The slumping gravity currents Silver Street, - PDF document

The slumping gravity currents Silver Street, up by axisymmetric spreading gravity current as initially given radial co-ordinate becomes so viscous forces, rather than forces, balance buoyancy forces. Using force for a current propagating under a free surface, inertia-buoyancy balance viscous-buoyancy balance is primarily vestigates a spreading, when balance between paper, Hoult above results by actually solving governing balancing forces. Arguing his analysis on motion in writes down gravity current. for axisymmetric spreading for two-dimensional spreading, volume per However, conservation argued previously by others conditicn, such one applied is controlled experience gained from condition is Froude number, suggests expressing determined from experiment. From a laws in a similarity calculation, these relationships valid only time has elapsed since be irrelevant. when viscosity dominates inertia, Hoult retains similarity variable variable for axisymmetric spreading, spreading, for two-dimensional spreading. (1.7b) The condition at the head is now somewhat unsatisfactory current needed cannot be postulated boundary-layer gradients are sniall compared gradients there. that as obtains (after correcting for surface-tension effects. a comprehensive series releasedfixed volumes ranging between channels. Amongst results, Keulegan gravity current resulting expression Keulegan's results compared. There agreement, either be discussed possible con- process from present paper out the physics involved physics. Keulegan's experiments carried out by gravity current becomes sufficiently gravity current and the head investigated experimentally simple model for The Froude number gravity current experimental results from analytic relationship relationships in in which dense fluid stage may inertial effects. stage as and Hoult then entirely absent. model does initial motion, during accelerates from predicted by model, is included. Nevertheless, physical principles involved predictions agree Gravity currents with large often propagate under an atmospheric effective lid pollutants are commonly pushed final example from mining practice, where explosive gas such as methane can escape simple model, which can be used Keulegan’s experi- presented in is based through a (see figures and the discussion in head is given gravity current steady-state experiments from a gravity current very much from figure be obtained steady-state gravity current The major particular extrapolation are equal areas, length. Substituting Substituting (’)(I, 1 Z, p/0*075H) (2.6) (2.7) = [Z, + g(g’3qH2/z0)*t] [I + O(g’3qH2t6/1;)+], where the slumping length 1, is the length of the gravity current when the fractional depth has has = t, = y(zt - Zi)/(q’3qH2)6], (2.8) where the slumping time t, is evaluated by setting 1 = 1, in (2.6), we obtain I = [@+ 1*78(g’p)*(t-t,)]3 (I, I I*) (2.9) - 1*47(g’q)ftf (t &#x Tj ;&#x/T1_;
1 ;&#xTf 0;&#x.17 ;&#xTc 6;&#x 0 0;&#x 11.; 33;�.63;3 2;E.3;٧ ;&#xTm 0; t,), (2.10) where 1, = (~~g‘v-~)+ is the length scale of the current when the viscous force for some time after the rate gravity current constant [and inferred from Further, as much from linear relationship approximately constant- be considered experiment, is encouraging. analysis in simple models used when considering axisymmetric currents. The equal-area assumption leads initial radius before release. The Froude-number are unaltered: length scale current is ships for propagating over over (‘)[R, R rs = (Q/0*075nH)f] (2.12) = [R, + +~-*(g’~&H~/Ri)*t] [ 1 + O(g’3&H2t6/R:)3], (2.13) and R = [rf+2.37na(g’Q)*(t-tt,)]* (r, R r*) (2.14) - 1.16(gf&)*t* (t $= ts), (2.15) where r, and r* are the radial equivalents for small times constant, and agreement between satisfactory, though free surface propagating over surface. This be discussed viscous-buoyancy phase our experiments, propagates over relationships derived (for propagation under are not viscosity is force on vertically averaged horizontal velocity volume conservation 14.9 14.9 30.0 0.50 30.0 0.50 44.0 0.34 14.9 45.2 0.33 29.8 29.8 10.2 44.2 0.23 The parameters similarity solution solution (’)(2.20) (2.21) Substituting (2.20) and (2.21) into (2.18), taking the first integral of the result and evaluating the single constant of integration by using (2.19), we determine that 1 = 1*41g’q3/v. (2.22) A similarity solution here. We which need propagating under above calculations buoyancy phase, experimental evaluation our experiments Plexiglas channel very kindly loaned us by The channel was filled 140 160 experiment number experiment number solid curves suitable density total depth sides was gravity current pre-set time intervals.? The parameters particular, experiments should like to thank collectively here. Almost every fluid dynamicist in one experiment gravity currents The ratio gravity currents predicted length in slumping phase I I The ratio line included suggested for initial fractional simple buoyancy- inertial relationship gravity current early stages is seen I I 500 750 1000 1250 5. Keulegan's memured two-dimensional gravity The solid curves, drawn for value viscous are important experimental results diverge from inviscid predictions. taken up further is evident, agreement between linear relationship measured values values + &(g'3qH2)it]G plotted as a function for all experiments. curves should should gradually decrease. gravity currents (cm) (cm3) 15.0 70.0 2 16.5 17.0 55.0 4 17.0 7 16.0 8 12.0 9 15.2 103.0 39 700 638 154.9 the actual except for 17) for viscous effects is completed. measured values values (‘)cf. (Z.lO),plotted as a function of t/t,. These values have a constant for small time (as in Hoult’s decrease again viscous forces overwhelm this point, documented previously by Hoult and plotted in predicted spreading relationship is in phase. The results experiments, Keulegan length. Differentiating relationship between slumping phase. Figure presents Keulegan’s original hand-drawn interpolated four experiments.? viscous effects are akin these in range where viscous effects slumping phase figure also was confirmed the data many other Keulegan’s experiments with experiments with relatively small values those with larger values. decrease in velocity experiments with smaller values consistently calculated for four gravity currents measured length nine axisymmetric gravity currents predicted length slumping phase measured length nine axisymmetric gravity currents Our experiments in a outlined above two-dimensional experiments. initial fractional depth had two-dimensional channel, axisymmetric experi- ments with initial fractional four typical agreement is be good. results in measured values evident, in The length our experiment viscous effects have presented rests two-dimensional rectangles or axisymmetric disks be described steady-state Froude number which agrees The appropriateness judged from four shadowgraph a two-dimensional slumping not yet is a intruding gravity current. tending towards second view is satisfyingly the third for a performed for a The results and this be discussed different surface tension fluids. While is only a our experiments considerable influence on We ascertained experiments with different surface tensions, keeping all by the experimental results for gravity current heads under a free surface contained in predictions. We this matter surface tension mentioned previously, been considered. occurs over short time importance. During subsequent inertial can increase, influence on gravity current appears in constant. The mixing during Longuet-Higgins for use his Plexiglas channel, for critical comments research has Procurement Executive Fluid Mech. Experiments on gravity current head. slicks on calm sea. spreading on Fluid Mech. engineer grapples nonlinear problems. The dynamics gravity current horizontal surface. release. The vertical lines seen on in the slumping phase gravity current release. The first two the third.