1 Introduction

Transition in supersonic boundary layers is of great practical importance as it has a critical effect on the system drag and thermal load of supersonic vehicles. Multiple families of instability modes may coexist in supersonic boundary layers^[1], including the first mode, which is of viscous origin, and the second mode (Mack mode), which is associated with inviscid instability. For Mach numbers above 4, two-dimensional (2D) second modes are the most unstable waves and thus may attain large amplitude first. Then, the finite amplitude 2D second mode leads to enhancement of oblique modes through nonlinear interactions, resulting in the onset of three-dimensionality of the flow, which is crucial for transition to turbulence. What kind of oblique modes can be enhanced and the mechanisms lying behind, namely, the selective mechanisms on enhancement of oblique modes, are fundamental problems regarding nonlinear regime of transition.

In incompressible boundary layers, when a 2D Tollmien-Schlichting(T-S) wave gains sufficient amplitude, oblique modes with frequency equal to or half that of the T-S wave can be excited. The corresponding mechanisms that lead to the enhancement of oblique waves are fundamental resonance (or K-type secondary instability)and subharmonic resonance (or H-type secondary instability), respectively, which have been confirmed by previous studies^[2-3]. In the supersonic flow, due to difficulty in performing stability experiments, relevant nonlinear mechanisms are still unclear.

Some of the investigations in supersonic boundary layers found that when the 2D second mode gains sufficient amplitude, it can significantly enhance the amplification of oblique waves with frequency equal to that of the 2D mode. It is a typical scenario of fundamental resonance caused by the K-type secondary instability mechanism. In direct numerical simulation (DNS) of a straight cone with a small angle of attack, Li et al.^[4-5] found the appearance of three-dimensional small-scale structures before transition was triggered. Liu^[6] attributed the three-dimensional small-scale structures appearing in the straight cone with a small angle of attack to the enhancement of oblique second mode waves caused by K-type secondary instability. A similar phenomenon was also found by Yu and Luo^[7-8] in flat-plate boundary layers. They also found the selective effect of the nonlinear interaction with respect to the spanwise wavenumber of the oblique modes. In experiments of a flared cone boundary layer performed in the quiet-flow wind tunnel of Purdue university^[9], streamwise streaks of low and high heating flux were found in the transition process. Numerical simulations of the same cone performed by Sivasubramanian and Fasel^[10] showed that streaks observed in the experiments may be induced by fundamental resonance (K-type instability) of the second mode, with the streaks of heating flux corresponding to the stationary streamwise vortex mode induced by fundamental resonance. These numerical and experimental results indicate that the second mode fundamental resonance is likely to be a viable transition route in supersonic boundary layers, which is characterized by streamwise streaks generated in the transition process.

Some other investigations showed the existence of a different transition route. By analyzing hot-wire data from transition experiments carried out in a conventional noisy-flow Mach 6.0 wind tunnel, Shiplyuk et al.^[11] and Bountin et al.^[12]identified the existence of a strong nonlinear interaction between the second mode and low-frequency first mode waves, which they believed could play a role in the transition process. In the numerical simulation performed by Dong and Luo^[13], by introducing a 2D second mode and oblique modes with lower frequency into a cone boundary layer, oblique first mode wave was found to be excited at downstream locations, then transition was triggered.Besides, streamwise streaks found in a quiet wind tunnel cannot be found in a noisy wind tunnel^[14], suggesting that in the noisy environment, transition is triggered through a route other than fundamental resonance. In recent wind tunnel experiments carried out by Zhu et al.^[15], they also found evidence of the second mode interacting with low-frequency disturbances. These investigations showed the possible existence of another transition route resulting from nonlinear interaction of 2D second mode with low-frequency first mode oblique waves. However, few relevant theoretical or numerical studies can be found in the literature. Gaponov and Terekhova^[16] performed analysis in the framework of weakly nonlinear stability theory, showing that the nonlinear interaction with the 2D second mode can exhibit an enhancement effect for amplification of the first mode oblique waves. Detailed numerical studies are still in need to further clarify this issue.

In the present research, nonlinear interactions of a finite amplitude 2D second mode with oblique modes in a broad band of frequencies and spanwise wavenumbers are simulated successively in a Mach 6.0 flat-plate boundary layer, in order to find out what kind of oblique modes can be enhanced by the nonlinear interaction and thus make contribution to the transition process.

Regarding nonlinear interactions of two modes, Wu and Stewart^[17] proposed the phase-lock mechanism in an incompressible flow. It is less restrictive than the subharmonic or fundamental resonant theory in that it can support enhancement of oblique modes in a broad frequency band. The theory shows that the2D mode with large amplitude can promote rapid growth of the oblique waves if the phase-lock condition is satisfied, i.e., the two modes have the same phase velocity. Analysis is also performed to explore whether the phase-lock mechanism is a relevant mechanism for the nonlinear interaction studied in this research.

