In joint work with Bernard J. Matkowsky, we have studied the development of complex spatial and temporal patterns in combustion. Gaseous mixtures do not always burn a uniform manner. In certain cases a smooth flame can break up into cells, characterized by ripples along the wave front. These cells can be stationary or move in a uniform manner or can exhibit more complex dynamics. It is believed that the development of cells is a stage in the transition from laminar to turbulent combustion. We have studied the development of complex spatial and temporal patterns in combustion, and in particular the development of patterns as parameters such as the Lewis number are varied. We have described traveling waves, periodically modulated traveling waves, quasi-periodically modulated traveling waves and chaotic modes of combustion. Currently, we are investigating the role of burner rotation on flame patterns.

Mathematically, combustion can be described by a system of highly
nonlinear partial differential equations. The numerical solution of
these equations is difficult because rapid changes in the
dependent variables (e.g., temperature and concentration of
different reactants) can occur over a narrow spatial region called
reaction zones. The location(s) of the reaction zones are not
known in advance and are often the objects of the computation.
Very high accuracy is required because the
resulting patterns are often highly sensitive to the pointwise values
of the computed solution. In order to obtain high resolution of
the reaction zone with a small computational effort, we have developed
an adaptive pseudo-spectral method. In this method we employ
a Chebyshev pseudo-spectral method across the reaction zone, together
with mappings that are chosen dynamically based on an error measure
of the solution. The effect of the mappings is to expand the reaction
zone region at the expense of other regions, or equivalently in the
mapped coordinate the solution varies more gradually and is easier to
compute. The method is applicable in areas other than combustion and
has been applied to problems in fluid dynamics and localization in
solid mechanics.

Thermonuclear astrophysical deflagrations occur in a variety of contexts, e.g., Type I X-ray bursts in neutron stars, the pre-ejection stage of classical novae, the pre-detonation stage of supernovae and burning occurring in the cores of main sequence stars. A characteristic of these deflagrations is that the Mach number is very small so that acoustic effects are generally not important. However, classical hydrodynamic codes require a timestep restriction based on the sound speed, thus severely limiting the timestep. As a result such codes are not practical, even on the most powerful current computers. Effective simulations require low-Mach number models in which sound waves are filtered out, thus allowing timesteps restricted only by the flow speeds.

Such methods have been commonly employed for terrestrial combustion, where generally the mean pressure to zeroth order is constant across the flame. For astrophpysical deflagrations, generally the zeroth order pressure (base pressure) is not constant across the flame but stratification in the base state leads to variations of several orders of magnitude. An expanded low Mach number methodology is required to account for this stratification. In joint work with Dr. David J. Lin and Ronald E. Taam of the Physics and Astronomy Department we have developed a low Mach number model which accounts for this stratification and used our model to provide the first multidimensional simulation simulation of thermonuclear shell flash on a neutron star. The multidimensional nature of the simulation of such a burst enables us to study not just the thermonuclear properties of the deflagration but also the associated convection pattern which typically varies on a more rapid timescale. Our simulation is able to encompass pre-burst, burst and post-burst behavior. The instantaneous log of the maximum energy generation rate (EGR) and the maximum Mach number M are shown over the time interval of the simulation in the figure below.

