NUMERICAL SIMULATION OF PULSATING INCOMPRESSIBLE VISCOUS FLOW IN ELASTIC TUBES. Eng. Osama Ali Abdelmonem El Banhawy

Transcription

1 NUMERICAL SIMULATION OF PULSATING INCOMPRESSIBLE VISCOUS FLOW IN ELASTIC TUBES By Eng. Osama Ali Abdelmonem El Banhawy A Thesis Submitted to the Faculty of Engineering at Cairo University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Engineering Mechanics FACULTY OF ENGINEERING, CAIRO UNIVERSITY GIZA, EGYPT 2015

2 NUMERICAL SIMULATION OF PULSATING INCOMPRESSIBLE VISCOUS FLOW IN ELASTIC TUBES By Eng. Osama Ali Abdelmonem El Banhawy A Thesis Submitted to the Faculty of Engineering at Cairo University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Engineering Mechanics Under the Supervision of Prof. Dr. Mohamed Samir Tosson. Professor of Engineering Mechanics Department of Engineering Mathematics and Physics Faculty of Engineering, Cairo University Dr. Amr Gamal Guaily. Assistant Professor Department of Engineering Mathematics and Physics Faculty of Engineering, Cairo University FACULTY OF ENGINEERING, CAIRO UNIVERSITY GIZA, EGYPT 2015 II

3 Engineer s Name: Eng. Osama Ali Abdelmonem El Banhawy Date of Birth: 14/8/1988 Nationality: Egyptian osama_elbanhawy@eng.cu.edu.eg Phone: Address: 4 th Quarter, New Damietta, Damietta Registration Date: 1/3/2012 Awarding Date:././2015 Degree: Master of Science Department: Engineering Mathematics and Physics Supervisors: Examiners: Prof. Mohamed Samir Tosson Dr. Amr Gamal Guaily Prof. Moustafa Saber Moustafa Abou-Dina (External examiner) (Math Dept., Faculty of Science, Cairo University) Prof. Mamdouh Abdelhamied Fahmy (Internal examiner) Porf. Mohamed Samir Tosson (Thesis main advisor) Dr. Amr Gamal Guaily (Thesis advisor) Title of Thesis: Numerical Simulation of Pulsating Incompressible viscous Flow in Elastic Tubes Key Words: Incompressible viscous flow, Finite element and Streamline Upwind / Petrov-Galerkin, Computational Fluid Dynamics, Fluid-Structure interaction Summary: The Streamline-Upwind/Petrove-Galerkin (SUPG) technique is used to study the behavior of incompressible viscous fluids in elastic tubes. The unsteady two dimensional Navier-Stokes equations along with the continuity equation are used for the simulation. The continuity equation is modified by adding an artificial viscosity term for the pressure to overcome the well-known problem of the continuity equation being a constraint equation rather than an evolution equation. The deformability of the boundary is accounted for by treating the boundary as a simply supported beam under transverse unsteady distributed load, namely the fluid pressure. The principle of minimum potential energy in elasticity in the case of bending in the regime of small deflections is used to model the boundary deflection. The current model is then used to study blood flow in elastic large arteries with an apparent degree of success. III

4 Acknowledgments I would like to express my sincere gratitude to Professor Mohamed Samir Tosson for his valuable guidance throughout the development stages of this thesis, I am very grateful to Dr. Amr Guaily for suggesting this research topic and for his help in building my foundation in numerical techniques and computer programming. Finally, I would like to thank all those who offered help in presenting this thesis especially Dr. Tamer Heshmat, Dr. Mohamed Elshabrawy and Eng. Waddaa Aboulatta. Eng. Osama El Banhawy IV

5 Table of contents ACKNOWLEDGMENTS...IV TABLE OF CONTENTS... V LIST OF FIGURES... VIII LIST OF SYMBOLS AND ABBREVIATIONS... X ABSTRACT... XV CHAPTER 1 : INTRODUCTION AND LITERATURE REVIEW Motivation Numerical Methods Finite Difference Methods Finite Element Methods The Incompressibility Constraint Mathematical Models of Elastic Boundaries Scope of the Current Work... 9 CHAPTER 2 : MATHEMATICAL MODELING AND NUMERICAL TECHNIQUE Introduction Fluid Mathematical Model Governing Equations The Incompressibility Constraint Boundary Conditions V

