Light refraction modeling for computational schlieren imaging in symmetrical flows

Abstract

A new approach is proposed for the calculation of refraction vector as a gradient of the phase shift when testing symmetrical flow with a parallel beam of light. It is assumed that the flow can be characterized by axial, conical or central symmetry. This is observed, for example, at supersonic flows around bodies of revolution. The density field between the bow shock wave and the body can be either calculated from the physics models or determined from the optical testing. The refraction vector modeling is based on polynomial representation of the radial density distributions. The confirmation of such representation is done by analysis of numerical data for various regimes of supersonic flows around bodies of revolution (sharp-tipped and blunt cones, spheres, bodies with ogive nose). Analytical formulas were obtained for refraction vector calculation in axial, conical and central symmetrical flows. For density field around a sphere in a supersonic flow at Mach number M=4, which was calculated from a physics model, refraction vector components were calculated for three different models. In the first model it is assumed that the phase change is taken place only on the bow shock wave. The second model uses a linear polynomial representation of the radial distributions based on density values calculated for the shock wave, symmetry axis and body surface. In the third model we used interpolation of the density values calculated in discrete points and least square method for polynomial representation. Results of the refraction vector calculations are illustrated by a set of graphics. The obtained formulas will be used further for the flow study using computational flow schlieren imaging technology. A recommendation is given on application of the formulas for this purpose.