4 – General Equation Discretization

Below we have the discretization of the general equation, and for the time being, we will omit de radius of the axisymmetric coordinates, so we have:






(4.1)
Integrating the equation in time and in x e h, in an elemental volume like that presented on figure 4.1, and remembering that we are using a fully implicit formulation, we have:

 
 






(4.2)

The coefficients and are presented bellow:
 
(4.3a)
 


(4.3b)
 

(4.3c)
 

(4.3d)
 


(4.3e)
 


(4.3f)
The coefficients present in the diffusive terms are given by:
 

(4.4a)
 


(4.4b)
 



(4.4c)
 


(4.4d)
 


(4.4e)
 


(4.4f)
 

(4.4g)
 


(4.4h)
and also:    
 


(4.5)



Figure 4.1 – Elemental volumes.

In order to evaluate the properties at the faces of the volumes we use the following equations:
 



(4.6a)
 




(4.6b)
 



(4.6c)
 



(4.6d)

The direct derivatives, which are part of the diffusive flux, are given by:
 

(4.7a)
 



(4.7b)
 



(4.7c)
 



(4.7d)

And the cross derivatives, approximated by central differences, are given by:
 



(4.8a)
 



(4.8b)
 



(4.8c)
 



(4.8d)

The interpolation equations given by and will be discussed in chapter 5.
The discretized differential equation is presented below:
(4.9)

The coefficients are given by:
 



(4.10)
 



(4.11a)
 



(4.11b)
 



(4.11c)
 



(4.11d)
 



(4.12a)
 



(4.12b)
 



(4.12c)
 



(4.12d)
 



(4.13a)

In natural convection problems we will use Boussinesq approximation, i. e., density varies with temperature only in the source term of momentum equation in y. It follows that:
 



(4.13b)
and,


    
  = Coefficient of thermal expansion;
  
  = Gravity acceleration;  
  = Reference field temperature.  

Pressure terms are evaluated using central differences, and when the Navier-Stokes equation is being considered they are given by:
 

(4.14a)
 



(4.14b)

And replacing the xx, hx, xyand hy metrics by their inverses, we have:
 


(4.15a)
 



(4.15b)

The equations (4.15) are used only for internal volumes of the domain, because, in spite of using fictitious volumes, the pressure terms of these volumes should not be considered in the calculations, therefore, at the boundary volumes we use a lower order approximation as follows:

For the faces of the domain:

East face:


 

(4.16a)
 



(4.16b)

West face:




 
 

(4.17a)
 



(4.17b)

North face:


   
 
(4.18a)
 



(4.18b)

South face:



 
 



(4.19a)
 



(4.19b)

For the corners of the domain:

Northwest corner:
 

(4.20a)
 



(4.20b)

Northeast corner:


    
 

(4.21a)
 



(4.21b)

Southeast corner:


    
 

(4.22a)
 



(4.22b)

Southwest corner:


    
 

(4.23a)
 



(4.23b)

In the linear system is being solved, equations (4.15) to (4.23) are added to the source term of the momentum equation. We should remark that in the source term of these equations we have a minus sign, as we could see in (4.1) and (4.2).

 
<< Back Index Next >>
 
© SINMEC/EMC/UFSC, 2001. All rights reserved.