High temperature non-equilibrium flow characteristics of impinging shock/flat-plate turbulent boundary layer interaction at Mach 8.42

CNEG

Chemical non-equilibrium gas

c

Mole fraction

DSS

Deflection separation shock

EF

Expansion fan

H

Specific total enthalpy (J/kg)

h₀

Total enthalpy (J/kg)

IS/FTBLI

Impinging shock/flat-plate turbulent boundary layer interaction

M_s

Molecular weight of species (g/mol)

N_m

Number of molecular species

Re

Reynolds number

RS

Reattachment shock

Sc

Schmidt number

SS

Separation shock

SZ

Separation zone

TCNEG

Thermochemical non-equilibrium gas

TPG

Thermally perfect gas

$T ref$

Reference temperature (K)

$T tr$

Translational-rotational temperature (K)

$T v$

Vibrational temperature (K)

The impinging shock/flat-plate turbulent boundary layer interaction (IS/FTBLI) is commonly observed in both internal and external flows of high-speed aircraft, such as the ramjet inlet,^1,2 the isolator,^3,4 and the two-stage-to-orbit (TSTO) reusable launch vehicle.⁵ When an oblique shock impinges on a flat-plate, a high adverse pressure gradient is generated to cause the flows inside the boundary layer to separate. The large separation bubble can result in serious issues, such as the inlet unstart,^6,7 low total pressure,⁸ buzz,⁹ and combustion instability,^10–12 in the internal flow channel of an air-breathing propulsion system. Moreover, the flow separation also contributes to increased flow instability,¹³ aerodynamic resistance, and thermal load on the aircraft's surface,¹⁴ necessitating enhanced requirements for the thermal protection mechanism of the aircraft. With increasing flight altitude and speed, the fluid dynamics problems involving shock wave, boundary layer, and the interaction between them become more complex, especially in extremely high-temperature flight environments.¹⁵ Therefore, it is crucial to allocate adequate attention to corresponding research in this field to understand the thorny matter.

To address the complex flow problem in the inner channel of an air-breathing engine working at the flight Mach number of 6, extensive research works^16–20 have been conducted on the interactions of the cowl shock wave or Mach 2 to 3 supersonic shock wave with the turbulent boundary layer. For instance, Zhang et al.¹⁶ performed numerical investigations to analyze the effect of expansion waves generated by the shoulder on the cowl shock/boundary layer interaction, where four types of interaction mechanisms between the expansion wave and shock wave were observed. Tong et al.¹⁷ utilized high-precision numerical simulations to study the interference between incident shocks with varying intensities and the turbulent boundary layer at Mach 2.25. They presented the characteristics of turbulence evolution during the recovery process. Due to actual engineering requirements, the influences of various wall conditions on IS/FTBLI appearing in supersonic flows were researched by Zhu et al.¹⁸ and Bernardini et al.¹⁹ With the aid of a wind tunnel, Li et al.²⁰ focused on the flow mechanism of the single and double incident shock/turbulent boundary layer interaction at Mach 2.73, where they analyzed the similarities and differences between the two kinds of interaction. On the other hand, various flow control methods were developed to effectively mitigate the flow problems arising from the supersonic IS/FTBLI. Active flow control techniques, such as the jet vortex generator²¹ and pulsed plasma actuators,^22,23 can adjust the control effect under different flow conditions. Conversely, passive flow control methods, like the boundary layer suction,²⁴ micro vortex generator,²⁵ and two-dimensional bump,²⁶ can suppress the flow separation in some specific flow fields. In the aforementioned studies, the total temperature of the incoming flow is relatively low. Hence, the calorically or thermally perfect gas can be used in the numerical simulation of the supersonic shock/boundary layer interaction.^{1–4,16–23}

Since the stagnation temperature of the free stream is larger than 2500 K in the flow of air-breathing aircraft above Mach 8, vibration relaxations and dissociation reactions of molecules can compete with the faster flow velocity in the channel, resulting in the vibrational and chemical non-equilibrium effects. These above non-equilibrium effects are also called high-temperature or thermochemical non-equilibrium effects,²⁷ and the calorically or thermally perfect gas assumption is no longer applicable here. Recently, some effort has been devoted to study the shock wave/flat-plate boundary layer interaction in the high Mach number or hypersonic flows. For example, based on the T4 Stalker Tube, Chang et al.²⁸ conducted an experiment on Mach 7 shock/flat-plate boundary layer in a flight-representative environment and studied the impact of the impinging shock strength and wall temperature on the boundary layer flow separation. The results from Chang et al.²⁸ revealed that stronger impinging shock, higher wall temperature, and lower unit Reynolds number would enlarge the extent of the flow separation. However, due to the strong shock wave, there are significant discrepancies compared to length scaling correlations for separation bubbles resulting from supersonic impinging shock/boundary layer interactions and hypersonic compression corner flows. It is important to note that in the experimental work by Chang et al.,²⁸ the maximum values of the stagnation temperature and wall temperature were 2334 and 675 K, respectively, indicating that high-temperature effects were not prominent. Nevertheless, this work provides valuable insights into the key physical behavior of the flow separation formed by hypersonic impinging shock/flat-plate boundary layer interactions.

