UNIVERSITY OF CALGARY. Stability of Detonation Wave under Chain Branching Kinetics: Role of the Initiation Step. Michael Anthony Levy A THESIS

Transcription

1 UNIVERSITY OF CALGARY Stability of Detonation Wave under Chain Branching Kinetics: Role of the Initiation Step by Michael Anthony Levy A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF SCIENCE GRADUATE PROGRAM IN MECHANICAL AND MANUFACTURING ENGINEERING CALGARY, ALBERTA December, 2014 c Michael Anthony Levy 2014

2 Abstract The linear stability analysis of planar detonation waves was studied under a chain branching reaction model. Such kinetics mimics the characteristics and reaction dynamics of hydrogen chemistry, and thus is the motivation of this study. The reaction model is applied to the steady one dimensional Zel dovich, von-neumann and Doering (ZND) detonation wave. It consists sequentially of a chain-initiation and chain-branching step both governed by Arrhenius rates, and followed by a temperature-independent chain-termination step where heat release is associated with. The stability analysis involves applying small perturbations to the reference ZND solution and observing for positive growth rates. The initiation activation energy was used as the varying parameter. The stability of the wave is mainly associated with the chain-branching reaction zone. The most challenging factor however, was formulating the downstream boundary condition and applying it far enough downstream where chemistry is more complete. The solution obtained is numerical, as it depends on results from the reference solution, which are only available numerically. Results shows a dominant, unstable mode that becomes non-oscillatory as initiation activation energy increases. Such behavior is consistent with predicted detonation cell mechanisms in literature. ii

3 Acknowledgments Firstly, I would like to thank my supervisor Dr. Luc Bauwens, for giving me the opportunity to do research in the form of a Masters degree. He provided guidance, support, encouragement and exposure to the scientific community throughout my study. His enthusiasm for research has helped me to overcome challenges that are associated with graduate studies. Secondly, I would like to thank family and friends who have shown great support throughout the study. Finally, I thank my colleagues who have helped me along the way with great advice and encouragement. The Research presented in this thesis is sponsored by the Natural Science and Engineering Research Council of Canada, and the Hydrogen Canada Strategic Network. iii

4 Table of Contents Abstract iii Acknowledgments iv Table of Contents v List of Figures vii List of Symbols viii 1 Introduction Background Chapman-Jouguet Theory The ZND Model Detonation Structure and Experimental Observation Linear Stability Analysis Numerical Simulations Initial Boundary Value Problem Formulation Linear Stability Analysis Conservation Equations Gas Dynamics Three-step Chemical Reaction Solution Domain and Boundary Conditions Normalizing and Dimensionless Formulation Reference ZND Solution Reference Solution Gas Dynamics: Rankine Hugoniot Formulation Chemical Kinetics Left Limit Right Limit Perturbation Solution Linear Stability Analysis Perturbed Conservation Laws Gas Dynamics Rate Equations Matrix Form The Independent Variable, Boundary Conditions Need for Boundary Conditions Left Boundary, N Right Boundary, N Approach Formulation Leading Order Higher order chemistry Effect of higher order in acoustics Higher order acoustics iv

5 5.3.7 Reconstructing Numerical Analysis Numerical Strategy Reference Solution Perturbation Solution Convergence Results ZND Solution Stability Conclusion References A Boundary Conditions Formulation for various Sections A.1 Higher Order Chemistry (section 4.4) A.2 Effects of Higher Order in Acoustics (section 4.5) A.3 Higher Order Acoustics (Section 4.6) v

6 List of Figures and Illustrations 1.1 Chapman-Jouguet detonation Wave Domains of detonation and deflagration solutions in the p v plane The Rayleigh line and Hugoniot curve The CJ or Tangency Solution The ZND detonation model Smoke-foil record of a planar detonation showing the trajectory of the triple point (Strehlow, 1984) Interferogram of a Mach reflection (White and Cary, 1963) Wave motion in a detonation cell ZND solution: Temperature vs x, for E I = 4.0, 6.0, 8.0, 12.0, 16.0 and 20.0 (from left to right) ZND solution: Mass Fractions λ 1 (dotted lines) and λ 2 (solid lines) vs x, for E I = 4.0, 6.0, 8.0, 12.0, 16.0 and 20.0 (from left to right) ZND solution: Residual Mass Fractions λ 1 vs x for large x, for E I = 4.0, 6.0, 8.0, 12.0, 16.0 and 20.0 (from left to right) Growth Rate (solid line) and Frequency (dotted line) vs Initiation Activation Energy in range E I = 0 to Growth Rate and Frequency vs Initiation Activation Energy. E I > vi

7 List of Symbols, Abbreviations and Nomenclature Symbol Definition A a c p c v E R e f k M m p Q q r T t u v h c ṁ matrix of elements a ij (x) for perturbation equations chain-branching parameter defined by Eq and small initiation parameter defined by (??) specific heat at constant pressure (dimensional) specific heat at constant volume (dimensional) activation energy Gas constant energy overdrive (M0 2 /M0CJ) 2 rate constant Mach number exponent in Frobenius solution pressure total heat release local heat release reaction rate temperature time longitudanal velocity transverse velocity enthalpy speed of sound mass flux per unit area vii

