1 Introduction

To develop vehicles capable of achieving hypersonic flight, it is crucial to accurately predict the transition position of the aircraft for the design of aerodynamic and thermal protection. In hypersonic flow, the temperature of the boundary layer increases rapidly, owing to the strong friction between the air and the wall of the aircraft, which causes the air in the boundary layer no longer to be a "calorically perfect gas". When the temperature exceeds 800 K, the molecular vibration of the air is excited, and the specific heat becomes a function of temperature. When the temperature further increases to approximately 2 000 K, the oxygen molecules begin to dissociate, followed by the nitrogen molecules at about 4 000 K. A further increase in the temperature will result in the ionization of the molecules and atoms. These chemical reactions alter the thermodynamic properties of air, and therefore change both the mean flow and disturbance characteristics of it. With respect to these real-gas effects, Chang et al.^[1] stated that it was "very important to account for the chemistry effect in the future transition prediction for hypersonic vehicle". Therefore, it is crucial to consider the effect of high-temperature chemical reactions when the flow stability of hypersonic boundary layers is analyzed.

In recent years, a perfect gas model has been used in the studies on hypersonic boundary-layer stability. Several unstable mode waves have been found with the perfect gas model. The mode with the lowest frequency is called the first mode or Tollmien-Schlichting wave. Besides the first mode, there are higher frequency modes coexisting in the boundary layer^[2]. These modes are called the (Mack) second, third, and higher modes. For moderate supersonic flow (the Mach number Ma is smaller than 4), the first mode is the dominant instability mode in the boundary layer. But for hypersonic flow, the second mode becomes the dominant instability, especially in cooled wall cases^[3]. Based on these findings, a few studies have been conducted to determine the effect of chemical reactions on these unstable modes in hypersonic boundary layers. The linear stabilities of a flat-plate boundary layer at the Mach numbers 10 and 15 were analyzed by Malik and Anderson with the assumption of chemical equilibrium^[4]. The thickness of the boundary layer clearly decreased and the temperature of the boundary layer decreased significantly when chemical reactions were considered. The chemical reactions were found to decrease the first-mode instability whereas to increase the second-mode instability. Hudson et al.^[5] analyzed the stability of a hypersonic flat-plate boundary layer at the Mach numbers 10 and 15 with three different flow models, i.e., chemical equilibrium, chemical non-equilibrium, and thermochemical non-equilibrium. The results revealed that the instability of the second-mode disturbance under the equilibrium flow did not differ greatly from that under the chemical non-equilibrium flow. This is because the change in flow is slower than the chemical reaction rate and the assumption of chemical equilibrium is valid in a general boundary layer. In addition to the stability analysis, considerable research based on direct numerical simulations has also been performed to investigate the effect of chemical reactions. Ma and Zhong^[6], Prakash and Zhong^[7], Mortensen and Zhong^[8] and Marxen et al.^[9-10] established direct numerical simulation tools with the consideration of chemical reactions to calculate the evolution of the disturbances under different conditions. The chemical reactions were found to exert a substantial effect on the growth rates of the first and second modes.

Although a small number of researchers have investigated the role of chemical reactions, no consistent conclusion has been reached regarding their effects on the transition position^[11]. Considering both the flow non-parallelism and the chemical reactions, Chang et al.^[1] studied the stability of the Mach 20 flow over a 6° wedge with the parabolized stability equation method. The results indicated that the second mode dominated the transition, whereas the chemical reactions made the flow more unstable and promoted the transition. Mortensen and Zhong^[12] analyzed the stability of hypersonic flow under the thermochemical non-equilibrium, and studied the hypersonic boundary-layer stability for a 7° half-angle blunt cone at Ma=15.99. The results demonstrated that, the real-gas effect increased the growth rate of the unstable wave and caused the unstable region to expand, resulting in a larger N-factor. Fan et al.^[13-14] studied the effect of variable specific heat on the stability and transition of the flat-plate boundary layer, and demonstrated that the transition position moved upstream when the variation of the specific heat of air was considered. However, other studies have suggested that chemical reactions might make the disturbances more stable and delay the transition. Johnson et al.^[15] used linear stability and the e^N method to predict the transition locations on the sharp cones in high-enthalpy flow. The chemical reactions were found to make the disturbances more stable. Wan et al.^[16] examined the effect of the viscosity coefficient and heat conduction coefficient on the basic flow and the stability of the boundary layer. They concluded that increased viscosity and decreased thermal conductivity increased the growth rate of the disturbance and promoted the transition.

