Similarity solution

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

Boundary-Layer Similarity Under an Axisymmetric, Gradient Wind Vortex

Abstract: We present similarity solutions for the mean boundary-layer profiles under an axisymmetric vortex that is in gradient wind balance; the similarity model includes the nonlinear momentum advection and curvature terms. These solutions are a generalization of the Ekman layer mean flow, which is the canonical boundary-layer basic state under a uniform, geostrophically-balanced flow. Near-surface properties such as inflow angle, surface wind factor, diffusive transport of kinetic energy into the surface layer and dissipational heating are calculated and shown to be sensitive to the choice of turbulence parameterization.

The hydrodynamic stability of ionization-shock fronts. Linear theory.

Abstract: The hydrodynamic stability of an ionization-shock front is investigated in the case where all regions are taken as isothermal and self-gravity is neglected. The evolving, unperturbed state is described by a similarity solution. A technique is developed which reduces the stability problem from a system of partial differential equations and associated boundary conditions to a coupled set of ordinary differential equations with time-dependent coefficients. This approach is shown to be valid as long as the perturbation wavelength is much greater than the thickness of the ionization-shock front. Perturbations are considered to arise from density inhomogeneities in the ambient medium. Numerical results along with an approximate analytic solution are given and show a new instability whereby all wavelengths greater than several recombination lengths grow without bound in an oscillatory manner. However, the wavelength with the fastest growth rate increases as the system evolves. A short discussion on the physical mechanism involved and several observational aspects, including a comparison with the morphology of the California nebula, is presented. The results suggest that this instability can produce irregular structures similar to the bright rims and elephant trunks seen in many diffuse nebulae.

Similarity solution of MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing

Abstract: Similarity analysis of diffusion of chemically reactive solute distribution in MHD boundary layer flow of an electrically conducting incompressible fluid over a porous flat plate is presented. The reaction rate of the solute is considered inversely proportional along the plate. Adopting the similarity transformation technique the governing equations are converted into the self-similar ordinary differential equations which are solved by shooting procedure using Runge-Kutta method. For increase of the Schmidt number the solute boundary layer thickness is reduced. Most importantly, the effects of reaction rate and order of reaction on concentration field are of conflicting natures, due to increasing reaction rate parameter the concentration decreases, but for the increase in order of reaction it increases. In presence of chemical reaction, the concentration profiles attain negative value when Schmidt number is large.

Evolution of clusters of sedimenting low-Reynolds-number particles with Oseen interactions

Abstract: A theoretical framework is developed to describe, in the limit of small but finite Re , the evolution of dilute clusters of sedimenting particles. Here, Re = aU /ν is the particle Reynolds number, where a is the radius of the spherical particle, U its settling velocity, and ν the kinematic viscosity of the suspending fluid. The theory assumes the disturbance velocity field at sufficiently large distances from a sedimenting particle, even at small Re , to possess the familiar source--sink character; that is, the momentum defect brought in via a narrow wake behind the particle is convected radially outwards in the remaining directions. It is then argued that for spherical clusters with sufficiently many particles, specifically with N much greater than O ( R 0 U /ν), the initial evolution is strongly influenced by wake-mediated interactions; here, N is the total number of particles, and R 0 is the initial cluster radius. As a result, the cluster first evolves into a nearly planar configuration with an asymptotically small aspect ratio of O ( R 0 U / N ν), the plane of the cluster being perpendicular to the direction of gravity; subsequent expansion occurs with an unchanged aspect ratio. For relatively sparse clusters with N smaller than O ( R 0 U /ν), the probability of wake interactions remains negligible, and the cluster expands while retaining its spherical shape. The long-time expansion in the former case, and that for all times in the latter case, is driven by disturbance velocity fields produced by the particles outside their wakes. The resulting interactions between particles are therefore mutually repulsive with forces that obey an inverse-square law. The analysis presented describes cluster evolution in this regime. A continuum representation is adopted with the clusters being characterized by a number density field ( n ( r , t )), and a corresponding induced velocity field ( u ( r , t )) arising on account of interactions. For both planar axisymmetric clusters and spherical clusters with radial symmetry, the evolution equation admits a similarity solution; either cluster expands self-similarly for long times. The number density profiles at different times are functions of a similarity variable η = ( r / t 1/3 ), r being the radial distance away from the cluster centre, and t the time. The radius of the expanding cluster is found to be of the form R cl ( t ) = A (ν a ) 1/3 N 1/3 t 1/3 , where the constant of proportionality, A , is determined from an analytical solution of the evolution equation; one finds A = 1.743 and 1.651 for planar and spherical clusters, respectively. The number density profile in a planar axisymmetric cluster is also obtained numerically as a solution of the initial value problem for a canonical (Gaussian) initial condition. The numerical results compare well with theoretical predictions, and demonstrate the asymptotic stability of the similarity solution in two dimensions for long times, at least for axisymmetric initial conditions.

Similarity analysis and non-linear wave propagation

Abstract: We characterize the most general 2 × 2 first order quasilinear hyperbolic systems in conservative form which are invariant with respect to the stretching group of transformation. The invariant solutions satisfy, under suitable conditions, an autonomous first order system of ordinary differential equations. A procedure is given to characterize the profile of the velocities of the strong and weak discontinuities.