Spectra and scaling in chemically reacting compressible isotropic turbulence

Numerical simulations are carried out to study the spectra and statistics in chemically reacting compressible homogeneous isotropic turbulence at turbulent Mach number $M_t$ from 0.1 to 1.0 and at Taylor Reynolds number ${\mathrm{Re}}_{\lambda }$ from 54 to 103 with solenoidal forcing. A single-step irreversible Arrhenius-type chemical reaction is implemented to evaluate the influence of chemical reaction on spectra and flow statistics. It is shown that in the situation of isothermal reactions, both the ratio of compressible kinetic energy to solenoidal kinetic energy $K^c/K^s$ and the ratio of compressible dissipation to solenoidal dissipation ${\epsilon }^c/{\epsilon }^s$ exhibit a $M_t^4$ scaling at low turbulent Mach numbers $M_t<0.4$. At $M_t⩾0.4$, $K^c/K^s$ and ${\epsilon }^c/{\epsilon }^s$ exhibit $M_t^2$ and $M_t^5$ scaling behaviors, respectively, and the flow is in strong acoustic equilibrium. The spectra of velocity, pressure, density, and temperature are nearly unaffected by the isothermal chemical reaction. In contrast, heat release in exothermal reactions significantly enhances the spectra of velocity and thermodynamic variables in a wide range of length scales. It is found that the spectra of pressure and compressible velocity satisfy the strong acoustic equilibrium relation at $M_t$ from 0.1 to 0.6, indicating that the acoustic mode dominates over the dynamics of compressible velocity and pressure. In the situation of exothermal reactions, $K^c/K^s$ and ${\epsilon }^c/{\epsilon }^s$ appear to be independent of turbulent Mach number. The normalized root-mean-square values of pressure, density, and temperature exhibit a $M_t^2$ scaling in the isothermal reactions and exhibit a $M_t$ scaling in the exothermal reactions.

Figure 1

Initial density field at the start of chemical reaction.

Figure 2

Temporal variation of turbulent timescale $T_T$ and reaction timescale $T_R$ (a) at $M_t=0.1$ in isothermal reactions ($\text{Da}=2$) for ${\text{Re}}_{\lambda }\approx 60$, (b) at $M_t=0.8$ in exothermal reactions ($\text{Da}=200$) for ${\text{Re}}_{\lambda }\approx 60$, (c) at $M_t=0.1$ and 1.0 in isothermal reactions ($\text{Da}=2$) for ${\text{Re}}_{\lambda }\approx 100$, and (d) at $M_t=0.6$ in exothermal reactions ($\text{Da}=200$, 3000) for ${\text{Re}}_{\lambda }\approx 100$.

Figure 3

Temporal variation of mass fraction of different species. (a, b) Product $P$ at $M_t=0.1\text{−}1.0$ in isothermal reactions ($\text{Da}=2$) at ${\text{Re}}_{\lambda }\approx 100$, (c, d) product $P$ at $M_t=0.1\text{–}0.8$ in exothermal reactions ($\text{Da}=200$) at ${\text{Re}}_{\lambda }\approx 100$, (e, f) reactant $A$ and product $P$ at $M_t=0.6$ in exothermal reactions ($\text{Da}=3000$) at ${\text{Re}}_{\lambda }\approx 60$ and ${\text{Re}}_{\lambda }\approx 100$.

Figure 4

Temporal variation of (a, b) reaction rate $W$ at $M_t=0.1\text{–}1.0$ in isothermal reactions ($\text{Da}=2$) at ${\text{Re}}_{\lambda }\approx 100$, (c, d) reaction rate $W$ at $M_t=0.1\text{–}0.8$ in exothermal reactions ($\text{Da}=200$) at ${\text{Re}}_{\lambda }\approx 100$, (e, f) mixing rate $G$ at $M_t=0.6$ in three reactions ($\text{Da}=2$, 200, 3000) at ${\text{Re}}_{\lambda }\approx 100$.

Figure 5

Temporal variation of compressible to solenoidal kinetic energy ratio (a) at $M_t=0.2$ in three reactions ($\text{Da}=2$, 200, 3000) for ${\text{Re}}_{\lambda }\approx 100$ and (b) at $M_t=0.6$ in two reactions ($\text{Da}=2$, 3000) for ${\text{Re}}_{\lambda }\approx 60$ and ${\text{Re}}_{\lambda }\approx 100$.

Figure 6

Temporal variation of compressible to solenoidal dissipation rate ratio (a) at $M_t=0.2$ in three reactions ($\text{Da}=2$, 200, 3000) at ${\text{Re}}_{\lambda }\approx 100$ and (b) at $M_t=0.6$ in two reactions ($\text{Da}=2$, 3000) at ${\text{Re}}_{\lambda }\approx 60$ and ${\text{Re}}_{\lambda }\approx 100$.

