Interaction of a planar reacting shock wave with an isotropic turbulent vorticity field

Linear interaction analysis (LIA) is employed to investigate the interaction of reactive and nonreactive shock waves with isotropic vortical turbulence. The analysis is carried out, through Laplace-transform technique, accounting for long-time effects of vortical disturbances on the burnt-gas flow in the fast-reaction limit, where the reaction-region thickness is significantly small in comparison with the most representative turbulent length scales. Results provided by the opposite slow-reaction limit are also recollected. The reactive case is here restricted to situations where the overdriven detonation front does not exhibit self-induced oscillations nor inherent instabilities. The interaction of the planar detonation with a monochromatic pattern of perturbations is addressed first, and then a Fourier superposition for three-dimensional isotropic turbulent fields is employed to provide integral formulas for the amplification of the kinetic energy, enstrophy, and anisotropy downstream. Transitory evolution is also provided for single-frequency disturbances. In addition, further effects associated to the reaction rate, which have not been included in LIA, are studied through direct numerical simulations. The numerical computations, based on WENO-BO4-type scheme, provide spatial profiles of the turbulent structures downstream for four different conditions that include nonreacting shock waves, unstable reacting shock (sufficiently high activation energy), and stable reacting shocks for different detonation thicknesses. Effects of the propagation Mach number, chemical heat release, and burn rate are analyzed.

Figure 1

Schematic view of a planar detonation wave interacting with a monochromatic shear wave in the fast-reaction limit [27] (left) and in the slow-reaction limit [32] (right).

Figure 2

Pressure perturbation $\mathcal{P}$ as a function of the relative incident shear angle ${\theta }_{cr}-\theta$ for $\zeta <1$ (left) and $\zeta >1$ (right). The gas properties are determined by $\gamma =1.4$ and $q=1$, and the overdrive degree is $M/M_{\text{CJ}}=1.5$, 2, and 5. Red points represent the neutral transmission conditions.

Figure 3

Dimensionless acoustic energy flux ${\varphi }_x$ as a function of the angle ${\theta }^{\prime }$. The gas properties are determined by $\gamma =1.4$ and $q=1$, and the overdrive degree is $M/M_{\text{CJ}}=1.5$, 2, and 5. The inset shows the same curves in a polar plot.

Figure 4

Total vorticity amplification ratio $\mathrm{\Omega }{sin}^2\theta$ (black lines) and ${\mathrm{\Omega }}_1{sin}^2\theta =R_d$ (thin gray lines) as a function of the incident shear angle $\theta$. The gas properties are determined by $\gamma =1.4$ and $q=1$, and the overdrive degree is $M/M_{\text{CJ}}=1.5$, 2, and 5. Red points represent the neutral transmission conditions.

Figure 5

Longitudinal (upper) and transverse (bottom) rotational velocity amplitudes ${\mathcal{U}}_r$ and ${\mathcal{V}}_{r}tan\theta$, respectively, as a function of the incident shear angle $\theta$. The gas properties are determined by $\gamma =1.4$ and $q=1$, and the overdrive degree is $M/M_{\text{CJ}}=1.5$, 2, and 5.

Figure 6

$L_a$ (dotted-dashed line), $T_a$ (dashed line), and $K_a$ (solid line) as a function of the overdrive parameter $M_0/M_{\text{CJ}}$ for $\gamma =1.4$, $q=1$ (red), and $q=0$ (black).

Figure 7

$L_r$ (dotted-dashed line), $T_r$ (dashed line), and $K_r$ (solid line) as a function of the detonation Mach number $M_0$ for $\gamma =1.4$, $q=0$ (black), and $q=1$ (red). Arrows represent the effect of small-scale disturbances on the dominant nonreactive results [32].

Figure 8

Shaded contours for the longitudinal (left panel) and transverse (right panel) kinetic energy ratios as a function of $q$ and $M_0/M_{\text{CJ}}$.

Figure 9

DNS for the longitudinal (upper panels) and transverse (lower panels) kinetic energy, scaled with preshock values, as a function of the streamwise direction for $M_0=1.5M_{\text{CJ}}$, $q=1$, and $\gamma =1.4$. Circles denote nearly complete depletion of reactant $Y=0.01$. Left panels display the reactive unstable scenario RU.

Figure 10

DNS for the turbulent kinetic energy $K$, scaled with preshock values, as a function of the streamwise direction for $M_0=1.5M_{\text{CJ}}$, $q=1$, and $\gamma =1.4$. Circles denote nearly complete depletion of reactant $Y=0.01$. The left panel displays the reactive unstable scenario RU.

Figure 11

$W_z$ (dotted-dashed line), $W_{\perp }$ (dashed line), and $W$ (solid line) as a function of the detonation Mach number $M_0$ for $\gamma =1.4$, $q=0$ (black), and $q=1$ (red).

Figure 12

DNS for the perpendicular (upper panels) and total (lower panels) enstrophy, scaled with preshock values, as a function of the streamwise direction for $M_0=1.5M_{\text{CJ}}$, $q=1$, and $\gamma =1.4$. Circles denote nearly complete depletion of reactant $Y=0.01$. Left panels display the reactive unstable scenario RU.

Figure 13

Instantaneous vorticity patterns for NR, RU, RS1, and RS2 obtained at the cut plane $y_t=\left(6.61\pi \right)/2$. Black contour lines on the reactive cases correspond to the isolines of nearly complete depletion of reactant $Y=0.01$.

Figure 14

Anisotropy factor $\mathrm{\Psi }$ as a function of the overdrive $M_0/M_{\text{CJ}}-1$ for nonreactive (black) and reactive (red) cases. Dotted lines represent the value of the rotational anisotropy factor ${\mathrm{\Psi }}_r$.

Figure 15

Sketch of the interaction of an initially planar detonation wave with a monochromatic vorticity field.

Figure 16

Temporal evolution of pressure perturbations behind the detonation front traveling at $M_0=1.5M_{\text{CJ}}$, for $q=1$ and $\gamma =1.4$, for two different upstream frequencies $k_x/k_y$.

Figure 17

Spatial distribution of pressure perturbations in the burnt gas for a detonation front traveling at $M_0=1.5M_{\text{CJ}}$, with $q=1$ and $\gamma =1.4$, and for two different upstream frequency ratios $k_x/k_y=1/4$ (left) and 1 (right).

Figure 18

Upper panel: entropic-density and longitudinal rotational-velocity perturbations as a function of $k_{y}x$ for $M_0=1.5M_{\text{CJ}}$, $q=1$, $\gamma =1.4$, and $\zeta =1.5$. Lower panel: two-dimensional vector field plot for rotational-velocity perturbations superposed to the isocontours of entropic-density disturbances for a detonation wave traveling at the same conditions.