{{short description|Solution to the Navier–Stokes equations}} thumb|300px|Projected streamlines of the Sullivan vortex on the axial <math>rz</math>-plane; <math>O</math> is the origin. 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. <ref>Roger D. Sullivan. (1959). A two-cell vortex solution of the Navier–Stokes equations. Journal of the Aerospace Sciences, 26(11), 767–768.</ref><ref>Donaldson, C. du P. and Sullivan, R. D.: 1960, ‘Examination of the Solutions of the Navier-Stokes Equations for a Class of Three-Dimensional Vortices. Part 1. Velocity Distributions for Steady Motion’, Aero. Res. Assoc. Princeton Rep. (AFOSR TN 60-1227).</ref> 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.<ref>Pandey, S. K., & Maurya, J. P. (2018). A Mathematical Model Governing Tornado Dynamics: An Exact Solution of a Generalized Model. Zeitschrift für Naturforschung A, 73(8), 753-766.</ref> 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.<ref>Morton, B. T. (1966). Geophysical vortices. Progress in Aerospace Sciences, 7, 145-194.</ref> Due to its multi-celled structure, the vortex is used to model tornadoes<ref>Gillmeier, S., Sterling, M., Hemida, H., & Baker, C. J. (2018). A reflection on analytical tornado-like vortex flow field models. Journal of Wind Engineering and Industrial Aerodynamics, 174, 10-27.</ref> and large-scale complex vortex structures in turbulent flows.<ref>Large-scale vortex structures in turbulent wakes behind bluff bodies. Part 1. Vortex formation</ref>
==Flow description== Consider the velocity components <math>(v_r,v_\theta,v_z)</math> of an incompressible fluid in cylindrical coordinates in the form<ref>Drazin, P. G., & Riley, N. (2006). The Navier–Stokes equations: a classification of flows and exact solutions (No. 334). Cambridge University Press.</ref>
:<math>v_r=- \alpha r + \frac{2\nu}{r} f(\eta),</math> :<math>v_z=2\alpha z\left[1-f'(\eta)\right],</math> :<math>v_\theta=\frac{\Gamma}{2\pi r}\frac{g(\eta)}{g(\infty)},</math>
where <math>\eta =\alpha r^2/(2\nu)</math> and <math>\alpha>0</math> is the strain rate of the axisymmetric stagnation-point flow. The Burgers vortex solution is simply given by <math>f(\eta)=0</math> and <math>g(\eta)/g(\infty)=1-e^{-\eta}</math>. Sullivan showed that there exists a non-trivial solution for <math>f(\eta)</math> from the Navier-Stokes equations accompanied by a function <math>g(\eta)</math> that is not the Burgers vortex. The solution is given by
:<math>f(\eta) = 3 (1-e^{-\eta}),</math> :<math>g(\eta)= \int_0^\eta t^3 e^{-t- 3\operatorname{Ei}(-t)} \, \mathrm{d} t</math>
where <math>\operatorname{Ei}</math> is the exponential integral. For <math>\eta\ll 1</math>, the function <math>g(\eta)</math> behaves like <math>g=e^{-3\gamma}(\eta+\eta^2+\cdots)</math> with <math>\gamma</math> being is the Euler–Mascheroni constant, whereas for large values of <math>\eta</math>, we have <math>g(\infty)=6.7088</math>.
The boundary between the inner cell and the outer cell is given by <math>\eta=2.821</math>, which is obtained by solving the equation <math>v_r=0.</math> Within the inner cell, the transition between the downdraft and the updraft occurs at <math>\eta=1.099</math>, which is obtained by solving the equation <math>\partial v_z/\partial r=0.</math> The vorticity components of the Sullivan vortex are given by
:<math>\omega_r=0,\quad \omega_\theta= - \frac{6\alpha^2}{\nu} rz e^{-\alpha r^2/2\nu}, \quad \omega_z=\frac{\alpha\Gamma}{2\pi\nu} \frac{\eta^3e^{-\eta- 3\operatorname{Ei}(-\eta)}}{g(\infty)}.</math>
The pressure field <math>p</math> with respect to its central value <math>p_0</math> is given by
:<math>\frac{p-p_0}{\rho} = - \frac{\alpha^2}{2}(r^2+4z^2) - \frac{18\nu^2}{r^2}(1-e^{-\alpha r^2/2\nu}) + \int_0^r \frac{v_\theta^2}{r}dr,</math>
where <math>\rho</math> is the fluid density. The first term on the right-hand side corresponds to the potential flow motion, i.e., <math>(v_r,v_\theta,v_z) = (-\alpha r,0,2\alpha z)</math>, whereas the remaining two terms originates from the motion associated with the Sullivan vortex.
==Sullivan vortex in cylindrical stagnation surfaces== Explicit solution of the Navier–Stokes equations for the Sullivan vortex in stretched cylindrical stagnation surfaces was solved by P. Rajamanickam and A. D. Weiss and is given by<ref>Rajamanickam, P., & Weiss, A. D. (2021). Steady axisymmetric vortices in radial stagnation flows. The Quarterly Journal of Mechanics and Applied Mathematics, 74(3), 367–378.</ref>
:<math>v_r=- \alpha \left(r-\frac{r_s^2}{r}\right) + \frac{2\nu}{r} f(\eta),</math> :<math>v_z=2\alpha z\left[1-f'(\eta)\right],</math> :<math>v_\theta=\frac{\Gamma}{2\pi r}\frac{g(\eta)}{g(\infty)},</math>
where <math>\eta=\alpha r^2/(2\nu)</math>,
:<math>f(\eta) = (3-\eta_s) (1-e^{-\eta}),</math> :<math>g(\eta)=\int_0^\eta t^3 e^{-t-(3-\eta_s) \operatorname{Ei}(-t)} \, \mathrm{d} t.</math>
Note that the location of the stagnation cylindrical surface is not longer given by <math>r=r_s</math>(or equivalently <math>\eta=\eta_s</math>), but is given by
:<math>\eta_{\operatorname{stag}} = 3 + W_0[e^{-3}(\eta_s-3)]</math>
where <math>W_0</math> is the principal branch of the Lambert W function. Thus, <math>r_s</math> here should be interpreted as the measure of the volumetric source strength <math>Q=2\pi \alpha r_s^2</math> and not the location of the stagnation surface. Here, the vorticity components of the Sullivan vortex are given by
:<math>\omega_r=0,\quad \omega_\theta= - \frac{2\alpha^2}{\nu}\left(3-\frac{\alpha r_s^2}{2\nu}\right) rz e^{-\alpha r^2/2\nu}, \quad \omega_z=\frac{\alpha\Gamma}{2\pi\nu} \frac{\eta^3 e^{-\eta+(\eta_s- 3)\operatorname{Ei}(-\eta)}}{g(\infty)}.</math>
==See also== *Kerr–Dold vortex
==References== {{Reflist|30em}}
Category:Vortices