Detonation in a relaxing gas and relaxational instability

8. V. K. Baev, E. A. Solovova, and P. K. Tret'yakov, in: Problems in Gas Dynamics [in Russian], ITPM, Novosibirsk (1975).

9. B. P. Leonov, S. V. Shteinman, and A. V. Kulikov, Fizo Goreniya Vzryva, ~, No. 4, 572 (1971).

i0. So L. Plee and A. M. Mellor, Combust. Flame, 32, No. 2, 193 (1978). ii. B. Lewis and G. Von Elbe, Combustion, Flames, and Explosions of Gases, Academic Press

(1961). 12. V. K. Baev, B. V. Boshenyatov, Yu. A. Pronin, et aZ., Fiz. Goreniya Vzryva, 17, No. 3,

72 (1981). 13. Mo E. Rudyak, Izv. Vyssh. Uchebn. Zaved., Aviatsion. Tekh., No. 3, 32 (1983). 14. R. B. Edel'man, S. S. Shmotolokha, and S. Slutskii, Raket. Tekh. Kosmn., 9, No. 7, 180

(1971). 15. V. L. Zimont and Yu. M. Trushin, Fiz. Goreniya Vzryva, ~,. No. i, 86 (1967). 16. A. S. Sokolik, Spontaneous Combustion, Flame, and Detonation in Gases [in Russian],

Izd. Akad. Nauk SSSR, Moscow (1960). 17. V. F. Sokolenko, R. S. Tyul'panov, and Yu. V. Ignatenko, Fiz. Goreniya Vzryva, [, No.

4, 566 (1971). 18. S. I. Baranovskii, in: Investigation of the Working Process in the Elements of Engines

and Power-Generating Devices with a Two-Phase Working Body [in Russian], G. N. Abramo- vich and I. A. Lepeshinskii (eds.), Moscow (1980).

19. J. M. Hoyt and J. J. Taylor, Trans. ASME, J. Fluids Eng., 101, No. 3, 304 (1979). 20. W. T. Pimbley and H. C. Lee, IBM J. Res. Develop., 21, No. i, 21 (1977). 21. H. Tsuju and J. Yamaoka, 13th Symp. (Int.) on Combustion, Pittsburgh (1971).

DETONATION IN A RELAXING GAS AND RELAXATIONAL INSTABILITY

N. M. Kuznetsov and V. A. Kopotev

One-dimensional detonation in a gas, whose internal degrees of freedom (not directly related with irreversible liberation of energy) are characterized by a finite relaxation time r, is studied theoretically. In the limiting cases of small and large values of the paramete

E ~/TQ (where TQ is the characteristic heating time) such a process transforms correspondin ly into the classical detonation (Zel'dovich-Neiman-Dering theory) or detonation with an en- dothermal stage of the reaction [i, 2]. A relaxing gas is essentially a variant of a system with finite reaction rates (system with a dispersion of the velocity of sound), whose detonation was studied in detail in [3-5], where the question of the selection of the velocity of detonation in such a medium was first posed and studied.

Kirkwood and Wood [3] showed, applying the method of characteristics to reacting gaseous mixtures (the fact that the characteristics propagate with the frozen velocity of sound was employed in the analysis), that in a one-dimensional flow the rarefaction wave can be smoothly joined to the stationary zone of the detonation wave (DW) only at the point (Jouguet where the velocity of the gas u relative to the DW equals the frozen (over the reactions) velocity of sound cf. It was shown in [4, 5], however, that when heat is liberated in a mono tonic fashion the velocity of the reaction products u in the stationary zone of the wave cannot exceed the value of the equilibrium velocity of sound Ce, for which the inequality Ce < cf always holds [6-8]. Investigation of nonmonotonic liberation of heat (the so-cal~ed "pathological" detonation) showed [i, 2, 6, 9, i0] that~ in this case, the Jouguet point with u = cf lies above the equilibrium detonation adiabat (where the system is in a nonequilibrium state) and the final velocity of the reaction products is greater than cf. The supersonic (with respect to cf) velocity of the reaction products can also be obtained as a result of the action of the boundary layer at the walls of the detonation tube on the flow [ii]. Thus, the difficulties associated with the non-single-valued nature of the velocity of sound were eliminated only in particular cases of detonation. In the general case, however, the questio~

Moscow. Translated from Fizika Goreniya i Vzgyva, Vol. 22, No. 5, pp. 75-86, September- October, 1986. Original article submitted August 2, 1984; revision submitted July 22, 1985.

0010-5082/86/2205-0563512.50 �9 1987 Plenum Publishing Corporation 563

of why the rarefaction wave, moving with the velocity cf, does not weaken the shock and how the selection of the detonation velocity occurs remained open.

Theoretical investigation of rarefaction waves in reacting media showed that only the forward front of the wave moves with the frozen velocity cf [12, 13]. The energy maximum occurs at some average part of the wave, whose velocity is less than cf. Based on this it was concluded qualitatively that the DW may not be attenuated for u < ~f also, and preference was given to choosing the equilibrium velocity of sound as the velocity of detonation (i.e., according to the condition at the Jouguet point, u = Ce, corresponding to the point at which the Rayleigh straight line touches the equilibrium detonation adiabat), as being the least inconsistent in this situation [6-8~ 14-17]. In many computational works on detonation (see, for example. [14-17]) the equality u = ce was used at the Jouguet point, although the experimental data in many cases [18-21] corresponded best to the condition u = cf. Additional difficulties arose in connection with the experiments of [18, 22, 23], in which DW velocities corresponding to u (at the Jouguet point) equal to the velocity of sound frozen not only according to the composition of the gas but also according to the energy of some internal degrees of freedom of the molecules were observed.

