shown in the past that this system of coupled PDEs is in fact an encoding of the entire Kadomtsev–Petviashvili (KP) hierarchy in terms so-called Miwa variables. Thus, in providing a Lagrangian description of this multidimensionally consistent system amounts to a new Lagrangian 3-form structure for the continuous KP system. A generalisation to the matrix (also known as non-Abelian) KP system is discussed.

Parametric electro‐mechanical dispersive behaviors of guided waves in the functionally graded piezoelectric annular plate

AbstractThis paper investigates the propagation characteristics of guided waves in functionally graded annular plates considering the electro‐mechanical coupling. In the orthogonal circular curvilinear coordinate system, the elastodynamic wave equation of guided waves is derived for the functional graded piezoelectric annular plate. The Chebyshev spectral elements are utilized to model the annular plate, and the governing wave equation is discretized. The convergence and accuracy of the proposed method are evaluated by comparing with the results in previous publications. The parametric study on dispersive characteristics of the functionally graded piezoelectric annular plate is conducted systematically for the influences of various volume fraction index, curvature, and geometry size. Some exciting results are discussed including mode separation, mode conversion, and cutoff frequency. This research can benefit the design and development of advanced smart materials and structures.

Parametric study of the Giesekus fluid flow in a curved duct with square cross section

In this paper, the log-conformation representation method (LCR) is applied in an orthogonal curvilinear coordinate system to study the Giesekus fluid flow in a curved duct. Derivations for evolution equations of LCR in this curvilinear coordinate system are presented. Secondary flow patterns and oscillation solutions are computed by using the collocation spectral method. The influence of a wide range of Dean number, Weissenberg number, and dimensionless mobility parameter α on fluid behaviors is studied. A six-cell secondary flow pattern is found under very low Dean number and relatively high Weissenberg number and α. Moreover, both Weissenberg number and α are able to facilitate the development of the secondary flow. In addition, simulations under critical Reynolds number for oscillation imply that Giesekus fluid flow with We≥0.1 is not able to retain a four-cell secondary flow pattern in a steady state, which is different from Newtonian fluids.

Graded infill design within free-form surfaces by conformal mapping

The infill design for free-form surfaces is challenging due to the mismatch of both the geometric feature and physical response between the curved surface on the macroscale and the regular-shaped infill unit cells on the meso/micro scale. Thus, this study proposes a novel optimization method to provide insight into the optimal design for the free-form surfaces consisting of micro-structured infills, where both the microstructural geometry and macroscopic distribution of the spatial-varying infills are optimized concurrently. In this method, a local level sets approach is proposed to generate a series of graded and connectable infill microstructures that are related to the volume fraction of each unit cell. The cubic polynomials interpolation is utilized to predict the effective properties for those graded infills rapidly. A computational conformal mapping technology is used to map the geometries between a 3D free-form surface and a regular 2D parameter space, such that the graded infills can be properly filled into the parameter domain, and then conformally mapped back onto the free-form surface to allow the infill unit cells to fit the curved surface without losing any geometric feature. Conformal shell finite element analysis defined in an orthogonal curvilinear coordinate system is derived in the lower-dimensional parameter domain to calculate the physical responses for the infill optimization during the conformal mapping. The infill optimization formulation for the free-form surface is established, which is recast as a 2D optimization problem defined in the parameter domain, aiming at minimizing the structural compliance subject to a global volume constraint. Thus, the dimension of the 3D free-form surface design problem is reduced. Several examples of the free-form surface demonstrate the features of the proposed method.

Dynamics Analysis of Penetration against Pebble-Concrete Target.

The classical continuum mechanics theory cannot sufficiently describe the effect of pebbles on projectile, which leads to a large calculation error. In this paper, an orthogonal curvilinear coordinate system is constructed, which effectively describes and perfects the normal cavity expansion theory. A couple stress theory based on the normal cavity expansion is proposed in which not only the tangential movements but also the rotations of the concrete medium are considered. According to the high-speed impact of pebble concrete, combined with dynamic equations and the FE simulation, the theoretical and simulation results of pebble particles scale on warhead resistance are compared. It is shown that, the larger the scale of pebble particles, the stronger the effect of rotation on the resistant force applied on the warhead.

