Convective Front Research Articles

Convection-front propagation and wavenumber selection

The propagation of convection front in a horizontal fluid layer heated from below is studied by the numerical integration of both the Newell-Whitehead-Segel amplitude equation and the full Boussinesq equations. It is shown that the propagation speed, before its steady-state value is reached, can lie within a wide range and vary, depending on the width of the front profile. Time-dependent regimes play an important role, resulting in different wavenumbers of the roll pattern formed behind the front. The rate of roll relaxation, or tendency toward the optimal (preferred) wavenumber, strongly depends on the width of the region occupied by the roll pattern and on the front-propagation speed. At large speeds the “freezing” of the wavenumbers is possible. The effect of the Rayleigh number R is mainly indirect, by the variation of the speed of the front, so that at lower R the relaxation is faster. The evolution of the whole pattern may be complicated because of the combined effect of two competing mechanisms: the role relaxation and the counteraction of the mass of other rolls. The available results of the studies of front propagation do not contradict the possibility of the existence of the preferred wavenumber.

Existence of stationary self-maintained combustion of porous and channel propellants

It is shown theoretically that in channel and porous propellants there exists a stationary self-maintained regime of propagation of a convective flame front. The existence of a spherical structure of the combustion zone results in the appearance of a qualitatively new propagation regime — subsonic relative to the condensed phase and supersonic relative to the gas phase ahead of the front. The final states of the reaction products lie on the branch of the detonation adiabat corresponding to weak detonation. It is shown that in a propellant with constant average density and thermal conductivity the velocity of the wave is variable and can be regulated by adjusting the size of the individual channels.

Propagation of weak disturbances in the combustion of compressible porous fuels

We have studied the properties of a system of equations describing the process of nonsteady propagation of a convective flame front through a porous compressible solid fuel, within the framework of a two-velocity model which takes into account the difference in pressure within the gas and the stresses in the condensed phase. We have determined a type of system of equations dependent on the differences in velocities and pressures in the phases. The continuous transition to earlier known solutions has been demonstrated for particular limit cases.

The solution of Burgers' equation for sinusoidal excitation at the upstream boundary

This paper generates an exact solution to Burgers' nonlinear diffusion equation on a convective stream with sinusoidal excitation applied at the upstream boundary, x = 0. The downstream boundary, effectively at x = 0% is assumed to always be far enough ahead of the convective front at x = Vst that no disturbance is felt there. The Hopf-Cole transformation is applied in achieving the analytical solution, but only after integrating the equation and its conditions in x to avoid a nonlinearity in the transformed upstream boundary condition. A very simple limiting solution valid for high Reynolds number is deduced from the exact solution. This approximate solution is found to be amenable to an elegant geometrical interpretation. This is in a style similar to Burgers' classical interpretation of the solution to the simpler problem for which the excitation is provided through the initial condition. The 'shocks' present in Burgers' classical solution develop with distance downstream of the excitation in the present work. Detailed results confirming the conclusions deduced by inspection of the solution formulae are computed and presented in the form of space-time plots. Evidence of period splitting in the x-variable at lower values of the Reynolds number is found in the numerical computations. This is a characteristic indicative of the onset of aperiodic chaotic response in many nonlinear dynamical systems. However, the computations are from approximate solution formulae valid for high Reynolds number; these formulae imply complete periodicity in time, for all values of the parameters. The correct interpretation is therefore unclear at this time, although the boundary-value problem appears to have the proper structure for chaotic behavior. Figure 1 depicts the motivating physical problem. A blunt-ended body of characteristic length g is placed in an otherwise undisturbed incompressible stream of velocity U. The flow separates from the after comer, forming a gas cavity, the constant pressure, Pc, of which can be presumed to be known in advance; tr denotes the cavitation number. The curvilinear coordinate along the separation streamline is x, originating at the separation point. The streamline extends an indefinite distance downstream, with the interest being the region near the separation point. The problem Reynolds number is ~ = U~/v, v being the fluid kinematic viscosity. Assuming high Reynolds number, the viscous boundary layer shed from the body at the separation point is representable as a vortex sheet coincident with the separation streamline. Under steady flow conditions, the strength of the vortex sheet, ~/, is very simply related to the cavitation number in meeting the dynamic requirement of pressure continuity across the vortex sheet:

