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Fundamentals of Gas Dynamics
Third Edition
Dr. Robert D. Zucker
Dr. Oscar Biblarz
Department of Mechanical and Aerospace Engineering
Naval Postgraduate School
Monterey, California, Usa
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Contents
Preface to Third Edition Xi
Preface to Second Edition Xiii
To the Student Xv
About the Companion Website Xix
1 Definitions and Fundamental Principles 1
1.1 Introduction 1
1.2 Units and Notation 2
1.3 Why we use Nondimensional Quantities 8
1.4 Thermodynamic Concepts for Control Mass Analysis 12
Review Questions 21
Review Problems 24
2 Control Volume Analysis—Part I 27
2.1 Introduction 27
2.2 Objectives 28
2.3 Flow Dimensionality and Average Velocity 28
2.4 Transformation of a Material Derivative to a Control
Volume Approach 31
2.5 Conservation of Mass 37
2.6 Conservation of Energy 39
2.7 Summary 48
Problems 50
Check Test 53
v3 Control Volume Analysis—Part II 55
3.1 Introduction 55
3.2 Objectives 55
3.3 Comments on Entropy 56
3.4 Pressure-Energy Equation 58
3.5 The Stagnation Concept 60
3.6 Stagnation Pressure-Energy Equation 64
3.7 Consequences of Constant Density 66
3.8 Momentum Equation 71
3.9 Summary 80
Problems 82
Check Test 88
4 Introduction to Compressible Flow 91
4.1 Introduction 91
4.2 Objectives 92
4.3 Sonic Speed and Mach Number 92
4.4 Wave Propagation 98
4.5 Equations for Perfect Gases in Terms of Mach Number 100
4.6 h–s and T–s Diagrams 107
4.7 Summary 108
Problems 109
Check Test 112
5 Varying-Area Adiabatic Flow 115
5.1 Introduction 115
5.2 Objectives 116
5.3 General Fluid with No Losses 117
5.4 Perfect Gases with Losses 123
5.5 The ∗ Reference Concept 127
5.6 Isentropic Table 129
5.7 Nozzle Operation 136
5.8 Nozzle Performance 144
5.9 Diffuser Performance 146
5.10 When γ is not Equal to 1.4 148
5.11 Beyond the Tables 148
5.12 Summary 152
Problems 153
Check Test 157
vi CONTENTS6 Standing Normal Shocks 159
6.1 Introduction 159
6.2 Objectives 160
6.3 Shock Analysis: General Fluid 160
6.4 Working Equations for Perfect Gases 163
6.5 Normal-Shock Table 167
6.6 Shocks in Nozzles 172
6.7 Supersonic Wind Tunnel Operation 178
6.8 When γ is not Equal to 1.4 180
6.9 (Optional) Beyond the Tables 182
6.10 Summary 183
Problems 184
Check Test 188
7 Moving and Oblique Shocks 191
7.1 Introduction 191
7.2 Objectives 192
7.3 Normal Velocity Superposition: Moving Normal Shocks 192
7.4 Tangential Velocity Superposition: Oblique Shocks 196
7.5 Oblique-Shock Analysis: Perfect Gas 202
7.6 Oblique-Shock Table and Charts 204
7.7 Boundary Condition of Flow Direction 206
7.8 Boundary Condition of Pressure Equilibrium 210
7.9 Conical Shocks 213
7.10 The Shock Tube 216
7.11 (Optional) Beyond the Tables 219
7.12 Summary 221
Problems 222
Check Test 227
8 Prandtl–Meyer Flow 229
8.1 Introduction 229
8.2 Objectives 229
8.3 Argument for Isentropic Turning Flow 230
8.4 Analysis of Prandtl–Meyer Flow 237
8.5 Prandtl–Meyer Function 241
8.6 Overexpanded and Underexpanded Nozzles 244
8.7 Supersonic Airfoils 249
8.8 Aerospike Nozzle 254
8.9 When γ is not Equal to 1.4 256
8.10 (Optional) Beyond the Tables 257
CONTENTS vii8.11 Summary 258
Problems 259
Check Test 264
9 Fanno Flow 267
9.1 Introduction 267
9.2 Objectives 267
9.3 Analysis for a General Fluid 268
9.4 Working Equations for Perfect Gases 275
9.5 Reference State and Fanno Table 280
9.6 Applications 285