Derivative of Spatial Variables in Orthogonal Curvilinear System with Respect to the Ones in Cartesian Coordinate System

Based on the transformation of the spatial variables between the orthogonal curvilinear coordinate system and the Cartesian coordinate system, the derivation of spatial variables in orthogonal curvilinear system to the ones in Cartesian coordinate system is derived. The derivatives of the spatial variables in cylindrical, spherical, and elliptic cylindrical coordinate system are derived respectively.

Asymptotic Modeling of a Viscous Laminar Flow Around Thin Airfoils: Resolution and Experimental Treatment in Case of Supersonic Flow

In this paper, starting from the conservation equations in a 2D orthogonal curvilinear coordinate system, an approached asymptotic model for perturbed potential flow equation is established. The study considers a high-speed steady laminar irrotational flow of a compressible and viscous fluid, over thin airfoils at low incidence in the standard atmosphere. This study is aimed to contribute to developing a fully two-dimensional general model that is efficiently applicable to industrial scale and scientific research, in a laminar flow, defined on thin geometries as it takes into consideration both compressibility and viscous effects. The singular perturbation model is solved in case of supersonic viscous flow, based on asymptotic and singular perturbation methods. Asymptotic matching of different layers is done through the triple deck theory. Experimental measurements have been conducted, with the AF300 supersonic wind tunnel, on a NACA 43013 airfoil model fitted with two pressure taps. The comparison of the local Mach numbers between the asymptotic model and the experimentation shows the accuracy of the model in case of laminar flow but diverge relatively as the flow becomes turbulent towards the trailing edge.

Thin liquid films on a rotary bell atomizer in surface-following coordinates

This article analyzes a published formulation of the Navier–Stokes equations cast into surface-following coordinates and provides some additional mathematical background to follow the article. Ubiquitous in the paint shops of automotive plants around the world, a high-speed rotary bell is succinctly described as a rapidly spinning concave axisymmetric surface with liquid paint supplied from a port coinciding with the center of rotation. The spinning surface transfers momentum to the paint film causing it to flow outward. Upon reaching the bell periphery, it is flung off, subsequently forming an atomized spray transferred to an automotive body through advection and electrostatics. Common analytical frameworks of rotating films were spherical or cylindrical coordinate systems where the wetted surface profile of the bell was constrained to follow a coordinate axis. This led to solutions for films modeled with conical, disk-like, or partial hemispherical profiles. An alternative was a more general case using a surface-following orthogonal curvilinear coordinate system along with its derived vector operators. In the unique case of a thin film, these results validated a simpler pattern found in common coordinate systems.

3D hygro-elastic shell model for the analysis of composite and sandwich structures

The present work proposes the study of the hygrometric loading effects in the static analysis of multilayered composite and sandwich plates and shells. The employed model is based on a general exact 3D shell theory valid for one-layered and multilayered sandwich and composite plates, cylinders and cylindrical/spherical shell panels. The employed 3D equilibrium equations are developed in an orthogonal mixed curvilinear coordinate system for spherical shells in order to obtain the other simpler geometries as particular cases. A layer-wise approach is employed and equilibrium equations are solved in the thickness direction via the exponential matrix method. The partial derivatives in the in-plane directions are exactly calculated by assuming simply-supported edges and harmonic forms for the variables. The presented 3D shell model is extended to hygro-elastic cases by considering moisture content amplitudes imposed at the external surfaces of the structures in steady-state conditions. The moisture content profile through the thickness of the structures can be included in the 3D shell model using three different methodologies: it can be calculated by solving the steady-state 3D or 1D version of the Fick diffusion law, or it can be “a priori” assumed as linear through the thickness. Therefore, the developed system includes three non-homogeneous second order differential equilibrium equations that are solved after the introduction of opportune mathematical layers to consider the curvature effects through the thickness. The reduction to a first order differential equation system is obtained by redoubling the number of variables. The exponential matrix method is employed to accurately define both general and particular solutions. The implemented 3D hygro-elastic shell model is firstly validated and then it is employed with confidence for new benchmarks where the effects of thickness ratio, geometry, lamination scheme, material, thickness layer and imposed moisture content values at the external surfaces are investigated for composite and sandwich plates and shells. Showed results demonstrate the importance of an accurate developing of the elastic part of the 3D shell model combined with an appropriate evaluation of the moisture content profile through the thickness. Assumed linear moisture content profiles are correct only for thin one-layered isotropic structures, the use of the 1D Fick diffusion law allows only the correct definition of the material layer effect, the use of its 3D version allows the correct definition of both material and thickness layer effects.

