Nonequilibrium shock-heated nitrogen flows using a rovibrational state-to-state method

A rovibrational collisional model is developed to study the internal energy excitation and dissociation processes behind a strong shock wave in a nitrogen flow. The reaction rate coefficients are obtained from the ab initio database of the NASA Ames Research Center. The master equation is coupled with a one-dimensional flow solver to study the nonequilibrium phenomena encountered in the gas during a hyperbolic reentry into Earth's atmosphere. The analysis of the populations of the rovibrational levels demonstrates how rotational and vibrational relaxation proceed at the same rate. This contrasts with the common misconception that translational and rotational relaxation occur concurrently. A significant part of the relaxation process occurs in non-quasi-steady-state conditions. Exchange processes are found to have a significant impact on the relaxation of the gas, while predissociation has a negligible effect. The results obtained by means of the full rovibrational collisional model are used to assess the validity of reduced order models (vibrational collisional and multitemperature) which are based on the same kinetic database. It is found that thermalization and dissociation are drastically overestimated by the reduced order models. The reasons of the failure differ in the two cases. In the vibrational collisional model the overestimation of the dissociation is a consequence of the assumption of equilibrium between the rotational energy and the translational energy. The multitemperature model fails to predict the correct thermochemical relaxation due to the failure of the quasi-steady-state assumption, used to derive the phenomenological rate coefficient for dissociation.

Figure 1

RVC model. Temperature (a) and $N$ mole fraction (b) evolution behind the shock wave with or without exchange processes [$p_{\infty }=0.1\mathrm{torr}$ and $u_{\infty }=10\mathrm{km}/\mathrm{s}$; in (a) unbroken line, $T$ with exchange; dashed line, $T^I$ with exchange; dotted-dashed line, $T$ without exchange; dotted line, $T^I$ without exchange; in (b) unbroken line, with exchange; dashed line, without exchange].

Figure 2

RVC model. Temperature (a) and $N$ mole fraction (b) evolution behind the shock wave with or without predissociation [$p_{\infty }=0.1\mathrm{torr}$ and $u_{\infty }=10\mathrm{km}/\mathrm{s}$; in (a) unbroken line, $T$ with predissociation; dashed line, $T^I$ with predissociation; dotted-dashed line, $T$ without predissociation; dotted line, $T^I$ without predissociation; in (b) unbroken line, with predissociation; dashed line, without predissociation].

Figure 3

RVC model. Comparative evolution of the translational, rotational, and vibrational temperatures behind the shock wave ($p_{\infty }=0.1$ torr). (a) Lines with symbols refer to the $u_{\infty }=6$ (km/s) case. In this case, the circles indicate $T$, squares $T^R$, and triangles $T^V$. The simple lines refers to the $u_{\infty }=4$ (km/s) case. In this case, the solid line indicates $T$, dashed-line $T^R$, and dot-dashed line $T^V$. (b) The $u_{\infty }=10$ (km/s) case. The solid line indicates $T$, dashed-line $T^R$, and dot-dashed line $T^V$.

Figure 4

RVC model. Comparative evolution of the N mole fraction (a) and rovibrational energy level distribution (b) behind the shock wave [$p_{\infty }=0.1\mathrm{torr}$ and $u_{\infty }=10\mathrm{km}/\mathrm{s}$; the symbols in (a) highlight the locations at which the population distributions, plotted in (b), are extracted].

Figure 5

RVC model. Evolution behind the shock wave of the population of the rovibrational energy levels lying close to the ground state ($p_{\infty }=0.1\mathrm{torr}$ and $u_{\infty }=10\mathrm{km}/\mathrm{s}$; the locations at which the distributions are plotted are the same as those of Fig. 4).

Figure 6

RVC model. Rotational energy level distribution for the ground and the first three excited vibrational states at $x=3.6\times {10}^{-3}\mathrm{m}$ behind the shock wave [$p_{\infty }=0.1\mathrm{torr}$ and $u_{\infty }=10\mathrm{km}/\mathrm{s}$; global view (a) and zoom around the ground state (b)].

Figure 7

RVC model. Translational and rotational temperature evolution behind the shock wave ($p_{\infty }=0.1\mathrm{torr}$ and $u_{\infty }=10\mathrm{km}/\mathrm{s}$; unbroken line, $T$; dashed line, $T^R$; line with circles, $T_0^R$; line with squares, $T_1^R$; line with triangles, $T_2^R$; line with diamonds $T_3^R$).

