Non orthogonal curvilinear coordinates pdf

Rans equations in curvilinear coordinates to begin with, the reynoldsaveraged navierstokes rans equations are presented in the familiar vector and cartesian tensor forms. An introduction to curvilinear orthogonal coordinates overview. Expression of strain tensor in orthogonal curvilinear. Curvilinear coordinates georgia institute of technology. In general, curvilinear coordinates allow the natural basis vectors hi not all mutually perpendicular to each other, and not required to be of unit length. Here, the curvilinear coordinates 12 3, are the familiar rz. Swift geophysical institute, university of alaska, fairbanks, alaska, 997757320, usa.

Lecture notes on mathematical methods mihir sen joseph m. Gradient, divergence and curl in curvilinear coordinates. Curvilinear coordinates and vector calculus 3 3 3 3 2 2 2 1 1 1 h u h u h. Note on curvilinear coordinates physics libretexts. It may also be described by a cylindrical coordinate system, which is a non cartesian coordinate system. Expression of strain tensor in orthogonal curvilinear coordinates 49 d surface nested in a cartesian coordinate system. I am beginning to study the mathematics of curvilinear coordinates and all textbooks and web sites do not have realistic examples of non othogonal systems.

For example, the threedimensional cartesian coordinates x, y, z is an orthogonal coordinate system, since its coordinate surfaces x constant, y constant, and z constant are planes that meet at right angles to one another, i. In this section we derive the expressions of various vector concepts in an orthogonal curvilinear coordinate system. A tensor operations in orthogonal curvilinear coordinate systems a. With riemanns insight, it grew to represent infinite space, and curvilinear coordinate systems were utilized in physical applications. A point or vector can be represented in any curvilinear coordinate system, which may be orthogonal or nonorthogonal. Pdf generalization of the second order vector potential. You can think of curvilinear coordinates as a generalized coordinate system, free of the straight line restraints that you see with a euclidean system. Generalization of the second order vector potential. This is useful because some problems do not fit ideally into cartesian x, y, z coordinates. Note on transformation to general curvilinear coordinates for maxwells.

Problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. Pdf tensor analysis and curvilinear coordinates researchgate. We therefore propose to examine the identity in a general curvilinear coordinate system,not necessarily orthogonal for which we assume that. We need to know all the lengths and all the angles between the. Examples of orthogonal coordinate systems include the. A set of coordinates u ux,y,z, v vx,y,z and w wx,y,z where the directions at any point indicated by u, v and w are orthogonal perpendicular to each other is referred to as a set of orthogonal curvilinear coordinates. Pdf orthogonal curvilinear coordinates mohamed sobhy. We will begin with the case that the coordinates are orthogonal. Following this theorem, we can separate the given vector. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. For example, the schrodinger equation for the hydrogen atom is best solved using spherical polar coordinates. Base and unit vectors in orthogonal curvilinear coordinates. Generalization of the second order vector potential formulation for arbitrary nonorthogonal curvilinear coordinates systems from the covariant form of maxwells equations.

The laplacian in curvilinear coordinates the full story peter haggstrom. Curvilinear coordinates are a coordinate system used for euclidean space where the coordinate lines think. Generalized curvilinear coordinates in hybrid and electromagnetic codes daniel w. Gradient, divergence, laplacian, and curl in non euclidean coordinate systems math 225 supplement to colleys text, section 3. Div, grad and curl in orthogonal curvilinear coordinates. Pdf strain gradient theory in orthogonal curvilinear. A tensor operations in orthogonal curvilinear coordinate. Generalization of the second order vector potential formulation for arbitrary nonorthogonal curvilinear coordinates systems from the covariant form of. Another reason to learn curvilinear coordinates even if you never explicitly apply the knowledge to any practical problems is that you will develop a far deeper understanding of cartesian tensor analysis. Piercey univserity of arizona november 21, 2007 abstract this paper will explore how to perform di. The equations are described with minimum use of tensor expressions arising from the coordinate transformation, to avoid unnecessary confusion for readers. An example of a curvilinear system is the commonlyused cylindrical coordinate system, shown in fig. Below is a summary of the main aspects of two of the most important systems, cylindrical and spherical polar coordinates. This cylindrical system is itself a special case of curvilinear coordinates in that the base vectors are always orthogonal to each other.

Fourier series expansion in a nonorthogonal system of. We shall say nothing further about nonorthogonal coordinate systems, for these find no application in conventional hydrodynamic problems. Tensor analysis and curvilinear coordinates xmission. The subscript i indicates a covariant quantity and the superscript i indicates a contravariant quantity. Curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved. In a non orthogonal coordinate system, there are two sets of basis vectors. By using this reference surface, the coordinates of any. Nonorthogonal systems are hard to work with and they are of little or no practical use.

Hypersonic threedimensional nonequilibrium boundary. Gradient, divergence, laplacian, and curl in noneuclidean. Each term in the transport equations is examined to see whether it. A quick overview of curvilinear coordinates 1 introduction curvilinear coordinate systems are general ways of locating points in euclidean space using coordinate functions that are invertible functions of the usual x icartesian coordinates. What are some examples of non orthoganal curvilinear coordinates so that i can practice on. Once an origin has been xed in space and three orthogonal scaled axis are anchored to this origin, any point in space is uniquely determined by three real numbers, its cartesian coordinates. In this note, we derive two formulas for the divergence and curl operators in a general coordinates system, whether orthogonal or not. Since in these systems lines of constant components e. Grad, div, and curl in orthogonal curvilinear coordinate systems. Curvilinear coordinates fujiun jiang october 11, 2010 i.

In addition, nbfc grid makes the application of boundary conditions more complex. Strain gradient theory in orthogonal curvilinear coordinates. Generalization of the second order vector potential formulation for arbitrary non orthogonal curvilinear coordinates systems from the covariant form of maxwells equations 401. An orthogonal system is one in which the coordinates arc mutually perpendicular. Note on transformation to general curvilinear coordinates. For example in lecture 15 we met spherical polar and cylindrical polar coordinates. To ensure that vector components can be treated in a simple way, a curvilinear coordinate system should reduce to a rectangular system when a sufficiently small region is considered, and it must not reduce to a net of obliquely inclined lines. Curvilinear coordinate an overview sciencedirect topics. Most of the coordinate systems we are interested in are orthogonal.

Curvilinear analysis in a euclidean space mechanical engineering. The laplacian in curvilinear coordinates the full story. Powers department of aerospace and mechanical engineering university of notre dame notre dame, indiana 465565637. Effectiveness of grid orthogonality on body fitted. Conclusion the formulae by ward and pendry 7, 9 for the maxwells curl equations in curvilinear non orthogonal coordinates are amended to allow for the marked di. The use of an orthogonal basis makes vector manipulations simpler than for non orthogonal. The graphics look ratty in windows adobe pdf viewers when not scaled up. However, other coordinate systems can be used to better describe some physical situations. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates.