A method for quantitative estimation of density variation in high-speed flow, which uses light as an interrogating tool, is described. The wavefront distortion of the interrogating beam induced by the compressible flow field is estimated quantitatively, in which the density gradient of the flow field is seen as refractive-index gradient by the probing beam. The distorted wavefront is measured quantitatively by using the cross-sectional intensities of the distorted wavefront along the optical axis. Iterative algorithms are developed using both deterministic (Gauss-Newton) and stochastic (ensemble Kalman filter) update strategies to recover unknown parameters such as the phase of the wavefront or the refractive index distribution in the flow directly from the measured intensities. With phase recovered in the first step, a ray tomography algorithm is used to obtain the refractive index and density distributions in the flow from the phase. Experiments are conducted to quantitatively visualize the shock-wave-induced flow field in a shock-tunnel facility. The reconstructed density cross sections, obtained using different reconstruction methods, are presented and compared with those obtained by solving the Navier-Stokes equation using computational fluid dynamic routines. It is observed that the iterative algorithms always outperform those depending on solution of the transport-of-intensity equation. In particular, when using the iterative algorithms, the stochastic search scheme outperforms the Gauss-Newton method.

A simple noninterferometric optical probe is developed to estimate wavefront distortion suffered by a plane wave in its passage through density variations in a hypersonic flow obstructed by a test model in a typical shock tunnel. The probe has a plane light wave trans-illuminating the flow and casting a shadow of a continuous-tone sinusoidal grating. Through a geometrical optics, eikonal approximation to the distorted wavefront, a bilinear approximation to it is related to the location-dependent shift (distortion) suffered by the grating, which can be read out space-continuously from the projected grating image. The processing of the grating shadow is done through an efficient Fourier fringe analysis scheme, either with a windowed or global Fourier transform (WFT and FT). For comparison, wavefront slopes are also estimated from shadows of random-dot patterns, processed through cross correlation. The measured slopes are suitably unwrapped by using a discrete cosine transform (DCT)-based phase unwrapping procedure, and also through iterative procedures. The unwrapped phase information is used in an iterative scheme, for a full quantitative recovery of density distribution in the shock around the model, through refraction tomographic inversion. Hypersonic flow field parameters around a missile-shaped body at a free-stream Mach number of ∼8 measured using this technique are compared with the numerically estimated values. It is shown that, while processing a wavefront with small space-bandwidth product (SBP) the FT inversion gave accurate results with computational efficiency; computation-intensive WFT was needed for similar results when dealing with larger SBP wavefronts.