Abstract: We investigate asymptotics of two-dimensional steady solutions simulating liquid flows in the steady hydraulic jumps and solitary waves. These asymptotics are obtained from two-dimensional integral conservation laws of mass, momentum and energy using the local hydrostatic approximation, which generalises the classical long-wave approximation. The application of the initial (zero-order) asymptotics leads to conservation laws of the Saint–Venant equations. In the case of a steady hydraulic jump, a family of stable discontinuous piecewise constant solutions is obtained. When constructing the first-order asymptotics for the liquid depth, we obtain the momentum conservation law for the Green–Naghdi model, from which follows an explicit formula for these asymptotics. Using this formula, we derive the first-order asymptotics for the horizontal and vertical liquid velocities, given respectively by quadratic and linear functions of the vertical coordinate. We show that in the case of the hydraulic jump, the first-order asymptotics are applicable only in its upstream part, where the fluid moves in the direction of the jump front. This means that such asymptotics cannot model the entire hydraulic jump. For the solitary wave, we obtain restrictions on the flow parameters such that the first-order asymptotics are correct only in certain sections of the wave. Therefore, we derive conditions under which these asymptotics can model the entire solitary wave. We compare the constructed asymptotics with the results of experimental modelling of the flows under consideration.

Cite: Kovyrkina O. , Ostapenko V.
Asymptotics of liquid flows in the hydraulic jumps and solitary waves
Journal of Fluid Mechanics. 2026. V.1042. A24 . DOI: 10.1017/jfm.2026.11987 Scopus OpenAlex

Identifiers:

≡ Scopus:	2-s2.0-105051218449
≡ OpenAlex:	W7211897399

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