2.1 Problem description

This study computes the deformation of a solid layer in a turbulent flow, as sketched in figure 1. A viscous and incompressible Newtonian fluid has mass density $\unicode[STIX]{x1D70C}_{f}$ , dynamic viscosity $\unicode[STIX]{x1D707}$ and kinematic viscosity $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}_{f}$ . The wall-friction velocity $u_{\unicode[STIX]{x1D70F}}$ is derived from $\unicode[STIX]{x1D70F}_{w}=\unicode[STIX]{x1D70C}_{f}u_{\unicode[STIX]{x1D70F}}^{2}$ with $\unicode[STIX]{x1D70F}_{w}$ the Reynolds-averaged wall shear stress for turbulent flow over a rigid wall. Because of the one-way coupling approach, the mean wall shear stress is not influenced by the fluctuating coating deformations. The Reynolds number is either $Re_{\unicode[STIX]{x1D70F}}=hu_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ for channel flow with half-height $h$ , or $Re_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D6FF}_{bl}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ for boundary-layer flow with boundary-layer height $\unicode[STIX]{x1D6FF}_{bl}$ .

The problem is two-dimensional in space: only horizontal and vertical deformations, velocities and stresses are considered, as is motivated in § 4.2.1. The fluctuating fluid stresses at the coating interface are the tangential stress $\unicode[STIX]{x1D70E}_{tc}(x,t)$ and the normal stress $\unicode[STIX]{x1D70E}_{nc}(x,t)$ , with the subscripts $c$ for the interface, $t$ for tangential stress and $n$ for normal stress. These stresses, which vary with the streamwise coordinate $x$ and with time $t$ , result in a deformation of the underlying coating. The horizontal and vertical coating displacements are denoted by $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D701}$ , with the corresponding velocities $u$ and $w$ .

The coating of thickness $\unicode[STIX]{x1D6FF}$ is linear, time-translation-invariant, homogeneous, isotropic and viscoelastic. It is attached at the bottom to a rigid wall, and it has infinite length in the streamwise direction to neglect the influence of coating boundaries and the associated reflections. The coating has mass density $\unicode[STIX]{x1D70C}_{s}$ , shear-wave speed $c_{s}$ , shear modulus $G=\unicode[STIX]{x1D70C}_{s}{c_{s}}^{\,2}$ , compressional-wave speed $c_{p}$ and compressional-wave modulus $\unicode[STIX]{x1D6F1}=\unicode[STIX]{x1D70C}_{s}{c_{p}}^{\,2}$ . These speeds and moduli are complex numbers, as explained below.

2.2 Surface stress and coating deformation as travelling waves

This subsection introduces the concept of travelling waves to describe the fluctuating surface stress and coating deformation. To simplify the analysis and to speed up the computations, four assumptions will be made regarding the surface stresses: (i) they are fully determined in space and time, thus ignoring any randomness, (ii) the stresses can be considered as travelling waves, which excludes growing or decaying stress disturbances, (iii) the set of frequencies is discrete and finite and (iv) each frequency corresponds with a single wavevector in the streamwise direction such that the spanwise dependence of the stresses can be neglected. Section 4.2.1 explains why the latter assumption is reasonable for turbulent flows. Given these assumptions, the fluctuating interface stresses can be expressed as a sum of $N_{m}$ streamwise-travelling spanwise-homogeneous waves:

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D748}_{\boldsymbol{c}}(x,t)=\mathop{\sum }_{m=1}^{N_{m}}\unicode[STIX]{x1D748}_{\boldsymbol{c},\boldsymbol{m}}\text{e}^{-\text{i}\left(k_{m}x-\unicode[STIX]{x1D714}_{m}t\right)},\end{eqnarray}$$

with the two-dimensional stress vector $\unicode[STIX]{x1D748}_{\boldsymbol{c}}=[\unicode[STIX]{x1D70E}_{tc}~\unicode[STIX]{x1D70E}_{nc}]^{\text{T}}$ , where $^{\text{T}}$ denotes the transpose. Each travelling wave (or mode) has number $m$ , complex amplitude $\unicode[STIX]{x1D748}_{\boldsymbol{c},\boldsymbol{m}}=[\unicode[STIX]{x1D70E}_{tc,m}~\unicode[STIX]{x1D70E}_{nc,m}]^{\text{T}}$ , (streamwise) wavenumber $k_{m}$ , angular frequency $\unicode[STIX]{x1D714}_{m}$ and wave speed $c_{m}=\unicode[STIX]{x1D714}_{m}/k_{m}$ . This paper considers only the deformation by stress fluctuations, so $\unicode[STIX]{x1D714}_{m}$ is non-zero.

Assuming that the coating response is linear and stable, the coating deformation will have the same temporal and spatial dependence as the stresses. For example, the surface stress $\unicode[STIX]{x1D748}_{\boldsymbol{c},\boldsymbol{m}}\exp \{-\text{i}(k_{m}x-\unicode[STIX]{x1D714}_{m}t)\}$ will generate the displacement $\unicode[STIX]{x1D743}_{\boldsymbol{m}}(z)\exp \{-\text{i}(k_{m}x-\unicode[STIX]{x1D714}_{m}t)\}$ , with $\unicode[STIX]{x1D743}_{\boldsymbol{m}}=[\unicode[STIX]{x1D709}_{m}~\unicode[STIX]{x1D701}_{m}]^{\text{T}}$ the complex displacement amplitude of mode $m$ . Because of linearity, the total response of the coating is simply a summation of the individual responses:

(2.2a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D743}(x,z,t)=\mathop{\sum }_{m=1}^{N_{m}}\unicode[STIX]{x1D743}_{\boldsymbol{m}}(z)\text{e}^{-\text{i}\left(k_{m}x-\unicode[STIX]{x1D714}_{m}t\right)},\quad \boldsymbol{u}(x,z,t)=\mathop{\sum }_{m=1}^{N_{m}}\boldsymbol{u}_{\boldsymbol{m}}(z)\text{e}^{-\text{i}\left(k_{m}x-\unicode[STIX]{x1D714}_{m}t\right)},\end{eqnarray}$$

where $\unicode[STIX]{x1D743}=[\unicode[STIX]{x1D709}~\unicode[STIX]{x1D701}]^{\text{T}}$ and $\boldsymbol{u}=[u~w]^{\text{T}}$ .

2.3 Viscoelastic deformation

This subsection summarizes the theory of viscoelastic deformations. Consider a solid particle at a position given by the Lagrangian coordinate vectors $\boldsymbol{x}$ in the undeformed medium and $\boldsymbol{X}$ in the deformed medium. The displacement or deformation vector $\unicode[STIX]{x1D743}$ with components $\unicode[STIX]{x1D709}_{i}$ is then given by $\unicode[STIX]{x1D743}=\boldsymbol{X}-\boldsymbol{x}$ . The particle’s velocity $u_{i}(\boldsymbol{x},t)$ is the time derivative of its actual position: $u_{i}=\unicode[STIX]{x2202}X_{i}/\unicode[STIX]{x2202}t=\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}/\unicode[STIX]{x2202}t$ . The equations of motion in the undeformed coordinates are (Chung Reference Chung2007):

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{s}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{ij}}{\unicode[STIX]{x2202}x_{j}}+\unicode[STIX]{x1D70C}_{s}f_{i},\end{eqnarray}$$

with stress tensor $\unicode[STIX]{x1D70E}_{ij}$ and body force $f_{i}$ . The present study is restricted to displacement fields that slowly vary in space (Lautrup Reference Lautrup2011): $|\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}(\boldsymbol{x},t)/\unicode[STIX]{x2202}x_{j}|\ll 1$ for all $i,j,\boldsymbol{x},t$ . This allows us to ignore density changes and nonlinear deformations. The corresponding strain tensor is Cauchy’s (infinitesimal) strain tensor $\unicode[STIX]{x1D716}_{ij}$ :

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D716}_{ij}=\frac{1}{2}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}}{\unicode[STIX]{x2202}x_{j}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{j}}{\unicode[STIX]{x2202}x_{i}}\right).\end{eqnarray}$$

For linear time-translation-invariant homogeneous isotropic media, the constitutive stress–strain relation can be written in integral form as (Robertsson, Blanch & Symes Reference Robertsson, Blanch and Symes1994; Carcione Reference Carcione2015):

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{ij}=\dot{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6EC}}}\ast \unicode[STIX]{x1D716}_{kk}\unicode[STIX]{x1D6FF}_{ij}+2\dot{\unicode[STIX]{x1D713}_{G}}\ast \unicode[STIX]{x1D716}_{ij},\end{eqnarray}$$

where the dot denotes a time derivative and the asterisk symbolizes convolution:

(2.6) $$\begin{eqnarray}f(t)\ast g(t)\equiv \int _{-\infty }^{\infty }\!f(\unicode[STIX]{x1D70F})g(t-\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}.\end{eqnarray}$$

The constitutive equation contains two relaxation functions, namely $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6EC}}(t)$ for dilatation and $\unicode[STIX]{x1D713}_{G}(t)$ for shear. The convolution expresses that the stress depends on the strain history (assuming causality).

When the boundary conditions and body forces of a viscoelastic problem are steady state harmonic functions of time, as was assumed in § 2.2, all field variables will have the same time dependence (Christensen Reference Christensen1982). Consider a harmonic time dependence of the form $\text{e}^{\text{i}\unicode[STIX]{x1D714}_{m}t}$ (cf. (2.2)), then the convolution with an arbitrary function $f(t)$ can be simplified: $f(t)\ast \text{e}^{\text{i}\unicode[STIX]{x1D714}_{m}t}=F(\unicode[STIX]{x1D714}_{m})\text{e}^{\text{i}\unicode[STIX]{x1D714}_{m}t}$ with $F(\unicode[STIX]{x1D714})={\mathcal{F}}\{f(t)\}=\int _{-\infty }^{+\infty }\!f(t)\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}\,\text{d}t$ the Fourier transform of $f(t)$ . Using this property, the viscoelastic stress–strain relation for mode $m$ becomes:

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{ij}=\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D714}_{m})\unicode[STIX]{x1D716}_{kk}\unicode[STIX]{x1D6FF}_{ij}+2G(\unicode[STIX]{x1D714}_{m})\unicode[STIX]{x1D716}_{ij},\end{eqnarray}$$

where $\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D714})={\mathcal{F}}\{\dot{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6EC}}}(t)\}$ and $G(\unicode[STIX]{x1D714})={\mathcal{F}}\{\dot{\unicode[STIX]{x1D713}_{G}}(t)\}$ are the complex dilatational and shear moduli (Tschoegl, Knauss & Emri Reference Tschoegl, Knauss and Emri2002; Carcione Reference Carcione2015). In the absence of body forces, the equations of motion (2.3) for mode $m$ then become the following viscoelastic-wave equations:

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{s}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D709}_{i}}{\unicode[STIX]{x2202}t^{2}}=(\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714}_{m})-G(\unicode[STIX]{x1D714}_{m}))\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{k}}{\unicode[STIX]{x2202}x_{k}}\right)+G(\unicode[STIX]{x1D714}_{m})\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D709}_{i}}{\unicode[STIX]{x2202}x_{j}^{2}},\end{eqnarray}$$

with compressional-wave modulus $\unicode[STIX]{x1D6F1}=\unicode[STIX]{x1D6EC}+2G$ . These equations are the same as for linear elasticity, except for the use of complex, frequency-dependent moduli: $G(\unicode[STIX]{x1D714})=|G(\unicode[STIX]{x1D714})|\text{e}^{\text{i}\unicode[STIX]{x1D719}_{G}(\unicode[STIX]{x1D714})}$ and $\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714})=|\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714})|\text{e}^{\text{i}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6F1}}(\unicode[STIX]{x1D714})}$ . The shear modulus has a magnitude $|G|$ , a loss angle $\unicode[STIX]{x1D719}_{G}$ and a loss tangent $\unicode[STIX]{x1D702}_{G}=\tan (\unicode[STIX]{x1D719}_{G})$ , and similarly for the compressional-wave modulus.

2.4 Coating model

A coating model is required to prescribe the mechanical coating properties, namely the frequency-dependent modulus magnitude and loss angle for both the shear and the compressional-wave modulus. Unless stated otherwise, the frequency dependence of the moduli is neglected, which is motivated by the observation that the maximum coating response occurs in a limited frequency range (cf. § 5.4, figures 15 and 16).

Accurate determination of the coating moduli requires simultaneous measurements of the coating response in shear and compression. In this way, researchers have measured the relation between the bulk modulus $K=\unicode[STIX]{x1D6F1}-(4/3)G$ and the shear modulus $G$ . Here we use the relations obtained by Pritz (Reference Pritz2009) that were validated for three solid polymeric materials. Specifically, the loss properties of the bulk and the shear modulus can be related through:

(2.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D702}_{K}}{\unicode[STIX]{x1D702}_{G}}=\frac{\tan \unicode[STIX]{x1D719}_{K}}{\tan \unicode[STIX]{x1D719}_{G}}=1-(2\unicode[STIX]{x1D708}_{pr})^{n},\quad n=2.3,\end{eqnarray}$$

where $\unicode[STIX]{x1D702}$ represents the loss tangent, $\unicode[STIX]{x1D719}$ the loss angle and $\unicode[STIX]{x1D708}_{pr}$ the (real part of) Poisson’s ratio. The above relation shows good agreement with experimental data for three different polymeric materials and $0.35\lesssim \unicode[STIX]{x1D708}_{pr}\lesssim 0.5$ . Equation (2.9) quantifies that the loss tangent of the bulk modulus is smaller than that of the shear modulus. The ratio between the magnitudes of the bulk and shear modulus was computed from:

(2.10) $$\begin{eqnarray}\left|\frac{K}{G}\right|=\left|\frac{2+2\unicode[STIX]{x1D708}_{p}}{3(1-2\unicode[STIX]{x1D708}_{p})}\right|,\quad \unicode[STIX]{x1D708}_{p}=\unicode[STIX]{x1D708}_{pr}(1-\text{i}\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D708}_{p}}),\end{eqnarray}$$

with the (complex) Poisson ratio $\unicode[STIX]{x1D708}_{p}$ , its real part $\unicode[STIX]{x1D708}_{pr}$ and its dissipation factor or loss tangent $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D708}_{p}}$ . The latter can be estimated from $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D708}_{p}}/\unicode[STIX]{x1D702}_{G}\approx 1-2\unicode[STIX]{x1D708}_{pr}$ when the medium is close to incompressible ( $\unicode[STIX]{x1D708}_{pr}\approx 0.5$ ) (Pritz Reference Pritz2009). The relation $\unicode[STIX]{x1D6F1}=K+(4/3)G$ together with the definitions $G=\unicode[STIX]{x1D70C}_{s}{c_{s}}^{\,2}$ and $\unicode[STIX]{x1D6F1}=\unicode[STIX]{x1D70C}_{s}{c_{p}}^{\,2}$ can finally be used to compute the ratio of the compressional-wave speed $c_{p}$ and the shear-wave speed $c_{s}$ :

(2.11a-c ) $$\begin{eqnarray}\frac{K}{G}=\left|\frac{K}{G}\right|\frac{\text{e}^{\text{i}\unicode[STIX]{x1D719}_{K}}}{\text{e}^{\text{i}\unicode[STIX]{x1D719}_{G}}},\quad \frac{\unicode[STIX]{x1D6F1}}{G}=\frac{K}{G}+\frac{4}{3},\quad \frac{c_{p}}{c_{s}}=\sqrt{\frac{\unicode[STIX]{x1D6F1}}{G}}.\end{eqnarray}$$

Hence, the (complex) ratio $c_{p}/c_{s}$ can be determined from the shear loss angle $\unicode[STIX]{x1D719}_{G}$ and the (real part of) Poisson’s ratio $\unicode[STIX]{x1D708}_{pr}$ . In this way, the shear and compression properties of the coating are related through a loss angle and a Poisson’s ratio.

