The physically realistic hypersonic flow in the continuum regime has outgrown the reach of the classic gas kinetic theory. From the basic approach of adopting the binary elastic collision, the internal degrees of freedom of gas particles in microscopic scale are not taken into consideration. In order to remedy the deficiency, the collision model needs to describe the interactions between particles by inelastic collisions. Meanwhile, the quantum phenomena impose formidable challenges both in concepts and basic understandings that are prohibitive for any rudimentary and systematic analytic development. For now, investigations are conducted mainly by interdisciplinary investigations with physics-based models derived from all the pertaining scientific disciplines integrating into aerodynamics.

The frame work of the most recent hypersonic flow research is built on accumulated multidisciplinary knowledge from the high-speed propulsion with chemical kinetics, and from the magnetohydrodynamics with electromagnetics. Nearly all hypersonic flows are immersed in a high enthalpy but weakly ionized air by shock wave compression. Although the electromagnetic effects are drastically altering the characteristics of flow medium, the magnitudes of all the remote-acting forces still remain at a perturbation level in comparison to aerodynamic inertia. The relative magnitude between aerodynamic and electromagnetic forces is measured by the magnetic Reynolds number, which is defined as *R*_{m} = *uL*/(*σμ*)^{−1} = *σμu*_{∞}*L* [19, 20, 35]. The value of this demarcation similitude is less than unity in most hypersonic flows, only in an MHD thruster its value reaches unity, and attains a value around one hundred in the controlled thermonuclear reaction [20].

The low magnetic Reynolds number approximation is fully justified for most hypersonic flows of aerodynamic interest. Based on the stipulated condition, the Faraday’s induction law can be decoupled from the interdisciplinary governing equations [19, 33]. The momentum exchange within the control volume is now balanced by the electrostatic force *ρ*_{e}*E* and Lorentz acceleration *J* × *B*, which are appended to the right-hand-side of the momentum conservation equation. The conservation of energy of the complete system needs to include the energy cascading and transmitting by quantum jumps from the translational mode to vibrational *Q*_{t, v} as well as the electronic *Q*_{t, e} modes. The Joule heating, *E* ⋅ *J* of the ionized air and the radiative heat transfers are also added to the conservation energy equation as the source terms. The effects of transport properties are reinforced by the gas kinetic theory, which is always the key coupling mechanism between conservation mass, momentum, and global energy equations.

The quantum jumps of the internal degrees of freedom between translational, vibrational, and electronic excitations are included to the conservation energy laws for each individual species. The identical species with different internal degrees of freedom are also identified as different chemical compositions; thus the vibrational energy conservation equations for polyatomic molecular species are included in the governing equations together with the energy conservation equation of the electronic excitation. The complete interdisciplinary governing equations acquire the following forms [19, 29];

$$ \partial {\rho}_i/\partial t+\nabla \cdot \left[{\rho}_i\left(u+{u}_i\right)\right]={dw}_i/ dt $$

(36)

$$ \partial \rho u/\partial t+\nabla \cdot \left(\rho u u+p-\tau \right)={\rho}_eE+\left(J\times B\right) $$

(37)

$$ \partial {\rho}_i{e}_{vi}/\partial t+\nabla \cdot \left[{\rho}_i\left(u+{u}_i\right){e}_{vi}+{q}_{vi}\right)\Big]=\left({dw}_{vi}/ dt\right){e}_{vi}+{Q}_{v,\Sigma} $$

(38)

$$ {\displaystyle \begin{array}{l}\partial {\rho}_i{e}_e/\partial t+\nabla \cdot \left[{\rho}_i\left(u+{u}_i\right){e}_e+u\cdot {p}_e+{q}_e\right)\Big]=\left({dw}_e/ dt\right){e}_e+E\cdot J\\ {}\kern2.5em +\left[{\rho}_eE+\left(J\times B\right)\right]\cdot \left(u+{u}_i\right)+{Q}_{e,\Sigma}\end{array}} $$

(39)

$$ \partial \rho e/\partial t+\nabla \cdot \left[\rho e u-\kappa \nabla T+\sum {\rho}_i{u}_i{h}_i+{q}_{rad}+u\cdot p+u\cdot \tau \right]+{Q}_{t,v}-{Q}_{t,e}=E\cdot J $$

