Nonsteady Gas Dynamics - Joseph Shepherd

Nonsteady Gas Dynamics

Joseph E. Shepherd

Graduate Aeronautical LaboratoriesCalifornia Institute of Technology

Pasadena, CA 91125

Revised October 31, 2020

Copyright c© California Institute of Technology, 2001-2019.

ForewordThe title of these notes is historical since they are based in large part on the class Ae237 – Non-steady Gasdynamics that has been offered as part of the graduate curriculum in Aeronautics atCaltech by the faculty in Graduate Aeronautical Laboratories since the 1960s. Brad Sturtevant,Anatol Roshko, Hans Hornung, and Joe Shepherd have all taught the class at various times. Thesenotes are a compilation of material from all of these instructors.

Despite the name, the class is an introduction to advanced ideas in wave propagation in com-pressible fluids of all types – not just gases. In the early years, the emphasis was on the gas dy-namics of perfect gases and approximate methods for treating shock wave interactions with otherwaves, contact surfaces, area changes in ducts, piston motion, and various aspects of shock tubeoperation. As the research interests of the faculty evolved, other topics such as acoustics in liquids,sonic boom propagation, shocks waves in inhomogeneous media, wave focusing, shock waves insolids, shock waves with phase changes, and detonation have been included.

i

[q, ‘-’---

N.A.C.A. Technical Memorandum No.505 Fig.1

.,—.

..

a =L_–

.— .——

Fig.1

.

Figure 1: Richard Becker’s conceptual schematic (Becker, 1922) of how shock waves are formed.This picture has become an icon of the compressible flow literature. Reproduced many times, thisis used as an illustration of the influence of wave amplitude on wave speed with higher amplitudedisturbances catching up with lower amplitude disturbances to form the shock front.

ii

Contents

1 Introduction 11.1 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Jump Conditions for Shock Waves 32.1 Unsteady Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Hugoniot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Rayleigh Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Graphical Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Steady Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5.1 Relation to Unsteady Versions . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Integral Equations for Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . 12

2.6.1 Contact Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Solution of Jump Equations 173.1 Results for Perfect and Ideal Gases . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Ideal gas with variable heat capacity . . . . . . . . . . . . . . . . . . . . . 193.1.2 Equilibrium chemically reacting mixtures . . . . . . . . . . . . . . . . . . 20

3.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.1 Iterative Solution with Density . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 Newton-Raphson Method in Temperature and Volume . . . . . . . . . . . 23

3.3 Weak Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Weak Shock Series – General Fluid . . . . . . . . . . . . . . . . . . . . . 273.3.2 Weak Shock Series – Perfect Gas . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Strong Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.1 Perfect Gas Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Causality and Entropy 334.1 Mass flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Geometry of Isentropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 Variation of Entropy on Hugoniot . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5 Slope of Hugoniot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.6 Geometric Relationship of Isentropes and Hugoniots . . . . . . . . . . . . . . . . 45

4.6.1 Relationship of Rayleigh Line, Hugoniot, and Isentrope . . . . . . . . . . 46

iii

4.6.2 Summary of Four cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.7 Hugoniot closeness to Isentrope for Weak Waves . . . . . . . . . . . . . . . . . . 48

4.7.1 Alternate Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.8 Graphical Interpretation of Entropy Generation . . . . . . . . . . . . . . . . . . . 52

5 Nonsteady Flow 555.1 One-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1.1 Courant-Fredrich-Lewy Condition . . . . . . . . . . . . . . . . . . . . . . 625.2 Simple Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Expansion Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3.1 Similarity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3.2 Expansion Fan in an Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Compression Waves and Shock Formation . . . . . . . . . . . . . . . . . . . . . . 725.5 Shock Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.5.1 Jump of Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Wave Interactions 836.1 Pressure-Velocity Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.1.1 Perfect Gas Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography 87

A Texts and References 91A.1 Elementary texts on Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . 91A.2 Graduate Level Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91A.3 Specialist Monographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92A.4 Mathematical Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92A.5 Shock Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93A.6 Handbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93A.7 Symposia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93A.8 Journals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94A.9 Detonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94A.10 Shock Waves in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94A.11 Explosions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96A.12 Numerical Simulation of Flows with Shock Waves . . . . . . . . . . . . . . . . . 97A.13 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

B Famous Numbers 99

C Essential Thermodynamic Relationships 103C.1 Thermodynamic potentials and fundamental relations . . . . . . . . . . . . . . . . 103C.2 Maxwell relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103C.3 Calculus identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103C.4 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104C.5 Various defined quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

iv

C.6 Specific heat relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106C.7 Sound Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106C.8 Gruneisen Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107C.9 Thermal Pressure Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

C.9.1 Fundamental Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 107C.10 Enthalpy, Energy and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107C.11 Perfect Gas Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

D High temperature gas thermodynamics 111D.1 Molar Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111D.2 Mass-specific Mixture Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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List of Figures

1 Richard Becker’s conceptual schematic (Becker, 1922) of how shock waves areformed. This picture has become an icon of the compressible flow literature. Re-produced many times, this is used as an illustration of the influence of wave am-plitude on wave speed with higher amplitude disturbances catching up with loweramplitude disturbances to form the shock front. . . . . . . . . . . . . . . . . . . . ii

2.1 Shock wave propagating at speed Us in a uniform flow moving at speed u1. Theupstream states is labeled 1 and the downstream state is labeled 2. . . . . . . . . . 3

2.2 Shock wave of velocity Us created by a steadily-moving piston of velocity upshown at two instants in time. The dashed box is control volume used in the com-putation of the relationships between states 1 and 2. . . . . . . . . . . . . . . . . . 5

2.3 (a) Schematic of Hugoniot locus and Rayleigh line showing graphical constructionof solution for state 2 given state 1 and wave speed. (b) Schematic of Rayleigh lineshowing interpretation of area underneath as internal energy increase. (c) Graphi-cal representation of enthalpy change across shock. (d) Partition of energy changeinto kinetic u2

p/2 and internal ∆e energy components. . . . . . . . . . . . . . . . . 92.4 Equivalent shock waves in two coordinate system. Lab coordinates on left, with

shock moving to right and fluid stationary ahead of shock. Shock-fixed coordinateson right with shock stationary and fluid moving from left to right. . . . . . . . . . 10

2.5 Control volume around a surface of discontinuity moving with an arbitrary veloc-ity. (a) Stationary observer (b) Observer moving with control volume. . . . . . . . 13

3.1 Property ratios across a shock wave in a perfect gas. . . . . . . . . . . . . . . . . . 193.2 Specific heat capacity as a function of temperature for some common species. . . . 193.3 Comparison of pressure, temperature, and density ratios for shock waves in air

computed with three models of various degrees of realism. . . . . . . . . . . . . . 213.4 Effect on initial pressure on pressure, temperature, and density ratios for shock

waves in air, from (from p. 131 Becker, 1968). . . . . . . . . . . . . . . . . . . . . 22

4.1 Allowed and forbidden regions for shock waves starting from state 1. . . . . . . . . 344.2 Illustrating the concept of causality for a shock driven by a piston. Acoustic dis-

turbances from shock overtake the shock wave from behind due to the subsonicnature of flow behind the shock. The shock is overtaking acoustic disturbancesahead, illustrating the supersonic nature of shock waves. . . . . . . . . . . . . . . 35

vii

4.3 Illustration of the meaning of causality for a shock wave. (a) Shock does notintersect the domain of dependence for point P . (b) Domain of influence of pointP must intersect shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Illustration of the relationship of Rayleigh lines and isentropes for both compres-sion and expansion discontinuities. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.5 Possible Hugoniot shapes. (a) Normal shape – shock strength increases monoton-ically along Hugoniot curve, unique solution for all shock strengths. (b) Excep-tional case – shock speed oscillates along Hugoniot, multiple solutions (2 and 2’)possible for a given shock speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.6 Variation of entropy along Hugoniot that satisfies the entropy variation condition(4.39). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.7 Possible configurations of isentropes and Hugoniots,H, corresponding to the fourcombinations of fundamental derivative, Γ, and Gruneisen coefficient, G. . . . . . 46

4.8 Illustration of the four possible cases for the geometry of isentropes, Hugoniots,and Rayleigh line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.9 Comparison of Hugoniot and isentrope for a perfect gas with γ = 1.4. . . . . . . . 494.10 Comparison of full expression for entropy jump with cubic term from weak shock

theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.11 Illustration of the irreversible portion of work in shock compression. . . . . . . . . 53

5.1 Geometry of characteristics passing through a point 3. The C+ characteristicpasses through points 1 and 3. The C− characteristic passes through points 2 and3. The Co characteristic is identical to the particle path (5.10). . . . . . . . . . . . 60

5.2 Schematic of approximate C+, C◦, and C− characteristic line segments used in themethod of characteristics integration scheme. . . . . . . . . . . . . . . . . . . . . 62

5.3 Schematic x-t diagrams illustrating the possible relationship between the domainof dependence of point 4 (the shaded region) and the points 1, 2, and 3 that are usedin the finite difference sheme to compute point 4. a) Unstable situation, domain ofdependence larger than span of difference stencil. b) Stable situation, domain ofdependence smaller than the span of the difference stencil.. . . . . . . . . . . . . . 63

5.4 Schematic x-t diagrams of simple waves and associated C+ and C− characteris-tics. (a) right-facing wave composed of C+ characteristics and crossing C− char-acteristics. (b) left-facing wave composed of C− characteristics and crossing C+

characteristics. Regions 1 and 2 are uniform, the shaded regions are the simplewaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.5 Geometry of characteristics illustrating twelve possible cases for simple waves asa function of the fundamental derivative Γ. . . . . . . . . . . . . . . . . . . . . . . 68

5.6 Schematic of piston-cylinder arrangement used to create an expansion fan (shadedregion) propagating to the left. The piston, initially located at x = 0, is impulsivelyaccelerated to a velocity up at time t = 0. The fluid is accelerated to the right by theexpansion wave and the wave propagates to the left. The wave is bounded on theleft by a uniform undisturbed region (1) and on the right by a uniformly movingregion (2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.7 Variation of properties in an ideal gas expansion wave, γ = 1.4 . . . . . . . . . . . 72

viii

5.8 Schematic of wave steepening, multivalued solution development, and shock for-mation (dashed line) due to the evolution of a compression wave in a fluid withΓ > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.9 Schematic of piston trajectory X(t) and associated C+ characteristics illustratingshock formation due to crossing of adjacent characteristics. . . . . . . . . . . . . . 74

5.10 Representative characteristics originating at the piston and in the uniform region. . 755.11 Relationship between the time t that a characteristic crosses a point in the region in

front of the piston as a function of the time tp that the characteristic left the piston.The contours represent different locations x in space. . . . . . . . . . . . . . . . . 76

5.12 Locus of minimum time t of adjacent characteristic crossings as a function of thetime tp that the characteristic left the piston. The minimum of this function corre-sponds to the shock formation time. . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.13 Shock S fitted into flowfield in order to reconcile inconsistent values on character-istics C+

1 and C+2 that intersect at point O with characteristic C−. . . . . . . . . . . 79

5.14 Schematic of pressure spatial profile after a shock has been fitted into flowfield. . . 79

6.1 Composite P (u) diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.2 Slopes of P (u) relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

ix

x

List of Tables

4.1 Possible cases to consider for isentropes in a general fluid Thompson (1971). . . . 45

5.1 Properties of right- and left-facing simple waves in a general compressible flow. . . 67

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Chapter 1

Introduction

Nonsteady gasdynamics is the study of wave propagation, wave interaction and the resulting mo-tion of a compressible fluid. The term “gasdynamics” is due to the historical roots of the subject inthe high-speed flow of gases, for which compressibility effects were first observed. In fact, liquidsand solids are compressible when the changes in pressure are sufficiently large, as observed withhigh explosive and high velocity impact. Both liquids and gases are fluids with no static strengthand solids can also behave like a fluid if the forces on the material exceed the yield stress.

Many of the ideas in the present notes are therefore applicable to gases, liquids, and solids.There are important exceptions that involve the strength of materials, such as elastic wave propa-gation and plasticity that we will not consider. In solids with small dynamic pressure changes, theelasticity and anisotropy of the material are always important. Tensile waves can cause cavitationin liquids, fracture and spallation in solids. At the microscopic level in solids, it is important toconsider the motions of atoms and the dynamics of dislocations leading to effects such as workhardening or strain rate dependence. Coverage of these topics can be found in the texts and refer-ence monographs listed in Appendix A under the heading Shock Waves in Solids.

The present notes are original but owe a substantial debt to the contributions of past researchers.A selection of the some of the text books, monographs and reference handbooks are listed inAppendix A.

1.1 Waves

The essential features of nonsteady compressible flow are propagating waves. For example, anexplosion in air, a bullet fired from a gun, an airplane moving faster than the speed of sound, anda high-speed turbulent flow will all create waves that we can hear and measure at a large distancefrom the source. These waves move through the fluid although sometimes it is more convenientto think of the fluid as passing through the wave. The goal of our study is to understand the typesof wave motion and to be able to predict how the waves are formed, propagate, and the associatedchanges that occur in the fluid,

Waves either expand or compress the fluid, changing the thermodynamic state (pressure P ,volume v, temperature T , etc) and set the fluid into motion, changing the fluid velocity u. Thetwo aspects are coupled through the conservation of mass, momentum and energy. This couplingis a consequence of the multiple roles of fluid pressure, velocity and energy. Pressure is both a

1

2 CHAPTER 1. INTRODUCTION

mechanical quantity, force per unit area, that can accelerate, compress or expand the fluid; and athermodynamics quantity, the statistical average of the forces due to molecular (or atomic) motionand interactions between molecules (or atoms), that opposes changes in fluid volume. The totalenergy of the fluid is represented by the sum of the kinetic energy associated with the fluid velocity1/2|u|2 and the internal energy e associated with the thermal (random) molecular motions andinteractions. Wave motion can result in a change in the total energy content e + 1/2|u|2 anda shift between kinetic energy and internal energy. We will consider the internal energy as athermodynamic property and for simple substances, this can be related to the pressure P andvolume v through an equation of state e(P, v) thereby providing a coupling between momentumand energy.

There are two types of waves: 1) compression; 2) expansion.Compression waves in ordinary fluids have the property that the front of the wave moves more

slowly than the rear so that rear of wave overtakes the frontShock waves are a consequence of the nonlinearity of compressible motion and occur in gases,

liquids, and solids. Analogs of shock waves occur in many other situations, for example, trafficflow and chromatography (Whitham, 1974). In general, whenever the speed of propagation of adisturbance is an increasing function of the disturbance amplitude, a shock wave is the probableoutcome. The nonlinearity of compressible motion in mechanical systems is due to the coupling ofthermodynamic properties like density, pressure, and sound speed to the speed of the flow behindthe wave.

Shock waves are commonly created by rapid energy addition, compression due to abruptlyturning a supersonic flow, the sudden impact of two solids or motion of a supersonic projectilethrough a fluid. In many cases, shock waves appear to be thin layers separating relatively uniformportions of the flow. The flow velocity and thermodynamic properties appear to suddenly changeor jump across the shock wave. Mathematically, this means that shock waves can be idealizedas surfaces of discontinuity embedded in the surrounding smooth flow. This creates substantialdifficulties in the mathematical and numerical analysis of compressible flows and a large amountof research has been dedicated to finding ways to fit or approximate shock waves into otherwisesmooth solutions to the equations of motion.

Shock waves can be thick enough to observe an macroscopic internal structure if the pres-sure is sufficiently low or if there are kinetic processes that limit the speed of transformationbetween states upstream and downstream of the wave. Vibrational relaxation, ionization, disso-ciation, combustion, thermal radiation, and physical phase changes all result in macroscopicallyobservable shock layers. Experiments have been carried out to examine many of these processesand the results have been analyzed and used by engineers, physicists, and chemists for scientificand technological purposes.

On the microscopic level, shock waves are all nonequilibrium in nature. A shock causes anabrupt change in the molecular velocity component aligned with the shock. This change perturbsthe equilibrium in the velocity distribution function for gases and liquids, and creates dislocationsin crystalline solids. Gas kinetic theory (solutions to the Boltzmann equation) and molecular dy-namics can be used to analyze the resulting relaxation to equilibrium. Experiments in low-pressuregases measuring the structure of the shock layer have been used to test models of the molecular in-teraction potential and approximate kinetic theory solutions. Observations in solids and liquids atthe molecular level is more difficult but the combination of experiments and numerical simulationshave provided insights.

Chapter 2

Jump Conditions for Shock Waves

We will start out thinking about the simplest situation of a planar shock wave moving with speedUs into a uniform state moving at speed u1 and there is a uniform state moving at speed u2 behindthe shock. The quantities such as pressure P , density ρ, and internal energy e describing the states

u2 US

u1

downstream upstream

shock12

Figure 2.1: Shock wave propagating at speed Us in a uniform flow moving at speed u1. Theupstream states is labeled 1 and the downstream state is labeled 2.

states upstream (by convention state 1) and downstream (by convention state 2) of a shock wave(see Fig. 2.1) are related by the conservation laws of elementary physics. The relationships areoften referred to as “jump conditions” since they specify the apparent jumps in properties ∆f = =f2 - f1, that occur across shock waves when they are considered as discontinuities. These jumpsare a function of the strength of the waves, usually measured in terms of the wave speed, the state1 upstream of the wave, and the equation of state for the substance. In general, it is the wave speedrelative to the upstream flow, Us - u1, that is important

We will derive the jump conditions by two approaches, first considering unsteady or propagat-ing shock waves and second by considering stationary waves. The two approaches are equivalentand the results can be related by the transformation in coordinates from a laboratory to a wave-fixedreference frame.

3

4 CHAPTER 2. JUMP CONDITIONS FOR SHOCK WAVES

2.1 Unsteady ApproachThe equations governing propagating shock waves can be derived simply by considering a basicthought experiment consisting of a piston moving along the axis of a tube. By considering thechange in quantities within the control volume over the duration ∆t = t1 - t2, we can determinerelationships between the properties of state 1 (upstream of shock) and those of state 2 (downstreamof the shock) as a function of the piston speed up or alternatively the shock speed Us. In this simpleexample, we will consider the piston and shock velocities to be constant and the states upstreamand downstream of the shock to be uniform.

The dashed lines in Fig. 2.2 indicate the control volume. If the constant cross-sectional area isA, then the change in the control between times t1 and t2 is

∆V = A∆L = A(Us − up)∆t (2.1)

Mass conservation The mass inside the control volume changes due to the increase in the sizeof the control volume

∆m = ρ2∆V (2.2)= ρ2A(Us − up)∆t (2.3)

The increase in mass inside the control volume is also equal to the mass that is “overtaken” by theshock wave

= ρ1AUs∆t (2.4)

Simplifying we have

ρ1Us = ρ2(Us − up) (2.5)

Momentum conservation The momentum inside the control volume changes due to the impulseapplied by the piston over the time interval ∆t

impulse = (P2 − P1)A∆t (2.6)

this must be equal to the change in momentum of the fluid that is “overtaken” by the shock waveand accelerated from u = 0 to u = up.

= ∆mup (2.7)= ρ1AUs∆tup (2.8)

Simplifying, we have

P2 = P1 + ρ1Usup (2.9)

2.1. UNSTEADY APPROACH 5

up Us

12 shoc

k

pist

on

up ∆t Us ∆t

t1

t2

L

L+∆L

ρ

t2t1up

x

t

x

1

2

shock

piston

partic

le path

t1

t2

Figure 2.2: Shock wave of velocity Us created by a steadily-moving piston of velocity up shown attwo instants in time. The dashed box is control volume used in the computation of the relationshipsbetween states 1 and 2.

Energy conservation The energy inside the control volume changes due to the work done by thepiston over the time interval ∆t


work done = P2Aup∆t (2.10)

this must be equal to the increase of energy of the fluid“ overtaken” by the shock wave. Both theinternal energy e and the kinetic energy u2/2 of the fluid increase.

= ∆m

(e2 +

u2p

2− e1

)(2.11)

= ρ1AUs∆t

(e2 +

u2p

2− e1

)(2.12)

Simplifying, we have

e2 +u2p

2= e1 + P2(v1 − v2) (2.13)

which can also be expressed as

h2 = h1 + up(Us −up2

) (2.14)

where the enthalpy is h = e + Pv.

Entropy The entropy of the shocked fluid (state 2) must be greater than or equal to that of theinitial fluid (state 1).

s2 ≥ s1 (2.15)

This is simply the Second Law of Thermodynamics expressed for this system assuming that theshock process is adiabatic. Since we have not specified any specific mechanism or structure forthe shock wave, it is not possible to explicitly calculate the entropy change at this stage of ourdiscussion.1

2.2 HugoniotThe results for mass, momentum, and energy conservation can be combined to eliminate the wavespeed Us and downstream flow speed up to obtain a single equation, the Hugoniot. In terms of theinternal energy e, the Hugoniot equation is

e2 − e1 =1

2(P2 + P1) (v1 − v2) . (2.16)

1The kinetic theory of gases can be used to explicitly compute the shock wave structure and determine the entropyproduction rate within a shock in a monoatomic gas. In the case of very weak shock waves, the Navier-Stokes equationscan be applied to compute the shock structure. We will see that this provides a method of estimating the thickness ofweak shock waves.

2.3. RAYLEIGH LINE 7

In terms of the enthalpy h, the Hugoniot equation is

h2 − h1 =1

2(P2 − P1) (v1 + v2) . (2.17)

or[h] = v1[P ] +

1

2[v][P ] (2.18)

For a single-phase substance in thermodynamic equilibrium the energy or enthalpy can be consid-ered function of the pressure P and specific volume v

e = e(P, v) and h = h(P, v) (2.19)

This means that the solution to (2.16) or (2.17) can be described as a curve PH(v) in the P -v plane.The curve is the locus of all possible downstream states 2 for a given upstream state 1. It is alsosometimes referred to as the shock adiabat which will be discussed further in section 2.6.

2.3 Rayleigh LineThe conservation of mass and momentum relationships can be combined to obtain a second equa-tion that related pressure and velocity states on either side of the shock. This relationship is knownas the Rayleigh line since it is a linear relationship for a fixed shock speed

P2 − P1

v2 − v1

= −(

Us

v1

)2

(2.20)

Using the definitions of the shock-fixed frame velocities (2.36–2.37) and mass conservation, thiscan also be written

P2 − P1

v2 − v1

= −(

w1

v1

)2

= −(

w2

v2

)2

(2.21)

which is useful in the later discussion on the geometry of the Rayliegh line relative to the isentropesand Hugoniot.

2.4 Graphical InterpretationsFor a fixed initial state, the Hugoniot can be visualized as a curve in the P -v plane that can beobtained by solution of the equation:

e(P, v) = e1 +1

2(P + P1)(v1 − v) (2.22)

and the Rayleigh line

P = P1 +

(Us

v1

)2

(v1 − v) (2.23)


both of which pass through the initial state 1 and the final state 2. For a given shock speed Us,the downstream state solution can be visualized as the intersection of the Rayleigh line and theHugoniot curve, shown in Fig. 2.3a.

The Hugoniot relationship can be written

∆e = −P∆v (2.24)

where P = (P2 + P1)/2. This can be visualized as the area under the Rayleigh line in the P -vplane, shown in Fig. 2.3b. An alternative visualization can be found using the enthalpy version ofthe Hugoniot, which can be written

∆h = v∆P (2.25)

where v = (v2 +v1)/2. This can be visualized as the area between the Rayleigh line and the P -axis,shown in Fig. 2.3c. The energy balance equation (2.13) can be written as

u2p

2=

1

2(P2 − P1)(v1 − v2) (2.26)

which can be visualized as the striped triangle in Fig. 2.3d. Note that for strong shock wavesP2 � P1, the change in internal energy is approximately equal to the change in kinetic energy

e2 − e1 ≈u2p

2(2.27)

as shown in Fig. 2.3d.An alternate explanation of the shock energy jump equation is to imagine that the shock can

be treated as a continuous series of states in a steady, inviscid, adiabatic flow. The flow within theshock can certainly be thought of as steady in the shock-fixed frame. However, so far we have notconsidered the structure of the shock and it is not at all clear that the inviscid approximation or evena continuum model is appropriate. So the following model should not be taken too literally, despitegiving the correct energy equation. Remember, internal energy is a state function and changes inenergy are independent of path and only depend on the initial and final states. We can thereforechoose any convenient path and get the correct answer.

The change in energy for a continuous (quasi-steady) adiabatic compression can be computedas

de = −Pdv (2.28)

and in a steady, quasi-one dimensional flow the momentum equation can be written

ρudu = −dP (2.29)

The mass conservation relation reduces to

m = ρu = constant (2.30)

2.4. GRAPHICAL INTERPRETATIONS 9

1

2

HR

v

P

1

2

∆e

HR

v

P

(a) (b)

1

2

∆h

HR

v

P

1

2

∆e

u2p2

HR

v

P

(c) (d)

Figure 2.3: (a) Schematic of Hugoniot locus and Rayleigh line showing graphical construction ofsolution for state 2 given state 1 and wave speed. (b) Schematic of Rayleigh line showing inter-pretation of area underneath as internal energy increase. (c) Graphical representation of enthalpychange across shock. (d) Partition of energy change into kinetic u2

p/2 and internal ∆e energycomponents.

and the momentum equation can be written

−m2dv = dP (2.31)


so that

e2 − e2 =

∫ 2

1

de (2.32)

=

∫ 2

1

PdP

m2(2.33)

=1

m2

P 22 − P 2

1

2(2.34)

=1

2(P2 + P1) (v1 − v2) (2.35)

which is equal to the results obtained previously (equation 2.16) from the moving control volumeanalysis with no assumptions about the nature of the shock wave structure.

2.5 Steady ApproachTransforming to the frame that is fixed with the shock, Fig. 2.4, define velocities w in the shock-fixed frame to be

w1 = Us − u1 (2.36)w2 = Us − u2 (2.37)

where we have generalized from the previous example by allowing a possible upstream velocity u1

(zero in the unsteady derivation of the jump conditions given previously) and replaced the pistonspeed by the downstream flow velocity u2.

shoc

k

shoc

k

lab frame shock frame

Us

11 22

up w1 w2

Figure 2.4: Equivalent shock waves in two coordinate system. Lab coordinates on left, with shockmoving to right and fluid stationary ahead of shock. Shock-fixed coordinates on right with shockstationary and fluid moving from left to right.