The thermal–chemical non-equilibrium effects of the hypersonic shock/laminar boundary-layer interaction in the flows over the double wedge^29,30 and double cone^31,32 have been investigated, whereas those of hypersonic IS/FTBLI flows have received limited attention. Recently, a couple of recent works utilized numerical simulations to study the real gas or chemical non-equilibrium effects of the hypersonic IS/FTBLI.^33,34 Brown³³ studied the uncertainty caused by real gas effects in Mach 7 and Mach 14 oblique shock/turbulent boundary layer interaction. Their findings revealed that, compared to the perfect gas, utilizing real gas in simulations can predict a smaller extent of the separation zone (SZ) and a higher static pressure peak value. The dissociation reaction of oxygen is exceptionally weak, and the mass fraction of oxygen atoms does not exceed 0.2%. With the help of a new solver CHARLIE, Volpiani³⁴ conducted the direct numerical simulation (DNS) of the Mach 5.6 impinging shock/boundary layer interaction in the chemical non-equilibrium effect while maintaining the plate surface at an isothermal temperature of 3042 K. The numerical results indicated that chemical reactions can change the flow features inside flows of the two-dimension laminar and three-dimensional turbulent boundary layer, and most of the chemical reactions occur after the interaction. The turbulent activity after shock reflection is more susceptible to chemical reactions.

Overall, the numerical analysis conducted by Brown³³ and Volpiani³⁴ focused on the influences of the real gas effect or chemical non-equilibrium effect on the hypersonic IS/FTBLI. However, the thermal non-equilibrium effect was disregarded in the past studies.^33,34 Since the high temperature aft of the stronger shock wave and inside the boundary layer will cause the out-of-equilibrium relaxation process of the molecular vibrational energy, the vibrational relaxations and chemical reactions should be considered to investigate the interference flow between the hypersonic shock and flat-plate turbulent boundary layer. Recent research by Chen et al.^35,36 addressed the thermal–chemical non-equilibrium effects on boundary layer stabilities in hypersonic and high-enthalpy flows. Passiatore et al.³⁷ showcased the high-fidelity flow patterns of the turbulent boundary layer in thermally out-of-equilibrium conditions at Mach 12.48 using DNS; their findings revealed that the predominance of vibrational non-equilibrium arises primarily from molecular nitrogen and is further maintained by the turbulent mixing. Moreover, Yu et al. established a multi-physics model to study the plasma flow of the Radio Attenuation Measurement-C-II vehicle in the thermochemical non-equilibrium effects³⁸ and the catalytic wall effect on the plasma sheath.³⁹ In one of our previous studies,⁴⁰ the interaction of the cowl shock with the ramp turbulent boundary layer has been observed in the high-temperature non-equilibrium flow field of a two-dimensional scramjet inlet at Mach 12, whereas no detailed discussion was made on flow interaction. Generally, little attention was paid to the thermal (or vibrational) and chemical non-equilibrium effects in the hypersonic IS/FTBLI flows, compared to the chemical non-equilibrium effect.^33,34

To bridge the aforementioned research gap, the current work intends to investigate thermal and chemical non-equilibrium flow characteristics in hypersonic IS/FTBLI flows, which have rarely been explored in previous studies. We conducted the numerical simulations focusing on the interaction between a Mach 8.2 oblique shock and a flat-plate turbulent boundary layer. The free-stream parameters ahead of the flat-plate align with the inviscid flow condition following the compression of the curved leading-edge shock, as outlined in Ref. 40. The overall structure of the remaining part of the article is as follows: first, the physical model, calculation details, validations, and mesh-convergence study are described in Sec. II. Then, the numerical results and discussions of the high-temperature flow fields of IS/FTBLI at Mach 8.42 are presented in Sec. III. Finally, the vital conclusions of the investigation are outlined in Sec. IV.

Figure 1 shows a schematic diagram of IS/FTBLI in the numerical experiment. The length of the flat-plate is L1 = 2.0 m. The flow deflection angle of the shock generator is 10°, the same as the total deflection angle of the external compression section of the Mach 12 two-dimensional inlet reported in Ref. 40. The horizontal and vertical distances between the leading edge of the shock generator and the leading edge of the flat-plate are L2 and H1, respectively. The vertical distance from the tail of the shock generator to flat-plate is H2.

The computational domain of the hypersonic IS/FTBLI flows is illustrated in Fig. 2. The boundary of the model is labeled as 1 being the inlet, 2 representing the wall surface, and 3 being the outlet; these three boundaries correspond to the blue, red, and green solid lines, respectively. The condition of the far field is employed at the inlet boundary, and the specific flow parameters are shown in Table I (the length of the flat-plate is the characteristic length in the dimensionless calculation). Assuming that all surfaces are adiabatic and viscous (non-slip), the present study will not consider complex phenomena, such as the thermal deformation and erosion. The outlet boundary is set to the condition of supersonic outflow, and all variables are extrapolated from the interior.

An enlarged view of the computational mesh near the interaction of the impinging shock wave with the flat-plate boundary layer is presented in Fig. 3. The grids in this region are structured as multi-block meshes. To meet the requirements of the shear stress transport (SST) k-ω turbulence models for the computational mesh, the grids on surfaces of the flat-plate and wedge are locally refined, with a first layer height of 1 × 10⁻⁶ m and a cell growth ratio of 1.15. This refinement can ensure that y⁺ values on all walls are less than 1.0. The definition of y⁺ is y⁺ = (ρu_τy_c)/μ, where u_τ and y_c are the friction velocity and the normal distance from the first off-wall centroid to the wall, respectively. The entire computational domain is discretized using 1950 × 300 nodes, with a total number of structured grids of approximately 5 × 10⁵.

