Similarity solution

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

Inertial gravity current in rectangular channels over a porous bottom: Asymptotic solutions

Abstract: We consider a high-Reynolds-number Boussinesq gravity current propagating in a channel above a permeable horizontal boundary. The current (of reduced gravity g ′ ) is released from a rectangular lock (of length x 0 and height h 0 ), and after an adjustment (slumping) stage is expected to enter into a similarity stage, on which we focus here, using a thin-layer shallow-water model. The classical analytical self-similar propagation solution predicts that the length of the current is given by x N ( t ) = K t 2 ∕ 3 (where K is a constant and t is time from release). The height h ( x , t ) and velocity u ( x , t ) display a similarity shape of the variable y = x ∕ x N ( t ) , y ∈ [ 0 , 1 ] ( x is the physical coordinate measured from the backwall of the lock). This solution, which is very useful in the analysis of gravity-current problems, is invalidated by the drainage effect into the porous bottom. Here we extend the classical similarity (basic) solutions by developing a perturbation (asymptotic) expansions about the basic solution. The expansion uses the small parameter λ which represents the ratio of the typical propagation time T = x 0 ∕ ( g ′ h 0 ) 1 ∕ 2 to the drainage time t B (a given property of the porous bottom). The perturbation terms can be calculated analytically, and we present the results of the first-order correction. This provides useful insights about the influence of the porous boundary, as compared with the classical similarity behavior: x N ( t ) is shorter, the profile of u ( y ) is deflected to lower values at the nose, and h ( y ) is reduced mostly at the tail. The deviation from the basic similarity solution increases like λ t . In addition, we show that the drainage influence is important in reducing the transition length from the inertial (inviscid) to the viscous regimes. We compared the analytical asymptotic leading-order solution with numerical finite-difference results for various values of λ , and found excellent qualitative agreement and fair quantitative agreement. We expect that higher-order terms will improve the accuracy of the new solution, but this additional extension was not performed in this work.

Viscoelastic inertial flow driven by an axisymmetric accelerated surface

Abstract: A similarity transform is used to analyse the flow of an upper-convected Maxwell fluid in an infinitely long cylinder whose surface has a velocity that increases in magnitude linearly with axial coordinate. Two types of problem are considered, the accelerated surface flow - when the surface velocity is outward towards the tube ends, and the decelerated surface flow - when it is inward. For the accelerated surface flow, the introduction of elasticity prevents the loss of similarity solution that occurs without elasticity at a Reynolds number ( Re ) of 10.25: with elasticity, solutions up to a Reynolds number of 95 were computed. As elasticity is introduced, normal stress gradients in an elastic boundary layer near the accelerated surface help offset inertially generated negative axial pressure gradients; with sufficient elasticity the turning point in the non-elastic solution family at Re = 10.25 disappears. For the decelerated surface flow, solutions could not be computed beyond a critical Re that depends on the level of elasticity considered, because at this critical Re , the axial velocity profile at the centreline becomes infinitely blunt.

A similarity solution for the flow and heat transfer over a moving permeable flat plate in an external free stream: case of strong injection

Abstract: The previous work of Bachok et al. (Heat Mass Transf. 47:1643–1649, 2011) on the forced convection heat transfer on an isothermal moving surface in an external free stream is extended to the case when fluid injection through the surface, characterized by the parameter γ, is large. The asymptotic solution derived in this limit shows that the boundary layer has a double region structure, with an inviscid inner region of thickness O(γ) and an outer shear layer. Some further aspects of the original problem not treated in Bachok et al. (Heat Mass Transf. 47:1643–1649, 2011) are discussed as well as the analogous problem for a constant surface heat flux, where relatively small injection rates are seen to give rise to large increases in the surface temperature.