Effect of discretization of viscous terms (FVM) for incompressible fluids on pressure

eltenedor · February 21, 2015, 18:51

According to Perić's book (Computational Methods for Fluid Mechanics 3rd. Ed. p. 158) when discretizing the viscous term of the incompressible Navier-Stokes equations

$\frac{\partial}{\partial x_j} \left[\mu \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \right]$

one can use the fact, that for flows with spatially invariant viscosity $\mu$ the second term will not contribute to the viscous term. This follows from the linearity of the derivation, interchange of the differentiation order and finally the application of the continuity equation for incompressible fluids

$\frac{\partial}{\partial x_j} \left( \mu \frac{\partial u_j}{\partial x_i} \right) = \mu \frac{\partial}{\partial x_j} \left( \frac{\partial u_j}{\partial x_i} \right) = \mu \frac{\partial}{\partial x_i} \left( \frac{\partial u_j}{\partial x_j} \right) = 0$

My solver application, which is based on Perić's CAFFA code, is able to choose between both approximations. I solve for a manufactured solution (2d Taylor-Green vortex) with Dirichlet boundaries and I experience a notable difference in the resulting discretization error depending on the decision to use the aforementioned simplification or not. Without the simplification the discretization error is about 10 times bigger. Even more confusing is the calculated pressure, which I attached. I don't understand why the pressure would look so different in the boundary control volumes.

Have you seen something similar before and can tell me, if there is something wrong with my code? The pressure has been extrapolated with a zero order profile at the boundaries.

FMDenaro · February 21, 2015, 20:04

as the identity requires Div v = 0, I suppose you can have some high residual in the continuity equation.
Furthermore, do not use the absolute pressure but its gradient as comparison terms

eltenedor · February 22, 2015, 05:46

I checked the residual of the continuity equation (right hand side of the pressure correction) and it is low (1.7269e-09). Furthermore continuity seems to be preserved locally, since each line of the linear system has an almost zero right hand side coefficient (maximum absolute value: 1.9118e-10).

Checking the pressure gradients shows the same strange behavior at the domain boundaries, I attached the results, the left plot represents the calculation using the simplification of the diffusive term.

What seems to be contradicting the fact that the residual of the continuity equation is low, can be seen in the third attachment. I calculated the additional contribution $\sum_{f} \left( \frac{\delta u_j}{\delta x_i}\right)_f n_j S_f$ for each cell ,

being the face normal unit vector and

being the surface area, and it doesn't vanish at all at certain domain boundaries.

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February 21, 2015, 20:04		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,897 Rep Power: 73	as the identity requires Div v = 0, I suppose you can have some high residual in the continuity equation. Furthermore, do not use the absolute pressure but its gradient as comparison terms