An asymptotic theory of incompressible turbulent boundarylayer flow over a small hump

Abstract

A rational asymptotic theory describing the perturbed flow in a turbulent boundary layer encountering a small two-dimensional hump is presented. The theory is valid in the limit of very high Reynolds number in the case of an aerodynamically smooth surface, or in the limit of small drag coefficient in the case of a rough surface. The method of matched asymptotic expansions is used to obtain a multiple-structured flow, along the general lines of earlier laminar studies. The leading-order velocity perturbations are shown to be precisely the inviscid, irrotational, potential flow solutions over most of the domain. The Reynolds stresses are found to vary across a thin layer adjacent to the surface, and display a singular behaviour near the surface which needs to be resolved by an even thinner wall layer. The Reynolds stress perturbations are calculated by means of a second-order closure model, which is shown to be the minimum level of sophistication capable of describing these variations. The perturbation force on the hump is also calculated, and its order of magnitude is shown to depend on the level of turbulence closure; a cruder turbulence model gives rise to spuriously large forces.

Break-away separation for high turbulence intensity and large Reynolds number

“…More generally, such a flow region is also found in other types of interacting fully developed turbulent boundary layers (see e.g. Melnik & Chow 1975;Sykes 1980;Agrawal & Messiter 1984). Therefore, determining the order of magnitude of δ in terms of Re not only completes the scaling of the boundary layer but also answers the question of the existence of that blending layer, and in turn the asymptotic structure of the flow near D.…”

“…In the first case, flow reversal in the predominantly inviscid small-defect region that reaches close to the wall requires a pressure rise O(1) acting on a relatively short streamwise distance, which contradicts the original assumption; in the second case, the transcendentally thin viscous sublayer prevents the formation of viscous-inviscid boundary layer interaction that is sufficiently strong to ensure a smooth continuation of the incident boundary layer into a separated shear layer (cf. Sykes 1980). In order to overcome this dilemma, two disparate routes to separation can be established: in the first situation, the assumption of a large velocity defect leads to a multi-layered flow structure that distinctly differs from that outlined here and in turn to the theory of turbulent marginal separation (Scheichl & Kluwick 2007a,b).…”

“…Beyond that, it might be useful for future self-consistent descriptions of, for example, turbulent boundary layer flows over wall-mounted obstacles (cf. Sykes 1980) or turbulent boundary layer/shock wave interaction (cf. Agrawal & Messiter 1984 and references therein) that include separation.…”

Numerical prediction of turbulent flow over a two‐dimensional ridge

“…When examining cases where the flow field boundaries retain continuity of the second spatial derivative and a decent numerical grid is used, the assumption that the grid lines are straight between two adjacent grid nodes is a reasonable one and in accordance with the discretization process of equation (2).…”

Aeolian Transport and Dune Formation

“…An analytical calculation of the shear stress perturbation due to a two dimensional hill has been performed first by Jackson and Hunt [8]. Later, the work has been extended to three dimensional hills and further refined [9,10,11,12]. The following discussion is mainly based on the work of [13].…”