1 Introduction

The boundary-layer receptivity has always been an important project in the field of fluid mechanics. Receptivity is the initial stage of the laminar-turbulent transition in the boundary layer, which determines the physical mechanisms of the entire transition process. To date, great progress has been made for the receptivity in two-dimensional boundary layers, while there are far fewer researches on the receptivity in three-dimensional boundary layers^[1-6]. Nevertheless, most of the natural and engineered laminar-turbulent transition occurs in the three-dimensional boundary layers. Therefore, the research of three-dimensional boundary-layer receptivity is more valuable in theory and application^[7].

It has been found by theoretical and experimental studies that, when the turbulent level is low, the laminar-turbulent transition in the three-dimensional boundary layer is dominated by steady cross-flow vortices; on the contrary, when the turbulent level is high, the transition is dominated by unsteady cross-flow vortices. According to the wind tunnel experiments, the influence of the free-stream sound is found to be very slight in the cross-flow transition, which can be ignored. Bippes^[8] discovered experimentally that the unsteady cross-flow vortices dominate the laminar-turbulent transition in the three-dimensional boundary layer when the upstream turbulent level T_u is greater than 0.15%; as the upstream turbulent level 0.15% < T_u < 0.20%, though the unsteady cross-flow vortices dominate the laminar-turbulent transition, the transition positions are delayed compared with those at the lower turbulent level case, and that is because the unsteady cross-flow vortices in the boundary layer have been weakened by the steady cross-flow vortices.

Early studies on the three-dimensional boundary-layer receptivity focused on the receptivity dominated by steady cross-flow vortices. It was until recently that the unsteady cross-flow vortices dominating receptivity began to be studied by numerical methods and experiments. For example, using the direct numerical simulation (DNS), Schrader et al.^[9-10] studied the three-dimensional boundary-layer receptivity to free-stream vortices, and found the unsteady cross-flow vortices in the boundary layer. Through the DNS and the parabolized stability equations (PSE), Tempelmann et al.^[11-12] researched the receptivity in a smooth swept-wing boundary layer, and discovered the unsteady cross-flow vortices as well. Borodulin et al.^[13] experimentally studied the exciting process of the unsteady cross-flow vortices under the interaction between free-stream vortices and wall localized roughness in the three-dimensional boundary layer.

In this paper, the three-dimensional boundary-layer receptivity to free-stream turbulence is studied by the DNS. The mechanism of exciting unsteady cross-flow vortices under the action of free-stream turbulence is investigated. And the relations of the three-dimensional boundary-layer receptivity with the level and the direction of free-stream turbulence, the non-parallel effect, the influence of the leading-edge stagnation point of the flat plate, and the variation of the back-swept angle are discussed separately. In a word, the in-depth research of this problem provides the theoretical basis for the prediction and control of the laminar-turbulent transition.

2 Fundamental equation and numerical method 2.1 Fundamental equations

The fundamental equations are the non-dimensional incompressible Navier-Stokes equations:

(1)

Here, the velocity is V = U + V', V' = (u, v, w)^T is the perturbation velocity, and the basic flow velocity U is obtained by solving the Navier-Stokes equations. p is the pressure. The Reynolds number is Re = (U_∞δ^∗)/υ, where δ^∗ is the displacement thickness of the boundary layer, U_∞ is the infinite upstream velocity, and υ is the kinematic viscosity coefficient.

To solve the fundamental equations (1) numerically, the time derivatives are discretized by a modified fourth-order Runge-Kutta scheme, the space derivatives are discretized by a compact finite difference scheme based on non-uniform meshes, and Fourier series expansions are applied in the z-direction; the pressure Helmholtz equation is solved by a third-order finite difference scheme based on non-uniform meshes. See Refs. [2] and [14-15] for the details of the discretization schemes.

2.2 Free-stream turbulence model

On the basis of Ref. [16], the free-stream turbulence model is established, which is written as follows:

(2)

in which

(3)

where , and ω = mκ₁; u_∞, v_∞, and w_∞ denote the fluctuating velocity components of the free-stream turbulence model in the x-, y-, and z-directions, respectively; , and represent the fluctuating velocity spectra of u_∞, v_∞, and w_∞, respectively; ϵ is the amplitude of free-stream turbulence; M, J, and N are the selected maximum mode numbers; κ₁, κ₂, and κ₃ are the fundamental wave numbers in the x-, y-, and z-directions, respectively; κ=(m²κ₁²+j²κ₂²+n²κ₃²) ^1/2. The streamwise wave number α=mκ₁, the normal wave number γ = jκ₂, and the spanwise wave number β = nκ₃. , and correspond to the one-dimensional energy spectrum E(κ) and the random phase angle σ.

