In fluid dynamics, the Sullivan vortex is an exact solution of the Navier–Stokes equations describing a two-celled vortex in an axially strained flow, that was discovered by Roger D. Sullivan in 1959. At large radial distances, the Sullivan vortex resembles a Burgers vortex, however, it exhibits a two-cell structure near the center, creating a downdraft at the axis and an updraft at a finite radial location. Specifically, in the outer cell, the fluid spirals inward and upward and in the inner cell, the fluid spirals down at the axis and spirals upwards at the boundary with the outer cell. Due to its multi-celled structure, the vortex is used to model tornadoes and large-scale complex vortex structures in turbulent flows.
Flow description Consider the velocity components ( v r , v θ , v z ) {\displaystyle (v_{r},v_{\theta },v_{z})} of an incompressible fluid in cylindrical coordinates in the form
v r = − α r + 2 ν r f ( η ) , {\displaystyle v_{r}=-\alpha r+{\frac {2\nu }{r}}f(\eta ),}
v z = 2 α z [ 1 − f ′ ( η ) ] , {\displaystyle v_{z}=2\alpha z\left[1-f'(\eta )\right],}
v θ = Γ 2 π r g ( η ) g ( ∞ ) , {\displaystyle v_{\theta }={\frac {\Gamma }{2\pi r}}{\frac {g(\eta )}{g(\infty )}},}
where η = α r 2 / ( 2 ν ) {\displaystyle \eta =\alpha r^{2}/(2\nu )} and α > 0 {\displaystyle \alpha >0} is the strain rate of the axisymmetric stagnation-point flow. The Burgers vortex solution is simply given by f ( η ) = 0 {\displaystyle f(\eta )=0} and g ( η ) / g ( ∞ ) = 1 − e − η {\displaystyle g(\eta )/g(\infty )=1-e^{-\eta }} . Sullivan showed that there exists a non-trivial solution for f ( η ) {\displaystyle f(\eta )} from the Navier-Stokes equations accompanied by a function g ( η ) {\displaystyle g(\eta )} that is not the Burgers vortex. The solution is given by
f ( η ) = 3 ( 1 − e − η ) , {\displaystyle f(\eta )=3(1-e^{-\eta }),}
g ( η ) = ∫ 0 η t 3 e − t − 3 Ei ( − t ) d t {\displaystyle g(\eta )=\int _{0}^{\eta }t^{3}e^{-t-3\operatorname {Ei} (-t)}\,\mathrm {d} t}
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