2.5 Summary of model and assumptions

This subsection summarizes the model and the underlying assumptions. The implication of the most important assumptions on the results is addressed in § 6. The deformation of a compliant coating in a turbulent flow is computed analytically using the one-way coupling method: turbulent flow stresses deform the compliant coating, but these coating deformations have negligible influence on the turbulent flow, as in the recent study by Zhang et al. (Reference Zhang, Wang, Blake and Katz2017). The turbulent surface stresses are expressed as a sum of streamwise-travelling and spanwise-homogeneous waves (cf. (2.1)). The amplitudes of the waves are obtained from point frequency spectra of turbulent stresses in flow over a rigid wall, as detailed in §§ 4.2 and 5.2. The compliant coating is considered to be a linear time-translation-invariant homogeneous isotropic viscoelastic medium (cf. (2.4), (2.7), (2.8)). It is attached at the bottom to a rigid wall, and it has infinite length in the streamwise direction. The frequency dependence of the viscoelastic moduli is neglected. The relations between the coating’s shear and compression properties are obtained from a model by Pritz (Reference Pritz2009) for solid polymeric materials.

3.1 Analytical solution and dimensionless parameters

This section considers the compliant wall deformation for a single travelling stress wave. Specifically, the stresses at the fluid–coating interface are:

(3.1a,b ) $$\begin{eqnarray}\left.\unicode[STIX]{x1D70E}_{13}\right|_{z=0}\equiv \unicode[STIX]{x1D70E}_{tc}=\unicode[STIX]{x1D70E}_{tc0}\text{e}^{-\text{i}\left(kx-\unicode[STIX]{x1D714}t\right)},\quad \left.\unicode[STIX]{x1D70E}_{33}\right|_{z=0}\equiv \unicode[STIX]{x1D70E}_{nc}=\unicode[STIX]{x1D70E}_{nc0}\text{e}^{-\text{i}\left(kx-\unicode[STIX]{x1D714}t\right)}.\end{eqnarray}$$

This stress wave has wavenumber $k$ , angular frequency $\unicode[STIX]{x1D714}$ , wavelength $\unicode[STIX]{x1D706}=2\unicode[STIX]{x03C0}/k$ , period $T=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}$ and wave speed $c=\unicode[STIX]{x1D714}/k=\unicode[STIX]{x1D706}/T$ . The wave amplitudes can be complex numbers to allow a phase difference between the tangential and the normal stress. Note that the subscript $m$ has been dropped for the remainder of this section. The wall underneath the coating is rigid, so the displacements should vanish at the coating–wall interface:

(3.2a,b ) $$\begin{eqnarray}\left.\unicode[STIX]{x1D709}\,\right|_{z=-\unicode[STIX]{x1D6FF}}=0,\quad \left.\unicode[STIX]{x1D701}\,\right|_{z=-\unicode[STIX]{x1D6FF}}=0.\end{eqnarray}$$

The viscoelastic equations (2.7) and 2.8 were solved using the Helmholtz decomposition (see appendix A, cf. Lamb (Reference Lamb1904), Chase (Reference Chase1991), Kulik (Reference Kulik2012), Zhang et al. (Reference Zhang, Wang, Blake and Katz2017)). The analytical solution was simplified by using $k$ , $\unicode[STIX]{x1D714}$ , $\unicode[STIX]{x1D70C}_{f}$ and derived parameters for non-dimensionalization:

(3.3a-e ) $$\begin{eqnarray}\widetilde{x}_{i}=kx_{i},\quad \widetilde{t}=\unicode[STIX]{x1D714}t,\quad \widetilde{\unicode[STIX]{x1D709}}_{i}=k\unicode[STIX]{x1D709}_{i},\quad \widetilde{u}_{i}=\frac{u_{i}}{c},\quad \widetilde{\unicode[STIX]{x1D70E}}_{ij}=\frac{\unicode[STIX]{x1D70E}_{ij}}{\unicode[STIX]{x1D70C}_{f}c^{2}}.\end{eqnarray}$$

The following dimensionless numbers appear in the analytical solution:

(3.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D70C}_{r}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{f}}{\unicode[STIX]{x1D70C}_{s}}}={\displaystyle \frac{\text{fluid density}}{\text{solid density}}},\\ \unicode[STIX]{x1D714}_{rs}={\displaystyle \frac{\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}}{c_{s}}}={\displaystyle \frac{\text{forcing frequency}}{\text{frequency of shear waves}}},\\ \unicode[STIX]{x1D714}_{rp}={\displaystyle \frac{\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}}{c_{p}}}={\displaystyle \frac{\text{forcing frequency}}{\text{frequency of compressional waves}}},\\ \unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}={\displaystyle \frac{\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D706}}}={\displaystyle \frac{\text{coating thickness}}{\text{wavelength of forcing}}},\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}_{r}$ is the density ratio. There are two frequency-related dimensionless numbers: $\unicode[STIX]{x1D714}_{rs}$ is relative to a typical shear-wave frequency, whereas $\unicode[STIX]{x1D714}_{rp}$ is relative to a typical compressional-wave frequency. The last parameter compares the coating thickness with the wavelength, the latter being the length scale that is introduced by the forcing. Some additional dimensionless numbers can be derived from the ones provided in (3.4):

(3.5a-c ) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FF}}=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}=\frac{|\unicode[STIX]{x1D714}_{rs}|}{|c_{rs}|},\quad c_{rs}=\frac{c}{c_{s}}=\frac{1}{\widetilde{c}_{s}}=\frac{\unicode[STIX]{x1D714}}{kc_{s}}=\frac{\unicode[STIX]{x1D714}_{rs}}{\widetilde{\unicode[STIX]{x1D6FF}}},\quad c_{rp}=\frac{c}{c_{p}}=\frac{1}{\widetilde{c}_{p}}=\frac{\unicode[STIX]{x1D714}}{kc_{p}}=\frac{\unicode[STIX]{x1D714}_{rp}}{\widetilde{\unicode[STIX]{x1D6FF}}}.\end{eqnarray}$$

The analytical solution provided in appendix A is fully determined by the following dimensionless numbers: $\widetilde{\unicode[STIX]{x1D70E}}_{tc0}$ , $\widetilde{\unicode[STIX]{x1D70E}}_{nc0}$ , $\unicode[STIX]{x1D70C}_{r}$ , $\unicode[STIX]{x1D714}_{rs}$ , $\unicode[STIX]{x1D714}_{rp}$ and $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}$ . Note that $\unicode[STIX]{x1D714}_{rs}$ and $\unicode[STIX]{x1D714}_{rp}$ can be calculated as follows:

(3.6a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{rs}=|\unicode[STIX]{x1D714}_{rs}|\text{e}^{\text{i}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D714}_{rs}}},\quad \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D714}_{rs}}=-\frac{\unicode[STIX]{x1D719}_{G}}{2},\quad \frac{\unicode[STIX]{x1D714}_{rp}}{\unicode[STIX]{x1D714}_{rs}}=\frac{c_{s}}{c_{p}}=\sqrt{\frac{G}{\unicode[STIX]{x1D6F1}}}.\end{eqnarray}$$

Using these relations and the coating model from § 2.4, the possibly complex numbers $\unicode[STIX]{x1D714}_{rs}$ and $\unicode[STIX]{x1D714}_{rp}$ can be computed from three real dimensionless numbers, namely $|\unicode[STIX]{x1D714}_{rs}|$ , $\unicode[STIX]{x1D719}_{G}$ and $\unicode[STIX]{x1D708}_{pr}$ . Hence, the analytical solution can also be obtained from these dimensionless numbers: $\widetilde{\unicode[STIX]{x1D70E}}_{tc0}$ , $\widetilde{\unicode[STIX]{x1D70E}}_{nc0}$ , $\unicode[STIX]{x1D70C}_{r}$ , $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}$ , $|\unicode[STIX]{x1D714}_{rs}|$ , $\unicode[STIX]{x1D719}_{G}$ and $\unicode[STIX]{x1D708}_{pr}$ . The purpose of the next subsection is to elucidate how the surface deformation depends on these dimensionless numbers, with a specific focus on the non-trivial dependence of the vertical surface displacement on $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}$ , $|\unicode[STIX]{x1D714}_{rs}|$ , $\unicode[STIX]{x1D719}_{G}$ and $\unicode[STIX]{x1D708}_{pr}$ .

3.2 Dimensionless parameter dependences

Before showing some results, it should be remarked that special care is required for the numerical computation of the interface quantities. The analytical solutions are fractions that contain sines and cosines of possibly large complex arguments. Hence, the numerator and denominator can become exponentially large, which might result in numerically calculated fractions that are completely wrong. For that reason, the functions to compute the interface quantities were equipped with statements to check the accuracy of the calculations. Specifically, the solid stress at the interface was computed from the numerically calculated coefficients that appear in the analytical solution. The computations, most of which were performed with Matlab using double precision, were considered sufficiently accurate when the difference between the thus obtained solid stress and the applied fluid stress was smaller than $10^{-6}$ . Otherwise, the function switched to variable precision arithmetic (vpa) in Matlab or the computations were performed in Maple. The Maple function was equipped with a similar accuracy check; the number of digits was doubled until sufficient precision was obtained.

Figures 2 and 3 show contours of the normal surface displacement as a function of $|\unicode[STIX]{x1D714}_{rs}|$ and $|c_{rs}|$ , which are related through $\widetilde{\unicode[STIX]{x1D6FF}}=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}=|\unicode[STIX]{x1D714}_{rs}|/|c_{rs}|$ (3.5). The two panels in figure 2 differ in the way the displacement is normalized, namely with coating thickness (a) and wavenumber (b). Figure 3 is the same as figure 2, except that the horizontal and vertical axes are linear instead of logarithmic, and that the panels correspond to two loss angles instead of two normalizations. The stress amplitudes were fixed at $\widetilde{\unicode[STIX]{x1D70E}}_{tc0}=0.0041$ and $\widetilde{\unicode[STIX]{x1D70E}}_{nc0}=0.0238$ , which were computed from $\widetilde{\unicode[STIX]{x1D70E}}=\unicode[STIX]{x1D70E}^{+}/(c^{+})^{2}$ with $\unicode[STIX]{x1D70E}^{+}$ from the r.m.s. values of the tangential and normal stress (cf. (4.5)) and $c^{+}=10$ . Note, however, that the magnitude of these stresses is not very relevant at this stage, since the figures should primarily facilitate a qualitative understanding of the dynamics.

The contour lines in figure 2 show a clear change of direction around $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}\approx 0.33$ , which is an important value, also according to other studies (Kulik et al. Reference Kulik, Lee and Chun2008; Kulik Reference Kulik2012; Zhang et al. Reference Zhang, Wang, Blake and Katz2017). Two interesting limits can be distinguished accordingly, namely (i) a long-wave limit and (ii) a short-wave limit (Kulik Reference Kulik2012; Vedeneev Reference Vedeneev2016). The long-wave limit corresponds with $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}\ll 1$ , so the wavelength is much larger than the coating thickness. On the other hand, $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}\gg 1$ indicates the short-wave limit, such that the wavelength is much smaller than the coating thickness. Both limits are described below in more detail.

Figure 2. The vertical surface displacement as a function of the dimensionless frequency $\left|\unicode[STIX]{x1D714}_{rs}\right|$ and the dimensionless convection velocity $\left|c_{rs}\right|$ . The other parameters are $\widetilde{\unicode[STIX]{x1D70E}}_{tc0}=0.0041$ , $\widetilde{\unicode[STIX]{x1D70E}}_{nc0}=0.0238$ , $\unicode[STIX]{x1D70C}_{r}=1$ , $\unicode[STIX]{x1D719}_{G}=10^{\circ }$ , $\unicode[STIX]{x1D708}_{pr}=0.45$ . Both panels are the same, except that the vertical displacement is normalized with the coating thickness (a) and with the wavenumber (b). The four square symbols indicate the dimensionless parameters for which figure 9 shows a displacement vector field.

The solution in the long-wave limit ( $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}\ll 1$ ) is derived in appendix A.2. The main outcome is presented here, namely the surface displacements for a single wave:

(3.7a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D709}_{c}}{\unicode[STIX]{x1D6FF}}=\frac{\tan (\unicode[STIX]{x1D714}_{rs})}{\unicode[STIX]{x1D714}_{rs}}\frac{\unicode[STIX]{x1D70E}_{tc}}{G},\quad \frac{\unicode[STIX]{x1D701}_{c}}{\unicode[STIX]{x1D6FF}}=\frac{\tan (\unicode[STIX]{x1D714}_{rp})}{\unicode[STIX]{x1D714}_{rp}}\frac{\unicode[STIX]{x1D70E}_{nc}}{\unicode[STIX]{x1D6F1}}.\end{eqnarray}$$

The displacements scale with $\unicode[STIX]{x1D6FF}$ : the coating thickness is the characteristic length scale in the long-wave limit, somewhat similar to the shallow-water limit for water waves. In the low-frequency limit (both $|\unicode[STIX]{x1D714}_{rs}|\ll 1$ and $|\unicode[STIX]{x1D714}_{rp}|\ll 1$ ), the displacements become independent of the frequency $\unicode[STIX]{x1D714}$ :

(3.8a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D709}_{c}}{\unicode[STIX]{x1D6FF}}=\frac{\unicode[STIX]{x1D70E}_{tc}}{G}=\unicode[STIX]{x1D70C}_{r}\,c_{rs}^{\,2}\,\widetilde{\unicode[STIX]{x1D70E}}_{tc},\quad \frac{\unicode[STIX]{x1D701}_{c}}{\unicode[STIX]{x1D6FF}}=\frac{\unicode[STIX]{x1D70E}_{nc}}{\unicode[STIX]{x1D6F1}}=\unicode[STIX]{x1D70C}_{r}\,c_{rp}^{\,2}\,\widetilde{\unicode[STIX]{x1D70E}}_{nc}.\end{eqnarray}$$

The normalized displacement then only depends on $c_{rs}^{\,2}$ (since also $c_{rp}^{\,2}\propto c_{rs}^{\,2}$ ), which corresponds to the horizontal contour lines in the top left corner of figure 2(a).

Apparent from figure 3(a) are lines for which the coating strongly responds to the external travelling-wave forcing. These lines, which we denote as resonances, correspond quite well with the dispersion curves that were obtained by Kulik et al. (Reference Kulik, Lee and Chun2008) for a similar viscoelastic solid. In agreement with their work and with Benschop & Breugem (Reference Benschop and Breugem2017), the resonances in the long-wave limit only occur when $|\unicode[STIX]{x1D714}_{rs}|\gtrsim 1$ . More specifically, for an elastic solid ( $\unicode[STIX]{x1D719}_{G}=0^{\circ }$ ) they occur when $\unicode[STIX]{x1D714}_{rs}$ and $\unicode[STIX]{x1D714}_{rp}$ are odd multiples of $\unicode[STIX]{x03C0}/2$ (cf. (3.7)), as indicated with the symbols on the top axes of figure 3. The resonances indeed coincide with these symbols when $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}\ll 1$ , which is equivalent to $|c_{rs}|\gg |\unicode[STIX]{x1D714}_{rs}|$ . The resonance at $\unicode[STIX]{x1D714}_{rp}=\unicode[STIX]{x03C0}/2$ becomes dominant in that limit, as the vertical displacement then solely depends on $\unicode[STIX]{x1D714}_{rp}$ (not clearly visible in figure 3 due to the limited range of the vertical axis). The strong resonances for an elastic solid are less pronounced or even absent for a viscoelastic solid (cf. figure 3 left versus right), which agrees with the findings from previous studies (Kulik Reference Kulik2012; Benschop & Breugem Reference Benschop and Breugem2017).

Figure 3. The vertical surface displacement as a function of the dimensionless frequency $\left|\unicode[STIX]{x1D714}_{rs}\right|$ and the dimensionless convection velocity $\left|c_{rs}\right|$ for two loss angles, namely $\unicode[STIX]{x1D719}_{G}=0^{\circ }$ (a) and $\unicode[STIX]{x1D719}_{G}=10^{\circ }$ (b). The other parameters are $\widetilde{\unicode[STIX]{x1D70E}}_{tc0}=0.0041$ , $\widetilde{\unicode[STIX]{x1D70E}}_{nc0}=0.0238$ , $\unicode[STIX]{x1D70C}_{r}=1$ , $\unicode[STIX]{x1D708}_{pr}=0.45$ . Panel (b) is the same as figure 2(a), except that the axes are linear instead of logarithmic. As in figure 2, the four square symbols in panel (b) indicate the dimensionless parameters for which figure 9 shows a displacement vector field.

