Table Of ContentHydrodynamics of Hypersonic Jets: Experiments and
Numerical Simulations
Marco Belan, Sergio de Ponte • Daniela Tordella •
Silvano Massaglia, Attilio Ferrari, Andrea Mignone
• Eberhard Bodenschatz
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Abstract are paramount: the Mach number and the jet to am-
Stars form in regions of the galaxy that are denser bient density ratio (Ferrari 1998; Reipurth and Bally
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M and cooler than the mean interstellar medium. These 2001). We have therefore devised anexperiment where
I regions are called Giant Molecular Clouds. At the be- these two parameters can be set to match the ones de-
h. ginning of their life, up to 105 − 106 years, stars ac- rivedfromobservationsofYSOjets(Belan et al.2004).
p crete matter from their rich surrounding environment In order to better understand the dynamical details
- and are origin of a peculiar phenomenon that is the of the jet propagation and evolution, we have carried
o
r jet emission. Jets from Young Stellar Objects (YSOs) out 3D numerical hydrodynamic simulations adopting
st areintensivelystudiedbythe astrophysicalcommunity boundary and injection conditions that were as close
a byobservationsatdifferentwavelengths,analyticaland as possible to the experimental ones. The numerical
[
numerical modeling and laboratory experiments. Indi- results show good agreementwith the laboratorydata,
1 cationsaboutthejetpropagationanditsresultingmor- namely: the jet head velocity is consistent for the two
v phologies are here obtained by means of a combined
situations, the presence of bright knots along the jet is
0
study ofhypersonicjets carriedoutboth inthe labora-
6 correctly reproduced, and non-axisymmetric perturba-
tory and by numerical simulations.
9 tions, seen in the heavy jet case, rise and grow in the
2 simulation to resemble the laboratory jet.
.
1 Observations show that stellar jets extend in space
1 Introduction
0 forverylongdistances,infacttheycanreachlengthsof
1
The physical parameters of astrophysical jets are im- hundreds, even thousands, jet radii. While the initial
1
: possible to reproduce in any Earth’s laboratory. We collimation,atthelaunchingregion,islikelytobemag-
v
therefore must limit ourselves to identify those param- netic, however at very large distances from the origin
i
X eters that arethe crucialonesin controllingthe overall stellar jets maintain their small free-jet opening angle,
r jet behavior and to rescale the laboratory simulations consistently with a Mach number ranging between 10
a
accordingly. Amongthemanyparametersthatdescribe and40. Hereweshowthatlaboratoryhydrohypersonic
thejetevolutionandpropagationfeatures,twoofthem jetsmaintaintheirstructure,atleastupto70jetradii.
The plan of the paper is the following: in Section 2
and3wepresenttheapparatussetupandthenumerical
MarcoBelan,SergiodePonte
code,Section4isdedicatedtothecomparisonofexper-
PolitecnicodiMilano-DipartimentodiIngegneriaAerospaziale,
Milano, Italy imental and numerical results, and the conclusions are
DanielaTordella drawn in Section 5.
PolitecnicodiTorino-DipartimentodiIngegneriaAerospaziale,
Torino, Italy
SilvanoMassaglia,AttilioFerrari,AndreaMignone 2 Experimental setup
Universita`diTorino-DipartimentodiFisicaGenerale,Torino,
Italy The jets under study are createdinside a modular vac-
EberhardBodenschatz uumvessel,withamaximumlengthof5mandadiame-
Max Planck Institute for Dynamics and Self-Organization - terof0.5m. Severalkindsofnozzlescanbemountedon
G¨ottingen, Germany thisvessel. Thevesseldiameterismuchlargerthanthe
2
diameterofthe jets, sothatthe lateralwallseffectsare account the effects of the boundary layer and heat ex-
limited. The required ambient inside the vessel is ob- changewiththenozzlewalls. Howevertheresultdiffers
tainedbymeansofasystemofvalveswhichsetsthede- by the well-known steady-state ideal relationship:
siredambientdensity(atpressuresinthe1.5to100Pa
γ−1
range)using a gasin generaldifferent fromthe jet gas. M2 = 2 p0 γ −1
Anelectrongunoperatingatverylowpressures,thanks γ−1 p
"(cid:18) a(cid:19) #
to a set of secondary pumps, and equipped with a de-
flection system, creates an electron sheet. This sheet by about 1%.