The steady approach uses a thin “pill-box” control volume around a small segment of the waveto apply the conservation relationship. Assuming that the volume can be made arbitrarily small sothat there is no internal storage, the steady conservation relationships (see Section 2.6 below) canbe applied to obtain the following form of the jump conditions:

2.5. STEADY APPROACH 11

ρ1w1 = ρ2w2 (2.38)P1 + ρ1w2

1 = P2 + ρ2w22 (2.39)

h1 +w2

1

2= h2 +

w22

2(2.40)

s2 ≥ s1 (2.41)

or defining [f ] ≡ f2 - f1

[ρw] = 0 (2.42)[P + ρw2

]= 0 (2.43)[

h+w2

2

]= 0 (2.44)

[s] ≥ 0 (2.45)

2.5.1 Relation to Unsteady VersionsThe generalization of the previous unsteady results to the case of a shock propagating into a movingstream can readily be made by using the correspondence

Us → w1 (2.46)Us − up → w2 (2.47)

up → [u] (2.48)

between a shock propagating into a stationary flow and the general problem of a shock propagatingin a moving stream. In figure 2.2, u1 = 0 and u2 = up, but in general u1 6= 0. The jump in lab framevelocity and the shock frame velocity are independent of the shock velocity and are related by

[w] = −[u] (2.49)

The mass conservation equation can be written

[w] = w1[v]

v1

(2.50)

or

[u] = −w1[v]

v1

(2.51)

The momentum equation can be written

P2 = P1 + ρ1w1[u] (2.52)

Compare with (2.9) to see how the previous expression has been generalized. The Rayleigh linedefinition was given earlier (2.21). The Hugoniot relation (2.17) is unchanged since it is indepen-dent of the reference frame.


2.6 Integral Equations for Jump ConditionsThe equations of motion can be written to apply to an (almost) arbitrary moving control volume.By restricting that control volume to be a thin “pill box” around a section of a discontinuity in aflow, the most general from of the jump conditions can be obtained.

The most general form of the integral conservation relations for a control volume moving withvelocity b are:Mass:

d

dt

∫V

ρdV +

∫∂V

ρ (u− b) · n dA = 0 (2.53)

Momentum:

d

dt

∫V

ρudV +

∫∂V

ρu (u− b) · n dA =

∫V

ρG dV +

∫∂V

T dA (2.54)

Energy:

d

dt

∫V

ρ

(e+|u|22

)dV +

∫∂V

ρ

(e+|u|22

)(u− b) · n dA =∫

V

ρG · u dV +

∫∂V

T · u dA−∫∂V

q · n dA (2.55)

Entropy:d

dt

∫V

ρsdV +

∫∂V

ρs (u− b) · n dA+

∫∂V

q

T· n dA ≥ 0 (2.56)

The surface traction forces areT = −P n + τ · n (2.57)

where τ is the viscous stress tensor. The heat flux q is determined by conduction due thermalconductivity k and thermal radiation qr

q = −k∇T + qr (2.58)

If the control volume is so thin that there is no internal storage then the volumetric integrals allvanish. Further, if the fluid is inviscid (τ = 0) and the flow is adiabatic (q = 0) we can simplifythese equations to be

∫∂V

ρ (u− b) · n dA = 0 (2.59)∫∂V

[ρu (u− b) · n + P n] dA = 0 (2.60)∫∂V

[ρ

(e+|u|22

)(u− b) + Pu

]· n dA = 0 (2.61)∫

∂V

ρs (u− b) · n dA ≥ 0 (2.62)

2.6. INTEGRAL EQUATIONS FOR JUMP CONDITIONS 13

lab frame shock frame

Figure 2.5: Control volume around a surface of discontinuity moving with an arbitrary velocity.(a) Stationary observer (b) Observer moving with control volume.

Since the control volume is assumed to be arbitrarily thin, the contributions to the area integralsonly come from two faces of the control volume (1) the upstream side and (2) the downstream side.∫

∂V

f dA = A(f2 + f1) (2.63)

The components of the velocity relative to the control volume can be defined using the velocitytriangles in Fig. 2.5. The components normal to the two faces are

w1 = −(u1 − b) · n1 (2.64)w2 = (u2 − b) · n2 (2.65)

and the components perpendicular to the faces are

v1 = (u1 − b) · t (2.66)

v2 = (u2 − b) · t (2.67)

Evaluating the integrals with these definitions, we have

ρ2w2 − ρ1w1 = 0 (2.68)ρ2u2w2 − ρ1u1w1 + P2n2 + P1n1 = 0 (2.69)

ρ2

(e2 +

u22

2

)− ρ1

(e1 +

u21

2

)+ P2n2 · u2 + P1n1 · u1 = 0 (2.70)

ρ2w2s2 − ρ1w1s1 ≥ 0 (2.71)

These equations can be further simplified to obtain the jump conditions equations given previ-ously (2.38-2.41). Adding b(ρ1w1 − ρ2w2) to both sides of (2.69) and taking the dot product withn1 = - n2 yields (2.39) and taking the dot product with the tangent t yields

v1 = v2 (2.72)


as long as w1 6= 0. This will be important in analyzing oblique shock waves. Another way ofstating this is that the velocity u only changes in the direction normal to the control volume

u2 − u1 = n1(w1 − w2) (2.73)

Adding P2w2-P1w1 to both sides of (2.70) yields

h2 +u2

2

2−(h1 +

u21

2

)= b · (u2 − u1) (2.74)

which demonstrates that the stagnation enthalpy h+ u2/2 is unchanged across a stationary shock.Using the relationships (2.64,2.65,2.72), (2.40) is obtained, which shows that in the shock-fixedframe the stagnation enthalpy h + w2/2 is always unchanged. The entropy inequality can besimplified to (2.41) by using (2.68).

2.6.1 Contact SurfacesA special case of a discontinuity is a contact surface or slip line which can occur between tworegions of a flow that originate from different sources or have been processed so that the fluidstates have different properties. Contact surfaces are important in compressible flow because theyoccur whenever wave interactions take place such as in many applications including shock tubes.

Unlike a shock, there is no normal component of velocity w1 = w2 = 0 so that the jump condi-tions reduce to

u2 · n1 = u1 · n1 (2.75)P2 = P1 (2.76)

for an inviscid flow—the most common approximation used in gas dynamics. The fluid on eachside of the interface can in general have different tangential velocities and thermodynamics prop-erties:

v2 6= v1 (2.77)ρ2 6= ρ1 (2.78)e2 6= e1 (2.79)s2 6= s1 (2.80)

In multi-dimensional flows, the differences properties across the interfaces lead to instabilities. Atangential velocity difference will cause a shear layer to develop which is subject to the Kelvin-Helmholtz instability. Differences in density across an interface subject to continuous or impul-sive acceleration can lead to Rayleigh-Taylor or Ritchmyer-Meshkov instabilities, respectively (M,2002).

In one-dimensional flows, the relationships across a contact surface reduce to P2 = P1 and u2 =u1. This continuity of pressure and velocity will be exploited to construct solutions to unsteady gasdynamics problems using representation in the pressure-velocity P -u plane. In two-dimensional

2.6. INTEGRAL EQUATIONS FOR JUMP CONDITIONS 15

flows, if the interfaces are idealized as being stable, the continuity of pressure and flow deflec-tion can be used to construct solutions in the pressure-flow deflection (P -θ) plane, a very usefulprocedure for pseudo-steady shock wave reflection and refraction.


Chapter 3

Solution of Jump Equations

The most common problem in shock physics is to determine the postshock state 2 given an initialstate 1 and shock velocity Us = w1. This is the direct problem of solving the shock jump conditions.On occasion, it is also of interest to determine the possible preshock states 1 that can correspondto a given postshock state 2. This is the inverse solution to the shock jump conditions.

For simple equations of state, it is possible to solve the jump conditions analytically. We willconsider several cases of this type, for example, the ideal gas and solids described by a linear Us-up relationship. However, for most realistic equations of state, numerical solutions of the jumpconditions are required.

First, we consider the simple analytic cases.

3.1 Results for Perfect and Ideal GasesThe most common special case of the jump conditions is that of a perfect gas, described by

Pv = RT or P = ρRT (3.1)

and

h = cpT cp =γR

γ − 1or e = cvT cv = cp −R =

R

γ − 1(3.2)

This model is useful for gas mixtures at intermediate temperatures and pressure, conditions underwhich the molecular interactions are negligible so that the compressibility factor Z is unity

Z =P

ρRTZ → 1 for an ideal gas (3.3)

and the chemical composition is fixed. The entropy of a perfect gas is

s− so = cp lnT/To −R lnP/Po = cv lnP/Po − cp ln ρ/ρo (3.4)

so that the sound speed is

a2 =

(∂P

∂ρ

)s

= γRT (3.5)

17

18 CHAPTER 3. SOLUTION OF JUMP EQUATIONS

Often is useful to combine (3.1) and (3.2) to eliminate temperature as a variable

h =γ

γ − 1

P

ρa2 = γ

P

ρ(3.6)

The Mach numbers upstream and downstream of the shock are defined to be

M1 =w1

a1

M2 =w2

a2

(3.7)

Substituting these into the jump conditions, the following relationships can be obtained.

[P ]

P1

=2γ

γ + 1

(M2

1 − 1)

(3.8)

[w]

a1

= − 2

γ + 1

(M1 −

1

M1

)(3.9)

[v]

v1

= − 2

γ + 1

(1− 1

M21

)(3.10)

[s]

R= − ln

Pt2Pt1

(3.11)

Pt2Pt1

=1(

2γ

γ + 1M2

1 −γ − 1

γ + 1

) 1γ−1

γ + 12 M2

1

1 +γ − 1

2M2

1

γγ−1

(3.12)

The Hugoniot equation for the perfect gas can be expressed

P2

P1

=

γ + 1

γ − 1− v2

v1

γ + 1

γ − 1

v2

v1

− 1(3.13)

Some alternative versions of the jump conditions that are also useful are:

P2

P1

= 1 +2γ

γ + 1

(M2

1 − 1)

(3.14)

=2γ

γ + 1M2

1 −γ − 1

γ + 1(3.15)

ρ2

ρ1

=γ + 1

γ − 1 + 2/M21

(3.16)

M22 =

M21 +

2

γ − 12γ

γ − 1M2

1 − 1(3.17)

These relationships are illustrated in Fig. 3.1 for γ = 1.4. Tables and charts for perfect gases withvarious values of γ can be found in texts and reports, an older reference that is common but stillvery useful is Staff (1953).

3.1. RESULTS FOR PERFECT AND IDEAL GASES 19

2

18642

1

1

Shock Mach Number

0

000

0

5

5

P2/P1

T2/T1

ρ2/ρ1M2

Figure 3.1: Property ratios across a shock wave in a perfect gas.

3.1.1 Ideal gas with variable heat capacity

In general, the heat capacity of molecular gases and gas mixtures (like air) is a function of temper-ature cp(T ) and the enthalpy h(T ) will be nonlinear function of temperature. In the simplest case

Table 1: Heats of formation for common combustion species

Substance Composition W ∆fHo

(g/mol) (kJ/mol)

Productswater, gas H2O(g) 18.0 -241.83water, liquid H2O(l) 18.0 -285.83carbon dioxide CO2(g) 44.0 -393.52carbon monoxide CO(g) 28.0 -110.53carbon, gas C(g) 12.0 716.67hydrogen chloride HCl(g) 36.5 -92.31ammonia NH3(g) 17.0 -45.90

Radicals and Intermediatesoxygen (atomic) O 16.0 249.17nitrogen (atomic) N 14.0 472.69hydrogen (atomic) H 1.0 218.00hydroxyl OH 17.0 38.99nitrogen oxide NO 30.0 90.29imidogen NH 15.0 376.56methyl CH3 15.0 145.69formyl HCO 29.0 43.51formaldehyde CH2O 30.0 -115.90hydrogen peroxide H2O2 34.0 -136.11hydroperoxo HO2 33.0 2.09

Hydrocarbon Fuels (gas phase)methane CH4 16.0 -74.85ethane C2H6 28.0 -84.68propane C3H8 44.0 -103.85butane C4H10 54.0 -126.15octane C8H18 114.0 -208.45

20

25

30

35

40

45

50

55

60

65

0 500 1000 1500 2000 2500 3000 3500 4000

Temperature (K)

Hea

t Ca

paci

ty C

p (J

/mol

K)

H2O

CO2

CO, N2, O2

H2

Ar

2Figure 3.2: Specific heat capacity as a function of temperature for some common species.

with no chemical reaction, usually referred to as frozen composition, the enthalpy of a gas mixture


with K species can be expressed as

h(T ) =K∑i=1

Yihi(T ) (3.18)

where Yi is the mass fraction of species i and hi is the specific enthalpy of species i. Values of prop-erties up to 2×104 K for species found in air are given in Appendix D. The P (ρ, T ) relationshipfor ideal gases with fixed composition is still the simple ideal gas law (3.1).

3.1.2 Equilibrium chemically reacting mixturesAn important case for ideal gases is a chemically reacting mixture, for which the species massfractions Yi will be a function of distance downstream of an initially nonreactive shock. Thetreatment of the non-equilibrium region where chemical reaction is taking place requires knowingthe types and rates of chemical reactions. However, if we are only interested in the final stateafter an equilibrium state has been established, then we can find state 2 by first obtaining theequilibrium composition Y eq(T, P ) using a chemical equilibrium solution method (Denbigh, 1981,van Zeggeren and Storey, 1970, Smith and Missen, 1991) and computing h as

h(T, P ) =K∑i=1

Y eqi (T, P )hi(T )

Note that although the individual species are ideal gases (internal energy and enthalpy only afunction of temperature), since the composition is in general a function of pressure, the mixtureenthalpy is a function of both pressure and temperature. The ideal gas law (3.1) is still valid butsince the composition is a function of pressure and temperature, so is the gas constant R.

Comparison of computations with the three models: perfect gas, frozen, and equilibrium, areshown in Fig. 3.3. The perfect gas results were obtained with Eqs. (3.8)-(3.10) and γ = 7/5. Thefrozen model was computed using realistic specific variation with temperature (Appendix D) anda numerical solution of the jump conditions (Reynolds, 1986) for air modeled as a mixture of N2,O2, Ar, and CO2 (Appendix B). The equilibrium model was also solved numerically (Reynolds,1986) and the post-shock state was a mixture of N2, O2, Ar, and CO2, dissociation products N,O, CO, and NO, positive ions N+, Ar+, O+, and NO+, and electrons. The composition of adissociated and ionized gas is function of pressure (Clarke and McChesney, 1964, Vincenti andKruger, 1965) with increasing dissociation with decreasing pressure, resulting in a dependence ofthe Hugoniot states on initial pressure as shown in 3.4. This is a type of real gas effect but isdistinct from the consideration of molecular interactions that occur in dense gases mentioned inthe subsequent section. The effects of dissociation and ionization behind strong shock waves in airis quite significant in hypersonic flow which occurs in the entry of a spacecraft or meteorite intothe atmosphere of earth or other planets with atmospheres. The aerodynamic heating and forceson the body may be strongly influenced by these chemical reaction effects which have been thesubject of extensive study by the aerodynamics community.

3.1. RESULTS FOR PERFECT AND IDEAL GASES 21

0

20

40

60

80

100

120

140

160

180

0 2 4 6 8 10 12

Shock Mach No.

pres

sure

ratio

EquilibriumFrozenPerfect gas

0

5

10

15

20

25

0 2 4 6 8 10 12

Shock Mach No.

tem

pera

ture

ratio


1

2

3

4

5

6

7

8

9

10

1 3 5 7 9 11

Shock Mach No.

dens

ity ra

tio


Figure 3.3: Comparison of pressure, temperature, and density ratios for shock waves in air com-puted with three models of various degrees of realism.


Figure 3.4: Effect on initial pressure on pressure, temperature, and density ratios for shock wavesin air, from (from p. 131 Becker, 1968).

3.2. NUMERICAL METHODS 23

3.2 Numerical MethodsThe most common predictive problem in shock physics is to numerically determine the postshockstate 2 given an initial state 1 and shock velocity Us = w1. There are many methods to accomplishthis task, two of these are given below.

In order to use these methods, an equation of state in the form e(P, ρ) or equivalently h(p, ρ)is required. This can be either an analytic formula, a set of tabulated data, or an algorithmicprocedure.

3.2.1 Iterative Solution with DensityOne convenient way to approach this problem is to rewrite the momentum and energy jump rela-tionships as a function of density ρ2

P2 = P1 + ρ1w21

(1− ρ1

ρ2

)(3.19)

h2 = h1 +1

2w2

1

[1−

(ρ1

ρ2

)2]

(3.20)

Using an assumed value of ρ2 = ρ∗2 and the initial state (P1, ρ1, h1, w1), Equations (3.19) and (3.20)are used to predict interim values of pressure and enthalpy, P ∗2 and h∗2. The enthalpy from (3.20)is then compared with the value obtained from the equation of state

h = h(P ∗2 , ρ∗2) (3.21)

to obtain an errorErr = h(3.21)− h(3.20) (3.22)

Depending on the sign of Err, a new value for ρ2 = ρ∗2 that will reduce the magnitude of Err isselected. Through repeated1 trials (iterations), Err can be reduced to less than a desired toleranceε. This formulation of the problem can be used with any equation of state that can be evaluatedto yield h(P, ρ); Mollier charts and tables that are widely available for many substances are wellsuited for this approach. This method is the basis of the postshock state solution in the originaldetonation structure program ZND developed by Shepherd (1986).

3.2.2 Newton-Raphson Method in Temperature and VolumeThe iterative solution with density requires a good initial guess for the density and care must betaken not to exceed the maximum density. The steep slope of the Hugoniot for strong shocksmakes this method unsuitable in those cases. We have found that a more robust method is to usea two-variable Newton-Raphson scheme with the variables temperature and specific volume. Thescheme presented below is an extension of the method used by Reynolds (1986) to solve the jumpconditions for a Chapman-Jouguet detonation.

1This is most conveniently carried out using one of the “canned” nonlinear root solvers available in standardlibraries of numerical subroutines (Press et al., 1986).


The momentum and energy jump conditions can be expressed as

H =

(h2 +

1

2w2

2

)−(h1 +

1

2w2

1

)(3.23)

P =(P2 + ρ2w

22

)−(P1 + ρ1w

21

)(3.24)

The exact solution to the jump conditions then occurs when bothH and P are identically zero. Wecan construct an approximate solution by simultaneously iterating these two equations untilH andP are less than a specified tolerance. An iteration algorithm can be developed by considering trialvalues of (T, v) for the downstream thermodynamic state 2 that are close to but not equal to theexact solution, (T2, v2). Expanding (3.24) and (3.23) in a Taylor series about the exact solution,

H(T, v) = H(T2, v2) +∂H∂T

(T − T2) +∂P∂v

(v − v2) + . . . (3.25)

P(T, v) = H(T2, v2) +∂P∂T

(T − T2) +∂P∂v

(v − v2) + . . . (3.26)

we can write this as a matrix equation(HP

)=

(∂H∂T

∂H∂v

∂P∂T

∂P∂v

)(δTδv

)(3.27)

where δT = T −T2 and δv = v−v2. This equation is used to compute corrections, δT and δv, to thecurrent values of (T, V ) and through successive applications, approach the true solution to withina specified error tolerance. At step i, we have values (T i, vi) that we can evaluate (3.23) and (3.24)to obtain Hi and P i; then we solve (3.27) for δT i and δvi and compute the next approximation tothe solution as

T i+1 = T i − δT ivi+1 = vi − δvi (3.28)

The corrections can be formally obtained by inverting the Jacobian

J =

(∂H∂T

∂H∂v

∂P∂T

∂P∂v

)(3.29)

and carrying out the matrix multiplication operation(δTδv

)= J−1

(HP

)(3.30)

This is equivalent to the Newton-Raphson method (Press et al., 1986) of solving systems of non-linear equations. The derivatives needed for the Jacobian might be computed analytically althoughit is often simpler to compute these numerically for complex equations of state or problems thatinvolve chemical equilibrium. This is the approach used by Reynolds (1986) in his implementationof this method for finding CJ states for detonations and the basis for the method used by Browneet al. (2008).

3.3. WEAK SHOCK WAVES 25

3.3 Weak Shock WavesAn important special case is when the “strength” of the shock wave becomes small. This occurswhen the shock Mach number approaches one and the jumps across the shock become small quan-tities compared to the upstream values. In this weak shock limit the shock speed approaches thesound speed

Us → a1 (3.31)

and the piston velocity approaches zero

up → 0 . (3.32)

The relationships between the shock speed, piston velocity, and jumps in other properties can bedetermined by linearizing the jump conditions, assuming that Us-a1 and up are small but nonzero.First consider the conservation of mass (2.5), which can be written exactly as

∆ρ (Us − up) = ρ1up (3.33)

Now take the weak shock limit (3.31) and (3.32) to obtain.

∆ρa1 ≈ ρ1up (3.34)

which can be written as

upa1

≈ ∆ρ

ρ1

(3.35)

From the momentum equation (2.9), the acoustic limit is

∆P ≈ ρ1a1up (3.36)

The quantity ρa is known as the acoustic impedance and is the one of the key physical properties(along with sound speed) in acoustic wave propagation computations. Combining the previous tworesults, we have that

∆P ≈ a21∆ρ (3.37)

From the energy jump equation, the acoustic limit is

∆e ≈ −P∆v (3.38)

Comparing this to the fundamental relationship of thermodynamics

∆e = T∆s− P∆v (3.39)

we see that the weak shock limit is isentropic, ∆s≈ 0 and the sound speed is given by the isentropicderivative.

a2 = lim∆ρ,∆P→0

∆P

∆ρ=

(∂P

∂ρ

)s

(3.40)


This shows that the theory of weak shock waves is identical to the theory of acoustic waves whichhas as a foundation the assumption of isentropic motion.

∆P =

(∂P

∂ρ

)s

∆ρ (3.41)

The isentropic nature of weak waves has broad implications for relationships between flow prop-erties. For example, the acoustic limit of the Rayleigh line is

limUs→a

∆P

∆v= −a

2

v2(3.42)

=

(∂P

∂v

)s

(3.43)

implying that the slope of the Rayleigh line and isentrope are very close for weak waves. The samelimit can be applied to the energy equation to find that

limUs→a

∆e

∆v= −P (3.44)

=

(∂e

∂v

)s

(3.45)

or equivalently

limUs→a

∆h

∆P= v (3.46)

=

(∂h

∂P

)s

(3.47)

These results indicate that weak shock waves are acoustic waves and can be approximated asisentropic processes. The nearly-isentropic nature of weak shock waves is explored in more depthin Section 4.

The relationship between pressure changes and velocities changes can be generalized to wavespropagating either to the right or the left and with an initial flow ahead of the wave. This general-ization is

∆u = u2 − u1 (3.48)∆P = ±ρa∆u (3.49)

where the + sign is for right-facing waves and the− sign is for left-facing waves. The correspond-ing generalization for density is

∆ρ

ρ= ±∆u

a(3.50)

In the acoustic limit, it is possible to have waves of compression and expansion without any re-strictions. Compression waves have ∆P > 0 and expansion waves have ∆P < 0. In fact, a very

3.3. WEAK SHOCK WAVES 27

common type of acoustic signal is musical tone which consists of alternating compressions andexpansions. If we generalize the weak shock limit as corresponding to acoustic waves of bothcompression and expansion, then (3.49) and (3.50) are valid for both negative and positive valuesof ∆P , ∆ρ and up. However, we will show in Section 4 that for finite amplitude waves, there arevery definite restrictions and expansion shocks can only exist under very special circumstances. Inordinary fluids, shock waves are always compression waves and expansion waves spread out ratherthan steepen to form shocks.

Note that the pressure jump can written in nondimensional terms as

Π =[P ]

ρa2 (3.51)

This scaling indicates that, for general fluid, the appropriate way to scale the pressure jump is withthe quantity ρa2 rather than P1 as is done for the simple perfect gas case. As we shall see in thenext section, this is essential in making estimates of the ”strength” of the shock wave. In gases,ρa2 = γP so that Π = ∆P/γP1. In liquids and solids, ρa2 can be many orders of magnitude largerthan P1.

3.3.1 Weak Shock Series – General FluidThe weak shock considerations in the previous section are only the lowest order approximation.The analysis of the wave propagation is greatly simplified by the acoustic approximation that allwaves travel at the isentropic speed of sound, independent of the wave amplitude. While theacoustic limit is an extremely useful limiting approximation for applications with weak shockslike sonic booms, water hammer, exhaust systems of internal combustion engines, and lithotripsy;essential aspects of shock wave behavior are completely missed by not including nonlinearity inthe form of a wave speed that depends on wave amplitude.

Even for weak shocks it is necessary to know the next order of approximation in order to beable to compute how the wave speed will depend on amplitude and find corrections to the acousticsolutions. Indeed, we will find that the nonlinearity is crucial in determining under what conditionsshock waves will form and what type of shock waves (compression or expansion) may exist.

In order to develop more general series expansions for pressure, density, and velocity jumpacross weak shock waves, we need to return to the exact jump conditions and carry out a systematicderivation of the relationships between the jumps and the wave speed. Our starting points are theexact mass and momentum jump conditions

[w]

a1

= M1[v]

v1

(3.52)

and

Π =v1[P ]

a21

(3.53)

or

Π = −M1[w]

a1

(3.54)


or

Π = −M21

[v]

v1

(3.55)

Instead of working directly with the energy jump relation, we will start with the equation of statein the form of a Taylor series about the initial state, using v(P, s) or P (v, s). In terms of volumewe have

v = v1 +

[(∂v

∂P

)s

(P − P1) +1

2

(∂2v

∂P 2

)s

(P − P1)2

]+

[1

6

(∂3v

∂P 3

)s

(P − P1)3 +

(∂v

∂s

)P

(s− s1)

]+ . . .

(3.56)

and in terms of pressure, we have

P = P1 +

[(∂P

∂v

)s

(v − v1) +1

2

(∂2P

∂v2

)s

(v − v1)2

]+

[1

6

(∂3P

∂v3

)s

(v − v1)3 +

(∂P

∂s

)v

(s− s1)

]+ . . .

(3.57)

From the previous analysis of the weak shock limit, we know that the entropy on the Hugoniotmust rise more slowly than first order in the jump in volume or pressure since in the limit of anacoustic disturbance, the flow is isentropic. In fact, we will show in Section 4 that the entropyincreases as (∆P )3 or (∆v)3 on the Hugoniot so that the last two terms in the Taylor series areof the same order. For weak shock waves, we can neglect these terms and only use the first threethree terms (second order expansion) of the approximate the insentrope to approximate the energyconservation relation on the Hugoniot.

Using the following definitions for the speed of sound a

a2 = −v2

(∂P

∂v

)s

(3.58)

and defining the fundamental derivative of gas dynamics (see Appendix C and Section 4)

Γ =a4

2v3

(∂2v

∂P 2

)s

=v3

2a2

(∂2P

∂v2

)s

(3.59)

we can rewrite the Taylor series expansions of the Hugoniot in the following compact forms.

