13 – Solid blocks

Elemental volumes that have different thermo physical properties from the rest of the domain constitute the solid blocks. We define a Solid Block as region in which there is no flow.

The method adopted in CFD Sinflow Library 1.0 to solve fields with solid blocks is to create different coefficients to the energy, momentum and pressure-velocity coupling equations.

Below we have, the solid blocks methodology according to the equations mentioned previously:

1. Momentum conservation equations

As mentioned previously, there is no flow inside solid blocks. Therefore, the momentum equations are given by:


 
(13.3)

So


    
  (13.4a)
 


(13.4b)
 


(13.4c)

And
 
 
 for momentum equation in x direction;
 



for momentum equation in y direction;

After assigning zero to velocities in the center of the volumes belonging to the solid blocks, we should assign zero to velocities on the faces of these volumes. So we attribute an infinite viscosity to these volumes. In the computational code this is done multiplying m by a factor of 10000.

The physical explanation to use this method is based on the fact that viscosity is related to the dynamic diffusivity coefficient, so the zero velocity inside the block is strongly diffused to surface.

Mathematically, we can see the infinite viscosity effect in the shear stress on the block’s surface:

 
(13.5)
Observing figure (13.1), we have, by the harmonic mean discussed in chapter (6) and considering mP << mW, meq:
 
(13.6)



Figure 13.1 – Numerical average X Real average.
We can see that even though the numerical derivative is equal to half the real, the real derivative is adjusted by the equivalent viscosity. This viscosity is twice the real. Therefore the shear stress on the surface approximates to its real value.

Continuing in the momentum conservation equation, the terms that involve pressure at volumes adjacent to the solid blocks are also modified. It is done one evaluation of these blocks in the same way that we evaluated the volumes adjacent to fictitious volumes. Eq. (4.16) to (4.19).


2. Pressure-velocity coupling
The influence of the solid blocks in the pressure-velocity coupling is seen at the coefficients at the faces of the volumes that belong to the solid blocks, there we attribute zero to the velocities. The advantage of this implementation is lies on the fact that when we calculate the velocity correction equation at the solid blocks, the velocities continue to be zero, and we do not have to execute further corrections to attribute zero to the velocities of solid blocks.

Energy conservation equation

When we are evaluating the temperature field, we have two types of solid blocks. In the first the temperature gradient is evaluated inside the solid block, and all the coefficients of this solid block, have the same expressions of the other elemental volumes of the domain.

For the second type, we prescribe a constant temperature. So, the energy equation for all solid blocks with constant temperatures are given by:


  (13.1)

hence


   
  (13.2a)
 


(13.2b)
 


(13.2c)

We also need that the temperature evaluated in the center of the block should be carried to the surface. In order to do this we do something similar to the technique employed in the momentum equations. Theoretically it can be achieved considering that the thermal diffusivity in the solid block is infinite. In SinFlow k is multiplied by a factor of 10000. Therefore the temperature inside the solid block is easily diffused to surface.

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