Nonlinear interactions in a rotating disk flow: From a Volterra model to the Ginzburg-Landau equation.

The physical system under consideration is the flow above a rotating disk and its cross-flow instability, which is a typical route to turbulence in three-dimensional boundary layers. Our aim is to study the nonlinear properties of the wavefield through a Volterra series equation. The kernels of the Volterra expansion, which contain relevant physical information about the system, are estimated by fitting two-point measurements via a nonlinear parametric model. We then consider describing the wavefield with the complex Ginzburg-Landau equation, and derive analytical relations which express the coefficients of the Ginzburg-Landau equation in terms of the kernels of the Volterra expansion. These relations must hold for a large class of weakly nonlinear systems, in fluid as well as in plasma physics. (c) 2000 American Institute of Physics.

[1]  N. Wiener,et al.  Nonlinear Problems in Random Theory , 1964 .

[2]  Petre Stoica,et al.  Decentralized Control , 2018, The Control Systems Handbook.

[3]  M. Rosenblatt,et al.  ESTIMATING THREE‐DIMENSIONAL ENERGY TRANSFER IN ISOTROPIC TURBULENCE* , 1982 .

[4]  Y. Kohama,et al.  Spiral vortices in boundary layer transition regime on a rotating disk , 1980 .

[5]  Mujeeb R. Malik,et al.  The neutral curve for stationary disturbances in rotating-disk flow , 1986, Journal of Fluid Mechanics.

[6]  Sheng Chen,et al.  Orthogonal least squares methods and their application to non-linear system identification , 1989 .

[7]  H. Moon,et al.  Transitions to chaos in the Ginzburg-Landau equation , 1983 .

[8]  M. B. Priestley,et al.  Non-linear and non-stationary time series analysis , 1990 .

[9]  R. Deissler.  Spatially growing waves, intermittency, and convective chaos in an open-flow system , 1987 .

[10]  A. Faller,et al.  A Numerical Study of the Instability of the Laminar Ekman Boundary Layer , 1966 .

[11]  Lennart Ljung,et al.  Nonlinear black-box modeling in system identification: a unified overview , 1995, Autom..

[12]  A. Dinklage,et al.  Numerical investigations on strong pattern selecting Eckhaus instabilities in neon glow discharges , 2000 .

[13]  D. Brillinger.  The identification of polynomial systems by means of higher order spectra , 1970 .

[14]  N. Aubry,et al.  Transition to turbulence on a rotating flat disk , 1994 .

[15]  M. Schetzen.  The Volterra and Wiener Theories of Nonlinear Systems , 1980 .

[16]  P. L. Gal,et al.  Experimental study of rotating disk instability. I. Natural flow , 1996 .

[17]  R. Lingwood,et al.  An experimental study of absolute instability of the rotating-disk boundary-layer flow , 1996, Journal of Fluid Mechanics.

[18]  Jerry P. Gollub,et al.  OSCILLATIONS AND SPATIOTEMPORAL CHAOS OF ONE-DIMENSIONAL FLUID FRONTS , 1997 .

[19]  Vincent Croquette,et al.  Nonlinear waves of the oscillatory instability on finite convective rolls , 1989 .

[20]  Vagn Walfrid Ekman,et al.  On the influence of the earth's rotation on ocean-currents. , 1905 .

[21]  P. L. Gal,et al.  BIORTHOGONAL DECOMPOSITION ANALYSIS AND RECONSTRUCTION OF SPATIOTEMPORAL CHAOS GENERATED BY COUPLED WAKES , 1998 .

[22]  Edward J. Powers,et al.  Experimental measurement of three‐wave coupling and energy cascading , 1989 .

[23]  N. Gregory,et al.  On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk , 1955, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[24]  Heinz Unbehauen,et al.  Structure identification of nonlinear dynamic systems - A survey on input/output approaches , 1990, Autom..

[25]  R. Lingwood,et al.  Absolute instability of the Ekman layer and related rotating flows , 1997, Journal of Fluid Mechanics.

[26]  Stephen A. Billings,et al.  Identi cation of nonlinear systems-A survey , 1980 .

[27]  T. Leweke,et al.  Determination of the parameters of the Ginzburg-Landau wake model from experiments on a bluff ring , 1994 .

[28]  T. Dudok de Wit,et al.  Identifying nonlinear wave interactions in plasmas using two-point measurements: A case study of Short Large Amplitude Magnetic Structures (SLAMS) , 1999 .

[29]  Sang-Won Nam,et al.  Application of higher order spectral analysis to cubically nonlinear system identification , 1994, IEEE Trans. Signal Process..

[30]  Michael Gaster,et al.  A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability , 1962, Journal of Fluid Mechanics.

[31]  P. L. Gal.  Complex demodulation applied to the transition to turbulence of the flow over a rotating disk , 1992 .

[32]  T. Kármán.  Über laminare und turbulente Reibung , 1921 .

[33]  P. L. Gal,et al.  Experimental study of rotating disk flow instability. II. Forced flow , 1996 .

[34]  St'ephane Zaleski,et al.  Identification of parameters in amplitude equations describing coupled wakes , 1996, chao-dyn/9601008.

[35]  Alexander M. Rubenchik,et al.  Hamiltonian approach to the description of non-linear plasma phenomena , 1985 .

[36]  A. Faller.  Instability and transition of disturbed flow over a rotating disk , 1991, Journal of Fluid Mechanics.

[37]  Paul Manneville,et al.  Dissipative Structures and Weak Turbulence , 1995 .

[38]  Nadine Aubry,et al.  Spatiotemporal analysis of complex signals: Theory and applications , 1991 .