Boiling flow parameter estimation from boundary layer data

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.

💡 Research Summary

The paper addresses the challenge of generating realistic synthetic phase‑aberration data for aero‑optic applications, where atmospheric turbulence and high‑speed flow around aircraft induce complex refractive‑index fluctuations. Traditional simulation tools rely on the Kolmogorov turbulence model and the Taylor frozen‑flow hypothesis, requiring physical parameters such as the outer scale L₀, Fried coherence length r₀, wind components (vₓ, vᵧ), and a “boiling coefficient” α. Because aero‑optic phenomena do not obey the isotropic Kolmogorov assumptions, these parameters are ill‑defined, making it difficult to configure the widely used boiling‑flow algorithm for realistic aero‑optic phase screens.

The authors propose a data‑driven method to estimate the five boiling‑flow parameters directly from measured turbulent‑boundary‑layer (TBL) phase‑aberration data. Their approach treats L₀ and r₀ not as strict physical quantities but as statistical descriptors of the measured spatial power spectral density (PSD). L₀ is set to the physical size of the measurement aperture (N·δ), avoiding the numerical singularity at zero frequency that occurs when L₀ → ∞. r₀ is obtained by computing a two‑dimensional PSD for each frame using an extended Welch method (mean removal, Hamming window, FFT) and fitting the resulting spectrum to the von Kármán PSD (the Kolmogorov form with an outer‑scale term).

Wind velocities (vₓ, vᵧ) and the boiling coefficient α are inferred from the temporal power spectral density (TPSD) of the phase‑gradient (slope) fields. The measured phase series is first differentiated in time to obtain slope images, then the TPSD of these slopes is calculated. In the boiling‑flow model, the TPSD exhibits peaks whose locations correspond to the advection speed (wind) and whose widths are governed by α. The authors formulate a joint nonlinear optimization that simultaneously aligns the peak locations with the wind components and matches the peak widths to determine α, extending the search range up to 200 Hz to accommodate the broader bandwidth of aero‑optic disturbances. This automated fitting avoids the manual tuning required by earlier peak‑fitter algorithms.

With the estimated parameters, synthetic phase screens are generated by the boiling‑flow recursion: each new screen is a weighted sum of a shifted previous screen (frozen‑flow term) and a fresh Kolmogorov screen (random boiling term). The implementation includes two practical enhancements: (1) generation of oversized screens to prevent phase‑wrapping artifacts caused by FFT‑based frequency shifts, followed by cropping to the desired aperture; (2) removal of tilt, tip, and piston from each screen to mimic the preprocessing applied to experimental aero‑optic data.

The authors evaluate the method on two TBL data sets. For the temporal statistics, the TPSD of the synthetic slopes matches the measured TPSD with an average relative error of 8–9 %, indicating that the estimated wind and α capture the dynamics of the highly convective flow. However, when comparing spatial statistics, the Kolmogorov structure function of the synthetic screens deviates from the measured one by more than 28 %. This discrepancy arises because the boiling‑flow algorithm preserves the Kolmogorov PSD at every time step, whereas aero‑optic phase aberrations exhibit non‑Kolmogorov, anisotropic spatial characteristics that the model cannot reproduce.

The study concludes that while boiling‑flow parameters can be efficiently inferred to reproduce temporal behavior of aero‑optic turbulence, the underlying model’s assumption of a Kolmogorov spatial spectrum limits its ability to capture the true spatial complexity of boundary‑layer phase aberrations. Future work is suggested to incorporate non‑Kolmogorov spectra, multi‑layer wind models, or alternative stochastic processes to improve spatial fidelity. Nonetheless, the presented parameter‑estimation framework offers a computationally inexpensive, largely automated tool for generating realistic time‑evolving phase screens, which could be valuable for system design, performance prediction, and algorithm testing in aero‑optic contexts.