2. Governing equations and numerical algorithms

An incompressible flow with free-stream velocity $U_\infty$ over a flat plate with half thickness $R$ is considered and all the following variables are non-dimensionalized by $U_\infty$ and $R$. In the inertial Cartesian coordinate system, $\tilde {\boldsymbol {x}}=({\tilde {x}, \tilde {y}, \tilde {z}})^\textrm {T}$ represents the streamwise, vertical and spanwise directions, respectively and the plate vibrates in the vertical direction with a velocity

(2.1)\begin{equation} \boldsymbol{v}_{w}=(0, A\sin(\omega_{w}t+\phi), 0)^\textrm{T}, \end{equation}

where $A, \omega _w$ and $\phi$ are the magnitude, frequency and phase of the vibration, respectively.

In the rest of this paper, we adopt a body coordinate system attached to the plate denoted as $\boldsymbol {x}=(x, y, z)^\textrm {T}$ (see figure 1). The variables in the body framework satisfy

(2.2a)\begin{gather} \boldsymbol{x}+\int_{0}^{t}\boldsymbol{v}_{w}\,\mathrm{d}t=\tilde{\boldsymbol{x}}, \end{gather}

(2.2b)\begin{gather}p=\tilde{p}, \end{gather}

(2.2c)\begin{gather}\boldsymbol{u}+\boldsymbol{v}_{w}=\tilde{\boldsymbol{u}}, \end{gather}

where $\boldsymbol {u}=(u,v,w)^\textrm {T}$ and $p$ are the velocity and pressure of the fluid in the body framework $\boldsymbol {x}$, respectively; and $\tilde {\boldsymbol {u}}$ and $\tilde {p}$ are the velocity and pressure in the inertia framework $\tilde {\boldsymbol {x}}$, respectively.

Figure 1. Schematic diagram of the computation domain, boundary conditions and the coordinate frameworks.

Then $\boldsymbol {u}$ and $p$ are solutions of the incompressible Navier–Stokes (NS) equations expressed in the body framework:

(2.3a)\begin{gather} \partial _{t} \boldsymbol{u}+\boldsymbol{u} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}+ \boldsymbol{\nabla}{p}- Re^{-1} \boldsymbol{\nabla}^{2}\boldsymbol{u}+\partial _{t}\boldsymbol{v}_{w}+\boldsymbol{v}_{w} \boldsymbol{\cdot} \boldsymbol{\nabla}\boldsymbol{u}=0, \end{gather}

(2.3b)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}=0, \end{gather}

where $Re$ is the Reynolds number based on $U_\infty$ and $R$. The last two terms on the left-hand side of (2.3a) are non-inertial terms induced by the vibration.

The velocity vector and the pressure can be decomposed as the sum of a base state $(\boldsymbol {U},P)$ and a perturbation state $(\boldsymbol {u}',p'), (\boldsymbol {u},p)=(\boldsymbol {U},P)+(\boldsymbol {u}',p')$. When the amplitude of the perturbation is sufficiently small, the development of perturbations can be described by the linearized NS equations

(2.4a)\begin{gather} \partial_{t}\boldsymbol{u}'+\boldsymbol{U} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}'+ \boldsymbol{v}_{w} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}'+\boldsymbol{u}' \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{U}+ \boldsymbol{\nabla}{p'}- Re^{-1} \boldsymbol{\nabla}^{2}\boldsymbol{u}'=0, \end{gather}

(2.4b)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}'=0, \end{gather}

which can be written compactly as $\partial _t \boldsymbol {u}'-\mathcal {L}(\boldsymbol {u}')=0$ with $\mathcal {L}$ denoting a linear operator depending on the base state.

To study the effects of free-stream disturbance on bypass transition, perturbations are introduced into the domain from the inflow boundary in the form of the velocity boundary condition of (2.4), and the initial condition of $\boldsymbol {u}'$ is set to zero. This inflow perturbation is a function of both space and time and can be decomposed as

(2.5)\begin{equation} \boldsymbol{u}'(\boldsymbol{x}_{{in}},t)=G(t)\boldsymbol{u}_{{in}}'(\boldsymbol{x}_{{in}}), \end{equation}

where $\boldsymbol {u}_{{in}}'(\boldsymbol {x}_{{in}})$ is the spatial dependence of the perturbation, $\boldsymbol {x}_{{in}}$ denotes the inflow boundary and $G(t)$ is a prescribed temporal function given by $(1-\textrm {e}^{-100 t^{2}}) (1-\textrm {e}^{-100 (T_o-t)^{2}})\textrm {e}^{\textrm {i}\omega t}$ in which the first and second terms ensure that the initial condition and boundary condition are compatible (Mao, Blackburn & Sherwin Reference Mao, Blackburn and Sherwin2013), and $\omega$ specifies the frequency of the inflow perturbation.

The amplification of the inflow perturbation in the boundary-layer flow can be described by

(2.6)\begin{equation} K(T_o)=\frac{\displaystyle\int_{\varOmega} \boldsymbol{u}'(T_o)\boldsymbol{\cdot} \boldsymbol{u}'(T_o)\mathrm{d}\varOmega}{\displaystyle\int_{\boldsymbol{x}_{{in}}} \boldsymbol{u}_{{in}}'\boldsymbol{\cdot} \boldsymbol{u}_{{in}}'\,\mathrm{d}S}, \end{equation}

where $\varOmega$ represents the upper half of the fluid domain and $T_o$ is the final time. Unless otherwise stated, $T_o$ is set to 180 in this work, as applied by Wang et al. (Reference Wang, Mao and Zaki2019). The lower half is symmetric with the upper half and this definition of $K$ takes into account the perturbation amplifications over the upper half only so as to concentrate the resolution on the upper boundary layer.

To quantify the impact of the wall vibration on the inflow perturbation developments, we compare the maximum value of $K$, denoted as $K_{{max}}$, for the vibrating and non-vibrating cases. Clearly this will result in a more robust evaluation of the vibration effect than adopting a random inflow noise. The optimal perturbation is regarded as the most critical component in the broadband free-stream disturbance. The consideration of optimal perturbation highlights the influence of vertical vibration of the wall on the variation and evolution of the key component of broadband disturbance. To compute this optimal value and the corresponding inflow perturbation, we define a Lagrangian functional:

(2.7)\begin{equation} {L}=K(T_o)-\frac{1}{T_o}\int_{0}^{T_o}\int_{\varOmega}[\boldsymbol{u}^{\dagger}\boldsymbol{\cdot}(\partial_t \boldsymbol{u}'-\mathcal{L}(\boldsymbol{u}')]\,\mathrm{d}\varOmega\,\mathrm{d}t, \end{equation}

where $\boldsymbol {u}^{{\dagger} }$ is the adjoint velocity, the first term on the right-hand side is the objective function to be maximized and the second represents a constraint that the perturbation satisfies the governing equations (2.4). Setting the variations of the Lagrangian functional to zero and after integration by parts, a set of adjoint equation is obtained,

(2.8a)\begin{gather} \partial_{t}\boldsymbol{u}^{\dagger}+(\boldsymbol{U}+\boldsymbol{v}_{w}) \boldsymbol{\cdot} \boldsymbol{\nabla}\boldsymbol{u}^{\dagger}- \boldsymbol{\nabla}\boldsymbol{U} \boldsymbol{\cdot}\boldsymbol{u}^{\dagger}- \boldsymbol{\nabla}{p}^{\dagger}+Re^{-1} \boldsymbol{\nabla}^{2}\boldsymbol{u}^{\dagger}=0, \end{gather}

