Set-valued solutions for non-ideal detonation

Transcription

1 Set-valued solutions for non-ideal detonation The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Semenko, R., L. M. Faria, A. R. Kasimov, and B. S. Ermolaev. Set-Valued Solutions for Non-Ideal Detonation. Shock Waves 26, no. 2 (December, 25): Springer Berlin Heidelberg Version Author's final manuscript Accessed Fri Feb 6 5:46:39 EST 28 Citable Link Terms of Use Detailed Terms Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.

2 Ë Ø¹Ú ÐÙ ÓÐÙØ ÓÒ ÓÖ ÒÓÒ¹ Ð ØÓÒ Ø ÓÒ ÊÓÑ Ò Ë Ñ Ò Ó ½ ÄÙ Þ Åº Ö ½ Ð Ò Êº à ÑÓÚ ½ ½ ÔÔÐ Å Ø Ñ Ø Ò ÓÑÔÙØ Ø ÓÒ Ð Ë Ò ÊÓÓÑ ¹¾¾¾ ¼¼ Ã Ò ÙÐÐ ÍÒ Ú Ö ØÝ Ó Ë Ò Ò Ì ÒÓÐÓ Ý Ì ÙÛ Ð ¾ ¹ ¼¼ Ë Ù Ö ÓÖ Ëº ÖÑÓÐ Ú ¾ ¾ Ë Ñ ÒÓÚ ÁÒ Ø ØÙØ Ó Ñ Ð È Ý ÊÙ Ò ÑÝ Ó Ë Ò ÃÓ Ý Ò ËØÖ Ø ÅÓ ÓÛ ½½ ÊÙ ØÖ Ø Ì Ü Ø Ò Ò ØÖÙØÙÖ Ó Ø Ý¹ Ø Ø ÓÙ ØÓÒ Ø ÓÒ ÔÖÓÔ Ø Ò Ò Ô Ó ÓÐ Ò ÖØ Ô ÖØ Ð Ö Ò ÐÝÞ Ò Ø ÓÒ ¹ Ñ Ò ÓÒ Ð ÔÔÖÓÜ Ñ Ø ÓÒ Ý Ø Ò ÒØÓ ÓÒ Ö Ø ÓÒ Ö Ø ÓÒ Ð Ò Ø ÐÓ ØÛ Ò Ø Ò Ø Ô ÖØ Ð º Ò Û ÓÖÑÙÐ Ø ÓÒ Ó Ø ÓÚ ÖÒ Ò ÕÙ Ø ÓÒ ÒØÖÓ Ù Ø Ø Ð Ñ Ò Ø Ø ÙÐØ Û Ø ÒÙÑ Ö Ð ÒØ Ö Ø ÓÒ ÖÓ Ø ÓÒ Ò ÙÐ Ö ØÝ Ò Ø Ö Ø Ú ÙÐ Ö ÕÙ Ø ÓÒ º Ï Ø Ø Ò Û Ð ÓÖ Ø Ñ Û Ò Ø Ø Û Ò Ø ÓÒ ÔÓ ÒØ ÔÔ Ö ÖÓÑ Ø ÓÛ Ø Ö Ü Ø ÓÒ ¹Ô Ö Ñ Ø Ö Ñ ÐÝ Ó ÓÐÙØ ÓÒ Ô Ö Ñ Ø Ö Þ Ý Ø Ö ÔÖ ÙÖ ÓÖ Ø ÑÔ Ö ØÙÖ Ø Ø Ò Ó Ø Ö Ø ÓÒ ÞÓÒ º Ì ÓÐÙØ ÓÒ Ø ÖÑ Ø¹Ú ÐÙ Ö µ ÓÖÖ ÔÓÒ ØÓ ÓÒØ ÒÙÓÙ Ô ØÖÙÑ Ó Ø ÒÚ ÐÙ ÔÖÓ Ð Ñ Ø Ø Ø ÖÑ Ò Ø ØÓÒ Ø ÓÒ Ú ÐÓ ØÝ ÙÒØ ÓÒ Ó ÐÓ ØÓÖº ½ ÁÒØÖÓ ÙØ ÓÒ Detonationisashockwaveinareactivemedium thatissustainedbythe energyreleased in chemical reactions initiated by the passage of the wave. Gaseous detonation propagating in a tube, in the interstitial space in a porous medium, or other obstructed environments is ÈÖ ÒØ Ö Ô ÖØÑ ÒØ Ó Å Ò Ò Å Ø Ñ Ø ÆÓÚÓ Ö ËØ Ø ÍÒ Ú Ö ØÝ ¼¼ ¼ È ÖÓ ÓÚ Ëغ ¾ ÆÓÚÓ Ö ÊÙ Ö Ñ Ñ ÐºÖÙ ÈÖ ÒØ Ö Ô ÖØÑ ÒØ Ó Å Ø Ñ Ø Å Ù ØØ ÁÒ Ø ØÙØ Ó Ì ÒÓÐÓ Ý Ñ Ö Å ÍË Ð Ö Ñ Øº Ù Ð Òº ÑÓÚ Ù Øº Ùº ÓÖÖ ÔÓÒ Ò ÙØ ÓÖ ÓÖ º ÖÑÓÐ ÚÝ ÓÓºÓÑ

3 subject to resistance due to heat transfer, boundary layers and shock reflections that arise attheinterfacesbetweenthegasandtheobstaclesorthewallsofthetube. Suchresistance leads to momentum and energy losses in the gas that can significantly affect the propagation of the detonation. In the presence of such losses, the detonation is termed non-ideal, and its propagation velocity is lower than in the case(termed ideal ) when there are no such losses. This drop in the detonation velocity is called the velocity deficit. Quantifying the velocity deficitintermsofthelossesisoneofthemajorproblemsindetonationtheoryanditsstudy goesbacktotheoriginalpaperbyzel dovich[24](seealsotheclassicbookbyzel dovichand Kompaneets[27]). An important feature observed in experiments on non-ideal detonation waves is their ability to propagate with velocities ranging from near sonic speed in the ambient gas, c a, to the ideal Chapman-Jouguet (CJ) value, D CJ [5, 6, 7, 2]. The physical mechanisms responsible for this self-sustained propagation of detonation depend on the extent and the nature of the losses. A particularly interesting aspect is the possibility that the frictional heating of the gas dominates over its heating by chemical reactions at low detonation velocities. We also mention a peculiar theoretical observation, first discussed in [24],thatinthepresenceoflosses,theflowinthedetonationreactionzonecanreverseits direction; that is, the velocity can become negative in the laboratory frame of reference, a situation typical for flames, but unusual for detonations. No experimental evidence for this prediction appears to exist. Since the early works mentioned above, this problem has been revisited many times by researchers seeking to understand the precise mechanisms of detonation propagation when the effects of losses are substantial. A particularly important issue, from a practical point of view, is the existence of detonation limits. The limit of detonation propagation is definedasathresholdintermsofsomecontrolquantity(e.g.,atubediameter)belowwhich the detonation cannot propagate. The problem of defining such limits is nontrivial and is complicated by the fact that detonation can and most often does propagate unsteadily. Although this fact makes the definitions of these limits based on the non-existence of stationary solutions somewhat limited in application, understanding the steady-state solutions isamajorfirststepinunderstandingthenatureofdetonationlimitsaswellasinbuilding a theory of unsteady detonations. Theoreticalworkontheroleoflossesindetonationswascontinuedin[3,4,25,26]. In [26], both convective and conductive heat losses were considered. Convective losses depend on the relative velocity between the gas and the particles/walls and they vanish with relative 2

