1 Introduction

Hypersonic aircraft is the key researched and developed flight tool on aerospace in future. However,many technical difficulties have not been overcome,and one of the key problems is the accurate prediction of transition position in boundary layer^[1]. Due to the effect of shock wave and viscous near to the body of aircraft,high temperature will appear when the aircraft flies at a hypersonic speed. The high temperature changes the physical and transport properties of the gas,and thus influences the basic flow and disturbance characteristics,eventually the transition position^[2].

Accurate prediction of the transition position is the critical factor for the exact calculation of aerodynamic drag and heat flux,which is also very important for the design of aerodynamic and thermal protection of hypersonic aircraft. As the air temperature becoming higher,the air molecules show signs of the vibration excitation,dissociation,ionization,etc. The real gas effect resulting from the high temperature is need to be considered in the calculation. Vibrationexcitation begins at about 600K,when temperature exceeds about 2 500K,oxygen molecules begin to dissociate while nitrogen dissociation begins at about 4 000K,and significant ionization takes place at about 9 000K^[3]. There are few researches about the prediction of transition position in hypersonic boundary layer considering the real gas effect of high temperature. By using the improved e^N method,Cao^[4] predicted the transition position of the hypersonic plane and sharp wedge boundary layer flow,but without considering the vibration excitation. Jia and Cao^[5] supposed that specific heat is variable,and when the temperature is in a range of 600K to 2 500K,the specific heat and temperature meet an analytical relationship. The stability of hypersonic boundary layer on a flat plate was investigated under this condition. Compared with the constant specific heat,research results show that the neutral curves and the maximal growth rate are influenced by the variable specific heat. Hence,it is worth researching the effect of variable specific heat on transition prediction of hypersonic flow. Fan et al.^[6] investigated the influence of variable specific heat on the transition position of hypersonic plane boundary layer. It was found that transition positions calculated by the variable specific heat are nearer to the leading edge than those by the constant specific heat,which suggested that the real gas effect should be taken into consideration when predicting the transition position for hypersonic flow. Though the hypersonic sharp wedge boundary layer is more close to the actual flight situation, up to now,there is no relevant research on this flow about the prediction of transition position with the consideration of vibration excitation.

When the vibration excitation of air molecules is significant,Fan^[7] developed a simple calcu- lation method for specific heat and suggested taking different values of the specific heat around the shock wave,i.e.,1.4 before the shock wave,9/7 after the shock wave with the assumption of full vibration excitation of air molecules. This method is applicable to strong shock wave like normal shock wave. In the hypersonic flow around the sharp wedge,the produced oblique shock wave is attached to the tip of sharp wedge and its strength is weaker than normal shock wave. Therefore,whether to consider different specific heats around shock wave is also the research content.

In this paper,the prediction of transition position of sharp wedge by considering variable specific heat is investigated to calculate the transition of actual situation more accurately. The oncoming flow parameters are taken as those values corresponding to the condition at a height of 40 km where the temperature 250.35K,the density 3.995 7×10⁻³ kg/m³,the dynamic viscosity 1.600 9×10⁻⁵ Pa·s,and the velocity of sound 317.19m/s,the Mach numbers of free stream are 6,7,and 8,respectively,and the half wedge angles of sharp wedge are 5,8,and 10 degrees, respectively. The transition positions of hypersonic sharp wedge boundary layer are predicted by using the improved e^N method. The computed results of variable specific heat are compared with those of constant specific heat in the same condition. Meanwhile,the applicable condition of taking different specific heats around shock wave is also investigated. 2 Calculation of basic flow

It is necessary to compute the basic flow when the improved e^N method is used to predict the transition. For the boundary layer after the shock wave of sharp wedge,it can be obtained by the two-dimensional laminar similarity solution with the parameters of gas after oblique shock wave. In this paper,the velocity and temperature profile of basic flow are calculated by taking variable and constant specific heat,respectively. In the interest of saving space,there only shows the result of Mach number 8 and half wedge angle 8 degrees with adiabatic wall in Fig. 1. Similar results can be found in the other cases.