The solution in the short-wave limit ( $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}\gg 1$ ) is derived in appendix A.3. The main outcome is presented here, namely the surface displacements for a single wave:

(3.9a ) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D709}}_{c} & = & \displaystyle \unicode[STIX]{x1D70C}_{r}\,c_{rs}^{\,2}\frac{\left(-2\sqrt{1-c_{rp}^{\,2}}\sqrt{1-c_{rs}^{\,2}}-\left(c_{rs}^{\,2}-2\right)\right)\text{i}\,\widetilde{\unicode[STIX]{x1D70E}}_{nc}+c_{rs}^{\,2}\sqrt{1-c_{rs}^{\,2}}\,\widetilde{\unicode[STIX]{x1D70E}}_{tc}}{d_{sw}},\end{eqnarray}$$

(3.9b ) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D701}}_{c} & = & \displaystyle \unicode[STIX]{x1D70C}_{r}\,c_{rs}^{\,2}\frac{c_{rs}^{\,2}\sqrt{1-c_{rp}^{\,2}}\,\widetilde{\unicode[STIX]{x1D70E}}_{nc}+\left(2\sqrt{1-c_{rp}^{\,2}}\sqrt{1-c_{rs}^{\,2}}+\left(c_{rs}^{\,2}-2\right)\right)\text{i}\,\widetilde{\unicode[STIX]{x1D70E}}_{tc}}{d_{sw}},\end{eqnarray}$$

(3.9c ) $$\begin{eqnarray}\displaystyle d_{sw} & = & \displaystyle 4\,\sqrt{1-c_{rp}^{\,2}}\sqrt{1-c_{rs}^{\,2}}-\left(c_{rs}^{\,2}-2\right)^{2}.\end{eqnarray}$$

$1/k\propto \unicode[STIX]{x1D706}$

$\unicode[STIX]{x1D6FF}$

$|\unicode[STIX]{x1D714}_{rs}|\propto \unicode[STIX]{x1D6FF}$

$c_{rs}=\unicode[STIX]{x1D714}/kc_{s}$

$\unicode[STIX]{x1D714}_{rs}=\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}/c_{s}$

$c_{rs}$

$\unicode[STIX]{x1D714}_{rs}$

$\unicode[STIX]{x1D6FF}$

$1/k$

The above expressions simplify in the limit that $c_{rs}\rightarrow 0$ at fixed $c_{s}/c_{p}=c_{rp}/c_{rs}$ :

(3.10a ) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D709}}_{c} & = & \displaystyle \frac{1}{2}\unicode[STIX]{x1D70C}_{r}\,c_{rs}^{\,2}\frac{\widetilde{\unicode[STIX]{x1D70E}}_{tc}+\text{i}\,\widetilde{\unicode[STIX]{x1D70E}}_{nc}\,(c_{s}/c_{p})^{2}}{1-(c_{s}/c_{p})^{2}},\end{eqnarray}$$

(3.10b ) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D701}}_{c} & = & \displaystyle \frac{1}{2}\unicode[STIX]{x1D70C}_{r}\,c_{rs}^{\,2}\frac{-\text{i}\,\widetilde{\unicode[STIX]{x1D70E}}_{tc}\,(c_{s}/c_{p})^{2}+\widetilde{\unicode[STIX]{x1D70E}}_{nc}}{1-(c_{s}/c_{p})^{2}}.\end{eqnarray}$$

$c_{rs}^{\,2}$

$c_{s}/c_{p}\rightarrow 0$

$\widetilde{\unicode[STIX]{x1D709}}_{c}=\unicode[STIX]{x1D70E}_{tc}/2G$

$\widetilde{\unicode[STIX]{x1D701}}_{c}=\unicode[STIX]{x1D70E}_{nc}/2G$

Resonances are also possible in the short-wave limit and they are especially strong for an elastic solid, cf. figure 3. The elastic resonances occur when the denominator of the expressions equals zero ( $d_{sw}=0$ , cf. (3.9)), which yields an expression that can be solved for $c_{rs}$ for a given wave-speed ratio $c_{p}/c_{s}=c_{rs}/c_{rp}$ . For the elastic solid with $\unicode[STIX]{x1D708}_{pr}=0.45$ (as in figure 3), that ratio equals $c_{p}/c_{s}=\sqrt{\unicode[STIX]{x1D6F1}/G}=\sqrt{2(1-\unicode[STIX]{x1D708}_{p})/(1-2\unicode[STIX]{x1D708}_{p})}=3.3$ and the equation $d_{sw}=0$ is solved by $c_{rs}=c/c_{s}=0.95$ . Figure 3(a) shows indeed a large coating response for $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}\gg 1$ and $c_{rs}=0.95$ . The corresponding waves are denoted as Rayleigh waves (Rayleigh (Reference Rayleigh1885), cf. the dispersion curves of Gad-el Hak et al. (Reference Gad-el Hak, Blackwelder and Riley1984)), which are surface waves that are well known in the field of seismology.

Figure 4. The vertical surface displacement as a function of the dimensionless coating thickness $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}$ for three loss angles $\unicode[STIX]{x1D719}_{G}$ (a) and three Poisson ratios $\unicode[STIX]{x1D708}_{pr}$ (b). The other parameters are $\widetilde{\unicode[STIX]{x1D70E}}_{tc0}=0.0041$ , $\widetilde{\unicode[STIX]{x1D70E}}_{nc0}=0.0238$ , $\unicode[STIX]{x1D719}_{G}=10^{\circ }$ (b), $\unicode[STIX]{x1D708}_{pr}=0.45$ (a). The graphs are independent of $\unicode[STIX]{x1D70C}_{r}$ due to the normalization of the vertical displacement. The curves are shown for four values of $|c_{rs}|$ (0.01, 0.05, 0.1, 0.2), but the difference is not visible because of the normalization.

The results for the long- and short-wave limits can be collapsed on a single curve when $|c_{rs}|\lesssim 0.2$ , see figure 4. The surface displacements are proportional to $\unicode[STIX]{x1D70C}_{r}\,c_{rs}^{\,2}$ in both limits, so similarity is observed when this factor is used for normalization. The normalized surface displacements only depend on $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}$ , $\unicode[STIX]{x1D719}_{G}$ and $\unicode[STIX]{x1D708}_{pr}$ when the stresses $\widetilde{\unicode[STIX]{x1D70E}}_{tc0}$ and $\widetilde{\unicode[STIX]{x1D70E}}_{nc0}$ are fixed. The dependence on $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}$ clearly reveals the long- and short-wave limits: the displacement is proportional to the coating thickness in the long-wave limit, whereas it is proportional to the wavelength in the short-wave limit. The peak response occurs at $\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D706}\approx 0.33$ , in agreement with other studies (Kulik et al. Reference Kulik, Lee and Chun2008; Kulik Reference Kulik2012; Zhang et al. Reference Zhang, Wang, Blake and Katz2017).

Although the loss angle has a pronounced influence on the displacement when resonances are present (figure 3), it has a much smaller influence when resonances are absent (figure 4 a). The vertical displacement is slightly affected by $\unicode[STIX]{x1D719}_{G}$ in the long-wave limit, which results from the fact that $|\unicode[STIX]{x1D6F1}|$ changes with $\unicode[STIX]{x1D719}_{G}$ for the coating model that is used (see § 2.4).

The influence of the (real part of) Poisson’s ratio $\unicode[STIX]{x1D708}_{pr}$ is depicted in figure 4 (b). Note that $\unicode[STIX]{x1D708}_{pr}$ was varied over a small range (from 0.4 to 0.5), as some of the expressions used for the coating model are only valid when the material is close to incompressible (see § 2.4). Poisson’s ratio has a significant influence on the vertical displacement in the long-wave limit: when $\unicode[STIX]{x1D708}_{pr}\rightarrow 0.5$ , the compressional-wave modulus $|\unicode[STIX]{x1D6F1}|\rightarrow \infty$ , such that the vertical displacement approaches zero (cf. (3.8)).

4.1 Viscous units

Since this study considers turbulent flow, viscous units are used for normalization:

(4.1a-f ) $$\begin{eqnarray}\unicode[STIX]{x1D70E}^{+}=\frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}_{f}u_{\unicode[STIX]{x1D70F}}^{2}},\quad k^{+}=\frac{k\unicode[STIX]{x1D708}}{u_{\unicode[STIX]{x1D70F}}},\quad c^{+}=\frac{c}{u_{\unicode[STIX]{x1D70F}}},\quad \unicode[STIX]{x1D714}^{+}=\frac{\unicode[STIX]{x1D714}\unicode[STIX]{x1D708}}{u_{\unicode[STIX]{x1D70F}}^{2}}=k^{+}c^{+},\quad \unicode[STIX]{x1D709}^{+}=\frac{\unicode[STIX]{x1D709}u_{\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D708}},\quad u^{+}=\frac{u}{u_{\unicode[STIX]{x1D70F}}}.\end{eqnarray}$$

Note that the viscous units are based on the flow over a rigid wall; see also the definition of $u_{\unicode[STIX]{x1D70F}}$ in § 2.1. It was shown in § 3.1 that the deformation by a single stress wave is fully specified with these dimensionless numbers: $\widetilde{\unicode[STIX]{x1D70E}}_{tc0}$ , $\widetilde{\unicode[STIX]{x1D70E}}_{nc0}$ , $\unicode[STIX]{x1D70C}_{r}$ , $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}$ , $|\unicode[STIX]{x1D714}_{rs}|$ , $\unicode[STIX]{x1D719}_{G}$ and $\unicode[STIX]{x1D708}_{pr}$ . The following relations demonstrate the conversion between $\,\widetilde{\,\,}\,$ and $^{+}$ normalization, as well as the computation of $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}$ and $|\unicode[STIX]{x1D714}_{rs}|$ from viscous quantities:

(4.2a-e ) $$\begin{eqnarray}\unicode[STIX]{x1D70E}^{+}=\left(c^{+}\right)^{2}\widetilde{\unicode[STIX]{x1D70E}},\quad \unicode[STIX]{x1D709}^{+}=\frac{\widetilde{\unicode[STIX]{x1D709}}}{k^{+}},\quad u^{+}=c^{+}\widetilde{u},\quad \unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}=\frac{\unicode[STIX]{x1D6FF}^{+}}{\unicode[STIX]{x1D706}^{+}},\quad |\unicode[STIX]{x1D714}_{rs}|=\frac{\unicode[STIX]{x1D714}^{+}\unicode[STIX]{x1D6FF}^{+}}{|{c_{s}}^{+}|}.\end{eqnarray}$$

When viscous units are used, the solution for a single stress wave is fully specified with nine dimensionless numbers. Four dimensionless variables are required to specify the fluid-stress properties, namely ${\unicode[STIX]{x1D70E}_{tc0}}^{+}$ , ${\unicode[STIX]{x1D70E}_{nc0}}^{+}$ , $c^{+}$ and $\unicode[STIX]{x1D714}^{+}$ , with the latter two related through $k^{+}=\unicode[STIX]{x1D714}^{+}/c^{+}$ . Note that these four variables are not independent, since the stresses ${\unicode[STIX]{x1D70E}_{tc0}}^{+}$ , ${\unicode[STIX]{x1D70E}_{nc0}}^{+}$ and the convection velocity $c^{+}$ depend on the frequency $\unicode[STIX]{x1D714}^{+}$ , as explained in § 4.2. Five dimensionless variables are required to specify five corresponding coating properties, namely $\unicode[STIX]{x1D70C}_{r}$ (density), $\unicode[STIX]{x1D6FF}^{+}$ (thickness), $|{c_{s}}^{+}|$ (stiffness), $\unicode[STIX]{x1D719}_{G}$ (viscoelasticity) and $\unicode[STIX]{x1D708}_{pr}$ (compressibility). Later on, the coating softness $|{c_{s}}^{+}|^{-1}$ will sometimes be used instead of the coating stiffness $|{c_{s}}^{+}|$ .

4.2 Turbulent surface stresses

4.2.1 Stress spectra and the assumption of spanwise homogeneity

This study employs the one-way coupling method, such that the turbulent flow stresses are not influenced by the coating deformation. Hence, it can be assumed that the coating is simply driven by the same stresses that the turbulent flow exerts on a rigid wall. Consider therefore a three-dimensional and time-dependent turbulent flow that is statistically homogeneous in the streamwise and spanwise directions, and statistically steady in time. Let $\unicode[STIX]{x1D70E}(x,y,t)$ denote a wall stress, where $x$ represents the streamwise, $y$ the spanwise and $t$ the temporal coordinate. The fluctuations of $\unicode[STIX]{x1D70E}$ can be quantified with a wavevector–frequency spectrum $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70E}}(k_{x},k_{y},\unicode[STIX]{x1D714})$ , with streamwise wavenumber $k_{x}$ , spanwise wavenumber $k_{y}$ and angular frequency $\unicode[STIX]{x1D714}$ (Hwang, Bonness & Hambric Reference Hwang, Bonness and Hambric2009):

(4.3a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70E}}(k_{x},k_{y},\unicode[STIX]{x1D714})=E_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D714})(c/\unicode[STIX]{x1D714})^{2}f_{\unicode[STIX]{x1D70E}}(\widetilde{k}_{x},\widetilde{k}_{y}),\quad \widetilde{k}_{x}=\frac{k_{x}c}{\unicode[STIX]{x1D714}},\quad \widetilde{k}_{y}=\frac{k_{y}c}{\unicode[STIX]{x1D714}},\end{eqnarray}$$

with one-sided point frequency spectrum $E_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D714})$ , convection velocity $c$ , normalized dimensionless wavevector spectrum $f_{\unicode[STIX]{x1D70E}}(\widetilde{k}_{x},\widetilde{k}_{y})$ and dimensionless wavenumbers $\widetilde{k}_{x}$ and $\widetilde{k}_{y}$ . Integration of the spectrum yields the mean square stress, which is equivalent to the square of the root-mean-square (r.m.s.) stress:

(4.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}_{rms}^{2} & = & \displaystyle \int _{0}^{\infty }\!\int _{-\infty }^{\infty }\!\int _{-\infty }^{\infty }\!\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70E}}(k_{x},k_{y},\unicode[STIX]{x1D714})\,\text{d}k_{x}\,\text{d}k_{y}\,\text{d}\unicode[STIX]{x1D714}\nonumber\\ \displaystyle & = & \displaystyle \underbrace{\int _{-\infty }^{\infty }\!\int _{-\infty }^{\infty }\!f_{\unicode[STIX]{x1D70E}}(\widetilde{k}_{x},\widetilde{k}_{y})\,\text{d}\widetilde{k}_{x}\,\text{d}\widetilde{k}_{y}}_{=\,1}\int _{0}^{\infty }\!E_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D714})\,\text{d}\unicode[STIX]{x1D714}=\int _{0}^{\infty }\!E_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D714})\,\text{d}\unicode[STIX]{x1D714}.\end{eqnarray}$$

Modelling of the turbulent surface stresses requires knowledge of the three-dimensional wavenumber–frequency spectra of the streamwise, spanwise and normal stresses at the wall. Insufficient knowledge of these three-dimensional spectra for the streamwise and spanwise shear stress was a first important reason for the assumption of spanwise homogeneity. Section 4.2.2 shows that the spectra for turbulent channel flow were obtained from Hu, Morfey & Sandham (Reference Hu, Morfey and Sandham2006), since that is – to our knowledge – one of the few references that also presents the shear-stress spectra, although only as function of frequency without reference to the streamwise and/or spanwise wavenumber.

