Suresh Sahu for productive discussions and assistance with the implementation of non-Newtonian models in the current computational framework. Part (b) is the magnified view of the mesh near the aneurysm neck.
Non-Newtonian model
In the sections that follow, we therefore provide a review of literature related to computational frameworks for incompressible flows with an emphasis on unstructured finite volume methods applied to non-Newtonian fluid flows. Nevertheless, Carreau family and shear-thinning power-law are commonly used to model the non-Newtonian behavior of blood [2-7].
Numerical approaches for non-Newtonian flows
18] develop a second order precisely structured collocated finite volume framework for the non-Newtonian power law currents using pseudo-compressibility technique. The solution variables are implicitly solved in the coupled way and its unstructured version can be found in [7] for the generalized Newtonian fluids.
Collocated versus staggered mesh framework
In addition, in curved offset meshes (applicable to complex geometries), a transformation of variables is required, which requires the discretization of Christoffel symbols, which is computationally expensive [28,29]. A hybrid staggered/non-staggered framework facilitates the implementation of boundary conditions in flows with complex boundaries in addition to balanced force calculations in incompressible multiphase flows [42].
Applications to hemodynamics
Three-dimensional investigations of hemodynamics in stenosis in [51] showed that an increase in the stenosis ratio leads to an increase in the pressure drop. In [55], it was described that the upper flow parameter decreases with increasing aneurysm size.
Aims of the thesis
Their results show that there is noticeable difference between the Newtonian and non-Newtonian models. The necessity of such an approach for incompressible flows in general and non-Newtonian flows in particular will be studied.
Organisation of the thesis
Applying the framework to understand unsteady non-Newtonian flows through these geometries and the role of pulsatility in flow characteristics, with the aim of relating flow parameters to clinical indices. In this chapter, we describe the detailed mathematical formulation of the incompressible non-Newtonian fluid flow solver based on the scaled/non-scaled hybrid framework.
Governing equations
However, the drawback of the power-law fluid model is that it does not describe the non-Newtonian behavior in the very low and very high shear rate regions by predicting infinite viscosity [6]. In these two non-Newtonian models, the Reynolds numbers are defined as Re=ρ∞U∞2−nLn∞/K and Re=ρ∞U∞L∞/µ∞, respectively, where K denotes the consistency coefficient of the power-law fluid model.
Hybrid staggered/non-staggered framework
- Discretisation of convective fluxes
- Discretisation of diffusive fluxes
- Discretisation of normal pressure gradient
- Temporal term discretisation and time integration
- Pressure correction equation
- Solution update
- Velocity reconstruction and gradient computation
- Boundary conditions
- Solution algorithm
Therefore, velocity reconstruction from the normal velocity Uf at the cell centroid is an essential feature of the hybrid staggered/non-staggered finite volume framework. Under the prescribed boundary conditions for the pressure field pb (see Figure 2.4(b)), the normal pressure gradient at the boundary is δp/δn|b according to Eq.
Summary
The main features of the framework and their implementation, including boundary conditions, are presented, and the framework will be evaluated on canonical problems later in the thesis. The velocity reconstruction approach is an essential part of the hybrid staggered/undistributed finite volume framework (described in Chapter 2) for physically accurate solutions.
IDeC(k)-GG algorithm for velocity reconstruction
- Reconstruction based on Gauss theorem
- Defect-corrected Gauss reconstruction
- Computing velocity gradients
- Determining face values in gradient computation
It follows that the accuracy of the nominal values will affect the accuracy of the calculated gradients. It is also clear that the nominal values of the velocity components are obtained from the center of gravity values, which in themselves are only orderly accurate.
Accuracy studies
These observations are in excellent agreement with the theoretical arguments presented in Theorems and demonstrate the effectiveness of the proposed IDeC(k)-GG reconstruction algorithm. The number of iterations for convergence is about 10 and is almost independent of the mesh topology and refinement.