8 X column vector of perturbations, ρ, u, p, λ 1, λ 2 x y z ɛ λ γ ρ u j i w σ κ ρ u v p λ 1 λ 2 T longitudinal coordinate transverse coordinate either N or N + 1, in boundary conditions heat release parameter, defined by Eq. initiation rate at the von Neumann point mass fraction ratio of specific heats total derivative density modes of solution for perturbed mass fractions arbitrary number 1 complex frequency Imaginary part of complex frequencies transverse wavelength perturbed density perturbed velocity in longitudinal coordinate perturbed velocity in transverse coordinate perturbed pressure perturbed mass fraction of reactants perturbed mass fraction of chain-branching species perturbed temperature a xy coefficients of perturbed ODE s, x and y are 1, 2, 3, 4... r xy symbols used in perturbed modes, x and y are 1, 2, 3, 4... viii

9 Subscripts 0 preshock state 1 reactant 2 chain-branching species B CJ I N T S chain-branching step Chapman-Jouguet point or value initiation step von Neumann point termination step instantaneous shock location Superscripts dimensional variable Perturbation in the chain-branching zone left- and right-going acoustic mode * end of chain-branching reaction ix

10 Chapter 1 Introduction Hydrogen s interest has been increasingly growing as a suitable addition to Canada s energy resources. It is being considered as a potential replacement to fossil fuel. Not only does it possess a high octane number, but also produces low emissions. With the continued interest in Fuel-Cell powered vehicles in recent years, hydrogen is projected to possibly become the preferred fuel for transportation. Though hydrogen possesses attractive features, it holds some serious disadvantages. One issue is that, it is not readily accessible, therefore requires another energy source to produce it. Another major concern are the numerous safety issues hydrogen has. It is highly flammable and detonable over a wide range of concentrations in comparison to hydrocarbons, and requires very little ignition energy. The current study is highly motivated by its detonable safety issues. A detonation wave is simple, a strong shock followed by exothermic chemical reactions. It consist of a wave traveling through a highly combustible medium. The unstable chemical reactions that occur behind the leading shock releases energy, this in-turn drives the wave forward. In the inviscid, non-conductive model, a discontinuity exist at the shock. Therefore, significant changes in temperature, pressure, density and velocity occurs across the shock. The pressure changes is usually the most influential and causes the most damages. Common accidental scenarios comes from the possibility of hydrogen leaking from a high pressure container (Lopez-Aoyagi, 2012), the most recent being the Fukushima nuclear disaster in Even, if the leak is detected early, it is potentially dangerous because of the wide flammability and detonability limits of hydrogen-air (AIAA G-90, 2004). The safety issues associated with the detonability of hydrogen is still poorly understood, and requires a more 1

11 in-depth study on its phenomenon before making it a widely use fuel. The detonation model that is described in this study, as a shock followed by chemical reactions, was determined independently by it s authors Zeldovich, Von Neumann and Doring (ZND). This self-propagating, one-dimensional wave is nearly always unstable and is not observed experimentally (Lopez-Aoyagi, 2012). The actual detonation wave exhibits an unsteady, multi-dimensional structure as a result of the instability of the planar structure (Strehlow, 1968b). Although the structure of the ZND wave can be easily developed from the one-dimensional Euler s conservation equations of mass, momentum and energy; the instability mechanism and its relationship with cellular structure and chemical kinetics is still sparsely understood (Lee and Stewart, 1990). Several studies have been carried out to gain better understanding of the ZND detonation wave with special interest being paid to the chemical kinetics. Erpenbeck (1962,1964) was the first to study the kinetics under a one-step Arrhenius reaction. He used a Laplace transform technique to analyze the behavior of small amplitude disturbances in plane steady detonation waves. A normal mode approach to address the linear instability problem was later carried out by Lee and Stewart (1990). Significant improvement in quality and accuracy of simulations were attained by Bourlioux, Majda and Royburd (1991a). They used a finite volume solution technique in conjunction with a simplified form of mesh refinement. Results were attained numerically, and found to be in close agreement with Lee and Stewart (1990). However, the study did not explore the very long time-propagation behavior. Therefore, in many cases the actual non-linear dynamical behavior for one-step Arrhenius pulsating detonation instability remains undetermined (Short and Quirk, 1996). This current study is focusing on hydrogen and oxygen as the reactive mixtures, which 2