4 velocity. In contrast, conductive losses persist in the absence of the relative velocity. In the relaxation zone far downstream of the lead shock, where the flow of the reaction products has sufficiently decelerated, the authors of[26] assumed that the heat losses corresponded to an approximately constant Nusselt number and thus were of the conductive type. Subsequently, theyreasonedthatthegasintheproductscooleddownduetotheconductiveheattransfer andreachedastateofrestfardownstreambutwiththesamepressureanddensityasthe upstream state with the fresh mixture. In contrast, the model we present here involves only convective heat losses, as a consequenceofwhichtheheattransferfromthegastotheparticlesvanisheswhenthegasstops. This is a reasonable assumption for detonations because the conductive losses occur on time scales that are much longer than the time scale of the detonation propagation. The temperature relaxation in the products can therefore be expected not to affect the detonation dynamics over the time scales of interest. We should note that our governing equations take thesamegeneralformasthosein[26,27]andincludethemomentumlossduetofriction, heat loss due to the temperature difference between the gas and the solid particles, and the effect of heating of the gas due to the friction. Convective heat transfer was also considered in[4]. To highlight the most important general features of detonation in systems with losses that have previously been found, we mention that, in а qualitative agreement with prior experimental research[5, 6, 7, 2], different regimes of detonation propagation have been theoretically identified in[3, 4]: ) When the detonation speed is below the ideal CJ value, but not substantially, the wavehasastructurethatisverysimilartothatoftheclassicalzndtheory(zel dovich[24], von Neumann[22] and Döring[7]; see also[, 4]) of a self-sustained detonation with its inherent transonic character in the post-shock flow and with a sonic point located at some distancefromtheshock. ½ 2) When the detonation speed drops significantly, often to around.6 of the ideal value, the sonic locus moves to infinity and, with a further decrease of the detonation velocity, disappears from the flow altogether. This structure, in which the post-shock flow is entirely subsonic in the shock-fixed frame, resembles piston-supported overdriven detonations. However, the latter gain energy from the piston and therefore propagate faster than CJ ½ ËÙ ØÓÒ Ø ÓÒ Ú Ò ÐÐ ÕÙ ¹ ØÓÒ Ø ÓÒ Ò Ø Ô Ø ½ º ÀÓÛ Ú Ö ÒÓÒ¹ Ð ØÓÒ Ø ÓÒ ÒÓØ ÒØ ÐÐÝ Ø ÒØ ÖÓÑ Ø Ð Ð Ð ØÓÒ Ø ÓÒ Ø Ø ÐÐ Ð ¹ Ù Ø Ò ÔÖÓ Ó Ó ¹Û Ú Ö Ø ÓÒ¹ÞÓÒ ÓÙÔÐ Ò º Ì ÔÖ Ò Ó ÐÓ Ó ÒÓØ Ò Ø Ò ØÙÖ Ó Ø Û Ú ÐÓÒ Ø Ö ÓÒ ÓÒ Ò Ñ ÒØ Ò Ø ÓÛº 3

5 detonations. In contrast, non-ideal detonations, with a subsonic post-shock flow, have no support and propagate at velocities that are substantially lower than the ideal CJ velocity. Becauseitsstructureismissingasonicpoint, itisnotaself-sustainedwave ¾ inthe sense of classical detonation theory. Without a sonic point, the wave evolution can be influenced by the downstream boundary conditions, while with a sonic point, the detonation dynamics are completely controlled by the processes occurring between the shock and the sonic point. In experiments, the velocity deficits are obtained by, for example, decreasing the initial pressure, p a, in the explosivemixture [5, 6, 7, 2]. In manycases, a decreasein p a to some critical value results in an abrupt decrease in the speed to subsonic velocities, which are associated with turbulent deflagration waves. In theoretical modeling, one can, in principle, look at the solutions that have velocities smaller than the ambient sound speed, as was done in [3, 4] and in subsequent works by the same authors and their collaborators. However, in suchcases, the structuredoes notcontainashockand the waveis subsonicrelativeto the state upstream. In this work, we limit the discussion to supersonic waves containing a shock as an inherent element of the structure. While the nature of self-sustained non-ideal detonations is relatively clear, that of the regimes with subsonic post-shock flow has led to some conflicting conclusions in the literature. In particular, the existence of steady solutions of the governing equations in this regime was questioned in [6]. Indeed, because the sonic point disappears from the flow, the classical self-sustained structure cannot be a solution of the governing equations. It therefore appears reasonable to conclude that no steady-state solutions exist in this case. In [3, 4], however, the authors have been able to construct the steady D-c f relations in the entire range of velocities from the ideal D = D CJ to the sonic velocity, D = c a (and evenfurtherbelow),where c a isthespeedofsoundintheupstreammixtureand c f isthe dimensionless drag coefficient. Instead of the sonic conditions, they applied u = at infinity and solved the appropriate boundary value problem. In this connection, a point that requires certain care in interpretation is the question of existence of steady-state solutions versus their stability. Numerical simulations in[6, 8] indicate that detonation with frictional momentumlossesisunstable; infact, itismoreunstableinthepresenceoflossesthanin their absence. Because of the instability, the steady-state solutions cannot be reached as long-time asymptotic solutions of an initial value problem of the underlying reactive Euler ¾ Ì Ö Ñ Û Ø ÖÑ Ó Ò Ö Ñ Ò ½ º ÀÓÛ Ú Ö Û ÒÓØ Ø Ø Ø ÓÛ Ò Ø Ó ÒÓØ ÓÒØ Ò ÓÒ ÔÓ ÒØ Ò ÓÒØÖ Ø ØÓ ØÝÔ Ð Ó Ò Ö Ñ Ò Ù ÝÒ Ñ Û Ó Ø Û Ø ØÖ Ò ÓÒ ÓÛ º 4