(40)

In Eqs. (36), (38), and (39), the production and depletion rates of each chemical species, and the same species but of different internal mods have been included. The detailed rate of change is described by Eq. (7). The specific internal energy of the air mixture including the electronic energy now becomes;

$$ \rho e=\sum \limits_{i\ne e}{\rho}_i\left({e}_i+u\cdot u/2\right)+\sum \limits_{i\ne e}{\rho}_i\varDelta {H}_i^o+\sum \limits_{i\ne e}{\rho}_i{e}_{vi}+\sum {\rho}_e\left({e}_e+{u}_e\cdot {u}_e/2\right) $$

(41)

The quantum transitions and energy redistribution among internal excitations *Q*_{t, v}, *Q*_{v, v}, *Q*_{t, e} are modeled including the relaxation phenomenon. The notations *Q*_{v, Σ} and *Q*_{e, Σ} in Eq. (38) and (39) represent all permissible quantum transitions for vibrational and electronic modes respectively. These models do not make any distinction between either by a single quantum number jump (ladder climbing theory) or the multiply quanta jump (big bang theory), but only evaluate the energy cascading among different species and quantum states.

The relaxation time scale between translational and vibrational mode by Landau et al. [46] is described by the principle of detailed balance;

$$ {Q}_{t,v}=\rho \left[{e}_v^{\ast }(T)-{e}_v\right]/\tau; \kern1em \tau =\left[{k}_1{T}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.} Exp{\left({k}_2/T\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]/p\left[1- Exp\left(-\raisebox{1ex}{${\Theta}_v$}\!\left/ \!\raisebox{-1ex}{$T$}\right.\right)\right] $$

(42)

The relaxation time scales among vibration-vibration quantum jumps are approximated by an outstanding correlation from experimental observations due to Millikan et al. [47] from a wide range of experimental data and by different experimental techniques.

$$ {Q}_{v,v}=\rho \left[{e}_v^{\ast }(T)-{e}_v\right]/\tau; \kern0.5em \tau =1.16\times {10}^{-3}{M_{12}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\Theta}_v^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left[\right({T}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-0.15{M}_{12}-18.42\Big] $$

(43)

where the M_{12} designates the reduced mass of the interacting species, *M*_{12} = *M*_{1}*M*_{2}/(*M*_{1} + *M*_{2}).

On the other hand, the quantum transition between electron-ion collisions is a pure empirical model to appear as [48];

$$ {Q}_{e,\mathtt{v}}=2\times {10}^{-16}\sum \limits_v{n}_e{n}_i{\omega}_{e,v}{P}_{1,0,v}\left\{{e}^{-\left(1.44{\omega}_{e,v}/ Te\right)}/\left[1-{e}^{-\left(1.44{\omega}_{e,v}/ Te\right)}\right]-{e}^{-\left(1.44{\omega}_{e,v}/ Tv\right)}/\left[1+{e}^{-\left(1.44{\omega}_{e,v}/ Tv\right)}\right]\right\} $$

(44)

where the empirical determined coefficient is *P*_{1, 0, V} = 0.45 × 10^{−10} exp − (10000/*T*_{e}).

There are numerous quantum jump models being applied from the dissociation biased vibration-vibration quantum transitions, vibration-electronic transition, electron-ion transition to the neural-electron collision [19, 23, 29]. Some of these models are developed from a rigorous theoretic consideration but others are simply devised by empirical means. Therefore, there are ample rooms for improvement as a promising new frontier for basic research by either experimental or the *ab initio* (from the beginning) approaches [18].

The high-enthalpy transport properties in a heterogeneous, nonequilibrium chemically reacting air mixture play a dominant role for momentum and energy transfer. The molecular viscosity, diffusion, and thermal conductivity originated from the nonequilibrium collision process, are the landmark achievements from the gas kinetic theory by describing it with an inter-molecular potential function [2, 20, 49]. These classic analytic results for collision cross sections and collision integrations are still valid but must be generated by different inter-molecular potentials according to the gas thermodynamic states [28, 29, 50].

