Statistical behavior of turbulent kinetic energy transport in boundary layer flashback of hydrogen-rich premixed combustion

A direct numerical simulation (DNS) database for boundary layer flashback of a premixed hydrogen-air flame with an equivalence ratio of 1.5 in a fully developed turbulent channel flow has been considered for this analysis. The nonreacting part of the channel flow is representative of the friction velocity based Reynolds number ${\text{Re}}_{\tau }=120$. A skeletal chemical mechanism with 9 chemical species and 20 reactions is employed for representing hydrogen-air combustion. In this work the flow configuration and the turbulence and flame characteristics are similar to those of Gruber et al. [J. Fluid Mech. 709, 516 (2012)]. The interaction between the flame structure and the turbulent flow has been investigated for boundary layer flashback for a comparison with the earlier work of Gruber et al. [J. Fluid Mech. 709, 516 (2012)]. The statistics of wall shear stress, turbulent kinetic energy, and its dissipation have been analyzed to probe the influence of the flame on the underlying turbulence in the channel flow configuration. Furthermore, the budgets for the individual terms in the turbulent kinetic energy transport equation have also been investigated at a given plane in the channel. It is found that the propagation of the flame into the upstream part of the fully developed turbulent boundary layer introduces a flow reversal in some regions upstream of the flame and these regions lead to negative wall shear stress. Interrogation of the DNS data for the budgets of the turbulent kinetic energy transport has revealed that the aforementioned local flow reversal regions have significant influences on the turbulent kinetic energy production, pressure dilatation, and pressure transport terms. It has been found that the flame propagation into the upstream reactants leads to some weak local compressibility effects as demonstrated by the changes in the pressure related terms in the turbulent kinetic energy transport equation. These results indicate that the pressure dilatation and turbulent transport due to pressure are the two dominant terms in the turbulent kinetic energy equation in the case of wall bounded flashback flames.

Figure 1

Computational grid used for the Direct Numerical Simulation shown on the $x\text{−}y$ midplane.

Figure 2

Computational domain in the channel flow region.

Figure 3

Mean velocity and Reynolds stresses in the turbulence generation region of the channel.

Figure 4

Instantaneous distributions of isosurfaces of the temperature at $1700K$ (colored in red) and second invariant of the velocity gradient tensor (colored by instantaneous vorticity normalized by $u_{\tau R}/h)$. Top left figure shows the isometric view, bottom left figure shows the side view, top right figure shows the instantaneous normalized vorticity and negative flow velocity regions (colored in green) and the bottom right figure shows the region near the top wall where dotted ellipses show the regions of flame generated turbulence.

Figure 5

Turbulence behavior in the in the flashback case. Top figure shows the wall shear stress normalized by ${\mu }_R{u}_{\tau R}/h$ on the top wall, black lines represent $0.1\le \tilde{c}\le 0.9$, regions demarcated by white lines represent negative wall shear stress/flow recirculation regions. Figure on the bottom shows the Favre averaged turbulent kinetic energy $\tilde{k}/u_{\tau R}^2$ on log scale (left) and its Favre averaged dissipation $\tilde{\epsilon }\times h/u_{\tau R}^3$ on log scale (right) on the $x\text{−}y$ plane at $z/h=2.5$. In figures on the bottom row the green lines indicate progress variable at $0.1\le \tilde{c}\le 0.9$.

Figure 6

The variation of turbulent kinetic energy (left) and turbulent dissipation (right) across the flame brush at different wall distances in the channel on the $x\text{−}y$ plane at $z/h=2.5$.

Figure 7

Reynolds stress profiles across the flame brush at different wall distances in the channel on the $x\text{−}y$ plane at $z/h=2.5$.

Figure 8

Lumley triangle on the plane of the invariants $\xi$ and $\eta$ of the Reynolds stress anisotropy tensor across the flame brush at different wall distances in the channel on the $x\text{−}y$ plane at $z/h=2.5$. $1C,2C$ and iso mean one-component limit, two-component limit, and isotropic, respectively.

Figure 9

Mean velocity gradient term $T_1$ normalized by ${\rho }_R{u}_{\tau R}^3/h$ on the $x\text{−}y$ plane at $z/h=2.5$. The green lines indicate progress variable at $0.1\le \tilde{c}\le 0.9$.

