Abstract
Sonic velocities of geologic fluids, such as volcanic magmas and geothermal fluids, can be as low as 1 m/s. Critical velocities in large rivers can be of the order of 1–10 m/s. Because velocities of fluids moving in these settings can exceed these characteristic velocities, sonic and supersonic gas flow and critical and supercritical shallow-water flow can occur. The importance of the low characteristic velocities of geologic fluids has not been widely recognized and, as a result, the importance of supercritical and supersonic flow in geological processes has generally been underestimated. The lateral blast at Mount St. Helens, Washington, propelled a gas heavily laden with dust into the atmosphere. Because of the low sound speed in this gas (about 100 m/s), the flow was internally supersonic. Old Faithful Geyser, Wyoming, is a converging-diverging nozzle in which liquid water refilling the conduit during the recharge cycle changes during eruption into a two-phase liquid-vapor mixture with a very low sound velocity. The high sound speed of liquid water determines the characteristics of harmonic tremor observed at the geyser during the recharge interval, whereas the low sound speed of the liquid-vapor mixture influences the fluid-flow characteristics of the eruption. At the rapids of the Colorado River in the Grand Canyon, Arizona, supercritical flow occurs where debris discharged from tributary canyons constricts the channel into the shape of a converging-diverging nozzle. The geometry of the channel in these regions can be used to interpret the flood history of the Colorado River over the past 103–105 years. The unity of fluid mechanics in these three natural phenomena is provided by the well-known analogy between gas flow and shallow-water flow in converging-diverging nozzles.
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References and footnotes
Approved for publication by Director, U.S. Geological Survey, January 23, 1986.
For example, see H.W. Liepmann and A. Roshko, Elements of Gasdynamics (Wiley, New York, 1957). For the sake of brevity, references in this paper are limited. The author has published detailed analyses of the three geologic problems discussed here in the references cited below, and the reader will find references to other relevant work in those papers.
This analogy was apparently first described by D. Riabouchinsky, C. R. Acad. Sci. 195, 998–999 (1932). For a more accessible reference, see Modern Developments in Gas Dynamics, edited by W.H.T. Loh (Plenum, New York, 1969), pp. 1–60.
For example, see P.A. Thompson, Compressible-fluid Dynamics (McGraw-Hill, New York, 1972), pp. 517–531.
In the spirit of emphasizing the similarity of the various flow fields discussed in this paper, the word “nozzle” will be used interchangeably with the words “flume”, “channel”, and “conduit”, and the word “contouring” will be used interchangeably with the word “eroding”.
“Low” pressure ratio in this context means that the reservoir pressure is less than about 2 times atmospheric pressure; see, for example, the tables of isentropic flow variables given in M.J. Zucrow and J.D. Hoffman, Gas Dynamics (Wiley, New York, 1976).
Although schematic illustrations of the structure of supersonic gas jets can be found in most textbooks on gas dynamics, collections of actual photographs are rare. One such collection is E.S. Love and C.E. Grigsby, NACA RM L54L31 (1955).
A “small” head difference means that the elevation difference between the two reservoirs should not exceed approximately one-third of the head of the source reservoir.
Collections of illustrations of subcritical flow accelerating to supercritical flow in a converging-diverging channel are rare. Some examples can be found in E. Preiswerk, NACA TM 935 (1940). Illustrations of supercritical flow in converging and diverging channels can be found in the four papers in High-velocity Flow in Open Channels: A Symposium, Paper 2434, Trans. ASCE 116, 265–400 (1951).
Reviewed in S.W. Kieffer, J. Geophys. Res. 82, 2895–2904 (1977).
S.W. Kieffer and J. Delany, J. Geophys. Res. 84, 1611–1620 (1979).