To study nonlinear interaction of instability modes, the parabolized stability equation (PSE) is an efficient research tool which can acquire comparable accuracy as DNS with considerably reduced computational resources^[18-19]. However, it will be shown in Section 2 that, when applied to simulate nonlinear interaction of two modes with different frequencies, the conventional PSE may lose its efficiency. The problem is solved in the present study by expanding the PSE into a transformed spectral space. The modified PSE is able to simulate nonlinear interaction of two modes with arbitrary frequencies efficiently.

2 Numerical method 2.1 Equations and numerical method of conventional PSE

Derivation of PSEs and numerical method applied to solve the equations are outlined in this subsection. The PSEs are derived from the disturbance equations. To derive the disturbance equations, the instantaneous flow quantities φ = [ρ, u, v, w, t] are divided into a mean laminar flow part φ= [ρ, u, v, w, t] and a disturbance fluctuation part φ' = [ρ', u', v', w', t']. Both the instantaneous flow quantities φ and the mean laminar flow quantities φ satisfy the Navier-Stokes equations. By proper combination of these two sets of equations, the disturbance equations for the fluctuation quantities φ' can be derived,

(1)

where x, y, and z represent the streamwise, wall-normal, and spanwise directions, respectively, and t denotes the time.Coefficient matrices Γ, A, B, C, D, V_xx, V_yy, V_zz, V_xy, V_xz, and V_yz are functions of mean flow quantities, while the vector F is composed of baseflow quantities and nonlinear terms of perturbations.

Assume the disturbance function φ' to be periodic in time and in the spanwise direction. Then, φ' is expanded into a truncated Fourier series,

(2)

where ω₀ and β₀ denote the fundamental temporal and spanwise wavenumbers, respectively, and M and N are the numbers of modes reserved for the truncated Fourier expansion.

can be further decomposed into a fast varying wave-like part described by a complex wavenumber α_mn and a slowly varying shape function

(3)

The wave decomposition in (3) is non-unique. A supplement condition is used here to remove the non-uniqueness,

(4)

where the superscript * represents the complex conjugate. This condition implies that the total energy contributed by the shape function is nearly independent of x, which assures that the shape function varies slowly along x.

Upon substitution of (2) and (3) into (1), we can get the governing equations for shape functions of the (m, n) Fourier mode. As the shape function varies slowly along x, several terms regarding the streamwise derivatives can be neglected by performing an order analysis for each term, which yields the nonlinear PSE for the mode (m, n),

(5)

where the matrices are given by

To compute the nonlinear terms for each mode, F is evaluated in the physical space, and then expanded into a truncated Fourier series using the fast Fourier transformation (FFT),

(6)

To close the problem, proper boundary conditions in the wall-normal direction are required. At the wall, no-slip and isothermal conditions are adopted,

(7)

At the far field, disturbances are assumed to be damped,

(8)

The boundary condition for the stationary mode (0, 0) (known as the mean flow distortion (MFD) mode^[18]) is an exception,

(9)

which allows the distorted mean flow to adjust itself in order to assure mass balance^[19].

To solve the problem numerically, (5) is discretized in a rectangular domain. The wall-normal derivatives are evaluated by a fourth-order central difference scheme. As the grid points are clustered near the wall in the y-direction, a proper coordinate transformation is required before the evaluation of the wall-normal derivatives, which can be found in Ref.^[20].

As (5) is of parabolized form, it can be solved by marching along the streamwise axis. The streamwise derivatives are evaluated by the second-order backward difference scheme. To initiate the streamwise marching procedure, eigenfunctions obtained from the quasi-parallel linear stability theory (LST) solution are adopted at the inlet.

At each marching step, the local streamwise wavenumber α_mn and the nonlinear forcing term are determined by an iteration procedure. The scheme of the iteration procedure can be found in the previous publications^{[19, 21]}.

To this end, description of the PSE method is completed. Any detail not shown here can be found in the previous publications^[20-21]. The computational code used in this research is an in-house code initially developed by Zhang and Zhou^[20]. A few modifications have been made to the code by Zhao et al.^[21] in order to stabilize the iterative procedure for calculation of nonlinear forcing terms. The code has been validated by DNS in several different cases, including self interaction of 2D finite-amplitude first mode and second mode^[20], fundamental resonance of second mode waves^[21], and self interaction of finite-amplitude cross-flow waves^[21].

2.2 PSE expanded in transformed spectral space

It will be shown in this section that, when applied to simulate nonlinear interaction of two modes with different frequencies, the conventional PSE may lose its efficiency. Consider that two modes with different frequencies and spanwise wavenumbers (ω₁, β₁) and (ω₂, β₂) are introduced as the fundamental modes. The fluctuation including the fundamental modes and harmonic modes generated by nonlinear interaction of the fundamental modes can be expressed as

(10)

where _{m, n} is the shape function of each mode, e.g., _{1, 0} is the shape function for the fundamental mode (ω₁, β₁), _{0, 1} corresponds to the fundamental mode (ω₂, β₂), and _{1, 1} represents the sum harmonic mode of the two fundamental modes.

