2.3 Vorticity conservation laws

The vorticity conservation laws can be viewed as constraints on the motion of the vorticity. These conservation laws can be used to monitor the accuracy of numerical computations or even to construct accurate numerical schemes.

The conservation laws can be derived from the vorticity equation. Poincaré [174] derived the conservation laws for two-dimensional flow of a homogenous incompressible viscous fluid in free space; they are,

$\displaystyle \frac{d}{dt}\int\int\; \omega(\vec{x},t) \; dx\,dy \; \equiv \frac{d \Gamma}{dt}$	$\textstyle =$	$\displaystyle 0$	(2.19)
$\displaystyle \frac{d}{dt}\int\int\; \vec{x}\,\omega(\vec{x},t) \; dx\,dy$	$\textstyle =$	$\displaystyle \vec{0}$	(2.20)
$\displaystyle \frac{d}{dt}\int\int\; \vec{x}\cdot\vec{x}\;\omega(\vec{x},t) \; dx\,dy$	$\textstyle =$	$\displaystyle 4\,\nu\,\Gamma \quad \ ,$	(2.21)

where $\nu$ is the kinematic viscosity of the fluid and $\Gamma$ is the total circulation. Howard [110] showed that these are the only conservation laws for such flows; they are also the conservation laws for the vorticity equation without convection (Stokes equation). Truesdell [225] derived various laws for the average motion of vorticity in flows over solid boundaries.

The above conservation laws have the following interpretation [14,122]: The first equation (2.19) states that the total circulation $\Gamma$ is conserved. Equation (2.20) implies that the average position of the vorticity (center of vorticity) does not change in time. Equation (2.21) is a measure of how fast a vortical region expands. For flows in which the linear and angular momenta are bounded, the equations (2.20) and (2.21) can also be interpreted as conservation of those two momenta respectively [122].