On complex-valued 2D eikonals. Part four: continuation past a caustic

Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude -- in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, comes on stage. In the present paper we explore such a principle in dimension $2.$ We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions.Comment: 48 pages, 15 figure

Computational and Experimental Comparison of Heat Transfer Characteristics of a Triple Row Impingement Channel at Large Impingement Heights

Impingement cooling is used in a variety of applications that require high localized convection heat transfer coefficients. Turbine airfoils can be cooled with high heat transfer impingement channels, local convection coefficients are the result of both the jet impact, as well as the channel flow produced from the exiting jets and the complex interaction between the jet, the crossflow and side walls. Numerous studies have investigated the effects of jet array and channel configurations on both target and jet plate heat transfer coefficients. However, it is important to understand how the jets behave at different impingement heights in order to properly design a near-wall cooling channel for a turbine airfoil. At large impingement heights, the jets have to travel further in order to impinge on the target plate, providing the cross flow more time to deflect it downstream. This downstream deflection reduces heat transfer; however, it provides uniform cooling throughout. The present study compares effects of jet-to-target wall distance and jet Reynolds number on local variation of the heat transfer coefficient. Results are presented for average jet based Reynolds numbers between 7,500 and 15,000. All experiments were carried out on an impingement channel with 3 jet holes per row, and a total of 15 streamwise rows, with X/D of 5, Y/D of 2, and Z/D of 6, 8 and 10, with a total channel width of Yc/D of 8. Results showed that the channel with a height of 6 diameters, and high Reynolds number yields highest heat transfer coefficients. The heat transfer profiles show very strong heat transfer in the first 4 rows of jets, rapidly decaying in the downstream direction due to crossflow degradation. CFD also showed the rapid degradation of heat transfer between the third and fourth row; although the magnitudes of heat transfer coefficients were upwards of 40% off from the experimental value. © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved