Thermochemical Nonequilibrium Effects Research Articles

Study on the Flow Characteristics of Double-Cone in Hypersonic Flows

Reusable spacecraft is one of the hot topics in the field of aerospace, attracting wide attention from researchers. Due to the high-Mach-number flight of re-entry, the air near the wall exhibits significant thermochemical nonequilibrium effects. Besides, the existing shock wave/boundary layer interaction (SWBLI) leads to severe aerodynamic heating issues. This study utilizes the two-temperature model to conduct simulations of hypersonic laminar flows around a canonical 25°/55° double-cone with low and high enthalpy. By varying the wall temperature and the aft cone angle, the evolution mechanism of the flow under the high-enthalpy and hypersonic freestream is explored. Our findings illustrate that while the thermodynamic nonequilibrium model closely reflects the separation zone evidenced in the experiments, it habitually overpredicts the peak heat transfer, particularly under high-enthalpy conditions. For the thermochemical nonequilibrium model, the flow field structure appears more uniform, with a reduced standoff distance of the detached shock. An incremental rise in wall temperature correlates with a proportional augmentation of the separation bubble, though its impact on the overall flow field is negligible. Increasing the aft cone angle intensifies the shock/shock interaction (SSI), transitioning from a lower-intensity Type VI to a more intense Type IV shock interplay. The examination reveals that the increase in temperature and cone angle amplifies the interaction between the separation region and shock waves, drastically escalating the peak heat transfer and fostering a secondary peak. The hypersonic flow of the double-cone demonstrates a multifaceted interaction of phenomena, notably SWBLI and SSI, with our analyses providing pivotal insights for the aerodynamic and thermal protection design of the high-Mach-number spacecraft.

Numerical study of shock-induced thermochemical nonequilibrium effects in a high Mach flow field

As Mach number increases, thermochemical nonequilibrium is recognized as potentially affecting the flow field structure, as well as mixing and combustion characteristics, where shock-induced thermochemical nonequilibrium is a common and crucial phenomenon in compressible flow fields. A numerical study of shock-induced thermochemical nonequilibrium effects within a high Mach flow field of the electre vehicle is conducted by employing a two-temperature model-based solver hy2foam. The validation through experimental and simulation data confirms that hy2foam coupled with Park's two-temperature model and Park's five-species mechanism correctly predicts the flow structure and nonequilibrium characteristics. Four regime cases of thermochemical equilibrium, thermal nonequilibrium, chemical nonequilibrium, and thermochemical nonequilibrium are designed for comparison. First, the mechanism of shock-induced nonequilibrium is revealed. The shock induces the thermal nonequilibrium to occur instantly, and then the equilibrium is reestablished by undergoing the relaxation process. However, chemical nonequilibrium works delayed after the shock, and the high temperature induced by the shock motivates deviation from the chemical equilibrium by turning on chemical reactions. Further comparison of the four cases reveals that thermodynamic nonequilibrium significantly affects both shock position and intensity. In contrast, chemical nonequilibrium only significantly affects the distance to the shock detachment. Furthermore, it is found that thermodynamic and chemical nonequilibria behave in a complex coupling relationship after the shock.

Surrogate multi-objective optimization of missile RCS performance in hypersonic rarefied flow of the near space

This study focuses on optimizing the lateral jet efficiency of THAAD-like (Terminal High Altitude Area Defense) missiles operating under hypersonic rarefied flow conditions. We employ the DSMC-QK algorithm to simulate the three-dimensional lateral jet flow field, accounting for thermochemical non-equilibrium effects. The analysis investigates how the force/momentum amplification coefficient varies with the angle of attack, jet pressure ratio, jet Mach number, and jet gas composition. Subsequently, we develop an artificial neural network (ANN) proxy model using the pyrenn toolbox, achieving an average prediction error of 0.866% and a maximum error of 1.60%. Utilizing this ANN model, we perform single- and multi-objective optimizations with a genetic algorithm to determine the optimal jet parameters. The results reveal that in multi-objective optimization, the proportion of helium in the jet gas composition increases, leading to a slight reduction in the force amplification coefficient but a substantial 61.4% decrease in the mass flow rate. This demonstrates that a judicious selection of jet gas composition can significantly reduce mass flow while maintaining high jet efficiency, thus achieving efficient lateral jet control.

