1 Introduction

Laminar-turbulence transition prediction is one of the unsolved problems in fluid mechanics. It is concerned by the engineers as the development of the hypersonic vehicle. The e^N method^[1] has been widely used to predict the transition location of the boundary layers in aerodynamic design. How to predict the amplification of the disturbance accurately plays a key role in the boundary layer instability analysis.

There are lots of procedures for obtaining the amplified rate of an unstable disturbance, such as the linear stability theory (LST)^[2] and the linear parabolized stability equations (PSE)^[3]. The LST is a conventional methodology for searching the spatial growth rate. With the quasi-parallel assumption in the laminar mean flow, the Orr-Sommerfeld (O-S) equation was solved using the LST. The dispersion relation of a disturbance can be obtained by solving a linear eigenvalue problem, and the negative imaginary part of the complex wavenumber α represents the spatial growth rate. Nevertheless, when the non-parallel effect of the base flow is strong (i.e., the local Reynolds number is lower), the LST loses its accuracy. Subsequently, the multiple-scale techniques were used^[4-6] to account for the non-parallel effects of the base flow. However, the complexity of this technique makes it difficult to apply to the engineering problems. The linear PSE is an alternative methodology for calculating the spatial growth rate in a non-parallel boundary layer. It drops the streamwise derivatives in the viscous terms to eliminate the elliptic characteristics of the linearized Navier-Stokes (N-S) equations. Because the PSE is parabolic in the streamwise direction, it can be solved by a marching procedure with higher efficiency. This approach was firstly proposed and applied in incompressible flows by Bertolotti and Herbert^[7]. Subsequently, it was applied in compressible boundary layers by Chang et al.^[8], Zhang and Zhou^[9], and Wang et al.^[10]. The PSE has become a popular approach in the stability analysis, but it is still not satisfactory. On one hand, the PSE is an initial-boundary problem. It means that the solution to the PSE depends on the initial information at the inlet. To obtain the dispersion relation of the disturbance downstream, one should give an accurate initial disturbance upstream that matches physics. Then, a marching procedure is adopted. Apparently, compared with the local eigenvalue problem, the process of specifying the initial condition increases the complexity of the PSE. On the other hand, the results of PSE are the evolution of a disturbance with a given frequency. To obtain the envelope of the N factor in the e^N method, one has to compute the evolutions of all disturbances with different frequencies and spanwise wavenumbers. However, for a local eigenvalue problem, the maximum N factor can be obtained simply by integrating the local maximum growth rate. Therefore, the PSE is inefficient and inconvenient in general three-dimensional (3D) boundary layer transition prediction.

A local methodology accounting for the non-parallel effects is required. Bertolotti and Herbert^[7] proposed a non-parallel approach in the incompressible boundary layer. They expanded the PSE locally with the first-order Taylor expansions. A local eigenvalue problem was obtained to describe the dispersion relations of a disturbance in a non-parallel boundary layer. This approach combines the advantages of the LST and the PSE. It not only can account for the non-parallel effects of the boundary layers but also can obtain the dispersion relations of the unstable disturbances directly without solving the evolution problem. However, the solving procedure for this eigenvalue problem lacks robustness because it requires convergence of two eigenvalues: the streamwise wavenumber (α) and its streamwise derivative (dα/dx). In addition, the convergence depends on a good initial value of dα/dx. For its poor robustness, this method was only used by Bertolotti to compute the initial condition of PSE. Without using the Taylor expansions, Chang^[11] proposed a non-parallel eigenvalue solution (NES) by combining the PSE at two near locations. Actually, this method was similar to the Taylor expansion of PSE by Bertolotti and Herbert^[7]. With two eigenvalues, the NES is still not robust, especially in a strong non-parallel boundary layer. For example, for the crossflow instability near the leading edge of a swept wing, a good guess of the initial value is crucial to convergence of the eigenvalue calculation. Recently, with the idea of Taylor expansions, Huang and Wu^[12] expanded the incompressible stability equations to the higher order expansions. In the expansions, an additional new assumption that the local streamwise wavenumber is constant is introduced for reducing the unknown variable dα/dx. Yu et al.^[13] also used this assumption to simplify the expansion of PSE and investigated the instability in the compressible flow. This method with only one eigenvalue is called the EPSE. Subsequently, the general forms of the nth-order EPSE were derived by Lu and Luo^[14] and applied to the crossflow instability analysis of a swept-wing boundary layer.