2.3 Computational domain and boundary conditions

Figure 1 shows the computational domain of the three-dimensional boundary-layer receptivity to free-stream turbulence: x∈[-200, 500], y∈[0, 14.39], z∈[-Z/2, Z/2], and Z=2π/κ₃. The leading-edge of the infinite flat-plate is at x=0. The back-swept angle Φ_BS is the angle between the direction of infinite upstream velocity U_∞ and the x-direction. The Reynolds number Re is set to 1 000. The computational meshes are 600×200×16 grids in the x-, y-, and z-directions respectively, and non-uniform meshes are utilized both in the x-and y-directions, which are concentrated at the near wall region and near leading-edge region so that the numerical results are more approximate to the actual flow. And the uniform mesh is adopted in the z-direction.

The boundary conditions are as follows:

The upper boundary: p=0; and the fluctuating velocity is given by (2).

The lower boundary: when x ≥ 0 and y=0, the no-slip condition is applied, i.e., u(x, 0) =v(x, 0) =w(x, 0) =0, and ; when x < 0 and y=0, all the variables apply symmetric conditions in the normal direction: .

The inflow boundary: the fluctuating velocity is also given by (2); .

The outflow boundary: ; the perturbation velocity is determined by the non-reflection outflow condition^[15].

The spanwise boundary: the periodic boundary condition is utilized.

2.4 Numerical validation

Solve the Navier-Stokes equations numerically to obtain the basic flow of the three-dimensional boundary layer, then the numerical solutions are compared with the theoretical solutions of the three-dimensional boundary layer. The maximum absolute errors of the velocity are less than 10^-5 order. It fully proves that the numerical method of the DNS in this paper has high precision, high resolution, and stability, which is a reliable computing means to study boundary-layer receptivity.

3 Numerical result and analysis 3.1 Three-dimensional boundary-layer receptivity to free-stream turbulence

First of all, the physical process of the three-dimensional boundary-layer receptivity to free-stream turbulence is studied. And the excited perturbation waves are extracted from the three-dimensional boundary layer by fast Fourier transform, then their dispersion relations, growth rates, and phase speeds are analyzed in detail. The amplitude of the free-stream turbulence at the outer edge of boundary layer tends to be steady after a long time calculation. Therefore, it can be defined as the free-stream turbulence level A_FST for the convenience of comparison, which is written as

(4)

Here, , and represent the time averaged square of the fluctuating velocity components of the free-stream turbulence in the x-, y-, and z-directions respectively. The non-dimensional frequency is defined as . Firstly, the three-dimensional boundary-layer receptivity to free-stream turbulence is investigated. The fundamental wave number κ₁=0.025, the maximum mode M=8, the spanwise wave number β=0.025, the back-swept angle is 45°, and the free-stream turbulence level A_FST=0.1%.

From the numerical results, a spatial sequence of wave packets of the streamwise, normal, and spanwise perturbation velocity has been discovered in the three-dimensional boundary layer at t=2 000 and propagates downstream in the S-direction, that is, the propagation direction of the excited wave packets, as shown in Fig. 2. According to Fig. 2, the angle between the S-direction and the x-direction is 5.5°; and the positions of the maximum and minimum of the excited wave packets of perturbation velocity are tracked at different time to approximately calculate the group speed of the wave packets, which is equal to 0.371.

Then, by fast Fourier transform, the distributions of the streamwise, normal, and spanwise perturbation velocity with frequency F=100 are extracted from the excited perturbation waves in the three-dimensional boundary layer at z=0, as shown in Fig. 3. From Fig. 3, the perturbation velocity distributions inside the boundary layer, which are excited by the interaction between the free-stream turbulence and the disturbances induced by the leading-edge of flat plate, are totally different from the perturbation velocity distributions of the free-stream turbulence outside the boundary layer. And it is also found that the wavelengths of the excited perturbation waves inside the boundary layer are relatively shorter than those outside the boundary layer. It illustrates that the free-stream turbulence can excite a sort of perturbation waves with the shorter wavelength in the three-dimensional boundary layer. Moreover, the normal perturbation velocity is smaller with one order than the streamwise and spanwise perturbation velocity.

