Orthogonal Curvilinear Coordinates Research Articles

Orthogonal mapping in two dimensions

A method for the generation of orthogonal boundary-fitted curvilinear coordinates for arbitrary simply- and doubly-connected domains is developed on the basis of the theory of quasi-conformal mappings of quadrilaterals and of previous work by Ryskin and Leal. The method has useful applications in orthogonal grid generation in two-dimensional and axi-symmetric domains and in the extension of rapid elliptic solvers and spectral methods to complex geometries. A new technique for the calculation of the conformal module of quadrilaterals is also presented.

A strategy to derive new internal coordinates by partitioning the internal configuration space according to invariance properties

A method to derive optimally orthogonal curvilinear coordinates for N-body systems is proposed. The invariance of certain subspaces under groups of linear transformations is employed to partition the configuration subspace into internal and external components. The construction is initially carried out locally by orthogonalizing typical group invariant vector fields. Integration is performed subsequently by means of integrating factors. Simple examples of orthogonal invariants illustrate the discussion.

Separation of variables and exact solution of the Dirac equation in nonstatic Minkowski spacetimes

Using a second-order formalism, a complete separation of variables in the Dirac equation for a free particle in nonstatic orthogonal curvilinear coordinates of the form t=f(u,v), x=g(u, v), y, z, is presented. It is shown that the Dirac equation is separable in eight nonequivalent coordinate systems where the Klein-Gordon equation separates. Exact solutions of the Dirac equation in the systems of coordinates obtained are presented.

ORTHOVEC: version 2 of the REDUCE program for 3-D vector analysis in orthogonal curvilinear coordinates

The revised version of ORTHOVEC is a collection of REDUCE 3.3 procedures and operations which provide a simple to use environment for the manipulation of scalars and vectors. Operations include addition, subtraction, dot and cross products, division, modulus, div, grad, curl, Laplacian, differentiation, integration, a · ∇ and Taylor expansion. Version 2 differs from the original in revised notation and extended capabilities.

Stress analysis of an elastic body with a wavy surface

The purpose of this paper is to investigate the stress distribution of a half-plane solid with a wavy surface under an applied concentrated load. Orthogonal curvilinear coordinates were built-up to describe the geometry. The complex potential of stress is determined following the conformai mapping technique. It is found that the stress at the surface is proportional to ϵ γ 2 , where ϵ 2π is the ratio of the amplitude to the wavelength of the boundary and γ is the distance from the load point. Near the loading point, surface tension is found when the load is applied at a crest; compressional-type stress is found when the load is applied at a trough. In general these results are quite different from those found for stress in a half plane with a flat surface under an applied concentrated load.

On the hybrid/mixed finite element method with energy constraints

An incremental variational formulation is presented for the large deflection analysis of the elastic-plastic continuum by considering the energy consistency of a nonlinear multivariable discrete system based upon incompatible trial functions. The functional is given in orthogonal curvilinear coordinates and the initial stress procedure has been introduced. Equilibrium imbalance correction is included to prevent drifting of the solution during the incremental process. An application to general shell analysis is given as the initial practice of the suggested formulation.

The effect of curvature on turbulence in stratified fluids

The influence of streamline curvature on small‐scale turbulence and vertical mixing in stratified fluids is the subject of this study. The roles of curvature and stratification in enhancing and suppressing turbulent mixing are explored using second‐moment closure for turbulence. Governing equations for second moments are expressed in generalized orthogonal curvilinear coordinates, from which, through a series of approximations, simplified expressions are derived for second moments in the limit of small streamline curvature. The governing equations are then used to obtain a quasi‐equilibrium turbulence model suited for application to atmospheric and oceanic mixed layers. A typical model application is illustrated by simulation of stratified flows over two‐dimensional, idealized mountains and valleys. The limit of local equilibrium is further invoked to derive semi‐analytical results for the enhancement and suppression of vertical turbulent mixing under the combined influence of stratification and curvature. It is shown that stabilizing curvature can drastically suppress turbulence even when the stratification is strongly destabilizing. Conversely, under strong stable stratification that would otherwise lead to total suppression of turbulence, destabilizing curvature can keep turbulence alive. Streamline curvature is also shown to significantly modify the Monin‐Obukhov similarity laws for momentum and heat fluxes in the constant flux region of the atmospheric boundary layer. Finally, the need for observational data on curvature effects on mixing in stratified flows either in the laboratory or in flows over topography in the oceans and the atmosphere is highlighted.

Separation of variables and exact solution to the Dirac equation in curvilinear orthogonal coordinates with cylindrical symmetry

In the present article, a complete separation of variables in the Dirac equation for a free particle is achieved in parabolic cylinder and elliptical coordinates. The resulting system of ordinary differential equations is solved and the asymptotic behavior of the spinor solution is discussed.

