Catastrophic Transitions in Reacting Turbulence:

Spontaneous Formation of Detonations

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This post was adapted from the cover article, which appeared in the Fall 2012 issue of the HPC Insights Magazine. Further description of the DDT mechanism discussed here can be found in this paper.

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On the morning of December 11, 2005, a violent explosion shook the London suburb of Hemel Hempstead. The incident occurred at the Hertfordshire Oil Storage Terminal, also known as the Buncefield complex, when the spill from one of the storage tanks led to the formation of a large fuel-vapor cloud. Subsequent ignition resulted in an exceptionally powerful explosion, considered by many accounts to be the largest in Europe since the end of the World War II. Thirty-five years earlier, almost to the day, on December 9, 1970, a similar explosion of an open-air vapor cloud, which formed as a result of a propane pipeline break, occurred in Port Hudson, Missouri. In both cases, the magnitude of the devastation was unprecedented for this type of industrial accidents.

Unlike these two examples, which are devastating but, fortunately, relatively rare, a different and incredibly more powerful kind of explosion is observed on a daily basis. These explosions, however, occur on cosmological distances and are an astronomical phenomenon known as the Type Ia supernovae (SN Ia). They result from the thermonuclear incineration of compact, degenerate, white dwarf stars. These stars are the end product of the evolution of normal stars such as the Sun. When formed in isolation, they lead a fairly peaceful existence slowly cooling after their nuclear fuel has been exhausted. However, in binary systems when a close-by stellar companion is present, white dwarfs can end their life in an extremely powerful and bright explosion, which fully disrupts the star in a matter of seconds and is capable of outshining the entire galaxy for the period of several days. Such brightness, along with the remarkable similarity of the majority of the explosions, has turned SN Ia in the last 15 years into an indispensible tool for measuring cosmological distances. This, in turn, led to the discovery of the accelerating expansion of the Universe and of the existence of dark energy.

SN Ia (along with their more massive counterparts, namely core-collapse supernovae) also play a crucial role in the evolution of Universe. They are crucibles, in which most of the elements surrounding us – from oxygen to iron – are formed in the process of explosive thermonuclear burning of carbon, contained in the progenitor white dwarf star. Without these astrophysical explosions both Earth, as well as life on it, would not exist.

These are the examples of some of the most powerful explosions on Earth (aside from nuclear ones) and in the Universe. While they might seem very different from each other, there is, in fact, a fundamental similarity of the key physical processes that power these events. In both cases, the explosion starts when a subsonic flame is ignited in the system. In an industrial explosion, this could occur due to some external factor, e.g., a spark. In a SN Ia, ignition is triggered by the increase in density and temperature in the stellar core as the white dwarf siphons matter from its stellar companion and grows in mass approaching the Chandrasekhar mass limit. Resulting energy release and the associated fluid expansion produce turbulent motions, which wrinkle and fold the flame, thus, accelerating burning significantly.

Turbulent flame acceleration alone, however, is often not sufficient to explain the observed properties of these explosions, primarily their power. The possible missing piece of the puzzle is the deflagration-to-detonation transition (or DDT), in which a subsonic flame develops into a detonation, a supersonic shock-driven reaction wave. The rate of energy release in a detonation is significantly higher resulting in a much more powerful explosion.

It must be emphasized, however, that it is far from certain whether DDT can indeed occur in the interior of a star or in an open-air vapor cloud. The reason for that is our lack of understanding of the DDT process in such unconfined systems. In fact, it is not even clear whether truly unconfined DDT is possible at all. For SN Ia, for instance, this prompted a search for alternative explosion models, which, however, are currently not as successful in explaining the observations as the model based on DDT. Therefore, elucidating the physics of DDT, as well as the conditions that can lead to it, is important for problems ranging from the safety of fuel storage and chemical processing facilities to the nature of the SN Ia phenomenon and of the enigmatic dark energy. Furthermore, better insight into the process of detonation formation is crucial for the development of the next generation of propulsion systems. This primarily concerns detonation-based engines, which hold promise to provide up to 25% increase in fuel efficiency.