However, the numerical procedure with the assumption dα/dx=0 in the EPSE method has some drawbacks, although it is easy to be solved with one unknown eigenvalue. For example, the mathematical analysis and the numerical calculation^[12] have demonstrated that two eigenmodes will be obtained by the EPSE simultaneously, which correspond to the multiple solutions to the eigenvalue problem. It leads to the difficulty determining which one should be selected in the transition prediction. It also implies that there is a paradox between the multiple solutions and physics that only one unstable disturbance can be found in a local two-dimensional flow with a given frequency. In addition, although the EPSE produces more accurate growth rate prediction than the LST, there is still a significant difference compared with the PSE prediction when non-parallelism is strong. A higher order Taylor expansion can reduce the difference, but the improvement is finite and the convergence becomes poor. These drawbacks lead to difficulties in the growth rate prediction of the unstable wave. Therefore, it is necessary to improve the EPSE. For this aim, an improved EPSE is proposed in this study. The instability of the boundary layer for the flat plate, the blunt cone, and the swept-wing is investigated to examine this new method.

2 Method 2.1 EPSE formula

Considering a general 3D compressible boundary layer, the flow is described in a general orthogonal curvilinear coordinates (x, y, z). The corresponding Lamé coefficients are h₁=1+k_xy, h₂=1, and h₃=1+k_zy, in which k_x and k_z are the streamwise and spanwise curvature at the surface, respectively. The coordinate variables are non-dimensionalized by the length scale L^*, where the superscript * indicates a dimensional quantity. The temperature T and the density ρ are scaled using the free-stream values T_e^* and ρ_e^*, respectively. The dimensionless pressure is p=p^*/(ρ_e^*U_e^*2).

In general, the instantaneous flow quantities can be expressed as

(1)

where ϕ = [ρ, u, v, w, T]^T is the mean-flow components, and ϕ = [ρ', u', v', w', T']^T is the perturbation quantities. Substituting Eq. (1) into the N-S equations and dropping the nonlinear terms in the equations, the linearized version of the N-S equation for the disturbance is obtained,

(2)

where the matrices Γ, A, B, C, D, V_xx, V_xy, V_xz, V_yy, V_yz, and V_zz are functions of the base flow quantities, and the coefficients are given in Appendix A. indicates the partial derivative, where i=1, 2, 3. x₁, x₂, and x₃ are the coordinates x, y, and z, respectively.

For a single wave, the perturbation can be written as

(3)

where is the disturbance shape function. α and β are the streamwise and spanwise wavenumbers, respectively, and ω is the frequency. Here, α is the slowly varying function of the streamwise coordinate x. x₀ is the x-coordinate of the inlet. Substituting Eq. (3) into Eq. (2) and neglecting the second derivatives in the x-direction, the parabolized stability equations are obtained,

(4)

where the matrices , and are given by

(5)

For simplicity, Eq. (4) can be rewritten as

(6)

where , and .

Both the mean flow and the shape function varying in the streamwise direction are taken into account in the PSE. Then, the local variation with respect to x in the vicinity of an arbitrary point x₀ can be expanded in Taylor series, namely,

(7)

Substituting Eq. (7) into Eq. (6), and neglecting the terms including (x - x₀)² and higher order, we can obtain the first-order expansion of PSE as follows:

(8)

where the streamwise derivatives of the matrices , , and are given by

(9)

In the matrix , the term containing dα/ dx is neglected.

The boundary conditions at the wall are taken to be

(10)

In the free stream, the Dirichlet boundary conditions are used,

(11)

Equation (8), together with the boundary conditions, constitutes an eigenvalue problem, which is called the (first-order) expansion of the PSE (EPSE). Note that in Eq. (9) and in the matrix , the variation of α in the streamwise direction is neglected, i.e., dα/dx=0. Thus, the EPSE has only one unknown eigenvalue β. Similar to the O-S equations, the homogeneous systems can be solved using an iteration method. For a fixed location x₀, the frequency ω, the wavenumber β, the eigenvalue α, and the eigenvectors and can be obtained by solving the EPSE.

In the EPSE, the complex streamwise wavenumber is defined as

(12)

where A is the amplitude of disturbances. represents a norm of A in the wall-normal direction. The term indicates the correction of the wavenumber by the distortion of the shape function in the streamwise direction. Generally, the maximum value of streamwise velocity in the wall-normal direction is selected to correct the complex wavenumber, i.e., . Thus, the physical growth rate is , and the physical streamwise wavenumber is .