12 in practice follows a chain branching model rather than a single-step Arrhenius reaction model. This theory was endorsed by Short and Quirk (1996). They explained that a large class of chemical reactions are not represented effectively by single step Arrhenius; rather a majority is influenced by chain-branching and proceed by a sequence of chain-initiation, chain-branching and chain-termination stages. The chain branching regime involves a chaininitiation step where reactants are converted to chain carriers, which can be free radicals or atoms. Next, in the chain-branching step, chain carriers are multiplied and accelerated through chemical reactions. Finally, reactants and free radicals are converted to products in the chain-termination step. Ficketts, Jacobson and Schott (1992) were one of the few researchers to recognize the difference between chain branching reaction mechanisms and its importance. They utilized the mechanism in an Arrhenius single-step thermal decomposition model in relation to the pulsating detonation instability problem. However, as noted by Abouseif and Toong (1992), it is difficult to gain any quantitative insight regarding the improvement of this chain-branching model over the one-step Arrhenius model. This is because no linear stability analysis was conducted in parallel to the computational study. In our study, we applied a three-step reaction model, which has been presented by Short and Dold (1996), Short and Quirk (1996), Trembly et al (2009), Lopez-Aoyagi et al (2012) and is also based on a generalization of the model of Gray and Yang (1965). The chain-initiation and chain-branching steps are represented by Arrhenius rates while the chain-termination step is represented by a constant rate. Heat release is only associated with the termination step. This three-step chain branching model is more realistic than a single-step model, especially for hydrogen-oxygen reactions. Not only because it resembles real chain branching; but also because, it allows the decoupling of energy release from initiation. A typical feature of this mechanism (particularly true for hydrogen) is that initiation is weak and slow compared with the other two steps (Alamo and Williams, 2005). Tremblay (2009) and Lopez-Aoyagi 3

13 (2012) shows the ZND profile consisting of a long initiation zone, a short main reaction zone and an even longer termination zone, as an asymptotic analysis of the steady wave structure in the limit of slow initiation rate. In this study, we developed a similar model, however, we focused on the role played by the initiation step. We also have a slightly longer chain branching step, as the boundary condition for the reaction zone is implemented further downstream. In the next chapter, the formulation of the initial-boundary value problem is shown. Chapter 3 will focus on the solution of the reference ZND solution. The perturbation linear stability analysis is presented in chapter 4. The crucial issue of boundary conditions is shown in chapter 5, while chapter 6 deals with the numerical methods employed. Chapter 7 will show results, while chapter 8 will discuss the conclusions. 4

14 1.1 Background Chapman-Jouguet Theory Ever since the fifteenth century, scientist have figured out that certain chemical compounds undergo unusual violent chemical decomposition when subjected to mechanical impact or shock. Abel (1869) however, was perhaps the first to measure the detonation velocity of explosive charges on guncotton. On the other hand, it was Berthelot (1881), Berthelot and Vieille (1883) who devised systematical measurements of the detonation velocity in a variety of gaseous fuels. This confirmed the existence of detonations in gaseous explosive mixtures. Later, Mallard and Le Chatelier (1863) used a drum camera to observe the transition from deflagration to detonation, thus demonstrating the possibility of two modes of combustion in the same gaseous mixture. Shortly after the discovery of the detonation phenomenon, Chapman (1889) and Jouguet (1904, 1905) independently formulated a theory that predicts the detonation velocity of an explosive mixture. Their theory was based on works of Rankine (1870) and Hugoniot (1887, 1889), who analyzed the conservation equations across a shock wave. The Chapman-Jouguet model represents a detonation wave as a discontinuity between burned and unburned fluid with instantaneous heat release due to chemical reactions. The theory is based on the assumptions that the flow is inviscid, one-dimensional, steady, and neglects mass and thermal diffusivity. As a result, the flow is isentropic everywhere except at the discontinuity. The steady one-dimensional system is governed by the conservation laws of mass, momentum, and energy. Based on the previous assumptions, and with the frame of reference attached to the wave, the following Rankine-Hugoniot relations are yielded: ρ 0 u 0 = ρ 1 u 1, p 0 + ρ 0 u 2 0 = p 1 + ρ 1 u 2 1, (1.1) 5

15 Figure 1.1: Chapman-Jouguet detonation Wave h 0 + u Q = h 1 + u2 1 2 (1.2) p, ρ, u and h represents pressure, density, velocity and enthalpy, respectively. Q is the total heat release. The subscript 0 refers to the pre-shock state while 1 refers to the post-shock state. We will consider the equation of state h(p, ρ), for both reactants and products. Assuming h = c p T, together with the perfect gas equation p = ρrt and relations c p c v = R, γ = c p /c v, the equation of state for sensible enthalpy can be written as: h = γ p γ 1 ρ (1.3) With initial conditions (p 0, ρ 0, h 0 ) specified, we shall have four equations (Eqs. 1.1 to 1.3), but five unknowns (p 1, ρ 1, u 1, h 1 and wave propagation speed u 0 ). Therefore, we require an additional equation. Combining the mass and momentum conservation equations, we obtain: ṁ = p1 p 0 v 0 v 1 (1.4) 6