intercepts the jet under test, and generatesa plane flu-
orescent section of the flow. These fluorescent zones
Table 1 Jets parameters in thepresent experiment
can be acquired by different kinds of cameras, includ-
Nozzle geometry DeLaval
ing intensifieddevices,depending onthe experimentto
Stagnation/ambient pr. ratio p0/pamb 300 to 2·105
be performed. The cameras and the electron gun can
Mach numberMmax 5 to 20
be mountedon different ports. The vesselincluding all Density ratio ρjet/ρamb 0.01 to 110
the available sections is shown in fig. 1. Reynolds,throat diameter based Ren 104 to 5·104
Reynolds,jet diameter based ReD up to 2·106
Reynolds,axial length based Rex >107
Severalnozzles in the Mach number range 5≤M ≤
20 are available, in this experiment a nozzle designed
to generate a M = 15 at the exit has been used. The
nozzle exit diameter is 2r = 0.07136 m. Slightly
0
Fig. 1 Sketchofthevacuumvesselincludingalltheavail-
different Mach numbers can be obtained by adjusting
able sections. The ports can be used for camera or elec-
thestagnation/ambientpressureratiop /p aroundthe
tron gun mounting. Ambient gas valve fittings and electri- 0 a
ideal isoentropic value 4.76·104. The present setup is
cal/service connectors are not shown.
sketchedin fig. 2, where the vesselhas a length of 2.46
m, and the camera (optical) window is 0.27 m wide.
Thissystemwasalreadyemployedtostudytheprop-
ertiesofhighlyunderexpandedjets,makinguseoftrun-
catednozzlesandcolorCCDcameras,detailedinforma-
tioncanbe foundinthe previousworksby Belan et al.
(2004, 2006); Belan, De Ponte and Tordella (2008).
In the present setup, by means of a fast piston, the
jet gas is compressedto stagnationpressures in the 0.1
to 0.7 MPa range, then it is accelerated by a de Laval
nozzleandentersthe vacuumvessel,mountedinthe 3-
modules configuration. The jet parameters attainable
by the experimental system are outlined in table 1.
The nozzle used in this experiment has been care-
fullydesignedtoperformnearlyisoentropicalgasaccel- Fig. 2 Typical setup for matched jets at M=15
erations and expansions, i.e. flows as close as possible
to the ideal conditions, but accounting for friction and In each run a pair of gases is selected for the jet
heat exchange effects. and the ambient. The available gases are helium, ar-
The nozzles were built at the CERN workshop of gon, xenon and air, only the first three ones are used
Geneva with surface accuracies in the order of 1µm. for the jet since the nozzle is designed for monoatomic
Under such conditions, the flow at the nozzle outlet gases. The jets meet the electron sheet under the opti-
has the design Mach number over a very wide cross- calwindow,andtheresultingimagesareacquiredbyan
section, excluding a thin boundary layer region at the intensified CMOS camera; besides visualizations, den-
walls,providedthatthestagnation-to-ambientpressure sity and structure velocities measurements are possible
ratio p /p maintained between inlet and outlet has a by image-processing.
0 a
well defined value. In this experiment, the stagnation- The physical time scale of jets τ can be defined as
to-ambientpressure ratio p /p is checkedby means of the ratio between the jet radius at the nozzle exit r
0 a 0
highaccuracypressuretransducers. The Machnumber and the speed of sound internal to the jet under nom-
isthenobtainedbyanumericalmethodthattakesinto inal conditions, i.e. when the nozzle works under ideal
3
1.0
0.8
nozzle
0.6
piston Q
0.4
Fig. 3 Longitudinal section of the piston-nozzle system.
0.2
The piston is shown at thestarting position
0.0
0 20 40 60 80 100
steady conditions, and is in the order of 0.1 to few ms. t
In the following, we will express time in units of τ and Fig. 4 A typical jet gas injection: dimensionless nozzle
lengths in units of r . mass flow versus dimensionless time. The mass flow Q is
0
The piston-nozzle system is shown in fig. 3. The jet normalized over thereference flow rate Qr which is yielded
by the same nozzle when operating in ideal steady condi-
gasiscompressedinachamberthatleadstothenozzle
tions. FortheHelium jet,Qr is0.43 g/s andfortheXenon
inlet. At the end of the compression run, owing to the
jet, Qr is 2.44 g/s. Time is normalized over the jet time
valves opening, the outflow increases to a maximum
unit τ (ratio between the jet radius at the nozzle exit and
value, then it diminishes as the gas contained in the the speed of sound internal to the jet). The piston output
chamber is used up. The mass flow Q is a function of valves begin to open at t = 0. In general, the jet hits the
time of the kind vessel end duringthe increasing phaseof Q(t)
Q=Q[p (t),f(t),g(t)],
0
The modular structureexploits ageneralframework
where p is the pressure in the piston chamber, f(t) for integrating a system of conservation laws, built on
0
is a factor function which accounts for the progressive modern Godunov-type shock-capturing schemes. The
opening of the valves, and g(t) is the function which discretization recipe involves three general steps: a
accounts for the natural decay in the amount of gas piecewisepolynomialreconstructionfollowedbytheso-
remaining in the compressionchamber after the piston lution of Riemann problems at zone interfaces and a
stop. This model, which is based on the piston-valve final evolution stage.
