On the Lighthill relationship and sound generation from isotropic turbulence

In 1952, Lighthill (Proc. Roy. Soc., A211, (1952)) developed a theory for determining the sound generated by a turbulent motion of a fluid. With some statistical assumptions, Proudman (Proc. Roy. Soc., A211, (1952)) applied this theory to estimate the acoustic power of isotropic turbulence. Recently, Lighthill established a simple relationship that relates the fourth-order retarded time and space covariance of his stress tensor to the corresponding second-order covariance and the turbulent flatness factor, without making statistical assumptions for a homogeneous turbulence. Lilley (Submitted to Theoretical and Computational Fluid Dynamics, 1993) revisited the Proudman''s work and applied the Lighthill relationship to directly evaluate the radiated acoustic power from isotropic turbulence. After choosing the time separation dependence in the two-point velocity time and space covariance based on the insights gained from direct numerical simulations, Lilley concluded that the Proudman constant is determined by the turbulent flatness factor and the second-order spatial velocity covariance. In order to estimate the Proudman constant at high Reynolds numbers, we analyzed a unique data set of measurements in a large wind tunnel and atmospheric surface layer that covers a range of the Taylor microscale based Reynolds number 2.0 X 10^3 = R = 12.7 X 10^3. Our measurements demonstrate that the Lighthill relationship is a good approximation, providing additional support to Lilley''s approach. The flatness factor is found between 2.7-3.3 and the second order spatial velocity covariance is obtained. Based on these experimental data, the Proudman constant is estimated to be 0.68-3.68.