A second reason for the assumption of spanwise homogeneity is the observation that spanwise coherent modes ( $k_{y}=0$ ) are most energetic. Indeed, the wavevector spectrum $f_{\unicode[STIX]{x1D70E}}(\widetilde{k}_{x},\widetilde{k}_{y})$ for wall pressure typically peaks at $\widetilde{k}_{x}=1$ and $\widetilde{k}_{y}=0$ (Hwang et al. Reference Hwang, Bonness and Hambric2009), which corresponds to $k_{x}=\unicode[STIX]{x1D714}/c$ and $k_{y}=0$ . The same is true for turbulent boundary-layer flow over a compliant coating, as is confirmed in appendix B with the wavevector spectrum of the vertical surface displacement that was obtained from the measurements described in § 5.1. A similar result was also reported for a direct numerical simulation (DNS) of turbulent channel flow over a compliant wall: the spanwise wavenumber spectra of wall pressure, wall displacement and wall velocity peak at the smallest spanwise wavenumber (Kim & Choi Reference Kim and Choi2014). In summary, the assumption of spanwise homogeneity was mainly motivated by insufficient knowledge of the three-dimensional shear-stress spectra and the predominant contribution of the spanwise-homogeneous mode to the stress and displacement r.m.s.

4.2.2 Stress amplitudes from stress spectra

Figure 5. Point spectra of the turbulent surface stresses (a) and the convection velocity (b) as functions of the angular frequency in viscous units. The spectra of the streamwise wall shear ( $\unicode[STIX]{x1D70E}_{tc}$ ) and wall pressure ( $\unicode[STIX]{x1D70E}_{nc}$ ) were obtained from a direct numerical simulation (DNS) by Hu et al. (Reference Hu, Morfey and Sandham2006) of a plane channel flow at $Re_{\unicode[STIX]{x1D70F}}=720$ . The convection velocity for two different Reynolds numbers was calculated from a model that Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009) derived from DNS data.

While § 4.2.1 motivated the specific choice for one wavevector ( $k_{x}=\unicode[STIX]{x1D714}/c$ , $k_{y}=0$ ), the frequency-dependent response still needs to be incorporated with use of frequency spectra. Figure 5(a) therefore shows the point spectra of the streamwise wall shear stress and the wall pressure as functions of the angular frequency at $Re_{\unicode[STIX]{x1D70F}}=720$ . The data were obtained from direct numerical simulations by Hu et al. (Reference Hu, Morfey and Sandham2006) of turbulent flow in a plane channel with rigid walls. Their data were interpolated to an equispaced set of $N_{m}=995$ frequencies ranging from $\unicode[STIX]{x1D714}^{+}=3.4\times 10^{-3}$ to $\unicode[STIX]{x1D714}^{+}=3.4$ with $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}=3.4\times 10^{-3}$ ; each symbol in the figure corresponds with one mode. The root-mean-square (r.m.s.) values of the stresses can be obtained by integration of the frequency spectra (cf. (4.4)), or analogously by summation of the discrete spectra (cf. (C 6a )):

(4.5a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{tc,rms}^{+}=\sqrt{\mathop{\sum }_{m=1}^{N_{m}}E_{\unicode[STIX]{x1D70E}_{tc},m}^{+}\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}}=0.41,\quad \unicode[STIX]{x1D70E}_{nc,rms}^{+}=\sqrt{\mathop{\sum }_{m=1}^{N_{m}}E_{\unicode[STIX]{x1D70E}_{nc},m}^{+}\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}}=2.38,\end{eqnarray}$$

where $E_{\unicode[STIX]{x1D70E}_{tc}}^{+}$ and $E_{\unicode[STIX]{x1D70E}_{nc}}^{+}$ are the point spectra for tangential and normal stress at the interface, respectively.

These spectra can be used to prescribe the amplitudes of the stress modes. Remember that each travelling-wave mode has a complex amplitude $\unicode[STIX]{x1D748}_{\boldsymbol{c},\boldsymbol{m}}^{^{+}}$ with two components:

(4.6a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{tc,m}^{+}=|\unicode[STIX]{x1D70E}_{tc,m}^{+}|\text{e}^{\text{i}\unicode[STIX]{x1D719}_{tc,m}},\quad \unicode[STIX]{x1D70E}_{nc,m}^{+}=|\unicode[STIX]{x1D70E}_{nc,m}^{+}|\text{e}^{\text{i}\unicode[STIX]{x1D719}_{nc,m}}.\end{eqnarray}$$

Unless stated otherwise, the phases $\unicode[STIX]{x1D719}_{tc,m}$ and $\unicode[STIX]{x1D719}_{nc,m}$ are assumed to be zero, since their influence on the coating deformation is small (cf. § 4.4.6). The stress amplitudes can be obtained from the stress spectra as explained in appendix C (cf. (C 6a )):

(4.7a,b ) $$\begin{eqnarray}|\unicode[STIX]{x1D70E}_{tc,m}^{+}|=\sqrt{2E_{\unicode[STIX]{x1D70E}_{tc},m}^{+}\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}},\quad |\unicode[STIX]{x1D70E}_{nc,m}^{+}|=\sqrt{2E_{\unicode[STIX]{x1D70E}_{nc},m}^{+}\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}}.\end{eqnarray}$$

4.3 Coating deformation computation

The following overview summarizes how the surface displacements were computed; the same procedure also applies to the surface velocities:

(4.8)

For a single mode $m$ with frequency $\unicode[STIX]{x1D714}_{m}^{+}$ , the stress amplitudes were obtained from the stress spectra (§ 4.2.2), the convection velocity from a semi-empirical model (§ 4.2.3) and the surface displacements and velocities from the analytical solution (§ 3.1). This procedure was followed for all $N_{m}$ modes, after which the r.m.s.-values and the point spectra were computed using (C 6a ):

(4.9a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D709}_{c,rms}^{+}=\sqrt{\mathop{\sum }_{m=1}^{N_{m}}\frac{1}{2}|\unicode[STIX]{x1D709}_{c,m}^{+}|^{2}},\quad E_{\unicode[STIX]{x1D709}_{c},m}^{+}=\frac{\frac{1}{2}|\unicode[STIX]{x1D709}_{c,m}^{+}|^{2}}{\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D714}^{+}},\end{eqnarray}$$

and similarly for $\unicode[STIX]{x1D701}_{c}$ , $u_{c}$ and $w_{c}$ .

Before considering the influence of the coating properties in detail, we derive a simplified equation for the r.m.s.-values of the surface displacements in the long-wave, low-frequency limit. Equation (3.8) reads in viscous units as:

(4.10a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D709}_{c}^{+}=\frac{\unicode[STIX]{x1D6FF}^{+}}{G^{+}}\unicode[STIX]{x1D70E}_{tc}^{+}=\frac{\unicode[STIX]{x1D70C}_{r}\,\unicode[STIX]{x1D6FF}^{+}}{({c_{s}}^{+})^{2}}\unicode[STIX]{x1D70E}_{tc}^{+},\quad \unicode[STIX]{x1D701}_{c}^{+}=\frac{\unicode[STIX]{x1D6FF}^{+}}{\unicode[STIX]{x1D6F1}^{+}}\unicode[STIX]{x1D70E}_{nc}^{+}=\frac{\unicode[STIX]{x1D70C}_{r}\,\unicode[STIX]{x1D6FF}^{+}}{({c_{p}}^{+})^{2}}\unicode[STIX]{x1D70E}_{nc}^{+}.\end{eqnarray}$$

Assuming that these relations hold for every mode $m$ , the r.m.s. then follows from (4.9), e.g.

(4.11) $$\begin{eqnarray}\left(\unicode[STIX]{x1D709}_{c,rms}^{+}\right)^{2}=\mathop{\sum }_{m=1}^{N_{m}}\frac{1}{2}|\unicode[STIX]{x1D709}_{c,m}^{+}|^{2}=\left(\frac{\unicode[STIX]{x1D6FF}^{+}}{|G^{+}|}\right)^{2}\mathop{\sum }_{m=1}^{N_{m}}\frac{1}{2}|\unicode[STIX]{x1D70E}_{tc,m}^{+}|^{2}=\left(\frac{\unicode[STIX]{x1D6FF}^{+}}{|G^{+}|}\,\unicode[STIX]{x1D70E}_{tc,rms}^{+}\right)^{2}.\end{eqnarray}$$

Hence, the r.m.s.-values for $\unicode[STIX]{x1D709}_{c}$ and $\unicode[STIX]{x1D701}_{c}$ can be easily derived from the r.m.s.-values of $\unicode[STIX]{x1D70E}_{tc}$ and $\unicode[STIX]{x1D70E}_{nc}$ :

(4.12a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D709}_{c,rms}^{+}=\frac{\unicode[STIX]{x1D70E}_{tc,rms}^{+}}{|G^{+}|}\,\unicode[STIX]{x1D6FF}^{+}=\frac{\unicode[STIX]{x1D70C}_{r}\unicode[STIX]{x1D6FF}^{+}}{|{c_{s}}^{+}|^{2}}\,\unicode[STIX]{x1D70E}_{tc,rms}^{+},\quad \unicode[STIX]{x1D701}_{c,rms}^{+}=\frac{\unicode[STIX]{x1D70E}_{nc,rms}^{+}}{|\unicode[STIX]{x1D6F1}^{+}|}\,\unicode[STIX]{x1D6FF}^{+}=\frac{\unicode[STIX]{x1D70C}_{r}\unicode[STIX]{x1D6FF}^{+}}{|{c_{p}}^{+}|^{2}}\,\unicode[STIX]{x1D70E}_{nc,rms}^{+}.\end{eqnarray}$$

The r.m.s.-values in the short-wave limit can be derived in a similar way from (3.9) and (4.9), but the resulting expressions are not so concise and therefore not reported here.

Figure 6. Contour plots of the root-mean-square displacement (a,b) and velocity (c,d) at the fluid–solid interface as functions of coating thickness ( $\unicode[STIX]{x1D6FF}^{+}$ ) and coating softness ( $|{c_{s}}^{+}|^{-1}$ ). The other parameters are $Re_{\unicode[STIX]{x1D70F}}=720$ , $\unicode[STIX]{x1D70C}_{r}=1$ , $\unicode[STIX]{x1D719}_{G}=10^{\circ }$ , $\unicode[STIX]{x1D708}_{pr}=0.45$ . The dashed lines are contour lines at integer powers of 10; the bold dashed line corresponds to $10^{0}$ . The white transparent area indicates the ‘instability region’, while the black transparent area corresponds to the ‘interaction region’. The four dots indicate the parameter values for which subsequent figures show the point spectra (figure 8), a displacement vector field (figure 9), the dependence on $\unicode[STIX]{x1D719}_{G}$ (figure 10), the dependence on $\unicode[STIX]{x1D708}_{pr}$ (figure 11) and the dependence on the phase angle of the stress waves (figure 12).

4.4 Influence of coating properties

4.4.2 Coating softness and thickness

The influences of coating softness and thickness are considered simultaneously, because they are related through the parameter $\unicode[STIX]{x1D714}_{rs}$ . Figure 6 shows contour plots for $\unicode[STIX]{x1D709}_{c,rms}^{+}$ , $\unicode[STIX]{x1D701}_{c,rms}^{+}$ , $u_{c,rms}^{+}$ and $w_{c,rms}^{+}$ as functions of coating thickness $\unicode[STIX]{x1D6FF}^{+}$ and softness $|{c_{s}}^{+}|^{-1}$ . Note that all the axes are logarithmic, so the dependence on a wide range of parameters is displayed. Typically, the coating deformation increases with increasing thickness and softness.

There are three regions in the contour plots for which the computed coating deformation might not be very accurate. The first region is designated as the ‘instability region’: the fluid–structure interaction can lead to instabilities (such as travelling-wave flutter or static-divergence waves) when $U_{\infty }/|c_{s}|$ exceeds a critical value, with $U_{\infty }$ the free-stream velocity in turbulent boundary-layer flow. According to experiments summarized in Gad-el Hak (Reference Gad-el Hak2002), that critical value might depend on the coating thickness: when the coating became thicker, the critical value decreased from approximately 12 to 4 for static-divergence waves and from 4 to 1 for travelling-wave flutter. A theoretical analysis by Duncan (Reference Duncan1986) for $\unicode[STIX]{x1D70C}_{f}/\unicode[STIX]{x1D70C}_{s}=1$ yielded a critical value of $U_{\infty }/|c_{s}|$ that decreased from approximately 3 for thin coatings and/or high damping to 1 for thick coatings and/or low damping. Based on these numbers, the instability region can be identified with $U_{\infty }/|c_{s}|\gtrsim 2$ . Note that the number 2 is nothing more than an estimate for the stability boundary, since the precise value depends on coating thickness and damping in a way yet to be determined. Duncan et al. (Reference Duncan, Waxman and Tulin1985) showed that the criterion for the onset of instability is actually given by $U_{\infty }/|c_{s}|=\sqrt{K\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x1D70C}_{f}}$ for a certain constant $K$ , which is equivalent to $\unicode[STIX]{x1D70C}_{f}U_{\infty }^{2}/\unicode[STIX]{x1D70C}_{s}|c_{s}|^{2}=K$ . This allows the interpretation of the instability criterion as a critical ratio of a characteristic fluid stress $\unicode[STIX]{x1D70C}_{f}U_{\infty }^{2}$ and a characteristic coating modulus $\unicode[STIX]{x1D70C}_{s}|c_{s}|^{2}=|G|$ . Hence, it might be more appropriate to include the density ratio in the criterion: the instability region then becomes $\sqrt{\unicode[STIX]{x1D70C}_{r}}\,U_{\infty }/|c_{s}|\gtrsim 2$ . Assuming that these results also apply to the bulk velocity $U_{b}$ in channel flow, then instabilities are expected to arise when $\sqrt{\unicode[STIX]{x1D70C}_{r}}\,U_{b}/|c_{s}|\approx 2$ . For figure 6 this amounts to $u_{\unicode[STIX]{x1D70F}}/|c_{s}|=|{c_{s}}^{+}|^{-1}\approx 0.11$ , since $\unicode[STIX]{x1D70C}_{r}=1$ and $Re_{\unicode[STIX]{x1D70F}}=720$ corresponds to $u_{\unicode[STIX]{x1D70F}}/U_{b}=0.053$ . The resulting instability region is marked with a white transparent area in figure 6.

The computed coating deformation is also not very reliable in a second region that is denoted as the ‘interaction region’: the significant coating deformation will influence the flow, which implies that the one-way coupling approach (as employed in this study) cannot be used reliably anymore. This region is differentiated by the requirement that at least one of the quantities $\{\unicode[STIX]{x1D709}_{c,rms}^{+},\unicode[STIX]{x1D701}_{c,rms}^{+},u_{c,rms}^{+},w_{c,rms}^{+}\}\gtrsim 1$ , as indicated with a black transparent area in figure 6. Note that this interaction region corresponds quite well to the area where resonances can be expected in the coating response. Figure 3 shows that resonances only occur when $|\unicode[STIX]{x1D714}_{rs}|=\unicode[STIX]{x1D714}^{+}\unicode[STIX]{x1D6FF}^{+}/|{c_{s}}^{+}|\gtrsim 1$ and $|c_{rs}|=(c/U_{b})\times (U_{b}/|c_{s}|)\gtrsim 1$ . The stress spectra of figure 5 indicate that the dominant frequencies are in the range $\unicode[STIX]{x1D714}^{+}\lesssim 1$ , while the maximum convection velocity is typically smaller than the bulk velocity ( $c/U_{b}\lesssim 1$ ), such that the resonances will appear for sufficiently thick and soft coatings with $\unicode[STIX]{x1D6FF}^{+}/|{c_{s}}^{+}|\gtrsim 1$ and $U_{b}/|c_{s}|\gtrsim 1$ . That part of the contour plots indeed shows anomalous behaviour, which is for instance apparent from the changed spacing and the wiggling of the contour lines.

Finally, the model results cannot be trusted in a third region which is called the ‘nonlinear region’, since it is characterized by nonlinear material behaviour that is not well represented by the linear coating model used in this study. This region, which is quantified with the criterion $\unicode[STIX]{x1D701}_{c,rms}^{+}/\unicode[STIX]{x1D6FF}^{+}=\unicode[STIX]{x1D701}_{c,rms}/\unicode[STIX]{x1D6FF}\gtrsim 0.01$ , is not sketched in figure 6 because it completely overlaps with the instability region and the interaction region.