Based on our simulations we conclude that: (1) The low Mach number approximation for highly stratified environments has been developed, verified, and implemented to study astrophysical deflagrations where large vertical pressure variations exist and the Mach number M is small. (2) This model has allowed the simulation of a 2D thermonuclear shell flash on a neutron star, starting from radiatively stable initial conditions approximately 1 second from the onset of the burst and following the burst through to the burst peak and then through the post-burst decay of the energy generation rate. (3) As the burst peak is approached, a vertically expanding convective layer of BĂ©nard-like cells naturally develops, and the vertical extent of the larger cells is found to be comparable to that of the convective layer. The convective layer expands to two pressure scale heights during the burst progression. (4) Even at their maximum values, convective flow speeds are substantially subsonic. The peak Mach number was below 0.15 for the parameters we considered. (For most of the computations the Mach number was actually significantly below 0.1.), while the deviation of the pressure from the zeroth order base state was always small, providing ex-post-facto justification of the assumptions leading to the low Mach number model. (5) As the convective layer expands, fuel is naturally mixed into the convective layer affecting the rate of nuclear energy generation and the temporal description of the outburst. Mixing within the layer is very efficient, however, at the convective layer boundaries, the less efficient mixing results in significant composition gradients. Penetration of the convective layer boundaries from both above and below on convective timescales is very limited and temporary. Thus, mixing occurs primarily on the relatively long nuclear reaction timescales as the convective layer expands. (6) Both sub- and superadiabatic regions are found within the convective layer, but it is always slightly superadiabatic on average. (7) Comparison with 1D models demonstrates that convection has a significant effect on the energy transport and compositional mixing.

Smoldering is a low temperature, self-propagating (albeit slowly) wave propagating through a flammable sample. Smolder waves derive their heat primarily through heterogeneous fuel oxidation. It is known that in many circumstances smolder waves spontaneously erupt into high temperature, rapidly propagating, flames - a phenomena with obvious important consequences for fire safety. While there have been many experimental studies of the transition from smoldering to flaming, the mechanisms leading to the transition have not been previously identified. In joint work with Bernard J. Matkowsky and Anatolya P. Aldushin of the Institute of Structural Macrokinetics and Materials Science in Russia, we have employed a simplified, 3-reaction model of smoldering and using simulations together with approximate analytical approaches, have determined mechanisms leading to the transition from smoldering to flaming and the effect of various physicochemical parameters as well as sample length on this transition.

We describe results for planar, forward smolder waves driven by a constant forced flow of gas containing oxidizer in the same direction as that of the smolder wave. The chemical kinetic scheme employed consists of three reactions, namely, fuel oxidation leading to char (exothermic and providing the heat necessary for smolder wave self-propagation), pyrolysis (endothermic conversion of fuel into char), and char oxidation. We showed that the char oxidation reaction hardly affects the characteristics of smolder wave propagation due to its small reaction rate. However, under appropriate conditions, it can act as the trigger for the transition to flaming due to its ability to self-accelerate. In particular, we introduced the concept of, and then computed, a quantity which we termed the flaming distance L_F. This is the distance that a steadily propagating smolder wave initiated at the gas flux inlet travels inside the porous medium before the char oxidation reaction spontaneously self-accelerates, resulting in an eruption of the temperature in the smolder front. That is, the smolder wave propagates for a relatively long latent period of time until it reaches a distance L_F. A transition to flaming then occurs. We showed that smolder waves propagating in porous samples of length L do (do not) exhibit a transition to flaming if L_F < L (L_F > L).

The flaming distance L_F depends on the physicochemical parameters of the sample as well as external parameters, such as the gas velocity and composition of the incoming gas, heat loss, etc. We have determined and published these dependencies. Analysis of the flaming distance is clearly important for theoretical understanding of the transition to flaming as well as for fire safety applications. An example of the dependence of the flaming distance on the strength of the gas flux is shown in the figure below.

Flaming distance vs. gas flux

SHS (Self Propagating High Temperature Synthesis) is a process for synthesis of various metallic alloys and ceramics. In the process, the reactants are ground into a powder and placed in a container which is generally cylindrical. The mixture is ignited at one end. Synthesis ensues as a self-sustaining high temperature combustion wave propagates along the cylinder converting reactants into the desired products. The SHS process enjoys a number of advantages over conventional technology, in which the sample is placed in a furnace and simply "baked" until it is "well done". The advantages include (i) simpler equipment, (ii) significantly shorter synthesis times, (iii) greater economy, since the internal energy of the chemical reactions is employed rather than the costly external energy of the furnace, (iv) greater product purity, due to volatile impurities being burned off by the very high temperatures of the propagating combustion wave, and (v) no intrinsic limit on the size of the sample to be synthesized, as exists in conventional technology.

