Supersonic Air Flow due to Solid-Liquid Impact

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by  Stephan Gekle

Supersonic Air Flow due to Solid-Liquid Impact
Stephan Gekle,1 Ivo R. Peters,1 Jos´e Manuel Gordillo,2 Devaraj van der Meer,1 and Detlef Lohse1
1 Department of Applied Physics and J.M. Burgers Centre for Fluid Dynamics,
University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
2 A´rea de Mec´anica de Fluidos, Departamento de Ingener´ıa Aeroespacial y Mec´anica de Fluidos,
Universidad de Sevilla, Avda. de los Descubrimientos s/n 41092, Sevilla, Spain
(Dated: January 21, 2010)
A solid object impacting on liquid creates a liquid jet due to the collapse of the impact cavity.
Using visualization experiments with smoke particles and multiscale simulations we show that in
0 addition a high-speed air-jet is pushed out of the cavity. Despitean impact velocity of only 1 m/s,
1 thisair-jetattainssupersonicspeedsalreadywhenthecavityisslightlylargerthan1mmindiameter.
0 Thestructureoftheairflowresemblescloselythatofcompressibleflowthroughanozzle–withthe
2
keydifference that here the“nozzle” is a liquid cavity shrinking rapidly in time.

n
a PACSnumbers: 47.55.D-,47.60.Kz,47.11.St,47.80.Jk
J

1
Takingastone andthrowingitontothe quiescentsur- A laser sheet (Larisis Magnum II, 1500mW) shining in
2
faceofalaketriggersaspectacularseriesofeventswhich from above illuminates a vertical plane containing the

] has been the subject of scientists’ interest for more than axis of symmetry of the system. A high-speed camera
n
a century [1–17]: upon impact a thin sheet of liquid (the (Photron SA1.1) records the motion of the smoke parti-
y
“crown splash”) is thrown upwards along the rim of the clesatupto15,000framesper second. Cross-correlation
d
- impacting object while below the water surface a large of subsequent images allows us to extract the velocity of
u cavity forms in the wake of the impactor. Due to the the smoke which faithfully reflects the actual air speed
l
f hydrostaticpressureofthesurroundingliquidthiscavity [18]. Our setup obeys axisymmetry and we use cylindri-
s. immediately starts to collapse and eventually closes in a cal coordinates with z = 0 the level of the undisturbed
c single point ejecting a thin, almost needle-like liquid jet. free surface.
i
s Justpriorto the ejectionofthe liquid jet the cavitypos- In the beginning of the process (see the snapshot in
y sesses a characteristic elongated “hourglass” shape with Fig.1 (a))air is drawninto the expandingcavitybehind
h
a large radius at its bottom, a thin neck region in the the impacting object with velocities of the order of the
p
center, and a widening exit towards the atmosphere. impact speed. In a later stage however, this downward
[
This shape is very reminiscent of the converging- flux is overcompensated by the overall shrinking of the

2 diverging (“de Laval”) nozzles known from aerodynam- cavity volume resulting in a net flux out of the cavity.
v
ics as the paradigmatic picture of compressible gas flow The cavity shape at the moment when the flow through
7
through, e.g., supersonic jet engines. In this Letter we the neck reverses its direction is illustrated in Fig. 1 (b).
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7 use a combination of experiments and numerical simula- Towards the end of the cavity collapse a thin and fast
3 tions to show that in addition to the very similar shape, air stream is pushed out through the cavity neck which
. alsothestructureoftheairflowthroughtheimpactcav- is illustrated in Fig. 1 (c). From images such as those
9
ity resembles closely the high-speed flow of gas through in Fig. 1 we can directly measure the air speed u up to
0
9 such a nozzle. Not only is the flow to a good approx- about 10 m/s as is shown in the inset of Fig. 2.
