6.3 Vorticity transport

For an incompressible fluid (with other simplifying assumptions), vorticity obeys the transport equation , Eq. (6.2 ). This is a typical advection-diffusion equation, similar to Eq. (2.65 ) for heat, which is expressed in terms of the local time derivative and advection in conservative form by

@! --- + r (u!) r2! = ! (ru) : @t |-------{z-------} |----{z----} |-------{z-------} advection diffusion source \relax \special {t4ht=
Notably, Eq. (6.1 ) does not include a term in eqn. This is in contrast with conservation of (linear) momentum which can redistribute a perturbation in eqn instantaneously across all the domain through eqn, as discussed in Sec. 2.22 .

Instead, like heat, vorticity evolves locally only, with a range of influence limited by advective and diffusive transport, as discussed on page 126 .

Advection of vorticity is clearly illustrated by the smoke ring shown in Sec. 2.11 . Diffusion occurs by viscous torques transferring angular momentum between fluid elements.

PICT\relax \special {t4ht=

The source of vorticity, eqn is due to vortices changing shape under the influence of a velocity gradient eqn. If a vortex is stretched, e.g. under shear as shown above, its radius decreases, so angular velocity, and thus eqn, increases. Similarly, eqn decreases if the vortex is compressed.

The vorticity transport equation

For incompressible flow with constant eqn (and zero, or constant, body force), vorticity obeys the following transport equation:

|--------------------------| D!-| ! (ru) r2! = 0 | -Dt------------------------| \relax \special {t4ht=
The equation is derived from momentum conservation of a homogeneous, incompressible, Newtonian fluid with eqn constant and ignoring body forces, Eq. (2.49 ). Noting that eqn due to Eq. (2.46 ) leads to
@u-+ u (ru) + r2u = rp: @t \relax \special {t4ht=
The vorticity equation is derived from the curl (eqn) of Eq. (6.3 ) combined with the vorticity definition Eq. (2.37 ). The first term is eqn.

Replacing eqn and eqn by eqn in Eq. (2.72d ) and applying Eq. (2.37 ) gives

 1- u ru 2 r(u u) u !: \relax \special {t4ht=
The curl of the second term in Eq. (6.3 ) is expressed by the curl of Eq. (6.4 ) with Eq. (2.75a ), Eq. (2.73a ), Eq. (2.75e ) and Eq. (2.46 ), leading to
r (u ru) u (r!) ! (ru): \relax \special {t4ht=
The curl of the third term eqn by Eq. (2.74f ). The curl of the fourth term eqn by Eq. (2.75a ).

Combining all the terms and applying the material derivative Eq. (2.14 ), eqn to Eq. (6.5 ), leads to Eq. (6.2 ).

Notes on CFD: General Principles - 6.3 Vorticity transport