It is crucial to study the effects of chemical reactions on flow stability. At present, it remains difficulties in choosing a convincing chemical non-equilibrium model. The transition position predicted by the chemical non-equilibrium model lies between those of the equilibrium model and the perfect gas model^[11]. In this study, air dissociation is assumed to be in the chemical equilibrium. To study how air dissociation affects the stability of the boundary layer, the results of two models, i.e., the air dissociation model and the perfect gas model, are compared. A stability analysis is conducted for an adiabatic flat plate at different Mach numbers and different flight heights. The instability of the dominant mode of the transition is analyzed, and the effect of air dissociation on the transition prediction is further studied with the e^N method.

2 Governing equations 2.1 Air dissociation model

If the gas flow is in the thermochemical equilibrium, the individual species in the mixture behave as perfect gases. However, the proportion of each component in the mixture depends on the pressure. Consequently, the enthalpy and the equilibrium specific heat of the mixture are functions of the temperature and pressure. Therefore, to obtain the thermodynamic parameters of the mixture at a particular temperature and pressure, the mass fraction of each species must first be determined. Several different models are available for simulating the air dissociation reaction, such as the 5-, 7-, 9-, 11-, and 13-species models. The 9-species reaction model considers the following species: O₂, N₂, O, N, NO, NO⁺, O⁺, N⁺, and e^-, whereas the eleven-species model considers O₂⁺ and N₂⁺. The partial pressure of each component p_i can be determined by solving the equations of chemical reaction equilibrium, Dalton's law of partial pressures, the element conservation law, and the electron conservation law^[17].

The relationship between the mole fraction and the pressure of the ith species is x_i = p_i/p, where x_i is the mole fraction, and p is the pressure of the mixture. The mass fraction of the ith species is

where M_i is the molecular weight of the ith species, and is the average molecular weight.

The enthalpy per unit mass of the mixture is where h_i represents the enthalpy of the ith species, and is a function of temperature. The specific heat at constant pressure is

(1)

where c_pf is the frozen specific heat at constant pressure.

For the mixture, the equation of state becomes p = ZρRT. R = R₀/M₀ is the special gas constant, where R₀ is the universal gas constant, and M₀ is the molecular weight of the perfect gas. Z = M₀/M represents the compression factor. For a perfect gas, Z = 1.

To select an appropriate chemical reaction model, the temperature dependence of the dimensionless specific heat at constant pressure is computed at the pressures of 1.013×10³ Pa, 1.013×10⁴ Pa, and 1.013×10⁵ Pa (see Fig. 1). It can be seen that when T < 10 000 K, the results of the 9- and 11-species models are the same. When the temperature of the boundary layer considered in this paper is less than 6 000 K, the 9-species dissociation model is accurate to calculate the thermodynamic quantities for the mixture. For this reason, the basic flow calculations and stability analysis are performed by using the 9-species model.

2.2 Basic flow

Considering air dissociation, with the Illingworth transformation

(2)

the similarity equations of two-dimensional (2D) flat-plate boundary-layer flow can be written as follows:

(3)

where the superscript ' indicates the partial derivative with respect to η. f is the dimensionless stream function, and

in which ψ satisfies

g is the dimensionless temperature defined by

The subscript e indicates a boundary-layer edge flow parameter. μ and κ are the viscosity and thermal conductivity coefficients, respectively.