9.7 Correlation with Shocks 290
9.8 Friction Choking 292
9.9 (Optional) How the Left-Hand-Side of Equation (9.40) Arose 296
9.10 When γ is not Equal to 1.4 296
9.11 (Optional) Beyond the Tables 297
9.12 Summary 298
Problems 300
Check Test 305
10 Rayleigh Flow 307
10.1 Introduction 307
10.2 Objectives 308
10.3 Analysis for a General Fluid 309
10.4 Working Equations for Perfect Gases 319
10.5 Reference State and the Rayleigh Table 323
10.6 Applications 326
10.7 Correlation with Shocks 330
10.8 Thermal Choking Due to Heating 334
10.9 When γ is not Equal to 1.4 338
10.10 (Optional) Beyond the Tables 338
10.11 Summary 339
Problems 341
Check Test 347
11 Real Gas Effects 349
11.1 Introduction 349
11.2 Objectives 350
11.3 What’s Really Going on 351
11.4 Semiperfect Gas Behavior and Development of the Gas Tables 354
11.5 Real Gas Behavior, Equations of State and, Compressibility
Factors 361
viii CONTENTS11.6 Variable-γ Variable-Area Flows 365
11.7 Variable-γ Constant-Area Flows 373
11.8 High-Energy Gas Lasers 375
11.9 Summary 377
Problems 380
Check Test 381
12 Propulsion Systems 383
12.1 Introduction 383
12.2 Objectives 384
12.3 Brayton Cycle 384
12.4 Propulsion Engines 394
12.5 General Performance Parameters, Thrust, Power, and Efficiency 412
12.6 Air-Breathing Propulsion Systems Performance Parameters 419
12.7 Air-Breathing Propulsion Systems Incorporating Real Gas
Effects 424
12.8 Rocket Propulsion Systems Performance Parameters 426
12.9 Supersonic Diffusers 431
12.10 Summary 434
Problems 435
Check Test 439
APPENDICES
A Summary of the English Engineering (EE) System of Units 441
B Summary of the International System (SI) of Units 445
C Friction-Factor Chart 449
D Oblique-Shock Charts (γ = 1.4) (Two-Dimensional) 451
E Conical-Shock Charts (γ = 1.4) (Three-Dimensional) 455
F Generalized Compressibility Factor Chart 459
G Isentropic Flow Parameters (γ = 1.4) (Including
Prandtl–Meyer Function) 461
H Normal-Shock Parameters (γ = 1.4) 473
I Fanno Flow Parameters (γ = 1.4) 483
J Rayleigh Flow Parameters (γ = 1.4) 495
K Properties of Air at Low Pressure 507
L Specific Heats of Air at Low Pressures 517
SELECTED REFERENCES 519
ANSWERS TO PROBLEMS 523
INDEX 53
Index
A
Absolute temperature scale, 2
Action, zone of, 99
Additive drag. See Pre-entry drag
Adiabatic flow. See also Isentropic flow
constant area (see Fanno flow)
varying area, 115–158
general, 117–122
of perfect gas
with losses, 123–126
without losses, 117–122
Adiabatic process, definition, 13
Aerospike nozzle, 254–256
Afterburner, 394–397
Airfoils
aerodynamic center, 251
drag, 253
lift, 251
subsonic, 249
supersonic, 249–254
Air tables
specific heat variation, 517–518
thermodynamic properties,
507–516
Area change, flow with. See Adiabatic flow
Area ratio, for isentropic flow, 173
Average gamma method. See Real gases
Average velocity, 28–31
B
Bernoulli’s equation, 68
Beyond the tables. See particular flows
(e.g., Fanno flow)
Body forces, 76
Boundary of system, 12
Brayton cycle, 384–394
basic closed cycle, 384–391
efficiency, 387–390
open cycle, 393–394
real cycles, 391–393
British thermal unit, 442, 443
Bulk modulus of elasticity, 96
By-pass ratio, 398–400
C
Capture area, 432, 439
Celsius temperature, 6
Center of pressure, of airfoils, 251
Centered expansion fan, 236. See also
Prandtl–Meyer flow
Fundamentals of Gas Dynamics, Third Edition. Robert D. Zucker and Oscar Biblarz.
