Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Experiments on Stability and Transition at Mach 3

Abstract: The results of an experimental study of stability, receptivity and transition of the flat-plate laminar boundary layer at Mach 3 are discussed. With a relatively low free-stream disturbance level (∼0.1%), spectra, growth rates and amplitude distributions of naturally occurring boundary layer waves were measured using hot wires. Physical (mass-flux) amplitudes in the boundary layer and free stream are reported and provide stability and receptivity results against which predictions can be directly compared. Comparisons are made between measurements of growth rates of unstable high-frequency waves and theoretical predictions based on a non-parallel, mode-averaging stability theory and receptivity assumptions; good agreement is found. In contrast, it was found that linear stability theory does not account for the measured growth of low-frequency disturbances. A detailed investigation of the disturbance fields in the free stream and on the nozzle walls provides the basis for a discussion of the source and the development of the measured boundary layer waves. Attention is drawn to the close matching in streamwise wavelengths for instability waves and the free-stream acoustic disturbances. It was also found that a calibration of the hot wire in the free stream yields a double-peak boundary layer disturbance amplitude distribution, as has been found by previous investigators, which is not consistent with the predictions of linear stability theory. This double peak was found to be an experimental anomaly which resulted from assumptions that are frequently made in the free-stream calibration procedure. A single-peak amplitude distribution across the boundary layer was established only when the hot-wire voltage was calibrated against the mean boundary layer profile. Finally, the late stages of transition, at a higher Reynolds number with a higher free-stream disturbance level, were explored. Calibrated amplitude levels are provided at locations where nonlinearities are first detected and where the mean boundary layer profile is first observed to depart from the laminar similarity solution. A qualitative discussion of the character of ensuing nonlinearities is also included.

Spread of Fluid Drops over a Horizontal Plane

Abstract: An asymptotic expansion procedure is presented for determining the two dimensional fluid motion of viscous thin films on a horizontal plane in the presence of gravity. The film surface height satisfies to the first order the nonlinear diffusion equation whose variable diffusion coefficient is proportional to the cube of the concentration. When this equation is applied to the fluid film of finite dimension, it has a similarity expressed in terms of an elementary function. The deformation of the fluid drops and their spread over the horizontal plane are asymptotically represented by the similarity solution. A similar procedure is available for the axisymmetric motion of the thin films. The results are compared with the two dimensional case.

A Coupled Variable Coefficient Modified KdV Equation Arising from a Two-Layer Fluid System

Abstract: A quite general coupled variable coefficient modified KdV (VCmKdV) equation in a two-layer fluid system is derived by means of the reductive perturbation method. Making use of the CK's direct method, some similarity reductions of the coupled VCmKdV equation are obtained and their corresponding group explanations are discussed. Some exact solutions of the coupled equations are also presented.

Application of the similarity method to the nonlinear Boltzmann equation

Abstract: A consequent application of the similarity method to the nonlinear Boltzmann equation leads to the general form of exact similarity solutions and allows a group-theoretic classification. Classes of similarity solutions depending very strongly on the source term but different from the Bobylev-Krook-Wu solution will be discovered and discussed.

Symmetry groups and similarity solutions for a free convective boundary-layer problem

Abstract: The free convective boundary-layer problem due to the motion of an elastic surface into an electrically conducting fluid is studied with group-theoretical methods. The symmetry groups admitted by the corresponding boundary value problem are obtained. Particular attention is paid on the group of scaling which provides the similarity solution of the problem. Also, the admissible form of the data, in order to be conformed to the obtained symmetries, is provided. Finally, with the use of the entailed similarity solution the problem is transformed into a boundary value problem of ODEs and is solved numerically.