behavior, gives the following for density, pressure and
velocity of the gas at the nozzle exit:
4 Results
ρ ∼Q(t)2/3, p ∼Q(t)2/3/γ,v ∼M Q(t)1/3,
jet jet jet
4.1 Experiments/numerical simulations comparisons
where M is the Mach number at the end of the nozzle.
Theresultinggasinjectioncurvehastheshapesketched Aslightlyunderdense(light)jet,Mach15,densityratio
in fig. 4. In this experiment two jets are considered, a 0.7 is shown in fig. 5.
lightHe jetflowinginaXe ambientandaheavyXe jet In this jet, dense knots form spontaneously on the
flowing in an air ambient. axis well before the impact to the vesselend. They are
visible both in the numerical and in the experimental
images. After the impact, the interaction with the re-
3 The numerical code PLUTO
flected flow does not spoil the axial symmetry of the
flow. The impact of this jet on the vessel end wall re-
The numerical code was developed for the solution of flects only a very small amount of matter, hence the
hypersonic flows in 1, 2, and 3 spatial dimensions and perturbation may be considered weak. It is also no-
different systems of coordinates (Mignone et al. 2007; ticeable that the inner part of the flow remains com-
Mignone2009). Thecodeprovidesamultiphysics,mul- pact and collimated after the impact. This is also an
tialgorithm modular environment particularly oriented interesting information, which could open the way to
towardthe treatmentofastrophysicalflowsinpresence other applications, in particular to perturbative near-
ofdiscontinuities. Differenthydrodynamicmodulesand linear experimental studies. The inertial effects, and
algorithms may be independently selected to properly the associated compressibility, must here be so power-
describe Newtonian, relativistic, MHD, or relativistic ful to inhibit the spatial growth of the jet, unlike what
MHD fluids.
4
Fig.5 Lightjet,HeliuminXenon. M=16.1,ambientpres-
surepa =4.0±0.1Pa, estimatednozzleexitvelocity=3200
±330 m/s, mean jet pressure at nozzle exit pj =pa±30%,
stagnation/ambient pressure ratio p0/pa = 7·104 ±30%.
Comparison of numerical simulations (density maps) and
visualizations (superpositions of scaled correlated frames).
The density maps are normalized to the unperturbed am- Fig. 6 Heavy jet, Xenon in Air. M=15, ambient pres-
bient value. The space unit is the exit radius of the nozzle sure pa = 9.95 ± 0.1 Pa, estimated nozzle velocity at
r0 =0.03568 m. The time unit for this light jet is τ =0.18 the exit = 560 ± 60m/s, mean jet pressure at the noz-
ms zle exit pj = pa±30%, stagnation/ambient pressure ratio
p0/pa = 4.76·104±30%. Comparison of numerical simu-
lations(densitymaps)andvisualizations(superpositionsof
happens inthe incompressiblesituation,wheresheared
scaledcorrelatedframes). Thedensitymapsarenormalized
flows generally grow fast in thickness.
to the unperturbed ambient value. The space unit is the
Anoverdense(heavy)jet,Mach15,densityratio100, exit radiusof thenozzle r0 =0.03568 m. The timeunit for
is shown in fig. 6 thisheavy jet is τ =0.96 ms
Here, there are overdense zones on the axis, they
are very close to each other, and are resolved only in
long term is remarkably largerthan that in fig. 5. The
the numerical simulation. For the heavy jet, the in-
reduction of the core diameter is clear (the jet head is
teraction with the reflected flow eventually produces a
outside the image on the right) and is also a measure
finite perturbation, which non linearly interacts with
of the spreading of the annular mixing layer that sur-
the outflow. Thisis sufficienttospoiltheaxialsymme-
rounds it. The mixing region is slow compared to the
try. However,itisnotsufficienttospoilthecollimation
flow in the core and is not visible in this figure. How-
of the jet. This suggests the possibility of a very large
ever,itcanbeobservedthroughconcentrationmeasure-
longitudinal extent of the heavy hypersonic jet and its
ments(Belan, De Ponte and Tordella(2008,2010)). In
related capability to transfer energy, momentum and
thisexamplethespreadinganglewithrespecttothejet
mass to large distances (onthe Earth,in space,in pos-
axis is about 8 degrees (Belan, De Ponte and Tordella
sible new applications).
