Chapter 6
Numerical Treatment of Spherical Shock
Waves
6.1 Introduction
The development of nuclear weapons heralded the need for numerical methods for
predicting the hydrodynamic effects of these devices outside the very strong shock
regime. The similarity solution to the intense point-source explosion in air only
applies to the early phases of the explosion where the pressures generated are very
much greater than the ambient air pressure. The blast wave becomes progressively
weaker at later stages of the expansion and the pressure behind the shock front will
eventually becomes comparable with the atmospheric pressure. The self-similar
solution no longer applies when the pressure drops below about 20 atmospheres
[1]. Consequently, it becomes necessary to take into account the counter-pressure
which has so far been neglected and, when this is included, the partial differential
equations describing the flow must be integrated numerically. Von Neumann, who
provided an analytical solution to the point-source strong shock problem, [2]
pioneered the application of numerical techniques for blast wave problems and
was instrumental in the development of high-speed computing machines for
performing numerical calculations.
As the blast wave travels further away from the point of detonation the over
pressure, p À p 0 , steadily decreases. Once the air has crossed the shock front and
been compressed, it expands again to a pressure even lower than the pre-shock
ambient pressure, p 0 ; this so-called “suction phase” is an important feature encountered in explosions. In Fig. 6.1 we illustrate the overpressure at four successive
times. In relation to the plot corresponding to t 4 it can be seen that the overpressure
has a negative value at some distance behind the shock front. During this phase, a
partial vacuum is created and the surrounding air is sucked in, which results in a
reversal in the air flow towards the centre as opposed to being pushed away from the
centre during the positive phase ( p>p 0 ). The duration of the negative phase is, in
general, larger than the positive phase and the air eventually returns to atmospheric
pressure. This particular feature of explosive behaviour cannot be accounted for with
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. Prunty, Introduction to Simple Shock Waves in Air, Shock Wave and High
Pressure Phenomena, https://doi.org/10.1007/978-3-030-63606-7_6
281
Numerical Treatment of Spherical Shock
Waves
6.1 Introduction
The development of nuclear weapons heralded the need for numerical methods for
predicting the hydrodynamic effects of these devices outside the very strong shock
regime. The similarity solution to the intense point-source explosion in air only
applies to the early phases of the explosion where the pressures generated are very
much greater than the ambient air pressure. The blast wave becomes progressively
weaker at later stages of the expansion and the pressure behind the shock front will
eventually becomes comparable with the atmospheric pressure. The self-similar
solution no longer applies when the pressure drops below about 20 atmospheres
[1]. Consequently, it becomes necessary to take into account the counter-pressure
which has so far been neglected and, when this is included, the partial differential
equations describing the flow must be integrated numerically. Von Neumann, who
provided an analytical solution to the point-source strong shock problem, [2]
pioneered the application of numerical techniques for blast wave problems and
was instrumental in the development of high-speed computing machines for
performing numerical calculations.
As the blast wave travels further away from the point of detonation the over
pressure, p À p 0 , steadily decreases. Once the air has crossed the shock front and
been compressed, it expands again to a pressure even lower than the pre-shock
ambient pressure, p 0 ; this so-called “suction phase” is an important feature encountered in explosions. In Fig. 6.1 we illustrate the overpressure at four successive
times. In relation to the plot corresponding to t 4 it can be seen that the overpressure
has a negative value at some distance behind the shock front. During this phase, a
partial vacuum is created and the surrounding air is sucked in, which results in a
reversal in the air flow towards the centre as opposed to being pushed away from the
centre during the positive phase ( p>p 0 ). The duration of the negative phase is, in
general, larger than the positive phase and the air eventually returns to atmospheric
pressure. This particular feature of explosive behaviour cannot be accounted for with
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. Prunty, Introduction to Simple Shock Waves in Air, Shock Wave and High
Pressure Phenomena, https://doi.org/10.1007/978-3-030-63606-7_6
281