Fig. 1. A supersonic exhaust jet trailing the Bell X-1 ... · Fig. 1. A supersonic exhaust jet trailing the Bell X-1 rocket plane, the first manned aircraft to break the sound barrier

Fig. 1. A supersonic exhaust jet trailing the Bell X-1 rocket plane, the first manned aircraft to break the sound barrier. Thebeaded appearance of the jet is due to a network of internal shock waves that repeatedly compresses and reheats the exhaust gas.(From “Orders of Magnitude’’ by F. W. Anderson, NASA Special Publication 4403, 1981.)

The results we present here were achievedin collaboration with Larry L. Smarr of theUniversity of Illinois and are based solely onnumerical simulations carried out on a Cray-1 computer at the Max Planck Institut furPhysik und Astrophysik in Garching beiMunchen. Working in West Germany, wewere constantly reminded of the dual defini-tion of a supersonic jet as NATO forcesmaneuvered overhead. This was strangelyappropriate, however, for the history of re-search on supersonic gas jets goes back nearlya century to the pioneering work of Ernst

I Mach at the behest of the Austrian military.

Ernst Machand the Military Connection

Back in the 1880s the concept of a shockwave was still brand new. In Europe thetheoretical properties of these nonlinearcompression waves were being hotly debatedby the likes of Riemann, Rankine, Rayleigh,and Hugoniot. It was still uncertain whetherit was energy or entropy that is conservedacross the surface of discontinuity created bya shock front (see Sidebar 1, “Shock Wavesversus Sound Waves”).

Meanwhile, artillery experts and ballisti-cians were groping for explanations of twocurious phenomena brought to light duringthe France-Prussian War. The first was thedouble report heard when a shell was fired athigh speed from an artillery piece; only asingle report was heard when a projectile wasfired at low speed. The second was the crater-like nature of the wounds inflicted by theFrench army’s new high-speed bullets. Ex-ploding bullets—expressly forbidden bytreaty—were suspected as having been used.

The Belgian ballistician Melsens proposedan explanation for the second phenomenon

Opening figure (previous page): Looking between the dark pressure boundaries of a calculated supersonic jet, we see a radio-brightness map for a long extragalactic jet extending south some quarter of a million light-years from radio galaxy 3C200 andterminating in a radio lobe. In this map, which was taken at the Very Large Array (VLA) near Socorro, New Mexico, by JackO. Burns, regions of high emission are depicted in red, intermediate emission in yellow, and low emission in blue. The jetpowering the radio lobe north of the radio galaxy is for some mysterious reason not seen. (Collage by Jim Cruz.)

40 Spring/Summer 1985 LOS ALAMOS SCIENCE

that so intrigued Ernst Mach that he set laboratory and fired airabout to confirm it experimentally. Melsens past it supersonically.conjectured that a high-speed projectile The supersonic air streampushes ahead of itself a considerable mass of was produced by allowingcompressed air, which may cause explosion- highly compressed air to escapelike effects on the body it strikes. In other from a reservoir through a col-words, Melsens was proposing the existenceof the bow shock that precedes a supersonic sentially the same apparatus is

Melsens was right.In 1886 Mach and his experimental col-

league Peter Salcher were the first to obtain

schlieren photographs of a bow shock. Theseand other gas shocks are easy to record

photographically because the abrupt changein density across a shock causes refractionanomalies in light transmitted through theshock. Mach correctly interpreted the bowshock as the envelope of sonic disturbancesoriginating at the projectile. (The formationof shocks through convergence of sonic dis-turbances is discussed in Sidebar 2, “Shocks,Rarefactions, and Contact Discontinuities, ”)He derived the famous equation sin a =c/v relating the Mach angle a (the openinghalf angle of the bow shock) to the projectilespeed v and the ambient sound speed c (seedrawing on this page).

Such a bow shock explains the first phe-nomenon. the double report: the first bang isproduced by the powder gases escaping fromthe muzzle of the gun, and the second bang isthe sonic boom associated with the bowshock of the supersonic projectile.

Ernst Mach is, of course, also remembered.for his principle of relativity (Mach’s prin-ciple), which states that physical forces existonly because of the motion of matter relativeto the rest of the matter in the universe.Mach’s belief in the relative nature of physi-cal laws must have played a role in the designof a novel setup for his ballistics experi-ments: he held the projectile fixed in the

LOS ALAMOS SCIENCE Spring/Summer 1985

tical reasons the nozzle could be nomore than a few centimeters in diameter.Mach quickly discovered that a projectileimmersed in this modestly sized supersonicair stream created a network of standingshock waves far more complex than thesingle bow shock he expected (see Fig. A inSidebar 3, “Steady-State Jets”). To quote the1889 discovery paper of Mach and Salcher,“The stream exhibited a customary appear-ance so remarkable, that a detailed study ofthe jet itself seemed worthwhile.” Such wasthe serendipitous discovery of the complexstructure of supersonic gas jets and the begin-ning of a systematic research program car-ried out by Ernst Mach’s son Ludwig todetermine the properties of the jets. Furtherresearch was also carried out by Emden andPrandtl around the turn of the century.

Steady-State Supersonic Jetsin the Modern Era

Interest in supersonic jets essentially van-ished for two generations until World War II,when military considerations again madethem the topic of research—this time at theAberdeen Proving Grounds by a group ofAmericans who were investigating the phe-nomenon known as gun flash.

When an artillery piece fires, gas from theburnt powder exits from the cannon muzzleat supersonic speeds (the cause of the first

bang) and produces for a brief instant thenetwork of oblique crisscrossed shocks thatMach had discovered. These standing shock

waves are set up by nonlinear radial oscilla-tions of the gas jet caused by a pressureimbalance at the nozzle orifice (in this casethe end of the muzzle). Of particular concernto the Aberdeen researchers was the occur-rence in the jet of a type of shock known as aMach disk. Mach disks, which are perpen-dicular to the jet flow and much strongerthan the oblique shocks, produce “gun flash”by reheating the powder gas.

Since these flashes could reveal the posi-tion of a gun to the enemy, Ladenburg, VanVoorhis, and Winckler were interested indesigning a muzzle to eliminate them. Theydiscovered that the crucial parameter is thepressure ratio at the nozzle, that is, the ratioof the pressure of the gas in the jet to thepressure of the ambient, or atmospheric, gas.As shown in Sidebar 3, pressure ratios farfrom unity produce Mach disks, and pressureratios near unity produce X-shaped “reg-ular” shock reflections. To eliminate Machdisks, one need only flare the muzzle walloutward so that the emerging gas expandsand repressurizes before reaching the at-mosphere.

continued on page 45

41

Shock Waves versus Sound Waves

SIDEBAR 1

Explosions, projectiles whizzing by atsupersonic speeds, high-speed colli-sions of solids—what do these phe-

nomena have in common?They all create very large changes in local

pressure over very short times, and theseviolent pressure changes self-steepen intoshock fronts, or shock waves. Unlike a soundwave, which is a small-amplitude com-pression wave that propagates at the localsound speed and leaves the state of the me-dium unchanged, a shock front is a nonlinearwave that abruptly changes the state of thesupersonically approaching gas. The gas isgenerally at a higher temperature and pres-sure after it has passed through the shock (or.equivalently, after the shock has passedthrough the gas). Moreover, the shock-heated gas moves subsonically with respectto the shock. As we describe below, the nar-row region defined as the shock front is aregion where thermodynamic processes arcirreversible.

The cartoon from Courant and Friedrichs’classic book on supersonic flow illustratesthe formation of a steep front in a discon-tinuous medium, namely, a train of skiers.The skiers, barreling single file down thenarrow run. pile up in a heap as first one skiergets wrapped around a tree, and then, beforehe can warn the skier behind him to slowdown, the next skier crashes into him, and soon. The pileup of human wreckage creates asteep front moving up the slope away fromthe tree analogous to a receding shock front.As in a continuous medium. formation of thefront depends critically on the fact that theflow of skiers is “supersonic” in the sensethat it is faster than the speed with which themedium, in this case the skiers, can respondto new boundary conditions. The high “pres-sure” of the skiers behind this front isanalogous to the change of state experiencedby shock-processed media.

About a hundred years ago Stokes,Earnshaw, Riemann, Rankine, Hugoniot,and Lord Rayleigh deduced the conditionsthat prevail at shock fronts in gases. Theframework they used to analyze this com-pressible flow are the equations of ideal gas

42

An example of a receding shock wave. From Supersonic Flow and Shock Waves byR. Courant and K. O. Friedrichs (New York:Interscience Publishers, Inc., 1948),

dynamics. These nonlinear equations de- within this context, infinitesimal pressurescribe conservation of mass, momentum, changes generate linear compression waves,and energy in an ideal fluid, a fluid in which better known as sound waves, that travelall changes in kinetic and internal energy aredue to pressure forces. Heat conduction and But what happens when the amplitude of aviscous stress are ignored and entropy is compression wave is finite?constant, so all thermodynamic changes are Figure A illustrates what happens. At timeadiabatic and reversible. It was known that, tl we have a finite-amplitude sound wave

Spring/Summer 1985 LOS ALAMOS SCIENCE

Supersonic Jets

P. Variations in pressure imply variations in

Thus each point on the waveform propagateswith its local sound speed, which is greater atthe peaks than in the troughs. With time thewaveform steepens (as shown at t2) a n deventually breaks (as shown at t3) to producemultiple values for the state variables of thegas, P, p, and T. Of course this prediction iswrong. Instead nature inserts a shock front(dashed line) just before the wave breaks,and the flow variables remain single-valued.

