1 Introduction

Transition prediction is very important for the design of aircrafts and aero-space vehicles. To solve the problem,one needs to investigate the evolution of disturbances in compressible boundary layers. The parabolized stability equations (PSEs)^{[1, 2, 3, 4, 5, 6, 7]} have been proven to be very useful for such problems^{[8, 9, 10]}. However,a careful study reveals that the conventional formulation of the nonlinear PSE method is not self-consistent. In the equation for those waves having the same wave number and frequency as one of the fundamental disturbances,the stream-wise wave number α_mn is kept to be a complex number,determined by the same way as those in the linear PSE. However,the wave number of the nonlinear driving terms of the equation,originating from nonlinear interactions of wave components,may have different imaginary parts from those of the fundamental wave. Moreover,in the equation for those waves whose wave number and frequency are different from those of the fundamental disturbances,the stream-wise wave numbers α_mn are obtained by means of algebraic operation from those of fundamental waves, and the nonlinear terms can also include terms having the same real part but different imaginary part for their stream-wise wave number as for α_mn. Take the incompressible boundary layer as an example. If there is a fundamental wave with frequency ω₁₁ and wave number α11, then through nonlinear interaction,higher harmonics will be generated,which includes terms with frequency 2ω₁₁ and wave number 2α₁₁,as well as terms with the same frequency but different wave number as 2α₁₁ +2iα_11i,i.e.,the real parts of their wave numbers are the same, but the imaginary parts are not. Because the number of such terms may become enormous as the order of nonlinear terms becomes higher and higher,to make the problem tractable,the real part of their wave number is determined by the phase-locked rule^[11],which is

where Re indicates the real part,the wave of mode (m₁,n₁) is one of the fundamental disturbances, and the imaginary parts of their wave numbers are simply deleted.

Obviously,the above treatment is not self-consistent in the wave number. To make it selfconsistent, a simple way is to let all wave numbers be real at the very beginning without the requirement that the shape functions of the fundamental disturbances should be normalized in their magnitude,and instead of that the growth and decay of the waves are manifested in the imaginary parts of their wave numbers,it is manifested simply in the growth or decay of the modulus of their shape functions. Correspondingly,the rules for the wave number iteration should also be modified.

In order to verify if the new formulation works,besides the merit of its self-consistency,the results from its applications to concrete problems should be verified,say,by comparison with those obtained by the corresponding direct numerical simulation (DNS).

The case study includes both linear and nonlinear PSEs. Although there is no problem of inconsistency in conventional linear PSEs,the comparison is still necessary as it is the starting point of the nonlinear PSEs. 2 Governing equations and numerical method

In the PSE method,the disturbance vector φ is expressed as

where is the vector shape function,x,y,and z are the stream-wise,normalwise, and span-wise coordinates,respectively,t is the time,ω is the frequency,α and β are the stream-wise and span-wise wave numbers,respectively,and m and n are the mode numbers.

The governing equations and the numerical method can be found elsewhere,for example in Ref. ^[7]. The nonlinear parabolized equations are written as

where m = 0,±1,±2,· · · ,n = 0,±1,±2,· · · , are the coefficients matrices, and is the vector nonlinear term,which is expressed as where F_mn is the Fourier component with the frequency m! and the span-wise wave number nβ of the sum of all nonlinear terms Fⁿ in the nonlinear parabolized equation,i.e.,

For the conventional PSE,usually one of the following iteration rules is used for the normalization of the shape functions:

where E represents the disturbance energy,and c indicates the complex conjugate.

In the self-consistent formulation of the PSE,the stream-wise wave number α is kept to be real,while the magnitude of the shape function can be varied. Therefore,the iteration rules should be modified correspondingly. For example,the first iteration rule (5) should be modified to be

where Im indicates the imaginary part. The difference between (5) and (9) is that the former implies that the modulus of the shape function is kept unchanged,while the latter only requires that the phase of the modulus of the shape function remains unchanged.

Also,the iteration rule (8) should be modified to be

3 Results and discussion

In this section,both the conventional PSE and the self-consistent PSE are used to compute the evolution of disturbances in a hypersonic boundary layer of a flat plate with Mach number of 6,and the results are compared with those from the DNS. The basic flow in the PSE computation is the same as that in the DNS.

The governing equations for the DNS are the compressible Navier-Stokes equations. A fifth-order upwind scheme and a sixth-order central scheme are used for the convective and viscous terms,respectively. A three-step and third-order Runge-Kutta scheme is used for time advancing. For details of the computational method,one can refer to Ref. ^[12]. After the steady basic flow is obtained by the DNS,the initial disturbance,which is a T-S wave obtained under the basic flow profile,is induced at a fixed location (x_δ = 100) in the computational domain. The computation ends up when the whole flow field becomes periodic with the period of the imposed T-S wave.

We first make the comparison for the linear PSE,which includes two cases,one with a relatively low Reynolds number,while the other with a large Reynolds number. The Reynolds number is based on the displacement thickness δ*₀ of the inlet of the computational domain. Then,the comparison is made for the nonlinear PSE.

For the linear cases,the initial amplitude of disturbance waves can be arbitrary for the PSE, but small enough for the DNS,and then simply multiply its result to make its initial amplitude the same as that in the PSE. For the nonlinear cases,the initial amplitude of disturbance waves should be the same for both the methods.

The non-slip and adiabatic boundary conditions are used at the wall.

The parameters of three cases are shown in Tables 1 and 2. All nondimensional quantities in this paper are based on the displacement thickness δ*₀ and the quantities of the oncoming flow,for which the temperature is T*_e = 79 K.

