Luc Bauwens

Detailed structure and propagation of three-dimensional detonations

Self-sustaining detonations exhibit a three-dimensional structure within the reaction zone which is much more complex than one-dimensional classical theory. While the structure of two-dimensional detonations is now fairly well understood, the details of the three-dimensional structure remain largely unknown. Numerical simulation, performed with the reactive Euler equations coupled to a single-step chemistry model, yields insight into the three-dimensional reaction zone structure. Simulations in a square channel show that, in the absence of wall losses, the three-dimensional transverse wave spacing is the same as in two dimensions. However, the transverse wave structure is much more intricate. Two perpendicular modes exist in this channel, and they are approximately one-quarter of a period out of phase. This phase shift accounts for the slapping wave phenomenon observed in experiments. The slapping wave imprints on smoke foils occur when the transverse wave collides with the wall. The curvature of the imprints is explained by the curvature of the slapping wave, which is convex toward the wall. The interaction of the two transverse waves results in a vorticity field that is much more complex than in two dimensions. In two dimensions, vorticity originates from a point singularity, which is now replaced by a line singularity that forms closed loops interconnecting the two sets of transverse waves. The interconnection changes after collisions, creating cuts and free edges on the sheet structure which otherwise remains globally interconnected. It is the vorticity in the part of the sheet initiating or ending along a free edge that rolls up into rings. The rings entrain unburned fluid behind the Mach stem, as in two dimensions. The vorticity system provides a strong coupling mechanism between the two orthogonal sets of transverse waves.

A numerical study of the mechanisms of self-reignition in low-overdrive detonations

Below a threshold in overdrive, both stability analysis and numerical simulations predict that one-dimensional detonations in high activation energy mixtures behave as a chaotic sequence of failures followed by reignition. Instead, less chaotic, cellular detonations almost invariably occur in experiments. Numerical simulation, based on the Euler equations with single step chemistry, shows that a ZND detonation initially fails in that regime. The detonation splits into a weaker shock, a surface discontinuity separating reacted from unreacted fluid, and a rarefaction wave. However, the detonation is eventually reignited by the explosion of a small gas pocket, in a process reminiscent of deflagration to detonation transition. In the fluid heated by the leading shock, the chemical reaction occurs slowly at first, but becomes faster as heat is released, until the pocket explodes. Small differences in initial temperature result in large enough differences in reaction time sufficient for one pocket of fluid to explode. In two dimensions, the explosion occurs earlier because an oblique shock structure develops which unevenly heats the fluid that passes through the leading shock. Hence, pockets that underwent more heating will explode sooner. As it moves upstream, the two-dimensional explosion, meets the leading shock and the detonation quickly develops a transverse wave structure.

Oscillating flow of a heat-conducting fluid in a narrow tube

Thermoacoustic refrigeration occurs in periodic flow in a duct with heat transfer within the fluid and to the tube. This study considers the periodic limit cycle with large pressure oscillations that is obtained in a tube when prescribed, phase-shifted, periodic velocities at the tube ends, at frequencies lower than acoustic eigenmodes, sweep a length comparable to the tube length. The temperature differences between the two ends are of arbitrary magnitude, heat transfer in the transverse direction within the fluid is assumed to be very effective and the thermal mass of the wall is large. The geometry is two-dimensional, axisymmetric, and conduction is accounted for, not only in the fluid, but also with and within the tube wall. A perturbation solution valid in a local near-isothermal limit determines the equilibrium longitudinal temperature profile that is reached at the periodic regime, the pressure field including longitudinal gradients, and the longitudinal enthalpy flux. Results are presented for tubes open at both ends and also with one end closed. In the latter case, a singularity occurs in the temperature at the closed end, with behaviour identical to Rotts result for acoustic flow with small pressure amplitude. Other new results obtained for tubes open at both ends show that when velocities at both ends are in opposite phase, internal singularities in the temperature profiles may occur.

Failure and reignition of one-dimensional detonations—The high activation energy limit

The structure of failed one-dimensional detonations is derived using a high activation energy analysis. On a time scale longer than the chemical time based upon the von Neumann temperature, high activation energy one-dimensional detonations break down into a weaker shock, a contact surface separating hot burned gases from a colder, unburned mixture, and an expansion wave. While this leading order solution is unaffected by chemistry, hence self-similar, the perturbation, accounting for the chemistry, which depends upon chemical times, is not. By accounting for the chemistry, the perturbation problem determines the delay until reignition of the detonation occurs. This problem is almost identical to the problem of initiation in the region between a shock and contact surface, which is created by the collision of two shock waves. The main difference between that problem and the current analysis is that the downstream boundary condition now consists of radiating acoustics into hot burned products, at the location of the surface discontinuity. When the Newtonian limit is applied, that is, for a ratio of the specific heats approaching unity, the hot spot at which reignition occurs approaches the location of the contact surface. The time and length to reignition are then found to vary exponentially with the activation energy of the mixture. However, the Newtonian limit is not a very realistic model, because it makes the interval between a Mach number of 1/√γ and 1 disappear: in this range of Mach numbers, adding energy to a steady flow lowers the temperature, hence the reaction rate.

Low Mach number analysis of idealized thermoacoustic engines with numerical solution

A model of an idealized thermoacoustic engine is formulated, coupling nonlinear flow and heat exchange in the heat exchangers and stack with a simple linear acoustic model of the resonator and load. Correct coupling results in an asymptotically consistent global model, in the small Mach number approximation. A well-resolved numerical solution is obtained for two-dimensional heat exchangers and stack. The model assumes that the heat exchangers and stack are shorter than the overall length by a factor of the order of a representative Mach number. The model is well-suited for simulation of the entire startup process, whereby as a result of some excitation, an initially specified temperature profile in the stack evolves toward a near-steady profile, eventually reaching stationary operation. A validation analysis is presented, together with results showing the early amplitude growth and approach of a stationary regime. Two types of initial excitation are used: Random noise and a small periodic wave. The set of assumptions made leads to a heat-exchanger section that acts as a source of volume but is transparent to pressure and to a local heat-exchanger model characterized by a dynamically incompressible flow to which a locally spatially uniform acoustic pressure fluctuation is superimposed.

