Abstract
Interesting analogies between shallow water dynamics and astrophysical phenomena have offered valuable insight from both the theoretical and experimental point of view. To help organize these efforts, here we analyze systematically the hydrodynamic properties of backwater profiles of the shallow water equations with 2D radial symmetry. In contrast to the more familiar 1D case typical of hydraulics, %already for {even in} isentropic conditions, a solution with minimum-radius horizon for the flow emerges, similar to the black hole and white hole horizons, where the critical conditions of unitary Froude number provide a unidirectional barrier for surface waves. Beyond these time-reversible solutions, a greater variety of cases arises, when allowing for dissipation by turbulent friction and shock waves (i.e., hydraulic jumps) for both convergent and divergent flows. The resulting taxonomy of the base-flow cases may serve as a starting point for a more systematic analysis of higher-order effects linked, e.g., to wave propagation and instabilities, capillarity, variable bed slope, and rotation.
Highlights
On January 16, 1630, Galileo Galilei sent a letter to Raffaello Staccoli, in which he stated: “(l’Idraulica), sempre da me tenuta per difficilissima e piena di oscurità” (Hydraulics, which I always reputed to be extremely difficult and obscure) [1].Browsing the recent literature on fluid analogies in astrophysics [2,3], Galileo may object that his quote does not refer to the darkness of the black holes, but to the mysteries of fluid dynamics, which still persist today
While most of the shallow water literature refers to 1D streams, here we focus on the role of the circular symmetry enforced by the continuity equation, which opens the possibility for convergent or divergent flows; we pay attention to the role of dissipation, provided by friction and, when present, by shock waves in the form of hydraulic jumps
The solutions of the shallow water equations present a variety of configurations, which besides their direct fluid dynamic interest may have useful implications as analogs
Summary
On January 16, 1630, Galileo Galilei sent a letter to Raffaello Staccoli, in which he stated: “(l’Idraulica), sempre da me tenuta per difficilissima e piena di oscurità” (Hydraulics, which I always reputed to be extremely difficult and obscure) [1]. The shallow water experiment and the collapse leading to a neutron star [19] are separated, after all, by only six orders of magnitude in size Within this context, our contribution aims at providing a systematic classification of the so-called backwater profiles (i.e., the elevation of the free surface as a function of the streamwise coordinate), possibly connected by shock waves (i.e., the hydraulic jumps), as solutions of the shallow water equations in circular symmetry. While most of the shallow water literature refers to 1D streams, here we focus on the role of the circular symmetry enforced by the continuity equation, which opens the possibility for convergent or divergent flows; we pay attention to the role of dissipation, provided by friction and, when present, by shock waves in the form of hydraulic jumps. While the interesting effects of capillarity and rotation are left for future work, we hope that the present analysis may be useful to provide a classification of base flows to analyze systematically the links between shallow water profiles and their astrophysical analogs
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
