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Register a new userOn the two separate decay time scales of a detonation wave modelled by the Burgers equation and their relation to its chaotic dynamics
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Added by
S.Sm Lau-Chapdelaine
Publication date
2020
fields
Physics
and research's language is
English
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This study uses a simplified detonation model to investigate the behaviour of detonations with galloping-like pulsations. The reactive Burgers equation is used for the hydrodynamic equation, coupled to a pulsed source whereby all the shocked reactants are simultaneously consumed at fixed time intervals. The model mimics the short periodic amplifications of the shock front followed by relatively lengthy decays seen in galloping detonations. Numerical simulations reveal a saw tooth evolution of the front velocity with a period-averaged detonation speed equal to the Chapman-Jouguet velocity. The detonation velocity exhibits two distinct groups of decay time scales, punctuated by reaction pulses. At each pulse, a rarefaction wave is created at the reaction fronts last position. A characteristic investigation reveals that characteristics originating from the head of this rarefaction take 1.57 periods to reach and attenuate the detonation front, while characteristics at the tail take an additional period. The leading characteristics are amplified twice, by passing through the reaction fronts of subsequent pulses, before arriving at the shock front, whilst the trailing characteristics are amplified three times. This leads to the two distinct groups of time scales seen in the detonation front speed.
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This work is devoted to the decay ofrandom solutions of the unforced Burgers equation in one dimension in the limit of vanishing viscosity. The initial velocity is homogeneous and Gaussian with a spectrum proportional to $k^n$ at small wavenumbers $k$ and falling off quickly at large wavenumbers. In physical space, at sufficiently large distances, there is an ``outer region, where the velocity correlation function preserves exactly its initial form (a power law) when $n$ is not an even integer. When $1<n<2$ the spectrum, at long times, has three scaling regions : first, a $|k|^n$ region at very small $k$ms1 with a time-independent constant, stemming from this outer region, in which the initial conditions are essentially frozen; second, a $k^2$ region at intermediate wavenumbers, related to a self-similarly evolving ``inner region in physical space and, finally, the usual $k^{-2}$ region, associated to the shocks. The switching from the $|k|^n$ to the $k^2$ region occurs around a wave number $k_s(t) propto t^{-1/[2(2-n)]}$, while the switching from $k^2$ to $k^{-2}$ occurs around $k_L(t)propto t^{-1/2}$ (ignoring logarithmic corrections in both instances). The key element in the derivation of the results is an extension of the Kida (1979) log-corrected $1/t$ law for the energy decay when $n=2$ to the case of arbitrary integer or non-integer $n>1$. A systematic derivation is given in which both the leading term and estimates of higher order corrections can be obtained. High-resolution numerical simulations are presented which support our findings.
Historical experimental testing of high-altitude nuclear explosions (HANEs) are known to cause severe and detrimental effects to radio frequency signals and communications infrastructure. In order to study and predict the impact of HANEs, tractable computational approaches are required to model the complex physical processes involved in the detonation wave physics. Modern reduced-order models (ROMs) can enable long-time and many-parameter simulations with minimal computational cost. However, translational and scale invariances inherent to this type of wave propagation problem are known to limit traditional ROM approaches. Specifically, dimensionality reduction methods are typically ineffective in producing low-rank models when invariances are present in the data. In this work, an unsupervised machine learning method is used to discover coordinate systems that make such invariances amenable to traditional dimensionality reduction methods. The method, which has previously been demonstrated on one-dimensional translations, is extended to higher dimensions and additional invariances. A surrogate HANE system, i.e. a HANE-ROM, with one detonation wave is captured well at extremely low-rank. Two detonation-waves are also considered with various amounts of interaction between the waves, with improvements to low-rank models for multiple wave quantities with limited interaction.
We address the question whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. For this purpose, we first solve the instanton equations using the Chernykh-Stepanov method [Phys. Rev. E 64, 026306 (2001)]. These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. Using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of Chernykh and Stepanov with remarkable agreement.
This work is devoted to the study of the decay of multiscale deterministic solutions of the unforced Burgers equation in the limit of vanishing viscosity. A deterministic model of turbulence-like evolution is considered. We con- struct the initial perturbation as a piecewise linear analog of the Weierstrass function. The wavenumbers of this function form a Weierstrass spectrum, which accumulates at the origin in geometric progression.Reverse sawtooth functions with negative initial slope are used in this series as basic functions, while their amplitudes are chosen by the condition that the distribution of energy over exponential intervals of wavenumbers is the same as for the continuous spectrum in Burgers turbulence. Combining these two ideas allows us to obtain an exact analytical solution for the velocity field. We also notice that such multiscale waves may be constructed for multidimensional Burgers equation. This solution has scaling exponent h=-(1+n)/2 and its evolution in time is self-similar with logarithmic periodicity and with the same average law L(t) as for Burgers turbulence. Shocklines form self-similar regular tree-like struc- tures. This model also describes important properties of the Burgers turbulence such as the self-preservation of the evolution of large scale structures in the presence of small scales perturbations.
Dynamics of ethylene autoignition and Deflagration-to-Detonation Transition (DDT) in a one-dimensional shock tube are numerically investigated using a skeletal chemistry including 10 species and 10 reactions. Different combustion modes are investigated through considering various premixed gas equivalence ratios (0.2 to 2.0) and incident shock wave Mach numbers (1.8 to 3.2). Four ignition and DDT modes are observed from the studied cases, i.e., no ignition, deflagration combustion, detonation after reflected shock and deflagration behind the incident shock. For detonation development behind the reflected shock, three autoignition hot spots are formed. The first one occurs at the wall surface after the re-compression of the reflected shock and contact surface, which further develops to a reaction shock because of the explosion in the explosion regime. The other two are off the wall, respectively caused by the reflected shock rarefaction wave interaction and reaction induction in the compressed mixture. The last hot spot develops to a reaction wave and couples with the reflected shock after a DDT process, which eventually leads to detonation combustion. For deflagration development behind the reflected shock, the wave interactions, wall surface autoignition hot spot as well as its induction of reaction shock are qualitatively similar to the mode of detonation after incident shock reflection, before the reflected shock rarefaction wave collision point. However, only one hot spot is induced after the collision, which also develops to a reaction wave but cannot catch up with the reflected shock. For deflagration behind the incident shock, deflagration combustion is induced by the incident shock compression whereas detonation occurs after the shock reflection.
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