Similarity solution

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

Diffusion flame in a two-dimensional, accelerating mixing layer

Abstract: A theoretical and computational study of a laminar, two-dimensional, compressible, mixing, reacting layer with a pressure gradient that accelerates the flow in the direction of the primary stream is performed. One objective is to analyze the problems of a new technology related to combustion occurring in an accelerating transonic flow. Potential exists for reduction in nitric oxide formation and improvement in engine efficiency and/or power/weight. A similarity solution is found that reduces the partial differential equations to a system of ordinary differential equations. The solution is found in terms of a new acceleration parameter for compressible flows. The parameter is also useful for nonreacting mixing layers and for reacting or nonreacting wall boundary layers. For a low Mach number, the parameter reduces to the classical incompressible parameter. A numerical solution to these equations was performed. In the presence of exothermic reaction and accelerating pressure gradients, there are nonmonotoni...

Axisymmetric Vortex Solution of Navier-Stokes Equation

Abstract: An exact solution of a viscous incompressible flow is presented for a general initial condition. This flow represents an axisymmetric shear layer superimposed on an irrotational straining flow. The solution incorporates the three representative features of vortex motion: stretching, convection and viscous diffusion. In a particular case of constant straining, the flow approaches to a steady state in which the above three effects are in equilibrium. However if the initial state is composed of the same amount of opposite vorticities, the shear layer disappears exponentially in time. Spectral analysis of the solution shows the cascade of vorticity fluctuations to smaller scales. The general solution includes the vortex solutions given by Oseen (1911) and by Burgers (1948), and partly overlaps the similarity solution found by Bellamy-Knights (1970).

Steady motion within a curved pipe

Abstract: The forcing of fluid through a tube, of uniform cross section $(\sim a)$ and coiled to form an arc of a circle of radius R$(\gg a)$, by the application of a steady constant pressure gradient along the tube is discussed for large values of the Dean number D. An asymptotic description is presented for the fully-developed laminar flow within a general symmetric section, in which there is outward centrifuging of the secondary motion in the inviscid core, supplemented by a faster viscous return motion near the pipewalls, while the downpipe velocity increases steadily across the core. A similarity solution of the viscous flow, with the associated core flow thereby being determined, is calculated for a triangular cross section, but for a rectangular tube analytical and numerical arguments are put forward that point fairly conclusively to the non-existence of an attached laminar motion, near the inside bend at least, at high Dean numbers. For if the pressure-gradient is imagined to vary like the mth power of distance across the section, then computed results indicate that local solutions of the boundary layer problem can be found only for $m>\frac{1}{2}$. The assertion is backed up by an analytical study for small positive values of $(m-\frac{1}{2})$. The dynamical properties in the neighbourhood of a flat outside bend, or of a 'pinched' inside bend, of a tube also need careful consideration, although these may be essentially passive flow regions and some account of the local features there can be given. Near a flat outside bend two possible flow models arise, depending on the local nature of the coreflow.

Propagation of strong converging shock waves in a gas of variable density

Abstract: The problem of a strong converging spherical (or cylindrical) shock collapsing at the centre (or axis) of symmetry is extended to take into account the inhomogeneity of a gaseous medium, the density of which is decreasing towards the centre (or axis) according to a power law. The perturbative approach used in this paper provides a global solution to the implosion problem yielding accurately the results of Guderley's similarity solution, which is valid only in the vicinity of the center/axis of implosion. The analysis yields refined values of the leading similarity parameter along with higher-order terms in Guderley's asymptotic solution near the center/axis of convergence. Computations of the flow field and shock trajectory in the region extending from the piston to the center/axis of collapse have been performed for different values of the adiabatic coefficient and the ambient density exponent.

Capillary Breakup of a Liquid Bridge: Identifying Regimes and Transitions

Abstract: Computations of the breakup of a liquid bridge are used to establish the limits of applicability of similarity solutions derived for different breakup regimes. These regimes are based on particular viscous-inertial balances, that is different limits of the Ohnesorge number $Oh$. To accurately establish the transitions between regimes, the minimum bridge radius is resolved through four orders of magnitude using a purpose-built multiscale finite element method. This allows us to construct a quantitative phase diagram for the breakup phenomenon which includes the appearance of a recently discovered low-$Oh$ viscous regime. The method used to quantify the accuracy of the similarity solutions allows us to identify a number of previously unobserved features of the breakup, most notably an oscillatory convergence towards the viscous-inertial similarity solution. Finally, we discuss how the new findings open up a number of challenges for both theoretical and experimental analysis.