Noise from Isotropic Turbulence

Abstract

Turbulence naturally returns to isotropy and often possesses locally isotropic regions. This paper presents a new method to predict noise from turbulence that has returned or is returning to isotropy. The Navier–Stokes equations are decomposed into anisotropic and isotropic turbulent components and corresponding radiating waves. An analytical solution is proposed for the radiating waves associated with isotropic turbulence that contains arguments involving the vector Green’s function of the Navier–Stokes equations and the multi-order two-point cross-correlation of the Navier–Stokes equations involving turbulent fluctuations. Using the theory of isotropic turbulence, the two-point cross-correlation of the Navier–Stokes equations is written as a vector-normalized, two-point cross-correlation multiplied by corresponding wave-number spectra of the structure functions. Composite structure functions of the field variables are adapted from canonical theory of isotropic turbulence. The vector-normalized two-point cross-correlation involves arguments of separation distance, wave number, and corresponding turbulent length and time scales. A solution of the vector Green’s function of the linearized Navier–Stokes equation for acoustic pressure is derived. A simple system of differential and algebraic equations is proposed to model the statistics of stationary and decaying isotropic turbulence, which are arguments of the model equation. Predictions of acoustic and turbulent statistics are compared with a wide variety of measurements and direct numerical simulations from various sources over a range of Reynolds numbers and initial scales. Predictions compare favorably with previous theories, direct numerical simulations, and measurements.