16 where v = 1/ρ is the specific volume and ṁ = ρu is the mass flux per unit area. This equation can also be written and interpreted as the Rayleigh line relation. p 1 = ṁ 2 (v 0 v 1 ) + p 0 (1.5) In the (p v) plane, we will have two regions of solution if the mass flux (ṁ) remains real. Therefore, if v 0 > v 1 (or ρ 0 < ρ 1 ) then p 1 > p 0, we have the compression solution for detonation waves; while, if v 0 < v 1 (or ρ 0 > ρ 1 ) then p 1 < p 0, and we have the expansion solution for deflagration waves. If we define x = v 1 /v 0 and y = p 1 /p 0, the detonation and deflagration region in the p v is shown in figure 1.2. Combining the Rayleigh line relation with the energy equation, we obtained the Rankine- Hugoniot equation: p 1 p 0 = γ + 1 2Q + ρ 0 /ρ 1 γ 1 γ + 1 ρ 0 /ρ 1 1 γ 1 (1.6) Plotting Eq. (1.6) in the (P, 1/ρ) plane, yields a hyperbola which is referred to as the Hugoniot curve. This curve is a function of the heat release, but independent of the mass flow rate. The slope of the Rayleigh line, however, is dependent on the mass flow rate. The Hugoniot curve represents the end states given the initial states, the heat release, and the ratios of specific heats. These end states belongs to both the Hugoniot curve and Rayleigh line, and therefore satisfies all three conservation laws. This model represents two solutions of mass flow rate, where the Rayleigh line becomes tangential to the Hugoniot curve as shown in figure 1.4. We obtained a minimum-detonation-velocity solution and a maximumdeflagration velocity solution. These tangency points are referred to as the upper CJ point (minimum entropy) and the lower CJ point (maximum entropy) respectively. At the CJ 7

17 Figure 1.2: Domains of detonation and deflagration solutions in the p v plane. points, the wave travels at the sonic speed. In selecting the detonation solution from the conservation laws, Chapman and Jouguet developed a criterion. Chapman (1889) found out that a minimum detonation velocity exists when the Rayleigh line is tangential to the Hugoniot curve. He found out that there are no solutions to the conservation equations at velocity lower than the minimum velocity, and there are two solutions for velocities higher than the minimum velocity. He then postulated, with the aid of experiments, that the correct solution must be the minimum-velocity or tangency solution. Jouguet (1904) later chose a solution that corresponds to the downstream state being sonic (relative to the wave). He also noted that this sonic solution also corresponds to a minimum value of the entropy change across the detonation wave. As a result, the sonic-flow or the minimum-entropy requirement can provide a criterion for selecting the solution from the conservation laws. Though there are similarities between the CJ detonation velocity and experimental measured 8

18 Figure 1.3: The Rayleigh line and Hugoniot curve. values, no theoretical justification was given by either Chapman or Jouguet. There are two solutions for detonation velocity greater than the minimum or CJ value. There is a upper strong (overdriven) or a lower weak (underdriven) detonation solution. For the strong solution, the flow velocity is subsonic behind the detonation, which results in an expansion wave penetrating the reaction zone and later attenuate the wave. Therefore, it is generally agreed upon that upper strong detonation is unstable for freely propagating detonations and can be ruled out from stability considerations. Lower weak detonation solution on the other hand, could not be easily discarded. Becker (1917, 1922a, 1922b) pointed out that entropy increase across a strong detonation, is higher than that across a weak detonation, and claimed that a strong detonation is the more probable solution. However, since the strong solution is unstable, he arrived at a conclusion similar to that of the CJ criterion; that the solution must be the sonic or be the minimum-velocity solution. 9

19 Figure 1.4: The CJ or Tangency Solution The ZND Model The work carried out by Chapman and Jouguet (CJ theory) to predict the detonation velocity is very important in understanding the detonation phenomenon; however, it lacks detailed transition through the structure of the detonation. Given initial states, the final states of detonation are obtained from the Hugoniot curve. The Rayleigh line describes the transition period from initial to final state. However, only the point where the Rayleigh line and Hugoniot curve intersects are usually of interest, since they suffice the relationship between upstream and downstream conditions. A model of the structure of a detonation wave is therefore required to described the transition zone. This model would specify the physical and chemical processes that are responsible for transforming the initial state to final state. There are several researchers from earlier years who proposed different structures of detonation propagation. Mallard and Le Châtelier (1881) stated that a detonation propagates as a sudden adiabatic compression wave that initiates a chemical reaction. Its propagation velocity should be comparable to the speed of sound of the products. Detonation is described 10

20 as a discontinuity supported by chemical reactions which are induced by this discontinuity (Vieille, 1899). Becker (1917) recognized the role of shock heating in the initiation of the chemical reaction in detonation waves. However, he believed that heat reaction from the reaction zone plays a part in initiating the the rapid chemical reactions in the shock-heated gases. He then, rejected the view that chemical reactions were brought to ignition temperatures by adiabatic compression alone, as postulated by Le Châtelier. The model for the detonation structure was accredited to work independently carried out by Zeldovich (1940), Von Neumann (1942) and Döring (1943) and is generally referred to as the ZND model. The ZND model is a continuation of the work of Chapman and Jouguet, and therefore, some assumptions from the CJ theory holds true. The flow is assumed to be steady and one-dimensional (as in the CJ theory). The shock is also seen as a discontinuous jump, with transport effects of diffusion, radiation, heat conduction, and viscosity neglected. There are improvements in the ZND model. Chemical reaction rates now being spatially resolved is a major improvement. The reaction rate is zero in front of the shock and finite behind until equilibrium conditions are reached. The reaction rate and thermodynamic properties in the reaction zone is governed by irreversible chemistry. 11