The above-described difficulties of the theory of detonation in a medium with dispersion of the velocity of sound (still, is this dispersion caused by the finite reaction rates or by the relaxation of the energy stored in the internal degrees of freedom of the molecules?) were often regarded as inconsistencies, inherent to the classical theory of detonation itself [7, 8, 24]. In this connection, later regimes of self-maintained (normal) and overdrivel one-dimensional detonation for arbitrary values of the parameter ~ werestudied. The basic qualitative features of such regimes were determined for the example of an ideal gas with relaxing vibrational energy of the molecules. The "topology" of the evolution, continuous as a function of ~, of the normal (Jouguet) and overdriven detonation regimes in the p-v plan, (the pressure, specific volume of the gas) as a function of = was investigated. It was shown that there exist values ~, and =** < ~, of the order of unity, such that for = < =, the liberation of heat in the translational degrees of freedom of the gas is nonmonotonic (analog of the "pathological" detonation) and at the Jouguet point (nonequilibrium) u = cf; for ~, > = > =~, the liberation of heat becomes monotonic, and the inequalities cf > u > ce hold at the Jouguet point (now equilibrium); for e < a** the liberation of heat remains monotonic at the equilibrium Jouguet point u = Ce also.

The velocity of normal detonation DJ in the limit = ~ ~ corresponds to the Rayleigh line being tangent to the "frozen" detonation adiabat Ff (the adiabat calculated with completely frozen vibrational degrees of freedom of the molecules). A decrease in = corresponds to a drop in DJ, which for = E =** assumes the minimum value corresponding to the Rayleigh line being tangent to the equilibrium detonation adiabat Fe. The question of the joining of the stationary zone of the DW to the rarefaction wave is solved. The procedure for calculating the parameters at the Jouguet point and the velocity of normal detonation in other gaseous systems with relaxation is discussed qualitatively. A new (relaxational) mechanism of instability of the DW is predicted. It is shown that when = > =**, the state at the end of the stationary zone of the wave depends infinitely strongly on its intensity; an infinitesimal change in the intensity results in a finite change in the pressure, temperature, and density at the boundary of the stationary zone. The boundary itself moves over a finite distance. Such a dependence leads to finite pressure drops in a direction perpendicular to the undisturbed flow and, therefore, to gasdynamic pulsations. In the case ~,~ < ~ < =, these pulsations, arising initially outside the boundaries of the stationary zone of the wave, propagate in a relaxing gas upward along the stationary zone, overtaking and deforming the forward front of the wave. The pulsations vanish in overdriven waves, whose intensity exceeds some critical value.

Equations of Stationary Flow

In a coordinate system fixed to the front of the wave, the equations describing the stationary flow of gas behind the shock have the following form in the standard notation:

ulv = Dlvo =--i, aVv + p = DVvo + po,

"f~ (-f~ - 1 ) - ' pv + E + uV2 + Q ( i - a ) ) = 7~ (-f~ - l ) -'poVo + DV2 + Q, dE/dt----(E~--E)/z, E=vpovo a t t - - - - O .

(i)

(2)

(3)

564

At the shock the time t = 0; the index 0 corresponds to the state in front of the shock; D is the velocity of the detonation wave; E and Ee are the specific vibrational energy and its equilibrium value; ~f, ye = [I + (Tf - 1)(7 + i)]/[i + (Tf - i)~] are the frozen (at E = 0) and equilibrium (at E = Ee) values of the adiabatic index of the gas; v is the number of vibrational degrees of freedom; Q is the specific heat of the chemical reaction. The nonnega- tive and monotonic (for t s 0) function ~(t/~Q) characterizes the kinetics of the irreversibl~ liberation of heat and by definition satisfies the relations

~ ( 0 ) =0, ~ ' = 0 at ~ = t . (4)

The value of Ee depends only on the temperature. For simplicity the classical approximation Ee = vpv and the constant quantity ~, identical for all oscillators, are used.

From (i) and (2) we obtain a quadratic equation for v:

A v ~ - - B v + c = O ,

A = t / 2 (? , + i ) (~,~- t ) - ' j - ' , B = ? f ( ? : - t)- ' /2Vo ( I + ~),

c (t) = Q~ - E + ~, (7, - i ) -tPoUo + DU2, s =- povo/D a.

The following solution of (5) corresponds to the flow behind the shock:

(s)

v = (B - ~ B ~ - & 4 c ) / 2 A . (6)

Formulas (6) and (3) determine v as a function of the time t only [with the help of (1), p and u can likewise be expressed as functions of t]. It is easy to obtain from (5) the de- rivative of v with respect to the time on the Rayleigh line:

dz~ = dc/dt = ?~- - I Q d ~ / d t - - dE~dr (7) dt B--2Av ?I ~ I ]2(vf-- v)

On any Rayleigh line at the poinfs v = vf 5 (1 + e)Tf(Tf + l)-iv0 and v = Ve 5 (i + e)Te(~e + l)-Iv0 the equalities u 2 = cf = ~ ~fpv and u 2 = Ce 2 ~ u respectively, hold. This can be seen from the relations

u ~- = c~ - f - (~'I + ! ) (~'j - ~,) v = c~ ~ / ~ (?, + 1) (~, - v ) v, ( 8 )

which follow from (i). For detonation, as a rule, ~ << 1 (strong DW). In this case, u = cf and u = ce hold with good accuracy on the isochores v = vf and v = Ve, respectively. For simplicity we assume below that the inequality g << 1 holds.

