1 Introduction

The mechanism of laminar-turbulent transition in the boundary layer is extremely complex, and generally involves five stages: receptivity, linear and nonlinear growth, secondary instability, and the formation of turbulence. Receptivity is the initial and also key stage in the entire process^[1]. Usually, it is classified into two classes: leading-edge receptivity triggered by the combined action of non-parallelism and free-stream disturbances; and local receptivity induced by the free-stream disturbances interacting with the surface roughness, blowing, or suction.

To better understand receptivity, Goldstein^[2] first showed a connection between the long-wavelength free-stream disturbances and the short-wavelength Tollmien-Schlichting (T-S) waves through the matched asymptotic expansions. A wavelength-reduction process was introduced to explain the connection that a decaying Lam-Rott^[3] asymptotic mode is excited by the free-stream disturbances and non-parallelism in the leading-edge region, then shortens the wavelength gradually, and evolves into a T-S wave eventually. Soon afterwards, a more efficient mechanism of boundary-layer receptivity was revealed by Ruban^[4] and Goldstein^[5], involving the mutual action of the convected disturbances and abrupt mean-flow distortion from localized inhomogeneity. For the case of vortical disturbances, Duck et al.^[6] formulated a similar local receptivity mechanism to generate the unstable T-S waves through the high-Reynolds-number approach. Then, Wu^[7] presented a second-order asymptotic expansion, which allows an arbitrary profile of vortices, to provide a local receptivity theory. The theoretical results matched well with Dietz's experimental data^[8]. Goldstein and Hultgren^[9] reviewed the receptivity theory and showed the differences in the receptivity coefficients between convected gusts and plane sound waves as an incident-angle function. Besides, Wu^[10] was the first to give the correct mathematical theory, which showed that excitation occurs near the neutral curve, that is, receptivity takes on a local character. He also put out that non-parallelism, which was ignored in some earlier work, plays a leading-order role in receptivity.

Heinrich and Kerschen^[11] and Fuciarelli et al.^[12] found that the leading-edge receptivity coefficients excited by the acoustic waves with an incident angle are higher than those in the case of zero-incidence angle from the numerical results and experimental data. Moreover, through the asymptotic method, Kerschen et al.^[13] studied the leading-edge receptivity induced by two-dimensional acoustic waves and convected gust, and the receptivity to convected gust is found to be four times of the acoustic receptivity when the incident angle is zero. Buter and Reed^[14] numerically studied the receptivity upon a flat plate with an elliptic leading, and the linear relation in the T-S wave amplitude with external disturbance intensity was confirmed under both symmetrical and asymmetrical conditions. Ricco and Wu^[15] extended Goldstein's approach^[2] to compressible boundary layers and found the viscous instability excited by highly oblique vortices not only in the subsonic case but also in the supersonic case. Recently, Goldstein and Ricco^[16] showed that at the moderate supersonic Mach number, both vortices and sound waves can excite oblique disturbances in the boundary layer which will evolve into modified Rayleigh-type instabilities. However, of all previous studies on leading-edge receptivity, the majority focuses on the leading-edge receptivity to vortical disturbances or sound waves, and neglects those to the free-stream turbulence (FST) that is more general in the nature. Usually, the FST is represented by a superposition of vortical disturbances, each of which is independent in the case of weak turbulence intensity. The existing studies on the receptivity of three-dimensional T-S waves concentrate on local receptivity. For example, Choudhari and Kerschen^[17] and Tadjfar and Bodonyi^[18] used the triple-deck equations to investigate the generating process of three-dimensional T-S waves which are excited by time-harmonic disturbances interacting with three-dimensional localized roughness, and amplified downstream into a wedge-shaped region. Würz et al.^[19] presented a numerical and experimental study of the excitation of three-dimensional T-S waves on a symmetric airfoil induced by three-dimensional acoustic-roughness interaction. The scattering of the sound waves on three-dimensional roughness was found to be more intensive than that on two-dimensional roughness. Moreover, Wu^[20] considered receptivity of three-dimensional T-S waves to three-dimensional vortical disturbance interacting with distributed roughness.

Most of the work for boundary-layer receptivity involved single-frequency waves, and only a few studies^[21] tried to determine the actual effects of the FST and extended the theories for single-frequency disturbances to broadband the FST. We have already studied the generating mechanism of two-dimensional T-S wave packets in Refs. [21]-^[24]. In this paper, further investigation is done towards leading-edge receptivity of three-dimensional T-S wave packets induced by the FST.

2 Formulation 2.1 Governing equations

In this work, the incompressible Navier-Stokes equations are introduced as the governing equations,

(1)

where the velocity V=U+V' is a superposition of the perturbation component V'=(u, v, w)^T and the basic flow U, and p denotes the pressure. The Reynolds number Re=(U_∞δ^*)/ν. The characteristic length is the displacement thickness of the boundary layer δ^* at the outlet boundary, the characteristic velocity is the free-stream velocity U_∞, and ν is the kinematic viscosity coefficient. In order to numerically solve the incompressible Navier-Stokes equations, a modified Runge-Kutta (R-K) scheme is introduced with the compact finite difference method and the Fourier expansions for temporal and spatial discretization. The details of the discretization method can be found in Refs. [21] and [23].