 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
Companion website: www.wiley.com/go/zucker/gas
535Choking
due to area change, 138
due to friction, 292–295
due to heat addition, 334–337
Clausius’ inequality, 58
Closed system, 12
Coefficient
of discharge, 146
of friction, 79, 292–295
of velocity, 146
Combustion chamber
efficiency, 395
heat balance, 395
Compressibility, 96
Compressibility factor, 10, 362–364
Compression shock. See Shock
Compressor
efficiency, 392
work done by, 390
Conical shocks, 213–216
charts, 455–458
Conservation
of energy, 13, 39–47
of mass, 37–39
Constant area adiabatic flow. See Fanno flow
Continuity equation, 37–39
Control mass, 12
Control surface, 12
Control volume, 12
Converging nozzle. See also Nozzle
with varying pressure ratio, 144–146
Converging‑diverging nozzle. See also Nozzle
isentropic operation, 136–143
with expansion waves outside, 289
with normal shocks inside, 175
with oblique shocks outside, 289
Comer flow. See Prandtl–Meyer flow
Critical points
first critical point, 172
second critical point, 173
third critical point, 143
Critical pressure, 138
Curved wall, supersonic flow past, 223, 245
Cycle, definition, 13. See also First law
D
Density, 5
Detached shock, 207–208
Diabatic flow. See Rayleigh flow
DeLaval nozzle. See Converging–diverging
nozzle
Diffuser, 122, 376, 394, 395, 397, 398
efficiency, 146
performance, 146–147
supersonic
oblique shock, 205
starting of fixed geometry, 433–434
in wind tunnels, 178–180
Dimension(s), 2
Dimensional analysis, 11
Dimensionless numbers, 10
Discharge coefficient, 146
Displacement work, 42
Disturbances, propagation of, 98–100
Drag
of airfoils, 253
pressure, 410
Duct flow
with friction (see Fanno flow)
with heat transfer (see Rayleigh flow)
E
Effective exhaust velocity, 426–428
Efficiency
combustion chamber, 395
compressor, 392
diffuser, 146–147
nozzle, 144–146
overall, 419
propulsive, 419
thermodynamic, 419
turbine, 392
Energy
internal, 13
for a perfect gas, 19
kinetic, 15
potential, 15
total, 15
Energy equation, 47–48
pressure–energy equation, 58–60,
64–66
stagnation pressure–energy equation, 64–66,
104–106
Engine. See Jet propulsion systems
English Engineering system. See Units
Enthalpy, definition, 16
for perfect gas, 19
stagnation, 60–64, 101–103
Entropy change
definition of, 20
evaluation of, 20
external (from heat transfer), 57
internal (from irreversibilities), 57
Equation of
536 INDEXcontinuity, 37–39
energy, 47–48
motion, 82
state, 7–8, 361–362
Equivalent diameter, 79, 284
Expansion fan, 236–237
Expansion wave, 236–237
Explosion, 192
External entropy change, 57
Euler’s equation, 59–60
F
Fanjet. See Turbofan
Fanno flow, 267–306
beyond the tables, 297–298
choking effects, 292–295
limiting duct length, 278, 283
relation to shocks, 290–292
tables, 280–285, 483–493
∗reference, 280–285
when γ 1.4, 296–297
working equations, 275–280
Fahrenheit temperature, 2
First critical, 142
First law of thermodynamics
for a cycle, 14
for process
control mass, 14–16, 37
control volume, 37
Flame holders, 406
Flow dimensionality, 28–31
Flow
with area change (see Adiabatic flow)
with friction (see Fanno flow)
with heat transfer
(see Rayleigh flow)
Flow work, 42–43
Fluid, definition, 7
Flux
of energy, 39
of mass, 36
of momentum, 72
Force, units of, 5
Forces
body, 76
surface, 76
Friction coefficient. See Friction factor
Friction factor
Darcy–Weisbach, 79
Fanning, 79. See also Moody diagram
Friction flow. See Fanno flow
Fuel–air ratio, 402, 409
G
Gas constant
individual, 8, 382
universal, 9
Gas dynamic laser, 375–377
Gas, perfect. See Perfect gas
Gas properties, tables of, 24
Gas tables
Fanno flow, 483–493
isentropic flow, 461–471
normal shock, 473–481
Rayleigh flow, 495–505
H
Heat, definition, 14
specific, 19
Heat exchanger, 385