Figure 4 gives the x-direction evolutions of the streamwise perturbation velocity components of the excited various frequency perturbation waves at y=0.66 and z=0, which is the maximum perturbation velocity position inside the boundary layer at frequency 100. According to the evolutions of the perturbation waves in Fig. 4, the peak (or valley) positions of the excited perturbation waves are tracked at different time to calculate the propagation speeds; moreover, the streamwise wave numbers (or wave lengths) and the growth rates are obtained by measuring the distance between the peaks (or valleys) and the neighbor peaks (or valleys), and the amplitude ratio of the peaks (or valleys) to the neighbor peaks (or valleys), respectively; in addition, those of other frequencies (25, 50, 75, 175, and 200) are also computed in the same way, and the results are shown in Table 1 and in Fig. 5. Note that the amplitude of the excited perturbation waves in the three-dimensional boundary layer A_CF is defined as follows:

(5)

where , and represent the time averaged square of the perturbation velocity components of the excited waves in the x-, y-, and z-directions, respectively.

Subsequently, as seen in Table 1 and Fig. 5, the numerical results of the dispersion relations, growth rates, and phase speeds of the excited perturbation waves are compared with the theoretical solutions of the linear stability theory (LST), and all of them are in good agreement. It is concluded that the free-stream turbulence is confirmed to be a mechanism to excite unsteady cross-flow vortices in the three-dimensional boundary layer, and the three-dimensional boundary-layer receptivity to free-stream turbulence does exist.

3.2 Effect of free-stream turbulence on receptivity

In this section, on account of the randomness and indeterminacy of free-stream turbulence, i.e., the level and the direction of free-stream turbulence shift uncertainly at any time, the roles of the level and the direction of free-stream turbulence in the three-dimensional boundary-layer receptivity are studied intensively. And the fundamental wave number κ₁=0.025, the maximum mode M=8.

Firstly, the effect of the free-stream turbulence level on the three-dimensional boundary-layer receptivity is studied. Figure 6 gives the relation between the maximum amplitude A_CF of the excited unsteady cross-flow vortex in the boundary layer and the amplitude of the imposed free-stream turbulence A_FST. A_CF is normalized by the maximum amplitude of the excited unsteady cross-flow vortices A₀ when A_FST=0.5%. When A_FST≤1.0% or A_FST > 1.0%, the relation between the excited unsteady cross-flow vortex amplitude and the free-stream turbulence level is linear or slightly less than the linear increase, respectively. These conclusions are in accord with the experimental data from Kurian et al.^[17] and the numerical results calculated by Schrader et al.^[10].

Secondly, in order to study the relation between the direction of free-stream turbulence and the three-dimensional boundary-layer receptivity, according to (2), the direction of free-stream turbulence is shifted by changing the streamwise wave number α, the normal wave number γ, and the spanwise wave number β. For simplicity, consider the influence of the projection of the direction of free-stream turbulence on the planes of Cartesian coordinate respectively, i.e., the angle θ between the x-direction and the projection of the direction of free-stream turbulence on the oxy plane, and the angle ϕ between the x-direction and the projection of the direction of free-stream turbulence on the oxz plane. Here, the receptivity coefficient K_I is defined as the ratio of the excited unsteady cross flow vortex amplitude A_CFI of a certain single frequency on the lower branch of the neutral stability curve in the boundary layer to the amplitude A_FS of a certain singe frequency disturbance among the free-stream turbulence whose frequency is equal to the frequency of the excited unsteady cross-flow vortex.

(6)

For the constant fundamental streamwise wave number κ₁=0.025 and the constant spanwise wave number of free-stream turbulence β=0, change the normal wave number γ to study the effect of the normal wave number γ or the angle θ on the three-dimensional boundary-layer receptivity. Figure 7 gives the relation between the excited receptivity coefficient K_I and normal wave number of free-stream turbulence γ or the angle θ. As the normal wave number or the angle θ grows, the excited receptivity coefficient K_I enhances gradually, and reaches a maximum around γ =0.05 or θ=16°; afterward, K_I decreases as the normal wave number or the angle θ continues to grow.

For the constant fundamental wave number κ₁=0.025 and the constant normal wave number γ = 0 of free-stream turbulence, just change the spanwise wave number to study the effect of the spanwise wave number β and the angle ϕ on the three-dimensional boundary-layer receptivity. Figure 8 gives the relation between the excited receptivity coefficient K_I and spanwise wave number of free-stream turbulence β or the angle ϕ. The excited receptivity coefficient K_I enhances gradually as the spanwise wave number or the angle ϕ increases. Besides, as shown in both Figs. 7 and 8, the lower the frequency is, the stronger the excited receptivity coefficient is, which agrees well with the numerical results of Schrader et al.^[9]. According to (2), it infers that the lower the streamwise wave number of free-stream turbulence is, the stronger the excited receptivity coefficient is as well.

In addition, the numerical results show that for the constant fundamental streamwise wave number of free-stream turbulence κ₁=0, whatever the normal wave number or spanwise wave number of free-stream turbulence changes, the unsteady cross-flow vortex cannot be excited in the three-dimensional boundary layer.

