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

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) widthby a transverse potential flow(yellow streamlines):

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 anadvection-diffusion problem (where Re is the Peclet number)

with the pressure given by .

Other steady vortex structures

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”

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.

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 approximationsin streamline coordinates

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

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

IIb. Vortex fishbones

• Generalize Burgers vortex sheet• Nontrivial dependence on Reynolds number • No singularities (convenient for testing numerics or in analysis)

IIc. Vortex wheels (or “churros”)

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.