For the high temperature and high speed flows in this study, a two-temperature model,⁴¹ including the translational-rotational temperature T_tr and the vibrational temperature T_v, is used to simulate the vibrational non-equilibrium processes of gas molecules (N₂, O₂, and NO). Generally, a single static temperature T is used to represent all the energetic modes for the thermal equilibrium gas model. Hence, to better compare, the translational-rotational temperature T_tr is same the static temperature T in this work, which is same to Refs. 29, 40, and 42. Moreover, the kinetics model of the chemical reaction⁴³ involving five species (N₂, O₂, NO, N, and O) and five chemical reaction processes is utilized to simulate the chemical non-equilibrium reactions in the air flows. A geometric-averaged temperature T_ave, expressed as T_ave = T_tr^α * T_v^1-α, is employed to model the coupling influence of the vibrational processes with chemical reactions and govern the chemical reaction rate computed by the Arrhenius formula. Notably, the value of α is 0.5 in dissociation reactions, whereas it should be equal 1 in other reactions (recombination, displacement). The two-dimensional compressible Navier–Stokes conservation equations in Cartesian coordinates are given as follows:

In Eqs. (3) and (4), e_v represents the vibrational energy of mixture. Q_t-v denotes the translational and vibrational energy relaxation term, which can be estimated by the Landau–Teller model.⁴⁴  E denotes the specific total energy and is expressed as $E = e + 1 2 u i × u i$⁠, e is the mixture internal energy and described as $e = ∑ s = 1 N s e s , e s = e t s + e r s + e v s + e ref s$⁠, $e t s$⁠, $e r s$⁠, _and $e v s$ denote the species energy in translational, rotational, and vibrational states. $e ref s$ represents the species formation energy at the reference temperature of 298.16 K. q_tr,i and q_ve,i represent the heat flux in the translational-rotational and vibrational modes, which are calculated by adopting Fourier's law [ $q t r , i = λ t r × ( ∂ T t r / ∂ x i )$⁠, $q v , i = λ v e × ( ∂ T v / ∂ x i )$].

For molecules, translational and rotational energies can be fully excited, each species internal energy is calculated as $e t s = 3 2 R s T , e r s = R s T , e v s = R s θ s exp ( θ s / T v ) − 1$⁠. Here, θ_s denotes the vibrational characteristic temperature of every molecular species and the values θ_s for O₂, N₂, and NO are 2273, 3393, and 2739 K, respectively. Yet, the internal energy for each atom species is calculated as $e t s = 3 2 R s T , e r s = 0 , e v s = 0$⁠.

The calculation in this study incorporates various complex phenomena, such as vibration relaxations, chemical reactions, strong shock waves, and turbulence flow. These factors can introduce a steeper problem than the calculation performed by Passiatore et al.,³⁷ which significantly contribute to the computational complexity of high-precision numerical simulations. Considering the calculation efficiency, the Reynolds averaged Navier–Stokes (RANS) method with the k-ω shear stress transport (SST) model introduced by Menter⁴⁵ is employed to model turbulence flows in the vicinity of the flat-plate surface. Additionally, to account for compressible effects, the turbulence model is augmented with the compressible effect correction proposed by Sarkar.⁴⁶ 

In this study, the three thermochemical models are applied for calculation, and their main features are shown in Table II. (1) Thermochemical non-equilibrium gas (TCNEG) is achieved using the standard model of vibrational relaxation⁴¹ and chemical reaction.⁴³ (2) Based on the TCNEG model, decreasing the vibrational relaxation time in the TCNEG model to infinitesimal ensures that the thermal state is in equilibrium, which can simulate chemical non-equilibrium gas (CNEG) flows. (3) Based on the CNEG model, thermally perfect gas (TPG) is obtained by decreasing chemical reaction rates to infinitesimal and often employed in predicting Mach 3 to 6 flows.^1,22,23 The CNEG and TPG model were compared in Refs. 33 and 34.

The compressible Reynolds averaged Navier–Stokes (RANS) equations mentioned above are solved by means of the finite volume method. The bulk viscosity and thermal conductivity are directly computed using Gupta-Yos' collision cross-section based on species and bulk viscosity fits.⁴⁷ A second-order point-implicit time scheme⁴⁸ with multi-grid acceleration and double time steps is employed to handle the coupling source terms of vibration energy, chemical reactions, and turbulence flow. Harten–Lax–van Leer contact (HLLC),⁴⁹ a nonlinear Riemann solver, is utilized to provide the upwind flux information. To prevent the introduction of new maxima and minima in the reconstruction, a multi-dimensional total variation diminishing (TVD)⁵⁰ approach with the continuous-type limiter is employed. Flow parameters at the inflow boundary, such as static pressure, static temperature, x-direction velocity, vibration temperature, species mass fractions, and turbulence quantities, are utilized for initializing all numerical calculations. The aforementioned transient state RANS equations described above are solved using a time-marching method, and the flow field obtained through numerical calculation is considered to be in a steady state when the residuals of each equation are less than 1.0 × 10⁻⁵. Such an calculation approach has been successfully employed to investigate the thermochemical non-equilibrium flows of a double wedge²⁹ in an environment with stagnation enthalpy of 8.0 MJ/kg, as well as a Mach 12 two-dimensional inlet³⁹ and a Mach 12 inward turning inlet.⁵¹ 

First, to validate the suitability of the current numerical code for thermochemical non-equilibrium flows, an evaluation is conducted using the results of a high enthalpy double cone experiment conducted by Holden et al.⁵² The inflow conditions for the double cone wind test are provided in Table III, with oxygen and nitrogen mass fractions of 0.235 and 0.765, respectively. Since the Reynolds number based on the total length of the double cone is 2.13 × 10⁴, the entire flow field is assumed to be laminar. The computational mesh and boundary conditions for the double cone are depicted in Fig. 4, and the x-axis is set as an axisymmetric condition. All solid walls are regarded as viscous, non-catalytic and maintained at a constant temperature of 300 K. Figure 4(b) illustrates the grid refinement near the wall, with a first layer grid height of 1.0 × 10⁻⁶ m near the cone surfaces. The total number of all structured grids amounts to 5.0 × 10⁵.

