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April 23, 2004, 02:38 
Equation in which coordinate system

#1 
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If I have to solve a flow problem for a cylindrical physical domain, in which coordinate system should I have the equations  rectangular cartesian or cylindrical coordinate? For example, if I want to study the natural convection problem for a fluid in a cylindrical container, should I write the NS and energy equations in cylindrical coordinate or can I write the equtions in cartesian coordinate ? For writing and solving the equations in cartesian coordinates for such situation, what are the consequences?


April 23, 2004, 05:22 
Re: Equation in which coordinate system

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Both Cylinderical and Cartesian systems are valid. But for treating BC, the former is obviously better.


April 23, 2004, 05:23 
Re: Equation in which coordinate system

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Cartesian coordinates are not adequate to the problem. If your flow has an axial symmetry you can make it two dimensional or even one dimensional if you use von Karman's method for integrating over the radius of the cylinder.


April 23, 2004, 05:44 
Re: Equation in which coordinate system

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Mihail,
My question is general. I do not mean that exploiting some symmetry, etc., I can convert it to a one or twoD problem. My point is that if I write the equations for cartesian coordinates and solve them, what consequences (out of physics) are expected. For example, there are certain terms specific for the cylindrical coordinates which arise because of the problem definition and geometry itself. The 1/r term(s), e.g., has a meaning. 

April 23, 2004, 06:25 
Re: Equation in which coordinate system

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Well, divergention and tension in Navier  Stokes equation don't look very friendly (something like 1/r(d(rVr)/dr)+1/r(d(Vfi)/dfi)+dVz/dz and so on) but it's worth trying it.


April 23, 2004, 08:44 
Re: Equation in which coordinate system

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If you are using finite differences on a cylindrical grid then you are forced to convert the equations to cylindrical coordinates.
If using a finite volume method on a cylindrical (or any other type of grid) you will obviously use the integral formulation of the equations. 

April 23, 2004, 11:35 
Re: Equation in which coordinate system

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If you intend to solve NS in cylindricalpolar coordinates, you will start out by writing the equations in polar coordinates. Of course in addition to different looking terms, there will be *more* terms, like the centrifugal term for the radial momentum balance. Also note, coordinate transformations, (and other Galilean transformations) do not affect the physics represented by the flow equations.
Solving eqns. in cylindrical polar may not be that easy for a pipe because 1/r blows up. Can be done with special basis functions tho, if anyone is interested. 

April 23, 2004, 14:21 
Re: Equation in which coordinate system

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If you can easily specify boundary conditions in cartesian coordinates, i think it's better to deal with this coordinate system : you don't have source terms and don't have singularity problem at symmetry axe. The 1/r before fluxes and the source term only arise because of the coordinates system change. There is not an "additional meaning" in the cylindrical coordinate formulation, i.e. all the terms are "include" in the cartesian formulation.


April 30, 2004, 10:33 
Re: Equation in which coordinate system

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good points of using cylindrical coordinates: (1) only two spacial variable (x, r) instead of three (x, y, z) in Cartesian coordinates. (2) as a consequence, you can save a lot of memory and run faster.
bad points: (1) extra source term (2) may encounter singularity at r=0. 

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