Density reconstruction from multi-directional quantitative Schlieren imaging using Poisson formulation

Abstract

This study introduces a framework for reconstructing steady effective line-of-sight-averaged density fields in compressible flows using multi-directional quantitative Schlieren imaging. The experimental setup utilizes a quantitatively calibrated monochrome weak-lens Schlieren configuration to acquire directional density gradient projections across four unique knife-edge orientations (0°, 41°, 93°, and −27°), achieving a high signal-to-noise ratio without requiring specialized optics. To solve the overdetermined inverse problem, the individual directional measurements are fused via least-squares estimation to recover the complete spatial gradient components (p=∂ρ/∂x, q=∂ρ/∂y). A Poisson equation ( ∇2ρ=f) is then formulated from the divergence of the estimated gradient and solved using the finite element method. The solution is constrained by mixed boundary conditions: homogeneous Neumann conditions upstream, strong Dirichlet conditions downstream, and sparse reference density values enforced as weak Dirichlet anchor points derived from static pressure measurements on the surfaces of the test geometry. A tunable regularization parameter ( λ) governs the enforcement strength of these anchor points, providing structural resilience against high noise levels and data incompleteness. Synthetic benchmarking on a known scalar field validated the algorithm’s robustness, demonstrating that high reconstruction accuracy is maintained even under severe noise contamination and sparse anchor distributions, with the optimal regularization parameter shifting toward smaller values as noise levels increase. The framework was applied to a transonic turbine cascade, where the reconstructed field recovered the principal aerodynamic features of the passage flow. Scalar-field accuracy was evaluated separately using synthetic benchmarks with known ground truth. The proposed methodology offers a scalable, versatile tool for quantitative density field visualization across a broad spectrum of flows containing significant pressure or temperature gradients.