Simulation of Transient Blood Flow in Models of Arterial Stenosis and Aneurysm

The Large Eddy Simulation (LES) technique with the Smagorinsky-Lilly dynamic subgrid model and two-equation Standard k-ω Transitional turbulence model are applied to investigate non-spiral and spiral blood flow through three dimensional models of arterial stenosis and aneurysm. A spiral pattern of blood flow is thought to have many beneficial effects on hemodynamics. Previous computational studies on spiral blood flow involve only steady spiral flow in a straight stenosed pipe without considering an upstream curved section of the artery. But a spiral pattern in the blood flow may exist due to the presence of an upstream curved section in the artery. On the other hand, pressure is generally considered a constant quantity in studies on pulsatile flow through either arterial stenosis or aneurysm; however, blood pressure is a waveform in a physiological flow. Although cosine-type or smooth regular stenoses are generally taken in investigations of blood flow in a three-dimensional model of arterial stenosis, in reality, stenoses are of irregular shape. Besides stenosis and aneurysm, another abnormal condition of the artery is the presence of stenosis with an adjacent aneurysm in the same arterial segment, especially in the posterior circulation. A study on (steady or pulsatile) flow through such arterial stenosis with an adjacent aneurysm in the same arterial segment is not available so far. Therefore, taking above things into consideration, thorough investigations of steady and unsteady pulsatile non-spiral and spiral blood flow in three-dimensional models of stenosis and aneurysm are needed to give a sound understanding of the transition-to-turbulence of blood flow due to stenosis and aneurysm and to study the the effects of spiral velocity on the transition-to-turbulence. The LES technique has mostly been used to investigate turbulent flow in engineering fields other than bio-fluid mechanics. In the last decade, LES has seen its excellent potential for studying the transition-to-turbulence of physiological flow in bio-fluid mechanics. Though the k-ω Transitional model is used in few instances, mainly LES is applied in this study. Firstly, investigations of steady non-spiral and spiral blood flow through threedimensional models of cosine-type regular stenosed tube without and with upstream curved segment of varying angles of curvature are performed by using the k-ω Transitional model and LES. A fully developed Poiseuille velocity profile for blood is introduced at the inlets of the models. To introduce a spiral effect at the inlet, onesixth of the bulk velocity is taken as the tangential velocity at the inlet along with the axial velocity profile there. Secondly, physiological pulsatile non-spiral and spiral blood flow through a three-dimensional model of a straight tube having cosine-type regular stenosis are investigated by using mainly LES. A two-equation k-ω Transitional model is also used in one non-spiral flow case. The first four harmonics of the Fourier series of pressure pulse are used to generate physiological velocity profiles at the inlet. At the outlet, a pressure waveform is introduced. The effects of percentage of area reduction in the stenosis, length of the stenosis, amplitude of pulsation and Womersley number are also examined. Thirdly, transient pulsatile non-spiral and spiral blood flow through a threedimensional model of irregular stenosis are investigated by applying LES and comparison is drawn between non-spiral flow through a regular stenosis and that through an irregular stenosis. Lastly, pulsatile non-spiral and spiral blood flow through a three-dimensional model of irregular stenosis with an adjacent post-stenotic irregular aneurysm in the same arterial segment are studied by applying LES and the k-ω Transitional model. The effects of variation in spiral velocity are also examined. The results presented in this thesis are analysed with relevant pathophysioloical consequences. In steady flow through the straight stenosed tube, excellent agreement between LES results for Re = 1000 and 2000 and the corresponding experimental results are found when the appropriate inlet perturbations are introduced. In the models with an upstream curved segment, no significant effect of spiral flow on any flow property is found for the investigated Reynolds numbers; spiral pattern disappears before the stenosis – which may be due the rigid wall used in the models and/or a steady flow at the inlet. The effects of the curved upstream model can be seen mainly in the maximum turbulent kinetic energy (TKE), the maximum pressure drop and the maximum wall shear stress (WSS), which in the curved upstream models generally increase significantly compared with the corresponding results in the straight stenosed tube. The maximum contributions of the SGS motion to the large-scale motion in both non-spiral and spiral flow through a regular stenosis, an irregular stenosis and an irregular stenosis with an adjacent post-stenotic irregular aneurysm are 50%, 55% and 25%, respectively, for the highest Reynolds number investigated in each model. Although the wall pressure and shear stress obtained from the k-ω Transitional model agree