Is symmetric tensor assumed in CFD?
 

Hello,


In CFD packages, is stress tensor assumed to be symmetric?


We know that in NS equation only linear momentum is considered, and the general form of NS equation does not assume that stress tensor is symmetric. Physically, if the tensor is asymmetric then there is torque on the microscopic volume m, and within its streamline in general it is subjected to the influence of:

1. gravity
2. normal stress
3. shear force ⇒ torque ⇒ angular acceleration mra (r assumed to be mean radius of m)

I think this understand is basically correct regarding torque: visualize a crowed square in which thousands of people, shoulder-to-shoulder, are moving toward a same direction (for example pilgrims in Mecca), then each body is subjected to torque from the neighbor, and in general he will not always orient toward the same direction during his course along the streamline. The orientation change is therefore due to the angular aabove, and his movement can always be decomposed into

1. linear translation ⇐ normal stress
2. rotation about his own axis ⇐ torque

therefore the assumption seems perfectly valid, especially when liquid swirl is encountered.



And although in general is a present, the accumulated mr⋅v(rotational) is small because temporarily (time) microscopically, and individual “rotation” not in-phase/aligned with its neighbors increases stress which soon acts to decelerate that.

However, if many places I read that the symmetry of stress is assumed, such as with Rutherford Aris’s Vectors, Tensors, and the Basic Equations of Fluid Mechanics, in section 6.41 and he also used a somewhat questionable term “polar fluid” to refer to fluid with torque/asymmetric stress.


I would like to know that in popular CFD packages Fluent, CFX , FloEFD and Solidworks Flow Simulation, is stress tensor assumed to be symmetric or not? I wonder if they do assume symmetric, then for viscous fluid symmetric shear stress implies a symmetric velocity field which is unrealistic for example for swirls, so the software will fail.


I just started learning this subject for a few days so the question might seem very basic to experienced users. Sincerely hope someone could resolve my puzzle!



Andy

Additionally, does CFD software solve for steady-state or transient NS equation? If steady state equation, like the time-averaged RANS is solved, then it is reasonably that the stress tensor is symmetric; if time evolving equation is solved, then I guess stress tensor in general should not be symmetric?

But even for steady-state flow, particles (or larger "parcel"s) still flow and most likely change orientation (the pilgrim queue can reach steady state, but people sitll chang orientation along the streamline). So what makes the symmetric stress tensor assumption valid?

You will find a convincing argument in Batchelor's "An Introduction to Fluid Dynamics", page 11.

Thanks, I have read that.

But isn't that for fluid at rest? What about the case where there are acceleration and rotation present, as discussed in the first post? Even when the solution is in steady-state, there is still velocity field and hence particles/parcels are not at rest.

According to Batchelors' reasoning

http://web.mit.edu/2.25/www/pdf/viscous_flow_eqn.pdf

No this is not for a fluid at rest (remember, at rest you have a shear stress tensor = 0!). Even when fluid particles are in motion, steady-state or not, you cannot have a torque applied to them in normal conditions (without magnetic fields). The question is not whether particles accelerate or rotate but whether there is any external torque to balance the local torque that a non-symmetric stress tensor would induce. As said in Batchelor's and in the pdf that FMDenaro sent, only magnetic fields can create such torques.