At first glance it follows from (7) that on the shock for any = there exists one singular point v = vf (where u = cf) and that in the case of normal detonation at this point the numerator must vanish (an analogous result was obtained in [3, 25]). On the other hand, if we first pass to the limit = ~ 0, assuming, according to [3], dE/dt = dEe/dt, then

dv ?~ -- i Qdo~/dt Od~/dt

and the singular point is located on the equilibrium isochore v = Ve. To clarify this para- dox it is necessary to investigate the solution of Eq. (7) as a function of the parameters j and ~.

Qualitative Analysis of the Integral Curves in the p-v Plane

In the p-v plane the trajectory of the point characterizing the state of the gas (we shall call it the image point) is a segment of the corresponding Rayleigh line. For a fixed heat-liberation function ~ and a fixed initial state, the trajectory of the image point is determined by the two parameters j and = only. We introduce on this trajectory the dimen- sionless Lagrangian time ~ m t/TQ. To solve Eq. (7) in the p-v plane it is convenient to rewrite (7) and (3) in the following manner:

u'= ~- ~ Q~'--E' ?,--IQ(~+=~.+~_l) (9) ~I+ ~j2(~ i-O= YI+ I =J~(~1-- ~) '

E' ==-'(E,-E), (i0)

S65

where T(j, v) ~ Q-lj2[v2(a + ~ - 1/2) -vv0(l + ~)(a + ~) + v0=(i/2 + ~a + ~)] + i; a 7f(Yf - i)-i; the prime indicates derivatives with respect to ~.

Figure 1 shows the equilibrium Fe (i.e., for E = Ee) and frozen Ff (for E = 0) detonation adiabats of the gas. For strong DW the isochores v = Ve and v = vf pass, respectively, through the equilibrium Je and frozen Jf Jouguet points (at these points the Rayleigh lines are tangent to the detonation adiabats Fe and Ff, respectively). The function T is positive below and negative above the equilibrium adiabat Fe; on the adiabat Fe, T = 0 (the same can be said for the function T- 1 and the equilibrium shock adiabat of the gas). The starting shock-compressed gas expands for Q~' > E' [v' > 0, see (9)], and the image point moves along the Rayleigh'line from top to bottom. [For sufficiently low values of = at the beginning of the irreversible process it may happen that Q~' < E'j and the stage of expansion in this case is preceded by a brief stage of compression, corresponding to the vibrational relaxation in a shock wave (SW). This stage, corresponding to the formation of an SW structure, is of no interest for the problem under study.] If in the process of such expansion the isochore v = vf for Q~' - E' > 0 is reached, then on it v' = = (the so-called locking of the flow occurs [8]), indicating that a stationary regime cannot exist. If the numerator of (9) vanishes first and the liberation of heat is not yet completed (~ ~ I), then in this case the expansion is replaced by compression. The point with the maximum value of v on this trajectory, corresponding to overdriving of the detonation, is a singular return point on the real curve [26]. In what follows we shall briefly call this point the return point. For a complex (for example, step) heat liberation function the motion of the image point along the Rayleigh line can be much more complicated: with a series of alternating expansions and compressions of the gas. In this case, for further analysis, only the return point with the highest value of v is important, and it is this point that we shall call the return point. We shall call the set of return points with fixed e and different values of j the return line La of the overdriven regimes (Le is defined only for them)~

For any ~ the lines L= lie above the equilibrium adiabat Fe. Indeed, the function T is negative and the numerator in (9) can change sign on the Rayleigh line from plus to minus, subsequently remaining negative until it reaches the image point of the adiabat Fe, only in this part of the p-v plane. With the help of Eqs. (9) and (I0) it can be shown that for fixe,

the return points (if they exist) with the largest value of v correspond to the lowest valu, of j and as ~ decreases the lines L= move downward in the p-v plane. The first assertion follows from the fact that on the trajectories with the lowest value of j (for fixed =) each value of v corresponds to a larger value of v' [see the second inequality (9)] and a lower value of E e and therefore of E' also [see (i0)]. As a result, the return point where Q~' = E' is reached for large values of v. The second assertion follows from the first assertion and from the fact that for fixed j small values of = correspond to return points with the lower value of v (since a decrease of v corresponds to an increase in E' along the trajectory

p

~l, Ao I

o ~'e ~" ~o v

Fig. i. Dependence of normal and overdriven detonation regimes on ~. Ff, L~ l, L~ 2, L~ 3 and Jf~ J1, J2, Je are the corresponding lines of return and the Jouguet points for

= ~, ~i, ~2, ~3 (~i > ~ > ~2 > ~ > ~).

566

of motion). On the Rayleigh line the entropy of the gas, neglecting the vibrational degrees of freedom, reaches a maximum at the point of intersection with the line L~. The temperature of the gas, however, moving down on the Rayleigh line, increases if v < Vm E 1/2(1 + s and drops if v > Vm. In what follows we shall call v = v m the isochore of the maximum temperature. For dp/dv the equality

-- v2dp/dv = u z (ii)

holds algng the Rayleigh line everywhere in accordance with (i). Normal detonation with fixec a corresponds to the regime jJ(a) with minimum j.

We shall now study the results of the qualitative analysis of Eq. (9).

Dependence of Normal and Overdriven Detonation Parameters on the Parameter a

I. In the limit a + ~ at the stage of liberation of heat E' = E = 0 everywhere (including on the line La). Therefore La practically coincides with the overdriven branch of the "frozen" adiabat Ff (see Fig. i). For overdriven detonation regimes (which correspond to Rayleigh lines passing above the point Jr) the image point first drops along the Rayleigh lin~ up to the adiabat Ff (over a characteristic time rQ), and then the reverse motion begins alon! it and in the limit t/~ ~ ~ stops on the overdriven branch Fe.

