Detuned secondary instabilities in three-dimensional boundary-layer flow

In this work we apply a formulation for capturing detuned secondary instabilities. This formulation, based on two-dimensional stability analysis coupled with a Bloch wave formalism originally described by Schmidt et al. [Phys. Rev. Fluids 2, 113902 (2017)], allows us to consider high-dimensional systems resulting from several repetitions of a spatially periodic unit by solving an eigenproblem of much smaller size. Secondary instabilities coupling multiple periodic units thus can be retrieved. The method is applied on the secondary stability of a swept boundary-layer flow subject to stationary cross-flow vortices. Two distinct detuned secondary instabilities are retrieved. The first one, obtained for a detuning factor $\epsilon =0.35$ and reaching a maximum growth rate for streamwise wave number ${\alpha }_v=0.087$, was already found in the work of Fischer and Dallmann [Phys. Fluids A: Fluid Dyn. 3, 2378 (1991)]. The second instability is obtained for streamwise independent modes and small detuning factor $\epsilon =0.08$. The corresponding mode presents large-wavelength oscillations, arising from a characteristic beating phenomenon. The physical origin of these two secondary instabilities has been investigated by varying the amplitude of the primary disturbance: for the latter instability, reminiscent of a type III mode, the unstable branch continuously deforms as the amplitude is increased, whereas a change of topology of the spectrum is observed for the ${\alpha }_v=0$ mode.

Figure 1

(a) Neutral curve [black solid line ${\omega }_i(\alpha ,\beta )=0]$ of the swept boundary layer for ${\text{Re}}_{\delta }=826$. The dashed line corresponds to stationary disturbances, ${\omega }_r(\alpha ,\beta )=0$. (b) Absolute value of the eigenfunctions for ${\text{Re}}_{\delta }=826, \alpha =0.0361$, and $\beta =0.4774 [\mathrm{\Psi }=arctan(\beta /\alpha )=89.9^{\circ }]$. The dashed lines are extracted from Fischer and Dallmann [9].

Figure 1

(a) Neutral curve [black solid line ${\omega }_i(\alpha ,\beta )=0]$ of the swept boundary layer for ${\text{Re}}_{\delta }=826$. The dashed line corresponds to stationary disturbances, ${\omega }_r(\alpha ,\beta )=0$. (b) Absolute value of the eigenfunctions for ${\text{Re}}_{\delta }=826, \alpha =0.0361$, and $\beta =0.4774 [\mathrm{\Psi }=arctan(\beta /\alpha )=89.9^{\circ }]$. The dashed lines are extracted from Fischer and Dallmann [9].

Figure 2

Secondary base flow $U_1(y,z_v)$ (a) and $W_1(y,z_v)$ (b) in the wave-oriented reference frame for one subunit. The wall-normal component $V_1$ is nonzero but is one order of magnitude smaller than $W_1$, thus it is not shown.

Figure 2

Secondary base flow $U_1(y,z_v)$ (a) and $W_1(y,z_v)$ (b) in the wave-oriented reference frame for one subunit. The wall-normal component $V_1$ is nonzero but is one order of magnitude smaller than $W_1$, thus it is not shown.

Figure 3

(a) Evolution of the secondary growth rate ${\sigma }_r$ (black circles) as a function of the streamwise wave number ${\alpha }_v$. A comparison is made with Fig. 8 of Fischer and Dallmann [9] (red symbols), where $\epsilon$ corresponds to the detuning factor of the Floquet analysis [9]. (b), (c) Spatial Fourier spectra of the energy of the most unstable mode for ${\alpha }_v=0$ (b) and ${\alpha }_v=0.087$ (c).

Figure 3

(a) Evolution of the secondary growth rate ${\sigma }_r$ (black circles) as a function of the streamwise wave number ${\alpha }_v$. A comparison is made with Fig. 8 of Fischer and Dallmann [9] (red symbols), where $\epsilon$ corresponds to the detuning factor of the Floquet analysis [9]. (b), (c) Spatial Fourier spectra of the energy of the most unstable mode for ${\alpha }_v=0$ (b) and ${\alpha }_v=0.087$ (c).

Figure 4

Full spectra of the secondary stability problem for ${\text{Re}}_{\delta }=826, A=0.0789, n=50$ and for ${\alpha }_v=0$ (a) and ${\alpha }_v=0.087$ (b). The full spectrum is constructed merging the $n$ spectra of the subsystems. The eigenvalues have been colored by the argument of their respective root of unity: $z\in [0,1]$ such as $\rho =exp\left(2i\pi z\right)$. The dashed line corresponds to the ${\sigma }_r=0$ line.

Figure 5

Three-dimensional views of the streamwise velocity component of the most unstable mode for different streamwise wave number ${\alpha }_v$ (a: ${\alpha }_v=0, n=25$; b: ${\alpha }_v=0.087, n=12$) and a number of visualized subunits $n$, chosen in both cases to give a good overview of the spatial characteristics of the mode. Isosurfaces correspond to $\pm 0.75u_{\max }$. Spanwise direction is normalized by the spatial period of a subunit $2\pi /\left|\right|\mathbf{k}\left|\right|$. Notice the beating phenomenon for ${\alpha }_v=0$.

Figure 6

Vectors: cross-flow velocity in the plane $(y,z_v)$ for a group of three subunits. The contour plot represents the streamwise component of the secondary base flow. The flow in the $z$ direction goes from left to right. (a) ${\alpha }_v=0$; (b) ${\alpha }_v=0.087$.

Figure 7

Absolute value of the streamwise component of the eigenfunction for ${\alpha }_v=0$ (a) and ${\alpha }_v=0.087$ (b). The solid contours depict the spanwise and wall-normal velocity gradients of the secondary base flow, respectively.

Figure 8

Spectra of the secondary stability problem for ${\text{Re}}_{\delta }=826, n=50, {\alpha }_v=0$ and for several amplitudes $A$. From left to right and top to bottom: $A=0.02, A=0.04, A=0.06$, and $A=0.07$. The dashed line corresponds to the ${\sigma }_r=0$ line.

Figure 9

Three-dimensional views of the streamwise velocity perturbation of the unstable modes at the exceptional point for ${\alpha }_v=0, A=0.06$, and $n=25$ subunits. Isosurfaces correspond to $\pm 0.75u_{\max }$. Spanwise direction is normalized by the spatial period of a subunit $2\pi /\left|\right|\mathbf{k}\left|\right|$. Notice the beating phenomenon for ${\alpha }_v=0$. (a) ${\rho }_{22}$; (b) ${\rho }_{35}$.

Figure 10

Same as Fig. 8 but for the ${\alpha }_v=0.087$ mode.

Figure 11

Three-dimensional views of the streamwise velocity perturbation of the most unstable mode for ${\alpha }_v=0.087$ and different amplitudes (a: $A=0.02$; b: $A=0.06$; c: $A=0.07$). The number $n=3$ of subunits visualized has been chosen to accommodate approximately one spatial period of the secondary mode ($\epsilon \approx 0.35$). Isosurfaces correspond to $\pm 0.75u_{\max }$. Black contours represent the secondary base flow. Spanwise direction is normalized by the spatial period of a subunit $2\pi /\left|\right|\mathbf{k}\left|\right|$.

Figure 12

Eigenvalue spectra for ${\text{Re}}_{\delta }=826, A=0.0789, {\alpha }_v=0.03$ (top row), and ${\alpha }_v=0.0087$ (bottom row) with a number of subunits (left) $n=50$ and (right) $n=80$. The dashed line corresponds to the ${\sigma }_r=0$ line.