2.1. Resolvent analysis

As discussed in McKeon & Sharma (Reference McKeon and Sharma2010) and McKeon et al. (Reference McKeon, Jacobi and Sharma2013), the resolvent formulation stems from the insight that the governing NSEs can be considered a forcing–response system with feedback. The conservative nonlinear terms, which serve to transfer energy across spectral space, are interpreted as the feedback forcing that acts on the linear terms of the NSEs to generate a velocity and pressure response. Importantly, it turns out that the linear terms behave like highly directional filters and so the flow response is (almost) insensitive to the nature of the nonlinear forcing. McKeon et al. (Reference McKeon, Jacobi and Sharma2013) argue that this directionality is partially responsible for the robust nature of wall turbulence, i.e. the presence of repeating, recognizable flow patterns across geometries, scales and operating conditions (NW cycle, hairpins, VLSMs). Given this directionality, a gain-based decomposition of the forcing–response transfer function is used to arrive at a low-order representation for the flow field.

For fully developed turbulent channel flow, spatial invariance and statistical stationarity ensure that Fourier modes are the most appropriate bases for decomposition in the streamwise direction $x$ , spanwise direction $z$ and time $t$ :

Here, $\boldsymbol{u}=[u,v,w]^{\text{T}}$ represents the streamwise ( $u$ ), wall-normal ( $v$ ) and spanwise ( $w$ ) velocity fields, and $p$ denotes the pressure field. Each wavenumber–frequency combination $\boldsymbol{k}=({\it\kappa}_{x},{\it\kappa}_{z},{\it\omega})$ represents a flow structure, or mode, with streamwise and spanwise wavelengths ${\it\lambda}_{x}=2{\rm\pi}/{\it\kappa}_{x}$ and ${\it\lambda}_{z}=2{\rm\pi}/{\it\kappa}_{z}$ propagating downstream at speed $c={\it\omega}/{\it\kappa}_{x}$ . (Note that, unless otherwise indicated, length scales are normalized with respect to the channel half-height $h$ and velocity scales are normalized with respect to the friction velocity $u_{{\it\tau}}$ . Following standard notation, a superscript $+$ is used to denote normalization with respect to $u_{{\it\tau}}$ and the kinematic viscosity ${\it\nu}$ .) The Fourier coefficients $\boldsymbol{u}_{\boldsymbol{k}}$ and $p_{\boldsymbol{k}}$ represent the wall-normal ( $y$ ) variation in the magnitude and phase of the velocity and pressure fields for each mode. The special case $\boldsymbol{u}_{\mathbf{0}}=[U(y),0,0]^{\text{T}}$ represents the mean velocity profile.

For the inhomogeneous wall-normal direction, the analysis seeks a gain-based decomposition. Specifically, the Fourier-transformed NSEs and continuity constraint are expressed as the following forcing–response system:

where $\boldsymbol{{\rm\nabla}}_{\boldsymbol{k}}=[\text{i}{\it\kappa}_{x},\partial /\partial y,\text{i}{\it\kappa}_{z}]^{\text{T}}$ and $\boldsymbol{{\rm\nabla}}_{\boldsymbol{k}}^{\text{T}}$ represent the gradient and divergence operators, respectively, and $\boldsymbol{f}_{\boldsymbol{k}}=(-\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u})_{\boldsymbol{k}}$ represents the nonlinear forcing. The forcing is mapped to a velocity and pressure response by the resolvent $\widetilde{\textstyle \pmb{\mathscr{H}}}_{\boldsymbol{k}}$ , which depends on the linear operator:

Here, $\mathit{Re}_{{\it\tau}}=u_{{\it\tau}}h/{\it\nu}$ is the friction Reynolds number and ${\rm\nabla}_{\boldsymbol{k}}^{2}=[-{\it\kappa}_{x}^{2}+\partial ^{2}/\partial y^{2}-{\it\kappa}_{z}^{2}]$ is the Fourier-transformed Laplacian. The first line in the operator on the right-hand side of (2.2) represents the momentum equations and the second line represents the continuity constraint.

Following McKeon & Sharma (Reference McKeon and Sharma2010), an SVD of the resolvent operator $\widetilde{\textstyle \pmb{\mathscr{H}}}_{\boldsymbol{k}}$ , discretized using a Chebyshev collocation method (§ 2.3), yields a set of orthonormal forcing (  $\boldsymbol{f}_{\boldsymbol{k},m}$ ) and response ( $[\boldsymbol{u}_{\boldsymbol{k},m},p_{\boldsymbol{k},m}]^{\text{T}}$ ) modes, ordered based on the input–output gain ( ${\it\sigma}_{\boldsymbol{k},m}$ ) under an $L^{2}$ energy norm. To enforce this energy norm, the primitive-variable resolvent in (2.2) is scaled as

or

Note that the discretized resolvent operator $\widetilde{\textstyle \pmb{\mathscr{H}}}_{\boldsymbol{k}}$ is a block matrix of dimension $[4\times 3]$ while the scaled resolvent operator $\widetilde{\textstyle \pmb{\mathscr{H}}}_{\boldsymbol{k}}^{S}$ is of dimension $[3\times 3]$ . The $[3\times 3]$ block diagonal matrices $\unicode[STIX]{x1D652}_{\boldsymbol{u}}$ and $\unicode[STIX]{x1D652}_{\boldsymbol{f}}$ incorporate numerical quadrature weights, which ensure that the SVD of the scaled resolvent operator,

where

yields forcing modes $\boldsymbol{f}_{\boldsymbol{k},m}=\unicode[STIX]{x1D652}_{\boldsymbol{f}}^{-1}{\it\phi}_{\boldsymbol{k},m}$ and velocity response modes $\boldsymbol{u}_{\boldsymbol{k},m}=\unicode[STIX]{x1D652}_{\boldsymbol{u}}^{-1}{\it\psi}_{\boldsymbol{k},m}$ with unit energy over the channel cross-section, i.e. the orthonormality constraints (2.7) translate into

where a superscript $\ast$ denotes a complex conjugate.

From (2.2)–(2.7), it can be seen that forcing in the direction of the $m$ th singular forcing mode with unit amplitude results in a response in the direction of the $m$ th singular response mode amplified by the singular value ${\it\sigma}_{\boldsymbol{k},m}$ , i.e. forcing $\boldsymbol{f}_{\boldsymbol{k}}=\boldsymbol{f}_{\boldsymbol{k},1}$ creates a response $[\boldsymbol{u}_{\boldsymbol{k}},p_{\boldsymbol{k}}]^{\text{T}}={\it\sigma}_{\boldsymbol{k},1}[\boldsymbol{u}_{\boldsymbol{k},1},p_{\boldsymbol{k},1}]^{\text{T}}$ . Luhar et al. (Reference Luhar, Sharma and McKeon2014a ) provide a more detailed description of how the primitive-variable resolvent shown in (2.2) is scaled to enforce the energy norm and how the pressure field is extracted from the unscaled operator.

Importantly, for $\boldsymbol{k}$ combinations energetic in natural turbulence, the resolvent operator tends to be of low rank. Only a limited number of inputs are highly amplified. Frequently, ${\it\sigma}_{\boldsymbol{k},1}\gg {\it\sigma}_{\boldsymbol{k},2}$ and so the velocity and pressure fields can be reasonably approximated by the first response modes, $\boldsymbol{u}_{\boldsymbol{k},1}$ and $p_{\boldsymbol{k},1}$ , respectively. Recent studies (McKeon & Sharma Reference McKeon and Sharma2010; McKeon et al. Reference McKeon, Jacobi and Sharma2013; Moarref et al. Reference Moarref, Sharma, Tropp and McKeon2013; Luhar et al. Reference Luhar, Sharma and McKeon2014a ) have shown that models based on this rank-1 approximation yield statistical and structural predictions consistent with previous observations for wall turbulence. The reader is referred to these studies for an expanded discussion of the amplification characteristics of the resolvent operator and the rank-1 approximation.