0 imation one-dimensional, but it even attains supersonic In order to determine the flow speed at even higher
: velocities. Nevertheless, the pressure inside the cavity is velocities we revert to multiscale numerical simulations.
v
i merely2%higherthanthesurroundingatmosphere. The Ournumericalmethodproceedsintwostages: anincom-
X keydifference,however,isthatinourcasethe“nozzle”is pressiblestageatthebeginningandacompressiblestage
r a liquid cavity whose shape is evolving rapidly in time – towards the end of the impact process. During the first
a
asituationforwhichnoequivalentexistsinthe scientific stage both air and liquid are treated as incompressible,
or engineering literature. irrotational,andinviscidpotentialfluids. Tosolveforthe
Our experimental setup consists of a thin circular disc flow field and to calculate the motion of the interface we
with radius R = 2 cm which is pulled through the use a boundary integral method (BIM) as described in
0
liquid surface by a linear motor mounted at the bot- [16]withextensionstoincludethe gasphase[22]. Atthe
tom of a large water tank [16] with a constant speed moment that the air flow through the neck reverses, see
of V = 1 m/s. To visualize the air flow we use small Fig. 1 (b), the simulation enters into the second, com-
0
glycerin droplets (diameter roughly 3 µm) produced by pressible stage: from now on only the liquid motion is
a commercially available smoke machine (skytec) com- computed by the incompressible BIM.
monlyusedforlighteffects intheatersanddiscotheques. To simulate the air flow in the second stage we need
Beforethe startofthe experimentthe atmosphereabove to take compressibilityinto accountmeaning that a sim-
thewatersurfaceisfilledwiththissmokewhichisconse- ple potential flow description is no longer possible. For-
quently entrained into the cavity by the impacting disc. tunately, at the end of the incompressible stage the air
2
velocityprofileis almostperfectly one-dimensionalalong
the axis of symmetry. We can therefore describe the gas
dynamics by the 1D compressible Euler equations [23]
inanalogyto gasflowingthrougha converging-diverging
nozzle. In the Euler equations we include two additional
terms accounting for the variation of the nozzle radius
in time and space [18]. For the numerical solution we
use a Roe scheme [23, 24] which is highly appreciated
for its computational efficiency and ability to accurately
capture shock fronts.
The two-way coupling between the gas and the liquid
FIG. 1: (a) After the impact of the disc an axisymmetric domainsis accomplishedvia(i)the interfacialshapeand
cavity is formed in its wake and air is entrained into this its instantaneous velocity which is provided by the BIM
cavity. (b)Duetohydrostaticpressurefrom thesurrounding and serves as an input into the gas solver and (ii) the
liquid the cavity starts to collapse and the air flow reverses pressure which is obtained from the solution of the Eu-
its direction. (c) As the collapse proceeds air is pushed out ler equations and serves as a boundary condition for the
of the shrinking cavity at very high speeds. In (a)–(c) we
BIM. Above the location of the initial free surface the
overlaid images ofthecavityshape(recordedwithbacklight)
surface pressure of the BIM remains atmospheric.
and images of the smoke particles (recorded with the laser
Combining our experiments with these numericalsim-
sheet and artificially colored in orange). In the latter, the
area illuminated by the vertical laser sheet is restricted by ulations leads to the main resultof this Letter contained
the minimum cavity radius [18]. A corresponding movie can inFig.2: thecollapsingliquidcavityactsasarapidlyde-
be found in [19]. formingnozzle,soviolentthattheairwhichispushedout
through the neck attains supersonic velocities (red line).
Our simulations show that the pressureinside the cavity
which is driving this flow is less than 1.02 atmospheres
(blue line). From the inset one can tell that our simu-
lations are in good agreement with the smoke measure-
400 1.05
15 ments overthe entire experimentally accessible range. It
is interesting to note that even towards the end of the
1.04
300 s] 10 process(when sonic velocities are reached)there is a net
m/ fluxofairupwardsthroughthecavity. Iftheprocesswas
[m/s]eck200 u [neck 05 11..0023 / pcava gwinoovbueolrdtnheedxvpemretceitcraethllyedibareiyrctttiohoenbsec.opTlulahspihsseendeootfuflttohowefttnhheuecsknueicntksdeerlrfelgionionenes
n p
u 5 10 15 20 25 1.01 the important role of the dynamics of the entire cavity.
r [mm]
100 neck Todeterminemorepreciselyatwhatpointtheairflow
1 throughtheneckbecomessonicweshowinFig.3(a)the
time evolution of the local Mach number, Ma = u /c
neck neck
0 0.99 (with the gas velocity u and the speed of sound c),
00 22 44 66 88 1100 1122 1144 neck
r [mm] for discs impacting at 1 and 2 m/s. We find that the
neck
speed of sound is attained at cavity radii as large as 0.5
mm for the lower impact velocity and 1.2 mm for the
FIG. 2: The speed of the gas flowing through the neck (red
higher impact velocity.