In regimes corresponding to Rayleigh lines passing below the point Jf, the flow is locke< on the isochore vf. Normal detonation corresponds to the Rayleigh line passing through the Jouguet point Jf; the image point, having reached the point Jf [where both the denominator an~ numerator of (9) vanish simultaneously] along the straight line, continues to move downward. For detonation in an unbounded medium (or sufficiently far away from the outer boundaries) further stationary motion, which is not supersonic, is terminated at the point Af on the equilibrium adiabat Fe. This limiting case is essentially the same as the case of nonmonotonic liberation of heat, first studied in [i]. The time dependence of v (and through it of other variables also) in the region of supersonic stationary flows is determined by the second root of Eq. (5) [I0]: v = (B + /B 2 - 4Ac)/2A.

The straight line A0J f is tangent to the adiabat Ef at the point Jf and the line of return coinciding with it (above Jr). At the point Jf the following equalities hold [see (8) and (ii)]:

- - u : d p / d v = u ~ = c~. ( 12 )

2. The c a s e o f f i n i t e v a l u e s o f a , s a t i s f y i n g a > a , , where a , i s o f o r d e r u n i t y ( s e e b e l o w ) . The l i n e o f r e t u r n La o f s u c h d e t o n a t i o n r e g i m e s l i e s be low F f and a l s o t e r m i n a t e s on t h e i s o c h o r e v f [ s e e ( 9 ) ] . But u n l i k e t h e c a s e s t u d i e d a b o v e , h e r e t h e e x c i t a t i o n o f t h e v i b r a t i o n a l d e g r e e s o f f r e e d o m i s a l r e a d y s i g n i f i c a n t . We s h a l l s t u d y one s u c h l i n e La~, c o r r e s p o n d i n g to some f i x e d v a l u e a = a 1 > a n ( s e e F i g . 1 ) . E v e r y t h i n g s a i d p r e v i o u s l y a b o u t t h e m o t i o n o f t h e image p o i n t a l o n g t h e R a y l e i g h l i n e , c o r r e s p o n d i n g t o o v e r d r i v e n and n o r m a l detonation and also lying below the Jouguet point (in this case this is the point Ji), also holds for a = a I. At the point Jl the equalities (12), for whose satisfaction the condition v(J l) = vf is sufficient [see (8) and (Ii)], also hold. Here v(Ji) is the value of v at the point Jm. Analogous notation is used below also. We draw through the point J1 the nonequilibrium detonation adiabat FjI with the fixed values E = E(JI) and ~ = ~(Jl). It is not difficult to see that E(Jl) < Ee(J1) and m(J1) ! i. This follows from the fact that the adiabat FjI lies above Fe, where Q~(Jl) - E(Jl) > Q - Ee(J1). It follows from the fact that the equalities (12) hold at the point Jl that the Rayleigh line AoJ1 is tangent to FjI. It can be shown that on the line of return L= I the value of Q~ - E does not exceed the value of Q~(JI) - E(Jl), since as j increases Ee and E' increase along the entire trajectory [see (9) and (i0)]. Thus, in the neighborhood of the point Ji the line La I lies between the adiabat FjI and the Rayleigh line, passing through J1- Therefore, all three lines studied (FJl, Lal, and AoJl) have a common point of tangency J1- The entropy of the system, neglecting the vibrational degrees of freedom, has at the Jouguet point Ji a maximum on the straight line A0J i. The boundary of the stationary zone of normal detonation is determined by the point A I .

As a decreases, the corresponding lines L= drop even lower. The velocity of normal detonation also decreases. At some sufficiently low values of = ~ a,.~ the line La, terminates

567

at the point J, of intersection of the equilibrium adiabat Fe and the isochore vf (see Fig. i) The line La, and the point J~, correspond to the lowest value of the parameter a for which in the normal detonation regime the velocity of the gas reaches the value cf at the Jouguet point. At the same time the image point drops along the Rayleigh line to the point J, (in this case this is the Jouguet point), where the stationary region of flow terminates.

3. The values of a < a, correspond to lines La lying below La,. Figure 1 shows for some a2 < =, the line of return La2, terminating on the adiabat Fe at the point J2. As already mentioned, La 2 cannot continue beyond Fe. At the point J2, v' and ~ vanish simultaneously and therefore [see (9)]

~'+ ~-- I =0. (13)

Using (13) it is not difficult to prove that at the point J2 the liberation of heat is completed, i.e., m : I. The latter equality also follows from the fact that the arbitrary monotonic function ~ [bounded only by the conditions (4)] can satisfy the relation (13) for an infinite set of values a, corresponding to an infinite set of points of the type J2 (lying on Fe), only for m : i, where by definition m' : O.