How does this happen? On a microscopiclevel large gradients in temperature and ve-locity at the front of the steepening wave

X—ct

Fig. A. Self-steepening of a finite-amplitude sound wave. In the regionwhere the state variables of the wave(here, pressure) would become multi-valued, irreversible processes dominateto create a steep, single-valued shockfront (vertical dashed line).


Inverse Density

Fig. B, Effects of the passage of a sound wave and of a shock wave. As a sound wavepasses through a gas, the pressure and density of the gas oscillates back and forthalong an adiabat (a line of constant entropy], which is a reversible path. Incontrast, the passage of a shock front causes the state of the gas to jump along anirreversible path from point 1 to point 2, that is, to a higher pressure, density, andentropy. The curve connecting these two points is called a Hugoniot, for it wasHugoniot (and simultaneously Rankine) who derived, from the conservation laws,the jump conditions for the state variables across a shock front, After passage of theshock, the gas relaxes back to point 3 along an adiabat, returning to its originalpressure but to a higher temperature and entropy and a lower density. The shockhas caused an irreversible change in the gas.

cause the irreversible processes of heat con-duction and viscous stress to dominate in aregion with a width equal to a few collisionmean free paths and to counteract the self-steepening process so that a single-valuedshock front forms. The net effect on a macro-scopic level is that mass, momentum, andenergy are conserved across the shock front,but entropy is not; it increases as relativekinetic energy is dissipated into heat throughatomic or molecular collisions.

In 1864 Riemann was the first to analyzewave-steepening within the context of idealgas dynamics. He mistakenly assumed thatentropy was conserved (in other words, thatall processes were adiabatic) across shockfronts as it is for finite-amplitude sound

waves. Later, Rankine, Rayleigh, andHugoniot showed that an adiabatic shockfront would violate conservation of energy,and therefore shock fronts must be non-adiabatic and irreversible. Figure B showsthe irreversible changes caused by the pas-sage of a shock front in contrast to the re-versible changes produced by a sound wave.To model these irreversible effects, the formof the Euler equations of gas dynamics wehave adopted must be modified as describedin Fig. 2 of the main text.

The dissipative nature of a shock frontimplies that it can maintain itself only inthe presence of a driving force. A simpleexample of a driven shock front is given in

43

Shock Waves, Rarefactions, andSIDEBAR 2

Contact Discontinuities

The nonlinear equations of ideal gasdynamics support three types ofnonlinear “waves”: shock fronts.

rarefactions, and contact discontinuities.Contact discontinuities are surfaces that

separate zones of different density and tem-perature. By definition such a surface is inpressure equilibrium, and no gas flows across

it. When, as often happens, the tangentialvelocity of the gas on one side of the surfacediffers considerably from that of the gas onthe other side, the surface is called a slipdiscontinuity. The boundary between a

supersonic jet and the ambient gas is anexample of a slip discontinuity.

The other two types of nonlinear wavesarise from abrupt changes in pressure. Shockfronts accompany compression of the me-dium, and rarefactions accompany ex-pansion of the medium. As a simple exampleof how these waves arise, consider the one-dimensional flow of gas in a small-diametercylindrical tube, which is fitted with a pistonatone end (x = 0) and closed at the other end.Figure A shows the time-dependent flow thatresults when the piston

Fig A. Compression in a Shock Tube

is suddenly pushed

into the cylinder and how a shock frontforms as sonic disturbances generated by thepiston converge. These sound waves travelalong trajectories called characteristic paths,or simply characteristics. In general, thecharacteristics of a set of hyperbolic partialdifferential equations are the paths alongwhich certain variables are conserved andare thus the paths along which informationtravels. The characteristics for the equationsof ideal gas dynamics are (1) the streamlinesalong which matter flows and entropy isconserved and (2) the PIUS and minus

Fig B. Expansion in a Shock Tube

Distance x

44

Characteristic Paths

Particle Paths


Supersonic Jets

SIDEBAR 2

characteristics along which informationtravels from one point to another by soundwaves, The so-called Riemann invariants,which are linear combinations of the fluidvelocity and sound speed, are conservedalong the plus and minus characteristics, Ingeneral, characteristics form a curvilinearcoordinate system that reduces a com-plicated set of nonlinear partial differentialequations to a coupled set of ordinary dif-ferential equations.

Since sonic disturbances travel in the gasat the local sound speed c, and since the gasimmediately ahead of the piston is movingwith the piston speed v piston(x), the soundwaves emanating from each point along thepiston path travel at a speed of vpiston (x) + c.Hence at each point x the characteristicmakes the same angle with the piston trajec-tory. As a result, the characteristics (red)emanating from the curved portion of thepiston path (the acceleration phase) convergeto form a shock front, (The bow shock of asupersonic projectile also forms where sonicdisturbances generated along the trajectoryconverge, except that in this case the sonicdisturbances move through a stationaryrather than a moving medium and thereforeall travel at speed c.) The shock front aheadof the piston travels along the tube at super-sonic speeds. Gas particles farther away fromthe tube remain stationary until the shockfront passes, and then they travel at a speedequal to the final constant speed of thepiston. The closer spacing of the particlepaths (dashed lines) gives graphic evidenceof shock compression.

Figure B shows the time-dependent flowthat results when the piston is suddenlywithdrawn from some position within thecylinder. As in Fig. A, characteristics drawnat a constant angle to the piston path rep-resent the path of sonic disturbances gener-ated in the gas immediately adjacent to thepiston. In this case the characteristics (blue)emanating from the curved portion of thepiston path diverge to form a rarefactionzone (gray). The greater separation of theparticle paths in the rarefaction zone in-


continued from page 41Other data on supersonic flows arc avail-

able from studies of jet and rocket pro-pulsion. Since these studies were aimed atdesigning more efficient, higher thrust en-gines, virtually all of them have focused onthe steady-state, or time-averaged, propertiesof the jet flow, as has our discussion so far.

Time-Dependent Jets in Nature

The time-dependent evolution of-jets be-comes important when we consider super-sonic “jets” in nature. One example is thesupersonic jet produced by the eruption of agas-rich volcano, such as the eruptions ofKrakatoa and Mount St. Helens. The blastwave that accompanies the formation ofsuch a volcanic jet produces widespread dev-astation. Experiments have shown that thesetransient supersonic jets have structuralproperties resembling those of steady-statejets.

Other examples are the numerous jetsemanating from astrophysical objects. In ourgalaxy protostars, old binary star systemsassociated with supernova remnants. andplanetary nebulae produce jets of’ ionizedgases. A number of these stellar jets haveknots of bright emission that remind one ofthe bright spots in the exhaust jet shown inFig. 1, Spectral lines from highly excitedatoms in the knots indicate that these brightspots are regions of high temperature,probably produced by shock heating. TheDoppler shifts of the spectral lines demon-strate that these jets arc still evolving withtime.

Our final example of time-dependent jets.and the ones that exhibit the most re-markable and mysterious stability, are theenormous jets emanating from quasars andother galaxies, These jets, known as radio jetsbecause most of them can be observed onlywith radio telescopes, range from thousandsto millions of light years in length, or aboutfifty to one hundred jet diameters. Many ofthem exhibit bright knots of radio emissionin the beam. A radio jet typically ends in a“hot spot”-a small region of intense radio

emission—embedded in a larger amorphouslobe of fainter emission (the opening illustra-tion is an exception).

The emission spectra of radio jets, whichare continuous and polarized, are producedby synchrotrons emission from a nonthermalpopulation of relativistic electrons gyratingin a magnetic field. This nonthermal compo-nent probably coexists with a diffuse thermalplasma in some uncertain mixture to formthe “stuff” of extragalactic radio sources. Weknow little else about the composition of thejets and the lobes and, in the absence ofspectral lines, have no direct measure of jetspeeds from Doppler shifts.

In a landmark paper, written in 1974 whenonly the radio lobes, and not the jets them-selves, had been observed, Roger Blandfordand Martin Rees interpreted the hot spots inthe lobes as the point of impact of a super-sonic gas jet impinging on the ambient me-dium as the jet advances. [n other words, theradio lobes are powered by fluid-like beams.or jets, originating at the centers of radiogalaxies. They suggested that the main struc-tural features of the jets could be explainedby ideal gas dynamics, that is, by thenonlinear differential equations of com-pressible fluid flow. Magnetic fields and com-plex plasma dynamics that might affect localregions could be ignored when consideringthe overall structure of a jet,

The fluid-beam hypothesis of Blandfordand Rees provides a simple physical expla-nation for radio observations and aframework for detailed model building, butwithin this framework the issue of stabilitybecomes paramount. We know from labora-tory experiments that Kelvin-Helmholtzshear instabilities grow rapidly on the bound-ary of supersonic gas jets and effectively de-celerate and disrupt the flow. How do ex-tragalactic jets propagate such great distanceswithout suffering the same fate? Are they orare they not, governed primarily by the equa-tions of gas dynamics’? If so, what are thecritical parameters determining the stabilityand the observed morphology of the flow?We undertook to answer these questions by

continued on page 50

45

SIDEBAR 3

Steady-StateJets

s upersonic gas jets are created in thelaboratory by allowing a highly com-pressed gas to escape through a nozzle

into the atmosphere or some other ambientgas. In contrast to astrophysical jets, labora-tory jets usually have high density ratios

mismatches at the source. Investigations oflaboratory jets have, until recently, focused

on their steady-state behavior rather than theinitial time-dependent behavior of interest inastrophysical contexts. Nevertheless, our un-derstanding of the steady-state structures ob-served in laboratory jets provides a point ofdeparture for interpreting our numericalsimulations of time-dependent jet flow.

