Similarity solution

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

Inverse method for the investigation of nonlinear instabilities associated with negative-energy modes

Abstract: Earlier work on explosively unstable similarity solutions of Hamilton's equations with Hamiltonians homogeneous of degree [ital N] and satisfying resonance conditions is applied to study the nonlinear stability of linearly stable equilibria with neighboring positive- and negative-energy waves. A multiple-time-scale expansion near equilibrium yields a Hamiltonian system of the assumed structure. In the inverse method an explosively unstable similarity solution is assumed and one solves for the coefficients of the terms in a Hamiltonian of some given structure. Through some general arguments and many examples one concludes that explosively unstable solutions occur generally for wide ranges of coefficient values. Hence the original equilibrium is nonlinearly unstable for wide ranges of interaction parameters.

Similarity Solution of Laminar Axisymmetric Jets With Effect of Viscous Dissipation

Abstract: In this work the similarity solution of the thermal boundary layer for a laminar axisymmetric jet with viscous dissipation is derived and solved numerically, using a fourth-order Runge-Kutta method. The results are compared with the existing solutions, in which viscous dissipation is neglected. After that the effects of Reynolds and Prandtl numbers of the incoming hot jet on the temperature profile of the jet is investigated. It is seen that inclusion of viscous dissipation term results in a more rapid heat exchange between the incoming jet and quiescent fluid than the case where the viscous dissipation term is neglected. Also, it is observed that by increasing the Reynolds and Prandtl numbers of the jet, the heat penetrates better in locations near the symmetry line, whereas in locations farther from the symmetry axis the penetration is less.

Self similarity and attraction in stochastic nonlinear reaction-diffusion systems

Abstract: Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of stochastic PDEs in a form to which stochastic centre manifold theory may be applied we resolve these issues in this class. For definiteness, a first example of self-similarity of the Burgers' equation driven by some stochastic forced is studied. Under suitable assumptions, a stationary solution is constructed which yields the existence of a stochastic self-similar solution for the stochastic Burgers' equation. Furthermore, the asymptotic convergence to the self-similar solution is proved. Second, in more general stochastic reaction-diffusion systems stochastic centre manifold theory provides a framework to construct the similarity solution, confirm its relevance, and determines the correct solution for any compact initial condition. Third, we argue that dynamically moving the spatial origin and dynamically stretching time improves the description of the stochastic similarity. Lastly, an application to an extremely simple model of turbulent mixing shows how anomalous fluctuations may arise in eddy diffusivities. The techniques and results we discuss should be applicable to a wide range of stochastic similarity problems.

On laminar wall jets

Abstract: The name “wall jet” is applied to a jet of fluid which spreads out over a surface, the fluid outside the jet being at rest. It may, for example, be produced by allowing a free jet to impinge normally on a plane surface. A discussion of the phenomenon on the basis of boundary layer theory, for both laminar and turbulent incompressible flow, has recently been given in a paper [1] by the author (1956). For laminar flow it was shown that a similarity solution exists in which the velocity profile does not vary along the jet length, and is the same for both radial and two-dimensional wall jets. In the present paper this similarity solution is obtained in a generalised form, with an additional length parameter governed by the distance which the jet profile takes to assume its final similar shape. Mr. N. Riley has shown that the similarity solution is also applicable when the fluid is compressible or heated, if it is assumed that the coefficient of viscosity is proportional to the temperature, and his results are briefly discussed, with his permission. Finally, consideration is given to perturbations of the basic similarity solution due to a small external flow, as might arise as a consequence of entrainment of fluid into the wall jet.