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[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. 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

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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   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[4]

 

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 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 gases with constant specific heats, we have  . The above set of equations cannot be solved analytically, but has to be integrated numerically. The solution has to be found for the range   subjected to the condition   at  

The function   is found to monotonically decrease from its value   to zero at a finite value of  , 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. Interior to the weak discontinuity   everywhere.

Location of the weak discontinuity (Mach wave)

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From the second equation described above, it follows that when  ,  . More precisely, as  , that equation can be approximated as[5]

 

As  ,   and   if   decreases as  . The left hand side of the above equation can become positive infinity only if  . Thus, when   decreases to the value  , the gas comes to rest (Here   is the sound speed corresponding to  ). Thus, the rarefaction motion occurs for   and there is no fluid motion for  .

Behavior near the weak discontinuity

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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   (the other possibility is   although it is of no interest here). If the gas is at rest  , then the disturbance speed is  . This is just a normal sound wave propagation. If however   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,  , where   is the specific heat ratio). This means that the above equation can be written as

 

whose solution is

 

where   is a constant. This determines   implicitly in the neighborhood of the week discontinuity where   is small. This equation shows that at  ,  ,  , but all higher-order derivatives are discontinuous. In the above equation, subtract   from the left-hand side and   from the right-hand side to obtain

 

which implies that   if   is a small quantity. It can be shown that the relation   not only holds for small  , but throughout the rarefaction wave.

Behavior near the detonation front

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First let us show that the relation   is not only valid near the weak discontinuity, but throughout the region. If this inequality is not maintained, then there must be a point where   between the weak discontinuity and the detonation front. The second governing equation implies that at this point   must be infinite or,  . Let us obtain   by taking the second derivative of the governing equation. In the resulting equation, impose the condition   to obtain  . This implies that   reaches a maximum at this point which in turn implies that   cannot exist for   greater than the maximum point considered since otherwise   would be multi-valued. The maximum point at most can be corresponded to the outer boundary (detonation front). This means that   can vanish only on the boundary and it is already shown that   is positive near the weak discontinuity,   is positive everywhere in the region except the boundaries where it can vanish.

Note that near the detonation front, we must satisfy the condition  . The value evaluated at   for the function  , i.e.,   is nothing but the velocity of the detonation front with respect to the gas velocity behind it. For a detonation front, the condition   must always be met, with the equality sign representing Chapman–Jouguet detonations and the inequalities representing over-driven detonations. The analysis describing the point   must correspond to the detonation front.

See also

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References

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  1. ^ Zeldovich, Y. B. (1942). On the distribution of pressure and velocity in the products of a detonation explosion, specifically in the case of spherical propagation of the detonation wave. Journal Experimental Theoretical Physics, 12(1), 389.
  2. ^ Zeldovich, Y. B., & Kompaneets, Alexander Solomonovich (1960). Theory of detonation. Academic Press, Section 23, pp. 279-284
  3. ^ Taylor, G. I. (1950). The dynamics of the combustion products behind plane and spherical detonation fronts in explosives. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 200(1061), 235-247.
  4. ^ Sedov, L. I., & Volkovets, A. G. (2018). Similarity and dimensional methods in mechanics. CRC press.
  5. ^ Landau, L. D., & Lifshitz, E. M. (1987). Fluid Mechanics Pergamon. New York, section 130, pp-496-499.