Heat transfer. See also Rayleigh flow
general, 14
Hydraulic diameter See equivalent diameter
Hypersonic flow, 10, 217
I
Impulse function. See Thrust Function
Incompressible flow, 71
Inlet. See Diffuser
Intercooling, 391
Internal energy, 13
for a perfect gas, 19
Internal entropy change, 57
International System. See Units
Irreversibility, 16
relation to entropy, 69–71
Isentropic flow, 115–158. See also Adiabatic
flow; Diffuser; Nozzle
area choking, 136–140
beyond the tables, 148–152
∗reference, 127–129
tables, 129–136, 461–471
when γ 1.4, 148
working equations, 123–126
Isentropic process
definition, 13
equations for perfect gas, 19–20
Isentropic stagnation state, 59–65
Isothermal process, 13
J
Jet. See also Coefficient
overexpanded, 224–249
underexpanded, 224–249
INDEX 537Jet propulsion systems. See also Pulsejet;
Ramjet; Rocket; Turbofan;
Turbojet; Turboprop
description of, 383–391
efficiency parameters, 434
power parameters, 434
real gas computer code, 424–425
thrust analysis, 412–419
Joule, 443, 446, 447
K
Kelvin temperature, 2, 446
Kilogram force, 5
Kilogram mass, 5, 446
Kinetic energy, 15
Kinematic viscosity, 7
Knudsen number, 10
L
Laminar flow, 30–21, 284
Length, units of, 2
Lift, 250. See also Airfoils
Limiting expansion angle, 250
Liquid. See Incompressible flow
Losses. See Internal entropy change
M
Mach angle, 92, 100
Mach cone, 92, 99
Mach line. See Mach wave
Mach number, 97–98
Mach wave, 100. See also Prandtl–Meyer flow
MAPLE code, see beyond the tables in particular
flows (e.g., Fanno flow)
Mass flow rate, 30, 38, 142
Mass, units of, 2. See also Conservation of mass;
Continuity equation
Mass velocity, 269, 299
Momentum equation, 71–80
Momentum flux, 72
Moody diagram, 283, 449–450
Motion. See Equation of motion
Moving shock waves, 193–196
N
Net propulsive thrust, 412, 416, 424
Newton force, 3, 443
Newton’s second law, 3, 80–81
Normal shock, 159–189
beyond the tables, 182–183
in ducts, 330–333, 342, 353, 408
entropy change, 167, 231–233
impossibility of expansion shock, 170
moving shocks, 193–196
in nozzles, 172–178
tables, 167–172, 473–481
velocity change across, 171
weak shocks, 233
when γ 1.4, 180–182
in wind tunnel, 178–180
working equations, 163–167
Normal stress. See Work
Nozzle, 121, 373, 376, 379, 403. See also
Converging nozzle;
Converging–diverging nozzle;
Isentropic flow
discharge coefficient, 146
efficiency, 144–146
operating characteristics, 142–145
overexpanded, 244–249
underexpanded, 244–249
velocity coefficient, 146
in wind tunnel, 178–180
O
Oblique shock, 191–228
beyond the tables, 219–221
charts, 204–206, 451–454
deflection angle, 198–200
detached, 207–209
equations for, 203–205
at nozzle outlet, 302
reflection from boundaries, 195
shock angle, 197–199
transformation from normal shock, 192–196
weak, 192, 233–234
One-dimensional flow
with area change (see Isentropic flow)
definition, 28
with friction (see Fanno flow)
with heat transfer (see Rayleigh flow)
Open system, 12
Overexpanded nozzle, 244–249
P
Perfect gas
definition of, 7, 21
enthalpy of, 16
entropy of, 17
equation of state, 7
internal energy of, 19
538 INDEXisentropic process, 23
polytropic process, 20–21
sonic velocity in, 92
Pipe flow. See Duct flow
Pitot tube, supersonic, 208–209
Polytropic process, 20–24
Potential energy, 15
Pound force, 2, 442
Pound mass, 2, 442
Power, 412–419
input, 418
propulsive, 418
thrust, 418
Prandtl–Meyer flow, 229, 230. See also
Isentropic flow
Prandtl–Meyer function, 241–244,
461–471
Prandtl number, 10
Pre-entry drag, 417
Pre-entry thrust, 416
Pressure drag, 415–417
Pressure–energy equation, 56, 58–60
Pressure, units, 5–6
absolute, 5
gage, 5
stagnation, 60–64, 69–71, 101–103
static, 60–63
Process, 15
Properties, 13
extensive, 13
of gases, 444, 448
intensive, 12
Propulsion systems. See Jet propulsion
