Laminar-turbulent transition in boundary-layer flows is of fundamental and practical importance in fluid mechanics. In the low-level free-stream turbulence condition,transition follows four processes: receptivity,linear development,nonlinear development,and breakdown^[1]. The nonlinear development of instability waves is crucial for understanding the transition mechanism.

In the nonlinear stability theory of incompressible boundary layers,weak nonlinear interactions may take place in several types such as the Craik^[2] or Herbert type^[3] and the Klebanoff type^{[4, 5, 6]}. The Craik or Herbert type involves the subharmonic interaction in which the frequencies of 3D disturbances are half of the 2D disturbances. The Klebanoff type refers to the nonlinear interaction between 2D and 3D disturbances with the same frequency. The secondary instability theory^{[7, 8, 9]} can explain the Klebanoff type and the Craik or Herbert type. The latter may also be explained by the theory of subharmonic resonant triad^{[10, 11]} .

Nevertheless,in supersonic boundary layers,the stability issue is more complicated because there exist first and second mode instabilities. It is therefore very difficult to directly generalize the weak nonlinear theory from incompressible to compressible flows. Recently,Sivasubramanian and Fasel^{[12, 13, 14]} performed some direct numerical simulations (DNS) in supersonic boundary layers. In these papers,the authors modeled a broad band natural transition by introducing a localized blowing and suction disturbance slot which consisted of a wide range of frequencies and wavenumbers. They confirmed that the fundamental resonance was much stronger than the subharmonic resonance in supersonic boundary layers on a flat-plate and a sharp cone. These simulations demonstrated that fundamental breakdown (Klebanoff breakdown) could be a path to transition in supersonic flows. Therefore,it is crucial to study the fundamental interactions in a supersonic boundary layer.

The numerical study in a supersonic boundary layer on a cone at a small angle of attack^{[15, 16]} found that the three dimensionality of the base flow makes a 2D second mode disturbance to acquire three-dimensionality,causing a second mode 3D disturbance with a high spanwise wavenumber to be rapidly amplified. A study in a supersonic flat-plate boundary layer found that wavepacket disturbances caused high spanwise wavenumber 3D disturbances to be amplified rapidly^[17]. Both studies indicate that a 2D second mode disturbance may selectively amplify a high spanwise wavenumber 3D disturbance. However,the three-dimensionality in both studies associated with the base flow and the spanwise modulated wavepacket,respectively,is relatively simply. In order to study more complex cases,the present paper simulates the evolution induced by multiple 3D disturbances,which have different or random amplitudes, and the evolution of a second mode 2D disturbance interacting with a pair of 3D disturbances in the later stage. 2 Governing equations and numerical method

Consider a flat-plate boundary layer. Let x,y,and z denote the coordinates in the streamwise,wall-normal,and spanwise directions,respectively,namely,x_i = (x,y,z). The equations are cast into the dimensionless form by using the velocity,temperature,and density at infinity and the local displacement thickness of the boundary layer at the inlet of the computation domain as the reference quantities. The non-dimensionalized compressible Navier-Stokes equations are written in the conservative form as

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where U denotes the conservation flux,∂E/∂x ,∂F/∂y ,and ∂G/∂z are nonlinear terms,and ∂E_v/∂x ,∂E_v/∂y, and ∂E_v/∂z are viscous terms. Details of these terms are given as follows:

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In the above formulations,p,T,and ρ are the pressure,temperature,and density,respectively,which satisfy the equation of state: . The velocity,viscous stress,and thermal conductivity along the x-,y-,and z-directions are u_i = (u,v,w),τ_ij,and q_i,respectively,where τ_ij and q_i are defined as

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with γ = 1.4 and P_r = 0.72. The viscosity µ is assumed to satisfy the Sutherland law. Ma_∞ is the free-stream Mach number.

Different terms of the equations are discretized by different methods. The nonlinear terms are separated into an upwind flux and a downwind flux by using the Steger-Warming’s splitting. Then,the 5th-order upwind scheme is applied for these nonlinear fluxes. The viscous terms are discretized by using the 6th-order central difference scheme. The standard 4th Runge-Kutta scheme is employed for the temporal discretization.

