1 Introduction

The stability and transition of compressible flows are very important in hypersonic aircraft design. Compared with the stability of incompressible or compressible flows at low Mach numbers,the stability of high speed flows is more complex. This is because that the compressibility complicates the governing equations significantly. In incompressible flows,there is a linear viscous instability mode,i.e.,the Tollmien-Schlichting (T-S) mode. However,there are often multiple unstable modes in compressible flows at high Mach numbers. In these unstable modes,the mode with the lowest frequency is called the first mode,which is considered as an extension of the T-S mode in incompressible flows. The other additional modes are often called the higher modes. Among these higher modes,the lowest frequency one,which is usually called the second mode,is particularly interesting because this mode is often the most amplified for the compressible boundary layers at higher Mach numbers. For example,the second mode instability is the least stable one in an adiabatic flat plate boundary layer at Ma=4.5. Mack^[1-2] found these higher modes first through a numerical method. Therefore,the second mode instability is also called the Mack mode instability.

Lots of factors affect the linear stability of high speed boundary layer flows,such as the wall temperature,the total temperature^[3],and the pressure gradient^[4]. Among these factors,the wall temperature has been concerned for over 20 years. Many researchers focused on the effects of the heat transfer on the transition prediction and flow control of high speed boundary layers. The effects of the wall temperature on the instability mode have been confirmed by many mathematical analyses,experiments,and numerical simulations. By the mathematical analyses,at high Reynolds numbers,Seddougui et al.5 found that wall-cooling might make the growth rates of the viscous T-S modes exceed the growth rates of the inviscid modes for moderate cooling and at any Mach number. Lysenko and Maslov6 experimentally investigated the disturbances in a cooled flat plate boundary layer when Ma∈[2, 4],and found that the first mode instability tended to be stabilized by the cooling wall,the second mode instability tended to be destabilized,and the cooling wall might lead to higher frequencies of the unstable second modes. Stetson et al.^[7] and Stetson and Kimmel^[8] performed a series of experiments on the instability of a hypersonic flow over a circular cone,and concluded that wall-cooling stabilized the first mode waves and destabilized the second mode waves. Mack^[9-10] gave a similar conclusion by a numerical technique,and predicted that relative supersonic disturbances would be found when the wall temperature was extremely low,e.g.,(T _w/T_ad=0.05). Malik^[11] investigated the role of the wall-cooling effect on the boundary layer transition,and suggested that the cooling effect destabilized the second mode which dominated the transition process of high Mach number flows. Masad et al.^[12] confirmed that the wall-cooling effect always stabilized the first mode and destabilized the second mode by considering the effect of real gas. Recently,the cooling effect on receptivity has been concerned^[13-14]. Some interesting phenomena are observed when the wall temperature is low enough. Mack^[15] found the exhibition of supersonic waves and that the upper neutral point of the Mack-mode instability could be determined by the generalized inflection point (GIP),and showed how the boundary layer′s profile was modified by the wall-cooling. Nevertheless,the reason why wall-cooling can destabilize the Mack-mode instability has not been completely understood.

In this study,we investigate the linear stability of a high speed flat-plate boundary layer,mainly focusing on the wall-cooling effect. The aim of this paper is to understand the wall-cooling effect on the Mack mode instability.

2 Methodology

A flat-plate is selected as the model for its simple shape. The reference length is

where x^* is the distance from the leading edge of the flat plate,and U_∈^*and v^* are the velocity and the viscosity of the freestream,respectively.

Generally,in a compressible flat-plate flow at Ma ＜2.2 with an adiabatic wall temperature,there are only the first mode instabilities. When the freestream Mach number reaches 2.2,the Mack mode instabilities can be found in the flow,while the instabilities of the flow is still dominated by the first modes. When the Mach number increases and reaches high enough (e.g.,Ma ＞ 4 with an adiabatic wall temperature),the second mode will become the most unstable mode in the flat plate boundary layer. In this study,the instability of a flat-plate boundary layer at varying wall temperatures and Mach numbers is analyzed through numerical methods. The freestream Mach number valued from 4 to 6 is mainly focused on since this region is the conventional condition behind the shockwave in hypersonic aircraft designing. The similarity solution is selected as the base flow. The linear stability theory analysis method is adopted. The linear stability theory (LST) is a main methodology to consider the growth and decay rates of normal modes in compressible flows for the free-flight and quiet wind-tunnel conditions,especially for the natural-transition-prediction. The linear inviscid stability equations are also selected,which is a good simplification in the compressible flows when a certain linear stability behavior is treated analytically. It is easy to derive the compressible Rayleigh equations,i.e.,the compressible linear inviscid stability equations with a streamwise parallel assumption. Because the viscous effect is neglected in the inviscid equations,there are two unknown terms left in the equations,i.e.,v′ and p′.