21 Figure 1.5: The ZND detonation model. The model has been formally described as a leading shock followed by chemical reactions. Reactants are heated and adiabatically compressed by the shock to ignition temperatures. Immediately behind the shock is the induction zone where reactants are converted to free radicals. This is a thermally neutral zone, where thermodynamic states of the mixture remains relatively constant. When sufficient concentrations of radicals are produced, chainbranching reaction takes place. In this reaction zone, chemical energy is released, which further increases temperature and conversely reduces the pressure and density of the mixture. Following the reaction zone is the termination zone where chemistry is usually complete and all reactants are converted to products. Due to the reduction of pressure in the reaction zone, an expansion wave is developed which moves backwards and in turn causes a forward thrust that propels the detonation. The detonation wave possesses the mechanism of self ignition and self propagation. The steady one-dimensional waves are nearly always unstable in all explosive mixtures and therefore are not observed experimentally. However, the ZND structure still serves as an important model where detailed chemical kinetics of explosive reactions can be studied under 12

22 the gas dynamic conditions that correspond to detonation processes (Lee, 2008) Detonation Structure and Experimental Observation The CJ theory and later ZND model provides a good framework in understanding detonation waves. It was not until the 1950 s and early 1960 s that experimental evidence proved that detonation waves are neither steady nor one-dimensional. Even though predictions from the ZND model agree closely with experimental results, the model cannot fully describe an actual detonation wave. Instabilities in the detonation was first observed by Campbell and Woodhead (1926). They identified the phenomenon of spinning detonations in smalldiameter tubes near the detonation limit. The instability scale of the detonation is of the order of the diameter tube, causing spinning detonations relatively easy to observe, even with limited resolution of early streak cameras. One of the most standard and useful technique in investigating the structure of detonation front and cell size is the soot-foil technique. Mach and Sommer (1877) used it in their study of spark discharges and interacting shock waves, but the discovery of the technique is accredited to Antolik (1875). He was the first to observe that the path of a triple-shock mach intersection can be recorded as a well-defined thin line on a soot-coated surface. It was Denisov and Troshin (1959) however, who first applied the technique through the study of unstable detonation structure. They observed transverse waves on the smoke-foils that were later interpreted as being the trajectory of triple points. It is believed that Denisov and Troshin work were inspired by earlier works of spinning detonation. Investigations later carried out by Strehlow (1967) confirmed that soot-foil track angles together with shock polars may be used to evaluate triple point configuration. 13

23 Figure 1.6: Smoke-foil record of a planar detonation showing the trajectory of the triple point (Strehlow, 1984). Sobbotin (1975), with the use of Schlieren images deduced that different mixtures produced different triple point structures. He confirmed that transverse waves are reactive in regular mixtures and nonreactive in irregular mixtures. Edward et al (1982) agreed with Sobbotin in their observation that there are islands of unburt gases that are isolated from the main wave after a triple point collision. Images of cellular structure of detonation waves at the reaction zone were produced by Pintgen (2000) using OH Planar Laser Induced Florescence (PLIF). From soot-foil technique, it became more clear that detonation waves are unsteady and multi-dimensional. White (1961) in the form of interferogram provided more evidence. He showed that detonation fronts are regular and becomes highly irregular after the front, continued through the reaction zone and beyond. It is shown in figure 1.7 below. 14

24 Figure 1.7: Interferogram of a Mach reflection (White and Cary, 1963). Figure 1.8 shows a simpler outline of a detonation cell that highlights the incident shock and mach stems. It also shows reflected shock and triple points. Figure 1.8: Wave motion in a detonation cell. Here the outline shows that the mach stem propagates much faster than the incident shock, and as a result the triple point moves into the incident shock (triple point is the intersecting point of incident shock, reflected shock and mach stem). As this happens, the mach stem grows and the incident shock shrinks until the two triple points collide. Immediately after the collision, an explosion occurs and the new mach stem is developed and the older mach stem become an incident wave. 15

25 1.1.4 Linear Stability Analysis Self-propagating one-dimensional ZND detonations are unstable and are not observed experimentally. The solution of steady one-dimensional conservation equations can be used to show laminar structures of ZND detonation waves. This solution enables the varying of activation energy, which is a direct relationship with temperature sensitivity, and consequently detonation stability. The most traditional and one of the most effective method of investigating detonation stability is to perform a linear stability analysis. It involves superimposing small time-dependent perturbations on the steady ZND solution and observe if there is a growth or decay with time. Special care must be taken when applying perturbations. Flow coming from upstream is supersonic as it hits the shock (frame of reference attached to shock). It bears little effect on the reaction zone and downstream conditions, but may affect the shock location and strength. On the contrary, downstream flows are usually sonic or subsonic and may penetrate the reaction zone, and consequently, the shock front. As a result, if the flow possesses multi-dimensional features, it is very likely that the shock front will exhibit similar multi-dimensional oscillation motion. The reference solution that is being investigated is the steady one-dimensional planar ZND wave. Typical solutions of ZND waves are time-independent and are easily linearized. The perturbation solution on the other hand, are time-dependent. The amplitudes are small and therefore can be represented in Fourier space as a linear combination of Fourier modes. The condition for a stable reference solution involves a negative growth rate of Fourier modes. This is indicated by the non-existence of solutions that grows with time. If the perturbation, however grows with time, the wave is recognized to be unstable. This type of analysis is called The Hydrodynamic Stability Analysis and is generally tedious and requires complicated numerical treatment. 16

