Similarity solution

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

Self-similar mean dynamics in turbulent wall flows

Abstract: This study investigates how and why dynamical self-similarities emerge with increasing Reynolds number within the canonical wall flows beyond the transitional regime. An overarching aim is to advance a mechanistically coherent description of turbulent wall-flow dynamics that is mathematically tractable and grounded in the mean dynamical equations. As revealed by the analysis of Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst. A, vol. 24, 2009, pp. 781–807), the equations that respectively describe the mean dynamics of turbulent channel, pipe and boundary layer flows formally admit invariant forms. These expose an underlying self-similar structure. In all cases, two kinds of dynamical self-similarity are shown to exist on an internal domain that, for all Reynolds numbers, extends from to , where is the kinematic viscosity, is the friction velocity and is the half-channel height, pipe radius, or boundary layer thickness. The simpler of the two self-similarities is operative on a large outer portion of the relevant domain. This self-similarity leads to an explicit analytical closure of the mean momentum equation. This self-similarity also underlies the emergence of a logarithmic mean velocity profile. A more complicated kind a self-similarity emerges asymptotically over a smaller domain closer to the wall. The simpler self-similarity allows the mean dynamical equation to be written as a closed system of nonlinear ordinary differential equations that, like the similarity solution for the laminar flat-plate boundary layer, can be numerically integrated. The resulting similarity solutions are demonstrated to exhibit nearly exact agreement with direct numerical simulations over the solution domain specified by the theory. At the Reynolds numbers investigated, the outer similarity solution is shown to be operative over a domain that encompasses of the overall width of the flow. Other properties predicted by the theory are also shown to be well supported by existing data.

Similarity solution of thermonuclear burn wave with electron and α-conductivities

Abstract: The process of non-uniform laser-driven DT plasma burning caused by the thermonuclear burn wave produced and propagating in plasma is investigated theoretically. The energy transfer from the burning plasma region to the remaining cold portion is assumed to be realized either by α-particles or by free electrons. A similarity solution of this problem has been obtained and, deriving from this solution, the conditions for "firing-up" non-uniform thermonuclear targets are defined.

Similarity solution for cavity expansion in thermoplastic soil

Abstract: Summary This paper presents an analytical solution for cavity expansion in thermoplastic soil considering non-isothermal conditions. The constitutive relationship of thermoplasticity is described by Laloui's advanced and unified constitutive model for environmental geomechanical thermal effect (ACMEG-T), which is based on multi-mechanism plasticity and bounding surface theory. The problem is formulated by incorporating ACMEG-T into the theoretical framework of cavity expansion, yielding a series of partial differential equations (PDEs). Subsequently, the PDEs are transformed into a system of first-order ordinary differential equations (ODEs) using a similarity solution technique. Solutions to the response parameters of cavity expansion (stress, excess pore pressure, and displacement) can then be obtained by solving the ODEs numerically using mathematical software. The results suggest that soil temperature has a significant influence on the pressure-expansion relationships and distributions of stress and excess pore pressure around the cavity wall. The proposed solution quantifies the influence of temperature on cavity expansion for the first time and provides a theoretical framework for predicting thermoplastic soil behavior around the cavity wall. The solution found in this paper can be used as a theoretical tool that can potentially be employed in geotechnical engineering problems, such as thermal cone penetration tests, and nuclear waste disposal problems.

Self-Similarity in the Outer Region of Adverse-Pressure-Gradient Turbulent Boundary Layers

Abstract: This paper presents a consistent theory of self-similarity and of equilibrium in the outer region of turbulent boundary layers that explains recent experimental findings on the subject, including new ones presented here. The theory is first presented in a general form where the outer scales are left unspecified and it is not assumed that the mean velocity defect and the Reynolds stresses share a common velocity scale. It is shown that the main results of the traditional similarity theory remain valid even in this case. Common outer scaling with the Zagarola-Smits length and velocity scales is then chosen. A new pressure gradient parameter is introduced to characterize the local effect of the pressure gradient in all flow conditions including strong adverse-pressure-gradient conditions. By analyzing several adverse-pressure-gradient flow cases, it is shown that self-similarity of the mean velocity defect profile is reached in all cases in localized but significant flow regions. The same is, however, not true of the Reynolds stress profiles. In agreement with the similarity analysis, the self-similar velocity defect profile is found to be a function of the pressure gradient and most flows studied here are only in an approximate state of equilibrium in the region of self-similar defect profiles despite the excellent collapse of the profiles.