Fuids - vorticity from viscocity

[julian]

If invisid flow starts with no vorticity then no vorticity will be produced. This can be understood
intuitively: we note that of the three types of force that can act on a cubic fluid
element, the pressure, body forces, and viscous forces, only the viscous shear forces are
able to give rotary motion. Hence if the viscous effects are nonexistent, vorticity cannot
be introduced.

Can this be derived from the Navier-Stokes equations for vorticity:

$\dfrac{\partial \vec{\omega}}{\partial t} + \vec{u} \cdot \nabla \vec{\omega} = \vec{\omega} \cdot \nabla \vec{u} + \nu \nabla^2 \vec{\omega}$

$\nabla \cdot \vec{u} = 0$?

 

[boneh3ad]

Sure. Note that the vorticity transport equation can be rewritten with the material derivative instead
$$\dfrac{\partial \vec{\omega}}{\partial t} + \vec{u}\cdot\nabla\vec{\omega} = \dfrac{D\vec{\omega}}{Dt} = \vec{\omega}\cdot\nabla\vec{u} + \nu\nabla^2\vec{\omega}.$$

For an inviscid flow, that last term is zero (or neglected), that leaves
$$\dfrac{D\vec{\omega}}{Dt} = \vec{\omega}\cdot\nabla\vec{u}.$$

So basically, this states that if the flow is irrotational initially, then $\vec{\omega}(t=0) = 0$, and therefore
$$\dfrac{D\vec{\omega}}{Dt} = 0,$$
meaning the flow remains irrotational forever. That does leave open the possibility that if the flow initially contains some vorticity, the vorticity can change over time and space.

If you want to take that further, consider a 2-D flow, in which case the $\vec{\omega}\cdot\nabla\vec{u}$ term drops out as well (it only only contains derivatives in the direction that is zero by the 2-D definition) and
$$\dfrac{D\vec{\omega}}{Dt} \equiv 0$$
regardless of initial conditions.

This whole concept is known as Kelvin's theorem.

 

[julian]

Thanks for that. I also want to know if you start with no vorticity at an instant in time, will the presence of the viscous term lead to the production of vorticity?

In 2D the vorticity equation is just the diffusion equation for the vorticity, and from what I know about that I would say that if there was no vorticity to start with there would be no production of vorticity.

Can vorticity only be produced from the non-slip condition for solid objects?

 

[boneh3ad]

Apologies for the late response; it's been a busy weekend.

Anyway, in theory, according to the vorticity transport equation, if you start with no vorticity you should never have any. However, this simply can't happen in most real flows subject to, as you said, the no-slip condition except in a few instances (e.g. Couette flow). Vorticity can also be "produced" through phenomena like free shear layers, but the no-slip condition is the big one.

 

[PJC01]

Yes, this can be derived from the Navier-Stokes equations for vorticity. The first term on the right-hand side represents the production of vorticity due to the stretching and tilting of existing vorticity by the velocity field. The second term represents the advection of vorticity by the velocity field. However, if the velocity field is zero (in the case of inviscid flow), then both of these terms will be zero. The third term represents the diffusion of vorticity due to viscosity, but if viscosity is zero, then this term will also be zero. Therefore, the equation reduces to:

\dfrac{\partial \vec{\omega}}{\partial t} = 0

This means that vorticity will not be produced or changed over time in the absence of viscous effects, as stated in the original content.