Nomenclature

Ma,  Mach number;

Re₁,  unit Reynolds number;

Re,   Reynolds number;

ρ,    density;

u,    stream-wise velocity;

v,    normal velocity;

T,    temperature;

p,    pressure;

φ,    cone half-angle;

γ,    specific heat ratio;

Pr,   Prandtl number;

t,    time;

x, y, axial and radial coordinates;

ξ, η, stream and normal coordinates;

s,    cone surface location;

U,   vector of the flow variable;

E,   convective flux in the axial direction;

F,   convective flux in the radial direction;

E_v,  viscous heat conduction flux in the axial direction;

F_v,  viscous heat conduction flux in the radial-direction;

M,   source term;

τ_ij, shear stress;

q_i,   heat flux;

e_s,   total internal energy;

μ,    viscosity coefficient;

κ,    heat conductivity coefficient;

A,    amplitude;

α,    stream-wise wavenumber;

ω,    angular frequency;

i,    imaginary unit;

c.c., conjugate complex.

Scripts

∞,    free stream;

w,    wall;

',    disturbance.

1 Introduction

The transition prediction of hypersonic boundary layers is of essential importance for the accurate computation of aerodynamic quantities such as skin friction and heat transfer for high-speed flying vehicles. The transition from a laminar to a turbulent state is due to the evolution and interaction of different disturbances in the boundary layer. In quiet environments, the transition process is dominated by the linear amplification of the boundary-layer mode, e.g., the first mode (Tollmien-Schlichting wave) and the second mode^[1-2], followed by nonlinear interaction and breakdown to turbulence. Different unstable boundary-layer modes amplify from different locations with different initial amplitudes. The determination of their initial amplitudes and locations relies on the understanding on how the free-stream disturbance triggers the unstable mode in the boundary layer, which is the so-called receptivity. Although receptivity has been attracting attention for a few decades^[3-7], the detailed mechanism remains to be inadequately understood, and is still far from being able to provide adequate information for practical transition prediction^[8].

In a uniform free stream, a general unsteady disturbance can be decomposed into three independent elementary constituents, i.e., acoustic wave, entropy wave, and vorticity wave^[9]. The acoustic wave can be further distinguished as fast or slow acoustic wave according to whether the acoustic wave propagates downstream or upstream relative to the free-stream flow. For high-speed flow, it has been universally acknowledged that the excitation of the boundary-layer mode is realized through the mechanism of synchronization^[10-19]. By synchronization, it means that the disturbance in the free stream and the boundary-layer mode have equal or very close wave lengths and phase speeds. As the pioneer of the theoretical work on receptivity, Fedorov and Khokhlov^[20-22] and Fedorov^[23] found that, at the leading edge of a flat plate, fast and slow acoustic waves can excite the fast and slow modes in the boundary layer, respectively. The fast mode has a faster phase velocity compared with the mean flow, which is always stable; while the slow mode has a smaller phase velocity, and can be unstable. Both fast and slow modes can excite the second mode through the intermodal exchange near the synchronization point when traveling downstream^[11-13]. However, the theoretical work has not consider the existence of shock and the co-existence of multiple types of disturbances behind the shock.

The shock ahead of the body actually plays a key role in generating the disturbances, which may eventually enter the boundary layer to trigger the unstable mode. Theoretical analysis^[24] shows that, when any type of free-stream disturbances strikes ahead of the shock in a free stream, all three types of disturbances may arise downstream. If they are of the same type as the incident one, they will be referred to as transmitted waves. Otherwise, they will be referred as generated waves. For the flow over a body with a relative big blunt leading edge (see Fig. 1), the shock is almost normal at the leading edge, and gradually changes to an oblique shock in the further downstream region, producing complicated disturbed field behind the shock. For the interaction of a plane slow acoustic wave with a bow shock, which we will show later in Subsection 5.1, three regions in terms of the acoustic wave can be distinguished behind the shock, i.e., (ⅰ) Zone 1, which is located near the leading edge where fast acoustic waves are generated, (ⅱ) Zone 2, downstream Zone 1, where the shock angle is beyond the critical angle^[25-26] to produce sound, (ⅲ) Zone 3, starting from a certain distance downstream the nose where the slow acoustic wave is transmitted. Except for the acoustic wave in Zones 1 and 3, entropy and vorticity waves can be generated in all the three regions. Various disturbances in the complicated disturbed field behind the shock may trigger unstable boundary-layer modes through various routes.

