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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 D. 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 ...
c with respect to the wave. Let v(r,t) be the radial velocity of the gas behind the wave, in a fixed frame. The detonation is ignited at t=0 at r=0. For t>0, the gas velocity must be zero at the center r=0 and should take the value v=D-c at the detonation location r=Dt. 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 ...
: \begin \frac + v\frac &= - \rho\left(\frac + \frac\right),\\ \frac + v \frac &= - \frac\frac,\\ \frac + v \frac &= 0 \end where \rho is the density, p is the pressure and s is the entropy. The last equation implies that the flow is isentropic and hence we can write c^2=d p/d \rho. 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 v(r,t)=v(\xi), p(r,t) = p(\xi),\, \rho(r,t) = \rho(\xi),\,c(r,t) = c(\xi), where \xi=r/t. The first two equations then become : \begin (\xi-v)\rho'/\rho &= v' + 2v/\xi,\\ (\xi-v) v' &= p'/\rho = c^2 \rho'/\rho \end where prime denotes differentiation with respect to \xi. We can eliminate \rho'/\rho between the two equations to obtain an equation that contains only v and c. Because of the isentropic condition, we can express \rho = \rho(c), \, p=p(c), that is to say, we can replace \rho^d\rho/dx with \rho^c'd\rho/dc. This leads to :\begin (\xi-v)\frac\frac c' &= v' + 2v/\xi,\\ \left frac-1\right' &= \frac. \end 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 \rho^d\rho/dc = 2/ \gamma-1)c/math>. The above set of equations cannot be solved analytically, but has to be integrated numerically. The solution has to be found for the range 0\leq \xi \leq D subjected to the condition \xi-v=c at \xi=D. The function v(\xi) is found to monotonically decrease from its value v(D) = c(D)-D to zero at a finite value of \xi, 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 v=0 everywhere.


Location of the weak discontinuity ( Mach wave)

From the second equation described above, it follows that when v=0, \xi=c. More precisely, as v\rightarrow 0, that equation can be approximated asLandau, L. D., & Lifshitz, E. M. (1987). Fluid Mechanics Pergamon. New York, section 130, pp-496-499. :(\ln v)' = 2c^2/ xi(\xi^2-c^2) As v\rightarrow 0, \ln v\rightarrow -\infty and (\ln v)'\rightarrow \infty if \xi decreases as v\rightarrow 0. The left hand side of the above equation can become positive infinity only if \xi\rightarrow c. Thus, when \xi decreases to the value \xi=c_0, the gas comes to rest (Here c_0 is the sound speed corresponding to v=0). Thus, the rarefaction motion occurs for c_0<\xi\leq D and there is no fluid motion for 0\leq \xi \leq c_0.


Behavior near the weak discontinuity

Rewrite the second equation as :v\frac = \frac\xi\left frac-1\right In the neighborhood of the weak discontinuity, the quantities to the first order (such as v,\,\xi-c_0,\,c-c_0) reduces the above equation to :v\frac(\xi-c_0) = (\xi-c_0) - (v+c-c_0). 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 v+c (the other possibility is v-c although it is of no interest here). If the gas is at rest v=0, then the disturbance speed is c_0. This is just a normal sound wave propagation. If however v is non-zero but a small quantity, then one find the correction for the disturbance propagation speed as v+c=c_0 + \alpha_0 v obtained using a
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion, where \alpha_0 is the Landau derivative (for
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 ...
, \alpha_0=(\gamma+1)/2, 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 ...
). This means that the above equation can be written as :v\frac(\xi-c_0) - (\xi-c_0) = \alpha_0 v whose solution is :\xi-c_0 = \alpha_0 v\ln (A/v) where A is a constant. This determines v(\xi) implicitly in the neighborhood of the week discontinuity where v is small. This equation shows that at \xi=c_0, v=0, dv/d\xi=0, but all higher-order derivatives are discontinuous. In the above equation, subtract v+c-c_0 from the left-hand side and \alpha_0v from the right-hand side to obtain :\xi-v-c = (\xi-c_0)-(v+c-c_0)=\alpha_0 v ln(A/v)-1/math> which implies that \xi-v>c if v is a small quantity. It can be shown that the relation \xi-v>c not only holds for small v, but throughout the rarefaction wave.


Behavior near the detonation front

First let us show that the relation \xi-v>c 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 \xi-v=c,\, v\neq 0 between the weak discontinuity and the detonation front. The second governing equation implies that at this point v' must be infinite or, d\xi/dv=0. Let us obtain d^2\xi/dv^2 by taking the second derivative of the governing equation. In the resulting equation, impose the condition \xi-v=c,\,v\neq 0,\, d\xi/dv=0 to obtain d^2\xi/dv^2 = -\alpha_0 \xi/c_0v\neq 0. This implies that \xi(v) reaches a maximum at this point which in turn implies that v(\xi) cannot exist for \xi greater than the maximum point considered since otherwise v(\xi) would be multi-valued. The maximum point at most can be corresponded to the outer boundary (detonation front). This means that \xi-v-c can vanish only on the boundary and it is already shown that \xi-v-c is positive near the weak discontinuity, \xi-v-c is positive everywhere in the region except the boundaries where it can vanish. Note that near the detonation front, we must satisfy the condition \xi-v=c,\, v\neq 0. The value evaluated at \xi=D for the function \xi-v, i.e., D-v(D) is nothing but the velocity of the detonation front with respect to the gas velocity behind it. For a detonation front, the condition D-v(D)\leq c(D) must always be met, with the equality sign representing Chapman–Jouguet detonations and the inequalities representing over-driven detonations. The analysis describing the point \xi-v=c,\, v\neq 0 must correspond to the detonation front.


See also

* Taylor–von Neumann–Sedov blast wave * Guderley–Landau–Stanyukovich problem


References

{{DEFAULTSORT:Zeldovich-Taylor flow Flow regimes Fluid dynamics Combustion Hyperbolic partial differential equations