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
An exact solution to the Navier-Stokes equations (Burgers, 1948)
Out-of-plane velocity (colors)
describes shearing half spaces:
Vorticity is confined to a
shear layer of O(1/Re) width
by a transverse potential flow
(yellow streamlines):
Other steady vortex structures
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
.
Conformal Mapping
1. Streamline coordinates (Boussinesq 1905):
Half of Burgers’ vortex sheet
2.
Streamline coordinates
Solutions are preserved for any conformal
mapping of “two-gradient” systems
(Bazant, Proc. Roy. Soc. A 2004):
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)
Asymptotic approximations
in streamline coordinates
Accurate numerical solutions by
conformal mapping inside the disk
New solutions: II. Vortex avenues
1. Analytic continuation of potential flow inside the disk
2. Continuation of non-harmonic concentration by circular reflection
• 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”)
IIa. Vortex eyes & butterflies
Conformally mapped vortex avenues
A pair of oppositely directed, parallel jets stabilized by a pair
of diverging (left) or converging (right) transverse flow dipoles.
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 NavierStokes 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.