In the present study, only the first-order EPSE is derived. For more details about its high order forms, one can see the general forms of the nth-order expansion of PSE (EPSEn) formulated by Lu and Luo^[14].

2.2 Improved EPSE

In Subsection 2.1, the traditional EPSE is derived with the assumption dα/dx=0. Thus, multiple eigenvalues are obtained by solving the EPSE. Specifically, for n-order (n=1, 2) expansion of the PSE, there exist n+1 different eigenvalues α, leading to n+1 different growth rates. Further discussion about the multiple eigenvalues is available from Huang and Wu^[12]. With the correction of the distortion of the shape function (see Eq. (12)), the discrepancy between these multiple eigenvalues can be reduced, and the two corrected wavenumbers can converge to the nearly same value. However, when non-parallelism of the base flow is relatively strong (like the oblique T-S waves and cross flow instabilities), the correction can only be partial and the discrepancy will be significant. It is difficult to choose the appropriate one in these solutions. What is more, the prediction of the EPSE will diverge from the PSE result, even losing its accuracy for the prediction. In order to reduce the multiple eigenvalues of the EPSE and improve the accuracy of the prediction for the instability waves, the improved EPSE (referred to as iEPSE hereafter) is proposed in this work.

Note that an assumption dα/dx=0 is made in the derivation of the EPSE in Eq. (9). This assumption is proposed considering that the variation of streamwise wavenumber is slow such that the variation of the dispersion relation can be absorbed in the variation of the eigenfunction. This variation of the dispersion relation then will be retrieved by correcting the eigenvalue (see Eq. (12)). However, the decomposition form in Eq. (3) requires that the variation of the eigenfunction is slow^[7]. Assuming dα/dx=0, the eigenfunction will change rapidly and the magnitude of O(1/Re) for ∂_xϕ may not be satisfied. Moreover, in the derivation of the EPSE, the terms of O(1/Re) are retained and the higher order terms are neglected. It means that for in Eq. (9), the terms of O(1/Re) should be retained. However, the term with the order O(1/Re) is neglected in the EPSE, which is apparently inappropriate.

Based on the above discussion, the iEPSE is derived. Its basic form is unchanged compared with the EPSE, except for the matrix,

(13)

in which there is one more term compared with Eq. (9). There is more than one term associated with dα/dx, such as the terms with V_xx and V_xz in , and V_xy in . However, the magnitude of these terms is O(1/Re), which is smaller than the term in with the order O(1). Therefore, these terms are neglected in the iEPSE.

The eigenvalue problem then becomes

(14)

where L is the linear operator. Unlike the traditional EPSE problem, this eigenvalue problem has two unknown eigenvalues, i.e., α and .

Note that in the PSE, both the shape function and the wavenumber depend on x. The oscillatory wave should be placed in α as much as possible to minimize the shape function variation in x. Therefore, referring to the approach in the PSE, a supplementary condition

(15)

is introduced, in which the norm of shape function can be selected as the maximum of the streamwise velocity in the wall normal direction or the kinetic energy integral in the wall normal direction. Together with this equation, Eq. (14) is solvable. Equations (14) and (15) form the basic equations of the iEPSE. Because the streamwise variation of the norm of the shape function has been absorbed in α, its growth rate σ = -α_i, and the physical streamwise wavenumber is α_r.

The idea of remaining ∂_xα in the present compressible 3D iEPSE is similar with the non-parallel eigenvalue problem proposed by Bertolotti and Herbert^[7] in the incompressible two-dimensional (2D) flow. Besides, the NES proposed by Chang^[11] contains the similar idea with two unknown eigenvalue α_i and α_i+1 at two near locations (i and i+1). However, the NES is not physical compared with the iEPSE. The NES is derived by combing the PSE at two near locations (i and i+1). Therefore, it is just the first-order approximation and it is difficult to be extended to high order equations. Moreover, the NES, even though the two locations are consecutive and near, is not a complete local problem. This is inconvenient in boundary layer stability analysis.

