8/3/2019 K. Bajer and H.K. Moffatt- Theory of Non-Axisymmetric Burgers Vortex with Arbitrary Reynolds Number

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THEORY OF NON-AXISYMMETRIC BURGERS VORTEX WITHARBITRARY REYNOLDS NUMBER

K . BAJERUniversity of Warsaw , Institute of Geophysicsu1. Pasteura 7, 02-093 Warszaw a, Polandwww.igf.fuw.edu.pl/"kb/WWWANDH.K . MOFFATTUniversity of Cambridge,Department of Applied Mathematics and Theoretical Physics,Silver Street, Cambridge, C B 3 9 E W , U Kwww.damtp.cam.ac.uk/user/tfd

Abstract. We develop an asymptotic theory of the steady state of a rec-tilinear vortex in linear straning flow. In the special case of axisymmetricstrain the solution is the familiar Burgers vortex. In the more general,non-axisymmetric situation the asymptotic theories were developed for lowReynolds number (Robinson & Saffman 1984) and for high Reynolds num-ber (Moffatt, Kida & Ohkitani 1994). In the present paper we develop a newexpansion in the parameter X which characterises the departure from ax-isymmetry. Hence we obtain an expansion valid uniformly for all Reynoldsnumbers and thus bridgeing the gap between the low and the high Reynoldsnumber theories. In practice the new expansion is useful when the non-axisymetric deformation of the vortex is not too large.

1. IntroductionG . I. Taylor (1938) recognised the fact that the competition between stretch-ing and viscous diffusion of vorticity must be the mechanism controlling thedissipation of energy in turbulence. A decade later Burgers (1948) obtainedexact solutions describing steady vortex tubes and layers in locally uniformstraining flow where the two effects are in balance. The discovery of the

193E. Krairse and K . Gersten (eds.),IUlMSymposium onDynamics of Slender Vortices, 193-202.0 998Kluwer Academic Publishers. Printed in the Netherlands.

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8/3/2019 K. Bajer and H.K. Moffatt- Theory of Non-Axisymmetric Burgers Vortex with Arbitrary Reynolds Number

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194 K. BAJ ER AND H.K. MOFFATTexact solutions stimulated the development of the models of the dissipativescales of turbulence as random collections of vortex tubes and/or sheets.

The intermittent nature of the vorticity field was observed in experi-ments by taking statistical measurements which indicated the existence ofthe small-scale localised structures (Townsend 1951). Only recently havethese structures been directly observed, first in the numerical simulations(see, for example Vincent & Meneguzzi 1991) and then in the laboratoryexperiments where a new visualisation technique was employed (Douady,Couder & Brachet 1991).

The Burgers vortex has axial symmetry unlikely to be found in realflows, hence the need to find solutions describing non-axisymmetric stretchedvortices. Let us consider incompressible fluid with viscosity U . We lookfor a steady state of a vortex having stream-function Q ( x , g ) , vorticityw = - ( V 2 Q ) z and total circulation r subjected to the ambient irrotationalstraining flow

U = ( a z , P y , y z ) , a + p + y = o , a < 0 , y > O (1)characterised by the parameter

which measures the departure from axisymmetry. Takingm, - ' , r/27r,yI'/27w to be the units of length, time, stream-function and vorticity re-spectively respectively we obtain the steady state equations in polar coor-dinates ( r , ),

(4 )w = -v Q,where Rr = F/27rv is the Reynolds number of the vortex and Lo,L1 arelinear operators,

LO = 1+ 2jr&+V2 ( 5 )~1 = 3 cos(20)ra , - 3 sin(20)80. ( 6 )

Robinson and Saffman (RS84) solved this equation numerically for 0