fractional step method in cylindrical coordinates, help please?

new_at_this · May 9, 2012, 13:31

I am trying to solve developing internal flow in a circular duct using a staggered grid and cylindrical navier stokes equations.

At the moment I am using a fractional step method with primitive variables but my code is currently wrong and I think it has to do with the pressure correction step and I think I might be using the wrong equations. Please let me know if you see anything wrong. My process is listed below.

fractional step for advection terms
$U^* = U^n-\Delta t(\frac{\partial UV}{\partial r}+\frac{\partial U^2}{\partial z})$
$V^* = V^n-\Delta t(\frac{\partial V^2}{\partial r}+\frac{\partial UV}{\partial z})$

fractional step for diffusion terms
$U^{**} = U^*+\frac{\Delta t}{Re}(\frac{1}{r}\frac{\partial U^{**}}{\partial r}+\frac{\partial^2 U^{**}}{\partial r^2} + \frac{\partial^2 U^{**}}{\partial z^2})$
$V^{**} = V^*+\frac{\Delta t}{Re}(\frac{1}{r}\frac{\partial V^{**}}{\partial r}+\frac{\partial^2 V^{**}}{\partial r^2} + \frac{\partial^2 V^{**}}{\partial z^2}-\frac{V^{**}}{r^2})$

pressure correction
$U^{n+1} = U^{**} - \Delta t(\frac{\partial P}{\partial z})$
$V^{n+1} = V^{**} - \Delta t(\frac{\partial P}{\partial r})$

where Pressure is calculated by
$\frac{1}{r}\frac{\partial P}{\partial r}+\frac{\partial^2 P}{\partial r^2}+\frac{\partial^2 P}{\partial z^2} = \Delta t(\frac{U^{**}}{r}+\frac{\partial U^{**}}{\partial r}+\frac{\partial V^{**}}{\partial z})$

new_at_this · May 10, 2012, 16:00

so I'm pretty sure i've narrowed it down to be an error with the poisson pressure solver for cylindrical coordinates.

Is there anyone out there who is familiar marker and cell cfd in cylindrical? or with the mit18086_navierstokes code written by benjamin seibold? I modified his code to cylindrical coordinates.

Also is there anyone out there who might consider looking at my code? I have been staring at it for so long that I might be missing something because I am so used to it.

If someone knows of a simple test case that I can run to debug my poisson solver, please let me know.

May 9, 2012, 13:31	fractional step method in cylindrical coordinates, help please?	#1
new_at_this Member Peter Join Date: Oct 2011 Posts: 52 Rep Power: 14	I am trying to solve developing internal flow in a circular duct using a staggered grid and cylindrical navier stokes equations. At the moment I am using a fractional step method with primitive variables but my code is currently wrong and I think it has to do with the pressure correction step and I think I might be using the wrong equations. Please let me know if you see anything wrong. My process is listed below. fractional step for advection terms $U^* = U^n-\Delta t(\frac{\partial UV}{\partial r}+\frac{\partial U^2}{\partial z})$ $V^* = V^n-\Delta t(\frac{\partial V^2}{\partial r}+\frac{\partial UV}{\partial z})$ fractional step for diffusion terms $U^{*} = U^+\frac{\Delta t}{Re}(\frac{1}{r}\frac{\partial U^{}}{\partial r}+\frac{\partial^2 U^{}}{\partial r^2} + \frac{\partial^2 U^{}}{\partial z^2})$ $V^{} = V^+\frac{\Delta t}{Re}(\frac{1}{r}\frac{\partial V^{}}{\partial r}+\frac{\partial^2 V^{}}{\partial r^2} + \frac{\partial^2 V^{}}{\partial z^2}-\frac{V^{}}{r^2})$ pressure correction $U^{n+1} = U^{} - \Delta t(\frac{\partial P}{\partial z})$ $V^{n+1} = V^{} - \Delta t(\frac{\partial P}{\partial r})$ where Pressure is calculated by $\frac{1}{r}\frac{\partial P}{\partial r}+\frac{\partial^2 P}{\partial r^2}+\frac{\partial^2 P}{\partial z^2} = \Delta t(\frac{U^{}}{r}+\frac{\partial U^{}}{\partial r}+\frac{\partial V^{*}}{\partial z})$

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May 10, 2012, 16:00		#2
new_at_this Member Peter Join Date: Oct 2011 Posts: 52 Rep Power: 14	so I'm pretty sure i've narrowed it down to be an error with the poisson pressure solver for cylindrical coordinates. Is there anyone out there who is familiar marker and cell cfd in cylindrical? or with the mit18086_navierstokes code written by benjamin seibold? I modified his code to cylindrical coordinates. Also is there anyone out there who might consider looking at my code? I have been staring at it for so long that I might be missing something because I am so used to it. If someone knows of a simple test case that I can run to debug my poisson solver, please let me know.