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[1][2] and G. I. Taylor in 1950,[3] although G. I. Taylor carried out the work in 1941 that being circulated in the British Ministry of Home Security.
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
be the radial velocity of the gas behind the wave, in a fixed frame.
The fluid motion is governed by the inviscid Euler equations[4] where
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 of the form
The first two equations then become where prime denotes differentiation with respect to
This leads to For polytropic gases with constant specific heats, we have
The above set of equations cannot be solved analytically, but has to be integrated numerically.
, where a weak discontinuity (that is a function is continuous, but its derivatives may not) exists.
The region between the detonation front and the trailing weak discontinuity is the rarefaction (or expansion) flow.
The left hand side of the above equation can become positive infinity only if
Rewrite the second equation as In the neighborhood of the weak discontinuity, the quantities to the first order (such as
) reduces the above equation to At this point, it is worth mentioning that in general, disturbances in gases are propagated with respect to the gas at the local sound speed.
In other words, in the fixed frame, the disturbances are propagated at the speed
This is just a normal sound wave propagation.
is non-zero but a small quantity, then one find the correction for the disturbance propagation speed as
obtained using a Taylor series expansion, where
is the Landau derivative (for ideal gas,
implicitly in the neighborhood of the week discontinuity where
from the right-hand side to obtain which implies that
between the weak discontinuity and the detonation front.
The second governing equation implies that at this point
by taking the second derivative of the governing equation.
In the resulting equation, impose the condition
reaches a maximum at this point which in turn implies that
The maximum point at most can be corresponded to the outer boundary (detonation front).
Note that near the detonation front, we must satisfy the condition