K. Richards, Rivers: Form and Process in Alluvial Channels (Methuen, London, 1983), p. 58.
As an example of the relative magnitudes of Froude and Mach numbers, consider order of magnitude estimates for Old Faithful and for the Mount St. Helens lateral blast. At Old Faithful, the exit velocity is ∼ 80 m/s (see text, Section IV D). As the hottest fluid (116–118 °C) ascends through the conduit to the exit plane, it becomes a two-phase mixture with about 4 weight percent vapor, for which the equilibrium sound speed is about 57 m/s at 0.8 bar atmospheric pressure at the elevation of Old Faithful. This gives a Mach number of ∼ 1.5, indicating that compressibility effects are important. An internal (densimetric) Froude number for the jet of Old Faithful can be calculated (see Section IV D). For Old Faithful, I take nominal values of jet velocity = u = 80 m/s, jet density = ϱo = 11.2 kg/m3 (decompression of 116°C water isentropically to 0.8 bar, 93 °C, 4 percent vapor), atmospheric density = ϱa = 0.7 kg/m3, and an equivalent axially symmetric conduit diameter = D = 1.1 m. With these parameters, the square root of the densimetric Froude number (which is the value to be compared with a Mach number) is 25. The jet is negatively buoyant because πo > ϱa. An internal Froude number for the Mount St. Helens lateral blast, considered as an incompressible density flow on an inclined plane, can be calculated from Fr = u/(g′d cos θ)1/2, where g′ = g (ϱa-ϱo)/ϱo (the absolute value of this quantity is taken); ϱo is the density of the jet; ϱa is the density of the atmosphere, assumed uniform; d is the flow thickness; and θ is the slope angle. For nominal parameters, I take a flow density of 100 kg/m3 (g′ = 9.7 m/s2), θ = 11 degrees, u = 100 m/s, d = 100 m. For these parameters, Fr ∼ 3.2. This represents a minimum estimate, because internal flow velocities may have been greater by a factor of two to three. Note that the observed velocity of the front of the blast (100 m/s) is nearly identical to the sound speed of a pseudogas laden with solid fragments at a mass ratio of 25/1 (see Ref. 37), so that M ∼ 1. These order-of-magnitude estimates demonstrate that quantitative models for the flow fields of Old Faithful and the Mount St. Helens lateral blast must eventually consider both compressibility and gravity effects.
The values of constriction used to obtain Fig. 6 were inferred by measuring the width of the surface water in the photo series of the 1973 U.S. Geological Survey Water Resources Division, one of which is shown in Fig. 5. From measurements of surface width, the constriction at Crystal Rapids in Fig. 5 is 0.33, and it plots as the left-most block in Fig. 6. However, for purposes of hydraulic modeling later in the discussion, it is necessary to assume an idealized cross section for the channel. A rectangular cross section is assumed. In this simplification, the “average” constriction used for modeling is generally less than that measured from air photos, because the shallow, slow flow across the debris fan, which shows in air photos of the water surface but accounts for only a small fraction of the total discharge, is ignored. In the case of Crystal Rapids, the model value is 0.25, so the reader should be alerted to this change in “shape parameter” when the modeling calculations are discussed. By either criterion used to determine the shape parameter, the channel at Crystal Rapids was more tightly constricted than at the older debris fans that formed before Glen Canyon Dam was emplaced.
Because of the common usage of cfs in hydraulics and by river observers, volume flow rates are given in both metric and English units throughout this paper.
Details given in S.W. Kieffer, J. Geol. 93, 385–406 (1985).
The backwater above Crystal Rapids extends as much as 3 km upstream and is affectionately dubbed “Lake Crystal” by river runners.
The past tense is used in this discussion because the events of 1983 modified Crystal Rapids, and these calculations are not appropriate to its current configuration.
The calculations described here attribute all changes in flow regime to lateral constriction, because of the assumption of constant specific energy. In all rapids there are changes in bed elevation that affect the total and specific energy of the flow and, therefore, affect the transition from subcritical to supercritical conditions. The words “subcritical” and “supercritical” as used in this section therefore apply to a large-scale condition of the rapid, not to local details, because these additional effects are not accounted for. Most rapids are weakly supercritical because of changes in bed elevation, even in the regime called “subcritical” in this section. There are also substantial small-scale irregularities in bed topography, such as ledges and rocks, that cause local supercritical flow. The behavior of the river in flowing around such obstacles is not included in this generalized discussion.
