Transient energy growth in channel flow with compliant walls

In this paper, we present a Lagrangian method for searching initial disturbances which maximize their total energy growth after a certain time horizon for linearized fluid-structure interaction problems. We illustrate this approach for the channel flow case with compliant walls. The walls are represented as thin spring-backed plates, the so-called Kramer-type walls. For nearly critical values of the control parameters (reduced velocity $V_R$ and Reynolds number $\text{Re}$), analyses for sinuous or varicose perturbations show that the fluid-structure system can sustain two types of oscillatory motions of large amplitude. The first motion is associated with two-dimensional perturbations that are invariant in the spanwise direction. For that case and a certain range of streamwise wavenumbers, the short-time dynamics of sinuous perturbations is driven by the nonmodal interaction between the Tollmien-Schlichting and the traveling-wave flutter (TWF) modes. The amplitude of the oscillation is found to increase with the reduced velocity, and the optimal gain exhibits larger values than its counterpart computed for a channel flow between rigid walls. For perturbations of varicose symmetry, the transient energy is rapidly governed by the unstable TWF mode without a clear low-frequency oscillation. The second type of motion concerns streamwise-invariant and spanwise-periodic perturbations. In that situation, it is found that perturbations of sinuous symmetry exhibit the largest amplification factors. For moderate values of the reduced velocity, $V_R=\mathcal{O}\left(1\right)$, the dynamics is the result of a simple superposition of a standing wave, due to traveling-wave flutter modes propagating downstream and upstream, and the roll-streak dynamics. The variations of these oscillations with the reduced velocity, spanwise wavenumber, and Reynolds number are then investigated in detail for the sinuous case.

Figure 1

Channel flow with infinite spring-backed flexible walls. (a) Schematic diagram showing the equilibrium state configuration and (b) wall deformation and coordinate system.

Figure 2

Validation of the time-marching linear solver for both direct and adjoint formulations for $\text{Re}=10000$, $\alpha =1$, $B_{*}=4$, $\mathrm{\Gamma }=2$, $d_{*}=0$, and $V_R=1$. Numerical parameters are set to $N=60$ and $\mathrm{\Delta }t=0.01$. The time evolution of the modulus of the wall vertical displacement for a given random initial perturbation is shown on a logarithmic scale. A linear regression provides a temporal growth rate $\sigma =0.031894$ for both direct and adjoint solvers. The temporal evolution of the most amplified linear eigenvalues obtained with the eigenvalue matrix solver gives $\sigma =0.031887$.

Figure 3

Same flow case as in Fig. 2. The envelope of the eigenfunctions computed either with the LDNS (in solid lines) or the eigenvalue solver (open circles) are shown.

Figure 4

$G$ as a function of the target time $\tau$ for $V_R=23.57$, $B_{*}=4$, $d_{*}=0.0071$, $\text{Re}=6667$, $\beta =0.2$, and $\alpha =0$. Results extracted from Hœpffner et al. [17] are also reported. Computations associated with the total energy norm and the fluid kinetic energy norm inside the domain are shown in black and blue, respectively.

Figure 5

$G$ as a function of the target time $\tau$ for $\text{Re}=6666$ and (a) $V_R=0.006$, (b) $V_R=0.45$, (c) $V_R=0.53$, and (d) $V_R=0.65$. For all panels, the streamwise wavenumbers are $\alpha =0.8$, 1, 1.2, 1.4, and 1.6 (from the left to the right). Red, sinuous symmetry; black, varicose symmetry.

Figure 6

$G$ as a function of the target time $\tau$ for $\text{Re}=6666$, $\alpha =1.2$, and $V_R=0.4,0.8$.

Figure 7

Isocontours of (a) $G_s^M$ and (b) ${\alpha }_s^M$ in the $(\text{Re},V_R)$ plane. The dashed line denotes the evolution of the critical Reynolds number with respect to the TS mode. The dash-dotted line represents the evolution of the critical reduced velocity with respect to the TWF mode.

Figure 8

Time evolution of the total energy for the optimal initial perturbation for $\tau =26$, 66, 133, 200, 266 (from the left to the right) and $\alpha =0.8$, 1, 1.2, 1.4. The Reynolds number is fixed to $\text{Re}=8300$ and the reduced velocity is set to $V_R=0.6$. In red, the envelope $G$ for the rigid case is represented.