As depicted in Fig. 5, it is observed that the static pressure increases rapidly, whereas the heat flux decreases quickly when the flow initiates separation. The peak values of the wall parameters are located at the position where the transmitted shock impinges on the second cone. In Fig. 5(a), the onset of the separation zone predicted by the TCNEG is closest the experimental result,⁵² but the results obtained from the CNEG deviate the most from the experiment. In addition, the value and position of the static pressure peak in the TCNEG flow are closer to the experiment than the other two results. As shown in Fig. 5(b), the TPG model predicts a peak value of the heat flux that largely deviates from the results of the experimental, TCNEG, and CNEG.

Table IV shows the deviation of the CNEG and TPG models from the TCNEG in the prediction of the double cone flow. Although the TPG model predicts the onset of the separation zone that is very close to the TCNEG's result, the relative errors of the static pressure and heat flux peak between the two are as high as 17.59% and 52.91%, respectively. Moreover, the differences in the flow separation and peak value between the CNEG and TCNEG models also are non-ignorable.

In generally, compared with the thermochemical equilibrium model (CNEG and TPG), the static pressure and heat flux on the cone surfaces, calculated using the TCNEG model, better match with the experimental results.⁵² Therefore, it can be argued that the aforementioned numerical method can accurately predict thermochemical non-equilibrium flows.

Then, a 34° wedge compression corner experiment⁵³ at Mach 9.22 is utilized to check whether the TCNEG model coupled with the k-ω SST model in this study can effectively predict the high Mach number turbulence flow. The free stream conditions of the test are provided in Table V, and the computational mesh and specific boundary conditions are illustrated in Fig. 6. Additionally, Fig. 7 presents the density gradient in the compression corner flow field. The leading edge shock resulting from the boundary layer on the horizontal wall intersects with the recompression shock near the ramp. The flow separation is found at the compression corner.

Comparisons between the wall parameters calculated using the TCNEG model coupled with the k-ω SST model and experimental results⁵³ are shown in Fig. 8. The wall non-dimensional parameters, including the static pressure and heat flux, can be obtained by using P_∞ and Q_∞, respectively. Figure 8(a) indicates that the pressure rise distribution obtained from the current computation at the starting position of flow separation aligns well with the experimental data.⁵³ However, numerical results reported by Zhang et al.,⁵⁴ Tu et al.,⁵⁵ and Jiang et al.⁵⁶ fail to estimate the length of separation accurately. Moreover, the peak static pressure value in the current matches the experimental data, whereas the above-reported results are 28.6% lower than the experimental value. In Fig. 8(b), the heat flux distribution obtained from our TCNEG coupled with the k-ω SST model shows no significant deviation from the experimental trend. Conversely, previous numerical works^54,56 fail to accurately capture the heat flux peak distribution. Therefore, it can be concluded that the numerical approach employed in this study better predicts the separation region in high Mach number turbulence flows compared to the reported numerical studies. Based on these validations, we are confident in utilizing the present numerical codes the TCNEG model coupled with the k-ω SST model with for the present investigation.

Furthermore, a mesh independent study is conducted to examine the sensitivity of the calculation results to different grids for the IS/FTBLI at Mach 8.42. Table VI presents the details of the different grids, while maintaining the grid growth ratio of 1.15 constant. In addition, the TCNEG model coupled with the k-ω SST turbulence model is employed to perform the numerical calculations. Figure 9 illustrates the static pressure distribution at the flat-plate surface predicted using three different calculation meshes. The wall static pressure distribution of grid2 is identical to that of grid3 but significantly deviates from grid1. Particularly in the region of the pressure peak, the static pressure computed using grid2 is 3.32% higher compared to grid1, and its peak location is closer to the downstream region. Hence, the computed result obtained from grid2 is considered convergent and suitable for further analysis in the subsequent study.

The turbulence flows in the three numerical results are all simulated by using the k-ω SST model, and the only difference between the three numerical calculations is the thermochemical model. Thus, the numerical result of the TCNEG model coupled with the k-ω SST model is represented using the TCNEG in the comparative analysis. Similarly, the other two results are described using the CNEG and TPG.

The flow parameters of the x = 1.25 m section in front upstream of the interaction of the impinging shock with the boundary layer are first analyzed. The specific location of this section is presented in Fig. 10. As shown in Fig. 11, the solid black line, dashed red line, and dotted green line represent the TCNEG, CNEG, and TPG, respectively. The dimensionless form of the static pressure and x-direction velocity is obtained by using T_∞ and U_∞. The translational-rotational energy partially remains unconverted into the vibrational energy in the TCNEG flow. In the thermal equilibrium flow, much more translational-rotational energy transferred to the vibrational mode is required to achieve the thermal equilibrium. Hence, the translational-rotational energy in the TCNEG flow is more than the thermal equilibrium flow. Macroscopically, the TCNEG model predicts a higher static temperature (T/T_∞) than thermal equilibrium gas (CNEG and TPG), as observed in Fig. 11(a). On the other hand, the CNEG model featuring endothermic dissociation reactions predicts the lower static temperature than the TPG model. The static temperature in TCNEG flow exceeds that in CNEG and TPG flows by 17.50% and 5.61%, respectively. As static temperature increases inside the boundary layer, the viscosity coefficient and kinetic energy loss also increase, leading to a lower x-direction velocity. Consequently, the x-direction velocity computed by the TCNEG is the lowest among these three results, as shown in Fig. 11(b). By comparing with the CNEG model, the x-direction velocity distribution inside the boundary layer obtained from the TCNEG model is closer to that of the TPG model, particularly in the outer layer of the boundary layer. This phenomenon occurs because the thermal equilibrium state and dissociation reactions in CNEG flow contribute to reducing the static temperature and increasing the flow velocity.