The streamwise coordinate x and the vertical coordinate y are nondimensionalized by using the boundary-layer displacement thickness δ^*. The velocity is scaled with the boundary-layer edge velocity U_e^*. The temperature T and the density ρ are scaled with the boundary-layer edge values T_e^* and ρ_e^*, respectively. The non-dimensional pressure is

The calculations of the viscosity and thermal conductivity coefficients of the mixture in the air dissociation model are based on the combination of Sutherland's formula and the Wilke mixing rule^[15-16]. That is, Sutherland's formula is used when the temperature is less than 1 000 K, while the Wilke mixing rule is used when the temperature is greater than 1 000 K.

For a perfect gas at a temperature below 800 K, the values of μ in C₁ and κ in C₂ can be calculated by using Sutherland's formula. In addition,

As a result, the similarity equations are the same as the traditional compressible Blasius equations.

The similarity equations are solved with the Runge-Kutta method.

2.3 Linear stability equations

For a small unsteady disturbance in two dimensions, the quantities in the unsteady flow can be expressed as follows:

(4)

where ϕ₀ = (ρ, u, v, T)^T is the vector of the mean-flow components, and ϕ'(x, y, t) is the vector of the fluctuating components. Substituting Eq. (4) into the Navier-Stokes (N-S) equations and linearizing about the mean flow yield the linear N-S equations for the disturbance.

In the linear stability theory, we assume that the mean flow is quasi-parallel, i.e., the mean flow variation in x is negligible. The perturbation can be written as the normal mode as follows:

(5)

where α and β are the streamwise and spanwise wave numbers, respectively, and ω is the frequency.

Substituting Eq. (5) into the linear N-S equation for the disturbance, we obtain the eigenvalue problem of the compressible linear stability equations as follows:

(6)

where L is the linear-parallel operator. The detailed expressions of this operator are given by Malik^[18].

The boundary conditions at the wall are taken to be In the free stream, the Dirichlet boundary condition is applied.

In the spatial problem, ω is real and corresponds to the frequency of the wave, while α is a complex wave number. The disturbance grows exponentially with x if the imaginary part of α is less than zero. In the temporal problem, α is real and corresponds to the wave number, while the frequency ω is complex. The disturbance grows exponentially with t if the imaginary part of α is greater than zero.

The stability calculations in the present study are formulated as a spatial stability problem. The growth rate of the disturbance is σ = -α_i. The phase speed of the disturbance is c_r=(ω/α)_r. The subscripts r and i indicate the real and imaginary parts, respectively.

The operator d(·)/dy in the linear stability equations is discretized by a fourth-order Malik scheme^[19]. The eigenvalue problem is solved by using an iterative procedure.

Compared with the perfect gas model, the stability equations considering air dissociation have the following differences:

(ⅰ) The constant-pressure specific heat in the energy equation is no longer a constant, but a function of pressure and temperature, i.e., c_p = c_p(p, T).

(ⅱ) The equation of state for the disturbance is changed to

which contains two more terms than that in the case of a perfect gas. The subscript 0 indicates a mean-flow component and the superscript ' indicates a fluctuating component.

(ⅲ) The calculations of the viscosity coefficient and thermal conductivity coefficient in the linear perturbation equation are consistent with the basic flow calculations.

2.4 Code validation

In the basic flow calculations, grid independence validation is performed. 401 grid points are distributed in the wall-normal computational domain, including at least 201 points in the boundary layer. The basic flow calculated by using this grid has been verified to be grid-independent.

