Fundamentals of Gas Dynamics

Fundamentals of Gas Dynamics
(2nd Edition)
V. Babu
Professor
Department of Mechanical Engineering
Indian Institute of Technology
Contents
Preface iii

1. Introduction 1
   1.1 Compressibility of Fluids . 1
   1.2 Compressible and Incompressible Flows . 2
   1.3 Perfect Gas Equation of State . 4
   1.3.1 Continuum Hypothesis . 5
   1.4 Calorically Perfect Gas 7
2. One Dimensional Flows – Basics 11
   2.1 Governing Equations . 11
   2.2 Acoustic Wave Propagation Speed 13
   2.2.1 Mach Number 16
   2.3 Reference States . 16
   2.3.1 Sonic State 17
   2.3.2 Stagnation State . 17
   2.4 T-s and P-v Diagrams in Compressible Flows 23
   Exercises . 28
3. Normal Shock Waves 31
   3.1 Governing Equations . 31
   3.2 Mathematical Derivation of the Normal Shock Solution . 33
   3.3 Illustration of the Normal Shock Solution on T-s and P-v
   diagrams . 36
   vvi Fundamentals of Gas Dynamics
   3.4 Further Insights into the Normal Shock Wave Solution . 41
   Exercises . 45
4. Flow with Heat Addition- Rayleigh Flow 49
   4.1 Governing Equations . 49
   4.2 Illustration on T-s and P-v diagrams . 50
   4.3 Thermal Choking and Its Consequences . 60
   4.4 Calculation Procedure 64
   Exercises . 67
5. Flow with Friction – Fanno Flow 69
   5.1 Governing Equations . 69
   5.2 Illustration on T-s diagram 70
   5.3 Friction Choking and Its Consequences . 75
   5.4 Calculation Procedure 75
   Exercises . 81
6. Quasi One Dimensional Flows 83
   6.1 Governing Equations . 84
   6.1.1 Impulse Function and Thrust 84
   6.2 Area Velocity Relation 86
   6.3 Geometric Choking 88
   6.4 Area Mach number Relation for Choked Flow 90
   6.5 Mass Flow Rate for Choked Flow 92
   6.6 Flow Through A Convergent Nozzle . 93
   6.7 Flow Through A Convergent Divergent Nozzle . 97
   6.8 Interaction between Nozzle Flow and Fanno, Rayleigh
   Flows . 111
   Exercises . 122
7. Oblique Shock Waves 127
   7.1 Governing Equations . 129
   7.2 θ-β-M curve . 131
   7.3 Illustration of the Weak Oblique Shock Solution on a T-s
   diagram 134
   7.4 Detached Shocks . 141Contents vii
   7.5 Reflected Shocks . 143
   7.5.1 Reflection from a Wall 143
   Exercises . 146
8. Prandtl Meyer Flow 149
   8.1 Propagation of Sound Waves and the Mach Wave 149
   8.2 Prandtl Meyer Flow Around Concave and Convex Corners 153
   8.3 Prandtl Meyer Solution 155
   8.4 Reflection of Oblique Shock From a Constant Pressure
   Boundary . 160
   Exercises . 163
9. Flow of Steam through Nozzles 165
   9.1 T-s diagram of liquid water-water vapor mixture . 165
   9.2 Isentropic expansion of steam 168
   9.3 Flow of steam through nozzles 171
   9.3.1 Choking in steam nozzles 173
   9.4 Supersaturation and the condensation shock . 179
   Exercises . 188
   Suggested Reading 191
   Table A. Isentropic table for γ = 1.4 193
   Table B. Normal shock properties for γ = 1.4 203
   Table C. Rayleigh flow properties for γ = 1.4 211
   Table D. Fanno flow properties for γ = 1.4 221
   Table E. Oblique shock wave angle β in degrees for γ = 1.4 231
   Table F. Mach angle and Prandtl Meyer angle for γ = 1.4 237
   Table G. Thermodynamic properties of steam, temperature table 243
   Table H. Thermodynamic properties of steam, pressure table 247
   Table I. Thermodynamic properties of superheated steam 251viii Fundamentals of Gas Dynamics
   Index 259
   About the author 263
   About the book 26
   Index
   Acoustic wave, 13, 149
   special case of normal shock
   wave, 36
   Afterburner, 117
   Area Mach number relation, 91
   Area velocity relation, 87
   Bow shock, 142
   Choking
   friction, 75
   geometric, 90
   thermal, 60
   Combustion wave, 61
   Combustor
   design issues, 57
   pressure oscillations, 62
