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Different Navier-Stokes equations used in COMSOL

I was using the incompressible navier stokes and found that the governing equation (first picture) applied by COMSOL is a little different from the normal navier stokes equation (second picture), the term for shear stress. When I using COMSOL to simulate the heat transfer through a rectangular enclosure filled by air, it seems that COMSOL's computation result is higher in terms of heat flux. However, it can be adjusted by applying Isotropic Diffusion. I have the following questions?

1. Is this situation caused by the different governing equation?
2.how to use the stabilization techniques? Changing streamline diffusion and crosswind diffusion can only affect the simulation slightly, while isotropic diffusion affect the simulation greatly, so which method I should employ?

Any help?

Thanks in advance.


3 Replies Last Post Oct 3, 2012, 12:01 PM EDT
Posted: 5 years ago Oct 3, 2012, 10:57 AM EDT
Plus:

The main difference between the equations is the stress term. In COMSOL the stress is equal to:
viscosity*(velocity gradient+ velocity gradient to the T power), while in Navier-stokes equation, the shear stress is equal to viscosity*velocity gradient. What does the velocity gradient to the T power stand for?
Plus: The main difference between the equations is the stress term. In COMSOL the stress is equal to: viscosity*(velocity gradient+ velocity gradient to the T power), while in Navier-stokes equation, the shear stress is equal to viscosity*velocity gradient. What does the velocity gradient to the T power stand for?

Posted: 5 years ago Oct 3, 2012, 11:57 AM EDT
That's not a power but the transpose of the velocity gradient tensor. The stress tensor is defined as tau = viscosity*(l+l^T) where "l" is the velocity gradient tensor and "^T" represent the transpose operator. That's the definition of the stress tensor.

Simplified expressions like the one you showed (viscosity*velocity_gradient) come in particular cases (e.g. u = [u(y),0,0]), where many terms in the velocity gradient tensor are zero and the tensor "becomes" a scalar.
That's not a power but the transpose of the velocity gradient tensor. The stress tensor is defined as tau = viscosity*(l+l^T) where "l" is the velocity gradient tensor and "^T" represent the transpose operator. That's the definition of the stress tensor. Simplified expressions like the one you showed (viscosity*velocity_gradient) come in particular cases (e.g. u = [u(y),0,0]), where many terms in the velocity gradient tensor are zero and the tensor "becomes" a scalar.

Posted: 5 years ago Oct 3, 2012, 12:01 PM EDT
The two NS equations you showed are exactly the same. Specifically T = nu*(nabla_u+nabla_u_transpose).

The difference you noticed when changing the stabilization technique comes because the isotropic diffusion is an inconsistent method, roughly speaking it means it is a stronger but rude stabilization technique - see COMSOL help. Use it only in cases where you can't stabilize the problem with other methods.
The two NS equations you showed are exactly the same. Specifically T = nu*(nabla_u+nabla_u_transpose). The difference you noticed when changing the stabilization technique comes because the isotropic diffusion is an inconsistent method, roughly speaking it means it is a stronger but rude stabilization technique - see COMSOL help. Use it only in cases where you can't stabilize the problem with other methods.

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