As shown in Fig. 1,for the velocity boundary layer,there is little effect of variable specific heat on the velocity profiles close to the outer of boundary layer,whereas the influence on the temperature profiles is obvious,especially near the wall. 3 Improved e^N method

According to Su^[1],compared with the traditional e^N method,which only considers the growth of disturbance,the improved e^N method takes the damping of disturbance into consideration. In the improved e^N method,the computational starting point is not taken to be at the neutral or zarf curve,but a position where the initial amplitude of the disturbance can be estimated. At a fixed position X very near to the leading edge,the estimated amplitude of disturbance is 0.000 3. In addition,we suggest that the transition position is at the place where the disturbance amplitude reaches 0.01.

Based on the linear stability theory,the disturbance can be written as

where φ represents u,v,T ,or p. x,y,and z are the flow,normal,and spanwise coordinates, respectively. i² = −1. In the spatial mode,ω is the real frequency,α and β are the respective flow and spanwise wave numbers,which are complex numbers,and the imaginary parts of them are the amplitude growth rates of disturbance. By taking formula (1) into the linearized disturbance align,the eigenvalue problem can be obtained,which is about the parameters ω, α,and β.

For the two-dimensional problem,N can be calculated from the following formula:

where A is the disturbance amplitude,and A0 is the initial value of A.

For the three-dimensional problem,the disturbance propagation direction is not parallel to the velocity direction at the outer layer of the boundary. According to the steepest descent method of complex function,Cebeci and Stewartson^[8] supposed that the wave propagation direction could be determined from tan ψ = −( ∂α/∂β )_r with the condition ( ∂α/∂β )_i = 0,here ψ is the angle between the wave propagation and flow directions. Hence,N can be calculated from the following formula:

In this paper,the initial amplitude of the disturbance A0 is 0.000 3,and thus,a value of N (N = 3.5) is used to estimate the transition position,which is obtained by the formula N = ln (0.01/A₀).

When the improved e^N method is used to predict the transition,the neutral curve and zarf curve with variables ω and Re would be calculated according to the basic flow. The results of Mach number 8,half wedge angle 8 degrees,and adiabatic wall with the variable and constant specific heat are shown in Fig. 2(a). The results of Mach numbers 6,7,and 8,half wedge angle 5 degrees,and adiabatic wall with the variable specific heats are shown in Fig. 2(b). The results of Mach number 8,half wedge angles 5,8,and 10 degrees,and adiabatic wall with the variable specific heat are shown in Fig. 2(c).

4 Results and analysis 4.1 Effect of different specific heats on transition position

To study the influence of variable and constant specific heats on transition position,the positions with the two cases of specific heat are calculated under different Mach numbers and half wedge angles. The N-X curves are calculated at any frequency for the first and second wave modes,and the ones shown in the figure are the curves for each mode in which position X is the nearest to the leading edge corresponding to the value of N as 3.5. The N-X curves are compared between the first and second mode waves,and the results are shown in Figs. 3 and 4. In the figure,v and c represent variable and constant specific heats,respectively,and the number is the value of T_w/T_e for isothermal wall.

As shown in Figs. 3 and 4,the corresponding positions of N reaching 3.5 are different under the same computing condition except different specific heats. No matter the wave is the first mode or the second mode,all transition positions decided by them under variable specific heat condition are nearer to the leading edge than constant results,and the influence on the second mode are much more obvious. When the Mach number is 6 or 7,and the half wedge angle is 5 degrees,the computing results remain the same. From the results,we can conclude that the transition prediction computing in the hypersonic boundary layer must take the variable specific heat factor into consideration. All the computations in the following are under variable specific heat condition without a special declaration. 4.2 Effect of Mach number on transition position

To study the influence of Mach number on transition position,the positions are calculated for Mach numbers 6,7,and 8 with adiabatic wall condition and half wedge angle 5 degrees. The N-X curves are compared between the first and second mode waves,as shown in Fig. 5.

Figure 5 illustrates the position X decided by the first mode wave becomes nearer to the leading edge with the increase of Mach number for adiabatic wall condition,and the similar result is observed more obviously for the second mode wave. These are different from the results of flat plate boundary layer with adiabatic wall condition,which the position X decided by the first mode wave becomes farther away from the leading edge with the increase of Mach number. Since the shock wave affects the Mach number and the air temperature,they would influence the transition position. 4.3 Effect of half wedge angle on transition position

To study the influence of half wedge angle on transition position,the positions are calculated for half wedge angles 5,8,and 10 degrees with Mach number 8 and adiabatic wall condition. The N-X curves are compared between the first and second mode waves,as shown in Fig. 6.

