Deformation of a compliant wall in a turbulent channel flow

Abstract

Interaction of a compliant wall with a turbulent channel flow is investigated experimentally by simultaneously measuring the time-resolved, three-dimensional (3D) flow field and the two-dimensional (2D) surface deformation. The optical set-up integrates tomographic particle image velocimetry to measure the flow with Mach–Zehnder interferometry to map the deformation. The Reynolds number is $Re_{\unicode[STIX]{x1D70F}}=2300$, and the Young’s modulus of the wall is 0.93 MPa, resulting in a ratio of shear speed to the centreline velocity ($U_{0}$) of 6.8. The wavenumber–frequency spectra of deformation show the surface motions consist of a non-advected low-frequency component and advected modes, some travelling downstream at approximately $U_{0}$ and others at ${\sim}0.72U_{0}$. The r.m.s. values of the advected and non-advected modes are $0.04~\unicode[STIX]{x03BC}\text{m}$$(0.004\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})$ and $0.2~\unicode[STIX]{x03BC}\text{m}$ ($0.02\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$), respectively, much smaller than the wall unit ($\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$), hence they do not affect the flow. Trends in the wall dynamics are elucidated by correlating the deformation with flow variables, including the 3D pressure distribution calculated by spatially integrating the material acceleration. Predictions by the Chase [J. Acoust. Soc. Am., vol. 89 (6), pp. 2589–2596] linear model are also calculated and compared to the measured trends. The spatial deformation–pressure correlations peak at $y/h\approx 0.12$ ($h$ is half channel height), the elevation of Reynolds shear stress maximum in the log-layer. Streamwise lagging of the deformation behind the pressure is caused in part by phase lag of the pressure with decreasing distance from the wall, and in part by material damping. Positive deformations (bumps) caused by negative pressure fluctuations are preferentially associated with ejections involving spanwise vortices located downstream and quasi-streamwise vortices with spanwise offset. Results of conditional correlations are consistent with the presence of hairpin-like structures. The negative deformations (dimples) are preferentially associated with positive pressure fluctuations at the transition between an upstream sweep to a downstream ejection.

Active and passive in-plane wall fluctuations in turbulent channel flows

“…These findings may explain why recent numerical studies consistently report either an increase in drag or a statistically unchanged flow field in the presence of travelling wave-like wall deformations (Xu et al 2003;Fukagata et al 2008;Rosti & Brandt 2017;Xia et al 2017). More recently, Zhang, Miorini & Katz (2015), Zhang et al (2017) have reported pioneering experiments that capture compliant wall deformations and turbulent velocity field in a water channel simultaneously. These experiments confirmed unequivocally the existence of travelling waves on the surface.…”

“…A water channel with L * 2 = 0.2 m height is considered, with intermediate Reynolds number, Re τ = 1000. Such flow conditions can be tested in various experimental facilities (Schultz & Flack 2013;Zhang et al 2017). The #40 parameter set (equivalent to #44) is used because this compliant wall model leads to drag reduction in both low and intermediate Reynolds number channel flows.…”

“…These findings contradict theoretical predictions (Duncan 1986;Kireiko 1990) and experimental results (Choi et al 1997) where 7 % drag reduction was measured on slender bodies coated with single-layer homogeneous viscoelastic material. Ongoing research, for instance, Rosti & Brandt (2017), Xia et al (2017), Zhang et al (2017), is aimed at understanding the interaction between turbulent boundary layers and passive wall motions in the context of compliant wall design.…”

Deformation of a linear viscoelastic compliant coating in a turbulent flow

“…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 (2017). The turbulent surface stresses are expressed as a sum of streamwise-travelling and spanwise-homogeneous waves (cf.…”

“…The latter was identified by a pronounced increase of the streamwise velocity fluctuations and the Reynolds stress, which suggests that turbulence was generated by the soft wall. Zhang et al (2017) investigated a compliant coating in a turbulent channel flow at Re τ = 2300. They report simultaneous measurements of the time-resolved, three-dimensional flow field (using particle image velocimetry (PIV)) and the twodimensional surface deformation (using Mach-Zehnder interferometry (Zhang, Miorini & Katz 2015)).…”