2.2 FST model

According to Ref. [25], the three-dimensional FST model is written as

(2)

where

Here, u_∞, v_∞, and w_∞ are the perturbation velocities of the FST in the x-, y-, and z-directions, respectively, and û_∞, , and are the velocity spectra. ε denotes the amplitude, and κ₁, κ₂, and κ₃ are the fundamental wave numbers. Hence, the wave numbers κ_x=mκ₁, κ_y=jκ₂, and κ_z=nκ₃, and the cross-stream wave number σ₁ and σ₂ are the random angles. The energy spectra tensors Φ₁ and Φ_P are

where

Here, C_d is the contraction ratio, κ_xo, κ_yo, and κ_zo are the wave numbers of the isotropic FST, and If C_d=1, the FST is isotropic. If Cd≠1, the FST is anisotropic. Each vortical component in Eq. (2) is independent, i.e., the effect is linear when ε is sufficiently small, but nonlinear effects come into play when ε exceeds a critical value.

2.3 Computational domain and boundary conditions

Figure 1 illustrates the computational domain in which the leading-edge receptivity is studied. The streamwise dimension is x∈[-200, 700], the leading is located at x=0, the normal dimension is y∈[0, 14.39], the spanwise dimension is z∈[-Z/2, Z/2], where Z=2π/κ₃=, and Re=1 000. Additionally, the grids are clustered in the vicinity of the leading and the near-wall region where flows vary sharply for the computational accuracy. Uniform meshes are arranged in the z-direction. 800×200×3 grids are arranged in the x-, y-, and z-directions.

At the wall boundary, the no-slip condition is utilized, and . If x<0 and y=0, u, w, and p are assumed to be symmetric about y=0, and v is assumed to be antisymmetric about y=0.

At the upper boundary, p=0. The perturbation velocities are given by the FST model.

At the inflow boundary, the perturbation velocities are given by the FST model.

At the outflow boundary, is valid when the excited waves do not reach the outlet boundary. To suppress the non-physical oscillations that may arise in the outflow boundary, the non-reflection condition is imposed. The governing equations of the non-reflection condition are

(3)

At the spanwise boundary, the periodic condition is applied naturally.

3 Results and analysis 3.1 Leading-edge receptivity induced by FST

The receptivity process to the three-dimensional isotropic FST is studied in this section. The root mean squares of the perturbation velocities outside the boundary layer tend to be steady after a long-time computation (t>500). For convenience of comparison, the FST intensity A_FST can be defined as

(4)

Here, and are the time averaged squares of the perturbation velocities of the FST. The dimensionless frequency is defined as F=(2πfν/U_∞²)×10⁶. The contraction ratio C_d=1. The fundamental wave numbers κ₁=0.025, κ₂=0.01, and κ₃=0.025 with M=8, J=1, and N=1. The FST intensity A_FST=0.1%.

The numerical results indicate that a group of three-dimensional waves has been excited in the boundary layer under the three-dimensional FST. For simplicity, Fig. 2 gives the contours of the streamwise velocity of the excited three-dimensional waves in the xz-plane. Based on Fig. 2, the propagation angle θ_g of the excited three-dimensional waves is ascertained by calculating the angle between the x-axis and the propagation direction S which is perpendicular to the wavefront, and it is equal to 6.81°.

According to Fig. 2, the spatial evolution of the excited perturbations can be plotted and shows a structure of wave packets. Figure 3 shows that the excited wave packets are gradually amplified in the S-direction. Subsequently, the group velocity of the excited wave packets is calculated and found to be 0.339 1.

By using the temporal fast Fourier transform, a group of three-dimensional waves with multiple frequencies and wave numbers is isolated from those wave packets. Figure 4 gives the evolutions of the unstable three-dimensional waves, which are the most concerned in the transition. On the basis of Fig. 4, the average streamwise wave numbers and phase speeds of the isolated waves can be calculated approximately. Those of the other frequencies are calculated with the same method. The obtained results and the solutions from the linear stability theory (LST) are both listed in Table 1, which shows good agreement. It indicates that the dispersion relations of the excited waves match perfectly with those of the T-S waves.

Then, we compare the numerical results of the amplitude variations of the unstable three-dimensional waves with those from the e^N method, and the DNS results are in accord with those of the e^N method, as seen in Fig. 5(a). The amplitude of the excited waves A_TS is defined as

(5)

where and are the time averaged squares of the velocities of the T-S waves. Moreover, Fig. 5(b) shows that the DNS results of the growth rates also agree well with the predictions of the LST.

The group of three-dimensional waves excited by the isotropic FST is confirmed to be T-S waves. It shows that the FST does, through interacting with the non-parallel boundary layer in the leading-edge region, generate T-S waves.