The coating softness has a pronounced influence on the deformation outside these three unreliable regions: figure 6 shows that the displacements and velocities are approximately proportional to $1/|{c_{s}}^{+}|^{2}\propto 1/|G|$ , the inverse of the shear modulus. This is in line with the observation that the displacement for relatively stiff coatings scales with $\unicode[STIX]{x1D70C}_{r}|c_{rs}|^{2}$ (cf. (3.8), (3.10) and figure 4). Using that the convection velocity is proportional to the bulk velocity ( $c\propto U_{b}$ ), this dimensionless number can also be interpreted as the ratio of a fluid stress and a coating modulus:

(4.13) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{r}|c_{rs}|^{2}=\frac{\unicode[STIX]{x1D70C}_{f}}{\unicode[STIX]{x1D70C}_{s}}\frac{c^{2}}{|c_{s}|^{2}}\propto \frac{\unicode[STIX]{x1D70C}_{f}}{\unicode[STIX]{x1D70C}_{s}}\frac{U_{b}^{2}}{|c_{s}|^{2}}=\frac{\unicode[STIX]{x1D70C}_{f}U_{b}^{2}}{|G|}.\end{eqnarray}$$

It thus follows that the surface displacements and velocities are linearly proportional to $\unicode[STIX]{x1D70C}_{f}U_{b}^{2}/|G|$ , in agreement with Rosti & Brandt (Reference Rosti and Brandt2017).

The influence of the coating thickness reveals the long-wave and short-wave behaviour discussed in § 3.2. This is more clearly demonstrated in figure 7(a), which shows the r.m.s. surface displacements as a function of coating thickness $\unicode[STIX]{x1D6FF}^{+}$ for a given coating softness $|c_{s}^{+}|^{-1}$ , together with the results in the long-wave, low-frequency and short-wave limits. For very thin coatings, the displacements increase proportional to $\unicode[STIX]{x1D6FF}^{+}$ (cf. (4.12)), which is characteristic of the long-wave response. For very thick coatings, the displacements become independent of the coating thickness $\unicode[STIX]{x1D6FF}^{+}$ , which identifies the short-wave limit. Section 5.3 derives criteria to check whether the coating response is in the long-wave limit, the short-wave limit or in between.

Figure 7. Root-mean-square surface displacements and velocities as functions of coating thickness ( $\unicode[STIX]{x1D6FF}^{+}$ ) for $Re_{\unicode[STIX]{x1D70F}}=720$ , $\unicode[STIX]{x1D70C}_{r}=1$ , $|{c_{s}}^{+}|^{-1}=0.02$ , $\unicode[STIX]{x1D719}_{G}=10^{\circ }$ , $\unicode[STIX]{x1D708}_{pr}=0.45$ . (a) Surface displacements, including the results in the long-wave, low-frequency (lw) and short-wave (sw) limits. (b) Surface displacements and velocities.

The relative importance of the horizontal and vertical displacements and velocities can be clarified with figure 7(b). The trends in this figure are typical of almost any value of $|{c_{s}}^{+}|^{-1}$ . In particular, the horizontal and vertical displacements generally have a comparable magnitude. In the long-wave limit, the horizontal displacements can be larger than the vertical displacements, especially for materials that are close to incompressible. In the short-wave limit, the vertical displacement is larger ( $\unicode[STIX]{x1D701}_{c,rms}^{+}=0.20$ ${>}$ $\unicode[STIX]{x1D709}_{c,rms}^{+}=0.097$ ), although the horizontal displacement still has a comparable magnitude. Similarly, the horizontal and vertical velocities are always comparable in magnitude, although the vertical component is clearly larger than the horizontal component in the short-wave limit ( $u_{c,rms}^{+}=1.2\times 10^{-3}$ versus $w_{c,rms}^{+}=5.6\times 10^{-3}$ ).

In comparing the displacements and velocities (figure 7(b)), two typical features can be noticed. First, the short-wave limit starts at a lower thickness for the velocities as compared to the displacements. Since the relations between the interface velocity and displacement for a certain mode $m$ are given by $u_{c,m}^{+}=\text{i}\unicode[STIX]{x1D714}_{m}^{+}\unicode[STIX]{x1D709}_{c,m}^{+}$ and $w_{c,m}^{+}=\text{i}\unicode[STIX]{x1D714}_{m}^{+}\unicode[STIX]{x1D701}_{c,m}^{+}$ , the higher frequencies are more prominent for the velocity, while the high frequencies are also the first to enter the short-wave regime. As a second observation, the interface velocity is smaller than the interface displacement when both are normalized in viscous units, which is typical for most parameters, especially in the short-wave limit. In that limit, the largest response comes from the longest wave with the lowest frequency ( $\unicode[STIX]{x1D714}^{+}\ll 1$ ), such that the relations $u_{c,m}^{+}=\text{i}\unicode[STIX]{x1D714}_{m}^{+}\unicode[STIX]{x1D709}_{c,m}^{+}$ and $w_{c,m}^{+}=\text{i}\unicode[STIX]{x1D714}_{m}^{+}\unicode[STIX]{x1D701}_{c,m}^{+}$ explain why $u_{c,rms}^{+}\ll \unicode[STIX]{x1D709}_{c,rms}^{+}$ and $w_{c,rms}^{+}\ll \unicode[STIX]{x1D701}_{c,rms}^{+}$ .

Figure 8. Point spectra of the surface displacements (a,c,e,g, logarithmic axes) and the surface velocities (b,d,f,h, linear axes) as functions of the angular frequency in viscous units for four parameter sets (see title inside panels, corresponding to the four dots in figure 6). The other parameters are $Re_{\unicode[STIX]{x1D70F}}=720$ , $\unicode[STIX]{x1D70C}_{r}=1$ , $\unicode[STIX]{x1D719}_{G}=10^{\circ }$ , $\unicode[STIX]{x1D708}_{pr}=0.45$ . The square symbols on the top axes of the panels indicate the frequencies for which figure 9 shows a displacement vector field. The long-wave (lw) and short-wave (sw) results are included in the top and bottom panels, respectively.

The type of coating response can be clarified with point spectra of the interface displacements and velocities, see figure 8. The displacement spectra (left, logarithmic axes) and velocity spectra (right, linear axes) are displayed for four different coatings, corresponding to the four dots in the contour plots of figure 6. For increasing coating thickness, the associated values of $\unicode[STIX]{x1D701}_{c,rms}^{+}$ are $5.5\times 10^{-4}$ , $6.9\times 10^{-2}$ , $0.99$ and $4.1\times 10^{-4}$ .

The first coating is very thin, which yields a characteristic long-wave response. The displacement spectra are the same as the stress spectra (figure 5), except for a mode-independent factor. The r.m.s. displacement is very well predicted by (4.12), which yields $\unicode[STIX]{x1D701}_{c,rms}^{+}=5.5\times 10^{-4}$ for $|c_{p}/c_{s}|=3.3$ . The velocity spectra are the displacement spectra multiplied with $(\unicode[STIX]{x1D714}^{+})^{2}$ , so the higher frequencies become more important.

The second coating has the same softness as the first one, but it is thicker by a factor 80. A resonance starts to appear at the higher frequencies, close to $\unicode[STIX]{x1D714}^{+}=2$ , $|\unicode[STIX]{x1D714}_{rs}|=3.5$ , $|c_{rs}|=1.1$ and $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}=0.51$ . Note that Kim & Choi (Reference Kim and Choi2014) also report the appearance of a resonance in the frequency spectra of wall displacement and wall velocity. The resonance makes the long-wave expressions for the surface displacements inaccurate: equation (4.12) returns $\unicode[STIX]{x1D701}_{c,rms}^{+}=4.4\times 10^{-2}$ , whereas the actual value is $\unicode[STIX]{x1D701}_{c,rms}^{+}=6.9\times 10^{-2}$ . The third coating has again the same softness, but it is approximately 6 times thicker than the second coating. The resonance now appears at lower frequencies, around $\unicode[STIX]{x1D714}^{+}=0.3$ , which corresponds with $|\unicode[STIX]{x1D714}_{rs}|=3.3$ , $|c_{rs}|=1.1$ , $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}=0.48$ . The long-wave prediction of $\unicode[STIX]{x1D701}_{c,rms}^{+}=0.28$ is again inaccurate compared to $\unicode[STIX]{x1D701}_{c,rms}^{+}=0.99$ .

The fourth coating is very thick and very stiff, which yields a response that is characteristic for the short-wave limit. The displacements are proportional to the wavelength, so the largest wavelengths (corresponding to the lowest frequencies) dominate the spectra. The spectra follow the short-wave predictions very well, except for a small difference at the first frequency, corresponding to $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}=0.35$ , which is just on the border of the short-wave region (cf. figure 2).

Figure 9 shows displacement vector fields for the four coatings just considered. The vector fields belong to the parameters that are indicated with a square in figures 2, 3(b) and 8. The selected frequencies belong to modes which make a large contribution to $w_{c,rms}^{+}$ (see figure 8). The first vector field corresponds to a low frequency and a long wavelength, as can be judged from the values of $|\unicode[STIX]{x1D714}_{rs}|$ and $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}$ in the figure title. The horizontal and vertical displacements are decoupled, although the ratio between both displacements stays the same due to the assumption of zero phase difference between the tangential and normal stresses.

The vector fields for the second and third coating belong to modes close to resonance, which is not only evident from figure 8, but also from figure 3(b). The vector fields resemble vortices that are separated by half a wavelength. The parameters for the second coating are such that the horizontal surface displacement is close to zero, whereas the vertical displacement is significant.

The fourth vector field is characteristic for the short-wave response: the coating is much thicker than the wavelength, namely a factor 14 for this specific case. Note that only the top part of the coating is shown, i.e. $1.5\unicode[STIX]{x1D706}$ of the $14\unicode[STIX]{x1D706}$ in total. The deformation wave, which is only present near the surface, has a characteristic penetration depth of the order of one wavelength.

4.4.3 Viscoelasticity

Figure 10 displays how the surface displacement depends on the viscoelastic properties of the coating, the loss angle $\unicode[STIX]{x1D719}_{G}$ in particular. The four panels correspond to the four coatings that have been introduced above. The streamwise displacement of the first coating is not affected by $\unicode[STIX]{x1D719}_{G}$ , since $\unicode[STIX]{x1D709}_{c,rms}^{+}$ in the long-wave limit only depends on the modulus magnitude $|G|$ , cf. (4.12). The vertical displacement is slightly affected by $\unicode[STIX]{x1D719}_{G}$ , as has been explained in the context of figure 4.

Figure 9. Displacement vector fields for four different sets of angular frequency ( $\unicode[STIX]{x1D714}^{+}$ ), coating thickness ( $\unicode[STIX]{x1D6FF}^{+}$ ) and coating softness ( $|{c_{s}}^{+}|^{-1}$ ) as given in the titles. The other parameters are $Re_{\unicode[STIX]{x1D70F}}=720$ , $\unicode[STIX]{x1D70C}_{r}=1$ , $\unicode[STIX]{x1D719}_{G}=10^{\circ }$ , $\unicode[STIX]{x1D708}_{pr}=0.45$ . The coating thickness and softness correspond to the dots in the contour plots of figure 6. The frequencies are indicated with black squares in figure 8, whereas black squares in figures 2 and 3 provide the corresponding dimensionless parameters. The vertical axis is stretched (a) or compressed (b,c,d) for clarity and compactness. Panels (a–c) show the full coating thickness, whereas panel (d) contains only the top part of the coating (namely 1.5 $\unicode[STIX]{x1D706}$ , while the coating is 14 $\unicode[STIX]{x1D706}$ thick).

Figure 10. Horizontal and vertical surface displacement as functions of the loss angle $\unicode[STIX]{x1D719}_{G}$ for four parameter sets (see title, corresponding to the four dots in figure 6). The other parameters are $Re_{\unicode[STIX]{x1D70F}}=720$ , $\unicode[STIX]{x1D70C}_{r}=1$ , $\unicode[STIX]{x1D708}_{pr}=0.45$ . The long-wave (lw) and short-wave (sw) results are included in panels (a) and (d), respectively.

The deformation of the second and third coatings is quite sensitive to the loss angle when it is close to zero, which is due to the presence of resonances in the coating response (cf. figure 8). The displacements are very large for $\unicode[STIX]{x1D719}_{G}=0^{\circ }$ , which is attributed to the very strong resonances for an elastic coating (cf. figure 3(a); Kulik (Reference Kulik2012), Benschop & Breugem (Reference Benschop and Breugem2017)). The resonances are less pronounced for a viscoelastic solid (cf. figure 3), such that the surface displacements do not vary much for $\unicode[STIX]{x1D719}_{G}\gtrsim 10^{\circ }$ .

The fourth coating is close to the short-wave limit, and the influence of the loss angle is small, in agreement with figure 4. Figure 10(d) shows that the short-wave predictions are slightly different from the actual values for the reason that the first frequency (which dominates the response) is just on the border of the short-wave region, as has been remarked in the context of figure 8. Overall, the surface displacements are relatively insensitive to changes in the loss angle as long as resonances in the coating response are absent.

4.4.4 Compressibility

Figure 11 displays how the surface displacement depends on the compressibility of the coating, the real part of Poisson’s ratio $\unicode[STIX]{x1D708}_{pr}$ in particular. The four panels correspond again to the four coatings that have been introduced above. The figures also show the deformations that result in the absence of shear stress, as will be discussed in the next subsection. The streamwise displacement of the first coating is not affected by the Poisson’s ratio, which is a consequence of the fact that $\unicode[STIX]{x1D709}_{c,rms}^{+}$ in the long-wave limit only depends on the shear modulus $G$ , independent of $\unicode[STIX]{x1D708}_{pr}$ (cf. (4.12)). In contrast, the Poisson’s ratio has a significant influence on the vertical displacement: when $\unicode[STIX]{x1D708}_{pr}\rightarrow 0.5$ , the compressional-wave modulus $|\unicode[STIX]{x1D6F1}|\rightarrow \infty$ , such that the vertical displacement approaches zero (cf. (4.12), figure 4). The influence of the Poisson’s ratio is small for the other coatings: there is a slight change of $\unicode[STIX]{x1D709}_{c,rms}^{+}$ , and $\unicode[STIX]{x1D701}_{c,rms}^{+}$ typically decreases with increasing $\unicode[STIX]{x1D708}_{pr}$ , while it does not approach zero.

Figure 11. Horizontal and vertical surface displacement as functions of the real part of Poisson’s ratio $\unicode[STIX]{x1D708}_{pr}$ for four parameter sets (see title, corresponding to the four dots in figure 6). The other parameters are $Re_{\unicode[STIX]{x1D70F}}=720$ , $\unicode[STIX]{x1D70C}_{r}=1$ , $\unicode[STIX]{x1D719}_{G}=10^{\circ }$ .

4.4.5 Influence of shear stress

To determine whether the shear stress has an important contribution to the surface deformation, figure 11 also shows the displacements that result in the absence of shear stress ( $\unicode[STIX]{x1D70E}_{tc}=0$ ). The influence of shear on the vertical displacement is marginal for all cases: the difference between the curves with and without shear stress is barely visible. The vertical displacement is thus predominantly determined by the normal stress, in agreement with similar observations by other researchers (Kulik Reference Kulik2012; Zhang et al. Reference Zhang, Wang, Blake and Katz2017).

The influence of the shear stress on the horizontal displacement is, however, more pronounced: $\unicode[STIX]{x1D709}_{c,rms}^{+}$ decreases in absence of shear for all cases. The reduction is most substantial in the long-wave limit, i.e. for very thin and stiff coatings: the horizontal deformation is then solely driven by the shear stress (cf. (4.12)), so $\unicode[STIX]{x1D709}_{c,rms}^{+}=0$ when shear is absent. For the other three cases shown in figure 11, the removal of shear results in a decrease of the horizontal displacement by a factor of approximately 1.7, 1.4 and 4.8.

Figure 12. Influence of the phase angle of the stress waves on the root-mean-square value of the surface displacements. The displacements were calculated for 1000 sets of random angles for all the 995 shear and pressure waves. The results are shown as histograms with 20 bins for four parameter sets (see title, corresponding to the four dots in figure 6). $P$ represents the probability to find a displacement within a certain bin; the sum of all the bar heights equals 1. The markers on the bottom axes correspond to stresses for which all phases are zero ( $\unicode[STIX]{x1D719}_{tc,m}=0=\unicode[STIX]{x1D719}_{nc,m}$ for all $m$ ), as is assumed throughout the paper. The other parameters are $Re_{\unicode[STIX]{x1D70F}}=720$ , $\unicode[STIX]{x1D70C}_{r}=1$ , $\unicode[STIX]{x1D719}_{G}=10^{\circ }$ , $\unicode[STIX]{x1D708}_{pr}=0.45$ .

