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
Yakov Borisovich Zeldovich (, ; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet people, Soviet Physics, physicist of Belarusians, Belarusian origin, who is known for his prolific contributions in physical Physical c ...
in 1942 and
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 ...
in 1950, although
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 ...
carried out the work in 1941 that being circulated in the British Ministry of Home Security. Since naturally occurring detonation waves are in general a
Chapman–Jouguet detonation wave, the solution becomes very useful in describing real-life detonation waves.
Mathematical description
Consider a spherically outgoing
Chapman–Jouguet detonation wave propagating with a constant velocity
. By definition, immediately behind the detonation wave, the gas velocity is equal to the local
sound speed
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. More simply, the speed of sound is how fast vibrations travel. At , the speed of sound in air is about , or in or one m ...
with respect to the wave. Let
be the radial velocity of the gas behind the wave, in a fixed frame. The detonation is ignited at
at
. For
, the gas velocity must be zero at the center
and should take the value
at the detonation location
. The fluid motion is governed by the inviscid
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 ...
:
where
is the density,
is the pressure and
is the entropy. The last equation implies that the flow is isentropic and hence we can write
.
Since there are no length or time scales involved in the problem, one may look for 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 ...
of the form
, where
. The first two equations then become
:
where prime denotes differentiation with respect to
. We can eliminate
between the two equations to obtain an equation that contains only
and
. Because of the isentropic condition, we can express
, that is to say, we can replace
with
. This leads to
:
For
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 ...
es with constant specific heats, we have