This can be illustrated especially conveniently for the example of the exponential function of heat liberation m : 1 - exp (-~), for which m' : 1 - m and the relation (13) holds only at the point m : 1 for any a (with the exception of the degenerate case = : i; see below). It is easy to prove graphically that w : 1 for an arbitrary function m at the point J2 if (13) is interpreted as the intersection of single-valued [by virtue of the (by definition) monotonic dependence ~(~)] functions w'(w) and ~(~) ~ (i - ~)/a on the segment 0 ~ ~ ~ i

If at the point J2 ~ = i, then the system is in an equilibrium thermodynamic State (E = Ee), since this point lies on Fe, where Qm - E = Q - Ee. The point J2 is at the same time the boundary of the region of stationary flow and corresponds to the minimum velocity of detonation for = = =2. In this sense J2 is the Jouguet point. It is evident from (8) that at the point J2 the inequalities

c~ < - - v2dp/du : u: < c? ( 1 4 )

ho ld . For motion a long the R a y l e i g h l i n e s , l y i n g below J2 [ i . e . , f o r j < j j ( a 2 ) ] , the image p o i n t i n t e r s e c t s t he e q u i l i b r i u m a d i a b a t and moves downward to the i s o c h o r e v f , where t he f low i s locked . This can be shown wi th t he he lp of Eqs. (9) and (10) , t a k i n g i n t o account t he f a c t t h a t t he f u n c t i o n ~ i s p o s i t i v e below t h e a d i a b a t Fe and t h a t f o r j < j j ( a 2) ( f o r f i x e d a) the volume f ( J 2 ) expands more r a p i d l y t han f o r j = j j ( a 2 ) .

We denote by a** the value of a at which the line La, * terminates at the equilibrium Jouguet point Je.

4. If ~ < ~**, then the line of return exists only for sufficiently slow liberation of heat at the concluding stage of vibrational relaxation.

Figure I shows as an example one of the lines of return L=3, corresponding to some =3 < .... , for the case of liberation of heat at a constant rate (m' = 1 for ~ < i, ~' = 0 for $ =

i~? The point at which the line of return terminates K lies between Je and the point M of intersection of the isochore Vm and the adiabat Fe. The lower the value of =, the closer the line of return lies to the equilibrium adiabat Fe and the closer the point K lies to the point M. The line La merges with Fe, and the point K reaches M only in the limit ~ = O.

For exponential liberation of heat the lines of return have a form analogous to L~ for 1 < a < ~,,. As e decreases, the point K moves toward M, reaching it when = = i. The line of return in this case completely merges with Fe and ceases to exist when = < i.

For all a satisfying the condition = < =**, the minimum velocity of detonation corresponds to passage of the Rayleigh line through the equilibrium Jouguet point Je, where

--v2dp/dv = u 2 = Ce 2.

The final state for j > JJe lies on the overdriven branch Fe, as in the case of the classical detonation. If j < JJe, the image point moves up to the isochore vf, where the flow is locked. It can be proved, however, that for such a motion on the isochore v = ve the derivatives with respect to $ of all quantities approach infinity as = ~ 0. In other words, the locking of the flow in the limit = + 0 is transferred from the isochore vf to the isochore ve.

568

It is interesting to note the character of the change in E and Ee along the Rayleigh lines, corresponding to the normal detonation regimes. As already mentioned, in the process of expansion the temperature of the ideal gas and therefore E e also increase when v < Vm and decrease when v > Vm (see the definition of Vm). In the case ~ > ~,, E < Ee everywhere in the region of stationary flow, including at the Jouguet point. The equilibrium state (E = Ee) is reached in the supersonic, relative to cf, flow, at points of the type A l as a result of the opposite motion of E and Ee - owing to the increase in E and the decrease in the value of Ee after v = Vm. For ~, > ~ > ~**, as before, in the region of stationary flow E < Ee, but equilibrium is realized at the Jouguet point (of the type J2).

For ~ < ~**, the instantaneous value of the energy E follows with a very short delay ("tracks") the equilibrium value Ee. For this reason, during the expansion of the gas, after the isochore v = Vm is crossed, when Ee begins to decrease, E also decreases, following Ee. According to (3), in this case, in the right-hand neighborhood of the isochore v = Vm an inversion of the vibrational energy occurs: the difference Ee - E changes sign, and with furth~ expansion E ~ Ee. For this reason, for any ~ < ~** on the Rayleigh line, passing through Je, Q~' - E' ~ 0, and the image point moves monotonically (v' ~ 0). The fact that there are n9 lines of return (i.e., the motion of the image point is monotonic) on the section v(K) < v < Ve in overdriven detonation regimes (j > JJe) for ~ < ~** is also explained by inversion of E [here v(K) is the value of v at the point K; see Fig. i].

The above analysis shows that the boundary of the stationary zone of normal detonation moves monotonically in the p--v plane along the equilibrium adiabat Fe from Af to the point Je as ~ decreases from infinity to zero.

The numerical values of ~, and ~** are determined by the specific form of the heat-liber~ tion function ~ and the parameters of the gas. For example, for liberation of heat at a constant rate v 0 = 862 cm3/g, P0 = 1 atm, ~f = 7/5, ~ = 1 (i.e., Ye = 9/7), and Q = 1000 cal/g, we have e, = 0.108 and ~** = 0.0742. In the case of exponential liberation of heat, for the parameters of the gas indicated above, ~, = 1.41 and ~** = 1.28. The velocities of normal detonation, corresponding to ~, and ~** (determined only by the parameters of the gas), in both cases of liberation of heat equal 2387.74 and 2384.95 m/sec, respectively.

