The flow field in the slender combustion chambers of solid propellant rockets

We analyse the near inviscid flow field generated in a slender non-axisymmetric cylindrical cavity by the gasification and combustion reactions of a surrounding solid propellant grain. These reactions are confined to a thin layer adjacent to the surface of the solid propellant, so that the flow is non-reacting in most of the cavity, and of the same form as the flow in slender ducts due to fluid injection through lateral porous walls. The non-reacting flow can be described in terms of self-similar solutions of the Navier -Stokes equations that we calculate numerically for star-shaped grain configurations, with Reynolds numbers only moderately large for the flow to remain laminar and steady. The self-similar flows for non-axisymmetric configurations show strong axial vortices that we analyze using the Euler simplified form of the flow equations for large Reynolds numbers, and the general solution of the Burgers equation for strained vortices that describes their viscous cores; a logarithmic singularity could then be encountered at their center line

Free convection from a point source of heat, and heat transfer from spheres at small Grashof numbers

A numerical description, based on the Boussinesq equations, is given for the steady free convection ¯ow due to a point source of heat and heated spheres. We begin with the non-dimensional formulation of the problem for the point source giving the numerical solution for a wide range of values of the Prandtl number, the remaining parameter. The analytical description of the temperature and ¯ow ®elds close to the point source includes constants that are evaluated numerically and are used to obtain the ¯ow ®eld around heated spheres for small Grashof numbers, and the correction of the Nusselt number, due to free convection, from its heat conduction value. The numerical solution of the free convection problem over a sphere at moderate and small Grashof numbers is carried out for Pr=0.72 and 7. Based on the small and large Grashof number descriptions, a correlation expression is proposed for the laminar ¯ow Nusselt number, covering all Grashof numbers

Laminar free convection induced by a line heat source, and heat transfer from wires at small Grashof numbers

The buoyancy-induced laminar flow and temperature _elds associated with a line source of heat in an unbounded environment are described by numerically solving the non-dimensional Boussinesq equations with the appropriate boundary conditions. The solution is given for values of the Prandtl number, the single parameter, ranging from zero to in_nity. The far-_eld form of the solution is well known, including a self-similar thermal plume above the source. The analytical description close to the source involves constants that must be evaluated with the numerical solution. These constants are used when calculating the free convection heat transfer from wires (or cylinders of non-circular shape) at small Grashof numbers. We _nd two regions in the flow _eld: an inner region, scaled with the radius of the wire, where the e_ects of convection can be neglected in _rst approximation, and an outer region where, also in _rst approximation, the flow and temperature _elds are those due to a line source of heat. The cases of large and small Prandtl numbers are considered separately. There is good agreement between the Nusselt numbers given by the asymptotic analysis and by the numerical analysis, which we carry out for a wide range of Grashof numbers, extending to very small values the range of existing numerical results; there is also agreement with the existing correlations of the experimental results. A correlation expression is proposed for the relation between the Nusselt and Grashof numbers, based on the asymptotic forms of the relation for small and large Grashof number

Structure of a flame front propagating against the flow near a cold wall

The flashback or propagation of premixed flames against the flow of a reacting mixture, along the low velocity region near a cold wall, is investigated numerically. The analysis, carried out using the constant density approximation for an Arrhenius overall reaction, accounts for the effects of the Lewis number of the limiting reactant. Flame front propagation and flashback are only possible for values of the near wall velocity gradient below a critical value. The flame propagation becomes chaotic for small values of the Lewis number

Free and forced convection around line sources of heat and heated cylinders in porous media

An analysis is presented for the steady, two-dimensional, free convection around line sources of heat and heated cylinders in unbounded saturated porous media. It is extended to account also for the effects of forced convection. The study is based on the Boussinesq equations, with the velocities calculated using Darcy's law. The analysis begins with the non-dimensional formulation and numerical solution of the problem of pure free convection around a line source of heat. When this analysis is extended to include the effects of forced convection, two parameters appear in the non-dimensional formulation: the non-dimensional value, V/sub infin/, of the free-stream velocity and its angle gamma of inclination with respect to the vertical. We first describe the asymptotic form of the solution for large and small values of the distance to the source. The far-field description, which is also applicable to the flow around heated cylinders, is needed to facilitate the numerical solution of the problem. It includes a thermal wake, aligned with the free stream, and an outer irrotational flow with a sink and a vortex at the line source. The temperature distribution near the source involves a constant A/sub 0/(V/sub infin/, gamma), to be calculated with the numerical solution of the complete problem, which is used in the evaluation of the heat transfer from heated cylinders when the Rayleigh and Peclet numbers are small compared with unity. In this case we find an inner region where heat conduction is dominant, and an outer region where the cylinder appears as a line source of heat. The asymptotic analysis is complemented with the numerical solution of the general problem for circular cylinders with a wide range of Rayleigh numbers and some representative values of V/sub infin/ and gamma. We give correlations for the Nusselt number in the limiting cases of pure free convection and pure forced convectio

