Non orthogonal correction for diffusion-like term in pressure equation

eltenedor · February 28, 2015, 17:26

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

and

. I am interested in the discretization of the term with the underbrace.

Normally one would assume

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:

$\left(\rho_f \overline{\mathbf{B}_f} \nabla p_f \right) \cdot \mathbf{S}_f \approx \rho_f \left( p_N - p_P \right) \frac{\overline{\mathbf{B}}_f \mathbf{S}_f \cdot \mathbf{S}_f} { \mathbf{d} \cdot \mathbf{S}_f } + \rho_f \left( \overline{\mathbf{B}_f} \nabla p_f \right) \left(\mathbf{S}_f - \frac{\mathbf{S}_f \cdot \mathbf{S}_f}{ \mathbf{d} \cdot \mathbf{S}_f } \mathbf{d}_{PF} \right)$

February 28, 2015, 17:26	Non orthogonal correction for diffusion-like term in pressure equation	#1
eltenedor New Member Fabian Gabel Join Date: Oct 2014 Location: Darmstadt Posts: 13 Rep Power: 12	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 and . I am interested in the discretization of the term with the underbrace. Normally one would assume 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: $\left(\rho_f \overline{\mathbf{B}_f} \nabla p_f \right) \cdot \mathbf{S}_f \approx \rho_f \left( p_N - p_P \right) \frac{\overline{\mathbf{B}}_f \mathbf{S}_f \cdot \mathbf{S}_f} { \mathbf{d} \cdot \mathbf{S}_f } + \rho_f \left( \overline{\mathbf{B}_f} \nabla p_f \right) \left(\mathbf{S}_f - \frac{\mathbf{S}_f \cdot \mathbf{S}_f}{ \mathbf{d} \cdot \mathbf{S}_f } \mathbf{d}_{PF} \right)$

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