The disturbance,i.e.,the solution of the linear stability equations,can be set as follows:

where (p′,,u′,,v′,w′,T′,p′)^T is the disturbance,and is the instability mode of the flow. α and β are two complex wave numbers. w means the complex frequency of a disturbance. Both the temporal mode and the spatial mode are investigated. The "temporal mode" means that the disturbance grows in time. In the temporal mode,the wave number is real,and the frequency is complex. The imaginary part w_i of w means the growth rate of the disturbance. When w_i ＞0,the disturbance is amplified. When w_i ＜0,the disturbance is decayed. The "spatial mode" means that the disturbance grows in space. It is similar to the temporal one except that the frequency w is often real and the streamwise wave number α is complex. When α_i＜0,the disturbance is amplified in space. To close the problem,we use the no-slip condition at the wall and the non-reflecting boundary conditions in the far field.

The linear viscous and inviscid stability equations are solved by the method used in Ref. ^[16]. The iteration method used in Ref. ^[17] is adopted to find the roots for its robustness. The solvers are validated and compared well with the results in Refs. ^[18] and ^[19]. The number of the wall-normal grid points is over 1 500 with over 700 points inside the boundary layer region. There is a uniform distribution outside the boundary layer,since it has a good effect on the capture disturbances in the freestream.

3 Results and discussion 3.1 Base flow

The main purpose of this work is to understand the effect of wall-cooling on the second mode instability. There is no difference on the governing equations between the different cases except the base flow profiles. It implies that the base flow profiles have an important effect on the instability behavior. The generalized inflection point (GIP),where (p U′)′=0,plays an important role in the compressible flow profile. Here,(′) means a y-direction derivative. Lees and Lin^[20] and Lees^[21] found that a compressible flow was inviscidly unstable if it had a GIP at $y_{\rm c}$,where U_yc ＞1-1/Ma.

In this section,we will evaluate the base flow profile first. The distribution of (p U′)′ with different wall temperatures at a given Mach number is displayed in Fig. 1,where $T_{\rm w}$ is the actual wall temperature,and T_ad is the adiabatic wall temperature (which is also called the recovery temperature). These distributions are consistent with the results of Ref. ^[22],which were obtained with different parameters. It can be seen that there are often two GIPs inside the boundary layer if we regard the point at y=0 as a GIP for the insulated case. When the surface temperature decreases,the distance between the two GIPs decreases. When the wall temperature reaches low enough,these two GIPs will cancel each other. The lower GIP,if exists,is always located in the boundary layer where U(y) ＜1-1/Ma,while the characteristic of the upper GIP always depends on the wall temperature. Generally,the upper GIP has a velocity greater than 1-1/Ma if the wall temperature is moderate. However,when the wall temperature is low enough,its velocity may be less than this value. According to the inviscid stability theory,the lower GIP has no effect on the inviscid stability whereas the upper GIP initiates the inviscid unstable disturbance. The location in the boundary layer where U(y)=1-1/Ma is very important for the instability,and we will analyze it in the following subsections.

The location of the GIP in the flat plate boundary layer varies with the Mach number and the wall temperature. Masad et al.12 gave the variations of the locations of GIPs with the surface temperature when Ma∈[2, 6]. Here,the distributions with a wider range of Mach numbers are given. These results match the results in Ref. ^[12] from Ma=2 to Ma=6 well. As can be seen in Fig. 2,the minimum surface temperature,i.e.,the critical surface temperature,eliminates the GIPs and reduces when the Mach number increases. It can be seen that the surface temperature at which the velocity of the upper GIP reaches 1-1/Ma also reduces.

3.2 Temporal mode

In earlier studies,it is found that the Mack mode instability can be destabilized by the surface cooling^{[6, 10, 12]}. Using the linear stability theory,we analyze the wall-cooling effect on the Mack mode instability for a deeper understanding. At first,we investigate the temporal mode.

