Abstract

For almost four decades, Hill’s “Model 4” [J. Fluid Mech. 88, 541 (1978) [CrossRef]  ] has played a central role in research and technology of optical turbulence. Based on Batchelor’s generalized Obukhov–Corrsin theory of scalar turbulence, Hill’s model predicts the dimensionless function $h(\kappa l_0,\mathrm{Pr})$ that appears in Tatarskii’s well-known equation for the 3D refractive-index spectrum in the case of homogeneous and isotropic turbulence, ${\mathrm{\Phi }}_n(\kappa )=0.033C_n^2{\kappa }^{-11/3}h(\kappa l_0,\mathrm{Pr})$. Here we investigate Hill’s model by comparing numerical solutions of Hill’s differential equation with scalar spectra estimated from direct numerical simulation (DNS) output data. Our DNS solves the Navier–Stokes equation for the 3D velocity field and the transport equation for the scalar field on a numerical grid containing ${4096}^3$ grid points. Two independent DNS runs are analyzed: one with the Prandtl number $\mathrm{Pr}=0.7$ and a second run with $\mathrm{Pr}=1.0$. We find very good agreement between $h(\kappa l_0,\mathrm{Pr})$ estimated from the DNS output data and $h(\kappa l_0,\mathrm{Pr})$ predicted by the Hill model. We find that the height of the Hill bump is $1.79{\mathrm{Pr}}^{1/3}$, implying that there is no bump if $\mathrm{Pr}<0.17$. Both the DNS and the Hill model predict that the viscous-diffusive “tail” of $h(\kappa l_0,\mathrm{Pr})$ is exponential, not Gaussian.

© 2015 Optical Society of America

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