Besides the routes initiated from the acoustic waves as we have mentioned before, there may exist another route to excite the unstable mode in a blunt cone boundary layer, i.e., through the disturbance in the entropy layer. An entropy layer is formed due to the large transverse entropy gradient behind the shock, which supports a group of instability with low growth rates at low frequencies^[27], i.e., entropy-layer instability. It has been observed in experiment^[28] that the perturbations excited in the entropy layer penetrate the boundary layer to generate fluctuations as the entropy layer is swallowed by the boundary layer. Therefore, a possible route in generating the unstable boundary-layer mode could be as follows: the disturbance is excited in the entropy layer near the nose region, which manifests itself as the entropy-layer mode. It penetrates into the boundary layer to trigger the boundary-layer mode evolved downstream. Although the direct numerical simulation (DNS) performed by Zhong and Ma^[15], Kara et al.^[16], and Balakumar^[18] reveals the stabilizing effect of the existence of the entropy layer on the boundary layer, the detailed effect of the entropy layer on the receptivity process still remains to be poorly understood.

Since the excitation of the unstable boundary-layer mode is actually the mixed consequence in response to various disturbances through various routes, a question arises naturally, i.e., which particular route plays a dominant role in the receptivity of the boundary layer. To solve this problem, we focus on two various routes in the receptivity of a hypersonic boundary layer over a blunt cone to the slow acoustic wave. One is through the disturbance in the entropy layer, and the other is through the transmitted slow acoustic wave which can excite the slow mode (the first mode directly due to the synchronization). Since the entropy-layer instability is located at a low frequency band, the low frequency unstable mode in the boundary layer is more likely to be excited. Therefore, only the first mode is considered in the current study. As mentioned before, the fast acoustic wave is able to excite the unstable mode via the fast mode. However, the efficiency is very low. Therefore, it is not considered.

The rest of the paper is organized as follows. In Section 2, we set up the computational model and introduce the numerical methods. In Section 3, we compute the base flow, and analyze the stability characteristics of the flow with the linear stability theory. In Section 4, a plane slow acoustic wave is introduced to strike the bow shock, and the response of the boundary layer is investigated. This case serves as the baseline to assess the efficiency of the two routes in the receptivity, which will be discussed in Section 5. In Section 6, a summary is given.

2 Governing equations and numerical methods 2.1 Computational model and flow parameters

We consider a blunt cone consisting of a straight cone and a spherical nose. The flow parameters are as follows:

where Ma_∞ is the Mach number, Re₁ is the unit Reynolds number, T_∞ is the temperature, r is the nose radius, φ is the cone half-angle, γ is the specific heat ratio, and Pr is the Prandtl number. They are the same as those in Ref. [16] except that the nose radius in the current paper is bigger in order to produce a stronger entropy layer. The coordinate systems and the computational domain are shown in Fig. 2, where (x, y) is the cylindrical coordinate system, and (ξ, η) is the body-fitted computational coordinate system. The stagnation point is the origin of both the coordinate systems.