To verify the basic flow calculation program and the linear stability analysis program with the consideration of air dissociation, basic flow calculations and instability analyses are performed. The Mach 10 boundary-layer case from Malik and Anderson^[4] is selected. The flow parameters include the free-stream Mach number Ma=10, the free-stream temperature T_e = 350 K, the flow pressure p_e =3 596 Pa, the unit Reynolds number Re_u = 6.6×10⁶ m^-1, and the adiabatic wall conditions. The comparisons between the present results and those obtained by Malik are presented in Fig. 2. Figure 2(a) shows the distribution of the basic flow profiles for the streamwise velocity and temperature. It can be seen that the highest temperature of the air reaches 3 150 K, and there are dissociation reactions in this case. The basic flow velocity and temperature profiles are consistent with Malik's results, indicating that the Blasius solution procedure for calculating the air dissociation is reliable. Figure 2(b) shows the comparison of the stability analyses. The growth rates and phase velocities of the second and third modes are in good agreement with Malik's results. The slight discrepancy between the two sets of data may be attributable to the differences in the gas properties such as viscosity and thermal conductivity.

When the temperature of the boundary layer is less than 800 K, the air is not dissociated. Therefore, the results of the dissociation model should be consistent with the results for a perfect gas. To test this, the two models are used to calculate an Ma=2.0 boundary layer with an adiabatic wall. The Reynolds number is 10⁵ and the flow parameters are taken for an altitude of 30 km (T_e=226.51 K). Figure 3 shows the comparisons of the basic flow velocity and temperature profiles calculated by using these two models. When the maximum temperature in the boundary layer is 1.66T_e (about 370 K), the vibrational energy of the gas molecules has not yet been excited. It can be seen that the results of the two models are consistent, which indicates that the calculation procedure for the air dissociation is applicable to a perfect gas under the conditions of low Mach number or low temperature.

3 Results and discussion 3.1 Flow conditions

In this study, several flow conditions are chosen to investigate the effects of air dissociation. The air parameters at various flight altitudes are listed in Table 1.

The flow conditions for several different Mach numbers at different altitudes are listed in Table 2.

3.2 Effects of air dissociation on the basic flow

To examine the effect of air dissociation on the basic flow profile, Fig. 4 presents the comparisons of the basic flow temperature profiles obtained by using the perfect gas and air dissociation models at Ma=8, 10, 12, and 15. Figure 4(a) shows a comparison of the temperature profiles at Ma=8. The temperature of the boundary layer obtained by using the perfect gas model is clearly higher than that obtained by using the air dissociation model. The maximum temperature of the boundary layer is approximately 2 300 K for the air dissociation model. At this temperature, the gas molecular vibration is excited, and the oxygen molecules starts to dissociate. The endothermic nature of this reaction leads to a decrease in the temperature of the entire boundary layer. Figure 4(b) shows a comparison of the temperature profiles at Ma=10. It can be seen that the consideration of air dissociation leads to a decrease in the thickness of the boundary layer and a decrease in the adiabatic wall temperature from 3 600 K to 2 500 K. In this case, the air dissociation exerts a substantial effect on the basic flow, and greatly reduces the temperature of the boundary layer. Apparent reductions in the temperature upon taking dissociation into account can also be observed in the temperature profiles at Ma=12 (see Fig. 4(c)) and Ma=15 (see Fig. 4(d)). By comparing the results at different Mach numbers, it can be seen that the difference in the temperature profiles between the two models increases with the increase in the Mach number. In addition, it should be noted that for the Ma=15 case, the temperature decreases significantly in the vicinity of y=0.2. This is because, at y=0.2, the temperature of air is approximately 3 000 K, and the specific heat of air reaches a maximum (see Fig. 1). This increase in the specific heat leads to an obvious decrease in the temperature.

Figure 5 shows the distributions of the generalized inflection points (GIPs) of the basic flow obtained by using the perfect gas and air dissociation models at various Mach numbers. It can be seen that, for the perfect gas model, there exists only one GIP in the boundary layer, excluding the GIPs at the outer edge of the boundary layer and at the wall. For the air dissociation model, when Ma=8 and 10, there is only one GIP in the boundary layer, which is consistent with the results obtained by using the perfect gas model. However, upon increasing the Mach number to 12 or 15, two additional GIPs appear in the boundary layer near the wall, which are not observed by using the perfect gas model. According to the inviscid stability analysis, the GIP is a necessary condition for the appearance of unstable modes, and plays an important role in instability. Instability analyses are performed for the basic flow profiles at Ma=12 and 15 with multiple GIPs. Besides the traditional first mode and the Mack mode, no other unstable modes are found. However, further research will be necessary in the future to determine whether there are new modes associated with the new GIPs or not. Overall, upon considering the dissociation of air, the basic flow is indeed modified.