26 There are several studies that were carried out in investigating stability analysis of ZND detonation waves dating back to Some of these works are presented below in a short synopsis. Zaidel (1961) was the first to perform a linear stability analysis. This was performed on a square wave solution in contrast to a ZND solution. One of his major assumptions was that no heat release took place within the induction zone, with the length taken from empirical model. Rather, the induction zone was described by the non-reactive acoustic equations. Due to the simplicity of the structure of this square wave solution, results showed that growth rate continues, even with increase frequencies. According to (Clavin et al, 1997), this did not seem reasonable. Erpenbeck (1962, 1964, 1967) used an initial value Laplace transform approach to analyze the stability of steady ZND solutions. His results did not exhibit the Zaidel paradox. He considered a single step Arrhenius finite rate chemical reactions and included multidimensional transverse perturbations. Due to analytical difficulties associated with the CJ point at the end of the CJ detonation wave; his results were limited to overdriven detonations. Buckmaster and Neves (1987, 1988) obtained similar results to Zaidel (1961) when they formulated an asymptotic approach for one-dimensional stability within high activation energies. The resulting spectrum consisted of infinite number of unstable oscillatory modes, which became more unstable as frequency increases. These results are expected, as the ZND solution approaches the square wave (as in Zaidel) in the limit of high activation energy. The major improvement in this analysis was the ability to determine the induction length. Lee and Stewart (1990) employed a normal-mode approach to the linear stability prob- 17

27 lem for the single step finite rate Arrhenius kinetics. The approach allowed the solution of unstable eigenfunction and its dispersion directly, through a two point numerical boundary value shooting method. This is called the Newton-Raphson method. A parametric study was performed to determine the evolution of the discrete spectrum while varying activation energy and overdrive. A continuous neutral stability zone was obtained. This boundary spectrum is where the solution is neither stable nor unstable, and the growth rate is zero along this neutral stability curve. Short (1996) derived an analytic dispersion relation to the square wave problem, but used a different asymptotic limit. The derivation was based on the Newtonian limit (where the ratio of specific heats is close to zero) and also within high activation energies. Short results resembles that of Buckmaster and Nevess (1987) with the Zaidel pathology. He later acknowledged the similarities in results between high activation single-step Arrhenius kinetics and the square wave structure. Short (1997a,b) later found out that disturbances with wavelengths longer than the heat release thickness yielded stable results. His analysis could not predict the behavior of modes with high frequency disturbances since the reaction zone could not be taken as a discontinuity. He postulated that these high frequency mode solutions can be determined using a proper asymptotic description. The spectrum would entail re-scaling of the characteristics disturbance parameters, considering the effects of the structure of the quasi-steady reaction zone due to the high frequencies. Short and Stewart (1998) investigated the single-step stability problem in two-dimension by applying a linear transverse perturbation. Parameters were varied to observe the wave structure. In this two-dimensional perturbation structure, oscillatory modes bifurcates into 18

28 two non-oscillatory modes, as transverse wave number exceeded a certain limit. Buckmaster and Ludford (1987) and later Bauwens et al (1998) developed various theories at this critical limit. Some of these theories stated that the detonation: failed without any oscillations, broke down into weaker shocks, formed a discontinuity at the surface separating burnt and unburnt mixture, and produced a rare-fraction wave moving downstream. Sharpe (1997) implemented a new normal mode approach to the linear stability problem of CJ detonations, by utilizing an asymptotic limit close to the sonic point. Linearizing the perturbation result in a set of ordinary differential equations from which analytical asymptotic solutions were found. These solutions were further used as the initial conditions for the numerical integration to the ordinary differential equations. This type of approach allowed the usage of higher activation energies. Short and Dold (1996) performed a linear stability analysis on the ZND detonation model using three-step kinetics with finite rates. Results obtained showed that unstable modes exist at lower frequencies. The most unstable oscillatory mode bifurcates into two non-oscillatory modes at higher frequency. Short and Quirk (1997b) performed a numerical simulation for the stability problem under three-step chain-branching kinetics. Although the three-step model is ideal enough to allow theoretical instabilities, results resembles fundamental dynamics of real chain-branching systems. Sharpe (1999a) performed linear stability analysis on both one-dimensional and two-dimensional pathological detonations. These detonations are characterized by two successive irreversible reactions, with the second of the two being endothermic. Adding perturbations to both 19

29 detonation analysis yielded results similar to that of CJ overdriven detonations with infinite length. Liang and Bauwens (2005a, 2005b, 2006) investigated a simple chain-branching chemical model that characterizes hydrogen-air chemistry, yielding three explosion limits. The chainbranching model includes four steps: initiation, chain branching, and two termination steps. This model resembles hydrogen chemistry Numerical Simulations The complexity of the detonation phenomenon has limited several experimental work. A very useful and effective method to obtain results of different aspects of the phenomenon is through numerical simulations. This method proved to be very efficient especially in understanding long term behavior of cellular structures of detonations. Several researchers have presented numerical studies on detonation ranging from one-dimensional single-step kinetics to multi-dimensional and multiple-steps kinetics. With the method of characteristics, Fickett and Wood (1996) investigated one-dimensional galloping detonations numerically. His results of pulsating instability concur with literature about linear stability of detonations. Short and Quirk (1996) also performed numerical investigations on one-dimensional detonation stability. Their work, however, is a slightly more complex model with three-step chain branching kinetics. Chain-branching crossover temperature was used as the bifurcation parameter. The simulation was carried out by employing a sophisticated adaptive mesh refinement algorithm from Quirk (1996). Lopez-Aoyagi et al. (2012) performed similar numerical task to Short and Quirk on steady one-dimensional detonations with three-step chain branching model. In their work, they used a secant method, where a carpet search was done to utilize interactive solutions that satisfies downstream conditions. Lee and Stewart (1990) approached linear stability similarly, but used a Newton- 20