Given the success of the rank-1 response modes in reproducing previous observations, the present paper employs them for the study of turbulence–compliant wall interactions. For the remainder of this paper, it is assumed that the velocity and pressure fields at each $\boldsymbol{k}$ correspond to the rank-1 responses: $[\boldsymbol{u}_{\boldsymbol{k}}(y),p_{\boldsymbol{k}}(y)]^{\text{T}}=[\boldsymbol{u}_{\boldsymbol{k},1},p_{\boldsymbol{k},1}]^{\text{T}}$ . The term ‘resolvent modes’ refers to these rank-1 velocity and pressure fields, while the terms ‘amplification’ and ‘gain’ are used interchangeably to refer to the corresponding singular values, ${\it\sigma}_{\boldsymbol{k},1}$ . The additional subscript 1 is dropped for notational convenience.

Assuming that the efficacy of any control scheme depends on its ability to suppress the generation of Reynolds stress (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002), the forcing–response interpretation of the NSEs suggests three possible mechanisms: (i) a reduction in the magnitude of the nonlinear forcing $\boldsymbol{f}_{\boldsymbol{k}}$ present in the flow or a change in the form of the forcing (e.g. towards lower-gain directions) such that the magnitude of the velocity response diminishes; (ii) a modification of the system such that the forcing–response gain diminishes; or (iii) a reduction in the Reynolds stress contribution from high-gain resolvent modes via a change in mode structure. While mechanism (i) clearly requires knowledge of the nonlinear coupling between resolvent modes, mechanisms (ii) and (iii) can be evaluated on a linear mode-by-mode basis. This mode-by-mode approach is adopted here.

In other words, the present paper considers only the effect of compliant walls on the shape ( $\boldsymbol{u}_{\boldsymbol{k}},p_{\boldsymbol{k}}$ ) and forcing–response amplification ( ${\it\sigma}_{\boldsymbol{k}}$ ) of the resolvent modes. It provides no information on the nonlinear interaction between resolvent modes, which would feed back via the quadratic nonlinearity to force the linearized system. A more complete model would require knowledge of the triadic coupling between all resolvent modes, which is outside the scope of this paper. However, even without this knowledge, the change in the gain and structure of the modes can provide significant insight into the problem, as evidenced by the success of the analogous rank-1 model for opposition control developed by Luhar et al. (Reference Luhar, Sharma and McKeon2014b ).

In addition to the nonlinearity, the mean velocity profile $U(y)$ in (2.3) is another point of connection between modes. Here, the discretized resolvent operator is constructed using a mean profile computed based on a turbulent viscosity model (Reynolds & Tiederman Reference Reynolds and Tiederman1967, see § 2.3). If the compliant wall has a significant influence on the flow, the mean velocity profile would no longer resemble the canonical form, which could lead to changes in the gain and structure of the resolvent modes. Again, in a complete model, the mean profile must be sustained by the total Reynolds stress generated by the resolvent modes to account for this feedback.

2.2. Compliant-wall effects

For the rigid-wall null case, the discretized resolvent operator in (2.2) is constructed using the standard no-slip boundary conditions at the lower ( $y=0$ ) and upper ( $y=2$ ) walls:

The effect of the compliant wall is introduced by changing the boundary conditions on velocity and pressure within the resolvent before computing the SVD (2.7). Specifically, for wall displacement ${\it\eta}(x,z,t)$ constrained to be in the vertical direction, the kinematic boundary conditions at the lower wall, $\boldsymbol{u}({\it\eta})=\partial {\it\eta}/\partial t$ , can be expressed as the following Fourier-transformed linearized Taylor series expansions:

where ${\it\eta}_{\boldsymbol{k}}$ represents the Fourier coefficient for the wall displacement at wavenumber–frequency combination $\boldsymbol{k}=({\it\kappa}_{x},{\it\kappa}_{x},{\it\omega})$ . The neglected quadratic terms are shown for reference. Physically, the kinematic boundary condition (2.10) estimates the streamwise velocity at $y=0$ by trying to satisfy the no-slip condition at the wall $y={\it\eta}$ . This means that the streamwise fluctuations must counter the effects of the mean flow, leading to the additional term ${\it\eta}_{\boldsymbol{k}}(\partial U/\partial y)_{0}$ . As will be shown later, this can result in relatively large magnitudes for $u_{\boldsymbol{k}}(0)$ . Further, (2.10) and (2.11) can be combined to yield the following expression: $\text{i}{\it\omega}u_{\boldsymbol{k}}(0)\approx v_{\boldsymbol{k}}(0)(\partial U/\partial y)_{0}$ . This means that the streamwise and wall-normal velocities have to be ${\rm\pi}/2$ out of phase near the wall.

Figure 1. Schematic showing a simple spring- and damper-backed compliant membrane and boundary conditions.

The dynamic boundary condition is expressed as a complex mechanical admittance $Y$ linking wall-normal velocity and pressure:

Similar boundary conditions apply at the top wall, $y=2$ , although, for walls with identical material properties, the sign of the admittance changes due to the differing symmetry of the pressure and wall-normal velocity fields with respect to the centreline, i.e.  $v_{\boldsymbol{k}}(0)/p_{\boldsymbol{k}}(0)=-v_{\boldsymbol{k}}(2)/p_{\boldsymbol{k}}(2)$ .

Clearly, the linearization of the kinematic boundary conditions (2.10)–(2.12) is a significant simplification since the neglected higher-order terms could be important for large wall displacements. However, retaining terms of quadratic or higher order in the fluctuations would again require a coupled nonlinear model allowing for interactions between resolvent modes, which is outside the scope of this paper, as noted earlier. Further, the linearized boundary conditions require an estimate of the mean shear at the wall (2.10), which is again assumed to correspond to the rigid wall case. (Note that this is similar to the compliant-wall boundary conditions imposed in linear stability analyses, see Carpenter Reference Carpenter, Bushnell and Hefner1990.) This assumption is not rigorously justified and breaks down if the compliant wall significantly alters the near-wall mean flow. As mentioned previously, in a more complete model, the mean flow must be consistent with the total Reynolds stress generated by the modes to enable a priori predictions of changes in the mean profile. This would allow for changes in the Reynolds stress contributions from the modes to be propagated through the system via an iterative process. Unfortunately, the convergence properties of this iterative process are not known and as such are the subject of ongoing research.

