The role of the spanwise correlation length in low-frequency sound radiation by turbulence/surface interaction in circular jets

The interaction between non-homogeneous turbulence carried by a non-planar mean field and the discontinuity created by an impermeable solid surface embedded within the flow causes an $O(1)$ increase in the low frequency sound above the background jet noise radiated by the turbulence itself. When the solid surfaces $S(\boldsymbol{y})$ are placed parallel to the level curves of the streamwise mean flow, $U(y_2,y_3)=const.$, generalized Rapid distortion theory (Ref) shows that the far-field pressure fluctuation, $p^\prime(\boldsymbol{x},t)$, at the spacetime field point $(\boldsymbol{x},t) = ({x}_1,{x}_2,{x}_3,t) $ in a three-dimensional Cartesian co-ordinate system with origin at the trailing edge, can be expressed in terms of the Green's function of the adjoint Rayleigh equation and a convected scalar field, $\tilde{\omega}_c$, defined by $D\tilde{\omega}_c /Dt = 0$ where$D /Dt \equiv \partial/\partial t + U(x_2, x_3) \partial/\partial x_1$ is the material derivative. Since this latter quantity is an arbitrary function of its arguments, its space-time spectrum ($\bar{S}$ below) can be related to the appropriate measured turbulence correlation function in the upstream undisturbed flow and used as the boundary condition to determine the downstream acoustic field. In reference 1\footnote{M. E. Goldstein, S. J. Leib \& M. Z. Afsar, {Rapid distortion theory on transversely sheared mean flows of arbitrary cross-section}, \textit{J. Fluid Mech.} (2019), vol. 881, pp. 551–584.}; it is shown that the acoustic spectrum ($I_\omega$) takes the form, $I_\omega \sim \int_u \int_{\tilde {u}} D(u,\tilde{u}) \bar{S}(u, \tilde{u};\omega) du d\tilde{u}$, where $D(u,\tilde{u})$ is a directivity factor and $u$ is the real part of a conformal mapping that transforms the nonrectangular cross sectional shape of an axisymmetric jet close to an external surface into a rectangular one for the subsequent Wiener-Hopf technique calculation. Our main contribution in this paper is to show that the low frequency decay of $I_\omega (\boldsymbol{x})$ is approximately the same as that obtained by taking $l_3 \rightarrow \infty$. The latter limit allows $\bar{S}$ to take a reduced analytical form. The large-$l_3$ limit corresponds to the allowing the spanwise correlation length to be infinite and therefore that the spectral function $\bar{S}$ is independent of spatial separation in the spanwise (or cross-stream) $3$-direction. We assess the importance of the spanwise correlation length in controlling the low frequency roll-off of $I_\omega (\boldsymbol{x})$ (i.e. its asymptotic value when the angular frequency, $\omega$, goes to zero starting from the peak sound. Our calculations below show that the maximum value and spatial structure of the integrand, $D(u,\tilde{u}) \bar{S}(u, \tilde{u};\omega)$ when $l_3 \rightarrow\infty$ (left figure) is almost identical to the $l_3 = O(1)$ case (right) at very low frequencies. The accompanying talk will discuss whether the structure of $\bar{S}$ at $l_3\rightarrow \infty$ can be used to approximate $I_\omega$ at more $O(1)$ values of spanwise turbulence length scale, $l_3$.