3.3 Effects of air dissociation on stability

In this paper, the three-dimensional (3D) unstable modes are not considered because the two-dimensional (2D) second mode is more unstable than the oblique one. Although the oblique first mode is often more unstable than the planar one, the growth rate of the first mode is much smaller than the dominant second mode in the hypersonic boundary layer. For this reason, the present study concerns only 2D disturbances.

Compared with the perfect gas model, the consideration of the dissociation of air simultaneously affects both the basic flow and the disturbances. To investigate the effects of these variations in the basic flow and the disturbances themselves on the stability, Fig. 6 compares the growth rate of the second mode wave at f=87 kHz and Ma=15 in three different cases. In the first case, both the basic flow and the disturbances are calculated by using the perfect gas model. In the second case, the basic flow is calculated by using the air dissociation model, while the disturbances are obtained without considering the air dissociation. In the third case, both the basic flow and the disturbances are calculated by using the air dissociation model. The comparison of the first and second cases reveals that the effect of air dissociation on the basic flow causes an increase in the maximum growth rate of the second mode and a narrower growth region. The comparison of the second and third cases shows that the effect of air dissociation on the disturbances causes an increase in the maximum growth rate of the second mode and an upstream shift of the growth region. These results demonstrate that the effects of air dissociation on the basic flow and the disturbances both lead to obvious changes in the instability, which has also been mentioned in Ref. [20]. Therefore, in high-temperature boundary-layer instability analyses, both of these effects should be taken into consideration. In the following calculations, the air model in the third case is adopted, which we call as the dissociation model.

Figure 7 presents the results of the linear stability analysis of the basic flow at Ma=8. Figure 7(a) shows the variation of the neutral curves of the disturbances along the streamwise direction. It can be seen that, for the perfect gas model, the unstable regions of the first and second modes merge downstream of Point A. The low-frequency region, i.e., the region with a frequency lower than that of Point A, corresponds to the first-mode instability region, whereas the high-frequency region corresponds to the second-mode instability region. For the air dissociation model, merging unstable regions occur at Point A'. Overall, upon considering the air dissociation, the unstable range of the wave becomes smaller. Figure 7(b) shows the growth rate curves of the disturbance at f=25 kHz along the streamwise direction. It can be seen that the two models afford considerably different results. Upon taking the air dissociation into account, the first mode becomes more stable whereas the second mode becomes more unstable.

Figure 8 shows the results of the linear stability analysis of the basic flow at Ma=10. The distributions of the first- and second-mode instability regions are similar to the results at Ma=8.

Figure 9 presents the results of the linear stability analysis of the basic flow at Ma=12. Figure 9(a) shows the variations of the neutral curves along the streamwise direction. It can be seen that there are two unstable regions for the perfect gas model. Besides the unstable regions of the first and second modes, there also exists an unstable region of the third mode. The unstable region with the lower frequency is formed by the first and second modes, which is similar to the results at Ma=8 and Ma=10. The unstable region (4.3 m < x < 6 m) at higher frequencies corresponds to the unstable region of the third mode, and its frequency band range is narrow compared with the lower frequency unstable region. For the air dissociation model, these two unstable regions merge at Point B'. Compared with the perfect gas model, the consideration of air dissociation causes the third-mode instability to occur upstream. Figure 9(b) shows the growth rate curves of the disturbance at f=55.65 kHz. It can be seen that when the air dissociation is considered, the maximum growth rates of the second and third modes are greater than those of the perfect gas model, and the unstable region of the third mode considerably deviates upstream.

