Help discretising diffusive terms

ski · December 17, 2009, 10:04

Hi,

I want to check that the method I'm attempting to implement is correct.

When evaluating the diffusive terms in the incompressible Navier-Stokes equations, (using the finite volume method on structured grids), for the U-momentum equation we have:

$\text{diffusive terms} = \frac{\partial}{\partial x} \left( \mu \frac{\partial U}{\partial x}\right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial U}{\partial y}\right) + \underbrace{\frac{\partial}{\partial x} \left( \mu \frac{\partial U}{\partial x}\right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial V}{\partial x}\right)}$

(hopefully no mistakes so far!!)

Now, do I treat the underbraced terms explicitly as a source, or implicitly?

Assuming the former, do I compute the value of the underbraced terms at the cell center and then multiply by the cell volume?

i.e. is there anything wrong with the following:

$\frac{\partial}{\partial x} \left( \mu \frac{\partial U}{\partial x}\right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial V}{\partial x}\right) = \mu \frac{\partial}{\partial x} \left( \frac{\partial U}{\partial x} + \frac{\partial V}{\partial y} \right) + \frac{\partial U}{\partial x}\frac{\partial \mu}{\partial x} + \frac{\partial V}{\partial x}\frac{\partial \mu}{\partial y}$

...Can I simply evaluate the above at the cell center (taking the first expression on the RHS as zero due to the continuity condition) and multiply by the volume?

Thanks very much in advance,

DoHander · December 17, 2009, 11:55

See the book of Anderson for an example of how you can discretize these terms using finite differences; or the book or Veersteg for a finite volume formulation.

Do

December 17, 2009, 10:04	Help discretising diffusive terms	#1
ski New Member AS Join Date: Jul 2009 Posts: 16 Rep Power: 16	Hi, I want to check that the method I'm attempting to implement is correct. When evaluating the diffusive terms in the incompressible Navier-Stokes equations, (using the finite volume method on structured grids), for the U-momentum equation we have: $\text{diffusive terms} = \frac{\partial}{\partial x} \left( \mu \frac{\partial U}{\partial x}\right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial U}{\partial y}\right) + \underbrace{\frac{\partial}{\partial x} \left( \mu \frac{\partial U}{\partial x}\right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial V}{\partial x}\right)}$ (hopefully no mistakes so far!!) Now, do I treat the underbraced terms explicitly as a source, or implicitly? Assuming the former, do I compute the value of the underbraced terms at the cell center and then multiply by the cell volume? i.e. is there anything wrong with the following: $\frac{\partial}{\partial x} \left( \mu \frac{\partial U}{\partial x}\right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial V}{\partial x}\right) = \mu \frac{\partial}{\partial x} \left( \frac{\partial U}{\partial x} + \frac{\partial V}{\partial y} \right) + \frac{\partial U}{\partial x}\frac{\partial \mu}{\partial x} + \frac{\partial V}{\partial x}\frac{\partial \mu}{\partial y}$ ...Can I simply evaluate the above at the cell center (taking the first expression on the RHS as zero due to the continuity condition) and multiply by the volume? Thanks very much in advance,

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Using source terms	jsm	Main CFD Forum	4	August 20, 2009 06:44
Source Terms in Momentum Balance	vidyaraja	Main CFD Forum	0	May 25, 2009 15:24
Question in definition of terms in solve	titio	OpenFOAM Running, Solving & CFD	0	March 19, 2009 16:02
Implicit or explicit treatment of convective terms	CH	Main CFD Forum	1	March 14, 2007 07:51
K-Epsilon model?	Brindaban Ghosh	Main CFD Forum	2	June 24, 2000 04:22

December 17, 2009, 11:55		#2
DoHander Senior Member Join Date: Nov 2009 Posts: 411 Rep Power: 19	See the book of Anderson for an example of how you can discretize these terms using finite differences; or the book or Veersteg for a finite volume formulation. Do