3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions

This paper presents the steady-state stress analysis of single-layered and multilayered plates and shells embedding Functionally Graded Material (FGM) layers under moisture conditions. This solution relies on an exact layer-wise approach; the formulation is unique despite the geometry. It studies spherical and cylindrical shells, cylinders, and plates in an orthogonal mixed curvilinear coordinate system (α, β, z). The moisture conditions are defined at the external surfaces and evaluated in the thickness direction under steady-state conditions following three procedures. This solution handles the 3D Fick diffusion equation, the 1D Fick diffusion equation, and the a priori assumed linear profile. The paper discusses their assumptions and the different results they deliver. Once defined, the moisture content acts as an external load; this leads to a system of three non-homogeneous second-order differential equilibrium equations. The 3D problem is reduced to a system of partial differential equations in the thickness coordinate, solved via the exponential matrix method. It returns the displacements and their z-derivatives as a direct result. The paper validates the model by comparing the results with 3D analytical models proposed in the literature and numerical models. Then, new results are presented for one-layered and multilayered FGM plates, cylinders, and cylindrical and spherical shells, considering different moisture contents, thickness ratios, and material laws.

Numerical evaluation of virtual mass force coefficient of single solid particles in acceleration

• Single solid sphere in acceleration is numerically simulated based on the vorticity-stream function formulation. • Particle acceleration changes significantly the surrounding flow field and the stress tensor on the particle surface. • Virtual mass force coefficient is evaluated from the numerical simulation. • A correlation of total drag coefficient with the Reynolds number and acceleration number is proposed for tentative use in particulate flow simulation. Virtual mass force is an indispensable component in the momentum balance involved with dispersed particles in a multiphase system. In this work the accelerating motion of a single solid particle is mathematically formulated and solved using the vorticity-stream function formulation in an orthogonal curvilinear coordinate system. The total drag coefficient was evaluated from the numerical simulation in a range of the Reynolds number ( Re ) from 10 to 200 and the dimensionless acceleration ( A ) between −2.0 to 2.0. The simulation demonstrates that the total drag is heavily correlated with A , and large deceleration even drops the drag force to a negative value. It is found that the value of virtual mass force coefficient ( C V ) of a spherical particle is a variable in a wide range and difficult to be correlated with A and Re. However, the total drag coefficient ( C DV ) is successfully correlated as a function of Re and A , and it increases as A is increased. The proposed correlation of total drag coefficient may be used for simulation of solid–liquid flow with better accuracy.