quite well with the corresponding LES results, the turbulent results obtained from the k-ω Transitional model differ significantly from the corresponding LES results – this shows unsuitability of the k-ω model for pulsatile flow simulation. Large permanent recirculation regions are observed right after the stenosis throat in both non-spiral and spiral flow, which in the model of a stenosis with an adjacent post-stenotic aneurysm are stretched beyond the aneurysm and the length of the recirculation regions increases with spiral velocity. This study shows that, in both steady and unsteady pulsatile flow through the straight tube model having either a stenosis (regular or irregular) or an irregular stenosis with an adjacent post-stenotic irregular aneurysm, the TKE rises significantly at some locations and phases if a spiral effect is introduced at the inlet of the model. However, the maximum value of the TKE in a high spiral flow drops considerably compared with that in a low spiral flow. The maximum wall pressure drop and shear stress occur around the stenosis throat during all the phases of the pulsatile cycle. In the model of a stenosis only, the wall pressure rises in the immediate post-stenotic region after its drop at the stenosis throat. However, in the model of a stenosis with an adjacent aneurysm, the wall pressure does not rise to regain its undisturbed value before the start of the last quarter of the aneurysm. The effects of the spiral flow on the wall pressure and WSS are visible only in the downstream region where they take oscillatory pattern. The break frequencies of energy spectra for velocity and pressure fluctuations from −5/3 power slope to −10/3 power slope and −7/3 power slope, respectively, are observed in the downstream transition-to-turbulence region in both the non-spiral and spiral flow. At some locations in the transition region, the velocity spectra in the spiral flow has larger inertial subrange region than that in non-spiral flow. The effects of the spiral flow on the pressure spectra is insignificant. Also, the maximum wall pressure drop, the maximum WSS and the maximum TKE in the non-spiral flow through the irregular stenosis rise significantly compared with the corresponding results in the non-spiral flow through the regular stenosis. When the area reduction in the stenosis is increased, the maximum pressure drop, the maximum WSS and the TKE rise sharply. As for the effects of the length of the stenosis, the maximum WSS falls significantly and the maximum TKE rises sharply due to the increase in the length of the stenosis; but the maximum pressure drop is almost unaffected by the increase in the stenosis length. The increase in the amplitude of pulsation causes both the maximum pressure drop and the maximum WSS to increase significantly under the inlet peak flow condition. While the increased amplitude of pulsation decrease the maximum TKE, it is nonetheless responsible for the sharp rise in the TKE found at some places in the transition-toturbulence region. The decrease in the Womersley number causes the maximum TKE to increase dramatically; however, the maximum pressure drop and the maximum WSS decrease slightly under the inlet peak flow condition as a result of the decrease in the Womersley number. The author does believe that the present study makes a breakthrough in understanding the non-spiral and spiral transient blood flows through arteries having a stenosis and a stenosis with an adjacent post-stenotic aneurysm. The findings of the thesis would, therefore, help the interested groups such as pathologists, medical surgeons and researchers greatly in gaining better insight into the transient non-spiral and spiral blood flow through models of arterial stenosis and aneurysm. Acknowledgements All the praises be to Allah, the Most Beneficent, the Most Merciful. I would like to express my gratitude to my supervisor, Dr Manosh C. Paul, for all his guidance, support and encouragement during this research. His stimulating comments and suggestions have been of immense benefit to me. The financial support of ORSAS and the school of Engineering of the University of Glasgow for this study is gratefully appreciated. I’m also grateful to Shahjalal University of Science and Technology, Sylhet, Bangladesh, for granting me study leave with pay. Grateful thanks are due to the members of IT Support Team of School of Engineering, especially Mr Walter Robinson and Mr Ken McColl for their constant effort to keep the system always running and help in fixing any problems in the computer systems. I would like to express my sincere thanks to all the friends and colleagues, Dr Md Mamun Molla, Mr Md Foyzul Haque, Mr Najeeb Yahya, Mr Ahmed Mahboob Ferdousi, Prof Syed Shamsul Alam, Dr Rizaul Islam, Mr Rafiqul Islam, Dr Bala Boyi Bukata, Mr Ali Saad Alzwayi, Mr Blaid Alganash and Mr Ahmed Al Waaly. Their spirited support and encouragement has been of great help to me throughout the relentless endurance test of my PhD. I am deeply indebted to my beloved parents, wife, son, siblings, maternal aunt and mother-in-law. I owe you all much more than I can possibly express. I am also grateful to all of my well-wishers for their invaluable advice and support that has helped me greatly to make it this far.