Particle and Field Descriptions of Fluid Motion

Fluctuations: Turbulent flows contain fluctuations in the dependent-field quantities (velocity, pressure, temperature, etc.) even when the flow's boundary conditions are steady. Turbulent fluctuations appear to be irregular, chaotic, and unpredictable

Nonlinearity: The momentum conservation equation contains the nonlinear advective-acceleration term, and even in ideal flows this nonlinearity causes pressure to depend on the square of the velocity. Turbulence represents an even further assertion of this nonlinearity, and occurs when the relevant nonlinearity parameter — the Reynolds number \( \mathrm{Re} \), the Rayleigh number \( \mathrm{Ra} \), or the inverse Richardson number \( \mathrm{Ri}^{-1} \) — comfortably exceeds a critical value. The enhanced nonlinearity of turbulence is evident in vortex stretching

Vorticity:Turbulence is characterized by fluctuating vorticity. A cross-section view of a turbulent flow typically appears as a diverse collection of streaks, strain regions, and swirls of various sizes that deform, coalesce, divide, and spin. Identifiable structures in a turbulent flow, particularly those that spin, are called eddies. Turbulence always involves a range of eddy sizes and the size range increases with \( \mathrm{Re}^{3/4} \). The characteristic size of the largest eddies is the width of the turbulent region; in a turbulent boundary layer this is the thickness of the layer

Dissipation: On average, the vortex stretching mechanism transfers fluctuation energy and vorticity to smaller and smaller eddies via nonlinear interactions, until velocity gradients become so large that fluctuation energy is converted into heat (i.e., dissipated) by the action of viscosity and the motions of the smallest eddies. Persistent turbulence therefore requires a continuous supply of energy from an imposed velocity or pressure difference to make up for this energy loss

Diffusivity: Within a turbulent flow, the prevalence of fluctuations and vortical overturning motions leads to mixing and diffusion of chemical species, momentum, and heat that is orders of magnitude faster than molecular transport in equivalent laminar flows that lack such fluctuations and vortical motions