2.2 Governing equations

The two-dimensional Navier-Stokes equations are used to obtain both the steady and unsteady flow fields. They can be written as the following non-dimensional conservative form:

(1)

where

In the above equations, t is the time, x and y are the axial and radial coordinates, respectively. U is the flow viable, and E and F are the convective fluxes in the axial and radial directions, respectively. E_v and F_v are the viscous heat conduction flux vectors in the axial and radial directions, respectively. The vector M is the source term associated with the axisymmetric geometry. The shear stress τ_ij are

The heat flux q_i are

e_s is the total internal energy given by

The state equation in the non-dimensional form is

The viscosity coefficient μ is calculated with Sutherland's formula. κ is the heat conductivity coefficient given by

The equation is made non-dimensional with the density ρ_∞, the velocity u_∞, and the temperature T_∞ of the oncoming flow as the reference quantities for their counterparts. ρ_∞u_∞² is for the pressure p, and the nose radius r is for the length. The Reynolds number is defined by

2.3 Boundary conditions

The upper boundary lies outside the shock, where the oncoming flow quantities are specified. At the outflow boundary, a 3rd-order extrapolation boundary condition is used. For the wall, the adiabatic wall condition is employed for the computation of the base flow. In the unsteady computation, the temperature fluctuation T′ is taken as zero on the wall^[15]. A plane slow acoustic wave is superimposed at the upper boundary, which can be expressed as follows:

(2)

The dispersion relation is

where A_∞ represents the amplitude, i is the imaginary unit, α and ω are the stream-wise wavenumber and the angular frequency, respectively, and c.c. refers to the conjugate complex.

2.4 Numerical methods

The governing equations are solved with the finite difference method. The 7-point stencil 6th-order central difference scheme is used for the discretization of the viscosity terms. The combination of the 5th-order accurate weighted essentially non-oscillatory (WENO) scheme^[29] and the 6-point stencil 5th-order upwind difference scheme is used for the discretization of the convection terms after the flux is split with the Lax-Friedrichs method. The WENO scheme is used in the vicinity of the shock, while the upwind difference scheme is used in the other region. The 3rd-order total variation diminishing (TVD) Runge-Kutta scheme is used for the time advancing.

The total x extent is taken to be 600, i.e., about 3 m in the dimensional form. 5 001×351 grid points are used in the computational domain. In the wall-normal direction, the meshes are stretched from the wall to the upper boundary. In the stream-wise direction, very fine meshes are used in the nose region, and become coarsening downstream.

2.5 Validation of the codes

The flow with the Mach number 7.99 over a blunt cone in Ref. [15] is used to validate our codes for the DNS and linear stability analysis. Figure 3 shows the comparison of the distributions of the wall temperature and pressure along the cone surface between our results and those from Ref. [15]. Figure 4 shows the comparison of the temperature and pressure profiles along the wall-normal direction at s=54, i.e., a distance of 54 times the nose radius from the leading edge along the cone surface. The agreement is excellent. Figure 5 shows the comparison of the spatial growth rate -α_i given by the linear stability theory (LST) between our result and that from Ref. [15]. Again, the agreement is good.

A sharp cone with the nose radius of 0.025 4 mm in Ref. [16] is used to validate our unsteady DNS and parabolized stability equation (PSE) codes. A plane slow acoustic wave with f=467 kHz is introduced at the upper boundary. The computation is performed until the whole flow field reached a periodic state. Figure 6 shows the comparison of the amplitude of the wall pressure fluctuations along ξ between our results and those from Ref. [16]. The PSE code is used to compute the amplitude amplification of the unstable mode with the same frequency. The initial amplitude of the mode as shown in Fig. 6 has been tuned to match the general trend of the growth given by the DNS. As before, the agreement is excellent.

3 Base flow and linear stability analysis 3.1 Base flow

Figure 7 shows the stream-wise velocity profiles of the base flow at different stream-wise positions. At ξ=20, the velocity u increases rapidly from 0.00 to 0.84 in a thin layer near the wall from η=0.00 to η=0.20, but then slowly increases to 0.99 at η=1.60, the value of the stream-wise velocity behind the shock. This behavior is attributed to the existence of the entropy layer. As the flow evolves downstream, the entropy layer merges with the developing boundary layer along the wall so that the effect of the entropy layer is weakening and the velocity profile gradually becomes typically Blasius boundary-layer-like.

