Chapter 16Compressible and Supersonic FlowVersion 0416.2.K.tex 23 February 2005Please send comments, suggestions, and errata via email to [email protected] or on paperto Kip Thorne, 130-33 Caltech, Pasadena CA 9112516.1 OverviewSo far, we have mainly been concerned with flows that are slow enough that they maybe treated as incompressible. We now consider flows in which the velocity approaches oreven exceeds the speed of sound and in which density changes along streamlines cannot beignored. Such flows are common in aeronautics and astrophysics. For example, the motionof a rocket through the atmosphere is faster than the speed of sound in air. In other words,it is supersonic. Therefore, if we transform into the frame of the rocket, the flow of air pastthe rocket is also supersonic.When the flow speed exceeds the speed of sound in some reference frame, it is not possiblefor a pressure pulse to travel upstream in that frame and change the direction of the flow.However, if there is a solid body in the way (e.g. a rocket or aircraft), the flow directionmust change. In a supersonic flow, this change happens nearly discontinuously, through theformation of shock fronts at which the flow suddenly decelerates from supersonic to subsonic.An example is shown in Fig. 16.1. Shock fronts are an inevitable feature of supersonic flows.In another example of supersonic flow, a rocket itself is propelled by the thrust created byescaping hot gases from its exhaust. These hot gases move through the exhaust at supersonicspeeds, expanding and cooling as they accelerate. In this manner the random thermal motionof the gas molecules is converted into an organised bulk motion that carries away negativemomentum from the rocket and pushes it forward.The solar wind furnishes yet another example of a supersonic flow. This high speed flowof ionized gas is accelerated in the solar corona and removes a fraction ∼ 10−14of the sun’smass every year. Its own pressure accelerates it to supersonic speeds of ∼ 400 km s−1. Whenthe outflowing solar wind encounters a planet, it is rapidly decelerated to subsonic speed bypassing through a strong discontinuity known as a bow shock, which surrounds the planet(Fig. 16.2). The bulk kinetic energy in the solar wind, built up during acceleration, is rapidlyand irreversibly transformed into heat as it passes through this shock front.12Fig. 16.1: Complex pattern of shock waves formed around a model aircraft in a wind tunnel withair moving ten percent faster than the speed of sound (i.e. with Mach number M = 1.1.) Imagefrom W. G. Vicenti; reproduced from Van Dyke 1982.Sun400 km s-1EarthBow ShockFig. 16.2: The supersonic solar wind forms a type of shock front known as a bow shock when itpasses by a planet.In this chapter, we shall study some properties of supersonic flows. After restating thebasic equations of compressible fluid dynamics (Sec. 16.2), we shall analyze three impor-tant, simple cases: quasi-one-dimensional stationary flow (Sec. 16.3), time-dependent onedimensional flow (Sec. 16.4), and normal adiabatic shock fronts (Sec. 16.5). In these sec-tions, we shall apply the results of our analyses to some contemporary examples, includingthe Space Shuttle (Box 16.1), rocket engines, shock tubes, and the mach cone, N-wave andsonic booms produced by supersonic projectiles and aircraft. In Sec. 16.6, we will developsimilarity-solution techniques for supersonic flows and apply them to supernovae, underwaterdepth charges, and nuclear-bomb explosions in the earth’s atmosphere.316.2 Equations of Compressible FlowIn Chap. 12, we derived the equations of fluid dynamics, allowing for compressibility. Weexpressed them as conservation laws for mass [Eq. (12.25)], momentum [Eq. (12.62)] andenergy [Eq. (12.66) with vanishing right-hand side by virtue of (12.67)], and an evolutionlaw for entropy [Eq. (12.69)]:∂ρ∂t+ ∇· (ρv) = 0 , (16.1)∂(ρv)∂t+ ∇ · (P g + ρv ⊗ v − 2ησ − ζθg) = ρge, (16.2)∂∂t(12v2+ u + φ)ρ+ ∇ · [(12v2+ h + φ)ρv − 2ησ · v − ζθv] = 0 , (16.3)∂(ρs)∂t+ ∇ · (ρsv) =2ησ : σ + ζθ2T. (16.4)Here σ : σ is index-free notation for σijσij.Some comments are in order. Equation (16.1) is the complete mass conservation equation(continuity equation) assuming that matter is neither added to nor removed from the flow.Equation (16.2) expresses the conservation of momentum allowing for one external force,gravity. Other external forces can be added. Equation (16.3), expressing energy conservation,includes a viscous contribution to the energy flux. If there are sources or sinks of fluid energy,then these must be included on the right-hand side of this equation. Possible sources of energyinclude chemical or nuclear reactions; possible energy sinks include cooling by emission ofradiation. We will incorporate the effects of heat conduction into the energy equation inthe next chapter. Equation (16.4) expresses the evolution of entropy, and will also needmodification if there are additional contributions to the energy equation. The right-handside of this equation is the rate of increase of entropy due to viscous heating. This equationis not independent of the preceding equations and the laws of thermodynamics, but is oftenmore convenient to use. In particular, one often uses it (together with the first law ofthermodynamics) in place of energy conservation (16.3).These equations must be supplemented with an equation of state in the form P (ρ, T )or P (ρ, s). For simplicity, we shall often focus on a perfect gas that undergoes adiabaticevolution with constant specific-heat ratio (adiabatic index) γ, so the equation of state hasthe simple form (Box 12.1 and Ex. 12.2)P = K(s)ργ. (16.5)Here K(s) is a function of the entropy per unit mass s and is thus constant during adiabaticevolution, but will change across shocks because the entropy increases in a shock (Sec. 16.5).The value of γ depends on the number of thermalized internal degrees of freedom of the gas’sconstituent particles (Ex. 16.1). For a gas of free particles (e.g. fully ionized hydrogen), itis γ = 5/3; for the earth’s atmosphere, at temperatures between about 10 K and 1000 K, itis γ = 7/5 = 1.4 (Ex. 16.1).4For such a gas, we can integrate the first law of thermodynamics (Box 12.1) to obtain aformula for the internal energy per unit mass,u =P(γ − 1)ρ, (16.6)where we have assumed that the internal energy vanishes as the temperature T → 0. Itwill
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