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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Vydáno v:	Studies in applied mathematics (Cambridge) Ročník 102; číslo 3; s. 221 - 266
Hlavní autoři:	Montgomery, P. J., Moodie, T. B.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Boston, USA and Oxford, UK Blackwell Publishers Inc 01.04.1999 Blackwell
Témata:	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
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Popis
Shrnutí:	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.
Bibliografie:	ArticleID:SAPM110 ark:/67375/WNG-965XN15J-F istex:385945D4DB49DB930CB7CA55DA0E19AB91609B84
ISSN:	0022-2526 1467-9590
DOI:	10.1111/1467-9590.00110