46

An Idealized Supersonic Jet

Figure,4 shows the characteristic structuretaken on by a slightly underexpanded super-sonic jet, that is, one for which the pressureof the gas at the nozzle orifice. P.. is slightlygreater than the ambient gas pressure, Pa. (Ajet is referred to as underexpanded if

pressure matched if K = 1.)This complicated axisymmetric structure

has several remarkable features. First, the jetboundary oscillates as the jet gas periodicallyoverexpands and reconverges in its attemptto match the ambient pressure. The gas con-tinually overshoots the equilibrium positionbecause the effects of the boundary arc com-

municated to the interior of the jet by soundwaves, which, by definition, travel moreslowly than the bulk supersonic flow. Thecharacteristic paths of the sound waves con-verge to form the second remarkable featureof the jet, the network of crisscrossed shockwaves, or shock diamonds (red lines). Thesestanding shocks alternate with rarefactionfans (blue lines). The gas in the jet interiorexpands and cools (shades of blue) as it flowsthrough the rarefaction fans and is com-pressed and heats (shades of red) as it passesthrough the shock diamonds. The figureclearly illustrates that the jet interior isalways out of step with the jet boundary. Forexample, the positions of greatest gas com-pression (dark red) do not coincide with the


Supersonic Jets

SIDEBAR 3

Temperature

Jet Boundary Streamline

Fig. A. Idealized steady-state structure of a slightly under- and the blue lines indicate the beginnings of rarefactionexpanded (Pn slightly greater than Pa) supersonic jet. The fans. The gas temperature varies according to the key. Blackred lines represent incident and reflected shocks (see Fig. B), streamlines follow the oscillating flow path of the jet gas.

Fig. B. Regular reflection in a slightly underexpanded jet. (blue) form rarefaction zones. Converging characteristicsThe pattern of crisscrossed shocks in Fig. A can be under- (red) from the boundary form the incident cortical shocks,stood in terms of characteristics. Diverging characteristics which reflect off the jet axis to form the reflected shocks.

positions of minimum jet diameter. Theblack streamlines in the figure indicate theflow paths of the gas. The gas bends outtoward the boundary as it passes throughrare faction fans and bends back toward theaxis as it passes through shock fronts.

Regular Reflections

Figure B shows how the shock structure in

Fig. A can be understood in terms of called incident shock reaches the jet axis itcharacteristics. As the gas leaves the nozzle, undergoes a regular reflection; that is, itit expands and a rarefaction fan (diverging forms a diverging shock. At the point whereblue characteristics) emanates from the this reflected shock reaches the jet boundary,nozzle orifice. The gas overexpands, and the it knocks the boundary outward, creating apressure of the ambient gas at the boundary, new rarefaction fan, and the process beginsacting like the piston in Sidebar 1, pushes the all over again. The geometric pattern of thejet gas back toward the axis, creating the red (x, y) characteristics that form the shockscharacteristics. These characteristics form a and rarefaction fans in this steady-state two.converging conical shock. When this so- dimensional flow is similar to that of the

LOS ALAMOS SCIENCE Spring/Summer 1985 47

Mach Disk

(1) Shock Triple Point

reflects at the perimeter of a Mach disk—astrong shock normal to the flow direction (sonamed because Ernst Mach was the first torecord its existence). The angle between theincident shock and the jet axis determinesthe type of reflection: small angles of in-cidence yield the regular reflections shown inFigs. A and B, and large angles of incidenceyield Mach reflections. When gas passesthrough a shock, its velocity component nor-

slows down the beam much more than passage through thesupersonic jet. A Mach reflection creates a shock triple point incident and reflected shocks (from point 1 through points 2where three shocks meet. Figures C1 and C2 show how and 3 to point 4'. As a result, a slip discontinuity is formedpassage through the Mach disk {from point 1 to point 4) {dashed line).

(x, t) characteristics for one-dimensionaltime-dependent flow described in Sidebar 2.

Mach Reflections

A striking change in flow structure occurswhen the pressure mismatch at the orifice is

for an underexpanded jet, the incident shock,rather than converging to a point on the axis,

48

mal to the shock is greatly reduced but itsparallel component remains unchanged.Thus shocks with large angles of incidencerelative to the flow axis are much more effec-tive at slowing down the flow than shockswith small angles of incidence. The criticalangle for transition from regular reflectionsto Mach reflections is approximately theangle that yields a sonic relative velocity forthe gas downstream of the shock.


Supersonic Jets

SIDE BAR 3

Fig. D, Realistic steady-state structure of an overexpanded overexpanded jet, shocks rather than rarefactions emanatefrom the nozzle orifice. Shear instabilities create a mixing

Mach reflections and regular reflections. Since this is an layer that grows until it reaches the beam axis.

A prominent feature of Mach reflections isthe emergence of a slip discontinuity (dashedline) from the shock triple point where theincident shock, reflected shock, and Machdisk meet, The flow velocity, density, andtemperature are discontinuous across thiscontact surface. This slip discontinuity arisesbecause. the thermodynamic pathwaythrough the incident and reflected shocksdoes not equal the pathway through the Machdisk.

In Fig. Cl we display two adjacentstreamlines, one on each side of the shocktriple point, and in Fig. C2 we display thecorresponding thermodynamic pathways.

Adjacent fluid elements at some initialpoint on either streamline have the samestate variables and the same total (kineticplus internal) specific energy, that is, energyper unit mass. This quantity is given by

1

where y is the ratio of the specific heat of the


fluid at constant pressure to the specific heatat constant volume.

By Bernoulli’s principle the total specificenergy remains constant along a streamline,Therefore, when the two adjacent fluid ele-ments arrive at points 4 and 4' in Fig. Clthey must still have the same total specificenergy. They must also have the same pres-sure because they are still adjacent. However,we see from Fig. C2 that their densities andentropies are different. The element at point4 has been shock-heated along a Hugoniot toa higher entropy and lower density than theelement that passed through the incident andreflected shocks along points 2 and 3 to point4'. The lower density of the fluid element atpoint 4 implies that its internal energy(which is proportional to P/p) is greater thanthat of the fluid element at point 4'.Bernoulli’s principle then implies that thefluid element at point 4 must have a cor-respondingly lower kinetic energy, and henceflow velocity, than the adjacent element atpoint 4’. A slip discontinuity results fromthis difference in flow velocities,

Laboratory Supersonic Jets

The jet structures shown in Figs. A, B, andCare idealizations in that real supersonic jetsdo not have sharp, stable boundaries butturbulent boundaries where jet and ambientgases mix. Figure D shows a more realisticsteady-state structure for an overexpandedlaboratory jet. Near the orifice, where thepressure mismatch is large, Mach reflectionsoccur, but farther downstream the reflectionsare regular. The mixing layer, which grows asa result of Kelvin-Helmholtz (shear) in-stabilities, progressively eats its way into thesupersonic core of the jet. When the mixinglayer reaches the axis of the jet. the flow issubsonic and fully turbulent. It is thensusceptible to twisting and bending motions,like a smokestack plume in a crosswind,

Strictly speaking the wave structures with-in the supersonic core are not steady sincethey are buffeted by the turbulent boundarylayer. However, their average positions,those in Fig, D, are well defined. ■

49

(a) Euler Equations for Compressible Flow

continued, from page 45solving the equations of gas dynamicsthrough numerical simulation.

Numerical Jet Laboratory

The numerical problem we set ourselves iswell defined. We start with the equations forcompressible fluid flow displayed in Fig. 2a.assume that at t = 0 the jet gas starts to flowcontinuously through a narrow orifice into aregion filled with quiescent gas as shown inFig. 2b, and compute the time-dependentflow numerically with finite-difference tech-niques. We assume that the flow is axisym-metric and nonswirling (that is, the gas haszero angular momentum) and thus reduce at h r e e - d i m e n s i o n a l p r o b l e m t o t w odimensions (R and Z). This reduction isnecessary to achieve reliable numerical re-sults with the computing power of a Cray-1.Moreover, since many extragalactic jets arcremarkably straight over most or all of theirlengths, axisymmetry is a reasonable as-sumption. The importance of nonaxisym-metnc effects is discussed toward the end ofthis article.

The interesting thing about this problem isthat it is scale-invariant, and therefore theresults of the calculation can be scaled fromlaboratory to cosmic dimensions by varyingthe only dimension that appears in the prob-lem, the jet diameter. Four scale-invariant(dimensionless) parameters determine the

ratio of the beam, or jet, gas density pb to theambient gas density pa,

the parameter K is the ratio of the jet gaspressure Pb to the ambient gas pressure Pa,

the parameter M, the Mach number. is theratio of the jet speed vb to the speed of soundin the jet gas cb,

50

Fig. 2. (a) The Euler equations of ideal compressible flow modified by source terms

shock front. These terms have zero value elsewhere so that farther away from theshock front all processes are reversible and adiabatic, in agreement with ex-perimental observations. (b) The computational grid and boundary conditions fornumerical simulation of axisymmetric pressure-matched jets. Gas enters thecomputational region through a small orifice on the axis. Ambient gas is subse-quently allowed to escape through the outer boundaries if required by advance ofthe jet.


