Analysis of turbulence by shadowgraph

Abstract

Shadowgraphs of turbulent flows are often employed to determine the statistical structure of the density fluctuations. Previous solutions for the optical response have employed geometric optics and the assumption that the flow thickness is small compared to its distance to the shadowgraph plate. In this paper the electromagnetic equations for the shadowgraph response are solved without the above restrictions and a response function is determined for arbitrary shadowgraph geometry. HE shadowgraph method has been extensively employed in the analysis of the turbulent wakes of models of hypersonic vehicles. These analyses13 have been based upon the work of Uberoi and Kovasznay4 who determined the optical response equation for the autocorrelation of irradiance (intensity) fluctuations of a plane optical wave traversing a turbulent dielectric. Uberoi and Kovasznay based their method upon a formula first derived by Weyl. 5 The Weyl formula is based upon the use of geometric optics and assumes that the region of dielectric turbulence is thin compared to its distance to the shadowgraph plate. This assumption is not in accord with practice. The assumption of a thin turbulent region was used by Weyl in the following sense: the individual rays follow straightline paths through the turbulent slab, but leave the region at angles determined by the line integral of the derivative of refractive index fluctuation through the medium. Irradiance fluctuations now occur because of the subsequent divergence of ray tubes outside the medium. The autocorrelation of the irradiance fluctuation is then easily shown to be proportional to the slab thickness times the square of the distance to the photographic plate. (It is also assumed that the distance to the plate is such that no caustics are formed. This assumption is discussed analytically in an appendix.) The Weyl procedure obviously does not account for fluctuations of the ray tube diameters within the turbulent medium. It is known6 in fact that within a turbulent medium the variance of irradiation fluctuations varies as the cube of the distance in the approximation of geometrical optics. In this paper, we provide a rigorous treatment beginning with the wave equation for the case of plane wave incident upon a plane slab of turbulent dielectric of arbitrary thickness d and obtain the autocorrelation of irradiation at a distance D > d from the front face. Using the Born approximation we obtain the result that the autocorrelation of irradiation is the sum of the autocorrelation of irradiation within the slab plus the autocorrelation due to ray tube fluctuations effected in back of the slab due to differences in angle of departure. The latter term, however, must be multiplied by a geometric factor, (D — d)/D, which accounts for slab thickness.