Vorticity equation explained

The vorticity equation of fluid dynamics describes the evolution of the vorticity of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity). The governing equation is:where is the material derivative operator, is the flow velocity, is the local fluid density, is the local pressure, is the viscous stress tensor and represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching.

The equation is valid in the absence of any concentrated torques and line forces for a compressible, Newtonian fluid. In the case of incompressible flow (i.e., low Mach number) and isotropic fluids, with conservative body forces, the equation simplifies to the vorticity transport equation:

D\boldsymbol\omega
Dt

=\left(\boldsymbol{\omega}\nabla\right)u+\nu\nabla2\boldsymbol{\omega}

where is the kinematic viscosity and

\nabla2

is the Laplace operator. Under the further assumption of two-dimensional flow, the equation simplifies to:
D\boldsymbol\omega
Dt

=\nu\nabla2\boldsymbol{\omega}

Physical interpretation

\frac + \nabla \cdot\left(\rho \mathbf u\right) &= 0 \\ \Longleftrightarrow \nabla \cdot \mathbf &= -\frac\frac = \frac\frac\end where is the specific volume of the fluid element. One can think of as a measure of flow compressibility. Sometimes the negative sign is included in the term.

Simplifications

Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to

D
Dt

\left(

\boldsymbol\omega
\rho

\right)=\left(

\boldsymbol\omega
\rho

\right)\nablau

Alternately, in case of incompressible, inviscid fluid with conservative body forces,

D\boldsymbol\omega
Dt

=

\partial\boldsymbol\omega
\partialt

+(u\nabla)\boldsymbol\omega=(\boldsymbol\omega\nabla)u

[1]

For a brief review of additional cases and simplifications, see also.[2] For the vorticity equation in turbulence theory, in context of the flows in oceans and atmosphere, refer to.[3]

Derivation

The vorticity equation can be derived from the Navier–Stokes equation for the conservation of angular momentum. In the absence of any concentrated torques and line forces, one obtains:

Du
Dt

=

\partialu
\partialt

+\left(u\nabla\right)u=-

1
\rho

\nablap+

\nabla\tau
\rho

+

B
\rho

Now, vorticity is defined as the curl of the flow velocity vector; taking the curl of momentum equation yields the desired equation. The following identities are useful in derivation of the equation:

\begin{align} \boldsymbol{\omega}&=\nabla x u\\ \left(u\nabla\right)u &=\nabla\left(

1
2

uu\right)-u x \boldsymbol\omega\\ \nabla x \left(u x \boldsymbol{\omega}\right) &=-\boldsymbol{\omega}\left(\nablau\right)+\left(\boldsymbol{\omega}\nabla\right)u-\left(u\nabla\right)\boldsymbol{\omega}\\[4pt] \nabla\boldsymbol{\omega}&=0\\[4pt] \nabla x \nabla\phi&=0 \end{align}

where

\phi

is any scalar field.

Tensor notation

The vorticity equation can be expressed in tensor notation using Einstein's summation convention and the Levi-Civita symbol :

\begin{align} D\omegai
Dt

&=

\partial\omegai
\partialt

+vj

\partial\omegai
\partialxj

\\ &=\omegaj

\partialvi
\partialxj

-\omegai

\partialvj
\partialxj

+eijk

1
\rho2
\partial\rho
\partialxj
\partialp
\partialxk

+eijk

\partial\left(
\partialxj
1
\rho
\partial\taukm
\partialxm

\right) +eijk

\partialBk
\partialxj

\end{align}

In specific sciences

Atmospheric sciences

In the atmospheric sciences, the vorticity equation can be stated in terms of the absolute vorticity of air with respect to an inertial frame, or of the vorticity with respect to the rotation of the Earth. The absolute version is

dη
dt

=\nablahvh-\left(

\partialw
\partialx
\partialv
\partialz

-

\partialw
\partialy
\partialu
\partialz

\right)-

1
\rho2

k\left(\nablahp x \nablah\rho\right)

Here, is the polar component of the vorticity, is the atmospheric density,,, and w are the components of wind velocity, and is the 2-dimensional (i.e. horizontal-component-only) del.

See also

Further reading

Notes and References

  1. Book: Fetter . Alexander L. . Walecka . John D. . Theoretical Mechanics of Particles and Continua . 2003 . Dover Publications . 978-0-486-43261-8 . 351 . 1st.
  2. Web site: K. P.. Burr . Marine Hydrodynamics, Lecture 9. MIT Lectures.
  3. Web site: Richard L. . Salmon . Lectures on Geophysical Fluid Dynamics, Chapter 4 . Oxford University Press; 1 edition (February 26, 1998) .