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 (=ρ_χ 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 (ρ_χL/ρ_mh))

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