University of Alberta
Edmonton, Alberta, Canada
Gravity currents resulting from the instantaneous release of a finite volume of dense fluid into a semi-infinite reservoir of less dense fluid overlying an inclined plane of small slope are examined both theoretically and numerically to investigate the importance of small slope on the flow.
Time-dependent evolution equations which describe the flow are derived from the Navier-Stokes equations for a constant density, inviscid, non-rotating fluid. Kinematic viscosity, surface tension, and entrainment between the fluid layers are assumed to be of secondary importance and neglected. A new condition imposed upon the lower layer permits a shallow-water theory which is a consequence of low aspect ratio flow. The equations are shown to be hyperbolic in nature, and the initial boundary value problem is shown to be well-posed via the method of localization.
Numerical results are obtained using a finite difference relaxation method which is conservative and resolves shocks without producing spurious oscillations. This method is applied to the developed system of four equations to predict the front speeds of gravity currents which result from various initial conditions and bottom slopes. These calculations are then compared to previous theoretical and experimental results.
Reference:
[1] Two-layer gravity currents with topography, by P.J. Montgomery and T.B. Moodie, to appear in Studies in Applied Mathematics.