Figure 7

Isosurfaces of instantaneous reaction rate (red, $W=\frac{1}{4}\langle W\rangle$) and velocity divergence (blue, $\theta =-2{\theta }^{\prime }$) isosurfaces with ${256}^3$ grid resolution in exothermal reactions ($\text{Da}=200$) for $t/\tau =0.8$ at ${\text{Re}}_{\lambda }\approx 100$ and different turbulent Mach numbers: (a, e) $M_t=0.2$; (b, f) $M_t=0.4$; (c, g) $M_t=0.6$; (d, h) $M_t=0.8$.

Figure 8

PDF of reaction rate $W$ at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$. (a) $M_t=0.1$, $\text{Da}=2$; (b) $M_t=0.1$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.

Figure 9

Temporal variation of compensated spectrum of velocity $E^u\left(k\right){\epsilon }^{-2/3}{k}^{5/3}$ for isothermal and exothermal reactions at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$. (a) $M_t=0.2$, $\text{Da}=2$; (b) $M_t=0.2$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.

Figure 10

Instantaneous compensated spectrum of velocity $E^u\left(k\right){\epsilon }^{-2/3}{k}^{5/3}$ at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$ for three reaction parameters. (a) $M_t=0.1\text{–}0.8$, $\text{Da}=200$, $t/\tau =34$; (b) $M_t=0.1$, $\text{Da}=200$, $t/\tau =0\text{–}34$; (c) $M_t=0.6$, $\text{Da}=3000$, $t/\tau =0\text{–}34$; (d) $M_t=0.6$, $\text{Da}=2,200,3000$, $t/\tau =34$.

Figure 11

Temporal variation of compensated spectrum of compressible velocity component $E^{u,C}\left(k\right){\epsilon }^{-2/3}{k}^{5/3}$ for an exothermal reaction ($\text{Da}=200$) at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$. (a) $M_t=0.2$, (b) $M_t=0.4$, (c) $M_t=0.6$, (d) $M_t=0.8$.

Figure 12

Instantaneous compensated spectrum of the compressible velocity component $E^{u,C}\left(k\right){\epsilon }^{-2/3}{k}^{5/3}$ at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$ for three reaction parameters. (a) $M_t=0.1\text{–}0.8$, $\text{Da}=200$, $t/\tau =34$; (b) $M_t=0.1\text{–}0.6$, $\text{Da}=3000$, $t/\tau =34$; (c) $M_t=0.2$, $\text{Da}=2$, 200, 300, $t/\tau =34$; (d) $M_t=0.6$, $\text{Da}=2$, 200, 300, $t/\tau =34$.

Figure 13

Temporal variation of compensated spectra of pressure, density, and temperature for exothermal reactions ($\text{Da}=200,3000$) at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$ at different turbulent Mach numbers: (a, b) $E^p\left(k\right){\epsilon }^{-2/3}{k}^{5/3}{\left(2\gamma {\rho }_0{p}_0\right)}^{-1}$, (c, d) $E^{\rho }\left(k\right){\epsilon }^{-2/3}{k}^{5/3}\gamma p_0{\left(2{\rho }_0^3\right)}^{-1}$, (e, f) $E^T\left(k\right){\epsilon }^{-2/3}{k}^{5/3}\gamma p_0{\left[2{\rho }_0{(\gamma -1)}^2{T}_0^2\right]}^{-1}$.

Figure 14

Instantaneous compensated spectrum of pressure $E^p\left(k\right){\epsilon }^{-2/3}{k}^{5/3}{\left(2\gamma {\rho }_0{p}_0\right)}^{-1}$ at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$ for three reaction parameters: (a) $M_t=0.1\text{–}0.8$, $\text{Da}=200$, $t/\tau =34$; (b) $M_t=0.6$, $\text{Da}=3000$, $t/\tau =0\text{–}34$; (c) $M_t=0.1$, $\text{Da}=2,200,3000$, $t/\tau =34$; (d) $M_t=0.6$, $\text{Da}=2,200,3000$, $t/\tau =34$.

Figure 15

Instantaneous spectrum of pressure dilatation $E^{p\theta }\left(k\right)$ at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$. (a) $M_t=0.1$, $\text{Da}=2$; (b) $M_t=0.1$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.

Figure 16

Compensated spectra of pressure $E^p\left(k\right){\epsilon }^{-2/3}{k}^{5/3}{\left(2\gamma {\rho }_0{p}_0\right)}^{-1}$, density $E^{\rho }\left(k\right){\epsilon }^{-2/3}{k}^{5/3}\gamma p_0{\left(2{\rho }_0^3\right)}^{-1}$, and temperature $E^T\left(k\right){\epsilon }^{-2/3}{k}^{5/3}\gamma p_0{\left[2{\rho }_0{(\gamma -1)}^2{T}_0^2\right]}^{-1}$ at $M_t=0.1\text{–}0.8$ for exothermal reactions ($\text{Da}=200,3000$) at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$. (a) $M_t=0.1$, $\text{Da}=200$; (b) $M_t=0.8$, $\text{Da}=200$; (c) $M_t=0.2$, $\text{Da}=3000$; (d) $M_t=0.6$, $\text{Da}=3000$.