Joining of the DetonationWave with the Rarefaction Wave

We shall briefly discuss the question of the joining of the stationary zone of the DW with the rarefaction wave. According to the classical theory of detonation, weak disturbance~ do not penetrate from the nonstationary zone into the stationary zone, because the velocity of the gas on the boundary of the stationary zone at the Jouguet point equals the velocity of sound. In a relaxing gas this condition is not single-valued owing to the dispersion of the velocity of sound. The maximum velocity of sound cf corresponds to the high-frequency limit. High-frequency perturbations do not penetrate upward along the flow of gas beyond the Jouguet point, generally speaking, only when = > ~**. In the case of more rapid relaxation (~ < =,), according to the inequality (14), the high-frequency perturbations can penetrate beyond the Jouguet point, leading to unique entropy losses. In the important case of detonation in an unbounded medium (or for a diameter of the charge significantly greater than the critical value) these losses can be significant only at the initial stage of the motion of the detonation wave. The position of the Jouguet point at this stage is nonstationary and is determined by the intensity of the liberation of heat and the indicated losses. Far from the plane of initiation the nonstationary supersonic flow "flattens out" and becomes isentropic (centered rarefaction wave, in which the gradients of the pressure and other quantities decrease as a function of time as t-z). In addition, the spectrum of the perturbations shifts into the region of low frequencies and the nonstationary flow no longer affects the structure of the stationary detonation complex, in complete agreement with the classical theor

Velocity of Normal Detonation and Parameters at the Jouguet Point for Othe r Systems with Relaxation

The characteristics of detonation regimes studied for the example of vibrational relaxation are qualitatively characteristic also of systems with chemical relaxation, for example with dissociation of inert components or ionization (in very strong detonation waves). For practical purposes, as a rule, accurate calculations of the velocity of normal detonation and of the parameters at the Jouguet point are required. To calculate the thermodynamic parameters at the equilibrium Jouguet point Je (the point at which the Ray!eigh line is tangent

569

to the equilibrium detonation adiabat) and the corresponding velocity of detonation it is sufficient to know only the equilibrium constants of the final detonation products. There exist a number of programs for calculating numerically the parameters at the point Je for different combinations of gas mixtures [14-17]. As can be seen from the analysis performed, however, the regime of normal detonation corresponds to the equilibrium Jouguet point Je only in the case when the relaxation is much more rapid than the rate of liberation of heat.

In real detonating media several relaxation processes, as a rule, occur behind the shock: rotational and vibrational relaxation, dissociation and ionization, heat conduction (in hetero geneous systems), and others. If the rate of the i-th relaxation process is much higher than the rate of liberation of heat (Ti << TQ, this condition usually holds for rotational relaxation), then the corresponding degrees of freedom may be regarded as populated in an equilibrium manner when determining the Jouguet point and the rate of detonation. The degrees of freedom for which Ti >> TQ, however, in the motion of the gas from the shock to the Jouguet point are frozen. These conditions can in many cases substantially facilitate the accurate calculation of detonation parameters.

The relaxational processes for which T and TQ are of the same order of magnitude present the greatest difficulties for the calculation of the normal velocity of detonation and the parameters at the Jouguet point. The effect of such processes on the detonation parameters can be estimated by calculating (for example, with the help of available programs) points of the type Jf and Je, corresponding to tangency of the Rayleigh lines to the "frozen" and equilibrium (with respect to the energy of the relaxation processes under study) detonation adiabats of the gas. (Detailed calculations performed in [14] demonstrated that dissociation has a very significant effect on the detonation parameters.) Such an estimate makes it possible to establish the limits of uncertainty of the parameters of normal detonation. It is not difficult to show that, in this case, the gasdynamic parameters at the Jouguet point will exhibit the highest uncertainty, while the detonation rate will exhibit a relatively lower uncertainty. If the obtained uncertainty in the detonation parameters is admissible in practical applications, then the calculations can be terminated here. Otherwise the Jouguet point and the detonation velocity can be found with as high an accuracy as required by calculating the line of return corresponding to the given relaxation process. This procedure is more laborious, requiring knowledge of the kinetic constants of the relaxation process and of reactions determining the liberation of heat on detonation. The results of this work can also be used to carry out a qualitative analysis of such calculations.

RelaxationalMechanism of Instability of the Detonation Front

A well-known reason for the instability of a plane detonation wave front is the strong temperature dependence of the rate of liberation of heat [27-30] (we shall call the corresponding instability the thermal instability). We shall examine below a different reason for the instability of a plane DW. From the above analysis it follows that, in the case

> ~**, there exists an infinitely strong dependence of the state (pressure, density, temperature) at the end of the stationary zone of the wave on the intensity of the wave: an infinitesimal change in the intensity produces a finite change in the pressure, density, and temperature on the boundary of the stationary zone. At the same time, a finite change occurs in the extent of the stationary zone. This dependence is determined by relaxation processes. Therefore, in what follows, we shall call the instability of a plane detonation front, to which, according to theoretical calculations, the indicated processes should lead, the relaxational instability.

An infinitesimal change in the intensity of the SW 6p relative to the SW of a self-main- taining detonation produces a finite displacement of the boundary of the stationary zone in the p-v plane upward or downward depending on the sign of 6p. If ~** < = < ~,, then when ~p > 0 the boundary of the stationary wave moves away from the Jouguet point of the type J2 (Fig. 2) upward into the point B 2 of the first intersection of the Rayleigh line with the equilibrium detonation adiabat Fe; when ~p < 0 motion along the Rayleigh line is possible (ignoring the condition of coupling of the stationary zone with the rarefaction wave) up to the intersection with the isochore v ~ vf. For = > =, and 6p > 0 the boundary moves along the Rayleigh line from the bottom to the top point of intersection with the equilibrium detonation adiabat. Such displacements of the boundary in the p-~1 plane are accompanied by a finite change in the pressure and density behind it. In reality, this can happen within some quite wide stream tube, surrounded by the undisturbed flow. (The stream tube must be wide so that over the displacement time of the boundary a significant lateral unloading does

570

1 \

~L J~

\

~ Ye ~r u 8 0

Fig. 2. Amplitude Ap of relaxational pulsations of the pressure: Ap = p(BI) - p(Ai) for ~ > ~, and Ap = p(B2) - p(X) for a, >

> ~.

not occur.) In addition, finite pressure drops will appear in the direction opposite to the main flow and, therefore, gasdynamic pulsations will also occur.

