Abstract
The study on blast waves in cold exponential atmospheres by Bach, Kuhl, and Oppenheim is extended to provide a uniformly valid numerical solution of a point explosion problem in isothermal exponential atmospheres with finite temperature at the centre. This is achieved initially by solving the equations of motion with the help of a perturbation technique which takes into account thermal radiation and heat conduction. Whereas the extended perturbation solution for the strong shock regime (i.e. short times after initiation) serves as a starting solution for the numerical integration, the solution using the method of characteristics is valid for the whole flow field since counterpressure effects and energy losses at the front are not neglected.
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Abbreviations
- a :
-
speed of sound
- e :
-
specific internal energy
- E 0 :
-
explosion energy
- E j :
-
blast wave energy
- f :
-
non-dimensional particle velocity
- g :
-
non-dimensional pressure
- g s :
-
acceleration of gravity
- h :
-
non-dimensional density
- H :
-
abreviation for the sum of higher order terms of the decay parameterλ
- I :
-
mass integral
- J :
-
energy integral
- K :
-
non-dimensional internal energy factor
- K C :
-
thermal conductivity
- M j :
-
blast wave mass
- n j :
-
geometrical factor
- p :
-
pressure
- q :
-
total heat flux
- Q :
-
total non-dimensional heat flux
- qC :
-
heat flux by conduction
- Q c :
-
non-dimensional heat flux by conduction
- qR :
-
heat flux by radiation
- Q R :
-
non-dimensional heat flux by radiation
- q s :
-
heat flux across the front
- r :
-
space coordinate
- R :
-
position of the shock front
- r 0 :
-
local dynamic length
- t :
-
time coordinate
- t 0 :
-
local dynamic time
- T :
-
absolute temperature
- u :
-
particle velocity
- W n :
-
shock propagation velocity
- x :
-
field coordinate (similarity variable)
- y :
-
front coordinate (reciprocal square of Mach number)
- z :
-
altitude, measured positive upward from the point of explosion
- α R :
-
Rosseland mean absorption coefficient
- β C :
-
temperature exponent of thermal conductivity
- β R :
-
temperature exponent of absorption coefficient
- Γ :
-
non-dimensional speed of sound factor
- Γ C :
-
non-dimensional conduction parameter
- γ H :
-
non-dimensional heat flux parameter
- γ R :
-
non-dimensional radiation parameter
- Δ :
-
characteristic scale height of the atmosphere
- θ :
-
non-dimensional temperature
- K :
-
specific heat ratio
- λ :
-
decay parameter of the front velocity
- ξ :
-
non-dimensional front coordinate
- ρ :
-
density
- σ :
-
Stefan-Boltzmann constant
- σ h :
-
scaling factor,σ h is positive for the upward direction;σ h is negative for the downward direction
- τ :
-
non-dimensional time scale
- Φ :
-
polar angle between the front radius and the vertical
- ω :
-
decay coefficient of the atmospheric density
- 0, 1, 2:
-
zeroth, first, second order
- a:
-
abient gas
- c:
-
condition at the centre
- i:
-
initial value
- n :
-
conditions immediately behind the shock front
- pp :
-
particle path
- η :
-
along characteristicη=const
- ξ :
-
along characteristicξ=const
- *:
-
non-dimensional variable
- .:
-
differentiation with respect to time
- ′:
-
differentiation with respect tox
References
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Gretler, W., Steiner, H. Blast waves in inhomogeneous atmospheres with counterpressure and heat transfer effects. Shock Waves 3, 83–94 (1993). https://doi.org/10.1007/BF02115888
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DOI: https://doi.org/10.1007/BF02115888