In order to simulate the disturbances by the conventional PSE, one has to expand the disturbances into the form of (2), which requires a proper fundamental frequency ω₀ and a spanwise wavenumber β₀ to satisfy the following equations:

(11)

We note that if ω₁/ω₂ is an irrational number, (11) cannot be satisfied. As for numerical simulations, all the quantities are approximated by float numbers of finite digits.Generally, we can find an approximate solution for the fundamental wavenumber (ω₀, β₀). However, the problem is that m₁, m₂, n₁, and n₂ can be very large, meaning that a huge number of Fourier modes may be needed to resolve (ω₁, β₁), (ω₂, β₂), and their harmonics. This can drastically increase the CPU time consumption.

The problem can be avoided by making a minor modification to the conventional PSE methodology, which was motivated by a transformation method proposed by Fang et al.^[22] in performing the pseudo-spectral DNS calculations. The transformation is

(12)

By substituting (12) into (10) and reserving only finite modes, an expansion similar to (2) can be derived,

(13)

The expansion and the inverse transformation are evaluated in the PSE by the FFT, as in the conventional PSE. The bases of the expansion e^{i(mθ ₁ + n θ ₂)} need to be independent of each other, which imposes constraints for the wavenumber of the fundamental modes. The constraint conditions for the general cases are a little complex and not shown here. As for the present research, in which the nonlinear interactions of a plane wave (ω₁>0, β₁=0) with an oblique wave (ω₂≥0, β₂ ≠ 0) are concerned, the bases are always independent of each other and no extra constraint conditions are needed.

_{m, n} can be decomposed into a fast varying wave-like part and a slowly varying shape function χ_{m, n}, as we have done in Subsection 2.1,

(14)

The nonlinear forcing term is also expanded to(θ₁, θ₂),

(15)

In the numerical simulation process, F is evaluated by χ', and then expanded to Fourier modes using the FFT as in Subsection2.1.

Substituting (13) and (15) into the disturbance equation (1), following the same procedure as Subsection 2.1, the governing equation of parabolized form for each shape function χ_{m, n} can be derived,

(16)

where the matrices are given by

The present governing equation is nearly the same as the original one. The only differences are the replacement of mω₀ by mω₁+nω₂ and nβ₀ by mβ₁+nβ₂ in the coefficient matrices. Numerical methods used to discretize the equations are also the same as the conventional PSE.

In the transformed spectral space, the mode (ω₁, β₁) is resolved as m=1, n=0 and the mode (ω₂, β₂) as m=0, n=1, regardless of the specific value of wavenumbers, so that the number of spectral modes needed is only determined by the accuracy requirement (M=N=7 is enough for most cases). Using the modified PSE, which is expanded in a transformed spectral space, nonlinear interaction of two modes with different wavenumbers can be efficiently simulated. Besides the binary wave interaction, wave-triad resonance can also be simulated effectively by the modified PSE. In a typical resonant triad, one wave of the wave-triad can be expressed as the combination or different forced mode of the other two waves. For validation, wave-triad resonance is simulated by the modified PSE, and the results are compared with those from the previous publications and those obtained by the conventional PSE in the next subsection.

2.3 Validation

For validation, the modified PSE expanded in the transformed spectral space is applied to simulate subharmonic resonance in a Mach 1.6 flat-plate boundary layer, a case from Chang and Malik^[23]. At the inlet, a 2D mode with the frequency F=0.4× 10^-4, the initial amplitude u_rms=0.5%, and a pair of oblique modes with frequency half that of the 2D mode, the spanwise wavenumber β/Re = ± 0.96 × 10 ^-4 is initiated. Results of the modified PSE are compared with those from Chang and Malik^[23] in Fig. 1. Good agreement is achieved for the nonlinear evolution of both the 2D mode and the oblique mode with subharmonic frequency, as shown in Fig. 1. This case shows that the modified PSE is able to predict nonlinear interaction of two modes with different frequencies.

In order to demonstrate the advantage of the modified PSE in dealing with nonlinear interaction between modes with different frequencies, another test case is performed. The flow parameters are the same as the first validation case. Subharmonic resonance with a small frequency detuning ∆F= 0.02× 10^-4 is concerned in this simulation. At the inlet, a 2D mode with the frequency F=0.4× 10^-4, the initial amplitude u_rms=0.5% (the same as the first case), and two oblique modes, i.e., F=0.18× 10^-4, β/Re = 0.96× 10 ^-4, and F=0.22× 10^-4, β/Re = -0.96 × 10 ^-4 are initiated. Both the conventional and modified PSEs are applied to simulate this case.