Coupled flow and heat transfer of power-law nanofluids on non-isothermal rough rotary disk subjected to magnetic field

We study the coupled flow and heat transfer of power-law nanofluids on a non-isothermal rough rotating disk subjected to a magnetic field. The problem is formulated in terms of specified curvilinear orthogonal coordinate system. An improved BVP4C algorithm is proposed, and numerical solutions are obtained. The influence of volume fraction, types and shapes of nanoparticles, magnetic field and power-law index on the flow, and heat transfer behavior are discussed. The obtained results show that the power-law exponents (PLE), nanoparticle volume fraction (NVF), and magnetic field inclination angle (MFIA) have almost no effects on velocities in the wave surface direction, but have small or significant effects on the azimuth direction. The NVF has remarkable influences on local Nusselt number (LNN) and friction coefficients (FC) in the radial direction and the azimuth direction (AD). The LNN increases with NVF increasing while FC in AD decreases. The types of nanoparticles, magnetic field strength, and inclination have small effects on LNN, but they have remarkable influences on the friction coefficients with positively correlated heat transfer rate, while the inclination is negatively correlated with heat transfer rate. The size of the nanoparticle shape factor is positively correlated with LNN.

High-Order Conformal Perfectly Matched Layer for the DGTD Method

Based on the discontinuous Galerkin time-domain (DGTD) method, a new conformal perfectly matched layer (PML) is introduced to truncate the 3-D open domain. Compared with the traditional (rectangular or cubic) PML, the conformal PML is a smooth convex shell, which significantly reduces the buffer space in the computational domain. In this article, we construct the conformal PML in an orthogonal curvilinear coordinate system. Conductivities are defined to absorb the outgoing waves, depending on the distance from the sampling point to the non-PML region and the principal curvature radii of the sampling point, which are calculated utilizing the Weingarten transformation. The analytical expression of 3-D conformal PML is derived with the complex coordinate stretching technique. Furthermore, to reduce the total degrees of freedom (DoFs) while maintaining accuracy, the hierarchical vector basis functions are chosen to discretize the conformal PML and physical region. Numerical results validate its good absorption performance.

A second‐order reduced multiscale method for nonlinear shell structures with orthogonal periodic configurations

AbstractThis article develops an efficient second‐order reduced multiscale (SORM) method to study the nonlinear shell structure with orthogonal periodic configurations. The heterogenous shell structure is periodically distributed in orthogonal curvilinear coordinate systems. At first, the nonlinear problems for the shell structure are introduced, and the detailed higher‐order nonlinear multiscale formulas based on the asymptotic homogenization approach are given including microscale unit cell functions, effective material parameter and the homogenized equation. Also, since it requires a large number of computation cost to solve the nonlinear multiscale problems by the traditional high‐order homogenization methods, the novel reduced order multiscale model is constructed. Further, according to the reduced‐order multiscale models and higher‐order nonlinear formulas, an effective SORM algorithm is provided for studying the nonlinear shell structures. The main characteristics of the proposed algorithm are that the novel reduced forms established to investigate the nonlinear shell structures and an efficient higher‐order homogenized solution evaluated by postprocessing that does not need higher‐order continuities of the homogenization solutions. Finally, according to some typical nonlinear examples including block structures, cylindrical shell and double‐curved shallow shell, the availabilities of the SORM algorithm are confirmed.

Generalization of ray tracing in symmetric gradient-index media by Fermat's ray invariants.

Ray tracing in gradient-index (GRIN) media has been traditionally performed either by using the analytical or numerical solutions to the Eikonal equation or by creating a layered medium where Snell's law is calculated in each layer. In this paper, an exact general method to perform ray tracing in GRIN media is presented based on the invariants of the system as stated by Fermat's principle when the media presents symmetries. Its advantage, compared with other methods reported in the literature, relies on its easy implementation. Besides the GRIN distribution and the initial conditions of the incident ray, once the invariants of the system are stated the resulting math is simple to solve and interpret. To benchmark the algorithm, ray tracing in typical cases of GRIN media is calculated, finding minimal discrepancies between the analytical solutions and our simulations. The used media are axial refractive index and parabolic index fiber and lenses with spherical gradient-index symmetry, such as: Luneburg's, Gutman's, generalized Maxwell's Fish-eye, Eaton's, and concentrator lenses. Our method can be further applied to distributions with symmetries associated with other common curvilinear orthogonal coordinate systems, in particular to those associated to the separability of the Helmholtz equation that would allow us to investigate wave optics in these GRIN media with the associated geometries.