Heating zone dynamics in in situ combustion

The integrodifferential equation of the quasisteady regime of a moving in situ combustion front is obtained and its exact solution is constructed in a particular case; the possibility of the heat generated at the combustion front being projected into the region ahead of the front is analyzed and the heating zone dynamics in the reservoir and the surrounding rock are investigated. In a number of studies of in situ combustion it is assumed that an increase in the water-air factor or, what amounts to the same thing, an increase in convection velocity in the reservoir leads to the total transfer of the heat into the region ahead of the combustion front [1–3]. In [3] the area of the heating zone ahead of the combustion front was calculated in accordance with the Marx-Longenheim model [4]. Below, on the basis of exact solutions of model problems it is shown that in the case of quasisteady Newtonian heat transfer between the surrounding medium, when the latter is assumed to be a thermal reservoir, i.e., maintain a constant temperature, this projection of heat is possible if the convection velocity exceeds the velocity of the combustion front. In the case of unsteady heat transfer in accordance with the Leverrier model there is no total projection of heat into the region in question; in the steady-state regime a limited heating zone, proportional in depth to the square of the difference of the convection and combustion front velocities, is formed ahead of the front.

Propagating convection fronts.

The evolution of periodic patterns of parallel convective rolls in a fluid heated from below that grow inwards from a sidewall after applying a lateral heat pulse is determined from numerical solutions of the Oberbeck-Boussinesq equations. Propagation speed and intensity profile of the front between homogeneous conductive and periodic convective state and the wave number selected under the front are determined for several Rayleigh numbers.

Propagation of convective combustion in two-phase systems with longitudinal porosity and transition into the underdriven detonation regime

It is well known that the rate of normal layerwise combustion of solid unitary fuels is relatively low and is determined by heat conduction in the condensed phase. In the case of the combustion of porous charges the flame can penetrate into cracks and pores and be strongly accelerated in longitudinal channels. This leads to the appearance of convective combustion regimes, in which the flame propagates with velocities three orders of magnitude higher than the velocity of the combustion of a monolithic fuel. In this paper a theoretical model is proposed for the nonstationary propagation of a convective flame front in porous solid fuel and in fuel with longitudinal channels, and the conditions for the appearance of convective combustion and the emergence of the process into a regime exhibiting the characteristics of pseudounderdriven (weak) detonation are studied.

Convective combustion regime in a deformable solid fuel with longitudinal channels

Convective combustion is initiated when hot reaction products flow into a channel from the open end. Thus the mechanism of propagation of the convective flame front is determined not by processes of heating subsequent fuel layers as a result of thermal conductivity but rather by convective heat transfer from hot gases flowing into the channel at high velocity. Rapid flame propagation leads to the occurrence of very high pressures in the channels which, under certain conditions, are able to initiate detonation of the fuel. In this paper the authors investigate the propagation of convective combustion in cylindrical channels of a compressible solid fuel and present a mathematical model depicting the transient problem of propagation of a convective flame front.

Wave propagation in the combustion of a low-concentration aerial suspension

In the general case the convective combustion of aerial suspensions is described by the equations of mechanics of multiphase media [1]. If the volume particle content is neglected and it is assumed that in the initial stage of convective front propagation the particles are stationary, and that during combustion their temperature is constant, then the equations for describing the combustion process reduce to the equations of gas dynamics for a distributed supply of heat and mass [2, 3]. The equations and model constant mass burning rate kinetics are used to solve the plane one-dimensional problem of the combustion of an aerial suspension in part of a region bounded on one side by a fixed wall. A small parameter proportional to the mass concentration and the heat value of the fuel is introduced. The method of matched asymptotic expansions [4] is used to construct a uniformly applicable first approximation. The solution obtained describes the wave propagation in aerial suspension combustion processes. The resulting pattern includes an inclined compression wave propagated with the speed of sound followed by a convective hot reaction product front whose propagation velocity is much less (in conformity with the small parameter introduced) than the speed of sound.

Convective burning in channels and cracks in solid propellants

The mathematical model and method of calculating the unsteady propagation of a convective flame front in cylindrical channels in solid fuel proposed in this paper take into account friction and interphase heat and mass transfer. The penetration of combustion into the channel comprises the process of ignition of the entrance zone exposed to the hot reaction products flowing into the channel and the propagation of the combustion front. The results of the calculations show that the channel surface is ignited by the flow of shocked gas. The presence of an ignition lag affects the distance from the shock wave to the ignition point, but has no significant influence on the propagation velocity of the front.

Nonstationary combustion regimes in porous powders

This article proposes a mathematical model of the process of convective combustion of porous powdered systems. Possible developments of convective combustion are analyzed, including the nonstationary two-front regime and the continuous combustion detonation transition regime. The planar one-dimensional motion of the porous powdered medium is considered in the presence of heterogeneous chemical reactions. A set of incompressible, but deformable particles are examined. The equations of the mechanics of heterogeneous reacting media are used to describe the motion and combustion of porous powdered media. It is determined that two convective combustion regimes are possible in powdered systems. At low ignition temperatures, a smooth transition from convective combustion to detonation occurs. At high ignition temperatures, the combustion-detonation transition does not occur, but a nonstationary two-front complex is formed, consisting of a porous frame compression wave and a convective combustion front, which supports the wave. It is shown that these fronts propagate at constant but different velocities.

