High-dimensional chaos can lead to weak turbulence


Journal Article

One of the outstanding unresolved questions of nonlinear dynamics is the relationship between chaos and turbulence. This is a deep and difficult question, not the least reason being that the definitions of "chaos" and "turbulence" are not universally agreed upon. Here we define chaos as the time history of a single descriptor of a deterministic dynamical system which undergoes a loss of temporal correlation with a change in some system parameter and that displays sensitivity to initial conditions. Turbulence is defined as the time history of the spatial distribution of a deterministic dynamical system which undergoes a loss of temporal and (subsequently) spatial correlation with a change in some system parameter(s). By analogy and numerical simulation it is argued that turbulence can be a consequence of multi-mode interaction of individually chaotic modes. The physical system used here is a fluttering panel in a supersonic airstream. am = modal amplitude coefficients D = panel stiffness (=Eh212(1-v2)) E = modulus of elasticity of panel material h = panel thickness k = dimensional foundation stiffness K = nondimensional foundation stiffness (=kL4/Dh) L = length of panel in direction of flow M = Mach number N = number of modes in series expansion of panel deflection Nfv/pa = dimensional applied inplane load Δp = dimensional static pressure differential across panel P = nondimensional static pressure differential across panel (=ΔpL4/Dh) q = dimensional dynamic pressure (=ρχU2/2) Rv = nondimensional inplane load (=Nfxpaa2/D) t = dimensional time T = period over which correlation is averaged U = dimensional flow velocity w = dimensional panel deflection W = nondimensional panel deflection (deflection/h) x = dimensional coordinate along panel α = inplane spring stiffness parameter λ = nondimensional dynamic pressure of flow over panel ( {Mathematical expression}) μ = mass ratio (ρχL/ρmh)) ν = Poisson's ratio ξ = nondimensional location along panel (x/L) Δξ = separation between points used in correlation function ξu = nondimensional correlation length ψ = correlation function © 1993 Kluwer Academic Publishers.

Full Text

Duke Authors

Cited Authors

  • Reynolds, RR; Virgin, LN; Dowell, EH

Published Date

  • December 1, 1993

Published In

Volume / Issue

  • 4 / 6

Start / End Page

  • 531 - 546

Electronic International Standard Serial Number (EISSN)

  • 1573-269X

International Standard Serial Number (ISSN)

  • 0924-090X

Digital Object Identifier (DOI)

  • 10.1007/BF00162231

Citation Source

  • Scopus