The transport coefficients of a single molecular species are obtained from the Maxwell-Boltzmann distribution by the Chapman-Enskog expansion [2, 20]. The molecular viscosity coefficient derived from the gas kinetic theory with the collision cross sections and collision integrals is,

$$ \mu =2.67\times {10}^{-5}\sqrt{M_iT}/{\sigma}_i^2{\Omega}^{\left(2,2\right)} $$

(45)

The molecular diffusion coefficients in most hypersonic flows are focused only on the ordinary diffusion generated by the gradient of species concentration. The coupled Dufour and Soret effects are omitted, which are consequences between the high pressure and temperature gradients on the viscosity. These detailed descriptions have been fully articulated by the Onsager reciprocal theorem [19, 29]. Again, by neglecting the ambipolar diffusion from the ionized air, only the binary diffusion coefficient is necessary for the high-enthalpy air,

$$ {d}_{i,j}=1.858\times {10}^{-3}\sqrt{T^3\left({M}_i+{M}_j\right)/{M}_i{M}_j}/{\sigma}_{i,j}^2{\Omega}^{\left(1,1\right)} $$

(46)

The classic molecular thermal conduction coefficients for the monatomic and polyatomic molecules are known;

$$ {\displaystyle \begin{array}{l}{\kappa}_{i,m}=1.989\times {10}^{-4}\sqrt{T/{M}_i}/{\sigma}_i^2{\Omega}^{\left(2,2\right)}\\ {}{\kappa}_{i,p}=2.519\times {10}^{-4}\sqrt{T/{M}_i}\in /{\sigma}_i^2{\Omega}^{\left(2,2\right)}\end{array}} $$

(47)

The intermolecular potentials vary from one gas species to another and according to their respective thermodynamic states, but the collision cross section of air mixture generally lies in between the range from 2 to 4 Angstroms (10^{− 8} cm). The typical volume of a gas molecule is approximately around 10^{− 23} per cubic centimeter at the standard atmospheric condition. Therefore, the intermolecular forces are really the short range forces acting on colliding particles. The most widely adopted molecular potential is the classic Lennard-Jones 6–12 potential [2],

$$ \varphi (r)=4{\varphi}_o\left[{\left(\sigma /r\right)}^{12}-{\left(\sigma /r\right)}^6\right] $$

(48)

This potential function has been universally applied for gas species up to a temperature of 2000 k. For a gas in a bounded state or only when the vibrational excitation is dominated, the Morse potential has been used [19, 48, 50],

$$ \varphi (r)={\varphi}_{eq}\left\{{e}^{\left[-2\left(c/\sigma \right)\left(r-{r}_{eq}\right)\right]}-2{e}^{\left[-2\left(c/\sigma \right)\left(r-{r}_{eq}\right)\right]}\right\};\kern0.5em {\varphi}_{eq}=\varphi \left({r}_{eq}\right) $$

(49)

In the dissociation temperature range, an exponential repulsive model is often adopted.

$$ \varphi (r)={\varphi}_o{e}^{-r/\rho } $$

(50)

For the ionized gas, the collisions mainly involve the ion-ion and electron-electron collision, and the screened Coulomb potential is utilized for the collision integration calculations [50];

$$ \varphi (r)=\left({e}^2/r\right){e}^{-r/d};\kern0.5em d=\left(\varepsilon \kappa T/{e}^2n\right) $$

(51)

The required collision integrals and cross sections for the high enthalpy hypersonic flow have been obtained by the Lenard-Jones potential for non-polarized molecules, and by a polarizability model for ion-neutral non-resonant collisions [50]. The transport properties generated from the gas kinetic theory for individual molecular species are used to determine the global property for a gas mixture by the Wilke’s mixing rule [29, 49].

The viscosity and thermal conductivity coefficients of ionized hypersonic flows are depicted by Figs. 16 and 17 respectively. Figure 16 presents the high-temperature air viscosity coefficient distributions covering a limited temperature range up to 20,000 k, and includes some bench-mark results by other formulations for the purpose of comparison. An inflection point of the viscosity coefficient in temperature appears at a value beyond 10,000 k, and decreases as the air temperature increases further. From the shared knowledge, the viscosity of the ionized air will gradually increase again beyond the displayed temperature range [51].