From Fig. 6,it can be seen that the position X decided by the first mode wave moves nearer to the leading edge with the increase of half wedge angle under Mach number 8. In addition, the one decided by the second mode wave moves nearer to the leading edge at first,but move farther away from the leading edge after that. Since the shock angle are influenced by both the Mach number of oncoming flow and the half wedge angle of sharp wedge,the Mach number and the air temperature after the shock wave are thereby affected. Therefore,the effects of half wedge angle on transition position are different for different Mach numbers. 4.4 Effect of wall temperature on transition position

To study the influence of wall temperature on transition position,the positions X are calcu- lated for different Mach numbers at different wall temperatures. The N-X curves are compared between the first and second mode waves,as shown in Fig. 7. The mark like “4.36 1-mode” in the figure means as follows: the number “4.36” is the value of T_w/T_e,and the “1-mode” represents the first mode.

From Fig. 7,it can be seen that the position X decided by the first mode wave becomes far away from the leading edge with the decrease of wall temperature under the same Mach number. In addition,the one decided by the second mode wave will be related to the Mach number,which moves father away from the leading edge for the Mach number 6,but moves nearer to the leading edge for the Mach numbers 7 and 8. 4.5 Comparison of results between sharp wedge and flat plate

The predicted results of transition position of sharp wedge,which are compared with the results of a flat plate of Fan et al.^[6],are shown in Tables 1 and 2.

Tables 1 and 2 give the transition positions for adiabatic and isothermal wall in different Mach numbers of oncoming flow,respectively. The results of both sharp wedge and flat plate, which are decided by the two modes,are shown in the tables with the relative difference between them,i.e.,(X_F − X_S)/X_F. The underline in the tables refers to the last transition position. From Table 1,it can be seen that the transition positions of flat plate boundary layer are about 20 meters which has been decided by both two modes. However,the transition positions of sharp wedge boundary layer are about 10 meters. The lager the Mach number of oncoming flow is,the closer the transition position of sharp wedge is to the leading edge than flat plate. It illustrates that the effect of oblique shock wave moves significant with increasing Mach number. From Table 2,it can be seen that the relative difference of transition positions between sharp wedge and flat plate becomes small with the decrease of wall temperature under the same Mach number,but the former is still nearer to the leading edge than the latter. In addition,for Mach number 7 with adiabatic wall,Mach number 6,T_w = 4.36 Te and Mach number 7,T_w = 7.4 Te with isothermal wall,the transition position of flat plate is decided by the second mode wave, while the one of sharp wedge is decided by the first mode wave.

As shown in Tables 1 and 2,for the Mach number 6,the transition positions of sharp wedge, which are decided by the second mode wave,are farther away from the leading edge than flat plate,but that is contrary to Mach numbers 7 and 8. In order to inspect the result,the positions are calculated for Mach numbers 6.25 and 6.5 with the adiabatic wall condition and compared with Mach numbers 6 and 7. The results are shown in Table 3.

As shown in Table 3,the transition positions of sharp wedge,which are decided by the second mode wave,are farther away from the leading edge than flat plate for the Mach number 6.25,but the gap between them is less obvious than the result for the Mach number 6. In addition,the transition positions of sharp wedge,which are decided by the second mode wave,are nearer to the leading edge than flat plate for the Mach number 6.5,but the gap between them is also less obvious than the result for the Mach number 7. Therefore,it can be concluded that at first the transition position of sharp wedge is far away from the leading edge than that of the flat plate,then gradually moves towards and will eventually become very close to the leading edge than that of flat plate as the Mach number increases from 6 to 7. 5 Shock wave relation with dual specific heats 5.1 Parameter relations around oblique shock wave with dual specific heats

In the common shock wave relation,the specific heats are the same around the shock wave with the assumption that the molecular vibrations in the air are completely non-excited. This assumption works well in the hypersonic flow with weak shock wave,but deviates from the actual situation for strong shock wave. According to Fan^[7],the specific heat before shock wave is 1.4 and the specific heat after the shock wave is related to the air temperature,i.e.,γ = γ(T ). The fundamental align of oblique shock wave is

The align of state for perfect gas is The geometrical relationship among deflection angle,shock angle,and velocity vector is where h₁ = c_p1T₁. The molecular vibrations of oxygen and nitrogen are excited when air temperature is between 600K and 2 500K. Therefore, and dh₂ = c_p2dT₂. Then, where T_ve is the characteristic temperature of air molecular vibrations. The shock angle β and the parameters v_n2,T₂,p₂,ρ₂ after the shock wave can be obtained by solving the above nonlinear aligns with Newton format. The relationship between the air temperature after shock wave T2 and the Mach number before shock wave Ma as well as the half wedge angle θ is shown in Table 4,which also gives a comparison with the results for the common shock wave relation. Figure 8 illustrates the relationship between the specific heat after shock wave and the Mach numbers before shock wave Ma as well as the half wedge angle θ.

