Length and Time Scales in Turbulent Combustion

Abstract

The different regimes of premixed turbulent combustion may well be illustrated in a diagram, initially proposed by R. Borghi, where the ratio of the turbulence intensity to the laminar flame speed is plotted over the ratio of the turbulence integral length scale to the flame thickness. In this diagram four different regimes of turbulent combustion are specified:

• 1. the regime of wrinkled flamelets,

• 2. the regime of corrugated flamelets,

• 3. the regime of distributed reaction zones and

• 4. the regime of the well-stirred reactor.

While the first and fourth regime, representing limiting conditions, have been analysed in the past rather successfully, the second and third regime describe a more intense interaction between turbulence and combustion. Using arguments based on Kolmogorov’s energy cascade, a new length scale is identified in each of these two intermediate regimes.

In the regime of corrugated flamelets, the Gibson scale ${{\rm{L}}_{\rm{G}}} = {\rm{v}}_{\rm{F}}^{3/{\rm{\varepsilon }}}$ is derived. Here v_F is the flame veloCity which by definition is equal to the characteristic turn-over veloCity of the eddy of size L_G. Eddies much larger than L_G which have a larger turn-over veloCity than v_F will convect the flame front within the flow field as if it was a passive surface. Eddies much smaller than L_G, having a smaller turn-over veloCity than v_F, are consumed by the flame front very rapidly and therefore cannot corrugate the front. The Gibson scale therefore is the lower cut-off of all scales that appear in the corrugated flame surface.

In the regime of distributed reaction zones the mean turbulent flame thickness is of the order of the quench scale δ_q, which is predicted to be proportional to ${\left( {{\rm{\varepsilon t}}_{\rm{q}}^3} \right)^{1/2}}$, where t_q=1/a_q is the quench time, a_q is the stretch rate at quenching of a premixed flame and ε the dissipation of turbulence. The quench scale presents the largest eddy within the inertial range, which is still able to quench the thin inner reaction zone of a premixed flame. Smaller eddies, inducing a larger stretch, will quench this thin layer more readily and therefore will try to homogenize the scalar field locally. Therefore a thickened flame front will appear. Larger eddies, inducing a weaker stretch that will not be able to quench the inner reaction zones, will only wrinkle this thickened flame front.