The mechanical admittance $Y$ dictates the relative phase and amplitude of the wall-normal velocity and the pressure at the wall, which can be estimated based on the structural properties of the wall. As an example, the dynamics of the simple spring- and damper-backed membrane shown in figure 1 can be expressed as

where

are the dimensionless mass, damping and spring coefficients, respectively, ${\it\rho}$ is the mass density of the fluid, ${\it\rho}_{w}$ is the mass of the wall per unit area, $k_{w}$ is the effective spring constant and $d_{w}$ is the damping ratio. From (2.14), it is easy to show that the mechanical admittance for this wall is

While (2.14) assumes that the membrane is of negligible stiffness and tension-free, both of these effects can be introduced via terms involving spatial derivatives. Specifically, the restoring effects of tension ( $C_{t}$ ) and stiffness ( $C_{s}$ ) would introduce additional terms proportional to $({\it\kappa}_{x}^{2}+{\it\kappa}_{z}^{2})$ and $({\it\kappa}_{x}^{4}+2{\it\kappa}_{x}^{2}{\it\kappa}_{z}^{2}+{\it\kappa}_{z}^{4})$ , respectively. In essence, this would result in a wavenumber-dependent effective spring constant $C_{ke}=C_{k}+C_{t}({\it\kappa}_{x}^{2}+{\it\kappa}_{z}^{2})+C_{s}({\it\kappa}_{x}^{4}+2{\it\kappa}_{x}^{2}{\it\kappa}_{z}^{2}+{\it\kappa}_{z}^{4})$ . Similarly, more complex walls involving viscoelastic coatings can be incorporated by solving a wave equation for the solid layer with appropriate boundary conditions (e.g. Gad-el-Hak Reference Gad-el-Hak2000).

Thus, the admittance $Y$ can be used to account for the effects of compliant walls with known material properties. However, instead of following the traditional approach involving a trial-and-error procedure with different wall properties, the present study considers the inverse problem: finding an optimal wall that leads to the most favourable effect on the turbulent flow structures that are known to be important in real flows. The concept of a complex wall admittance has been used previously (Landahl Reference Landahl1962), perhaps most notably by Sen & Arora (Reference Sen and Arora1988), who employed a similar inverse approach to study the stability of laminar boundary layer flows over compliant surfaces.

In this paper, a simple gradient-free pattern search is used to find the optimal $Y$ (Hooke & Jeeves Reference Hooke and Jeeves1961), with a particular focus on resolvent modes resembling the NW cycle and VLSMs. Optimality in this context is defined in two different ways: (i) the compliant wall that leads to the greatest reduction in forcing–response gain ( ${\it\sigma}_{\boldsymbol{k}}$ ) relative to the rigid-wall case, indicating mode suppression, and (ii) the compliant wall that leads to the greatest reduction in the channel-integrated Reynolds stress contribution from the resolvent mode:

where $\text{Re}(\cdots )$ denotes the real component. The weighing factor $(y-1)$ is included per the decomposition developed by Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002), who showed that the friction coefficient in turbulent channel flow comprises a laminar component inversely proportional to the Reynolds number, defined using the bulk-averaged velocity, and a turbulent component proportional to the weighted Reynolds stress integral shown in (2.17). For channels with compliant walls, the drag coefficient also comprises an additional term that depends on the wall deformation. However, the integrated Reynolds stress still plays a major role (Nakanishi, Mamori & Fukagata Reference Nakanishi, Mamori and Fukagata2012). Note that (2.17) represents a forcing-normalized quantity; it assumes unit forcing along the first forcing mode, such that the velocity response at that wavenumber–frequency combination is given by ${\it\sigma}_{\boldsymbol{k}}\boldsymbol{u}_{\boldsymbol{k}}(y)$ .

Keep in mind that the above optimization of wall properties is only valid for a single wavenumber–frequency combination. As discussed in § 3.4, the optimal admittance identified for one mode could very well have adverse effects on resolvent modes at other wavenumber–frequency combinations. Further, the admittance of the same wall also changes across spectral space. For instance, the simple second-order model (2.16) shows that $Y=f({\it\omega})$ . Accounting for the effects of tension and stiffness would also introduce wavenumber dependencies, $Y=f({\it\kappa}_{x},{\it\kappa}_{z})$ .

2.3. Numerical implementation

The discretized resolvent operator (2.2) is constructed using a spectral collocation method on Chebyshev points. The differentiation matrices are computed using the MATLAB differentiation matrix suite developed by Weideman & Reddy (Reference Weideman and Reddy2000). The mean velocity profile in the resolvent operator $U(y)$ is obtained based on the turbulent viscosity model proposed by Reynolds & Tiederman (Reference Reynolds and Tiederman1967):

where ${\it\nu}_{T}$ is normalized by ${\it\nu}$ . The parameter ${\it\kappa}$ is the von Kármán constant, and ${\it\alpha}$ appears in van Driest’s mixing length equation. The values for these parameters are taken to be ${\it\kappa}=0.426$ and ${\it\alpha}=25.4$ , based on the optimization performed by Moarref & Jovanovic (Reference Moarref and Jovanovic2012) to fit the mean profile obtained from channel-flow DNS at $\mathit{Re}_{{\it\tau}}=2000$ (Hoyas & Jimenez Reference Hoyas and Jimenez2006). The majority of the results presented in this paper correspond to this Reynolds number.

Unlike in a pipe, the SVD of the channel-flow resolvent operator generally yields pairs of structurally similar response modes with near-identical singular values but differing symmetry along the channel centreline (Moarref et al. Reference Moarref, Sharma, Tropp and McKeon2013). To avoid this pairing and to make the computation more efficient, the grid is restricted to $N$ Chebyshev points in the lower half-channel with user-specified symmetry across the centreline. All of the results presented in this paper correspond to resolvent modes with antisymmetric $u_{\boldsymbol{k}}$ , $w_{\boldsymbol{k}}$ , $p_{\boldsymbol{k}}$ fields and symmetric $v_{\boldsymbol{k}}$ . The differentiation matrices for the velocity and pressure fields are modified to account for these symmetry conditions (see Trefethen Reference Trefethen2000).

A grid resolution study showed that the singular values converge to within $O(10^{-4})$ for $N\geqslant 100$ , which is the minimum resolution employed in this study. For $N\geqslant 100$ , the singular values for symmetric and antisymmetric modes were also found to differ by less than $O(10^{-4})$ at the wavenumber–frequency combinations considered in this paper. Similarly, the gradient-free pattern search procedure to find the optimal $Y$ has a specified tolerance of $O(10^{-4})$ . As a rough estimate of computational expense, each resolvent evaluation (i.e. setting up the operator with the appropriate boundary conditions and performing the SVD) takes approximately 0.1 s on one core of a laptop for $N=100$ and 0.5 s for $N=200$ .

3.1. Near-wall modes

The dynamically important NW cycle is characterized by coherent structures with streamwise length scale ${\it\lambda}_{x}^{+}\approx 1000$ , spanwise length scale ${\it\lambda}_{z}^{+}\approx 100$ and characteristic propagation speed $c^{+}\approx 10$ (Robinson Reference Robinson1991). As a result, this section considers the design of compliant walls that interact favourably with the resolvent mode corresponding to $\boldsymbol{k}=({\it\kappa}_{x},{\it\kappa}_{z},c^{+})=(12,\pm 120,10)$ at $\mathit{Re}_{{\it\tau}}=2000$ . The wavenumbers are normalized with respect to the channel half-height, such that the streamwise and spanwise wavelengths for this mode are ${\it\lambda}_{x}^{+}=2{\rm\pi}\mathit{Re}_{{\it\tau}}/{\it\kappa}_{x}\approx 1050$ and ${\it\lambda}_{z}^{+}=2{\rm\pi}\mathit{Re}_{{\it\tau}}/{\it\kappa}_{z}\approx 105$ , respectively. Note that $\mathit{Re}_{{\it\tau}}=2000$ was chosen because it represents the upper limit for which DNS results are readily available (Hoyas & Jimenez Reference Hoyas and Jimenez2006), and the lowest Reynolds number at which the VLSMs considered in the following section become prominent (Smits et al. Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011). Resolvent modes resembling the NW cycle at the lower Reynolds numbers typical of DNS studies with compliant walls (e.g.  $\boldsymbol{k}\approx (1,10,10)$ at $\mathit{Re}_{{\it\tau}}=140{-}180$ ) exhibit very similar behaviour to that described below.