[v]

v1

= −Π + ΓΠ2 +O(Π)3 (3.60)

or as

Π = − [v]

v1

+ Γ

([v]

v1

)2

+O ([v])3 (3.61)

3.4. STRONG WAVES 29

Substituting (3.60 or 3.61) into (3.52 and 3.54) and linearizing, the following expansions of thejump conditions can be derived:

− [w]

a1

= Π− Γ

2Π2 +O(Π)3 (3.62)

M1 = 1− Γ

2

[w]

a1

+O

([w]

a1

)2

(3.63)

M1 = 1 +Γ

2Π +O(Π)2 (3.64)

M2 = 1− Γ

2Π +O(Π)2 (3.65)

[a]

a1

= (Γ− 1)Π +O(Π)2 (3.66)

M1 − 1 = 1−M2 +O(Π2) (3.67)

These are valid for weak shock waves in an arbitrary fluid, see Thompson (1983) for a morecomplete discussion.

3.3.2 Weak Shock Series – Perfect GasIf we specialize to perfect gases for which

Γ =γ + 1

2, (3.68)

we can use the analytical solutions (3.8-3.10) for the jump conditions and the Hugoniot equation(2.18) to obtain the following series expansions for weak shocks.

[P ]

P1

=2γ

γ + 1(M2

1 − 1) (3.69)

[w]

a1

= − 2

γ + 1(M2

1 − 1) +O((M21 − 1)2) (3.70)

[v]

v1

= − 2

γ + 1(M2

1 − 1) +O((M21 − 1)2) (3.71)

[T ]

T1

=2(γ − 1)

γ + 1(M2

1 − 1) +O((M21 − 1)2) (3.72)

1−M22 = M2

1 − 1 +O((M21 − 1)2) (3.73)

Note that the expansion is in powers of M21 -1 rather than M1-1.

3.4 Strong WavesThe opposite situation from a weak shock wave is a strong shock wave, which is the limit

Us � a1 (3.74)


The density ratio approaches a constant value for strong waves if the value of the Gruneisen coef-ficient reaches an asymptotic value G∞ as Us � a1. From the subsequent discussion on the slopeof the Hugoniot, we will show in section 4 from (4.49) that the limiting density ratio will be

limUs→∞

ρ2

ρ1

=2 +G∞G∞

(3.75)

for an arbitrary substance. Many simple models of fluid behavior do exhibit a limiting densityratio. However, in actual practice the ideal strong shock limit is seldom achieved due the variouschemical transformations that take place due to the large increases in temperature across a strongshock wave. These are transformation processes such as phase changes, dissociation into smallermolecules and atomic species, and ionization.

For an ideal strong shock, the pressure approaches

limUs→∞

P2 ∼2

G∞ + 2U2s (3.76)

and the enthalpy approaches

limUs→∞

h2 ∼G∞(G∞ + 4)

(G∞ + 2)2U2s (3.77)

3.4.1 Perfect Gas CaseFor a perfect gas, G = G∞ = γ - 1 and the strong shock limit is

limUs→∞

ρ2

ρ1

=γ + 1

γ − 1(3.78)

limUs→∞

M2 =

√2

γ − 1(3.79)

limUs→∞

P2

P1

∼ 2γ

γ + 1M2

s (3.80)

limUs→∞

T2

T1

∼ 2γ(γ − 1)

(γ + 1)2M2

s (3.81)

or alternatively, since P2� P1, making P1 irrelevant as a scaling parameter

limUs→∞

P2 ∼2

γ + 1ρ1U2

s (3.82)

Similarly, h2� h1 and T2� T1 so that more useful approximate energy jump relations are

limUs→∞

h2 ∼2γ

(γ + 1)2ρ1U2

s (3.83)

3.4. STRONG WAVES 31

or

limUs→∞

T2 ∼2(γ − 1)

R(γ + 1)2ρ1U2

s (3.84)

Note that you can expand the density ratio in inverse powers of M2s for a strong shock and come

up with series solutions for finite but large shock strength that improve on these simple estimates.


Chapter 4

Causality and Entropy

There are constraints on the possible downstream states that can reached from a given upstreamstate in a shock wave. These constraints determine the type of shock waves that can exist and thegeometrical relationships between the Rayleigh line, Hugoniot, and isentropes. The constraintsare:

1. The mass flux through the shock wave is a real quantity

2. The flow is supersonic upstream and subsonic downstream

3. Entropy has to increase across the shock

4. The downstream state has to be single valued

We will consider each of these conditions in turn and show how they constraint the possible statesthan can exist downstream.

4.1 Mass fluxThe mass flux through the shock wave is a physical property and can be represented by a positivenumber. This constrains the slope of the Rayleigh line:

−∞ ≤ ∆P

∆v= −m2 ≤ 0 (4.1)

Graphically this can be represented by the allowed and forbidden quadrants in Fig. 4.1.

4.2 CausalityShock waves are local phenomena. The fluid directly ahead of the shock does not know of theshock’s existence until the shock reaches that location. In other words, a shock moves supersoni-cally relative to the fluid upstream. At the same time, the shock is not independent of what happensin the region downstream and disturbances in the region behind the shock can catch up to the shock.

33

34 CHAPTER 4. CAUSALITY AND ENTROPY

1allowed

allowed

forbidden

forbidden

P

v

R

Figure 4.1: Allowed and forbidden regions for shock waves starting from state 1.

So the shock moves subsonically relative to the downstream fluid. Since these conditions have todo with the flow of information, they are known as causality conditions:

w1 > a1 upstream (4.2)w2 < a2 downstream (4.3)

The concept of causality is illustrated in Fig. 4.2, which shows the disturbances from the pistonovertaking the shock wave, which means that up + a > Us or a > Us − up, which is equivalent tothe condition given in (4.3). Acoustic disturbances travel along characteristic lines in a uniformflow and the slope of the characteristics are

dx

dt= u+ a on C+ (4.4)

dx

dt= u− a on C− (4.5)

which correspond to waves moving downstream C+ and upstream C−. The acoustic waves movingfrom the piston to the shock front travel along the C+ characteristics in Fig. 4.2. A simple inter-pretation is possible in the case of regular fluids: the acoustic waves move more slowly ahead of

4.2. CAUSALITY 35

t

x

piston

shock

characteristic C+

USa+uPuP a

Figure 4.2: Illustrating the concept of causality for a shock driven by a piston. Acoustic distur-bances from shock overtake the shock wave from behind due to the subsonic nature of flow behindthe shock. The shock is overtaking acoustic disturbances ahead, illustrating the supersonic natureof shock waves.

the shock than behind since there is no flow ahead of the shock in this case and the sound speedahead of the shock is lower than the sound speed behind the shock if Γ > 1. From the definition ofsound speed and Γ (see Appendix C), we find that

∂a

∂v

)s

= −av

(Γ− 1) , (4.6)

or in terms of denisty

∂a

∂ρ

)s

=a

ρ(Γ− 1) . (4.7)

Materials with positive nonlinearity, Γ > 1, have a sound speed that increases with increasingdensity so that the characteristic slope is higher behind shock waves than ahead of them with theconsequence that the higher amplitude (larger ρ or P ) disturbances travel faster than the loweramplitude disturbances resulting in wave steepening and the formation of shock waves (see theimage from Becker reproduced in Fig. 1). The causality conditions are satisfied for values of Γ <1 also but require carefully considering the changes in the downstream stream with Γ. For example,


we will show below that if Γ < 0, expansion shocks are possible which means that the pressurewill decrease across the shock wave and the sound speed will increase.

The characteristics can be visualized as the limiting speeds with which information can movefrom one point in the flow to another. The region of a flow that can influence events at a pointis the domain of dependence, shown for point P in Fig. 4.3a as the shaded area bounded by thecharacteristics C+ and C−. The solution at point P depends on the flow with the shaded region.Another way of expressing this is that only acoustic disturbances originating in the shaded regioncan effect the solution at point P . The upstream causality condition for the shock dictates that ashock reaching point P must not have passed through the domain of dependence, i.e., a shock’sinfluence at a point is not felt until it has reached and passed by that point.

Events at a point P ′ will influence the flow within the domain of influence bounded by thecharacteristics C+ and C− passing through P ′. The downstream causality conditions for the shockmeans that the the domain of influence (Fig. 4.3b) for point P ′ must eventually intersect the shock.The domain of influence and dependence for a point P are not symmetric with respect to inversionthrough the point P since the characteristic slopes (u± a) are affected by the nonuniformity of theflow near P . In the case of a point being crossed by a shock wave, there can be a marked differencebetween upstream and downstream states, resulting in very asymmetric regions of influence anddependence. On the other hand, for a point in a quiescent region, the characteristics are straightlines and the domains of influence and dependence are symmetric with respect to inversion throughthe point of interest.

P

t

x

C+ C−

shock

P’

t

x

C+

C−

shock

(a) (b)

Figure 4.3: Illustration of the meaning of causality for a shock wave. (a) Shock does not intersectthe domain of dependence for point P . (b) Domain of influence of point P must intersect shock.

These conditions determine the relationship between the Rayleigh line and isentropes at theupstream and downstream states. The slope of the Rayleigh line in the P -v plane is

4.3. GEOMETRY OF ISENTROPES 37

∆P

∆v=P2 − P1

v2 − v1

= −w21

v21

Upstream (4.8)

= −w22

v22

Downstream (4.9)

The slope of the isentrope is

∂P

∂v

)s

= −a21

v21

Upstream (4.10)

= −a22

v22

Downstream (4.11)

This gives the following relationships between the slopes

Upstream 0 >∂P

∂v

)s

≥ ∆P

∆v(4.12)

Downstream 0 >∆P

∆v≥ ∂P

∂v

)s

(4.13)

These results are independent of the properties of the fluid and only depend on the causality con-ditions. In particular, these hold for both compression and expansion discontinuities as showngraphically in Fig. 4.4.

P

v

Rayleigh line

S1

S2 > S1

1

2

P

v

Rayleigh line

S1

S2 > S1

1

2

expansion discontinuity

compressiondiscontinuity

Figure 4.4: Illustration of the relationship of Rayleigh lines and isentropes for both compressionand expansion discontinuities.

4.3 Geometry of IsentropesThe causality condition shows that the geometry of the isentropes plays a critical role in determin-ing the types of solutions that are acceptable as downstream states in shock waves. We know that


the slope of the isentropes is always negative and can be characterized by the sound speed a(∂P

∂v

)s

= −a2

v2(4.14)

The ordering of the isentropes in the P -v plane can be determined by finding the sign of thederivatives (

∂s

∂P

)v

>< 0 (4.15)

and (∂s

∂v

)P

>< 0 (4.16)

From Maxwell’s relationships (Appendix C) we have(∂s

∂P

)v

= −(∂T

∂v

)s

(4.17)

which through manipulation of the derivatives (Appendix C) can be written in terms of the coeffi-cient of thermal expansion

α =1

v

(∂v

∂T

)P

(4.18)

the isothermal compressibility

KT = −1

v

(∂v

∂P

)T

(4.19)

and specific heat capacity

cv = T

(∂s

∂T

)v

(4.20)

so that (∂s

∂P

)v

=Tα

KT cv. (4.21)

In general, elementary considerations about the stability of matter to small changes in pressure ortemperature imply that

cv > 0 (4.22)

4.3. GEOMETRY OF ISENTROPES 39

and

KT > 0 (4.23)

so that the sign of (4.15) depends completely on the coefficient of thermal expansion, which canbe either negative or positive as determined by the details of the atomic structure of the material.(

∂s

∂P

)v

> 0 if α > 0 (4.24)(∂s

∂P

)v

< 0 if α < 0 (4.25)

This can also be written in terms of a single parameter, the Gruneisen coefficient G which has thesame sign as α and is defined as

G = v

(∂P

∂e

)v

(4.26)

=v

T

(∂P

∂s

)v

(4.27)

so that we can write the derivative (4.15) as(∂s

∂P

)v

=v

GT. (4.28)

The derivative with respect to volume (4.16) can be transformed into(∂s

∂v

)P

=cpαTv

(4.29)

which has the same sign as (4.15). For ideal gases, we have that

α =1

T> 0 (4.30)

KT =1

P> 0 (4.31)

so that (∂s

∂P

)v

> 0 (ideal gas) (4.32)

The curvature of the isentropes can be characterized by a nondimensional quantity Γ, which wasdefined earlier (3.59) (

∂2P

∂v2

)s

=2a2

v3Γ (4.33)

(4.34)


The shape and ordering of the isentropes will be completely determined by the three parametersa, Γ, and G (or equivalently α). The isentropes can be sketched on the P -v plane (Fig. 4.8) usingthe geometrical interpretations of slope, curvature and ordering which can also be represented forsmall changes in volume and pressure by the Taylor series expansion (3.57) of the isentrope P (v, s)about the state (v1, s1).

P = P1 −a2

v2(v − v1) + Γ

a2

v3(v − v1)2 +

GT

v(s− s1) + . . . (4.35)

4.4 Variation of Entropy on HugoniotThe relationship of the isentropes to the Hugoniot is constrained by the Second Law of Thermody-namics which requires that the entropy of the downstream state not be less than that of the upstreamstate

s2 ≥ s1 .

For finite strength waves, the inequality is strict

s2 > s1

and we can make an even stronger statement by showing that the entropy increases monotonicallywith increasing shock strength, as measured by the shock velocity. First consider the Hugoniotrelationship in the following form

e2 − e1 =1

2(P + P1)(v1 − v) (4.36)

and differentiate considering state 2 as variable with state 1 fixed.

de2 =v1 − v2

2dP2 −

P2 + P1

2dv2 (4.37)

Now eliminate energy as a variable by applying the fundamental relationship of thermodynamicsto the downstream state

de2 = T2ds2 − P2dv2 (4.38)

for a fixed upstream state. Using the expression for the Rayleigh line (2.21) with shock speed U =w1, algebraic manipulation results in

Tds2 =1

2

(v1 − v2

v1

)2

dU2 (4.39)

This shows that increasing the shock velocity always results in increasing the entropy of state 2and vice versa.

s2(U′) > s2(U) if U′ > U (4.40)

However, this does not assure that the entropy increases monotonically along the Hugoniot sinceit does not rule out the situation shown in Fig. 4.5b. Further restrictions on the thermodynamicproperties are necessary in order to rule this out. The key requirement is that the curvature ofisentropes crossing the Hugoniot do not change sign. A comprehensive discussion of this is givenby Menikoff and Plohr (1989) who prove what they refer to as the Bethe-Weyl theorem.

4.4. VARIATION OF ENTROPY ON HUGONIOT 41

1

2

HR

v

P

1

2

2’ HR

v

P

(a) (b)

Figure 4.5: Possible Hugoniot shapes. (a) Normal shape – shock strength increases monotonicallyalong Hugoniot curve, unique solution for all shock strengths. (b) Exceptional case – shock speedoscillates along Hugoniot, multiple solutions (2 and 2’) possible for a given shock speed.

1. The isentropes with entropy greater than the entropy of the initial state, s > so, cross theHugoniot at least once.

2. If Γ > 0 on an isentrope, it crosses the Hugoniot only once.

(a) If s > so, then v < vo and w2 < a2

The Bethe-Weyl theorem assures us that as long as the fluid is reasonably well-behaved, i.e.there is no phase transition or chemical reaction and Γ > 0 in the region of interest, then theHugoniot does not double back on itself and only compression shocks with subsonic downstreamconditions can exist so that the Hugoniots must be of the form shown in Fig. 4.5a. However, thebehavior shown in Fig. 4.5b may be created when the Hugoniot crosses a phase transition regionor if chemical reaction takes place within the shock front.

The behavior shown in Fig. 4.5b creates an issue of interpretation - which of the two solutions2 or 2’, will be realized in a given situation? A multivalued solution is clearly not physical anda single shock jump to state 2’ will be unstable and as discussed in the following section, a wavewith a postshock pressure equal to 2’ will split spontaneously split into two waves traveling in thesame direction. In the case of Fig. 4.5b, the solution 2’ also violates the causality condition (seeSection 4.5) with a supersonic downstream state.

The Hugoniot curve PH(v) in Fig. 4.5a is a single-valued function of volume but there isa range in volume for which the curve in Fig. 4.5b is multi-valued. More importantly, for theRayleigh line shown, there are two downstream solutions (2, 2’) possible. As the shock speed isfurther increased, the points 2 and 2’ will coalesce and the shock speed will have a maximum atthe point where the Rayleigh line and Hugoniot line are tangent. In order to satisfy the entropycondition (4.39), the entropy must vary on the Hugoniot curve as shown in Fig. 4.6, with ds/dv <


0 between point 1 and A, > 0 between A and B and > 0 states beyond point B. This implies that

P

v

H

1

A

B

S

R1

R2

R3

Figure 4.6: Variation of entropy along Hugoniot that satisfies the entropy variation condition(4.39).

the entropy has a maximum at point B. Entropy extrema on Hugoniot are commonly found in caseswhere there is chemical reaction or phase change from a metastable state. These points are knownas Chapman-Jouguet (CJ) points and play an important role in the theory of detonation waves.

The rate of change of entropy with volume on the Hugoniot can be determined by using thederivative of the Hugoniot (4.37) and the fundamental relation of thermodynamics and interpretingall derivatives as being taken along the Hugoniot.(

∂s

∂v

)H

=∆v

2T

[∆P

∆v−(∂P

∂v

)H

](4.41)

The terms inside the square brackets represent the difference in the slopes of the Hugoniot andRayleigh line. For compression shocks, ∆v < 0, and we can verify that (4.41) is consistent withthe arrangement of slopes shown in Fig. 4.6.

4.4.1 Stability

The sign and magnitude of the terms within the square brackets in (4.41) play an important role inthe theory of the stability of shock waves to small disturbances. There are two (related) instabilities:1) a single planar shock front can split into two planar waves traveling at different speeds andpossibly, opposite directions; 2) an initially planar shock front may become wrinkled or corrugatedand spontaneously emit acoustic waves. These are referred to (see Griffiths et al., 1975, Bates andMontgomery, 2000) as D’yakov-Kontorovich instability after the first researchers to theoreticallyanalyze this problem. Instability is predicted to depend on the values of the parameter A, the

4.5. SLOPE OF HUGONIOT 43

negative of the slope of the Rayleigh line divided by the slope of the Hugoniot.

A = m2

(∂v

∂P

)H

(4.42)

= −∆P

∆v

/(∂P

∂v

)H

(4.43)

The wave splitting type of instability is predicted to occur when

A < −1 (4.44)

or

−∆P

∆v> −

(∂P

∂v

)H

(4.45)

This means that the branch of the Hugoniot above state B in Fig. 4.6 is unstable. A theoreticaldiscussion of wavesplitting is discussed in Menikoff and Plohr (1989) and experimental examplesare given in Thompson et al. (1986), Zel’dovich and Raizer (1966). The wave corrugation type ofinstability is predicted to occur when

A >1−M2

2 (1 + v1/v2)

1−M22 (1− v1/v2)

(4.46)

The corrugation instability has been observed for very strong shocks in experiments in argon andcarbon dioxide by Griffiths et al. (1975) and in computations for a fluid modeled by the van derWaals equation of state by Bates and Montgomery (2000). Shocks in perfect gases are completelystable for all values of M1 > 1 according to the D’yakov-Kantorovich criteria.

4.5 Slope of Hugoniot

The relationship of the Hugoniot to the isentropes requires determining the slope of the Hugoniot.This can be accomplished by expanding internal energy e(P, v) as a function of pressure andvolume.

de =

(∂e

∂P

)v

dP +

(∂e

∂v

)P

dv (4.47)

Using thermodynamic relationships (Appendix C), we can write the coefficients in terms of theGruniesen coefficient G

de =v

GdP −

[v

G

(∂P

∂v

)s

+ P

]dv (4.48)


Equating this to the expression (4.37) obtained by differentiating the Hugoniot and solving for theslope, we have

(∂P

∂v

)H

=

(∂P

∂v

)s

+G

2v∆P

1 +G

2v∆v

(4.49)

An independent relationship between the Hugoniot and isenrope slopes can be obtained by ex-panding P (v, s) on the Hugoniot(

∂P

∂v

)H

=

(∂P

∂v

)s

+

(∂P

∂s

)v

(∂s

∂v

)H

(4.50)

This can be simplified by using the thermodynamic relations to read(∂P

∂v

)H

=

(∂P

∂v

)s

+GT

v

(∂s

∂v

)H

(4.51)

Note that the unsubscripted variable v, G, and the slope of the isentrope are to be evaluated at thedownstream conditions (2) in this equations. Eq. (4.51) indicates how the sign of G and the rate ofchange of entropy along the isentrope determines if the slope of the isentrope is larger or smallerthan the slope of the Hugoniot. For substances with G > 0, the slope of the Hugoniot will besmaller (larger) than the slope of the isentrope when the entropy derivative ds/dv < 0 (> 0). Thisis shown in Fig. 4.8.

Using the expression (4.49) derived previously for the slope of the Hugoniot, we can rewrite(4.41) as

(∂s

∂v

)H

=∆v

2T

∆P

∆v−(∂P

∂v

)s

1 +G∆v

2v2

(4.52)

which can also be written as

(∂s

∂v

)H

=∆v

2Tv22

a22 − w2

2

1 +G∆v

2v2

(4.53)

This brings out a crucial connections between the flow speed (subsonic vs supersonic) downstreamof the shock, the Gruniesen parameter and the variation of entropy along the Hugoniot. As long asthe denominator is positive, which is true when

−∆v

vG < 2 , (4.54)

4.6. GEOMETRIC RELATIONSHIP OF ISENTROPES AND HUGONIOTS 45

a subsonic downstream state (satisfying causality) will result in the term inside the brackets beingpositive so that the entropy

∆s =

(∂s

∂v

)H

∆v =(∆v)2

2v22T

a22 − w2

2

2 +G∆v

v2

> 0 (4.55)

increases along the Hugoniot for both compression (∆v < 0) and expansion (∆v > 0) shocks.

4.6 Geometric Relationship of Isentropes and HugoniotsMost “normal” fluids, including equilibrium gases, liquids, and solids away from phase boundarieshave

Γ > 0 and G > 0

For example, this is always true for perfect gases since it is straightforward to compute

Γ =γ + 1

2(4.56)

G = γ − 1 (4.57)

For modest strength shocks away from phase boundaries, liquids and solids typically have positivevalues of Γ and G. Most metals have a value of G ≈ 2. Fluids with Γ < 0 are termed fluidswith negative nonlinearity or more whimsically, “funny” fluids, since the usual rules of compress-ible flow in gases are turned topsy-turvy in these substances. The situations under which Γ < 0are discussed in Thompson (1971) and the possibility of expansion shock waves is examined inThompson and Lambrakis (1973).

The behavior of a compressible fluid is almost entirely determined by the signs of Γ and G.With this characterization of the isentropes, we will find that there are four possible cases to con-sider:

Table 4.1: Possible cases to consider for isentropes in a general fluid Thompson (1971).

Case A Γ > 0 G > 0 Normal fluid, compression shocksCase B Γ > 0 G < 0 Unusual fluid, compression shocksCase C Γ < 0 G > 0 Abnormal fluid, expansion shocksCase D Γ < 0 G < 0 Abnormal fluid, expansion shocks

These four cases are illustrated in Fig. 4.7. By considering the variation of entropy along theHugoniot as discussed in the previous section, model Hugoniots can be drawn in for each case.Equation (4.39) shows, independent of the sign of Γ, entropy increases with increasing shockspeed. We have made use of this in constructing each of the Hugoniots in Fig. 4.7 so that theHugoniot crosses the isentropes in the direction of increasing entropy in all cases. In these figures,we have also anticipated the result of Section 4.7 below, in which we show that the isentropes andHugoniot curves at state 1 are tangent up to third order.


S1

S2

S3

S4

dS > 0

G > 0

Γ > 0

H

Case A

v

P

S1

S2

S3

S4

dS > 0

G < 0

Γ > 0H

Case B

v

P

(a) (b)

S1 S2 S3 S4

dS > 0 G > 0

Γ < 0

HCase C

v

P

S1S2S3S4

dS > 0 G < 0

Γ < 0

HCase D

v

P

(c) (d)

Figure 4.7: Possible configurations of isentropes and Hugoniots, H, corresponding to the fourcombinations of fundamental derivative, Γ, and Gruneisen coefficient, G.

4.6.1 Relationship of Rayleigh Line, Hugoniot, and IsentropeIt is also possible to draw Rayleigh lines connecting initial (1) and final (2) states on the Hugoniots.In order to do this properly it is necessary to know the relative slopes of the Rayleigh line, Hugo-niot, and isentrope at states 1 and 2. This can be done with the aid of the earlier results (4.12–4.13)and (4.41). (

∂s

∂v

)H

=∆v

2T

(∆P

∆v−(∂P

∂v

)H

)At state 1, the slope of the Hugoniot (denotedH here) is the same as the isentrope so that 0 > I =H ≥ R for all cases. That is, the Rayleigh line slope is more negative than that of the isentropes

4.6. GEOMETRIC RELATIONSHIP OF ISENTROPES AND HUGONIOTS 47

or the Hugoniot. Downstream the situation is more complex and depends on the individual case.For all cases, we have 0 > R ≥ I. For cases A and B, Γ > 0, we have compression shocks and(

∂s

∂v

)H< 0 (4.58)

for entropy to increase with increasing shock strength (decreasing volume). This means that Equa-tion (4.41) implies that R > H at the downstream state for both cases A and B. The relationshipbetween I andH can be determined by using (4.51).(

∂P

∂v

)H

=

(∂P

∂v

)s

+GT

v

(∂s

∂v

)H

For case A, G > 0 implies that I >H so that

R > I > H Case A, downstream state (4.59)

For case B, G < 0, so (4.51) implies thatH > I and

R > H > I Case B, downstream state (4.60)

For cases C and D, Γ < 0, so we have expansion shocks (∆v > 0) and(∂s

∂v

)H> 0 (4.61)

for entropy to increase with increasing shock strength (increasing volume). This means that Equa-tion (4.41) implies thatR >H at the downstream state for both cases C and D. For case C, G > 0,and (4.51) implies thatH > I so that

R > H > I Case C, downstream state (4.62)

For case D, G < 0, and (4.51) implies that I >H and

R > I > H Case D, downstream state (4.63)

These relationships are illustrated for all for cases in Fig. 4.8.

4.6.2 Summary of Four casesCase A - Compression shock For a given ∆p, an isentropic wave gives the largest volume re-duction, single shock gives the least (less than multiple shocks). At the downstream point, theisentrope I lies between HugoniotHo and Rayleigh lineR.Case B - Compression shock For a given ∆p, a single shock gives greater volume reduction thanmultiple shocks. At the downstream point, the Hugoniot H′ lies between the Rayleigh line R andthe isentrope I.Case C - Expansion shock For a given ∆p, a single shock gives a greater volume increase thanmultiple shocks. An isentropic wave gives smallest volume increase. At the downstream point, theHugoniotH′ lies between the isentrope I and Rayleigh lineR.


G > 0

Γ > 0

G < 0

Γ < 0

IoI

HoRo

O

O

O

O

P

PP

P

v

vv

v

Figure 4.8: Illustration of the four possible cases for the geometry of isentropes, Hugoniots, andRayleigh line.

Case D - Expansion shock For a given pressure decrease, a single shock gives less volume in-crease than multiple shocks. The largest volume increase is given by an isentropic wave. At thedownstream point, the isentrope I lies betweenR andH′ at downstream point.