The variations of the maximum growth rates of the Mack mode instabilities with wall temperature for four Mach numbers and three freestream temperatures are shown in Fig. 3,which are obtained by the linear viscous and inviscid analysis. It can be seen that the Mack-mode instability is destabilized when the wall temperature reduces,which is consistent with the former findings. However,when the wall temperature reaches an extreme cold level (about 0.2 times of the adiabatic wall temperature in Fig. 3(b)),an reverse effect of the wall cooling on the Mack mode instability will exist,i.e.,the Mack mode instability will be stabilized by the wall-cooling. We define this extreme low value as the reversed wall temperature. It is obvious that the reversed wall temperature always decreases with the increase in the Mach number. Especially,this value tends to be zero at Ma=7 (see Figs. 3(a)--3(d)). The reversed wall temperature is also a function of the freestream temperature. It can be seen in Figs. 3(b),3(e),and 3(f) that when the freestream temperature increases,the reversed wall temperature tends to be a lower value. Though this variation is the same as the one with the Mach number,the reversed wall temperature is insensitive to the freestream temperature compared with the Mach number. In other words,this value has a deeper connection with the compressibility rather than the gas property. It is fortunate that this extreme-low wall temperature is so low that it is often not concerned in the aircraft designing. Yet,it is still important sometimes,especially in nuclear fusion researches. For example,the recovery temperature is about 6 600 K when Ma=4.5 and T∞=1 500 K. It means that T_w/T_ad is about 0.045 when T _w=300 K. Then,the Mack mode instability will be stabilized by the wall-cooling effect. Moreover,it can be seen in Fig. 3 that the growth rate becomes less and less when the Reynolds number decreases. It suggests that the viscosity only has a damping effect on the Mack mode instability,and the characteristics of the Mack mode instability can be described more clearly by the linear inviscid stability analysis. Therefore,the inviscid linear analysis is adopted below for its efficiency.

Two dimensional (2D) Mack mode waves are considered to examine the wall-cooling effect because the 2D Mack mode waves are the most unstable in a compressible flat plate boundary layer at higher Mach numbers. The inviscid growth rate w_i versus the streamwise wave number α with different wall temperatures at Ma=4.5 is shown in Fig. 4. The surface temperatures are from 1.0T_ad to 0.1T_ad. As can be seen,the maximum amplified rates vary with the cooling wall effect as the above results. Additionally,when the wall temperature decreases,the range of the unstable wave numbers expands and the wave numbers of the upper and lower neutral waves grow to higher values. These results are in good agreement with the ones of the pervious experiments6. It can also be seen that the diagram has another branch (Branch B) if the surface wall temperature is low enough,e.g.,the region from α=0.476 for the wall temperature T_w=0.3T_ad. As a result,two local extremum growth rates can be observed. The greater extremum value is in Branch A,and the lower extremum value is in Branch B.

Figure 5 plots the phase velocity c_r versus the streamwise wave number α at two typical wall temperatures,i.e.,T_w=0.3T_ad and T_w=1.0T_ad. It can be seen that the phase velocities of the amplified waves in Branch A are between 1 and 1-1/Ma,while the phase velocities in Branch B are less than 1-1/Ma. The phase velocity of the lower neutral solution is always equal to 1,and the phase velocities of other regular neutral solutions are greater than 1. For the adiabatic surface temperature,the phase velocity of the upper neutral solution is 0.862 9,which is just equal to the velocity of the upper GIP inside the boundary layer. However,there is no upper neutral solution in the Tw=0.3T_ad case.

As mentioned previously,there are often two GIPs inside a compressible Blasius boundary layer flow. It has been known that the upper GIP contributes to an inviscid instability while the lower GIP has no effect on the instability. Through an asymptotic analysis,Lees and Lin20 gave a proof that the mean flow at the critical layer (which is the location of the upper GIP) equals the phase velocity of the neutral wave. As can be seen from Fig. 4,the Mack mode instabilities always have two neutral solutions with an adiabatic wall for a high speed flat plate boundary layer. According to Lee′s proof^[21],the disturbance at the upper neutral point has a phase velocity c=U(y_c),where y_c is the position of the upper GIP (see Fig. 5(b)). In other words,the upper neutral solution is associated with the critical layer. Besides,there still remains a lower neutral point of the Mack mode instability,whose phase velocity equals 1. Fedonov23 proposed that the phase velocity c=1 meant that the fast mode waves were synchronized with the entropy waves and the vorticity waves during the receptivity process. Actually,we can give a proof (see Appendix A) that a necessary condition for the existence of a neutral wave with a unique phase velocity between the maximum mean flow and the minimum mean flow is that the location where U=c is the position of the GIP of the mean flow or the normal disturbance v=0. According to this theorem,it can be inferred that the lower neutral solution is associated with the edge of a boundary layer for the reason that its location is also a GIP and the velocity of this GIP just equals 1. Until now,in the adiabatic wall temperature case,the relationship between the two neutral solutions and the boundary layer profile GIPs has been clarified. Figure 6 illustrates these relations briefly. As can be seen,the two neutral points are indicated by the two dashed lines in Fig. 6(a),the associated phase velocities are marked by the dashed lines in Fig. 6(b),and finally the associated GIPs are illustrated by the dashed lines in Fig. 6(c). From the shadow area in Fig. 6,it can be inferred that the region between the boundary layer edge and the critical layer (if exists) determines the Mack mode instability. Due to these important internal relations,some aspects of the wall cooling effect on the Mack mode instabilities can be concluded. Recalling to Fig. 2,it is noted that the velocity of the upper GIP decreases with the decrease in the wall temperature,which means that the associated phase velocity of the upper neutral point will decrease. On the contrary,the phase velocity of the lower neutral point remains unit since it is associated with the edge of the boundary layer,where the velocity always equals 1. Therefore,a wider phase velocity range between the two neutral points will be expected. In other words,the Mack mode will be destabilized by the wall cooling effect if the critical layer exists.