The null-case velocity field associated with these NW-type resolvent modes is shown in figure 2. As expected for Fourier modes, the velocity field shows regions of alternating positive and negative velocity with length scales ${\it\lambda}_{x}^{+}$ and ${\it\lambda}_{z}^{+}$ . Consistent with known features of the NW cycle, the velocity field in the spanwise–wall normal plane (see $x^{+}=0$ ) shows counter-rotating quasi-streamwise vortices localized just above the critical layer $y_{c}^{+}\approx 15$ , where the mode speed matches the local mean velocity, $U(y_{c}^{+})=c^{+}$ . The velocity field in the streamwise–wall normal plane (see $z^{+}=0$ ) exhibits alternating prograde (with the mean shear) and retrograde rotation. The wall-pressure field lags the wall-normal velocity field by ${\approx}{\rm\pi}/2$ , such that regions of maximum (minimum) wall pressure coincide with regions of increasing (decreasing) wall-normal velocity.

Figure 2. Velocity structure for the NW-type mode $\boldsymbol{k}=({\it\kappa}_{x},{\it\kappa}_{z},c^{+})=(12,\pm 120,10)$ at $\mathit{Re}_{{\it\tau}}=2000$ over a rigid wall. The black-and-white isosurfaces show negative and positive wall-normal velocities at 80 % of maximum absolute value. The shading at the wall represents the normalized wall-pressure field. Note that the velocity structure shown includes contributions from both the ${\it\kappa}_{z}=120$ and ${\it\kappa}_{z}=-120$ modes.

The amplitude, phase and Reynolds stress profiles shown in figure 3(a–c) provide further insight into the structure of this mode. The streamwise velocity field peaks at the critical layer while the wall-normal velocity peaks at a location slightly farther from the wall. The streamwise and wall-normal velocity are ${\approx}{\rm\pi}$ out of phase at the critical layer, which means that the Reynolds stress contribution from this mode (figure 3 c) also peaks near $y_{c}^{+}$ . As discussed in McKeon & Sharma (Reference McKeon and Sharma2010), these features are typical of high-gain resolvent modes.

Figure 3. Profiles showing the wall-normal variation in (a,d,g) amplitude and (b,e,h) phase for the streamwise velocity (solid lines), wall-normal velocity (dashed lines) and pressure fields (fine dashed lines) for the resolvent modes resembling the NW cycle. (c, f,i) The normalized Reynolds stress contribution. Panels (a–c) represent the rigid-wall case, (d–f) represent the optimal wall in terms of singular-value suppression ( $Y=1.92+0.55\text{i}$ ), and (g–i) represent the optimal wall in terms of Reynolds stress reduction ( $Y=2.23+1.55\text{i}$ ).

Note that the wall-normal velocity and pressure fields exhibit a near-constant ${\rm\pi}/2$ phase difference. Luhar et al. (Reference Luhar, Sharma and McKeon2014a ) show that this is because the pressure field associated with the resolvent modes arises primarily from the so-called fast source terms in the Poisson equation, which represent the linear interaction between the mean shear and wall-normal velocity, i.e.  ${\rm\nabla}_{\boldsymbol{k}}^{2}p_{\boldsymbol{k}}\approx -2\text{i}{\it\kappa}_{x}v_{\boldsymbol{k}}(\partial U/\partial y)$ . It can be shown that the slow source term in the pressure Poisson equation corresponds to the divergence of the nonlinear forcing, $\boldsymbol{{\rm\nabla}}_{\boldsymbol{k}}\boldsymbol{\cdot }\boldsymbol{f}_{\boldsymbol{k}}$ , which tends to be near-zero for the resolvent modes. In other words, the most amplified forcing fields tend to be divergence-free. However, despite the fact that the resolvent pressure fields do not contain contributions from the nonlinear slow terms, they reproduce many previous observations (e.g. local pressure minima under hairpin heads, pressure maxima associated with regions where sweeps meet ejections, etc., Luhar et al. Reference Luhar, Sharma and McKeon2014a ).

The pattern-search optimization procedure indicates that a compliant wall with admittance $Y=1.92+0.55\text{i}$ leads to the greatest reduction in singular value for this mode. Specifically, this wall yields a 32 % reduction in gain (i.e. the ratio of compliant- to rigid-wall singular values is ${\it\sigma}_{\boldsymbol{k}c}/{\it\sigma}_{\boldsymbol{k}0}=0.68$ ). As shown in figure 3(d–f), this reduction in singular value is not accompanied by a significant change in mode structure. The only major distinction is that the magnitude of the pressure field at the wall is much lower over the compliant wall (figure 3 d, fine dashed line). Perhaps this reduction in the magnitude of the wall-pressure field indicates a link to the vorticity-flux control method proposed by Koumoutsakos (Reference Koumoutsakos1999), who showed that a suppression of the spanwise vorticity flux from the wall led to a significant (up to 40 %) drag decrease in channel-flow DNS at $\mathit{Re}_{{\it\tau}}=180$ . Since the planar gradients of the wall-pressure field ( $\text{i}{\it\kappa}_{x}p_{\boldsymbol{k}}(0)$ and $\text{i}{\it\kappa}_{z}p_{\boldsymbol{k}}(0)$ ) are directly proportional to the flux of streamwise and spanwise vorticity from the wall, the reduction in wall pressure suggests a reduction in the vorticity flux. However, keep in mind that a direct comparison between the present results and the vorticity flux control method is not strictly appropriate. The controller developed by Koumoutsakos (Reference Koumoutsakos1999) only considered the flux of spanwise vorticity and so the resulting flow structures were more similar to the spanwise-constant travelling waves discussed in § 3.4. Finally, note that the normalized Reynolds stress contribution from this mode does not change significantly relative to the rigid-wall case (figure 3 c, f). However, for identical forcing strengths, the 32 % suppression in mode amplitude over the compliant wall would translate into a near-50 % reduction in the actual Reynolds stress generated by this mode ( $\propto {\it\sigma}_{\boldsymbol{k}}^{2}$ ; see (2.17)).

A compliant wall with admittance $Y=2.23+1.55\text{i}$ is found to be optimal based on the lowest channel-integrated Reynolds stress criterion (2.17). Although this wall does not yield a substantial reduction in gain ( ${\it\sigma}_{\boldsymbol{k}c}/{\it\sigma}_{\boldsymbol{k}0}=0.90$ ), it does result in a significant modification of the flow structure (figure 3 g–i). Instead of peaking at or near the critical layer, the magnitude of the velocity fields is largest at the wall. Further, the phase relationship between the streamwise and wall-normal velocity is modified substantially relative to the rigid-wall case (figure 3 h). The velocity fields are approximately $\pm {\rm\pi}/2$ out of phase across $y_{c}^{+}\approx 0{-}40$ , leading to a significant reduction in the normalized Reynolds stress (figure 3 i). Physically, this change in mode structure means that the quasi-streamwise vortices observed for the rigid-wall case, which are important for Reynolds stress and turbulent kinetic energy production via the lift-up mechanism, are almost entirely suppressed, and structures resembling rollers are generated near the wall (figure 4).

Figure 4. Velocity structure for the NW-type mode with the optimal Reynolds-stress-reducing wall. The black-and-white isosurfaces show negative and positive wall-normal velocities at 80 % of maximum absolute value. The shading on the compliant wall (deflection not to scale) represents the normalized pressure field.