To compare the differences among these unstable modes, Fig. 10 shows the shape functions of the first mode (f=20 kHz), second mode (f=40 kHz), and third mode (f=70 kHz) at x=4 m and Ma=12 determined by using the air dissociation model. For the supersonic boundary layer, the first mode corresponds to the Tollmien-Schlichting wave mode in the incompressible boundary layer, whereas the high-frequency unstable modes, such as the second and third modes, correspond to the Mack modes. The second mode is the most unstable among all the three modes. It can be seen that these three modes exhibit significantly different shape functions. This indicates that, although the instability regions of different modes may merge at the downstream location, the disturbances at different frequencies still belong to different modes.

Figure 11 presents the results of the linear stability analysis of the basic flow at Ma=15. Figure 11(a) shows the variations of the neutral curves of the unstable wave along the streamwise direction. In contrast to the results at Ma=12, all the unstable regions merge in the downstream region of x>1.5 m for both the perfect and dissociation models. Figure 11(b) shows the growth rate curves of the waves at f=87 kHz. It can be seen that the first mode is damped. For the perfect gas model, the unstable regions of the first and second modes merge at Point A, while the unstable regions of the second and third modes merge at Point B. For the air dissociation model, the second and third modes merge at Point B'. Since the first mode remains consistently stable throughout the entire flow field, the unstable region is formed only by the second and third modes. The comparisons of these results demonstrates that when the air dissociation is taken into account, the maximum growth rates of the second and third modes are greater than those of the perfect gas model, although the unstable region of the second mode is clearly narrower.

From the comparisons of the neutral curves at Ma=8, 10, 12, and 15, we can see that the two gas models indicate a similar effect of the Mach number on the instability. When the Mach number increases, the unstable third-mode region gradually appears, and eventually merges with the second-mode unstable region. The consideration of the air dissociation affords a first-mode unstable region that is clearly smaller than that of the perfect gas model. When the Mach number increases, the first-mode unstable region gradually disappears, while the second- and third-mode unstable regions move to lower frequencies upon the considered air dissociation. Since the third mode is amplified upon the consideration of air dissociation, it appears earlier with the increase in the Mach number.

3.4 Effects of air dissociation on the transition prediction

The e^N method^[21] is one of the laminar-to-turbulence transition prediction tools in engineering. The N-factor is determined by integrating the growth rate from the lower branch of the neutral curves, i.e.,

(7)

The N-factor envelope (N_max) is the calculated maximum N-factor for any frequency, i.e.,

(8)

Figure 12 shows the comparisons of the N-factor envelopes obtained by using the perfect gas and air dissociation models at Ma=8, 10, 12, and 15. It can be seen that, when Ma=8, 10, or 12, the N-factor envelope obtained from the air dissociation model is higher than that obtained from the perfect gas model. However, when Ma=15, the N-factor envelope of the former is lower. This is because that, in the case of Ma=8, 10, or 12, the maximum growth rate of the second mode in the air dissociation model is greater than that in the perfect gas model, while the width of the growth region remains similar, which results in a higher integral value of N. However, in the case of Ma=15, although the maximum growth rate of the second mode in the air dissociation model is still greater, the growth region is substantially smaller, resulting in a lower integral value. It can also be seen that the N-factor envelope decreases with the increase in the Mach number for the two models.

For smooth-surfaced geometries in low-disturbance environments, the N-factor corresponding to the transition occurs in the range from 8 to 11^[22-23]. In the case of Ma=8 (see Fig. 12(a)), it can be seen that if N=8 is selected as the transition N-factor, the transition position obtained from the perfect gas model is x=18.3 m, whereas that obtained from the air dissociation model is x=12.8 m. Therefore, the predicted transition position moves upstream by about 5.5 m when air dissociation is considered. In the case of Ma=10 (see Fig. 12(b)), the transition position shifts upstream by approximately 5.4 m. In the case of Ma=12 (see Fig. 12(c)) and Ma=15 (see Fig. 12(d)), the N-factors obtained from both the perfect gas and air dissociation models are less than 8, which can hardly lead to the transition for an aircraft with a finite length.

