Low Reynolds Number Turbulent FSI and its Applications in Biological Flows

Abstract

Fluid-structure interactions (FSIs) of fluid-conveying collapsible tubes produce rich physiologically significant phenomena in many biological systems. Despite the significant progress made in recent years, the physical mechanisms responsible for the onset of self-excited oscillations of collapsible tubes remain unclear. To study nonlinear dynamic behaviors of collapsible tubes, a numerical framework based on the immersed boundary-lattice Boltzmann method (IB-LBM) for the simulation of FSI in collapsible tubes is developed, and then applied to investigate the physical mechanisms behind self-excited oscillations in a two-dimensional (2D) collapsible channel and a three-dimensional (3D) collapsible tube.

In the proposed numerical framework, the lattice Boltzmann method (LBM) is employed to solve the fluid dynamics, while the structural equations for the 2D collapsible channel and 3D collapsible tube are solved by the finite difference method (FDM) and finite element method (FEM), respectively. The immersed boundary method (IBM) is adopted for the fluid and structural coupling. A power-law non-Newtonian fluid model is used to model the non-Newtonian fluid, and large eddy simulation (LES) is used to capture the turbulence in the FSI system.

The nonlinear dynamics of a two-sided collapsible channel flow is first investigated. The stability of the hydrodynamic flow and collapsible channel walls are examined for a wide range of Reynolds number (100 < Re < 3000), structure-to-fluid mass ratio (0.3 < M < 20), external pressure (1 < Pe < 10) and wall thickness (0.01D < h < 0.1D). Chaotic behaviors of the collapsible channel flow are characterized and possible routes to chaos as well as physical mechanisms responsible for the onset of self-excited oscillations are identified. Nonlinear and rich dynamic behaviors of the collapsible system are newly observed. Specifically, the system experiences a supercritical Hopf bifurcation leading to period-1 limit cycle oscillations as the Reynolds number increases. The existence of chaotic behavior of the collapsible channel walls is confirmed by a positive dominant Lyapunov exponent and a chaotic attractor in the velocity-displacement phase portrait of the mid-point of the collapsible channel wall. Chaos in the system can be reached via period-doubling and quasi-periodic bifurcations. In addition, it is found that symmetry breaking is not a prerequisite for the onset of self-excited oscillations, but symmetry breaking induced by large mass ratio and external pressure may lead the system into a chaotic state. Unbalanced transmural pressure, wall inertia and shear layer instabilities in the vorticity waves have been found to induce the onset of the self-excited oscillations of the collapsible system. The period-doubling, quasi-periodic and chaotic oscillations are closely associated with vortex pairing and merging of adjacent vortices, and interactions between the upper and lower vorticity rows.

The IB-LBM solver is applied to examine the nonlinear dynamics of unsteady flows in 3D collapsible tubes. The stability of the hydrodynamic flow and collapsible tube walls are examined for a range of Reynolds numbers (100 < Re < 1000) from laminar to turbulent. The effects of non-Newtonian rheology on the dynamic behaviors of the collapsible system are examined with a power-law model, and LES is used to model turbulent effects. For the effects of Reynolds number, it is found that the periodic vortex shedding downstream the throat of the elastic tube is responsible for small-amplitude and quasi-periodic self-excited oscillations developed at Re=200. During these oscillations, two regions of wall-thickening are developed near the end of downstream elastic tube (non-dimensional axial distance z/R=7.25) due to compressive stress concentrations, suggesting the potential for fatigue failures at these two regions. A small secondary buckling pattern develops at the downstream end of the elastic tube which has not been observed in the 2D model. The reverse-flow region is open and fills the entire symmetry plane with the major axis in the 3D model while it is closed and cannot fill the entire height of the channel in the 2D model. For the effects of non-Newtonian fluid, it is found that shear-thickening fluids tend to stabilize the collapsible system while shear-thinning fluids will trigger the onset of self-excited oscillations. The deformation of the tube decreases as the power-law index n increases. The tube snaps from a fully constricted state at n=1.6 to an unconstricted state at n=1.7.

For turbulent flows at Re=1000, flow bifurcation occurs and the system settles into large-amplitude and quasi-periodic self-excited oscillations which are regular and repetitive. These oscillations are caused by the periodic shedding of vortices downstream of the throat of the elastic tube. The shed vortices feed back periodic perturbations through the elastic tube wall. At two monitor points placed in the downstream rigid tube, small secondary oscillations were found in the time history of pressure and streamwise velocity, which are caused by two jets merging together and interactions at the middle section of the downstream rigid tube.