Recall that the above results were obtained with linear approximations to the boundary conditions (2.10)–(2.12). Since the velocity magnitudes peak at the wall (figure 3), the boundary conditions clearly play a major role in dictating mode behaviour. Hence, the use of linearized boundary conditions is a significant limitation of the present approach, which needs to be carefully evaluated against results from simulations employing more accurate boundary conditions (e.g. involving immersed boundaries or coordinate transformations) or from experiments as they become available.

The results presented thus far suggest that it might be possible to suppress the dynamically important NW cycle using compliant walls. However, it is important to keep in mind that the admittance $Y$ must reflect the physical properties of the wall. The simple spring–damper model (2.16) suggests that the optimal admittances identified in this section, $Y=1.92+0.55\text{i}$ for mode suppression and $Y=2.23+1.55\text{i}$ for Reynolds stress reduction, would require negative damping coefficients, i.e.  $\text{Re}(Y)>0$ corresponds to $C_{d}<0$ . This is further confirmed by the contour maps shown in figure 5. Only compliant walls with $\text{Re}(Y)>0$ lead to a reduction in singular value (figure 5 a). A reduction in Reynolds stress does appear possible with $\text{Re}(Y)<0$ , despite the increase in gain (see $Y\approx -0.1+2\text{i}$ ; figure 5 b). However, this result must be treated with caution because the higher-rank resolvent modes not considered in this paper may become important as the singular values increase relative to the rigid-wall case, and the Reynolds stress contribution from these higher-rank modes could offset the Reynolds stress reduction from the rank-1 modes. In other words, the rank-1 approximation might fail in the modified flow.

Figure 5. Contours showing the ratio of (a) compliant- to rigid-wall singular values ${\it\sigma}_{\boldsymbol{k}c}/{\it\sigma}_{\boldsymbol{k}0}$ and (b) channel-integrated Reynolds stress $RS_{\boldsymbol{k}c}/RS_{\boldsymbol{k}0}$ across a range of wall admittances, $Y$ , for resolvent modes resembling the NW cycle. The optimal admittances identified based on the pattern-search procedure are marked with a white $+$ . The channel-integrated Reynolds stress is defined in (2.17). The dashed contour line corresponds to a ratio of 1, i.e. no change.

Since the NW cycle is a dominant feature of low-Reynolds-number flows, the results presented in this section may also explain why previous DNS studies have met with limited success. The present analysis suggests that the positive-damping walls typically tested in DNS are unlikely to interact favourably with the NW cycle (to our knowledge, no previous simulations have considered negative damping coefficients). Optimal wall admittances with $\text{Re}(Y)>0$ are also indicative of a situation where the wall-normal velocity is, at least partially, in phase with the wall-pressure field (figure 4). This phase relationship is consistent with suggestions made in previous studies based on heuristic arguments (e.g. Xu et al. Reference Xu, Rempfer and Lumley2003; Fukagata et al. Reference Fukagata, Kern, Chatelain, Koumoutsakos and Kasagi2008) that the velocity fluctuations need to be in phase with the wall pressure for drag reduction.

Finally, it is also important to keep in mind that a wall with negative damping is, strictly speaking, no longer passive. It results in a net energy transfer into the flow. However, negative damping does not necessarily imply that the wall requires an external energy input. It may be possible to identify materials that are passive globally but exhibit negative damping over a limited bandwidth.

3.2. Very-large-scale motions

This section considers the effect of compliant surfaces on modes with $\boldsymbol{k}=(1,\pm 10,16)$ at $\mathit{Re}_{{\it\tau}}=2000$ . Similar to the VLSMs (Monty & Chong Reference Monty and Chong2009; Smits et al. Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011), these modes represent flow structures of streamwise and spanwise length scales ${\it\lambda}_{x}\approx 6h$ ( ${\it\lambda}_{x}^{+}\approx 12\,600$ ) and ${\it\lambda}_{z}\approx 0.6h$ ( ${\it\lambda}_{z}^{+}\approx 1260$ ), propagating downstream at $c^{+}=16$ . Despite the roughly 10-fold increase in wavelength, figure 6(a–c) shows that the rigid-wall structure for this resolvent mode is very similar to that for the NW mode considered in the previous section. The streamwise velocity, wall-normal velocity and Reynolds stress again peak at or near the critical layer, which is now located at $y_{c}^{+}\approx 94$ , and there is a near-constant ${\rm\pi}/2$ phase difference between $v_{\boldsymbol{k}}$ and $p_{\boldsymbol{k}}$ .

Figure 6. Profiles showing the wall-normal variation in (a,d,g) amplitude and (b,e,h) phase for the streamwise velocity (solid lines), wall-normal velocity (dashed lines) and pressure fields (fine dashed lines) for the resolvent modes resembling VLSMs. (c, f,i) The normalized Reynolds stress contribution. Panels (a–c) represent the rigid-wall case, (d–f) represent the optimal wall in terms of singular-value suppression ( $Y=-2.04-0.44\text{i}$ ), and (g–i) represent the optimal wall in terms of Reynolds stress reduction ( $Y=-3.10-0.39\text{i}$ ).

However, unlike the modes resembling the NW cycle, compliant walls with positive damping interact favourably with these larger-scale modes. This is illustrated by the predicted performance maps shown in figure 7; regions of singular value and Reynolds stress reduction are primarily confined to admittances with $\text{Re}(Y)<0$ . The pattern search suggests that the optimal wall admittances for gain and Reynolds stress reductions for this mode are $Y=-2.04-0.44\text{i}$ and $Y=-3.19-0.39\text{i}$ , respectively. Both admittances lead to similar reductions in singular value and changes in mode structure (see figure 6 d–f and g–i). The best gain-reducing wall leads to ${\it\sigma}_{\boldsymbol{k}c}/{\it\sigma}_{\boldsymbol{k}0}=0.52$ , while the optimal Reynolds-stress-reducing wall leads to ${\it\sigma}_{\boldsymbol{k}c}/{\it\sigma}_{\boldsymbol{k}0}=0.56$ .

Figure 7. Contours showing the ratio of (a) compliant- to rigid-wall singular values ${\it\sigma}_{\boldsymbol{k}c}/{\it\sigma}_{\boldsymbol{k}0}$ and (b) channel-integrated Reynolds stress $RS_{\boldsymbol{k}c}/RS_{\boldsymbol{k}0}$ across a range of wall admittances, $Y$ , for resolvent modes resembling VLSMs. The optimal admittances identified based on the pattern-search procedure are marked with a white $+$ . The dashed contour line corresponds to a ratio of 1, i.e. no change.

The mode structure over the compliant walls is modified such that the streamwise velocity (solid lines) is largest at the wall, while the magnitude of the wall-normal velocity (bold dashed lines) exhibits local maxima above and below the critical layer. The local peak in $v_{\boldsymbol{k}}$ below the critical layer is also accompanied by a $\pm {\rm\pi}/2$ phase shift between the wall and the critical layer (see bold dashed lines, figure 6 e,h). Note that the normalized Reynolds stress contribution from the compliant-wall resolvent modes is significantly lower compared to the rigid-wall case. Figure 6(i) suggests that the compliant wall may even lead to a reversal in the sign of the Reynolds stress.

The rigid-wall flow structure associated with this large-scale mode is broadly similar to that shown in figure 2, but with vastly different length scales. Figure 8 shows how the wall optimized for Reynolds stress alters this physical structure. The compliant wall again leads to a suppression of the quasi-streamwise vortices associated with Reynolds stress production. However, unlike the NW modes, the wall-normal velocity and pressure fields are not in phase at the wall. Regions of positive $p_{\boldsymbol{k}}$ coincide with regions of negative $v_{\boldsymbol{k}}$ (i.e. the wall moves downwards under high pressure), as expected for $\text{Re}(Y)<0$ .

