Taylor–von Neumann–Sedov blast wave

The blast wave was described by a self-similar solution independently by G. I. Taylor, John von Neumann and Leonid Sedov during World War II.

[1][2] G. I. Taylor was told by the British Ministry of Home Security that it might be possible to produce a bomb in which a very large amount of energy would be released by nuclear fission and asked to report the effect of such weapons.

[3] Exactly at the same time, in the United States, John von Neumann was working on the same problem and he presented his results on June 30, 1941.

[6] von Neumann published his results in August 1947 in the Los Alamos scientific laboratory report on "Blast wave" (PDF).

[8][9] In the second paper, Taylor calculated the energy of the atomic bomb used in the Trinity (nuclear test) using the similarity, just by looking at the series of blast wave photographs that had a length scale and time stamps, published by Julian E Mack in 1947.

[10] This calculation of energy caused, in Taylor's own words, 'much embarrassment' (according to Grigory Barenblatt) in US government circles since the number was then still classified although the photographs published by Mack were not.

Taylor's biographer George Batchelor writes This estimate of the yield of the first atom bomb explosion caused quite a stir... G.I.

was mildly admonished by the US Army for publishing his deductions from their (unclassified) photographs.

[11] Consider a strong explosion (such as nuclear bombs) that releases a large amount of energy

This will create a strong spherical shock wave propagating outwards from the explosion center.

The self-similar solution tries to describe the flow when the shock wave has moved through a distance that is extremely large when compared to the size of the explosive.

At these large distances, the information about the size and duration of the explosion will be forgotten; only the energy released

To a very high degree of accuracy, then it can be assumed that the explosion occurred at a point (say the origin

cannot be neglected since the density jump across strong shock waves is finite as a direct consequence of Rankine–Hugoniot conditions.

is quite close to unity, thereby demonstrating (for this problem) the quantitative predictive capability of the dimensional analysis in determining the shock-wave location as a function of time.

is a constant, the density immediately behind the shock wave is not changing with time, whereas

The gas motion behind the shock wave is governed by Euler equations.

For an ideal polytropic gas with spherical symmetry, the equations for the fluid variables such as radial velocity

, the solutions should approach the values given by the Rankine-Hugoniot conditions defined in the previous section.

The following non-dimensional self-similar variables are introduced,[13][14] The conditions at the shock front

Solving these differential equations analytically is laborious, as shown by Sedov in 1946 and von Neumann in 1947.

Due to self-similarity, it is clear that not only the total energy within a sphere of radius

that is valid for ideal polytropic gas leads to The continuity and energy equation reduce to Expressing

only using the relation obtained earlier and integrating once yields the solution in implicit form, where The constant

The asymptotic behavior of the central region can be investigated by taking the limit

The entire mass of the gas which was initially spread out uniformly in a sphere of radius

The pressure ratio also drops rapidly to attain the constant value

It becomes clear if the above forms are rewritten in dimensional units, The velocity ratio has the linear behavior in the central region, whereas the behavior of the velocity itself is given by As the shock wave evolves in time, its strength decreases.

The governing equations are then integrated numerically, as was done by H. Goldstine and John von Neumann,[15] Brode,[16] and Okhotsimskii et al.[17] Furthermore, in this stage, the compressing shock wave is necessarily followed by a rarafaction wave behind it; the waveform is empirically fiited by the Friedlander waveform.

The analogous problem in cylindrical geometry corresponding to an axisymmetric blast wave, such as that produced in a lightning, can be solved analytically.