(2010)).
An example of hypersonic jet with a completely
different behaviour is reported in fig. 7, where an
4.2 Results analysis
underdense jet having Mach number and density ra-
tio similar to the ones of the jet in fig. 5 is shown Thevelocitiesofflowstructures(jet’shead,knots,trav-
(Belan, De Ponte and Tordella (2008, 2010)). In this eling waves..) can be measured by means of standard
case, the jet is created by means of a truncated noz- correlation techniques applied to the pixel matrices of
zle, and is far from being nearly isoentropic. It is in consecutive images.
fact characterized by a strong normal shock close to For the light jet of fig. (5), in the simulations the
the nozzle exit,and bya lateralspreadingwhich inthe head (bow shock) velocity grows from 440±3m/s to
5
Fig.9 ’Frozen’imagesofaheavyjet,XeinAratM=13.5,
obtainedbysuperimposedframesinthetimeranges0<t<
20τ and 60τ <t<70τ
the numerical simulations. Figure 10 compares longi-
Fig. 7 Helium jet in an Argon medium: highly un-
tudinal 2D cuts, at t = 14, of the light jet simulation
derexpanded jet, maximum Mach number = 16, stagna-
obtained employing the actual lateral boundary con-
tion/ambientpressureratiop0/pa =304,densityratioafter
thenormal shock ∼0.15. ditions with the one obtained with outflow conditions,
Differencesinthedensitydistributionmorphologiesare
visible at the jet’s head, that appears slightly narrower
706 ± 3m/s in the 2 < t < 16 time interval (before
in the free-boundary simulation, while the head’s posi-
the impactto the vesselend) due to the increaseinthe
tionsdifferbyafewpercentinthetwocases. Theshock
mass flow. The experimental head velocity at t = 15
reflectionatx∼36isabsentinthefree-boundarycase.
measuredbyimagecorrelationis490±140m/s. Forthe
heavy jet of fig. (6), the jet head velocity grows from
130±3m/sto270±3m/sinthe2<t<9timeinterval,
and the experimental value at t=8 is 140±30m/s.
When the velocity of a structure is known, two im-
ages in succession containing the same structure can
also be superimposed after shifting the second one by
the displacement of the structure under study. Then,
this procedure can be repeated, and a spatial recon-
struction of a part of the jet structure larger than the
camera window can be done by superimposing parts of
adjacentframes(’slices’). Clearly,thereconstructionof
jet morphology on large scales has a physical meaning
Fig. 10 Comparison between simulations with and with-
onlyifthechangesinstructurepropertiesareslowwith
out lateral boundaries at t=14
respect to the interframe time.
Twoimagereconstructionsforalightjetandaheavy
jet, obtained by correlated frames superposition, are
showninfigures8 and9. Thesefiguresareconstructed
5 Conclusions
overcomingthe validity rangeofthe technique in order
to obtain single pictures visualizing a whole jet or a
Our experiment highlights the following aspects:
largepartofit,theresultingimagesmustbeconsidered
I) - the collimation of near-isoentropic jets over dis-
as’frozen’,asifthejetwerepassingthroughthecamera
tances much larger that the initial radius.
window without changes in its morphology.
II) - the jet persistence (which, for the light jet, is
mantained even after the impact with the vessel end-
wall).
III) - the presence of a cocoon surrounding the un-
derdense jet, while this is not visible for the overdense
Fig. 8 ’Frozen’ image of the light jet analyzed in figure one.
5, Helium in Xenon at M=16.1, obtained by superimposed IV) - the spontaneous formation of knots along the
frames in thetime range 0<t<66τ jetaxis(before the impactofthe jet to the vesselend).
These aspects refer both to numerical simulations
The effect ofthe lateralboundaries ofthe vesselcan andto experiments ruled by Newtoniandynamics, and
be estimated by changing the boundary conditions in do not need of the confining effect associated to the
presence of magnetic fields.
6
Some of these aspects are common to stellar jets, in
particularthejetconfinementandpersistenceoverlong
distances, in units of the initial radius.
7
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