Thermo-chemical nonequilibrium effects on combustion characteristics of a transverse jet in the scramjet

As the Mach number increases, the local static temperature gradually approaches the characteristic gas vibration/dissociation temperature, and significant thermo-chemical relaxation processes occur. The thermo-chemical nonequilibrium phenomenona could affect the mixing and combustion process and become a factor that must be considered for scramjet design. In this paper, a large eddy simulation solver based on the two-temperature model is developed. The transfer terms between vibrational and trans-rotational energy are modeled by the Landau–Teller formulation and the vibrational-dissociation coupling is considered by adopting the Park model. Three sets of simulations based on the Hyshot Ⅱ engine configuration with different models were performed simultaneously to validate numerical methods and analyze results. The results reveal that the shock, cross-injection of fuel, and Prandtl-Meyer flow induce significant nonequilibrium regions. The vibration relaxation process in the jet mixing region is frozen, the vibration temperature is much higher than the trans-rotational temperature, and the extra energy is converted from the vibration mode to the trans-rotational mode, resulting in the increase of the trans-rotational temperature and the advance of auto-ignition in the nonequilibrium condition. Considering the vibration-dissociation coupling, the higher vibration temperature promotes dissociation. It inhibits the generation of active radicals in the ignition process, weakening the enhancement effect of nonequilibrium on auto-ignition. In the expanded nozzle, the concentrations of molecules and trans-rotational temperature in nonequilibrium cases are slightly lower than those in the equilibrium case due to the enhancement of dissociation.

Numerical investigation of thermochemical non-equilibrium effects in Mach 10 scramjet nozzle

Abstract High-temperature non-equilibrium effects are prominent in scramjet nozzle flows at high Mach numbers. Hence, the thermochemical non-equilibrium gas model incorporating the vibrational relaxation process of molecules in the hydrocarbon-air reaction is developed to numerically simulate the flow of a hydrocarbon fuel scramjet nozzle at Mach 10. Besides, the results computed by the models of the thermally perfect gas, chemically non-equilibrium gas, and thermally non-equilibrium chemically frozen gas are applied for comparative studies. Results indicate that chemical non-equilibrium effects are more significant for the flow-field structure and parameters compared to thermal non-equilibrium effects. Meanwhile, vibrational relaxation and chemical reactions interact in the flow-field. The heat released from the chemical reactions in the flow-field of the thermochemical non-equilibrium gas model makes the thermal non-equilibrium effects weaker compared to the thermally non-equilibrium chemically frozen gas model; the chemical reactions in the thermochemical non-equilibrium gas model are more intense than in the chemically non-equilibrium gas model. Due to the slow relaxation of vibrational energy, the thermal non-equilibrium models predicted nozzle thrust lower than the thermal equilibrium models by approximately 1.11% to 1.33%; when considering the chemical reactions, the chemical non-equilibrium models predicted nozzle thrust higher than the chemical frozen models by approximately 7.30% to 7.54%. Hence, the structural design and performance study of the high Mach numbers scramjet nozzle must consider thermochemical non-equilibrium effects.

Numerical investigations on flow and heat transfer characteristics of a high-enthalpy double-cone in thermal and chemical non-equilibrium for hypersonic propulsion

There are strong interests in hypersonic propulsion communities in predicting the flow and heat transfer characteristics over a hypersonic vehicle. However, this is quite challenging due to the emergence of thermal and chemical out-of-equilibrium effects as prominent in a high-temperature flow. In this work, we conduct 2D numerical investigations on the hypersonic double cone flow in a surrounding of the stagnation enthalpy of 15.23 MJ/kg by solving the laminar Navier–Stokes equation systems with the vibrational non-equilibrium processes and seven species air reactions. We then conduct the sensitivities analyses on the flow and heat transfer features, as different transport models and different thermochemical models are applied. It is shown that the aerodynamic heat is more sensitive to the transport models than the flow, and the transport coefficients computed by the Gupta-Yos model are more suitable for predicting the double cone flow at the high enthalpy. On the other hand, the flow field of the high enthalpy double cone is more sensitive to the thermochemical models than the heat transfer on the wall excluding the peak position. Behind the detached shock, chemical reactions are found to enhance thermal or vibrational out-of-equilibrium effects. However, vibrational non-equilibrium effects are revealed to weaken the degree of molecule dissociation reactions. Two flow separations occur in the flow field computed by using the thermal non-equilibrium gas models, but the flows of thermal equilibrium gas models separate only once. The maximum values of wall parameters without chemical reactions are found to deviate greatly from the experimental measurements. The thermochemical out-of-equilibrium gas model can predict the separation region size closer to the experiment than the two reported numerical results. The wall static pressure and heat flux are also shown to well match the experimental results. The present work should shed lights on better understanding thermochemical non-equilibrium effects of complex flow phenomenon like the shock wave/boundary layer interaction in hypersonic propulsion with high enthalpy surroundings.