To simulate the nonlinear interaction by the conventional PSE, the fundamental frequency is set as ω₀=0.02× 10^-4, and the fundamental spanwise wavenumber is set as β₀= 0.96 ×10^-4. The 2D fundamental mode can be resolved as mode (20, 0) and the oblique modes as (9, 1) and (11, -1). In order to reserve the fourth-order nonlinear harmonics for self interaction of the fundamental modes, a total of 20× (2×4+1) ²=1620 modes are needed. To simulate the interaction by the modified PSE, the three modes are resolved as m=1, n=0, m=0, n=1, and m=1, n=-1, respectively. Only a total of (2×4+1) ²=81 modes are needed.The results obtained by the conventional PSE and the modified PSE are compared in Fig. 2. As shown in the figure, with the small frequency detuning, the evolution of the oblique modes is similar to the previous case. It is clear that the results calculated by the modified PSE fall exactly on those of the conventional PSE. With 100×500 grid points adopted in the simulation, the CPU time cost for the conventional PSE is more than 5 hours, while the CPU time cost for the modified PSE is only about 10 minutes. This test case shows a great advantage of the modified PSE in efficiency when dealing with interaction of modes with different frequencies.

3 Results and discussion

In the current study, nonlinear interactions of 2D second mode wave and oblique waves with lower frequency are simulated in a Mach 6.0flat-plate boundary layer. A self-similar solution of the compressible flat-plate boundary layer is adopted as the baseflow.The boundary layer edge temperature is T_e=80 K, the wall is set to be isothermal as T_w=6.5T_e, and the Reynolds number defined by the displacement boundary layer thickness δ_d^* at the inlet is Re=2 × 10⁴. Flow quantities are non-dimensionalized by the inlet boundary layer thickness and free-stream quantities. The streamwise axis x denotes the non-dimensional distance from the leading edge. The disturbance amplitude is defined as the maximum streamwise fluctuation velocity in the wall-normal direction. The computational domain is of rectangular shape, with the streamwise axis x ranging from 91.28 to 300.0 and the vertical axis y ranging from 0 to30.0. The streamwise grid is equally spaced with totally 200 grid points adopted, while in the vertical direction, a total of 300 grid points are clustered near the wall with the smallest mesh size being 0.02δ_d^*. A grid convergence study is performed by doubling the mesh points in both the x-and y-directions, which will be shown below in this section.

Stability characteristics of the baseflow are explored by the LST.Growth rate contours of 2D instability waves and oblique waves with the spanwise wavenumber β=1.0 are shown in Fig. 3. The high-frequency bands in Figs. 3(a) and 3(b) are the unstable area of the second mode, for which the 2D waves (see Fig. 3(a)) have larger growth rate than the oblique modes (see Fig. 3(b))^[1].The low-frequency bands in Figs. 3(a) and 3(b) are the unstable area of the first mode waves, for which the oblique modes have larger growth rate than the 2D instability waves. The growth rate contour plot in the βω -plane at x=200 is shown in Fig. 3(c), which clearly shows that the high-frequency second mode waves acquire the maximum growth rate at β=0, while the oblique first mode waves acquire the maximum growth rate at about β=0.7. Overall, the growth rate of the first mode is smaller than the second mode.

In supersonic boundary layers of large Mach numbers, the 2D second modes have the largest growth rate (see Fig. 3) and thus may attain large amplitude first. Then, 2D second modes of finite amplitude interact with other modes, leading to enhancement of oblique modes, which is crucial to transition. The general forms of the interactions are 2D second modes with large amplitude interacting with oblique modes with low amplitude, which are simulated numerically in the present study. The frequency of the 2D second mode (referred to as W_2d) simulated in this research is set to be ω₁=1.5, and the initial amplitude is set to be A_2d=2×10^-3. The area of the computational zone is shown in Fig. 3(a) by the dashed line. Before the nonlinear interactions of W_2d with oblique mode are studied, the nonlinear self evolution of W_2d is calculated by the PSE. Amplitude evolution of the 2D second mode and its harmonic modes are shown in Fig. 4, together with the linear evolution of W_2d calculated by the linear PSE. Initially, the second mode W_2d is amplified according to its linear growth rate. The second-order harmonic mode (SHM) and the steady MFD mode are amplified rapidly. As the amplitude of W_2d gets larger, the amplification of W_2d is enhanced by the nonlinear effect. The second mode W_2d achieves its maximum amplitude A_max=0.059 at x=210, around its upper-branch neutral point. Then, W_2d begins to decrease at a larger decay rate than the value predicted by the linear PSE. The SHM of W_2d also begins to decay rapidly, while the amplitude of MFD mode varies by a much smaller decay rate. Results calculated by a refined grid with doubled grid points in both the x-and y-directions are also shown in Fig. 4. The results of the two sets of grid agree well with each other, indicating that the grid used in this paper is fine enough to converge the results.

