Boiling flow parameter estimation from boundary layer data

TL;DR

This work addresses aero-optics phase aberrations by proposing a data-driven method to estimate Boiling Flow parameters from boundary-layer phase data. It fits spatial statistics through $(L_0,r_0)$ and temporal statistics through $(v_x,v_y,lpha)$ to measured phase data, producing phase screens that closely match the temporal dynamics of phase slopes with typical errors around 8–9%. However, the method struggles to reproduce the spatial statistics, with Kolmogorov-based structure functions showing substantial mismatches (errors >28%), indicating the need for a more general spatial model for aero-optic phase screens. The approach is computationally efficient and validated on two turbulent boundary layer data sets, offering a practical path to aero-optics data-driven simulations while highlighting limits in current spatial-statistics modeling for anisotropic aero-optic effects.

Abstract

Atmospheric turbulence and aero-optic effects cause phase aberrations in propagating light waves, thereby reducing effectiveness in transmitting and receiving coherent light from an aircraft. Existing optical sensors can measure the resulting phase aberrations, but the physical experiments required to induce these aberrations are expensive and time-intensive. Simulation methods could provide a less expensive alternative. For example, an existing simulation algorithm called boiling flow, which generalizes the Taylor frozen-flow method, can generate synthetic phase aberration data (i.e., phase screens) induced by atmospheric turbulence. However, boiling flow depends on physical parameters, such as the Fried coherence length r0, which are not well-defined for aero-optic effects. In this paper, we introduce a method to estimate the parameters of boiling flow from measured aero-optic phase aberration data. Our algorithm estimates these parameters to fit the spatial and temporal statistics of the measured data. This method is computationally efficient and our experiments show that the temporal power spectral density of the slopes of the synthetic phase screens reasonably matches that of the measured phase aberrations from two turbulent boundary layer data sets, with errors between 8-9%. However, the Kolmogorov spatial structure function of the phase screens does not match that of the measured phase aberrations, with errors above 28%. This suggests that, while the parameters of boiling flow can reasonably fit the temporal statistics of highly convective data, they cannot fit the complex spatial statistics of aero-optic phase aberrations.

Figures (3)

• Figure 1: This figure shows side-by-side images from the measured phase aberration data (left) and synthetic phase screens (right). Figure \ref{['fig: F06 Image']} shows images for data set F06 and Fig. \ref{['fig: F12 Image']} shows images for F12. The x- and y-axes show $x/d$ and $y/d$ (respectively), where $d$ is a sub-aperture spacing; $x/d$ and $y/d$ indicate the pixel index along each axis. The pixel values in each image are the phase values $\phi(x/d,y/d)$, quantized according to the color-bar on the right of each image pair. These images illustrate that boiling flow can generate phase screens on the same scale as the measured aero-optic phase aberrations and the images show close resemblance.
• Figure 2: This figure shows images of the anisotropic structure function of the measured phase aberration data (left) and synthetic phase screens (right). Figure \ref{['fig: F06 Structure Function']} shows the results for data set F06 and Fig. \ref{['fig: F12 Structure Function']} shows the results for F12. The inputs to the structure function are $(x/d, y/d)$, where $d$ is the sub-aperture spacing; these input values are then the pixel distances between two grid locations. The structure function values are interpolated (using bi-linear interpolation) at the center of each pixel; the images are quantized according the color-bars on the right of each image. These results show that the contours of the structure function of the phase screens are circles, while the contours of the measured data's structure function are ellipses. Therefore, boiling flow does not accurately match the spatial statistics of measured aero-optic phase aberrations.
• Figure 3: This figure shows temporal power spectral density (TPSD) plots of the measured data (blue) and synthetic data (orange). We show the TPSD of both the deflection angle in streamwise direction of the flow, $\theta_x$ (left), and the phase aberrations $\phi$ (right). Figures \ref{['fig: F06 TPSD of Deflection Angle']} and \ref{['fig: F06 TPSD of Phase']} show the TPSD of $\theta_x$ and $\phi$ for data set F06, while Figs. \ref{['fig: F12 TPSD of Deflection Angle']} and \ref{['fig: F12 TPSD of Phase']} show the TPSD of $\theta_x$ and $\phi$ for data set F12 (respectively). Temporal frequency $f$ (Hz) is plotted on the $x$-axis and the TPSD value $S(f)$ (energy/sec) is plotted on the $y$-axis. These plots illustrate that boiling flow reasonably matches the TPSD of $\theta_x$ at all frequencies and closely matches the TPSD of $\phi$ at frequencies above 20 kHz. However, it does match the TPSD of $\phi$ at frequencies below 20 kHz, nor does it match the lowest frequencies of either TPSD calculation. Thus, while our algorithm's direct estimation of the flow velocities $(v_x, v_y)$ fits the temporal statistics of the slopes of convective data, the three parameters $(v_x, v_y, \alpha)$ are unable to model the long-range temporal statistics of the measured phase aberrations.