Approved for publication by Director, U.S. Geological Survey, January 23, 1986.
For example, see H.W. Liepmann and A. Roshko, Elements of Gasdynamics
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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
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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).
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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.
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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.
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