ALBERT

Description: Solutions to Long's equation for a stably stratified incompressible fluid traversing a mountain range are obtained for various terrain shapes and amplitudes when the horizontal scale is large compared to the vertical wavelength. Nonlinear lower and upper (radiative) boundary conditions are utilized and found to have a strong influence on the wave structure at large amplitudes. The results for symmetric and asymmetric mountain profiles reveal that the wave amplitude and wave drag are significantly enhanced for mountains with gentle windward and steep leeward slopes. These results confirm and explain those obtained by Raymond (1972) using a different solution method. Several results obtained by Smith (1977) from perturbation analysis are also confirmed and extended to large amplitudes. The methods are also applied to investigate the nonlinear nature of the singularity predicted by linear theory for flow over a step. © 1979, Cambridge University Press

Description: Nearly all analytical models of lock-exchange flow are based on the shallow-water approximation. Since the latter approximation fails at the leading edges of the mutually intruding fluids of lock-exchange flow, solutions to the shallow-water equations can be obtained only through the specification of front conditions. In the present paper, analytic solutions to the shallow-water equations for non-Boussinesq lock-exchange flow are given for front conditions deriving from free-boundary arguments. Analytic solutions are also derived for other proposed front conditions - conditions which appear to the shallow-water system as forced boundary conditions. Both solutions to the shallow-water equations are compared with the numerical solutions of the Navier†"Stokes equations and a mixture of successes and failures is recorded. The apparent success of some aspects of the forced solutions of the shallow-water equations, together with the fact that in a real fluid the density interface is a free boundary, shows the need for an improved theory of lock-exchange flow taking into account non-hydrostatic effects for density interfaces intersecting rigid boundaries. © 2011 Cambridge University Press.

Description: For the uniform flow past a semi-infinite flat plate subject to a blowing velocity profile equal to C(Uv/x),½ the conventional boundary-layer approximations break down as C approaches 0middot;6192. Here, we consider the structure of the flow for large Reynolds numbers R when C exceeds this critical value. It is shown that, for C 〉 0·6192, a region containing injected fluid O(R-1/3)) in thickness forms directly above the plate. To a first approximation the flow in this region is inviscid and the pressure a function of x only. This blowing region is separated from the free stream by a free shear boundary layer of thickness O(R-½). Thus the flow domain consists of three distinct regions which interact to yield a similarity solution valid for large values of Rx. This solution is then extended to higher order by expanding the stream function in each region in powers of (Rx)-1/3 and evaluating the first four terms in the resulting series using standard matching techniques. Finally, more general blowing profiles which also lead to boundary-layer ‘blow off’ are considered and an expression, valid far downstream of boundary-layer detachment, is derived for the position of the streamline separating the injected fluid from that of the free stream. For the case of uniform blowing the blowing region takes on the shape of a wedge, indicating that no solution can exist for the corresponding external flow if the plate is truly semi-infinite. © 1972, Cambridge University Press. All rights reserved.

Description: According to classical boundary-layer theory, when two uniform parallel streams are brought into contact at large Reynolds number (R) the location of the dividing streamline remains indeterminate to O(R-1/2) if both streams are subsonic and semi-infinite in extent. It is demonstrated here that this indeterminacy is a fundamental property of such a system which cannot be resolved, as Ting (1959) proposed, by balancing the pressure across the viscous mixing region to higher order in R. © 1972, Cambridge University Press. All rights reserved.

