Physica 23D (1986) 72-74 North-Holland, Amsterdam

VORTEX DYNAMICS AND SINGULARITIES IN THE FLUID EQUATIONS

Alain PUMIR* Laboratory of Atomic and Sofid State Physics, Cornell University, Ithaca, N Y 14853, USA

A pointwise, infinite stretching is found in a simulation of the Biot-Savart equation for a vortex tube in three dimensions. The relevance of this result to the incompressible Euler and Navier-Stokes equations is discussed.

This study deals with one of the challenging phenomena in 3-dimensional incompressible fluid dynamics, namely the stretching of the vortex lines by the flow. This stretching is responsible for many theoretical difficulties. For example, it has proven to be impossible so far to show that the solutions of the 3-dimensional Navier-Stokes equations do not blow-up when the viscosity is small enough, even in the absence of any external forces. In this respect, the 2-dimensional situation is simpler due precisely to the absence of any vorticity stretching. Experimentally, it is suspected that the vortex stretching is responsible for the violent intermittency observed in many turbulent flows.

On the basis of a numerical simulation of a simplified evolution equation (the velocity field is computed with a Biot-Savart formula, with a suitable cutoff), we find a finite time, pointwise blow-up of the vorticity. The evolution equation reads:

dr(O,t) dt

F 4~r

x f (dr/dO') A (r (O)-r(O' ) )dO'

{ ( , . ( e ) - + o (e) + ' / = '

(1)

where o measures the core-size of the slender

*Permanent address: D-PhG/SPT, CEN Saclay, L'Orme- des-Merisiers F-91191, Gif-sur-Yvette, France.

tube, and acts as a cutoff. This equation can be rigorously deduced in the limit of an infinitely thin tube (namely a << R c, where R e is the radius of curvature of the filament). The core-size is varied according to the equation:

0 21 ds/dO [ = cst,

where 0 is a Lagrangian parameter, and s is the arclength along the curve. The equation for the core-size insures the local conservation of the volume of the vortex tube. It is an approximation, appealing for describing the evolution of the fila- ment when the time scale of the r(O, t) is of the same order as the time scale of the motion inside the core. However, a pathology of this model is that it does not conserve energy. With these as- sumptions, the vorticity scales like 1 /0 2.

Starting from a simple, nonplanar curve, a pair- ing occurs and leads to a smooth curve composed of two anti-parallel filaments [1]. This new curve stretches very rapidly and generates smaller and smaller scales. Typically, two anti-parallel arcs of filament grow like two anti-parallel expanding circles, until an instability develops, and folds the curve into smaller pieces. These smaller pieces start stretching, and the process keeps repeating itself. The radius of curvature is always signifi- cantly larger than the core-size, and the distance between the two filaments remains of the same order as the core-size. The minimum of the square of the core-size decreases according a linear law: o 2 _ ( t . - t), suggesting a divergence of the vorticity like 1/(t* - t) (with possibly logarithmic

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A. Pumir / Vortex dynamics and singularities in the fluid equations 73

corrections) [2]. This result also follows from naive dimensional analysis of the vorticity equation.

A local approximation to the Biot-Savart for- mula is also obtained which gives some physical insights on the mechanisms that govern the stretching process after pairing. It allows too, a better analytic understanding of the numerical results. The key observation is that the contribu- tion to the velocity at a point on the filament due to remote parts of the paired filaments nearly cancel out. This leads to an expansion of the Biot-Savart integral as a power series of 1 / R c.

The resulting equations involve terms originating from the local induction effects, together with a term representing the effect of the rotation in- duced by each filament on the other. These equa- tions can be integrated numerically and give fur- ther support for the existence of a pointwise collapse, obeying the same laws as the one found for the Biot-Savart equations [3]. A simplified version of these equations has been presented in a previous paper [2]. The two filaments are de- scribed by a mean position, R , and a difference vector, O, and evolve according to:

0R ( 2 ) O R tgt = 02+202 pA Os '

O-'t= --~-A--~-s2 In 1+~-~o 2 (2)

2 ) O0 p 2 + 2 o 2 0 A ' ~ •

It can be shown that 02/0 2 decreases like 1 / ( - I n o). This means that the cores will end up overlapping when o goes to 0 and casts some doubts on the validity of the model. This point will be examined in more detail below. It is also possible to show that if the curve remains smooth and confined to a finite fraction of space, then, the length of the curve has to become infinite in a finite time. In any case, something has to happen; again, our numerical results rather suggest a point- wise collapse.

As already pointed out, there are several rea- sons why the Biot-Savart model may not be appli- cable to real vortex tubes near the singularity if their cores deform excessively. In order to in- vestigate this question, we use a perturbation analysis and try to build a solution of the fluid equations out of our numerical solutions. Our asymptotic expansion, that exploits the small ratio e = o/11 c observed in the numerical simulations, does not completely rule out a secular deforma- tion of the core, that we have not succeeded in bounding. Clearly, the cores most eventually de- form. The energy of a quasi 2-dimensional dipole with constant shape scales as /"EL. When the stretching proceeds, the length L increases, and energy conservation constraint forces a violation in the assumption of constant shape. Crude ana- lytic estimates suggest that the energy constraint is not too stringent, until the filament has undergone a significant amount of stretching. Moreover, in the case of the local approximation, the total arclength has a square root cusp when the first singularity appears. In order to simulate the inter- nal motion of the core, one can try to replace one filament by a bundle of parallel filaments. The results of the simulation look very similar to the results of the initial Biot-Savart calculation, thus supporting the validity of our results.

If the collapse does proceed at least partway, according to the Biot-Savart approximation for an inviscid fluid, it is not obvious that the effect of viscosity can prevent it [2]. The viscosity, which is commonly believed to wipe out any perturbation smaller than a 'dissipation scale' can play a sub- tler role in this problem. A naive equation for the core size in the presence of viscosity is:

do 2 d In [ds /dO I d---T = v - o 2 dt (3)

The first term in the right-hand side of (3) represents fattening of the core due to viscous diffusion. The other term is due to the stretching;

74 A. Pumir / Vortex dynamics and singularities in the fluid equations

a crude estimate of it leads to

0' 2 d in [ d s / d O I dt - rf(0", p) ,

where f is a dimensionless function of o and p. As v and F have the same dimension and the ratios between the different lengths do not vary much, (3) can be approximated by d o 2 / d t = v -

F × cst. On the basis of this naive argument, the viscosity may not be able to prevent the collapse. A more rigorous analysis of the strained 2-dimen- sional Navier-Stokes equations confirms that the usual Laplacian dissipation is only marginally able to control the stretching.

A complementary numerical simulation of the full fluid equations allows one to identify different ways the core can be destroyed. When e is not small enough shocks may appear on the tube, or the core can be deformed into thin sheets, that roll-up. Such mechanisms are the most likely im- pediments to the collapse [3].

In any case we have worked out a situation where some strong stretching effects do occur. The

relevance of our particular flow to the experimen- tal situation is rather unclear, but this study sheds some light on an important feature of 3-dimen- sional, incompressible hydrodynamics.

Acknowledgements

It is a pleasure to thank E.D. Siggia for the collaboration which has lead to this study. I am also indebted to R. Kerr for having made his code available to us, and for his collaboration to parts of this work. I also want to thank the CNLS of the Los Alamos National Laboratory for its hospital- ity. I have been supported by the CNRS, France, and by the Department of Energy Grant No. DE-AC02-83ER13044.

References

[1] E.D. Siggia, Phys. Fluids 28 (1985) 794. [2] E.D. Siggia and A. Pumir, Phys. Rev. Lett. 55 (1985) 1749. [3] In preparation.