A simple model to estimate turbulent density fluctuation and associated optical distortion over hydro-dynamically rough surfaces

Abstract

Here we develop an analysis to estimate root-mean-square (RMS) turbulent density fluctuation over hydro-dynamically rough plates and cones for compressible flows. This information is then used to estimate standard aero-optical quantities such as the mean square random phase error and the Strehl ratio. To compute the density fluctuation, a closed-form analytical model has been developed using classical (inner-law) boundary layer results. The model estimates local compressible (adiabatic) skin friction, equivalent (Van Driest) log-law velocity profiles and boundary layer thickness and, via the Crocco–Busemann energy integral (assuming for simplicity adiabatic conditions) temperature fluctuations. Finally, using state and appropriate closures for the turbulent pressure fluctuation and Reynolds stresses, we arrive at an RMS turbulent density fluctuation result. This result is compared to experimental and semi-empirical models and shows reasonable agreement. The density fluctuation is of particular interest since it can be directly related to estimate the local refractive index associated with the flow through the Gladstone–Dale constant. The refractive index field provides the basis for aero-optical modeling of a particular system, with a particular focus on beam path bending, the mean square random phase error and the related Strehl ratio.