(2.8b)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}^{\dagger}=0, \end{gather}

where $p^{{\dagger} }$ is the adjoint pressure and the initial condition for the adjoint velocity is given by

(2.9)\begin{equation} \boldsymbol{u}^{\dagger}(T_o)=\frac{2\boldsymbol{u}'(T_o)}{\displaystyle\int_{\boldsymbol{x}_{{in}}} \boldsymbol{u}_{{in}}'\boldsymbol{\cdot} \boldsymbol{u}_{{in}}'\,\mathrm{d}S} . \end{equation}

The gradient of the Lagrangian functional with respect to the inflow perturbation is given by

(2.10)\begin{equation} \boldsymbol{\nabla}_{\boldsymbol{u}'_{{in}}}L=\frac{-2\displaystyle\int_{\varOmega}\boldsymbol{u}'(T_o) \boldsymbol{\cdot} \boldsymbol{u}'(T_o)\mathrm{d}\varOmega}{\displaystyle\left(\int_{\boldsymbol{x}_{{in}}} \boldsymbol{u}_{{in}}'\boldsymbol{\cdot} \boldsymbol{u}_{{in}}'\,\mathrm{d}S\right)^{2}}\boldsymbol{u}_{{in}}'+ \frac{1}{T_o}\int_{0}^{T_o}({p}^{\dagger}\boldsymbol{n}- Re^{-1} \boldsymbol{\nabla}_{\boldsymbol{n}}\boldsymbol{u}^{\dagger})G\,\mathrm{d}t, \end{equation}

where $\boldsymbol {n}$ is the unit outward norm of the inflow boundary.

A gradient-based iterative optimization algorithm is used to find the optimal inflow perturbation. When the gradient of the Lagrangian with respect to the inflow perturbation vanishes, the obtained inflow perturbation is the optimal one generating the most energetic perturbations in the boundary-layer flow, and the corresponding $K$ takes the maximum value $K_{{max}}$. Further details of the algorithm can be found in Mao et al. (Reference Mao, Blackburn and Sherwin2013).

3. Computation set-up

The computation domain is shown in figure 1. The inflow boundary locates at $x=-20$ and the outflow boundary is at $x=200$. In the wall-normal direction, the domain spans from $y=-40$ to $y=40$. The leading edge of the two-dimensional (2-D) plate is located at $(0,0,0)^\textrm {T}$. The streamwise length of the plate is $l=200$ for most of the studies and is extended to 300 for the investigation of transition, and its half-thickness is used as the characteristic length. In the spanwise direction, a complex Fourier decomposition is applied to the perturbation velocity:

(3.1)\begin{equation} \boldsymbol{u}'(x,y,z)=\sum_{\beta=0}^{\infty}\boldsymbol{u}'_{\beta}(x,y)\textrm{e}^{\textrm{i}\beta z} . \end{equation}

The base flow is a 2-D solution of the NS equations (2.3). It is unsteady owing to the periodic vibration of the plate and snapshots of the velocity field are saved after reaching a periodic state when solving the 2-D NS equations. A Lagrangian interpolation is used to reconstruct the base flow $U$ at every time step when solving (2.4) and (2.8) (Mao, Sherwin & Blackburn Reference Mao, Sherwin and Blackburn2011). In the linear regime, perturbations with different spanwise wavenumbers are decoupled and can be simulated independently. A spectral element method is used to discretize the governing equations in the $x$–$y$ plane (Karniadakis & Sherwin Reference Karniadakis and Sherwin2007). In this work, the domain is decomposed into 3835 elements and each element is further decomposed into a $(N+1)(N+1)$ grid, where $N$ represents the polynomial order of the spectral discretization.

A summary of boundary conditions is given in table 1. Zero-Neumann boundary conditions are used for the pressure term on all boundaries, except for the outflow boundary, where the pressure is set to zero. The boundary conditions for the adjoint variables ensure that the variances of the Lagrangian functional are zero (Blackburn & Sherwin Reference Blackburn and Sherwin2004; Mao et al. Reference Mao, Blackburn and Sherwin2013).

Table 1. Summary of boundary conditions.

The convergence tests for grid resolution are conducted by varying the polynomial order $N$ in each element in the $x$–$y$ plane. The resolution in the spanwise direction is only relevant to three-dimensional (3-D) studies and is presented in § 5. Figure 2 shows the convergence of $K_{{max}}$ at $\beta =1.4$ and $Re=800$, the largest Reynolds number considered in this work. At $N=7$, the optimal gain $K_{{max}}$ converges to three significant digits and this polynomial order will be used in all the following linear and nonlinear calculations.

Figure 2. Convergence of $K_{{max}}$ with respect to the polynomial order $N$ at $(Re, \beta )=(800, 1.4)$, and various $\omega$.

4. Linear perturbation analyses

Adopting the algorithms presented in § 2, we conduct the linear perturbation analyses to investigate the development of the optimal inflow perturbations over the vibrating plate. First, the base flow around the vibrating flat plate is presented.

4.1. The base flow

The base flow obtained by integrating (2.3) over enough cycles of the plate vibration until reaching periodic states at $Re=800,\ A=0.01$ and $\omega _{w}=2{\rm \pi} /180$ is shown in figure 3. Here the phase of vibration $\phi$ only shifts the initial phase of the base flow. Without loss of generality, $\phi =0$ is adopted. Later in § 4.3, effects of $\phi$ are discussed. The vibration leads to a sinusoidal variation of the angle of attack, which can be visualized from the instantaneous flow fields at different phases as shown in figure 3(a,b). In figure 3(c), the mean velocity averaged over a period for the vibrating plate almost overlaps with the steady velocity profile for a fixed plate and the Blasius solution, indicating that the plate vibration does not change the mean boundary-layer profile over the parameters studied. It is also observed that the instantaneous velocity profile remains Blasius-like and shifts around the steady solution. The evolution of the boundary-layer thickness $\delta$, which reaches $\delta =2.27R$ at $x=200$, also shows that the small vertical vibration in this study has little effect on the mean boundary-layer flow (see figure 3e). As the pressure on the windward side is larger than that on the leeward side, pressure on the upper surface increases or decreases from the mean value when the plate is moving upward or downward (see figure 3d) and the largest variance can be found around the leading edge. We show the pressure distribution along the upper surface, while a similar pressure field is developed on the lower surface and is not presented here. The flow is therefore subject to alternating favourable and adverse pressure gradients around the leading edge in one vibration cycle. Further downstream of the leading edge, the mean flow undergoes small favourable pressure gradients while instantaneously, negative pressure gradient exists over almost half of the cycle. Though the vibration amplitude of the plate is small, it produces remarkable change of flow with time. Base flows at other $Re,\ A$ and $\omega _{w}$ considered in this work are also featured by sinusoidally varying angles of attack and time-dependent pressure gradients, and are not shown here for brevity.

Figure 3. Base flow at $(Re, A, \omega _{w})=(800, 0.01, 2{\rm \pi} /180)$. (a), (b) Contours of streamwise velocity and streamlines around the leading edge at $t={\rm \pi} /(4\omega _{w})$ and $t=3{\rm \pi} /(4\omega _{w})$, respectively. (c) Comparisons of velocity profiles at $x=100, \eta$ is defined as $y/\sqrt { Re\cdot {x}}$. (d) Pressure coefficient $C_p=p-p_\infty$, where $p_\infty$ is the non-dimensionalized pressure in the far field, along the upper surface of the plate. (e) Development of the boundary-layer thickness along the flat plate.

