Numerical investigation of the coupling of vibrational nonequilibrium and turbulent mixing using state-specific description

In flows where the relaxation rate of molecular vibrational energy to equilibrium is comparable to the flow through timescales, the presence of turbulence can alter the mixing and equilibration processes. To understand the coupling between mixing and vibrational relaxation, a novel state-specific species model is solved in a background turbulent flow. The method is applied to mixing of two nitrogen streams at different static temperatures. The relaxation rates for each state are computed using quasiclassical trajectory analysis. The rates obtained from this study were used to first study relaxation to equilibrium in a constant volume bath. Results indicate that the thermal relaxation process is not linear over the range of conditions tested and exhibits quasisteady behavior with the higher energy levels relaxing first, followed by a slower relaxation of the lower energy levels. The state-specific model is then used to study the interaction of turbulent mixing and relaxation process in a turbulent mixing layer of two nitrogen streams at different static temperatures. The direct numerical simulation shows that gas compressibility effects impact the translational energy through flow acceleration and deceleration while the vibrational energy remains constant, triggering vibrational nonequilibrium. Also, the vibrational state populations are significantly affected by turbulence. In certain locations in the jet, the population from the direct calculation is several orders of magnitude different than that based on a Boltzmann distribution at the local vibrational temperature. These results show that considering details of the molecular populations in different vibrational states is important in a range of high enthalpy flows.

Figure 1

Directly calculated inelastic scattering rates [units for the rate are ${\mathrm{cm}}^3$/(mol s)]: (a) $T=2000\mathrm{K}$; (b) $T=4000\mathrm{K}$

Figure 2

Interpolated and extrapolated inelastic scattering rates [units for the rate are ${\mathrm{cm}}^3$/(mol s)]: (a) $T=2000\mathrm{K}$; (b) $T=4000\mathrm{K}$.

Figure 3

Relative error of Arrhenius fit rates compared to directly calculated rates.

Figure 4

${\mathcal{R}}_{vij}$ for the first 9 levels ($v=\left[08\right]$) normalized by its peak values at a temperature of 4000 K. Dark blue corresponds to a maximum depletion, and dark red to a maximum replenishment of ${\varphi }_v$.

Figure 5

Vibrational state populations versus time in constant volume for varied initial conditions; (a) $T_v^0=2000$ K, $T_0=4000$ K; (b) $T_v^0=4000$ K, $T_0=2000$ K.

Figure 6

Vibrational energy versus time in constant volume for varied initial conditions; (a) $T_v^0=2000$ K, $T_0$ varied; (b) $T_v^0=4000$ K, $T_0$ varied.

Figure 7

Vibrational energy relaxation timescales compared to Millikan and White's empirical model [31].

Figure 8

Isocontour of mixture fraction $Z_{\mathrm{mix}}=0.5$ colored by streamwise velocity.

Figure 9

Snapshots of the magnitude of density gradient in ${\mathrm{kg}/\mathrm{m}}^4$ inside the turbulent planar jet.

Figure 10

Spatial resolution in the (top) $x$ and (bottom) $y$ directions.

Figure 11

Snapshots of departure between $e_v$ and ${e_v}^{*}$ [%]. Red/blue indicate a locally vibrationally over-/under-excited population.

Figure 12

Snapshot of compressibility factor $\mathcal{C}$ [%]. Red/blue indicate a locally compressed/expanded flow.

Figure 13

Scatter plot of relative difference of $e_v$ against the mixture fraction $Z_{\mathrm{mix}}$ (left) and the compressibility factor (right) colored by local realization count (blue is lowest, yellow is highest). Plots obtained for case 1.

Figure 14

Snapshots of departure between $e_v$ and ${e_v}^{*}$ [%] for case 2. Red/blue indicate a locally vibrationally over-/under-excited population.

Figure 15

Snapshot of compressibility factor $\mathcal{C}$ [%] for case 2. Red/blue indicate a locally compressed/expanded flow.

Figure 16

Scatter plot of relative difference of $e_v$ against the mixture fraction $Z_{\mathrm{mix}}$ (left) and the compressibility factor (right) colored by local realization count (blue is lowest, yellow is highest). Plots obtained for case 2.

Figure 17

Snapshots of vibrational state population number densities ${\varphi }_v$ for levels $v\in \left[1249\right]$ from top to bottom for the (left) cold jet and (right) hot jet.

Figure 18

(Black dots) Distribution of state populations ${\varphi }_v$ for $v\in \left[012345\right]$ with (dashed blue) ${\varphi }_v(T=2000K)Z_{\mathrm{mix}}+{\varphi }_v(T=4000K)(1-Z_{\mathrm{mix}})$ and (dashed red) the Boltzmann number fractions $f_v(v,T)$. Only case 1 is shown.

Figure 19

(Left) Instantaneous snapshots of ${\mathcal{E}}_v$ [%] for $v\in \left[1239\right]$ from top to bottom. (Right) Realizations of ${\mathcal{E}}_v$ [%] for $v\in \left[1239\right]$ from top to bottom with $Z_{\mathrm{mix}}$.