http://orcid.org/0000-0002-0734-9569
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ABSTRACT
A new front condition for gravity currents in a two-layer shallow water framework with a free surface is developed by applying the theory of the weak formulation to a first order system of four partial differential equations written in conservation form. The Rankine–Hugoniot jump conditions are applied and a relation between the velocity and the height of the shock is obtained. Laboratory experiments on lock-release non-Boussinesq gravity currents are performed and the front propagation and the free surface displacement are evaluated. Front velocities, predicted by the present mathematical model, are compared to both laboratory measurements and the predictions of the Benjamin formula, which is based on the rigid lid approximation. The new jump condition is in a good agreement with the laboratory measurements, while the Benjamin formula exhibits an overestimation of the shock velocity in the non-Boussinesq cases. In addition, the free surface displacement near the shock predicted by the new model is in agreement with the laboratory measurements.
ORCID
C. Adduce http://orcid.org/0000-0002-0734-9569
Notation
| b | = | tank width (m) |
| c | = | shock velocity (m s−1) |
| = | dimensionless shock velocity (–) | |
| = | dimensionless squared celerity (–) | |
| = | infinitesimal volume of dense fluid per unit length (m2) | |
| = | infinitesimal volume of ambient fluid per unit length (m2) | |
| = | source vector | |
| = | flux vector | |
| g | = | gravity acceleration (m s−2) |
| = | reduced gravity (m s−2) | |
| = | lock height (m) | |
| = | height of dense fluid (m) | |
| = | height of ambient fluid (m) | |
| H | = | total height (m) |
| = | set of test functions having compact support in Ω | |
| L | = | tank length (m) |
| r | = | density ratio (–) |
| R | = | lock aspect ratio (–) |
| t | = | time (s) |
| = | unknown vector | |
| = | mean velocity of dense fluid (m s−1) | |
| = | upstream mean velocity of dense fluid (m s−1) | |
| = | upstream dimensionless mean velocity of dense fluid (–) | |
| = | mean velocity of ambient fluid (m s−1) | |
| = | upstream mean velocity of ambient fluid (m s−1) | |
| = | upstream dimensionless mean velocity of ambient fluid (–) | |
| = | vector upstream of the shock | |
| = | vector downstream of the shock | |
| = | control volume per unit length (m2) | |
| x,y | = | spatial coordinates (m) |
| = | lock length (m) | |
| α | = | height ratio (–) |
| = | free surface superelevation (m) | |
| = | dimensionless free surface superelevation (–) | |
| Γ | = | shock curve |
| = | heavier fluid density (kg m−3) | |
| = | lighter fluid density (kg m−3) | |
| = | parametric representation of the shock curve | |
| Ω | = | domain in the x,t plane |
| = | sub-domains of Ω divided by Γ |