30 Raphson iterative root search instead. Two-dimensional and three-dimensional numerical modeling are more complex and are usually done with single-step schemes. Taki and Fujiwara (1981) performed numerical study on transverse waves, to observe how the type and number of perturbations affects its waves structure. Earlier study by Taki and Fujiwara (1978) concluded that, independently from initial perturbation, a fix number of triple points could be achieve from a given channel height and gas mixtures. By improving the induction parameter, Oran et al. (1981, 1982) developed a model that provided more accurate energy release simulations of hydrogen-air and methane-air detonations. Kailasanath et al. (1985) adapted the same technique as Oran et al. (1981) and provided a systematic approach in determining characteristic cell size. They found a direct relationship with channel height, characteristic cell size, and triple points. Whenever the channel height is larger than the cell size, more triple points are produced around the detonation cell. Even though multi-dimensional modelings are usually associated with single-step kinetics, Laing and Bauwens (2005) performed a more complex four-step chain branching kinetics model of two-dimensional transverse waves. Williams et al. (1996a) used single-step Arrhenius kinetics and presented well resolved threedimensional results. While results in three dimensions are more complex, they suggested that the interaction between transverse waves are nearly equivalent to a two-dimensional model super-imposed in the two transverse directions. However, the resulting vorticity fields are more intricate than in two-dimensions. Quirk (1994) and later Sharpe and Falle (2000) also formulated single-step three-dimensional models with varying resolutions. 21

31 Chapter 2 Initial Boundary Value Problem Formulation 2.1 Linear Stability Analysis A linear stability analysis is performed on the steady ZND reference solution. The analysis involves adding small time-dependent perturbation series to the solution of the initial boundary value problem and observe for growth. The perturbation problem is linearized in the limit of diminishing amplitudes. As a result, since the reference solution is time-independent, the perturbation problem becomes an initial boundary value problem that can be expressed in Fourier space in time. Solution to the perturbation problem is constructed by linear combination of Fourier modes, which can be obtained through a series of eigenvalues. Complex frequencies depicting time-dependency of Fourier modes are sought after. Characteristics of the complex solution (real and imaginary parts) will therefore explain the stability mechanism of the reference solution. The imaginary part of the solution contributes an oscillatory component and explains the frequencies of each mode. The real component, however, tells whether or not the mode grows in time. If the eigenvalues are greater than zero, or exhibits positive real parts, then the solution grows in time and is therefore deemed unstable. The next section of this chapter will look at the conservation equations, including the gas dynamics equations, the chemical kinetics equations, and the heat release parameter that links the gas dynamics to the chemistry. Subsequent sections will discuss the solution domain and associated boundary conditions. This chapter is ended by discussing the normalizing parameters that makes the problem dimensionless. 22

32 2.2 Conservation Equations Gas Dynamics The detonation wave being considered is described by the reactive, non-conductive, invicid Euler s conservation equations of mass, momentum, and energy. Some of the key assumptions includes: diffusion processes being negligible and the reactive mixture consisting of an ideal gas with constant specific heats. The equations in their dimensional form is shown below: (ρu) t ρ t + (ρu) = 0 + (ρu : u + p) = 0 (ρe) t + (ρuh) = 0 (2.1) The thermodynamic properties are related by the equations of state. e = C v T + u u 2 q, p = ρrt (2.2) where ρ is density, u is velocity, p is pressure, T is temperature, and t is time. The variable e, denotes energy, including internal energy, kinetic energy, and partial heat release q due to combustion. γ = C P /C V is the ratio of specific heats and R = C P C V is the gas constant. The link between gas dynamics and chemical kinetics is entirely due to the contribution of q Three-step Chemical Reaction The three-step chain branching kinetic model is considered to be a good prototype, especially for hydrogen chemistry. It was first used by Kapila (1978) and later used by Short and Dold (1996), Short and Quirk (1996), Tremblay (2009) and Lopez Aoyagi et al (2012). The model consists of an initiation step described by a stiff Arrhenius rate, a chain-branching step also described by an Arrhenius rate, and a termination step which is described by a constant 23