4.7 Hugoniot closeness to Isentrope for Weak Waves

The analysis of weak shock waves (Section 3.3) shows that the compression process is almostisentropic when the shock wave is weak. In the limit of vanishing strength Ms→ 1, the Hugoniot,Rayleigh line and isentrope all have the same slope as state 2 approaches state 1. The closenessof the Hugoniot and the isentrope are shown in Fig. 4.9. The precise fashion in which the entropychange depends on the shock strength can be derived by considering the change in properties veryclose to the initial state 1.

Consider a series expansion of P (s, v) in the neighborhood of the initial state

P (s, v) = P (s1 + ∆s, v1 + ∆v) (4.64)

4.7. HUGONIOT CLOSENESS TO ISENTROPE FOR WEAK WAVES 49

0.0

1.0

2.0

3.0

4.0

5.0

0.4 0.5 0.6 0.7 0.8 0.9 1.0

v2/v1

P 2/P

1

Hugoniotisentrope

Figure 4.9: Comparison of Hugoniot and isentrope for a perfect gas with γ = 1.4.

P (s, v) = P1 +

(∂P

∂s

)v

∆s+

(∂P

∂v

)s

∆v +1

2

(∂2P

∂v2

)s

(∆v)2+

1

6

(∂3P

∂v3

)s

(∆v)3 +O(∆v)4 +O(∆s)2

(4.65)

In general, the coefficient of the entropy term(∂P

∂s

)v

= GT

v(4.66)

does not vanish, and since we know that the entropy increases with volume (4.41) along the Hugo-niot, we can not neglect this term. However, examination of (4.41) reveals that

limUs→a1

(∂s

∂v

)H

= 0 (4.67)

indicating that for weak shocks, the change in entropy ∆s is higher order in specific volume ∆vthan the changes in pressure with specific volume. In other words, ∆s = O(∆P )n where n ≥ 2.In fact, the dependence is even weaker and as we show below, ∆s = O(∆P )3. The expansion wehave written above anticipates this result. We will now show that this is the case by comparing theisentrope and the Hugoniot expansions around the initial state.


On the isentrope ∆s = 0, and equation 4.65 is a power series in volume that approximates theisentrope near the initial state.

P (s1, v) = P1 +

(∂P

∂v

)s1

∆v +1

2

(∂2P

∂v2

)s1

(∆v)2 +1

6

(∂3P

∂v3

)s1

(∆v)3 + . . . (4.68)

Now consider the variation of pressure on the Hugoniot by expanding PH about state 1.

PH(v) = P1 +

(∂P

∂v

)H

∆v +1

2

(∂2P

∂v2

)H

(∆v)2 +1

6

(∂3P

∂v3

)H

(∆v)3 + . . . (4.69)

Examining the expression (4.49) for the slope of the Hugoniot, we see that(∂P

∂v

)H

=

(∂P

∂v

)s1

at state 1 (4.70)

To determine the remainder of the terms, consider how entropy varies along the Hugoniot (4.41)

T

(∂s

∂v

)H

=1

2(P − P1) +

1

2(v1 − v)

(∂P

∂v

)H

(4.71)

We have already observed that at state 1, we have P = P1 and v = v1 so that(∂s

∂v

)H

= 0 at state 1 (4.72)

Differentiate (4.71) with respect to volume and evaluate at state 1 to obtain(∂2s

∂v2

)H

= 0 at state 1 (4.73)

Differentiate (4.71) twice with respect to volume and evaluate at state 1 to obtain

T

(∂3s

∂v3

)H

=1

2

(∂2P

∂v2

)H

at state 1 (4.74)

This implies that at state 1, the expansion for entropy change on the Hugoniot can be written

∆s =1

12T

(∂2P

∂v2

)H

(∆v)3 + . . . (4.75)

Substituting this back into (4.65), we see that the expansion of PH(v) and P (v, s1) agree to thesecond order term, but differ in the coefficient of the third order term. This implies that(

∂2P

∂v2

)H

=

(∂2P

∂v2

)s

at state 1 (4.76)

It is possible to also do this computation by using an expansion in terms of pressure, in which casewe would obtain

∆s =1

12T

(∂2v

∂P 2

)H

(∆P )3 + . . . (4.77)

4.7. HUGONIOT CLOSENESS TO ISENTROPE FOR WEAK WAVES 51

The final result can be written in nondimensional form by using the definition of Γ (3.59) as

T [s]

a21

= −1

6Γ

([v]

v

)3

+ . . . or =1

6ΓΠ3 + . . . (4.78)

A consequence of this equation is that only compression shocks are possible for Γ > 0 and onlyexpansion shocks are possible for Γ < 0.

The cubic expansion and the full expression for the entropy jump in a perfect gas are comparedin Fig. 4.10.

volume jump[v]v

entrop

yjump

[s]

R

Cubic

0.00

0.00

0.10

0.10

0.02

0.20 0.30 0.40 0.50

0.04

0.06

0.08

Figure 4.10: Comparison of full expression for entropy jump with cubic term from weak shocktheory.

4.7.1 Alternate Derivation

Consider the entropy change across a weak shock in a homogeneous single phase fluid. From thegeneral shock jump relations we have

∆e = −P∆v =P0 + P

2∆v (4.79)

which we can rewrite as

∆e = −∆P

2∆v − P0∆v (4.80)


Now proceed to obtain Taylor-series expansions of both sides of this equation. First, consider theleft-hand side and take e= e(s, v)

∆e =∂e

∂v

∣∣∣∣0

∆v +∂2e

∂v2

∣∣∣∣0

(∆v)2

2+∂3e

∂v3

∣∣∣∣0

(∆v)3

6+ · · ·+

(∂e

∂S

)v

∆S + . . . (4.81)

and we have

de = Tds− Pdv (4.82)

so that(∂e

∂v

)s

= −P (4.83)

Hence

∆e = −P0∆v −(∂P

∂v

)s

(∆v)2

2−(∂2P

∂v2

)s

(∆v)3

6+ · · ·+ T0∆s+ . . . (4.84)

Now consider P = P (s, v) and expand as a Taylor series

∆p =∂p

∂v∆v +

∂2p

∂v2

(∆v)2

2+ · · ·+ ∂p

∂s∆s (4.85)

substitute back into the Hugoniot

−P0∆v − ∆P

2∆v = −p0∆v −

(∂P

∂v

)s

(∆v)2

2−(∂2P

∂v2

)s

(∆v)3

4+ · · · −

(∂P

∂s

)v

∆s∆v

2+ . . .

(4.86)

Now equate the expansions for the left and right-hand sides

T0∆s = − 1

12

(∂2P

∂v2

)s

(∆v)3 = −Γ

6a2

0

(∆v

v0

)3

(4.87)

This is identical to the previous result that the Hugoniot is isentropic to second order at the upstreampoint.

4.8 Graphical Interpretation of Entropy GenerationFrom the Hugoniot relations, we have that the energy change across the shock is

e2 − e1 = −1

2P∆v (4.88)

4.8. GRAPHICAL INTERPRETATION OF ENTROPY GENERATION 53

which as illustrated in Fig. 2.3b, is the area under the Rayliegh line. The energy change canpartitioned into a “reversible” and an “irreversible component” by considering the fundamentalrelation of thermodynamics

Tds = de+ pdv (4.89)

which can be integrated to yield

T∆s︸︷︷︸irreversible

= ∆e−∫s1

Pdv︸︷︷︸reversible

(4.90)

The area associated with the irreversible portion of the energy change is illustrated in Fig. 4.11 asthe shaded area labeled T∆s.

P

v

1

2

T∆s

HRI

Figure 4.11: Illustration of the irreversible portion of work in shock compression.


Chapter 5

Nonsteady Flow

Compressible flows can be represented as smooth regions connected by jumps or discontinuitiesin the from of shock waves or contact surfaces. The usual approximations for the smooth part ofthe flow is that of an adiabatic and inviscid fluid. The equations of motion in this approximationare usually referred to as the Euler Equations. These can be written in conservation form as:

∂ρ

∂t+ ∇ · (ρu) = 0 (5.1)

∂ρu∂t

+ ∇ · (ρuu) = −∇P (5.2)

∂

∂tρ

(e+

u2

2

)+ ∇ ·

(ρu(h+

u2

2)

)= 0 (5.3)

∂s

∂t+ ∇ · (us) ≥ 0 (5.4)

In addition, these equations must be supplemented by equation of state information. For mostpurposes, an incomplete equation of state

P = P (e, ρ) (5.5)

is sufficient to close the system and the complete thermodynamic potential does not have to beknown.

The conservation form is very useful for the purpose of deriving algorithms for numericalsimulation and also when considering the behavior of the solutions around singular surfaces suchas shocks or contact discontinuities. Conservation form for a two-dimensional flow in Cartesiancoordinates can be written in condensed notation as

∂W

∂t+∂Fx

∂x+∂Fy

∂y= S (5.6)

for nonreacting flows S = 0, the vector W of conserved variables and the flux vector F are

W =

ρρuρvE

Fx =

ρu

ρu2 + Pρuv

(E + P )u

Fy =

ρvρvu

ρv2 + P(E + P )v

(5.7)

55

56 CHAPTER 5. NONSTEADY FLOW

The total energy E is defined to be the sum of internal and kinetic energies

E = ρ

(e+

u2 + v2

2

)(5.8)

Most modern numerical algorithms use (5.6) as a starting point.For analytical computation, it is more convenient to rewrite the equations in terms of the con-

vective or substantial derivativeD

Dt=

∂

∂t+ u ·∇· (5.9)

This can be interpreted as differentiation along a particle path which is the trajectory xp(t) a fluidparticle or element follows. The particle path is defined by

Particle path:dxpdt

= u(xp, t) (5.10)

In terms of the substantial derivative, the Euler equations can be written:

Mass:

1

ρ

Dρ

Dt= −∇ · u (5.11)

Momentum:

ρDuDt

= −∇P (5.12)

Energy:

ρD

Dt

(h+

u2

2

)=∂P

∂t(5.13)

Entropy

Ds

Dt≥ 0 (5.14)

Note that for a smooth flow, the entropy equation is an equality

Ds

Dt= 0 (5.15)

This is equivalent to stating the entropy is constant along a particle path

s = so alongdx

dt= u (5.16)

The fundamental relation of thermodynamics can be written in terms of the enthalpy

dh = Tds+dP

ρ

5.1. ONE-DIMENSIONAL FLOWS 57

and applied to each fluid element using the convective derivative

Dh

Dt= T

Ds

Dt+

1

ρ

DP

Dt(5.17)

On an isentrope, this yields

ρDh

Dt=

DP

Dt(5.18)

which is identical to subtracting the dot product of u with (5.12) from (5.13). So we concludethat in isentropic regions of a nonsteady flow that the energy and entropy equations are equivalent.Therefore in this situation, we can replace the energy equation with entropy equation. Since theentropy equation can be written

s(P, ρ) = so ondx

dt= u (5.19)

we can differentiate this to obtain the equivalent form

DP

Dt= a2 Dρ

Dt(5.20)

This leads to the alternative version of the Euler equations which involves only P , u and s asvariables.

DP

Dt+ ρa2∇ · u = 0 (5.21)

ρDuDt

+ ∇P = 0 (5.22)

Ds

Dt= 0 (5.23)

This has to supplemented with an equation of state relationship ρ(P, s) so that the density andsound speed can be computed at every point in the flow.

5.1 One-Dimensional FlowsIn order to make further progress, we will specialize to planar, one-dimensional situations, forwhich the equations reduce to:

∂P

∂t+ u

∂P

∂x+ ρa2∂u

∂x= 0 (5.24)

ρ∂u

∂t+ ρu

∂u

∂x+∂P

∂x= 0 (5.25)(

∂

∂t+ u

∂

∂x

)s = 0 (5.26)

Multiplying (5.25) by a and adding and substracting from (5.24) we obtain two equivalent equa-tions for P and u (

∂

∂t+ (u+ a)

∂

∂x

)P + ρa

(∂

∂t+ (u+ a)

∂

∂x

)u = 0 (5.27)


and (∂

∂t+ (u− a)

∂

∂x

)P − ρa

(∂

∂t+ (u− a)

∂

∂x

)u = 0 (5.28)

and one equation for s (∂

∂t+ u

∂

∂x

)s = 0 (5.29)

These equations are in what is known as characteristic form, which can be more concisely writtenas

1

ρa

dP

dt± du

dt= 0 along C±:

dx

dt= u± a (5.30)

ds

dt= along C◦:

dx

dt= u (5.31)

The paths C± are the characteristics and represent that paths along which small distrubances(acoustic waves) will propagate in a flow. We will show below that these paths also determinethe flow of information for large disturbances,

Now consider defining a function F such that

dF =dP

ρaF =

∫dP

ρa(5.32)

With this definition, we can write the characteristic version of the equations as

d

dt(F ± u) = 0 on C± :

dx

dt= u± a (5.33)

The functions F ± u are known as the Riemann invariants, P and Q, defined as

F + u = P F − u = Q (5.34)

These quantities are referred to as invariants since (5.33) implies that P = constant on C+ and Q= constant on C−.

In general, the integral F depends on the thermodynamic path taken in evaluation. In additionto pressure variations, we know that other thermodynamic properties may vary in a manner that isnot known ahead of time. In particular, even though the flow is isentropic (5.31), the entropy ofdifferent fluid elements may differ, so that in general, the integral F will not be unique.

However, there is one very important special case for which the integral F is uniquely de-fined. That occurs when the entropy is constant and uniform in an region, a condition known ashomoentropic. When the flow is homoentropic, knowledge of only one thermodynamic property issufficient to completely determine all other thermodynamic properties. Therefore the integrationcan be carried out along an isentrope

F (P ) =

∫ P

Po

dP ′

ρaon s = so (5.35)


where Po is some reference state chosen for convenience. The resulting function F (P ) is uniquelyrelated to all the other thermodynamic variables since on the isentrope s = so we can always findthermodynamic functions ρ(P ; so), T (P ; so), a(P ; so), etc.

The homoentropic situation is very convenient for analysis of problems since it is possible tofind a number of exact solutions to nonsteady flow problems using characteristic considerations.In the remainder of this section, we will assume a homoentropic situation.

For a homoentropic fluid with a constant value of Γ, F can be computed by directly integrating(5.32). This can be most readily carried out by using the thermodynamic relations to transform theintegrand

dF =dPρa

=

(∂P

∂a

)s

daρa

(5.36)

and using one of the alternate definitions of the fundamental derivative

ρa

(∂a

∂P

)s

= Γ− 1 (5.37)

to obtain

F =

∫da

Γ(a)− 1(5.38)

The special case of constant Γ can be integrated to obtain

F =a

Γ− 1Constant-Γ substance (5.39)

A subset of this is the important special case of the perfect gas Γ = (γ + 1)/2

F =2a

γ − 1perfect gas (5.40)

The geometry of the characteristics passing through a point (3) are shown in Fig. 5.1. Ifthe values of the Riemann invariants passing through a point like (3) are known, then the state(u, ρ, P, a, . . .) of the flow at (3) can be completely determined. From the definition (5.34) of theRiemann invariants, we have

u =1

2(P −Q) F (a) =

1

2(P +Q) (5.41)

This can be used to compute the solution at a point (3) by using the data at earlier points (1) and (2)to find the value of the Riemann invariants on the characteristics passing through (3). The Riemanninvariant P on C+ is determined by the values at the state (1)

P = F1 + u1 where F1 = F (a1) (5.42)

and the value of the Riemann invariant Q on C− is determined by the values at the state (2)

Q = F2 − u2 where F2 = F (a2) (5.43)


C+ C−Co

1 2

3

t

x

Figure 5.1: Geometry of characteristics passing through a point 3. The C+ characteristic passesthrough points 1 and 3. The C− characteristic passes through points 2 and 3. The Co characteristicis identical to the particle path (5.10).

then we can compute

u3 =1

2(u1 + u2) +

1

2(F1 − F2) (5.44)

F3 =1

2(F1 + F2) +

1

2(u1 − u2) (5.45)

These results are exact and independent of the shape of the characteristics.By approximating the slope of characteristics as constant, which should be a reasonable approx-

imation over sufficiently small intervals of time, and linearizing the Riemann invariant relations,a numerical scheme known as the method of characteristics can be devised for finding a solutiona time t + ∆t if the solution at time t is known. The method of characteristics can be used to de-vise numerical simulations of simple gas dynamic situations, particularly shock-free flow, and withsome effort can be extended to treat more complex flows with shock waves and chemical reactions.The method of characteristics was the first technique developed for obtaining a purely numericalsolution to gas dynamics problems. However, the method of characteristics is very cumbersomefor complex flows in two and three space dimensions and today, has been almost completely re-placed by finite-difference and finite-volume methods. Despite this, some ideas from the method ofcharacteristics do play an important role in numerical simulation and it is important to understandthe concept.

Consider the general problem of a flow in which the entropy is not uniform and may change asa function of time. Writing P (ρ, s) and differentiating along a particle path, we have

dP =

(∂P

∂ρ

)s

dρ+

(∂P

∂s

)ρ

ds (5.46)


Using thermodynamic definitions, we can rewrite this as

dP

dt= a2 dρ

dt+GT

v

ds

dt(5.47)

where G is the Gruneisen parameter; G = γ -1 for a perfect gas and more generally

G =T

v

(∂P

∂s

)v

(5.48)

We can redrive the characteristic equations by combining this result with (5.24) and (5.25) toobtain (

∂

∂t+ (u+ a)

∂

∂x

)P + ρa

(∂

∂t+ (u+ a)

∂

∂x

)u =

GT

vs (5.49)(

∂

∂t+ (u− a)

∂

∂x

)P − ρa

(∂

∂t+ (u− a)

∂

∂x

)u =

GT

vs (5.50)(

∂

∂t+ u

∂

∂x

)P − a2

(∂

∂t+ u

∂

∂x

)ρ =

GT

vs (5.51)

where (∂

∂t+ u

∂

∂x

)s = s (5.52)

Consider integrating this set of equations from time tn to tn+1. The approximate characteristics(line segments) and associated points at these two times are shown in Figure 5.2.

Integrating each equation in time along the associated characteristic and approximating ρa, a2,and GT/v as constants, we have

P4 − P1 + ρa (u4 − u1) =GT

v(s4 − s1) (5.53)

P4 − P3 − ρa (u4 − u3) =GT

v(s4 − s3) (5.54)

P4 − P2 − c2 (ρ4 − ρ2) =GT

v(s4 − s2) (5.55)

(5.56)

If we assume that the mechanisms that generate entropy are specified, then s4 can be found byintegration of the entropy equation

s4 = s2 +

∫ tn+1

tn

s dt (5.57)

In particular, if the flow is isentropic, but not homoentropic, then s =0 and

s4 = s2 (5.58)

Note that the flow may be nonuniform, so that in general s1 6= s2 or s3. With s4 known, we havethree equations in the three unknowns: P4, u4, and ρ4. This simple linear system can be readilysolved to determine state 4. Carrying this out for a whole sequence of points at time tn+1, thesolution can be advanced from tn to tn+1.


3

4

1 2

C+ Co C−

tn

tn+1

t

x

Figure 5.2: Schematic of approximate C+, C◦, and C− characteristic line segments used in themethod of characteristics integration scheme.

5.1.1 Courant-Fredrich-Lewy ConditionThe Courant-Fredrich-Lewy or CFL condition is an essential restriction on any numerical integra-tion scheme for a hyperbolic set of equations such as the Euler equations. The idea is most simplyunderstood by considering the points {1, 2, 3} to be three points from a set {x1, x2, . . . , xn} with auniform spacing ∆x = xi+1 - xi. The region bounded by the characteristics C+ and C− was intro-duced in Section 4 as the domain of dependence of point 4. Now consider the situation shown inFigure 5.3a where a finite difference scheme uses information from points 1, 2, and 3 to computevalues at point 4. The domain of dependence for point 4, as determined by the characteristics,is greater than the span of the set of points or stencil used to compute the state. It is possible toshow that the numerical scheme will be unstable in this situation (see Ritchmyer and Morton orLevesque). If we decrease the time increment ∆t, we have the situation shown in Figure 5.3b,where the span of the points used in the difference scheme is greater than the domain of depen-dence. This situation is stable. From the geometry, the stability or CFL criterion must be

∆t <∆x

cwhere c = max(|u+ a|, |u− a|) (5.59)

5.2. SIMPLE WAVES 63

t

t

t+∆t

∆x

1 2 3

4

x

t

t

t+∆t

∆x

1 2 3

4

x

(a) (b)

Figure 5.3: Schematic x-t diagrams illustrating the possible relationship between the domain ofdependence of point 4 (the shaded region) and the points 1, 2, and 3 that are used in the finitedifference sheme to compute point 4. a) Unstable situation, domain of dependence larger thanspan of difference stencil. b) Stable situation, domain of dependence smaller than the span of thedifference stencil..

5.2 Simple Waves

Consider a smooth wave propagating to the right and bounded by uniform, homoentropic regionsof the flow, as shown in Fig. 5.4a. The wave is bounded by two C+ characteristics and crossed byC− characteristics that all have the same value of the Riemann invariant Q

Q = F − u = constant in a right-facing simple wave (5.60)

Differentiating and using dF = ρadP , we find that changes in pressure and velocity in the waveare related by

dP = ρa du right-facing simple wave (5.61)

Considering the situation in the left-facing simple wave, Fig. 5.4b, the C+ characteristics all havethe same value of the invariant P

P = F + u = constant in a left-facing simple wave (5.62)

Differentiating, we find that changes in pressure and velocity in the wave are related by

dP = −ρa du left-facing simple wave (5.63)


For each direction of propagation, there are two cases, compression and expansion waves. Forsmall-amplitude waves, the differential relationships will approximately hold for small but finitechanges in pressure and velocity, ∆P and ∆u, so that:

∆P = ±ρa∆u

{right-facing waveleft-facing wave (5.64)

These relations are the basis of the acoustic treatment of simple waves. The quantity ρa is knownas the acoustic impedance and plays an essential role in wave propagation problems.

1

2

C+2 C+

1

C−

t

x

1 2

C−1

C−2

C+

t

x

(a) (b)

Figure 5.4: Schematic x-t diagrams of simple waves and associated C+ and C− characteristics.(a) right-facing wave composed of C+ characteristics and crossing C− characteristics. (b) left-facing wave composed of C− characteristics and crossing C+ characteristics. Regions 1 and 2 areuniform, the shaded regions are the simple waves.

5.2.1 CharacteristicsThe geometry of the characteristics in simple waves are shown in Figure 5.4 for two illustrativecases of left and right-facing waves. The characteristics in the uniform regions 1 and 2 are straight,parallel lines since u and a have constant values in the uniform regions.

The left facing wave, Figure 5.4a is bounded by the two characteristics, C+1 and C+

2 and theright-facing by C−1 and C−2 . In a right-facing wave, the C+ characteristics are straight and the C−

characteristics curve; in a left-facing wave, the situation is reversed: the C− characteristics arestraight and C+ characteristics curve. Consider the case of the right-facing wave, the left-facingwave case can be readily deduced using the analogous reasoning and interchanging the roles of theC+ and C− characteristics. On a given C+ characteristic, the value of P is constant and all the C−

characteristics which cross this particular C+ originate in a uniform region and have a common

5.2. SIMPLE WAVES 65

value Q1 of the Riemann invariant. From the previous considerations (5.41) using about Riemanninvariants P and Q to obtain the flow properties, we conclude that

F (a) =1

2(P +Q1) (5.65)

and

u =1

2(P −Q1) (5.66)

are both constant on the C+ characteristics. This implies that the slope of the C+ characteristic

dxdt

= u+ a (5.67)

is constant and the flow properties such as u, a, P , ρ are all fixed along a given C+ characteristic.In other words, the solution is propagated along the C+ characteristic without any variation invalues of the properties. This property plays an important role in our subsequent discussion ofwave steepening, Section 5.4.

Within the wave, Q = Q1 = constant and we can differentiate (5.66) to find that find that

dP = 2 du , (5.68)

the value of the invariant P changes directly in proportion to change of velocity in the wave,

P = P1 + 2(u− u1) . (5.69)

The variation in slope of the C+ characteristics within the wave can be found by also using theconstancy of Q within the wave. Differentiating the slope of C+, we have

d(dx

dt

)C+

= du+ da (5.70)

From the constancy of Q, we have

dQ = dF − du (5.71)

0 =dPρa− du (5.72)

so that on C−, we have

dP = ρa du (5.73)

From the definition of the Fundamental Derivative

dPρa

=da

Γ− 1(5.74)


so that

d(

dxdt

)C+

=Γ

ρadP (5.75)

or

d(

dxdt

)C+

= Γ du (5.76)

The variation in the C− characteristic slope can be obtained in a similar fashion

d(

dxdt

)C−

= (2− Γ) du (5.77)

or

d(

dxdt

)C−

=(2− Γ)

ρadP (5.78)

The left-facing simple wave can be treated in completely analogous fashion. A summary ofthe relationships between changes in pressure and all other properties in simple waves is given inTable 5.1.

The geometry of the characteristics depends on type of wave: compression dP > 0 (+) orexpansion dP < 0 (-); the direction of propagation: left-facing or right-facing; and, the value of Γ.Inspection of Table 5.1 shows that there are four distinct cases as far as the value of Γ influences thesolution: Γ> 2; 2 > Γ> 1; 1 > Γ> 0; 0 > Γ. The value Γ = 1 is clearly significant with respect tothe sound speed behavior but as far as the characteristic geometry is concerned, the region 2 > Γ>0 behaves in a common fashion. For the determination of the characteristic behavior, there are 12cases to consider. These twelve cases are illustrated in Figure 5.5. The most important distinctionis between cases with Γ less than zero and greater than zero. As we learned in Section 4, fluidswith Γ < 0 may have expansion shock waves since the entropy generation can be positive, at leastfor weak waves.

Perfect Gases The special but important case of a perfect gas has

Γ =γ + 1

2

and 5/3 ≥ γ ≥ 1, so that 4/3 ≥ Γ ≥ 1, therefore only one of the three case distinctions for Γare significant and there are only 4 possible wave configurations. Two of these are illustrated inFigure 5.4. The cases illustrated are both expansion waves, which are spreading out as the wavepropagates. This is characteristic of gases, the head of an expansion wave travels faster than the tailand the wave spreads out. This can also be deduced from Table 5.1 by examining the expressionfor how the slope of the characteristics change with pressure. The opposite is true of compressionwaves in gases, the tail travels faster than the head, which leads to the wave steepening and shockwave formation. This is discussed further in Section 5.4.

5.3. EXPANSION WAVES 67

Table 5.1: Properties of right- and left-facing simple waves in a general compressible flow.