6 equations. This, however, does not imply that the steady-state solutions of the equations do not exist. In order to establish the existence of the steady-state solutions, one must be able to solve the appropriate boundary value problem for the time-independent governing equations. Importantly, the proper statement of such a boundary value problem does not necessarily involve sonic conditions. We discuss this point in detail in the subsequent sections of this paper. In this work, we show that the steady-state solutions of the governing reactive Euler equationswithheatandmomentumlosseshaveapeculiarnatureinthatthe D-c f dependenceisnolongeracurve,butratheraset-valuedfunction. Thatis,whenthesoniclocus disappears from the flow, at a given loss coefficient, there exists a continuous range of detonation velocities, all corresponding to steady-state solutions of the governing equations. Effectively, D, as an eigenvalue of the nonlinear eigenvalue problem, is found to have both discrete and continuous spectra. This is in contrast with prior results, in which either the steady-statesolutionappearsnottoexist[6]orthatthereisaz-shapedcurveinthe D-c f plane,givingafiniteanddiscretesetofsolutionsforagiven c f [3]. Wepointoutthata continuous spectrum is found in a number of problems involving self-similar solutions of the Euler equations of gasdynamics(without chemical reactions)(see, e.g.,[5]). Another contribution of our work is a new formulation of the governing equations that completely avoids the difficulty of integrating through the sonic locus. The latter is a wellrecognizedproblemindetonationtheory[,3,26]. Wehavefoundanewsetofdependent variables in which the governing equations are no longer singular at the sonic point. This finding results in a numerically well-conditioned integration of the detonation structure in the entire flow region. A generalization of the algorithm to incorporate other loss factors and multiple reactions is possible and is introduced in[]. Apossibilityof the flowreversalinthe reactionzone[24] wasalsobrieflydiscussedin [26], where the authors continued the analysis of the problem and found that, with the farfield conditions they used, the convective heat transfer between the gas and the tube walls (or solid particles) was insufficient to cause the flow reversal. However, with the inclusion of conductive heat losses, the flow reversalwas indeed found. The authors of [4] did not discuss this possibility even though convective heat transfer was also an essential element of their analysis. Our present finding is that, with convective heat transfer, the flow reversal necessarily occurs in the regime where the flow behind the shock is entirely subsonic. In contrast, no flow reversal is possible with only friction losses. We emphasize, however, that 5

7 our far-field conditions are different from those of[26]. The remainder of this paper is organized as follows. In Section 2, we introduce the reactive Euler equations that include momentum and heat exchange terms. The Rankine- Hugoniot conditions are also introduced in this section. In Section 3, we discuss the steadystate solutions and introduce a change of dependent variables that removes the sonic singularities from the equations. In Section 4, we calculate various solutions when both the friction and heat loss terms are present and show that the detonation speed versus loss coefficient is a set-valued function. Concluding remarks are offered in Section 5. ¾ ÓÚ ÖÒ Ò ÕÙ Ø ÓÒ Consider a detonation in a perfect gas that fills the interstitial space in a packed bed of solid particles of diameter d. The particles are assumed to be immobile and inert; their only role is to exchange momentum and energy with the detonating gas. The chemical reaction in the gas is described globallyas Reactant Product. The progressof this reaction is measuredbyvariable, λ, that variesfrom atthe shockto in the fully burntgas. The reactionrateisassumedtobegiveninthearrheniusform, ( ω = k ( λ) exp E ), () pv where E is the activation energy, p is the pressure and v is the specific volume. The perfectgasequationofstateisgivenby e i = pv γ, (2) where γ istheconstantratioofspecificheats. With these modeling assumptions, the governing reactive Euler equations, further incorporating the heat and momentum exchange between the gas and particles, are as follows. The continuity equation is ρ t + (ρu) x =, (3) where ρ = /v and u are the gas density and velocity, respectively. The gas momentum equation is u t + uu x = ρ p x f ρφ, (4) 6

8 or in conservation form, (ρu) t + ( p + ρu 2) x = f φ, (5) wherethedragforceduetothesolidparticlesisassumedtotaketheform[2], ( f = A s ρ b + b ) 2 u u, (6) Re with A s = 4φ d p, d p = 2φd 3( φ), Re = d p u ν, (7) where φ is the porosityof the packedbed, i.e., the fractionof spaceoccupied by the gas, d is the particle diameter, Re is the Reynolds number, ν is the gas kinematic viscosity and b, b 2 arenumericalconstants. Inthe calculationsbelow, wetake φ =.4, b =.75and b 2 = [2]therebyassuminghighReynoldsnumberfrictionlosses. The energy equation can be written as ( ) uf h p t + up x + γpu x = (γ )Qρω + (γ ). (8) φ It contains contributions due to: ) the chemical energy release(the first term, wherein Q istheheatrelease),2)theworkdonebythefrictionforces(thesecondterm,involving uf) and 3) the heat transfer between the gas and particles (the last term, involving h). The heatexchangerate, h,isassumedtobegivenbynewton slaw, h = A s α s (T T s ), (9) inwhichtheparticletemperatureisdenotedby T s andtheheatconductioncoefficient, α s, is calculated from[2] α s = λ g Nu d p, () where Nu = a + a 2 Re m isthenusseltnumber, λ g istheheatconductivityofthegasand a, a 2, marenumericalconstants. Wetake a =, a 2 =.425and m = [2], thereby assuming purely convective losses. The validity of this approximation for heat transfer in flows in packed beds of solid particles is justified in[2]. A slightly different heat exchange lawwasusedin[4,24],whichusedthestagnationtemperature T +u 2 /2insteadof T in(9). We have found that the general nature of the solutions is unaffected by this change even though some of the small details are. 7

9 In conservation form, the energy equation becomes (ρe) t + (ρu (e + pv)) x = h φ, () where e = e i + u2 2 pv λq = γ + u2 λq. (2) 2 Note that the friction workterm uf does not appear in the conservationform of the full energy equation because obstacles are motionless in the laboratory reference frame (see [8, 9]). The internal energy equation in a non-conservation form is (e i ) t + u(e i ) x + p ρ u x = ωq + uf h ρφ, where the work by friction appears explicitly on the right-hand side. The reaction rate equation is λ t + uλ x = ω, (3) or in conservation form, (ρλ) t + (ρuλ) x = ρω. (4) We note that these equations with appropriate modifications of the loss terms can also be used to describe gaseous detonation in rough tubes. The ideas that follow are expected to carry over to such a setting without significant changes. Across the shock-discontinuity surface, the following Rankine-Hugoniot conditions hold: D[ρ] + [ρu] =, (5) D[ρu] + [p + ρu 2 ] =, (6) D[ρe] + [ρu(e + pv)] =, (7) D[ρλ] + [ρuλ] =. (8) Here, [j] = j a j s denotesajumpofquantity j acrosstheshock, (ρ a, u a, p a, λ a )denotes theambientstateaheadoftheshockand (ρ s, u s, p s, λ s )isthestateimmediatelyafterthe shock. For a perfect gas, the shock conditions can be solved explicitly and the following 8

