D. R. Kassoy

Variable viscosity effects on the onset of convection in porous media

The onset of convection in a horizontal, isotropic, water‐saturated porous medium is considered. The temperature difference between the top and bottom is as large as 250 °C. The effects of an eightfold variation in kinematic viscosity are included. The critical Rayleigh number is found to be substantially reduced from the classical value although the associated wavenumber is nearly the same. Neutral mode streamline and isotherm patterns are considerably distorted in the vertical direction in distinction to the symmetric patterns found in the constant viscosity classical calculation.

Three‐dimensional natural convection motion in a confined porous medium

Weakly nonlinear analysis is used to calculate the possible two‐ and three‐dimensional convection patterns in a rectangular parallelepiped of saturated porous media when the horizontal dimensions are integral multiples of the vertical dimension. A two‐term expansion for the Nusselt number is found for values of the Rayleigh number close to the critical values. It is shown that the two‐dimensional roll configurations transfer heat more effectively than does the three‐dimensional pattern of motion when the Rayleigh number is just above the critical value.

On the direct initiation of a plane detonation wave

It is assumed that energy is transferred at a rapid rate through a plane wall into a spatially uniform and initially stagnant combustible gas mixture. This action generates a shock wave, just as it does in an inert mixture, and also switches on a significant rate of chemical reaction. The Navier-Stokes equations for plane unsteady flow are integrated numerically in order to reveal the subsequent history of events. Four principal time domains are identified, namely ‘early’, ‘transitional’, ‘formation’ and ‘ZND’. The first contains a conduction-dominated explosion and formation of a shock wave; in the second interval the shock wave is responsible for the acceleration of chemical activity, which becomes intense during the ‘formation’ period. Finally a wave whose structure is in essence that of a ZND detonation wave emerges.

The Thermal Explosion Confined by a Constant Temperature Boundary:I—The Induction Period Solution

The time history of a spatially varying thermal explosion in a vessel with constant wall temperature is considered. A one-step irreversible, high activation energy reaction of the Arrhenius type is assumed to occur in a rigid, nondiffusing, combustible material. The induction period equations are solved numerically for a system confined to a slot-like region. A stiff-equation integrator is employed to delineate the nature of the thermal runaway process. A well defined hot spot is observed to form in the vicinity of the symmetry line. A precise description of the hot spot development is given in terms of an asymptotic theory valid close to the explosion time. The solution is constructed in terms of a slowly varying conduction-controlled outer region surrounding a much smaller zone in which the relatively rapid chemical kinetics determine how the hot spot therein develops. This analytical solution describes the final phase of the spatially varying induction period thermal runaway process, which cannot be ob...

Onset of natural convection in a box of water‐saturated porous media with large temperature variation

The onset of convection in a box of water‐saturated porous media is considered. The effects of viscosity variation due to large temperature differences are included. Critical conditions for three box geometries which are representative of the problem are reported: a thin vertical slab, a flat box, and a tall slender column. The critical Rayleigh number is found to be substantially reduced from the value predicted by constant viscosity calculation. The horizontal planform of the preferred cellular motion is found to be independent of the temperature difference, being a function of the aspect ratios of the box only. The liquid motion is faster near the lower hot surface since the viscosity decreases with depth. The case of very tall slender columns led to the study of a fourth‐order ordinary differential equation with a turning point at the boundary.

On the Evolution of Plane Detonations

Numerical solutions of the Navier–Stokes equations for the plane one-dimensional unsteady motion of a compressible, combustible gas mixture are used to follow the history of events that are initiated by addition of large heat power through a solid surface bounding an effectively semi-infinite domain occupied by the gas. Plane Zel’dovich–von Neumann–Doring detonations eventually appear either at the precursor shock (which exists in every set of circumstances) or in the regions, occupied by an unsteady induction-domain and an initially quasi-steady fast-flame, that lie behind the precursor shock.

The induction period of a thermal explosion in a gas between infinite parallel plates

A study is made of the initiation of gas dynamical processes during the induction period of a high activation energy supercritical thermal explosion in a reactive gas confined between two infinite parallel plates. Solutions are assumed to vary spatially only in the direction perpendicular to the plates. The detailed development of the density, velocity, temperature, and fuel mass fraction fields during the induction period is separated into three phases. During the first phase, occurring on the acoustic time scale of the vessel, conduction-dominated boundary layers generate an acoustic field in a nondissipative interior core region. The second phase, occuring on the conduction time scale of the vessel, is characterized by a pointwise competition between reaction-generated heat release, conduction, and compression. The Frank-Kamenetskii criteria dividing super- and subcritical systems is found to be the same as that for rigid explosive materials. In a supercritical system, δ >δcrit, the third phase, of extremely limited duration, is dominated by the development of a tiny self-focusing hot spot embedded within a nearly invariant conduction-dominated field filling most of the vessel. The rapid gas expansion in the hot spot is the source of further, more dramatic gas dynamical processes. The thermal runaway time for the gas system is found to be reduced by about 25% from that for an analogous rigid explosive over a wide range of values of the Frank-Kamenetskii parameter δ.

Acoustically generated vorticity in an internal flow

A mathematical model is formulated to describe the initiation and evolution of intense unsteady vorticity in a low Mach number ( M ), weakly viscous internal flow sustained by mass addition through the sidewall of a long, narrow cylinder. An O ( M ) axial acoustic velocity disturbance, generated by a prescribed harmonic transient endwall velocity, interacts with the basically inviscid rotational steady injected flow to generate time-dependent vorticity at the sidewall. The steady radial velocity component convects the vorticity into the flow. The axial velocity associated with the vorticity field varies across the cylinder radius and in particular has an instantaneous oscillatory spatial distribution with a characteristic wavelength O ( M ) smaller than the radius. Weak viscous effects cause the vorticity to diffuse on the small radial length scale as it is convected from the wall toward the axis. The magnitude of the transient vorticity field is larger by O ( M −1 ) than that in the steady flow. An initial-boundary-value formulation is employed to find nonlinear unsteady solutions when a pressure node exists at the downstream exit of the cylinder. The complete velocity consists of a superposition of the steady flow, an acoustic (irrotational) field and the rotational component, all of the same magnitude.

Convection fluid dynamics in a model of a fault zone in the earth's crust

Faulted regions associated with geothermal areas are assumed to be composed of rock which has been heavily fractured within the fault zone by continuous tectonic activity. The fractured zone is modelled as a vertical, slender, two-dimensional channel of saturated porous material with impermeable walls on which the temperature increases linearly with depth. The development of an isothermal slug flow entering the fault at a large depth is examined. An entry solution and the subsequent approach to the fully developed configuration are obtained for large Rayleigh number flow. The former is characterized by growing thermal boundary layers adjacent to the walls and a slightly accelerated isothermal core flow. Further downstream the development is described by a parabolic system. It is shown that a class of fully developed solutions is not spatially stable.

Unsteady vorticity generation and evolution in a model of a solid rocket motor

The cylindrical, axisymmetric Navier- Stokes equations are solved numerically to study the generation and evolution of vorticity in an injection-induced transient shear flow. An initially steady internal flowfield driven by constant sidewall injection is disturbed by positive transient sidewall injection, which simulates the unsteady mass input from propellant burning variations. The disturbance amplitude is as large as that of the steady sidewall injection to ensure that nonlinear effects influence the vorticity field evolution. Initial value solutions show that relatively intense vorticity is generated at the sidewall and eventually fills the cylinder with a rotational flow. Although the pressure response is essentially that found in acoustic stability theory, the axial and radial velocity components contain large local radial velocity gradients that cannot be predicted from acoustic theory alone.