Therefore, the base flow field can be divided into three regions starting from the wall to the shock, i.e., the boundary-layer region, the entropy-layer region, and the outer-layer region (see the entropy contour in Fig. 8). The empirical criteria proposed in Ref. [30] are used to identify the boundary-layer edge (the interface between the entropy layer and the boundary layer) and the entropy-layer edge (the interface between the outer layer and the entropy layer). The entropy-layer edge almost overlaps the shock near the nose region, and deviates at about x=18. It is getting closer to the boundary-layer edge gradually downstream, indicating that the effect of the entropy layer on the velocity profiles (see Fig. 7) is weakening.

3.2 Linear stability analysis

According to the LST, the physical quantities in the disturbed flow can be considered as the base flow superimposed by a small amplitude disturbance, which can be written as q(ξ, η, t) = q₀(ξ, η, t) + q'(ξ, η, t), where q stands for any flow variable such as the velocity, the temperature, the density, and the pressure, and q₀ and q' represent the basic flow and the disturbance, respectively. The small amplitude disturbance can be written in the form of a travelling wave as q'(ξ, η, t) = , where is the eigen-function. For the spatial stability analysis, ω is real, while α is complex, and -α_i represents the growth rate of the disturbance. α along with can be solved from the eigenvalue problem of the so-called Orr-Sommerfeld equation for a given ω and ξ.

With the LST, a group of unstable entropy-layer modes is identified. The eigen-functions of the stream-wise velocity and temperature at ξ=30 for ω=1.1 are shown in Fig. 9. Both the maximums of the modulus lie above the boundary-layer edge. The phase velocity of the neutral mode of entropy-layer instability and the stream-wise velocity at the generalized inflection point (GIP) are depicted along ξ (see Fig. 10). The good agreement indicates that the entropy-layer instability is of an inviscid nature.

The neutral curves of the entropy-layer mode and the first mode are shown in Fig. 11. The second mode can also be found in a higher frequency region, which is not shown here. It can be seen that the unstable entropy-layer mode only exists near the nose region with the frequency lower than 0.6, and the first mode is stable until ξ becomes larger than 200. In order to investigate the effect of the entropy-layer mode on the excitation of the first mode, the free-stream disturbance, i.e., a plane slow acoustic wave in our consideration, is chosen to have a frequency of 1.1, i.e., about 33 kHz, to maximize the amplitude amplification of the first mode within the computational domain, as we will show later in Section 4. Note that the entropy-layer mode with a frequency of 1.1 is stable.

4 Receptivity to free-stream slow acoustic wave

A plane slow acoustic wave given by Eq. (2), where A_∞=10^-6 and ω=1.1, is introduced at the upper boundary of the computational domain. The computation is performed until the whole disturbed flow field reaches a periodic state. Figure 12 shows the contour of the density perturbations, where the shock and the entropy-layer edge are marked with dark solid lines. It can be seen that, in the outer layer, the slow acoustic wave is transmitted across the shock and large fluctuations appear in the entropy layer. Near the entropy-layer edge, the phenomenon of "beating" can be splotted, which is caused by the mismatch of the phase velocity of the entropy-layer disturbance with that of the transmitted slow acoustic wave. The phase velocity of the former is nearly 1, while the latter is about 0.8, which we will show later in Fig. 17.

In order to clearly show the disturbance excited in the boundary layer downstream, the contour of the density perturbation is plotted in the ξη-plane (see Fig. 13). It can be seen that another group of "beating" appears near the boundary-layer edge, starting from ξ=120, which is caused by the mismatch of the phase velocity of the entropy-layer mode with that of the excited boundary layer mode. The location, where the "beating" appears, infers the exciting place of the boundary-layer mode, i.e., the first mode, which is around ξ=120.