10 expressions give the post-shock state in terms of the upstream state, p s p a = 2γ γ + M2 a γ γ +, (9) u s = 2 Ma 2, c a γ + M a (2) ρ s = (γ + )M2 a ρ a 2 + (γ )Ma 2, (2) λ s =. (22) Here M a = D/c a isthedetonationmachnumberwithrespecttotheupstreamstate. ËØ Ý¹ Ø Ø ÓÐÙØ ÓÒ In this section, we analyze the existence of the steady-state solutions of the governing equations and calculate their properties. As always in detonation theory, the speed of the wave is unknown a priori and must be determined as a solution of a nonlinear eigenvalue problem. Thegoalisthentocomputethesteadytraveling-wavestructureanditsspeed. Inorderto do that, one must carefully pose the mathematical boundary-value problem by prescribing appropriate equations and boundary conditions. Because of the nature of the problem at hand, two apparently distinct situations can arise, which can however be put in a unified framework. In most of the literature on detonation theory, the interest is in finding a self-sustained structure, which by definition means that the propagating shock reaction-zone complex contains an embedded sonic locus. This sonic locus provides acoustic confinement that precludes any downstream influence on the detonation dynamics, thus making the detonation wave effectively an autonomous process whose dynamics are driven by the finite region betweentheshockandthesoniclocus[2]. Tocomputethestructureofthisfiniteregion, one requires knowledge of the governing equations, shock conditions and appropriate conditions at the sonic point. Knowledge of the flow conditions downstream of the sonic locus is not required. Another structure that has been extensively investigated in the past is that of overdriven detonation. This wave is obtained by providing downstream support in the form of a piston ÆÓØ Ø Ø Ò ÓØ Ö Ý Ø Ñ Ö Ý Ö ÒØ Ñ Ø Ñ Ø Ð ÑÓ Ð Û Ú Ò Ð ¹ Ù Ø Ò Û Ø ÓÙØ Ø ÔÖ Ò Ó ÓÒ ÐÓÙ º Ü ÑÔÐ Ö Ñ Ò ÓÐ ØÓÒ Û Ö Ø Ô Ý Ð Ñ Ò Ñ ÒÚÓÐÚ Ò Ð ¹ Ù Ø Ò ÔÖÓÔ Ø ÓÒ Ó Ø Û Ú Ö Ö ÒØ ÖÓÑ Ø Ó Ò ØÓÒ Ø ÓÒ º ÇÙÖ Ù Ó Ø Ø ÖÑ ÓÒ Ø ÒØ Û Ø Ø ØÖ Ø ÓÒ Ð ÒØ ÖÔÖ Ø Ø ÓÒ Ò ØÓÒ Ø ÓÒ Ø ÓÖݺ 9

11 pushing the flow in the direction of the lead shock and thus preventing the formation of the sonic confinement. In overdriven detonations, the entire region between the shock and the piston is subsonic. To determine the structure, knowledge of the flow condition at the piston is necessary. This condition is simply that the flow velocity at the piston location is thesameasthepistonvelocity. Aboundaryvalueproblemisthusposedthatrequiresthe construction of a smooth solution connecting the shock state with the state at the piston. Thus, we find that, in general, the problem of calculating the detonation structure(selfsustained or not) is posed as follows: determine a traveling shock-wave solution of the reactive Euler equations subject to quiescent equilibrium conditions upstream of the shock and appropriate conditions at an infinite distance downstream of the shock. If the detonation is self-sustained, the boundary-value problem is divided into two sub-problems: ) finding the structure between the shock and the sonic point, which is independent of the state downstream of the sonic locus, and 2) finding the flow structure downstream of the sonic locus,whichisdeterminedbythestateatthesoniclocusaswellasbytheconditionsatan infinite distance downstream of the shock. In the problem at hand, it is required to determine a traveling-wave solution of governing equations(3),(5),()and(4),whichconsistsofaleadshockfollowedbyasmoothreaction zone, subject to a quiescent non-reacting state ahead of the shock and the equilibrium state in the burnt products at infinite distance from the shock. From the governing equations, it followsthat the equilibrium state is necessarilyat λ = and u =. No otherpossibility exists, because the equilibrium requires that the right-hand sides of(5),() and(4) vanish atinfinity,whichcanonlyhappenat λ = and u =. ApplicationoftheRankine-Hugoniot conditions reduces the problem domain to a half-line between the shock and the downstream infinity. This formulation of the boundary value problem is general in the sense that the self-sustained solution is part of it, but solutions without a sonic point are also admitted. To proceed, we rescale the governing equations with respect to the upstream state, denoted by a subscript a: ˆρ = ρ/ρ a, ˆp = p/p a, u a = p a v a, û = u/u a, Ê = E/u 2 a, ˆQ = Q/u 2 a, ˆT = TR/ (p a v a W) = ˆp/ˆρ,and ˆD = D/u a. Thelengthscaleischosentobethe half-reactionzone length, l /2, of the ideal planar detonation, such that ˆx = x/l /2. The timeisrescaledas ˆt = t u a /l /2. Inthenewdimensionlessvariables,thegoverningequations retain their form. However, the dimensionless loss terms become(dropping the hats after the rescaling):

12 f = c f ρ u u, c f = 6b ( φ) l/2 d, (23) h = c h u (T ), c h = 9a 2( φ) 2 φ ltl /2 l v d. (24) Here, l t = λ g W/R p a ρ a and l v = ν/ p a /ρ a are the characteristic thermal and viscous lengthscales,respectively. Thedimensionlesscoefficients, c f and c h,whicharenowjusttwo parameters measuring the effects of the momentum and heat losses, respectively, depend on the ratios of various length scales that are characteristic to the relevant transport processes. For c f,itistheratioofthelengthofthereactionzoneandtheparticlediameter. For c h, it is a combination of four length scales: the viscous scale, l v, the thermal scale, l t, the reactionscale, l /2,andtheparticlediameter, d. Wenote,however,that c h =.5a 2 ( φ) c f b φ l t =.5a 2 ( φ) λ g W l v b φ Rν (25) is independent of the reaction or the particle diameter, but depends on the gas viscosity and heat conduction coefficient. We alsonote that c f is proportionalto l /2 /d, which equals, roughly, the number of particles that fit inside the reaction zone of the ideal detonation. Thus,smallvaluesof c f correspondtoparticlesthataremuchlargerthanthethicknessof the reaction zone in the ideal detonation. Thetraveling-wavesolutionofthegoverningequationsissoughtintheform U = U() = (ρ, u, p, λ) T (),where = x Dt.Substitutioninto(3),(5),()and(4)yields(withthe primes denoting the derivative with respect to ): (ρ(u D)) =, (26) (p + ρ(u D)u) = f φ, (27) (ρ(u D)e + pu) = h φ, (28) (u D)λ = ω. (29) Thissystemofequationsmustbesolvedsubjecttothejumpconditionsattheshock, = : ρ () = ρ s (D), u () = u s (D), p () = p s (D), λ() =,

