Model for Shock Wave Chaos
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Citation Kasimov, Aslan R., Luiz M. Faria, and Rodolfo R. Rosales. Model for Shock Wave Chaos. Physical Review Letters 110, no. 10 (March 2013). © 2013 American Physical Society.
As Published http://dx.doi.org/10.1103/PhysRevLett.110.104104
Publisher American Physical Society
Version Final published version
Citable link http://hdl.handle.net/1721.1/79877
Model for Shock Wave Chaos
Aslan R. Kasimov* and Luiz M. Faria Division of Computer, Electrical, and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
Rodolfo R. Rosales Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 12 February 2012; published 8 March 2013) 2 We propose the following model equation, ut þ 1=2ðu uusÞx ¼ fðx; usÞ that predicts chaotic shock waves, similar to those in detonations in chemically reacting mixtures. The equation is given on the half line, x<0, and the shock is located at x ¼ 0 for any t 0. Here, usðtÞ is the shock state and the source term f is taken to mimic the chemical energy release in detonations. This equation retains the essential physics needed to reproduce many properties of detonations in gaseous reactive mixtures: steady traveling wave solutions, instability of such solutions, and the onset of chaos. Our model is the ﬁrst (to our knowledge) to describe chaos in shock waves by a scalar ﬁrst-order partial differential equation. The chaos arises in the equation thanks to an interplay between the nonlinearity of the inviscid Burgers equation and a novel forcing term that is nonlocal in nature and has deep physical roots in reactive Euler equations.
DOI: 10.1103/PhysRevLett.110.104104 PACS numbers: 47.40.Nm, 02.30.Jr, 05.45. a, 47.40.Rs
Shock waves arise in a wide range of physical phe- shock, and one acoustic wave that propagates toward the nomena: gas dynamics, shallow water ﬂows, supernovae, shock. These waves interact nonlinearly with each other stellar winds, trafﬁc ﬂows, quantum ﬂuids, and many and with the shock and can amplify due to the chemical others [1–3]. The theory of shock waves has a rich history energy release. It is these interactions that are responsible beginning with the fundamental contributions by Riemann for the detonation-wave dynamics, wherein the shock in the mid-19th century. Nevertheless, due to the complex- oscillates in a periodic or chaotic fashion . The main ity of the underlying governing equations, many open questions in understanding the physical mechanism of questions remain, especially in problems involving these oscillations are: How does this wave ampliﬁcation dynamical interactions of shock waves with magnetic occur, and which wave interactions are responsible for the and gravitational ﬁelds, chemical reactions, and radiation. chaos? Do all four families of waves in the reactive Euler Our focus in this Letter is on a fascinating feature of equations have to be accounted for or is there a simpler shock wave propagation in chemically reacting media: mechanism? shock wave chaos, in which the shock speed oscillates These difﬁcult questions are still largely open. However, chaotically. This phenomenon occurs in gaseous detona- the model that we propose here suggests that the mecha- tions as seen in numerical simulations of reactive Euler nism is in fact rather simple. We show that the complex equations [4,5]. In detonations, a shock wave propagates in dynamics of one-dimensional detonations are captured by a combustible medium (for example, a gaseous mixture of considering only two wave families: very fast waves hydrogen and air), ignites the medium, and is sustained by reﬂecting off the shock into the reaction zone and slow the energy released in the burning mixture. During the waves moving from the reaction zone toward the shock; chemical reaction, the mixture temperature rapidly rises moreover, the fast waves can be assumed to be inﬁnitely and, because the gas expansion is slow, the resulting pres- fast —the model still retains all the essential features. sure buildup gives rise to strong compression waves. These The inﬁnite-speed assumption leads to a single equation of waves can reach the shock because the ﬂow immediately a very unusual type: a nonlocal ﬁrst-order hyperbolic behind the shock is subsonic. The detonation-shock propa- partial differential equation. Its solutions provide a strong gation is sustained by those pressure waves in the reaction indication that there exists a simple mechanism hidden in zone that reach the shock. the Euler equations that is responsible for the complex The classical model of detonation, pioneered by behavior of reactive shocks in gas dynamics. Zel’dovich, von Neumann, and Do¨ring (ZND model, Before introducing our model, we recall that Burgers  [2,6]), is based on a system of reactive Euler equations in proposed his equation, ut þ uux ¼ uxx (subscripts t and x one dimension, that consists of equations for the balance of indicate partial derivatives and 0 is a viscosity coefﬁ- mass, momentum, energy, and the fuel concentration. This cient) in the hope of capturing the essential nature of system describes four families of waves: one acoustic and turbulence with a simple and tractable model. Following two material-entropic waves that propagate away from the a similar idea, Fickett [9,10] and Majda  introduced