To confirm the excitation of the first mode, the shape functions of the disturbances from the DNS are obtained by performing the Fourier analysis for the disturbed flow field, which are compared with the eigen-functions given by the LST for the first mode at ξ=105, 120, 150 (see Figs. 14 and 15). The thin black line shows the location of the boundary-layer edge. It can be seen that, in the boundary-layer region, the shape functions from the DNS are getting closer to the eigen-functions given by the LST from ξ=120 to ξ=150, confirming that the first mode is excited.

The excitation of the first mode can also be verified by the phase velocity of the wall pressure fluctuations given by the DNS in comparison with that of the first mode given by the LST (see Fig. 16). The phase velocity of the entropy-layer mode is also shown in the figure. The phase velocity is about 0.65 at the leading edge, which is very close to that of the fast acoustic wave behind the shock by the Rankine-Hugoniot normal shock relations. Then, the curve increases to about 1 due to the influence of the entropy-layer mode. Starting from ξ=120, the phase velocity matches perfectly with that of the first mode given by the LST, indicating that the first mode is excited, and becomes the dominant disturbance downstream. The fluctuations of the curve are due to the co-existence of all types of the disturbances in the flow field.

The receptivity coefficient is defined as the ratio of the pressure amplitude on the wall at the neutral location to that of the free-stream disturbance. The reason that using the amplitude at the neutral location rather than the exciting location, i.e., ξ=120, is to make the receptivity coefficient applicable to the traditional e-N method, which accumulates the growth rate starting from the neutral point to predict transition. Since the PSE has the advantage of taking the non-parallelism effect of the boundary layer into account, and is able to provide more accurate evolution of boundary-layer mode compared with the LST, the PSE is used to compute the amplitude evolution of the first mode by specifying the solutions by the LST at ξ=160. The initial amplitude is tuned by making the general trend of the amplitude amplification overlap that from the DNS as much as possible (see Fig. 17). Following the amplitude amplification curve up to the neutral location, the amplitude at the neutral location can be obtained. Then, we can obtain the receptivity coefficient as 0.224. It should be noted that, from Fig. 17, the disturbance given by the DNS is growing, starting from ξ=120 despite the fact that the PSE shows that the first mode should be stable until ξ=200. This phenomenon has also been observed in the stability and transition experiments^[31-33] for sharp plates, which is attributed to the external forcing and can be explained by the forcing theory^[34].

5 Assessment of various routes in receptivity

So far, we have shown where the first mode is excited through receptivity. However, which is the dominant route that the external disturbance may take to trigger the unstable boundary-layer mode is still not clear. In this section, we will first analyze the details of the disturbances generated/transmitted downstream due to the interaction of a plane slow acoustic wave with a bow shock, and then investigate two routes of receptivity in detail.

5.1 Linear interaction with the shock

Since the free-stream disturbance is small, the interaction with a shock can be treated linearly by solving the linearized disturbed Rankine-Hugoniot equations, for which readers can refer to Refs. [24] and [26]. The bow shock, which has been obtained in Section 3, can be treated as a locally plane shock wave when the wave lengths of the oncoming acoustic wave and all the generated/transmitted waves downstream the shock are much smaller compared with the curvature radius of the bow shock, which is the case in our study. By neglecting the reflections of the waves from the wall, the wave vector and wave amplitude of the generated/transmitted disturbances downstream the shock can be readily determined^[26]. Figure 18 shows the variations of the phase velocities of the fast acoustic wave, slow acoustic wave, and entropy/vorticity wave behind the shock along x. Three regions are shown (see Fig. 19). In Zone 1 where 0 < x < 0.9, the flow is subsonic, and the fast acoustic wave and the entropy/vorticity wave are generated. In Zone 2 where 0.9 < x < 23, no acoustic wave can be found, and only the entropy/vorticity wave is generated. Zone 3 contains the remaining part of the shock, i.e., x > 23, where the slow acoustic wave is transmitted, and the entropy/vorticity wave is generated.

