P
Particle paths, 298, 301
Particle velocity, 34, 37, 39, 45, 47, 48, 56, 57,
59, 66, 70, 84, 109, 116, 127, 129,
136, 153, 154, 158–162, 164, 165,
168–172, 174, 179–185, 187,
191–194, 207, 210, 221, 284, 285,
290, 292–294, 296, 298, 300, 302,
305, 309–311
Piston generated shock wave, 152–157
Piston motion with small velocity, 177, 180,
182–187
Plane geometry
energy balance equation, 17–21
mass conservation, 13–15
momentum equations, 15–17
Plane shocks
numerical examples (see Numerical
solution)
numerical procedure, 132
discrete form of the equations, 150, 151
finite difference representation, 145–151
plane wave motion, differential
equations, 145, 146
numerical techniques, need for, 131, 132
stability, difference equations, 151
time-marching procedure, 131
Point source solution, 218, 219, 252, 256, 259,
291, 295
Positive characteristics, 73–79, 82, 83, 86
Pressure ratio for weak shocks, 117–118
Principle of equipartition of energy, 10
R
Rankine-Hugoniot equations
conservation laws, 92
Mach number, 103
parameters, sides of shock, 96
pressure and density changes, weak shock,
97–98
relations, 97
strong shock, 108
Rarefaction wave, 49, 189–191, 195, 207
Reflected shock, 164
conservation equations, 110
density ratio, 111
discontinuity, 111
latter equation, 111, 115
moving and stationaryshocks, 110
plane shock wave propagation, 89
pressure ratio, 111, 113
quadratic equation, 112
relationship, Mach numbers, 114
speed, 115
temperature ratio, 113
velocity, 111
Riemann invariants
areas, 59
breaking time, 71, 72
continuity equation, 60
first-order partial differential equations,
61–66
latter equation, 48
local particle velocity, 59
local velocity of sound, 61
method of characteristics, 59
momentum equation, 60
nonlinear distortion, 69–72
nonlinear equations, 66–69
one-dimensional isentropic fluid flow
equations, 59
piston moves into tube
negative characteristics, 81, 82
positive characteristics, 82, 83
time, shock wave, 84
uniform acceleration, 81
piston withdrawal
constant speed, 77–80
location, 74
piston path in xt-plane, 74
positive and negative characteristics, 74
straight line, 76
surface, 76
uniform region, 75
positive and negative characteristics, 74
Rotational degrees of freedom, 10
S
Second law of thermodynamics, 11–13,
102, 108
Self-similar solution, 217, 219, 281
Separation of variables method, 186
Shock decay, 188–205
Shock front
density distribution, 227
derivatives, 226, 227
on the similarity solution, 218
pressure in atmospheres, 232, 245
pressure-time relationship, 233, 234, 236
radius and time, 249
radius of, 220, 230, 231
strong shock conditions, 258
velocity of, 220, 221
Shock path, 190, 191, 195, 200, 301
342
Index
Particle paths, 298, 301
Particle velocity, 34, 37, 39, 45, 47, 48, 56, 57,
59, 66, 70, 84, 109, 116, 127, 129,
136, 153, 154, 158–162, 164, 165,
168–172, 174, 179–185, 187,
191–194, 207, 210, 221, 284, 285,
290, 292–294, 296, 298, 300, 302,
305, 309–311
Piston generated shock wave, 152–157
Piston motion with small velocity, 177, 180,
182–187
Plane geometry
energy balance equation, 17–21
mass conservation, 13–15
momentum equations, 15–17
Plane shocks
numerical examples (see Numerical
solution)
numerical procedure, 132
discrete form of the equations, 150, 151
finite difference representation, 145–151
plane wave motion, differential
equations, 145, 146
numerical techniques, need for, 131, 132
stability, difference equations, 151
time-marching procedure, 131
Point source solution, 218, 219, 252, 256, 259,
291, 295
Positive characteristics, 73–79, 82, 83, 86
Pressure ratio for weak shocks, 117–118
Principle of equipartition of energy, 10
R
Rankine-Hugoniot equations
conservation laws, 92
Mach number, 103
parameters, sides of shock, 96
pressure and density changes, weak shock,
97–98
relations, 97
strong shock, 108
Rarefaction wave, 49, 189–191, 195, 207
Reflected shock, 164
conservation equations, 110
density ratio, 111
discontinuity, 111
latter equation, 111, 115
moving and stationaryshocks, 110
plane shock wave propagation, 89
pressure ratio, 111, 113
quadratic equation, 112
relationship, Mach numbers, 114
speed, 115
temperature ratio, 113
velocity, 111
Riemann invariants
areas, 59
breaking time, 71, 72
continuity equation, 60
first-order partial differential equations,
61–66
latter equation, 48
local particle velocity, 59
local velocity of sound, 61
method of characteristics, 59
momentum equation, 60
nonlinear distortion, 69–72
nonlinear equations, 66–69
one-dimensional isentropic fluid flow
equations, 59
piston moves into tube
negative characteristics, 81, 82
positive characteristics, 82, 83
time, shock wave, 84
uniform acceleration, 81
piston withdrawal
constant speed, 77–80
location, 74
piston path in xt-plane, 74
positive and negative characteristics, 74
straight line, 76
surface, 76
uniform region, 75
positive and negative characteristics, 74
Rotational degrees of freedom, 10
S
Second law of thermodynamics, 11–13,
102, 108
Self-similar solution, 217, 219, 281
Separation of variables method, 186
Shock decay, 188–205
Shock front
density distribution, 227
derivatives, 226, 227
on the similarity solution, 218
pressure in atmospheres, 232, 245
pressure-time relationship, 233, 234, 236
radius and time, 249
radius of, 220, 230, 231
strong shock conditions, 258
velocity of, 220, 221
Shock path, 190, 191, 195, 200, 301
342
Index