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transformation from curvilinear space

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Old   April 5, 2003, 23:47
Default transformation from curvilinear space
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I am working on a project and need to transform a vector field of a curvilinear grid to a 2D Cartesian computation grid. I tried to implement the algorithm in the papaer "Visualizaing Flow Over Curvilinear Grid Surfaces Using Line Integral Convolution" by Forssell. But the result it wrong. I guess I did something wrong, but isn't it just multiply the physical-space (curvilinear grid) velocity vectors by the inverse of Jacobian matrix? Or is there any other way to do it?

I know it should be an entry level question here. But since I am new to this, so somebody please help me!

Thank you!

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Old   April 6, 2003, 23:37
Default Re: transformation from curvilinear space
Ananda Himansu
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when using 2D curvilinear coordinates in a 2D euclidean space, there are several possibilities for representation of vectors and (higher-order) tensors. a vector can be represented by its (a) physical rectangular cartesian components relative to a globally defined orthonormal rectangular cartesian basis, (b) contravariant components relative to a covariant basis, (c) covariant components relative to a contravariant basis, or (d) physical curvilinear components relative to an orthonormal basis tangential to an orthogonal curvilinear coordinate system. note that (d) is valid only for orthogonal curvilinear coordinates such as polar and rotated rectangular cartesians, and can be obtained in these cases by scaling the components of type (b) or (c) appropriately.

it is my guess that you have components of type (d), and wish to convert them to those of type (a). this is not quite as simple as multiplying by the inverse of the jacobian matrix. the required relations can be found in most texts on vector and tensor analysis. i recommend that you consult one of the few books that explicitly show the basis vectors. i find explicit display of the basis vectors to be of help when converting among representations. a useful reference in this regard is the book "Numerical Grid Generation" by Thompson, Warsi and Mastin. you can find the link to a free downloadable copy of this book under the "Books" link at the top of the current cfd-online webpage, under the "Grid Generation" link on the "Books" webpage.
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Old   April 7, 2003, 00:46
Default Re: transformation from curvilinear space
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Hi Ananda,

Thank you so much for your response. I will go check out the book your recommended!

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