Figure 17 depicts the thermal conductivity coefficients of high temperature air plasma spanning the temperature range up to 30,000 k [52]. All the presented numerical results include a wide variation of physics-based models and exhibit a considerable amount of disparities from each other. Although there is substantial progress in the kinetic theory of gases for evaluating transport properties by the rational intermolecular potentials, yet the calculated transport coefficients still yield an unacceptable difference of 16.3% to 24.2% in viscosity and thermal conductivity for high-temperature ionized air. The conclusion is reached by examining the transport properties using different sets of collision integrals or cross sections in the temperature range from 300 to 30,000 k [19, 29].

The initial values and boundary conditions have been well established for the hypersonic interdisciplinary equations [19, 22, 23, 29]. The only implicit assumption for the species concentrations on the solid surface is either the non-catalytic or fully catalytic model. However, the mostly widely used condition is simply letting the chemical reaction be controlled by the local thermodynamic condition which is identical to the immediately adjacent domain. In other words, the species concentration on the surface is assumed to be the same as the oncoming stream. Sometimes, such a boundary condition is referred to as super-catalytic; technically, it is assumed that the solid surface is fully catalytic. The other limiting condition is the non-catalytic condition in which the gradient of the species concentration is set to zero.

In describing an ablating surface, the majority of research efforts have been concentrated on the interface boundary conditions either from physical observations or by unique insights. However, by the Reynolds transport theory, the formulation on the interface boundary conditions has been derived from the eigenvector structure of the governing equations [19]. The ejection velocity of the pyrolysis gas and the release vapor rate of the sublimated material on the ablating surface are required as the input. Therefore the details of gaseous motion through the porous ablating material must be relegated to the research results from a specific ablative material [53, 54].

The interdisciplinary conservation equations, Eqs. (36) through (40), have been routinely applied for simulating the high-enthalpy hypersonic flows. The processes for evaluating the ionized air, nonequilibrium quantum chemical-physics, and radiative energy transfer have been developed by physics-based models. However, the shortcomings of these state-of-the-art models for the nonequilibrium high-temperature chemical kinetic models are known to be woefully limited in their physical fidelity. Nevertheless, the interdisciplinary formulation has been successfully applied to a large group of interplanetary and low-earth reentry simulations, supersonic combustion for Scramjet, and the ion engines performance for engineering purpose [18, 19, 22, 23, 27, 29].

The typical results by the hypersonic interdisciplinary equations with the nonequilibrium quantum chemical-physics models and the transport properties that are derived from the gas kinetic theory are depicted in Figs. 18 and 19. Figure 18 presents the computational results of the entire flow field of the FIRE II space vehicle on a multi-block grid system, solving the radiation rate equation with the spectral line-by-line technique for the emission and absorption coefficients [29, 53]. These simulations duplicate the reentry trajectory at the time of 1636 s after an initial recording point. At this instance, the vehicle travels at a speed of 11.31 km/s, the free stream density and temperature are *ρ*_{∞} = 0.857 × 10^{7}*g*/*cm*^{3}, and *T*_{∞} = 210*k*, and the space vehicle surface temperature is assigned a value of 810 k.

On the left-hand-side of the composite figure displays the translational temperature of the air and velocity contours over the space vehicle; the maximum temperature over 25,400 k is confined in the stagnation point region and followed by a rapid expansion over the juncture of the forebody, the afterbody, and continuing into the wake. The vibrational temperatures of nitrogen (T_{v1}) and oxygen (T_{v2}) are given by the right-hand-side of the figure. Both molecular species attain a maximum vibrational temperature of 12,000 k in the thin bow shock layer, but there are distinctive differences between vibrational temperatures of these two species downstream. The vibrational temperature of oxygen molecules decreases continuously toward the wake similar to the translational temperature, but at a 100 times greater value. The vibrational temperature of nitrogen molecules, however, reveals a second peak temperature region in the near wake due to the recompression and nonequilibrium chemical reaction.

The computational simulations of the Stardust sample return capsule generated by the interdisciplinary hypersonic governing equations are exhibited in Fig. 19 for the complete flowfield and the chemical composition in the shock layer. The phenolic impregnated carbon ablator (PIC) has been installed on the front face of the capsule for thermal protection, which increases the complexity for the physical-based computational simulations. Many two and three-dimensional engineering analyses haven been performed for the fastest man-made object in a hypersonic earth reentry [22, 23, 25, 29, 53, 54]. From the trajectory analysis, the reentry vehicle glides at a shallow oscillating angle around eight degrees, which requires the computational simulation conducting in three-dimensional space. The computational results are presented at an altitude of 59.8 km with the reentry velocity of 11.137 km/s, and the ambient conditions are characterized by a density of *ρ*_{∞} = 2.34 × 10^{−7}*g*/*cm*^{3} and a temperature of *T*_{∞} = 238.5*k*.