Right-Facing Wave Left-Facing Wave

dudPρa

−dPρa

da (Γ− 1)dPρa

(Γ− 1)dPρa

dρdPa2

dPa2

dQ 0 2dPρa

dP 2dPρa

0

d(

dxdt

)C+

ΓdPρa

−(2− Γ)dPρa

d(

dxdt

)C−

(2− Γ)dPρa

−ΓdPρa

5.3 Expansion WavesExpansion waves in fluids with Γ > 0 can be created by the same piston-cylinder apparatus ofFigure 2.2 that was used to derive the shock jump conditions. In order to create an expansionwave, the piston must be moved out of the fluid rather than into it. This is shown in Figure 5.6 fora left-facing expansion produced by a piston that is moved suddenly to the right.

Since the piston is moved impulsively, there is no length or time scale - only a velocity scaleup. This is a common situation in gas dynamics and implies that the solution for the flow mustdepend only on a similarity variable η = x/a1t or x/upt. This assumes that the piston is initiallylocated at x = 0 and that the motion starts at t =0. Otherwise for a piston located at xo and startingmotion at to, the correct similarity variable would be η = (x−xo)/a1(t− to). For an ideal gas, thismeans that the pressure has the nondimensional solution of the form

P

P1

= f(η, γ) (5.79)

and that the upstream and downstream states are related by an expression of the form

P2

P1

= f(up/a1, γ) (5.80)


Direction ∆P Γ > 2 2 > Γ > 0 0 > Γ

Right +1

2

t

x

1

2

t

x1

2

t

x

Left +

1

2t

x

1

2t

x

1

2t

x

Right -1

2

t

x1

2t

x

1

2t

x

Left -

1

2

t

x

1

2t

x

1

2t

x

Figure 5.5: Geometry of characteristics illustrating twelve possible cases for simple waves as afunction of the fundamental derivative Γ.

Similar considerations hold for the other variables such as density ρ/ρ1, sound speed a/a1 andvelocity u/a1. This situation is a specific instance of the general case of self-similar behavior,in which the solution depends on combination of space and time rather than each individually.This situation frequently arises in inviscid, nonreacting gas dynamic problems because there areno intrinsic length or time scales in the equations themselves.

5.3.1 Similarity Solutions

In general, unsteady solutions to the Euler equations depend on space and time. For example, asolution for the pressure can be expressed as

P = P (x, t) (5.81)

This equation is not dimensionally homogeneous and more insight can be gained by formulatingnondimensional variables and using the principles of dimensional analysis to rewrite the solutionin dimensionally correct form. Consider a situation in which there is a characteristic length L,time T , initial density ρ1, and sound speed a1. Considering the equations of motion, we can define


nondimensional variablesρ

ρ1

u

a1

P

ρ1a21

x

L

t

T(5.82)

and a parameter that characterizes the relationship between the time and length scales

a1T

L(5.83)

You can verify that Equations (5.1)-(5.3) can be rewritten in nondimensional form using thesedefinitions. Other nondimensional parameters that characterize the equation of state: γ, G, Γ, etc,will also come into the solution. Using these variables, we can, for example, write the solution tothe pressure as

P

ρ1a21

= f

(x

L,t

T,a1T

L, γ, . . .

)(5.84)

This type of solution is appropriate for a situation such as forced oscillations inside of a finite-length tube. The length of the tube is the obvious choice for L and the period of the forcedoscillation can be used for T .

For a problem in which there is no imposed length scale, the parameter L can be definedin terms of T and a characteristic speed such as a1, L = a1T . This would be appropriate for asituation such as forced oscillations in a semi-infinite tube. The parameter a1T/L = 1 is no longermeaningful in this situation and the solution can be written

P

ρ1a21

= f

(x

a1T,t

T, γ, . . .

)(5.85)

Finally, if there is no imposed length or time scale, the only possibility is to use a1t itself to definea length scale or equivalently, x/a1 to define a time scale and the solution must have the form

P

ρ1a21

= f

(x

a1t, γ, . . .

)(5.86)

This is the form of solution we are seeking for the situation considered in Figure 5.6

5.3.2 Expansion Fan in an Perfect GasThe C− characteristics in the expansion wave shown in Figure 5.6 form a fan extending out fromthe origin. For this reason, the similarity solution for an expansion wave is known as an expansionfan. The form of the solution can be obtained by either formally subsituting the similarity trans-formation into the equations and applying the boundary condition or else by some simple physicalconsiderations. The slope of the C− characteristics within the fan is x/t so that

x

t= u− a − a1 ≤ x/t ≤ up − a2 (5.87)

The variation of a within the wave can be related to u by considering the Rieman invariant on theC+ characteristic shown in Figure 5.6.

P =2a

γ − 1+ u =

2a1

γ − 1=

2a2

γ − 1+ up (5.88)


Figure 5.6: Schematic of piston-cylinder arrangement used to create an expansion fan (shadedregion) propagating to the left. The piston, initially located at x = 0, is impulsively accelerated to avelocity up at time t = 0. The fluid is accelerated to the right by the expansion wave and the wavepropagates to the left. The wave is bounded on the left by a uniform undisturbed region (1) and onthe right by a uniformly moving region (2).

where we have evaluated the invariant in region 1 (u = 0, a = a1) and in region 2 (u = up, a = a2).From (5.88), we find that sound speed a varies linearly with u within the wave

a = a1 −γ − 1

2u (5.89)


or

a

a1

= 1− γ − 1

2

u

a1

(5.90)

Substituting into (5.87), we have the similarity solution for velocity

u

a1

=2

γ + 1

(1 +

x

a1t

)(5.91)

and for sound speed

a

a1

= 1− γ − 1

γ + 1

(1 +

x

a1t

)(5.92)

The other quantities can be found by using the isentropic relations for a perfect gas

T

T1

=

(a

a1

)2P

P1

=

(a

a1

) 2γγ−1 ρ

ρ1

=

(a

a1

) 2γ−1

(5.93)

The corresponding results can be obtained for a right-facing expansion wave by similar reasoning,interchanging the roles of C+ and C− characteristics.

a

a1

= 1 +γ − 1

2

u

a1

(5.94)

u

a1

=2

γ + 1

(1− x

a1t

)(5.95)

In this case, uwill be negative since the piston will be moving to the left in order to generate a wavemoving to the right. All other properties can be computed using the isentropic thermodynamicrelationships.

Upstream State Motion, Non-similar Solutions The previous results can be generalized tocases in which an expansion wave propagates into a moving upstream state. In order to obtainthe relationship between u and a, we only have to consider the invariants on the characteristic fam-ily that connects the upstream and downstream states. It is not necessary to assume a self-similarsituation or to make piston-cylinder constructions but only to imagine that there is a well-definedwave with upstream (1) and downstream (2) states that can be connected with a characteristic.From the invariant relation, the sound speed can be determined from the jump in velocity [u] =u-u1 as

a

a1

= 1± γ − 1

2

[u]

a1

{right-facingleft-facing (5.96)


0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

[u]/a1

a2/a1

T2/T1

P2/P1

2/1

Figure 5.7: Variation of properties in an ideal gas expansion wave, γ = 1.4

5.4 Compression Waves and Shock FormationFrom Figure 5.5, it is apparent the characteristics making up a compression wave in a fluid withΓ > 0 are converging and will ultimately intersect. This is a consequence of the tail of the wavetraveling faster than the head, due to the increase in characteristic slope with pressure

d(

dxdt

)C+

=Γ

ρadP

The intersection or crossing of the characteristics in the same family is equivalent to the solu-tion becoming multi-valued since there will now be multiple, inconsistent values for the Riemanninvariant on the family of characteristics that intersect: C+ for a right-facing wave, C− for a left-facing wave. Another way to look at this was discussed in the previous section on flow propertieswithin a simple wave, the solution is propagated along the characteristics without change. There-fore, intersection of characteristics with different properties will result in the solution becomingmulti-valued. This is shown in Figure 5.8 and also in Figure 5.9 for the situation of a compressionwave being produced in a tube by a piston moving with a prescribed speed up = dX/dt = X(t).It is not necessary to actually have a moving piston in order to obtain a compression wave thatevolved into a shock - this is simply a convenient way of visualizing how the compression wave iscreated.

The crossing of characteristics of the same family is an unphysical situation since there canonly be single value for pressure, density, etc, at a point in space. The remedy is to insert a shockwave to replace the overlapping characteristics. The location and strength of the shock wave aredetermined by requiring that the resulting solution be consistent with the jump conditions. The

5.4. COMPRESSION WAVES AND SHOCK FORMATION 73

up(t)

t1 t2 t3

x

P

Figure 5.8: Schematic of wave steepening, multivalued solution development, and shock formation(dashed line) due to the evolution of a compression wave in a fluid with Γ > 0.

evolution of the pressure profile shown in Figure 5.8 suggests the terminology wave steepening todescribe the evolution of a smooth compression wave to a profile containing a shock. The profilewill have a vertical slope

∂P

∂x=∞ (5.97)

just prior to shock formation. In the absence of viscous effects and scattering, compression wavesin gases will invariably steepen to form shock waves if a sufficiently long propagation distance isavailable. Viscous effects and atmospheric turbulence can have a dramatic effect on compressionwaves propagating in the atmosphere and a well-defined shock is not always the outcome.

It is clear from examining Figure 5.9 that there is a particular time that the shock forms - itdoes not form immediately when the piston begins to move but at a later time and some distancein front of the piston. The exact location of the shock formation depends on the piston trajectoryX(t) as well as the fluid properties. The time and location of shock formation can be predictedby examing the geometry of the C+ characteristics that originate at the piston and determiningwhen and where adjacent characteristics first cross. This computation is carried out here using theperfect gas model but can be readily extended to the case of an aribitrary fluid.

Refering to Figure 5.10, there is a characteristic C+1 separating the uniform, undisturbed region

to the right from the compression wave located to the left. All of the C− characteristics orginate inthe undisturbed region, a representative characteristic is shown intersecting the piston X(t) at thepoint where the characteristic C+

2 originates. All of the C− characteristics have the same Riemanninvariant

Q =2

γ − 1a1 (5.98)

so that this situation is a simple wave as long as the compression is smooth, i.e., before the shockforms. The values of velocity and sound speed (for an ideal gas) are determined from the invariants


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

x (distance)

t(tim

e)

Figure 5.9: Schematic of piston trajectoryX(t) and associatedC+ characteristics illustrating shockformation due to crossing of adjacent characteristics.

Q and P as discussed previously

u =1

2(P −Q) (5.99)

a =γ − 1

4(P +Q) (5.100)

so that the slope of the C+ characteristic will be

dxdt

= u+ a (5.101)

=γ + 1

4P − 3− γ

4Q (5.102)

In the smooth part of the compression, all C− characteristics have the same value (5.98)of Q, sothat the slope of a given C+ characteristic with a given value of P will be constant. Therefore theC+ characteristics emerging from the piston are straight lines until crossing occurs.

The characteristic C+2 has a slope that is greater than that of C+

1 due to the increase of soundspeed and velocity associated with the compression and motion of the fluid created by the move-ment of the piston. From the previous discussion on the right-facing wave (Section 5.2.1) we canapply the method of characteristics at the piston surface and use u = X to compute the rate ofchange of pressure at the piston

dP

dt= ρaX (5.103)

5.4. COMPRESSION WAVES AND SHOCK FORMATION 75

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

x (distance)

X(t)

(xp, tp)

(x, t)

C−

C+1

C+2

t(tim

e) O

Figure 5.10: Representative characteristics originating at the piston and in the uniform region.

as well as explicit relationships between piston speed and thermodynamic properties

a

a1

= 1 +γ − 1

2

X

a1

(5.104)

and because the flow is still smooth and homoentropic

ρ

ρ1

=

(1 +

γ − 1

2

X

a1

)2/(γ−1)

(5.105)

P

P1

=

(1 +

γ − 1

2

X

a1

)2γ/(γ−1)

(5.106)

The values of velocity and sound speed on a given C+2 are constant and the same as those values

at the piston so that the characteristic slope is(dx

dt

)C+

= u+ a = up + ap . (5.107)

As discussed previously, the sound speed at the piston ap can be found be evaluating the invariantQ on the C− characteristic at the point x= X(tp) where it reaches the piston

Q =2

γ − 1a1 =

2

γ − 1ap − up (5.108)


and the fluid velocity at the piston is up = X so that the slope of C+2 is(

dx

dt

)C+

= a1 +γ + 1

2X(tp) (5.109)

This demonstrates explicitly how the slope of the C+ characteristics increase directly in proportionto the increase in piston speed due to the change in both sound speed and flow velocity. This ismanifested as a decrease in the co-slope (dt/dx) in the x-t diagram, leading ultimately to thecrossing of C+ characteristics that originate from the piston at different times.

A point (t, x) on the characteristic C+2 can be located by geometric construction

x−X(tp)

t− tp= a1 +

γ + 1

2X(tp) (5.110)

Given a piston trajectory X(t), this expression determines the time t when the C+ characteristicoriginating from the piston at time tp reaches the location x. Rearranging (5.110), we have

t = tp +x−X(tp)

a1 + γ+12X(tp)

(5.111)

An example of this function is shown in Figure 5.11 for a cubic piston trajectory, X ∼ t3, and aselection of locations x. The actual piston trajectory and some representative C+ characteristicswere shown earlier in Fig. 5.9. The initial sound speed a1 has been set to a value of unity for thepurposes of making these plots.

tp (time characteristic left piston)

t(tim

e)

0.00.0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1.0

1.0

Figure 5.11: Relationship between the time t that a characteristic crosses a point in the region infront of the piston as a function of the time tp that the characteristic left the piston. The contoursrepresent different locations x in space.

5.5. SHOCK FITTING 77

The interpretation of Figure 5.11 is that each of the contours intercepts the t axis at the locationx in space labeling that contour. The line t = tp at 45◦ represents the trajectory of the pistonitself. For example, the 6th contour from the top intercepts the t axis at 0.35. This means that thiscontour corresponds to x = 0.35. Tracing this contour up and to the right, we see that the time tmonotonically increases with increasing tp. This means that characteristics that leave the pistonlater in time will arrive at the location x = 0.35 later in time. Ultimately the piston itself reachesthis location at a time of t = tp = 0.7 where this contour intercepts the piston location. We noticethat contours t(tp) are single valued until x > 0.63. For example, the contour for x = 0.75 (2ndfrom top) is clearly multiple-valued - a line of contant t will intercept the contour several timesfor some values 0.80 < t < 0.85. This indicates that C+ characteristics have crossed and a shockmust have formed by this point in space.

The point of incipient shock formation must be located where adjacent characteristics cross.This can only happen when characteristics leaving the piston at a slightly different times tp reachthe same location at a given time t

t(tp) = t(tp + dtp) = t(tp) +dt

dtpdtp +O(dtp)

2 (5.112)

For this to be true as dtp → 0, then we must have

dt

dtp

)tc

= 0 (5.113)

This implicitly defines the time tc of adjacent characteristic crossing. This time will in general bemultivalued, as can be deduced from the envelop of crossings that can be seen in Figure 5.9. Forthe perfect gas case, the crossing time can be computed explicitly to be

tc = tp +2

(γ + 1)X(tp)

[a1 +

γ − 1

2X(tp)

](5.114)

A plot of the crossing time tc as a function of the time tp is shown in Figure 5.12. As shown, thereis a minimum value of tc at a time t∗p ≈ 0.358. This implies a time of first shock formation of t∗c ≈0.78 at a location x∗c ≈ = 0.65. The time t∗p can be determined precisely by using the condition

dtcdtp

)t∗p

= 0 (5.115)

For the perfect gas case, this requires solving the resulting implicit equation

γ(X(t∗p)

)2

=...X(t∗p)

[a1 +

γ − 1

2X(t∗p)

](5.116)

for t∗p and substituting into (5.114) to obtain t∗c .

5.5 Shock FittingOnce the characteristics cross, Figure 5.13, a shock must be inserted to maintain a single-valuedsolution, as shown in Figure 5.14. The location and strength of the shock is determined by requiring


tp (time characteristic left piston)

t(tim

e)

0.00.0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1.0

1.0

Figure 5.12: Locus of minimum time t of adjacent characteristic crossings as a function of the timetp that the characteristic left the piston. The minimum of this function corresponds to the shockformation time.

that upstream and downstream states implied by the characteristics also satisfy the jump conditions.In general, this requires an iterative solution since the insertion of a shock in the flowfield changesthe value of the invariant on the characteristic that crosses the shock (labeled C− in Figure 5.13).

In the special case of a weak shock, it is possible to determine the shock location explicitlywithout any iteration. This is a consequence of the very small jump in the invariant on the charac-teristic that crosses a weak shock. In fact, we will show that the jump in the invariant is third orderin the wave strength. This result seems reasonable since we know that the Hugoniot and isentropeagree up to third order in the shock strength.

Consider the situation shown in Figure 5.13. This is a generic configuration since the shockslope is always intermediate to the upstream and downstream characteristic slopes due to thecausality conditions:

Us >a1 + u1 (5.117)Us <a2 + u2 (5.118)

For this situation, the jump conditions for the shock are most useful when written in terms of thevolume and pressure jumps. Conservation of mass across the shock can be expressed

[u] = − [v]

v1

w1 (5.119)


C +1

C +2

C1−

O

shock

t

x

C2−

Figure 5.13: Shock S fitted into flowfield in order to reconcile inconsistent values on characteristicsC+

1 and C+2 that intersect at point O with characteristic C−.

Us

1

2

x

P

Figure 5.14: Schematic of pressure spatial profile after a shock has been fitted into flowfield.

and the conservation of momentum is

[P ]

ρ1a21

= −M2s

[v]

v1

(5.120)


Since we are considering weak waves, we can use the isentrope v(P ) instead of the full energyequation. Since we are interested in an analytical solution, the Taylor series solution up through2nd order in the pressure jump is the appropriate approximation.

[v]

v1

= −Π + ΓΠ2 + . . . (5.121)

where we have use the notation

Π =[P ]

ρ1a21

(5.122)

Substituting the isentrope approximation into the momentum equation, we have

M2s ≈

1

1− ΓΠ(5.123)

linearizing, we have

Ms ≈ 1 +Γ

2Π (5.124)

Using the definition of the shock Mach number

Ms =Us − u1

a1

(5.125)

we have

Us ≈ u1 + a1 +Γ

2

[P ]

ρ1a1

(5.126)

Now recall that for a simple right-facing wave, the change in pressure is related to the change incharacteristic slope by

d (u+ a) = ΓdP

ρa(5.127)

which can be integrated for small changes in P, consistent with weak shock waves, to obtain thechange in the C+ characteristic slope across the shock

[u+ a] = Γ[P ]

ρa(5.128)

Using this relationship, the shock velocity can be written as

Us ≈1

2[(u1 + a1) + (u2 + a2)] (5.129)

This is the key result: For weak waves, the shock travels at the average of the upstream anddownstream chracteristic speeds. Unlike the previous results on weak waves, the key result (5.129)is only correct up to 2nd order.


An Integral Approach to Shock Fitting Another way to look at shock fitting is to work directlywith the approximate invariant on the characteristic that crosses the shock. In example shown inFig. 5.13, this is C− and we have

Q1 ≈ Q2 (5.130)

or

F1 − u1 ≈ F2 − u2 (5.131)

Using the definition of F , we can write this as

[u] =

∫ 2

1

dP

ρa(5.132)

Changing the variable of integration to volume and rewriting the jump in velocity using (5.119),we have

0 ≈∫ 2

1

[ρa− ρ1w1] dv (5.133)

This is our approximate integral solution to the problem of fitting a weak shock. Interpretingintegration as finding the area under a specified curve, we can view the shock fitting condition asrequiring equal areas of ρa above and below the line ρ1w1. This is one example of an equal arearule that can be derived for shock fitting.

5.5.1 Jump of InvariantThe invariant that crosses the shock jumps due to the small but nonzero change in entropy across ashock wave. The jump can be evaluated by using

[Q] = Q2 −Q1 =

∫ 2

1

dP

ρa− [u] (5.134)

In general, it will be necessary to use the jump conditions and evaluate the jump numerically. Forweak waves, it is possible to compute the first term in a series expansion in shock strength. Inorder to do this, it will be necessary to compute the expansions for the Hugoniot and the isentropethrough third order.

First we will evaluate the integral in (5.134). Using the definition of sound speed, this can berewritten ∫ 2

1

dP

ρa=

∫ √−(∂v

∂P

)s

dP (5.135)

and √−(∂v

∂P

)s

=v

a(5.136)


expanding in powers of ∆P , we have√−(∂v

∂P

)s

=v1

a1

(1− 2ΓΠ +

K

2Π2 + . . .

)1/2

(5.137)

where

K = −a6

v4

(∂3v

∂P 3

)s

(5.138)

Computing the square root of the series expansion (see Appendix C), we have√−(∂v

∂P

)s

=v1

a1

(1− ΓΠ + (

K

4− Γ

2)Π2 + . . .

)(5.139)

and integrating term by term, we find that∫ 2

1

dP

ρa= a1

(Π− Γ

2Π2 +

1

6(K

2− Γ2)Π3 + . . .

)(5.140)

Second, we will determine the jump in velocity using the series expansion of the Hugoniot.One form of the Rayleigh line relationship is (exactly)

[u] =√−[P ][v] (5.141)

which can be written as

[u] = a1

√Π

(− [v]

v1

)(5.142)

The volume jump on the Hugoniot is

[v]

v1

= −Π + ΓΠ2 +1

6(GΓ−K)Π3 + . . . (5.143)

Computing the square root of this series, we find

[u]

a1

= Π

[1− Γ

2Π− 1

4

(GΓ−K

3+

Γ2

2

)Π2 + . . .

](5.144)

Finally, the first and second results can be combined to obtain

[Q] =a1Γ

12

[G− Γ

2

]Π3 + . . . (5.145)

As promised, the jump is third order in the shock strength for a weak wave. Interestingly enough,although we expanded both series through third order, the coefficient K involving the third deriva-tive of volume with respect to pressure does not appear in the final result.

Chapter 6

Wave Interactions

A number of situations in one-dimensional unsteady flow can be treated by considering compositesolutions that consist of uniform flow regions separated by simple waves, which can be shocks,expansions or in some situations, steady flow through area changes. For perfect gases, analyticalresults can be obtained for certain cases; for real gases or fluids, graphical or iterative solutions canbe readily obtained. Examples of situations that can be treated include impact, reflection of wavesfrom interfaces, resolution of discontinuities in pressure and velocity and the idealized analysis ofshock tubes, shock tunnels and expansion tubes.

6.1 Pressure-Velocity Relations

The jump conditions for shock waves and the similarity solutions for isentropic expansions can beexpressed as a relationship between the pressure change ∆P and the velocity change ∆u acrossthe wave. The results can be plotted on a pressure-velocity or P -u diagram to facilitate graphicalillustration of various situations. The locus of solutions P (u) is a continuous curve connectingstates created by shock and expansion waves of the moving in the same direction into the sameuniform state, see Fig. 6.1. Shock waves and expansions moving to the right are indicated by

→S

(shock) and→E (expansion); to the left by

←S (shock) and

←E (expansion).

6.1.1 Perfect Gas Relations

The shock jump conditions (3.8) and (3.9) can be combined to obtain the following relationshipbetween the absolute change in velocity across the shock ∆u = |[u]| and the jump in pressure acrossthe shock ∆P = |[P ]|

∆P

P1

= γ∆u

a1

γ + 1

4

∆u

a1

+

√(γ + 1

4

∆u

a1

)2

+ 1

, (6.1)

83

84 CHAPTER 6. WAVE INTERACTIONS

0

5

10

15

20

-4 -2 0 2 4

P/P1

u]/a1

S

E E

S

expansion

Figure 6.1: Composite pressure-velocity diagram for right and left facing shock and expansionwaves for a perfect gas, γ = 1.4. Space-time diagrams are shown illustrating each case of wavepropagation into a common stationary state 1.

or equivalently

∆u

a1

=

1

γ

∆P

P1√1 +

γ + 1

2γ

∆P

P1

. (6.2)

The relationship between pressure and velocity change across an expansion wave can be computedfrom the Riemann invariants (5.90) and (5.94) and the perfect gas isentropic relationships (5.93)

∆P

P1

=

[1− γ − 1

2

∆u

a1

]2γ/(γ−1)

− 1 . (6.3)

Note that the maximum value of the velocity change is achieved with expansion to P = 0 (gasescannot sustain tension) ,

∆umax =2

γ − 1a1 . (6.4)

6.1. PRESSURE-VELOCITY RELATIONS 85

For small changes in pressure and velocity across the waves, both shock and expansion waverelationships reduce to the acoustic relation

lim∆u→0

[P ] = ±ρ1a1[u] , (6.5)

with the + sign referring to right-facing waves and the − sign referring to the left-facing waves;and ρa is the acoustic impedance. Expansion of the expressions (6.2) and (6.3) confirms that theexpansion and shock wave P (u) curves have a common slope

lim∆u→0

dP

du= ±ρ1a1 , (6.6)

and are continuous at the point P = P1, [u] = 0, as shown in Fig. 6.1.

0

10

20

30

40

0.0 1.0 2.0 3.0 4.0

P/P1

u]/a1

S12

S23

1

2

Figure 6.2: Pressure-velocity diagram showing the P (u) relationships for two right-facing shockwaves for a perfect gas, γ = 1.4. The inital slopes (acoustic impedance) of the two wave curves areshown as dashed lines.

The slope of the P (u) curve increases monotonically with increasing shock strength; differen-tiating (6.2), we obtain

dP

du= ρ1a1

[1 +

γ + 1

2γ

∆P

P1

]3/2

1 +γ + 1

4γ

∆P

P1

≥ ρ1a1 (6.7)

so that the slope of the P (u) relation for finite strength waves is always larger than the acousticimpedance of state 1. On the other hand, the acoustic impedance ρa at other states on the wave

86 CHAPTER 6. WAVE INTERACTIONS

curve is larger than the slope of the P (u) relation

dP

du= ρa

1 +γ + 1

2γ

∆P

P1

1 +γ + 1

4γ

∆P

P1

√√√√√√√1 +γ − 1

2γ

∆P

P1

1 +∆P

P1

≤ ρa (6.8)

for 1 < γ < 2; equality occurs for [u] = 0. The implication of this is that a shock wave→S23

overtaking a shock wave→S12 will have a wave curve that originates from state 2 with a larger slope

and lies above the curve originating from state 1. This situation and the initial slopes (acousticimpedance) of the two wave curves are shown in Fig. 6.2.

Bibliography

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S. Glasstone and P.J. Dolan. The Effects of Nuclear Weapons. United States Department of Defenseand Department of Energy, third edition, 1977. UF767 .E33 1977.

R. W. Griffiths, R. J. Sandeman, and H. G. Hornung. The stability of shock waves in ionizing anddissociating gases. Journal Physics D: Applied Physics, 8:1681–1691, 1975. 42, 43

L. V. Gurvich, I. V. Veyts, and C. B. Alcock. Thermodynamic Properties of Individual Substances,volume I. Hemisphere Pubishing, 4 edition, 1989. Parts 1 and 2. 111

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R.J. Kee, M.E. Coltrin, and P. Glarborg. Chemically Reacting Flow. John Wiler & Sons, 2003.111

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John H. S. Lee. The Detonation Phenomenon. Cambridge University Press, 2008.

