Wave-number-frequency spectrum for turbulence from a random sweeping hypothesis with mean flow

Area of Science:

• Fluid Dynamics
• Turbulence Theory
• Statistical Mechanics

Background:

• Turbulent flows exhibit complex energy dynamics across various scales.
• Understanding turbulence requires models that capture both spatial and temporal correlations.
• Existing models like Taylor's frozen-in flow and Kraichnan's random sweeping hypothesis offer partial insights.

Purpose of the Study:

• To derive an energy spectrum model for turbulent flows in wave-number-frequency space.
• To incorporate both mean flow and Kraichnan's random sweeping hypothesis into a unified model.
• To provide a tool for distinguishing between Taylor's frozen-in flow and random sweeping effects in experimental data.

Main Methods:

• Derivation of the energy spectrum using Kraichnan's random sweeping hypothesis with added mean flow.
• Multiplication of the instantaneous energy spectrum by a Gaussian frequency distribution.
• Analysis of the model's properties in both wave-number-frequency and real space.

Main Results:

• The derived model yields an energy spectrum in wave-number-frequency space.
• The model incorporates two adjustable parameters: mean flow velocity and sweeping velocity.
• The power-law index of the wave-number spectrum is invariant in the frequency spectrum, independent of velocity choices.

Conclusions:

• The model naturally integrates Taylor's frozen-in flow and random sweeping approximations.
• The model can differentiate between Taylor's frozen-in flow and random sweeping effects in time-resolved turbulence data.
• The study discusses space-time velocity correlations and compares the model to the elliptic model.