4.2. Amplification of inflow perturbations

The optimal inflow perturbations and the gain $K_{{max}}$ are then calculated following the algorithms outlined in § 2. It is noted that if the plate does not vibrate, the linear dynamics can be further simplified by studying only the half of the domain with symmetric/asymmetric boundary conditions at $y=0$ (Wang et al. Reference Wang, Mao and Zaki2019). However, in the present investigation, the full domain is considered to accommodate the vibration effect, even if the resolution is concentrated on the upper half.

Effects of the Reynolds number on the optimal gain $K_{{max}}$ are shown in figure 4, where the contours of $K_{{max}}$ over a wide range of spanwise wavenumbers and perturbation frequencies are compared.Over the Reynolds numbers considered, the global maximum of $K_{{max}}$ across the range of $\beta$ and $\omega$ studied corresponds to the steady inflow perturbation, similar to the result for a fixed plate (Monokrousos et al. Reference Monokrousos, Åkervik, Brandt and Henningson2010). As the Reynolds number increases from $Re=300$ to $Re=500$ and $Re=800$ (figures 4a,b and 6c), the boundary layer's response to disturbances becomes more energetic, and the optimal spanwise wavenumber at which the gain of the steady perturbation reaches maximum increases from 0.87 to 1.14 and 1.4, respectively. By taking into account the normalized wavenumber,

(4.1)\begin{equation} \beta'=\beta/\sqrt{Re} , \end{equation}

the three Reynolds numbers give rise to optimal wavenumbers $\beta '_{{max}}=\{4.95,5.10,5.02\}\times 10^{-2}$, indicating that the optimal spanwise wavenumber at $\omega =0$ is insensitive to $Re$ after the appropriate normalization (Luchini Reference Luchini2000; Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001).

Figure 4. Contours of $K_{{max}}$ for optimal inflow perturbations at $(\omega _{w}, A)=(2{\rm \pi} /180, 0.01)$ with (a) $Re=300$ and (b) $Re=500$. The red dashed line marks the vibration frequency of the flat plate and the red solid line illustrates the optimal spanwise wavenumber at different $\omega$. The black circle represents the global maximum point of $K_{{max}}$. Here $\beta '$ is a normalized spanwise wavenumber.

Actually, the optimal spanwise wavenumbers at other perturbation frequencies are also insensitive to the Reynolds number. The red solid curves in figure 4, which connect the optimal spanwise wavenumber at different perturbation frequencies, are extracted and plotted compactly in figure 5. With the same vibration frequency and amplitude, the curves at different Reynolds numbers show a similar shape. The optimal spanwise wavenumber increases slowly when $\omega$ is small and then rises almost linearly for larger $\omega$. Over the range of perturbation frequencies considered $(\omega <0.15)$, the curves can be fitted as the Logistic functions, which can be approximately unified as $\omega =C_2+(C_1-C_2)/(1+(\beta '_{{max}}/C_3)^\sigma )$, where $C_1=0.0508,\ C_2=0.0975,\ C_3=0.122$ and $\sigma =2.39$ for the present case. Later, we show that this $Re$ insensitive parameter $\beta '_{{max}}$ strongly depends on the vibration of the plate. From figures 4 and 6(c), we also find that the optimal gain $K_{{max}}$ decreases with $\omega$, manifesting the shear sheltering mechanism of the boundary layer to high-frequency disturbances. As shown later, this nearly monotonic decrease does not hold when the plate vibrates at high frequencies. After clarifying the Reynolds number effects, we focus on $Re=800$ in the following analyses.

Figure 5. Variation of the optimal spanwise wavenumber at different Reynolds numbers.

Figure 6. Effects of vibration amplitude on the amplification of the optimal inflow perturbations. (a) Contours of $K_{{max}}$ at $(Re, \omega _{w}, A)=(800, 2{\rm \pi} /180, 0)$. (b) Contours of $K_{{max}}$ at $(Re, \omega _{w}, A)=(800, 2{\rm \pi} /180, 0.005)$. (c) Contours of $K_{{max}}$ at $(Re, \omega _{w}, A)=(800, 2{\rm \pi} /180, 0.01)$. (d) Variation of the global maximum of $K_{{max}}$ with vibration amplitude $A$. In contours (a) to (c), the red dashed line marks the vibration frequency of the flat plate and the red solid line connects the optimal spanwise wavenumbers at different $\omega$. The black circle represents the point of the global maximum of $K_{{max}}$.

The influences of vibration amplitudes on the amplification of optimal inflow perturbations are presented in figure 6. Under the range of parameters considered, a stronger vibration results in weaker amplification of the inflow perturbations. The global maximum of $K_{{max}}$, which is obtained at spanwise wavenumber $\beta =1.4$ and frequency $\omega =0$, decreases from 6850 to 4936 and 3580 when the vibration amplitude increases from 0 (non-vibrating) to 0.005 and 0.01, as shown in figure 6(a–c). The variation of the global maximum of $K_{{max}}$ with the vibration amplitude at given frequencies is shown in figure 6(d). It confirms that the global maximum amplification of the inflow perturbation decreases monotonically with the vibration amplitude $A$. Compared with the fixed plate case, the reduction of the global maximum gain is up to 60 % at $A=0.0145$, suggesting that the lift-up mechanism would be suppressed owing to the plate vibration.

The amplification of optimal inflow perturbations at $\beta =1.4$ for different vibration amplitudes is extracted and presented in figure 7(a). It shows that the amplification of the optimal perturbation reduces faster as $\omega$ increases for higher vibration amplitudes. Above a critical frequency, the inflow perturbation does not enter the boundary layer and develops in the free stream. If ignoring the viscous diffusion, the corresponding perturbation growth is $K_{{max}}=To/2=90$. This critical frequency is defined as the cut-off frequency, which decreases with the vibration amplitude $A$ as can be seen in figure 7(a). This observation suggests that the vibration strengthens the sheltering effect of the boundary layer to high-frequency free-stream disturbance (it will be shown later that this is true only at low vibration frequencies). In addition to the cut-off frequency, the vibration also clearly alters the optimal spanwise wavenumber (see figure 7b), which has been shown to be insensitive to $Re$ in figure 5.

Figure 7. Effects of vibration amplitude on the cut-off frequency and optimal spanwise wavenumber. (a) Variation of $K_{{max}}$ with $\omega$ at $(Re, \beta )=(800, 1.4)$. The vertical lines denote the position of $\omega =\omega _{w}=2{\rm \pi} /180$. (b) Variation of $\beta '_{{max}}$ with $\omega$.