As the wall temperature decreases,the velocity of the upper GIP and the phase velocity of the upper neutral point decrease. However,when the velocity of the GIP velocity reduces to less than 1- 1/Ma,the upper neutral point will disappear,and a new branch will be found. The Rayleigh equations can be adopted to explain this phenomenon. The Rayleigh equation in terms of the disturbance pressure can be simplified as follows:

As a result,for a compressible flow,the existence of higher inviscid unstable modes depends on the region,where the relative Mach number is larger than 1. Here,the relative Mach number is defined by

where c is the phase velocity of the wave,and a is the local acoustic velocity. Compared with the base flow inflections in Fig. 1,the diagram for a extremely low wall temperature (T_w=0.3T_ad or T_w=0.4_ad) has no GIP inside the boundary layer,while the second mode instability still exists.

When the wall temperature is T_w=0.3T_ad,there is always a region of the relative supersonic flow . Therefore,the second mode instability is not eliminated by the wall-cooling even there is no GIP inside the boundary layer (see Fig. 7). Additionally,when the phase velocity of the inviscid amplified wave c_r ＜1-1/Ma (Branch B),the wave outside the boundary layer will be also relative supersonic to the base flow. It makes the disturbance equation become a wave equation outside the boundary layer. The solution is associated with an outgoing wave,known as the relative supersonic disturbance. When the wave number α increases,this disturbance is always relative supersonic,and the relative supersonic region outside the boundary layer can always be found and becomes larger and larger. It means that a neutral solution can never be found in Branch B,i.e.,the upper neutral point is cancelled. In contrast,the unstable disturbances with the phase velocity range 1-1/Ma≤ c_r≤ 1,associating with Branch A,has only a relative supersonic flow inside the boundary layer,and the flow outside the boundary layer is relative subsonic or sonic,known as the subsonic or sonic disturbances.

When the wall temperature is not low enough,i.e.,the upper GIP inside the boundary layer has a velocity greater than 1-1/Ma,and the associated phase velocities c_r of the unstable waves are between 1 and 1-1/Ma,all the unstable disturbances are relative subsonic,and two neutral solutions can be found. Conversely,when the wall temperature is too low,i.e.,the velocity of the upper GIP is less than 1-1/Ma or even cancelled and the associated phase velocity c_r is less than 1-1/Ma,the diagram of the unstable waves will be divided into two branches,i.e.,the subsonic branch and the supersonic branch. The eigenfunctions of the inviscid subsonic mode and the supersonic mode are plotted in Fig. 8. It is obvious that there is a divergence between the eigenfunctions of the subsonic and supersonic unstable disturbances. For the subsonic disturbance,the eigenfunction always decays outside the boundary layer exponentially,whereas the eigenfunction of the supersonic one is a wave-like solution outside the boundary layer. This divergence is caused by the difference between the relative Mach numbers outside the boundary layer,i.e., for the subsonic disturbances and for the supersonic disturbances. Particularly,the wall temperature at which the velocity of the upper GIP equals 1-1/Ma becomes a critical point (see Fig. 2). When the wall temperature is the critical value,the diagram of the unstable waves will be slipped by a neutral solution which is associated with a sonic wave in the freestream,i.e.,the relative Mach number equals 1 outside the boundary layer. The amplified rate of the relative supersonic branch (Branch B) is often smaller than the relative subsonic one (Branch A). It is a concern if the wall temperature is extremely low. Fortunately,as shown in Fig. 9,the supersonic branch is often immersed as a result of the viscous damping effect.