Table VII presents the nominal thickness of boundary layer δ (δ is the boundary-layer thickness at 99% of the edge velocity) near the flat-plate at x = 1.25 m section, and its dimensionless form is calculated by using H1. The TCNEG flow exhibits the thickest turbulent boundary layer on the flat-plate surface, whereas the CNEG flow has the thinnest boundary layer (with the fullest velocity-profile).

Figure 12 shows the distributions of static temperature, Mach number, specific heat ratio (Gamma), and x-direction velocity along the x = 1.25 m section downstream of the impinging shock wave. In Fig. 12(a), the boundary layer on the wedge surface has a thickness that is thickest in the TCNEG flow, but thinnest in the CNEG flow, which is similar to the characteristics of the boundary layer on a flat-plate. It is important to note that the specific heat ratio calculated by the TCNEG model experiences a slight drop after the impinging shock wave, while it plunges in the other two results (CNEG and TPG). Compared to the results of the other gas models, the TCNEG model computes a higher value of the specific heat ratio due to the lower vibrational temperature. The reasons for this difference can be seen in the  Appendix A. Furthermore, the differences in the x-direction velocity variations predicted by the three gas models are small in both viscous and inviscid flow regions. Based on the relation between the sound speed c and Mach number Ma (c² = Gamma·R·T and Ma = v/c, where R denotes the gas constant), the Mach number within the boundary layer predicted by the TCNEG model is the lowest, but that in the CNEG flow is the highest, as shown in Fig. 12(b).

Due to the specific heat ratio with a higher value and the thicker boundary layer, the larger impinging shock angle predicted from the TCNEG model results in a shock position at the x = 1.25 m section that differs by approximately 2.81% from the other two gases (CNEG and TPG). As the downstream flow parameters behind the impinging shock wave mainly depend on the shock wave angle when the inflow conditions are consistent, changes in the parameters following shock wave compression require further analysis. In the mainstream area downstream of the impinging shock, as illustrated in Fig. 12(b), the distributions of the static temperature and Mach number computed by the two thermal equilibrium gases (CNEG and TPG) are identical due to the comparable position of the impinging shock. However, in the same zone, the static temperature predicted by the TCNEG model is roughly 4.84% higher than that predicted by the thermal equilibrium gases, and its Mach number is about 4.24% lower than that of the thermal equilibrium gases. It is, thus, evident that the thermochemical non-equilibrium effects significantly influence the flow parameters behind the oblique shock wave in the turbulent flow of the wedge.

Figure 13 exhibits the contours of the static temperature predicted by the three thermochemical models. The high temperature mainly exists in the turbulent boundary layer near the wall. From Figs. 13(a)–13(c), the static temperatures within the separation bubble estimated by the TCNEG and CNEG models are lower than the TPG models due to the dissociation reactions. For the TCNEG flow shown in Fig. 13(c), the static temperature inside the separation zone exceeds 4000 K, which is sufficient to induce molecular vibration. As shown in Fig. 13(d), as a result of vibrational non-equilibrium relaxations, the vibrational temperature behind the impinging shock increases slowly, as opposed to the abrupt increase observed in static temperature [Fig. 13(c)]. The lower the flow velocity, the longer the characteristic time in the local flow. There is enough time for more vibrational energy to be excited in the separation zone, which is characterized by the low flow velocity. Thus, the highest vibrational temperature is found inside the separation bubble, as shown in Fig. 13(d).

From a macroscopic perspective, the energy exchange between translational-rotational and vibrational modes can be quantified by the corresponding temperature variations, as mentioned in Ref. 37. Figure 14 displays the contour of the discrepancy between the static and vibrational temperatures within the TCNEG flow field. Vibrational non-equilibrium effects exhibit a significant amplification from the mainstream toward the outer layer of the boundary layer, as well as after the compression of the impinging shock, separation shock (SS), and reattachment shock (RS). Importantly, the most pronounced vibrational non-equilibrium effects occur following the intersection of the impinging shock with the separation shock and within the convergence region of compression waves during flow reattachment.

Within the expansion fan (EF) region, the static temperature experiences a rapid decrease while the vibrational temperature undergoes minor changes. Consequently, an over-excited vibrational non-equilibrium effect³² emerges, characterized by a vibrational temperature surpassing the static temperature. This phenomenon occurs due to the dominance of vibrational excitation over translational-rotational energy in this region. Following the compression of the reattachment, the vibrational non-equilibrium effect intensifies again. Subsequently, a low-pressure zone arises near the wall subsequent to the intersection of expansion waves and the reattachment shock, leading to an additional over-excited vibrational non-equilibrium phenomenon.

