Worthington jets are a familiar sight in light rain upon puddles and ponds. These narrow vertical jets are formed by the radial collapse of the liquid "craters" produced by the impacting rain drops. Such jets can also be generated by super-critically forcing the standing Faraday waves on a liquid surface and have recently been cast in the formalism of physical singularities to investigate the role of the inertial focusing and the influence of surface tension on their strength. Zeff et al. propose that during the collapse of the free-surface cavity, the surface develops a curvature singularity which is dominated by inertia and surface tension, hence generating high-energy vertical jets. We have discovered that similar narrow jets occur even for granular materials, where surface tension is absent. This new phenomenon suggests that a singularity in the surface tension force is not needed to produce such jets and raises the question whether the inertial focusing is the sole mechanism. . . .published or submitted for publicationis peer reviewe

Shen, Amy Q.

Thoroddsen, Sigurdur T.

English

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Granular jets

When a solid sphere impacts on a deep layer of granular medium, it generates an ejecta sheet and a transient axisymmetric crater. The gravity-driven radial collapse of this crater generates a pressure spike, as the cavity closes up. This pressure spike drives up a narrow granular jet along the axis of
symmetry. The maximum height of the jet is found to depend on the impact velocity, gravity as well as the effective viscosity of the granular medium, through a simple product of the Reynolds and Froude numbers. The presence of such granular jets, where surface tension is absent, may help pinpoint the role of surface tension for similar liquid jets.published or submitted for publicationis peer reviewe