Different oblique modes (referred to as W_3d), varying in a broad band of frequencies and spanwise wavenumbers, are explored in this research to study the selective effect of the nonlinear interaction. The initial amplitude of the oblique mode is set at a low value A_3d=1×10^-5. Evolution of different oblique modes in the boundary layer distorted by the second mode and its harmonic modes will be simulated successively using the modified PSE expanded in the transformed spectral space in the next part of this section. In the modified PSE, the 2D second mode is resolved as m=1, n=0, and the oblique mode, regardless of the specific value of its frequency, is resolved as m=0, n=1. In the following numerical simulations, spectral modes are reserved to resolve the 2D second mode and higher harmonics are M=7 (see(13)), while one spectral mode (N=1) is reserved to resolve the oblique mode, considering the fact that higher harmonics of the oblique mode are too small to have any noticeable influence on the nonlinear interaction process. The number of spectral modes adopted here has been verified by numerical simulations with double spectral modes (not shown here).

3.1 Overview

Nonlinear evolution of the 2D second mode W_2d and an oblique first mode is simulated in this subsection to provide an overview of the nonlinear interaction process. In this simulation, we chose the frequency of the oblique mode W_3d as ω_3d=0.5, and the spanwise wavenumber as β_3d=1.0. Nonlinear amplitude evolution of W_2d and W_3d simulated by the modified PSE is shown in Fig. 5. The oblique mode W_3d is initially amplified according to its linear growth rate at upstream locations, where the amplitude of W_2d is low. When the amplitude of W_2d exceeds 0.04, the nonlinear interaction shows impact on the evolution of W_3d. Interestingly, the nonlinear interaction firstly exhibits a weak suppressing effect on W_3d, and then shows significant enhancement on W_3d. As the amplitude of W_2d decreases, the nonlinear effect on the evolution of W_3d also weakens. At downstream of x=260, the amplification rate of W_3d falls back to the value predicted by the linear PSE, as shown in Fig. 5.

Based on the disturbance equations, the nonlinear effect acted on the oblique modes W_3d is caused by the inhomogeneous forcing terms, which are generated by interaction of different harmonic modes. Of all the modes in the process of nonlinear interaction, the 2D second mode W_2d has the largest amplitude. Therefore, most contributions to the nonlinear forcing term of W_3d are likely to be made by W_2d.There are three main ways that W_2d can generate inhomogeneous forcing terms of W_3d. (ⅰ) By generating the MFD mode W₀ (W_2d, W_2d^*).Interaction of the MFD mode with W_3d generates the inhomogeneous forcing terms F(W₀, W_3d). F(W_m, W_n) denotes the inhomogeneous forcing terms generated by modes W_m and W_n which are acted on the oblique mode W_3d. The superscript * denotes complex conjugation. (ⅱ) By generating a mode with the difference frequency W_D(W_2d, W_3d^*) through interaction of W_2d and W_3d. Then, W_D interacts with W_2d generating the inhomogeneous forcing terms F(W_2d, W_D^*). (ⅲ) By generating a mode with the sum frequency W_S(W_2d, W_3d) through interaction of W_2d and W_3d.Then, W_S interacts with W_2d generating the inhomogeneous forcing terms F(W_S, W_2d^*).

By reserving only some selected modes in the PSE, different nonlinear interaction mechanisms can be isolated, as have been applied in the previous studies^[24-25]. In order to determine the key nonlinear interaction that leads to the enhancement of W_3d, only the equations of the mode W_3d and W_D are solved in the PSE, while the fluctuation quantities of W_2d are loaded from the previous fully coupled PSE calculation, and all the other modes are suppressed to zero. The results are compared with evolution of W_2d, W_3d, and W_D calculated by the fully coupled PSE in Fig. 6. The evolution of W_3d when only W_2d, W_3d, and W_D are reserved agrees well with the original results calculated by the fully coupled PSE, suggesting that the nonlinear enhancement effect on W_3d results mainly from the nonlinear interactions of modes W_2d, W_3d, and their difference modeW_D. More specifically, a difference frequency W_D(W_2d, W_3d^*) is generated by the interaction of W_2d and W_3d. Then, the forced difference mode W_D interacts with W_2d generating the nonlinear forcing terms F(W_2d, W_D^*) that promotes the amplification of W_3d. In the process, the MFD mode, the mode with the sum frequency, and other harmonic modes only have a minor effect on the enhancement of the oblique mode.

It is shown that, amplification of the oblique mode is significantly enhanced by the 2D second mode with certain amplitude. The enhancement effect is accomplished by the nonlinear interaction of the second mode, the oblique mode, and a forced mode with the difference frequency.

