The propagation of twodimensional and ... （将二维和传播u2026u2026）.pdf

43 J. Fluid Mech. (1982),vol. 121,pp. 43-58 Printed in Grcat Britain The propagationoftwo-dimensionaland axisymmetric viscous gravity currentsoverarigidhorizontalsurface By HERBERT E.HUPPERT Department ofApplied MathematicsandTheoreticalPhysics, Silver Street, CambridgeCB3SEW (Received2 September 1981andin revisedform 23March 1982) The viscous gravity current that results when fluid flows along a rigid horizontal surfacebelow fluid of lesserdensity is analysedusing a lubrication-theory approxi- mation. Itisshownthat theeffectonthegravity currentofthemotion intheupper fluidcanbe expressedasacondition ofzero shearontheunknown upper surfaceof the gravity current. With the supposition that the volume of heavy fluid increases withtimelikeF,whereaisaconstant,asimilaritysolutiontothegoverning nonlinear partial differential equations is obtained, which describes the shape and rate of propagation ofthe current. Theviscoustheory is shownto be valid for t t, when a -ca,andfort -4t,when a a,,wheret, isthetransition timeatwhichtheinertial and viscous forcesare equal, with a,=$ for atwo-dimensional current and a, =3 for an axisymmetric current. The solutions confirm the functional forms for the spreading relationships determined for a =1 in the preceding paper by Didden Maxworthy (1982),aswell asevaluatingthe multiplicativefactorsappearing in the relationships. The relationshipscompareverywellwith experimental measurements ofthe axisymmetric spreadingof silicone oilsinto air for a =0and 1.There is also very good agreement between the theoretical predictions and the measurements of the axisymmetric spreading of salt water into fresh water reported by Didden Maxworthy and by Britter (1979).The predicted multiplicative constant is within 10Yoofthat measured by Didden Ma