The contour maps of $K_{{max}}$ for different vibration frequencies are presented in figures 6(c), 8(a) and 8(b). The global maximum of $K_{{max}}$ decreases as the vibration frequency $\omega _w$ increases. Compared with low-frequency vibration cases (figure 6), the high-frequency vibration of the plate apparently induces a second local peak on the contour plot of $K_{{max}}$. The first peak remains at $\omega =0$ with a reduced value as compared with the stationary plate result, whereas the second appears at $\omega =\omega _w$. This new peak can be considered as a manifestation of the resonance effect of the flow perturbations and the vibrating plate and is the focus of the rest of this work. In fact, there is always a resonance effect as long as the plate vibrates. For the lower-frequency case ($\omega _w=2{\rm \pi} /180$ in figure 6b,c), the resonance peak is submerged in the main contour so it is not clearly visible. Figure 7(a) shows that the curves of $K_{{max}}$ present a small local peak at $\omega =\omega _w=2{\rm \pi} /180$, manifesting a weak resonance effect for a low-frequency vibrating plate. The profile of the normalized optimal spanwise wavenumber (marked by the red line in figure 8) shows that the optimal $\beta$ increases and then decreases with $\omega$ when there is a resonance induced local maximum of perturbation amplification. At the resonant frequency, the optimal spanwise wavenumber equals that at zero perturbation frequency $(\beta =1.4)$, indicating a similar mechanism of perturbation amplifications (and streak generations) at the resonance frequency and zero perturbation frequency. This will be further illustrated in § 4.3. To summarize, a higher magnitude of the plate vibration results in a lower perturbation amplification and lower cut-off frequency, whereas a higher vibrating frequency results in clear resonance with the perturbation development.

Figure 8. Effect of vibration frequency on the amplification of optimal inflow perturbations. (a) Contours of $K_{{max}}$ at $(Re, \omega _{w}, A)=(800, 2{\rm \pi} /60, 0.01)$. (b) Contours of $K_{{max}}$ at $(Re, \omega _{w}, A)=(800, 2{\rm \pi} /45, 0.01)$. The red dashed line marks the vibration frequency of the flat plate and the red solid line connects the optimal spanwise wavenumbers at different $\omega$. The black circle represents the point of local maximum of $K_{{max}}$. The black square point marks the resonant point.

4.3. Resonance between the vibrating plate and perturbations

The details of the influences of resonance on the perturbation amplifications are plotted in figure 9. The amplification of the optimal inflow perturbations shows a downward trend owing to the shielding mechanism of the boundary layer, as can be seen in figure 9. Further, $K_{{max}}$ fluctuates owing to the plate vibration and several peak values (marked by solid dots) appear at the harmonics of $\omega _w$, manifesting the resonance phenomena. Therefore, the curves can be interpreted as the superposed effects of the boundary-layer sheltering to high-frequency perturbations and resonance between perturbations and the plate. One of the consequences is that the cut-off frequency becomes higher when resonance happens, indicating that higher-frequency free-stream perturbations are admitted by the boundary layer owing to the plate vibration. Within the scope of parameters considered, the resonance-induced gain is still smaller than that at zero frequency. For example, the gain at $\omega =\omega _w=2{\rm \pi} /60$ is about 36 % of the gain at $\omega =0$. We focus on the resonance-induced gain because it allows higher-frequency perturbations to enter the boundary layer, which might alter the bypass transition process on the vertically vibrating plate. For instance, the resonance-induced gain is 1.46 times of the gain of stationary case at the same perturbation frequency $\omega =2{\rm \pi} /60$.

Figure 9. Variation of $K_{{max}}$ with $\omega$ at $(Re, \beta )=(800, 1.4)$ for different $\omega _{w}$. The dots denote the resonant points. The vertical lines denote the position of $\omega =\omega _{w}$.

To further examine the resonance effect, the velocity profiles of the optimal inflow perturbation at various $\omega _w$ are shown in figure 10. The inflow perturbations are scaled so that they have the same energy or norm, which is defined as

(4.2)\begin{equation} \|\boldsymbol{u}_{{in}}'\|=\sqrt{\int_{\boldsymbol{x}_{{in}}} \boldsymbol{u}_{{in}}' \boldsymbol{\cdot} \boldsymbol{u}_{{in}}'\,\mathrm{d}S} . \end{equation}

As the velocity is optimized to induce maximum perturbation gain on the upper half of the domain, the perturbation velocity reduces to trivial levels for $y>5$ or $y<-1$. The inflow velocity perturbation is decomposed into streamwise $(u')$, wall-normal $(v')$ and spanwise $(w')$ components. The velocity profiles show that the optimal perturbations are mainly composed of wall-normal and spanwise components, although there is a weak streamwise one (see figure 10a,b). The streamwise vorticity perturbation formed by the wall-normal and spanwise components tilts the base shear profile and leads to the generation of streamwise perturbation via the lift-up mechanism. Therefore, if there is no leading edge, the optimal inflow velocity should be concentrated on the wall-normal and spanwise components only. However, the leading edge tilts the inflow perturbations and therefore a streamwise velocity perturbation component appears in the optimal profile and will contribute to the streamwise vorticity downstream of the leading edge (Wanderley & Corke Reference Wanderley and Corke2001; Schrader et al. Reference Schrader, Brandt, Mavriplis and Henningson2010). Figure 10(a) also shows that the optimal disturbance of the stationary case is concentrated at $0 < y < 2$ and associated with the boundary layer on the upper surface. As a result of the vertical vibration of the wall, the optimal perturbation of the vibrating case is less localized as compared with that of the stationary case (see figure 10d). In the resonance regime, the velocity profile over the vibrating plate consists of more layers in the vertical direction than that over the stationary plate, as shown in figure 10(b,e). This multi-layer structure will be shown later to be critical to activate the resonance. Above the cut-off frequency, the velocity profiles in figure 10(c,f) share similar shapes. The wall-normal component $(v')$ is small whereas $u'$ becomes the dominant component, which can be attributed to the ineffective lift-up mechanism above the cut-off frequency.

Figure 10. Velocity profile of the optimal inflow perturbation at $(Re, \beta )=(800, 1.4)$: (a), (b) and (c) perturbation components for a non-vibrating plate at $\omega =0,\ \omega =2{\rm \pi} /60$ and $\omega =0.5$, respectively; (d)–(f) same parameters as (a)–(c) but for the vibrating plate with $\omega _w=2{\rm \pi} /60$.

In figure 11, we further acquire the responses of the boundary-layer flow to the optimal inflow perturbations by applying the optimal perturbations at the inflow and integrating the linearized NS equations (2.4). Again, the inflow perturbations are normalized to have a unit norm. As mentioned previously, the steady inflow perturbations penetrate the boundary layer and activate the lift-up mechanism to generate elongated streamwise velocity streaks in the boundary layer while the plate vibration reduces the magnitude of the streaks as can be inferred from figure 6. This is confirmed by the streak profiles shown in figure 11(a,b), where the plate is stationary and vibrating, respectively. At $\omega =2{\rm \pi} /60$, which is still below the cut-off frequency, short streaks with alternating signs in the streamwise are induced by the optimal inflow perturbation over a stationary plate (see figure 11c). When the plate vibrates and resonance occurs, the sign of the streaks becomes invariant and a single streak is generated (see figure 11d). This effective generation of strong streak reflects the large amplification of the optimal inflow perturbation (shown in figure 10e) illustrated by the local peak of $K_{{max}}$ in figure 9. The mechanism to generate such elongated streaks will be explored later. The high-frequency perturbations over the cut-off frequency can hardly penetrate the boundary layer and mainly reside in the free stream while the plate vibration induces the vertical motion of the perturbation (see figure 11e,f). This observation agrees with the shear sheltering effect on high-frequency perturbations revealed by Zaki & Saha (Reference Zaki and Saha2009). As the responses of the boundary layer to disturbances with frequencies above the cut-off value are weak, such high-frequency optimal inflow perturbations will not be considered further. Please note that we only show the evolution of the optimal perturbations at three perturbation frequencies $\omega ={0,2{\rm \pi} /60,0.5}$ in figure 11. In fact, the inflow perturbation with frequency above $\omega =2{\rm \pi} /60\approx 0.105$ is still allowed inside the boundary layer because it is below the cut-off frequency $\omega =0.37$.