Figure 8. Velocity structure for the VLSM-type mode with the optimal Reynolds-stress-reducing wall. The black-and-white isosurfaces show negative and positive wall-normal velocities at 80 % of maximum absolute value. The shading on the compliant wall (deflection not to scale) represents the normalized pressure field.

Figure 9. Optimal wall admittances, in terms of singular-value suppression, for modes of varying speed $c^{+}=5{-}20$ . Each marker represents an increment of ${\rm\Delta}c^{+}=0.2$ . The darkness of the marker indicates the degree of suppression. Also shown are the optimal admittances for Reynolds stress reduction for the NW and VLSM-type modes considered in §§ 3.1 and 3.2 (marked with RS).

The fact that walls with positive damping are predicted to interact favourably with the larger VLSM-type modes might explain why previous experiments have met with more success than numerical simulations. In general, the experiments (Lee et al. Reference Lee, Fisher and Schwarz1993; Choi et al. Reference Choi, Yang, Clayton, Glover, Atlar, Semenov and Kulik1997) have been carried out at higher Reynolds numbers, where these larger-scale flow structures play a more prominent role. Since the VLSMs are known to have an organizing influence on the near-wall turbulence (Marusic, Mathis & Hutchins Reference Marusic, Mathis and Hutchins2010), they may also serve as a pathway for compliant walls to influence the entire flow.

3.3. Effect of mode speed

As a sensitivity analysis, this section evaluates the effects of compliant walls on resolvent modes with varying speed $c^{+}$ . The speed $c^{+}$ determines the wall-normal localization of the mode (McKeon & Sharma Reference McKeon and Sharma2010). Slower-moving resolvent modes are located close to the wall and so they have stronger source terms in the pressure Poisson equation (recall that the fast source term $\propto v_{\boldsymbol{k}}\partial U/\partial y$ and the mean velocity gradient is larger close to the wall). Luhar et al. (Reference Luhar, Sharma and McKeon2014a ) show that, everything else being equal, the stronger source terms and wall proximity translate into a larger wall-pressure signature for slower-moving resolvent modes. As the mode speed increases, the flow structure associated with the resolvent modes moves further away from the wall and the magnitude of the wall-pressure field decreases. Above some threshold speed, the modes detach from the wall entirely and have a near-zero wall-pressure field. This threshold speed tends to be wavelength-dependent; longer modes have a larger wall-normal extent, which means that they detach from the wall at higher mode speeds.

The above observations suggest that relatively rigid walls with low $|Y|$ are likely to interact only with slower-moving modes with high fluctuating wall pressure, while softer walls might be able to interact with modes further away from the wall. However, it may be impossible to influence detached modes with a compliant wall. This is confirmed by the data shown in figure 9, which plots the optimal admittance for gain reduction for modes with the same wavelengths as the NW- and VLSM-type structures considered in the previous two sections, $({\it\kappa}_{x},{\it\kappa}_{z})=(12,120)$ and $({\it\kappa}_{x},{\it\kappa}_{z})=(1,10)$ , but mode speed varying from $c^{+}=5$ to $c^{+}=20$ . The magnitude of the optimal admittance, $|Y|$ , is smallest for the slowest modes and increases with increasing mode speed. In other words, stiffer walls are optimal for slow-moving modes with high-amplitude wall-pressure fields, while softer walls are better suited for faster modes further away from wall. Note that there is a turning point near $c^{+}\approx 19$ for the longer ${\it\kappa}_{x}=1$ (figure 9, ○) modes and $c^{+}\approx 13$ for the ${\it\kappa}_{x}=12$ modes (figure 9, $+$ ), above which the optimal $|Y|$ decreases and the effectiveness of the compliant surfaces diminishes (i.e. the markers get lighter). Consistent with the mode detachment hypothesis, it may not be possible to design walls that can interact with these fast-moving structures.

Figure 9 also provides a measure of the sensitivity of the optimal wall admittances on mode parameters. The fact that there are no sharp jumps in the optimal $Y$ with increasing $c^{+}$ indicates that the optimal wall admittances are unlikely to be very different for resolvent modes similar to those identified in the previous two sections. This is important because the previous two sections assume that the NW cycle and VLSMs are adequately characterized by resolvent modes at one specific wavenumber–frequency combination. However, in reality, the NW cycle and the VLSMs represent regions that are energetic in spectral space. Hence, resolvent modes with slightly different wavelengths and speeds are equally viable candidates to serve as models for the NW cycle and VLSMs.

Finally, note that there is a consistent trend in the optimal pressure–velocity phase relationship, $\angle Y=\angle v_{\boldsymbol{k}}(0)-\angle p_{\boldsymbol{k}}(0)$ , with mode speed. Perhaps the most interesting aspect of this trend is that the optimal admittance for the larger $({\it\kappa}_{x},{\it\kappa}_{z})=(1,10)$ modes is also characterized by $\text{Re}(Y)>0$ (i.e.  $C_{d}<0$ ; see (2.16)) for $c^{+}\lesssim 14$ . It may be the case that the negative-damping requirement discussed in § 3.1 is restricted to slower-moving modes. These issues will be explored further in future publications.

3.4. Comparison with previous DNS and overall efficacy of optimal walls

So far, this paper has focused primarily on the interaction between compliant walls and individual resolvent modes that resemble flow structures that are energetic in natural wall turbulence. However, it is important to keep in mind that the compliant wall could also have detrimental effects elsewhere in spectral space. This is illustrated by the recent DNS results obtained by Kim & Choi (Reference Kim and Choi2014) at $\mathit{Re}_{{\it\tau}}=140$ . The softest compliant wall tested by Kim & Choi (Reference Kim and Choi2014, termed case II) led to the generation of large-amplitude quasi-two-dimensional (2D) waves of wavelength ${\it\lambda}_{x}\approx 2.4h$ propagating downstream at a speed roughly 0.3 times the centreline velocity ( $c^{+}\approx 5$ ). These waves, which are clearly not energetic in rigid-wall turbulent channel flow, led to a substantial increase in the total drag. Kim & Choi (Reference Kim and Choi2014) attributed the generation of the quasi-2D waves to a wall-resonance effect.

Figure 10 shows that the framework developed in this study can predict this resonant response with minimal computation. Figure 10(a) shows the rigid-wall singular values for 2D ( ${\it\kappa}_{z}=0$ ) resolvent modes at $\mathit{Re}_{{\it\tau}}=140$ as a function of the streamwise wavenumber and wave speed. Figure 10(b) shows the singular values predicted by the present analysis over the case II compliant wall of Kim & Choi (Reference Kim and Choi2014). Since Kim & Choi (Reference Kim and Choi2014) also employed a simple mass–spring–damper mechanical model for the compliant wall, the admittances were estimated using (2.16) to arrive at these predictions. The mass coefficient for the case II compliant wall was $C_{m}=2$ , while the spring and damping coefficients, normalized by 1.5 times the bulk velocity ( ${\approx}21u_{{\it\tau}}$ ), were $C_{k}^{\ast }=1$ and $C_{d}^{\ast }=0.5$ , respectively. This translates into $C_{k}\approx 440$ and $C_{d}\approx 10.5$ with the $u_{{\it\tau}}$ normalization shown in (2.15). The resonant frequency for this wall is ${\it\omega}_{r}={\it\omega}_{n}\sqrt{1-2{\it\zeta}^{2}}=14.7$ . Here, ${\it\omega}_{n}=\sqrt{C_{k}/C_{m}}$ is the undamped natural frequency of the wall and ${\it\zeta}=C_{d}/(2\sqrt{C_{k}C_{m}})$ is the damping factor.