3.2 Selective effect with respect to frequency of oblique modes

Unlike most of the previous investigations^{[8, 26]}, in which only a few oblique modes with a fixed frequency or several limited frequencies were taken care of, oblique modes covering a rather broad band of frequency are simulated in this subsection, with the help of the modified PSE (see Section 2). The motivation of these simulations is to form a general picture of the behavior of oblique modes with different frequencies under the influence of finite amplitude second mode wave and to check the possible existence of selective effect with respect to frequency of oblique modes.

In each simulation, an oblique mode is initiated at the inlet together with the 2D second mode W_2D. For different simulations, the spanwise wavenumber of the oblique mode keeps constant as β_3d=1.0, while the frequency of the oblique mode varies from 0.1 to 1.5 with the fixed frequency step0.1. Before nonlinear simulations are performed, linear amplitude evolution of the oblique modes predicted by the linear PSE is shown in Fig. 7. For clear demonstration, amplitude evolution of oblique mode waves with different frequencies is shown in the same figure as a contour plot. Both the first mode and the second mode oblique waves are considered here. Based on the LST analysis results shown in Fig. 3(b), we can tell that in most part of the computational zone, oblique modes with the frequency ω_3d higher than 1.2 belong to the second mode instability waves, while oblique modes with the frequency lower than 1.2 belong to the oblique first mode waves.

Nonlinear amplitude evolution of oblique waves W_3d under the influence of finite amplitude second mode W_2d and evolution of the forced difference mode W_D is shown in Fig. 8. At upstream of x=180, the oblique mode W_3d evolves linearly. Nonlinear impacts set in at around x=180, where the forced difference modes W_D are amplified to the same amplitude level as the oblique modes W_3d. The nonlinear interaction firstly exhibits a weak suppressing effect on W_3d, and then shows an enhancement effect on W_3d. An overall enhancement effect is achieved for all the oblique modes at downstream locations. However, the enhancement effect is not equal for oblique waves with different frequencies, resulting in two amplitude peaks in Fig. 8(a). One peak is located in a narrow frequency band ω_3d∈ [1.4, 1.5], while the other peak is distributed in a broad frequency band ω_3d∈ [0.2, 0.8]. The amplitude of forced mode W_D also exhibits two peaks in Fig. 8(b), one of which is located in the frequency band ω_D∈ [0, 0.1], and the other is located in the region ω_D∈[0.2, 0.7].

Amplitude of the oblique mode ω_3d=1.5 gains great enhancement due to the nonlinear interaction, exhibiting a peak in Fig. 8(a). The corresponding forced difference mode W_D with the frequency ω_D=0 is a stationary streamwise vortex mode, which gains even larger amplitude than W_3d, growing into the dominant three-dimensional structure in the flow. This is consistent with the results of previous research^{[8, 26]} that the oblique modes with the frequency equal to the 2D second mode can be significantly promoted and a stationary streamwise vortex mode with large amplitude can be generated in the process. This type of nonlinear interaction is known as the fundamental resonance. It is interesting to note that, the oblique modes with the frequency ω_3d∈ [1.4, 1.5] can also be significantly enhanced, indicating that the fundamental resonance can be effective with a small frequency shift.

Another peak of the oblique modes amplitude lies in the frequency band ω_3d∈ [0.2, 0.8], which corresponds to the first mode unstable region. This peak is more distributed than the peak caused by the fundamental resonance, with all oblique modes in this region significantly enhanced. This broad peak is similar to the experimental findings of Bountin et al.^[12], in which they identified nonlinear interaction between the most amplified second mode with a broad band of low frequency waves corresponding to the first mode unstable region (see Figs. 2 and 3 in their paper). They inferred that this was caused by subharmonic resonance with frequency detuning. However, no distinct difference is found in the results that can separate the oblique modes of any specific frequency from other modes except the minor quantity difference, which seems to suggest a more general kind of interactions which do not require a specific frequency ratio between the oblique mode and the 2D mode. The resonant interaction theory of Gaponov and Terekhova^[16] and the theory of phase-lock by Wu and Stewart^[17] can be possible explanation for the interactions found here.

The oblique modes with the frequency ω_3d∈[0.9, 1.3] are not significantly amplified. Instead, the corresponding forced difference modes lie in the frequency band ω_D∈ [0.2, 0.6], and attain amplitude larger than that of the oblique mode W_3d, exhibiting a peak in Fig. 8(b). Note that the forced difference modes are also three-dimensional modes. Therefore, the enhancement of them can make contributions to transition. This is consistent with the previous numerical simulations performed by Dong and Luo^[13]. In their research, the first mode oblique mode was found to be excited by the difference interaction of a 2D second mode and an oblique mode with lower frequency, which lead to transition at downstream locations.

