Implementation of theta BC in cylindrical coordinate system FVM

abhi084 · June 2, 2016, 06:01

I am working on cylindrical coordinate system using FVM. I am solving the 3d unsteady diffusion equation using r,theta and z coordinates. Whereas I am able to apply the r and z BC's, I am unable to understand how to correctly apply the theta boundary condition. The equation is

$\rho$ c $\frac{dT}{dt}$ = $\frac{1}{r}$ $\frac{d}{dr}$ $kr\frac{dT}{dr}$ + $\frac{1}{r}$ $\frac{d}{d\phi}$ $\frac{kdT}{rd\phi}$ + $\frac{d}{dz}$ $\frac{kdT}{dz}$

P.S.: The equation has partial differentials, couldn't find out how to write them here....

The theta BC is

T(r,0,z,t)=T(r,2 $\pi$ ,z,t) and
$\frac{dT}{d\phi}$ (r,0,z,t)= $\frac{dT}{d\phi}$ (r,2 $\pi$ ,z,t)

Please help me in understanding how to implement this BC in the above equation using FV Method. Also if u guys need it u can assume the wall temperatures as fixed temperatures (Dirichlet BC, since my problem is implementation of theta BC.)
Any help is appreciated.

FMDenaro · June 2, 2016, 11:28

First, in a FV formulation you have to work with the conservative form of the equation in such a matter to discretize the fluxes of the balanced varables.

Then, the azimuthal direction has the natural periodic condition f(0)=f(2*pi), if you work in the physical domain you see easily this fact.

abhi084 · June 2, 2016, 11:56

Quote:

Originally Posted by FMDenaro

First, in a FV formulation you have to work with the conservative form of the equation in such a matter to discretize the fluxes of the balanced varables.

Then, the azimuthal direction has the natural periodic condition f(0)=f(2*pi), if you work in the physical domain you see easily this fact.

Yes tats what I'm asking physically these boundary conditions make sense as teperature at same CV centre can't have 2 distinct values at 0 and 2*pi angle and same goes for flux which comes from the second theta BC. What I mean to ask is how to implement these in my program. Like when I do with r and z, I get the discretized equation in the form of
a $_{p}$ T $_{p}$ = $\sum$ a $_{b}$ T $_{b}$ + ap0*T_{old}

What I mean is how to put the theta BC in conservation form of above diffusion equation, (similar to r and z???) , but i'm unable to understad how to do that.

Thank you for your help and any suggestion is appreciated.

June 2, 2016, 06:01	Implementation of theta BC in cylindrical coordinate system FVM	#1
abhi084 New Member Abhi Join Date: Dec 2014 Posts: 28 Rep Power: 11	I am working on cylindrical coordinate system using FVM. I am solving the 3d unsteady diffusion equation using r,theta and z coordinates. Whereas I am able to apply the r and z BC's, I am unable to understand how to correctly apply the theta boundary condition. The equation is $\rho$ c $\frac{dT}{dt}$ = $\frac{1}{r}$ $\frac{d}{dr}$ $kr\frac{dT}{dr}$ + $\frac{1}{r}$ $\frac{d}{d\phi}$ $\frac{kdT}{rd\phi}$ + $\frac{d}{dz}$ $\frac{kdT}{dz}$ P.S.: The equation has partial differentials, couldn't find out how to write them here.... The theta BC is T(r,0,z,t)=T(r,2 $\pi$ ,z,t) and $\frac{dT}{d\phi}$ (r,0,z,t)= $\frac{dT}{d\phi}$ (r,2 $\pi$ ,z,t) Please help me in understanding how to implement this BC in the above equation using FV Method. Also if u guys need it u can assume the wall temperatures as fixed temperatures (Dirichlet BC, since my problem is implementation of theta BC.) Any help is appreciated.

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June 2, 2016, 11:28		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	First, in a FV formulation you have to work with the conservative form of the equation in such a matter to discretize the fluxes of the balanced varables. Then, the azimuthal direction has the natural periodic condition f(0)=f(2*pi), if you work in the physical domain you see easily this fact.