The unsteady disturbed flow filed, which has been obtained in Section 4, can be used to verify the theoretical analysis of the disturbances. The density amplitude, which is induced by all types of disturbances (including the acoustic wave and the entropy wave, as the vorticity wave has no density perturbation) behind the shock, is compared with the density fluctuations just behind the shock at a certain time instant in the DNS. The result is shown in Fig. 20, where the amplitude of the density fluctuations induced solely by the acoustic wave in Zones 1 and 3 is also depicted. Excellent agreement can be found in Zones 1 and 3 except in the vicinity at the critical locations x=0.9 and x=23, where the theoretical analysis fails to provide reasonable results, as discussed in Ref. [25]. It can also be seen that the acoustic wave actually contributes substantially all to the density fluctuations in Zones 1 and 3, and so does it to the velocity fluctuations which is not shown here. Small discrepancy can be spotted in Zone 2 between the DNS and the theoretical analysis, which is due to the absolute small values of the density fluctuations in this region.

According to the generated/transmitted disturbances as discussed above, there may exist a few possible routes where the disturbances may take to excite the unstable boundary-layer mode. First, near the leading edge, the disturbance in the entropy layer may excite the first mode when evolving downstream. Second, the transmitted slow acoustic wave may interact with the boundary layer, and directly induce the first mode through the mechanism of synchronization. Since the fast acoustic wave can only be generated in a very small region, i.e., Zone 1, and it has been proved significantly inefficient^{[13, 17-18]} in receptivity, it is not considered in the current paper.

5.2 Various routes of receptivity

A sub-domain is taken out from the whole computational domain together with the already obtained base flow in Section 3 (see Fig. 21). The inlet of the sub-domain is taken as ξ=30, and the outlet is ξ=600. The disturbance from the disturbed flow field in the DNS in Section 4 at the inlet and the upper boundary are denoted by D1 and D2, respectively. According to the theoretical analysis in Subsection 5.1, the disturbances at ξ=30 are induced as the consequence of multiple waves including the entropy layer mode in the entropy layer and the fast mode in the boundary layer, which is generated by the fast acoustic wave near the leading edge. As the disturbances travel downstream, the entropy-layer mode is getting dominant since the fast mode decays significantly. The disturbance in the outer layer (see Subsection 3.1) at ξ=30, which is represented as d1 for easy reference, is induced by the transmitted slow acoustic wave, the entropy wave, and the vorticity wave. At the upper boundary of the sub-domain, the transmitted slow acoustic wave dominates.

To investigate the routes of receptivity, three cases of the DNS are performed on the sub-domain with different initial disturbances (see Table 1). D1, D2, and d1 are recorded from the already obtained disturbed flow filed in Section 4, starting from the same time instant for a whole time period. The main purpose of Case 1 is to validate the method of using the sub-domain by reproducing the receptivity process. In Case 2, the disturbance in the outer layer is removed from D1 before it is introduced at the inlet of the sub-domain. Therefore, the main disturbance is the entropy-layer mode, by which we show the first route of receptivity, i.e., through the entropy-layer instability. In Case 3, the initial disturbances include the transmitted slow acoustic wave D2 and the disturbance in the outer layer at the inlet, i.e., d1. The dominant disturbance is the transmitted slow acoustic wave, by which we show the second route of receptivity, i.e., through the synchronization of the first mode.

All the computations are performed until the whole disturbed flow field reaches a periodic state. Figure 22 shows the wall pressure fluctuations from the simulation for Case 1, in comparison with the previous results given in Fig. 17. It can be seen that the receptivity process is exactly reproduced, which is certainly with the same place of the excitation and receptivity coefficient, which further confirms that the main contributors of receptivity have been considered in our computation.