Figure 17

Normalized spectra of the residual density (a, b) and residual temperature (c, d) at turbulent Mach numbers $M_t=0.1\text{–}0.8$ and at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$ for exothermal reactions ($\text{Da}=200,3000$) at $t/\tau =34$.

Figure 18

Normalized spectra of pressure and compressible velocity component at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$ for exothermal reactions ($\text{Da}=200,3000$): (a) $M_t=0.1$, (b) $M_t=0.2$, (c) $M_t=0.4$, (d) $M_t=0.6$.

Figure 19

(a) Ratio of compressible to solenoidal kinetic energy $K^c/K^s$; (b) ratio of compressible component to solenoidal component of kinetic energy dissipation ${\epsilon }^c/{\epsilon }^s$ as a function of $M_t$.

Figure 20

Normalized rms values of pressure, density, and temperature at different turbulent Mach numbers and Taylor Reynolds numbers: (a) $\text{Da}=2$; (b) $\text{Da}=200$; (c) $\text{Da}=3000$.

Figure 21

Initial density field at the start of chemical reactions. $ini{t}_1$: homogeneous distribution of $Y_A$ and $Y_B$, $Y_A=0.5$, $Y_B=0.5$; $ini{t}_2\sim ini{t}_4$: gray slab $Y_A=0.99$, $Y_B=0.01$, white slab $Y_A=0.01$, $Y_B=0.99$.

Figure 22

Temporal variation of mass fraction of product $P$ at $M_t=0.1$ and 0.8 at ${\text{Re}}_{\lambda }\approx 60$ for isothermal ($\text{Da}=2$) and exothermal ($\text{Da}=200$) reactions. (a) $M_t=0.1$, $\text{Da}=2$; (b) $M_t=0.1$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.

Figure 23

Temporal variation of mass fraction of product $P$ at $M_t=0.1$ and 0.8 at ${\text{Re}}_{\lambda }\approx 60$ for isothermal ($\text{Da}=2$) and exothermal ($\text{Da}=200$) reactions, log-log scale. (a) $M_t=0.1$, $\text{Da}=2$; (b) $M_t=0.1$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.

Figure 24

Temporal variation of reaction rate $W$ at $M_t=0.1$ and 0.8 at ${\text{Re}}_{\lambda }\approx 60$ for isothermal ($\text{Da}=2$) and exothermal ($\text{Da}=200$) reactions. (a) $M_t=0.1$, $\text{Da}=2$; (b) $M_t=0.1$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.

Figure 25

Temporal variation of reaction rate $W$ at $M_t=0.1$ and 0.8 at ${\text{Re}}_{\lambda }\approx 60$ for isothermal ($\text{Da}=2$) and exothermal ($\text{Da}=200$) reactions, log-log scale. (a) $M_t=0.1$, $\text{Da}=2$; (b) $M_t=0.1$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.

Figure 26

Temporal variation of reaction timescale $T_R$ at $M_t=0.1$ and 0.8 at ${\text{Re}}_{\lambda }\approx 60$ for isothermal ($\text{Da}=2$) and exothermal ($\text{Da}=200$) reactions, log-log scale. (a) $M_t=0.1$, $\text{Da}=2$; (b) $M_t=0.1$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.

Figure 27

Temporal variation of compressible to solenoidal kinetic energy ratio $K^c/Ks$ at $M_t=0.1$ and 0.8 at ${\text{Re}}_{\lambda }\approx 60$ for isothermal ($\text{Da}=2$) and exothermal ($\text{Da}=200$) reactions, log-log scale. (a) $M_t=0.1$, $\text{Da}=2$; (b) $M_t=0.1$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.

Figure 28

Temporal variation of compressible to solenoidal dissipation rate ratio ${\epsilon }^c/{\epsilon }^s$ at $M_t=0.1$ and 0.8 at ${\text{Re}}_{\lambda }\approx 60$ for isothermal ($\text{Da}=2$) and exothermal ($\text{Da}=200$) reactions, log-log scale. (a) $M_t=0.1$, $\text{Da}=2$; (b) $M_t=0.1$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.

Figure 29

Instantaneous compensated spectra of velocity $E^u\left(k\right){\epsilon }^{-2/3}{k}^{5/3}$ at $M_t=0.1$ and 0.8 at ${\text{Re}}_{\lambda }\approx 60$ for isothermal ($\text{Da}=2$) and exothermal ($\text{Da}=200$) reactions at $t/\tau =0.7$. (a) $M_t=0.1$, $\text{Da}=2$; (b) $M_t=0.1$, $\text{Da}=200$; (c) $M_t=0.8$, $\text{Da}=2$; (d) $M_t=0.8$, $\text{Da}=200$.