5.1 Experiments

The deformation of a compliant coating on a rigid wall underneath a turbulent boundary-layer flow was studied experimentally. Preliminary results were presented by Delfos et al. (Reference Delfos, Greidanus, Charruault and Westerweel2017) and Greidanus et al. (Reference Greidanus, Delfos and Westerweel2017), whose work was continued to allow a comparison with the model proposed in this paper. Three coatings were produced in-house, applied to a rigid plate and tested in a water tunnel. Three quantities were measured, namely the flow velocity with planar particle image velocimetry (PIV, Adrian & Westerweel (Reference Adrian and Westerweel2011)), the drag force on the plate with a force balance and the vertical coating displacement with high-speed background-oriented schlieren (BOS, Raffel Reference Raffel2015). Below, we provide a short description of the water tunnel, the flow, the drag force, the three coatings and the deformation measurements.

The water tunnel has an optically fully accessible test section that has a length of 2 m and an inlet with a cross-sectional area of $300\times 300~\text{mm}^{2}$ . The top wall of the test section can be replaced to mount the test plates with a surface area of $1998\times 297~\text{mm}^{2}$ . The PIV and BOS measurements were performed at 1.7 m downstream of the test section’s entrance.

The flow properties are listed in table 1, e.g. the free-stream velocity $U_{\infty }$ was varied from about 0.9 to $5.4~\text{m}~\text{s}^{-1}$ . The velocity profiles, as measured by PIV, approximately satisfy $u/U_{\infty }=(z/\unicode[STIX]{x1D6FF}_{bl})^{1/8}$ , with streamwise velocity $u$ , the vertical distance to the surface $z$ , and the boundary-layer thickness $\unicode[STIX]{x1D6FF}_{bl}$ (note the difference with the coating thickness $\unicode[STIX]{x1D6FF}$ ). The momentum thickness $\unicode[STIX]{x1D703}$ was computed by integration of the velocity profiles, and the boundary-layer thickness then followed from the relation $\unicode[STIX]{x1D6FF}_{bl}=(45/4)\unicode[STIX]{x1D703}$ for velocity profiles with a power $1/8$ . The thus obtained boundary-layer thickness follows quite well the power law $\unicode[STIX]{x1D6FF}_{bl}=0.057U_{\infty }^{-1/7}$ , which was henceforth used to compute the boundary-layer thickness for a given free-stream velocity. From now on, the superscript $^{o}$ is used to denote quantities that are normalized with outer units ( $U_{\infty }$ and $\unicode[STIX]{x1D6FF}_{bl}$ ):

(5.1a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D714}^{o}=\frac{\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}_{bl}}{U_{\infty }},\quad k^{o}=k\unicode[STIX]{x1D6FF}_{bl},\quad c^{o}=\frac{c}{U_{\infty }}.\end{eqnarray}$$

Drag measurements were performed both on smooth rigid plates and on coated plates. The force balance measured the total drag on the whole test plate, from which the plate-averaged shear stress $\langle \unicode[STIX]{x1D70F}_{w}\rangle _{}$ was obtained. The local shear stress $\unicode[STIX]{x1D70F}_{w}$ (at the location of the PIV and BOS measurements) was estimated from $\unicode[STIX]{x1D70F}_{w}=(6/7)\langle \unicode[STIX]{x1D70F}_{w}\rangle _{}$ (Greidanus et al. Reference Greidanus, Delfos and Westerweel2017), and the local wall-friction velocity then followed from $u_{\unicode[STIX]{x1D70F}}=\sqrt{\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C}_{f}}$ . A fit to the thus obtained shear velocity yields the power law $u_{\unicode[STIX]{x1D70F}}=0.041U_{\infty }^{0.91}$ for the smooth rigid wall. Like in the analytical model, viscous scaling is based on the shear velocity of the smooth-wall flow. Next, three different Reynolds numbers can be defined:

(5.2a-c ) $$\begin{eqnarray}Re_{\unicode[STIX]{x1D70F}}=\frac{\unicode[STIX]{x1D6FF}_{bl}u_{\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D708}},\quad Re_{\unicode[STIX]{x1D6FF}}=\frac{\unicode[STIX]{x1D6FF}_{bl}U_{\infty }}{\unicode[STIX]{x1D708}},\quad R_{T}=\frac{\unicode[STIX]{x1D6FF}_{bl}/U_{\infty }}{\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}^{2}}=\frac{Re_{\unicode[STIX]{x1D70F}}^{2}}{Re_{\unicode[STIX]{x1D6FF}}},\end{eqnarray}$$

where $R_{T}$ represents the ratio of the outer layer to inner layer time scale that appears later in the analytical model. Note that table 1 provides the range of the numerical values for the free-stream velocity, boundary-layer thickness, shear velocity and three Reynolds numbers.

Table 1. Properties of the turbulent boundary-layer flow in the experiments.

The properties of the three coatings are given in table 2: all coatings have the same thickness of 5 mm and approximately the same mass density. In contrast, the moduli of the coatings are significantly different, with coating 1 being the softest and coating 3 the stiffest. The frequency-dependent shear modulus ( $G=G_{1}+\text{i}G_{2}$ ) of the three coatings was measured using a commercial rheometer (ARES-G2, TA Instruments) with a parallel plate geometry of 25 mm in diameter. Specifically, the storage modulus $G_{1}$ and the loss modulus $G_{2}$ were measured in a frequency range of $\unicode[STIX]{x1D714}=0.1~\text{rad}~\text{s}^{-1}$ to $\unicode[STIX]{x1D714}=100~\text{rad}~\text{s}^{-1}$ at $20\,^{\circ }\text{C}$ and 0.5 % strain. The measurements were limited to $100~\text{rad}~\text{s}^{-1}$ , which is the maximum frequency that the rheometer could reach. In theory, modulus values at higher frequencies can be obtained with use of the time–temperature superposition (TTS) principle. In practice, however, measurements below room temperature resulted in vapour condensation on the measurement facility and the samples, such that the obtained values were considered unreliable. The table lists the values at $100~\text{rad}~\text{s}^{-1}$ , as that turned out to be the frequency closest to the most dominant frequency in the coating response. There is one exception: the loss properties at $100~\text{rad}~\text{s}^{-1}$ for coating 1 are $G_{2}=0.20$ kPa and $\unicode[STIX]{x1D719}_{G}=8.1^{\circ }$ , while the table provides somewhat larger values, as is motivated in the context of the displacement spectra in § 5.4. While the shear modulus of the three coatings is of the order of kPa, the bulk modulus is of the order of GPa, such that the coatings are practically incompressible: $\unicode[STIX]{x1D708}_{pr}\approx 0.5$ .

Small height variations of the optically transparent coating were measured with the background-oriented schlieren (BOS) method of Moisy, Rabaud & Salsac (Reference Moisy, Rabaud and Salsac2009), which they call a synthetic schlieren method. A high-speed camera was used for time-resolved measurements of the optical distortion of a random dot pattern, placed behind the coating, due to refraction of light at the deforming coating interface. The displacement field of the dots was determined from digital image correlation (DIC, Adrian & Westerweel Reference Adrian and Westerweel2011) between the deformed and the undeformed dot pattern. The dot displacements were related to the interface slope, and the vertical interface displacement was subsequently obtained from spatial integration. This method was applied at approximately 180 $\times$ 180 points within a field of view of size $100\times 100~\text{mm}^{2}$ . The measurement signals contain 2000 time samples with a measurement frequency $f$ , which was fixed at $f=1200$ Hz for coating 1. It was increased in linear proportion to the free-stream velocity for coatings 2 and 3, namely from $f=200$  Hz at $U_{\infty }=0.87~\text{m}~\text{s}^{-1}$ to $f=1240$  Hz at $U_{\infty }=5.39~\text{m}~\text{s}^{-1}$ .

Table 2. Properties of the coatings employed in the experiments, together with the dimensionless numbers that the analytical model requires to predict the coating deformation.

The long-wave coating deformation could not be reliably measured. Specifically, long waves result in displacement fields that are almost spatially uniform, whereas plate vibrations also result in such uniform fields. As independent measurements of the plate position were not performed, the distinction between long-wave coating deformations and plate vibrations could not be made. Therefore, coating deformation waves with a wavelength larger than the length of the field of view (i.e. $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{max}=100$  mm) could not be quantified reliably.

Since long waves could not be accurately measured, the same is true for low frequencies. Specifically, the minimum frequency that can be measured relates to the maximum wavelength according to $\unicode[STIX]{x1D714}_{min}=2\unicode[STIX]{x03C0}c/\unicode[STIX]{x1D706}_{max}$ or $\unicode[STIX]{x1D714}_{min}^{o}=2\unicode[STIX]{x03C0}c^{o}/\unicode[STIX]{x1D706}_{max}^{\circ }$ . Given that $\unicode[STIX]{x1D706}_{max}^{\circ }=\unicode[STIX]{x1D706}_{max}/\unicode[STIX]{x1D6FF}_{bl}$ increases from 1.7 to 2.2, and assuming that $c^{o}=c/U_{\infty }\approx 0.75$ (Delfos et al. Reference Delfos, Greidanus, Charruault and Westerweel2017), it follows that $\unicode[STIX]{x1D714}_{min}^{o}$ varies between 2.8 and 2.1. The low-frequency response ( $\unicode[STIX]{x1D714}\lesssim \unicode[STIX]{x1D714}_{min}$ ) is most likely attributable to non-advected features that could result from resonances or reflected waves associated with the finite length of the compliant wall (Zhang et al. Reference Zhang, Wang, Blake and Katz2017). Indeed, the response below $\unicode[STIX]{x1D714}_{min}^{o}$ partially results from pronounced tunnel vibrations at $f=5$ Hz (independent of flow velocity), which corresponds with a dimensionless frequency $\unicode[STIX]{x1D714}^{o}$ that decreases from 2.1 to 0.26 when the flow velocity increases.

The focus of the present study is on advected phenomena, as in the study by Zhang et al. (Reference Zhang, Wang, Blake and Katz2017). For that reason, the measured surface displacement was filtered to exclude frequencies below $\unicode[STIX]{x1D714}_{min}^{o}$ . First, the displacement point spectra were computed using fast Fourier transform (FFT): the time signal was written as $\unicode[STIX]{x1D701}(t)=\sum _{m}\unicode[STIX]{x1D701}_{m}\exp (\text{i}\unicode[STIX]{x1D714}_{m}t)$ , the amplitudes $\unicode[STIX]{x1D701}_{m}$ were obtained from the FFT and the point spectra followed from (C 6a ). Next, frequencies below $\unicode[STIX]{x1D714}_{min}^{o}$ were removed from the spectra. To smooth the quite spiky spectra, a filtered time signal was reconstructed from an inverse FFT, this time signal was cut in 40 pieces of 50 time samples each and a spectrum was computed for each piece using the FFT. Finally, the spectra were averaged over all pieces and all points within the field of view. For all coatings and flow velocities, the r.m.s. of the filtered data varied between 83 and 98 % of the r.m.s. for the unfiltered data.

5.2 Analytical model

The analytical model requires the turbulent stress spectra, the convection velocity and the frequency-dependent coating properties as input, as described below in more detail. Several models for the turbulent pressure spectra exist, as reviewed by Hwang et al. (Reference Hwang, Bonness and Hambric2009). According to their review, the model by Goody (Reference Goody2004) provides the best overall prediction of the spectra. It is given by:

(5.3) $$\begin{eqnarray}E_{\unicode[STIX]{x1D70E}_{nc}^{+}}^{^{o}}=\frac{E_{\unicode[STIX]{x1D70E}_{nc}}(\unicode[STIX]{x1D714})U_{\infty }}{\unicode[STIX]{x1D70F}_{w}^{2}\unicode[STIX]{x1D6FF}_{bl}}=\frac{C_{2}(\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}_{bl}/U_{\infty })^{2}}{\left[(\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}_{bl}/U_{\infty })^{0.75}+C_{1}\right]^{3.7}+\left[C_{3}(\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}_{bl}/U_{\infty })\right]^{7}},\end{eqnarray}$$

with model parameters $C_{1}=0.5$ , $C_{2}=3.0$ and $C_{3}=1.1R_{T}^{-0.57}$ . The spectrum is defined as $E_{\unicode[STIX]{x1D70E}_{nc}^{+}}^{^{o}}=E_{\unicode[STIX]{x1D70E}_{nc}^{+}}U_{\infty }/\unicode[STIX]{x1D6FF}_{bl}$ , which is the spectrum $E_{\unicode[STIX]{x1D70E}_{nc}^{+}}$ normalized in outer units, where $E_{\unicode[STIX]{x1D70E}_{nc}^{+}}=E_{\unicode[STIX]{x1D70E}_{nc}}/\unicode[STIX]{x1D70F}_{w}^{2}$ represents the spectrum of $\unicode[STIX]{x1D70E}_{nc}^{+}$ . Note that $E_{\unicode[STIX]{x1D70E}_{nc}^{+}}^{^{o}}$ depends on $\unicode[STIX]{x1D714}$ in the ratio $\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}_{bl}/U_{\infty }=\unicode[STIX]{x1D714}^{o}$ , which is the frequency normalized with a characteristic frequency of the outer layer. The ratio of $C_{1}$ to $C_{3}$ determines the size of the overlap range, which depends on the ratio $R_{T}$ of the outer layer to inner layer time scale (cf. (5.2)).

Figure 13(a) displays the Goody spectrum at four different Reynolds numbers that are typical for the experiments. The spectra collapse for low frequencies because outer scaling is used, while inner scaling will collapse the spectra at high frequencies. The extent of the overlap range increases with increasing Reynolds number. Hwang et al. (Reference Hwang, Bonness and Hambric2009) provide more information about the use of different scales to collapse the spectra in different frequency regions. The Goody spectrum at $Re_{\unicode[STIX]{x1D70F}}=720$ is very similar to the channel-flow spectrum of Hu et al. (Reference Hu, Morfey and Sandham2006) (figure 5), except that the latter does not include the very low frequencies for which the Goody spectrum exhibits a decay when $\unicode[STIX]{x1D714}\rightarrow 0$ .

Figure 13. Pressure spectrum (a) and convection velocity (b) as functions of frequency for four Reynolds numbers, using outer scaling. The pressure spectrum was obtained from the analytical relation provided by Goody (Reference Goody2004) for turbulent boundary-layer flow (5.3). The convection velocity was calculated from a model that Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009) derived from DNS data of turbulent channel flow.

An important characteristic of the spectrum is its integral, which provides a measure of the root-mean-square (r.m.s.) of the fluctuations. Goody (Reference Goody2002) obtained a relation (their equation 8) that should describe the variation of the mean square pressure with $R_{T}$ for the Goody spectrum. His relation, however, does not yield the numerical values that were reported in table 3 from his paper, which explains why we propose a slightly different relation. The Goody spectrum was integrated numerically for a range of $R_{T}$ between $10$ and $10^{3}$ , using (C 6b ) for logarithmically distributed frequencies in the range $\unicode[STIX]{x1D714}^{o}=10^{-4}$ to $\unicode[STIX]{x1D714}^{o}=10^{4}$ . The resulting r.m.s.-values were fitted to the following curve:

(5.4) $$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D70E}_{nc,rms}^{+})^{2}=0.0309+0.745(\ln (R_{T}))^{2},\end{eqnarray}$$

which is the same relation as in Goody (Reference Goody2002), except for the different coefficients. Given that $R_{T}$ in the experiments varied between 83 and 300 (table 1), the corresponding pressure r.m.s. $\unicode[STIX]{x1D70E}_{nc,rms}^{+}$ ranges from 3.8 to 4.9.

