Burgers vortex

Azimuthal velocity and ${\displaystyle z}$-component vorticity in a Burgers' vortex

The flow for the Burgers vortex is described in cylindrical ${\displaystyle (r,\theta ,z)}$ coordinates. Assuming axial symmetry (no ${\displaystyle \theta }$-dependence), the flow field associated with the axisymmetric stagnation point flow is considered:

${\displaystyle v_{r}=-\alpha r,}$

${\displaystyle v_{z}=2\alpha z,}$

${\displaystyle v_{\theta }={\frac {\Gamma }{2\pi r}}g(r),}$

where ${\displaystyle \alpha >0}$ (strain rate) and ${\displaystyle \Gamma >0}$ (circulation) are constants. The flow satisfies the continuity equation by the two first of the above equations. The azimuthal momentum equation of the Navier–Stokes equations then reduces to[3]

${\displaystyle r{\frac {d^{2}g}{dr^{2}}}+\left({\frac {\alpha r^{2}}{\nu }}-1\right){\frac {dg}{dr}}=0}$

where ${\displaystyle \nu }$ is the kinematic viscosity of the fluid. The equation is integrated with the condition ${\displaystyle g(\infty )=1}$ so that at infinity the solution behaves like a potential vortex, but at finite location, the flow is rotational. The choice ${\displaystyle g(0)=0}$ ensures ${\displaystyle v_{\theta }=0}$ at the axis. The solution is

${\displaystyle g=1-\exp \left(-{\frac {\alpha r^{2}}{2\nu }}\right).}$

The vorticity equation only gives a non-trivial component in the ${\displaystyle z}$-direction, given by

${\displaystyle \omega _{z}={\frac {\alpha \Gamma }{2\pi \nu }}\exp \left(-{\frac {\alpha r^{2}}{2\nu }}\right).}$

Intuitively the flow can be understood by looking at the three terms in the vorticity equation for ${\displaystyle \omega _{z}}$,

${\displaystyle -\alpha r{\frac {d\omega _{z}}{dr}}=2\alpha \omega _{z}+{\frac {\nu }{r}}{\frac {d}{dr}}\left(r{\frac {d\omega _{z}}{dr}}\right).}$

The first term on the right-hand side of the above equation corresponds to vortex stretching which intensifies the vorticity of the vortex core due to the axial-velocity component ${\displaystyle v_{z}=2\alpha z}$. The intensified vorticity tries to diffuse outwards radially due to the second term on the right-hand side, but is prevented by radial vorticity convection due to ${\displaystyle v_{r}=-\alpha r}$ that emerges on the left-hand side of the above equation. The three-way balance establishes a steady solution. The Burgers vortex is a stable solution of the Navier–Stokes equations.[4]

${\displaystyle y}$-component velocity and ${\displaystyle z}$-component vorticity in a Burgers' vortex layer

Burgers vortex layer or Burgers vortex sheet is a strained shear layer, which is a two-dimensional analogue of Burgers vortex. This is also an exact solution of the Navier–Stokes equations, first described by A. A. Townsend in 1951.[6] The velocity field ${\displaystyle (v_{x},v_{y},v_{z})}$ expressed in the Cartesian coordinates are

${\displaystyle v_{x}=-\alpha x,}$

${\displaystyle v_{z}=\alpha z,}$

${\displaystyle v_{y}=U\operatorname {erf} \left({\frac {{\sqrt {\alpha }}x}{\sqrt {2\nu }}}\right),}$

where ${\displaystyle \alpha >0}$ is the strain rate, ${\displaystyle v_{y}(+\infty )=U}$ and ${\displaystyle v_{y}(-\infty )=-U}$. The value ${\displaystyle 2U}$ is interpreted as the vortex sheet strength. The vorticity equation only gives a non-trivial component in the ${\displaystyle z}$-direction, given by

${\displaystyle \omega _{z}={\frac {2U}{\sqrt {2\pi }}}{\sqrt {\frac {\alpha }{\nu }}}\exp \left(-{\frac {\alpha x^{2}}{2\nu }}\right).}$

The Burgers vortex sheet is shown to be unstable to small disturbances by K. N. Beronov and S. Kida[7] thereby undergoing Kelvin–Helmholtz instability initially, followed by second instabilities[8][9] and possibly transitioning to Kerr–Dold vortices.