13 asgivenby(9 22)andthefar-fieldconditionsat =,whichare u = and λ =. The only other condition, besides the requirement that the solution of the system satisfies these boundary values, is that it be smooth and bounded throughout the domain <. As wewill seebelow, this requirementisessentialin ordertodetermine the valuesof the detonation speed, D, given all the parameters of the problem. In the ideal case, when f = h =, system(26 29)can be integrateddirectly to yield the ideal detonation solution, v (λ) = /ρ (λ) = u (λ) = DCJ 2 γ ( γ + D CJ [ p (λ) = + D2 CJ γ + = ˆ λ γ + D 2 [ CJ γ + DCJ 2 D2 CJ γ ] λ γ ( + DCJ 2 ), (3) ) + λ + D2 CJ γ + D 2 CJ, (3) ] λ, (32) u (λ) D CJ ω dλ. (33) The ideal solution is chosen to be of the self-sustained nature (i.e., the CJ solution) with thesonicpointat = and λ =. Then, D CJ = γ + 2 (γ2 )Q + 2 (γ2 )Q. (34) The pre-exponential factor that imposes the half-reaction length of unity is found from k = ˆ /2 u (λ) D CJ dλ. (35) ( λ) exp ( Eρ (λ)/p (λ)) This value of k is used in the calculations of the non-ideal detonation when f and h are non-zero. Indimensionalterms,iftheratefunctionisgivenas k ( ) ( λ)exp Ẽ/ T (the tilde denotes a dimensional quantity), then the dimensional length of the region over which halfofthereactantisburnt,is ˆ /2 l/2 = ũ (λ) ( D CJ k ( λ) exp Ẽ/ T )dλ = ũa k ˆ /2 u (λ) D CJ ũa dλ = k. (36) ( λ) exp( E/T) k Therefore, k = l /2 k/ũa, which identifies the characteristic length scale in terms of the mixture properties. 2

14 The continuity equation can be integrated even in the non-ideal case to result in ρ = D D u. (37) To proceed with the solution of the remaining equations, we rewrite(26 29) in matrix form: (A DI)U = G, (38) where U = ρ u p λ, A = u ρ u /ρ γp u u, G = F H ω, Iistheunitmatrixand F = f ρφ, ( ) uf h H = (γ )Qρω + (γ ). (39) φ The acoustic eigenvalues of A are s = u + c and s 2 = u c, where c = γpv is the sound speed. Their corresponding left eigenvectors are l = (,, c/γp, ) and l 2 = (,, c/γp, ). Left-multiplying(38)by l,weobtain, (s D)(u + c γp p ) = F + c H. (4) γp Similarly,left-multiplying(38)by l 2,weobtain, (s 2 D)(u c γp p ) = F c H. (4) γp From(4,4),itfollowsthat du d = [ F + c γp H 2 u + c D + F c dp d = γp 2c γp H u c D [ F + c γp H u + c D F c γp H u c D ], (42) ]. (43) Theimportanceofthisformoftheequationsisthatitmakesevidentthatthesolution is not necessarily regularif the denominators of (42) or (43) vanish somewhere at <. Note that at =, they cannot vanish due to the Laxconditions. Because u D is the 3

15 flow velocity relative to the shock, which must be negative for the shock propagating to the right,itfollowsthat u c Dcannevervanish. However, u + c Dcanvanish. Ifitdoes, wemustrequirethatatthesamepoint(whichisthesonicpoint) F + ch/γp = aswell. This is basically the statement of the generalized CJ condition [9, 23] and this condition isnecessarytodeterminethevalueof D whentheflowcontainsasonicpoint. Itmustbe emphasizedthattheexistenceofthesonicpointisnotnecessaryifonecanfindasmooth solution that connects the shock state with the far-field equilibrium conditions. Irrespective ofthepresenceofthesonicpoint,thefundamentalproblemisthatoffindingasmoothand bounded solution of the boundary value problem at hand. Another important consequence of(42 43) is that the only possible equilibrium solution at = isthatwhichhas u ( = ) =. Indeed,settingtheright-handsidesofboth equations to zero, we obtain that in equilibrium both F and H must vanish. Therefore, from F =,weobtain u = and,asaconsequenceofthisandof λ =,italsofollowsthat H =. Nootherpossibilityexists. Ifasoniclocusdenotedby exists,thenatthatpointbydefinition, u + c D =. Requiringthat F + ch/γpvanishat aswell,oneobtainsthegeneralizedcjconditions, ( c (γ ) Qρ ω(λ ) + u ) f h f =, (44) γp φ ρ φ u + c D =. (45) This is a system of equations for D and λ that in principle allows one to find the full solution. However, in practice it is difficult to solve numerically. To illustrate the required calculations, we assume that all parameters of the problem aregiven. Then,theonlyunknownsare Dandthesolutionprofiles. Toremindthereader, inordertodetermine D in theidealcase,onefindsthefull solutionasafunctionof λ as p = p (λ, D), u = u (λ, D)and ρ = ρ (λ, D)(givenby(3 33)),whichdependparametrically on D (λ as a function of is found implicitly by integrating (33)). The solution is then substitutedintothecjconditions(44 45). Theselatterintheidealcasearesimply ω = and u + c D = ;therefore, λ = andthesecondconditionyieldsthedesiredresult for D,asgivenby(34). In the non-ideal case, however, the solution profiles are unavailable analytically and must be determined numerically. The procedure of finding D is in principle, to guess the value of D, integrate the ordinary differential equations(ode) for p and u in the λ-variable from 4

16 the shock until λ such that (44 45) are satisfied. This is done iteratively by varying D until the sonic conditions are satisfied. Generally, the sonic point has a saddle character and, therefore, the integration process just described is extremely ill-conditioned. For this reason, it is often preferable first to identify the sonic locus, then linearize the governing ODE system in its neighborhood, step out analytically by a small step and then integrate numerically back to the shock. This procedure is better conditioned and has been used in thepast. Yet,itisnotagoodchoiceforourproblemeither. Onereasonisthatthesonic locus is unknown a priori, so that linearization about it does not eliminate the algorithmic complexityofthesearchfor Dandthelocusitself. Thesecondandmoreseriousreasonis that the method fails when the solution contains no sonic locus. The latter is a possibility that cannot be ignored as one does not in general know a priori whether or not a sonic point exists in the solution. To resolve the difficulties mentioned in the previous paragraph, we propose a change of dependent variables that completely eliminates the singular behaviour from the governing equations. To proceed with this change, note that(4 4) can be written as du d + c dp γp d = F + c γp H u + c D, (46) du d c dp γp d = F c γp H u c D. (47) In this system, only the first equation contains a potential singularity. Note further that c γp/ρ γp = D u = γp Dγp. With this substitution, the left-hand sides of(46,47) become 2 D u d d ( ) p D u ±, γd which motivates the introduction of new variables, r = p D u γd, (48) s = p D u + γd. (49) Goingbackto uand pintermsof rand siseasybecause r + s = 2 D uand r s = 2 p/γd. Itisinterestingtocompute u + c Dintermsofthenewvariables. Thespeed 5

