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Geologic Nozzles GEOLOGIC NOZZLES Susan Werner Kieffer U.S. GeologicalSurvey Flagstaff,Arizona Abstract. Sonicvelocities of geologicfluids, such as low soundvelocity. The high soundspeed of liquid water volcanic magmas and geothermal fluids, can be as low determines the characteristics of harmonic tremor observed as 1 m/s. Critical velocities in large rivers can be of the at the gyeserduring the rechargeinterval, whereas the low order of 1-10 m/s. Becausevelocities of fluidsmoving in sound speed of the liquid-vapor mixture influencesthe these settingscan exceed these characteristicvelocities, fluid flow characteristicsof the eruption. At the rapidsof sonicand supersonicgas flow andcritical and supercritical the Colorado River in the Grand Canyon, Arizona, the shallow-waterflow can occur. The importanceof the low channel is constrictedinto the shape of a converging- characteristicvelocities of geologic fluids has not been divergingnozzle by debrisflows that enter from tributary widely recognized,and as a result, the importanceof canyons. Both subcriticaland supercriticalflow occur supercriticaland supersonicflow in geologicalprocesses within the rapids. The transportcapacity in the rapidscan has generallybeen underestimated.The lateral blast at be so great that the fiver contoursthe channelto a charac- Mount St. Helens, Washington,propelled a gas heavily teristicshape. This shapecan be usedto interpretthe flood laden with dust into the atmosphere.Because of the low historyof the Colorado River over the past 103-105 years. sound speed in this gas (about 100 m/s), the flow was The unity of fluid mechanics in these three natural internallysupersonic. Old Faithful Geyser,Wyoming, is a phenomena is provided by the well-known analogy converging-divergingnozzle in which liquid water refilling betweengas flow and shallow-waterflow in converging- the conduit during the recharge cycle changesduring divergingnozzles. eruptioninto a two-phaseliquid-vapor mixture with a very 1. INTRODUCTION apparentlyfirst describedby Riabouchinsky[ 1932] (a more recentand accessiblereference is Loh [1969, pp. 1-60]). This paper was written for the symposium The analogy is semiquantitativeand was thus widely "Perspectivesin Fluid Mechanics"held at the California exploredin the earlydays of wind tunneldevelopment. In Institute of Technology, January 10-12, 1985. It is modem times the analogyhas been primarily a teaching specifically an essay of the author's perspective on tool [e.g., Thompson,1972, pp. 517-531] and has never geologicfluid flow problems,rather than an exhaustive been used by geologiststo explain large-scalenatural review. For the sakeof brevity,references in thispaper are phenomena.The purposeof thispaper is to showthe basis limited. The authorhas published derailed analyses of the for invoking nozzle-flow theory for interpretationof threegeologic problems presented, and the readerwill find complexgeologic events and to providea perspectiveon referencesto otherrelevant work in the citedpapers. geologicproblems in which the importanceof supercritical and supersonicflow hasbeen underestimated. 2. GEOLOGIC NOZZLES A major reason that geologic events have not been viewed from the particularperspective of fluid mechanics An eruption of Old Faithful geyser, a flood on the presentedhere is the subdivisionof fluid mechanicsand its Colorado River, and a lateral blast from Mount St. Helens applied fields into the specialtiesof compressibleand do not, at first glance,appear to be related. A geographic incompressibleflow, for example, aeronauticsversus map of the locationsof thesethree places certainly does hydraulics. This subdivisionarises from the need to not reveal any underlying geologic unity (Figure 1). simplifythe complexmomentum and continuityequations However, a fluid-dynamical unity is revealed when the in orderto solve mostpractical problems. In vectorform, "locations"are shown insteadon schematicdiagrams of the momentumequation for a viscousfluid moving in a gas flowing througha nozzle or of shallowwater flowing gravitational field under the influence of a pressure througha flume (Figure 2) (for example,see Liepmann gradient is complex because of dimensionality and andRoshko [1957, p. 127]). The analogybetween the flow nonlinearity: Du fields for compressiblegas and shallow water was P•i- =-VP + [V. x]+ pg (1) This paperis not subjectto U.S. copyright. Reviewsof Geophysics,27, 1 / February1989 pages 3-38 Publishedin 1989 by the AmericanGeophysical Union. Paper number 89RG00093 e3e 4 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer: GEOLOGIC NOZZLES where p is the fluid density,D/Dt is the materialderiva- tive, u is the fluid velocity,VP is thepressure force acting on the fluid, V. x is the viscousforce, and g is the accelera- tion of gravity. The