Assessment of continuum breakdown for chemically reacting wake flows

This work aims at predicting continuum breakdown in high-speed, chemically reacting wake flows using breakdown parameters obtained from the generalized Champan-Enskog (GCE) theory. Previous efforts have addressed the mathematical development of these mechanism driven breakdown parameters and their utility in predicting continuum breakdown in high Mach number compressing flows. Here we present a specieswise perturbation parameter, based on the 2-norm in Hilbert space of the first-order GCE perturbation, which can be used to assess the extent to which the underlying equilibrium Maxwell-Boltzmann (MB) distribution of a given species has been perturbed. This parameter is derived for the one-, two-, and three-temperature models, and is applicable to multicomponent, chemically reacting flowfields. All transport mechanisms that can lead to distortion of the specieswise equilibrium MB distribution function—translational, rotational and vibrational heat fluxes, mass and thermal diffusion fluxes, and stress tensor components including bulk viscosity and relaxation pressure terms—are simultaneously incorporated into this perturbation parameter that can be computed for each species. This parameter and the mechanism-based GCE breakdown parameters are used to assess continuum breakdown in the forebody and wake region of a cylinder subjected to hypersonic flow. The influence of altitude, freestream velocity, and cylinder surface reactions on continuum breakdown is analyzed. Regions of continuum breakdown are observed in the shock, boundary layer, as well as in the wake. Most notably, the surface chemistry at the cylinder wall forebody can lead to the formation of breakdown regions in the wake that are detached from the cylinder surface.

Figure 1

The computational domain indicated by temperature contours. The black lines represent locations along which continuum breakdown assessment is performed.

Figure 2

Near wall breakdown parameter profiles along $\theta =0^{\circ }$. The left column shows ${\mathrm{B}}_{\mathrm{GCE}}$ [solid line computed from Eqs. (15) and (18)] and $\parallel {\varphi }^s\parallel$ [dashed lines computed from Eq. (16)], and the right column indicates the GCE parameters (for most significant mechanisms) for species with large perturbations. The grey regions highlight zones of continuum breakdown predicted by both $\parallel {\varphi }^s\parallel$ and ${\mathrm{B}}_{\mathrm{GCE}}$. (a) Noncatalytic wall: breakdown region, (b) noncatalytic wall: breakdown mechanism, (c) finite rate air-silica wall model: breakdown region, (d) finite rate air-silica wall model: breakdown mechanism, (e) fully catalytic wall: breakdown region, and (f) fully catalytic wall: breakdown mechanism.

Figure 3

Breakdown parameter profiles along $\theta ={135}^{\circ }$. The left column shows ${\mathrm{B}}_{\mathrm{GCE}}$ [solid line computed from Eqs. (15) and (18)] and $\parallel {\varphi }^s\parallel$ [dashed lines computed from Eq. (16)], and the right column indicates the GCE parameters (for most significant mechanisms) for species with large perturbations. The grey regions highlight zones of noncontinuum predicted by both $\parallel {\varphi }^s\parallel$ and ${\mathrm{B}}_{\mathrm{GCE}}$, and the blue areas are breakdown regions indicated only by the former. (a) Noncatalytic wall: breakdown region, (b) noncatalytic wall: breakdown mechanism, (c) finite rate air-silica wall model: breakdown region, (d) finite rate air-silica wall model: breakdown mechanism, (e) fully catalytic wall: breakdown region, and (f) fully catalytic wall: breakdown mechanism.

Figure 4

Streamlines overlayed on contours colored by velocity, indicating regions of high velocity gradients along the envelope of the recirculation zone, forming a region of local noncontinuum.

Figure 5

Near wall breakdown parameter profiles along $\theta ={150}^{\circ }$. The left column shows ${\mathrm{B}}_{\mathrm{GCE}}$ [solid line computed from Eqs. (15) and (18)] and $\parallel {\varphi }^s\parallel$ [dashed lines computed from Eq. (16)], and the right column indicates the GCE parameters (for most significant mechanisms) for species with large perturbations. The grey regions highlight zones of continuum breakdown predicted by both $\parallel {\varphi }^s\parallel$ and ${\mathrm{B}}_{\mathrm{GCE}}$. (a) Noncatalytic wall: breakdown region, (b) noncatalytic wall: breakdown mechanism, (c) finite rate air-silica wall model: breakdown region, (d) finite rate air-silica wall model: breakdown mechanism, (e) fully catalytic wall: breakdown region, and (f) fully catalytic wall: breakdown mechanism.