Over the past 50 years, significant experimental and theoretical progress has been made in understanding the mechanism of DDT in confined systems, such as closed channels. In this case, however, there is a significant simplification. Burning in a closed space naturally leads to pressure increase and the formation of shock waves, which are driven by hot expanding burning products. These shock waves can eventually become strong enough to ignite a detonation. In contrast, in an unconfined system, there is no obvious way to form shock waves of sufficient strength. Furthermore, turbulence that can develop in the course of the explosion in the interior of a white dwarf or in the fuel-vapor cloud is subsonic and, thus, it cannot itself form shocks or accelerate the flame to supersonic speeds.

The question then arises:

Can a highly subsonic flame interacting with the highly subsonic turbulence spontaneously develop a supersonic detonation without any assistance from the confining effect of external walls, boundaries, or obstacles?

Answering this question using numerical modeling presents a number of challenges. The main difficulty is associated with a broad dynamical range of scales involved in the problem. For instance, an open-air vapor cloud can reach hundreds of meters in size, while the characteristic burning scale (flame width) can be less than a millimeter (see animation above). The dissipative (Kolmogorov) scale of the fast turbulence that develops in the explosion can be even smaller.

This disparity of scales is further exacerbated in SN Ia. While the thermonuclear burning scale is typically similar to that of chemical flames (fractions of a millimeter to centimeters), the overall size of a star is thousands of kilometers. This is illustrated in a figure below, which shows the characteristic size and structure of a turbulent thermonuclear flame formed inside of an exploding white dwarf star, as as well as the typical range of spatial and temporal scales, temperatures, and densities encountered in the course of a SN Ia explosion. This is truly an example of combustion on an extreme scale! Modeling such a system from first principles, while resolving all relevant scales, is not feasible neither now nor in the foreseeable future.

In addition to the multiscale nature of those problem, there is also a wide variety of physical processes involved. These include complex nuclear or chemical reactions, thermal conduction and species diffusion, complex equation of state, radiation transport, etc. An attempt to include a detailed description of all these processes would typically make the cost of any three-dimensional (3D) computation prohibitive.

Therefore, in our approach, we attempted to find the simplest, yet realistic, setting that would exhibit a spontaneous transition to a detonation. Most importantly, this requires fully resolving the flame width in a 3D computation in order to avoid using any model descriptions of burning that could introduce significant uncertainties into the end result. The resolution requirement associated with this constrains the maximum practical physical dimensions of the computational domain. Since the acceleration of the flame through its interaction with turbulence is an essential part of the overall process, accurate generation of the turbulent flow field is also important. Turbulence is typically stirred on large scales, which cannot be captured in the simulation. Therefore, in our calculations, homogeneous, isotropic, Kolmogorov-type turbulence is generated on the computational grid using a spectral method. Finally, we used a simplified, single-step, Arrhenius-type reaction kinetics calibrated to represent stoichiometric hydrogen-air and methane-air mixtures. In particular, this reaction mechanism produces realistic speeds and widths of both flames and detonations and is much more efficient computationally than complex reaction networks.

The primary numerical tool for this work was the code Athena-RFX, developed in collaboration with Dr. T. Gardiner (Sandia National Laboratories). This is the reactive flow extension of the astrophysical magnetohydrodynamic code Athena, initially developed by Drs. T. Gardiner and J. Stone (Princeton University). The code is fixed grid, finite volume, higher order, Godunov-type and is massively parallel with excellent scalability demonstrated up to 100,000 CPU cores. Over the year, it has been successfully deployed at the majority of the US Department of Defense Supercomputing Resource Centers (DSRC) on a variety of computational platforms.

Since the conditions required for the onset of DDT were not known a priori, a survey of a large parameter space was required. This involved varying the type of the reactive mixture, which primarily affected the characteristic burning speed, as well as the system size and turbulence intensity. Calculations typically had the computational grid size in the range from initially modest 256 x 256 x 8096 cells (0.5 billion cells) to extreme 2048 x 2048 x 8096 cells (32 billion cells) more recently (see animation on the main Research page here). Since we use a fully compressible, explicit numerical solver and prior to the development of a detonation the flow in the system is highly subsonic, the overall number of time-steps per calculation can be substantial often in the range 100,000 – 500,000 or, equivalently, great than 10¹⁶ cell-steps! The total CPU cost of such large calculations can easily exceed 10-20 million CPU hours. Overall, numerous calculations were required to fully sample the parameter space.