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PHYSICS OF FLUIDS VOLUME 13, NUMBER 1 JANUARY 2001Granular jetsS. T. Thoroddsen and Amy Q. ShenDepartment of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign,Urbana, Illinois 61801-2935~Received 9 June 2000; accepted 4 October 2000!When a solid sphere impacts on a deep layer of granular medium, it generates an ejecta sheet anda transient axisymmetric crater. The gravity-driven radial collapse of this crater generates a pressurespike, as the cavity closes up. This pressure spike drives up a narrow granular jet along the axis ofsymmetry. The maximum height of the jet is found to depend on the impact velocity, gravity as wellas the effective viscosity of the granular medium, through a simple product of the Reynolds andFroude numbers. The presence of such granular jets, where surface tension is absent, may helppinpoint the role of surface tension for similar liquid jets. © 2001 American Institute of Physics.@DOI: 10.1063/1.1328359#Worthington jets1 are a familiar sight in light rain uponpuddles and ponds. These narrow vertical jets are formed bythe radial collapse of the liquid ‘‘craters’’ produced by theimpacting rain drops.2–6 Such jets can also be generated bysupercritically forcing the standing Faraday waves7,8 on aliquid surface and have recently been cast in the formalismof physical singularities2,3 to investigate the role of the iner-tial focusing and the influence of surface tension on theirstrength.We have discovered that similar narrow jets occur evenfor granular materials, where surface tension is absent.The sequence in Fig. 1 shows the generation of a granu-lar jet resulting from the impact of a solid lead sphere onto adeep flat layer of granular medium consisting of sphericalglass beads. The impacting sphere produces a deep cylindri-cal cavity in the sand, which subsequently collapses radiallyunder the gravity-induced ‘‘hydrostatic’’ pressure. The sandconverges axisymmetrically towards the center of the cavity.Due to the relatively small compressibility of the granularmedium, the radial velocity diverges as 1/r when the cavitycloses up. This inertial focusing9 produces a large dynamicpressure spike driving up the sand in a narrow jet along theaxis of symmetry. The granular jets are quite narrow, beingcomparable in shape to the energetic liquid jets. The jet inFig. 1~d! is, however, about 40 grain-diameters wide, wherethe grain size clearly would provide an ultraviolet cutoff inthis process, thus supporting a continuum viewpoint of theflow. The porosity of the sand medium may result in an evenmore pronounced inertial focusing than in the liquid case, asthe gas caught on the axis of symmetry can escape betweenthe grains.A similar setup using the impact of a solid sphere wasused by Hogrefe et al.4 to generate jets in a liquid.10 Thegranular jets described here should clarify the relative impor-tance of surface tension and inertial focusing in the liquidcase.The tuning of the granular jet does not depend on thelayer depth5 as in the fluids case, however, the granular jetheight is strongly dependent on the grain size of the granular41070-6631/2001/13(1)/4/3/$18.00medium ~see Fig. 2!. We find that the highest jets occur byusing the finest granular medium for the same impact veloc-ity.We should mention that vertically vibrated thin granularlayers have been shown to develop ‘‘oscillons’’ that areheaps emanating from the surface after the closing ofcraters,11 but those are much less energetic than the jets stud-ied here.Our experiments may also give insights into the consti-tutive properties of flowing granular media at high shearrates. The quantity we use to characterize the strength of thejetting event is the maximum height attained by the jet H j .The absence of surface tension makes dimensional analysismore tractable here than in the liquid case. The only otherphysical quantities of importance are: the sphere diameterDb , the impact velocity U, gravity g, along with materialproperties: the density of the sand rs, the sphere material rb ,and finally the effective viscosity of the granular media me .Here we assume that the grain size ds only enters the prob-lem through the effective viscosity. The dimensional analysisshows that the jet height should follow an unknown functionof only three dimensionless parameters:H j /Db5F~rr ,Re,Fr!,i.e., a density ratio rr5rb /rs , Reynolds number Re5rsUDb /me , and a Froude number Fr5U/AgDb. We havestudied this relation by keeping rr constant while varying Uand me independently, as shown in Fig. 3. The effective vis-cosity of flowing granular media remains an active topic ofresearch12–17 and is far from fully characterized. We use theresults of Savage et al.13–15 based on their shear cell experi-mental results and the kinetic theory of granular materials,me.2rsds2U/Db.In the current granular flow during the impact, the granu-lar volume fraction is not fixed and the flow may be gov-© 2001 American Institute of Physics5Phys. Fluids, Vol. 13, No. 1, January 2001 Granular jetsFIG. 1. Granular jet generated by theimpact of a lead sphere (U55.5 m/s,Db51.34 cm, rb511.5 g/cm3) onto adeep layer of granular medium, con-sisting of spherical glass beads 0.080mm in diameter. The medium sits in acircular glass container 12.2 cm deepand 18.8 cm in diameter. ~a! t5218ms; ~b! t525 ms; ~c! t592 ms; ~d! t5160 ms; ~e! t5260 ms. The upwardspeed of the granular jet is about one-third of the sphere impact velocity.erned more by compressive rather than shear stresses, there-fore, the above model is only a rough approximation.Furthermore, note that with the above definition of theeffective granular viscosity, the Reynolds number simply re-duces to a geometric factor Re5 12(Db /ds)2, which is a con-stant for each grain size. The effective dynamic viscosityfrom the above formulas varies here in a range between 4and 200 times that of water. The corresponding Reynoldsnumbers are between 1200 and 14 000.Dimensional analysis does not provide the form of thefunction F . However, using the above definition we find thatthe normalized jet height collapses using a simple product ofthe Froude and Reynolds numbers, as shown in Fig. 3.Instead of using the Reynolds number one could simplyreplace it by the dimensionless ratio Db /ds in the abovedimensional analysis. This would, however, obscure the ef-FIG. 2. The maximum height attained by the granular jet versus the releaseheight of the lead sphere. Granular media consist of spherical glass beadswith four different mean diameters: ds50.080 mm (1), 0.118 mm (n),0.176 mm (j), and 0.275 mm (s). The glass has a density of 2.4860.05 g/cm3. The material was poured slowly between two identical glasscontainers and leveled using a metallic ruler dragged over the surface. Thiswas done before every impact, to avoid any effects of compaction, whichcan greatly alter the results for the finest grains.fects of frictional dissipation, which clearly plays a crucialrole in this process. Ignoring friction, one can construct asimple energy balance between the kinetic energy of the im-pacting sphere }rbDb3U2 and the potential energy of the jetcolumn }rsgds2H j2, where we assume the jet thickness isproportional to ds . This produces the following scaling:H j /Db}rrFrARe, which does not collapse the data in Fig. 2.In other words, were one to increase the grain diameter ds bya factor of 10 ~i.e., reduce Re by 100! for the same impactconditions, then the frictionless flow would predict a 10-foldreduction in H j , whereas the experiments suggest a 100-foldreduction. This shows clearly that viscous forces play a rolein the granular jetting, as well as gravity and inertia.High-speed video clips showing the granular jet can beviewed at our web site http://www.tam.uiuc.edu/Faculty/Thoroddsen/GranularJet.htmlFIG. 3. The dimensionless plot of the maximum height attained by thegranular jet, versus the jetting parameter Fr3Re. Granular media consist ofspherical glass beads with four different mean diameters: ds50.080 mm(1), 0.118 mm (n), 0.176 mm (j), and 0.275 mm (s).6 Phys. Fluids, Vol. 13, No. 1, January 2001 S. T. Thoroddsen and A. Q. Shen1A. M. 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Fluid Mech. 127, 103 ~1983!.9G. Birkhoff, D. P. MacDougall, E. M. Pugh, and G. I. Taylor, ‘‘Explosiveswith lined cavities,’’ J. Appl. Phys. 19, 563 ~1948!.10A. Prosperetti and H. N. Oguz, ‘‘The impact of drops on liquid surfacesand the underwater noise of rain,’’ Annu. Rev. Fluid Mech. 25, 577~1993!.11P. B. Umbanhowar, F. Melo, and H. L. Swinney, ‘‘Localized excitation ina vertically vibrated granular layer,’’ Nature ~London! 382, 793 ~1996!.12R. A. Bagnold, The Physics of Blown Sand and Desert Dunes ~Methuen,London, 1941!.13C. K. Lun, S. B. Savage, D. J. Jeffrey, and N. Chepurniy, ‘‘Kinetic theo-ries for granular flow: Inelastic particles in Couette flow and slightly in-elastic particles in a general flowfield,’’ J. Fluid Mech. 140, 223 ~1984!.14S. B. Savage, ‘‘Flow of granular materials,’’ Theoretical and AppliedMechanics ~Elsevier, New York, 1989!, pp. 241–266.15S. B. Savage and M. 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Granular jets

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