The development of G6rtler vortices in wall jet flow over curved surfaces is considered in both the linear and nonlinear growth r~gimes. It is shown, using asymptotic methods based on the largeness of the wavenumber of the vortices, that this hydrodynamic instability is prone to occur more readily on concave rather than convex surfaces. It is found that after passing the position of neutral stability, the flow develops a surprising structure quite unlike that produced in the Blasius boundary-layer. Once the flow is into the unstable regime, the effect of increasing the G6rtler number is to move the vortices away from the wall. §1. Introduction
We shall consider some stability characteristics of G6rtler vortices in wall-jet flow over a curved surface. Most previous investigations have been limited to the study of this centrifugal instability in a Blasius boundary layer, where the monotonicity of the velocity profile requires that the surface be concave in order for the instability to become manifest. With the wall-jet velocity profile, however, flow over plates of either curvature can produce vortices. Glauert [1] gave the solution to the boundary-layer equations for the wall jet, which is found when, for example, a jet of air impinges on a surface. Carpenter et al. [2] demonstrated experimentally how G6rtler vortices may be found over Coanda surfaces. Such surfaces have industrial applications in waste-gas flares.
G6rtler [3] considered the stability of incompressible boundary layers over a slightly curved surface, and showed how a secondary flow consisting of vortices aligned with the principal direction of motion can be supported. His equations were solved approximately and produced the result that boundary layers on concave curved walls are unstable at sufficiently high flow speeds. The stability of the flow depends on the G6rtler parameter, G, which is related to the Reynolds number of the flow and the local concave or convex curvature of the wall.
Hfimmerlin [4] solved G6rtler's equations exactly and found the physically unacceptable result that the critical G6rtler number, Go, below which no instability could be found, corresponded to vortices of infinite wavelength. By retaining some higher order curvature terms, he re-derived the equations [5], the solution of which yielded a critical G6rtler number at non-zero wavenumber.
Numerous authors have considered modifications to Hhmmerlin's equations. Herbert [6] and Floryan and Saric [7] gave reviews of previous work, and the latter authors presented results concerning the effect of suction on the vortex mechanism. Floryan [8] has discussed the stability of the wall jet to G6rtler vortices, mentioning the result that this velocity field admits the instability on convex walls.
However, all the above authors used parallel-flow approximations, and their results disagree for vortices with small wavenumbers. Here, by 'parallel-flow', we mean that some of the terms in the governing equations have been neglected or replaced by simplified terms in a manner which does not reflect the structure of the G6rtler problem correctly. For example, Floryan [8] assumes that all disturbance quantities vary as exp(/3x) where x is the downstream co-ordinate, and/3 is the downstream spatial growth rate. Hall [9] showed that the parallel-flow theories had no mathematical justification except for high wavenumbers. He developed a formal asymptotic expansion of the appropriate linear stability equations based on the smallness of the wavelength of the imposed disturbance, and showed that the position for neutral stability in this high wavenumber r6gime depends on the form and location of the initial disturbance from the mean flow. It was shown that the vortices form in a region of thickness O(el/2), where 2~re is the (small) wavelength of the vortices under consideration, centred on the point where gffy is a local maximum, that is, the point at which Rayleigh's [10] stability criterion is most violated.
Stuart [11] and Watson [12] showed how non-linear effects could be taken into account close to the conditions of neutral stability under linear theory for plane poiseuille flow and for Couette flow. However, their approach required the correction to the mean flow to be an order of magnitude smaller than the mean flow itself. Hall [13] has shown that this is not the case for the G6rtler problem. He considered the weakly non-linear development of a locally neutrally stable vortex. He showed how in an e-neighbourhood of the point of neutral stability according to linear theory, X, say, the downstream velocity component due to the vortex is of the same magnitude as that of the mean flow. At an asymptotically large distance downstream of X,, it was shown how the 'correction' to the mean flow becomes an order of magnitude larger than the mean flow itself, for the Blasius boundary layer. At this stage, the vortex flow extends beyond the boundary layer into the free stream.
The full, linear, partial-differential equations were solved numerically by Hall [14] for vortices of O(1) wavenumber. An initial disturbance was imposed on the flow at some location and its downstream development was examined. The growth rate of the disturbance, based on a non-dimensional energy function associated with the flow, was found to depend on the form of the initial disturbance and the place at which it had been introduced to the flow. Hall concluded that no unique neutral curve for the G6rtler instability exists. He did, however, show that the different neutral curves merged to form one curve in the high wavenumber regime and that this was the curve which parallel-flow theorists had produced. This confirms the idea that such theories are only valid in this regime, and are of little use elsewhere in connection with the G6rtler problem.
Hall and Lakin [15] gave an account of the fully non-linear problem, using Hall [13] as a starting point. They showed how the region of vortex activity increases to an O(1) depth. The way in which the vortex disturbance in the core decays away in two bounding shear layers was given, and asymptotic solutions for the initial and ultimate forms of the instability were calculated. At an O(1) distance from the point where the vortices start to grow, a numerical calculation was performed in order to find the upper and lower bounds for the region of disturbance activity. The mean flow was found to be so changed by the presence of the vortex that it bears no relationship to the unperturbed state; indeed, the vortex was shown to drive the mean flow.
We shall use the approach of H~ill [9] to show how the right-hand branch of the neutral curve may be generated for wall-jet flow. We show how our results compare with those of Floryan [8] who used a normal mode approach. It is found that the third term in the asymptotic expansion of the neutral curve depends on the position in the flow where the requirement for neutral stability is applied. We develop the non-linear stability theory for