Mathematical model of density current development
[ jeudi 14-12-2006 17:00
| anglais ]Mathematical model of density current development
Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia
Two-dimensional gravity currents flowing down a slope have been investigated theoretically. The gravity currents are generated by a constant mass flux of dense fluid on the top of the slope. The flow patterns are described and analyzed by the three-layer shallow water equations taken into account the mixing and entrainment processes. The multi-layer shallow water approximation is used, in which a high entrainment zone, where mixing between the dense bottom flow (the core) and ambient fluid occurs, is treated as the intermediate layer. The simple mathematical model describing the evolution of the intermediate turbulent layer in three-layer flow over topography have been developed (Liapidevskii, 2004). Within the model the entrainment rate of fluid from the homogeneous layer can be found by the energy consideration without employing any empirical functions.
The analytical solution gives the boundaries and intensity of the mixing layer in a steady-state flow over an incline as well as the nonstationary structure of the density flow head. After the lower boundary of the mixing layer approaches the bottom, entrainment in the bottom layer deceases abruptly and vertical mass transfer from the bottom layer is only due to wave instability of the interface and roll wave generation (Boudlal & Liapidevskii, 2002). The well-known hypothesis on the density current head velocity Df = 1.5(M)1/3 for the constant intensity of the buoyancy source M = bhouo is supported by the model for different angles of the channel inclination. The relations at the head of the gravity current express very clear statement that the flow behind the front is critical. They give the analogy between the gas dynamics processes (normal detonation) and let formulate proper boundary conditions in nonstationary calculation of gravity currents.
The two-layer structure of the bottom gravity current (the undiluted core and the turbulent intermediate layer in which the density drops very quickly) is used to explain the details of the splitting process during the interaction of the head of gravity current with a pycnocline.
Liapidevskii V.Yu. “Mixing layer on the lee side of an obstacle” (2004), J. Appl. Mech. Tech. Phys., 45(2), 199-203.
Boudlal A. & Liapidevskii V., “Stability of roll waves in open channel flows” (2002), CR Mécanique, 303, 291-295.
Britter R.E. & Linden P.F. “The motion of the front of a gravity current traveling down an incline” (1980), J. Fluid Mech., 99, 531-543.