2.3 Solving procedure of iEPSE

Generally, the Newton iteration is used to solve the problem of two eigenvalues. However, as mentioned by Chang^[11], the problem is less robust because it requires convergence of two eigenvalues. The lack of robustness also results from the inconsistent initial guess of dα/dx=0. Without the Newton iteration, an improved iteration procedure is proposed in this paper. In the solving procedure of the PSE, the shape function is obtained by solving the PSE with an initial α. Then, the correction of α is made by the conservation relation (see Eq. (15)). Compared with the PSE, the iEPSE is very similar except that it is a local eigenvalue problem. It means that the conservation relation (Eq. (15)) can also be adopted to correct α. Following this idea, the iteration relation for α is constructed as follows:

(16)

The marching approach for solving the iEPSE can be outlined as follows:

Step (ⅰ) α⁰ is initialized by using the eigenvalue of the LST.

Step (ⅱ) dα/dx and ϕ are obtained by solving Eq. (14).

Step (ⅲ) A new α is calculated by using the iteration relation (16).

Step (ⅳ) Repeat Steps (ⅱ)-(ⅲ) until the streamwise wavenumber satisfies the convergence condition |α_iⁿ - α_i^{n - 1}| < ε. Hence, most of the oscillation in the shape function is contained in α.

The initial guess of dα/dx is important for the start of the iteration procedure. In the traditional approach, this value is given at first to start the whole solving process. Nevertheless, when the baseflow is non-parallel compared with the unstable perturbation, the solving process may be divergent if the initial dα/dx is set to be zero. It may come from the fact that zero is often not in the radius of convergence of the real dα/dx in this case. On the contrary, in our new solving procedure, different from a given initial dα/dx in Newton iteration, it is solved in Step (ⅱ) and an α given by the LST is used to start. With this procedure, it can be seen that the eigenvalue problem of dα/dx is the one order correction of the zeroth-order streamwise wavenumber α. As a result, it will be more likely to lead to a converged solution even with an initial guess of dα/dx=0 in Step (ⅱ).

2.4 Discussion about advantage of iEPSE

By comparing with the equations in the traditional EPSE and the iEPSE, one can find that they both contain the correction procedure for the eigenvalue α. However, there is an essential difference between these two methods. In the traditional EPSE, this correction is a post procession. It means that the streamwise variation of the shape function is unrestricted in the solving procedure of the eigenvalue problem. As a result, the wave components in the shape function are retained. In other words, the assumption that the shape functions and the wavenumbers vary slowly in the streamwise direction is not satisfied in some cases. However, the iEPSE introduces the conservation relation to normalize the shape function. This restriction is contained in the solving procedure of the eigenvalue problem, thus ensuring that the streamwise variation of the shape function is as slow as possible. As a result, all of the wave components in the shape function are placed in α. This is apparently more in line with the assumption of the PSE.

In addition, the Taylor expansion introduces more degrees of freedom in the system so that it leads to more eigenmodes. Ideally, these mathematical eigenmodes will converge to the same value with the correction procedure in the traditional EPSE. However, this is nearly impossible in practice, especially in the strong non-parallel flow. However, for the iEPSE, with the constraint of the conservation relation, the eigenvalue problem only produces one eigenmode. This is apparently more physical.

With these two advantages, the iEPSE is more accurate and convenient in the stability analysis and the transition prediction. In the next section, the performance of the iEPSE will be examined in three common cases.

3 Cases and validation 3.1 Flat plate case

To validate the iEPSE, the linear instability in a flat plate boundary layer is investigated at first. Yu et al.^[13] proposed the EPSE method and examined it in flat-plate cases. The same cases are investigated by using the iEPSE approach in our study. The flow parameters are shown in Table 1. The subscript e indicates a boundary layer edge flow parameter. Two types of the unstable modes are investigated by the iEPSE. The parameters of the disturbances are listed in Table 2.

