Turbulence Decay of a Shear-Thinning Fluid in Pipe Flow

Abstract

The human body is the subject of several interesting phenomena and blood flow comes under that category. The main motivator for this thesis is the flow of blood in aneurysms. An aneurysm is a sudden expansion of an artery, with large expansion angles causing an adverse pressure gradient and leading to flow separation. Several studies have shown the transitional nature of the flow in aneurysms, and it has been seen that the variation of the wall shear stress from cycle-to-cycle is one of the major reasons for the growth of aneurysms, and possibly leading to their rupture at a later stage. It has also been noticed that the turbulent kinetic energy (TKE) does not decay with the mean flow kinetic energy and the periodic kinetic energy of the cardiac cycle. Therefore, it is intersting to research the turbulent decay to see how the flow in an aneurysm can be affected. However, the number of variables in an aneurysm are too high to effectively characterize this, and therefore, a simplified geometry was chosen. Pipe flow is a good choice to start with, as it can mimic the wall-bounded nature of an aneurysm. It also has a well-defined statistically steady turbulent state from where the decay of turbulence can be studied. Additionally, blood being a non-Newtonian fluid, makes it interesting to study the effects of shear-thinning.

To this extent, a Direct Numerical Simulation (DNS) study has been carried out using a higher-order spectral element method code. First, statistics for fully-developed pipe flow are compared with existing results. To a fully-developed turbulent state, a deceleration is applied to bring the flow to a steady, laminar state. The decay of the turbulent quantities is monitored during this process. Comparison studies are undertaken to study the influence of the ramp rate, the dependence of the decay on the initial Reynolds number, and the variation of the results between Newtonian and generalized Newtonian fluids. A modelling approach using RANS has also been undertaken to see if only studying the mean flow is sufficient to characterize the decay.

It is seen that two regimes of decay exist -- a power-law decay based on turbulent scaling, and an exponential viscous decay. The power-law decay is further divided into two stages -- one before the saturation of the integral length scale, and one after the saturation of the length scale -- with the maximum length scale being set by the diameter of the pipe. The exponential model has been validated using the hypothesis of Skrbek (2008). The point of divergence from the power-law to the exponential decay has been hypothesized here. It is seen that for all the cases studied, the point of divergence occurred at $Re_ au = 60$. It is noticed that the decay is independent of the ramp rates when they are applied at time on the order of magnitude of 1 Eddy Turnover Time (ETT). The decay does show a dependence on the initial Reynolds number and the reasons for this are hypothesized. The RANS modelling used was found to be insufficient due to the inability of the RANS model to gauge the size of the domain. For generalized Newtonian fluids, it is noticed that the decay rate increases with shear-thinning. The results obtained are discussed in the context of an aneurysm. Based on the diameter, length and flow rate of the aneurysm, it can be hypothesized at which stage of decay the flow is, and based on this, it has been discussed whether using a non-Newtonian modelling approach is more beneficial than using a Newtonian approach for the decay.