Different stages of flame acceleration from slow burning to Chapman-Jouguet deflagration

Numerical simulations of spontaneous flame acceleration are performed within the problem of flame transition to detonation in two-dimensional channels. The acceleration is studied in the extremely wide range of flame front velocity changing by 3 orders of magnitude during the process. Flame accelerates from realistically small initial velocity (with Mach number about ${10}^{-3}$) to supersonic speed in the reference frame of the tube walls. It is shown that flame acceleration undergoes three distinctive stages: (1) initial exponential acceleration in the quasi-isobaric regime, (2) almost linear increase in the flame speed to supersonic values, and (3) saturation to a stationary high-speed deflagration velocity. The saturation velocity of deflagration may be correlated with the Chapman-Jouguet deflagration speed. The acceleration develops according to the Shelkin mechanism. Results on the exponential flame acceleration agree well with previous theoretical and numerical studies. The saturation velocity is in line with previous experimental results. Transition of flame acceleration regime from the exponential to the linear one, and then to the constant velocity, happens because of gas compression both ahead and behind the flame front.

Figure 1

Time dependence of scaled velocity $U_{tip}/U_f$ for $\text{Re}=6.67$ is shown until full-developed CJ detonation. The dashed vertical line indicates that the range of the quantitative study is limited by the moment of explosion onset.

Figure 2

A flame shape in a tube with nonslip at the walls; $\text{Re}=6.67$, scaled time instant $U_{f}t/R=4.57$.

Figure 3

Time dependence of scaled velocity $U_{tip}/U_f$ for different values of Reynolds number: $\text{Re}=6.67;10;13.3$. Each velocity graph is shown until the explosion onset. The lines without markers reproduce the phenomenological formula Eq. (14).

Figure 4

The acceleration rate $\sigma$ versus the Reynolds number Re. The solid line presents the theoretical results of Ref. [20]. The empty and the filled markers present the simulation result of Ref. [20] and of the present work, respectively.

Figure 5

Time dependence of the scaled acceleration $aR/U_f^2$.

Figure 6

The scaled acceleration $aR/U_f^2$ versus the Reynolds number Re.

Figure 7

Scaled density distribution along the channel axis for $\text{Re}=6.67$ at different time instants. Time instants are equally spaced in the range of $\left(0–3.8\right)U_{f}t/R$. The plot emphasized by bold corresponds to $U_{f}t/R=3.28$.

Figure 8

Scaled velocity distribution along the channel axis for $\text{Re}=6.67$ at the same time instants as in Fig. 7.

Figure 9

The velocity profile $u_z$ at the cross-section just ahead of the flame front scaled by the amplitude $U_{max}$. The solid lines shows the theoretical result Eq. (9) and the Poiseuille profile. The markers correspond to the simulation results for $\text{Re}=6.67$, at the time instants $U_{f}t/R=0.47;2.58;4.8$ (triangles, circles and crosses). The arrows illustrate the direction of the flow.

Figure 10

(Color online) 3D representation of the flow velocity $u_z$ for $\text{Re}=6.67$, at the time instants $U_{f}t/R=0.7;3.12;3.94;4.71$, figures (a), (b), (c), and (d), respectively.

Figure 11

Increase in the scaled density $\rho /{\rho }_0$ versus time just ahead of the flame front at the channel axis and at the walls. The dashed line presents the theoretical result Eq. (10).

Figure 12

Increase in the scaled density $\rho /{\rho }_0$ just ahead of the flame front at the channel axis and at the walls versus Mach number ${\text{Ma}}_{tip}=U_{tip}/c_s$. The dashed line presents the theoretical result Eq. (10).

Figure 13

The sketch of the grid with variable resolution used in numerical simulations.

Figure 14

Position of the flame tip versus time for different values of the mesh size and Reynolds number. Midthick lines correspond to $\text{Re}=6.67$, thin lines to $\text{Re}=10.0$, and thick lines to $\text{Re}=13.3$