For the flat plate boundary layer, the first mode and the second mode are the typical streamwise instabilities. Firstly, the second mode instability is calculated. This mode is often the most unstable mode in a hypersonic boundary layer. When the Reynolds number is high, the boundary layer tends to be parallel so that the non-parallel effect can be ignored. In this case, the results of the iEPSE and EPSE should be similar and they should agree with the results of the PSE. To validate it, the instability analysis is performed by using the LST, EPSE, iEPSE, and PSE methods. Figure 1 shows the instability analysis results of the base flow profile at x=16 m for a high Reynolds number case. The eigenvalue -α_i, growth rate σ, and correction of -α_i are compared between the EPSE and iEPSE. For the first-order EPSE, two eigenvalues can be obtained simultaneously. The branch with larger -α_i is labeled with EPSE_upper, while the other one is labeled with EPSE_lower. In Fig. 1(a), there is a discrepancy between the eigenvalues of the upper and lower branches of the EPSE. This discrepancy is corrected by the distortions of the shape functions (shown in Fig. 1(c)), and their growth rates (shown in Fig. 1(b)) eventually converge to be unique. For the iEPSE, only one eigenvalue is obtained in the solution to the eigenvalue problem and dα_i is zero. Because the distortion of the shape function has been absorbed in α, its growth rate is the same as the eigenvalue -α_i. Because of the quasi-parallelism of the baseflow, both approaches can give similar prediction on the unstable disturbances.

The physical growth rates and the amplitudes of the disturbances are often concerned in the instability analysis. Figure 2 compares the amplitude and growth rate variations of the wave from x=16.0 m to 16.4 m predicted by the LST, PSE, EPSE, and iEPSE. In this case, the amplitude of the disturbance is amplified about 75 times. A perfect agreement can be observed among the amplitudes of the LST, the traditional EPSE, the iEPSE, and the PSE. For the growth rate curves shown in Fig. 2(b), the maximum percentage difference between these results is only about 0.3%. This is because that the streamwise location is far away from the leading edge of the flat plate and non-parallelism is weak. These results indicate that the iEPSE is available for the weak non-parallelism case.

To validate the results of the iEPSE in the strong non-parallelism flow, a low Reynolds number case is computed. In this case, the first mode is used as the benchmark. The streamwise wavenumber of this mode is often very small. It means that the wavelength of the first mode is rather long compared with that of the second mode. As a result, the scale difference between the disturbances and the baseflow is small, i.e., non-parallelism is strong in this case. Additionally, an oblique first mode is chosen. On one hand, it is often the most unstable wave for the first mode in the flow field. On the other hand, the oblique mode often has a smaller wavenumber than the planar one. Figure 3 compares the amplitudes and the growth rates of the oblique first mode predicted by different stability analysis approaches at ω = 0.4. Two spanwise wavenumbers are selected: β = 0.3 and β=0.4. Their results are plotted in Fig. 3. Figures 3(a) and 3(b) are the results of the disturbance with β=0.3, and Figs. 3(c) and 3(d) are the results of β=0.4. For the disturbance with β=0.3, the amplitude of the disturbance is amplified about 21 times for the PSE calculation but only 14 times for the LST. The two branches of the traditional EPSE can be obtained, and the amplitudes (or growth rates) of them are both less than the results of the LST although the upper one is closer to the LST. As discussed above, the non-parallel effect of the base flow is significant. As a result, the PSE here is the criterion for checking the local prediction approaches. It can be seen that the iEPSE has an excellent prediction on the amplitude and the amplified rate, while the prediction of the traditional EPSE is less than the PSE significantly. The similar results can also be observed for the β=0.4 case. It suggests that the iEPSE is a good local choice for calculating the evolutions of the streamwise instabilities in the non-parallel boundary layers.

Compared with the amplitudes and the growth rates, the shape functions of the unstable disturbances are vital for examining the accuracy of the prediction tools. Figure 4 shows the pressure shape functions and the wall-normal velocity shape functions for the disturbance with β=0.4 at x=0.45 m. The enlarged view in this figure shows the distribution in the boundary layer. A discrepancy in the amplitude of the shape functions can be seen between different methods. Compared with the traditional EPSE method, the iEPSE results are closer to those of the PSE. This is identical with the growth rate results shown in Fig. 3. It indicates that when non-parallelism is strong, the traditional EPSE can hardly improve the prediction accuracy compared with the LST, but the iEPSE works excellently.

3.2 Swept wing flow case

The boundary layer instability on a swept wing is very challenging to predict accurately due to its strong non-parallelism. For this typical 3D boundary layer, the crossflow instability dominates the unstable mode in the leading edge region^[15].

For the traditional EPSE, the cross flow instability was analyzed by Lu and Luo^[14] using the EPSE method. It was demonstrated that the EPSE is more accurate than the LST in predicting the growth rate of the cross flow mode. However, a discrepancy between the EPSE prediction and the PSE result still exists, even with a second-order expansion of the PSE (ESPE2). To validate the performance of the iEPSE, the same flow condition is taken in this study. The swept wing is an NLF(2)-0415 airfoil with the chord length c=1.29 m, the attack angle θ=-4°, and the swept angle Λ=45°. The detailed flow parameters are given in Table 3. The evolution of the stationary cross flow mode with ω=0 and the travelling wave mode with ω=0.1 is computed.

