High-dimensional chaos can lead to weak turbulence

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 (=Nfx^paa²/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.

Duke Scholars

Author Lawrence N. Virgin Thomas Lord Department of Mechanical Engineering and Materia ...

Author Earl H. Dowell Thomas Lord Department of Mechanical Engineering and Materia ...