The dynamics of hydraulic jumps is described by hyperbolic partial differential equations with source terms due to frictional losses and channel-bed variations. The analysis of hyperbolic systems has traditionally been focused on systems without non-linear source terms, as the nonlinearities in the hyperbolic operators themselves are rather intricate, and play an important role in the nature of the solutions. Significantly less attention has been paid to the role of nonlinear source terms such as those arising in the shallow-water equations and in the reactive Euler equations. Such source terms are responsible for a rich variety of phenomena, including the complex dynamical features of detonation shock fronts. In this project, we investigate the role of the nonlinear wave interactions arising from source terms in the shallow-water system, which may be responsible for the formation of polygonal hydraulic jumps.
This project concerns the dynamics of peculiar flow structures that may emerge from the circular hydraulic jump. When a vertical jet impinges on a flat solid surface, the jet fluid spreads radially in a thinning film until reaching a critical radius at which the film thickness increases dramatically in what is termed a `hydraulic jump'. In certain parameter regimes, despite the axisymmetric source conditions, striking asymmetric flows emerge, including polygonal hydraulic jumps, the explanation for which remains elusive. Our combined theoretical, numerical and experimental project will be focused towards rationalizing these subtle flows by developing the mathematical analogy between hydraulic jumps and detonation shock fronts. A new area of mathematical analysis will be initiated and explored.
When a vertical fluid jet strikes a horizontal plate, the fluid spreads radially, giving rise to a fluid layer that thins until reaching a critical radius at which the fluid height jumps. The resulting abrupt change in fluid depthis called the hydraulic jump, whose radius is determined by a delicate balance between fluid inertia, gravity and surface tension. Quite remarkably, in a certain parameter regime, this jump transforms from a circular to a polygonal form (Figure 1). The rationale for this subtle jump structure has eluded explanation now for 15 years. The proposed research was directed towards rationalizing the polygonal hydraulic jump through an integrated theoretical and experimental approach. A number of surprising results emerged. Investigations of discontinuous solutions (for example, hydraulic jumps) in water-wave problems have a long and rich history, and are of great interest in terms of both applications and theory. Hydraulic jumps arise at a large scale in a variety of natural settings, including tidal flows inchannels and rivers, flood flows in natural disaster areas, atmospheric and oceanic currents. From a theoretical point of view, the subject of hydraulic jumps is a rich topicin fluid mechanics and has been the inspiration for many important mathematical developments.It has long been recognized that shallow water equations are analogous to the equations of classical gas dynamics. The central goal of the proposed research was to see if this analogy could be used to rationalize the polygonal hydraulic jump. The first component of the theoretical investigation was an application of geometric shock dynamics to the stability of the hydraulic jump. This approach demonstrated that the jump is indeed potentially unstable, witha complicated behavior prescribed by the governing fluid parameters. The governing equation indicates the existence not only of circular and polygonal jumps, but of time-dependent traveling-wave solutions, a cascadeof period-doubling transitions and the onset of chaos. The possibility of chaotic shock dynamics revealed through this study has bearing not only on the hydraulic jump, but on analogous gaseous detonation flows. The experiments undertaken at MIT highlighted the role of the toroidal `roller' vortex downstream of the jump in prompting the polygonal instability (Figure 2). It has been noted previously that the circular hydraulic jump suffers a polygonal instability only when the jump is adjoined by a toroidal vortex. Moreover, our experiments suggest that similar polygonal instabilities may accompany the toroidal vortex adjoining a plunging jet, even in the absence of a hydraulic jump. This study underscores the importance of thetoroidal vortex in this class of problems, and has motivated two independent but complementary theoretical investigations. The resulting relatively simple analysis elucidates a new and potentially critical mechanism for instability. The second experimental focus was on the spinning hydraulic jumps discovered through this study and their associated downstream flow structure (Figure 1). Our analysis indicates that the downstream flow, wherein striking spiral waves of surface elevation arise, bears a close resemblance to spiral arms in galaxies. The resemblance may be more than superficial, in which case it may ultimately shed light on the mechanisms of spiral arm formation and persistence in spiral galaxies, an enduring puzzle in astronomy.