curve) as a function of the shrinking cavity neck taken from
In a steady state one could expect from the (com-
the fully compressible simulations. The main plot demon-
strates that sonic speeds are attained with the cavity pres- pressible) Bernoulli equation that these very high air
sure (blue curve) being less than 2% higher than the atmo- speeds would cause a greatly diminished air pressure in
spheric pressure. The enlargement (inset) shows that the the neck region. Despite the unsteadiness of our sit-
numerical scheme (red curve) agrees very well with the ex- uation, the data presented in Fig. 3 (b) indeed shows
perimentally measured velocity (black diamonds; the hole in that the pressure p decreases significantly once the
neck
the data between rneck = 16 mm and 22 mm is due to mea- neck has shrunk to a diameter of roughly 4 mm (for the
surementuncertaintiesatlow absolutevelocities [18]). Slight
1 m/s impact) while before that point it is practically
non-axisymmetric perturbations [20, 21] in the experimental
atmospheric throughout. Classical steady-state theory
setup may be responsible for the somewhat slower air speed
[25]for a converging-divergingnozzlepredicts that when
of the experiment as compared to the simulation. One can
Ma =1 the pressure at the neck reaches a minimum
clearly see the inversion of the flow direction from negative neck
(into thecavity) to positive (out of thecavity) velocities. value of
γ−1 −γ/(γ−1)
p /p = 1+ =0.53 (1)
neck a
(cid:18) 2 (cid:19)
3
(a) −4.5
1 4 5(b)
(c) m]
neck −2 (a) 1 2 3 zneck −4.6z [c
a 0.5
M
0.5 1 1.5 2
−4 Ma
00 r5neck 1[m0m]15 z [cm] −6 zneck (c) −−55..42
z
a 1(b) 1mm −8 stag −5.6z [cm]
p
/ck0.8 −10 −5.8
e
pn −6
0 0.4 0.8 −0.01 0 0.01
0.6
Ma Ma
0 5 10 15
rneck [mm] FIG. 4: (a) The vertical air velocity normalized by the local
speedofsoundMa=u/casafunctionoftheverticalposition
(the corresponding cavity image is shown in the middle) for
FIG. 3: (a) The evolution of the local Mach number at the
r = 0.9 mm: the profile exhibits a sharp peak approxi-
cavity neck for different impact speeds (red: 1 m/s, blue: 2 neck
mately at the height of the neck. (b) A close-up of the zone
m/s). For the 2 m/s impact speed sonic flow is attained at
aroundtheneckillustratesthesteepeningofthevelocitypro-
a cavity radius of 1.2 mm. (b) The pressure at the neck di-
files towards pinch-off (numbers1-5 correspond to neck radii
minishes due to Bernoulli suction as the neck radius shrinks
between 0.9 mm (number 1, bright red) and 0.5 mm (num-
and air is forced to flow faster and faster. The minimum
ber 5, dark brown)) and the development of the shock front
pressureliesatabout0.6pa whichisattainedwhentheMach
at roughly 0.1 ms before pinch-off. The neck position z
number reaches unity. (c) The experimental image shows a neck
corresponding to curve 5 is shown by the dashed line. (c) A
pronounced kink at the neck which is not captured by the
close-up of thearea below theneck shows thelocation of the
smoothly roundedcurvepredicted bythesimulation without
air(cyanline). Onlytheinclusionofaireffectsintothesimu- gas flow stagnation point zstag (dashed line).
lations(redline)isabletoreproducethekinkedshapecaused
bythelowairpressureattheneckaswellastheshapeofthe
cavity above theneck.
downwards by the moving disc. An interesting conse-
quence of this competition between cavity expansion at
the bottom and cavity shrinking in the neck is the exis-
with pa the atmospheric pressure and γ = 1.4 the isen- tence of a stagnation point with u=0 as can readily be
tropic exponent. As shown in Fig. 3 (b) our situation – observedinFig. 4 (a) andits magnificationin Fig.4 (c).
although highly unsteady – exhibits a similar behavior
As can be seen in the inset of Fig. 5, the distance
withp ≈0.6p astheMachnumberbecomesoforder
neck a between the neck and the stagnation point is no larger
unity.
