Inertial Ranges in Two-Dimensional Turbulence
Abstract
Two-dimensional turbulence has both kinetic energy and mean-square vorticity as inviscid constants of motion. Consequently it admits two formal inertial ranges, E(k) is approx. (epsilon to the 2/3 power) k to the -5/3 power and E(k) is approx. (eta to the 2/3 power) k to the -3rd power, where epsilon is the rate of cascade of kinetic energy per unit mass, eta is the rate of cascade of mean-square vorticity, and the kinetic energy per unit mass is the integral from 0 to infinity E(k)dk. The -5/3 range is found to entail backward energy cascade, from higher to lower wavenumbers k, together with zero vorticity flow. The -3 range gives an upward vorticity flow and zero energy flow. The paradox in these results is resolved by the irreducibly triangular nature of the elementary wavenumber-interactions. The formal -3 range gives a nonlocal cascade and consequently must be modified by logarithmic factors. If energy is fed in at a constant rate to a band of wavenumbers approx. k sub i and the Reynolds number is large, it is conjectured that a quasisteady state results with a -5/3 range for k << k sub i and a -3 range for k >> k sub i, up to the viscous cut-off. The total kinetic energy increases steadily with time as the - 5/3 range pushes to ever-lower k, until scales the size of the entire fluid are strongly excited. The rate of energy dissipation by viscosity decreases to zero if kinematic viscosity is decreased to zero with other parameters unchanged.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1967
- Accession Number
- AD0653111
Entities
People
- Robert H. Kraichnan