Deformation of a linear viscoelastic compliant coating in a turbulent flow

Abstract

We investigate the deformation of a linear viscoelastic compliant coating in a turbulent flow for a wide range of coating parameters. A one-way coupling model is proposed in which the turbulent surface stresses are expressed as a sum of streamwise-travelling waves with amplitudes determined from the stress spectra of the corresponding flow over a rigid wall. The analytically calculated coating deformation is analysed in terms of the root-mean-square (r.m.s.) surface displacement and the corresponding point frequency spectra. The present study systematically investigates the influence of five coating properties namely density, stiffness, thickness, viscoelasticity and compressibility. The surface displacements increase linearly with the fluid/solid density ratio. They are linearly proportional to the coating thickness for thin coatings, while they become independent of the thickness for thick coatings. Very soft coatings show resonant behaviour, but the displacement for stiffer coatings is proportional to the inverse of the shear modulus. The viscoelastic loss angle has only a significant influence when resonances occur in the coating response, while Poisson’s ratio has a minor effect for most cases. The modelled surface displacement is qualitatively compared with recent measurements on the deformation of three different coatings in a turbulent boundary-layer flow. The model predicts the order of magnitude of the surface displacement, and it captures the increase of the coating displacement with the Reynolds number and the coating softness. Finally, we propose a scaling that collapses all the experimental data for the r.m.s. of the vertical surface displacement onto a single curve.

On the interaction of a compliant wall with a turbulent boundary layer

“…The next discussion compares the one-dimensional temporal spectra of deformation to predictions by two one-dimensional linear models, namely those of Chase (1991) and Benschop et al . (2019). The experimental results for all the present speeds are presented in figure 12.…”

“…In Benschop et al . (2019), the comparison is based on the experimental frequency spectrum. However, the dependence on wavenumber is modelled as advection at a frequency-dependent speed, which is assumed to decrease with increasing frequency.…”

“…(2007) at different Reynolds numbers: ( a ) normalized by inner variables and ( b ) normalized by the outer variables.

Figure 14.The amplitude of deformation (normalized by ) in response to pressure perturbations modulated by the pressure spectra of Goody (2004), as predicted by: ( a ) the Chase (1991) model and ( b ) the Benschop et al . (2019) models. Inclined dotted lines refer to advection bands at U 0 = 5.9 m s −1 .

Figure 15.Comparison of the advected (0.46 U 0 < ω/k < 0.86 U 0 ) measured power spectra with the predictions by the Chase and Benschop models.

…”

“…This procedure has been subsequently used by Benschop et al . (2019) to characterize and model the deformation of compliant walls over surfaces with varying properties. Their findings and model predictions are compared to the present results in §§ 4 and 6 of this paper.…”

“…Examples of relevant models are provided in Duncan (1986), Chase (1991), and recently in Benschop et al . (2019). As discussed in § 2, following our prior work, we start by utilizing the linear one-dimensional model introduced by Chase (1991) to predict the response of a visco-elastic material to pressure perturbation with prescribed amplitude and frequency.…”