Shock reflection, 205, 206
Shock strength, 116, 118, 120–123, 176, 191,
194, 200–203, 205
Shock tube, 174–179, 299
Shock wave interactions, 174, 204–212
Shock waves
conservation equations (see Conservation
equations)
intense explosion, 217, 218
piston generated (see Piston generated
shock wave)
steady-state plane shock wave, 138
time and place, formation of discontinuity
conditions, 54
piston moving with uniform accelerated,
54–57
piston moving with velocity, 54–57
Shock wave zone, 191, 194, 200, 201, 203
Short duration piston motion, 188, 191–199,
201, 202, 205, 207
Similarity solution, 281, 292
density, 294
pressure, 292
specification, initial conditions, 294
velocity, 293
Small amplitude disturbances
ambient values, 34
approximation, 38
continuity equation, 33
isentropic conditions, 34, 35
isentropic flow, 33, 37
momentum equation, 33
one-dimensional wave equations, 35
ordinary sound waves, 33
perfect gas, 37
perturbation procedure, 37
pressure and temperature perturbations, 35
relationship, 37
speed, 33, 36
wave profile, 37
wave travel, 37
Sound speed ratio for weak shocks, 121
Sound wave parameters, 38–41
Sound waves
small amplitude disturbances (see Small
amplitude disturbances)
Specific volume, 4, 15, 20, 21, 98, 99, 132, 133,
136, 137, 141–144, 157, 188, 191,
206, 207, 284, 286–289
Speed of sound, 33
Spherical geometry
continuity equation
accumulation of mass, 13
area, 13
coordinate system, 25
element of mass, 13
latter equation, 14
mass flow rate, 13
volume elements, 13, 14
equation of energy conservation, 32–33
equation of motion (Newton’s law), 30–32
Spherical shock waves, 33
analytical solutions (see Analytical
solutions)
as the initial condition (see Similarity
solution)
at successive times, 282
conservation equations, 288
density for small values of η, 241, 242
difference equations, 288, 290, 291
energy of the explosion, 229–231
numerical integration, 227, 294–299, 301
numerical solution, 291, 292
pressure, 231–233
pressure-time relationship, for a fixed point,
233, 234, 236
route to an analytical solution, 264–269
(see also Shock front)
temperature
at any point, 233
in the central region, 242, 243
wasted energy, 243–250
Strong shocks, 98, 102, 108, 129
“Suction phase”, 281
T
Taylor’s analysis, very strong shocks
continuity equation, 224
dimensional parameters, 219
energy equation, 225, 226
momentum equation, 222–224
pressure, gas velocity and density, 222
self-similar solution, 219
shock front
radius of, 220
Rankine-Hugoniot equations, 221
velocity of, 220, 221
Taylor’s analytical approximations
for density, 234, 235, 237–241
for pressure, 234, 235, 237–241
for velocity, 234, 235, 237–241
Taylor’s numerical analysis, 256
Temperature ratio for weak shocks, 104, 115,
120, 176
Thermodynamics of perfect gases
Index
343
Shock strength, 116, 118, 120–123, 176, 191,
194, 200–203, 205
Shock tube, 174–179, 299
Shock wave interactions, 174, 204–212
Shock waves
conservation equations (see Conservation
equations)
intense explosion, 217, 218
piston generated (see Piston generated
shock wave)
steady-state plane shock wave, 138
time and place, formation of discontinuity
conditions, 54
piston moving with uniform accelerated,
54–57
piston moving with velocity, 54–57
Shock wave zone, 191, 194, 200, 201, 203
Short duration piston motion, 188, 191–199,
201, 202, 205, 207
Similarity solution, 281, 292
density, 294
pressure, 292
specification, initial conditions, 294
velocity, 293
Small amplitude disturbances
ambient values, 34
approximation, 38
continuity equation, 33
isentropic conditions, 34, 35
isentropic flow, 33, 37
momentum equation, 33
one-dimensional wave equations, 35
ordinary sound waves, 33
perfect gas, 37
perturbation procedure, 37
pressure and temperature perturbations, 35
relationship, 37
speed, 33, 36
wave profile, 37
wave travel, 37
Sound speed ratio for weak shocks, 121
Sound wave parameters, 38–41
Sound waves
small amplitude disturbances (see Small
amplitude disturbances)
Specific volume, 4, 15, 20, 21, 98, 99, 132, 133,
136, 137, 141–144, 157, 188, 191,
206, 207, 284, 286–289
Speed of sound, 33
Spherical geometry
continuity equation
accumulation of mass, 13
area, 13
coordinate system, 25
element of mass, 13
latter equation, 14
mass flow rate, 13
volume elements, 13, 14
equation of energy conservation, 32–33
equation of motion (Newton’s law), 30–32
Spherical shock waves, 33
analytical solutions (see Analytical
solutions)
as the initial condition (see Similarity
solution)
at successive times, 282
conservation equations, 288
density for small values of η, 241, 242
difference equations, 288, 290, 291
energy of the explosion, 229–231
numerical integration, 227, 294–299, 301
numerical solution, 291, 292
pressure, 231–233
pressure-time relationship, for a fixed point,
233, 234, 236
route to an analytical solution, 264–269
(see also Shock front)
temperature
at any point, 233
in the central region, 242, 243
wasted energy, 243–250
Strong shocks, 98, 102, 108, 129
“Suction phase”, 281
T
Taylor’s analysis, very strong shocks
continuity equation, 224
dimensional parameters, 219
energy equation, 225, 226
momentum equation, 222–224
pressure, gas velocity and density, 222
self-similar solution, 219
shock front
radius of, 220
Rankine-Hugoniot equations, 221
velocity of, 220, 221
Taylor’s analytical approximations
for density, 234, 235, 237–241
for pressure, 234, 235, 237–241
for velocity, 234, 235, 237–241
Taylor’s numerical analysis, 256
Temperature ratio for weak shocks, 104, 115,
120, 176
Thermodynamics of perfect gases
Index
343