In many instances gas does not play a significant role in the reaction in which case it is referred to as gasless combustion and the combustion waves referred to as solid flames. It is well known that solid flames do not always propagate in a uniform fashion. Rather, instabilities can develop along the flame front. Several different types of instabilities have been observed. Pulsating planar instabilities occur when there is no transverse structure to the flame front, but the front speed and temperature pulsate in a periodic or quasiperiodic fashion. Since the actual product produced depends on the temperature of the combustion wave, such instabilities are manifested by striations of different products in the sample. Two-dimensional instabilities have also been observed and analyzed when it is assumed that combustion only occurs on the surface of the sample. Such instabilities are generally associated with the formation of hot spots which exhibit various dynamics on the surface, e.g., propagating in a helical fashion along and around the sample. Again, they can lead to nonuniformities in the synthesized product and in fact some materials can only be synthesized by the hot temperatures associated with the spots.

While 3D instabilities, i.e., instabilities which depend on the cylindrical radial coordinate in addition to the angular and axial coordinate, have been analyzed theoretically, computational descriptions of these instabilities have been limited due to the cost of the 3D computations. In addition, these instabilities are most difficult to observe experimentally since the sample must be dissected to ascertain the interior structure.

In joint work with Dr. Jang K. Park and Bernard J. Matkowsky we have developed a discrete rod model to simulate 3D disturbances. In this model the cylindrical sample is replaced by a cylindrical array of rods connected via heat transfer. The heat transfer simulates the transverse Laplacian, providing a qualitative description of 3D waves, limited of course by the number of rods employed. While the rod model can only provide a qualitative description of the full 3D model, we have compared aspects of the full 3D model with the rod model. We showed that in both cases the uniformly propagating solution can lose stability as the Zeldovich number (related to the activation energy of the reaction) passes through a critical value. Furthermore, both models exhibit the same dispersion relation provided parameters of the models are suitably identified.

Simulations of rod models have provided a relatively detailed description of 3D solid flame instabilities. We showed that there were three basic kinds of disturbances - radial and spin modes in addition to the planar pulsating mode. Radial modes are characterized by modes with no angular dependence so that the flame temperature and speed depends on the cylindrical radius r but not the cylindrical angle. As the front temperature pulsates one can think of a radial mode as characterized by different rings "firing" at different times. In contrast spin modes involve one or more hot spots which spin around the cylindrical axis. In the context of the rod model one can think of rods firing in an angular pattern, e.g., for a fixed radius one rod fires then its neighbor on the same disk fires and so forth. The planar pulsating mode exhibits no transverse structure and is the same for both the rod model and the full 3D model. For sufficiently large Zeldovich number the planar pulsating mode is stable for small diameter samples but not for large diameter samples.

We studied the evolution of these modes as the cylindrical radius R increased. This parameter is important for the SHS process as it is generally desirable to synthesize large diameter samples. The large diameter regime is also the regime where computations of the full 3D problem are most expensive. Our computations have shown that radial and spin modes are the building blocks of virtually all of the 3D modes that we have found. In general, however, the modes are not periodic but quasiperiodic. The quasiperiodic modes were shown to be combinations of periodic spin and radial modes which do not exist stably by themselves. We have identified the different combinations, leading to periodic or quasiperiodic solutions, which occur as R increases.

Furthermore, we have been able to obtain information
for very large diameter samples. In particular, we found
the very surprising result that as R increases, while
all computed solutions exhibit complex spatial behavior,
the frequency content
approaches that of the pulsating planar solution which
is generally not stable for large R but can be stable for small R.
Thus, our computations suggest that while the spatial structure
of solid flames for large diameter samples can be very complex,
the temporal structure is simply that of copies of the pulsating
planar solution which would be observed for small R.

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Last modified: February 8, 2007