Use of a variable parameter test-cell for the study of latent-heat solar walls

A paraffin Trombe wall with double glazing, is studied experimentally with the help of a test-cell allowing several working conditions to be checked. The efficiency of the wall itself is shown to be very similar whether thermocirculation is on or not. But an important additional contribution comes from the glazing in the first case. The analysis of the results points out that a controlled air circulation would give much better results than the two cases considered. Indeed the overheating due to thermocirculation can be cancelled by some controlled air circulation only, and the important convective front losses observed at the begining of the night have to be decreased by getting the heat back to the room through forced air circulation.

Transition from convective combustion to detonation for an aerocolloidal suspension of a unitary fuel

The need to investigate the transition of aerocolloidal suspensions of unitary fuels {solid propellants and high explosives) from convective combustion to detonation has been recognized lately in connection with safety-engineeri ng problems. Various techniques exist for the initiation of combustion and detonation of aerocolloidal suspensions. In this article we discuss the initiation of combustion by ignition without an increase in the pressure. A distinctive feature of unitary-fuel systems is the liberation of a large quantity of gaseous reaction products during the combustion process. These products form a convection front, which moves under the action of the pressure drop created in the gas liberation process, entraining new particles in the combustion process. We have previously [ 1, 2] developed an asymptotic theory of the initial stage of propagation of convective combustion in aerocolloidal suspensions, during which time the velocity of motion is strongly subsonic and the particles are still unable to be entrained in the motion of the gas. In this case the condition of uniformity of the pressure p =p(t) (or homobaricity) holds in the region of the hot gases, and the gas flow is a simple Riemann wave ahead of the convection front. Within the framework of this approach it is possible to determine the law governing the motion of the convective combustion front up to the time of formation of a weak shock in the flow. In this study we obtain a numerical solution of the problem of transition from convective combustion to detonation for act.colloidal suspensions within the framework of the complete system of equations of the mechanics of multiphase reacting media [3]. We describe the evolution of convective combustion of aerocolloidal suspensions up to the formation of a strong shock in the flow, behind which particle combustion takes place, followed by the transition of this shock into the steady Chapman-Jouguet detonation regime [4]. We note a recent attempt [5] to analyze the combustion-to-de tonation transition in a porous solid propellant under shock excitation and an investigation [6] of the transition to steady detonation in a gas-droplet medium as the result of a point explosion. Fundamental Equations; Statement of the Problem. Let us consider the planar one-dimensional motion of a monodisperse aerocolloid in the presence of a heterogeneous chemical reaction. We assume for simplification that the chemical reaction is initiated by heating of the particle surface to the dissociation temperature T s (p) and continues as an equilibrium process at a temperature equal to T s, so that all the heat admitted to the particle is used for its gasification and the chemical reaction obeys the simple equation A---B, where A and B are the symbols for the chemical elements; the thermodynamic properties of the reaction products and the gas carrier are the same, the gas is calorically ideal, and the particles are incompressible. The equations of motion of the aerocolloids under the stated assumptions have the form [2, 3]

Convective combustion of aerosuspensions in fuel that contains the oxidant

The convective combustion of porous gunpowder and high explosives is an intermediate stage in the transition from layered combustion to detonation [1, 2]. The theory of convective combustion of such systems is developed in [3–6]. It has now become necessary to analyze the possibility of convective combustion of aerosuspensions. The present paper develops the theory of the combustion of such systems on the basis of an analysis of the equations of gas dynamics with distributed supply of mass and heat; the problem of nonstationary motion of a convective combustion front is formulated. In the homobaric approximation [7], when the pressure is assumed to be spatially homogeneous, an analytic solution to the problem is found; this determines the law of motion of the front and the distribution of the parameters that characterize the gas and the particles in the combustion zone. Necessary conditions for the transition from convective combustion to explosion are obtained.

Ignition of Solid Propellant Crack Tip under Rapid Pressurization

Under rapid chamber pressurization rates (~ 105 atm/s or higher), the closed end of a solid propellant crack was observed to ignite prior to the arrival of the convective ignition front. Tests were conducted in an inert crack with propellant only at the tip to eliminate some of the possible mechanisms for crack-tip ignition. Ignition delay time, defined as the time lag between the arrival of the pressure wave at the tip and the subsequent onset of emission of luminous light from the propellant surface, has been measured as a function of chamber pressurization rate. A theoretical model has been developed to explain the tip ignition phenomena. The model considers: a one-dimensional transient heat conduction equation for the solid phase; one-dimensional, unsteady mass and energy conservation equations for the gas phase near the crack tip; and an experimentall y observed pressure-time trace near the crack-tip region in place of the gas phase momentum equation. Both experimental and theoretical results indicate that the ignition delay time decreases as the pressurization rate is increased. Theoretically calculated ignition delay times are in good agreement with the experimental data. Based upon this agreement, ignition near the crack-tip region is considered to be caused by the enhanced turbulent transport of energy in the gas phase, which is driven by strong compression waves.

