Influid dynamics,Batchelor vortices, first described byGeorge Batchelor in a 1964 article, have been found useful in analyses of airplane vortex wake hazard problems.^[1][2]

The Batchelor vortex is an approximate solution to theNavier–Stokes equations obtained using aboundary layer approximation. The physical reasoning behind this approximation is the assumption that the axial gradient of the flow field of interest is of much smaller magnitude than the radial gradient.
The axial, radial and azimuthal velocity components of the vortex are denoted${\displaystyle U}$,${\displaystyle V}$ and${\displaystyle W}$ respectively and can be represented in cylindrical coordinates${\displaystyle (x,r,\theta )}$ as follows:

The parameters in the above equations are

• ${\displaystyle U_{\mathrm{\infty }}}$, the free-stream axial velocity,
• ${\displaystyle W_0}$, the velocity scale (used for nondimensionalization),
• ${\displaystyle R_0}$, the length scale (used for nondimensionalization),
• ${\displaystyle R=R(t)=\sqrt{R_0^2+4\nu t}}$, a measure of the core size, with initial core size${\displaystyle R_0}$ and${\displaystyle \nu }$ representing viscosity,
• ${\displaystyle q}$, the swirl strength, given as a ratio between the maximum tangential velocity and the core velocity.

Note that the radial component of the velocity is zero and that the axial and azimuthal components depend only on${\displaystyle r}$.
We now write the system above in dimensionless form by scaling time by a factor${\displaystyle R_0/W_0}$. Using the same symbols for the dimensionless variables, the Batchelor vortex can be expressed in terms of the dimensionless variables as

where${\displaystyle a=U_{\mathrm{\infty }}/W_0}$ denotes the free stream axial velocity and${\displaystyle Re}$ is theReynolds number.

If one lets${\displaystyle a=0}$ and considers an infinitely large swirl number then the Batchelorvortex simplifies to theLamb–Oseen vortex for the azimuthal velocity:

${\displaystyle W_{\mathrm{\Theta }}(r)=\frac{\mathrm{\Gamma }}{2\pi r}\left(1-e^{-r^2/R_c^2}\right)}$

where${\displaystyle \mathrm{\Gamma }}$ is the circulation.

• Continuous spectra of the Batchelor vortex (Authored by Xueri Mao and Spencer Sherwin and published byImperial College London)

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