Figure 11. Side view of isosurfaces of streamwise perturbation $u'$ at $(Re, t)=(800, 180)$: (a,b) $\omega =0$; (c,d) $\omega =\omega _{w}$ and (e,f) $\omega =0.5$. (a), (c) and (e) Non-vibrating plate and (b), (d) and (f) vibrating plate at $\omega _{w}=2{\rm \pi} /60$. The dashed line marks the boundary-layer thickness $\delta$, which is the height from the wall where the streamwise velocity reaches 99 % of the free-stream velocity.

Comparing the present outcome with the case of transversely oscillating wall considered by Hack & Zaki (Reference Hack and Zaki2012), the transverse oscillation introduces the Stokes shear layer to the boundary-layer flow, which leads to a substantial reduction of free-stream disturbance energy entering the boundary layer. In addition to the stronger shear sheltering effect, the Stokes shear layer reorients the shear vector and imposes a new restriction on the set of penetrating modes such that the orientation of modes subject to weak sheltering effect deviates from the streamwise direction. For our case of vertically vibrating flat plate, the vibration introduces the periodic variation of the angle of attack, which leads to the time harmonic pressure gradient on the wall. As the vertical vibration in our work is small, this pressure gradient is only significant around the leading edge and becomes negligible on the flat plate far from the leading edge. The shear layer on the vertically vibration wall is thus inappreciably changed by the time harmonic pressure gradient on the plate. However, the shear sheltering of the boundary-layer flow is still influenced by the vertical wall motion because it allows synchronized inflow perturbation entering the boundary layer.

To inspect the effect of plate vibration on the resonance of the fluid flow more closely, the evolution history of the perturbation is shown in figure 12. Note that over the optimization time $T_o=180$, the inflow perturbation (and the plate vibration) presents three cycles. At $t=3T_o/12$, the streamwise velocity perturbation upstream of the leading edge consists of a positive and a negative component, matching the profile of the inflow perturbation in figure 10(e). Then the free-stream perturbation is stretched and tilted by the curved mean-flow streamlines near the leading edge (see figure 3a,b). When the inflow disturbance moves closer to the leading edge, the upward/downward movement of the plate imposes a negative/positive angle of attack, which shifts the inflow perturbation towards the lower/upper side of the plate. As a result, the inflow disturbance is staggered. As the frequency and phase the plate vibration and flow oscillation match each other, both the negative and positive components of the streamwise velocity perturbation enter the upper side of the plate and the negative (positive) part is in (out of) the boundary layer at $t=3T_o/12$. Then at $t=5T_o/12$, the positive perturbation on the lower array is shifted to the lower side of the plate, while the negative perturbation on the upper array enters into the upper side. In other words, the inflow disturbance is appropriately staggered that only the same sign of perturbation can remain inside the boundary layer. At $t=7T_o/12$, the negative components of the streamwise velocity on the lower array enters the upper side of the plate again, completing a vibration cycle started at $t=3T_o/12$. At $t=9T_o/12$ and $t=11T_o/12$, as the negative streamwise velocity is continuously convected downstream, an elongated streak is formed on the upper side. The streamwise vorticity shown in the right column of figure 12 presents similar staggered motion as the streamwise velocity. As the sign of the streamwise vorticity inside the upper boundary layer remains the same, it has the capacity to amplify the streaks to a magnitude much larger than the non-resonance cases. At this point of view, the multi-arrays of inflow disturbance is necessary to produce a sign-invariant perturbation inside the boundary layer and thus induce an energetic streaky response, which is exactly the case of optimal inflow perturbation shown in figure 10(e). The negative streamwise vorticity on the lower array, which enters into the upper side of the plate due to the downward movement of the plate (see at $t=3T_o/12, 7T_o/12, 11T_o/12$), is named as $\xi _{x1}'$. The negative streamwise vorticity on the upper array, which enters into the upper side of the plate because of the upward movement of the plate (see at $t=5T_o/12, 9T_o/12$), is denoted as $\xi _{x2}'$. In § 5, we show that $\xi _{x1}'$ and $\xi _{x2}'$ have profound effects on the distortion of low-speed streaks.

Figure 12. Evolution of resonant inflow perturbations (a) $u'$ and (b) streamwise vorticity $\xi _x'$ at $\omega =\omega _{w}=2{\rm \pi} /60$. The dashed line marks the boundary-layer thickness on the upper side, $\delta$. The dashed box marks the incoming $\xi _{x2}'$ and the dash-dotted box marks the incoming $\xi _{x1}'$.

The pattern of the staggering, illustrating a synchronization of the motion of the plate and the perturbation, is shown in figure 13, where the black arrows represent the angle of attack as the disturbances move around the leading edge. This effect increases monotonically with the vibration amplitude $A$. For steady inflow perturbations, the vibrating plate shifts the streamwise vorticity to lower/upper side of the plate when the plate is moving upwards/downwards. Therefore, the free-stream streamwise vorticity and velocity entering the upper side of the boundary layer are broken into segments, preventing the continuous amplification of the streaks, and subsequently the lift-up mechanism is weakened by the plate vibration. As a result, the amplification of steady free-stream perturbations in the boundary layer over a vibrating surface reduces at higher vibration magnitude $A$ as observed in figure 6(d). In fact, the synchronization of the motion of the plate and the perturbation can also be activated when the vibration frequency is half of the perturbation frequency. Therefore, there are peaks at $0.5\times \omega _w$ in figure 9, though the peak values are small.

Figure 13. Schematic plot of the staggering effect of vibrating plate at resonant frequency. The black arrows represent the angle of attack as the disturbances move around the leading edge.

From the discussion given previously, clearly the resonance depends on the match of frequencies and phases between the vibration and inflow disturbances. When the match is broken, the synchronized staggering disappears and disturbances are disordered on the upper part of the plate. Figure 14 shows the response of boundary-layer under different initial phases of the vibrating plate. The optimal inflow perturbation shown in figure 10(e) is applied on the inflow boundary when we integrate the linearized NS equations. The perturbation vorticity is tilted more obviously by the leading edge when the initial phase of vibration is $\phi ={\rm \pi} /4$ (see figure 14a), compared with the results shown in figure 12(b) when the initial phase is zero. Although the perturbation is more inclined, the streamwise vorticity in the upper boundary layer is still sign-invariant. In fact, the resonance effect is robust and can be found over a wide range of phases (around $\phi =\pm {\rm \pi}/4$) because the inflow disturbances can tilt and deform to climb over the leading edge. When the phase difference is increased to $\phi ={\rm \pi} /2$, the perturbation is cut by the vibrating leading edge and tilted severely (see 14b). The inflow disturbance is staggered inappropriately at this case and thus both positive and negative streamwise vorticity enter the boundary layer. The opposite phase $\phi ={\rm \pi}$ gives rise to continuous positive streamwise vorticity inside the boundary-layer because now it is the other array of inflow perturbation that remains inside the boundary layer, which is confirmed by comparing the perturbation around the leading edge (shown in figure 14c). It is also worth noting that at those mismatched phases, the inflow perturbation is no longer the optimal solution, while the optimal one ensures a phase match and the synchronized motion by slightly adjusting its frequency around the vibration frequency.