systems
Propjet. See Turboprop
Pulsejet, 409–410
R
Ramjet, 406–409
Ram pressure ratio. See Total-pressure recovery
factor
Rankine temperature, 6
Rayleigh flow, 307–348
beyond the tables, 338–339
choking effects, 334–337
limiting heat transfer, 285, 298
relation to shocks, 330–334
∗reference, 323–326
tables, 323–326, 495–505
when γ 1.4, 338
working equations, 319–323
Real gases, 349–382
compressibility factor, 361–364
equilibrium flow, 353–354
equations of state, 361–362
frozen flow, 353–354
gas tables, 354–360 (see also Air tables)
microscopic structure, 352
types of molecules, 352
types of motion, 352
properties from equations, 360
variable gamma method, 365–375
constant area, 373–375
variable area, 365–373
Reflection of waves
from free boundary, 259
from physical boundary, 259
Regenerator, 391, 392
Reheat, 391, 394
Reversible, 16
Reynolds number, 268
Reynolds transport theorem, 36
derivation of, 31–36
Rocket, 426–431
Roughness, pipe or wall
absolute, 283–284
relative, 283–284
S
Second critical, 173
Second law of thermodynamics, 16–17
Shaft work, 41
Shear stress. See Work, done by
Shock. See Normal shock; Oblique shock;
conical shock
Shock tube, 216–219
SI. See Units
Silence, zone of, 99–100
Slug mass, 4
Sonic velocity
in any substance, 299
in perfect gas, 299
Specific fuel consumption,
422–424
Specific heats, 16
Specific impulse, 427–431
Speed of sound. See Sonic velocity
Spillage, 335, 432, 437
Stagnation enthalpy, 60–64, 101–103
Stagnation pressure, 81, 104–106
Stagnation pressure–energy equation, 64–66,
104–106
INDEX 539Stagnation reference state, 107–108
Stagnation temperature, 83, 103
Static conditions, 63, 103
State, 13
perfect gas equation of, 7
Steady flow, 29
Streamline, 31
Streamtube, 31
Stress, work done by. See Work
Subsonic flow, 100–101
Supersonic flow, 100, 101
compared with subsonic, 109–112
Supersonic inlet. See Diffuser
Supersonic nozzle. See Nozzle
Supersonic wind tunnel, 178–180
Surface forces, 76
Swallowed shock, 432
System
control mass, 12
control volume, 12
T
Tables. See Gas tables, Air tables
Temperature
scales, 6
stagnation, 83, 103
static, 83
Thermal efficiency of cycles, 419
Thermodynamic properties. See Properties
Thermodynamics
first law for cycle, 14
for control volume, 59
for process, 14, 39
second law, 16–17
zeroth law, 14
Third critical, 142
Three-dimensional flow, 28
Thrust coefficient, 429–430
Thrust function, 305, 414
Thrust of propulsive device, 415
Time, units of, 2
Total enthalpy, 61–62
Total pressure, 66, 71, 103
Total-pressure recovery factor, 146, 409, 431, 435
Total temperature, 107, 240
Transonic flow, 10, 122
Turbine
efficiency, 392
work done by, 390
Tunnel. See Supersonic wind tunnel
Turbofan, 397–404
Turbojet, 394–397
Turboprop, 404–406
Turbulent flow, 31, 284
Two-dimensional flow, 28
U
Underexpanded nozzle, 244–249
Units
conversion factors, 443, 447
English Engineering, 2, 441–444
International System (SI), 2, 445–448
Universal gas constant, 9
V
Variable gamma method. See Real gases
Varying-area adiabatic flow. See Adiabatic flow
Velocity coefficient, 146
Velocity, sonic, 92–94
effective exhaust, 426–427, 440
Venturi, 142
Viscosity, 7
of gases, 444, 448
W
Wall
flow past curved, 230, 262
friction force, 296
reflection of waves from, 247–249
Wave. See Acoustic waves; Mach wave;
Prandtl–Meyer flow; Reflection of
waves; Shock
Weak shocks, 221
Wedge, supersonic flow past, 206, 245. See also
Airfoils; Oblique shock
Wetted area, 296
When γ 1.4. See particular flow
(e.g., Fanno flow)
Wind tunnel, supersonic,
178–180
Wings. See Airfoils
Work
definition of, 14
done by normal stresses, 41
done by shear stresses, 41
shaft, 41
Z
Zeroth law of thermodynamics, 22
Zone of action, 92, 99
Zone of silence, 92, 99
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