Studies using IDeC(k)-GG and IDeC(1)-GG
- Two-dimensional channel flow
- Lid driven cavity flow
- Backward facing step in a channel
- Flow past circular cylinder
It is interesting to note that the IDeC(1)-GG reconstruction predicts an incorrect vortex structure and a larger difference in the size of corner vortices compared to the IDeC(k)-GG reconstruction. It is clear from Table 3.2 as well as Figure 3.14 that the predictions of the recirculation zone lengths from the IDeC(1)-GG reconstruction are not comparable to the calculations of Choi and Barakat [13], in contrast to the results using IDeC (k)- GG -reconstruction which shows a reasonable agreement.
Summary
Method used Drag coefficient, Recirculation length Cd IDeC(1)-GG IDeC(k)-GG IDeC(1)-GG IDeC(k)-GG. However, the IDeC(1)- GG approach, without error correction, over-predicts the drag coefficient and under-predicts the recirculation length, compared to the calculated results of Sivakumar et.al.
Accuracy study
Temporal accuracy
Spatial accuracy
Poisson equation and preconditioning
A note on discrete conservation
In this section, we investigate the discrete conservation of mass and momentum for the shifted/non-shifted finite volume approach introduced in this thesis. We consider the two-dimensional channel flow of the shear-thinning (n = 0.8) power-law fluid, for which the analytical solutions can be readily derived for the fully developed axial velocity profile.
Lid-driven cavity flow
The parameters of the Carreau-Yasuda fluid model are listed in Table 4.1, where the Reynolds number is defined as Re=ρUlidH/µ0. It is not surprising to see this agreement, since the Carreau-Yasuda fluid behaves like a power-law fluid for a large value of W i(W i1 ).
Two-dimensional channel flow
Following the studies in Hao and Chao [70] showing that for a given set of the model parameters with n = 0.3678, the Carreau-Yasuda model resembles the power-law model with n = 0.708, we use this Carreau model to provide an indirect comparison and validation of the implementation of the Carreau model in our framework. 71] have shown that the ratio of the pressure drop for the Newtonian fluid model to that of the Carreau-Yasuda fluid model remains approximately constant.
Backward facing step flow
Our results in Table 4.2 also support this fact and closely agree with the ratio of 0.33 reported in [71] for the same flow and model parameters.
Flow past a circular cylinder
A computational domain size with the center of the cylinder at (10,10) is discretized using a hybrid mesh with triangular cells around the circular cylinder and quadrilateral cells in the rest of the domain. Figure 4.24(a) and Table 4.5: Average drag coefficients (c¯d) of unsteady flow of non-Newtonian fluid over stationary circular cylinder at Re= 100 and Re= 140. b) present the history of CdandCl for stationary (β = 0) and rotating cylinder (β = 1.57) respectively at the two given Reynolds numbers using the Newtonian fluid model.
Summary
In this chapter, the internal computational framework has been used to investigate steady blood flow in canonical configurations of aneurysm and stenosis. Therefore, in this chapter we try to understand the stable hemodynamics in aneurysm and stenosis by investigating different non-Newtonian models and geometries.
Problem set-up and computational details
The W SS value changes sign from positive on the upper wall of the parent artery to negative inside the aneurysm due to reverse flow (recirculation). The parameters of fluid models of fluid models are tabulated in Table 5.1 to mimic the rheology of blood [2, 4, 6].
Effect of fluid models on double stenoses
It can be argued that for shear rates that persist in the taper, the apparent viscosity of the Carreau-Bird fluid is closest to the Newtonian model. Consequently, the "equivalent" Reynolds number1 would be largest for a power-law fluid and smallest for a Carreau-Yasuda fluid.
Role of degree of stenoses on hemodynamics
In the flow recirculation region, the apparent viscosity becomes high due to the low shear rate and increases with increasing taper ratio. It is easy to see from this table that the pressure drop increases with increasing taper ratios.
