Similarity solution

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

Evolution of a wave packet into vortex loops in a laminar separation bubble

Abstract: A laminar boundary layer develops in a favourable pressure gradient where the velocity profiles asymptote to the Falkner & Skan similarity solution. Flying-hot-wire measurements show that the layer separates just downstream of a subsequent region of adverse pressure gradient, leading to the formation of a thin separation bubble. In an effort to gain insight into the nature of the instability mechanisms, a small-magnitude impulsive disturbance is introduced through a hole in the test surface at the pressure minimum. The facility and all operating procedures are totally automated and phase-averaged data are acquired on unprecedently large and spatially dense measurement grids. The evolution of the disturbance is tracked all the way into the reattachment region and beyond into the fully turbulent boundary layer. The spatial resolution of the data provides a level of detail that is usually associated with computations.Initially, a wave packet develops which maintains the same bounded shape and form, while the amplitude decays exponentially with streamwise distance. Following separation, the rate of decay diminishes and a point of minimum amplitude is reached, where the wave packet begins to exhibit dispersive characteristics. The amplitude then grows exponentially and there is an increase in the number of waves within the packet. The region leading up to and including the reattachment has been measured with a cross-wire probe and contours of spanwise vorticity in the centreline plane clearly show that the wave packet is associated with the cat's eye pattern that is a characteristic of Kelvin–Helmholtz instability. Further streamwise development leads to the formation of roll-ups and contour surfaces of vorticity magnitude show that they are three-dimensional. Beyond this point, the behaviour is nonlinear and the roll-ups evolve into a group of large-scale vortex loops in the vicinity of the reattachment. Closely spaced cross-wire measurements are continued in the downstream turbulent boundary layer and Taylor's hypothesis is applied to data on spanwise planes to generate three-dimensional velocity fields. The derived vorticity magnitude distribution demonstrates that the second vortex loop, which emerges in the reattachment region, retains its identity in the turbulent boundary layer and it persists until the end of the test section.

Fluid flow due to a stretching cylinder

Abstract: The fluid flow outside of a stretching cylinder is studied. The problem is governed by a third‐order nonlinear ordinary differential equation that leads to exact similarity solutions of the Navier–Stokes equations. Because of algebraic decay, an exponential transform is used to facilitate numerical integration. Asymptotic solutions for large Reynolds numbers compare well with numerical results. The heat transfer is determined.

An Introduction to Gravity Currents and Intrusions

Abstract: Introduction Classification The Navier-Stokes equations Non-stratified ambient currents Shallow-water (SW) formulation for high-Re flows Motion of the interface and the continuity equation One-layer model A useful transformation The full behavior by numerical solution Dam-break stage Similarity solution The validity of the inviscid approximation The steady-state current and nose jump conditions Benjamin's analysis Jump condition Box models for 2D geometry Fixed volume current with inertial-buoyancy balance Inflow volume change Two-layer SW model Introduction The governing equations Boussinesq system in dimensionless form Jumps of interface for H < 2 Energy and work in a two-layer model Axisymmetric currents, SW formulation Governing equations A useful transformation The full behavior by numerical solution Dam-break stage Similarity solution The validity of the inviscid approximation Some comparisons Box models for axisymmetric geometry Fixed volume current with inertial-buoyancy balance Inflow volume change Effects of rotation Axisymmetric case Rotating channel Buoyancy decays: particle-driven, porous boundary, and entrainment Particle-driven currents Axisymmetric particle-driven current Extensions of particle-driven solutions Current over a porous bottom Axisymmetric current over a porous bottom Entrainment Non-Boussinesq systems Introduction Formulation Dam-break and initial slumping motion The transition and self-similar stages Summary Lubrication theory formulation for viscous currents 2D geometry Axisymmetric current Current in a porous medium II Stratified ambient currents and intrusions Continuous density transition Introduction The SW formulation SW results and comparisons with experiments and simulations Dam break Critical speed and nose-wave interaction Similarity solution The validity of the inviscid approximation Axisymmetric and rotating cases SW formulation SW and NS finite-difference results The validity of the inviscid approximation The steady-state current Steady-state flow pattern Results Comparisons and conclusions Intrusions in 2D geometry Introduction Two-layer stratification Linear transition layer Rectangular lock configurations Cylindrical lock in a fully linearly-stratified tank Similarity solution Non-symmetric intrusions Intrusions in axisymmetric geometry Introduction Two-layer stratification Fully linearly-stratified tank, part-depth locks Box models for 2D geometry Fixed volume and inertial-buoyancy balance S = 1, inflow volume change Box models for axisymmetric geometry Fixed volume and inertial-buoyancy balance S = 1, inflow volume change Lubrication theory for viscous currents with S = 1 2D geometry Axisymmetric geometry Energy Introduction 2D geometry Axisymmetric geometry SW equations: characteristics and finite-difference schemes Characteristics Numerical solution of the SW equations Navier-Stokes numerical simulations Formulation A finite-difference code Other codes Some useful formulas Leibniz's Theorem Vectors and coordinate systems

Spherically symmetric similarity solutions of the Einstein field equations for a perfect fluid

Abstract: Spherically symmetric space-times which admit a one parameter group of conformal transformations generated by a vectorξ μ such thatξ μ;v +ξ v;μ =2g μv are studied. It is shown that the metric coefficients of such space-times depend essentially on the single variablez=r/t wherer is a radial coordinate andt is the time. The Einstein field equations then reduce to ordinary differential equations. The solutions of these equations are analogous to the similarity solutions of the classical theory of hydrodynamics. In case the source of the field is a perfect fluid whose specific internal energy is a function of temperature alone, the solution of the field equations is uniquely determined by specifying data on the time-like hypersurfacez=constant and is a similarity solution. The problem of fitting a similarity solution to another solution of the field equations across a shock described by the hypersurfacez=constant is treated. A particular similarity solution for whichw=3p obtains is shown to describe a Robertson-Walker space-time. This solution is fitted to a special static solution of the Einstein field equations which has a singularity atr=0. The resulting solution of the Einstein field equations is shown to be regular everywhere except atr=0≧t and the shock. The special Robertson-Walker metric is also fitted to a particular class of collapsing dust solutions (which are also similarity solutions) across a shock. The resulting solution is regular everywhere except atr=t=0 and on the shock.

The bifurcation of liquid bridges

Abstract: Details of the shape of the liquid bridge joining a nascent water drop to its parent body are presented for times before, after and at the instant of bifurcation when the drop is created and also when the secondary droplet is formed. After the instant of bifurcation there is ‘unbalanced’ surface tension which gives an impulse to the rest of the fluid causing strong surface deformations. The major point of this work is to draw attention to the strong up–down asymmetry at each bifurcation point. The geometric similarity at each bifurcation instant supports the conjecture that the flow converges to just one similarity solution of the type described by Keller & Miksis (1983) in which only surface tension and inertia are important. Features of the flow before and after bifurcation are discussed.