Figure 14. Contour of streamwise vorticity $\xi _x'$ under different initial phases (a) $\phi ={\rm \pi} /4$, (b) $\phi ={\rm \pi} /2$ and (c) $\phi ={\rm \pi}$. The optimal inflow perturbation at $(Re, \omega , \omega _w)=(800, 2{\rm \pi} /60, 2{\rm \pi} /60)$ is imposed on the inflow. The dashed line marks the boundary-layer thickness, $\delta$.

5. Resonance-induced bypass transition

As discussed in the previous linear study, the optimal inflow perturbations in resonance with the vibrating plate produce the elongated, high-amplitude streamwise streaks whose further nonlinear development can be expected to lead to bypass transition as revealed by numerous work in flow around stationary plates. In this section, the 3-D DNS is performed to examine the breakdown of streaks and transition to turbulence activated by the resonant inflow perturbation. The computational domain is extended to $x=300$ to incorporate the inception of bypass transition and the spanwise width $l_z$ of the domain is set as $13.46$ (about 5.93 times of the boundary-layer thickness $\delta$ at $x=200$) to contain three pairs of the optimal inflow waves. Then the computational domain is decomposed into 120 Fourier modes in the spanwise direction. The norms of inflow perturbations $(\|\boldsymbol {u}'_{in}\|)$ are scaled to 0.1 and 0.12, which correspond to maximum inflow perturbation velocities $6.4\times 10^{-2}$ and $7.6\times 10^{-2}$, respectively. We have also confirmed that the streaky response of boundary layer is similar to the result of linear calculation when the norm of inflow perturbation is small ($\|\boldsymbol {u}'_{in}\|=0.001$, not shown here).

Figure 15 shows the nonlinear response of the boundary-layer with $\|\boldsymbol {u}'_{in}\|=0.12$. The streamwise vorticity strongly interacts with the streamwise streaks and the low-speed streaks are apparently lifted away from the surface whereas the high-speed streaks are shifted towards the surface owing to a nonlinear lift-up mechanism (Mao et al. Reference Mao, Zaki, Blackburn and Sherwin2017). The low-speed streaks have reached the edge of the boundary layer and are thus exposed to free-stream disturbances out of the boundary layer, as shown in figure 15(a). In the present case, the inflow perturbation is fixed at frequency $\omega =2{\rm \pi} /60$ and in the absence of high-frequency free-stream disturbances, the streaks do not undergo secondary instabilities such as meandering motions. However, the low-speed streaks on the vibrating plate are distorted in the vertical direction, which was not observed on fixed flat plate flow (Wang et al. Reference Wang, Mao and Zaki2019), whereas the high-speed streaks show a smooth profile. The distortion of low-speed streaks is attributed to the vibration of the plate and the unsteady nature of the perturbation. As has been shown in figure 12, the incoming unsteady streamwise vorticity $\xi _x'$ is staggered by the vibrating plate to form the continuous $\xi _x'$ and this continuity reduces in the vertical direction. In other words, the underlying sign-invariant $\xi _x'$ is composed of continuous vorticity inside the boundary layer and become more unsteady or streamwise dependent around the edge of the boundary layer. As the low-speed streaks are lifted away from the wall by the nonlinear lift-up effect, they are exposed to the inhomogeneous streamwise vorticity around the edge of the boundary layer. The inhomogeneous vorticity at the top the boundary layer further stretches the low-speed streaks in the wall-normal direction. As a result, the low-speed streaks are streamwisely distorted. On the other hand, the high-speed streaks are shifted towards the wall by the nonlinear lift-up effect, so they are less affected by the streamwisely inhomogeneous vorticity at the top of the boundary layer and, thus, are not deformed.

Figure 15. Nonlinear evolution of the inflow perturbations with time $t$. (a) 3-D view of isosurfaces of streamwise perturbation velocity $u'=0.1$ and $u'=-0.1$ at $(Re, \omega , \omega _{w}, \|\boldsymbol {u}'_{in}\|)=(800, 2{\rm \pi} /60, 2{\rm \pi} /60, 0.12)$. The isosurfaces are colored by blue and red for $u'=-0.1$ and $u'=0.1$, respectively. (b) Contours for $u'$ extracted at $x=150$. The dashed line in (b) marks the boundary-layer thickness, $\delta$.

The streamwise location of the distortion changes periodically with time owing to the convection effect. This can be also observed in figure 15(b), where the profiles of streaks over one vibrating period at $x=150$ are illustrated. At $t=270$, the nonlinear lift-up mechanism already manifested itself that the low-speed streaks are lifted above the high-speed streaks and show the mushroom shape. Then at $t=285$, owing to the inhomogeneous streamwise vorticity (locally strong at some positions) at the top of the boundary layer, the low-speed streaks are distorted significantly whereas the high-speed streaks deform slightly. At $t=300$ and $t=315$, the low-speed streaks are further stretched in the wall-normal direction by the inhomogeneous streamwise vorticity, resulting in a slender cross-section of the low-speed streaks. At $t=330$, the locally strong vorticity is convected downstream and the streaks have redeveloped to the mushroom shape again.

The streamwise development of the magnitude of the streaks is further studied to quantify this inhomogeneity induced distortion. According to the previous studies (Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001; Brandt et al. Reference Brandt, Schlatter and Henningson2004), the streak amplitude can be defined as

(5.1)\begin{equation} Am(x,t)=\frac{1}{2}\left[\mathop{max}\limits_{y,z}({u}(x,y,z,t)-{U}(x,y,t))- \mathop{min}\limits_{y,z}({u}(x,y,z,t)-{U}(x,y,t))\right], \end{equation}

where $u(x,y,z,t)$ is the total streamwise velocity in the presence of streaks and $U(x,y,t)$ is the streamwise velocity of the base flow. It is also a function of time $t$ because of the vibration. Variations of streak amplitude along $x$ direction and the contours of $\xi _x'$ at $t=480$ are shown in figure 16. The continuous tilting of the base flow by $\xi _x'$ gives rise to a trend of amplification of streaks in streamwise direction with strong fluctuations. When the inflow perturbations is strong with $\|\boldsymbol {u}'_{in}\|=0.12$ and the plate is vibrating, the global maximum of streaks is located at $x=125$ and its amplitude is 42.5 %. Further downstream, the streaks decay under the viscous effect. The fluctuation of streaks on the vibrating plate can, thus, be classified into three zones: growing $(x<64)$, mature $(64 < x < 127)$ and decay $(x>127)$. In the growing zone, $A_m$ grows quickly and the streamwise wavelength of streak fluctuation is approximately half of the wavelength of the inflow perturbation $(0.5\times {2{\rm \pi} }{U_\infty }/\omega =30)$. The explanation of this wavelength is given later. In the mature zone, the magnitude of the streaks reaches maximum with the oscillation frequency reducing. In the decay zone, the viscosity dissipation results in a downward trend of the streak amplitude and the fluctuation of streak amplitude is also smaller. Similar evolution of $A_m$ is found when the norm of the inflow perturbation is scaled to 0.1. The difference is that the streaks decay slower and the global maximum of $A_m$ is obtained at larger $x$ around 170. To summarize, even if the streaks show an invariant sign owing to the synchronized motion revealed in § 4.3, their magnitude is periodically modulated by the vibration of the plate. For the non-vibrating case, the growing zone is smaller $(x<30)$ and the decay zone is longer $(x>90)$.

