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Taylor–von Neumann–Sedov blast wave (or sometimes referred to as Sedov–von Neumann–Taylor blast wave) refers to a blast wave induced by a strong explosion. The blast wave was described by a
self-similar solution In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar ...
independently by
G. I. Taylor Sir Geoffrey Ingram Taylor Order of Merit, OM Royal Society of London, FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, who made contributions to fluid dynamics and wave theory. Early life and education Tayl ...
,
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
and Leonid Sedov during
World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...
.


History

G. I. Taylor Sir Geoffrey Ingram Taylor Order of Merit, OM Royal Society of London, FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, who made contributions to fluid dynamics and wave theory. Early life and education Tayl ...
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. Taylor presented his results on June 27, 1941. Exactly at the same time, in the
United States The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 ...
,
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
was working on the same problem and he presented his results on June 30, 1941. It was said that Leonid Sedov was also working on the problem around the same time in the
USSR The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...
, although Sedov never confirmed any exact dates. The complete solution was published first by Sedov in 1946. von Neumann published his results in August 1947 in the Los Alamos scientific laboratory report on , although that report was distributed only in 1958. Taylor got clearance to publish his results in 1949 and he published his works in two papers in 1950. In the second paper, Taylor calculated the energy of the atomic bomb used in the
Trinity (nuclear test) Trinity was the first detonation of a nuclear weapon, conducted by the United States Army at 5:29 a.m. MWT (11:29:21 GMT) on July 16, 1945, as part of the Manhattan Project. The test was of an implosion-design plutonium bomb, or "gadg ...
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. 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 George Keith Batchelor FRS (8 March 1920 – 30 March 2000) was an Australian applied mathematician and fluid dynamicist. He was for many years a professor of applied mathematics in the University of Cambridge, and was founding head of the ...
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''.


Mathematical description

Consider a strong explosion (such as nuclear bombs) that releases a large amount of energy E in a small volume during a short time interval. This will create a strong spherical
shock wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
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 E will have influence on how the shock wave evolves. To a very high degree of accuracy, then it can be assumed that the explosion occurred at a point (say the origin r=0) instantaneously at time t=0. The shock wave in the self-similar region is assumed to be still very strong such that the pressure behind the shock wave p_1 is very large in comparison with the pressure (atmospheric pressure) in front of the shock wave p_0, which can be neglected from the analysis. Although the pressure of the undisturbed gas is negligible, the density of the undisturbed gas \rho_0 cannot be neglected since the density jump across strong shock waves is finite as a direct consequence of
Rankine–Hugoniot conditions The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, describe the relationship between the states on both sides of a shock wave or a combustion wave (deflagration or detonation ...
. This approximation is equivalent to setting p_0=0 and the corresponding sound speed c_0=0, but keeping its density non zero, i.e., \rho_0\neq 0. The only parameters available at our disposal are the energy E and the undisturbed gas density \rho_0. The properties behind the shock wave such as p_1,\,\rho_1 are derivable from those in front of the shock wave. The only non-dimensional combination available from r,\,t,\,\rho_0 and E is :r\left(\frac\right)^. It is reasonable to assume that the evolution in r and t of the shock wave depends only on the above variable. This means that the shock wave location r=R(t) itself will correspond to a particular value, say \beta, of this variable, i.e., :R= \beta \left(\frac\right)^. The detailed analysis that follows will, at the end, reveal that the factor \beta 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. The propagation velocity of the shock wave is :D=\frac=\frac=\frac\left(\frac\right)^ With the approximation described above,
Rankine–Hugoniot conditions The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, describe the relationship between the states on both sides of a shock wave or a combustion wave (deflagration or detonation ...
determines the gas velocity immediately behind the shock front v_1, p_1 and \rho_1 for an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
as follows :v_1 = \fracD, \quad p_1 = \frac\rho_0 D^2, \quad \rho_1= \rho_0 \frac where \gamma is the
specific heat ratio In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volu ...
. Since \rho_0 is a constant, the density immediately behind the shock wave is not changing with time, whereas v_1 and p_1 decrease as t^ and t^, respectively.