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90 BIBLIOGRAPHY

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Appendix A

Texts and References

The topic of compressible flow and shock waves, particularly in ideal gases, has been covered bymany authors from various points of view, mathematical, physical, and chemical. I haven’t triedto catalog all of the possible texts that touch on these subjects but I have just listed a selection ofthose that students might find useful. Unfortunately, specialist textbooks and monographs have ashort lifetime in print. Some classic texts have been reprinted in recent years but many fine bookshave gone out of print and are difficult to find.

A.1 Elementary texts on Compressible FlowMany textbooks on fluid mechanics include a section on compressible flow that treats shock wavesin perfect gases at an elementary level. Some representative texts include:

J. D. Anderson. Modern Compressible Flow: with a Historical Perspective. McGrawHill, 2003.QA911 .A6 2003.

M. Saad. Compressible Fluid Flow. Prentice Hall, 1993. QA911 .S13 1993 (Out of print).

A.2 Graduate Level TextbooksThe classics that focus on perfect gases are:

H. W. Liepmann and A. Roshko. Elements of Gasynamics. Wiley, New York, 1957. (Available asa Dover Paperback)

A. H. Shapiro. The Dynamics and Thermodynamics of Compressible Fluid Flow. Ronald PressCo., NY, 1953-54. QA911 .S497 (Out of print).

A somewhat broader perspective is given in:

P. A. Thompson. Compressible Fluid Dynamics. McGraw-Hill, New York, 1972. Out of printfrom McGraw-Hill, but reprinted privately and available from the Rensselaer Union Bookstore,RPI, Troy, NY. TEL 518 276 6555.

A little appreciated text that covers the same ground as Liepmann and Roshko and much of Thomp-son but is much more concise is:

91

92 APPENDIX A. TEXTS AND REFERENCES

E. Becker. Gas Dynamics. Academic Press, 1968. QC168 .B4313 (Out of print).

The Landau-Lifshitz text on Fluid Mechanics has a very brief but useful section on compressibleflow and shock waves.

L. D. Landau and E. M. Lifshitz. Fluid Mechanics. Wiley, 1959. QA901 L283.

A concise but broad introduction to waves in fluid that covers similar subjects as Whitham butfrom a less mathematical viewpoint is

J. Lighthill. Waves in Fluids. Cambridge University Press, 1978.

A.3 Specialist MonographsThe most comprehensive discussion of the physics and chemistry of shock waves in gases andsolids remains the masterpiece:

Ya. B. Zel’dovich and Yu. P. Raizer. Physics of Shock Waves and High-Temperature Hydrody-namic Phenomena, volume 1 and 2. Wiley, NY, 1966. Thankfully, now available from Dover inpaperback.

For the non-steady flow of gases, a fairly complete reference is:I.I. Glass and J. P. Sislian. Nonstationary flows and shock waves. Oxford University Press, 1994.QC168.85.S45 G55 1994.

The fundamental equations of high-temperature reacting gas dynamics and application to shockwaves are covered by

V. G. Vincenti and C. H. Kruger. Introduction to Physical Gas Dynamics. Wiley, 1965. QC168.V55 (Reprinted by R.E. Kreiger).

J.F. Clarke and M. McChesney. The Dynamics of Real Gases. Butterworths, 1964. QC168 .C56(Out of print).

A.4 Mathematical TreatmentsThe classical mathematical reference book on shock waves is

R. Courant and K. O. Friedrichs. Supersonic Flow and Shock Waves. Interscience, 1948. QA930.C6.

A wide-ranging treatment of waves and nonlinear effects is given by

G. B. Whitham. Linear and Nonlinear Waves. Wiley Interscience, 1974. QA927 .W48.

The numerous Russian contributions to blast wave and unsteady flow theory are discussed by

V. P. Korobeinikov. Problems of Point-Blast Theory. AIP, 1991. Translation of 1985 Russianpublication: Zadachi teorii tochechnogo vzryva.

Hard to find, difficult to read but full of interesting analytical solutions is:

A.5. SHOCK TUBES 93

K. I. Stanyukovich. Unsteady motion of continuous media. Pergamon Press, 1960. QC168 .S8131960.

The classic discussion about similarity solutions for blast waves is given in:

L.I. Sedov. Similarity and dimensional methods in mechanics. Academic Press, 1959.

A.5 Shock Tubes

The technology of shock tubes has not changed much since the 1960s although diagnostic methodshave substantially advanced due to laser, light sensor, and electronic imaging developments. Twoclassic texts on shock tubes are:

J. N. Bradley. Shock Waves in Chemistry and Physics. Wiley, 1962. QC168 B64.

A. G. Gaydon and I. R. Hurle. The shock tube in high-temperature chemical physics. Chapmanand Hall, 1963.

A comprehensive German-language reference is:

H. Oertel. Stossrohre: Theorie, Praxis, Anwendungen. Springer, 1966. QC168 .03.

A.6 Handbooks

There is a three volume handbook on shock waves that has entries on many topics.

Handbook of Shock Waves. Volumes 1, 2 and 3.. Editors: Gabi Ben-Dor, Ozer Igra and TovElperin. Elsevier, 2001.

A.7 Symposia

International Symposium on Shock Waves (ISSW). Held on odd-numbered years. Primarily cov-ers studies in gases and shock tube technology. Proceedings are published with short (6 page)summaries of research.

APS Shock Compression Science Meetings. Held on odd years, proceedings with short papers arepublished by AIP.

International Symposium on Detonation. Held every fourth year, the most recent was in 2004.Sponsored by the DoD and DOE Laboratories in USA. Proceedings are published with long formatpapers.

International Colloquium on the Dynamics of Explosions and Reactive Systems (ICDERS) Heldon odd years.

Symposium on High Dynamic Pressure.


A.8 JournalsShock wave discussions are found spread over the scientific periodicals with a concentration injournals specializing in fluid mechanics. Some major titles include: Physics Fluids, Journal ofFluid Mechanics, Experiments in Fluids, Journal of Applied Physics, AIAA Journal.

Springer publishes a speciality journal Shock Waves which contains both original contributions andreview articles.

The Russian Academy of Science continues to publish Combustion, Explosion and Shock Wavesas an outlet mainly for scientists in the FSU.

A.9 DetonationA discussion of high explosive detonation from a practicing engineer’s perspective is given by:

P. W. Cooper. Explosives Engineering. VCH, 1996.

More in-depth discussions are given in the compilation of:

J. A. Zukas and W.P. Walters, editors. Explosive Effects and Applications. High Pressure ShockCompression of Condensed Matter. Springer, 1995.

The classic reference on detonation is:

Ya. B. Zel’dovich and A. S. Kompaneets. Theory of Detonation. Academic Press, NY, 1960.English translation of original Russian. Out of print and in many ways out of date.

A more up to date theoretical treatment is given by:

W. Fickett and W. C. Davis. Detonation. University of California Press, Berkerely, CA, 1979.Now available as a Dover paperback.

A extensive reference that emphasizes the phenomenology for gaseous detonations isJohn H.S. Lee The Detonation Phenomenon. Cambridge University Press, 2008.

Articles on detonations can be found in the two Springer monographs series described below.

A.10 Shock Waves in SolidsThere is a series Springer Series in Shock Wave and High Pressure Phenomena which are variouslytitled, the volumes published as High Pressure Shock Compression of Solids are edited volumeswith contributions that cover many different topics in shock compression.

1. L.L. Altgilbers, M.D.J. Brown, I. Grishnaev, B.M. Novac, I.R. Smith, I. Tkach, and Y. Tkach:Magnetocumulative Generators (2000)

2. T. Antoun, D.R. Curran, G.I. Kanel, S.V. Razorenov, and A.V. Utkin: Spall Fracture (2003)

3. J. Asay and M. Shahinpoor (Eds.): High-Pressure Shock Compression of Solids (1993)

A.10. SHOCK WAVES IN SOLIDS 95

4. S.S. Batsanov: Effects of Explosion on Materials:Modification and Synthesis Under High-Pressure Shock Compression (1994)

5. R. Cheret: Detonation of Condensed Explosives (1993)

6. L. Davison,D.Grady, and M. Shahinpoor (Eds.):High-Pressure Shock Compression of SolidsII - Dynamic Fracture and Fragmentation (1996)

7. L. Davison, Y. Horie, and T. Sekine (Eds.): High-Pressure Shock Compression of Solids V -Shock Chemistry with Applications to Meteorite Impacts (2003)

8. L. Davison, Y. Horie, and M. Shahinpoor (Eds.):High-Pressure Shock Compression of SolidsIV - Response of Highly Porous Solids to Shock Loading (1997)

9. L. Davison and M. Shahinpoor (Eds.): High-Pressure Shock Compression of Solids III -Shock Wave and High Pressure Phenomena (1998)

10. A.N. Dremin: Toward Detonation Theory (1999)

11. Y.Horie, L. Davison, and N.N. Thadhani (Eds.):High-Pressure Shock Compression of SolidsVI - Old Paradigms and New Challenges (2003)

12. G. Ben-Dor Shock Wave Reflection Phenomena (2007)

13. R. Graham: Solids Under High-Pressure Shock Compression (1993)

14. J.N. Johnson and R. Cheret (Eds.): Classic Papers in Shock Compression Science (1998)

15. V. I. Kedrenskii: Hydrodynamics of Explosion.

16. V.F. Nesterenko: Dynamics of Heterogeneous Materials (2004)

17. M. Suceska: Test Methods of Explosives (1993)

18. J.A. Zukas andW.P.Walters (Eds.): Explosive Effects and Applications (1998)

19. G.I. Kanel, S. V. Razorenov, and V.E. Fortov: Shock-Wave Phenomena and the Properties ofCondensed Matter (2004)

20. V.E. Fortov, L.V. Altshuler, R.F. Trunin, and A.I. Funtikov: High-Pressure Shock Compres-sion of Solids VII - Shock Waves and Extreme States of Matter (2004)

21. L.C. Chhabildas, L. Davison, Y. Horie (Eds.): High-Pressure Shock Compression of SolidsVIII - The Science of High Velocity Impact (2005)

22. L. Davidson. Fundamentals of Shock Wave Propagation in Solids. (2008)

23. C. Needham. Blast Waves. (2010)

24. D. Grady. The Fragmentation of Rings and Shells (2006)


Springer publishes a set of volumes Shock Wave Science and Technology Reference Librarywhich have encyclopedia style articles reviewing various topics, mainly concerned with shock ordetonation waves in solids. The available volumes include:

Vol. 1 Multiphase Flows (2007)Vol. 2 Solids I (2007)Vol. 3 Solids II (2009)Vol. 4 Heterogeneous Detonation (2009)Vol. 5 Nonshock Initiation of Explosives (2010)Vol. 6 Detonation Dynamics (2012)

The results of decades of work on measuring Us-up curves by researchers at Los Alamos arecompiled as tabulated data and plots in:

S. P. Marsh, editor. LASL Shock Hugoniot Data. Univ. Calif. Press, 1980. QC307 .L3 1980.

A recent monograph (although the methods and results are primarily from 1960-1990) on testmethods for shock waves in solids and results for selected materials is:

W.M. Isbell. Shock waves : measuring the dynamic response of materials. Imperial College Press,2005. TA418.32 .I83 2005.

The classic introduction to elastic wave propagation in solids is:

H. Kolsky. Stress Waves in Solids. Dover Publications, Inc., 1963.

An comprehensive introduction to shock effects on materials with an emphasis on strain rate ef-fects, plasticity and fragmentation is:

M. A. Meyers. Dynamic behavior of materials. John Wiley & Sons, 1994.

A modern textbook covering both experimental and analytical methods in a style that is conduciveto self-study is:

J. W. Forbes. Shock Wave Compression of Condensed Matter: A Primer. Springer, 2012.

A.11 ExplosionsThere are several monographs specifically discussing the problems of explosions in air or waterand the resulting shock waves.

W. E. Baker. Explosions in air. U. Texas Press, 1973. QC168 .B33 1973.

W. E. Baker, P. A. Cox, P. S. Westine, J. J. Kulesz, and R. A. Strehlow. Explosion Hazards andEvaluation. Elsevier, 1983. Out of print. The sections on blast waves and structural response arestill very useful.

R. H. Cole. Underwater Explosions. Princeton Univ. Press, 1948. QC151 .C6 1948.

S. Glasstone and P.J. Dolan. The Effects of Nuclear Weapons. Third edition, 1977. UF767 .E331977.

P.D. Smith and J.G. Hetherington. Blast and Ballistic Loading of Structures. Butterworth Heine-mann, 1994.

A.12. NUMERICAL SIMULATION OF FLOWS WITH SHOCK WAVES 97

A.12 Numerical Simulation of Flows with Shock WavesThere are a number of text books on numerical simulation methods that describe techniques forflows with shock waves. Randy Leveque has written a good introduction and has made available afree software package CLAWPACK that is extremely useful for education and research.

R.J. Leveque. Finite volume methods for hyperbolic problems. Cambridge University Press, 2002.

A.13 AcousticsAcoustics is linearized compressible flow and many results from acoustics are useful in interpretingthe results of experiments and computations with weak shock waves. The most comprehensive(and biblical in length) theoretical treatment is Morse and Ingard, but in the style of mathematicalphysics. Pierce is more accessible but also rigorous and the most easy to read text is Kinsler et al.

P. M. Morse and K. Uno Ingard. Theoretical Acoustics. McGraw-Hill, 1968.

A. D. Pierce. Acoustics. An Introduction to Its Physical Principles and Applications. McGraw-Hill, 1981.

L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders. Fundamentals of Acoustics. JohnWiley and Sons, 3rd edition, 1982.

http://www.amath.washington.edu/~claw/


Appendix B

Famous Numbers

Fundamental Physical Constants

co speed of light in a vacuum 2.998×108 m/sεo permittivity of the vacuum 8.854×10−12 C2/kg-mµo permeability of the vacuum 4π×10−7 H/mh Planck constant 6.626×10−34 J-sk Boltzmann constant 1.381×10−23 J/KNo Avogadro number 6.022×1023 molecules/mole charge on electron 1.602×10−19 Camu atomic mass unit 1.661×10−27 kgme electron mass 9.109×10−31 kgmp proton mass 1.673×10−27 kgG universal gravitational constant 6.673×10−11 m3/kg-s2

σ Stefan-Boltzmann constant 5.670×10−8 W/m2K4

Astronautics

go gravitational acceleration at earth’s surface 9.807 m/s2

RE radius of earth 6378 kmME mass of earth 5.976×1024 kgMS mass of sun 1.99×1030 kgau mean earth-sun distance 1.496×108 km

mass of moon 7.349×1022 kgmean earth-moon distance 3.844×105 km

99

100 APPENDIX B. FAMOUS NUMBERS

Gases

R Universal gas constant 8314.5 J/kmol-K8.3145 J/mol-K82.06 cm3-atm/mol-K

1.9872 cal/mol-Kmechanical equivalent of heat 4.186 J/calvolume of 1 kmol at 273.15 K and 1 atm 22.41 m3

number of molecules at 298.15 K and 1 atm 2.46×1025 m3

collision frequency at 273.15 K and 1 atm 4.3×109 s−1

mean free path in N2 at 273.15 K and 1 atm 74 nm

Consistent with the 1998 CODATA adjustment of the funadamental physical constants.For the most recent values, see NIST Reference on Units and Uncertainty.

http://physics.nist.gov/cuu/Reference/contents.html

101

Our Atmosphere

composition (mol fractions)0.7809 N2

0.2095 O2

0.0093 Ar0.0003 CO2

Sea level

P pressure 1.01325×105 Paρ density 1.225 kg/m3

T temperature 288.15 Kc sound speed 340.29 m/sR gas constant 287.05 m2/s2-KW molar mass 28.96 kg/kmolµ viscosity (absolute) 1.79×10−5 kg/m-sk thermal conductivity 2.54×10−3 W/m-Kcp heat capacity 1.0 kJ/kg-K

30 kft

P pressure 3.014×104 Paρ density 0.458 kg/m3

T temperature 228.7 Kc sound speed 303.2 m/s

Based on the U.S. Standard Atmosphere, Minzer et al. (1975).

Unit Conversions

1 m ≡ 3.28 ft0.3048 ft ≡ 1 m1 lb (force) ≡ 4.452 N1 lb (mass) ≡ 0.454 kg1 btu ≡ 1055 J1 hp ≡ 745.7 W1 hp ≡ 550 ft-lbf /s1 mile (land) ≡ 1.609 km1 mph ≡ 0.447 m/s1 mile (nautical) ≡ 1.852 km

For on-line units conversions, see NIST Links.

http://physics.nist.gov/cuu/Reference/unitconversions.html

102 APPENDIX B. FAMOUS NUMBERS

Appendix C

Essential Thermodynamic Relationships

C.1 Thermodynamic potentials and fundamental relations

energy e(s, v)

de = T ds− P dv (C.1)enthalpy h(s, P ) = e+ Pv

dh = T ds+ v dP (C.2)Helmholtz f(T, v) = e− Ts

df = −s dT − P dv (C.3)Gibbs g(T, P ) = e− Ts+ Pv

dg = −s dT + v dP (C.4)

C.2 Maxwell relations

(∂T

∂v

)s

= −(∂P

∂s

)v

(C.5)(∂T

∂P

)s

=

(∂v

∂s

)P

(C.6)(∂s

∂v

)T

=

(∂P

∂T

)v

(C.7)(∂s

∂P

)T

= −(∂v

∂T

)P

(C.8)

C.3 Calculus identities

F (x, y, . . . ) dF =

(∂F

∂x

)y,z,...

dx+

(∂F

∂y

)x,z,...

dy + . . . (C.9)

103

104 APPENDIX C. ESSENTIAL THERMODYNAMIC RELATIONSHIPS

(∂x

∂y

)f

= −

(∂f

∂y

)x(

∂f

∂x

)y

(C.10)

(∂x

∂f

)y

=1(∂f

∂x

)y

(C.11)

C.4 Power seriesTaylor’s series (single variable)

f(x+ ∆x) = f(x) +dfdx

∆x+1

2

d2f

dx2(∆x)2 + · · ·+ 1

n!

dnfdxn

(∆x)n + . . . (C.12)

Taylor’s series (two variables)

f(x+ ∆x, y + ∆y) = f(x, y) +∂f

∂x∆x+

∂f

∂y∆y

+1

2!

(∂2f

∂x2(∆x)2 + 2

∂2f

∂x∂y(∆x)(∆y) +

∂2f

∂y2(∆y)2

)+

1

3!

(∂3f

∂x3(∆x)3 + 3

∂3f

∂x2∂y(∆x)2(∆y) + 3

∂3f

∂x∂y2(∆x)(∆y)2 +

∂3f

∂y3(∆y)3

)+ . . . (C.13)

Binomial series

(1+x)n = 1+nx+n(n− 1)

2!x2 +

n(n− 1)(n− 2)

3!x3 + . . .+

n!

(n−m)!m!xm+ . . .+xn (C.14)

Consider a series of the form

y = ax+ bx2 + cx3 + dx4 + . . . (C.15)

The Reversion of the series is

x = Ay +By2 + Cy3 +Dy4 + . . . (C.16)

A =1

a(C.17)

B = −b2

a(C.18)

C =1

a5

(2b2 − ac

)(C.19)

D =1

a7

(5abc− a2d− 5b3

)(C.20)

C.4. POWER SERIES 105

Consider a series of the form

y = a+ bx+ cx2 + dx3 + . . . (C.21)

The square of the series is

y2 = A+Bx+ Cx2 +Dx3 + . . . (C.22)A = a2 (C.23)B = 2ab (C.24)C = b2 + 2ac (C.25)D = 2 (ad+ bc) (C.26)

The square root of the series is

√y = A+Bx+ Cx2 +Dx3 + . . . (C.27)

A = a1/2 (C.28)

B = a1/2 b

2a(C.29)

C = a1/2

(c

2a− b2

8a2

)(C.30)

D = a1/2

(d

2a− bd

4a2+

b3

16a3

)(C.31)

The reciprocal of the series is

1

y= A+Bx+ Cx2 +Dx3 + . . . (C.32)

A =1

a(C.33)

B = −1

a

b

a(C.34)

C =1

a

(b2

a2− c

a

)(C.35)

D =1

a

(2bc

a2− d

a− b3

a3

)(C.36)


C.5 Various defined quantities

specific heat at constant volume cv ≡(∂e

∂T

)v

= T

(∂s

∂T

)v

(C.37)

specific heat at constant pressure cp ≡(∂h

∂T

)P

= T

(∂s

∂T

)P

(C.38)

ratio of specific heats γ ≡ cpcv

(C.39)

coefficient of thermal expansion α ≡ 1

v

(∂v

∂T

)P

(C.40)

isothermal compressibility KT ≡ −1

v

(∂v

∂P

)T

(C.41)

isentropic compressibility Ks ≡ −1

v

(∂v

∂P

)s

=1

ρa2(C.42)

C.6 Specific heat relationships

KT = γKs or(∂P

∂v

)s

= γ

(∂P

∂v

)T

(C.43)

cp − cv = −T(∂P

∂v

)T

(∂v

∂T

)2

P

(C.44)

(∂Cv∂v

)T

= T

(∂2P

∂T 2

)v

(C.45)

(∂Cp∂P

)T

= −T(∂2v

∂T 2

)P

(C.46)

C.7 Sound Speed

sound speed squared a2 ≡(∂P

∂ρ

)s

(C.47)

sound speed squared a2 = −v2

(∂P

∂v

)s

(C.48)

sound speed squared a2 =

(∂h

∂ρ

)P

1

ρ−(∂h

∂P

)ρ

(C.49)

sound speed squared a2 =1

ρKs

=γ

ρKT

(C.50)

C.8. GRUNEISEN COEFFICIENT 107

C.8 Gruneisen Coefficient

G ≡ v

(∂P

∂e

)v

(C.51)

=vα

cvKT

(C.52)

=vα

cpKs

(C.53)

= − vT

(∂T

∂v

)s

(C.54)

=T

v

(∂P

∂s

)v

(C.55)

Relationship to entropy derivatives (∂s

∂P

)v

=v

TG(C.56)(

∂s

∂v

)P

=a2

TvG(C.57)

C.9 Thermal Pressure Coefficient(∂P

∂T

)v

=α

KT

(C.58)

C.9.1 Fundamental Derivative

Γ =a4

2v3

(∂2v

∂P 2

)s

(C.59)

=v3

2a2

(∂2P

∂v2

)s

(C.60)

=1

a

(∂ρa

∂ρ

)s

(C.61)

= 1 +a

v

(∂a

∂P

)s

(C.62)

(C.63)

C.10 Enthalpy, Energy and EntropyEnergy e(T, v)

de = CvdT +

[T

(∂P

∂T

)v

− P]

dv (C.64)


Enthalpy h(T, P )

dh = CpdT +

[v − T

(∂v

∂T

)P

]dP (C.65)

Entropy s(T, P )

ds =CpT

dT −(∂v

∂T

)P

dP (C.66)

Entropy s(T, v)

ds =CvT

dT +

(∂P

∂T

)v

dv (C.67)

C.11 Perfect Gas RelationshipsDefined by using ideal gas relationship for P (ρ, T )

P = ρRT (C.68)

R = R/W (C.69)

and a constant value of the ratio of specific heats γ, which is the single parameter needed tocompute all other properties. The fundamental derivative and Gruneisen coefficient are:

Γ =γ + 1

2(C.70)

G = γ − 1 (C.71)

Coefficient of thermal expansion

α =1

T(C.72)

Isothermal compressibility

KT =1

P(C.73)

Isentropic compressibility

Ks =1

γP(C.74)

Specific heat:

Cp =γR

γ − 1(C.75)

Cv =R

γ − 1(C.76)

C.11. PERFECT GAS RELATIONSHIPS 109

Internal energy and enthalpy:

e = CvT (C.77)h = CpT (C.78)

Entropy:

s− so = Cp ln(T/To)−R ln(P/Po) (C.79)s− so = Cv ln(T/To) +R ln(v/vo) (C.80)s− so = Cv ln(v/vo)− Cp ln(P/Po) (C.81)

Sound speed:

a =√γRT (C.82)

Relationships on an isentrope

P/Po = (a/ao)2γ/(γ−1) (C.83)

ρ/ρo = (a/ao)2/(γ−1) (C.84)

T/To = (a/ao)2 (C.85)

T/To = (P/Po)(γ−1)/γ (C.86)

ρ/ρo = (P/Po)−1/γ (C.87)


Appendix D

High temperature gas thermodynamics

The data given in the attached tables are based on the compilation of thermodynamic propertiesfor ideal gases given by Gurvich et al. (1989). The properties were computed using basic ideasof statistical mechanics (Maczek, 1998) and partition functions that are as accurate as possible(Gurvich et al., 1989, McBride and Gordon, 1992). The properties of the elemental compoundsare documented further in McBride et al. (1993) for high temperatures and for a very large numberof compounds at lower temperatures (less than 6000 K) in Chase (1985).

Print-on-paper data tabulations have been replaced by interactive web sites for the large part.The most useful of these are:

NIST Chemistry Webbook

NASA Thermodynamic Database and Chemical Equilibrium Computation

D.1 Molar PropertiesThe tabulated data are given in molar units in terms of the thermodynamic properties discussed be-low. The thermodynamics of chemically reacting flow and details of these properties are discussedby Denbigh (1981), Kee et al. (2003).

Specific heat CP

This is the specific heat for each species defined per mol of substance.

Cp,i(T ) = (∂Hi/∂T )P (J/mol-K)

The molar average specific heat of a gas mixture is

Cp =K∑i=1

XiCp,i(T )

where Xi is the mole fraction of species i

Xi = ni/n n =K∑i=1

ni

where ni is the number of moles (or molecules) of species i.

111

http://webbook.nist.gov/chemistry/

http://cea.grc.nasa.gov/

112 APPENDIX D. HIGH TEMPERATURE GAS THERMODYNAMICS

Standard heat of formation ∆fHo

This is the heat of reaction for the formation of one mole of the species of interest from the elementsin their most stable state at standard thermodynamic conditions, 298.15 K and 1 bar. It is also, bydefinition, the enthalpy at the standard state T = T o = 298.15 K.

∆fHoi ≡ Hi(T

o) = Hi(298.15K)

For the elements in their most stable state, the standard heat of formation is, by definition, zero.

∆fHoi (stable elements) ≡ 0

Enthalpy difference H −H(T o)

This is the difference in enthalpy between the temperature of interest and the standard state.

Hi(T )−Hi(To) =

∫ T

T oCp,i(T

′) dT ′ (kJ/mol)

In order to find the total enthalpy for any species i, the heat of formation and the specific enthalpydifference must be added together.