Orthogonal Curvilinear Coordinates and Angular Velocity

Orthogonal Curvilinear Coordinates and Angular Velocity

Impact buckling of orthotropic shells of revolution with allowance for geometric nonlinearity

Numerous experiments have established the presence of axisymmetric and nonaxisymmetrie stages in the buckling of elastic cylindrical and conical shells subjected to axial impact. It has been shown [8, 10, 14] that metallic, glassplastic, and polymer shells become unstable due to the intensive development of nonaxisymmetric strains caused by geometric shape flaws. Until recently, calculations of the nonaxisymmetric buckling of cylindrical shells in a geometrically nonlinear formulation have been performed only by the Bubnov--Galerkin method in lower-level approximations [5]. In [6], a numerical solution was obtained for the problem of the dynamic axial compression of a metallic cylindrical shell while retaining a large number of terms in the expansion for nonaxisymmetric deflection. The authors of [2, 4] formulated and numerically solved a problem concerning the impact buckling of flawed isotropic cylindrical and conical shells within the framework of the Kirchhoff--Love hypothesis with allowance for boundary, wave, and nonlinear effects associated with the coupling of the axisymmetric mode of instability and one of the nonaxisymmetric modes. The authors of [3] solved the problem of the impact buckling of an isotropic cylindrical shell with allowance for the interaction of the axisymmetric mode and a finite number of nonaxisymmetric modes. The 9 method in [2] was used in [7] to study the stability and strength of laminated orthotropic shells. The goal of the present investigation is to use the findings in [2-4] as a basis for developing a method of numerically analyzing the impact buckling of orthotropic shells of revolution with initial shape flaws. We also want to study the interaction of nonaxisymmetric modes of buckling. We will examine the axial impact buckling of thin orthotropie shells of revolution of constant thickness h referred to orthogonal curvilinear coordinates ,~ and/3 which coincide with the lines of curvature of the middle surface of the shell. The equations of motion of the shell will be written in the form [13]

Large displacement and rotational formulation for laminated shells including parabolic transverse shear

The paper presents an approach for a general laminated shell geometry describable by orthogonal curvilinear coordinates. The theory includes a through-the-thickness parabolic distribu- tion of transverse shear stress.-Additionally, a simplified approach that allows large displacements and rotations is incorporated. The theory is cast into a displacement-based finite element formula- tion and then specialized to a cylindrical shell geometry. The theory is then applied to the problem of a transversely loaded isotropic deep arch, and results show a slightly more flexible response compared with published results that are based upon inextensible assumptions. This problem also indicates that the usual locking associated with shell elements is apparently eliminated.

Rayleigh-Love wave coupling in an azimuthally anisotropic medium.

We present the generalized representation of the equations of motion for an elastic solid in a concise vector form with arbitrary orthogonal curvilinear coordinates and no assumption on symmetry of elastic moduli. For the particular case of the Cartesian coordinates, this representation leads to the generalized y-method for eigenvalues of surface wave dispersion in an anisotropic plane-stratified medium. The generalized y-method is the integration method to find eigenvalues of surface wave dispersion by iterating numerical depth-integration of first-order ordinary differential equations that are derived from the generalized representation of the equations of motion and Hookean law. Based on the generalized y-method, we present some numerical results for dispersion curves and azimuthal variations of surface wave velocities to fully display Rayleigh-Love wave coupling in Kawasaki's (1986) azimuthally anisotropic model for the upper mantle beneath the Pacific ocean. When Rayleigh-Love wave coupling takes place in a particular period range between a pair of nearby modes, surface waves display the following distinct singularities: (1) a difference of phase velocities of the pair of the modes is smaller than about 0.5 km/s, (2) a pair of dispersion curves of group velocities cross each other, (3) polarizations of particle motion directions are twisted along the pair of dispersion curves from Rayleigh- to Love-types and from Love- to Rayleigh-types. These sigularities are dependent on the relative depth- and azimuthal-distribution of the upper mantle anisotropy. When Rayleigh-Love wave coupling does not take place, the first-order perturbation theory works well.

First-order covariant treatment of the influence of space charge and external fields on finite-emittance relativistic beams of charged particles in generalized curvilinear coordinates

A first-order covariant treatment of space-charge effects on relativistic beams of charged particles under the influence of applied fields of arbitrary forms is presented. The beams and fields are assumed to fulfill no particular symmetry conditions, and curvilinear orthogonal coordinates are used. Relative to a given distribution function, a suitable choice of both metric tensor and optical axis allows for the selection of appropriate coordinates, reducing the overall transverse motion of a charged particle to a pair of independent motions along the principal axes of a tensor that is related to both geometry and field. Detailed formulas applying to a microcanonical distribution in an eight-dimensional phase space are worked out, and the envelope equations for a monokinetic beam are carried out explicitly. The distribution function is shown to separate into the product of a stationary distribution in transverse-coordinate space and momentum space by using factors carrying the relativistic corrections. As a result, the envelope equations are formally identical to the customary equations describing the envelope of a nonrelativistic beam under the influence of merely transverse forces. A few numerical applications are finally presented; the behavior of beams moving through a magnetic prism and through a cycloidal analyzer is displayed graphically.

