CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > Main CFD Forum

Non orthogonal correction for diffusion-like term in pressure equation

Register Blogs Members List Search Today's Posts Mark Forums Read

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   February 28, 2015, 16:26
Default Non orthogonal correction for diffusion-like term in pressure equation
  #1
New Member
 
Fabian Gabel
Join Date: Oct 2014
Location: Darmstadt
Posts: 13
Rep Power: 11
eltenedor is on a distinguished road
I am having trouble deriving the discretized form of the pressure equation for a collocated variable arrangement on unstructured grids using a finite volume method as it is presented in

Code:
@article{darwish09, 
title = "A coupled finite volume solver for the solution of incompressible flows on unstructured grids ", 
journal = "Journal of Computational Physics ", 
year = "2009"
} 
@article{mangani14,
author = {Mangani, L. and Buchmayr, M. and Darwish, M.},
title = {Development of a Novel Fully Coupled Solver in OpenFOAM: Steady-State Incompressible Turbulent Flows in Rotational Reference Frames},
journal = {Numerical Heat Transfer, Part B: Fundamentals},
year = {2014},
}
The starting point is a semi-discretized form of the pressure equation

\underbrace{
 \sum_f \rho_f \left( - \overline{\mathbf{B}_f} \nabla p_f \right) \cdot 
\mathbf{S}_f}_{\text{diffusion-like term}
}
+
\sum_f \rho_f \overline{\mathbf{v}_f} \cdot \mathbf{S}_f
=
\sum_f \rho_f \left( - \overline{\mathbf{B}_f} \overline{\nabla p_f} \right) \cdot \mathbf{S_f},

where \mathbf{B}_P =diag(\frac{V_P}{a_P^u}, \frac{V_P}{a_P^v}, \frac{V_P}{a_P^w}) and the overline denotes linear interpolation between values of the neighboring cells P and N. I am interested in the discretization of the term with the underbrace.

Normally one would assume a_P^u = a_P^v = a_P^w and proceed without problems, but there are situations in which this is assumption is incorrect, for example if different under relaxation factors are used for the different momentum balances or, which is why I am interested in this, a problem is solved involving wall boundaries.

I am aware that for wall boundaries there exist discretizations that avoid the formulation of different diagonal contributions for each momentum balance, but I think a fully coupled solution approach should take this into account.

Problems surge, when I try to discretize the diffusion-like operator from the pressure equation. Normally those terms are discretized as

\left(\nabla \phi\right) \cdot \mathbf{S}_f
\approx
\frac{\phi_N - \phi_P}{ | \mathbf{d}_{NP} | }  | \Delta | 
+ \left(\nabla \phi\right)_f \left(\mathbf{S}_f - \Delta \right)

=\left( \phi_N - \phi_P \right) 
\frac{\mathbf{S}_f \cdot \mathbf{S}_f}{ \mathbf{d} \cdot \mathbf{S}_f }  
+ \left(\nabla \phi\right)_f \left(\mathbf{S}_f - \frac{\mathbf{S}_f \cdot \mathbf{S}_f}{ \mathbf{d} \cdot \mathbf{S}_f } \mathbf{d}_{PF} \right)

using the "over-relaxed" correction approach (s.a. Jasak's Phd Thesis). What I can't understand is the derivation of the discretization for the pressure equation, since it involves an additional (diagonalized) operator, \mathbf{B}_f. Can someone explain how to derive the result presented in the above mentioned paper:


\left(\rho_f \overline{\mathbf{B}_f} \nabla p_f \right) \cdot \mathbf{S}_f
\approx
\rho_f \left( p_N - p_P \right)
\frac{\overline{\mathbf{B}}_f \mathbf{S}_f \cdot \mathbf{S}_f}
{ \mathbf{d} \cdot \mathbf{S}_f } 
+ \rho_f \left( \overline{\mathbf{B}_f} \nabla p_f \right)
\left(\mathbf{S}_f - \frac{\mathbf{S}_f \cdot \mathbf{S}_f}{ \mathbf{d} \cdot \mathbf{S}_f } \mathbf{d}_{PF} \right)
eltenedor is offline   Reply With Quote

Reply

Tags
discretization, non orthogonal

Thread Tools Search this Thread
Search this Thread:

Advanced Search
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Non-linearity Pressure Equation -- PISO algorithm gdeneyer OpenFOAM Programming & Development 1 August 23, 2012 05:19
Timestep and Pressure Correction Relationship in SIMPLE rks171 Main CFD Forum 23 May 4, 2012 01:04
changing the coefficients of pressure correction Noel Phoenics 1 April 7, 2009 08:54
BCs for Pressure Correction Equation (SIMPLE) Bharath Somayaji Main CFD Forum 1 March 1, 2006 06:12
Preconditioning for Pressure Correction Equation cfd101 Main CFD Forum 1 February 23, 2006 14:34


All times are GMT -4. The time now is 15:18.