Autoignition of hydrogen/air mixtures by a thin catalytic wire

The autoignition of the catalytic reaction of hydrogen/air mixtures on thin palladium wires is analyzed in this paper. The reduced heterogeneous kinetics is modeled with a mechanism that includes the dissociative adsorption of both reactants, together with three surface reactions of the Langmuir-Hinshelwood type, and the desorption reaction of the adsorbed product, H2O(s). We show that for the description of the ignition conditions, this mechanism can be simplified to a single overall surface reaction involving the temperature and gas concentrations of the reactants at the surface of the wire. The resulting overall rate for the surface reaction has been tested with existing experimental results, after describing the transport of heat and reactants, by natural convection, in the gas phase for a wide range of Rayleigh numbers. The critical conditions for the catalytic ignition have been deduced using high activation energy asymptotics for the desorption kinetics of the most efficiently adsorbed reactant, H(s)

Existence conditions and drift velocities of adiabatic flame-balls in weak gravity fields

Combining activation energy asymptotics, suitable scalings and numerical methods, we study how flame-balls move under the action of the free convection that they themselves generate in the presence of a weak, uniform gravity field. Attention is focused on steady configurations (in a suitable reference frame), on an isolated flame-ball of size comparable to what is obtained in the absence of gravity, and on deficient reactants that are characterized by a low Lewis number. For the sake of simplicity, we consider an adiabatic combustion process, in the sense that the radiative exchanges are neglected. This work provides one with: (a) a description of the free-convection field around the flame-ball, along with an asymptotic estimate of the drift velocity; (b) a relationship between the flame-ball radius, strength of gravity and physico-chemical properties of the reactive premixture; (c) extinction conditions, caused by the net effect of heat extraction from the flame-ball to its surroundings by the free-convection field. Hints on generalizations currently under consideration are also given

The anchoring of gaseous jet diffusion flames in stagnant air

A numerical analysis is presented to describe the structure of the attachment region of gaseous jet diffusion flames in stagnant air. For the typical Reynolds numbers encountered in applications, the region of flame attachment, in the near wake at the injector rim, is small compared with the injector radius. The local flow, which we consider to be laminar, quasi-steady and two-dimensional, is determined by the wall value of the fuel stream velocity gradient and the chemical reaction time or residence time in the premixed flame of a stoichiometric mixture of the fuel stream and required amount of air. When the Karlovitz number, or product of the velocity gradient and the flame residence time, grows above a critical value the flame is lifted off the injector

Heat propagation from a concentrated energy source in a gas

This paper investigates the heat propagation process in a gas from concentrated energy sources with deposition times, t/sub d/', of the order of the characteristic acoustic time, t/sub a/', across the region where the temperature will be increased by a factor of order of unity. Heat propagation takes place by two different neatly defined spatial regions of comparable size. Around the source, we find a conductive region of very high temperature where the spatial pressure variations are negligible. The edge of the resulting strongly heated low- density region appears as a contact surface that acts as a piston for the outer flow, where the pressure disturbances, of order of the ambient pressure in the distinguished regime t/sub d/' ~ t/sub a/' considered here, generate a shock wave that heats up the outer gas as it propagates outwards. The mass and energy balances for the conductive region provide a differential equation linking its pressure with the velocity of its bounding contact surface, which is used, together with the jump conditions across the shock, when integrating the Euler equations for the outer compressible flow. Solutions for the heating history are obtained for point, line and planar sources for different values of the ratio t/sub d/'/t/sub a/', including weak sources with t/sub d/' Gt t/sub a/' and very intense sources with t/sub d/' Lt t/sub a/'. The solution determines in particular the temperature profile emerging as the pressure perturbations become negligible for times much larger than the acoustic tim