To elucidate the role played by surface tension on the formation and on the structure of a circular hydraulic jump, the results from three different approaches are compared: the shallow-water (SW) equations without considering surface tension effects, the depth-averaged model (DAM) of the SW equations for a flow with a parabolic velocity profile, and the numerical solutions of the full Navier-Stokes (NS) equations, both considering the effect of surface tension and neglecting it. From the SW equations, the jump can be interpreted as a transition region between two solutions of the DAM, with the jump's location virtually coinciding with a singularity of the DAM's solution, associated with the inner edge of a recirculation region near the bottom. The jump's radius and the flow structure upstream of the jump obtained from the NS simulations practically coincide with the results from the SW equations for any flow rate, liquid properties, and downstream boundary conditions, being practically independent of surface tension. However, the structure of the flow downstream of the jump predicted by the SW equations is quite different from the stationary flow resulting from the NS simulations, which strongly depends on surface tension and on the downstream boundary conditions (radius of the disk). One of the main findings of the present work is that no stationary and axisymmetric circular hydraulic jump is found from the NS simulations above a critical value of the surface tension, which depends on the flow conditions, fluid properties, and downstream conditions. Published under license by AIP Publishing. https://doi.Published under license by AIP Publishing FIG. 6. Stationary liquid film profiles (dimensional) obtained numerically from the NS equations for three liquids and different flow rates (see the main text), for jets of different diameters impinging on two disks with different radius, 80 mm (a) and 120 mm (b), with and without considering the effect of surface tension (except for water, which is only plotted for σ = 0 because no stationary flow is reached when using its actual surface tension σ = 0.072 N/m). The temporal evolution of the different flows is given in videos 1-10 of the supplementary material.

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Many axisymmetric vortex cores are found to have an external azimuthal velocity v, which diverges with a negative power of the distance r to their axis of symmetry. This singularity can be regularized through a near-axis boundary layer approximation to the Navier-Stokes equations, as first done by Long for the case of a vortex with potential swirl, v∼r−1. The present work considers the more general situation of a family of self-similar inviscid vortices for which v∼rm−2, where m is in the range 0 n< m < 2. This includes Longs Vortex for the case m =1. The corresponding solutions also exhibit self-similar structure, and have the interesting property of losing existence when the ratio of the inviscid near-axis swirl to axial velocity (the swirl parameter) is either larger (when 1m < 2) or smaller (when 0m < 1) than an m-dependent critical value. This behaviour shows that viscosity plays a key role in the existence or lack of existence of these particular nearly inviscid vortices and supports the theory proposed by Hall and others on vortex breakdown. Comparison of both the critical swirl parameter and the viscous core structure for the present family of vortices with several experimental results under conditions near the onset of vortex breakdown show a good agreement for values of m slightly larger than 1. These results differ strongly from those in the highly degenerate case m =1.

The dam-break flow problem in the shallow-water approximation on an inclined bed for arbitrary slopes of the bottom is considered. An analytical solution for the spreading of the water fronts at the initial stages is given. A self-similar solution asymptotically valid at large time is also found. For intermediate times the problem is solved numerically by the method of characteristics.

A spatial, viscous stability analysis of Poiseuille pipe flow with superimposed solid body rotation is considered. For each value of the swirl parameter ͑inverse Rossby number͒ LϾ0, there exists a critical Reynolds number Re c (L) above which the flow first becomes convectively unstable to nonaxisymmetric disturbances with azimuthal wave number nϭϪ1. This neutral stability curve confirms previous temporal stability analyses. From this spatial stability analysis, we propose here a relatively simple procedure to look for the onset of absolute instability that satisfies the so-called Briggs-Bers criterion. We find that, for perturbations with nϭϪ1, the flow first becomes absolutely unstable above another critical Reynolds number Re t (L)ϾRe c (L), provided that LϾ0.38, with Re t →Re c as L→ϱ. Other values of the azimuthal wave number n are also considered. For Re ϾRe t (L), the disturbances grow both upstream and downstream of the source, and the spatial stability analysis becomes inappropriate. However, for ReϽRe t , the spatial analysis provides a useful description on how convectively unstable perturbations become absolutely unstable in this kind of flow.