Batchelor vortex

The Batchelor vortex is an approximate solution to the Navier-Stokes equations obtained using a boundary 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:

${\displaystyle {\begin{array}{cl}U(r)&=U_{\infty }+{\frac {W_{0}}{(R/R_{0})^{2}}}e^{-(r/R)^{2}},\\V(r)&=0,\\W(r)&=qW_{0}{\frac {1-e^{-(r/R)^{2}}}{(r/R_{0})}}.\end{array}}}$

The parameters in the above equations are

• ${\displaystyle U_{\infty }}$, 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

${\displaystyle \left\lbrace {\begin{array}{cl}U(r)&=a+\displaystyle {{\frac {1}{1+4t/Re}}e^{-r^{2}/(1+4t/Re)}},\\V(r)&=0,\\W(r)&=q\displaystyle {\frac {1-e^{-r^{2}/(1+4t/Re)}}{r}},\end{array}}\right.}$

where ${\displaystyle a=U_{\infty }/W_{0}}$ denotes the free stream axial velocity and ${\displaystyle Re}$ is the Reynolds number.

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

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

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

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