17 ofsoundis γp p c = ρ = γ γd (D u) = γ r2 s 2, 4 wherewetookintoconsiderationthat r > s atany p >. Therefore, (r + s)2 u + c D = + γ r2 s = γ 4 (r + s)(r µs), (5) where µ = γ + γ. (5) Thusthesoniccondition u + c D = isreducedtoasimplelinearequationintermsof thenewvariables, rand s, r µs =. (52) Equations(42 43)leadtoanewsystemofODEfor r, sand λ, dr d = γ 2ρc D u(u c D) ds d = ρc (D u)(r µs) dλ d = [ Qρω + hφ ( u + [ Qρω + hφ ( u c ) ] f G(r, s, λ), (53) γ φ ) ] f g(r, s, λ) φ r µs, (54) c γ ω L(r, s, λ). (55) u D Singularityiscontainedonlyin(54),butitisnowofaverysimpleforminthenewvariables. By introducing another change of variables, we can completely remove the singularity from the system. Let z = (r µs) 2 (56) toreplaceoneofthevariables,say s = µ ( r ± z ). (57) Beforeproceedingfurther,itisimportanttonotethattheinversionofthevariablesfrom r, z to r, sisnot single-valued. Thechangefromonebranchofthetransformationtoanother happens exactly at the sonic point, which is now located at z =, where the Jacobian of the transformation vanishes. The positive sign in(57) corresponds to the subsonic flow in the reference frame of the shock, while the negative sign corresponds to the supersonic flow. Indeed, at the shock u + c D = (γ )(r + s)(r µs)/4 > and since r + s = 6

18 2 D u <,then r µsshouldbe < inthesubsonicregionbetweentheshockandthe sonic point. This amounts to selecting the positive sign in(57). Downstream of the sonic point, the sign switches to negative. Byremovingthe svariablefrom(53 55)infavorof z,weobtainanewsystem: dr d = G ±(r, z, λ), (58) dz d = ±2 zg ± (r, z, λ) 2µg ± (r, z, λ), (59) dλ d = L ±(r, z, λ), (6) which is now completely free of singularity. Here, G ± and g ± are the same functions as G and g in (53 54)with the ± sign indicating the branch chosenin (57). Of course, the transonic character of the solutions cannot just disappear from the equations. There is deep physics behind the existence of the sonic point in the solution that must remain represented somehowintheequations. Itindeeddoesremainintheformofaswitchfromonebranch of the square root to another when passing through the sonic point. Equations(58 6) must be solved subject to the shock conditions, r () = r s, (6) z() = (r s µs s ) 2, (62) λ() =, (63) where r s = ps D u s γd, (64) s s = ps D u s + γd, (65) and u = at =,whichamountsto r ( ) = D z ( ) = D + p ( ) γd, (66) p ( ) γd. (67) 7

19 As was mentioned previously, the cases where the solution contains no sonic point and therefore the entire flow downstream of the shock remains subsonic should not be excluded fromconsideration. Condition u ( ) = leadstoconditionson rand zgivenby(66 67). However, the latter depend on p ( ), which cannot be enforced, in general. If the solution contains a sonic point, then the regularization conditions(44 45) become simply dz d = at z =. (68) Inthiscase,theequilibriumconditionsat = muststillbeenforced,buttheyarenow required only for the determination of the solution downstream of the sonic locus. If the latter does not exist, the system(58 6) must be solved subject to three conditions at the shock,(6 63), and the conditions at =. This set of boundary conditions is expected to be complete on physical grounds in order to yield a well-defined solution of the problem. The solution is not necessarily unique, as in previous studies of closely related problems. A typical result is a multi-valued solution for the detonation speed, D, as a function of, for example,thefrictioncoefficient, c f,giventhatalltheotherparametersarefixed. However, in contrast to the previous work, we find that the boundary-value problem as posed above yields set-valued solutions at certain parameters. In other words, a continuous range of solutions with D varying in some interval can exist for a given mixture and given loss conditions. Thus, the nonlinear eigenvalue problem at hand can have not only a discrete but also a continuous spectrum. º½ ÆÙÑ Ö Ð Ö ÙÐØ Ì ÒØ Ö Ø ÓÒ Ð ÓÖ Ø Ñ Thechangeofdependentvariablesto rand zisveryhelpfulasiteliminatesthenumerical difficulties that are otherwise inevitable in the integration of the original system (53-55). Such difficulties have previously been avoided in simpler situations(for example, when the governing equations can be reduced to a single ODE), by integrating from the sonic point to the shock [26, 3, ]. This algorithmworkswell if the sonicpoint location is known a priori. If, however,the location of the sonic point depends on the shock speed and other parameters, which one is attempting to find, then the algorithm becomes unwieldy, even though still valid. 8

20 Figure illustrates the difficulty. If we choose to integrate from the sonic locus, the algorithm proceeds as follows. Assume that all the parameters of the problem are given suchthatweneedtodetermine D andthesolutionprofiles. First,weneedtoidentifythe soniclocuswhere(44 45)musthold. Giventhat ρ = D/ (D u), p = ρc 2 /γand T = c 2 /γ, thesetwoequationscanbewrittenintermsof u, c, λ and D. Onecaneliminate,say, D using(45)andobtainoneequationintermsof u, c and λ bysubstituting D = u + c into (44). To proceed, one has to make a guess on two sonic variables and then find the third one by solving(44). Only then would the sonic state be known completely, after which onecanproceed,asusual,withanumericalintegrationoutofthesonicpointbacktothe shock, then check the shock conditions, and if they are not satisfied, iterate on the guesses of the two sonic state variables. Tounderstandthisprocedureindetail, letusrewrite(44)in termsof M = u /c, c and λ as (γ )Qω(λ ) = c h(γ )(c 2 γ) M c γφ(m + ) c fc 2 M ((γ )M ) M, φ from which we obtain λ = c hc (c 2 γ) M Qkγφ(M + ) + c fc 3 M ((γ )M ) M. (69) (γ )Qkφ Themaindifficultyinthealgorithmjustdescribedistechnical theneedtoiterateontwo sonicvariables(m and c )asopposedtooneinsimplersituations. Itisclearthataguess for M and c makessenseonlyif(69)yields λ intheintervalfrom to. Inordertogive thereaderanideaoftherangeof M and c thatmustbeguessed,infig. weplotthe domainsinthe M c planewhere(69)hassolutionswith λ [, ]. Thecoloredlines aretheiso-linesof λ rangingfromtowithanincrement.. Thesearchalgorithmfor the sonic locus can, in principle, yield the solution anywhere within the area covered by all theiso-linesfor λ. Themoststrikingpropertyoftheseregionsisthepresenceof highly intricate features, especially at negative velocities when both heat and friction losses are present(fig. b)). Thus, one would have to scan this complicated region for possible locations of the sonic state variables M and c in order to proceed with the integration toward the shock. We note that when only momentum loss is present, negative velocity at the sonic point is impossible(fig. a)). 9

21 Figure: Theregionsinthe M -c spacewherethesoniclocuscanexist: a)inthepresence of only friction losses, c f =., c h =, and b) when both heat and friction losses are present, c f =., c h =.5c f. Thelinesinthefiguresaretheiso-linesof λ takenin [, ] with an increment.. Incontrast,intermsofournewvariables,thesoniclocusisfixedanddeterminedexactly by z = z = at any parameters. Moreover, the singularity is no longer present in the system. More precisely, the singularity is reflected by the requirement that the branch of the square root of z must change as one crosses the sonic locus. This procedure is numerically benign. Thus, we solve (58-6) with the corresponding boundary and sonic conditions, provided the sonic point exists. The non-singular character of the system allows ustocalculatethesolutionwithhighaccuracy. ThegoalistofindsuchaDthatthesolution of equations(58-6) satisfies the boundary conditions at the shock and at = and to studythedependenceofthedetonationspeedonthelossfactors c f and c h assumingthat other parametersarefixed. If there is a sonic point, we requirethat the sonic conditions 2