As shown in Table 4,when the Mach number and the half wedge angle is small,the tem- perature after shock wave calculated from shock wave relation with dual specific heats has little difference with the one calculated from the common shock wave relation. However,the difference of air temperatures after the shock wave calculated from two different shock wave relations increases a lot for the larger Mach number and half wedge angle. Figure 8 illustrates that the specific heat after shock wave shows little changes for small Mach number and half wedge angle and has an evident variation for large Mach number and half wedge angle. Hence, we need to investigate the effect of using shock wave relation with dual specific heats on the transition position of hypersonic boundary layer. 5.2 Effect of shock wave relation with dual specific heats on transition

Based on the above analysis,the case with the maximum Mach number and half wedge angle, i.e.,Ma = 8 and θ = 10◦,is studied under adiabatic wall condition. Three different conditions are chosen to predict transition position of the hypersonic boundary layer: common shock wave relation and constant specific heat in the boundary layer,common shock wave relation and variable specific heat in the boundary layer,and shock wave relation with dual specific heats and variable specific heat in the boundary layer. The results are shown in Fig. 9. (cc,cv,and vv marked in the figure represent the first,second,and third conditions,respectively.)

From Fig. 9,it can be seen that the predicted transition position using the common shock wave relation and constant specific heat in the boundary layer is nearer to the leading edge compared with the other two conditions. Combining Table 4 and Fig. 8,it can be concluded that it is unnecessary to use the shock wave relation with dual specific heats under all the studied cases,but it needs to consider the variation of specific heat in the boundary layer. 5.3 Effect of shock wave relation with dual specific heats on transition under high Mach number

To study the influence of shock wave relation with dual specific heats on the transition position of hypersonic sharp wedge boundary layer under high Mach number,the case with Mach number 20,half wedge angle 8 is studied under adiabatic wall condition. At the same time,two different circumstances are also calculated with the same oncoming flow,half wedge angle,and wall condition. (i) Specific heat adopted the same value before and behind the shock. (ii) According to Fan^[7],the specific heat before the shock wave is 1.4,and the specific heat after the shock wave is 9/7 with the assumption that the molecular vibrations of oxygen and nitrogen are completely excited,when c_p = 4.5R and c_v = 3.5R. The transition positions of these three circumstances are predicted with the improved e^N method. The N-X curves are compared between the first and second mode waves,as shown in Fig. 10.

From Fig. 10,it can be seen that the transition position calculated by considering different specific heats around shock wave is nearer to the leading edge than the one calculated with the constant specific heat around shock wave for the first and second modes,as well as the one considering that the air molecules are completely excited. Therefore,there is of significance to use shock wave relation with dual specific heat under high Mach number flow or normal shock wave,such as the hypersonic reentry blunt bodies. 6 Conclusions

(i) When the Mach number of oncoming flow is not high enough,such as below 10,and the half wedge angle of sharp wedge is not large enough,such as below 10 degrees,the temperature increment of gas is small after the shock wave. Then,there is no need to consider taking different specific heats around oblique shock wave. However,when the Mach number of oncoming flow is high enough,such as above 20,it needs to take different specific heats around oblique shock wave.

(ii) In our calculation cases,the transition positions of sharp wedge computed with consid- ering variable specific are nearer to the leading edge than constant results. Compared with the results of flat plate,the transition position in wedge flow is much nearer to the leading edge. The relative differences between them can range from 30 to 50 percent in most cases.

(iii) The oblique shock wave directly affects the Mach number and the air temperature after the shock wave. Thus,its effect on transition position needs to have a comprehensive consideration.

(iv) The transition position of wedge boundary layer decided by the first or second mode wave is dependent of the Mach number of oncoming flow and half wedge angel of sharp wedge and wall condition.