Influence of thermochemical non-equilibrium effects on non-uniform entrance scramjet nozzle

The high Mach number scramjet nozzle flow has prominent non-uniform characteristics and high-temperature non-equilibrium effects. Thus, the thermochemical non-equilibrium gas model was developed for simulating flows in the non-uniform and uniform entrance of a Mach 10 hydrocarbon fuel scramjet nozzle. The results show that the non-uniform entrance has a higher non-uniform degree in the flow-field compared to the uniform entrance. Additionally, the expansion degree inside the nozzle is greater and the thermal non-equilibrium effects downstream of the nozzle exit are stronger. The non-uniform entrance condition has a more intense chemical reaction near the nozzle exit but has less impact on the flow parameters. For the nozzle performance, under the influence of the non-uniform entrance, the greater gas expansion degree inside the nozzle results in a lower pressure acting on the upper wall, and then the nozzle thrust decreases by 1.68%. Hence, the structural design and performance optimization of the high Mach number scramjet nozzle must consider entrance non-uniform characteristics and thermochemical non-equilibrium effects.

Navier-Stokes characteristic boundary conditions for high-enthalpy compressible flows in thermochemical non-equilibrium

This fundamental study presents Navier-Stokes characteristic boundary conditions (NSCBCs) for high-enthalpy hypersonic flows in thermochemical non-equilibrium. In particular, the relevant locally one-dimensional inviscid (LODI) relations are derived within a two-temperature framework for high-enthalpy hypersonic flows undergoing finite-rate thermochemical processes, including air dissociation and vibrational relaxation. Using these LODI relations, a set of NSCBCs are proposed and later demonstrated in canonical test cases, including the interaction of homogeneous isotropic turbulence with a shock wave subject to high-enthalpy thermochemical non-equilibrium effects.

A Boltzmann model equation synchronously involving molecular internal energy, dissociation and recombination effects for multicomponent gases

On the basis of previous work, we propose a phenomenological computable Boltzmann model equation synchronously involving polyatomic molecular internal energies, dissociation and recombination effects for multicomponent gases in this paper. Here, the distribution functions of each species have nine independent variables, which are time, physical space, molecular velocity, rotational energy and vibrational energy, if this component is a polyatomic gas. In this model equation, the concept of ‘chemical reaction rate’ is introduced in the chemical collision term to characterize the affect of chemical reactions. In order to make this complex model equation being ‘computable’, reduced distribution functions are adopted to get rid of two independent variables about rotational energy and vibrational energy of polyatomic component. We prove the conservation of this model equation, and then numerically solve it by the framework of gas-kinetic unified algorithm (GKUA), which is one kind of discrete velocity method (DVM). Here, non-catalytic boundary condition is adopted. Finally, a simple hypersonic flow around a cylinder with Mach number being 24.85 is numerically simulated to verify this model equation. In this case, four different components and eight chemical reactions are considered. It is shown that, this proposed Boltzmann model equation has the potential to be used to simulate thermochemical non-equilibrium effects of hypersonic flows covering various regimes.

Influences of thermochemical non-equilibrium effects on Type III shock/shock interaction at Mach 10