Non-axisymmetric Burgers' vortices emerge in non-axisymmetric strained flows. The theory for non-axisymmetric Burgers's vortex for small vortex Reynolds numbers ${\displaystyle Re=\Gamma /(2\pi \nu )}$ was developed by A. C. Robinson and Philip Saffman in 1984,[4] whereas Keith Moffatt, S. Kida and K. Ohkitani has developed the theory for ${\displaystyle Re\gg 1}$ in 1994.[10] The structure of non-axisymmetric Burgers' vortices for arbitrary values of vortex Reynolds number can be discussed through numerical integrations.[11] The velocity field takes the form

${\displaystyle v_{x}=-\alpha x+u(x,y),}$

${\displaystyle v_{y}=-\beta y+v(x,y),}$

${\displaystyle v_{z}=\gamma z}$

subjected to the condition ${\displaystyle \gamma =\alpha +\beta }$. Without loss of generality, one assumes ${\displaystyle \alpha >0}$ and ${\displaystyle \gamma >0}$. The vortex cross-section lies in ${\displaystyle xy}$ plane, providing a non-zero vorticity component in the ${\displaystyle z}$ direction

${\displaystyle \omega _{z}={\frac {\partial u}{\partial y}}-{\frac {\partial v}{\partial x}}.}$

The axisymmetric Burgers' vortex is recovered when ${\displaystyle \alpha =\beta =\gamma /2}$ whereas the Burgers' vortex layer is recovered when ${\displaystyle \alpha =\gamma }$ and ${\displaystyle \beta =0}$.

Explicit solution of the Navier–Stokes equations for the Burgers vortex in stretched cylindrical stagnation surfaces was solved by P. Rajamanickam and A. D. Weiss.[12] The solution is expressed in the cylindrical coordinate system as follows

${\displaystyle v_{r}=-\alpha \left(r-{\frac {r_{s}^{2}}{r}}\right),}$

${\displaystyle v_{z}=2\alpha z,}$

${\displaystyle v_{\theta }={\frac {\Gamma }{2\pi r}}P\left(1+{\frac {\alpha r_{s}^{2}}{2\nu }},{\frac {\alpha r^{2}}{2\nu }}\right),}$

where ${\displaystyle \alpha >0}$ is the strain rate, ${\displaystyle r_{s}\geq 0}$ is the radial location of the cylindrical stagnation surface, ${\displaystyle \Gamma >0}$ is the circulation and ${\displaystyle P}$ is the regularized gamma function. This solution is nothing but the Burger's vortex in the presence of a line source with source strength ${\displaystyle Q=2\pi \alpha r_{s}^{2}}$. The vorticity equation only gives a non-trivial component in the ${\displaystyle z}$-direction, given by

${\displaystyle \omega _{z}={\frac {\alpha \Gamma }{2\pi \nu {\tilde {\Gamma }}(1+\alpha r_{s}^{2}/2\nu )}}\left({\frac {\alpha r^{2}}{2\nu }}\right)^{\alpha r_{s}^{2}/2\nu }\exp \left(-{\frac {\alpha r^{2}}{2\nu }}\right)}$

where ${\displaystyle {\tilde {\Gamma }}}$ in the above expression is the gamma function. As ${\displaystyle r_{s}\rightarrow 0}$, the solution reduces to Burgers' vortex solution and as ${\displaystyle r_{s}\rightarrow \infty }$, the solution becomes the Burgers' vortex layer solution. Explicit solution for Sullivan vortex in cylindrical stagnation surface also exists.

In 1959, Roger D. Sullivan extended the Burgers vortex solution by considering the solution of the form[13]

${\displaystyle v_{r}=-\alpha r+{\frac {1}{r}}f(\eta ),}$

${\displaystyle v_{z}=2\alpha z\left[1-{\frac {f'(\eta )}{2\nu }}\right],}$

${\displaystyle v_{\theta }={\frac {\Gamma }{2\pi r}}{\frac {g(\eta )}{g(\infty )}},}$

where ${\displaystyle \eta =\alpha r^{2}/(2\nu )}$. The functions ${\displaystyle f(\eta )}$ and ${\displaystyle g(\eta )}$ are given by

${\displaystyle f(\eta )=6\nu (1-e^{-\eta }),}$

${\displaystyle g(\eta )={\frac {\nu }{\alpha }}\int _{0}^{\eta }t^{3}\exp \left[-t-\operatorname {Ei} (-t)\right]\,\mathrm {d} t}$

where ${\displaystyle \operatorname {Ei} }$ is the exponential integral. For Burgers vortex ${\displaystyle v_{r}<0}$, ${\displaystyle v_{\theta }>0}$ and ${\displaystyle v_{z}/z>0}$ are always positive, Sullivan's result shows that ${\displaystyle v_{r}>0}$ for ${\displaystyle \eta \leq 2.821}$ and ${\displaystyle v_{z}/z<0}$ for ${\displaystyle \eta \leq 1.099}$. Thus Sullivan vortex resembles Burgers vortex for ${\displaystyle \eta >2.821}$, but develops a two-cell structure near the axis due to the sign change of ${\displaystyle v_{z}/z}$.

• Kerr–Dold vortex