Stability and the transition to turbulence in the flow through conduits with compliant walls

Abstract

The mechanisms of destabilisation of the flow through soft-walled channels/tubes are qualitatively different from those in rigid-walled conduits. The stability depends on two dimensionless parameters, the Reynolds number $(\rho _f V_f h_f/ \mu _f)$ and $\varSigma = (\rho _f G h_f^{2}/\mu _f^{2})$ , where $\rho _f$ and $\mu _f$ are the fluid density and viscosity, $h_f$ and $V_f$ are the fluid length and velocity scale and $G$ is the wall elasticity modulus. There is an instability at zero Reynolds number when the dimensionless parameter $\varGamma = (\mu _f V_f / h_f G)$ exceeds a critical value. The low-Reynolds-number instability of the Couette flow past a compliant surface is well understood, and has been confirmed in experiments, but that in a pressure-driven flow is not completely understood. Two modes of instability at high Reynolds number have been predicted: the inviscid mode with an internal viscous layer, for which the transition Reynolds number scales as ${Re}_t \propto \varSigma ^{1/2}$ ; and the wall mode instability with a viscous layer at the wall, for which ${Re}_t \propto \varSigma ^{3/4}$ . The wall mode instability has been observed in experiments at Reynolds number as low as 300 in a soft-walled tube and as low as 100 in a channel with one compliant wall, though the scaling of the transition Reynolds number differs from the theoretical prediction due to substantial wall deformation. Though the flow after transition shares many of the characteristics of hard-wall turbulence, it differs in significant ways, suggesting that soft-wall turbulence is a separate class distinct from hard-wall turbulence.