The key conclusion that emerged from these models was that subsonic turbulent flames are inherently susceptible to the formation of a detonation even in the absence of any confining factors, such as walls or boundaries. Upon reaching a critical burning velocity, the flame develops a catastrophic runaway process illustrated in a figure below, which shows DDT in a stoichiometric methane-air mixture interacting with fast turbulence. The characteristic turbulent velocity at the scale of the domain width is ~36 m/s, or ~10 percent of the sound speed. Shortly after ignition, once the turbulent flame becomes fully developed, pressure begins to rise throughout the volume of the flame accelerating it and forming the leading planar global shock. Shock waves, repeatedly generated within the flame, coalesce at the leading shock front amplifying it until the detonation is ignited (panels e-i) (also see the animation above).

The underlying physical cause of the spontaneous pressure increase is the development of a supersonic flow of burning products downstream of the flame. In the reference frame co-moving with the flame, fuel enters the flame with the speed equal to the flame-burning velocity. Products leave the flame with a much higher velocity due to the overall fluid expansion caused by heating. This means that at a certain subsonic flame speed, the product velocity will become equal to the speed of sound. At this point, any pressure increase as a result of burning cannot be propagated upstream by pressure waves, which will cause an overpressure to form within the flame volume. Such overpressure compresses and heats up the fuel, which, in turn, accelerates burning and further increases the outflow velocity of the burning products. This promotes pressure confinement and sets off the runaway process, which ultimately leads to a detonation.

The critical threshold, at which this process begins, is known as the Chapman-Jouguet (CJ) deflagration speed, the theoretical maximum speed for the steady flame propagation. Laminar flames, both chemical and thermonuclear, never reach such high speeds. Turbulent flames, in contrast, can become sufficiently fast. When they exceed this threshold, their steady-state propagation is indeed no longer possible.

What do these calculations performed using a simplified, highly idealized reaction model, teach us about more realistic systems? For instance, how can these results apply to SN Ia explosions, in which thermonuclear flames are governed by very different physics?

Fundamentally, chemical turbulent flames on Earth and thermonuclear turbulent flames in SN Ia share many similarities in terms of both their structure and their response to the action of turbulence. Two visualizations below illustrate this similarity for the case of a hydrogen-air flame and a thermonuclear carbon-burning deflagration. Note the differences in the characteristic time scales shown in the upper parts of the animations.

Recently, we have performed calculations of DDT using multi-step hydrogen-air chemistry and thermonuclear carbon burning. Those simulations confirmed findings discussed above and obtained using a highly idealized single-step chemistry discussed above. In particular, the animation below shows spontaneous formation of a strong shock wave by a fast turbulent thermonuclear deflagration, which exceeds the critical speed of a CJ deflagration.

The detailed structure of a thermonuclear flame during the early and late stages of the evolution can be seen more clearly in these visualizations:

These results were, by no means, the end of the story but rather a promising starting point. They showed that spontaneous DDT is indeed possible in unconfined systems, such as open-air fuel-vapor clouds or the interior of a star. Furthermore, they specify a precise condition for the flame speed required for DDT to occur. One of the major outstanding problems, however, is that it is not known how to predict reliably the speed of a turbulent flame formed by the turbulent flow field, which may exist in a given practical situation. This is particularly true for highly unsteady flows encountered in the course of DDT. Turbulent flame models can be developed on the basis of ab initio 3D simulations, such as the ones discussed here. However, their validity has to be verified through calculations that use much larger ratios of the domain size to the characteristic burning scale along with more realistic turbulent flow fields. In particular, calculations must allow for the turbulence (possibly, inhomogeneous and anisotropic) to form self-consistently, rather than through the artificially imposed mechanisms. Such next generation of calculations will by far exceed the scale of the simulations performed to-date and will require the next generation of exascale-class supercomputing platforms. Development of such novel models is one of the main topics of my research.

Acknowledgements

I am deeply grateful to the staff of the DoD HPCMP Data Analysis and Assessment Center, in particular Drs. Vu Tran, Chris Lewis, and Miguel Valenciano, for their assistance with data visualization.