The order of magnitude of the amplitude of the pressure pulsations is determined by the pressure drop Ap on the Rayleigh line, corresponding to the normal detonation, between the points of its top intersection with the equilibrium detonation adiabat [v = v(B2)] and with the isochore v = vf in the case =,, < = < =,, or between the top [v = v(Bz)] and bottom [v = v(A1)] points of intersection of the Rayleigh line with the equilibrium detonation adiabat for a > a, (see Fig. 2):

.o A Ap = ].~ v - - - - - [p(d) - -po lAv/[Vo_V(y) l ,

(15) Av = ( v ( A 1 ) _ _ ~ ( B I ) ' ~ > a * "

Here p(J) and v(d) are the pressure and specific volume at the corresponding douguet points. The value of the specific volume v(J) is bounded by the conditions

v~ ~ u ( J ) ~ vi (16)

and s h i f t s f rom t h e t o p b o u n d a r y (16 ) t o t h e b o t t o m b o u n d a r y when t h e p a r a m e t e r a d e c r e a s e s .

Using an order-of-magnitude estimate of Ap, we obtain

A v ~ v j - u e , v o - u ( J ) ~ V o - V ~ . (17 )

For strong DW in an ideal gas with relaxation of vibrational energy

?~--7~ I vj - - ve = (~I § 1) (?e + t) ~o, vo - - ve = Ve + 7 ~o. ( 1 8 )

Substitution of (17)" and (18) into (15) gives

A p l p ( J ) ~ ( y f - 7~)l(gJ + i ) . ( 19 )

Fo r a d i a t o m i c o n e - c o m p o n e n t g a s a p / p ( J ) = 0 . 0 5 . Fo r p o l y a t o m i c g a s e s i t can be two t o t h r e e times higher. The amplitude of the pulsations decreases when the concentration of the relaxin components in the detonating mixture of gases decreases.

The time of the shifts in the boundary of the stationary flow equals, in order of magnitude, the relaxation time ~. [Strictly speaking, this estimate is valid for perturbations which are not too small. It can be shown that in the limit 6p/p(J) + 0 the displacement time increases in proportion to in(p(J)/16p[).] The minimum radius r of the stream tube for which over this time the pressure does not have enough time to equalize in a direction transverse to the flow is determined by the relation

571

r = c z = C~TQ ~ ~.d,

where c is the velocity of sound at the Jouguet point (c - u); d is the characteristic width of the DW. The quantity r is the characteristic scale of the transverse pulsations (at least at the beginning of their development) for the instability mechanism under study, since the probability of an initial perturbation decreases very rapidly as r increases. As a increases, the characteristic transverse scale of the pulsations also increases relative to the width of the detonation wave.

The finite uncertainties and pulsations under study arise beyond the Jouguet point of the unperturbed flow. The characteristic pulsational frequencies are of the order of T -I. Disturbances propagating with the "frozen" velocity of sound cf predominate in the frequency spectrum of such pulsations. They can propagate upstream, reaching the shock front and stron~ ly deforming it, if the velocity of the gas at the Jouguet point is less than cf (because of the comparatively large gradients and amplitudes of the pressure in the pulsations their velocity of propagation can even be somewhat higher than the velocity of sound). This condition for the velocity at the Jouguet point holds in a finite range of values of the parameter a** < ~ < ~, where in order of magnitude it equals unity.% In the case ~ > a,, when at the Jouguet point u = cf, the pulsations occur only beyond the Jouguet point, without penetrating upstream and without affecting the shock. In addition, the relaxational pulsations encompass only part of the stationary region of the flow from the Jouguet point of the type Jl to the point of the type AI on the equilibrium adiabat Fe (see Fig. 2). Such a pulsating zone is stationary only on the average over time. As a increases the zone of pulsations expands and when =~.~ < = < =, it encompasses the entire stationary region from the shock up to the Jouguet point ~of the type J2)- The relaxational instability vanishes for sufficiently intense overdriving of the DWo The randomly appearing disturbance Ap, characterized by the amplitude (19), will not lead to an overdriven detonation wave in the "regime" j < jj (see Fig. 2), if the pressure of the overdriven wave at the point v = v(J) satisfies the relation

p-p(1)> a.• a=i,

From here and from (19) we find for the limiting degree of overdriving~ relaxationally sepa, rating the stable and unstable waves,

I --I~ ?!--?e D--Dj ?~--7. i~- v !+ ~' Dj 2 (~t ~ t)"

CONCLUSIONS

It is well known that the one-dimensional structure of DW is encountered relatively rarely in practice. As a rule, detonation waves have a three-dimensional pulsating structure (triple configurations [I0, 24]). There is no question that the one-dimensional model of detonation studied here cannot and does not pretend to describe the nature and structure of three-dimensional pulsations caused by the thermal or relaxational instability of the wave- front. In the case of a pulsating detonation the proposed model, like the classical theory (limit of small a), refers to time-averaged gasdynamic quantities and to the selection rule for the velocity of detonation. In addition, the one-dimensional model can be used (see the section concerning the relaxational instability) as a starting point for the analysis of the instability of DW.