Figure 10

The variation of mean velocity gradient term $T_1$ normalized by ${\rho }_R{u}_{\tau R}^3/h$ across the flame brush (top left), leading edge of the flame (top right) and trailing edge of the flame (bottom) at different locations away from the wall on the $x\text{−}y$ plane at $z/h=2.5$.

Figure 11

Direction cosines between the eigenvectors of $\partial {\tilde{u}}_i/\partial x_j$ and $-\widetilde{u_i^{″}{u}_j^{″}}$ across the flame brush at different $y/h$ locations on the $x\text{−}y$ plane at $z/h=2.5$.

Figure 12

Mean pressure gradient term $T_2$ normalized by ${\rho }_R{u}_{\tau R}^3/h$ on the $x\text{−}y$ plane at $z/h=2.5$ (left). Relative mean pressure normalized by ${\rho }_R{u}_{\tau R}^2$ on the $x\text{−}y$ plane at $z/h=2.5$ (right). Green lines indicate progress variable at $0.1\le \tilde{c}\le 0.9$.

Figure 13

Mean pressure gradient term $T_2$ normalized by ${\rho }_R{u}_{\tau R}^3/h$ across the flame brush at different locations away from the wall on the $x\text{−}y$ plane at $z/h=2.5$.

Figure 14

Pressure dilatation term $T_3$ normalized by ${\rho }_R{u}_{\tau R}^3/h$ on the $x\text{−}y$ plane at $z/h=2.5$. The green lines indicate progress variable at $0.1\le \tilde{c}\le 0.9$.

Figure 15

Pressure dilatation term $T_3$ normalized by ${\rho }_R{u}_{\tau R}^3/h$ across the flame brush at different locations away from the wall on the $x\text{−}y$ plane at $z/h=2.5$.

Figure 16

Molecular diffusion and dissipation contribution $T_4$ (top left), $T_{41}$ (top right), $T_{42}$ (bottom left), and $T_{43}$ (bottom right). All terms are shown on the $x\text{−}y$ plane at $z/h=2.5$ and are normalized by ${\rho }_R{u}_{\tau R}^3/h$. The green lines indicate progress variable at $0.1\le \tilde{c}\le 0.9$.

Figure 17

Molecular diffusion and dissipation contribution normalized by ${\rho }_R{u}_{\tau R}^3/h$ across the flame brush at different locations away from the wall on the $x\text{−}y$ plane at $z/h=2.5$.

Figure 18

Pressure transport term $T_5$ normalized by ${\rho }_R{u}_{\tau R}^3/h$ on the $x\text{−}y$ plane at $z/h=2.5$. The green lines indicate progress variable at $0.1\le \tilde{c}\le 0.9$.

Figure 19

Pressure transport term $T_5$ across the flame brush normalized by ${\rho }_R{u}_{\tau R}^3/h$ at different locations away from the wall on the $x\text{−}y$ plane at $z/h=2.5$.

Figure 20

Turbulent transport term $T_6$ normalized by ${\rho }_R{u}_{\tau R}^3/h$ on the $x\text{−}y$ plane at $z/h=2.5$. The green lines indicate progress variable at $0.1\le \tilde{c}\le 0.9$.

Figure 21

Turbulent transport term $T_6$ across the flame brush normalized by ${\rho }_R{u}_{\tau R}^3/h$ at different locations away from the wall on the $x\text{−}y$ plane at $z/h=2.5$.

Figure 22

Total budget of the turbulent kinetic energy transport equation within the flame brush at different locations away from the wall on the $x\text{−}y$ plane at $z/h=2.5$. All the terms are normalized by ${\rho }_R{u}_{\tau R}^3/h$.

Figure 23

The behavior of $\overline{\rho u_i^{″}{u}_j^{″}{u}_j^{″}}\times \partial \tilde{k}/\partial x_i$ within the flame brush at different locations away from the wall on the $x\text{−}y$ plane at $z/h=2.5$. All the terms are normalized by ${\rho }_R{u}_{\tau R}^5/h$.