Ignition between a shock and a contact surface: Influence of the downstream temperature

Slow chemistry in a finite zone of reactive mixture behind an inert shock results in the formation of a hot spot and eventually triggers ignition close to the surface where the shock originated, which, depending upon the specific scenario, could consist of a piston, a contact surface, or a flame. The current study assumed high activation energy and a ratio of the specific heats close to unity. The various scenarios can then be translated into different values of the acoustic reflection coefficient or of the ratio of the speeds of sound across the contact surface, with the piston being equivalent to a zero speed of sound downstream. Thus this study proceeds as a generalization of the work of Blythe and Crighton, who studied piston-induced ignition, for arbitrary downstream acoustic reflection coefficients. It is found that in all cases except for the piston, ignition occurs at a small distance from the contact surface. Initially, a shockless supersonic wave appears, moving toward the contact surface at a speed initially infinite but decreasing. For a ratio of the speeds of sounds below a critical value equal to the inverse of the Chapman-Jouguet (CJ) speed, this wave eventually reaches the contact surface at a speed still above the CJ value. But above this threshold, the speed reaches the CJ value, a shock appears, and the wave proceeds as a strong CJ wave. Finally, for higher temperatures behind the contact surface, a spatially uniform thermal, explosion eventually occurs in the largely unburned mixture ahead of the CJ wave before it reaches the contact surface. Thus, a variety of possible scenarios leading to detonation were uncovered. The last and strongest one is particularly applicable to flames, and it provides for a transition mechanism in the region immediately in front of the flame.

Oscillating flames: multiple-scale analysis

A complete multiple-scale solution is constructed for the one-dimensional problem of an oscillating flame in a tube, ignited at a closed end, with the second end open. The flame front moves into the unburnt mixture at a constant burning velocity relative to the mixture ahead, and the heat release is constant. The solution is based upon the assumption that the propagation speed multiplied by the expansion ratio is small compared with the speed of sound. This approximate solution is compared with a numerical solution for the same physical model, assuming a propagation speed of arbitrary magnitude, and the results are close enough to confirm the validity of the approximate solution. Because ignition takes place at the closed end, the effect of thermal expansion is to push the column of fluid in the tube towards the open end. Acoustics set in motion by the impulsive start of the column of fluid play a crucial role in the oscillation. The analytical solution also captures the subsequent interaction between acoustics and the reaction front, the effect of which does not appear to be as significant as that of the impulsive start, however.

THERMOACOUSTICS : TRANSIENT REGIMES AND SINGULAR TEMPERATURE PROFILES

The stationary, periodic solution to the problem of oscillating flow of a conducting fluid, in a duct closed at one end and with periodical mass flow rate at the other end, is known to exhibit a singularity in the mean temperature at the closed end when transverse conduction in the fluid and the duct wall is taken into account, but longitudinal conduction is neglected. For instance, a solution of that type was originally suggested by Gifford and Longsworth as a prototype for pulse-tube refrigeration. Whether these stationary singular solutions are physical or mere theoretical curiosities depend upon the existence of a scenario leading to such a limit cycle. To address that question, a transient theory is formulated, using the narrow duct approximation. The results show that at least for constant fluid thermal conductivity, all singular profiles generated by the mechanism under study are linearly stable. For round tubes, the temperature profile is shown to evolve, from an arbitrary initial value, toward an...

Interface Loss in the Small Amplitude Orifice Pulse Tube Model

Linearized analysis of the orifice pulse-tube refrigerator yields simple expressions and key concepts such as the enthalpy flux. However, except for the irreversibility that occurs at the orifice, the linear model is ideal, and it yields the theoretical Kittel coefficient of performance, equal to the temperature ratio. One of the major losses, and one of the least-well understood, that this device shares with Stirling refrigerators, is the interface loss. This loss is due to the abrupt transition between the cold end or freezer, which is an approximately isothermal space, and the tube, in which the flow is approximately isentropic. In the linearized analysis, the loss appears as a second order correction. It is found to be strongly dependent upon the pressure amplitude and of the cosine of the phase angle.

Modeling Pulse Tube Coolers with the MS*2 Stirling Cycle Code

The MS*2 Stirling Cycle Code implements a numerical model originally designed to analyze Stirling cycle engines and refrigerators on a PC. It can also be used to create usefully accurate models of pulse tube cryocoolers. The technique requires preparation of higher and lower resolution files from which a hypothetical file of infinite resolution can be projected. By varying heat transfer area assigned to the cold heat exchanger, different cooling loads are applied. From the resulting equilibrium cold end temperatures, load curves are developed. Separate determination of load, including load represented by losses not modelled by the code (principally, conduction in pulse tube and regenerator housing) leads to an estimate of temperatures achievable at various assumed loads by reference to the load curve. Output of the code includes projections of pressure drop loss in the system, enthalpy flow in the regenerator, PV work input in the compressor and temperatures, heat flows, mass flows as well as mean Reynolds, Nusselt and Mach numbers for each control volume. Material options include several working fluids and regenerator materials. Regenerator configuration options include screens, spheres and parallel plates of any density and element size. Phase shift between flows at the compression piston and orifice is an adjustable input.