Shock/shock interactions (SSIs) often lead to high thermal loads. To understand the influences of thermochemical non-equilibrium effects on the Type III SSI, four gas models based on the assumption of thermal perfect gas (TPG), thermal equilibrium chemical non-equilibrium gas (TECNG), thermal non-equilibrium chemical frozen gas (TNCFG), and thermochemical non-equilibrium gas (TCNEG) are employed to simulate the Type III SSI at Mach 10. The intersection of incident shock and bow shock is determined by dimensionless intercept of 0.05 and 0.1. It is found that the chemical non-equilibrium effects significantly lift up the impingement position of shear layer by reducing the standoff distance of bow shock. As a result, the peaks of wall pressure and heat flux calculated by TECNG and TCNEG model are higher than TPG and TNCFG model, respectively. Type IIIa SSI occur in the flows calculated by the TECNG and TCNEG model for the case with a dimensionless intercept of 0.1. The peaks of wall pressure and heat flux calculated by TECNG model are both over four times higher than those of TPG model. The thermal non-equilibrium effects slightly increase the standoff distance of bow shock and lower the impingement position of shear layer. In addition, the thermal non-equilibrium reduces the angle to the horizontal direction of shear layer by increasing the specific heat ratio.

Simulation of High-Enthalpy Turbulent Shock Wave/Boundary Layer Interaction Using a RANS Approach

In the era of space exploration, the scientific community is strongly focusing on the analysis of hypersonic flows in the presence of shock wave/boundary layer interaction. In these conditions, the flow field presents a complex shock structure due to the interaction of different shock waves with the boundary layer. The strong adverse pressure gradient makes the boundary layer separate, giving rise to a separation bubble. In the reattachment zone, the temperature can reach very high values, inducing thermochemical non-equilibrium effects. This research field is recently achieving more and more relevance in aerospace research, as the analysis of turbulent shock wave/boundary layer interaction so far has been mainly focused on perfect gas flows. In this manuscript, a Reynolds averaged Navier–Stokes (RANS) approach is considered, the shear stress transport (SST) model being coupled with the multitemperature approach proposed by Park to investigate thermochemical non-equilibrium effects in hypersonic turbulent shock wave/boundary layer interaction. The first part of the manuscript is devoted to the validation of the solver, and results for low enthalpy flat plate and compression ramp flows are presented. The numerical results are shown to be in good agreement with numerical solutions and experimental measurements. Afterward, the free stream conditions are modulated to make non-equilibrium relevant and analyze a reacting flow.

Thermochemical non-equilibrium flow characteristics of high Mach number inlet in a wide operation range

The high-temperature non-equilibrium effect is a novel and significant issue in the flows over a high Mach number (above Mach 8) air-breathing vehicle. Thus, this study attempts to investigate the high-temperature non-equilibrium flows of a curved compression two-dimensional scramjet inlet at Mach 8 to 12 utilizing the two-dimensional non-equilibrium RANS calculations. Notably, the thermochemical non-equilibrium gas model can predict the actual high-temperature flows, and the numerical results of the other four thermochemical gas models are only used for comparative analysis. Firstly, the thermochemical non-equilibrium flow fields and work performance of the inlet at Mach 8 to 12 are analyzed. Then, the influences of high-temperature non-equilibrium effects on the starting characteristics of the inlet are investigated. The results reveal that a large separation bubble caused by the cowl shock/lower wall boundary layer interaction appears upstream of the shoulder, at Mach 8. The separation zone size is smaller, and its location is closer to the downstream area while the thermal process changes from frozen to non-equilibrium and then to equilibrium. With the increase of inflow Mach number, the thermochemical non-equilibrium effects in the whole inlet flow field gradually strengthen, so their influences on the overall work performance of the high Mach number inlet are more obvious. The vibrational relaxation or thermal non-equilibrium effects can yield more visible influences on the inlet performance than the chemical non-equilibrium reactions. The inlet in the thermochemical non-equilibrium flow can restart more easily than that in the thermochemical frozen flow. This work should provide a basis for the design and starting ability prediction of the high Mach number inlet in the wide operation range.

Stabilization of hypersonic boundary-layer instability using porous coatings under thermochemical nonequilibrium

In this study, the suppression of the instability of the hypersonic boundary-layer transition under thermochemical nonequilibrium (TCNE) using a porous coating configuration is examined through linear stability theory (LST) and direct numerical simulation (DNS). Fedorov's admittance model is applied in LST to account for the porous coating effect. The model is modified accordingly such that the flow condition simulated under the TCNE assumption is considered. The results are compared with the case under the calorically perfect gas (CPG) assumption. Conditions of Mach 20-6 degree wedge flow and Mach 45-10 degree wedge flow are chosen for this study. In general, the second mode instability is destabilized and shifts upstream under the TCNE effect. Through the comparison between the DNS and LST results in the presence of a porous coating, it is concluded that the effect of suppressing the instability is very similar under both the CPG and TCNE assumptions. The TCNE effect has minimal impact on the acoustic attenuation of the instability within the cavity. Furthermore, the use of the admittance models in LST can fairly predict the trend of disturbance growth in DNS if the porous coating region is sufficiently long, validating the feasibility of extending the application of the admittance model to a different thermochemical state in addition to the CPG assumption. Examination of the perturbation contours under different state assumptions reveals the absence of second-mode wave structure in the region of microcavities, confirming the stabilization of hypersonic boundary-layer instability.

