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 wavenumber 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, wavenumber, 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.

Radiation from large-scale structures within a turbulent shear layer

“…The acoustic analogy and acoustic source modeling are partly based on the approach of Miller. 34 The decomposition of the flow field is similar to that of Liu 12 and Morris. 27 Our model of the instability waves is represented by an infinitely differentiable and integrable basis function that can be summed to represent more complex waves.…”

“…The resultant right-hand side is considered as the source of noise and the left-hand side is an operator that propagates acoustic radiation. The right-hand side sources are then transformed using the newly proposed approach of Miller 34 to involve scales of turbulence in time and space. We now have a set of equations that can be solved through a convolution of the vector source term with the vector Green's function of the linearized Navier-Stokes equations.…”

“…As the sound field is dominated by waves from large-scale structures, the refraction effects within the shear layer are small. Miller 34 shows the analytical solution for the linearized Navier-Stokes vector Green's function within a quiescent environment. This assumption implies that the refraction effects upon sound propagation are neglected.…”

“…The resultant right-hand side is considered as the source of noise and the left-hand side is an operator that propagates acoustic radiation. The right-hand side sources are then transformed using the newly proposed approach of Miller 34 to involve scales of turbulence in time and space.…”

“…We find an analytical solution for the acoustic radiation through the use of an acoustic analogy developed by Miller. 34 The arguments of the acoustic analogy involve the two-point cross-correlation of quantities associated with the base flow, instability waves, and spectra of the field variables. The instability waves are modeled with a newly proposed basis function.…”