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 Apurva Shukla April 9, 1999 01:30

Volume Calculation In Cylindrical Co-ordinate

Hello, I am working on Turbomachinery Flows and looking out for information on Surface & Volume calculation in Cylindrical Co-ordinate system, as I will working in Cylindrical Co-ordinate system. Please provide me the source like books, journals etc. from where I colud get the information.

 Patrick Godon April 12, 1999 12:14

Re: Volume Calculation In Cylindrical Co-ordinate

Hi, there. As a start let me suggest you to first look a some good books just to find the exact form of the equations in a cylindrical coordinate system. I am thinking mainly at 2 good books where you will find treatment of compressibility of the flow, viscosity and even more (such as energy treatment etc..). The first one is: Landau and Lifshitz (1959), Fluid Mechanics, Pergamon press (a classic reference for all branches of fluid mechanics). The original work is in Russian. The other book is excellent for all the small details of the equations: Tassoul, Theory of rotating stars, Princeton Series in astrophysics (Princeton, NJ, 1978). mainly Appendix B there. You should also find there a good treatment of the Coriolis and centrifugal terms if you wish to solve in the rotating frame of reference. This is a good start to get the necessary background to understand the form of the equations. PG.

 Apurva Shukla April 12, 1999 14:46

Re: Volume Calculation In Cylindrical Co-ordinate

Hello,

Thank For Your Response Actually, I have the governing equations in Cylindrical Coordinates; which I got from "Transport Phenonmena" by Bird and Stewart, and from "Lakshminarayana's" book on Turbomachinerys. I will refer your suggested books( if available in our library). Actually, my problem is that, if I give r,theta,z coordinate of the 8 corner points of a cell, how to calculate cell area & cell volume; which you know are very essential to be calculated accurately in CFD. I am bit confussed about this, as i have to start coding, I am looking for above information. If you have got any information or idea about it Sir, please inform me

Apurv

 Patrick Godon April 12, 1999 15:05

Re: Volume Calculation In Cylindrical Co-ordinate

Hi. I guess you want to have curved cells, like the geometry of the problems. THen the volume of each cell of limit r1<r2, z1<z2 and theta1<theta2 is obtained by substracting the relevant cylinders from each others and dividing by the number of angles (theta's): V=pi*(z2-z1)*(r2**2-r1**2)/(2*pi/(theta2-theta1)) , or if h=z2-z1 (height of each cell) and Nt=2*pi/(theta2-theta1) is the number of cells in the angular dimension then: v=pi*h*(r2**2-r1**2)/Nt . The surface of each cell should be also straightforward to derive by substracting and adding the relevant surfaces. Is that all the question, or am I missing something? PG.

 Patrick Godon April 12, 1999 15:14

Re: Volume Calculation In Cylindrical Co-ordinate

It seems that my message somehow got cut, maybe because of some mathematical characters. Assume the corner of the curved cell to be r1, r2, z1, z2, theta1 and theta2, where the index 2 refers to larger quantities. Let us denote h=z2-z1 and Nt=2*pi/(theta2-theta1) then the volume of each cell is obtained by substracting the relevant cylinders and by dividing the result by the number of cell in the angular direction. Namely: V=pi*h*(r2**2-r1**2)/Nt. In the same way the surface of each can be calculated easily. PG.

 Apurva Shukla April 13, 1999 08:08

Re: Volume Calculation In Cylindrical Co-ordinate

Sir, Thanks for your response. I agree with your method for calculation of volume, but for surface calcultions where we have the area expressed in terms of unit vectors in r, z & theta direction; how will we proceed. For cartesian coordintes I use the method given by "Duckowitch" in Journal of Computational Physics, 1977. I was actually looking for any similar method, because in that case I can have 8 different values of theta for eight corners, only one method which I came across was to use isoparametric transformation, but the method is quite complex, requiring complex coordinate transformations. This is my problem. I am really in a fix, other thing which I though was to calculate volume in cartesian coordinate ( though I will land in error especially near the centres) and for surface area express the vector in terms of Sin(theta) or Cos(thetha) for transforming from cartesian to cylindrical. This method will not have accuracy, so only I was looking for alternate method.

Apurv

 Patrick Godon April 13, 1999 14:44

Re: Volume Calculation In Cylindrical Co-ordinate

I am not sure I got what you meant when you mentioned eight different values for the angles of a volume. In any case the surfaces of the volume element in a cylindrical symmetry (using the same notation as before) would be: front: 2*pi*r2*h/Nt ; back: 2*pi*r2*h/Nt ; upper and lower surfaces would be equal: (pi*r2**2-pi*r1**2)/Nt ; and the side panels would just be h*(r2-r1).

 Patrick Godon April 14, 1999 08:45

Re: Volume Calculation In Cylindrical Co-ordinate

correction: back 2*pi*r1*h/Nt (r1 instead of r2). Sorry for the missprint.

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