High-dimensional chaos can lead to weak turbulence

Abstract

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 temporaland (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. a m = modal amplitude coefficients D = panel stiffness (=Eh 212(1−v 2)) E = modulus of elasticity of panel material h = panel thickness k = dimensional foundation stiffness K = nondimensional foundation stiffness (=kL 4/Dh) L = length of panel in direction of flow M = Mach number N = number of modes in series expansion of panel deflection N fv/pa = dimensional applied inplane load Δp = dimensional static pressure differential across panel P = nondimensional static pressure differential across panel (=ΔpL 4/Dh) q = dimensional dynamic pressure (=ρχ U 2/2) R v = nondimensional inplane load (=N fx 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 ( $$\begin{gathered} \hfill \\ \left( { = 2qL^3 /D\sqrt {M^2 - 1} } \right) \hfill \\ \end{gathered} $$ ) 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