Finally, the relations of the propagation directions and speeds of the excited unsteady cross-flow vortex wave packets with spanwise wave numbers of free-stream turbulence are discussed. As shown in Table 2, the propagation direction of the excited unsteady cross-flow vortex wave packets whose spanwise wave number equals zero, is in accordance with the x-direction; furthermore, the angle ϕ_CF between the x-direction and the propagation direction of the unsteady cross-flow vortex wave packets increases gradually as the spanwise wave number of free-stream turbulence rises. Also seen in Table 2, the propagation speed of the excited unsteady cross-flow vortex wave packets increases slightly with the spanwise wave number of free-stream turbulence.

3.3 Effect of non-parallel and back-swept angle on three-dimensional boundary-layer receptivity

In this section, the effect of the non-parallel and back-swept angle on the three-dimensional boundary-layer receptivity is studied. The selected fundamental wave number, the level, and the maximum mode of the free-stream turbulence are κ₁=0.025, A_FST=0.5%, and M=8, respectively. At first, the effect of the non-parallel on the three-dimensional boundary-layer receptivity is studied. The inflow Reynolds number of the computational domain is defined as Re_x = U_∞l/υ, where l is the streamwise distance from the leading-edge.

Figure 9 gives the receptivity coefficient K_I of cross-flow vortices with different frequencies varying with Re_x. In the range of Re_x less than 200², the receptivity coefficient K_I changes rapidly. It is found that the stronger non-parallel effect is, when the inflow boundary of the computational domain is closer to the leading-edge stagnation point of flat plate, the stronger three-dimensional boundary-layer receptivity is triggered, and it turns to be the strongest when the computational domain includes the leading-edge stagnation point of flat plate. In the range of Re_x between 200² and 400², as Re_x increases, K_I declines slowly; and when Re_x is greater than 400², K_I decreases very slowly and almost stabilizes, for the non-parallel effect has been weak. Moreover, as seen in Fig. 9, the lower the disturbance frequency is, the more intense the excited receptivity coefficient is.

At last, the effect of the back-swept angles on the three-dimensional boundary-layer receptivity is also studied. Figure 10 shows the variation of the excited receptivity coefficient in the three-dimensional boundary layer with the back-swept angle. The excited three-dimensional boundary-layer receptivity coefficient rises as the back-swept angle increases; and reaches maximum when the back-swept angle is about 50°; subsequently, it decreases rapidly as the back-swept angle continues to increase.

4 Conclusions

Three-dimensional boundary-layer receptivity to free-stream turbulence is studied by the DNS, and the following conclusions are obtained:

(ⅰ) Under the free-stream turbulence, a group of unsteady cross-flow vortices have been discovered in the three-dimensional boundary layer. And the numerical results of their dispersion relations, growth rates, and phase speeds are in accord with the theoretical solutions of the LST, which confirms that free-stream turbulence is a mechanism to excite three-dimensional boundary-layer receptivity.

(ⅱ) When the free-stream turbulence level A_FST≤1.0% or A_FST > 1.0%, the amplitude of the excited unsteady cross-flow vortices in the boundary layer increases linearly or it is slightly less than the linear relation.

(ⅲ) The effect of the direction variation of the free-stream turbulence on the three-dimensional boundary-layer receptivity is considered. In other words, the variation of the angle θ and ϕ is considered. As θ or ϕ grows, the excited receptivity coefficient K_I in the three-dimensional boundary layer is similar to a downward parabola or increases gradually tendency, respectively. And K_I reaches a maximum around θ=16°.

(ⅳ) As the spanwise wave number of free-stream turbulence increases, the propagation direction of the excited unsteady cross-flow vortex wave packets deviates gradually from the x-direction, and its propagation speed increases slightly. In addition, when the spanwise wave number is equal to zero, its propagation direction is in line with x-direction.

(ⅴ) The stronger non-parallel effect is, when the inflow boundary of the computational domain is closer to the leading-edge stagnation point of flat plate, the stronger three-dimensional boundary-layer receptivity is triggered. Besides, when the computational domain includes the leading-edge stagnation point of flat plate, the strongest three-dimensional boundary-layer receptivity is going to be inspired. And the lower single frequency disturbance among the various frequency free-stream turbulence is, the more intense the excited receptivity coefficient is.

(ⅵ) The three-dimensional boundary-layer receptivity coefficient rises as the back-swept angle increases; and reaches the maximum when the back-swept angle Φ_BS is about 50°; afterward, it decreases as the back-swept angle continues to increase.