Speaker
Description
Background to the study
Thermal-chemical kinetics for non-equilibrium flows can be accurately represented through state-to-state (StS) models, where each molecular energy state is treated as a distinct pseudo-species. However, StS modeling requires a significantly larger number of variables than multi-temperature models, as it involves hundreds of internal energy states and thousands of kinetic processes, resulting in high computational demands.
This study introduces a new computational approach to simplify chemical kinetics in nonequilibrium hypersonic flows by identifying and reducing the number of dominant reaction mechanisms for a given flow condition. By recognizing that chemical reactions and molecular transport operate at vastly different time scales, it is possible to separate fast reactions that reach local equilibrium from those that impact the local flow dynamics.
Maas and Pope proposed the Intrinsic Lower Dimensional Manifold (ILDM) as a reduction technique to simplify the chemical kinetics of combustion problems in subsonic conditions. This method is based on the principle of timescale separation, assuming that fast time scales have equilibrated, resulting in the creation of a lower dimensional space governed by slow reactions. The ILDM method simplifies the conservation equations for a spatially homogeneous, adiabatic, and isobaric system. By linearizing the solution around an initial state with small perturbations, the problem can be formulated as an ordinary differential equation. The solution is then expressed as a linear combination of exponential terms, with the eigenvalues of the Jacobian matrix determining the time scale of the mechanism and the eigenvectors indicating the associated response direction.
Nevertheless, the assumptions of adiabatic and isobaric conditions are not representative of the conditions experienced in a reactive hypersonic nonequilibrium flow, such as those occurring in the shock layer or along a stagnation line.
In this paper, the constancy of pressure and energy is not assumed; instead, they are bounded by the conservation equations along an isentropic path. A generalized eigenvalue problem is proposed based on the linearized reactive Euler equations with Park's two-temperature model. The specific case of the eigenvalue problem corresponding to a stationary disturbance is analysed, unveiling the relationship between lower-order dimensional manifolds and the reaction mechanisms investigated. The methodology discussed can be applied to identify the relevant reaction mechanisms and reduce the computational cost of nonequilibrium hypersonic simulations for spacecraft re-entry applications.
Methodology
The conservation equations for a chemically reacting, non-conducting, inviscid one-dimensional flow incorporating vibrational relaxation are expressed below in vector conservative form, where $U$ is the vector of conserved variables, $F$ is the inviscid flux vector and the source term $W$ is the vector of production rates.
$\frac{\partial U}{\partial t}+\frac{\partial F}{\partial x}=W$
The instantaneous flow is defined as the sum of the mean state variables $\hat{Q}$ and their fluctuating disturbance $\hat{Q}$, where the latter is expressed as a harmonic wave of angular frequency $\omega$ and wave number $\nu$.
$ Q=\bar{Q}+\hat{Q} e^{j(\omega t+ \nu x)}$
The linearization of the governing equations around the equilibrium state is justifiable for small perturbations, yielding the formulation given as follows, which is of analogous type as a generalised eigenvalue problem $(B\xi=\lambda A\xi)$:
$\left(j \omega \frac{\partial U}{\partial Q} - \frac{\partial W}{\partial Q} \right) \hat{Q} = -j \nu \frac{\partial F}{\partial Q} \hat{Q}$
In the case of $\omega=0$, a stationary disturbance is obtained. The eigenvalue problem thus reduces to \autoref{eq:firstorder}, yielding eigenvalues of zeros for the conservation laws of mass, momentum and energy. The remaining non-zero eigenvalues $s$ are due to the presence of chemical source terms and describe the relaxation process towards equilibrium of the perturbed flow.
$ \frac{\partial W}{\partial Q} \hat{Q} = j \nu \frac{\partial F}{\partial Q} \hat{Q}$
The first-order linear differential equation has the straightforward exponential solution given below, where the initial conditions are given by the jump in properties across the shock $\Delta Q$. The physical meaning of the eigenvalues $s$ clearly appears to define the length-scale over which the relaxation process occurs for a flow that is travelling at a finite speed, whereas the right-eigenvectors $V_R$ define the characteristic direction along which the relaxation occurs. The components with large, negative, $s$ are in equilibrium, while those with small $s$ evolve slowly and govern the flow.
$ q(x)= \sum V_{R} e^{-s x} V_{L} \Delta Q $
Results and Conclusions
The paper will present a manifold analysis based on rate Jacobian for mechanism reduction at hypersonic conditions subject to dissociation, ionization and relaxation through a shock. The methodology is applied to 5-species and 11-species Park 2-Temperature model to demonstrate that a reduced number of independent mechanisms is sufficient to simplify and accurately describe the flow relaxation. Calculations of such reduced mechanisms for Earth re-entry scenarios are presented at different altitudes and speeds. The extracted reduced mechanisms are detailed and their validity is investigated. Comparisons are drawn with commonly used mechanisms for air using reduced species sets.
Summary
This study presents a new computational approach to simplify chemical kinetics in non-equilibrium hypersonic flows. Traditional state-to-state (StS) models are highly accurate but computationally intensive due to the large number of variables. By focusing on dominant reaction mechanisms and separating fast, locally equilibrated reactions from slower, flow-impacting ones, the complexity and computational cost of the problem are reduced.
Building on the Intrinsic Lower Dimensional Manifold (ILDM) technique, this study removes the assumption of an isobaric and adiabatic system and enables the reduction of chemical kinetics for hypersonic scenarios. The methodology uses a generalized eigenvalue problem derived from linearized reactive Euler equations with Park's two-temperature model. This identifies key reaction mechanisms and simplifies the modelling process.
Applied to 5-species and 11-species models, the approach accurately describes post-shock flow relaxation with fewer mechanisms. This reduction significantly reduces computational costs while maintaining accuracy, offering a practical solution for hypersonic flow simulations.