High Enthalpy Non-Equilibrium Expansion Effects in Turbulent Flow of the Conical Nozzle

High enthalpy stagnation gas can be converted into hypervelocity flow through the contraction—expansion nozzle. The enthalpy flow in the nozzle can be divided into three regions: an equilibrium region, a non-equilibrium region, and a frozen region. The stagnation gas with a total enthalpy of 13.4 MJ/kg is used to analyze the thermochemical non-equilibrium effects. At the selected conditions, the effects of a conical nozzle under different expansion angles of the expansion section, curvature radius of the throat, throat radius, and convergence angle of the convergent section are investigated. Based on the Spalart–Allmaras one-equation turbulence model with the Catris–Aupiox compressibility correction, a multi-block solver for axisymmetric compressible Navier–Stokes equations is applied to simulate the thermochemical non-equilibrium flow in several high enthalpy conical nozzles. The multi-species two-temperature equation is employed in the calculation. The results reveal three interesting characteristics: Firstly, the thermochemical non-equilibrium effects are sensitive to the maximum expansion angle and throat radius but not to the radius of throat curvature and contraction angle. Secondly, as the maximum expansion angle decreases and the throat radius increases, the flow approaches equilibrium state. When the maximum expansion angle decreases from 12° to 4°, the freezing temperature decreases from 2623 K to 2018 K. When the throat diameter increased from 10 mm to 30 mm, the freezing temperature decreased from 2442 K to 2140 K. Finally, the maximum expansion angle and throat radius not only affect the position of the freezing point but also the flow field parameters, such as temperature, Mach number, and species mass fraction.

Shock impingement on a transitional hypersonic high-enthalpy boundary layer

Shock-wave boundary layer interactions are of utmost importance for both aeronautical and aerospace applications. The dynamics of high-speed compressible turbulent boundary layers becomes tightly coupled with strong gradients of thermodynamic properties, leading to an increase of thermomechanical loads. In the contextual presence of hypersonic and high-enthalpy regimes, thermochemical nonequilibrium effects must be considered as well. This work is a first step toward the fundamental understanding of this configuration.

Thermochemical non-equilibrium and electromagnetic effects of double-cone in hypervelocity flow

Hypervelocity flows generated over a 25°/55° double-cone are numerically investigated, characterized by a complex shock wave/boundary layer interaction. Based on the low magnetic Reynolds number assumption, the coupled model is established for the thermochemical non-equilibrium flow field and the externally applied magnetic field. The aim of the study is on one hand to evaluate the high temperature effects and on the other to investigate the electromagnetic effects of such an interaction. The results indicate that the thermochemical non-equilibrium effects become relevant and significantly affect the flow structure and surface properties in the case of high-enthalpy flows. The performance of shock wave/boundary layer interaction flows magnetohydrodynamic control is mainly determined by the Lorentz force near the triple point and the separated flow region. Furthermore, the modification of the flow field due to the applied magnetic field causes the mitigation of the aerodynamic heating. This is enhanced with the increase of magnetic field strength.

Numerical simulation of hypersonic thermochemical nonequilibrium flows using nonlinear coupled constitutive relations

To predict aeroheating performance of hypersonic vehicles accurately in thermochemical nonequilibrium flows accompanied by rarefaction effect, a Nonlinear Coupled Constitutive Relations (NCCR) model coupled with Gupta’s chemical models and Park’s two-temperature model is firstly proposed in this paper. Three typical cases are intensively investigated for further validation, including hypersonic flows over a two-dimensional cylinder, a RAM-C II flight vehicle and a type HTV-2 flight vehicle. The results predicted by NCCR solution, such as heat flux coefficient and electron number densities, are in better agreement with those of direct simulation Monte Carlo or flight data than Navier-Stokes equations, especially in the extremely nonequilibrium regions, which indicates the potential of the newly-developed solution to capture both thermochemical and rarefied nonequilibrium effects. The comparisons between the present solver and NCCR model without a two-temperature model are also conducted to demonstrate the significance of vibrational energy source term in the accurate simulation of high-Mach flows.

