The Growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

The nonlinear development of Gortler vortices in growing boundary layers

Most studies of triple flames in counterflowing streams of fuel and oxidizer have been focused on the symmetric problem in which the stoichiometric mixture fraction is 1/2. There then exist lean and rich premixed flames of roughly equal strengths, with a diffusion flame trailing behind from the stoichiometric point at which they meet. In the majority of realistic situations, however, the stoichiometric mixture fraction departs appreciably from unity, typically being quite small. With the objective of clarifying the influences of stoichiometry, attention is focused on one of the simplest possible models, addressed here mainly by numerical integration. When the stoichiometric mixture fraction departs appreciably from 1/2, one of the premixed wings is found to be dominant to such an extent that the diffusion flame and the other premixed flame are very weak by comparison. These curved, partially premixed flames are expected to be relevant in realistic configurations. In addition, a simple kinematic balance is shown to predict the shape of the front and the propagation velocity reasonably well in the limit of low stretch and low curvature.

Influences of stoichiometry on steadily propagating triple flames in counterflows

On the critical conditions for pool-fire puffing

Structures with multiple flames, nonpremixed and premixed, in strained flow representative of turbulent combustion are analyzed with reacting, viscous, three-dimensional counterflows. There can be ...

Combustion with Multiple Flames under High Strain Rates

It has been reasoned that the structures of strongly cellular flames in very lean mixtures approach an array of flame balls, each burning as if it were isolated, thereby indicating a connection between the critical conditions required for existence of steady flame balls and
those necessary for occurrence of self-sustained premixed combustion. This is the starting assumption of the present study, in which structures of near-limit steady sphericosym-metrical flame balls are investigated with the objective of providing analytic expressions for critical combustion conditions in ultra-lean hydrogen-oxygen mixtures diluted with N2 and water vapor. If attention were restricted to planar premixed flames, then the lean-limit mole
fraction of H2 would be found to be roughly ten percent, more than twice the observed flammability limits, thereby emphasizing the relevance of the flame-ball phenomena.
Numerical integrations using detailed models for chemistry and radiation show that a onestep chemical-kinetic reduced mechanism based on steady-state assumptions for all
chemical intermediates, together with a simple, optically thin approximation for water-vapor radiation, can be used to compute near-limit fuel-lean flame balls with excellent accuracy. The previously developed one-step reaction rate includes a crossover temperature that determines in the first approximation a chemical-kinetic lean limit below which combustion cannot occur, with critical conditions achieved when the diffusion-controlled radiation-free
peak temperature, computed with account taken of hydrogen Soret diffusion, is equal to the crossover temperature. First-order corrections are found by activation-energy asymptotics in a solution that involves a near-field radiation-free zone surrounding a spherical flame sheet,
together with a far-field radiation-conduction balance for the temperature profile. Different scalings are found depending on whether or not the surrounding atmosphere contains water vapor, leading to different analytic expressions for the critical conditions for flame-ball
existence, which give results in very good agreement with those obtained by detailed numerical computations.

Flammability conditions for ultra-lean hydrogen premixedcombustion based on flame-ball analyses

This paper investigates the steady axisymmetric structure of the cold boundary-layer flow surrounding fire whirls developing over localized fuel sources lying on a horizontal surface. The inviscid swirling motion found outside the boundary layer, driven by the entrainment of the buoyant turbulent plume of hot combustion products that develops above the fire, is described by an irrotational solution, obtained by combining Taylor's self-similar solution for the motion in the axial plane with the azimuthal motion induced by a line vortex of circulation . The velocity description provided is useful in mathematical formulations of localized fire-whirl flows, providing consistent boundary conditions accounting for the ambient swirl level.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/1645AC06C1FF6B180BA594341DBA3C65/S0022112020005133a.pdf/div-class-title-a-model-for-the-constant-density-boundary-layer-surrounding-fire-whirls-div.pdf

A model for the constant-density boundary layer surrounding fire whirls

Non-Boussinesq stability analysis of natural-convection gaseous flow on inclined hot plates

Near-limit H2-O2-N2 combustion in nonpremixed counterflow mixing layers

Abstract Numerical computations employing the relevant 9-step detailed chemistry are used to characterize the different combustion modes emerging in mixing layers separating nitrogen-diluted counterflowing planar streams of hydrogen and oxygen. Attention is focused on high degrees of dilution, resulting in near-limit flames, with peak temperatures close to the crossover temperature. A bifurcation diagram is presented in a plane, having the stoichiometric mixture fraction and normalized strain rate as coordinates, that identifies six different combustion regimes involving four different flame types, namely, diffusion-flame sheets, advancing and retreating edge flames, multiple flame tubes, and single isolated flame tubes. Multiple-tube flame configurations vary from small, round, widely separated flame strings at high strain rates to wide, flat, densely packed flame strips, with narrow flame-free gaps between them, at lower strain rates, and they are steady and stable in various arrays over a continuum of tube-separation distances. The observed flame behavior exhibits hysteresis in a certain range of parameters, with the structure that is established depending on the ignition mechanism, as it also does at high strain rates, and a continuum of different stable steady-state flame configurations exists, each accessed from a different initial condition.

Prabakaran Rajamanickam

Papers

Influences of stoichiometry on steadily propagating triple flames in counterflows

A model for the constant-density boundary layer surrounding fire whirls

Non-Boussinesq stability analysis of natural-convection gaseous flow on inclined hot plates

Near-limit H2-O2-N2 combustion in nonpremixed counterflow mixing layers

Near-limit H 2 -O 2 -N 2 combustion in nonpremixed counterflow mixing layers