Supersonic Jets

(a)

Fig. 3. A comparison of computed and measured steady-state flow fields in an

velocity vectors at every third grid point in a 60-by-180 calculation grid. Theexperimental results in (b), which were adapted from R. Ladenburg et al., PhysicalReview 76(1949):662, show density contours and the triple-shock configuration.Note the good agreement between calculation and experiment on the position andgeometry of the embedded shock waves.

the jet,

where vR and vZ are the components of thebeam velocity in the R and Z directions.

For extragalactic jets it is reasonable toreduce the number of parameters to two. TO

mimic the small observed opening angles ofthese jets. we start with a perfectly collimated

that the jet gas pressure equals the ambient


gas pressure (K= 1 ) because x-ray data fromsatellites tell us that hot thermal gas pervadesintergalactic space and is adequate in mostcases to pressure-confine the jets. The re-

solution space that has scarcely been ex-plored in the laboratory because of thedifficulties of achieving very high Machnumbers (M > 10) and very low density

tions corresponding to arbitrary values of-

these parameters is one of the great advan-tages of numerical simulation.

Solution of the two-fluid problem posedby jet flow depends critically on the motionof the interface separating the two fluids.Fluid interfaces are generally subject toKelvin-Helmholtz instabilities, which aredriven by shear motion parallel to the inter-face, and by Rayleigh-Taylor instabilities,which are driven by accelerations normal tothe interface. Instabilities lead to mixing ofthe two fluids and can modify the jet flowsignificantly.

Techniques for tracking interfaces are aspecialty at Los Alamos and Livermore.where complicated multi-fluid problemshave been handled on large computers forthree decades. Although multi-fluid prob-lems are abundant in astrophysics as well,they have usually been approximated as one-fluid problems. Consequently, the use of in-terface-tracking techniques is somewhat of anovelty in numerical astrophysics, althoughin this application it is essential.

We adapted an interface-tracking tech-nique developed by James M. LeBlanc andJames R. Wilson at Livermore. This calcula-tion is Eulerian; that is, the calculation grid isfixed. The interface between the two fluids isdefined by those computational zones withfractional fluid volumes. The fractional vol-umes are carried along by the bulk flow, andtheir consequent redistribution on the grid isthen used to construct an approximategeometry of the displaced fluid interface, Al-though an approximate treatment, the tech-nique is robust and very effective at prevent-ing an undesirable consequence of mostEulerian computations, the artificial “blend-ing” of disparate materials at moving bound-aries.

We tested the code through comparisonswith experiment. Figure 3 shows one suchcomparison, in which we attempted toreproduce not only the general shape of theflow, but also the position and geometry ofthe embedded shock waves. The surprisinglygood agreement gives us confidence that wecan reproduce accurately the steady-flowstructures measured in laboratory supersonicjet experiments and, with perhaps some luck,the transient jet structures as well.

51

Supersonic Jets

Fig. 5. Laser-schlieren photographs of the time evolution of a low-density super-sonic jet. The jet was created by release of 3 joules of energy in a spark, whichcaused ejection of a plasma pulse into ambient air. Shown here, top to bottom, areshort-exposure photographs taken 15, 35, and 80 microseconds after the pulse andat a very late stage in the evolution of the jet. The short exposure time reveals thetransient behavior of the flow and emphasizes the turbulent structure of the jetcocoon. The bow shock preceding the jet can be seen as an expanding arc in the firstthree frames; it has moved out of view in the last frame. The length of the areaviewed is 4.5 centimeters. (Reproduced with the permission of J. A. Cavolowskyand A. K. Oppenheim, Department of Mechanical Engineering, University ofCalifornia, Berkeley.)


the beam speed vb, which means that thebeam gas constantly overtakes the jet head.Rather than accumulate there, the incomingbeam gas continuously escapes into the co-coon after being shock-decelerated anddeflected sideways at the working surface.Surprisingly, the cocoon gas actually flowsbackward toward the inlet. Backflow is asimple consequence of mass conservationand is encountered on earth when supersonicwater jets are used to drill into rocks.

The relative velocity between the back-ward flowing cocoon gas and the ambient gasat the cocoon boundary is substantial. but itis subsonic for the jet shown in Fig. 4. Conse-quently, the cocoon boundary is subject tothe Kelvin-Helmholtz shear instability.which overturns and mixes the cocoon andambient gases, as can be seen in Figs. 4b and4c. AS a result of the instability, only a lobe ofcocoon gas is left intact at the jet head in Fig.4d. The central supersonic beam neverthe-less appears to propagate without significantmixing at its boundary. We can verify thatthis is indeed the case with the aid of addi-tional plots that elucidate the flow field in thejet (Fig. 7).

Details of the Flow Field

In Fig. 7a a novel mode of display revealsthe large-scale structures of the flow field for

ematically distinct parts of the flow insideand outside the jet are assigned differentcolors. Within the jet boundary (black) wecan identify the central supersonic beam

coon gas (white), and the forward flowing ( vZ

> 0) cocoon gas (yellow). Outside the jet wecan identify the undisturbed ambient gas(dark blue) and the bow shock (red). Thebeam consists by definition of those regionswhere the local kinetic energy flux ½ pvZ

3 isat least 5 percent of its maximum value.Aside from a few yellow patches within thebeam, which wc shall comment on below, wecan follow the supersonic channel uninter-rupted to the very end of the jet.

53

Fig. 6. Flow schematic of the jet head in the rest frame of the working surface. Thespeed of advance W of the working surface is determined by equating the rampressure, or momentum flux, of the beam to the ram pressure of the ambient gas ina frame in which the working surface is stationary. In this frame the beam speed isvb — W and the speed of the ambient gas is — W. The expression derived for W/vb

much more slowly than the beam gas. However, the beam gas does not collect at theworking surface but is shock decelerated and deflected sideways to form a low-density backward flowing cocoon around the forward moving beam.

The stability of the beam boundary is ver-ified in Fig, 7b, where we plot the azimuthal

positive vorticity are green, and regions witha large negative vorticity are yellow. Thebeam boundary, a region of high positivevorticity (shear). can be seen as a stable greenlayer, just inside the black jet boundary, thatcontinues beneath the forward cocoon lobeup to the working surface. Note the coinci-dence of the beam boundary as defined bythe two complementary techniques of Figs.7a and 7b. At the end of the jet, the shearlayer is carried into the cocoon, where itbreaks up into isolated vortices. The ex-istence of vortices in the cocoons of diffusejets is one of our general findings and ac-counts for the forward flowing cocoon gasindicated in yellow in Fig. 7a.

In addition to positive vorticity, Fig. 7breveals that a substantial amount of negative

54

vorticity is being created within the beam.What is the mechanism responsible for this’?The answer is given in Fig. 7c, where regions

colored blue, and regions of large negative

rarefaction zones, and the red regions en-compass shock waves (see Sidebars 2 and 3).We see within the beam a well-developednetwork of crisscrossing shock waves sepa-rated by strong rarefaction zones. Thenegative vorticity is generated by the obliqueshock waves. which are highly nonuniformin strength because of their cylindrical con-vergence toward the axis. The terminal shockat the working surface can also be seen in Fig.7C just behind the leading contact surface.which is more or less normal to the incomingbeam. This leading contact discontinuityseparates the jet from the ambient gas. Itpropagates into the ambient gas, and no gas

Fig. 7. Computer-graphic diagnosticsof the flow field at t = 60 for the (M=3,n = 0.1) jet of Fig. 4. (a) Thekinematic plot depicts cinematicallydistinct regions in different colors. Thebeam boundary (black) encloses thebeam (green), which by definition con-sists of those regions for which thekinetic energy flux ½pv3 exceeds 5 per-cent of the global maximum of thatquantity. The backward flowing (vZ <O) and forward flowing (vZ > 0) cocoonregions are shown in white and yellow,respectively. The undisturbed ambientgas is dark blue, and the bow shock isred. (b) The vorticity plot depicts re-gions of large positive and negative

and yellow, respectively, and revealsthe vortex sheet at the beam boundary(green) and the shear layer at the co-coon boundary (yellow). (c) Thedivergence plot depicts regions of largepositive and negative divergence of the

tively, and reveals a network of internalshock waves (red) and rarefactionzones (blue). (d) The temperature plotreveals the alternate heating and cool-ing of the gas by shock and rarefactionwaves, respectively.

flows across it in either direction.A color-coded temperature plot (Fig. 7d)

clearly shows that the effect of the embeddedshock waves is to compress and heat thebeam gas. Knots of shock-heated gas can beseen downstream of each oblique shock in-tersection. Such knots frequently appear insupersonic rocket plumes as bright patches,which are due to enhanced thermal emissionor to chemiluminescence.

Although the shock pattern in our simula-tion of a pressure-matched jet bears re-markable similarities to the shock wavescharacteristic of the overexpanded andunderexpanded jets produced in the labora-tory (see Sidebar 3), there is a vital difference.In the laboratory jets the shock pattern isstationary, whereas in the pressure-matchedjet it moves downstream at a speed equal to asignificant fraction of the beam speed. Weshall return to this point later.

Spring/Summer 1985 L0S ALAMOS SCIENCE

Airbrush illustrations in this article by Jim Cruz. . . .

