Showing papers on "Multiple-scale analysis published in 1980"

The radiation of sound by the instability waves of a compressible plane turbulent shear layer

Abstract: The problem of acoustic radiation generated by instability waves of a compressible plane turbulent shear layer is solved. The solution provided is valid up to the acoustic far-field region. It represents a significant improvement over the solution obtained by classical hydrodynamic-stability theory which is essentially a local solution with the acoustic radiation suppressed. The basic instability-wave solution which is valid in the shear layer and the near-field region is constructed in terms of an asymptotic expansion using the method of multiple scales. This solution accounts for the effects of the slightly divergent mean flow. It is shown that the multiple-scales asymptotic expansion is not uniformly valid far from the shear layer. Continuation of this solution into the entire upper half-plane is described. The extended solution enables the near- and far-field pressure fluctuations associated with the instability wave to be determined. Numerical results show that the directivity pattern of acoustic radiation into the stationary medium peaks at 20 degrees to the axis of the shear layer in the downstream direction for supersonic flows. This agrees qualitatively with the observed noise-directivity patterns of supersonic jets.

A Comparison of Perturbation Methods for Nonlinear Hyperbolic Waves

Abstract: Publisher Summary This chapter discusses the comparison of perturbation methods for nonlinear hyperbolic waves. It discusses five numbers of perturbation methods including the method of renormalization, the method of strained coordinates, the analytic method of characteristics, the method of multiple scales, and the Krylov–Bogoliubov–Mitropolsky method. These techniques are applied to nonlinear acoustic waves propagating in thermoviscous fluids. For lossless, oppositely traveling, one-dimensional waves, a first-order uniformly valid expansion can be obtained by using any of these techniques provided the waves do not mutually interact in the body of the medium. This is so if the waves are periodic or pulses. If the mutual-interaction terms are not negligible, only the method of renormalization, strained coordinates, and characteristics can be used. For dissipative media, it is not clear how one can use the method of renormalization, strained coordinates, and characteristics to determine an approximate solution when the dissipation term is the same order as the nonlinear term. For conservative multidimensional waves, a combination of the methods of renormalization and matched asymptotic expansions appear to be the most powerful. In applying the analytic method of characteristics to multi-dimensional waves, one can obtain uniformly valid expansions only if the characteristic surfaces are chosen appropriately. For waves that are short compared with the radii of curvature of the wave fronts, choosing the appropriate characteristics does not present any difficulty because the linear solution displays the role of the characteristic and geometric rays.

On the boundary value problems for a class of ordinary differential equations with turning points

Abstract: In this paper we study the boundary value problems for a class of ordinary differential equations with turning points by the method of multiple scales. The paradox in [1] and the variational approach in [2] are avoided. The uniformly valid asymptotic approximations of solutions have been constructed. We also study the case which does not exhibit resonance.