Elastic turbulence in von Karman swirling flow between two disks

We discuss the role of elastic stress in the statistical properties of elastic turbulence, realized by the flow of a polymer solution between two disks. The dynamics of the elastic stress are analogous to those of a small-scale fast dynamo in magnetohydrodynamics, and to those of the turbulent advection of a passive scalar in the Batchelor regime. Both systems are theoretically studied in the literature, and this analogy is exploited to explain the statistical properties, the flow structure, and the scaling observed experimentally. The following features of elastic turbulence are confirmed experimentally and presented in this paper: (i) The rms of the vorticity (and that of velocity gradients) saturates in the bulk of the elastic turbulent flow, leading to the saturation of the elastic stress. (ii) The rms of the velocity gradients (and thus the elastic stress) grows linearly with Wi in the boundary layer, near the driving disk. The rms of the velocity gradients in the boundary layer is one to two orders ...

[1]  G. Batchelor.  Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity , 1959, Journal of Fluid Mechanics.

[2]  P. Gennes.  Coil-stretch transition of dilute flexible polymers under ultrahigh velocity gradients , 1974 .

[3]  R. Bird.  Dynamics of Polymeric Liquids , 1977 .

[4]  J. Stokes.  Swirling flow of viscoelastic fluids , 1998 .

[5]  R. Armstrong,et al.  Observations on the elastic instability in cone-and-plate and parallel-plate flows of a polyisobutylene Boger fluid , 1991 .

[6]  Turbulent decay of a passive scalar in the Batchelor limit: Exact results from a quantum-mechanical approach , 1998, physics/9806047.

[7]  T. Burghelea,et al.  Statistics of particle pair separations in the elastic turbulent flow of a dilute polymer solution , 2004 .

[8]  T. Sridhar,et al.  A filament stretching device for measurement of extensional viscosity , 1993 .

[9]  A. Groisman,et al.  Mechanism of elastic instability in Couette flow of polymer solutions: Experiment , 1998 .

[10]  Passive scalar evolution in peripheral regions. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  J. Lumley.  ON THE SOLUTION OF EQUATIONS DESCRIBING SMALL SCALE DEFORMATION , 1972 .

[12]  N. Lawson,et al.  Swirling flow of viscoelastic fluids. Part 1. Interaction between inertia and elasticity , 2001, Journal of Fluid Mechanics.

[13]  V. Steinberg,et al.  Single-polymer dynamics: Coil-stretch transition in a random flow , 2004, nlin/0404045.

[14]  Wim van Saarloos,et al.  Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids. , 2004, Physical review letters.

[15]  J. Pinton,et al.  Characterization of Turbulence in a Closed Flow , 1997 .

[16]  M. Chertkov,et al.  Polymer stretching by turbulence , 1999, Physical review letters.

[17]  Victor Steinberg,et al.  Efficient mixing at low Reynolds numbers using polymer additives , 2001, Nature.

[18]  R. Larson,et al.  A transition occurring in ideal elastic liquids during shear flow , 1988 .

[19]  V. Lebedev,et al.  Spectra of turbulence in dilute polymer solutions , 2002, nlin/0207008.

[20]  Ronald G. Larson,et al.  A purely elastic transition in Taylor-Couette flow , 1989 .

[21]  L. Taillefer,et al.  New features in the vortex phase diagram of YBa2Cu3O7- delta , 1997 .

[22]  T. Burghelea,et al.  Mixing by polymers: experimental test of decay regime of mixing. , 2004, Physical review letters.

[23]  R. Larson.  Instabilities in viscoelastic flows , 1992 .

[24]  D. Tritton,et al.  Physical Fluid Dynamics , 1977 .

[25]  Imperial College London,et al.  Simulations of the Small-Scale Turbulent Dynamo , 2003, astro-ph/0312046.

[26]  A. Fouxon,et al.  Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[27]  M. Chertkov,et al.  INTERMITTENT DISSIPATION OF A PASSIVE SCALAR IN TURBULENCE , 1998 .

[28]  Decay of scalar turbulence revisited. , 2002, Physical review letters.

[29]  Eric S. G. Shaqfeh,et al.  Purely elastic instabilities in viscometric flows , 1996 .

[30]  T. Burghelea,et al.  Validity of the Taylor hypothesis in a random spatially smooth flow , 2005 .

[31]  Lebedev,et al.  Statistics of a passive scalar advected by a large-scale two-dimensional velocity field: Analytic solution. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[32]  Victor Steinberg,et al.  Chaotic flow and efficient mixing in a microchannel with a polymer solution. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  Gareth H. McKinley,et al.  Spiral instabilities in the flow of highly elastic fluids between rotating parallel disks , 1994, Journal of Fluid Mechanics.

[34]  O. Cadot,et al.  The statistics of power injected in a closed turbulent flow: Constant torque forcing versus constant velocity forcing , 2003 .

[35]  E. M. Lifshitz,et al.  CHAPTER I – IDEAL FLUIDS , 1959 .

[36]  Ronald G. Larson,et al.  A purely elastic instability in Taylor–Couette flow , 1990, Journal of Fluid Mechanics.

[37]  Robert A. Brown,et al.  Instability of a viscoelastic fluid between rotating parallel disks: analysis for the Oldroyd-B fluid , 1993, Journal of Fluid Mechanics.

[38]  Victor Steinberg,et al.  Elastic turbulence in curvilinear flows of polymer solutions , 2004, nlin/0401006.

[39]  V. Lebedev,et al.  Turbulence of polymer solutions. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  Turbulent dynamics of polymer solutions , 1999, Physical review letters.

[41]  A. Groisman,et al.  Elastic turbulence in a polymer solution flow , 2000, Nature.

[42]  S. Owley,et al.  Simulations of the Small-scale Turbulent Dynamo , 2008 .

[43]  Elastic vs. inertial instability in a polymer solution flow , 1998 .