Self-similar solution

The gas motion behind the shock wave is governed by
Euler equations In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
. For an ideal
polytropic gas In astrophysics, a polytrope refers to a solution of the Lane–Emden equation in which the pressure depends upon the density in the form P = K \rho^ = K \rho^, where is pressure, is density and is a constant of proportionality. The consta ...
with spherical symmetry, the equations for the fluid variables such as radial velocity v(r,t), density \rho(r,t) and pressure p(r,t) are given by :\begin \frac + v \frac &= - \frac\frac,\\ \frac + \frac &= - \frac,\\ \left(\frac+ v\frac\right)\ln \frac &=0. \end At r=R(t), the solutions should approach the values given by the Rankine-Hugoniot conditions defined in the previous section. The variable pressure can be replaced by the sound speed c(r,t) since pressure can be obtained from the formula c^2=\gamma p/\rho. The following non-dimensional self-similar variables are introduced, :\xi = \frac, \quad V(\xi) = \frac, \quad G(\xi) = \frac, \quad Z(\xi) = \frac. The conditions at the shock front \xi=1 becomes :V(1) = \frac, \quad G(1)=\frac, \quad Z(1) = \frac. Substituting the self-similar variables into the governing equations will lead to three ordinary differential equations. Solving these differential equations analytically is laborious, as shown by Sedov in 1946 and von Neumann in 1947. G. I. Taylor integrated these equations numerically to obtain desired results. The relation between Z and V can be deduced directly from energy conservation. Since the energy associated with the undisturbed gas is neglected by setting p_0=0, the total energy of the gas within the shock sphere must be equal to E. Due to self-similarity, it is clear that not only the total energy within a sphere of radius \xi=1 is constant, but also the total energy within a sphere of any radius \xi<1 (in dimensional form, it says that total energy within a sphere of radius r that moves outwards with a velocity v_n=2r/5t must be constant). The amount of energy that leaves the sphere of radius r in time dt due to the gas velocity v is 4\pi r^2\rho v(h+v^2/2)\mathrmt, where h is the
specific enthalpy Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant extern ...
of the gas. In that time, the radius of the sphere increases with the velocity v_n and the energy of the gas in this extra increased volume is 4\pi r^2 \rho v_n(e+v^2/2)\mathrmt, where e is the
specific energy Specific energy or massic energy is energy per unit mass. It is also sometimes called gravimetric energy density, which is not to be confused with energy density, which is defined as energy per unit volume. It is used to quantify, for example, st ...
of the gas. Equating these expressions and substituting e=c^2/\gamma(\gamma-1) and h=c^2/(\gamma-1) that is valid for ideal polytropic gas leads to :Z = \frac. The continuity and energy equation reduce to :\begin \frac - (1-V) \frac &= - 3V\\ \frac - (\gamma-1) \frac &= -\frac. \end Expressing \mathrmV/\mathrm\ln\xi and \mathrm\ln G/\mathrmV as a function of V only using the relation obtained earlier and integrating once yields the solution in implicit form, :\begin \xi^5 &= \left frac(\gamma+1)V\right \left\^\left frac(\gamma V-1)\right,\\ G &= \frac\left frac(\gamma V-1)\right\left\^\left frac(1-V)\right \end where :\nu_1= -\frac,\quad \nu_2 = \frac, \quad \nu_3 = \frac,\quad \nu_4 = -\frac, \quad \nu_5 = - \frac. The constant \beta that determines the shock location can be determined from the conservation of energy :E=\int_0^R\rho ^2/2+c^2/\gamma(\gamma-1)\pi r^2\mathrmr to obtain :\beta^5 \frac\int_0^1 G ^2/2+Z/\gamma(\gamma-1)xi^4\mathrm\xi = 1. For air, \gamma=7/5 and \beta=1.033. The solution for \gamma=7/5 is shown in the figure by graphing the curves of \rho/\rho_1=G(\gamma-1)/(\gamma+1), v/v_1 = \xi V(\gamma+1)/2, p/p_1=\xi^2GZ(\gamma+1)/(2\gamma) and T/T_1=\xi^2Z(\gamma+1)^2/ \gamma(\gamma-1) where T is the temperature.


