Blast wave

A blast wave in fluid dynamics is the pressure and flow resulting from the deposition of a large amount of energy in a small very localised volume. The flow field can be approximated as a lead shock wave, followed by a 'self-similar' subsonic flow field.

History

The classic flow solution — the so-called "similarity solution" — was independently devised by Geoffrey Ingram Taylor [Taylor, Sir Geoffrey Ingram, "The formation of a blast wave by a very intense explosion. I. Theoretical discussion," "Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences", Vol. 201, No. 1065, pages 159 - 174 (22 March 1950).] and John von Neumann [Neumann, John von, "The point source solution," "John von Neumann. Collected Works," edited by A. J. Taub, Vol. 6 [Elmsford, N.Y.: Permagon Press, 1963] , pages 219 - 237.] during World War II. After the war, the similarity solution was published by three other authors — L. I. Sedov [Sedov, L. I., "Propagation of strong shock waves," "Journal of Applied Mathematics and Mechanics", Vol. 10, pages 241 - 250 (1946).] , R. Latter [Latter, R., "Similarity solution for a spherical shock wave," "Journal of Applied Physics", Vol. 26, pages 954 - 960 (1955).] , and J. Lockwood-Taylor [Lockwood-Taylor, J., "An exact solution of the spherical blast wave problem," "Philosophical Magazine", Vol. 46, pages 317 - 320 (1955).] — who had discovered it independentlyBatchelor, George, "The Life and Legacy of G. I. Taylor", [Cambridge, England: Cambridge University Press, 1996] , pages 202 - 207.] .

Applications

Bombs

In response to an inquiry from the British MAUD Committee, G. I. Taylor estimated the amount of energy that would be released by the explosion of an atomic bomb in air. He postulated that for an idealized point source of energy, the spatial distributions of the flow variables would have the same form during a given time interval, the variables differing only in scale. (Thus the name of the "similarity solution.") This hypothesis allowed the partial differential equations in terms of r (the radius of the blast wave) and t (time) to be transformed into an ordinary differential equation in terms of the similarity variable $frac\{r^\{5\}\; ho\_\{o\{t^\{2\}E\}$ ,

where $ho\_\{o\}$ is the density of the air and $E$ is the energy that's released by the explosion [Discussion of similarity solutions, including G. I. Taylor's:http://en.wikipedia.org/wiki/Buckingham_Pi_theorem] [Derivation of G. I. Taylor's similarity solution:http://www.atmosp.physics.utoronto.ca/people/codoban/PHY138/Mechanics/dimensional.pdf] [Discussion of G. I. Taylor's research, including his similarity solution:http://www.deas.harvard.edu/brenner/taylor/physic_today/taylor.htm ] . This result allowed G. I. Taylor to estimate the yield of the first atomic explosion in New Mexico in 1945 using only photographs of the blast, which had been published in newspapers and magazines. The yield of the explosion was determined by using the equation: $E\; =\; left(frac\{\; ho\_\{o\{t^2\}\; ight)left(frac\{r\}\{C\}\; ight)^5$ ,

where $C$ is a dimensionless constant that is a function of the ratio of the specific heat of air at constant pressure to the specific heat of air at constant volume. In 1950, G. I. Taylor published two articles in which he revealed the yield E of the first atomic explosion [Taylor, Sir Geoffrey Ingram, "The formation of a blast wave by a very intense explosion. II. The atomic explosion of 1945," "Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences", Vol. 201, No. 1065, pages 175 - 186 (22 March 1950).] , which had previously been classified and whose publication therefore caused a great to-do.

Astronomy

This so called Sedov-Taylor solution has become useful in astrophysics, i.e. for quantitative estimation of the outcome from supernova-explosions. The Sedov-Taylor expansion is also known as 'Blast Wave' phase, which is an adiabatic expansion phase in the life cycle of supernova. The temperature of the material in supernova shell decreases with time, but the internal energy of the material is always 72% of E₀, the initial energy released. This is helpful for Astrophysicists in predicting the behavior of supernova remnants.

The radius R of the blast wave is given as,

R = 14 (E₀/n)^1/5 t^2/5 pc

where, E is the initial energy, t is the age n is the surrounding medium density

The shock temperature is also given as,

T = 1.0×10¹⁰(E₀/n) R⁻³ K [Exploring the X-ray Universe, Philip A. Charles, Frederick D. Seward]

Further reading

* Cathy J. Clarke & Bob Carswell; "Principles of Astrophysical Fluid Dynamics", Cambridge University Press (2007), Chapter 8. ISBN 978-0521853316

References

External links

* [http://www.deas.harvard.edu/brenner/taylor/handouts/bomb/bomb.html "The formation of a blast wave by a very intense explosion"] G. I. Taylor's solution