Two-layer Gravity Currents with Topography

Two‐dimensional and time‐dependent gravity currents involving the initial release of a fixed volume of heavy fluid over a gradually sloping bottom and underlying a layer of lighter fluid are considered. The equations which describe the resulting two‐layer flow are derived from the Navier–Stokes equa...

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Veröffentlicht in:Studies in applied mathematics (Cambridge) Jg. 102; H. 3; S. 221 - 266
Hauptverfasser: Montgomery, P. J., Moodie, T. B.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Boston, USA and Oxford, UK Blackwell Publishers Inc 01.04.1999
Blackwell
Schlagworte:
Exact sciences and technology
Fluid dynamics
Fundamental areas of phenomenology (including applications)
Laminar flows
Low-reynolds-number (creeping) flows
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Physics
Sciences and techniques of general use
Initial boundary value problem
Gravity current
Equation of motion
Initial value problem
Well posed problem
Jacobi matrix
Froude number
Finite element method
Incompressible flow
Conservation law
Boundary value problem
Navier Stokes equation
Classification
Relaxation method
Stratified flow
ISSN:0022-2526, 1467-9590
Online-Zugang:Volltext
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Beschreibung
Zusammenfassung:Two‐dimensional and time‐dependent gravity currents involving the initial release of a fixed volume of heavy fluid over a gradually sloping bottom and underlying a layer of lighter fluid are considered. The equations which describe the resulting two‐layer flow are derived from the Navier–Stokes equations for a constant density, inviscid, nonrotating fluid, neglecting kinematic viscosity, surface tension, and entrainment between the layers. A new addition to the theory is introduced in the form of a forcing term in the lower layer horizontal momentum equation which is incorporated to produce the characteristic structure typical of such gravity currents in the laboratory. This delaying term is restricted to the front of the gravity current, and as such is shown to be valid under conventional shallow‐water scaling assumptions. The hyperbolic character of the equations of motion is shown, a simple numerical test for hyperbolicity is derived from theoretical considerations, and these results are related to the stability Froude number of the flow. Well‐posedness of the initial boundary value problem is proven via localization of the equations, and the discussion is extended to a two‐point boundary value problem with examples of steady‐state and traveling wave solutions given for a bottom surface of constant slope. Numerical results are obtained by using a recently developed finite‐difference relaxation scheme for conservation laws, sufficiently modified herein to include spatial variability and forcing terms, which approximates the material interface at the front of the lower fluid layer as a shock. The effects of slope and the delaying force are investigated numerically to determine their theoretical importance, and the range of expected values is compared to published experimental results. Some calculations for the temporal evolution of the flow are produced that display the phenomenon of rear wall separation for nonzero slopes.
Bibliographie:ArticleID:SAPM110
ark:/67375/WNG-965XN15J-F
istex:385945D4DB49DB930CB7CA55DA0E19AB91609B84
ISSN:0022-2526
1467-9590
DOI:10.1111/1467-9590.00110