As done in the flat-plate boundary layers, the amplitudes and the growth rates are investigated in the swept-wing boundary layer. Figure 5 shows the amplitudes and the growth rates of the stationary instability obtained by different methods. As can be seen from Fig. 5(b), at the downstream location x=0.4, both the parallel and non-parallel stability prediction results show excellent agreement owing to the weak non-parallel effect. However, at the upstream location, there is a significant discrepancy among these results. The lower branch of the traditional EPSE has a lower growth rate than that of the PSE though it gives a more accurate result than that of the LST. On the contrary, the upper branch has a larger growth rate (resulting in a larger amplitude) than that of the PSE. This leads to a confusion to select which of them as the EPSE prediction. If the most unstable branch, i.e., the upper branch, is selected as the eigenmode to predict the instability, it is more conservative. For the iEPSE approach, there is only one branch of the growth rate curve and the confusion of the selection is nonexistent. Moreover, it shows excellent agreement with the PSE results, even though non-parallelism of the base flow is strong at the leading edge.

The travelling mode case has very similar results. Figure 6 compares the amplitudes and the growth rates of the travelling wave with ω = 0.1 obtained by different methods. In this case, the growth rate is larger than those of the stationary mode case, but the trends of the amplitude and the growth rate curves are similar. The results of iEPSE are more accurate than those of the traditional EPSE. It indicates that the iEPSE is more effective than the traditional EPSE in predicting the amplification of the cross flow instability in compressible boundary layers, not only for the stationary mode but also for the travelling one.

The traditional EPSE can be improved by a higher expansion process. In the swept wing stability analysis by Lu and Luo^[14], it has been shown that the prediction of the EPSE2 is more accurate than that of the EPSE. Here, the iEPSE is evaluated compared with the EPSE2. Figure 7 shows the amplitudes and growth rates predicted by the EPSE2^[14], iEPSE, and PSE. Although the amplitude and the growth rate predicted by the EPSE2 seem satisfied, the results of the iEPSE match the PSE prediction better (which is clear in the enlarged view). The percentage difference between the results of the EPSE2 and the PSE is about 2.5%. The difference can be reduced to 0.6% by the iEPSE. It seems that there is only little difference between the iEPSE and EPSE2. Nevertheless, from the EPSE to the EPSE2, the scale of the matrix in the numerical computation will be enlarged from 2 × 2 to 3× 3 for a higher Taylor expansion. It is inconvenient in the instability analysis. Moreover, the EPSE2 obtains three branches of the growth rate curves, i.e., three different eigenvalues, bringing more trouble on the selection of a proper eigenvalue to compute the physical growth rate.

Besides the compressible cross flow instability, the incompressible cross flow instability is also investigated. The model is also the NLF(2)-0415 airfoil. The flow parameters are the same as those of Haynes and Reed^[16] and are listed in Table 4. The basic flow data for the stability analysis were provided by Jing. The non-parallel effect is stronger than that in the compressible flow for the crossflow modes. The stability for the flow is analyzed by using the iEPSE, and the comparison of the amplitudes and the growth rates of the stationary vortex with ω=0 and β=0.52 is plotted in Fig. 8. As can be seen, the growth rate curve of the upper branch of the traditional EPSE deviates significantly from the PSE results. The lower branch of the traditional EPSE gives a slightly larger prediction than the LST results. It means that the traditional EPSE nearly has no improvement compared with the LST prediction in this case. However, the result of the iEPSE shows excellent agreement with the PSE prediction. The improvement is apparently remarkable.