The Working Surface

We now turn our attention to the workingsurface itself and inquire how a supersonicflow can be deflected sharply backward tosupply the cocoon with gas. The detailedstructure of the working surface (Fig. 8) re-veals a complex network of nonlinear wavesand flow patterns that propagates as acoherent whole at the head of the jet.

Quite generally, the terminal shock systemlooks just like the triple-shock structureformed by Mach reflection in steady-statejets (see inset (a) in Fig. 8 and Sidebar 3),except that the structures downstream of theMach disk (the reflected shock and rarefac-tion zones) are swept back by the oncomingambient gas. Upon passing through theMach disk, beam gas is virtually brought torest with respect to the leading contact dis-continuity, which is itself advancing into theambient gas. The Mach disk converts thedirected kinetic energy of the beam into in-ternal energy, thereby creating a region ofvery high pressure and temperature betweenthe Mach disk and the contact discontinuity.It is generally believed that this region cor-responds to the hot spots observed in theradio lobes of extragalactic jets. The shockedhigh-pressure gas within this zone can onlyescape sideways, being confined longi-tudinally by the ram pressure of the oppos-ing beam and ambient streams. Once beyondthe radius of the beam, however, a fluid

element will be blown back into the cocoonby the ram pressure of the ambient gas,which is there not balanced by the ram pres-sure of the beam. As the gas expands into thecocoon, its pressure decreases until it reachesthe ambient pressure. Inset (b) in Fig. 8 tracesthe irreversible thermodynamic evolution ofa fluid element following the streamline inFig. 8 that passes through the Mach disk andinto the cocoon. A key consequence of theirreversible nature of the terminal shockfront is that the cocoon gas is generally lessdense than the beam gas after pressure bal-ance has been re-established.

Vorticity is generated at the shock triplepoint in the form of a slip discontinuity (thedashed line in Fig. 8), is carried into thecocoon, and contributes to the formation of alarge-scale toroidal vortex surrounding thebeam. The streamline in Fig. 8 that flowsabove the shock triple point circles thevortex to produce a forward flowing streamof high-pressure cocoon gas. This high-pres-sure environment in the cocoon drives theoblique incident-shock component of thebeam’s terminal triple shock. This intimaterelation between the flow structures at theworking surface and the adjacent cocoonconstitutes a nonlinear self-consistent solu-tion to the problem of flow deceleration anddeflection in supersonic jets.

How can we be sure that our numericaltechniques are giving us the correct solutionto this time-dependent problem when no

Fig. 8 The working surface of a super-sonic jet. The terminal shock structurein a calculated isodensity contour plotof the (M = 3, n = 0.1) jet of Figs. 4and 7 has been highlighted to reveal itsresemblance to the triple-shock con-figuration of a Mach reflection. Theincident shock, the Mach disk, and thereflected shock are dark red, and therarefaction fan is dark blue. This ter-minal shock structure is similar to theMach reflection in a steady-state over-expanded (K< 1) jet shown in inset (a)(and to the Mach reflection in asteady-state underexpanded jetpresented in Sidebar 3), except that thereflected shock and rarefaction fan areswept back by the pressure of the on-coming ambient gas. The gas tempera-ture varies from low (blue) to high(red) according to the scale shown inSidebar 3. Inset (b) shows the thermo-dynamic evolution of a gas element asit travels along a streamline throughthe Mach disk (point 1 to point 2) andaround to the vortex in the cocoon(point 3). By the time the gas elementreaches point 3, its pressure matchesthat of the beam and ambient gases, butits density is much lower than beforebeing shock-heated by the terminalshock. (This irreversible process wasdiscussed in Sidebar 1.)

56


Supersonic Jets

analytic solutions or experimental data existfor comparison? In principle, we cannot besure; in practice, however, we can verify,through a convergence study, that our simu-lation captures the important physical struc-tures and processes. Figure 9 displays theresults of a convergence study in which weinvestigated the effects of numerical tech-niques and grid resolution on the structure ofa (M = 6,n = 0.1 ) jet. Scanning the figurefrom top to bottom, one sees the flow fieldcome into focus as the accuracy and capabil-ity of the algorithms are improved and as thegrid resolution in each dimension is doubledand then doubled again. Note that all of thekey flow features (beam, cocoon, workingsurface, and embedded shocks) revealed atthe high resolution of Fig. 9e also appear inthe solution computed at half that resolution,Fig. 9d. The same cannot be said of Fig. 9c.

We used the resolution of Fig. 9d for oursimulations but increased the number ofzones by a factor of 4 to extend the spatialdomain beyond that of Fig. 9. Many ex-tragalactic jets are very long, with length-to-width ratios of up to 100. Our 75-by-640 gridallows us to study jets with lengths of up to40 jet diameters—well within the range ofobserved extragalactic jets. Figure 9, as wellas the consistency of Fig. 8 with “textbook”supersonic flow structures, gives us con-fidence that our numerical simulations areproducing reliable results.

We have delved into the structural detailsof a single jet to illustrate the similarities and

differences between laboratory jets and oursimulations. In the remainder of this articlewe explore how both jet structure and sta-bility change as the parameters M and n arevaried.

We did numerical simulations corre-sponding to thirteen points in the (M, n) pa-rameter space. We varied M between 1.5 and12 and n between 0.01 and 10. Jet power,

roughly four orders of magnitude betweenthe lowest power, (M= 1.5, n = 1) jet and thehighest power (M = 12,n = 0.01) jet. Al-though jets of even lower density and higherMach number may be relevant to astro-physics, our survey has revealed the impor-tant t rends that seem appl icable toastrophysical objects. It should be noted thatour survey covers a range of parameters thatfar exceeds what has been investigated in thelaboratory. Laboratory jets rarely exceedMach 3 and are almost exclusively moredense than the ambient gas. As we shall see,the density ratio—a parameter that is seldomvaried systemat ical ly in laboratorystudies—has a profound effect on jetmorphology and stability.

Jet Morphology

Figure 10 tells the story of jet morphologyin snapshots of our simulated jets arrangedaccording to their values of M and n. All thejets are pictured at the same length.

Two general trends are apparent, Scanning

Fig. 9 Effects of resolution (dimensions offinite-difference grid) and algorithmicaccuracy on jet structure, as revealed by isodensity contours, for a supersonic jet

first-order-accurate computation shows little internal structure of the jet andexcessive smearing of the jet boundary. (b) A low-resolution, second-order-accurate computation shows a sharpening of internal structure and boundary. (c)A low-resolution, second-order-accurate computation with interface trackingshows a sharply defined jet boundary but little change in internal structure. (d) Acomputation with the same algorithms as in (c) but twice the resolution revealsconsiderable internal structure, including the central supersonic beam, internalshock waves, the cocoon, and waves at the boundary. (e) A computation with thesame algorithms as in (c) and (d) but twice the resolution of (d) shows sharplyfocused internal structure with qualitatively the same features as in (d).


the third column from high to low densityratios, we see a marked thickening of the jet.A jet with a low density ratio (n < 1) builds athick cocoon of backward flowing gas arounditself as it propagates, whereas a jet with ahigh density ratio (n > 1) is just a nakedbeam, lacking a cocoon altogether. Scanningthe third row from low Mach number to highMach number, we see that the spatial extentof the cocoon increases, At low Machnumbers the cocoon is but a lobe of materialat the head of the jet, whereas at high Machnumbers the cocoon surrounds the centralsupersonic beam along its entire length. Thecombined effect of these trends is that onlyjets with low density ratios and high Machnumbers possess extensive cocoons.

These qualitative trends can be under-stood with the help of a few simple conceptsand equations. As mentioned earlier, the co-coon is supplied by gas that flows through thejet head at a rate proportional to v b – W ..

Equation 1 tells us that this rate is small in

on kinematic grounds alone one would ex-pect an inverse correlation between densityratio and the size of the cocoon.

This inverse correlation is enhanced bythe shock heating of the jet gas at the ter-minal shock. The amount of shock heating at

the relative Mach number MW is defined as(vb – W)/cb. With the help of Eq. 1 and thedefinition of M, we find that MW, = M/ (1 +

thus the amount of shock heating, is largestwhen n is smallest. The greater the shockheating, the more the gas flowing into thecocoon must expand to match the ambientpressure. Thus, a small n implies not only alarge rate of flow into the cocoon but also agreater expansion of the shock-processed gas.The result, in agreement with our simula-tions, is a cocoon with a relatively largevolume and low density.