Effect of fluid models on intracranial aneurysm
Effect of aneurysm size on hemodynamics
Summary
It was found that increasing aneurysm size results in higher W SS at the aneurysm neck, while it decreases in the aneurysm dome. In [47], it is stated that Newtonian modeling of blood results in an overestimation of the shear stress in the aneurysm dome wall.
Problem set-up and computational details
The low and high P I waveforms corresponding to the diseased state (healthy state ranges from 0.54 to 2.8) of the artery are constructed from the baseline pulsatile velocity profile by adjusting a, b, and c [85, 86]. Important context flow parameters are instantaneous flow patterns, instantaneous wall shear stress, mean wall shear stress (W SSmean), and Oscillating Shear Index (OSI).
Effect of fluid models on double stenoses
Furthermore, the location of peakOSI in the case of Carreau-Yasuda fluid model is shifted away from the two constrictions in contrast to the other two fluid models. In addition, it is reported that "the highly localized distribution of low and oscillatory shear stress along the walls strongly correlates with the foci of atheroma1 in the human left coronary artery" in [48].
Effect of degree of stenoses on hemodynamics
On the other hand, the mild stenosis behaves similarly to the case of a constant flow with a relatively low value of OSI compared to the other two configurations. However, this study can only provide limited information, and more realistic waveforms are needed to shed light on the mechanics behind atherosclerosis and.
Effect of fluid models on intracranial aneurysm
The distribution of W SSmean on the wall of the aneurysm using the two fluid models is shown in Figure 6.12 and is consistent with the distributions of immediate W SS. Importantly, the non-Newtonian model calculates higher mean values in the neck region of the aneurysm compared to the Newtonian model.
Effect of aneurysm size on hemodynamics
In addition, the high hemodynamic stress at the aneurysm neck site may also cause the neck to expand. Our results indicate higher value of W SS means the aneurysm neck region which is consistent with the result in [87].
Effect of flow waveform on aneurysm
First (1) and second (2) rows represent the high and low P I cases, respectively, using the Newtonian model. Third (3) and fourth (4) rows indicate for the high and low P I cases respectively using the non-Newtonian model.
Summary
The overall second-order accuracy of the flow solver is also dependent on the order of accuracy of the velocity reconstruction approach. In addition, the implications of aneurysm size (aspect ratio: height to neck) are more significant than the different fluid models on the hemodynamics of the intracranial aneurysm.
Future directions of research
Numerical investigation of non-Newtonian flows through double constrictions by an unstructured finite volume method. A Newton-Krylov finite volume algorithm for power law non-Newtonian fluid flow using pseudo-compressibility technique.
Grid topologies for the accuracy studies
L ∞ norm of global error in velocity with uniform grid refinement on the
L ∞ norm of error in velocity and gradient calculation on distorted mesh
L ∞ norm of error in gradients on hybrid mesh when calculated using (a)
Convergence history of IDeC(k)-GG reconstruction on hybrid and skewed
Triangulated and quadrilateral (at the center) meshes used for the com-
Velocity profile plot of downstream using IDeC(k)-GG and IDeC(1)-GG
Hybrid mesh used for the computation. Part (b) shows the zoom in view
The streamline pattern obtained using the two reconstruction approaches. 45
An enlarge view of the hybrid mesh in the computational domain sur-
Variation of apparent viscosity with shear rate for the fluids considered
Streamline patterns obtained using the Carreau-Yasuda non-Newtonian
Wall shear stress distributions on the top wall for the different stenosed
Streamline patterns inside the aneurysm dome using the different fluid
Wall shear stress variation on the aneurysm wall
Streamline patterns in the different sized aneurysm
Wall shear stress distributions on the aneurysm wall
Pulsatile velocity waveforms over a cycle prescribed at the inlet of the
Streamline patterns at four time instants of a cycle using the different
Temporal change of wall shear stress at the post-stenotic point (x = 5)
Average wall shear stress variation on the top wall comprising the two
OSI distribution of the multiple constricted two-dimensional channel
Streamline patterns for the three geometries of the double stenoses