In summary, nonlinear interactions of the 2D second mode with oblique modes show a selective effect for the frequency of oblique modes. Two types of oblique waves are preferred by the finite amplitude 2D second mode. (ⅰ) The oblique mode with the frequency close to that of the 2D second mode can be amplified as a result of fundamental resonance. The nonlinear process is characterized by the large amplitude stationary streamwise vortex induced by the forced difference mode. (ⅱ) Oblique modes with low-frequency can be enhanced either through nonlinear interaction of the 2D second mode, the oblique mode, and the forced difference mode, or through nonlinear enforcement by interaction of 2D second mode and oblique modes with lower frequency. This category of nonlinear interaction is characterized by nonlinear interaction of second mode with a broad band of low-frequency oblique modes. Both types of oblique modes have potential of being amplified by nonlinear interactions and playing a role in transition.

In a quiet-flow wind tunnel, due to the very low free-stream disturbance level, the spectrum of the disturbance in the boundary layer tends to be concentrated in a narrow frequency band around the frequency of the second mode^[9]. Oblique modes with frequency close to the second mode are amplified as a result of fundamental resonance, generating a large amplitude stationary streamwise vortex mode. The stationary streamwise vortex mode then results in the streaks found in the experiments^[9] performed in the quiet tunnel at Purdue university. In a noisy-flow wind tunnel, the inflow disturbances are distributed in a very broad frequency band with a large composition lying in the low-frequency first mode unstable region^[11]. In this case, strong interactions between low-frequency first mode oblique waves and finite amplitude second mode exist in the flow, as detected in the experiments performed in a noisy-flow tunnel by Shiplyuk et al.^[11] and Bountin et al.^[12]. The interactions lead to great enhancement of low-frequency oblique modes, which may play a role in promoting transition.

Based on the analysis above, we infer that the two types of oblique waves, significantly enhanced by nonlinear interactions with the second mode, may have a close relationship with the two different types of transition routes found in the experiments.

3.3 Selective effect with respect to spanwise wavenumber of oblique modes

Nonlinear interaction of the 2D second mode with oblique modes of different spanwise wavenumbers is simulated in order to explore the selective effect with respect to the spanwise wavenumber of oblique modes. As in Subsection 3.2, results are presented in a contour plot of oblique mode amplitude, with x along the horizontal axis and β_3d (the spanwise wavenumber of the oblique mode)along the vertical axis. Two sets of representative results(ω_3d=0.5, 1.5) are shown in Fig. 9 with linear results located in the left column and nonlinear results in the right column. Linearly, oblique modes with the frequency ω_3d=0.5 are unstable in the spanwise wavenumber range of β_3d∈ [0.2, 2.2], as shown in Fig. 9(a).Nonlinear effect comes into play at around x=180, resulting in an amplitude peak in the spanwise wavenumber range of [ 0.8, 2.0 ] at around x=270. Even the linearly stable oblique modes with large spanwise wavenumbers (β_3d∈[2.2, 3.8]) are amplified as a result of the nonlinear interaction. The spanwise wavenumber of the most amplified oblique mode is shifted from β_3d=0.9 to β_3d=1.2. Results of oblique first modes with other frequencies (ω_3d=0.3, 0.7) (not shown here) are similar to Figs. 9(a) and 9(b).Oblique second modes with the frequency ω_3d=1.5, shown in Figs. 9(c) and 9(d), are a little different. Linearly, oblique second modes with small β_3d are more amplified than those with large β_3d. Nonlinear interactions with the 2D wave also lead to enhancement for the second mode oblique waves, resulting in an amplitude peak in the spanwise wavenumber region [0.5, 1.5] at around x=230. Under the impact of nonlinear interactions, the most amplified spanwise wavenumber is shifted from β_3d=0 to β_3d=1.

To demonstrate the relationship between the nonlinear enhancement effect with the spanwise wavenumber, linear and nonlinear amplification factors (amplitude normalized by the inlet amplitude)of oblique modes with the frequency ω_3d=0.3, 0.5, 0.7 at x=270 and oblique modes with the frequency ω_3d=1.5 at x=230 (the maximum amplitude location) are shown in Fig. 10 as a function of spanwise wavenumber. The net enhancement of the oblique wave caused by the nonlinear interaction is defined here as the nonlinear amplitude of the oblique mode divided by the linear amplitude, which is also shown in Fig. 10. Linearly, the maximum amplification factor for oblique modes with the frequency ω_3d=0.3, 0.5, 0.7, 1.5 are achieved at β_3d=0.7, 0.9, 1.0, 0.0, respectively. Nonlinear interaction with the 2D second mode leads to an enhancement effect for oblique first modes (ω_3d=0.3, 0.5, 0.7) with relative large spanwise wavenumbers in the range β_3d∈[0.6, 3.8], while for the oblique second mode (ω_3d=1.5), the enhancement effect is achieved in the whole spanwise wavenumber range (β_3d∈ [0.2, 3.8]).