The N-factors obtained from the perfect gas and air dissociation models for various Mach numbers at x=10 m are summarized in Table 3. It can be seen that when Ma=8, 10, or 12, the N-factor obtained from the air dissociation model is greater. The difference between the N-factors obtained from the two models gradually decreases with the increase in the Mach number. When Ma=15, the N-factor obtained from the perfect gas model is greater.

As mentioned above, when Ma=12, the third mode begins to become unstable. To examine the contribution of the third mode to the N-factor, Fig. 13 compares the integrals of the N-factor obtained from the air dissociation model. It can be seen that for a particular frequency, owing to the presence of the third mode, the N-factor continues to increase downstream. However, compared with the second mode, the growth rate of the third mode is much lower, and the amplified range of the N-factor is finite. Consequently, the third mode does not greatly affect the N-factor envelope. Therefore, in the e^N transition prediction at high Mach numbers, the effect of the third mode is finite, and the second mode dominates the transition.

In Fig. 14, the N-factor envelopes obtained from the two gas models for various Ma at an altitude of 40 km are compared. It can be seen that, when Ma=8, 10, or 12, the N-factor envelope is higher upon the air dissociation. However, when Ma=15, the results exhibit the opposite trend. The N-factor decreases gradually with the increase in the Mach number. These trends agree with those at an altitude of 30 km. In addition, for an altitude of 40 km, the N-factor envelope is relatively low at the Mach numbers from 8 to 15.

To study the effects of air dissociation on the transition position at different altitudes, the N-factor envelopes are calculated for Ma=10 at 20 km, 30 km, and 40 km, and the results are presented in Fig. 15. It can be seen that the N-factors obtained from the two models both decrease with the increase in the flight altitude. At the same height, the air dissociation model affords greater N-factors than the perfect gas model. If N=8 is taken as an indicator for the onset of transition, at an altitude of 20 km, the perfect gas and air dissociation models would yield the transition onset locations of 3.8 m and 3.0 m, respectively. The same trends can be observed at the altitudes of 30 km and 40 km. In other words, the effect of air dissociation on the transition position follows the same trend at different heights.

4 Conclusions

In this paper, the effects of air dissociation on the flow instability and transition prediction in hypersonic adiabatic flat-plate boundary layers are studied. The air dissociation is assumed to be a chemical equilibrium process, and the 9-species model is applied to calculate the thermodynamic parameters. The following conclusions are obtained:

(ⅰ) Air dissociation is found to be able to decrease the first-mode unstable region. The maximum growth rate of the second mode increases, and the unstable region moves to a lower frequency. The high-frequency third mode appears earlier upon the consideration of air dissociation, and its unstable region moves to a lower frequency. With the increase in the Mach number, the first-mode unstable region gradually disappears, while the unstable regions of the second and third modes gradually merge into one.

(ⅱ) When Ma < 12, the N-factor envelope is increased upon the consideration of air dissociation, which results in an earlier transition onset. However, when Ma>12, the N-factor envelope decreases, owing to the apparent decrease in the second-mode unstable region due to the air dissociation. As a result, the transition will be delayed. The N-factor envelope is also found to gradually decrease with the increase in the Mach number.

(ⅲ) At an altitude of 30 km and a Mach number of 8 or 10, the N-factor is greater, which may lead to transition. However, when Ma=12 or 15, the N-factor is so small that transition can hardly occur. Compared with the second mode, the low growth rate of the third mode has a negligible effect on the transition prediction. In the altitudes from 20 km to 40 km, when the flight height decreases, the N-factor envelope becomes higher and the transition is promoted. At an altitude of 40 km, it is difficult for the transition to occur, owing to the relatively small N-factors in the Mach numbers from 8 to 15.

(ⅳ) It is worth mentioning that new GIPs are observed in the basic flow profiles at Ma=12 and 15 upon the consideration of air dissociation, although no other unstable modes are detected. Further study is required to assess the effects of these new GIPs on the instability.