Figure 8

Comparison of the VC and RVC models. Translational and vibrational temperature (a) and N mole fraction (b) evolution behind the shock wave [$p_{\infty }=0.1\mathrm{torr}$ and $u_{\infty }=10\mathrm{km}/\mathrm{s}$; in (a) unbroken line, $T$ RVC; dashed line, $T^V$ RVC; dotted-dashed line, $T$ VC; dotted line, $T^V$ VC; in (b) unbroken line, RVC; dashed line, VC].

Figure 9

RVC and MT model comparison. Temperature and N mole fraction evolution behind the shock wave [$p_{\infty }=0.1\mathrm{torr}$ and $u_{\infty }=10\mathrm{km}/\mathrm{s}$; in (a) unbroken line, $T$ RVC; dashed line, $T^R$ RVC; dotted-dashed line, $T^V$ RVC; line with circles, $T$ MT; line with squares, $T^R$ MT; line with triangles, $T^V$ MT; in (b) unbroken line, RVC; dashed lines, MT].

Figure 10

RVC and MT model comparison. Mass production (a) and energy transfer (b) term evolution behind the shock wave [$p_{\infty }=0.1\mathrm{torr}$ and $u_{\infty }=10\mathrm{km}/\mathrm{s}$; in (a) unbroken line, RVC; dashed line, MT; in (b) unbroken line, ${\Omega }_{N_2}^{\mathit{T}-\mathit{R}}$ RVC; dashed line, ${\Omega }_{N_2}^{\mathit{T}-\mathit{V}}$ RVC; dotted-dashed line, ${\Omega }_{N_2}^{\mathit{D}-\mathit{R}}$ RVC; dotted line, ${\Omega }_{N_2}^{\mathit{D}-\mathit{V}}$ RVC; line with circles, ${\Omega }_{N_2}^{\mathit{T}-\mathit{R}}$ MT; line with squares ${\Omega }_{N_2}^{\mathit{T}-\mathit{V}}$ MT; line with triangles, ${\Omega }_{N_2}^{\mathit{D}-\mathit{R}}$ MT; line with diamonds, ${\Omega }_{N_2}^{\mathit{D}-\mathit{V}}$ MT].

Figure 11

RVC and MT model comparison. Temperature evolution behind the shock wave switching off dissociation [$p_{\infty }=0.1\mathrm{torr}$ and $u_{\infty }=10\mathrm{km}/\mathrm{s}$; unbroken line, $T$ RVC; dashed line, $T^R$ RVC; dotted-dashed line, $T^V$ RVC; line with circles, $T$ MT; line with squares, $T^R$ MT; line with triangles, $T^V$ MT].

Figure 12

RVC model. Time evolution of the macroscopic dissociation rate coefficient [33] (unbroken line, $T=10000\mathrm{K}$; dashed line, $T=15000\mathrm{K}$; dotted-dashed line, $T=20000\mathrm{K}$; line with circles, $T=30000\mathrm{K}$; line with squares, $T=40000\mathrm{K}$).

Figure 13

Estimation of the phenomenological rate coefficient for dissociation $k^{\mathrm{Df}}$ in a 0D isothermal reactor, behind the normal shock wave [unbroken line, $k^{\mathrm{Df}}$ based on the RVC population distribution and temperature field [Eq. (43)]; dashed line, $k_{\mathrm{QSS}}^{\mathrm{Df}}$ based on the RVC temperature field; and dotted-dashed line, $k_{\mathrm{QSS}}^{\mathrm{Df}}$ based on the MT temperature field].

Figure 14

Estimation of the phenomenological rate coefficient for dissociation $k^{\mathrm{Df}}$, behind the normal shock wave [unbroken line, $k^{\mathrm{Df}}$ based on the RVC population distribution and temperature field [Eq. (43)]; dashed line, $k_{\mathrm{QSS}}^{\mathrm{Df}}$ based on the RVC temperature field; and dotted-dashed line, $k^{\mathrm{Df}}$ based on the VC population distribution and temperature field].

Figure 15

Cumulative distribution function for the RVC (a) and VC (b) models [in (a) unbroken line, $X_N=5\%$; dashed line, QSS ($X_N=50\%$); dotted-dashed line, $X_N=$ LTE; in (b) unbroken line, $X_N=5\%$; dashed line, $X_N=10\%$; dotted-dashed line, $X_N=25\%$; line with circles, $X_N=50\%$; line with squares, LTE].