   Compressibility, 1
   isentropic, 3
   isothermal, 2
   limit, 4
   Conservative form, 12
   Continuum limit, 6
   Convergent divergent nozzle, 97
   establishment of flow in
   illustration in T-s diagram,
   98
   Convergent nozzle
   establishment of flow in, 93
   illustration in T-s and P-v
   diagram, 94
   Corner
   concave, 129, 153
   convex, 155
   Critical pressure
   nozzle
   steam, 174
   Critical state, 166
   Detached shock, 141
   Dryness fraction
   definition, 167
   Enthalpy, 9
   Equilibrium
   metastable, 180
   Expansion fan, 155
   nozzle exit, 96
   reflection from constant pressure
   boundary, 161
   260 Fundamentals of Gas Dynamics
   Fanno curve, 74
   Fanno flow, 69
   changes in properties, 74
   illustration in T-s diagram, 74
   interaction with nozzle flow, 111
   sonic state, 76
   Friction
   effect of, 69
   factor, 70
   H-curve
   Normal shock wave, 44
   Rayleigh flow, 57
   sonic state, 59
   Heat addition, 49
   Impulse function, 84
   Intake
   capture area, 137
   critical mode, 137
   supersonic, 107, 136
   Internal energy
   distribution, 8
   modes, 7
   Isentropic process, 13
   illustraion in T-s and P-v
   diagram, 27
   relation, 20
   steam, 170
   Knudsen number, 6
   Mach
   angle, 133, 153
   cone, 152
   number, 3
   reflection, 145
   wave, 133, 152
   flow across, 155
   Mass flow rate
   choked, 92
   Normal shock, 31
   entropy change, 35
   illustration in P-v diagram, 44
   illustration in T-s and P-v
   diagram, 37
   loss of stagnation pressure, 35
   moving, 39
   special case of oblique shock,
   133
   strength, 35
   Oblique shock, 127
   θ − β − M relation, 131
   flow deflection angle, 129
   from nozzle exit, 139
   reflection
   from constant pressure
   boundary, 160
   from wall, 143
   Over expanded flow, 103
   Perfect gas
   calorically perfect, 7
   Equation of state, 4
   alternative forms, 5
   Specific heats, 9
   Prandtl Meyer
   angle, 159
   flow, 153Index 261
   Quasi one-dimensional, 83
   Ramjet engine
   inlet interaction, 63
   intake, 136
   schematic, 29
   Rankine-Hugoniot equation, 44
   Rayleigh curve, 54
   Rayleigh flow, 49
   changes in properties, 53
   illustration in P-v diagram, 57
   ilustration in T-s diagram, 54
   interaction with nozzle flow, 114
   sonic state, 64
   Rayleigh line
   normal shock, 42
   sonic state, 59
   Reference states, 17
   sonic state, 17
   stagnation state, 18
   Saturation
   pressure, 166
   temperature, 166
   Scramjet engine
   inlet interaction, 63
   intake, 136
   Second law
   differential form, 12
   Entropy change for a process, 13
   TdS relation, 12
   Shock
   condensation, 183
   Shock angle, 129
   Shock diamond, 141
   Slip line, 145, 154
   Sonic state, 17, 54
   Fanno flow, 76
   illustration in T-s, P-v diagram,
   27
   nozzle
   steam, 174
   nozzle flow, 90
   Rayleigh flow, 64
   Rayleigh line, 59
   Speed of sound, 15
   mixture of gases, 16
   Perfect gas, 16
   reacting flow, 16
   steam, 170
   Stagnation density, 20
   Stagnation pressure
   changes in, 22
   definition, 19
   loss, 35, 57, 74, 127, 133
   Stagnation state
   definition, 18
   illustration in T-s, P-v diagram,
   27
   Stagnation temperature
   changes in, 21
   definition, 19
   Strong oblique shock, 132
   Supersaturated flow
   degree of supercooling, 182
   degree of supersaturation, 182
   Wilson line, 183
   Thrust
   loss of, 96
   momentum, 85
   pressure, 85262 Fundamentals of Gas Dynamics
   Under expanded flow, 62, 75, 96,
   103
   Velocity triangle, 129, 156
   Wave angle, 129
   Weak oblique shock, 132
   Wilson line, 183
   Wind tunnel
   supersonic, 105

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