A similar argument, but now with n heldfixed, suggests that larger Mach numbers im-ply greater shock heating, greater pressuredifferences between the working surface and

59

Fig. 10. Dependence of the flow morphology of pressure- in gray, the undisturbed ambient gas in dark gray, and thematched supersonic jets on the dimensionless parameters M disturbed ambient gas in light gray. The simulations revealand n. In these kinematic plots the supersonic beam and the an inverse dependence of cocoon thickness on density ratio nbow shock are shown in black, the backward flowing (vZ < O) and a direct dependence of cocoon length on Mach numbercocoon gas in white, the forward flowing (VZ > O) cocoon gas M.

the ambient gas, and thus cocoons with are produced by the Kelvin-Helmholtz in-larger volumes. stability. At low Mach numbers the com-

Shear instabilities on the cocoon boundary bined effects of low pressure differ-tend to limit the extent of cocoons for jets ence and instability-induced flow resistancewith low Mach numbers, The gas in the serve to localize the cocoon gas to a lobe nearcocoon generally flows back toward the the head of the jet.source, as indicated by the white regions in Can we use these morphological trends toFigure 10. This backflow is resisted by cor- infer the physical conditions in extragalacticrugations in the cocoon boundary, such as radio sources? In principle, yes, but translat-

60

ing from a fluid flow field to a radio surface-brightness distribution is not straight-forward. The main uncertainty is the quan-titative relationship between synchrotronemissivity and thermal energy in the flowfield. In the simplest model one assumes thata small but constant fraction of the thermalenergy density is equally divided between theradiating relativistic electrons and the mag-

Spring/Summer 1985 L0S ALAMOS SCIENCE

Supersonic Jets

netic field, One then can derive a completelylocal prescription relating the total synchro-trons emissivity to the thermal pressure. This

by applying this prescription to the workingsurface and have reproduced many featuresobserved in the hot spots of radio lobes,However, mapping lobes and cocoons willrequire a more detailed emission model thattakes into account the sources and sinks ofthe energetic electrons, in particular, theradiative and adiabatic expansion losses ex-perienced by a parcel of synchrotron-emit-ting plasma as it flows from the working


surface into the cocoon.When observed at low frequencies, power-

ful double radio sources generally exhibitextensive cocoons with mean diameters farin excess of their hot-spot diameters, whichwe may take as measures of the beamdiameters. The morphological trends dis-cussed above imply very large Mach

with the thickest, most extensive cocoonsshould be the most powerful. in agreementwith observation,

Although we understand the trends in

morphology on a qualitative level, our at-tempts to quantify these effects analyticallyhave not been successful Nonlinear andtime-dependent effects are just too com-plicated. For example, the terminal-shockstructure is complex and time-dependent,which implies that the cocoon gas is highlyinhomogeneous, These inhomogeneitics leadto mass motions (vortices) within the cocoonthat perturb the central beam and ultimatelycontribute to the unsteadiness of the ter-minal shock. In the face of such complex andcoupled effects, numerical simulations arcthe best predictive tool we have for the struc-ture of time-dependent jets.

61

Jet Stability

Numerical simulation is also a powerfultool for studying the effects of instabilities intime-dependent jets. This approach comple-ments the traditional analytic tool of linearperturbation theory, which completelycharacterizes all modes of instability only inthe small-amplitude limit. In contrast tolinear theory, our simulations show whichmodes grow to large amplitudes and howthey ultimately affect the flow. We now havesome tanta l iz ing h in ts tha t Kelvin-Helmholtz instabilities. which disrupt labo-ratory jets with low Mach numbers, may bereduced or totally absent in the high-Mach-number jets emerging from radio galaxiesand quasars. In addition, our simulationsconfirm recent analytic predictions of theexistence in supersonic jets of a new family ofmodes of the Kelvin-Helmholtz instabilityand characterize for the first time thenonlinear behavior of the new modes. Tomotivate our discussion. we review the keyresults gleaned from analytic studies of jetstability,

Linear Stability Theory

A simple analysis of the growth of axisym-metric, or pinch, modes of instability insupersonic beams is given in Sidebar 4,“Pinch Instabilities and the BernoulliEffect.” This analysis predicts that the rela-tion

(2)

defines a stability boundary in the (M,n)

This analysis applies to a purely longitudinalmode, that is, one that introduces no trans-verse structure within the perturbed channel.Such a mode corresponds to the fundamentalaxisymmetric mode of instability}.

In 1981 Attilio Ferrari and coworkers inBologna determined from perturbation the-ory that a whole family of unstable (that is.capable of growth) higher order modes exists

62

Pinch Instabilities—SIDEBAR 4

and the Bernoulli Effect

w hat happens to a fluid beamwhen its flow is slightly con-stricted? Does the flow resist the

constriction and remain stable? Or does theconstriction grow and pinch off the flow?Here we use a simple argument based onBernoulli’s principle to derive the conditionsunder which supersonic flow is susceptible topinch instabilities, or, in other words, is un-stable to longitudinal constrictions.

To establish the basic mechanism thatdrives a pinch instability, we consider first inFig. Al the steady flow of an incompressiblefluid, such as water, in a pipe with a localizedconstriction. The velocity and pressure of thefluid far from the constriction are vO and P o,respectively. At steady state the mass flux,pvA, is constant along the length of the pipe,Since the fluid is incompressible (that is, p isconstant) the fluid velocity v must increasewhen the cross-sectional area A decreases,The Bernoulli principle states that along afluid streamline the total specific energy Eremains constant:

Consequently, if p remains constant and vincreases, the pressure P in the vicinity of theconstriction must decrease. Figure A2 showswhat happens if the channel wall is replacedby a static background fluid of uniform pres-sure Po. Since the beam pressure P near theconstriction is less than the background pres-sure P o, the channel boundary will moveinward, further constricting the flow. By theabove reasoning further constriction leads toeven higher velocities, lower pressures. andeven further constriction.

Precisely this sequence of events is respon-sible for pinch instabilities in subsonic jets,

In principle, an initial constriction ofarbitrarily small amplitude will be amplifiedby the Bernoulli effect until the constrictioncompletely pinches off the flow. In actuality,the ambient fluid following the channelboundary inward at the point of constrictionis swept into the channel and carried down-stream by the flow as in Fig. A3. As a result,pinch instabilities contribute to the mixing ofa subsonic jet with its environment.

When the beam is supersonic, two othereffects come into play. First and most impor-tant. the compressibility of the fluid can nolonger be ignored. Thus the constancy of themass flux no longer implies an increase invelocity with a decrease in area. Instead thedensity increases so rapidly as the area de-creases that the velocity decreases also. Ageneral mathematical expression relating ve-locity change to area change in a flow channelof slowly varying cross section is

where c is the speed of sound in the fluid. Wesee that if the flow is subsonic (v < c), itsvelocity increases (dv > 0) at a constriction(dA < 0) just as it does in an incompressiblefluid, whereas if the flow is supersonic (v> c),its velocity decreases (dv < 0) at a constric-tion (dA < 0).

A naive application of the Bernoulli prin-ciple would predict that all supersonic beamsare stable to the pinch instability, since byEqs. 1 and 2 the pressure now increases at aconstriction and thus resists furtherpinching. This prediction is incorrect be-cause we have failed to take into account theeffects of the motion of the boundary in thedirection of the flow. As shown in Fig. B1, theinstability created when one fluid moves past


Supersonic Jets

Fig. A. Incompressible Flow

SIDEBAR 4

another (the Kelvin Helmholtz instability) isa convective instability; that is, the waveformof the perturbed boundary is carried down-stream with some velocity v.

Figure B2 shows the flow in a frame mov-ing at velocity v’. In this frame the beamflows through the constriction with a relativespeed of vb — v', and the ambient gas flowspast the constriction with a relative speed of—v’. Now consider the Bernoulli effect in thetwo flow channels shown in Fig. B2—oneinside the jet boundary and one surroundingit. To guarantee pinching. we require that atthe constriction the pressure decrease insidethe jet boundary and increase outside the jetboundary. This will be the case if both the jetgas and the ambient gas move subsonicallywith respect to the constriction, since, fromEqs. 1 and 2, only for relative Mach numbers(ratios of flow velocities in the moving frameto sound speed) less than unity will the chan-nel area and the pressure vary in the samedirection. Therefore, a sufficient conditionfor instability is that vb - v' < cb and v' < ca,where cb and ca are the speeds of sound in thebeam gas and the ambient gas, respectively.Adding the two inequalities eliminates theunknown v’ and yields the condition

(3)

Thus, on very simple physical grounds in-stability requires that the jet speed be subsonicwith respect to the sum of the internal andexternal sound speeds.

To express this condition in terms of thedimensionless parameters Mach number Mand density ratio n, we eliminate ca by invok-ing the pressure balance between the jet and

(4)

This relationship, which was first deduced byDavid Payne and Haldan Cohn from a dis-persion analysis of the linearized equationsof motion, thus follows simply from the factthat the growth of pinch instabilities requiressubsonic flow of both beam gas and ambientgas relative to the moving constriction. ■


at all Mach numbers greater than unity.These so-called reflection modes derive theirname from a special property of a supersonicshear surface, namely, for certain angles ofincidence, a sound wave will be amplifiedupon reflection from the surface. A reflectionmode instability in a supersonic beam fol-lows from the repeated amplification ofperturbations reflected back and forth be-tween the channel walls. A recent analysis ofthe reflection modes by David Payne andHaldan Cohn has shown that at the resonantwavelength of each mode, the wave fronts ofthe perturbation meet the beam boundary atthe Mach angle. Thus, the reflection modesare largely transverse in character, in con-trast to the fundamental mode. Figure 11shows the structure, as computed by Payneand Cohn with linear theory, of the funda-mental pinch instability and of the first andsecond reflection pinch instabilities.

A key finding of Payne and Cohn is thatEq. 2 defines a transition boundary (Fig, 12)in the dominant mode of instability: for M <

the fundamental mode exists, in agreement

reflection modes dominate. This boundary isindicated in Fig. 12 by the upper dashed line.Note that the majority of our simulated jets(solid circles) reside in the reflection-moderegime; in particular, five cases straddle thetransition boundary. We can therefore inves-tigate whether indeed a transition in the mo-dal character of nonlinear pinch instabilitiesoccurs, and if so, what effect the transitionhas on jet propagation.