continuityequation for mass is (or generally simpler but is still difficult to apply in a (a) geometricallycomplicated problem: Dp_ Dt - -p(V.u) (2) where(V ß u) is the divergenceof velocity. In many casesthese two importantequations can be considerably simplified by considerationof the fluid propertiesor the boundaryconditions of the problem. For example,if pressurechanges are relativelysmall, compres- sibilitycan be neglectedso thatp - constand V ß u = 0. Such an assumptionunderlies all of hydraulics, and geologistswith interestsin hydraulicsor relatedgeomor- phic problemstypically divergeat an early stageof their educationfrom advancedstudies of compressiblefluid dynamics. Alternatively,in manyflows the pressuregradient may be great enough that compressibilityis importantbut gravity is not; g ~ 0. This latter conditionis assumedin mostof gas dynamics,and becauseof the prominentrole of gravity in mostgeologic processes, few geologistsare exposedto a rigorousgas dynamics curriculum. 129 ø 125 ø 121 ø 117 ø 113 ø 109 ø 105 ø Figure 2. Four diagramsof behaviorof gas flowing througha , / /?,...,,-'--4.. • • t converging-divergingflume. Vector velocitiesare indicatedby I "•/"r-•. CANADA arrows. At the exit of the nozzle (or channel),on the right, the 45 ø ;,, structureof the flow field in the departing fluid is shown ';i /-'L.*-o,,./ - schematicallyby the medium shading. Shock and rarefaction waves (alternatively,positive and negativenormal and oblique hydraulicjumps) are indicatedby the lightestshading. 41 ø /"•' "7"-• J• / / ! Althoughthe subjectsof nozzle gas dynamicsand of T--'-- shallow-water hydraulics evolve from very different 37 ø - 'x i i , approximationsto the conservationequations, important '\ /' _,,øy•. I' , conceptscommon to both subjectshave beenrecognized because, when reduced to suitable nondimensional variables,the conservationequations in the two subjects 33 ø becomeidentical. (Readersfamiliar with thisidentity can skip directlyto the last paragraphof this sectionand to section3.) Examinefirst the massand momentumequations for a 29 ø EXICO perfect gas. For simplicity, assume that the flow is quasi-one-dimensionalalong a coordinatedirection x. The equationsof massand momentumconservation for flow of 500 k m'"N. a compressiblegas are 25 ø -'''''' i 121ø 117ø 113ø 109ø 105ø 101ø 8p 8p 8u (3) Figure 1. Index map of the geographiclocations of Crystal •)t+ U•xx+P•xx =0 Rapids (Grand Canyon, Arizona), Old Faithful Geyser (Yel- lowstone National Park, Wyoming), and Mount St. Helens 8u 8u ! 8P= 0 (4) (Washington). Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWSOF GEOPHYSICS ß 5 For a perfectgas and isentropicflow, within the contextof all of the simplifyingassumptions, flow of gas in a nozzle and flow of shallow water in a PV = RT (5) flume are governedby the same conservationequations. Identical flow fields therefore occur when the proper P W = P0V0 •' = const (6) nondimensional variables are considered. The illustrations in Figure 2 represent the flow whereP is pressure,V is volume,R is the gasconstant, T is conditionsat different ratios of upstream(reservoir) and temperature,and ¾ is the ratio of specific heats (the downstream(atmosphere or tailwater) conditions. If the isentropicexponent). The subscriptzero indicatesa figure is interpretedas representinga crosssection of a referencestate (typically one wherethe fluid is at rest with horizontalnozzle throughwhich gas is flowing from left to velocityu = u0 = 0). For a perfectgas, (3)-(6) canbe right, the different parts of the figure representthe flow combinedto give field from differentpressures (P1, P2,'") in the left reservoirsto lower pressuresin the right reservoirs. The 3U 3U •tP0T--2 30 -- 0 (7) low-pressurereservoirs are assumedto be infinitely large •}t+ u•-•+ •0•P • and are therefore not shown to scale. Alternatively,if Figure 2 is interpretedas representing For water flowing from one infinite reservoir into a map view of a horizontalchannel in which shallowwater anotherwith lower head,the equationsof motionthat can is flowing from left to right, the differentparts of the figure be directlycompared with (3) and (7) are representthe flow field from reservoirsof different depth. 3h 3h 3u The driving energy for the flow is the elevated depth of i•t+ u•-• + h• =0 (8) water in the left reservoircompared with the right. The waterhas a potentialenergy H r (calledthe headand generallyexpressed asa depth),indicated as H l, H2 ,..., in 3u+ 3u 3h (9) i•t U•xx+g• =0 the partsof Figure2. With this introduction, let us reexamine the sense in whereh is the water depth. In theseequations, and in the which eachof the geologicproblems mentioned above is a figuresin this paper,it is assumedthat in a verticalcross nozzleproblem. In the spiritof emphasizingthe similarity section(specified
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