Hi(T ) = ∆fHoi + (H(T )−H(T o))i

The reference temperature T o is, by convention, taken to be 298.15 K. The molar enthalpy of anideal gas mixture is

H(T ) =K∑i=1

XiHi(T )

Standard entropy So

This is the temperature dependent portion of the molar specific entropy .

Soi (T ) =

∫ T

0

CPi(T′)

T ′dT ′ (J/mol-K)

This is equivalent to the entropy of a pure substance evaluated at the standard pressure P o = 1 bar.At any other pressure P , the entropy of a pure substance will be

Si(T, P ) = Soi (T )− R ln(P/P o)

The molar entropy of an ideal gas mixture is

S(T, P ) =K∑i=1

XiSi(T, Pi) =K∑i=1

Xi

(Soi − R ln(Pi/P

o))

where Pi = XiP is the partial pressure of the ith component. An alternate formulation of this ruleis

S =K∑i=1

XiSoi − R

K∑i=1

Xi lnXi − R ln(P/P o)

D.2. MASS-SPECIFIC MIXTURE PROPERTIES 113

Standard Gibbs energy Go

This is the temperature dependent portion of the Gibbs energy.

Goi (T ) ≡ Hi − TSoi ≡ µo (kJ/mol)

This is equivalent to the Gibbs energy of a pure substance evaluated at the standard pressure P o.The actual Gibbs energy of a pure substance at any pressure can be determined by evaluating:

Gi(T, P ) = Goi (T ) + RT ln(P/P o)

The Gibbs energy of a mixture can be evaluated as

G(T, P ) =K∑i=1

XiGi(T, Pi) =K∑i=1

Xi

(Goi + RT ln(Pi/P

o))

The chemical potential of a species i in a mixture can be evaluated as

µi = µoi + RT ln(Pi/Po)

D.2 Mass-specific Mixture Properties

The mass-specific properties can be obtained from the molar properties and the molar mass (molec-ular weight)Wi of each species

cp,i = Cp,i/Wi hi = Hi/Wi si = Si/Wi

The mixture properties can be obtained in the same fashion but using the mixture molar massinstead of the species values

W =K∑i=1

XiWi W =

(K∑i=1

YiWi

)−1

cp = Cp/W h = H/W s = S/W

where the mass fractions are

Yi =Wi

W Xi

and the mixture properties can also be written as

cp =K∑i=1

Yicp,i h =K∑i=1

Yihi etc.


References1. Gurvich, L. V., Veyts, I. V., and Alcock, C. B. 1989 Thermodynamic Properties of Individual

Substances, Fourth Edition, Hemisphere Pubishing. Volume I, Parts 1 and 2.

2. McBride, B. J., and Gordon, S. 1992 Computer Program for Calculating and Fitting Ther-modynamic Functions NASA Reference Publication 1271.

3. McBride, B. J., Gordon, S., and Reno, M. A. 1993 Thermodynamic Data for Fifty ReferenceElements NASA Technical Paper 3287.

4. Chase, M. W. et al. 1985 JANAF Thermochemical Tables, Third Edition, Parts I and II,J. Phys. Chem. Ref. Data 14, Supplement No. 1.


Thermodynamic Properties of Ar

Molar mass W (g/mol) 39.95Enthalpy of formation ∆fH

◦ = H(298.15) (kJ/mol) .00

T CP H −∆fH◦ S◦ G◦

(K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.1 20.78 .00 154.83 -46.16500.0 20.78 4.20 165.57 -78.59

1000.0 20.78 14.59 179.98 -165.391500.0 20.78 24.98 188.40 -257.632000.0 20.78 35.37 194.38 -353.402500.0 20.78 45.76 199.02 -451.793000.0 20.78 56.15 202.81 -552.283500.0 20.78 66.55 206.01 -654.504000.0 20.78 76.94 208.79 -758.224500.0 20.78 87.33 211.24 -863.245000.0 20.78 97.72 213.43 -969.415500.0 20.78 108.11 215.41 -1076.636000.0 20.78 118.50 217.22 -1184.796500.0 20.92 128.93 218.89 -1293.827000.0 20.96 139.41 220.44 -1403.667500.0 20.94 149.89 221.88 -1514.248000.0 20.87 160.34 223.23 -1625.538500.0 20.78 170.76 224.50 -1737.469000.0 20.69 181.12 225.68 -1850.019500.0 20.62 191.45 226.80 -1963.13

10000.0 20.58 201.75 227.85 -2076.8010500.0 20.59 212.04 228.86 -2190.9811000.0 20.66 222.35 229.82 -2305.6511500.0 20.81 232.71 230.74 -2420.7912000.0 21.05 243.17 231.63 -2536.3912500.0 21.37 253.78 232.50 -2652.4213000.0 21.80 264.57 233.34 -2768.8813500.0 22.34 275.60 234.17 -2885.7614000.0 22.98 286.92 235.00 -3003.0514500.0 23.73 298.59 235.82 -3120.7515000.0 24.59 310.67 236.64 -3238.8715500.0 25.56 323.20 237.46 -3357.3916000.0 26.63 336.25 238.29 -3476.3316500.0 27.81 349.85 239.12 -3595.6817000.0 29.07 364.07 239.97 -3715.4517500.0 30.42 378.94 240.83 -3835.6518000.0 31.84 394.50 241.71 -3956.2918500.0 33.32 410.78 242.60 -4077.3619000.0 34.84 427.82 243.51 -4198.8919500.0 36.40 445.63 244.44 -4320.8820000.0 37.97 464.23 245.38 -4443.33


Thermodynamic Properties of Ar+


◦ = H(298.15) (kJ/mol) 1526.6



298.1 20.98 .00 166.38 1476.97500.0 21.90 4.33 177.44 1442.18

1000.0 22.77 15.60 193.04 1349.141500.0 22.32 26.88 202.19 1250.172000.0 21.95 37.94 208.56 1147.402500.0 21.65 48.84 213.42 1041.863000.0 21.42 59.60 217.35 934.133500.0 21.26 70.27 220.64 824.624000.0 21.16 80.87 223.47 713.574500.0 21.10 91.43 225.96 601.215000.0 21.06 101.97 228.18 487.665500.0 21.02 112.49 230.18 373.066000.0 20.95 122.99 232.01 257.516500.0 20.93 133.45 233.68 141.087000.0 20.91 143.91 235.23 23.857500.0 20.90 154.37 236.68 -94.138000.0 20.90 164.82 238.03 -212.818500.0 20.89 175.26 239.29 -332.159000.0 20.88 185.71 240.49 -452.099500.0 20.88 196.15 241.61 -572.62

10000.0 20.87 206.58 242.69 -693.7010500.0 20.86 217.02 243.70 -815.3011000.0 20.86 227.45 244.67 -937.4011500.0 20.85 237.88 245.60 -1059.9712000.0 20.84 248.30 246.49 -1182.9912500.0 20.84 258.72 247.34 -1306.4513000.0 20.84 269.14 248.16 -1430.3213500.0 20.84 279.56 248.94 -1554.6014000.0 20.86 289.98 249.70 -1679.2614500.0 20.88 300.42 250.43 -1804.3015000.0 20.92 310.87 251.14 -1929.6915500.0 20.97 321.34 251.83 -2055.4416000.0 21.04 331.84 252.50 -2181.5216500.0 21.13 342.38 253.14 -2307.9317000.0 21.26 352.98 253.78 -2434.6617500.0 21.41 363.64 254.40 -2561.7018000.0 21.60 374.39 255.00 -2689.0518500.0 21.83 385.25 255.60 -2816.7019000.0 22.11 396.23 256.18 -2944.6519500.0 22.44 407.37 256.76 -3072.8820000.0 22.83 418.69 257.33 -3201.41


Thermodynamic Properties of H


◦ = H(298.15) (kJ/mol) 218.0



298.1 20.78 .00 114.70 183.77500.0 20.78 4.20 125.45 159.44

1000.0 20.78 14.59 139.85 92.701500.0 20.78 24.98 148.28 20.532000.0 20.78 35.37 154.26 -55.182500.0 20.78 45.76 158.90 -133.513000.0 20.78 56.15 162.69 -213.943500.0 20.78 66.55 165.89 -296.104000.0 20.78 76.94 168.67 -379.764500.0 20.78 87.33 171.11 -464.715000.0 20.78 97.72 173.30 -550.835500.0 20.78 108.11 175.28 -637.986000.0 20.78 118.50 177.09 -726.086500.0 20.81 128.90 178.76 -815.057000.0 20.81 139.31 180.30 -904.827500.0 20.79 149.71 181.73 -995.338000.0 20.77 160.10 183.08 -1086.548500.0 20.75 170.48 184.33 -1178.399000.0 20.73 180.85 185.52 -1270.869500.0 20.73 191.21 186.64 -1363.90

10000.0 20.75 201.58 187.70 -1457.4910500.0 20.78 211.96 188.72 -1551.6011000.0 20.84 222.37 189.69 -1646.2011500.0 20.92 232.81 190.61 -1741.2812000.0 21.03 243.29 191.51 -1836.8112500.0 21.16 253.84 192.37 -1932.7813000.0 21.32 264.46 193.20 -2029.1713500.0 21.50 275.17 194.01 -2125.9814000.0 21.70 285.97 194.79 -2223.1814500.0 21.92 296.87 195.56 -2320.7715000.0 22.15 307.89 196.31 -2418.7315500.0 22.38 319.02 197.04 -2517.0716000.0 22.62 330.27 197.75 -2615.7716500.0 22.85 341.64 198.45 -2714.8217000.0 23.06 353.12 199.14 -2814.2117500.0 23.25 364.70 199.81 -2913.9518000.0 23.41 376.36 200.46 -3014.0218500.0 23.52 388.10 201.11 -3114.4119000.0 23.58 399.88 201.74 -3215.1219500.0 23.57 411.67 202.35 -3316.1420000.0 23.47 423.43 202.94 -3417.47


Thermodynamic Properties of H+


◦ = H(298.15) (kJ/mol) 1536.0



298.1 20.78 .00 108.93 1503.56500.0 20.78 4.20 119.68 1480.40

1000.0 20.78 14.59 134.08 1416.541500.0 20.78 24.98 142.51 1347.252000.0 20.78 35.37 148.49 1274.432500.0 20.78 45.76 153.13 1198.983000.0 20.78 56.15 156.92 1121.443500.0 20.78 66.55 160.12 1042.164000.0 20.78 76.94 162.90 961.394500.0 20.78 87.33 165.34 879.325000.0 20.78 97.72 167.53 796.095500.0 20.78 108.11 169.52 711.826000.0 20.78 118.50 171.32 626.606500.0 20.78 128.90 172.99 540.527000.0 20.78 139.29 174.53 453.647500.0 20.78 149.68 175.96 366.018000.0 20.78 160.07 177.30 277.698500.0 20.78 170.46 178.56 188.729000.0 20.78 180.85 179.75 99.149500.0 20.78 191.25 180.87 8.98

10000.0 20.78 201.64 181.94 -81.7210500.0 20.78 212.03 182.95 -172.9511000.0 20.78 222.42 183.92 -264.6711500.0 20.78 232.81 184.85 -356.8612000.0 20.78 243.21 185.73 -449.5112500.0 20.78 253.60 186.58 -542.5913000.0 20.78 263.99 187.39 -636.0813500.0 20.78 274.38 188.18 -729.9814000.0 20.78 284.77 188.93 -824.2614500.0 20.78 295.16 189.66 -918.9115000.0 20.78 305.56 190.37 -1013.9115500.0 20.78 315.95 191.05 -1109.2716000.0 20.78 326.34 191.71 -1204.9616500.0 20.78 336.73 192.35 -1300.9717000.0 20.78 347.12 192.97 -1397.3017500.0 20.78 357.51 193.57 -1493.9418000.0 20.78 367.91 194.16 -1590.8718500.0 20.78 378.30 194.73 -1688.0919000.0 20.78 388.69 195.28 -1785.6019500.0 20.78 399.08 195.82 -1883.3720000.0 20.78 409.47 196.35 -1981.41


Thermodynamic Properties of H2


◦ = H(298.15) (kJ/mol) .00



298.1 28.83 .00 130.66 -38.96500.0 29.29 5.89 145.75 -66.98

1000.0 30.16 20.68 166.21 -145.531500.0 32.35 36.33 178.87 -231.982000.0 34.19 52.98 188.44 -323.892500.0 35.73 70.47 196.24 -420.123000.0 37.04 88.67 202.87 -519.943500.0 38.16 107.48 208.67 -622.854000.0 39.13 126.80 213.83 -728.504500.0 39.98 146.59 218.48 -836.595000.0 40.75 166.77 222.74 -946.915500.0 41.43 187.32 226.65 -1059.276000.0 42.05 208.20 230.29 -1173.526500.0 42.25 229.28 233.66 -1289.527000.0 42.18 250.40 236.79 -1407.147500.0 41.90 271.43 239.69 -1526.278000.0 41.43 292.27 242.38 -1646.808500.0 40.81 312.83 244.88 -1768.629000.0 40.07 333.06 247.19 -1891.659500.0 39.24 352.89 249.33 -2015.78

10000.0 38.35 372.29 251.32 -2140.9510500.0 37.42 391.24 253.17 -2267.0811000.0 36.47 409.71 254.89 -2394.1111500.0 35.52 427.70 256.49 -2521.9612000.0 34.59 445.23 257.98 -2650.5812500.0 33.69 462.30 259.38 -2779.9213000.0 32.83 478.92 260.68 -2909.9413500.0 32.02 495.13 261.91 -3040.5914000.0 31.27 510.95 263.06 -3171.8414500.0 30.59 526.41 264.14 -3303.6415000.0 29.96 541.55 265.17 -3435.9715500.0 29.40 556.39 266.14 -3568.8016000.0 28.90 570.96 267.07 -3702.1016500.0 28.45 585.30 267.95 -3835.8617000.0 28.05 599.43 268.79 -3970.0417500.0 27.68 613.35 269.60 -4104.6418000.0 27.32 627.10 270.37 -4239.6418500.0 26.97 640.68 271.12 -4375.0119000.0 26.60 654.07 271.83 -4510.7519500.0 26.20 667.27 272.52 -4646.8420000.0 25.73 680.26 273.18 -4783.27


Thermodynamic Properties of H2O


◦ = H(298.15) (kJ/mol) -241.78



298.1 33.58 .00 188.80 -298.09500.0 35.21 6.92 206.50 -338.12

1000.0 41.29 26.00 232.71 -448.501500.0 47.33 48.23 250.65 -569.542000.0 51.67 73.04 264.90 -698.552500.0 54.72 99.69 276.78 -834.053000.0 56.84 127.61 286.95 -975.043500.0 58.30 156.42 295.83 -1120.784000.0 59.36 185.85 303.69 -1270.704500.0 60.21 215.74 310.73 -1424.345000.0 60.98 246.04 317.11 -1581.325500.0 61.75 276.72 322.96 -1741.366000.0 62.56 307.80 328.37 -1904.216500.0 63.33 339.28 333.41 -2069.677000.0 63.90 371.10 338.12 -2237.567500.0 64.29 403.15 342.55 -2407.748000.0 64.54 435.36 346.70 -2580.078500.0 64.66 467.67 350.62 -2754.419000.0 64.68 500.01 354.32 -2930.659500.0 64.63 532.34 357.81 -3108.69

10000.0 64.51 564.62 361.13 -3288.4410500.0 64.35 596.84 364.27 -3469.7911000.0 64.17 628.97 367.26 -3652.6811500.0 63.97 661.00 370.11 -3837.0312000.0 63.76 692.93 372.83 -4022.7712500.0 63.57 724.77 375.42 -4209.8313000.0 63.39 756.51 377.91 -4398.1713500.0 63.23 788.16 380.30 -4587.7314000.0 63.10 819.74 382.60 -4778.4614500.0 63.00 851.27 384.81 -4970.3215000.0 62.92 882.75 386.95 -5163.2615500.0 62.87 914.19 389.01 -5357.2516000.0 62.84 945.62 391.01 -5552.2616500.0 62.83 977.04 392.94 -5748.2517000.0 62.83 1008.45 394.81 -5945.1917500.0 62.83 1039.86 396.64 -6143.0518000.0 62.81 1071.27 398.41 -6341.8218500.0 62.78 1102.67 400.13 -6541.4519000.0 62.70 1134.05 401.80 -6741.9319500.0 62.58 1165.37 403.43 -6943.2420000.0 62.37 1196.61 405.01 -7145.35


Thermodynamic Properties of H2O+


◦ = H(298.15) (kJ/mol) 981.5



298.1 33.68 .00 195.35 923.25500.0 35.77 6.99 213.22 881.88

1000.0 42.22 26.48 239.97 768.001500.0 47.77 49.03 258.19 643.242000.0 52.15 74.05 272.55 510.442500.0 55.66 101.04 284.58 371.083000.0 58.50 129.60 294.99 226.133500.0 60.78 159.45 304.19 76.294000.0 62.52 190.30 312.42 -77.904500.0 63.64 221.87 319.86 -236.005000.0 63.97 253.81 326.59 -397.645500.0 64.03 285.81 332.69 -562.486000.0 63.99 317.82 338.26 -730.246500.0 63.87 349.79 343.38 -900.677000.0 63.67 381.67 348.10 -1073.557500.0 63.42 413.45 352.49 -1248.718000.0 63.13 445.09 356.57 -1425.998500.0 62.82 476.58 360.39 -1605.249000.0 62.48 507.90 363.97 -1786.349500.0 62.14 539.06 367.34 -1969.17

10000.0 61.81 570.05 370.52 -2153.6510500.0 61.49 600.87 373.53 -2339.6611000.0 61.18 631.54 376.38 -2527.1511500.0 60.90 662.06 379.09 -2716.0212000.0 60.65 692.44 381.68 -2906.2212500.0 60.42 722.71 384.15 -3097.6813000.0 60.23 752.87 386.52 -3290.3513500.0 60.07 782.94 388.79 -3484.1814000.0 59.94 812.95 390.97 -3679.1214500.0 59.84 842.89 393.07 -3875.1415000.0 59.77 872.80 395.10 -4072.1815500.0 59.72 902.67 397.06 -4270.2216000.0 59.69 932.52 398.95 -4469.2316500.0 59.66 962.35 400.79 -4669.1717000.0 59.63 992.18 402.57 -4870.0117500.0 59.60 1021.99 404.30 -5071.7318000.0 59.54 1051.77 405.98 -5274.3018500.0 59.45 1081.52 407.61 -5477.6919000.0 59.32 1111.22 409.19 -5681.9019500.0 59.12 1140.83 410.73 -5886.8820000.0 58.85 1170.33 412.22 -6092.62


Thermodynamic Properties of N


◦ = H(298.15) (kJ/mol) 472.6



298.1 20.78 .00 153.28 426.92500.0 20.78 4.20 164.03 394.80

1000.0 20.78 14.59 178.43 308.771500.0 20.80 24.99 186.87 217.302000.0 20.77 35.38 192.85 122.302500.0 20.80 45.77 197.48 24.683000.0 20.95 56.20 201.29 -75.043500.0 21.29 66.75 204.54 -176.524000.0 21.82 77.52 207.41 -279.524500.0 22.55 88.61 210.02 -383.895000.0 23.45 100.10 212.45 -489.515500.0 24.47 112.07 214.73 -596.316000.0 25.53 124.57 216.90 -704.226500.0 26.68 137.63 218.99 -813.207000.0 27.63 151.22 221.01 -923.207500.0 28.40 165.23 222.94 -1034.198000.0 29.01 179.59 224.79 -1146.138500.0 29.49 194.22 226.57 -1258.979000.0 29.86 209.06 228.26 -1372.689500.0 30.13 224.06 229.88 -1487.22

10000.0 30.32 239.18 231.43 -1602.5510500.0 30.45 254.37 232.92 -1718.6411000.0 30.53 269.62 234.34 -1835.4611500.0 30.57 284.90 235.69 -1952.9712000.0 30.59 300.19 237.00 -2071.1412500.0 30.60 315.49 238.24 -2189.9613000.0 30.59 330.78 239.44 -2309.3813500.0 30.58 346.08 240.60 -2429.3914000.0 30.58 361.37 241.71 -2549.9714500.0 30.59 376.66 242.78 -2671.1015000.0 30.60 391.96 243.82 -2792.7515500.0 30.62 407.26 244.83 -2914.9116000.0 30.65 422.58 245.80 -3037.5716500.0 30.69 437.92 246.74 -3160.7117000.0 30.73 453.27 247.66 -3284.3117500.0 30.76 468.64 248.55 -3408.3618000.0 30.79 484.03 249.42 -3532.8518500.0 30.79 499.43 250.26 -3657.7719000.0 30.76 514.82 251.08 -3783.1119500.0 30.69 530.18 251.88 -3908.8520000.0 30.56 545.49 252.65 -4034.98


Thermodynamic Properties of N+


◦ = H(298.15) (kJ/mol) 1882.



298.1 21.28 .00 159.78 1834.23500.0 20.95 4.26 170.68 1800.79

1000.0 20.83 14.69 185.15 1711.411500.0 20.80 25.09 193.59 1616.582000.0 20.80 35.49 199.57 1518.222500.0 20.85 45.90 204.22 1417.223000.0 20.96 56.35 208.03 1314.133500.0 21.13 66.88 211.27 1209.294000.0 21.36 77.50 214.11 1102.934500.0 21.62 88.24 216.64 995.235000.0 21.89 99.11 218.93 886.335500.0 22.15 110.12 221.03 776.336000.0 22.35 121.25 222.97 665.326500.0 22.56 132.48 224.76 553.397000.0 22.74 143.81 226.44 440.587500.0 22.89 155.21 228.02 326.968000.0 23.02 166.69 229.50 212.588500.0 23.14 178.23 230.90 97.489000.0 23.23 189.83 232.22 -18.319500.0 23.31 201.46 233.48 -134.74

10000.0 23.38 213.14 234.68 -251.7810500.0 23.45 224.85 235.82 -369.4111000.0 23.50 236.58 236.91 -487.5911500.0 23.55 248.34 237.96 -606.3112000.0 23.59 260.13 238.96 -725.5412500.0 23.64 271.94 239.93 -845.2713000.0 23.68 283.77 240.85 -965.4613500.0 23.72 295.62 241.75 -1086.1214000.0 23.77 307.49 242.61 -1207.2114500.0 23.81 319.38 243.45 -1328.7215000.0 23.86 331.30 244.25 -1450.6515500.0 23.91 343.25 245.04 -1572.9716000.0 23.97 355.22 245.80 -1695.6816500.0 24.03 367.22 246.54 -1818.7717000.0 24.09 379.24 247.25 -1942.2217500.0 24.15 391.30 247.95 -2066.0218000.0 24.22 403.40 248.64 -2190.1718500.0 24.28 415.52 249.30 -2314.6519000.0 24.35 427.68 249.95 -2439.4719500.0 24.42 439.87 250.58 -2564.6020000.0 24.48 452.09 251.20 -2690.04


Thermodynamic Properties of N2


◦ = H(298.15) (kJ/mol) .00



298.1 29.12 .00 191.58 -57.12500.0 29.59 5.91 206.72 -97.45

1000.0 32.68 21.46 228.14 -206.681500.0 34.76 38.36 241.82 -324.372000.0 36.00 56.08 252.01 -447.932500.0 36.70 74.28 260.12 -576.033000.0 37.07 92.73 266.85 -707.833500.0 37.29 111.32 272.58 -842.724000.0 37.48 130.01 277.57 -980.294500.0 37.71 148.81 282.00 -1120.205000.0 37.96 167.73 285.99 -1262.215500.0 38.20 186.77 289.62 -1406.136000.0 38.29 205.90 292.95 -1551.786500.0 38.20 225.00 296.00 -1699.037000.0 38.59 244.18 298.85 -1847.757500.0 39.38 263.65 301.53 -1997.858000.0 40.51 283.61 304.11 -2149.278500.0 41.94 304.22 306.61 -2301.959000.0 43.60 325.59 309.05 -2455.869500.0 45.45 347.85 311.46 -2610.99

10000.0 47.42 371.06 313.84 -2767.3210500.0 49.48 395.28 316.20 -2924.8311000.0 51.58 420.55 318.55 -3083.5211500.0 53.67 446.86 320.89 -3243.3812000.0 55.72 474.21 323.22 -3404.4112500.0 57.68 502.56 325.53 -3566.5913000.0 59.53 531.87 327.83 -3729.9413500.0 61.23 562.07 330.11 -3894.4214000.0 62.75 593.07 332.37 -4060.0414500.0 64.07 624.78 334.59 -4226.7815000.0 65.15 657.10 336.78 -4394.6315500.0 65.97 689.89 338.93 -4563.5516000.0 66.52 723.02 341.04 -4733.5516500.0 66.78 756.36 343.09 -4904.5817000.0 66.73 789.75 345.08 -5076.6317500.0 66.36 823.04 347.01 -5249.6518000.0 65.66 856.06 348.87 -5423.6318500.0 64.63 888.65 350.66 -5598.5119000.0 63.25 920.63 352.36 -5774.2719500.0 61.52 951.84 353.98 -5950.8620000.0 59.46 982.10 355.52 -6128.24


Thermodynamic Properties of N+2


◦ = H(298.15) (kJ/mol) 1509.