For the first case,the initial disturbance is a two-dimensional unstable T-S wave of the first mode obtained by the linear stability theory (LST) with the frequency ω_δ = 0.840 0,the stream-wise wave number α_δ = 0.905 6 − 0.003 955i,and the span-wise wave number β_δ = 0. The initial disturbance is imposed at x_δ = 100. The disturbance amplitude for the DNS is taken to be 7.5 × 10⁻⁹.

The iteration rule (8) is used for the complex wave number,and the rule (10) is used for the real wave number.

Figure 1 shows the disturbance amplitude development obtained by the DNS,the linear PSE, and the LST. For the PSE,the stream-wise velocity and the temperature of the disturbance at x₁ are expressed as

and A_u = is the amplitude of the stream-wise disturbance velocity.

The figure shows that the amplitudes obtained by both the conventional and self-consistent PSEs are close to the DNS result,obviously better than those from the LST. It is due to the nonparallel effect of the boundary layer,which is not trivial for the case of small Reynolds numbers. There is a slight difference between the results from the conventional and self-consistent PSEs, and the latter is a little closer to the DNS result.

Figures 2 and 3 show the profiles and at x_δ = 150 and 250. Again,the results from the PSE computations are close to the results from the DNS and are apparently better than those from the LST,and the result from the self-consistent PSE is slightly closer to the result from the DNS than that from the conventional PSE.

Figure 4 shows the real part α_δr of the wave number α_δ. The results from the two PSE methods are very close to each other,but the result from the LST is appreciably different.

The second is also for the linear PSE,but the Reynolds number is much larger,i.e.,Re_δ = 10⁵. The initial disturbance is a two-dimensional unstable T-S wave of the first mode with the frequency ω_δ = 1.342 and the stream-wise wave number α_δ = 1.470 − 0.009 075i.

Figure 5 shows the development of the disturbance amplitude. The results from both the PSE computations for complex α and real α are close to the DNS result,while the result from the LST is obviously different from the DNS one.

Figures 6 and 7 show the comparisons of the profiles at x_δ = 300 and 500. At x_δ = 300,the profiles for the two PSE computations agree quite well with the DNS results. However,both of them are slightly different from the DNS results at x_δ = 500. The amplitude of the PSE for the real wave number α is slightly larger than that of the DNS,while the result from the PSE for complex α is on the opposite side. The result from the LST is significantly different from that from the DNS,but the shapes of the profile bear similarity to those of the DNS.

Figure 8 shows the real part α_δr of the wave number α_δ. The results of the two cases,i.e.,the PSE computations for complex and real α,are very close to each other. The figure also shows the LST result,which is close to the PSE results for x_δ < 250 and x_δ > 450,but far from those of the PSE from x_δ = 250 to 450. In general,the LST results of this large Reynolds number case are closer to the linear PSE results than those in the previous case,which is apparently due to that the non-parallel effect becomes weaker when the Reynolds number becomes larger.

The third case is for the nonlinear PSE. The free-stream parameters are the same as those in the first case,but the two-dimensional T-S wave as the initial disturbance is different. Its frequency is ω_δ = 0.700 0,and the stream-wise wave number is α_δ = 0.764 3 − 0.001 598i. In order to have the nonlinear effects,the initial amplitude cannot be too small and is taken to be 0.003 75. The initial disturbance is denoted as the component (1,0),implying that its frequency is ω_δ and its span-wise wave number is 0. The total components in the computation include terms from (−7,0) to (7,0).

Figure 9 shows the comparison of the disturbance amplitudes obtained by the DNS and the nonlinear PSE. The wave of order (1,0) is the fundamental wave. The wave of order (0,0) (i.e., the mean flow distortion) and the wave of order (2,0) (i.e.,the harmonic wave) are generated by nonlinear effects. For the fundamental wave,the results of two nonlinear PSE computations are very close to that of the DNS,and both of them are slightly larger than that of the linear PSE,implying that the nonlinear effect enhances the growth of fundamental wave,but the effect is not appreciable in this case. For the second harmonic wave and mean flow distortion, the results of two PSE calculations are close to each other,but they are somewhat different from that of the DNS.

Figures 10 and 11 show the comparison of the profiles at x_δ = 150 and 230,respectively. For the fundamental wave,the results of two nonlinear PSE computations agree very well with the DNS result. For the mean flow distortion and harmonic wave,there are slight differences between the nonlinear PSE results and the DNS results.

Figure 12 shows the real part α_10δr of the wave number α_10δ. The results of the two nonlinear PSE cases are very close to each other. Both of them are different from the linear PSE result because of the nonlinear effects.

The above three case studies show that the self-consistent PSE method does work,and its accuracy is at least the same,or sometimes a little larger,as compared with those of the conventional PSE method. 4 Conclusions

(i) The conventional nonlinear PSE method is not self-consistent. Therefore,a self-consistent PSE method is proposed. The rules for the wave number iteration for the fundamental waves have to be modified accordingly.

(ii) The comparison of the results from both the conventional and self-consistent PSEs with those obtained by the DNS shows that there is no essential difference for the numerical results by either method. The reason might be that the imaginary parts of the wave numbers are very small compared with its real parts in the test cases. However,anyway,the self-consistent formulation works and is better from the theoretical point of view.

Acknowledgements The corresponding author acknowledges that the second author has essen- tially an equal contribution to this work,and both of them are grateful to Professor Heng ZHOU for his constant encouragement and useful suggestions.