Instabilityin the Nonlinear Regime

How do wc detect an instability in a nu-merical simulation’! Figure 13 illustrates thegrowth of a reflection-mode instability in the

Figure 13a is a color-coded plot of gas pres-sure for this jet at the same instant in itsevolution as that depicted in Fig. 7. Thecontinuity of color at the orifice indicatesthat we arc indeed fulfilling our pressure-

Fig. 11. Structure of the fundamental and first and second reflection pinchinstabilities in a (M= 5,n = 0.1) supersonic beam as predicted by linear pertur-bation theory. Dark and light shades of gray correspond to regions of positive andnegative pressure about the mean. Whereas the fundamental pinch mode is purelylongitudinal, the nth reflection mode possesses n nulls in the radial perturbationfunction. Adapted from David G. Payne and HaIdan Cohn, Astrophysical Journal291(1985):655-667.

●

0.1

t

Fig. 12. Transition boundaries in (M, n) parameter space. The transition boundary

analytically determined (see Sidebar 4). The approximate location of the line ofdemarcation between jets with cocoons and those without cocoons (naked beams)was determined from a visual inspection of the jets displayed in Fig. 10. The solidcircles indicate the initial (M, n) values for those jets.


Fig. 14. High-resolution simulations in the nonlinear re- stability in a (M= 1.5, n = 1) beam leads to constriction ofgime for supersonic jets on both sides of the transition the beam boundary (dark line) and entrainment of tonguesboundary between the fundamental mode and reflection of ambient gas. Isopressure contour lines reveal the forma-modes of the Kelvin-Helmholtz pinch instability. At t = 0 tion of normal shock fronts upstream of the channel con-the beam extends across the computational domain, flowing striations. (b) Growth of a fundamental pinch instability inin at the left with fixed parameters and flowing out at the a (M = 3, n = 10) beam establishes a mixing layer at theright. The initial jet radius is resolved into twenty computa- beam boundary. The isodensity contour lines reveal that thistional zones. A perturbation is applied at the inlet by mixing layer rapidly spreads into the supersonic core. (c)continuously admitting beam gas with a l-percent pressure Growth of the first reflection pinch instability in a (M = 3, nexcess (that is, K = 1.01). The subsequent evolution of the = 1) beam causes only a mild undulation of the beambeam is computed by using a numerical fluid-interface boundary and, as revealed by the isopressure contour lines,technique to track the beam boundary until the instability is excites a network of oblique internal and external shockwell established. (a) Growth of a fundamental pinch in- waves.

matching boundary condition (K = 1). We can now understand the origin of the stabilities. This evolution is traced in Fig,Nevertheless, large-amplitude pressure fluc- network of oblique shock waves that appears 13b, a plot of the axial pressure distributiontuations appear within a few beam diameters in the beam in Fig. 7a—it is the result of the within the jet as a function of time. To orientdownstream of the orifice, caused by the oblique acoustic disturbances (reflection- the reader. we point out that the color dis-presence there of embedded rarefactions and mode instabilities) pictured in Figs. 11b and continuity along the diagonal line from lowershock waves (see Fig. 7a). These fluctuations 11c, which have grown to finite amplitude left to upper right is the space-time trajectoryare not tied to the orifice, in marked contrast and steepened into shock fronts. The super- of the bow shock as it is driven into theto the pressure fluctuations in laboratory ficial resemblance of the resulting shock ambient gas (which has uniform pressure) bysupersonic jets, which are excited with large network to the diamond-shaped steady-state the advance of the jet head. The sharp jumpamplitudes at the nozzle by the pressure shock pattern displayed in Sidebar 3 breaks from blue and green to orange and red tracesmismatch there and subsequently decay down when wc investigate the time evo- the motion of the terminal shock front. Thedownstream, lution of nonlinear reflection-mode in- red diagonal ridges trace the movement of


Supersonic Jets

External Shock Knot

Fig. 15. The shock-wave structure of a nonlinear reflection- The pattern created by the saturated oscillation has amode pinch instability and the resulting temperature dis- wavelength of about 2.5 beam diameters and travels down-tribution. Temperature values are depicted according to an stream at a speed vknot that is a significant fraction of thearbitrary scale in which hot is white and cold is dark gray. beam speed.

the high-pressure knots produced by theembedded shock network and reveal a re-markably well-ordered space-time flow. Thispattern indicates that the reflection mode is aconvective instability, that is. one that growsin amplitude as the perturbation is con-vected downstream.

Payne and Cohn suggest that the sta-tionary shock diamonds and high-pressureknots arising in underexpanded (K > 1 )supersonic jets are a signature of the reflec-tion mode at marginal stability, that is, at astate of zero growth rate and zero phasevelocity. Zero growth implies zero damping,and this may well account for the remarkablelongevity of the oscillations observed in theplumes of some rocket engines (see Fig. l).Of course, the irreversible processes of shockheating and turbulence do introduce somedamping, and the growth of the turbulentmixing layer at the jet boundary ultimatelyovercomes the oscillations, as shown in Fig.D of Sidebar 3.

An ever-broadening mixing layer is con-spicuously absent in Fig. 7a; the clumps ofmaterial above and below the beam are whatis left of the cocoon as it mixes with theambient gas (see Fig. 4). In addition, thenearly constant thickness of the green vortexlayer in Fig. 7b shows very little spreading ofthe beam boundary. Reflection-mode insta-bilities thus seem to be far less disruptive


than the fundamental-mode instability.To further investigate this interesting re-

sult, we performed high-resolution simula-tions of supersonic beams with parameterson both sides of the transition boundary. Theresults of these calculations (Fig. 14) supportthe general conclusion that boundary mixingis greatly reduced in supersonic beams satis-

cal study of the transition from fundamentalmode to reflection mode is in progress.

The key features of the saturated reflectionmode are summarized in Fig. 15. The mildundulation of the jet boundary is coupled tothe network of oblique internal shock waves.Two subtle but important differences be-tween this wave pattern and the steady-statejet depicted in Fig. A of Sidebar 3 arc ( 1 ) thereflection of shocks from the boundary asshocks rather than rarefactions and (2) thetransmission of a shock into the gas externalto the beam. The external shock wave is themechanism by which the energy of relativemotion amplifies the internal reflected shock.Since there is no relative motion betweenambient gas and boundary corrugations inunderexpanded supersonic jets, no externalshock wave is excited in such jets,

We observe the growth of nonlinear reflec-tion modes in all but one of the jets lyingwithin the parameter space below the in-stability transition boundary and above the

boundary for cocoon formation (lowerdashed line in Fig. 12). In the one exception,a (M= 1.5,n = 0.1) jet, the shock waves weretoo weak to be detected numerically. Sincethe pattern wavelength of the saturated os-cillation is about 2.5 beam diameters in theMach 3 and Mach 6 jets and only slightlygreater at Mach 12, the dependence ofwavelength on Mach number is quite weak.This result is consistent with perturbationtheory, which predicts a maximum growthrate at wavelengths near the beam circumfer-ence. The speed of advance of the patternsvaries between 10 and 50 percent of the beamspeed and is consistent with the density-ratioscaling given in Eq. 1. This is not surprisingbecause ram-pressure balance in a framecomoving with the oblique shocks de-termines the advance speed of the patterns,just as it determines the advance speed of theworking surface. Quite generally, therefore,the high-pressure knots advance with ap-proximately the same speed as the jet head,as can be seen in Fig, 13b.

In principle, cocoon-shrouded beams arealso subject to reflection-mode instabilities,We find, however, that cocoon dynamicsputs its own imprint on the internal structureof the beam. At high Mach numbers thebackflow within the cocoon is also super-sonic, and components of this momentumflux normal to the beam drive oblique shock

67

waves into the beam (Fig. 16). The perturba-tions are associated with toroidal vorticeswithin the cocoon that drift back toward thesource, The oblique shocks they excite pos-sess a less regular pattern than shocks in-duced by reflection-mode instabilities, andconsequently the high-pressure knots are alsoless regular,

Astrophysical Implications

Knots of enhanced emission, frequentlyregularly spaced, are observed in numerousextragalactic radio jets (Fig. 17). Thenonlinear reflection-mode instability is anattractive candidate mechanism for produc-ing these knots because of our finding thatthis instability is ubiquitous in highly super-sonic jets. The high-pressure knots createdby the instability would appear as regions ofenhanced emission because of the P2 scalingof the synchrotrons emissivity mentionedpreviously.

A prediction of our model is motion of theemission knots. In principle, a measurementof their apparent motion with respect to thegalactic nucleus could be used to infer the jetspeed and density ratio. Unfortunately, thetime scales associated with the evolution ofkiloparsec-scale radio jets are too great forthis motion to be detectable. Some parsec-scale extragalactic jets and stellar jets alsoexhibit regular strings of emission knots, andthe motion of these knots is detectable withcurrent radio and optical telescopes, Futurehigh-resolution observations of stellar jetswith the Hubble Space Telescope and ofcompact radio jets with the the Very LargeArray of radio telescopes can put our modelto the test.

Returning to the large-scale radio jets andtheir remarkable ability to propagate withoutdisruption over intergalactic distances, wepropose that the strings of emission knotsmay be a signature of nature’s way out of thestability problem. This may sound con-tradictory, since the reflection modes thatproduce the knots are a kind of instability.However, like many other instabilities in

Fig. 16. Interaction of backward flowing cocoon gas (white) with forward flowingsupersonic beam gas produces oblique internal shock waves in a (M = 12, n = 0.01)

boundary (heavy line).

