1. Exact solutions of the Navier-Stokes equations having steady vortex structures M. Z. Bazant † and H. K. Moffatt ‡ † Department of Mathematics, M IT ‡ DAMTP, University of Cambridge

2. Burgers’ Vortex Sheet Out-of-plane velocity (colors) describes shearing half spaces: An exact solution to the Navier-Stokes equations (Burgers, 1948) Vorticity is confined to a shear layer of O(1/Re) width by a transverse potential flow (yellow streamlines):

3. We seek solutions to the steady 3d Navier-Stokes equations for 2d vortex structures stabilized by planar potential flow where the non-harmonic out-of-plane velocity satisfies an advection-diffusion problem (where Re is the Peclet number) with the pressure given by. Other steady vortex structures

4. Conformal Mapping 1. Streamline coordinates (Boussinesq 1905): 2.Solutions are preserved for any conformal mapping of “two-gradient” systems (Bazant, Proc. Roy. Soc. A 2004): Half of Burgers’ vortex sheet Streamline coordinates New solution: a “vortex star”

5. New solutions: I. Mapped vortex sheets Nonuniformly strained “wavy vortex sheets” A “vortex cross” with 3 stagnation points For each f(z), these “similarity solutions” have the same isovorticity lines for all Reynolds numbers.

6. Towards a Class of Non-Similarity Solutions… “The simplest nontrivial problem in advection-diffusion” An absorbing cylinder in a uniform potential flow. Maksimov (1977), Kornev et al. (1988, 1994) Choi, Margetis, Squires, Bazant, J. Fluid Mech. (2005) Accurate numerical solutions by conformal mapping inside the disk Asymptotic approximations in streamline coordinates

7. New solutions: II. Vortex avenues An exact steady solution for a cross-flow jet Vorticity is pinned between flow dipoles at zero and infinity (uniform flow) Nontrivial dependence on Reynolds number (“clouds to “wakes”) 1. Analytic continuation of potential flow inside the disk 2. Continuation of non-harmonic concentration by circular reflection

8. IIa. Vortex eyes & butterflies A pair of oppositely directed, parallel jets stabilized by a pair of diverging (left) or converging (right) transverse flow dipoles. Conformally mapped vortex avenues

9. IIb. Vortex fishbones Generalize Burgers vortex sheet Nontrivial dependence on Reynolds number No singularities (convenient for testing numerics or in analysis)

10. IIc. Vortex wheels (or “churros”)

11. Conclusions New exact solutions demonstrate how shear layers can be stabilized by transverse flows Interesting starting point for further analysis of the Navier- Stokes equations (stability, existence,…) Possible applications to transverse jets in fuel injectors and smokestacks (e.g. can extend jets into cross flow by introducing a counter-rotating vortex pair in the inlet pipe) See also our poster (#83) in the 2005 Gallery of Fluid Motion.