Figure 6

Near wall breakdown parameter profiles along $\theta ={180}^{\circ }$. The left column shows ${\mathrm{B}}_{\mathrm{GCE}}$ [solid line computed from Eqs. (15) and (18)] and $\parallel {\varphi }^s\parallel$ [dashed lines computed from Eq. (16)], and the right column indicates the GCE parameters (for most significant mechanisms) for species with large perturbations. The grey regions highlight zones of continuum breakdown predicted by both $\parallel {\varphi }^s\parallel$ and ${\mathrm{B}}_{\mathrm{GCE}}$. (a) Noncatalytic wall: breakdown region, (b) noncatalytic wall: breakdown mechanism, (c) finite rate air-silica wall model: breakdown region, (d) finite rate air-silica wall model: breakdown mechanism, (e) fully catalytic wall: breakdown region, and (f) fully catalytic wall: breakdown mechanism.

Figure 7

Contours of ${\varphi }_{\text{max}}$ [Eq. (17)] indicating regions of continuum breakdown in the flowfield, for the (a) noncatalytic and (b) fully catalytic wall. For a breakdown criterion of 0.05, red regions represent zones with continuum breakdown. The inlay shows a magnified version of the shock and near wall breakdown region in the vicinity of the stagnation line.

Figure 8

Breakdown region around the shock and near the wall, along the stagnation line. The left and right columns shows the GCE parameters computed from Eq. (15) (for most significant mechanisms) for species with large perturbations, for Case 1 with 8600 m/s velocity, and Case 3 with 6600 m/s velocity, respectively. The grey regions highlight zones of continuum breakdown.

Figure 9

Breakdown region in the wake region for $\theta ={135}^{\circ }\mathrm{and}{180}^{\circ }$. The left and right columns shows the GCE parameters computed from Eq. (15) (for most significant mechanisms) for species with large perturbations, for Case 1 with 8600 m/s velocity, and Case 3 with 6600 m/s velocity, respectively. The grey regions highlight zones of continuum breakdown.

Figure 10

Contours of ${\varphi }_{\text{max}}$ [Eq. (17)] indicating regions of continuum breakdown in the flowfield, for the cases with freestream velocity (a) 8600 m/s (b) 6600 m/s. For a breakdown criterion of 0.05, red regions represent zones with continuum breakdown. The inlay shows a magnified version of the shock and near wall breakdown region in the vicinity of the stagnation line.

Figure 11

Breakdown region at the shock and near the cylinder surface along the stagnation line at an altitude of: (a) 60 km, (b) 80 km. The plots show the GCE parameters computed from Eq. (15) (for most significant mechanisms) for species with large perturbations. The grey regions highlight zones of continuum breakdown.

Figure 12

Breakdown region in the wake region for $\theta ={135}^{\circ },{150}^{\circ }\mathrm{and}{180}^{\circ }$. The left and right columns shows the GCE parameters computed from Eq. (15) (for most significant mechanisms) for species with large perturbations for Case 2 with altitude of 60 km, and Case 4 with altitude of 80 km, respectively. The grey regions highlight zones of continuum breakdown.

Figure 13

Contours of ${\varphi }_{\text{max}}$ [Eq. (17)] indicating regions of continuum breakdown in the flowfield, for the cases with altitude (a) 60 km (b) 80 km. For a breakdown criterion of 0.05, red regions represent zones with continuum breakdown. The inlay shows a magnified version of the shock and near wall breakdown region in the vicinity of the stagnation line.

Figure 14

Contours of ${\varphi }_{\text{max}}$ [Eq. (17)] [top] and ${\mathrm{Kn}}_{\mathrm{GLL}}$ [Eq. (4)] [bottom] indicating regions of continuum breakdown in the flowfield, for (a) fully catalytic wall (b) noncatalytic wall. For a breakdown criterion of 0.05, red regions represent zones with continuum breakdown.

Figure 15

Breakdown profiles in solid line- ${\varphi }_{\text{max}}$ [Eq. (17)], and dashed line- ${\mathrm{Kn}}_{\mathrm{GLL}}\text{[Eq}$. (4)] along: (a), (b) $\theta ={135}^{\circ }$; (c), (d) $\theta ={165}^{\circ }$. This is shown for (a), (c) fully catalytic and (b), (d) noncatalytic wall. The gray areas are region of continuum breakdown predicted by both ${\varphi }_{\text{max}}$ and ${\mathrm{Kn}}_{\mathrm{GLL}}$, and the areas shaded blue are additional regions of noncontinuum predicted by ${\varphi }_{\text{max}}$.