Figure 9

Time evolution of the total energy for the optimal initial perturbation for $\tau =26$, 66, 133, 200, 266 (from the left to the right) and $\alpha =0.8$, 1, 1.2, 1.4. The Reynolds number is fixed to $\text{Re}=8300$ and the reduced velocity is set to $V_R=0.7$. In red, the envelope $G$ for the rigid case is represented.

Figure 10

Spectrum for $\text{Re}=8300$, $V_R=0.7$, and $\alpha =1$.

Figure 11

(a) Circular frequency ${\omega }_r$ versus the streamwise wavenumber $\alpha$ and (b) temporal amplification rate ${\omega }_i$ versus $\alpha$ for TS and TWF modes and $\text{Re}=8300$, $V_R=0.7$.

Figure 12

Total energy evolution for the optimal initial perturbation projected onto different subsets of eigenmodes for $\text{Re}=8300$, $\alpha =1$, $\tau =26$, and $V_R=0.6,0.7$. The envelope $G$ for the rigid case is also reported.

Figure 13

Time evolution of the total energy for $\text{Re}=6666$, $\beta =0.5$, and $V_R=0.4$, 0.8, and 1.2 and sinuous initial optimal perturbations computed for $\tau$ varying from 14 to 1820. The envelope $G$ for the rigid case is also reported.

Figure 14

Time evolution of the total energy for $\text{Re}=6666$, $\beta =0.5$, and $V_R=0.4$, 0.8, and 1.2 and varicose initial optimal perturbations computed for $\tau$ varying from 14 to 1820. The envelope $G$ for the rigid case is also reported.

Figure 15

(a) Spectrum at $\beta =0.5$, $V_R=1.2$, and $\text{Re}=6666$. (b) Time evolution of the total energy growth for the initial optimal perturbation computed for $\tau =14$.

Figure 16

Time evolutions for the total energy growth for $\beta =0.5$, $V_R=1.2$, $\text{Re}=6666$, and $\tau =14$. (a) LDNS are compared with ROMs for $m=50$ eigenmodes and the two TWF modes (referenced as $m=2$). (b) Total energy curves computed with the ROM for $m=50$ excluding the TWF modes are compared to the rigid-wall case.

Figure 17

Time evolutions for the total energy growth for $\beta =0.5$, $V_R=1.2$, $\text{Re}=6666$, and $\tau =14$. The evolution of $E\left(t\right)$ for the initial perturbation restricted to the summation of the pair of streamwise vortices and the standing waves is compared to $E\left(t\right)$ associated with the optimal perturbation.

Figure 18

Cross sections of the optimal perturbation for $\beta =0.5$, $\text{Re}=6666$, $V_r=1.2$, and $\tau =14$ in the $(z,y)$ plane for $tk=t_i+k/8\mathrm{\Delta }T$ with $k$ varying from 0 to 7 and $t_i=8.7$. Vectors for the cross-stream components and isocontours of the streamwise velocity fields are shown.

Figure 19

Cross sections of the optimal perturbation projected onto the two TWF modes for $\beta =0.5$, $\text{Re}=6666$, $V_r=1.2$, and $\tau =14$ in the $(z,y)$ plane for $tk=t_i+k/8\mathrm{\Delta }T$ with $k$ varying from 0 to 7 and $t_i=8.7$. Vectors for the cross-stream components and isocontours of the streamwise velocity fields are shown.

Figure 20

Cross sections of the optimal perturbation projected onto 50 eigenmodes excluding the two TWF modes for $\beta =0.5$, $\text{Re}=6666$, $V_r=1.2$, and $\tau =14$ in the $(z,y)$ plane for $tk=t_i+k/8\mathrm{\Delta }T$ with $k$ varying from 0 to 7 and $t_i=256$. Vectors for the cross-stream components and isocontours of the streamwise velocity fields are shown.

Figure 21

Amplitude of the oscillations. (a) Distribution of $\mathrm{\Delta }{E}^M$ with $V_R$ for various $\beta$ and $\text{Re}=16000$. (b) Distribution of $A$ as a function of $\beta$ for $\text{Re}=2000$, 6666, and $16000$.

Figure 22

Frequency of the standing wave as a function of $\beta$ and $V_R$. Dashed lines denote the frequency computed with the added-mass model.