The calculations of the base flow are carried out through two steps. First,the base flow is simulated in a long streamwise region by taking the Blasius similarity solution as the initial state. Second,the base flow of the computation domain is cut out from the above long flow field,and then the final base flow is computed until the steady state is met. For the calculations of the perturbed flow,the development of disturbances are simulated by superimposing certain disturbances on the base flow at the inlet. 3 Verification of numerical simulation scheme

The program code for the present numerical simulation is validated by repeating the calculations of Mayer et al.^[18]. In Ref. ^[18],the Mach number at the free-stream was 3,the free-stream temperature was 103.6 K at the infinity,and the unit Reynolds number at the inflow was 2.181× 106 m⁻¹. In addition,for the base flow,the wall temperature was set to the adiabatic wall temperature,which was obtained from the compressible similarity solution.

The spatial and temporal evolution of small disturbances in a boundary layer is governed by the linear stability theory (LST). In LST,it is assumed that disturbance u′ follows the following propagating wave form:

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where α = α_r +iα_i is the complex streamwise wavenumber,β is the spanwise wavenumber,and ω is the frequency. The negative value of α_i is the streamwise amplification rate. Disturbances that neither grow nor decay are referred to as neutral.

Figure 1(a) depicts the neutral curve at 0.258 m downstream of the leading edge of the flat plate,and Fig. 1(b) shows the streamwise amplification rates obtained by LST and Mayer etal.^[18]. In Fig. 1,the inverse triangles represent the simulation results from the current study while the solid lines represent the results obtained from Mayer et al.’s work^[18]. As illustrated, good agreement can be observed between Mayer et al.’s results and the simulations proposed in the current research.

Fig. 1 Comparison of results of current paper with those of Mayer et al.’s obtained by LST

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Disturbances are introduced through a blowing and suction slot located nearly between 0.394 m and 0.452 m. Figures 2(a) and 2(b) compare the disturbance amplitude curves for the early and late nonlinear stages of the oblique breakdown. The notation [h,k] is used to identify a mode according to its frequency h and spanwise wavenumber k. The results denoted by symbols are obtained from the code by our research group and the lines represent Mayer et al.’s calculations. Figure 2 indicates that the streamwise development of the disturbances obtained by the current research mainly coincides with Mayer et al.’s. Hence,the agreement suggests that the code by our research group is credible.

Fig. 2 Comparisons of streamwise development of maximum u-velocity disturbance for different spanwise wavenumbers

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In our computation,the free-stream Mach number is taken to be 4.5,the gas parameter values are chosen to be those at 5 km altitude,the free-stream temperature is T_∞ = 255.7 K,the wall isothermal temperature is T_w=900.5 K,and the Reynolds number is R_e = 1.12 × 105. The inlet of the computation domain locates at 3.43 m downstream from the leading edge.

The simulation is carried out with the initial condition consisting of 2D and 3D disturbances, all components of the disturbances have an identical frequency. Hence,the disturbance at the inlet has the form

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where ω is the frequency,A_2d is the initial amplitude of the 2D disturbance, is the corresponding eigenvector,A_3d is the initial amplitude of the 3D disturbance,Cc means the conjugate,and φ(y,z) represents the spatial distribution of the 3D disturbance. Simulations are to be performed for three forms of 3D disturbances,i.e.,(i) multiple pairs of 3D second modes, (ii) random disturbances,and (iii) a pair of 3D second modes. Choosing the first two forms of disturbances aims at studying the selective amplification phenomenon of 3D disturbances. This phenomenon is caused by the varying amplitude and disturbance forms of the second unstable mode. The last form is introduced to study the nonlinear development of the preferred 3D disturbances and second mode disturbance. The amplitude of the 2D disturbance is A_2d = 0.01 and the frequency is ω = 1.75 which is close to the most unstable second mode. 5 Results and discussion

To understand the selective amplification of 3D disturbances by a second mode unstable disturbance,multiply pairs of 3D disturbances are introduced firstly. The specific form is taken as

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where N is 50,the fundamental spanwise wavenumber of 3D disturbances β₀ = 0.08,and is the corresponding eigenvector.

In Fig. 3,contours of the streamwise velocity of the disturbance u′ in the xz-plane and contours of |u′| in the xβ-plane obtained by the Fourier transform are displayed for A_3d = 5.0 × 10₋₄ and A_3d = 5.0 × 10₋₅.

Fig. 3 Velocity distributions of multiple pairs of 3D disturbances

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As illustrated,for both initial amplitudes,the spanwise small-scale 3D structures appear downstream and 3D disturbances are rapidly amplified for a band of spanwise wavenumbers β in the range of 1 to 3. As the amplitude of 3D disturbances changes,the small-scale 3D structures still arise downstream,while the starting location shifts. The smaller the initial amplitude of 3D disturbances is,the later the occurrence location of the small-scale 3D structures becomes.

