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 independently by
G. I. Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as ...
,
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
and
Leonid Sedov
Leonid Ivanovich Sedov (russian: Леонид Иванович Седов; 14 November 1907 – 5 September 1999) was a leading Soviet expert on hydro- and aerodynamics and applied mechanics.
In 1930 Sedov graduated from the Moscow State Universit ...
during
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...
.
History
G. I. Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as ...
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 (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
,
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
was working on the same problem and he presented his results on June 30, 1941. It was said that
Leonid Sedov
Leonid Ivanovich Sedov (russian: Леонид Иванович Седов; 14 November 1907 – 5 September 1999) was a leading Soviet expert on hydro- and aerodynamics and applied mechanics.
In 1930 Sedov graduated from the Moscow State Universit ...
was also working on the problem around the same time in the
USSR
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nationa ...
, 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) 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
Grigory Isaakovich Barenblatt (russian: Григо́рий Исаа́кович Баренблат; 10 July 1927 – 22 June 2018) was a Russian mathematician.
Education
Barenblatt graduated in 1950 from Moscow State University, Department of Mec ...
) 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 De ...
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
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 med ...
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
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
) instantaneously at time
.
The shock wave in the self-similar region is assumed to be still very strong such that the pressure behind the shock wave
is very large in comparison with the pressure (atmospheric pressure) in front of the shock wave
, which can be neglected from the analysis. Although the pressure of the undisturbed gas is negligible, the density of the undisturbed gas
cannot be neglected since the density jump across strong shock waves are 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
and the corresponding sound speed
, but keeping its density non zero, i.e.,
.
The only parameters available at our disposal are the energy
and the undisturbed gas density
. The properties behind the shock wave such as
are derivable from those in front of the shock wave. The only non-dimensional combination available from
and
is
:
.
The solution is assumed to be a function only of the above variable. The shock wave location
will correspond to some value denoted here by
, i.e.,
:
The propagation velocity of the shock wave is
:
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
,
and
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 a ...
as follows
:
where
is the
specific heat ratio. Since
is a constant, the density immediately behind the shock wave is not changing with time, whereas
and
decrease as
and
, respectively.
Self-similar solution
The gas motion behind the shock wave is governed by
Euler equations
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are 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 ...
. For an ideal polytropic gas with spherical symmetry, the equations for the fluid variables such as radial velocity
, density
and pressure
are given by
:
At
, 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
since pressure can be obtained from the formula
. The following non-dimensional self-similar variables are introduced,
:
.
The conditions at the shock front
becomes
:
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
and
can be deduced directly from energy conservation. Since the energy associated with the undisturbed gas is neglected by setting
, the total energy of the gas within the shock sphere must be equal to
. Due to self-similarity, it is clear that not only the total energy within a sphere of radius
is constant, but also the total energy within a sphere of any radius
(in dimensional form, it says that total energy within a sphere of radius
that moves outwards with a velocity
must be constant). The amount of energy that leaves the sphere of radius
in time
due to the gas velocity
is
, where
is the
specific enthalpy
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant p ...
of the gas. In that time, the radius of the sphere increases with the velocity
and the energy of the gas in this extra increased volume is
, where
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, sto ...
of the gas. Equating these expressions and substituting
and
that is valid for ideal polytropic gas leads to
:
The continuity and energy equation reduce to
:
Expressing
and
as a function of
only using the relation obtained earlier and integrating once yields the solution in implicit form,
:
where
:
The constant
that determines the shock location can be determined from the conservation of energy
:
to obtain
:
For air,
and
. The solution for
is shown in the figure by graphing the curves of
,
,
and
where
is the temperature.
Asymptotic behavior near the central region
The asymptotic behavior of the central region can be investigated by taking the limit
. 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
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
. The temperature ratio follows from the ideal gas law; 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
:
Remember that the density
is time-independent whereas
which means that the actual pressure is in fact time dependent. 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
:
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
becomes comparable to
(more precisely, when
). At this later stage of the evolution,
(and consequently
) cannot be neglected. This means that the evolution is not self-similar, because one can form a length scale
and a time scale
to describe the problem. The governing equations are then integrated numerically, as was done by H. Goldstine and
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
, Brode, and Okhotsimskii et al.
Cylindrical line explosion
The analogous problem in cylindrical geometry corresponding to an axisymmetric blast wave can be solved analytically. This problem was solved independently by
Leonid Sedov
Leonid Ivanovich Sedov (russian: Леонид Иванович Седов; 14 November 1907 – 5 September 1999) was a leading Soviet expert on hydro- and aerodynamics and applied mechanics.
In 1930 Sedov graduated from the Moscow State Universit ...
, 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.]
See also
*
Guderley–Landau–Stanyukovich problem Guderley–Landau–Stanyukovich problem describes the time evolution of converging shock waves. The problem was discussed by G. Guderley in 1942 and independently by Lev Landau and K. P. Stanyukovich in 1944, where the later authors' analysis was p ...
*
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. Tayl ...
References
{{DEFAULTSORT:Taylor-von Neumann-Sedov blast wave
Fluid dynamics
Equations of fluid dynamics