If the part of amplification factor caused by the linear effect is removed from the nonlinear amplification factor, we get the net enhancement effect caused by the nonlinear interaction. As shown in Fig. 10 by the dashed dot line, for the oblique mode(ω_3d=0.3, 0.5), the net nonlinear enhancement is the most effective for oblique modes with the spanwise wavenumber around β_3d=2.0. It is interesting that the oblique mode (β_3d=2.0) which undergoes the largest net nonlinear enhancement effect is not the oblique mode (β_3d∈ [1.1, 1.2]) that is most amplified. This is because the total amplification of the oblique mode is a combination effect of the linear amplification and the nonlinear enhancement effect. For the oblique mode ω_3d=0.7, the net nonlinear enhancement is the most effective for a larger spanwise wavenumber β_3d=2.4, while the maximum amplification factor is achieved at β_3d=1.4. For the second mode oblique waves (ω_3d=1.5), the maximum amplification factor is achieved at β_3d=1.0, and the net nonlinear enhancement effect exhibits the peak at β_3d=1.4. An extra peak of net nonlinear enhancement effect can be found at a much larger spanwise wavenumber β_3d=3.4, which is barely amplified, as shown by the nonlinear amplification factor.

Based on the results in Figs. 9 and 10, we determine the spanwise wavenumber of the most enhanced oblique modes. Inspired by the phase-lock theory proposed by Wu and Stewart^[17] in the incompressible flow, the phase velocity c_phx (defined by c_phx=ω/ α) of the 2D second mode W_2d and the oblique modes with different spanwise wavenumbers are calculated by the LST. According to the phase-lock theory, the most enhanced oblique waves have the phase velocity equal to that of the 2D mode. In Fig. 11, we compare the phase velocity of the 2D mode with oblique modes undergoing the largest net nonlinear enhancement effect. For the two cases with lower frequencies (ω_3d=0.3, 0.5), the phase velocity of the oblique wave undergoing the largest nonlinear enhancement effect(β_3d=2.0) coincides well with that of the 2D second mode downstream of x=180, while the phase velocities of oblique waves with other spanwise wavenumbers are either larger or smaller than that of the 2D second mode, which supports the phase-lock theory. However, for the other two cases with larger frequencies(ω_3d=0.7, 1.5), noticeable deviation can be found between the phase velocity of the 2D second mode and the oblique mode undergoing the largest nonlinear enhancement (β_3d=2.4 in Fig. 11(c) and β_3d=1.4 in Fig. 11(d)). Instead, the phase velocity of other oblique modes shows better agreement with that of the 2D second mode, which does not support the phase-lock theory. The present results cannot give a definite conclusion on whether the phase-lock theory is a relevant mechanism here. Since this is not the main scope of the present research, we would like to leave it for future study.

4 Conclusions

By expanding the PSE in a transformed spectral space, the PSE is enabled to simulate nonlinear interaction of two modes with arbitrary frequencies effectively. Using the modified PSE, nonlinear interactions of a second mode wave with large amplitude and oblique modes with various frequencies and spanwise wavenumbers are simulated in a Mach 6.0 flat-plate boundary layer, focusing on the selective effect of the nonlinear interaction with respect to frequency and spanwise wavenumber of oblique waves.

It is found that once the 2D second mode attains some certain magnitude, the amplification of oblique modes in a broad band of frequencies and spanwise wavenumbers can be significantly enhanced.The nonlinear enhancement effect is shown to be accomplished by nonlinearly generating a forced mode with difference frequency, which in turn interacts with the 2D mode to generate nonlinear forcing effect on the oblique waves.

The nonlinear interaction shows a selective effect with respect to the frequency of oblique waves, making two types of oblique modes more likely be amplified with the presence of finite amplitude 2D second mode. They are (ⅰ) oblique modes with the frequency close to that of the 2D second mode, which correspond to the fundamental resonance. During the nonlinear process, a large amplitude stationary streamwise vortex mode can be forced. (ⅱ) The other is low-frequency first mode oblique waves, which are enhanced through nonlinear interaction with the 2D second mode. If the spectrum of the inflow disturbances focuses on the frequency of the second mode, fundamental resonance is more likely to take control of the nonlinear process. Otherwise, if the spectrum of the inflow disturbance is more distributed, low-frequency oblique first modes may be the most amplified oblique modes and play a role in the transition process. This is a possible explanation on how the differences of inflow disturbance spectral distribution lead to different transition phenomena in a quiet-flow wind tunnel and a noisy-flow wind tunnel.

The nonlinear interaction also shows a selective effect with respect to spanwise wavenumbers of oblique waves. The spanwise wavenumbers of the oblique wave preferred by the nonlinear interaction at several different frequencies are also determined by numerical simulations. It is found that the oblique modes undergoing the strongest nonlinear enhancement effect are not the most amplified mode, as the total amplification of the oblique modes is a combination effect of the linear amplification and the nonlinear enhancement.