Figure 15 presents the static temperature, vibrational temperature, and their differences on the vertical cross-section at x = 1.25 m. As shown in Fig. 15(a), between y = 0.025 m and y = 0.012 m, the increase rate of the vibrational temperature is lower than that of the static temperature, meaning that here is a gradual strengthening of vibrational non-equilibrium effects. From y = 0.012 m to y = 0.005 m, the increase rate of the vibrational temperature exceeds that of the static temperature, leading to a gradual reduction in vibrational non-equilibrium effects. In the vicinity of y = 0.005 m, the flow in this region originates from a weak vibrational non-equilibrium region near the leading edge, and the thermal state approaches equilibrium as the flow progresses. The flow between y = 0.005 m and the wall arises from the severe vibrational non-equilibrium region near the wall at the tip of the leading edge. Here, the thermal state requires a longer flow distance to reach equilibrium, leading to a higher static temperature than the vibrational temperature.

As shown in Fig. 15(b), following the impinging shock compression, the static temperature rapidly increases and then remains unchanged in the mainstream region. However, the vibrational temperature gradually increases along the vertical section, gradually weakening the vibrational non-equilibrium effect.

Figure 16 displays the contours of the mass fraction for oxygen and nitrogen molecules in the TCNEG and CNEG flow fields. The mainstream exhibits mass fractions of 0.23 for oxygen and 0.77 for nitrogen. Due to chemical reactions, the molecular content decreases in the high-temperature zone. In the turbulent boundary layer along the shock generator wall, the high temperature induces molecular dissociation. These dissociation reactions continue as the flow enters the expansion fan region. Due to the thicker boundary layer, the dissociation area near the flat-plate is more extensive than that of the shock generator wall. Notably, the mass fraction of oxygen and nitrogen molecules reaches its lowest point in the separation region, indicating the occurrence of intense chemical reactions. This can be attributed to the reduced flow velocity and extended residence time in the separation zone, which enhance the extent of chemical reactions.

A lower mass fraction of the molecular means a stronger chemical reaction. The comparison of Fig. 16(a) with Fig. 16(b) indicates that the chemical reaction of the oxygen molecule predicted by the TCNEG model is weaker than the CNEG model inside the flat-plate boundary layer upstream of the separation zone. However, inside the separation region, the chemical reactions in the TCNEG flow field are more pronounced and occur over a wider area than the CNEG flow field. The reason for the above discrepancy should be that the high vibrational temperature within the separation zone promotes the chemical reaction, and it is shown in Fig. 13(d). Owing to its higher dissociation temperature (4000 K), the reduction in the nitrogen molecule in the TCNEG and CNEG flow fields is less than that of the oxygen molecule, as seen in Figs. 16(c) and 16(d).

Figure 17 presents the distribution of species mass fraction along the x = 1.25 m section in the flat-plate boundary layer. As depicted in Fig. 17(a), the mass fractions of oxygen and nitrogen molecules exhibit a gradual decrease from the outer edge of the boundary layer toward the wall. In Fig. 17(b), the mass fractions of atomic nitrogen and oxygen atoms exhibit an increase, while the mass fraction of nitrogen atom alone is approximately 0.015% in both reaction flow fields, indicating its negligible contribution. Based on the aforementioned observations, the reduction in the mass fraction of oxygen molecule can be attributed to its dissociation reaction. However, the decrease in nitrogen molecule mass fraction is primarily due to the production of nitric oxide rather than the dissociation of the nitrogen molecule. This finding is consistent with the results reported in Ref. 37.

In Fig. 17(a), on the wall of the flat-plate, the mass fractions of oxygen molecules in the TCNEG and CNEG flow fields are about 0.208 and 0.192, respectively. This indicates that 9.57% of oxygen molecules dissociate. The mass fractions of nitrogen molecules in the two flow fields are roughly 0.764 and 0.765, respectively, which are close to the results reported in Ref. 37. In Fig. 17(b), the mass fraction of the oxygen atom along the normal direction in the TCNEG flow field is lower than that in the CNEG flow field. This indicates a less intense dissociation reaction of the oxygen molecule in the former. In contrast, the mass fractions of the nitric oxide in the TCNEG flow field are higher than that in the CNEG flow field, meaning a more intense production reaction of the nitric oxide in the former.

Overall, it can be inferred that chemical non-equilibrium reactions involving oxygen dissociation and the production of atomic nitrogen occur within the boundary layer. However, the dissociation reaction of nitrogen remains frozen.

Figure 18 illustrates the distribution of the mass fraction of oxygen and nitrogen molecules close to the flat-plate wall along the x-direction. In the TCNEG flow field, the mass fraction of the oxygen molecule experiences a rapid decrease after flow separation, followed by a gradual increase in the range of x = 1.355–1.430 m. Due to the influence of the compression wave behind the flow reattachment position, the rise in static temperature enhances the dissociation reaction, leading to a second decrease in the mass fraction of the oxygen molecule. In the subsequent expansion region, the reduction in static temperature weakens the dissociation reaction, increasing in the oxygen molecule content behind x = 1.51 m. The behavior of the nitrogen molecule follows a similar trend to that of the oxygen molecule. Inside the separation region, the minimum mass fractions of the oxygen and nitrogen molecules in the TCNEG flow field are approximately 0.124 and 0.73, respectively. Notably, the content of the oxygen molecule reaches its minimum at x = 1.355 m, which is lower than the valley value behind the reattachment position. This indicates a more severe dissociation reaction of the oxygen molecule within the separation region predicted by the TCNEG.

The changing trend of molecule content in the CNEG flow field is similar to that in the TCNEG flow field. However, there is no noticeable drop in content after the reattachment position. Surprisingly, within the separation region, the content of the oxygen molecule in the TCNEG flow field is higher than that in the CNEG flow field. This is primarily due to the presence of increased vibrational energies, which promote chemical reactions here. Moreover, this phenomenon persists during the compression wave forming a reattachment shock.