Figure 10. Singular values for spanwise-constant ( ${\it\kappa}_{z}=0$ ) modes as a function of streamwise wavenumber and wave speed at $\mathit{Re}_{{\it\tau}}=140$ . (a) The rigid-wall singular values and (b) the singular values for the case II compliant wall tested by Kim & Choi (Reference Kim and Choi2014). The $+$ marker represents the length and velocity scale for the quasi-2D travelling wave observed in the DNS. The solid black line shows the resonant frequency of the wall ${\it\omega}_{r}$ . Note the log scaling for the shading.

The compliant-wall singular values clearly show a region of high amplification around ${\it\omega}_{r}$ extending from $({\it\kappa}_{x},c^{+})\approx (3,5)$ to $({\it\kappa}_{x},c^{+})\approx (1.5,9)$ , which does not exist for the rigid-wall case. The singular values over the compliant wall are as much as 70 times larger relative to the rigid case. The greatest change in singular value relative to the rigid-wall case falls at ${\it\kappa}_{x}=2.8$ and $c^{+}=5.25$ , which corresponds closely to the wavelength and speed ( ${\it\kappa}_{x}=8/3$ and $c^{+}=4.78$ ) of the quasi-2D travelling wave observed in DNS. The slight differences in wavenumber and speed could be attributed to three different reasons: (i) the uncertainty in $C_{k}$ and $C_{d}$ arising from the bulk velocity to friction velocity translation of these dimensionless coefficients; (ii) the use of the unperturbed mean velocity profile in the resolvent operator; and (iii) the linearized boundary conditions at the wall. The latter two sources of uncertainty are significant because the DNS results show a substantial increase in drag over the compliant wall, which is likely to be accompanied by a change in the mean velocity profile (not reported by Kim & Choi Reference Kim and Choi2014), and the 2D waves excited had amplitudes in excess of 10 viscous units.

Figure 11 shows the flow structure and $v{-}p$ relationship associated with this resonant resolvent mode. The streamwise velocity for this mode is confined to a very small region close to the wall ( $y^{+}<10$ ). Above this location, the velocity field comprises a predominantly up–down motion. This is consistent with the DNS results, which showed that the streamwise velocity field at $y^{+}=14$ above the compliant wall was dominated by streaky structures elongated in the streamwise direction (figure 8 in Kim & Choi Reference Kim and Choi2014). Further, the predicted phase relationship between the wall-normal velocity and pressure at the wall (figure 11 b) is also similar to the DNS observations (figure 7 in Kim & Choi Reference Kim and Choi2014).

Figure 11. (a) Velocity structure for the resolvent mode corresponding to wavenumber–frequency combination $({\it\kappa}_{x},{\it\kappa}_{z},c^{+})=(8/3,0,5)$ at $\mathit{Re}_{{\it\tau}}=140$ . This represents roughly the quasi-2D travelling waves observed in DNS by Kim & Choi (Reference Kim and Choi2014). The shading on the compliant wall (deflection not to scale) represents the normalized pressure field. (b) The phase relationship between wall-normal velocity and pressure at the wall.

Although qualitative in nature, the comparison presented above indicates that the framework developed in this paper is a useful first-order test of material properties before implementation in more computationally intensive numerical simulations or experiments. Recall that each resolvent evaluation only takes ${\approx}0.1~\text{s}$ on one core of a laptop. The contour maps shown in figure 10 were estimated at 100 linearly spaced intervals in $c^{+}$ and 50 log-spaced intervals in ${\it\kappa}_{x}$ , and as such required approximately $2\times 500~\text{s}$ of computation time on one core of a laptop (without any significant effort towards making the computation efficient). Of course, the above computations were limited by the constraint ${\it\kappa}_{z}=0$ . A true a priori evaluation requires a sweep across the entire spectral space. However, this is still inexpensive relative to numerical simulations, especially so given the spectral sparsity observed recently in wall turbulence (Bourguignon et al. Reference Bourguignon, Tropp, Sharma and McKeon2014).

The high-amplitude resonant modes observed in DNS suggest that compliant walls optimized to suppress modes resembling the NW cycle and the VLSMs could lead to detrimental effects at other wavenumber–frequency combinations. If the increased drag contribution from these other modes outweighs the decrease from the NW- and VLSM-type modes, the compliant wall would have a negative net influence on performance. To test whether this is the case, the effect of the optimal gain-reducing walls identified in §§ 3.1 and 3.2 on modes elsewhere in spectral space is considered next.

Assuming a dimensionless mass coefficient of $C_{m}=2$ , it can be shown based on (2.16) that the optimal wall for the NW-type modes ( $Y=1.92+0.55\text{i}$ ) requires a high stiffness coefficient $C_{k}=28\,817$ and a negative damping coefficient $C_{d}=-0.48$ . Similarly, the optimal wall for the VLSM-type modes ( $Y=-2.04-0.44\text{i}$ ) requires $C_{k}=510$ and $C_{d}=0.47$ . Note that the resonant frequencies of the optimal walls correspond very closely to the temporal frequencies for the modes resembling the NW cycle ( ${\it\omega}_{r}=120.03$ , ${\it\omega}=120$ ) and the VLSMs ( ${\it\omega}_{r}=15.97$ , ${\it\omega}=16$ ). This is to be expected given that the amplitude of the compliant-wall response is likely to be largest (i.e. largest $|Y|$ ) at or near the resonant frequency (2.16). This large wall response would translate into a bigger impact on the turbulent flow structure.

Figure 12 shows how the optimal walls affect the amplification of resolvent modes with ${\it\kappa}_{z}=0$ , ${\it\kappa}_{z}=10$ and ${\it\kappa}_{z}=120$ as a function of streamwise wavenumber and mode speed at $\mathit{Re}_{{\it\tau}}=2000$ . As expected, the compliant walls have the largest effect, positive or negative, on modes with frequencies close to ${\it\omega}_{r}$ . Further, the impact of the compliant wall is restricted to a smaller region of spectral space around ${\it\omega}_{r}$ as the spanwise wavelength decreases ( ${\it\kappa}_{z}$ increases; compare figure 12 a,b with figure 12 e, f) and the mode speed $c^{+}$ increases (compare $c^{+}\approx 4$ with $c^{+}\approx 12$ in figure 12 d). This is because, everything else being equal, larger-scale resolvent modes that are slow-moving and localized close to the wall have a much larger wall-pressure signature compared to smaller, faster modes farther from the wall (Luhar et al. Reference Luhar, Sharma and McKeon2014a ). The higher-amplitude wall-pressure field yields a larger wall response, which compensates for the decrease in $|Y|$ as the temporal frequency of the modes moves away from ${\it\omega}_{r}$ . This scale and speed dependence may also explain why the compliant wall optimized for the NW-type modes (figure 12 a,c,e) has a smaller reach in spectral space compared to the wall optimized for the VLSM-type modes (figure 12 b,d,f). The higher resonant frequency for the wall optimized for NW-type modes means that it interacts preferentially with modes that are faster or shorter in $x$ ( ${\it\omega}=c^{+}{\it\kappa}_{x}\approx {\it\omega}_{r}$ ). These modes are likely to have a smaller wall-pressure signature and so only interact significantly with the compliant wall at conditions close to resonance.