The density perturbation contours from the simulation for Case 2 are shown in Fig. 23. Downstream the inlet, the disturbance in the entropy layer decays slowly when traveling downstream. As shown in Fig. 23(b), the disturbance in the entropy layer penetrates the boundary layer to excite the boundary-layer mode. As before, the variation of the phase velocity of the wall pressure fluctuations along ξ (see Fig. 24) indicates that the excitation of the first mode by the entropy-layer mode occurs at a place close to the neutral point, i.e., ξ=200, rather than ξ=120. The wall pressure fluctuations are displayed in log scale in Fig. 25, compared with the result given by the PSE. It can been seen that, without the forcing of the external wave in the outer layer, the disturbances in the boundary layer start to grow at the neutral point. The obtained receptivity coefficient is 0.014, one order of magnitude smaller compared with the case shown in Section 4 (or Case 1). Therefore, the entropy-layer mode does not play a key role in exciting the first mode.

Similar to Fig. 23, Fig. 26 shows the density perturbation contours obtained from the simulation for Case 3. Similarly, the disturbance in the boundary layer is excited downstream. The phase velocity of the wall pressure fluctuations (see Fig. 27) is close to that of the first mode from the inlet to ξ=120. Beyond that, it matches with the first mode perfectly, indicating the excitation of the latter. Similar to Fig. 25, Fig. 28 shows the wall pressure fluctuations in comparison with the result given by the PSE. As in Case 1, the disturbances in the boundary layer start to grow before the neutral point. The obtained receptivity coefficient is 0.253, which has the same order as that in the case shown in Section 4 (or Case 1). Since all the input disturbances for the three cases start at the same time instant, the instantaneous wall pressure fluctuations obtained at a certain time instant for Cases 2 and 3 can be superimposed, resulting in a receptivity coefficient of 0.233, which is close to the value obtained in Section 4, i.e., 0.224. The small difference is attributed to the errors introduced by taking d1 out of D1. Therefore, the excitation by the transmitted slow acoustic wave plays a key role in the receptivity of a blunt cone boundary layer to slow the acoustic wave.

6 Summary and conclusions

The receptivity of a hypersonic boundary layer over a blunt cone to a plane slow acoustic wave is investigated with the DNS in this paper. The simulations are performed for both steady base flow and unsteady disturbed flow field.

For the body with a relative big blunt leading edge, an entropy layer is formed due to the large transverse entropy gradient behind the shock, which supports a group of instability at low frequencies with an inviscid nature. On the basis of the understanding on the disturbances generated/transmitted behind the shock, two routes of receptivity are investigated in detail. One is through the entropy-layer instability wave, and the other is through the synchronization of the transmitted slow acoustic wave of the first mode. The results show that, the entropy-layer mode is able to generate the first mode as the entropy layer merges with the developing boundary layer downstream, but the receptivity efficiency is very low. In contrast, The transmitted slow acoustic wave plays a key role in generating the first mode as it has a close phase velocity with the latter. The slow acoustic wave is transmitted from a certain distance downstream the nose, which is different from the receptivity problem of the boundary layer over a sharp cone or a sharp leading edge plate, where the receptivity is shown to occur very close to the leading edge^{[11-14, 17-18]}.

Besides, as shown in Fig. 11, the frequency band of the unstable first mode covers that of the entropy-layer mode. In order to investigate the effect of the entropy-layer mode on the receptivity of boundary layer, only the first mode is considered in the current paper. Although in hypersonic boundary layers, the second mode has a larger growth rate and is usually considered as the dominant boundary-layer mode, its initial amplitude observed is much smaller than that of the first mode^[35-36]. In a very recent experiment^[37], the low frequency disturbances in the boundary layer are observed to have larger amplitude than that of the high frequency disturbances, even with the linear stability analysis, the latter should be dominant in the transition process. Therefore, whether the first or second mode is the main cause leading to the final transition depends on the competition of the receptivity process and the eigen-mode amplification.

Acknowledgements The first author and the third author are grateful to Prof. Heng ZHOU of Tianjin University for the stimulating discussion and tremendous support on this work. The third author would like to thank Prof. Xuesong WU of Imperial College for his continuous encouragement and inspiring discussion. Special thanks to Dr. Lei ZHAO from Tianjin University for generously providing his DNS codes for the current computations.