Similarity solution

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

Laminar flow analysis in a pipe with locally pressure-dependent leakage through the wall

Abstract: The paper analyzes the problem of the leaky pipe (or a porous-walled pipe), namely the laminar flow of a pure fluid that takes place in a pipe, the wall of which is composed of a porous material. This configuration is inspired by some watering systems or by the cross-flow (or tangential) filtration configuration for membrane separation or capillary flow. It assumes that the leakage through the wall (or permeate) results from the pressure difference between both sides of the pipe wall, and is here modeled by the Starling–Darcy law. The inner pressure along the pipe behaves accordingly with two competitive features: the viscous pressure drop competing against the pressure increase due to pipe axial flow deceleration. It is long known that both features compensate at a critical value, R t i s o , of the transverse Reynolds number R t (based on transpiration velocity); this corresponds to the only situation where the pressure remains uniform along the channel. The case with uniform leakage–known as Berman flow–possesses a similarity solution due to Yuan and Finkelstein (1956) [2] for the pipe configuration. The paper is aimed at extending the latter study to a non-uniform leakage depending linearly on local pressure. First, the similarity solution is revisited. Its expansion in a series of R t allows us to propose a hierarchy of new ordinary differential equations (ODEs), that extend–to small or moderate R t –the linear ODE proposed for the limit case R t = 0 by Regirer (1960) [25] . As by-products, we propose approximate analytical solutions that solve the problem of the leaking pipe with increasing accuracy in the weakly non-linear case (WNL) (i.e. for small and moderate R t ). Finally, the validity of ODEs and WNL solutions is numerically checked with respect to flow simulations in the Prandtl approximation.

Turbulent secondary flows in wall turbulence: vortex forcing, scaling arguments, and similarity solution

Abstract: Spanwise surface heterogeneity beneath high-Reynolds number, fully-rough wall turbulence is known to induce a mean secondary flow in the form of counter-rotating streamwise vortices—this arrangement is prevalent, for example, in open-channel flows relevant to hydraulic engineering. These counter-rotating vortices flank regions of predominant excess(deficit) in mean streamwise velocity and downwelling(upwelling) in mean vertical velocity. The secondary flows have been definitively attributed to the lower surface conditions, and are now known to be a manifestation of Prandtl’s secondary flow of the second kind—driven and sustained by spatial heterogeneity of components of the turbulent (Reynolds averaged) stress tensor (Anderson et al. J Fluid Mech 768:316–347, 2015). The spacing between adjacent surface heterogeneities serves as a control on the spatial extent of the counter-rotating cells, while their intensity is controlled by the spanwise gradient in imposed drag (where larger gradients associated with more dramatic transitions in roughness induce stronger cells). In this work, we have performed an order of magnitude analysis of the mean (Reynolds averaged) transport equation for streamwise vorticity, which has revealed the scaling dependence of streamwise circulation intensity upon characteristics of the problem. The scaling arguments are supported by a recent numerical parametric study on the effect of spacing. Then, we demonstrate that mean streamwise velocity can be predicted a priori via a similarity solution to the mean streamwise vorticity transport equation. A vortex forcing term has been used to represent the effects of spanwise topographic heterogeneity within the flow. Efficacy of the vortex forcing term was established with a series of large-eddy simulation cases wherein vortex forcing model parameters were altered to capture different values of spanwise spacing, all of which demonstrate that the model can impose the effects of spanwise topographic heterogeneity (absent the need to actually model roughness elements); these results also justify use of the vortex forcing model in the similarity solution.

On a similarity solution in the theory of unsteady marginal separation

Abstract: The present paper deals with initial value problems associated with the high Reynolds number asymptotic theory of unsteady, marginally separated boundary layer flows. In particular, the subsonic planar flow case is treated. Special emphasis is placed on solutions which blow up within finite time. As is well-known, steady solutions of the underlying equations only exist up to a critical value of the crucial parameter which controls the conditions leading to localized boundary layer separation. Our numerical analysis shows that any blow-up solution finally approaches a unique structure, entirely independently of the choice of initial data, sub- or super-critical flow conditions, and, if present at all, the type of forcing. Further support for the existence of a self-similar, unique blow-up structure is gained from asymptotic analysis.