Asymptotic behavior near the central region

The asymptotic behavior of the central region can be investigated by taking the limit \xi\rightarrow 0. From the figure, it can be observed that the density falls to zero very rapidly behind the shock wave. The entire mass of the gas which was initially spread out uniformly in a sphere of radius R is now contained in a thin layer behind the shock wave, that is to say, all the mass is driven outwards by the acceleration imparted by the shock wave. Thus, most of the region is basically empty. The pressure ratio also drops rapidly to attain the constant value p_c. The temperature ratio follows from the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
; since density ratio decays to zero and the pressure ratio is constant, the temperature ratio must become infinite. The limiting form for the density is given as follows :\frac \sim \xi^, \quad \frac\rightarrow p_c, \quad \frac \sim \xi^ \quad \text \quad \xi\rightarrow 0. Remember that the density \rho_1 is time-independent whereas p_1\sim t^ which means that the actual pressure is in fact time dependent. It becomes clear if the above forms are rewritten in dimensional units, :\rho \sim r^t^, \quad p\rightarrow p_c t^, \quad T \sim r^t^ \quad \text \quad r\rightarrow 0. The velocity ratio has the linear behavior in the central region, :\frac \sim \xi \quad \text \quad \xi\rightarrow 0 whereas the behavior of the velocity itself is given by :v \sim r t^ \quad \text \quad r\rightarrow 0.


Final stage of the blast wave

As the shock wave evolves in time, its strength decreases. The self-similar solution described above breaks down when p_1 becomes comparable to p_0 (more precisely, when p_1\sim \gamma+1)/(\gamma-1)_0). At this later stage of the evolution, p_0 (and consequently c_0) cannot be neglected. This means that the evolution is not self-similar, because one can form a length scale (E/p_0)^ and a time scale (E/p_0)^/c_0 to describe the problem. The governing equations are then integrated numerically, as was done by H. Goldstine and
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
, Brode, and Okhotsimskii et al. Furthermore, in this stage, the compressing shock wave is necessarily followed by a rarefaction wave behind it; the waveform is empirically fitted by the Friedlander waveform.


Cylindrical line explosion

The analogous problem in cylindrical geometry corresponding to an axisymmetric blast wave, such as that produced in a
lightning Lightning is a natural phenomenon consisting of electrostatic discharges occurring through the atmosphere between two electrically charged regions. One or both regions are within the atmosphere, with the second region sometimes occurring on ...
, can be solved analytically. This problem was solved independently by Leonid Sedov, A. Sakurai and S. C. Lin.Lin, S. C. (1954). Cylindrical shock waves produced by instantaneous energy release. Journal of Applied Physics, 25(1), 54-57. In cylindrical geometry, the non-dimensional combination involving the radial coordinate r (this is different from the r in the spherical geometry), the time t, the total energy released per unit axial length E (this is different from the E used in the previous section) and the ambient density \rho_0 is found to be :r\left(\frac\right)^.


See also

* Guderley–Landau–Stanyukovich problem *
Zeldovich–Taylor flow Zeldovich–Taylor flow (also known as Zeldovich–Taylor expansion wave) is the fluid motion of gaseous detonation products behind Chapman–Jouguet detonation wave. The flow was described independently by Yakov Zeldovich in 1942 and G. I. Taylo ...
* Becker–Morduchow–Libby solution


References

{{DEFAULTSORT:Taylor-von Neumann-Sedov blast wave Fluid dynamics Equations of fluid dynamics