The velocity topology around the Stardust capsule is described by streamline traces on the right-hand-side of the composite presentation in Fig. 19. The extremely thin shock layer over the forebody of the capsule stands out. The rapid expansions are also noted at the junction of the forebody and afterbody, as well as, in the base region of the capsule. The vortical formation in the near wake region transits from two counter-flowing recirculation zones and merges into a biased far-wake structure as the capsule traveling along the trajectory. The basic flow structure follows the rule of the classic aerodynamic theory, however, the air composition in the shock layer and in the near wake is drastically different from the perfect gas model. The air composition in the shock layer indicates the profound and rapid changes by nonequilibrium chemical reactions. In the base region, the air composition by mixing and recombining chemical reactions has affected the local heat transfer drastically.

The air composition within the shock layer and along the stagnation streamline is depicted on the right-hand-side of Fig. 19. The maximum translational temperature attains a value as high as 22,750 k and drops to a value of 3250 k in the far wake region. The composition is displayed in the molar fraction of the air mixture, and a total of eighteen species of the ablating gas are presented. The presence of free electrons and a nearly equal amount of positively charged oxygen and nitrogen atoms reveal the air is ionized. The molar fraction of the oxygen molecules is nearly completely depleted in the shock layer and the species of C_{3}, HN, and C^{+} are presented as a tracing amount of 10^{− 4} or lower. The rapid recombination and the ablating reactions of different species are noticed in the ablating zone (x < 0.2 cm).

In spite of these engineering accomplishments, only a limited amount of results from the hypersonic interdisciplinary governing equations have been verified by comparing with the flight and the ground-based experimental data. For the latter, an exemplified collection of the high-enthalpy hypersonic flows experimental facilities and their measurement techniques can be found in reference [55,56,57]. Nevertheless, the simulated computational results still can duplicate the essential physics to be retained for the engineering purpose. A substantial amount of basic research efforts are urgently and critically needed to build on the basic knowledge of a lasting scientific value for the high enthalpy hypersonic flows.

Some promising basic research activities have been initiated to address the pacing item in hypersonic flow research by combining the theoretical knowledge with the support from the high performance computational technology [18, 19]. The *ab initio* or the first principle approach offers the opportunity for a scientific breakthrough, which is based on the Born-Oppenheimer approximation by separating the wave function of the nuclei from the electron [58]. These solutions are focused on the energy distribution by quantum mechanics. The Schrödinger equation is first solved for a single electron then adds the internucleus repulsive electronic energy to get the total internal energy of a molecule. In order to examine these phenomena, the entire potential energy surfaces (PES) must be constructed for chemical-physics interactions [58, 59]. In fact, the PES is a plot of the collective nuclei and electronic energy versus the molecule geometric coordinates*.* In other words, the PES provides a visualizing and understanding of the relationship between potential energy and molecular geometry. Therefore, the *ab initio* approach actually determines the structure, the energy of molecules, and the transition states involving chemical reactions. Impressive accomplishments have been achieved for constructing the critical potential surface, which enables the prediction of the critical point for transition [18, 58,59,60].

An example of the potential energy surface is displayed by Fig. 20. The contours of the PES are constructed from 3326 *ab initio* solutions by solving the Schrödinger equation for a N-N bond nitrogen molecule that shows the double barriers of two transition states connected by a shallow energy well [59]. The two barriers are symmetric with respect to the interchange of two nitrogen atoms, and the bond distance is less than 3 Angstrom, and the calculated bond angle is fixed at 119^{o}. The inset of this figure illustrates the partition of atoms A and B either in a heteronuclear or a homonuclear molecule [58]. These atoms are separated by a zero-flux surface S that passes through the critical saddle point on the bond path C. According to the critical point theory by Eyring, it’s the location for transition to occur [60]. This type of incisive knowledge is absolutely crucial for advancing our knowledge by basic research for high-enthalpy hypersonic flows.