22 z = z = aresatisfiedwithinmachineprecision. Ifthereisnosonicpoint,theconditionat theleftboundaryofthedomainisimposedthatboth uand λareclosetozerowithin. Anychangesin c f withinthemachineprecisionresultinthechangesof uand λ atinfinitywithin,sothatrequiringevensmallererrorsisnotfeasible. Thenewnumericalalgorithmworksasfollows. Wefirstfix E, Q, γandtheratio c f /c h. Thelatterisessentiallyaconstantforagivenmixture,asseenfrom(25). Then,forevery given D [c a, D CJ ], we lookfor the c f that providesasmooth solution to our boundary valueproblem. Here c a,thesoundspeedintheambientstate,isequalto γ inourscales. The calculations show that there are two possible cases. If the detonation speed is less than D CJ, but greater than some value D s, which is constant for the given set of parameters, thenthereisasonicpointinsidethereactionzone. Ontheotherhand,if D (c a, D s ),then there exist smooth solutions that satisfy the boundary conditions and remain subsonic from the shock down to =. This effect of the sonic-point disappearance in a closely-related problem was discovered in[3, 4]. z a) z b) z c) 3 2 ǫ ǫ Figure2: Profilesof zintheshootingmethodfor D =.8D CJ andvarious c f : a) c f =.; b) c f =.6;c) c f =.7. Here E = 3, Q = 2, γ =.2and c h =. The possible outcomes of the numerical searchfor c f for a given D are shown in Fig. 2. Thefiguresarecomputedbyshootingfromtheshocktothesonicpointwithvarious c f. The Matlab ode5s solverwith a reference tolerancevalue of 3 is used. In Fig. 2a), onecanseetheprofileof zfor c f thatistoolarge. Here, zreachesitsminimumvalue,but thisvalueispositive. At lessthantheminimumpoint, z beginstogrowwithoutbound. Fortheexactsolution,weexpect ztovanishattheminimumpoint. Hence,tomeasurethe closenessof the computed profileto the exactsolution, wedefine the errorfunction, ǫ, to be the minimum value of z. On the other hand, if c f is too small, then z will reachzero withapositiveslope,asinfig. 2b). Thismeansthatcondition(68)isnotsatisfiedand u willblowupbecauseoftheinfinitederivativeatthesonicpoint. Inthiscase,wedefine ǫ toequaltothederivativeof zatthesonicpoint. Thenwefindtheminimumvalueof ǫ(c f ) usingmatlab sfminsearch functionwithatolerancevalueof 3. Theminimumvalue 2

23 of ǫ = correspondstofig. 2c). Thiscaseistheonethatsatisfiescondition(68)andthe corresponding c f isthepropervalueforthegivendetonationspeed. Wepointoutthatatthisvalueof c f,therearetwowaysinwhichthesolutioncanbe continuedacrossthesonicpoint. Wecaneitherstayonthesamebranchof z,suchthat theflowdownstreamofthesonicpointremainssubsonic,orswitchtothesecondbranchof zandobtainasupersonicsolutiondownstream. ThesetwopossibilitiesareshowninFig. 3. This figure shows that for subsonic-subsonic transition, Fig. 3b), the velocity profile is continuousbutnotsmooth. Moreimportantly,inthiscase ztendstoinfinityand utendsto Das,unlikethesubsonic-supersoniccaseshowninFig. 3a). Thismeansthatsuch non-smooth solutions cannot satisfy the boundary conditions at = and thus must be discarded. 3 a) 3 b) 2 2 u z u z Figure 3: Profiles of u and z at D =.7D CJ, c h =, E = 2, Q = 2, γ =.2 for: a) subsonic-to-supersonic and b) subsonic-to-subsonic transitions across the sonic point(here, uand zarenotinthesamescale). For D < D s,suchnumericalalgorithmshowsthatthereexistsno c f thatgives ǫ(c f ) =. However, smooth and bounded subsonic solutions still exist, i.e., z remains positive in (, ) and u( ) =. Such solutions are investigated below. The shooting from the shock gives three possibilities, as shown in Fig. 4a). If c f is too large, then z passes its minimum point and then grows to infinity (solid line). If c f is too small, then z reaches the zero value with a non-zero slope, which means the blow-up of velocity(dash-dot line). However,insteadofthe behaviourseeninfig. 2c), hereweobtainthedashedlinein the middle,whichisthesolutionwithaboundedvalueof zat =. Thissolutionispossible onlyinthecasewhen u( ) =,sinceitistheonlyequilibriumstateforequations(58-6). ItisimportanttonotethatthekinksinthecurvesinFig. 4b)indicateasmooth,butrapid change in the solution. It occurs due to the exponentially sensitive terms in the governing equations. 22

24 z a) z 5 b) Figure4: Profilesof zintheshootingprocessat D =.4D CJ, c h =, Q = 2, γ =.2and various c f : c f =.(solid), c f =.22(dash), c f =.(dash-dot);a) E = 5andb) E = 3. º¾ ËÓÐÙØ ÓÒ Ò Ø ÔÖ Ò Ó ÓØ Ø Ò Ö Ø ÓÒ ÐÓ If D s < D < D CJ,thesteady-statesolutionpassesthroughasonicpoint,whereconditions (68) must be satisfied. Imposition of these conditions yields the exact value of the detonationspeedforthegivenlossterms. Thepresenceofasonicpointisaspecialfeaturethat constrains the solutions to a particular form, essentially independent of the boundary conditionsatinfinity. If c a < D < D s,thesonicpointandhencetheconstraintdisappearsand the nature of the solutions changes since now the downstream boundary condition determines the detonation speed[3, 4]. The solutions connect the Rankine-Hugoniot conditions atthe shock, =, andsomeappropriateconditionsat =. Inthe referenceframe of the shock, the flow remains subsonic throughout. As a consequence of the loss of the sonicconstraints,whichwhenpresentpindownthesolutiontoyieldaunique c f foragiven D, a degree of freedom is gained. The only condition at = that can justifiably be imposedis u =,whichleavesoneconditiontheretobefree. Thiscaneitherbepressure or temperature; the density at infinity is the same as in the ambient gas, which follows from the continuity equation(37). As a result of this additional degree of freedom, the case c a < D < D s yields a set-valued solution in which the pressure at = is allowedto varyin some range. That is, giventhe shockconditions and u ( ) =, for anychosen D (c a, D s ),thereexistsacontinuousrangeof c f (atfixed c f /c h )thatyieldsasmoothand bounded solution of(58-6). The transition from the single-valued to set-valued solutions is shown in Fig. 5a). For the existence of the set-valued solutions, the pressure at infinity cannot simply be arbitrary. For a fixed D < D s, as we vary c f, the flow variables p and T at = vary as well (but u = and λ = remain the same). The range of the correspondingvariationof p ( ) is shown in Fig. 5b). It is important to keep in mind 23

