8.2.3 – Symmetry condition

The symmetry boundary condition is generically defined as follows:
 

(8.2.3.1)

if f is a scalar or a vector parallel to the boundary; 
 

(8.2.3.2)

if f is a scalar or a vector perpendicular to the symmetry boundary.


When we are analysing the symmetry condition and considering as unknowns the u and v Cartesian components of the velocity vector we notice that the momentum equation in x direction the h axis must be parallel to y axis, where the condition is applied and the velocity u is perpendicular to the boundary of symmetry. In an analogous way, for the momentum equation in y direction the x must be parallel to the x-axis, where the boundary condition is being applied (by definition the velocity v must be perpendicular to the symmetry boundary).

Now, if we look at (8.2.3.1), and defining a normal derivative equal to zero at a h surface, we have:

 


(8.2.3.3)

And the normal to surface x is:
 



(8.2.3.4)

So the fictitious volumes equations are given by:
 
East face:


Momentum conservation equation in x direction
 


(8.2.3.5)

Logo
  (8.2.3.6a)
  (8.2.3.6b)
  (8.2.3.6c)
  (8.2.3.6d)

Momentum conservation equation in y direction
 



(8.2.3.7)
So
 


(8.2.3.8a)
 
(8.2.3.8b)
 
(8.2.3.8c)
  (8.2.3.8d)
  (8.2.3.8e)
 


(8.2.3.8f)
West face:


Momentum conservation equation in x direction
 

(8.2.3.9)

So
  (8.2.3.10a)
  (8.2.3.10b)
  (8.2.3.10c)
  (8.2.3.10d)

Momentum conservation equation in y direction
 



(8.2.3.11)

So
 
(8.2.3.12a)
 
(8.2.3.12b)
 
(8.2.3.12c)
 
(8.2.3.12d)
 
(8.2.3.12e)
 
(8.2.3.12f)
 
North face:


Momentum conservation equation in x direction
 



(8.2.3.13)

So
 



(8.2.3.14a)
 
(8.2.3.14b)
 
(8.2.3.14c)
 
(8.2.3.14d)
 


(8.2.3.14e)
 
(8.2.3.14f)

Momentum conservation equation in y direction
 


(8.2.3.15)

So 
  (8.2.3.16a)
  (8.2.3.16b)
  (8.2.3.16c)
 

(8.2.3.16d)

 
South face:


Momentum conservation equation in x direction
 



(8.2.3.17)

So
 


(8.2.3.18a)
 
(8.2.3.18b)
 
(8.2.3.18c)
 
(8.2.3.18d)
 
(8.2.3.18e)
 
(8.2.3.18f)

Momentum conservation equation in y direction
 

(8.2.3.19)

So
 

(8.2.3.20a)
  (8.2.3.20b)
  (8.2.3.20c)
  (8.2.3.20d)
 
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