The mean flow profiles are obtained by conducting the statistical average of the flow field within one time period and spanwise regional scale (see Figs. 4(a) and 4(b)). The profiles of the velocity and temperature of the mean flow both have inflection points at the later streamwise location (x = 510). The appearance of the inflection point may generate more unstable disturbances. Figure 4(c) shows the neutral curves obtained by analyzing the mean flow by using LST at certain streamwise locations. It can be seen that the unstable region of the first mode enlarges to the region consisting of disturbances with high spanwise wavenumbers, and the unstable region of the second mode enlarges to the region consisting of disturbances with high frequencies.

Fig. 4 Profiles of mean flow and neutral curves obtained by LST at several streamwise locations

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Consider some random disturbances with the same frequency as the 2D disturbance. At the inlet,the form of the disturbances is

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The amplitude is A_3d = 5.0 × 10⁻⁶. The disturbances are random disturbances rather than unstable modes at the inlet. Therefore,the disturbances except the 2D fundamental disturbance component are much more disordered and can better reflect the selectivity of 3D disturbances. Figure 5 displays the contours of the disturbance velocity u′ in the xz-plane and the disturbance velocity |u′| in the xβ-plane. As is shown,the spanwise small-scale 3D structures still arise downstream,and the 3D disturbances in the spanwise wavenumber band from 1 to 3 are amplified rapidly.

Fig. 5 Velocity distributions of random disturbances

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Whether the 3D disturbances introduced at the inlet are random disturbances or instability modes,the spanwise wavenumbers of rapidly growing 3D disturbances are in the range of 1 to 3. This phenomenon shows a highly selective enhancement of 3D disturbances by the second mode disturbance in the supersonic flow. According to the weak nonlinear theory in hydrodynamic stability,this outcome is somewhat consistent with the secondary instability theory,which states that when the amplitude of the 2D fundamental disturbance reaches a certain level,some 3D components will grow rapidly and selectively. In the paper,the 2D disturbance is a growing wave. When the amplitude of the 2D disturbance reaches a certain level,the 3D disturbances with the same frequency will be rapidly amplified in a certain wavenumber span.

In order to study the nonlinear evolution of disturbances attributed to the secondary instability mechanism,a 2D fundamental disturbance and a pair of small amplitude 3D second modes are introduced at the inlet,and the form of the 3D disturbance is

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The amplitude is A_3d = 10₋₄,and the wavenumber is β = 1.8. The contours of the disturbance velocities u′ and |u′| are exhibited in Fig. 6. The 3D disturbances grow rapidly downstream and a small-scale 3D structure appears in the flow. The spanwise wavenumber of this structure is β = 1.8,which coincides with the spanwise wavenumber of the initially introduced 3D modes.

Fig. 6 Velocity distributions of 2D fundamental disturbance and pair of 3D disturbances

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Figure 7 depicts the streamwise development of the skin-friction coefficient. As shown in Fig. 7,the skin-friction coefficient grows rapidly in the later downstream location.

Fig. 7 Streamwise development of skin-friction coefficient with β = 1.8

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Figure 8 displays the results for β = 3. These high wavenumber disturbances are amplified rapidly,leading to the appearance of a small-scale spanwise structure. The curve of the streamwise development of the skin-friction coefficient is plotted in Fig. 9 and the skin-friction coefficient grows downstream.

Fig. 8 Velocity distributions of 2D fundamental disturbance and pair of 3D disturbances

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Fig. 9 Streamwise development of skin-friction coefficient with β = 3

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The above results indicate that the 3D disturbances in a finite span of wavenumbers may be amplified rapidly under the given 2D disturbance. In our simulations,the wavenumber is relatively large in this span. Therefore,spanwise small-scale structure occurs obviously. This phenomenon will not appear neither for the first mode disturbances in a supersonic boundary layer nor for the unstable waves in an incompressible boundary layer. 6 Conclusions

In a supersonic flat-plate boundary layer,a second mode 2D fundamental disturbance and some 3D disturbances with the same frequency are introduced at the inlet of the computation domain. As the 2D fundamental wave evolves downstream and its amplitude reaches a certain level,some high spanwise wavenumber 3D disturbances are preferred to grow rapidly and selectively. Under the calculation condition in this paper,the wavenumber of the rapidly amplified 3D disturbances ranges from 1 to 3. The rapid growth of the 3D disturbances may be attributed to the secondary instability mechanism.