L
Lagrangian equations
in plane geometry with artificial viscosity
continuity equation, 133, 137
equation of energy conservation, 135,
136
equation of motion, 134, 135
in spherical geometry
continuity equation, 285–287
energy equation, 287
momentum equation, 283–285
total blast energy, 282
Linear ramp, 159
Loud sounds, 41, 180
M
Mach number
acoustic speed, 102, 103
density ratio, 103
entropy changes across a shock, 105–108
momentum equation, 104
pressure, density and temperature ,
ratios,104
pressure ratio, 109
Rankine-Hugoniot equation, 103
relationship, 104, 105
side of shock, 102
weak shock, 104
Mass conservation, 285, 286
Method of characteristics, 59, 61
Molecular motion, 17
Momentum equation, 283–285
N
Negative characteristics, 73–75, 77–79,
81, 82, 86
Newton’s law of motion, 30–32, 134
Nonlinear distortion
breaking time, 71
characteristic line, 70
initial condition, 70, 71
inviscid form, Burger’s equation, 69
multivalued solution, 70
time, wave profile, 70
Normalized Lagrangian radius
density, 296
particle velocity, 295, 296
pressure, 294, 297
Normalized particle velocity, 285, 293
Normal shock waves
conservation equations (see Conservation
equations)
entropy change of gas, 99–102
fluid motion
air flow, 109
limit, 103
sonic velocity, 109
velocity, 108
formation
compression front, 52
disturbances and associated
compression waves, 53
disturbance speed, 51
molecular mean-free paths, 52
uniformly accelerated piston, 52
velocity, sound waves, 52
match number (see Match number)
Rankine-Hugoniot equations (see RankineHugoniot equations)
reflected shock (see Reflected shock)
relationships, 89
thermodynamic relations, 93
weak shock, 94–96
Numerical instability, difference
equations, 151
Numerical solution
finite difference expressions, 146–150
linear ramp, 157
oscillations, 151
piston generated shock wave, 152–157
piston motion according to the law,
159–160
piston withdrawal generating expansion
wave, 170–173
shock tube, 174–179
initial conditions, 177
theoretical vs. numerical results,
177–179
spherical shock waves, point source
solution, 291
tube closed at end, reflected shock, 160,
162–165, 167, 168
value of κ, for the artificial viscosity, 169
N-wave, 202, 204, 205
O
One-dimensional wave equations, 35
Ordinary differential equations
(ODEs), 264
Overtaking shock waves, 207–210
Index
341
Lagrangian equations
in plane geometry with artificial viscosity
continuity equation, 133, 137
equation of energy conservation, 135,
136
equation of motion, 134, 135
in spherical geometry
continuity equation, 285–287
energy equation, 287
momentum equation, 283–285
total blast energy, 282
Linear ramp, 159
Loud sounds, 41, 180
M
Mach number
acoustic speed, 102, 103
density ratio, 103
entropy changes across a shock, 105–108
momentum equation, 104
pressure, density and temperature ,
ratios,104
pressure ratio, 109
Rankine-Hugoniot equation, 103
relationship, 104, 105
side of shock, 102
weak shock, 104
Mass conservation, 285, 286
Method of characteristics, 59, 61
Molecular motion, 17
Momentum equation, 283–285
N
Negative characteristics, 73–75, 77–79,
81, 82, 86
Newton’s law of motion, 30–32, 134
Nonlinear distortion
breaking time, 71
characteristic line, 70
initial condition, 70, 71
inviscid form, Burger’s equation, 69
multivalued solution, 70
time, wave profile, 70
Normalized Lagrangian radius
density, 296
particle velocity, 295, 296
pressure, 294, 297
Normalized particle velocity, 285, 293
Normal shock waves
conservation equations (see Conservation
equations)
entropy change of gas, 99–102
fluid motion
air flow, 109
limit, 103
sonic velocity, 109
velocity, 108
formation
compression front, 52
disturbances and associated
compression waves, 53
disturbance speed, 51
molecular mean-free paths, 52
uniformly accelerated piston, 52
velocity, sound waves, 52
match number (see Match number)
Rankine-Hugoniot equations (see RankineHugoniot equations)
reflected shock (see Reflected shock)
relationships, 89
thermodynamic relations, 93
weak shock, 94–96
Numerical instability, difference
equations, 151
Numerical solution
finite difference expressions, 146–150
linear ramp, 157
oscillations, 151
piston generated shock wave, 152–157
piston motion according to the law,
159–160
piston withdrawal generating expansion
wave, 170–173
shock tube, 174–179
initial conditions, 177
theoretical vs. numerical results,
177–179
spherical shock waves, point source
solution, 291
tube closed at end, reflected shock, 160,
162–165, 167, 168
value of κ, for the artificial viscosity, 169
N-wave, 202, 204, 205
O
One-dimensional wave equations, 35
Ordinary differential equations
(ODEs), 264
Overtaking shock waves, 207–210
Index
341