Figure 19 shows the numerical schlieren diagram, which represents the density gradient predicted by the three thermochemical gas models. In Fig. 19(a), the interaction between the impinging shock (IS) generated by the 10° wedge and the turbulent boundary layer adjacent to the flat-plate gives rise to the features of the flow separation. These include the formation of a separation zone (SZ), a separation shock (SS), and a reattachment shock (RS). The horizontal portion connected to the wedge corresponds to a zero-gradient exit, resulting in the emergence of a significant expansion fan (EF) region. The separation shock intersects with the impinging shock, deflecting downward and forming a deflected separation shock (DSS). Subsequently, the DSS bends upward due to the interference caused by the expansion fan before ultimately intersecting with the reattachment shock.

Comparing Fig. 19(a) with Figs. 19(b) and 19(c), it is clear that the TCNEG model predicts a larger extent of the separation region than the thermal equilibrium models (CNEG and TPG), and the former is located closer to the upstream zone. This can be attributed to the presence of a thicker flat-plate boundary layer and a stronger impinging shock in the TCNEG flow field. In contrast, the CNEG flow field exhibits a thinner flat-plate boundary layer than the TPG flow field, with a more pronounced endothermic effect resulting from dissociation reactions within the separation region. Consequently, the CNEG model predicts a smaller scale for the separation region. The size of the separation region and the angle of the separation shock are positively correlated, thus influencing the proximity of the intersection between the deflected separation shock and the reattachment shock to the downstream area. Accordingly, the intersection position predicted by the TCNEG model is closest to the downstream zone, whereas the CNEG model yields the closest intersection position to the upstream zone. Interestingly, the deflected separation shock predicted by the CNEG model exhibits an almost horizontal orientation due to the weak separation shock. Based on these findings, it can be argued that the high-temperature non-equilibrium effect noticeably impacts the characteristic shock wave structures within separated flows.

To qualitatively analyze the characteristics of turbulent flow separation in the high-temperature non-equilibrium effects, Fig. 20 presents the distribution of the wall shear stress in the x direction in the vicinity of the interaction region. This parameter is defined as τ-x = (μ_L + μ_T) * (∂u/∂y). The flow separation and reattachment occur at the most upstream and downstream positions where the wall shear stress crosses zero. The TCNEG model, CNEG model, and TPG model predict the separation position at x = 1.303 m, 1.343 m, and 1.320 m, respectively. The reattachment positions are determined to be at x = 1.404 m, 1.414 m, and 1.412 m, respectively.

Upstream of the separation region, the TCNEG flow exhibits the smallest velocity gradient owing to the presence of a thicker boundary layer. This results in a lower wall shear stress than the other models. The TCNEG model predicts the largest impinging shock angle, leading to the closest proximity of its interaction location to the upstream zone. Consequently, compared to CNEG and TPG, the flow separation is more likely to occur and the onset position of the separation bubble is closer to the upstream zone in the TCNEG flow fields. Although the position of the impinging shock predicted by the CNEG model is similar to that of the TPG model, the velocity gradient within the CNEG flow field is smaller. Additionally, the energy-consuming dissociation reactions occur in the CNEG flow field. Consequently, the onset position of the separation bubble predicted by the CNEG model is closer to the downstream zone than the TPG model. Notably, the reattachment position predicted by the TCNEG model differs from the other models. However, the difference in reattachment position between the two thermal equilibrium gas models (CNEG and TPG) is approximately 0.14%.

The stronger impinging shock wave will produce the higher adverse pressure gradient. Moreover, the fluid momentum is lower in the thicker boundary layer. Hence, the TCNEG model predicts the maximum scale of the separation zone among the three flow fields, as shown in Fig. 21. Specifically, the length of the separation region in the TCNEG flow field is approximately 43.52% and 9.54% larger than that in the CNEG and TPG flow fields. The presence of a thermal equilibrium state and dissociation reactions in the CNEG flow helps to decrease the static temperature, which is beneficial for inhibiting the flow separation. However, only dissociation reactions occur in the TCNEG flow, and only the thermal equilibrium state exists in the TPG flow. As a result, the CNEG model estimates the minimum scale of the separation bubble among the three numerical results. By comparing the results from all three models, it can be seen that the size of the separation bubble in the TCNEG flow field is close to that in the TPG flow field but largely differs from the CNEG flow field.

In general, the position of flow separation is more susceptible to high-temperature non-equilibrium effects than the reattachment position. By comparing with the TPG flow field, chemical non-equilibrium reactions with the endothermic action reduce the flow separation extent in the CNEG flow field, which is consistent with the results reported by Brown³³ and Volpiani.³⁴ As stated in Ref. 37, both thermal and chemical non-equilibrium effects must be considered in the high Mach number flow. Due to the out-of-equilibrium vibrational state, the static temperature and specific heat ratio in the TCNEG flow are higher than the two thermal equilibrium (CNEG and TPG) flows. Consequently, the interaction of the impinging shock with the flat-plate boundary layer in the TCNEG flow showcases significant differences from the thermal equilibrium flows (CNEG and TPG).

Figure 22 illustrates the distribution of wall static pressure in the vicinity of the interaction region, with several significant positions marked by blue dash-dot lines: the separation position, platform pressure, reattachment position, and static pressure peak. The wall pressure rise in the separation zone caused by the IS/FTBLI can be divided into two stages: flow separation and flow reattachment. The static pressure peak, resulting from the reattachment shock, is followed by a sudden drop due to the reflection of the expansion wave caused by the tail of the shock generator. This finding is similar to the one reported in Refs. 20, 57, and 58. Since the reattachment shock possesses high intensity, the pressure rise caused by the reattachment flow is approximately seven times higher than that of the separation stage. The notable disparity in pressure rise between the two stages is also observed in Ref. 28.

