Energy in a nuclear explosion

We next turn to a famous example worked out by the eminent British fluid dynamicist G. I. Taylor.¹ In a nuclear explosion there is an essentially instantaneous release of energy E in a small region of space. This produces a spherical shock wave, with the pressure inside the shock wave thousands of times greater that the initial air pressure, which may be neglected. How does the radius R of this shock wave grow with time t? The relevant governing variables are E, t, and the initial air density , with dimensions , [t]=T, and . This set of variables has independent dimensions, so n=3, k=3. We next determine the exponents:

 

so that

 

with the solution a=1/5, b=-1/5, c=2/5. Therefore

 

with C an undetermined constant. If we plot the radius R of the shock as a function of time t on a log-log plot, the slope of the line should be 2/5. The intercept of the graph would provide information about the energy E released in the explosion, if the constant C could be determined. By solving a model shock-wave problem Taylor estimated C to be about 1; he was able to take de-classified movies of nuclear tests, and using his model, infer the yield of the bombs [6]. This data, of course, was strictly classified; it came as a surprise to the American intelligence community that this data was essentially publicly available to those well versed in dimensional analysis.

Footnote:

¹ Taylor's name is associated with many phenomena in fluid mechanics: the Rayleigh-Taylor instability, Saffman-Taylor fingering, Taylor cells, Taylor columns, etc.