Influence of corner layers in the variational determination of bubble solutions of the constrained Allen–Cahn equation

A steady-state bubble solution to the constrained mass conserving Allen–Cahn equation in a two-dimensional domain is constructed in the limit of small diffusivity. The solution is asymptotically constant inside a circle of radius rb centred at some unknown location x0 and has a sharp interface at the bubble radius that allows for a transition to a different asymptotically constant state outside the bubble. In a study by M. J. Ward (Metastable bubble solutions for the Allen–Cahn equation with mass conservation. SIAM J. Appl. Math. 56, 1996, 247–1279), the bubble centre was determined by a limiting solvability condition. The solution found by Ward suggests the existence of a corner type boundary layer where the normal derivative of the bubble solution readjusts to satisfy the no-flux condition at the boundary of the domain. This work is concerned with the details of the readjustment. A variational approach similar to the one of W. L. Kath, C. Knessl and B. J. Matkowsky (A variational approach to nonlinear singularly perturbed boundary-value problems. Stud. Appl. Math. 77, 1987, 61–88) shows the formation of a corner layer (for the derivative of the solution) which influences as a high-order correction the available determination of the bubble centre. This corner layer describes to leading order the readjustment of the level lines of the bubble to lines parallel to the boundary of the container; moreover, it provides to leading order a smooth solution across the corner layer.

Flow along streamwise corners revisited

In this paper we investigate the three-dimensional laminar incompressible steady flow along a corner formed by joining two similar quarter-infinite unswept wedges along a side-edge. We show that a four-region construction of the potential flow arises naturally for this flow problem, the formulation being generally valid for a corner of an arbitrary angle (π−2α), including the limiting cases of semi- and quarter-infinite flat-plate configurations. This construction leads to five distinct three-dimensional boundary-layer regions, whereby both the spanwise length and velocity scales of the blending intermediate layers are O(δ), with Re−1/2 [Lt ] δ [Lt ] 1, Re being a reference Reynolds number supposed to be large. This reveals crucial differences between concave and convex corner flows. For the latter flow regime, the corner-layer motion is shown to be mainly controlled by the secondary flow which effectively reduces to that past sharp wedges with solutions being unique and existing only for favourable streamwise pressure gradients. In this regime, the corner-layer thickness is shown to be O(Re−0.5+α/π/δ2α/π), −½π [les ] α [les ] 0, which is much smaller than O(Re−1/2) for concave corner flows.Crucially, our numerical results show conclusively that, for α ≠ 0, closed streamwise symmetrically disposed vortices are generated inside the intermediate layers, confirming thus the prediction made by Moore (1956) for a rectangular corner, which has so far remained unconfirmed in the literature.For almost planar corners, three-dimensional corner boundary-layer features are shown, as in (Smith 1975), to arise when α ∼ O(1/ln Re). On the other hand, we show that the flow past a quarter-infinite flat plate would be attained when both values of the streamwise pressure gradient and external corner angle (π+2α) become O(1/ln Re) or smaller.Numerical results for all these flow regimes are presented and discussed.

Flow past wing-body junctions

The three-dimensional laminar flow past a junction formed by a thin wing protruding normally from a locally flat body surface is considered for wings of finite span but short or long chord. The Reynolds number is taken to be large. The leading-edge interaction for a long wing has the triple-deck form, with the pressure due to the wing thickness forcing a three-dimensional flow response on the body surface alone. The same interaction describes the flow past an entire short wing. Linearized solutions are presented and discussed for long and short two-dimensional wings and for certain three-dimensional wings of interest. The trailing-edge interaction for a long wing is different, however, in that the three-dimensional motions on the wing and on the body are coupled together and in general the coupling is nonlinear. Linearized properties are retrieved only for reduced chord lengths. The overall flow structure for a long wing is also discussed, including the traditional three-dimensional corner layer, which is shown to have an unusual singular starting form near the leading edge. Qualitative comparisons with experiments are made.

Streamwise Corners with Curvature Independent of Streamwise Distance

SummaryA two parameter approximate solution is presented for the flow along a streamwise corner having large, possibly infinite, curvature in its region of transition between the two quarter infinite planes which form its asymptotes. The two parameters are the angle between the asymptotes and a quantity proportional to the ratio of the local two-dimensional boundary layer thickness and the radius of curvature of the corner at the symmetry plane. Relative to the corner of infinite curvature (sharp corner) a finite curvature always tends to modify the solution towards that for a flat plate. This implies that for corner angles less than 180° the corner layer thickness in the symmetry plane is less than that for the sharp corner of the same angle and the shear stress is higher, the converse holding for angles greater than 180°. The flow in planes normal to the free stream direction is rather complex and typically there is a reversal in its direction within the symmetry plane.