thanroughly5mmpriortocavityclosure. Nevertheless,
InFig.3(c)weillustratehowthislowpressuregivesus the pressure at the stagnation point equals the overall
ahandletoobservetheconsequencesofthesupersonicair pressure inside the cavity which is roughly atmospheric
speed in our experiments: despite the air being three or- during the whole process (see Fig. 2). Recalling that
dersofmagnitudelessdensethanwater,itisabletoexert p ≈ 0.6p this results in a tremendous vertical pres-
neck a
asignificantinfluenceevenontheshapeoftheliquidcav- suregradientwhichofcourseaffects the dynamics ofthe
ity providedthat its speed is high enough[26, 27]. From cavitywall: theflowofairissostrongthatitcandragthe
the experimental image it is clear that the free surface liquid along resulting in an upward motion of the cavity
close to collapse no longer possesses a smoothly rounded neck just before the final collapse. That this effect is in-
shape but instead shows a significant increase in curva- deed presentin the simulations can be seen fromthe red
ture at the minimum (a “kink”). While this feature is line in Fig. 5. For comparison, the cyan curve demon-
notpresentinasimulationneglectingtheinfluenceofair strates that a single fluid simulation neglecting the air
as those in [16], the inclusion of air effects allows us to dynamics would predict a monotonously decreasing po-
capture quite accurately the cavity shape observed ex- sition. The experimental data however is in quantitative
perimentally. This gives strong evidence that in the ex- agreement with the compressible simulations. Together
periment the air indeed becomes as fast as predicted by with the cavity shape shown in Fig. 3 (c) these results
the simulations and produces a Bernoulli suction effect constituteanimpressive–albeitindirect–demonstration
strong enough to deform the cavity. ofthecredibilityofournumericalpredictionsdespitethe
The positive sign of u (see Fig. 2) indicates that fact that, understandably, it is not possible to directly
neck
the gas flow is directed upwards at the neck. At the measure (super-)sonic air speeds with our smoke setup.
same time, the air at the bottom of the cavity is pulled Furthermore they show that the perfectly axisymmetric
4
4 ities [20, 21] inevitably destroy the axisymmetry of the
0
m] system.
m −5
3 z [neck−10 necLkoo(skeinegFmig.or4e(cbl)o)seolnyeafitntdhsetvhealtocititypopsrsoefissleesaabodvisectohne-
−
m] 2 ag−15 tinuous jump: the signature of a shock front developing
m zst−20 in the air stream. While such a shock front is a com-
z [c 1 0 r 5 [mm]10 monphenomenoninsteadysupersonicflows,hereweare
− neck able to illustrate its development even in our highly un-
ck steady situation when the gas velocity passes from sub-
e
zn 0 to supersonic.
In conclusion, we showed that the air flow inside the
−1
impact cavity formed by a solid object hitting a liquid
surface attains supersonic velocities. We found that the
−2 veryhighairspeedscanbereachedeventhoughthepres-
0 2 4 6 8 10 12
sure inside the cavity is merely 2% higher than the sur-
r [mm]
neck roundingatmosphere. Thisisduetothehighlyunsteady
gas flow created by the rapidly deforming cavity. We il-
FIG.5: Theverticalpositionofthecavityneckrelativetothe lustrated how the air affects the cavity shape close to
final closure height zc as a function of the shrinking neck ra- the final collapse in two different ways: (i) the initially
diusfromexperiment(blackdiamonds),simulationswith(red
smoothly curved neck shape acquires a kink which can
line) and without (cyan line) air dynamics. The experimen-
be attributed to a Bernoulli suction effect and (ii) the
tal data is in quantitative agreement with the compressible
initially downward motion of the neck reverses its direc-
simulations, while clearly the simulation neglecting air fails
to capture the upward motion of the minimum induced by tion and starts to travel upwards. The quantitatively
thelarge pressuregradient between thestagnation point and consistent observation of both effects in numerics and
the cavity neck. Experimental error bars are determined by experiment makes us confident that our rather involved
the number of vertically neighboring pixels all sharing the numerical procedure truthfully reflects reality.
same minimum radius. The inset shows the approach of the
WethankA.Prosperetti,J.Snoeijer,andL.vanWijn-
stagnation point to the neck.
gaardenfordiscussions. Thisworkispartoftheprogram
of the Stichting FOM, which is financially supported by
NWO. JMG thanks the financial support of the Span-
approach of the simulations is justified and, therefore, ishMinistry ofEducationunder projectDPI2008-06624-
thatsupersonicgasvelocitiesarereachedbeforeinstabil- C03-01.
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