Coarse-grained modeling of thermochemical nonequilibrium using the multigroup maximum entropy quadratic formulation

This work addresses the construction of a reduced-order model based on a multigroup maximum entropy formulation for application to high-enthalpy nonequilibrium flows. The method seeks a piecewise quadratic representation of the internal energy-state populations by lumping internal energy levels into groups and by applying the maximum entropy principle in conjunction with the method of moments. The use of higher-order polynomials allows for an accurate representation of the logarithm of the distribution of the low-lying energy states, while preserving an accurate description of the linear portions of the logarithm of the distribution function that characterize the intermediate- and high-energy states. A comparison of the quadratic and the linear reconstructions clearly demonstrates how the higher-order reconstruction provides a more accurate representation of the internal population distribution function at a modest increase in the computational cost. Numerical simulations carried out under conditions relevant to hypersonic flight reveal that the proposed model is able to capture the dynamics of the nonequilibrium distribution function using as few as three groups, thereby reducing the computational costs for simulations of nonequilibrium flows.

Figure 1

Nonequilibrium state-to-state population distribution function (blue triangles) and a seven-group quadratic piecewise reconstruction (orange diamonds).

Figure 2

Different types of reconstruction functions estimating the actual distribution function in each group; constant (solid blue line), linear (dashed red line), and quadratic (dotted-dashed pink line); symbols denote the actual distribution function.

Figure 3

Comparison of the state population distributions obtained using the state-to-state (triangles; blue at 0.25 ns and green at 0.$1\mu \mathrm{s})$ and two-group maximum entropy linear (black circles) and quadratic (yellow diamonds) models. (a) $T=10000\mathrm{K}$; (b) $T=20000\mathrm{K}$.

Figure 4

Comparison of the state population distributions obtained using the state-to-state (triangles; blue at 0.25 ns and green at 0.$1\mu \mathrm{s})$ and three-group maximum entropy linear (black circles) and quadratic (yellow diamonds) models. (a) $T=10000$ K; (b) $T=20000\mathrm{K}$.

Figure 5

State-specific dissociation rate constants for rovibrational states of ${\mathrm{N}}_2$.

Figure 6

Percentage error in mean and variance of the PMF estimated using three linear and quadratic groups. (a) Error in mean and (b) error in variance of the distribution function.

Figure 7

Mole fraction profiles zoomed in during dissociation.

Figure 8

Total internal energy.

Figure 9

State population distribution obtained from a one-group maximum entropy quadratic model compared with the actual distribution from the state-to-state model.

Figure 10

Time evolution of the number of moles in each group and the group energy. Symbols represent the quadratic model and solid lines represent the full state-to-state model. (a) Group number of moles; (b) group internal energy.

Figure 11

Time evolution of the group internal temperature. Symbols represent the quadratic model and solid lines represent the full state-to-state model.

Figure 12

Time evolution of quadratic parameters using three groups. (a) $\widehat{\beta }$; (b) $\widehat{\gamma }$.