Numerical Analysis of a Linearized One-Dimensional Hydraulic Jump Model

In this paper, a one-dimensional linear hydraulic jump problem is numerically analyzed. The stability of the solution is investigated, the investigation is based on the known formulas for the linear problem's solution, which express it in the terms of the model parameters. A system of calculations, that allows to separate the regions of stability and instability and calculate their common boundary - the surface of neutral stability, is constructed. For the cases of stability and neutral stability, the principal asymptotic characteristics of the hydraulic jump are calculated: the front asymptotics and, accordingly, the first oscillation frequency. The results are compared with the available numerical results for the solution of the general (nonlinear) problem.

Keywords

hydraulic jump; shallow water theory; shear shallow water theory; shock wave; shock wave front.

References

1. Teshukov V.M. Gas-Dynamic Analogy for Vortex Flows with a Free Boundary. Applied Mechanics and Technical Physics, 2007, vol. 48, no. 3, pp. 8-15. (in Russian)
2. Richard G.L., Gavrilyuk S.L. A New Model of Roll Waves: Comparison with Brock's Experiments. Journal of Fluid Mechanics, 2012, vol. 698, pp. 374-405.
3. Richard G.L., Gavrilyuk S.L. The Classical Hydraulic Jump in a Model of Shear Shallow-Water Flows. Journal of Fluid Mechanics, 2013, vol. 725, pp. 492-521.
4. Delis A.L., Guillard H., Tai Yih-Chin. Numerical Simulation of Hydraulic Jumps with Shear Shallow Water Model. Journal of Computational Mathematics, 2018, vol. 4, p. 319-344.
5. Ivanova K.A., Gavrilyuk S.L. Structure of the Hydraulic Jump in Convergent Radial Flows. Journal of Fluid Mechanics, 2019, vol. 860, p. 441-464.
6. Semenko E.V. The Analytic Description of Hydraulic Jump in the Linear Theory of the Shear Shallow-Water Flows. Physics of Fluids, 2019, vol. 31, article ID: 016101. DOI: 10.1063/1.5072772