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Numerical Methods for Boundary-Layer-Type Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
Triple-deck theory provides the equations and boundary conditions needed to match the solutions in each of the three regions. The results are only valid for laminar flows where Re → ∞, so in a sense are of limited practical value. These equations are frequently solved numerically using viscous–inviscid interaction procedures (Jobe and Burggraf, 1974).
The Speed and Temperature of an Edge Flame Stabilized in a Mixing Layer: Dependence on Fuel Properties and Local Mixture Fraction Gradient
Published in Combustion Science and Technology, 2020
The configuration adopted in this study is the edge flame established in the laminar mixing layer of two merging streams, one containing fuel and the other oxidizer, separated upstream by an infinitesimally thin splitter plate. The mathematical problem requires solving the reactive Navier-Stokes equations in an infinite domain that represents the immediate vicinity of the plate trailing edge, ensuring that the far field asymptotic behavior in the wake region is properly captured. A qualitative description in the global context of the triple-deck theory for large Reynolds number was given by Liñán (1994), and further exploited by Lu and Matalon (2016). The problem is quite laborious, particularly if one is interested in performing a parametric study while varying key physico-chemical parameters. Most studies have therefore adopted some simplifications. Using a global single-step chemical Kurdyumov and Matalon (2002, 2004) examined the problem assuming that the velocity of the incoming streams is constant and uniform, and remains unperturbed throughout the combustion field by treating the density constant. They analyzed the flame structure and examined the stability of steady solutions for various Lewis numbers (assumed the same for the fuel and oxidizer), in the presence/absence of volumetric heat losses. In a followup study they reexamined the problem (Kurdyumov and Matalon 2007) retaining the constant density approximation, but to better describe the flow field they solved the Navier-Stokes equations throughout the entire domain assuming that far upstream the incoming flow is of constant strain rate (an appropriate approximation of the boundary layer flow along the plate). Lu and Matalon (2016) extended these results by considering the effect of unequal strain rates in the fuel and oxidizer streams. They examined the structure and dynamic properties of the edge flame over a wide range of flow conditions for low and high Lewis numbers and determined criteria for flame stabilization and blow-off. More recently they studied the effect that thermally-active splitter plates have on flame stabilization and on the conditions leading to liftoff and blow-off (Lu and Matalon 2019a, 2019b). Two regimes of flame stabilization were identified; a thermal-interaction regime where there is appreciable heat loss from the edge flame to the splitter plate, and a freely-standing regime where the edge flame lifts off considerably and the plate plays no role on its stabilization. For high Lewis numbers, the edge flame remains practically attached to the splitter plate and gets blown off when the flow rate becomes excessively large. For low Lewis numbers, the edge flame moves away from the splitter plate and stabilizes at a distance which increases substantially when increasing the flow rate; the lifted flame may either remain stationary or undergo sustained oscillations, at or relative to a well-defined position. Representative solutions corresponding to attached and lifted flames were also reported by Fernández-Tarrazo, Vera, and Liñán (2006) by specifying the flow and mixture conditions and assuming that the incoming streams of constant speed are injected into the combustion field through porous walls.