Next to the pressure spectrum, the analytical model also requires a shear-stress spectrum. To the authors’ knowledge, an equivalent of the Goody spectrum for shear-stress fluctuations does not exist. However, the contribution of the shear stress to the vertical displacement is presumably marginal (cf. § 4.4.5). To check this assumption, two computations were performed for all three coatings, namely one without shear stress ( $E_{\unicode[STIX]{x1D70E}_{tc}}=0$ ) and another with $E_{\unicode[STIX]{x1D70E}_{tc}}=0.28E_{\unicode[STIX]{x1D70E}_{nc}}$ . The factor 0.28 was chosen because that is the maximum of the ratio $E_{\unicode[STIX]{x1D70E}_{tc}}/E_{\unicode[STIX]{x1D70E}_{nc}}$ for the channel flow spectra shown in figure 5. The results (shown later) indicate that the contribution of shear to the vertical surface displacement is indeed marginal.

Another important ingredient of the analytical model is the convection velocity, which was estimated from the semi-empirical model proposed by Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009) (cf. § 4.2.3). That model prescribes how the convection velocity in a turbulent channel flow depends on the wavenumber and the Reynolds number. The model parameters were derived mostly from DNS at $Re_{\unicode[STIX]{x1D70F}}\approx 550$ and $Re_{\unicode[STIX]{x1D70F}}\approx 950$ . To the authors’ knowledge, a comparable model for turbulent boundary-layer flow does not exist, which is why we assumed that the model by Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009) can also be used to estimate the convection velocity for turbulent boundary-layer flow in the range of $Re_{\unicode[STIX]{x1D70F}}=2100$ –8500. Figure 13(b) displays $c/U_{\infty }$ for four different Reynolds numbers. The convection velocity scales with $U_{\infty }$ at the lowest frequencies (except for $Re_{\unicode[STIX]{x1D70F}}=2000$ ), with $U_{\infty }$ the characteristic velocity for the outer layer. In contrast, the highest frequencies scale in viscous units, with a characteristic velocity of $c^{+}=10$ (cf. figure 5), such that $c/U_{\infty }$ decreases from 0.40 to 0.35 when the Reynolds number increases.

The model also requires the frequency-dependent coating properties, namely the shear and compressional-wave modulus. The previous section already explained that the complex shear modulus could only be measured for $\unicode[STIX]{x1D714}$ between 0.1 and $100~\text{rad}~\text{s}^{-1}$ , which corresponds with $\unicode[STIX]{x1D714}^{o}=6.7\times 10^{-3}$ to $6.7$ at $Re_{\unicode[STIX]{x1D70F}}=2.1\times 10^{3}$ and $\unicode[STIX]{x1D714}^{o}=8.4\times 10^{-4}$ to $0.84$ at $Re_{\unicode[STIX]{x1D70F}}=8.5\times 10^{3}$ . From figure 13(a) we estimate that the frequencies in the range $\unicode[STIX]{x1D714}^{o}=10^{-1}$ to $10^{2}$ are most relevant, but the shear moduli data do not span this range. As it is unknown how the storage and loss moduli should be extrapolated to higher frequencies, the model calculations were performed with frequency-independent mechanical properties. Table 2 lists the dimensionless coating properties that the analytical model requires, namely $\unicode[STIX]{x1D70C}_{r}$ , $\unicode[STIX]{x1D6FF}^{+}$ , $|{c_{s}}^{+}|^{-1}$ , $\unicode[STIX]{x1D719}_{G}$ and $\unicode[STIX]{x1D708}_{pr}$ .

Finally, the numerical computations were performed in a way comparable to what has been described above. For each Reynolds number, the corresponding spectra $E_{\unicode[STIX]{x1D70E}_{tc}}(\unicode[STIX]{x1D714}^{o})$ and $E_{\unicode[STIX]{x1D70E}_{nc}}(\unicode[STIX]{x1D714}^{o})$ , convection velocity $c(\unicode[STIX]{x1D714}^{o})$ , coating thickness $\unicode[STIX]{x1D6FF}^{+}$ and coating softness $|{c_{s}}^{+}|^{-1}$ were calculated. A set of 1000 frequencies was distributed logarithmically over the range $\unicode[STIX]{x1D714}^{o}=10^{-3}$ – $10^{3}$ , and the response for each individual frequency was calculated in Maple. This procedure was followed for all Reynolds numbers for which experimental data were available, namely 18 Reynolds numbers for coating 1, 11 for coating 2 and 10 for coating 3.

5.3 Long-wave versus short-wave coating response

This subsection derives criteria to check whether the expected coating response is in the long-wave limit, the short-wave limit or in between, since this will facilitate the interpretation of the results that are presented in the next subsection. Based on figure 7, three regions can be distinguished in the coating response, dependent on the coating thickness:

(5.5) $$\begin{eqnarray}\left.\begin{array}{@{}rl@{}}\unicode[STIX]{x1D6FF}\lesssim \unicode[STIX]{x1D6FF}_{lw},\quad & \text{long-wave coating response};\\ \unicode[STIX]{x1D6FF}_{lw}\lesssim \unicode[STIX]{x1D6FF}\lesssim \unicode[STIX]{x1D6FF}_{sw},\quad & \text{combined long- and short-wave coating response};\\ ~\unicode[STIX]{x1D6FF}\gtrsim \unicode[STIX]{x1D6FF}_{sw},\quad & \text{short-wave coating response}.\end{array}\right\}\end{eqnarray}$$

The long-wave response is characterized by an r.m.s. displacement that linearly increases with the coating thickness, while the short-wave response yields a displacement that is independent of the coating thickness.

Two length scales have been introduced to separate the three different coating responses: $\unicode[STIX]{x1D6FF}_{lw}$ is the maximum coating thickness for a long-wave response, and $\unicode[STIX]{x1D6FF}_{sw}$ is the minimum coating thickness for a short-wave response. Figures 2 and 4 are helpful to distinguish the long- and short-wave responses for a single travelling stress wave: $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}=0.33$ (equivalent to $k\unicode[STIX]{x1D6FF}=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}\approx 2$ ) is taken as an approximate value to separate the long- and short-wave regions. However, the coating deformation in a turbulent flow is the result of the coating response to a whole spectrum of stress waves. The longest stress waves might induce a long-wave response, whereas the shortest waves could excite a short-wave response. In what follows, the coating response is denoted long wave (short wave) when all the relevant stress waves in the spectrum generate a long-wave (short-wave) response. The wording ‘relevant waves’ is used to denote those waves that make a significant contribution to the square of the stress r.m.s., i.e. the integral of the spectrum.

The coating response is denoted ‘long wave’ when all the relevant stress waves in the spectrum generate a long-wave response. In other words, we require that $k\unicode[STIX]{x1D6FF}\lesssim 2$ for all wave modes, which is guaranteed when the shortest wave (or highest wavenumber and frequency) satisfies this requirement. For the highest frequencies, the pressure spectrum scales in inner units and it decays quickly with $(\unicode[STIX]{x1D714}^{+})^{-5}$ (Goody Reference Goody2004). An estimate for the highest relevant frequency is $\unicode[STIX]{x1D714}_{max}^{+}\approx 2$ (cf. the spectrum in figure 5). The corresponding convection velocity is $c^{+}=10$ , such that $k_{max}^{+}\approx 2/10$ . The relation $k\unicode[STIX]{x1D6FF}=k^{+}\unicode[STIX]{x1D6FF}^{+}\lesssim 2$ applied to $k_{max}$ yields the following long-wave criterion:

(5.6a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}u_{\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D6FF}^{+}\lesssim 10\quad \text{or}\quad \unicode[STIX]{x1D6FF}\lesssim \unicode[STIX]{x1D6FF}_{lw}=10\frac{\unicode[STIX]{x1D708}}{u_{\unicode[STIX]{x1D70F}}},\quad \text{long-wave coating response}.\end{eqnarray}$$

The coating response is denoted ‘short wave’ when all the relevant stress waves in the spectrum generate a short-wave response. In other words, we require that $k\unicode[STIX]{x1D6FF}\gtrsim 2$ for all wave modes, which is guaranteed when the longest wave (or lowest wavenumber and frequency) satisfies this requirement. For the lowest frequencies, the pressure spectrum scales in outer units and it rises as $(\unicode[STIX]{x1D714}^{o})^{2}$ (Goody Reference Goody2004). An estimate for the lowest relevant frequency is $\unicode[STIX]{x1D714}_{min}^{o}\approx 10^{-1}$ (cf. the spectrum in figure 13). The corresponding convection velocity is $c^{o}=0.75$ , such that $k_{min}^{o}\approx 2/15$ . The relation $k\unicode[STIX]{x1D6FF}=k^{o}\unicode[STIX]{x1D6FF}^{o}\gtrsim 2$ applied to $k_{min}$ yields the following short-wave criterion:

(5.7a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D6FF}_{bl}}=\unicode[STIX]{x1D6FF}^{o}\gtrsim 15\quad \text{or}\quad \unicode[STIX]{x1D6FF}\gtrsim \unicode[STIX]{x1D6FF}_{sw}=15\unicode[STIX]{x1D6FF}_{bl},\quad \text{short-wave coating response}.\end{eqnarray}$$

These criteria can also be applied to channel flow. Figures 6 and 7 show that $\unicode[STIX]{x1D6FF}^{+}\lesssim 10$ is a good estimate for the long-wave response as characterized by displacements that linearly increase with $\unicode[STIX]{x1D6FF}^{+}$ . In case the half-channel height $h$ is used instead of the boundary-layer thickness $\unicode[STIX]{x1D6FF}_{bl}$ , the short-wave criterion reads $\unicode[STIX]{x1D6FF}/h=\unicode[STIX]{x1D6FF}^{+}/h^{+}\gtrsim 15$ . Since $h^{+}=hu_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}=Re_{\unicode[STIX]{x1D70F}}=720$ , the short-wave requirement becomes $\unicode[STIX]{x1D6FF}^{+}\gtrsim 15Re_{\unicode[STIX]{x1D70F}}=10\,800$ . Figures 6 and 7 show indeed that the displacements become independent of the coating thickness when $\unicode[STIX]{x1D6FF}^{+}\gtrsim 10^{4}$ .

Next, the above criteria are applied to the experiments reported in § 5.1. The coating thickness in inner units ( $\unicode[STIX]{x1D6FF}^{+}$ ) ranges from 180 to 950, while it varies in outer units between $\unicode[STIX]{x1D6FF}^{o}=0.085$ and 0.11, such that the coating deformation in the experiments is neither a long-wave nor a short-wave response. A long-wave response at the highest flow velocity requires a coating that is approximately 100 times thinner, namely $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{lw}=0.053$  mm. In contrast, a short-wave response at the highest flow velocity is obtained when the coating is approximately 140 times thicker, namely $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{sw}=0.68~\text{m}$ . This example illustrates that a long-wave response requires very thin coatings, whereas a short-wave response demands very thick coatings. The ratio of $\unicode[STIX]{x1D6FF}_{sw}$ and $\unicode[STIX]{x1D6FF}_{lw}$ increases with the Reynolds number:

(5.8) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}_{sw}}{\unicode[STIX]{x1D6FF}_{lw}}=1.5Re_{\unicode[STIX]{x1D70F}},\end{eqnarray}$$

with $\unicode[STIX]{x1D6FF}_{sw}/\unicode[STIX]{x1D6FF}_{lw}=1.3\times 10^{4}$ for the highest velocity in the experiments.

When $\unicode[STIX]{x1D6FF}_{lw}\lesssim \unicode[STIX]{x1D6FF}\lesssim \unicode[STIX]{x1D6FF}_{sw}$ , the coating response is a combination of long-wave and short-wave behaviour: the lowest frequencies (or longest waves) in the spectrum induce a long-wave response, whereas the highest frequencies (or shortest waves) excite a short-wave response. For the experiments reported above, we want to distinguish the long- and short-wave parts of the spectrum with a criterion based on $\unicode[STIX]{x1D714}^{o}$ . Long- and short-wave behaviour is again separated by $k\unicode[STIX]{x1D6FF}=k^{o}\unicode[STIX]{x1D6FF}^{o}=2$ or $\unicode[STIX]{x1D714}^{o}=k^{o}c^{o}=2c^{o}/\unicode[STIX]{x1D6FF}^{o}$ . As all three coatings have the same thickness, this relation yields a value for $\unicode[STIX]{x1D714}^{o}$ that only depends on the Reynolds number. Specifically, $\unicode[STIX]{x1D714}^{o}$ decreases from 9.5 at $Re_{\unicode[STIX]{x1D70F}}=2.1\times 10^{3}$ to 6.6 at $Re_{\unicode[STIX]{x1D70F}}=8.5\times 10^{3}$ , and it follows quite well the power law $\unicode[STIX]{x1D714}^{o}=70.6Re_{\unicode[STIX]{x1D70F}}^{-0.261}$ . From now on, the value of $\unicode[STIX]{x1D714}^{o}=7$ is taken as an approximate value to separate long- and short-wave scaling in the spectra, as indicated by a vertical line in figure 13 and other figures that follow.

5.4 Comparison of experiments and analytical model

Figure 14 shows the experimentally measured root-mean-square (r.m.s.) of the vertical interface displacement $\unicode[STIX]{x1D701}_{c}$ (normalized using viscous units) as a function of the Reynolds number for the three coatings. The displacement increases with increasing $Re_{\unicode[STIX]{x1D70F}}$ (or flow speed), in line with the experimental finding that tangential wall-displacement fluctuations increase with increasing Reynolds number (Srinivas & Kumaran Reference Srinivas and Kumaran2017). The surface displacement also increases with decreasing $|G|$ (or coating stiffness), in agreement with other studies (Kim & Choi Reference Kim and Choi2014; Rosti & Brandt Reference Rosti and Brandt2017). Coating 1, which is the softest coating, displays a sudden increase of the displacement around $U_{\infty }=4.5~\text{m}~\text{s}^{-1}$ or $Re_{\unicode[STIX]{x1D70F}}=7500$ , with a corresponding drag increase beyond that of a smooth plate (Greidanus et al. Reference Greidanus, Delfos and Westerweel2017). At $U_{\infty }=4.5~\text{m}~\text{s}^{-1}$ , the r.m.s. of the vertical displacement is 2.4 viscous units, which agrees with the presumption that the two-way coupling becomes important when the r.m.s. displacement is of the order of a viscous wall unit. The speed of $U_{\infty }=4.5~\text{m}~\text{s}^{-1}$ corresponds with $\sqrt{\unicode[STIX]{x1D70C}_{r}}\,U_{\infty }/|c_{s}|=3.7$ , which is close to the values at which other researchers have found the onset of instabilities, namely $\sqrt{\unicode[STIX]{x1D70C}_{r}}\,U_{\infty }/|c_{s}|\approx 1$ to 3 (Duncan Reference Duncan1986) and $\sqrt{\unicode[STIX]{x1D70C}_{r}}\,U_{\infty }/|c_{s}|\approx 1$ to 12 (Gad-el Hak Reference Gad-el Hak2002).

Figure 14. The measured and modelled vertical surface displacement as functions of the Reynolds number for three different coatings mentioned in table 2. As explained in the text, the figures display $\unicode[STIX]{x1D701}_{c,rms}^{+}$ for the experiments and $0.35\unicode[STIX]{x1D701}_{c,rms}^{+}$ for the model. Two model results are shown, one with shear ( $E_{\unicode[STIX]{x1D70E}_{tc}}=0.28E_{\unicode[STIX]{x1D70E}_{nc}}$ ) and another without shear ( $E_{\unicode[STIX]{x1D70E}_{tc}}=0$ ). The continuous line represents a simple analytical relation, with $\unicode[STIX]{x1D70E}_{nc,rms}^{+}$ estimated from (5.4).

Figure 14 also shows the r.m.s. displacement as obtained from the analytical model. Note that the figure plots $\unicode[STIX]{x1D701}_{c,rms}^{+}$ for the experiments, but $0.35\unicode[STIX]{x1D701}_{c,rms}^{+}$ for the model. The factor 0.35 was chosen to have a close match with the experimental data, which indicates that the present model seems not able to accurately predict the numerical value of the surface displacement. This should not be surprising, however, for several reasons. First, the model is restricted to spanwise-homogeneous and streamwise-travelling deformations. Second, the pressure spectra were not measured, so we cannot check whether the Goody spectrum is an accurate representation of the actual pressure fluctuations at the wall. Third, the convection velocity was estimated from a semi-empirical model based on channel-flow data at relatively low Reynolds numbers; there is no guarantee that this model is also accurate for boundary-layer flow at higher Reynolds numbers. Finally, the frequency dependence of the shear modulus was neglected, as the available data were not sufficient to span the whole range of relevant flow frequencies. In summary, a perfect match between the analytical and experimental data cannot be expected because of the modelling assumptions and limited knowledge of the experimental conditions.