Cross-flow vortices and their secondary instabilities in hypersonic and high-enthalpy boundary layers

Compared to the streamwise instability, the cross-flow instability in high-enthalpy flows has received relatively less attention, but the latter is of vital importance in the flow transition for practical configurations. This work aims to investigate the cross-flow primary and secondary instabilities in hypersonic and high-enthalpy boundary layers, considering thermochemical non-equilibrium (TCNE) effects. The numerical tools adopted include a high-order shock-fitting solver, nonlinear parabolized stability equations and secondary instability theory (SIT). The flow over a swept parabola is calculated at a free-stream Mach number of 16. It is found that TCNE has a destabilizing effect on the cross-flow mode with a non-catalytic wall. Two important non-dimensional parameters are summarized to explain this effect. One is the ratio between the wall and boundary-layer edge temperatures, and the other is the cross-flow Mach number. Due to nonlinear effects, the stationary cross-flow vortices evolve and exhibit the classic rollover structures as in lower-speed flows. Two different disturbance energy norms are used in the energy budget analysis to classify the secondary cross-flow instability modes. The results from SIT highlight the importance of type-IV modes in TCNE flows at the downwash region of the vortex. The type-IV modes arise with the combined contribution from the wall-normal (on top and trough of the vortex) and spanwise (in the downwash region) production terms. The type-I mode is dominant in the calorically perfect gas case with an adiabatic wall, whereas the type-IV mode has the largest growth rate in the TCNE cases irrespective of wall temperature variation.

Thermochemical non-equilibrium effects on hypersonic wavecatcher intake at Mach 12

The thermal and chemical non-equilibrium effects on the flow structures and performance of a hypersonic inward-turning intake, for the airbreathing scramjet propulsion, are numerically studied at Mach 12. Three gas models of 1) thermally perfect gas (TPG), 2) chemical non-equilibrium gas (CNEG), and 3) thermochemical non-equilibrium gas (TCNEG), are performed. The influence of wall temperatures is analyzed, with isothermal and adiabatic wall conditions. Wall temperature has a significant effect on the mass capture coefficient, dropping by almost 6% from Tw = 300 K to adiabatic cases. The dissociation of air occurs mainly in the boundary layer near the wall. As the wall temperature decreases, so does the degree of dissociation inside the intake. For adiabatic cases, the results show that the degree of dissociation with TCNEG model is higher than that of the CNEG model. The degree of oxygen dissociation with TCNEG model reaches 54.0% in the symmetry plane, which is 6.3% higher than the value predicted by CNEG model. For the isothermal case at Tw = 2000 K, only a few dissociated oxygen atoms are observed in the isolator with TCNEG and CNEG model, and it is much less than that of adiabatic case. It is suggested that the thermochemical non-equilibrium effect is significant for a Mach 12 intake at adiabatic wall conditions.

Thermochemical nonequilibrium effects on high-enthalpy double-wedge flows

A hypersonic laminar flow over double wedges with a fixed forward angle of 15° and varied aft angles is studied using computational fluid dynamics and global stability analysis (GSA) at a free-stream Mach number of 12.82 and a total enthalpy of 21.77 MJ/kg. The specific total enthalpy is high enough to trigger evident vibrational excitation and air chemistry. To assess the effects of thermal and chemical nonequilibrium, three different thermochemistry models of air are considered, including frozen, thermal nonequilibrium, and thermochemical nonequilibrium gases. Two-dimensional base-flow simulations indicate that the onset of incipient and secondary separation is insensitive to the inclusion of thermochemistry, although the size of the separation region is substantially reduced. GSA is then performed on the base flows and identifies a three-dimensional stationary global instability beyond a critical aft angle, which is also insensitive to thermochemical nonequilibrium. The criterion of the global stability boundary established for the supersonic flow over compression corners in a calorically perfect gas in terms of a scaled deflection angle [Hao et al., “Occurrence of global instability in hypersonic compression corner flow,” J. Fluid Mech. 919, A4 (2021)] is, thus, extended to high-enthalpy conditions.