Similarity solution

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

A diffusive Boussinesq plume

Abstract: -An axisymmetric Boussinesq plume in a uniform ambient fluid is considered. The classical similarity solution of Morton et al. (1956) is modified to account for diffusive losses of buoyancy flux and momentum flux. This leads to a buoyancy flux and a momentum flux that both tend to zero at infinite height. The mass flux at infinity will tend to a finite value that depends on the diffusion parameters for buoyancy and momentum.

Travelling-wave similarity solutions for an unsteady shear-stress-driven dry patch in a flowing film

Abstract: We investigate unsteady flow of a thin film of Newtonian fluid around a symmetric slender dry patch moving with constant velocity on an inclined planar substrate, the flow being driven by a prescribed constant shear stress at the free surface of the film (which would be of uniform thickness in the absence of the dry patch). We obtain a novel unsteady travelling-wave similarity solution which predicts that the dry patch has a parabolic shape and that the film thickness increases monotonically away from the dry patch.

Similarity Investigation of Thermosolutal Mixed Convection in a Saturated Porous Medium

Abstract: This contribution investigates the thermosolutal convection phenomenon around a thin wall, vertically immersed in a fluid saturating a porous medium of non-uniform permeability, taking into account the thermal conditions of the wall which is exposed to a fluid suction/injection. The conservation equations with boundary conditions have been transformed by the similarity method into a set of nonlinear differential equations. The resulting equations are numerically resolved using a fifth-order Runge-Kutta scheme coupled with the shooting technique. A graphical and physical interpretation of the found results as a function of the control parameters was performed. It is noticed that the heat transfer rate and the mass transfer rate at wall, are intensified for free convection and for significant permeability.

Symmetries and similarities of the zero-pressure-gradient turbulent boundary

Abstract: The symmetries and similarities of the zero-pressure-gradient turbulent boundary layer (ZPGTBL) are investigated to derive the full set of similarity variables, to derive the similarity equations, and to obtain a higher-order approximate solution of the mean velocity profile. Previous analyses have not resulted in all the similarity variables. We perform a symmetry analysis of the equations for ZPGTBL using Lie dilation groups, and obtain local, leading-order symmetries of the equations. The full set of similarity variables were obtained in terms of the boundary layer parameters. The friction velocity was shown to be the outer-layer velocity scale. The downstream evolution of the boundary thickness and the friction velocity is obtained analytically. The dependent similarity variables are written as asymptotic expansions. By asymptotically matching the expansions, an approximate similarity solution up to the third order in the overlapping layer are obtained. These results are obtained from first principles without any major assumptions and a turbulence model. The similarities and differences between ZPGTBL and turbulent channel flows in terms of the similarity equations, the gauge functions and the approximate solutions are discussed. In particular, the leading-order expansions are identical for ZPGTBL and channel flows, supporting the notion of universality of the near-wall layer. In addition, the logarithmic friction law for ZPGTBL is accurate to all orders while it is only accurate at the leading order in channel flows. The results will help further understand ZPGTBL and the issue of universality of the near-wall layer in wall-bounded turbulent flows.