0031-9007=13=110(10)=104104(5) 104104-1 Ó 2013 American Physical Society week ending PRL 110, 104104 (2013) PHYSICAL REVIEW LETTERS 8 MARCH 2013 simple analog models for detonations in the hope of gain- where f ¼ q vanishes for > s. The shock speed, ing some insight into their behavior. Fickett’s model is a V ¼ d =d , follows from the Rankine-Hugoniot shock 1 2 modiﬁcation of the Burgers’ equation and takes the form: condition , V½u þ2 ½u ¼0, where the brackets ½ denote the jump of the enclosed quantity across the shock 1 2 ut þ ðu þ q Þx ¼ 0; t ¼ !ð ; uÞ; (1) (the value behind minus the value ahead). Since ½u ¼us 2 2 2 and ½u ¼us, it follows that V ¼ us=2. where u is the primary unknown mimicking density, tem- Next, we change coordinates to the shock-ﬁxed frame, perature, or pressure, ! is a given chemical rate function, introducing x ¼ sðtÞ and t ¼ . Then, Eq. (3) yields and q>0 is a constant measuring the total chemical the following nonlocal partial differential equation: energy release (i.e., q is a fraction of energy released at 1 u þ ðu2 uu Þ ¼ fðx; u Þ; any given time). The chemistry here is represented by the t 2 s x s (4) reaction reactants ! products, with measuring a normal- ized concentration of the reaction products; it varies from which must be solved in x 0 and t 0. Here, usðtÞ¼ ¼ 0 at the shock to ¼ 1 in the products. Even though uð0;tÞ, the boundary value of the solution, is not prescribed Fickett’s model has been shown to reproduce some of the a priori, but follows by solving Eq. (4), as explained below. features of detonations [9,10,12], the key unstable charac- The function f is chosen such that it mimics the typical ter of detonations was not seen in this model until behavior of the reaction rate in reactive Euler equations, Radulescu and Tang  extended it to a two-step chemi- with a maximum some distance away from the shock cal reaction with an inert induction zone followed by an (see, e.g., Ref. , p. 47). The location of this maximum energy-releasing reaction zone. is chosen to depend sensitively on the shock—state a To underscore the physical origins of our model, we common feature in the reactive Euler equations, where show now that it is closely related to the theory of the reaction rate depends exponentially on the temperature Rosales and Majda , which is (in contrast to the as ! expð E=RTÞ, with E the activation energy and R ad hoc models of Fickett and Majda) based on weakly the universal gas constant. nonlinear asymptotic approximations of the Euler equa- Equation (4) is thus a model for the reaction zone of a tions. In Ref. , the following reduced system for the detonation in coordinates attached to the leading shock. By evolution of detonations was derived: construction, this shock is located at x ¼ 0 at any time. We also assume that the shock satisﬁes the usual Lax entropy 1 2 u þ ðu Þ ¼ q ; ¼ !ð ; uÞ; (2) conditions , such that the characteristics from both 2 sides of the shock converge on the shock. In characteristic du=dt ¼ fðx; u Þ dx=dt ¼ where and are time and space, respectively, and u is a form, Eq. (4) is written as s along u u =2 u ¼ 0 temperaturelike variable. The reaction rate ! was chosen s . Assuming that ahead of the shock, it u > 0 to be of the ignition-temperature type with !>0 if u>0 follows that s guarantees the Lax conditions. x ¼ 0 and ! ¼ 0 if u<0. The validity of the model derivation is, Importantly, no boundary condition at is needed, x<0 however, not restricted to this type of rate function. The as the characteristics from are outgoing, i.e., dx=dtj ¼ðu u =2Þ ¼ u =2 > 0 u key difference between the Fickett-Majda models and x¼0 s x¼0 s . Note that s u ¼½u Eq. (2) is that the rate equation in Eq. (2) involves the measures the shock strength, because s . space derivative as opposed to the time derivative in Eq. (1). The most unusual mathematical feature of Eq. (4) is that u ðtÞ Physically, this means that in Eq. (1), the rate equation does it contains the boundary value of the unknown, s . This not propagate any waves, while in Eq. (2), waves are is in fact the main reason for the observed complexity of propagated instantaneously by the second equation. the solutions and has a simple physical interpretation: the x ¼ 0 For Eq. (2), consider a shock at ¼ sð Þ, moving into a boundary information from is propagated instanta- x<0 uniform state ahead of the shock, where u ¼ 0 and where neously throughout the solution domain, , while there there is no reaction, ! ¼ 0. Assume that the reaction is is a ﬁnite-speed inﬂuence propagating from the reaction triggered by the shock, and it is such that at <