Figure 12. Shaded contours showing the singular-value ratio ${\it\sigma}_{\boldsymbol{k}c}/{\it\sigma}_{\boldsymbol{k}0}$ as a function of wavenumber ${\it\kappa}_{x}$ and speed $c^{+}$ at (a,b)  ${\it\kappa}_{z}=0$ (spanwise constant), (c,d)  ${\it\kappa}_{z}=10$ ( ${\it\lambda}_{z}^{+}\approx 1250$ ) and (e, f)  ${\it\kappa}_{z}=120$ ( ${\it\lambda}_{z}^{+}\approx 100$ ). White regions denote an increase in mode amplification over the compliant wall. Darker regions denote suppression. Note the log scaling for the shading. The dashed contour lines indicate the magnitude of the singular values above the compliant wall, ${\it\sigma}_{\boldsymbol{k}c}$ . (a,c,e) The compliant wall optimized for modes resembling the NW cycle ( ${\it\kappa}_{x}=12$ , ${\it\kappa}_{z}=120$ , $c^{+}=10$ ; white circle in panels e and f). (b,d,f) The optimal compliant wall for VLSM-type modes ( ${\it\kappa}_{x}=1$ , ${\it\kappa}_{z}=10$ , $c^{+}=16$ ; white circle in panels c and d). The solid black lines show the resonant frequency of the walls ${\it\omega}_{r}$ .

Note that there are sharp transitions in the effectiveness of the compliant walls around the resonant frequency. In general, modes with ${\it\omega}<{\it\omega}_{r}$ are further amplified by the compliant wall, while modes with ${\it\omega}>{\it\omega}_{r}$ are suppressed (although this appears to reverse for $c^{+}\gtrsim 15$ ). Thus, modes with length and velocity scales very similar to the assumed parameters for NW cycle or VLSMs may actually be further amplified over the optimal compliant walls. This result suggests that it may be better for the overall performance to design compliant walls that are slightly detuned and resonant away from the spectral region of interest. More generally, the above results suggest that it is insufficient to optimize compliant-wall properties for a single wavenumber–frequency combination; the optimization must be performed over a wider region in spectral space. This is similar to the approach employed in designing compliant walls for transition delay, wherein compliant-wall properties are optimized against all the mode types that the fluid and wall can concurrently support (see e.g. Dixon, Lucey & Carpenter Reference Dixon, Lucey and Carpenter1994).

3.5. More realistic walls and surface instabilities

Much of the discussion thus far has been based on the simple mass–spring–damper wall model. As illustrated by figure 12, this results in fluid–structure interactions that are primarily dependent on frequency. In addition, these dynamics are local; the wall does not communicate in the streamwise or spanwise directions. This means that the wall cannot support wave propagation and is non-dispersive. Practically constructed walls such as the elastic and viscoelastic surfaces tested in experiment not only support waves but also have an embedded natural length scale (e.g. layer thickness). This allows for spatio-temporal matching between the fluid and solid motions, leading to much stronger interactions. The experiments performed by Gad-el-Hak and co-workers (Gad-el-Hak, Blackwelder & Riley Reference Gad-el-Hak, Blackwelder and Riley1984; Gad-el-Hak Reference Gad-el-Hak1986) show that such elastic and viscoelastic layers support two different kinds of surface instabilities under laminar and turbulent boundary layers: static divergence and travelling wave flutter. The so-called static divergence waves are very slow-moving ( ${<}5\,\%$ of free-stream velocity) with long wavelengths and large amplitudes, while travelling-wave flutter tends to have shorter wavelengths and lower amplitudes but propagates much faster, at speeds comparable to the free shear-wave speed in the solid. Figure 13(a) shows that the resolvent framework qualitatively predicts similar effects.

Figure 13. Shaded contours showing the singular-value ratio ${\it\sigma}_{\boldsymbol{k}c}/{\it\sigma}_{\boldsymbol{k}0}$ as a function of wavenumber ${\it\kappa}_{x}$ and speed $c^{+}$ for spanwise-constant ${\it\kappa}_{z}=0$ modes. Both panels correspond to walls optimized to suppress the VLSMs (cf. figure 12 b): (a) a compliant wall in streamwise tension $C_{tx}$ with spring constant $C_{k}=0$ , and (b) a mass–spring–damper wall with mass ratio $C_{m}=200$ . White regions denote an increase in mode amplification over the compliant wall. Darker regions denote suppression. Note the log scaling for the shading. The dashed contour lines indicate the magnitude of the singular values above the compliant wall, ${\it\sigma}_{\boldsymbol{k}c}$ . The solid black lines show the free wave speed, $c_{w}^{+}$ , for the tensioned wall in (a) and the resonant frequency ${\it\omega}_{r}$ in (b).

Specifically, figure 13(a) shows the effect of a compliant wall optimized to suppress VLSMs ( $Y=-2.04-0.44\text{i}$ at ${\it\kappa}_{x}=1$ , ${\it\kappa}_{z}=10$ and $c^{+}=16$ ) on spanwise-constant modes, similar to the results shown in figure 12(b). However, the wall is assumed to be in streamwise tension $C_{tx}=510$ rather than on a spring support. As a result, the effective spring constant for the wall is $C_{ke}=C_{tx}{\it\kappa}_{x}^{2}$ and the admittance is $Y=\text{i}{\it\omega}(C_{tx}{\it\kappa}_{x}^{2}-C_{m}{\it\omega}^{2}-\text{i}C_{d})^{-1}$ . Note that this wall can support streamwise-propagating waves. Relative to the rigid-wall case, there are two regions of extremely high amplification over the compliant wall: for very long slow-moving modes ( ${\it\kappa}_{x}\lesssim 1$ and $c^{+}\lesssim 8$ ) and for shorter modes ( ${\it\kappa}_{x}=2{-}10$ ) at speeds close to the undamped wave speed for the wall, $c_{w}^{+}\approx \sqrt{C_{tx}/C_{m}}=16$ . It can be argued that the first class of modes are similar to static divergence waves and the second class of resonant modes are similar to travelling-wave flutter (see also Riley et al. Reference Riley, Gad-el-Hak and Metcalfe1988; Sen & Arora Reference Sen and Arora1988). Keep in mind that the tensioned wall does not have a natural length scale and so the wavelengths of the high-amplification regions are probably determined by the channel height.

For more quantitative comparisons with experiments, the resolvent formulation needs to be extended to account for boundary conditions that more accurately reflect elastic and viscoelastic coatings. This is the subject of ongoing work. Even so, direct comparisons with the observed high-amplitude static divergence waves may be difficult given that the framework developed in this paper is limited to linearized boundary conditions.

Finally, note that the preceding results are based on an assumed mass coefficient $C_{m}=2$ . While $C_{m}\sim O(1)$ is appropriate for liquid flows over compliant walls, $C_{m}\sim O(10^{3})$ for gas flows over walls with realistic material properties. Compliant walls are unlikely to be effective for systems with such large mass ratios because the magnitude of the admittance decreases sharply with increasing $C_{m}$ (2.16) away from resonance, resulting in much smaller wall responses to the fluid pressure perturbations. This is illustrated by the results shown in figure 13(b). Again, this figure shows the effect of a wall optimized to suppress the VLSMs on spanwise-constant modes. However, the mass ratio here is $C_{m}=200$ . Unlike the results shown in figure 12(b), this wall has a minimal effect on resolvent modes with frequencies far from ${\it\omega}_{r}$ . In other words, the wall is effectively rigid at this high- $C_{m}$ limit, suggesting that the fluid and solid impedances must at least match approximately for the compliant surface to have a significant effect on the flow.