298.1 29.13 .00 197.64 1450.38500.0 29.76 5.93 212.80 1408.83

1000.0 33.10 21.65 234.46 1296.491500.0 35.32 38.74 248.29 1175.612000.0 37.66 56.99 258.78 1048.752500.0 39.95 76.40 267.43 917.143000.0 42.08 96.91 274.90 781.513500.0 43.97 118.43 281.53 642.374000.0 45.60 140.84 287.51 500.094500.0 47.02 164.00 292.97 354.955000.0 48.28 187.83 297.99 207.195500.0 49.52 212.28 302.65 57.026000.0 50.92 237.38 307.02 -95.416500.0 52.57 263.26 311.16 -249.967000.0 54.00 289.91 315.11 -406.537500.0 55.21 317.22 318.88 -565.048000.0 56.20 345.08 322.47 -725.388500.0 56.96 373.38 325.90 -887.489000.0 57.50 402.00 329.17 -1051.259500.0 57.82 430.84 332.29 -1216.63

10000.0 57.92 459.79 335.26 -1383.5210500.0 57.82 488.73 338.09 -1551.8711000.0 57.51 517.57 340.77 -1721.5811500.0 57.01 546.21 343.32 -1892.6112000.0 56.34 574.55 345.73 -2064.8812500.0 55.50 602.52 348.01 -2238.3213000.0 54.51 630.02 350.17 -2412.8713500.0 53.39 657.00 352.21 -2588.4714000.0 52.16 683.39 354.13 -2765.0514500.0 50.83 709.14 355.93 -2942.5715000.0 49.44 734.21 357.63 -3120.9715500.0 48.01 758.58 359.23 -3300.1916000.0 46.56 782.22 360.73 -3480.1816500.0 45.12 805.14 362.14 -3660.9117000.0 43.72 827.34 363.47 -3842.3117500.0 42.40 848.87 364.72 -4024.3618000.0 41.18 869.76 365.89 -4207.0218500.0 40.11 890.07 367.01 -4390.2419000.0 39.22 909.90 368.06 -4574.0119500.0 38.56 929.34 369.07 -4758.3020000.0 38.15 948.50 370.04 -4943.08


Thermodynamic Properties of NO


◦ = H(298.15) (kJ/mol) 91.25



298.1 29.86 .00 210.72 28.43500.0 30.49 6.06 226.23 -15.80

1000.0 33.98 22.23 248.50 -135.021500.0 35.71 39.69 262.63 -263.012000.0 36.72 57.82 273.06 -397.042500.0 37.28 76.34 281.32 -535.713000.0 37.58 95.06 288.15 -678.133500.0 37.78 113.90 293.95 -823.694000.0 38.00 132.84 299.01 -971.964500.0 38.28 151.91 303.50 -1122.615000.0 38.64 171.14 307.56 -1275.395500.0 39.03 190.56 311.26 -1430.106000.0 39.34 210.15 314.67 -1586.606500.0 40.01 229.98 317.84 -1744.737000.0 40.83 250.19 320.84 -1904.417500.0 41.75 270.83 323.68 -2065.548000.0 42.74 291.95 326.41 -2228.078500.0 43.76 313.58 329.03 -2391.949000.0 44.77 335.71 331.56 -2557.099500.0 45.73 358.34 334.01 -2723.48

10000.0 46.64 381.43 336.38 -2891.0810500.0 47.44 404.96 338.67 -3059.8511000.0 48.14 428.86 340.90 -3229.7411500.0 48.72 453.08 343.05 -3400.7312000.0 49.15 477.55 345.13 -3572.7812500.0 49.43 502.20 347.14 -3745.8513000.0 49.56 526.96 349.09 -3919.9113500.0 49.54 551.74 350.96 -4094.9214000.0 49.36 576.47 352.76 -4270.8614500.0 49.03 601.07 354.48 -4447.6715000.0 48.57 625.48 356.14 -4625.3315500.0 47.99 649.62 357.72 -4803.7916000.0 47.30 673.45 359.23 -4983.0316500.0 46.52 696.90 360.68 -5163.0117000.0 45.68 719.95 362.05 -5343.7017500.0 44.81 742.58 363.36 -5525.0618000.0 43.94 764.76 364.61 -5707.0518500.0 43.10 786.52 365.81 -5889.6619000.0 42.33 807.87 366.95 -6072.8519500.0 41.69 828.87 368.04 -6256.6020000.0 41.21 849.59 369.09 -6440.88


Thermodynamic Properties of NO+


◦ = H(298.15) (kJ/mol) 990.6



298.1 29.12 .00 198.20 931.58500.0 29.57 5.91 213.33 889.91

1000.0 32.66 21.44 234.74 777.381500.0 34.75 38.34 248.41 656.392000.0 36.01 56.06 258.60 529.542500.0 36.71 74.26 266.71 398.143000.0 37.09 92.72 273.45 263.053500.0 37.33 111.33 279.18 124.864000.0 37.54 130.04 284.18 -16.014500.0 37.76 148.87 288.61 -159.235000.0 38.01 167.81 292.61 -304.555500.0 38.21 186.87 296.24 -451.786000.0 38.24 205.99 299.57 -600.746500.0 37.83 224.98 302.61 -751.297000.0 37.95 243.91 305.41 -903.317500.0 38.56 263.02 308.05 -1056.688000.0 39.60 282.54 310.57 -1211.348500.0 41.00 302.67 313.01 -1367.239000.0 42.72 323.59 315.40 -1524.349500.0 44.70 345.44 317.76 -1682.63

10000.0 46.89 368.33 320.11 -1842.1010500.0 49.23 392.36 322.45 -2002.7411000.0 51.68 417.58 324.80 -2164.5511500.0 54.18 444.05 327.15 -2327.5412000.0 56.69 471.76 329.51 -2491.7012500.0 59.15 500.72 331.88 -2657.0513000.0 61.51 530.89 334.24 -2823.5813500.0 63.75 562.21 336.61 -2991.2914000.0 65.79 594.61 338.96 -3160.1914500.0 67.61 627.97 341.30 -3330.2515000.0 69.16 662.18 343.62 -3501.4815500.0 70.40 697.08 345.91 -3673.8716000.0 71.28 732.52 348.16 -3847.3916500.0 71.77 768.30 350.36 -4022.0217000.0 71.82 804.22 352.51 -4197.7417500.0 71.40 840.04 354.58 -4374.5218000.0 70.48 875.54 356.58 -4552.3118500.0 69.00 910.43 358.50 -4731.0919000.0 66.94 944.44 360.31 -4910.8019500.0 64.27 977.27 362.02 -5091.3820000.0 60.95 1008.60 363.60 -5272.79


Thermodynamic Properties of O


◦ = H(298.15) (kJ/mol) 249.1



298.1 21.91 .00 161.04 201.12500.0 21.24 4.34 172.17 167.39

1000.0 20.92 14.86 186.76 77.231500.0 20.85 25.30 195.23 -18.412000.0 20.82 35.71 201.22 -117.602500.0 20.85 46.12 205.87 -219.413000.0 20.94 56.57 209.68 -323.333500.0 21.09 67.08 212.92 -429.004000.0 21.30 77.67 215.75 -536.184500.0 21.54 88.38 218.27 -644.695000.0 21.80 99.21 220.55 -754.405500.0 22.05 110.17 222.64 -865.216000.0 22.26 121.25 224.57 -977.026500.0 22.48 132.44 226.36 -1089.757000.0 22.65 143.72 228.03 -1203.357500.0 22.79 155.08 229.60 -1317.778000.0 22.90 166.51 231.07 -1432.948500.0 22.98 177.98 232.46 -1548.829000.0 23.05 189.49 233.78 -1665.399500.0 23.10 201.02 235.03 -1782.59

10000.0 23.15 212.59 236.21 -1900.4010500.0 23.20 224.18 237.34 -2018.8011000.0 23.24 235.78 238.42 -2137.7411500.0 23.29 247.42 239.46 -2257.2112000.0 23.34 259.07 240.45 -2377.1912500.0 23.40 270.75 241.40 -2497.6613000.0 23.46 282.47 242.32 -2618.5913500.0 23.54 294.22 243.21 -2739.9714000.0 23.62 306.01 244.07 -2861.7914500.0 23.71 317.84 244.90 -2984.0415000.0 23.81 329.72 245.70 -3106.6915500.0 23.92 341.66 246.49 -3229.7316000.0 24.03 353.64 247.25 -3353.1716500.0 24.14 365.68 247.99 -3476.9817000.0 24.24 377.78 248.71 -3601.1517500.0 24.35 389.93 249.41 -3725.6918000.0 24.44 402.12 250.10 -3850.5618500.0 24.51 414.36 250.77 -3975.7819000.0 24.56 426.63 251.43 -4101.3319500.0 24.59 438.92 252.06 -4227.2120000.0 24.58 451.21 252.69 -4353.40


Thermodynamic Properties of O+


◦ = H(298.15) (kJ/mol) 1568.



298.1 20.78 .00 154.94 1522.38500.0 20.78 4.20 165.69 1489.92

1000.0 20.78 14.59 180.09 1403.071500.0 20.79 24.98 188.52 1310.772000.0 20.78 35.37 194.50 1214.952500.0 20.78 45.76 199.14 1116.493000.0 20.78 56.15 202.93 1015.953500.0 20.83 66.55 206.13 913.664000.0 20.92 76.99 208.92 809.894500.0 21.08 87.48 211.39 704.805000.0 21.34 98.09 213.62 598.545500.0 21.72 108.85 215.68 491.216000.0 22.25 119.83 217.59 382.886500.0 22.89 131.12 219.39 273.647000.0 23.57 142.73 221.11 163.517500.0 24.28 154.69 222.76 52.538000.0 24.99 167.01 224.35 -59.258500.0 25.70 179.68 225.89 -171.819000.0 26.38 192.70 227.38 -285.139500.0 27.04 206.06 228.82 -399.18

10000.0 27.66 219.73 230.23 -513.9410500.0 28.23 233.71 231.59 -629.4011000.0 28.75 247.96 232.91 -745.5311500.0 29.21 262.45 234.20 -862.3112000.0 29.60 277.16 235.45 -979.7212500.0 29.93 292.04 236.67 -1097.7613000.0 30.19 307.08 237.85 -1216.3913500.0 30.39 322.23 238.99 -1335.6014000.0 30.51 337.45 240.10 -1455.3714500.0 30.58 352.73 241.17 -1575.6915000.0 30.58 368.02 242.21 -1696.5415500.0 30.54 383.30 243.21 -1817.9016000.0 30.44 398.55 244.18 -1939.7416500.0 30.31 413.74 245.11 -2062.0717000.0 30.16 428.86 246.02 -2184.8517500.0 29.99 443.89 246.89 -2308.0818000.0 29.81 458.84 247.73 -2431.7418500.0 29.65 473.71 248.55 -2555.8119000.0 29.51 488.50 249.33 -2680.2819500.0 29.42 503.23 250.10 -2805.1420000.0 29.39 517.93 250.84 -2930.37


Thermodynamic Properties of O2


◦ = H(298.15) (kJ/mol) .00



298.1 29.37 .00 205.12 -61.16500.0 31.08 6.09 220.67 -104.25

1000.0 34.88 22.70 243.55 -220.851500.0 36.50 40.56 258.01 -346.452000.0 37.85 59.16 268.70 -478.242500.0 38.99 78.38 277.27 -614.803000.0 39.99 98.13 284.47 -755.283500.0 40.87 118.34 290.70 -899.114000.0 41.66 138.98 296.21 -1045.864500.0 42.38 159.99 301.16 -1195.235000.0 43.03 181.35 305.66 -1346.955500.0 43.58 203.00 309.79 -1500.826000.0 44.02 224.91 313.60 -1656.686500.0 44.29 247.00 317.13 -1814.387000.0 44.36 269.17 320.42 -1973.777500.0 44.27 291.33 323.48 -2134.768000.0 44.03 313.41 326.33 -2297.228500.0 43.67 335.34 328.99 -2461.059000.0 43.22 357.06 331.47 -2626.189500.0 42.68 378.54 333.79 -2792.50

10000.0 42.09 399.74 335.97 -2959.9510500.0 41.46 420.63 338.01 -3128.4411000.0 40.80 441.19 339.92 -3297.9311500.0 40.12 461.42 341.72 -3468.3512000.0 39.45 481.31 343.41 -3639.6312500.0 38.78 500.87 345.01 -3811.7413000.0 38.14 520.10 346.52 -3984.6313500.0 37.53 539.01 347.95 -4158.2514000.0 36.95 557.63 349.30 -4332.5614500.0 36.41 575.97 350.59 -4507.5315000.0 35.91 594.04 351.81 -4683.1415500.0 35.46 611.89 352.98 -4859.3416000.0 35.06 629.52 354.10 -5036.1116500.0 34.71 646.96 355.17 -5213.4317000.0 34.40 664.23 356.21 -5391.2817500.0 34.13 681.36 357.20 -5569.6318000.0 33.89 698.36 358.16 -5748.4718500.0 33.68 715.25 359.08 -5927.7819000.0 33.49 732.05 359.98 -6107.5519500.0 33.32 748.75 360.85 -6287.7620000.0 33.14 765.36 361.69 -6468.39


Thermodynamic Properties of O+2


◦ = H(298.15) (kJ/mol) 1172.



298.1 30.67 .00 205.36 1110.44500.0 30.85 6.17 221.16 1067.26

1000.0 34.11 22.44 243.58 950.531500.0 35.80 39.95 257.76 824.982000.0 36.80 58.12 268.21 693.372500.0 37.36 76.68 276.49 557.133000.0 37.68 95.45 283.33 417.123500.0 37.92 114.35 289.16 273.974000.0 38.21 133.38 294.24 128.094500.0 38.66 152.59 298.76 -20.185000.0 39.32 172.07 302.87 -170.615500.0 40.23 191.95 306.66 -323.006000.0 41.37 212.34 310.21 -477.226500.0 43.78 233.63 313.61 -633.187000.0 46.10 256.11 316.94 -790.827500.0 48.26 279.70 320.20 -950.118000.0 50.24 304.34 323.38 -1111.018500.0 52.00 329.91 326.48 -1273.489000.0 53.51 356.30 329.49 -1437.479500.0 54.75 383.38 332.42 -1602.95

10000.0 55.69 411.00 335.25 -1769.8810500.0 56.33 439.01 337.99 -1938.1911000.0 56.65 467.27 340.62 -2107.8511500.0 56.67 495.62 343.14 -2278.7912000.0 56.39 523.89 345.54 -2450.9712500.0 55.81 551.96 347.84 -2624.3213000.0 54.96 579.66 350.01 -2798.7813500.0 53.86 606.88 352.06 -2974.3014000.0 52.53 633.48 354.00 -3150.8314500.0 51.01 659.38 355.82 -3328.2815000.0 49.35 684.47 357.52 -3506.6215500.0 47.58 708.71 359.11 -3685.7816000.0 45.77 732.04 360.59 -3865.7116500.0 43.96 754.47 361.97 -4046.3517000.0 42.22 776.01 363.26 -4227.6617500.0 40.62 796.72 364.46 -4409.6018000.0 39.24 816.67 365.58 -4592.1118500.0 38.16 836.01 366.64 -4775.1619000.0 37.46 854.89 367.65 -4958.7419500.0 37.23 873.54 368.62 -5142.8120000.0 37.58 892.22 369.56 -5327.35


Thermodynamic Properties of OH


◦ = H(298.15) (kJ/mol) 39.35



298.1 29.88 .00 183.71 -15.44500.0 29.48 5.98 199.03 -54.20

1000.0 30.69 20.92 219.69 -159.441500.0 32.99 36.86 232.59 -272.692000.0 34.71 53.81 242.33 -391.522500.0 36.02 71.50 250.22 -514.723000.0 37.03 89.77 256.88 -641.543500.0 37.85 108.50 262.66 -771.464000.0 38.54 127.60 267.76 -904.084500.0 39.14 147.03 272.33 -1039.135000.0 39.65 166.73 276.48 -1176.355500.0 40.05 186.67 280.28 -1315.556000.0 40.29 206.76 283.78 -1456.586500.0 40.28 226.91 287.00 -1599.287000.0 40.09 247.01 289.98 -1743.547500.0 39.74 266.97 292.74 -1889.238000.0 39.28 286.73 295.29 -2036.258500.0 38.71 306.23 297.65 -2184.499000.0 38.06 325.42 299.85 -2333.879500.0 37.35 344.27 301.89 -2484.31

10000.0 36.60 362.76 303.78 -2635.7310500.0 35.83 380.87 305.55 -2788.0711000.0 35.05 398.59 307.20 -2941.2611500.0 34.27 415.92 308.74 -3095.2512000.0 33.52 432.87 310.18 -3249.9912500.0 32.78 449.44 311.54 -3405.4213000.0 32.09 465.66 312.81 -3561.5113500.0 31.43 481.53 314.01 -3718.2214000.0 30.81 497.09 315.14 -3875.5114500.0 30.25 512.35 316.21 -4033.3415000.0 29.73 527.35 317.23 -4191.7115500.0 29.25 542.09 318.19 -4350.5616000.0 28.83 556.61 319.11 -4509.8916500.0 28.44 570.92 320.00 -4669.6717000.0 28.08 585.05 320.84 -4829.8817500.0 27.74 599.00 321.65 -4990.5018000.0 27.43 612.80 322.43 -5151.5218500.0 27.11 626.43 323.17 -5312.9219000.0 26.78 639.90 323.89 -5474.6919500.0 26.42 653.20 324.58 -5636.8120000.0 26.02 666.32 325.25 -5799.27


Thermodynamic Properties of OH+


◦ = H(298.15) (kJ/mol) 1289.



298.1 29.19 .00 182.72 1234.67500.0 29.33 5.90 197.83 1196.14

1000.0 31.61 21.05 218.73 1091.461500.0 34.18 37.53 232.07 978.582000.0 35.94 55.09 242.16 859.922500.0 37.21 73.39 250.32 736.743000.0 38.27 92.27 257.20 609.823500.0 39.31 111.66 263.18 479.694000.0 40.45 131.60 268.50 346.754500.0 41.74 152.14 273.34 211.275000.0 43.16 173.36 277.81 73.475500.0 44.63 195.31 281.99 -66.496000.0 45.97 217.97 285.93 -208.486500.0 47.25 241.30 289.67 -352.387000.0 48.01 265.13 293.20 -498.117500.0 48.34 289.24 296.52 -645.558000.0 48.29 313.41 299.64 -794.608500.0 47.93 337.47 302.56 -945.169000.0 47.30 361.29 305.29 -1097.139500.0 46.47 384.74 307.82 -1250.41

10000.0 45.48 407.74 310.18 -1404.9210500.0 44.36 430.20 312.37 -1560.5611000.0 43.17 452.09 314.41 -1717.2711500.0 41.93 473.37 316.30 -1874.9512000.0 40.68 494.02 318.06 -2033.5512500.0 39.44 514.05 319.69 -2192.9913000.0 38.24 533.47 321.22 -2353.2213500.0 37.09 552.29 322.64 -2514.1914000.0 36.00 570.56 323.97 -2675.8514500.0 35.00 588.31 325.21 -2838.1415000.0 34.08 605.58 326.38 -3001.0515500.0 33.25 622.40 327.49 -3164.5216000.0 32.50 638.84 328.53 -3328.5316500.0 31.83 654.92 329.52 -3493.0417000.0 31.23 670.68 330.46 -3658.0417500.0 30.69 686.16 331.36 -3823.5018000.0 30.18 701.37 332.22 -3989.3918500.0 29.68 716.34 333.04 -4155.7119000.0 29.17 731.05 333.82 -4322.4219500.0 28.62 745.50 334.57 -4489.5320000.0 27.99 759.66 335.29 -4656.99


Thermodynamic Properties of CO2


◦ = H(298.15) (kJ/mol) -393.4



298.1 37.13 .00 213.76 -457.19500.0 44.62 8.30 234.85 -502.58

1000.0 54.32 33.39 269.25 -629.321500.0 58.21 61.61 292.08 -769.972000.0 60.45 91.33 309.17 -920.462500.0 61.63 121.89 322.80 -1078.573000.0 62.23 152.87 334.10 -1242.883500.0 62.63 184.09 343.72 -1412.394000.0 63.08 215.51 352.11 -1586.394500.0 63.72 247.20 359.57 -1764.345000.0 64.58 279.26 366.33 -1945.855500.0 65.59 311.80 372.53 -2130.586000.0 66.53 344.84 378.28 -2318.306500.0 68.46 378.58 383.68 -2508.817000.0 70.52 413.32 388.83 -2701.947500.0 72.65 449.11 393.77 -2897.608000.0 74.80 485.97 398.52 -3095.688500.0 76.92 523.91 403.12 -3296.109000.0 78.97 562.88 407.58 -3498.789500.0 80.89 602.85 411.90 -3703.65

10000.0 82.67 643.75 416.10 -3910.6610500.0 84.27 685.49 420.17 -4119.7311000.0 85.66 727.98 424.12 -4330.8111500.0 86.84 771.12 427.96 -4543.8312000.0 87.80 814.79 431.67 -4758.7412500.0 88.52 858.88 435.27 -4975.4813000.0 89.00 903.26 438.75 -5193.9913500.0 89.26 947.84 442.12 -5414.2214000.0 89.30 992.49 445.37 -5636.0914500.0 89.14 1037.10 448.50 -5859.5715000.0 88.80 1081.60 451.51 -6084.5715500.0 88.32 1125.88 454.42 -6311.0616000.0 87.72 1169.90 457.21 -6538.9716500.0 87.04 1213.59 459.90 -6768.2617000.0 86.33 1256.93 462.49 -6998.8617500.0 85.64 1299.92 464.98 -7230.7318000.0 85.02 1342.58 467.39 -7463.8318500.0 84.54 1384.96 469.71 -7698.1019000.0 84.26 1427.15 471.96 -7933.5219500.0 84.26 1469.27 474.15 -8170.0520000.0 84.62 1511.47 476.28 -8407.66


Thermodynamic Properties of CO


◦ = H(298.15) (kJ/mol) -110.5



298.1 29.14 .00 197.63 -169.44500.0 29.81 5.93 212.82 -210.99

1000.0 33.16 21.69 234.51 -323.341500.0 35.13 38.80 248.36 -444.262000.0 36.28 56.68 258.64 -571.122500.0 36.91 75.00 266.82 -702.563000.0 37.24 93.54 273.58 -837.713500.0 37.45 112.22 279.33 -975.974000.0 37.63 130.99 284.35 -1116.924500.0 37.85 149.86 288.79 -1260.225000.0 38.10 168.84 292.79 -1405.635500.0 38.31 187.95 296.43 -1552.956000.0 38.36 207.12 299.77 -1702.026500.0 37.98 226.18 302.82 -1852.687000.0 38.24 245.21 305.64 -2004.807500.0 39.04 264.51 308.30 -2158.298000.0 40.30 284.33 310.86 -2313.098500.0 41.94 304.87 313.35 -2469.149000.0 43.88 326.32 315.80 -2626.449500.0 46.04 348.79 318.23 -2784.94

10000.0 48.36 372.38 320.65 -2944.6710500.0 50.77 397.17 323.07 -3105.6011000.0 53.21 423.16 325.49 -3267.7411500.0 55.62 450.37 327.91 -3431.0912000.0 57.96 478.77 330.33 -3595.6512500.0 60.17 508.31 332.74 -3761.4113000.0 62.20 538.91 335.14 -3928.3813500.0 64.03 570.48 337.52 -4096.5514000.0 65.61 602.90 339.88 -4265.9014500.0 66.92 636.04 342.20 -4436.4215000.0 67.92 669.76 344.49 -4608.1015500.0 68.60 703.91 346.73 -4780.9016000.0 68.93 738.30 348.91 -4954.8116500.0 68.90 772.78 351.03 -5129.8017000.0 68.51 807.14 353.09 -5305.8317500.0 67.75 841.23 355.06 -5482.8818000.0 66.61 874.83 356.96 -5660.8818500.0 65.10 907.77 358.76 -5839.8219000.0 63.23 939.87 360.47 -6019.6319500.0 61.01 970.94 362.09 -6200.2720000.0 58.45 1000.82 363.60 -6381.70


Thermodynamic Properties of C


◦ = H(298.15) (kJ/mol) 716.6



298.1 20.84 .00 158.08 669.44500.0 20.80 4.20 168.84 636.35

1000.0 20.79 14.60 183.25 547.921500.0 20.79 24.98 191.68 454.042000.0 20.97 35.42 197.68 356.632500.0 21.27 45.97 202.39 256.583000.0 21.63 56.69 206.30 154.383500.0 22.00 67.60 209.66 50.374000.0 22.35 78.69 212.62 -55.214500.0 22.65 89.94 215.27 -162.205000.0 22.89 101.33 217.67 -270.445500.0 23.06 112.81 219.86 -379.836000.0 23.16 124.37 221.87 -490.276500.0 23.25 135.98 223.73 -601.677000.0 23.32 147.62 225.45 -713.977500.0 23.38 159.30 227.06 -827.118000.0 23.45 171.00 228.57 -941.028500.0 23.53 182.75 230.00 -1055.679000.0 23.64 194.54 231.35 -1171.009500.0 23.78 206.39 232.63 -1287.00

10000.0 23.97 218.33 233.85 -1403.6210500.0 24.20 230.37 235.03 -1520.8411000.0 24.47 242.53 236.16 -1638.6411500.0 24.80 254.85 237.25 -1757.0012000.0 25.17 267.34 238.32 -1875.8912500.0 25.59 280.02 239.35 -1995.3113000.0 26.04 292.93 240.36 -2115.2413500.0 26.53 306.07 241.36 -2235.6714000.0 27.05 319.46 242.33 -2356.5914500.0 27.58 333.12 243.29 -2478.0015000.0 28.11 347.04 244.23 -2599.8815500.0 28.63 361.23 245.16 -2722.2316000.0 29.13 375.67 246.08 -2845.0416500.0 29.59 390.35 246.98 -2968.3117000.0 29.98 405.25 247.87 -3092.0217500.0 30.29 420.32 248.75 -3216.1818000.0 30.49 435.52 249.60 -3340.7718500.0 30.57 450.79 250.44 -3465.7819000.0 30.49 466.06 251.25 -3591.2019500.0 30.22 481.25 252.04 -3717.0320000.0 29.74 496.25 252.80 -3843.24


Thermodynamic Properties of Electrons

Molar mass W (g/mol) 5.486×10−4

Enthalpy of formation ∆fH◦ = H(298.15) (kJ/mol) .00



298.1 20.78 .00 20.87 -6.22500.0 20.78 4.20 31.61 -11.61

1000.0 20.78 14.59 46.02 -31.431500.0 20.78 24.98 54.44 -56.692000.0 20.78 35.37 60.42 -85.482500.0 20.78 45.76 65.06 -116.893000.0 20.78 56.15 68.85 -150.403500.0 20.78 66.55 72.05 -185.644000.0 20.78 76.94 74.83 -222.384500.0 20.78 87.33 77.28 -260.425000.0 20.78 97.72 79.47 -299.625500.0 20.78 108.11 81.45 -339.856000.0 20.78 118.50 83.26 -381.046500.0 20.78 128.90 84.92 -423.097000.0 20.78 139.29 86.46 -465.947500.0 20.78 149.68 87.89 -509.538000.0 20.78 160.07 89.24 -553.818500.0 20.78 170.46 90.50 -598.759000.0 20.78 180.85 91.68 -644.309500.0 20.78 191.25 92.81 -690.42

10000.0 20.78 201.64 93.87 -737.1010500.0 20.78 212.03 94.89 -784.2911000.0 20.78 222.42 95.85 -831.9811500.0 20.78 232.81 96.78 -880.1412000.0 20.78 243.21 97.66 -928.7512500.0 20.78 253.60 98.51 -977.7913000.0 20.78 263.99 99.33 -1027.2513500.0 20.78 274.38 100.11 -1077.1114000.0 20.78 284.77 100.87 -1127.3614500.0 20.78 295.16 101.60 -1177.9715000.0 20.78 305.56 102.30 -1228.9515500.0 20.78 315.95 102.98 -1280.2716000.0 20.78 326.34 103.64 -1331.9316500.0 20.78 336.73 104.28 -1383.9117000.0 20.78 347.12 104.90 -1436.2117500.0 20.78 357.51 105.50 -1488.8118000.0 20.78 367.91 106.09 -1541.7118500.0 20.78 378.30 106.66 -1594.9019000.0 20.78 388.69 107.21 -1648.3619500.0 20.78 399.08 107.75 -1702.1120000.0 20.78 409.47 108.28 -1756.12

Documents

Nonsteady Gas Dynamics - Joseph Shepherd