33 rate. The three-step model is advantageous to the single-step Arrhenius model, since it allows the decoupling of heat release from initiation, which is typically unavoidable with a simple single-step model. Hydrogen chain-branching chemistry usually associates heat release with termination and a similar approach is taken in this model, where heat release is associated with the termination step only. The kinetics are primarily described by the mass fractions of reactants, λ 1 and chain-branching species, λ 2 and is shown below: λ 1 t + u λ 1 = r I r B (2.3) λ 2 t + u λ 2 = r I + r B r T (2.4) where subscripts I, B and T represents initiation, chain-branching and termination respectively. The reaction rates are described as follows: r I = λ 1 k I exp E I RT, r B = ρλ 1 λ 2 k B exp E B RT, r T = λ 2 K T (2.5) where E is the activation energy. Rate constants of initiation and chain-branching k I and k B are represented by their respective activation energy and temperature, k I = exp E I /RT I and k B = exp E B /RT B. Using Q as the total heat release, and the product mass fraction as 1 λ 1 λ 2, the partial heat release, q that links gas dynamics to chemistry is described as: q = (1 λ 1 λ 2 )Q (2.6) 2.3 Solution Domain and Boundary Conditions The solution domain is taken to be infinite in space and semi-infinite in time. In the stability analysis, initial conditions can be seen as consisting of the reference solution to which an arbitrary perturbation of diminishing small amplitude is added. This infinite domain in space is therefore transformed to a more realistic domain where desired results can be 24

34 obtained. This solution however, requires intense boundary applications, since frequency tends to increase considerably at boundary limits. Coming from the left, x, a supersonic incoming flow of unburnt mixture is specified, with Mach number M 0, density ρ 0, pressure p 0, hence temperature p 0 /R T 0, and mass fractions λ 1 = 1 and λ 2 = 0. The frame of reference is set as such so that x is parallel to the flow. No matter how slow chemical kinetics may be, if the flow comes from infinity, it will be fully burnt when reaching a finite location. In order to retain a meaningful problem, the usual approach (Lee and Stewart, 1990) is to turn on the chemical reaction at the shock, for instance by specifying a (small) cut-off temperature. In the perturbation problem however, the shock may not necessarily be at zero due to small amplitude oscillations. As a result, we need to account for these motions when formulating the left boundary conditions. As we move towards complete combustion at the exit, x +, the left-going Reimann variant is set to zero assuming an infinitely long duct. However, there is on-going acoustics towards the right. This causes the solution to oscillate more rapidly in the newly formed domain. The boundary condition is therefore applied at some finite location close to complete combustion, but we will need to correct for the difference as chemistry is not quite complete. 2.4 Normalizing and Dimensionless Formulation Various options have been used in scaling the current problem. Short and Dold (1996) used the post shock state as the reference. This has one crucial advantage; the boundary separates the region where the chain-branching explosion takes place, from the conditions at which it does not, precisely at a dimensionless T B = 1. Here, however, there is great value in comparing the effect of overdrive (or shock strength) in a fixed, specified preshock mixture. This problem has been made dimensionless scaling density and temperature by 25

35 their preshock (x ) values (represented by a subscript 0), velocity by the preshock speed of sound (c 0 ), the heat release by the preshock speed of sound squared, and finally, pressure by γ times the preshock pressure. As mentioned above, chemistry is taken to be negligible at the conditions corresponding to the left boundary conditions, where the variables are dimensional. Time is scaled by the termination rate so that t = t/k T. Space is scaled by c 0 t/k T. Then, dimensionless variables are: ρ = ρ ρ 0, T = T, c 0 = γ p 0, T 0 ρ 0 u = ũ c 0, p = p γ p 0, Q = Q c 0 2 (2.7) This leaves the conservation laws above unchanged, while equations of state and mass fraction equations are in complete description: in which and e = (ρu) t ρ t + (ρu) = 0 + (ρu : u + p) = 0 (ρe) t + (ρuh) = 0 (2.8) p (γ 1)ρ + u u 2 q, h = e + p ρ, γp = ρt, q = (1 λ 1 λ 2 )Q (2.9) λ 1 t + u λ 1 = r I r B, λ 2 t + u λ 2 = r I + r B r T (2.10) r I = λ 1 k I exp E I RT, r B = ρλ 1 λ 2 k B exp E B RT, r T = λ 2 K T (2.11) with k I = exp E I T I, k B = exp E B T B (2.12) 26

36 Chapter 3 Reference ZND Solution 3.1 Reference Solution The profile of the steady one-dimensional ZND wave will be formulated in this chapter. The three-step chain branching kinetic model discussed in the previous chapter is implemented. The purpose of this study was to identify how stability results affect the main reaction zone of this reference solution. The formulation will be based on applying a right boundary condition far enough downstream of the reaction zone, where chemistry is close to completion. The solution involves using the three fundamental set of equations: gas dynamics, chemical kinetics, and the heat release parameter that links the two. 3.2 Gas Dynamics: Rankine Hugoniot Formulation The previous chapter describes an initial boundary value steady one-dimensional solution, in which the reactive Euler s equations (2.1) becomes uniform across the wave. This yields the Rankine Hugoniot equations: ρu = M 0, M 0 u + p = M γ (3.1) γp (γ 1)ρ + u2 2 q = 1 γ 1 + M (3.2) with chemistry being characterized by the rate equations: dλ 1 dx = r I r B, u dλ 2 dx = r I + r B r T (3.3) Reactants mass fraction is maximum at the shock λ 1 = 1, but no chain-branching species are 27