I am currently studying the vorticity dynamics from revolving wings. My current research requires that I study the vorticity dynamics using the relative vorticity equation. I am looking for a better mathematical understanding of the relative vorticity equation. Specifically, what the nature of each term behaves like (for example a source, sink, flux etc.).
In order to define the relative vorticity, I used the planetary vortex $2 \mathbf \Omega$ to arrive at the relative vorticity equation and $\mathbf\Omega$ is not a function of time. $$\mathbf \omega'=\mathbf\omega-2 \mathbf\Omega$$ $$\mathbf u'=\mathbf u-\mathbf\Omega\times\mathbf R$$ $$\frac{\partial \mathbf \omega'}{\partial t}=-(\mathbf u'\cdot \nabla)\mathbf\omega'+(\mathbf \omega'\cdot \nabla)\mathbf u'+(2\mathbf \Omega\cdot \nabla)\mathbf u'+\nu\nabla^2\mathbf\omega'$$
- I know that the $-(\mathbf u'\cdot \nabla)\mathbf\omega'$ term can usually, be understood as a flux of $\mathbf \omega'$.
- Also, I believe the $\nu\nabla^2\mathbf\omega'$ is best understood as a dissipative term.
I am more confused about how to describe $(\mathbf \omega'\cdot \nabla)\mathbf u'+(2\mathbf \Omega\cdot \nabla)\mathbf u'$. One of my professors thinks they might be best described as a source or sink since both change the direction of the vortex lines. For instance, $(2\mathbf \Omega\cdot \nabla)\mathbf u'$ will "tilt" the planetary vortex into the direction of the velocity.
The term $\boldsymbol\omega \cdot \nabla\mathbf{u}$ in the general vorticity equation accounts for the production of vorticity due to vortex stretching.
Roughly speaking, as an element of incompressible fluid is stretched in an axial direction it will be compressed in the radial direction to conserve mass. If there is an axial component of vorticity present, it will be amplified as a consequence of conservation of angular momentum. An analogy is a spinning ice skater increasing (decreasing) rotational speed by moving the arms closer to (further from) the rotational axis.
You may have observed how water entering a drain can develop a swirling component. A simple model is the axial extensional flow
$$u_r = - \frac{\alpha}{2} r, \,\,\, u_\theta = 0, \,\,\, u_z = \alpha z.$$
which satisfies the equation of continuity $\nabla \cdot \mathbf{u} = 0.$
If $\alpha > 0$, then the axial component of velocity increases with $z$, stretching the fluid elements in that direction while they are compressed in the radial direction by the inwardly directed radial velocity. If, initially, there is no azimuthal velocity component, one can develop -- causing the swirling motion down the drain -- as small distubances introduce vorticity which gets amplified by the vortex stretching mechanism.
Vortex tilting, as you describe it, is a related phenomenon which leads to amplification of vorticity in one direction as vortex tubes initially aligned in the orthogonal directions are tilted and bent by velocity gradients.
Consider the term $\boldsymbol\omega \cdot \nabla\mathbf{u}$ in terms of Cartesian components when there is initially one non-zero vorticity component $\omega_z$. We have
$$\boldsymbol\omega \cdot \nabla \mathbf{u} = \underbrace{\omega_z \frac{\partial u_x}{\partial z} \mathbf{e}_x + \omega_z \frac{\partial u_y}{\partial z}\mathbf{e}_y}_{\text{vortex tilting}} + \underbrace{\omega_z \frac{\partial u_z}{\partial z}\mathbf{e}_z}_{\text{vortex stretching}}.$$
The third term accounts for amplification of the z-component due to vortex stretching in the z-direction. The first and second terms are associated with vortex tilting. Gradients of the x- and y-components of velocity are now appearing as source terms in the equations for the x- and y- components of vorticity. The gradients cause vortex lines to bend or tilt bringing the initially z-aligned vorticity field into other directions.