Figure 16. (a) Variation of streak amplitude $Am$ with $x$ and (b) contours of $\xi _x'$ at $t=480$. The dash-dotted box marks $\xi _{x1}'$ and the dashed box marks $\xi _{x2}'$. The optimal inflow perturbations are obtained at $(Re, \omega , \omega _{w})=(800, 2{\rm \pi} /60, 2{\rm \pi} /60)$. The contours from top to down correspond to cases at $\|\boldsymbol {u}'_{in}\|=0.12$, on vibrating plate, $\|\boldsymbol {u}'_{in}\|=0.1$, on vibrating plate, and $\|\boldsymbol {u}'_{in}\|=0.12$, on non-vibrating plate.

The inhomogeneous distribution of $\xi _x'$ is shown in figure 16(b) to explain the fluctuation of $A_m$. In § 4.3, we have classified the streamwise vorticity on the upper side of the plate into two components, $\xi _{x1}'$ and $\xi _{x2}'$, representing the streamwise vorticity entering the upper boundary layer when the plate is moving downward and upward, respectively. For clarity, in the first two rows of figure 16(b) (the nonlinear counterpart of figure 12b), $\xi _{x1}'$ and $\xi _{x2}'$ are marked by dash-dotted box and dashed box, respectively. The fluctuation of $A_m$ shown in figure 16(a) is in accordance with the inhomogeneity of $\xi _x'$ shown in figure 16(b), which indicates the influence of inhomogeneous $\xi _x'$ on the nonlinear lift-up mechanism. Take the vibrating case with $\|\boldsymbol {u}'_{in}\|=0.12$ as an example, the streak amplitude $A_m$ is plotted in figure 16(a) as the red solid line and the streamwise vorticity is shown in the top row of figure 16(b). These two figures show that the local peak of streak amplitude at $x\approx 125$ is in accordance with the locally strong $\xi _x'$ at $x\approx 125$. Similar features can also be seen at $x\approx 185$ and $x\approx 245$. Later, we discuss the fluctuation of the streak amplitude in different zones in details. In the growing zone, both $\xi _{x1}'$ and $\xi _{x2}'$ are strong and capable of amplifying the streak. Therefore, for the vibrating plate, the fluctuation of $A_m$ in this zone is due to the alternation of $\xi _{x1}'$ and $\xi _{x2}'$. Thus, the streamwise wavelength of $A_m$ is half of the streamwise wavelength of the inflow perturbation. For the non-vibrating plate, the fluctuation is attributed to the alternation of positive and negative streamwise vorticity (unsteadiness of the inflow perturbation). In the mature zone, $\xi _{x2}'$ on the vibrating plate is decaying and, thus, the alternation of $\xi _{x1}'$ and $\xi _{x2}'$ does not lead to any sharp increase of $A_m$ for the vibrating case. For the stationary case, the pattern of the streamwise vorticity is unchanged but the streak amplitude has reached the global peak value and starts to decrease. In the decay zone, the weaker streamwise vorticity under the viscous effect results in smaller fluctuation of $A_m$. For the vibrating plate, the wavelength of fluctuation in decay zone is close to the wavelength of inflow perturbations $(2{\rm \pi} {U}_\infty /\omega =60)$ because only $\xi _{x1}'$ can lead to a sudden increase of $A_m$ in this zone, whereas $\xi _{x2}'$ is weak and confined inside the boundary layer. For the non-vibrating plate, the wavelength of fluctuation in the decay zone remains unchanged because the pattern of $\xi _x'$ is the same as that in the growing zone and mature zone (see figure 16(b), the last row) and the streak amplitude reduces to less than $0.05$ at $x=300$.

The periodical distortion of streaks might lead to intermittent secondary instabilities. As discussed in the previous section, the lifted low-speed streaks are prone to secondary instabilities seeded by high-frequency disturbances in the free stream. In order to activate the secondary instability of streaks and transition to turbulence, high-frequency disturbances are added to the optimal inflow perturbations to perturb the flow. The high-frequency disturbances consist of Fourier waves with the form

(5.2)\begin{equation} u'_h=\sum_{j=2}^{J}\sum_{m=0}^{M}{\sum_{n=0}^{N}}{\eta_{jmn}\sin(\gamma_{n}\cdot{y}) \cos(\beta\cdot{m}\cdot{z})\textrm{e}^{\textrm{i}j\omega{t}}}, \end{equation}

where $m\cdot \beta$ is the spanwise wavenumber of the $m$th spanwise wave and $\gamma _{n}$ is the arbitrary wall-normal wavenumber of the $n$th wall-normal wave. The high-frequency disturbances consist of 48 spanwise waves and 15 wall-normal waves $(M=48,\ N=15)$. Here $\eta _{jmn}$ is the random amplitude of the high-frequency noise satisfying $\sqrt {\sum {\eta _{jmn}^2}}=0.015$, and $j\omega$ is the disturbance frequency. The frequency of the inflow perturbation that gives rise to the velocity streaks is $\omega =2{\rm \pi} /60$. Here the high-frequency disturbances are made up of 72 modes $(J=72)$. The nonlinear response of the boundary layer is traced by performing 3-D DNS and the results are shown in figure 17. For comparison, the response of the boundary-layer on stationary plate (figure 17a) as well as response to smaller-amplitude perturbations (figure 17b) are also plotted. The corresponding velocity perturbations are shown in figure 17(d–f).

Figure 17. Top view of isosurfaces of instantaneous flow fields after adding higher-frequency inflow disturbance :(a, d) $(A, \omega , \omega _{w}, \|\boldsymbol {u}'_{in}\|)=(0, 2{\rm \pi} /60, 2{\rm \pi} /60, 0.12), \ (\textit{b}, \ \textit{e})$ $(A, \omega , \omega _{w},$ $(\|\boldsymbol {u}'_{in}\|)=(0.01, 2{\rm \pi} /60, 2{\rm \pi} /60, 0.1)$ and (c,f) $(A, \omega , \omega _{w}, \|\boldsymbol {u}'_{in}\|)=$$(0.01, 2{\rm \pi} /60,$ $2{\rm \pi} /60, 0.12)$. (a)–(c) Isosurfaces of $\lambda _2=\{-0.01, 0.005, 0.002\}$ colored by the full velocity, using the isosurfaces of streamwise velocity $u=0.8$ as the background. (d)–(f) Isosurfaces of streamwise perturbation $u'=\pm 0.12$.

