Two-point velocity average of turbulence: statistics and their implications

TL;DR

This paper investigates the statistical properties of the two-point velocity average in turbulence, revealing its scale-dependent behavior and implications for understanding energy distribution across scales.

Contribution

It introduces the analysis of the velocity average in turbulence, showing its scale-dependent features and deriving an exact energy budget equation for it.

Findings

Velocity average satisfies an exact scale-by-scale energy budget.

Flatness factor of velocity average varies universally with scale.

Velocity average is not independent of scale, contradicting previous assumptions.

Abstract

For turbulence, although the two-point velocity difference u(x+r)-u(x) at each scale r has been studied in detail, the velocity average [u(x+r)+u(x)]/2 has not thus far. Theoretically or experimentally, we find interesting features of the velocity average. It satisfies an exact scale-by-scale energy budget equation. The flatness factor varies with the scale r in a universal manner. These features are not consistent with the existing assumption that the velocity average is independent of r and represents energy-containing large-scale motions alone. We accordingly propose that it represents motions over scales >= r as long as the velocity difference represents motions at the scale r.