25 thatinfig. 5b), c f isnotfixed,butrathervariesintherangeseeninfig. 5a)foranygiven D. Aninterestingfeatureofthepressureatinfinityisthatitsmaximumvalueappearsto coincidewiththemomentwhenthesoniclocusisexactlyatinfinity. Thisisthecaseinall of the calculations that we have performed. As the velocity decreases further, the pressure at infinity(hence the temperature there as well) begins to drop. Therefore, the numerical calculations indicate that the onset of the set-valued solutions coincides with the maximum temperature(or pressure) in the products..8 a).8 b).6 D D C J.4 D s D C J c a D C J.6 D D C J.4 D s D C J c a D C J c f p Figure 5: a) The dependence of the detonation speed, D, on the loss factor, c f ; b) the pressureatinfinity, p ( ),forthe D-c f dependenceshownina). Inbothcases E = 3. In Fig. 5 and all of the calculations below the following parameters are fixed (unless indicatedotherwise): Q = 2, γ =.2and c h =.4c f. For the sake of completeness, we analyze the structure of the solutions on the functionvalued branch where they correspond to the self-sustained detonations. One of the main problems in the analysis is to understand the relative role of the three driving physical processes: chemical reaction, friction and heat transfer. While the roles of the chemical reaction and heat transfer are relatively simple, the former always contributing energy and thelatteralwaystakingitaway,theroleoffrictionismorecomplex. Thefrictionterm, f, changes its sign with velocity, which can result in momentum gain as opposed to momentum losswhen u <. Moreimportantly,however,thefrictionresultsinheatingofthegasand thus actsasaheat source. This is seenin the energyequation(8). Here, weanalyzethe role of these terms in some detail by evaluating their contributions to the fluid acceleration, du/d, and to the temperature gradient, dt/d. Ifonewritestheequationfor u ()from(42)asfollows [ du d = (u D) 2 (γ )Qω c 2 (γu D) f + ρφ ] (γ ) ρφ h, (7) 24

26 thenitbecomesclearthatthebehaviourof udependsonthecompetitionbetweentheheat release,frictionandheatloss. Recallthat f = c f ρu u, h = c h u (T )andthat c h = βc f forsomefixed β (β =.4inmostofourcalculations). Itisinterestingthat γu Dinthefrictiontermin(7)cantakeeithersign. Toevaluate itattheshock,weusethejumpcondition(2), ( γu s D = c a γ u ) [ ] s (γ )M 2 M a = c a 2γ a. (7) c a (γ + )M a Therefore, γu s D < if Ma 2 < 2γ/ (γ )and γu s D > otherwise. Inthetailofthe reactionzone,where uissmall, γu D <. Thus,forsufficientlystrongdetonations,the factor γu Dchangesitssignsomewhereinthereactionzone. Forsufficientlylarge M a,we obtainthatneartheshock (γu D)f > and,therefore,since (u D) 2 c 2 < nearthe shock,thefrictiontermleadstoavelocitydecreaseasdoestheheatreleaseterm. Theeffect of the heat loss term is the opposite. At small enough detonation Mach numbers, however, theroleofthefrictiontermisreversedbecausethesignof γu Dchangesneartheshock. Thereversalofthesignof γu Dseemstooccurwellabovetheturningpointofthe D c f curve and hence no significant qualitative change to the solution results. However, we have not explored the whole range of possibilities and the consequences of this sign change. In the cases we considered, there seems to be no significant effect. Intheset-valuedpartofthesolution,theflowisfullysubsonic,i.e., (u D) 2 c 2 < everywhere. If the term inside the brackets on the right-hand side of(7) is also negative, then u >, such that u decreases as tends to. Suppose now that u reaches its zerovalue. At that point, both f and h vanish, but ω is still non-zero. Therefore, u will continue decreasing. Once u becomes negative, the h term retains its sign, i.e., it remains positive,butthe f termnowswitchessign. Insteadofactingwiththeheatlossterm,which it did for u >, now the friction term acts with the reactionterm. As the magnitude of the negative velocity keeps increasing, the friction and heat loss terms in(7) balance the diminishing reaction rate term at some point. Downstream of that point, it is the loss terms that dominate, while the reaction term stops playing any significant role. Since the sign of u is nowreversed, ubeginstoincreasetowardzero. Thetrendisasymptotic, sinceat small u,theright-handsideof(7)isessentiallyproportionalto u,suchthat u = near infinity is a(half-) stable fixed point. Theequationfortemperatureisalsorevealing. Using T = p v + pv, v = (D u)/d 25

27 and(42-43)tocalculate p and u,wefindthat dt d = (γ )Dv γ ( M 2 ) M 2 c 2 [ ( γm 2 ) ( Qω h ) ( + γu ] ρφ D M2) Df. (72) ρφ Here, M = (D u)/cisthelocalmachnumberoftheflow. Becausethefactorinfrontof thebracketsin(72)isalwaysnegative,theriseofthetemperatureinthereactionzoneis due to the positive terms in the brackets. Depending on whether M 2 is smallerorlarger than /γ, the heat release term is seen to contribute to the temperature increase or decrease, respectively. The latter happens when the cooling by expansion dominates over heating, a situation well known in ideal detonation theory. This dual role of heat releaseleads to a temperature maximum in the reaction zone. The heat loss term is seen to havean effect opposite to that of the heat release. Note that the rolesof chemicalheat releaseand the heattransferarealwaysoppositeandtheyswitchsignswhere γm 2 switchessigns. When u >,thefrictiontermleadstothelocalheatingofthegasif γum 2 /D > and to cooling it otherwise. Because f = c f ρu u, we obtain that γum 2 /D > and f <,when u. Therefore,inthiscase,thefrictioncoolsthegaslocally. Thatis,this temperature decrease is not that of a fluid particle. For the latter, its rate of temperature change is udt/d, the contribution to which by friction remains positive when u changes sign. Immediately at the shock, the jump conditions yield γu s D = γu s/c a = 2γ Ma 2 D/c a γ + Ma 2 and Alittlebitofalgebrashowsthat γu s D M2 s = Ms 2 = 2 + (γ )M2 a 2γMa 2 (γ ). [ 4γ (γ + )Ma 2 (2γMa 2 Ma 4 (γ )) ] 3 (2 γ) Ma 2 +. (73) 4 Thus,thesignoftheleft-handsideiscontrolledbytheexpressioninthebracketswhichis positiveforreasonable γ. Then,intheregionimmediatelyaftertheshock,where u >, the friction term always heats the gas. Again, the important question here is to determine Ï Ø t = Ma 2 > Û Ò Ø Ø Ma 4 3(2 γ) M 2 4 a + = t γ t + 4 3γ Û ÔÓ Ø Ú Ø 4 4 < γ < 4/3º 26