Integral methods for shallow free-surface flows with separation

Abstract

We study laminar thin film flows with large distortions in the free surface using the method of averaging across the flow. Two concrete problems are studied: the circular hydraulic jump and the flow down an inclined plane. For the circular hydraulic jump our method is able to handle an internal eddy and separated flow. Assuming a variable radial velocity profile like in Karman-Pohlhausen's method, we obtain a system of two ordinary differential equations for stationary states that can smoothly go through the jump where previous studies encountered a singularity. Solutions of the system are in good agreement with experiments. For the flow down an inclined plane we take a similar approach and derive a simple model in which the velocity profile is not restricted to a parabolic or self-similar form. Two types of solutions with large surface distortions are found: solitary, kink-like propagating fronts, obtained when the flow rate is suddenly changed, and stationary jumps, obtained, e.g., behind a sluice gate. We then include time-dependence in the model to study stability of these waves. This allows us to distinguish between sub-and supercritical flows by calculating dispersion relations for wavelengths of the order of the width of the layer.

An accurate and comprehensive model of thin fluid flows with inertia on curved substrates

“…Newton iteration then finds the steady solutions for fluid thickness η(r) and mean velocityū(r) shown in figure 10(a). The flow spreads out in a supercritical thin (η ≈ 1) fast flow before Bohr et al (1996), see also Watanabe, Putkaradze & Bohr (2003). Our model expressed in depth-averaged quantities resolves such non-trivial internal flow structures.…”

The Origin of Cores and Density Profiles of Gaseous Baryonic Structures

“…2 Already from this one sees that a spherical solution, in equation (3), must have a different radial structure for the inner and outer solutions (because the only solutions are , Ϫ2), a p 1 and hence one may expect to find different density profiles in the central and outer regions. We point out that such a phenomenon as the simultaneous existence of two flow patterns is rather common in hydrodynamics; the simplest may be the hydraulic jump that is observed as a several centimeter large circular ring in any kitchen sink, when the flowing water goes from a profile to a constant (see Hansen et al 1997 and1/r Watanabe, Putkaradze, &Bohr 2003). We emphasize that a nonzero viscosity appears to be a necessary condition for the existence of these specific solutions, even though a fundamental understanding of how the microscopic physics (viscosity) can determine the macroscopic properties (the general flow pattern) is still generally missing in hydrodynamics.…”

The hydraulic jump and the shallow-water equations

“…However, the traditional technique used in literature [5,7,8,[15][16][17] has been to vertically average these equations to obtain an evolution equation for the height-profile. The evolution equation is usually derived using an additional self-similar assumption on the velocity-profile.…”