LITERATURE CITED

i. Ya. B. Zel'dovich and S. B. Ratner, Zh. Eksp. Teor. Fiz., ill, No. i, 170 (1941). 2. Ya~ B. Zel'dovich, Zh. Eksp. Teor. Fiz., 12, No. 9, 389 (1942). 3. J. G. Kirkwood and W. W. Wood, J. Chem. Phys., 22, No. ii, 1915 (1954). 4. W. W. Wood and J. G. Kirkwood, J. Chem. Phys., 25, No. 6, 1276 (1956). 5. W. W. Wood and J. G. Kirkwood, J. Chem. Phys., 29, No. 4, 957 (1958). 6. W. W. Wood and Z. W. Salsburg, Phys~ Fluids, ~, No. 4, 549 (1960). 7. F. A. Williams, Theory of Combustion [Russian translation], Nauka, Moscow (1971). 8. F. Bartl'me, Gas-Dynamics of Combustion [in Russian], ~nergoizdat, Moscow (1981). 9. J. J. Erpenbeck, Phys. Fluids, ~, No. 4, 481 (1961).

?The width of this range of a increases as the heat capacity of the relaxing degrees of freedom increases.

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I0.

ii. 12. 13. 14. 15. 16. 17. 18.

19 20 21 22 23 24 25 26 27 28 29 30

B. V. Voitsekhovskii, V. V. Mitrofanov, and M. E. Topchiyan, Structure of the Detonation Front in Gases [in Russian], Siberian Branch, Academy of Sciences of the USSR, Novo- sibirsk (1963). J. A. Fay, Phys. Fluids, ~, No. 3, 283 (1959). W. W. Wood and F. R. Parker, Phys. Fluids, ~, No. 3, 230 (1958). V. N. Arkhipov, Zh. Prikl. Mekh. Tekh. Fiz., No. 4, 40 (1962). C. L. Eisen, R. A. Gross, and T. J. Rivlin, Combust. Flame, ~, No. 2, 137 (1960). S. Eizen, R. Gross, and T. Rivlin, Vopr. Raket. Tekh., No. i, 20 (1961). Yu. A. Nikolaev and M. E. Topchiyan, Fiz. Goreniya Vzryva, 13, No. 3, 393 (1977)o F. J. Zeleznik and S. Gordon, ARS J., 32, 606 (1962). G. B. Kistiakowsky and P. Ho Kydd, J. Chem. Phys., 22, 1940 (1954); 23, 271 (1955); 25, 824 (1956). R. E. Duff, H. T. Knight, and J. Rink, J. Chem. Phys., 25, 1301 (1956). R. E. Duff, H. T. Knight, and J. Rink, Phys. Fluids, l, 393 (1958). R. A. Gross and A. K. Oppenheim, ARS J., 29, 173 (1959). H. M. Peek and R. G. Thrap, J. Chem. Phys., 26, 740 (1957). B. Levitt and D. F. Hornig, J. Chem. Phys., 36, 219 (1962)o E. S. Shchetinkov, Physics of Combustion of Gases [in Russian], Nauka, Moscow (1965)o S. R. Brinkley and Jo M. Richardson, 4th Symp. (Int.) on Combustion (1953). Encyclopedia of Mathematics [in Russian], Vol. 4, SE, Moscow (1984). K. I. Shchelkin, Zh. Eksp. Teor. Fiz., 36, 600 (1959). K. I. Shchelkin, Dokl. Akad. Nauk SSSR, 160, 1144 (1965). R. M. Zaidel' and Ya. B. Zel'dovich, Zh. Prikl. Mekh. Tekh. Fiz., No. 6, 59 (1963). J. J. Erpenbeck, 9th Symp. (Int.) on Combustion (1963).

CRITERIA OF OCCURRENCE OF "CENTRAL-ZONE"-TYPE MACROINHOMOGENEITIES

IN THE SHOCK-WAVE LOADING OF POROUS MEDIA

N. A. Kostyukov and G. E. Kuz'min

The shock-wave loading of porous media and the attendant physical processes are of great interest from both scientific and practical points of view. In the practical sense, one of the usual requirements for the final product (semifinished product or specimen) is its uniforr ity. However, many investigators have noted that the use of the most common methods of shock loading of porous media often produce nonuniform physicochemical changes over the volume of the product [I-Ii]o The occurrence of the macroscopic (relative to the characteristic dimen- sion of the porous medium) inhomogeneity termed the "central zone" is a particularly acute problem. This inhomogeneity is either a cavity with fused edges or a compacted region in which the structure and properties are significantly different from the structure and properties of adjacent regions. This problem is seen fairly often in the loading of axisymmetric and plane specimens under conditions of the collision of shock waves (SW) (Fig~ i) or reflec- tion from interfaces with a denser medium.

The conclusions reached by various authors as to the loading parameters responsible for the occurrence of such inhomogeneities are applicable only to relatively narrow ranges of initial density and geometric dimensions of the loaded medium and sometimes contradict one another. Many studies have lacked a physical model of the phenomena occurring during shock loading and subsequent unloading. Experimental data describing the structure and properties of the final product is of little help in constructing suitable models and, thus, in explain- ing the mechanism by which these inhomogeneities occur. Since it is just this type of data that is now prevalent, there is as yet no reliable theory which can be used to predict the conditions under which specimens with a structure that is uniform over their entire volume will be obtained. Instead, we presently have a set of different semiempirical criteria which can be used only to solve a limited range of specific problems.

Novosibirsk. Translated from Fizika Goreniya i Vzryva, Vol. 22, No. 5, pp. 87-96, Sep ~ tember-October, 1986. Original article submitted July 30, 1985.

0010-5082/86/2205-0573512.50 O 1987 Plenum Publishing Corporation 573

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Detonation in a relaxing gas and relaxational instability