In the PSE, there is often a transient growth if the disturbance introduced at the inlet does not match the physical local solution. As a result, less divergence with the real solution leads to less transient growth region. The pressure disturbance profiles at x=0.1 obtained by the LST, EPSE, and iEPSE are plotted in Fig. 9(a). It can be seen that the disturbance shapes predicted by different stability analysis approaches are different. In order to investigate which method is proper for the local non-parallel stability analysis, different profiles obtained by different methods shown in Fig. 9(a) are introduced at the inlet. Then, the marching calculations are performed by the PSE and the growth rates are compared in Fig. 9(b). It can be seen that the amplified rates are different near the inlet, but they eventually tend to the same one downstream of x=0.2. The discrepancy among these results indicates an adjustment process of the disturbance evolution. Note that if the inlet profile is obtained from the iEPSE shape function, the growth rate curve is smooth and identical with that of the iEPSE. It indicates that the eigenvalue and eigenfunction generated by the iEPSE produce a smaller transient effect than the traditional EPSE solution, and it is more accurate to provide the initial value for the evolution calculation of the disturbance. In addition, it also suggests that the iEPSE can give a proper evaluation of the growth of the unstable crossflow mode wave.

3.3 Blunt cone flow case

The blunt cone flow is a simplified model for a hypersonic aircraft. The instability analysis for the unstable wave is performed for this model by using the iEPSE. The parameters are selected from Kara et al.^[17]. It is a hypersonic flow over a 5-degree straight cone at a free-stream Mach number of 6.0. The nose radius is 0.05 inch, and the Reynolds number based on this nose radius is 32 500. The wall temperature condition is adiabatic. Figure 10 compares the boundary layer profiles at several streamwise locations from Kara's results^[17] and the present direct numerical simulation (DNS) results. For each location, excellent agreement can be observed, indicating that the DNS solver for the base flow is reliable.

The stability analysis is performed by using the LST, EPSE, iESPE, and PSE for this base flow. In this case, the most unstable wave is the second mode. The second mode instabilities with two frequencies are computed. The comparison of the growth rates predicted by different approaches is shown in Fig. 11. For the lower frequency wave of ω=2.5, the unstable region is located downstream compared with the higher frequency (ω=3.0) wave. It means that the non-parallel effect of the flow for the ω=2.5 case is weaker than that of ω=3.0. It can be seen from Fig. 11(a) that although the upper branch of the EPSE shows good agreement with the PSE results, the prediction of the iEPSE is much closer to the PSE results, especially in the region 0.56 < x < 0.66. For the case with ω=3.0, the discrepancy between the LST and PSE is more obvious owing to the strong non-parallel effect. In this case, the prediction of the iEPSE is in excellent agreement with that of the PSE, while the traditional EPSE results have significant differences. It indicates that the iEPSE is more accurate than the traditional EPSE for the second mode prediction in the blunt cone flow.

4 Conclusions

In conclusion, the iEPSE approach for predicting the instabilities of a non-parallel boundary layer is proposed. In the new method, the eigenvalue problem is solved for the eigenvalue dα/dx with an initial α, and the correction of α is performed with the conservation relation, which is similar with the α iteration in the PSE. It improves the accuracy and reduces the additional eigenvalue in the traditional EPSE. The iEPSE method is applied in several boundary layer instability analyses, such as in the hypersonic flat plate flow, in the compressible and incompressible swept wing flows, and in hypersonic blunt cone flow. The results show that in the weak non-parallelism region, both the results of the iEPSE and EPSE show good agreement with that of the PSE. In the strong non-parallelism region, the results of the iEPSE are in excellent agreement with the PSE results and its accuracy is significantly higher than the traditional EPSE. In addition, the iEPSE is more physical and its solving procedure is more convenient than the NES method. It suggests that the iEPSE is an excellent choice for the local stability analysis in the non-parallel flows. As a local stability analysis tool, the iEPSE can be easily applied in the transition prediction of the general 3D boundary layers which will be performed in the future.

Acknowledgements

The authors would like to thank Xuezhi LU and Zhenrong JING for valuable discussion and for providing some of the DNS data.

Appendix A

The non-zero elements of the matrices Γ, A, B, C, D, V_xx, V_xy, V_xz, V_yy, V_yz, and V_zz in Eq. (2) are given as follows. Here, the Lamé coefficients are h₁=1+k_xy, h₂=1, and h₃=1+k_zy with k_x and k_z representing the streamwise and spanwise curvature, respectively. In all the terms, , where i=1, 2, 3 and j=1, 2, 3. γ is the ratio of specific heat. μ is the viscosity coefficient, and κ is the thermal conductivity coefficient.

For the matrix Γ,

For the matrix A,

For the matrix B,

For the matrix C,

For the matrix D,

For the matrices V_xx, V_xy, V_xz, V_yy, V_yz, and V_zz,