Fig, 17. Radio emission knots in the extragalactic jet 1919+479. Regions of highradio brightness are shown in white. The bright spot to the upper right is theunresolved radio “core” (the source of the jet), which coincides with the nucleus ofa giant elliptical galaxy. The onset of large-scale bends in the jet signals itsincipient disruption. The radio jet is roughly 300,000 light-years long. (Un-published VLA image courtesy of J. O. Burns.)


Supersonic Jets

fluids and plasmas, the reflection modes ap-pear to saturate at finite amplitudes and dominimum damage to the flow.

The enhanced boundary stability of thereflection modes is almost certainly exag-gerated by our two-dimensional simulations.It is nevertheless known experimentally, butnot understood, that the spreading rate ofsupersonic shear layers drops precipitouslybeyond Mach numbers of about 5. Anotherimportant feature of reflection modes is theexistence of oblique internal shock waves.Their oblique character allows the postshockaxial flow speed to remain supersonic eventhough the component normal to the shock issubsonic, These two effects-enhancedboundary stability and minimal flow de-celeration—make the reflection mode abenign instability in terms of jet disruption.

In very long jets these small effects mayadd up to a big effect and ultimately disruptthe flow. Quite often the signature of incip-ient disruption is the onset of large-scalewiggles in the jet, as can be seen in Fig. 17.This kind of behavior, which is observedexperimentally in fluid jets and magneticallyconfined plasma columns, is a consequenceof the growth of nonaxisymmetric in-stabilities to amplitudes at which nonlineareffects become important. These so-calledgarden-hose instabilities are fundamentallythree-dimensional in character and thereforecannot be studied within the context of two-dimensional axisymmetric numericalsimulations. However, two-dimensional

Fig. 18. Numerical simulation of thegrowth of a kink instability in a two-dimensional “slab’’ jet with M = 3 andn = 10. The variable plotted is density.The instability is excited at the inlet byimposing on the beam a sinusoidallyvarying transverse velocity with anamplitude equal to 5 percent of thebeam’s axial velocity. The frequency ofthe perturbation corresponds to the

divided uniformly into forty zones.


simulations of a planar “slab” jet do allow usto study the effects of kinking on the flow.

We performed such a simulation for a slabjet with parameters in the ordinary-moderegime, namely M = 3 and n = 10. As shownin Fig. 18, the amplitude of the oscillationsgrows rapidly, thereby destroying the direc-tionality of the flow. A similar calculation fora slab jet with values of M and n in thereflection-mode regime also shows growth ofkinks, but the kinks grow much more slowlyand are of longer wavelength. As we mightexpect, oblique internal shock waves accom-pany the bends. These shocks become quite

produce high-pressure knots at the bend.This is a likely explanation for the origin ofthe bright emission knot at the bend in theradio jet shown in Fig. 17.

Both the axisymmetric and nonaxisym-metric instabilities derive from the interac-tion between a moving fluid stream and aconfining medium. Although the pressure ofthe intergalactic gas appears adequate to con-fine the vast majority of radio jets, evidenceis accumulating that the jets emerging fromdistant quasars are too highly pressurized tobe confined by their local environment. Mag-netic confinement, which is often invoked toexplain these observations, would requirethat the jets carry substantial currents. As wehave no unambiguous evidence for the pres-ence or absence of such currents in radio jets,

magnetic confinement remains just ahypothesis. It is curious and perhaps signifi-cant that Payne and Cohn find that theproperties of the linear reflection modes arequite insensitive to the confinement mecha-nism, be it pressure or magnetic forces. If thisfinding holds also in the nonlinear regime,strings of emission knots may well be a uni-versal signature of confined supersonic jets.

OutlookIn this article, and in our research, we have

emphasized the phenomenology of a set ofbasic physical laws operating in anastrophysical context and the dependence ofthis phenomenology on the underlyingparameters in order to uncover trends inmorphology and stability that may aid in theinterpretation of extragalactic radio sources.Far from being detailed source models, ournumerical simulations are rather thought-experiments carried out by a computer thatshow us the consequences of self-consistencyin a nonlinear, multidimensional, time-de-pendent context. An important outcome ofour investigation is the discovery of large-scale coherent structures associated with thesupersonic beam, the working surface, andthe cocoon and of the essential role thesestructures play in contributing to the phe-nomenological trends.

We foresee future research in the physics

of extragalactic jets proceeding along two

Further Reading

R. Courant and K. O, Friedrichs. Supersonic Flow and Shock Waves. New York: IntersciencePublishers, Inc., 1948.

H. Reichenbach. “Contributions of Ernst Mach to Fluid Mechanics.” Annual Review of FluidMechanics 15(1983):1-28.

IR. Ladenburg, C. C. Van Voorhis, and J. Winckler, “Interterometric Studies of Faster Than SoundPhenomena. Part II. Analysis of Supersonic Air Jets.” Physical Review 76(1949):662-677.

R. D. Blandford and M, J. Rees. “A ‘Twin-Exhaust’ Model for Double Radio Sources.” MonthlyNotices of the Royal Astronomical Society 169(1974):395-415.

Roger D, Blandford, Mitchell C. Begelman, and Martin J. Rees. “Cosmic Jets.” Scientific American,May 1982, pp. 124-142.

complementary tracks: 1) further phenome-nological studies aimed at systematically in-vestigating the new dynamical effects thatarise when magnetic forces, bulk relativisticmotion, and turbulence are incorporatedinto two- and three-dimensional models, aswell as understanding the impact of in-homogeneities, pressure stratifications, andcrosswinds in the intergalactic environment;and 2) the development of detailed sourcemodels aimed at bridging the gap betweensimulation and observation by incorporatingself-consistent synchrotrons emission modelsinto the dynamical calculations and followedby mapping of the resultant flow field forobservable quantities such as the distrib-ution of radio brightness, degree of polariza-tion, spectral index, and magnetic field orien-tation. Only through such a twin-track ap-proach will we be able to select betweencompeting hypotheses concerning the com-position, internal processes, state of motion,and ultimately the origin of extragalacticradio sources and their associated jets, whichhave fascinated and continue to fascinatetwentieth-century astronomers and astro-physicists. The increase in resolving power ofour respective tools—radio telescopes andcomputers—is continuing, if not accelerat-ing, and promises exciting research on intothe twenty-first century and progress towardunderstanding these spectacular cosmiceruptions. ■

Norman J. Zabusky, “Computational Synergetic.” Physics Today, July 1984.


Supersonic Jets

Michael L. Norman, Larry Smarr, Karl-Heinz A. Winkler, and Michael D. Smith. “Structure andDynamics of Supersonic Jets.” Astronomy and Astrophysics 113(1982):285-302.

Larry L. Smarr, Michael L. Norman, and Karl-Heinz A. Winkler. “Shocks, Interfaces, and Patternsin Supersonic Jets. ”In Fronts, Interfaces, and Patterns—Proceedings of the Third Annual Conferenceof the Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico, May2-6, 1983. Edited by Alan R. Bishop, Laurence J. Campbell, and Paul J. Channell. Physica12D( 1984):83-106.

Michael L. Norman, Larry Smarr, and Karl-Heinz A. Winkler. “Fluid Dynamical Mechanisms forKnots in Astrophysical Jets.” In Numerical Astrophysics. Edited by J. Centrella, J. LeBlanc, and R.Bowers. Portola Valley, California: Jones and Bartlett, 1985.

Michael L. Norman, Karl-Heinz A. Winkler, and L. Smarr. “Knot Production and Jet Disruption viaNonlinear Kelvin-Helmholtz Pinch Instabilities.” In Physics of Energy Transport in ExtragalacticRadio Sources—Proceedings of NRAO Workshop No. 9, Green Bank, West Virginia, August 1984,pp. 150-167. Edited by Alan H. Bridle and Jean A. Eilek. Green Bank, West Virginia: National RadioAstronomy Observatory, 1985.

David G. Payne and Haldan Cohn. “The Stability of Confined Radio Jets: The Role of ReflectionModes.” Astrophysical Journal 291 ( 1985):655-667.

Michael L. Norman joined the Laboratory in 1984 as a staff member in the Advanced Concepts andPlasma Applications Group of the Applied Theoretical Physics Division. He received his B.S. inastronomy from the California Institute of Technology in 1975 and his M.S. in 1976 and Ph.D. in1980 in engineering and applied sciences from the University of California, Davis. While earning hisPh. D., he performed numerical calculations of protostar formation. From 1980 until he joined theLaboratory in 1984 he was a staff member at the Max Planck Institut fur Physik und Astrophysik inMunich where he did the work described in this article. Mike is also an Adjunct Assistant Professorin the Physics and Astronomy Department of the University of New Mexico and is directing theresearch of a number of graduate students on extragalactic jets.

Karl-Heinz A. Winkler earned his Dr. rer. nat. in physics from the Georg-August UniversitatGottingen in 1976. He then became a staff member at the Max Planck Institut fur Physik undAstrophysik in Munich until August 1984, where he did research on radiation hydrodynamics, starformation, and supersonic jets. In 1978 he became a consultant to Los Alamos and Livermore. Karl-Heinz joined the Laboratory as a staff member in the Applied Theoretical Physics Division duringthe summer of 1984. He is designing numerical methods and applying them to interesting problemsin physics and astrophysics.


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Fig. 1. A supersonic exhaust jet trailing the Bell X-1 ... · Fig. 1. A supersonic exhaust jet trailing the Bell X-1 rocket plane, the first manned aircraft to break the sound barrier