3.2 Relation between leading-edge receptivity and three-dimensional FST intensity

As the fundamental wave numbers κ₁=0.025, κ₂=0.01, and κ₃=0.025 with M=8, J=1, and N=8 are selected, the effect of three-dimensional FST intensity on leading-edge receptivity is studied with the spanwise wave number β chosen to be 0.025, 0.050, 0.100, 0.150, and 0.200, where β=nκ₃ with n=1, 2, …, N. Figure 6 gives the relation between the FST intensity A_FST and the initial amplitudes of the excited T-S wave packets A_TSP/A₀ defined at x = 200 where the amplitude begins to amplify^[26]. A₀ is the initial amplitude of the excited T-S wave packets at A_FST=0.5%. Figure 6 shows that when A_FST≤1.0%, the initial amplitudes increase linearly with the FST intensity. When A_FST>1.0%, the increase is slightly greater.

3.3 Relation between leading-edge receptivity and FST direction

The actual turbulent flow has the nature of being random. Therefore, it is meaningful to study the effect of the FST directions on leading-edge receptivity. The fundamental wave numbers of the three-dimensional FST are set to be κ₁=0.025, κ₂=0.040, and κ₃=0.025 with M=8, J=1, N=8, and A_FST=0.1%. For the purpose of changing the FST directions L, the streamwise, normal, and spanwise wave numbers are varied separately. For convenience of analysis, we define θ as the angle between the FST direction L and the xy-plane, and φ as the angle between the FST direction L and the xz-plane, as shown in Fig. 7. θ is positive as the projection of L on the z-axis is positive, and φ is positive as the projection of L on the y-axis is positive.

First, as the typical spanwise wave numbers 0.025, 0.050, 0.100, 0.150, and 0.200 are chosen, the influence of the angle φ of the FST direction on the leading-edge receptivity is calculated here. Figure 8 shows the relationship between the initial amplitudes A_TSP and the angle φ. As the angle φ increases, the initial amplitudes become large gradually and reach the maximum at φ=-20°. Subsequently, the initial amplitudes fall down as the angle φ continues to increase.

Secondly, with the typical normal wave numbers -0.080, -0.040, 0.000, 0.040, and 0.080, the influence of the angle θ of the FST direction on the leading-edge receptivity is also studied. Figure 9 gives variation of the initial amplitudes A_TSP with the angle θ. Figure 9 shows a rapid increase in the initial amplitudes with the angle θ.

In the following, the effects of the FST directions on the propagation directions and group velocities of the T-S wave packets are studied. First, we keep θ=0° and change the angle φ individually to investigate the effects on the propagation directions and group velocities of the excited T-S wave packets, which are listed in Table 2. The results show that the variation in the angle φ of the FST direction hardly changes the propagation directions and group velocities of the T-S wave packets. Then, we keep φ=0° and change the angle θ individually, and the results are given in Table 3. It shows that as the angle θ keeps increasing, the propagation direction of the excited T-S wave packets deviates from the x-axis gradually, and the angle θ_g of propagation direction increases simultaneously. However, the group velocities of the excited T-S wave packets do not vary with the angle θ, and maintain roughly one-third of the free-stream velocity.

3.4 Leading-edge receptivity to anisotropic FST

Most of the FST in engineering is anisotropic, but the studies on the anisotropic FST have been rarely reported so far. Hence, we study the leading-edge receptivity to anisotropic FST. As the fundamental wave numbers are set to be κ₁=0.025, κ₂=0.040, and κ₃=0.025 with M=8, J=1, N=8, and A_FST=0.1%, the contraction ratio C_d is varied to change the anisotropic degree of the FST. That is, the greater the contraction ratio C_d is, the stronger the anisotropic degree of the FST is. On the contrary, the smaller the contraction ratio C_d is, the weaker the anisotropic degree is. Figure 10 gives the variation of the initial amplitudes of excited T-S wave packets A_TSP with the contraction ratio. It shows that the initial amplitude increases rapidly with the contraction ratio. This is because a greater anisotropic degree of the FST can induce more intense leading-edge receptivity.

4 Conclusions

The leading-edge receptivity to isotropic and anisotropic FST is studied by direct numerical simulation, and the conclusions are presented as follows.

(ⅰ) Under the isotropic or anisotropic FST, three-dimensional T-S waves are generated in the flat-plate boundary layer.

(ⅱ) When the FST intensity is less than or equal to 1.0%, the initial amplitude of the excited T-S wave packets increases linearly with the FST intensity. When it is greater than 1.0%, the increase is greater.

(ⅲ) As the propagation angle φ of FST increases, the initial amplitudes of the excited T-S wave packets grow gradually and reach the maximum at around φ=-20°. Subsequently, the initial amplitudes decrease. An increase in the initial amplitudes with the angle θ is found.

(ⅳ) The group velocities of excited three-dimensional T-S wave packets are almost invariable with the angle θ of the FST. However, the propagation direction of the excited T-S wave packets deviates gradually from the x-axis with the increase in the angle θ. Additionally, the variation of the angle φ of the FST does not change the propagation directions and group velocities of the excited T-S wave packets.

(ⅴ) The stronger receptivity is induced with the greater anisotropic degree of the FST.