Figure 14 shows the analytical results both for cases with shear and without shear. Adding shear has only a marginal influence on the vertical displacement, in agreement with the results presented in § 4.4.5. The vertical displacement thus appears to be mainly driven by pressure fluctuations, which is why the analytical results in subsequent figures correspond to an absence of shear, simply denoted as MOD. The model predicts quite well the increase of the displacement with flow speed and coating softness, especially for coatings 2 and 3. The prediction for coating 1 deviates from the experiments, but a closer agreement would be obtained when $0.23\unicode[STIX]{x1D701}_{c,rms}^{+}$ for the model is compared with $\unicode[STIX]{x1D701}_{c,rms}^{+}$ from the experiments, which shows that the factor 0.35 introduced above is not universal. There is also a slight deviation for coating 3 at the lower Reynolds numbers, which is most likely caused by insufficient accuracy to precisely measure the correspondingly small displacements.

The theoretical framework of the previous sections can be used to propose a scaling for the vertical surface displacement. In § 5.3 it was shown that the coating response to the complete spectrum of stress waves is neither long wave nor short wave, such that there is no preference for scaling with either the coating thickness $\unicode[STIX]{x1D6FF}$ or the wavelength $\unicode[STIX]{x1D706}$ . The coating thickness was used to scale the vertical surface displacement (as in figure 4), since that is a fixed quantity, whereas the wavelength varies with the spectrum. In § 3.2 it was shown that the displacement in the long-wave and short-wave limits is proportional to $\unicode[STIX]{x1D70C}_{r}\,c_{rs}^{2}\,\widetilde{\unicode[STIX]{x1D70E}}=\unicode[STIX]{x1D70E}/G$ , while it was demonstrated in § 4.4.5 that the vertical displacement is primarily determined by the normal stress. Hence, the scaling $\unicode[STIX]{x1D701}_{c,rms}/\unicode[STIX]{x1D6FF}=0.031\unicode[STIX]{x1D70E}_{nc,rms}/|G|$ is proposed, where the factor 0.031 is a fit parameter that captures the influence of all the unknown factors that have been mentioned above. This theoretical relation can be rewritten in viscous units as $\unicode[STIX]{x1D701}_{c,rms}^{+}=0.031\unicode[STIX]{x1D70C}_{r}\unicode[STIX]{x1D6FF}^{+}\unicode[STIX]{x1D70E}_{nc,rms}^{+}/|{c_{s}}^{+}|^{2}$ . Note that $\unicode[STIX]{x1D70E}_{nc,rms}^{+}$ is not available from the experiments, but it was estimated from (5.4).

Figure 15. (a) The measured vertical surface displacement for all three coatings as a function of the normalized stress, with $\unicode[STIX]{x1D70E}_{nc,rms}^{+}$ estimated from (5.4). (b) The measured and modelled point spectra of the vertical surface displacement as functions of the angular frequency in outer units for coating 1. The figure shows nine spectra in the one-way coupling regime, namely for $Re_{\unicode[STIX]{x1D70F}}$ from $3.6\times 10^{3}$ to $7.0\times 10^{3}$ . The spectra at the lowest $Re_{\unicode[STIX]{x1D70F}}$ are not shown because of the difficulty in accurately measuring very small displacements.

Figure 14 demonstrates that this simple analytical relation predicts the experimental data quite well. Furthermore, figure 15(a) confirms that it can be used to collapse the experimental data for the one-way coupling regime onto a single line. That figure also corroborates the assumption that the coatings behave as linear solids, since $\unicode[STIX]{x1D701}_{c,rms}/\unicode[STIX]{x1D6FF}<0.01$ for most measurements. Nonlinear solid behaviour might become relevant for coating 1 at the highest Reynolds numbers: the displacement reaches $\unicode[STIX]{x1D701}_{c,rms}^{+}=14$ at $Re_{\unicode[STIX]{x1D70F}}=8.5\times 10^{3}$ , which corresponds to $\unicode[STIX]{x1D701}_{c,rms}/\unicode[STIX]{x1D6FF}=0.015$ .

Figure 15(b) shows the point spectra of the vertical surface displacement for coating 1, both for the experiments and the analytical model. The spectrum is defined as $E_{\unicode[STIX]{x1D701}_{c}^{+}}^{^{o}}=E_{\unicode[STIX]{x1D701}_{c}^{+}}U_{\infty }/\unicode[STIX]{x1D6FF}_{bl}$ , which is the spectrum $E_{\unicode[STIX]{x1D701}_{c}^{+}}$ normalized in outer units, where $E_{\unicode[STIX]{x1D701}_{c}^{+}}=E_{\unicode[STIX]{x1D701}_{c}}u_{\unicode[STIX]{x1D70F}}^{2}/\unicode[STIX]{x1D708}^{2}$ represents the spectrum of $\unicode[STIX]{x1D701}_{c}^{+}$ . Each spectrum is normalized with $(\unicode[STIX]{x1D701}_{c,rms}^{+})^{2}$ , which is the integral of the spectrum. The figure displays the normalized spectra for 9 different Reynolds numbers, ranging from $Re_{\unicode[STIX]{x1D70F}}=3.6\times 10^{3}$ to $7.0\times 10^{3}$ . The spectra at the lowest Reynolds number are not shown because of the difficulty to accurately measure very small displacements. Furthermore, the spectra for $Re_{\unicode[STIX]{x1D70F}}\gtrsim 7\times 10^{3}$ are also excluded, since the significant drag increase in this Reynolds number range indicates the invalidity of the one-way coupling assumption.

Table 3. Parameters corresponding to the maxima in the point spectra of the vertical surface displacement. A parameter range is indicated, as the peak values vary with the Reynolds number. The Reynolds number range is the same as for the spectra shown in figures 15(b) and 16.

A clear peak in the experimental spectra is observed: when the Reynolds number increases, the peak becomes narrower and higher, while it shifts from $\unicode[STIX]{x1D714}^{o}\approx 9.1$ to higher frequencies ( $\unicode[STIX]{x1D714}^{o}\approx 12.4$ ) and then returns to $\unicode[STIX]{x1D714}^{o}\approx 10$ . Note that these trends are quite well reproduced by the analytical model. Table 3 lists the parameters for which the spectra exhibit a peak. The values of $|\unicode[STIX]{x1D714}_{rs}|\approx 3$ and $|c_{rs}|\approx 1$ indicate that the narrowing peak corresponds to a resonance (cf. figure 3), which is significantly influenced by the loss properties of the material (cf. § 4.4.3). However, the viscoelastic properties of coating 1 are unknown for the relevant frequency range: $\unicode[STIX]{x1D714}^{o}=10$ corresponds to a radial frequency between $330~\text{rad}~\text{s}^{-1}$ ( $Re_{\unicode[STIX]{x1D70F}}=3.6\times 10^{3}$ ) and $1190~\text{rad}~\text{s}^{-1}$ ( $Re_{\unicode[STIX]{x1D70F}}=8.5\times 10^{3}$ ), while the shear modulus could only be measured up to $\unicode[STIX]{x1D714}=100~\text{rad}~\text{s}^{-1}$ . The shear loss modulus $G_{2}$ showed an increase near $\unicode[STIX]{x1D714}=100~\text{rad}~\text{s}^{-1}$ , namely from 36 Pa at $10~\text{rad}~\text{s}^{-1}$ to 197 Pa at $100~\text{rad}~\text{s}^{-1}$ , with a corresponding increase of $\unicode[STIX]{x1D719}_{G}$ from $1.4^{\circ }$ to $8.1^{\circ }$ . Hence, one might expect that $\unicode[STIX]{x1D719}_{G}>8.1^{\circ }$ for the frequency with the maximum coating response. To check the influence of the loss angle, the spectra according to the model were computed for $\unicode[STIX]{x1D719}_{G}=8^{\circ },10^{\circ },15^{\circ },20^{\circ },25^{\circ },30^{\circ },40^{\circ }$ . The magnitude of the peak response decreased with increasing loss angle (as expected from figure 3), but the shape of the spectra was more or less unchanged. A fairly good agreement between the modelled and measured spectra was obtained for $\unicode[STIX]{x1D719}_{G}=20^{\circ }$ , which is the value that was assumed for all frequencies.

Figure 16. The measured and modelled point spectra of the vertical surface displacement as functions of the angular frequency in outer units for coating 2 (a) and coating 3 (b). Panel (a) shows 10 spectra for $Re_{\unicode[STIX]{x1D70F}}$ from $3.6\times 10^{3}$ to $8.5\times 10^{3}$ . Panel (b) displays six spectra for $Re_{\unicode[STIX]{x1D70F}}$ from $5.5\times 10^{3}$ to $8.3\times 10^{3}$ . The spectra at the lowest $Re_{\unicode[STIX]{x1D70F}}$ are not shown because of the difficulty in accurately measuring very small displacements.

Figure 16 shows the normalized displacement spectra of the experiments and the model for coating 2 and 3 at various Reynolds numbers. The spectra collapse well for the model and reasonably well for the experiments. The model reproduces two important spectral properties, namely a decay for $\unicode[STIX]{x1D714}^{o}\gtrsim 15$ and a peak response around $\unicode[STIX]{x1D714}^{o}=7$ . The experimental spectra for coating 2 also peak around $\unicode[STIX]{x1D714}^{o}=7$ for the lower Reynolds numbers, but the peak shifts to approximately $\unicode[STIX]{x1D714}^{o}=15$ for the three highest Reynolds numbers. This might be an indication of fluid–structure interaction, as the vertical surface displacement is close to $\unicode[STIX]{x1D701}_{c,rms}^{+}=1$ . The experimental spectra for coating 3 also exhibit a peak for $\unicode[STIX]{x1D714}^{o}\approx 7$ at the highest Reynolds numbers. The unfiltered spectra do not show a clear peak for the lower Reynolds numbers, so the peak around $\unicode[STIX]{x1D714}^{o}=4$ in the filtered spectra results from the removal of the low frequencies by the filtering procedure explained in § 5.1. Note that Zhang et al. (Reference Zhang, Wang, Blake and Katz2017) have also found a large response for $\unicode[STIX]{x1D714}h/U_{0}\lesssim 4$ (channel half-height $h$ , channel centreline velocity $U_{0}$ ), which they attribute to non-advected features with nearly zero phase speed.

Figure 17 shows the point spectra for coatings 1 and 3 on a logarithmic scale, which is useful to investigate the scaling of the spectra. Long-wave scaling applies for $\unicode[STIX]{x1D714}^{o}\lesssim 7$ (cf. § 5.3), and the relevant length scale is the coating thickness $\unicode[STIX]{x1D6FF}$ , independent of the frequency. As a result, the stress and displacement spectra should exhibit the same scaling with frequency. Indeed, the modelled stress and displacement spectra both scale with $\unicode[STIX]{x1D714}^{2}$ for $\unicode[STIX]{x1D714}^{o}\lesssim 10^{-2}$ (cf. figures 13 and 17). The scaling is less clear for $10^{-2}\lesssim \unicode[STIX]{x1D714}^{o}\lesssim 7$ , which is caused by the transition in the stress spectra from $\unicode[STIX]{x1D714}^{2}$ to $\unicode[STIX]{x1D714}^{-0.7}$ scaling (figure 13), and the transition in the coating response from long-wave to short-wave behaviour (figure 4). Long-wave scaling of the experimental spectra could not be confirmed because of the difficulty in measuring low frequencies (§ 5.1).

Figure 17. Scaling of the modelled point spectra of the vertical surface displacement for coating 1 (a) and coating 3 (b). The spectra are shown for the same range of Reynolds numbers as in figures 15 and 16.

Short-wave scaling appears when $\unicode[STIX]{x1D714}^{o}\gtrsim 7$ , and the relevant length scale is the wavelength $\unicode[STIX]{x1D706}\propto \unicode[STIX]{x1D714}^{-1}$ . The scaling of the displacement spectra for mode $m$ can be derived as follows:

(5.9) $$\begin{eqnarray}E_{\unicode[STIX]{x1D701}_{c},m}\propto |\unicode[STIX]{x1D701}_{c,m}|^{2}\propto \left|\frac{\unicode[STIX]{x1D70E}_{nc,m}}{\unicode[STIX]{x1D714}}\right|^{2}\propto \unicode[STIX]{x1D714}^{-2}E_{\unicode[STIX]{x1D70E}_{nc},m}.\end{eqnarray}$$

Hence, the stress spectra scalings of $\unicode[STIX]{x1D714}^{-0.7}$ and $\unicode[STIX]{x1D714}^{-5}$ (figure 13) become $\unicode[STIX]{x1D714}^{-2.7}$ and $\unicode[STIX]{x1D714}^{-7}$ for the displacement spectra of materials with frequency-independent mechanical properties, as is indeed confirmed by figure 17 for the modelled spectra. The experimental spectra for coating 1 show a more rapid decay than $\unicode[STIX]{x1D714}^{-2.7}$ , which might result from the expected increase of the coating stiffness with increasing frequency. The spectral decay for coating 3 seems closer to $\unicode[STIX]{x1D714}^{-2.7}$ , while the measurement frequency is too low to resolve the spectrum for $\unicode[STIX]{x1D714}^{o}\gtrsim 10^{2}$ . The spectra for coating 3 are very similar in shape to the displacement spectrum of Zhang et al. (Reference Zhang, Wang, Blake and Katz2017) (their figure 10), except that they kept the low frequencies that we filtered out. In addition, the frequency dependence of their spectrum was estimated to be approximately $\unicode[STIX]{x1D714}^{-2.4}$ for $4\lesssim \unicode[STIX]{x1D714}h/U_{0}\lesssim 20$ , which is close to the $\unicode[STIX]{x1D714}^{-2.7}$ scaling shown in figure 17.

Table 3 lists the dimensionless parameters for which the modelled and measured spectra exhibit a peak response. The model typically predicts a maximum for $\unicode[STIX]{x1D714}^{o}$ around 7 and $\unicode[STIX]{x1D6FF}_{r\unicode[STIX]{x1D706}}$ close to 0.33, in agreement with other researchers that also found a peak response for $\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D706}\approx 0.33$ (Kulik et al. Reference Kulik, Lee and Chun2008; Kulik Reference Kulik2012; Zhang et al. Reference Zhang, Wang, Blake and Katz2017). The modelled and measured convection velocities differ by a factor of 2, namely $c^{o}\approx 0.35$ – $0.40$ (model) versus $c^{o}\approx 0.70$ – $0.80$ (experiments). A model for low Reynolds number turbulent channel flow was used to estimate the convection velocity in a turbulent boundary-layer flow at higher Reynolds numbers, which presumably explains this difference.

To check the influence of the convection velocity on the displacements, another approach was attempted, namely a frequency-independent convection velocity (as in Zhang et al. (Reference Zhang, Wang, Blake and Katz2017)). Specifically, the displacement spectra were computed for $c^{o}=0.75$ and compared with the spectra from figure 17. There was no influence on the lower frequencies ( $\unicode[STIX]{x1D714}^{o}\lesssim 0.4$ , for which the convection velocity was already 0.75 $\,U_{\infty }$ ), but the response at the higher frequencies was significantly larger. For example, the maximum response for coating 3 occurred between $\unicode[STIX]{x1D714}^{o}=13$ ( $Re_{\unicode[STIX]{x1D70F}}=5.5\times 10^{3}$ ) and 38 ( $Re_{\unicode[STIX]{x1D70F}}=8.3\times 10^{3}$ ), clearly contrasting with the experimental spectra. Also, a frequency-independent convection velocity does not capture the differences in scaling for the inner and the outer layer. Hence, the model of Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009) was still used for the model results presented here, as it is (to our knowledge) the best model that is currently available.