Similarity solution

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

Effect of surfactants on free-surface turbulent flows

Abstract: In two earlier papers, we studied the statistical and mechanistic structure of the turbulent boundary layer under a stress-free (clean) free surface. Findings there, such as the presence of inner and outer surface layers, are very much the direct result of the absence of shear stresses at the surface. The latter condition is easily lost when the surface is contaminated and surface elasticity varies with space and time. In this paper we consider the effect of surfactant on features of the free-surface turbulent flow. We perform direct numerical simulations of the Navier–Stokes equations subject to surfactant-laden free-surface boundary conditions for varying Reynolds and Marangoni numbers and low Froude numbers. As expected, the Marangoni effect decreases the horizontal turbulence intensity and normal vorticity at the surface. The direct effect on the turbulent kinetic energy is an increase in the dissipation and viscous diffusion and a decrease in the production near the surface relative to the clean case. The most prominent effect of the presence (of even a small amount) of surfactant is the drastic reduction in the surface divergence and the associated sharp decrease of up- and downwelling at the surface which has direct implications to near-surface turbulent transport. The observed surfactant effects on turbulent kinetic energy budget can be attributed to the generation of Marangoni vorticity at the free surface by approaching hairpin vortices. The Marangoni effect has also a direct effect on the boundary-layer structure, causing an increase of the thickness of the boundary layer and in the maxima of the mean shear near the surface. For moderate values of the Marangoni number, up-/downwelling effectively vanishes and the flow approaches a state independent of the Marangoni number. Guided by these results and to obtain theoretical insight, we develop a similarity solution for the mean flow. The analytic solution agrees well with the numerical data and provides precise measures for the multi-layer structure of the boundary layer. Based on the theoretical model, we derive scaling laws for the thickness of the inner and the outer boundary layers, which are also confirmed by numerical simulations.

Unsteady turbulent buoyant plumes

Abstract: We model the unsteady evolution of turbulent buoyant plumes following temporal changes to the source conditions. The integral model is derived from radial integration of the governing equations expressing the conservation of mass, axial momentum and buoyancy. The non-uniform radial profiles of the axial velocity and density deficit in the plume are explicitly described by shape factors in the integral equations; the commonly-assumed top-hat profiles lead to shape factors equal to unity. The resultant model is hyperbolic when the momentum shape factor, determined from the radial profile of the mean axial velocity, differs from unity. The solutions of the model when source conditions are maintained at constant values retain the form of the well-established steady plume solutions. We demonstrate that the inclusion of a momentum shape factor that differs from unity leads to a well-posed integral model. Therefore, our model does not exhibit the mathematical pathologies that appear in previously proposed unsteady integral models of turbulent plumes. A stability threshold for the value of the shape factor is identified, resulting in a range of its values where the amplitude of small perturbations to the steady solutions decay with distance from the source. The hyperbolic character of the system allows the formation of discontinuities in the fields describing the plume properties during the unsteady evolution. We compute numerical solutions to illustrate the transient development following an abrupt change in the source conditions. The adjustment to the new source conditions occurs through the propagation of a pulse of fluid through the plume. The dynamics of this pulse are described by a similarity solution and, by constructing this new similarity solution, we identify three regimes in which the evolution of the transient pulse following adjustment of the source qualitatively differ.

Similarity Solution of Overland Flow on Pervious Surface

Abstract: Similarity solutions for the overland flow-infiltration problem are obtained by use of a physically-based mathematical model. The mathematical model is founded on the kinematic equations of overland flow and the Green-Ampt formulation of the infiltration process. The governing equations of surface flow and infiltration are coupled and rewritten in terms of four physically-based nondimensional parameters. Constant values of these parameters imply hydraulic similarity of an infinite number of different overland flow-infiltration situations. An implicit finite difference technique is employed to solve the governing equations. The similarity approach is verified by comparisons of similarity solutions with the results of a theoretically sophisticated mathematical model for the cases of Columbia sandy loam, Yolo light clay, and Guelph loam. Also, a group of peak discharge prediction charts are developed based on the similar solutions of the overland flow-infiltration problem and presented to emphasize the feasibility of the similarity approach. The use of these charts is illustrated by a practical application section included in the paper.

Wall jet flows of Glauert type: Heat transfer characteristics and the thermal instabilities in analytic closed forms

Abstract: Heat transfer characteristics of the traditional wall jet flows subject to various thermal boundary conditions including isothermal surface, prescribed temperature, constant heat flux, prescribed heat flux, adiabatic surface and thermally convective surface are documented in analytic closed forms. Heat dissipation has been also included where the similarity energy equation could structurally adjust to this specific term (for the adiabatic case, this term has been necessarily included). In all the studied cases, both the transpiration velocity and moving wall conditions are allowed to exist (where applicable) in such a way, being consistent with the Glauert integral constraint and subject to the context of exponentially decaying wall jet flows. In particular, it is analytically proved (even without having the closed form solutions) that for the Glauert original case, a surface with a prescribed temperature in the form of T w ( x ) = m x − 1 4 + T ∞ serves zero contribution to heat transfer at the wall (the Induced Heat Shield). More precisely, the value − 1 4 as the prescribing parameter is the Surface Heat Transfer Stopping Point and below this point, heat transfer phenomenon falls from a usual physical interpretation, expressing spectrums of thermal instabilities through a hyper geometric function. Furthermore, it is argued that a normalized similarity temperature in the form of θ ( η ) = T − T ∞ m x n − 1 4 results in the appearance of an uninterruptable singularity within the heat transfer phenomenon for the Glauert case subject to a rigorous physical ground; and as an immediate practical consequence, there is no physically-valid similarity solution for an adiabatic surface or more precisely, for energy equation with heat dissipation consideration. It should be mentioned that the concept of Induced Heat Shield is discussed in here for the first time (to our knowledge) which may inspire a concealed fact regarding the restrictions of the thermal similarity solutions. Therefore, it is hopeful that heat transfer phenomenon in the Glauert type wall jet flows and the associated physical representations can be better understood by the present research.