Burning to detonation transition in picric acid

Picric acid (PA) shows DDT behavior which follows the physical model originally proposed for 91/9 RDX wax. However, in the more porous charges, the convective flame front fails to propagate continuously during the predetonation period. Detonation occurs, but only after a significant period of several hundred μ sec after failure of the convective flame front. The uncommonly long relative time to detonation found for PA is attributed entirely to the ignition and burning processes which precede the first detected compressive (PC) wave. The curve of predetonation length vs % theoretical maximum density (TMD) is U-shaped as are the reported curves for other porous neat explosives. PA is the least sensitive explosive which has exhibited DDT under self-gas loading in our apparatus.

Predictions of vigorous ignition dynamics for a packed bed of solid propellant grains

This work represents the key results of a research project to predict the pressure wave propagation and flame spreading in a porous propellant charge ignited at one end by a high explosive initiator. The predictions attempted to model controlled laboratory tests with large grain M30 propellant confined in a cylinder and ignited by detonating an explosive disc at the open end. It was predicted that the magnitude of the initiator mass discharge rate determines the convective flame front speed and the pressure wave propagating into the propellant bed.

Studies in the transition from deflagration to detonation in granular explosives—III. Proposed mechanisms for transition and comparison with other proposals in the literature

The experimental data and proposed mechanisms reported in the literature for studies in the transition from deflagration to detonation in porous explosives have been reviewed. Earlier proposed mechanisms identified the convective flame front as the driving force leading to formation of the precursor shock. More recent experimental results for 91/9 RDX/wax [1] show, however, that the driving force for shock formation originates near the point of ignition, behind the advancing convective flame front. Using the concept of an accelerating pressure buildup in the ignition region as the driving force for precursor shock formation (similar to cast explosive results) and pertinent literature permeability data, it is shown that most of the experimental results for porous systems (e.g., variation of predetonation column length with compaction and particle size) can be explained. Consequently, variations in deflagration transition mechanisms resulting from changing compaction have been proposed and discussed at length. The importance of energy per unit volume in the solid phase in producing the accelerated pressure buildup in the ignition region is illustrated with results from an RDX/wax series.

Studies in the transition from deflagration to detonation in granular explosives—I. Experimental arrangement and behavior of explosives which fail to exhibit detonation

The purpose of this paper is to describe apparatus and instrumentation whereby the sequence of events in confined burning of explosives can be followed. These events are pressure buildups, formation of a convective flame front, formation of a detonation front, or a pressure rupture of the confining tube. Furthermore, although a convective front has been observed in every reported case of transition from deflagration to detonation in granular explosives, we find that it also occurs (at about the same driving pressures and traveling at about the same velocity) in explosives unable to exhibit a transition in this apparatus. Data are presented from the confined burning of ammonium picrate at several compactions and of 95/5 TNT/Wax. Burning these relatively shock-insensitive explosives produces a mild pressurization of the confining tube which leads, after a period of instability, to the formation of a convective flame front and, eventually to tube rupture. The rapid pressure buildup necessary for a transition to detonation is absent. Confined burning of explosives which do exhibit transition is described in a companion paper.

Studies in the transition from deflagration to detonation in granular explosives—II. Transitional characteristics and mechanisms observed in 91/9 RDX/Wax

The apparatus and instrumentation described in the previous paper have been used to observe the transitional behavior of burning to detonation for a 91/9 RDX/wax mixture. Transitions to detonation were observed for all densities between 67 and 95% TMD. The sequence of observed events following ignition of the explosives charge was: stable propagation of a convective flame front, compaction of the more porous burning charges, formation of a postconvective (compressive) wave in the ignition region, subsequent coalescence of compressive waves some distance beyond the ignition region, and the formation of a detonation wave 1 to 2 cm beyond the intersection of the convective and postconvective fronts. It is proposed that the shock to detonation sequence starts in the region of coalescence of compressive waves. The velocity of the convective flame front was observed to increase with increasing charge density in agreement with earlier results from ammonium picrate. The predetonation column length as a function of charge compaction exhibited a minimum; such minima have been reported by other workers for PETN, RDX, and HMX.