The instability is also investigated at a higher freestream Mach number Ma=6. Figure 10 plots the amplified rates w_i versus α with different wall temperatures at Ma=6. The distributions are very similar to the ones in the Ma=4.5 case. It is obvious that there are relative supersonic branches in the flow. Figure 11 shows the growth rate w_i and the phase velocity c_r versus the wave number α of the unstable Mack-mode instability at two typical wall temperatures. It is also found that there is a corresponding relation between the Mack mode and the boundary layer profile. It suggests that the mechanism of the instability is the same as that in the Ma=4.5 case.

3.3 Spatial mode

The temporal stability analyses revealed the features of the wall-cooling effect on the Mack mode instability at a fixed Reynolds number. In the cases with different wall temperatures in the spatial theory,the numerical results indicate that the spatial analyses are the same as the temporal analyses. Figure 12 shows the spatial growth rates -α_i versus FRe of the amplified disturbances with different wall temperatures,where F=2π f^*v_∞^*/U_∞^*2 is the normalized frequency of the disturbance. It can be seen that the Mack mode instability is still destabilized first by the wall-cooling effect as the temporal analyses if the critical layer exists. When the wall temperature is extremely low,the wall cooling effect will stabilize the boundary layer. A relative supersonic branch (Branch B) can also be found when the wall temperature is low enough. Especially,Fig. 13 plots the phase velocities c_r versus FRe of the unstable Mack-mode instabilities at T_w=0.3T_ad and T_w=1.0T_ad. It seems that the phase velocity of the lower neutral solution is close to 1 and the phase velocity of the upper solution of Branch A is close to the velocity of the critical layer (at T_w=1.0T_ad) or 1-1/Ma (at T_w=0.3T_ad). It suggests a similar rule that the amplified Mack mode waves are associated with the boundary layer as the inviscid temporal analyses. Therefore,the range of the unstable Mack-mode disturbances becomes wider with the same variation between the boundary layer edge and the critical layer as the surface is cooling. Then,the Mack-mode instability is destabilized by the wall-cooling effect if the critical layer is found in the boundary layer.

4 Conclusions

In summary,we have investigated the wall-cooling effect on the Mack-mode instability of a high speed flat plate boundary layer by a numerical methodology. The analyses of the Mack mode instability by the viscous and inviscid linear stability equations indicate the conclusions as follows:

(i) The Mack mode instability is often destabilized by the wall-cooling effect except when the wall temperature is extreme low to the recovery temperature. This result has a little difference from the formers′ preconization. There is a reversed wall temperature,from which the wall-cooling effect can stabilize the Mack-mode wave. The reversed wall temperature is related to the freestream condition. As the Mach number or freestream temperature increases,this reversed wall temperature tends to be zero. In nuclear fusion research,this difference is important because T_w/T_ad may be close to this extreme value. However,the formers′ conclusion is still suitable for the aircraft designing as the wall cooling effect is moderate in that field.

(ii) It is inferred that the base flow is changed by the wall-cooling and the second mode instability is affected indirectly. The critical layer is associated with the upper neutral solution of the inviscid second mode wave. It is in good agreement with the asymptotic analyses in Refs. ^[20] and ^[21]. Moreover,it is found that the edge of the boundary layer is associated with the lower inviscid neutral Mack-mode instability and the region between the edge and the critical layer can determine the feature of the inviscid second mode instability. When the wall is cooling,the velocity of the critical layer decreases while the velocity of the boundary layer edge remains unit. Therefore,the velocity range between the critical layer and the boundary layer edge becomes wider,and makes the associated phase velocity of the two neutral points of the Mack mode wider. It means that more disturbances become unstable and the Mack-mode instability will be destabilized if the critical layer exists.

(iii) Additionally,supersonic disturbances will be found in the flow if the velocity of the upper GIP is less than 1-1/Ma or the GIPs are cancelled with the decrease in the surface temperature.

Our results provide a deeper understanding of the wall-cooling effect on the Mack-mode instability in the compressible boundary layer at a higher freestream Mach number. However,future work will concentrate on the mechanism of the stabilized effect of the wall cooling when the wall temperature is extreme cold. Additionally,the results are obtained only by numerical methods,and should be explored by a mathematical analysis in the future.

Acknowledge The authors would thanks to Dr. Jun GAO for his contribution to the solver of the inviscid stability equations.