The streaks on the stationary plate do not breakdown, and they are finally attenuated by viscous effects in the absence of secondary instabilities (see figure 17d). In the previous studies by Andersson et al. (Reference Andersson, Brandt, Bottaro and Henningson2001) and Zaki & Durbin (Reference Zaki and Durbin2006), the instability of streaks is related to the inflectional point of the velocity profile. For the sinuous mode, the instability of streaky boundary layer is related to the spanwise shear induced by the streaks, whereas for the varicose mode, the instability is in connection with the wall-normal shear (Saric Reference Saric1994; Asai et al. Reference Asai, Minagawa and Nishioka2002). The low- and high-speed streaks alternately appear in the streamwise direction and overlap with each other inside the boundary layer (the heads of the low-speed streaks lay above the tails of the high-speed streaks and vice versa) when the plate is stationary, causing strong wall-normal shear. Therefore, the streaks on the stationary plate are prone to the varicose-type instability. Vaughan & Zaki (Reference Vaughan and Zaki2011) argued that the inner mode (varicose type) was stable in low-amplitude, higher-frequency streaks. In our study, the normalized frequency $(F=10^6\omega /\sqrt {U_\infty {x}/\nu })$ of streaks at $x=300$ is $F=214$, which is outside of the most energetic frequency range $(0<F<100)$ (Jacobs & Durbin Reference Jacobs and Durbin2001; Vaughan & Zaki Reference Vaughan and Zaki2011) and the amplitude of the streak is small $(A_m=0.05)$. Therefore, the inner mode instability of the streaks on the stationary plate is still too weak to arouse the streaks to breakdown and form the turbulent spots. In another study with the same geometry and Reynolds number as those in the present work, bypass transition was achieved by combining the inflow perturbations and a body forcing (Wang et al. Reference Wang, Mao and Zaki2019).

The secondary instabilities of streaks are successfully stimulated by the high-frequency disturbances for the vibrating plate flow (see figure 17b,e). The meandering motions of streaks are spotted at $x\approx 95$, 150, 200, 250 and 300 and the meandering becomes more intense as $x$ increases, indicating that the secondary instability of streaks is intermittent and growing in every period. The interruption of the secondary instabilities of streaks is attributed to the inhomogeneity of streamwise vorticity around the edge of the boundary layer, which breaks the mushroom shape of the low-speed streaks periodically. From the $\lambda _2$ (Jeong & Hussain Reference Jeong and Hussain1995) plot at $x\approx 250$ and 300, the meandering motions of streaks are asymmetric, suggesting that the secondary instability is of the sinuous type (see figure 17c). The meandering does not appear at the position where a local peak of streak amplitude appears. Instead, it appears at the positions where the cross-section of the streak is of mushroom shape. The streak amplitudes at the positions where the meandering is observed $(x=95, 150, 200, 250, 300)$ are 0.31, 0.34, 0,35, 0.35, 0.29 at perturbation level $\|\boldsymbol {u}'_{in}\|=0.1$. The streak amplitude around 300 is smaller, but the meandering motion of the streaks becomes stronger (see figure 17e). In fact, it has been found that the streaky boundary layer could be more unstable when the cross-section of low-speed streaks are altered into trapezoidal shape (wide at the top and narrow at the bottom), even though its amplitude is decreased slightly (Wang et al. Reference Wang, Mao and Zaki2019). The result presented previously agrees with this observation.

The streaks finally break down and the transition is triggered as the perturbation level is increased to $\|\boldsymbol {u}'_{in}\|=0.12$. The meandering motions of the streaks occur at $x\approx 95$, 150 and 200 and the corresponding streak amplitudes are 0.36, 0.38 and 0.37. The secondary instability after $x=200$ is strong enough to sustain itself even if the cross-section of the streaks is slightly stretched (see the meandering motions of the streaks at $x=210$ and 220). While for the smaller perturbation level, the development of instability is interrupted when the cross-section of the mushroom shape of the streak is broken. For the streaky boundary layer over the stationary plate perturbed by steady perturbation, the meandering occurs at $x=101$ where $A_m$ is 0.35 (Wang et al. Reference Wang, Mao and Zaki2019). In comparison with their result, although the onset of secondary instability is observed at a similar streamwise location and streak amplitude (the meandering motions of the streaks are spotted at $x=95$ where $A_m=0.36$), the transition process over the vibrating plate is prolonged. The breakdown of streaks and the onset of transition are quick after the burst of secondary instability for stationary plate flow, while for the vibrating plate, the streaks undergo several rounds of instabilities before the onset of transition.

The cross-sections of the streaky boundary layer at $x=95$, 150, 200 and 230 are shown in figure 18 to illustrate the details of the secondary instabilities. In general, it is clear that the cross-section of the streak is of mushroom shape when the secondary instabilities are stimulated. At $t=95$, the low-speed streaks are lifted above the high-speed streaks and the periodicity in the spanwise direction is broken because of the secondary instabilities. The secondary instability at this position is weak and interrupted by the inhomogeneous nonlinear lift-up, so the spanwise periodicity of the low-speed streaks is recovered downstream. At $x=150$, as the low-speed streaks are lifted higher owing to the lift-up, the influence of the inhomogeneous nonlinear lift-up becomes stronger and, thus, the low-speed streaks are further distorted. The distortion of the streaks is mainly in spanwise direction, manifesting the spanwise shear correlation of the sinuous-type instability. The secondary instability of streaks continues to develop at $x=200$. The non-periodicity and the meandering motions of the streaks is so strong that the inhomogeneous nonlinear lift-up downstream cannot interrupt the spanwise distortion of the streaks. Further downstream at $x=230$, the streaks break down as has been observed in figure 17.

Figure 18. Contour for $u'$ at various streamwise locations after adding higher-frequency noise, (a) $x=95$, (b) $x=150$, (c) $x=200$ and (d) $x=230$. The result is obtained at $(t, \omega , \omega _w, \|\boldsymbol {u}'_{in}\|)=(480, 2{\rm \pi} /60, 2{\rm \pi} /60, 0.12)$.

The skin friction coefficients $(C_f)$ for cases shown in figure 17 are plotted in figure 19. The black solid curve represent mean $C_f$ for the periodic base flow, which is close to the theoretical laminar skin friction coefficient. Over half of the gap between the base flow $C_f$ and the theoretical laminar $C_f$ at larger streamwise location can be directly attributed to the pressure gradient downstream of the leading edge, because the present streamwise force should be $\tau _w-{\partial {p}}/{\partial {x}}$, where $\tau _w$ is the wall shear. For the non-vibrating plate $(A=0)$, the skin friction coefficient is increased slightly compared with the base case at $x\approx 50$ because of the presence of streamwise velocity streaks. Further downstream, the streaks decay and, thus, $C_f$ remains at the laminar value for the rest of the plate. For the vibrating plate, the elongated streaks perturbed by weaker inflow disturbance $(\|\boldsymbol {u}'_{in}\|=0.1)$ also cause an increase of the skin friction coefficient. However, the streak amplitude is insufficient to trigger the inception of transition and, thus, the value of $C_f$ falls down when the streaks are attenuated by viscosity. Once the base flow is perturbed by the strong resonant inflow perturbation $(\|\boldsymbol {u}'_{in}\|=0.12)$, the skin friction coefficient shoots up at $x\approx 230$, representing the onset of transition, before converging to the turbulent $C_f$ curve. Compared with the stationary plate results (Nolan & Zaki Reference Nolan and Zaki2013), the skin friction coefficient over the vibrating plate takes long distance to shoot up, indicating that the transition stage covers a larger area. Therefore, the vibration of the plate staggers the resonant inflow perturbation to produce the elongated streamwise streaks inside the boundary layer, which creates the precondition for the bypass transition. However, the inhomogeneous nonlinear lift-up caused by the vibration acts as a hindrance in the secondary instability excitation process.

Figure 19. Skin friction coefficients $(C_f)$ for various cases. Higher-frequency disturbances to the optimal inflow perturbation is added.