S.W. Kieffer, J. Volc. Geoth. Res. 22, 59–95 (1984).
C.O. McKee, D.A. Wallace, R.A. Almond, and B. Talais, Geol. Survey, Papua New Guinea Mem. 10, 63–84 (1981).
F. Birch and G. Kennedy, in Flow and Fracture of Rocks, Geophys. Monogr. Ser. 16, 329–336 (1972).
Summarized in S.W. Kieffer and J. Westphal, EOS: Trans. Amer. Geophys. Union 66 (46), 1152 (1985); details in preparation.
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W. Hentschel, master's thesis (Göttingen, 1979); also Fortsch. Akustik, DAGA'80, Berlin, p. 415–418.
The geometry of the conduit determines whether the conduit should be considered open-ended or closed-ended. Since no probe has demonstrated a large reservoir at the bottom of the conduit, it is treated here as a closed pipe. Resonant frequencies of an open pipe would be half of those calculated from this formula.
J.S. Turner, J. Fluid Mech. 26, 779–792, 1966; C.J. Chen and W. Rodi, Vertical Turbulent Buoyant Jets — A Review of Experimental Data (Pergamon Press, New York, 1980).
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F.J. Moody, Trans. ASME J. Heat Transfer 87, 134–142, 1965, as reviewed by Wallis (Ref. 30, p. 48).
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S. Fuller and J. Schmidt, Yellowstone in Three Seasons (Snow Country Publications, Yellowstone National Park, Wyoming, 1984).
R.C. Leet, J. Geophys. Res., in press; contains review of other models for volcanic tremor.
Many detailed articles about Mount St. Helens can be found in P.W. Lipman and D.R. Mullineaux, eds., The 1980 Eruptions of Mount St. Helens, Washington, U.S. Geological Survey Prof. Paper 1250 (U.S. Gov't. P.O., Washington, 1981).
J.G. Moore and W.C. Albee, in Ref. 35. The 1980 Eruptions of Mount St. Helens, Washington, U.S. Geological Survey Prof. Paper 1250 (U.S. Gov't. P.O., Washington, 1981)
S.W. Kieffer, pp. 379–400 in Ref. 35. The 1980 Eruptions of Mount St. Helens, Washington, U.S. Geological Survey Prof. Paper 1250 (U.S. Gov't. P.O., Washington, 1981)
See, for example, M. Malin and M. Sheridan, Science 217, 637–640, 1982; J. Eichelberger and D. Hayes, J. Geophys. Res. 87, 7727–7738; or J. Moore and C. Rice, in Explosive Volcanism: Inception, Evolution, and Hazards, edited by F.M. Boyd (Nat. Acad. Press, Washington, 1984), Chap. 10.
JANNAF [Joint Army, Navy, NASA, Air Force] Handbook of rocket exhaust plume technology, Chemical Propulsion Information Agency Publication 263, Chap. 2 (1975).
S.W. Kieffer and B. Sturtevant, J. Geophys. Res. 89, 8253–8268 (1984).
S.W. Kieffer, in Explosive Volcanism: Inception, Evolution, and Hazards, edited by F.M. Boyd (Nat. Acad. Press, Washington, 1984), Chap. 11.
M.J. Davis and E.J. Graeber, EOS: Trans. Amer. Geophys. Union 61, 1136 (1980).
S.W. Kieffer and B. Sturtevant, Erosional furrows formed during the lateral blast at Mount St. Helens, May 18, 1980, submitted to J. Geophys. Res.
Comprehensive reviews of this work can be found in L. Wilson and J.W. Head III, Nature 302, 663–669 (1983), and in R.S.J. Sparks, Bull. Volcanol. 46 (4), 323 (1983).
Review articles on Ionian volcanism can be found in D. Morrison, Satellites of Jupiter (U. Arizona Press, Tucson, 1982), including S.W. Kieffer, Chap. 18; see also Ref. 41.
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Kieffer, S.W. (1988). Geologic nozzles. In: Coles, D. (eds) Perspectives in Fluid Mechanics. Lecture Notes in Physics, vol 320. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0021122
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DOI: https://doi.org/10.1007/BFb0021122
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