In contrast to the separation position, the static pressure plateau is positioned closer to the reattachment position, resulting in a noticeable asymmetry in the shape of the separation bubble. Unlike the high Mach number IS/FTBLI, the pressure rise ratio between flow reattachment and flow separation is approximately 1–2 in the supersonic IS/FTBLI.^20,57,58 The static pressure plateau is observed roughly midway between the separation and reattachment positions. These analyses indicate that the primary flow characteristics of the high Mach number IS/FTBLI differ from those of the supersonic IS/FTBLI.

Next, we will discuss the influence of high-temperature non-equilibrium effects on the wall static pressure in the vicinity of the flow separation. During the separation stage, the trends of static pressure change predicted by the three thermochemical models are generally similar. However, at the plateau position, there are differences in the static pressure values among the TCNEG, CNEG, and TPG models, with P/P_∞ values of 2.66, 2.47, and 2.57, respectively. That is to say, the value of the static pressure plateau is positively correlated with the size of the separation zone, which is consistent with the result reported in Ref. 59. The TCNEG model predicts the static pressure peak to be closest to the upstream area, whereas the two thermal equilibrium models (TPG and CNEG) yield similar predictions, depending on the reattachment shock illustrated in Fig. 19. The reattachment shock predicted by the TCNEG model is interfered with stronger expansion waves, leading to a lower peak value of static pressure compared to the other two models. The peak value of the wall pressure predicted by the TCNEG model is 2.11% and 3.40% lower than the TPG and CNEG, respectively. In the downstream zone of the peak, the wall static pressure computed by the CNEG model closely aligns with the TPG model. In Fig. 22, the black dashed line represents the inviscid pressure ratio computed by the shock wave reflection theory. Notably, the peak value calculated by the TCNEG model, rather than the thermal equilibrium models (CNEG and TPG), agrees with the inviscid pressure ratio. As the impinging shock angle in turbulent flow is greater than that in the inviscid flow, the reattachment location occurs upstream of the inviscid pressure jump.

In the present work, we utilized a two-dimensional RANS solver to investigate the high-temperature non-equilibrium flows of a Mach 8.42 shock impingement on the flat-plate. Our numerical results revealed the influence of the thermal and chemical non-equilibrium effects on the impinging shock/flat-plate boundary layer interaction and the turbulence flow separation. The vital conclusions are outlined below:

1. Before the interaction of the shock with the flat-plate boundary layer, the impinging shock angle is larger, and the turbulence boundary layer exhibits greater thickness in the thermochemical non-equilibrium flows compared to the thermal equilibrium flows (TPG and CNEG).

2. The vibrational non-equilibrium effects become pronounced after the impinging shock intersects with the separation shock, as well as in the convergence area of compression waves during the flow reattachment. Moreover, the vibrationally over-excited non-equilibrium phenomenon is observed in the expansion fan region formed by the tail of the shock generator.

3. Within the separation zone, the chemical reactions in the TCNEG flow field are more pronounced and occur over a wider area than the CNEG flow field. The chemical non-equilibrium reactions mainly include oxygen dissociation and atomic nitrogen production. These reactions intensify after the flow separation and reattachment shock.

4. The position of flow separation is more susceptible to high-temperature non-equilibrium effects than the reattachment position. In the thermochemical non-equilibrium flow, the separation bubble is located closer to the upstream zone, and its scale is larger compared to the TPG and CNEG flows.

5. The wall pressure rise of the high Mach number IS/FTBLI exhibits distinct differences from those of the supersonic IS/FTBLI. The peak value of the wall pressure predicted by the TCNEG model agrees well with the inviscid pressure ratio, and it is 2.11% and 3.40% lower than the TPG and CNEG, respectively.

The first author would like to thanks the China Scholarship Council (Grant No. 202206840048) for the financial support, who is now visiting the University of Canterbury and conducting study in Professor Dan Zhao's group in New Zealand. Dan Zhao would like to thank University of Canterbury for the financial support with Grant No. 452DISDZ. Moreover, this work was co-supported by the Training Fund for Excellent Doctoral Candidates of Nanjing University of Science and Technology, XXX National Key Laboratory Fund (Grant No. 2022-XXXX-LB-020-01), Key laboratory of hypersonic aerodynamic force and heat technology, AVIC Aerodynamics Research Institute (Grant No. XFX20220104), China Postdoctoral Science Foundation (Grant No. BX20200070), and the Fundamental Research Foundation of the Central Universities (Grant No. 2022CDJXY-012).

The authors have no conflicts to disclose.

Chun liang Dai: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Bo Sun: Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Dan Zhao: Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Shengbing Zhou: Formal analysis (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Changsheng Zhou: Project administration (equal); Resources (equal); Supervision (equal). Yanjin Man: Project administration (equal); Resources (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

As shown in Fig. 23, there is a negative correlation between the specific heat ratio for the oxygen and nitrogen molecules and the vibrational temperature. Due to the vibrational non-equilibrium relaxation, the vibrational temperature in the TCNEG flow is lower than the vibrational equilibrium (CNEG and TPG) flow. Hence, the value for the specific heat ratio calculated by the TCNEG model is higher than that obtained from the vibrational equilibrium model (CNEG and TPG).