Description: The problem of uniform flow past a flat plate whose surface has a constant velocity λU opposite in direction to that of the mainstream is considered for large values of the Reynolds number R. In a previous communication (Klemp & Acrivos 1972) it was shown that, if the region of reverse flow which is established next to the plate as a consequence of its motion is O(R−1/2) in thickness, the appropriate laminar boundary-layer equations have a solution provided λ ≤ 0·3541. Here the analysis is extended to the range λ 〉 0·3541, which cannot be treated using a conventional boundary-layer approach. Specifically, it is found that for λ 〉 0·3541 the flow consists of three overlapping domains: (a) the external uniform flow; (b) a conventional boundary layer with reverse flow for xs 〈 x 〈 1, where xs, refers to the point of detachment of the ψ = 0 streamline and x = 1 is the trailing edge of the plate; and (c) an inviscid collision region in the neighbourhood of xs, having dimensions O(R−1/2) in both the streamwise and the normal direction, within which the reverse moving stream collides with the uniform flow, turns around and then proceeds downstream. It is established furthermore that xs = 0 for 0 ≤ λ ≤ 1 and that xs 〈 0 for λ 〉 1. Also, detailed streamline patterns were obtained numerically for various λ's in the range of 0 ≤ λ ≤ 2 using a novel computational scheme which was found to be more efficient than that previously reported. Interestingly enough, the drag first decreased with λ, reached a minimum at λ = 0.3541, and then increased monotonically until, at λ = 2, it was found to have attained essentially the value predicted from the asymptotic λ → ∞ similarity solution available in the literature. Thus it is felt that the present numerical results plus the two similarity solutions for λ = 0 and for λ → ∞ fully describe the high-R steady flow for all non-negative values of λ. © 1976, Cambridge University Press. All rights reserved.

Description: According to classical boundary-layer theory, uniform flow past a semi-infinite wedge, inclined at a negative angle ½Πβ to the direction of the free stream, does not separate unless β ≤-0·1988. It has been assumed, therefore, that, in the range -0·1988 〈 β 〈 0, the flow within the boundary layer is represented by the Falkner-Skan equation, which, as was shown by Stewartson (1954), has two admissible solutions. All such solutions for p 〈 0 appear to be somewhat unsatisfactory, however, because they require an adverse pressure gradient, which, by becoming infinite as the corner of the wedge is approached, could lead to separation even if, β 〉 -0·1988. In addition, the structure of the high Reynolds number flow for β 〈 -0·1988 has remained, to date, unresolved. We present here a fundamentally different solution to this classical problem which eliminates the singularity in the potential region by allowing the flow to separate at the leading edge of the inclined surface. The associated flow field is then characterized by an essentially uniform free stream flowing over an inviscid and, to a high approximation, irrotational region of reverse flow in which the velocity is of O(R-½) in magnitude, R being the Reynolds number. Mixing of these two streams is confined to a free shear boundary layer, of O(R-½) in thickness, extending downstream from the leading edge and parallel to the direction of the undisturbed main flow. Finally, an additional boundary layer, of O(R-½) in thickness, is shown to exist between the separated region and the surface of the wedge. Owing to the absence of a characteristic length in the problem, similar solutions to the appropriate equations describing the flow in each region are obtained and are valid for all β 〈 0, provided that the Reynolds number is sufficiently large. The analysis is then extended to higher order in R to increase its range of validity and to demonstrate that the proposed structure of the flow field remains self-consistent. Although the solution is developed only for a semi-infinite wedge with β 〈 0, it is believed that certain of its features may be of value in the analysis of other problems involving high Reynolds number separated flows. © 1972, Cambridge University Press. All rights reserved.

Description: If a region of reverse flow remains confined within a boundary layer the conventional boundary-layer equations should continue to apply downstream of the point of detachment of the surface streamline (Ω = 0). Nevertheless, standard numerical techniques fail in the presence of backflow since these methods become highly unstable and, in addition, neglect the upstream flow of information. A procedure for numerically integrating the boundary-layer equations through a region of reverse flow which takes downstream influence into account is therefore presented. This method is then applied to the problem of uniform flow past a parallel flat plate of finite length whose surface has a constant velocity directed opposite to that of the main stream. Although singularities occur at both the point of detachment (xs) and reattachment (xr) of the Ω = 0 streamline, this integration technique provides a solution which ceases to apply only in the close proximity of these singular points. From this solution it is evident that, throughout a large portion of the separated region, the flow is strongly affected by conditions near xr, thereby demonstrating the importance of allowing information to be transmitted upstream in a region of backflow. Near (xs), however, it is found that, in spite of the presence of reverse flow, the solution has a self-similar form in this particular example. © 1972, Cambridge University Press. All rights reserved.