Formula of Laminar Flame Speed Coupled with Differential Equation for Temperature in Low-Mach-Number Model

TL;DR

This paper derives an analytical formula for laminar flame speed considering non-zero temperature gradients and compression effects in low-Mach-number combustion, enhancing understanding of flame dynamics and transition to detonation.

Contribution

It introduces a coupled analytical approach for flame speed and temperature distribution accounting for non-zero temperature gradients and compression zones, extending previous models.

Findings

Derived an analytical formula for laminar flame speed.

Showed the influence of compression on flame dynamics.

Provided numerical and analytical solutions for temperature distribution.

Abstract

The non-monotonic profile of temperature is to be considered in the context of combustion inside tubes or thermonuclear flames, which may accelerates to become detonation waves. This transition is known as deflagration-to-detonation transition. Focusing on this point, we extend the study of adiabatic flames, which assumes zero temperature gradient at the burned-side edge of a flame front, to the case of non-zero temperature gradient by introducing compression zone behind the front. Such a treatment has been performed in the previous work based on the ignition temperature approximation. In that case, the effect of gas compression on flames was described by asymptotic solutions with respect to small squared Mach numbers: for example, in the compression zone, inner solutions are expressed by use of Lambert $W$ function. However, due to the aim of obtaining analytical solutions inside a flame front, the temperature dependence of reaction term was ignored and, as a result, the cold boundary difficulty was not treated seriously. This issue is resolved in this study by use of large activation energy asymptotics with the reaction term expressed by the Arrhenius type. The temperature distribution is computed numerically in the reaction zone and calculated analytically in the compression zone. The burning-rate eigenvalue problem is also considered to derive a formula of laminar flame speed, which is coupled with differential equation for temperature in the reaction zone. Then, the dependence of laminar flame speed on thermal-diffusive parameters, accompanied with hydrodynamic properties brought by the compression effect, becomes clear. The non-zero gradient condition of temperature on the burned side of a flame front is obtained for non-zero values of Mach number.