Vector 33: A reduce program for vector algebra and calculus in orthogonal curvilinear coordinates

This paper describes a package with enables REDUCE 3.3 to perform algebra and calculus operations upon vectors. Basic algebraic operations between vectors and between scalars and vectors are provided, including scalar (dot) product and vector (cross) product. The vector differential operators curl, divergence, gradient and Laplacian are also defined, and are valid in any orthogonal curvilinear coordinate system. The package is written in RLISP to allow algebra and calculus to be performed using notation identical to that for operations. Scalars and vectors can be mixed quite freely in the same expression. The package will be of interest to mathematicians, engineers and scientists who need to perform vector calculations in orthogonal curvilinear coordinates.

Periodic viscous flow through a circular hole.

Periodic viscous flows through a circular hole driven by a fluctuating far field pressure are numerically studied. The time-dependent incompressible N-S equations formulated with orthogonal curvilinear coordinates are solved by using a finite difference method. Numerical examples compared with the potential theory predictions are presented. The acoustic impedances of a circular hole are calculated for various combinations of the pressure/viscous force ratio, frequency and the hole edge thickness.

THREE-DIMENSIONAL LAMINAR BUOYANT FLOW IN A HORIZONTAL CIRCULAR CYLINDER WITH DIFFERENTIALLY HEATED END WALLS AT HIGH RAYLEIGH NUMBERS

A numerical study is made of natural convection inside a horizontal cylinder with differentially heated ends. The geometric aspect ratio L/R is taken to be 2.0, and the Rayleigh number varies from 10 4 to 10 7 . The approach is based on primitive-variable finite differences discretized in generalized orthogonal coordinates. The governing equations are given in curvilinear orthogonal coordinates and the corresponding finite-difference control-volume formulation is presented. The resulting flow and temperature fields are compared to those known for other aspect ratios and Rayleigh numbers. Overall heat transfer data have also been obtained and compared with those from several approximate theories.

Solution of the Neumann pressure problem in general orthogonal coordinates using the multigrid technique

The multigrid (MG) technique has been advanced for use with Neumann boundary-value problems in clustered curvilinear orthogonal coordinates. This comprises an important step in the analysis of incompressible flow using the velocity-pressure formulation of the Navier-Stokes equations. The finite-difference representation of the problem and the formulation of the restriction and coarse-grid correction operators are examined in detail. Maintaining consistency between these and the integral constraint associated with the Neumann problem is found to be critical for the success of the MG technique. The influence of the smoothing operator is examined by employing Gauss-Seidel, alternating-direction implicit, and strongly implicit techniques. The MG procedure enhances the efficiency of fine-grid solutions of the Neumann problem by a factor of 3 to 14, depending on the type of smoothing operator employed and the values of the problem parameters.

ORTHOVEC: A REDUCE program for 3-D vector analysis in orthogonal curvilinear coordinates

ORTHOVEC is a collection of REDUCE procedures and operations which provide a simple to use environment for the manipulation of scalars and vectors. Operations include addition, subtraction, dot and cross products, division, modulus, div, grad, curl, Laplacian, differentiation, integration, a · ▽ and Taylor expansion.

An approach to the numerical solution of two-dimensional problems of plate and shell theory with variable parameters

Here Zi = 2~ (a, ~) (a I ~ a ~ a2; ~, ~ ~ ~ ~_) are the desired resolving functions, r is a linear function in ~kzj/~flk; fi(~ ' fl) are free terms and ~, fl are orthogonal curvilinear coordinates. For open shells (plates) boundary conditions on the contour lines ~ = coast; fi = coast are appended. For shelis closed in one direction the boundary conditions'in this direction are replaced by periodicity conditions. Before going over to an exposition of the proposed approach, let us examine the fundamental aspects of the method of integral relations in application to the solution of the calls of problems under consideration (2, 3, 9). The solution of the boundary value problem consists of two stages. In the first stage the initial scheme of the partial differential equations is approximated by a system of ordinary differential equations satisfying the boundary conditions on a part of the shell contour.

Pressure drop during laminar oblique flow through in-line tube assemblies

The pressure drop values occurring during laminar, oblique, fully-developed flow through inline tube assemblies of pitch-to-diameter ratios from 1.25 to 2.00 are calculated in detail. The inclination angle Θ of the mean velocity vector to the tube axes is varied in the range 0° ⩽ Θ ⩽ 90°. New information is provided on the influence of inclination angle on pressure drop. The method of calculation is based on a finite-difference solution of the momentum and continuity equations expressed in curvilinear-orthogonal coordinates. The method is validated by comparing the results with existing theoretical and experimental data in the limiting cases of purely axial ( Θ = 0°) and purely transverse ( Θ = 90°) flow, since such data are not available for 0° < Θ < 90°.