6 2.3 Elastic Boundary Potential Energy Numerical Technique and Discretization Method Finite Element Model of The Fluid Flow Discretization of Finite Element Equations of Fluid Finite Element Model of The Elastic Boundary Discretization of Finite Element Equation of The Elastic Boundary Time Marching Scheme Solution Algorithm CHAPTER 3 : VALIDATION CASES Introduction Lid-Driven Cavity Problem (LDCP) Problem Definition Boundary and Initial Conditions Results Flow Over a Circular Cylinder (FOCC) Problem Definition Boundary and Initial Conditions Results CHAPTER 4 : PULSATING BLOOD FLOW (PBF) Introduction Problem Definition Boundary and initial conditions VI

7 4.4 Pulsating Blood Flow in a Rigid Artery (PBFRA) Pulsating Blood Flow in an Elastic Artery (PBFEA) CHAPTER 5 : SUMMARY AND CONCLUSIONS Summary Conclusion Future work APPENDIX Contour Integral Calculations BIBLIOGRAPHY VII

8 List of figures Figure 2-1: Problem definition Figure 2-2: Simply supported beam Figure 2-3: Galerkin and Petrov-Galerkin weighting functions Figure 2-4: SUPG and Galerkin weighting functions for node A Figure 2-5: Quadrilateral element in Cartesian and local coordinates Figure 2-6: Typical four-node element Figure 2-7: Flow chart of the solution algorithm Figure 3-1: LDCP-Problem description Figure 3-2: LDCP -Effect of the aspect ratio and Reynolds number.([39]) Figure 3-3: LDCP-Computational domain Figure 3-4: LDCP-Pressure isocontours for Figure 3-5: LDCP-Streamlines for Figure 3-6: LDCP-Image of the cavity flow at taken from Pakdel et al[39] Figure 3-7: LDCP- the -component velocity at =0.5 for Figure 3-8: LDCP-Streamlines at different dimensionless times: (a) t =1, (b) t =10 for Figure 3-9: LDCP-Hirsch s solution at different times :(a) t =1, (b) t =10 for. [40] Figure 3-10: LDCP- the -component velocity at =0.5 and = 10 for Figure 3-11: LDCP- Mesh-independent solution test using the -component velocity at =0.5 for Re Figure 3-12: LDCP-Solution at t=2 for different Re :(a)500, (b)1000, (c)5000,(d) Figure 3-13: FOCC-Physical domain description Figure 3-14: FOCC-Different flow regimes(taken from[42]) Figure 3-15: FOCC- Computational domain and the grid VIII

9 Figure 3-16: FOCC- Zoom on the mesh around the cylinder Figure 3-17: FOCC-Velocity vectors of steady symmetric vortices for = Figure 3-18: FOCC-Steady symmetric vortices streamlines for = Figure 3-19: FOCC-Pressure isocontours of symmetric vortices for = Figure 3-20: FOCC-Streamlines of shedding vortex for =500: (a) at =0, (b) at = Figure 3-21: FOCC-Pressure contours of shedding vortex for =500: (a) at =0, (b) at = Figure 3-22 FOCC-Streamlines of Heshmat s solution for =500: (a) at =0, (b) at = [45] Figure 4-1: PBF- Carotid artery model Figure 4-2: PBF-Finite element mesh of the Carotid artery model Figure 4-3: PBF - The pulsating volumetric flow rate versus time Figure 4-4: PBFRA- Pulsating inlet flow rate and pressure Figure 4-5: PBFRA-Volumetric flow rate at both S1 and S Figure 4-6: PBFRA - Pressure distribution at both locations S1 and S Figure 4-7: PBFRA -Velocity profiles at different axial locations at different time points Figure 4-8: PBFRA -Velocity distribution at different time points Figure 4-9: PBFEA - Boundary deflection and boundary pressure at different axial locations Figure 4-10: PBFEA Pressure field in the boundary layer blood flow Figure 4-11: PBFEA Velocity field in the boundary layer blood flow Figure 4-12: PBFEA-Pressure gradient along the boundary at different time points of the pressure wave Figure 4-13: PBFEA Pressure wave at B1 for rigid and deformable wall Figure 4-14: PBFEA Wall shear stress at B3 for rigid and deformable wall at different numbers.67 IX

10 List of Symbols and Abbreviations Symbol / Abbreviation Description, Cartesian Co-ordinates, Dimensionless Cartesian Co-ordinates Time Dimensionless Time Gradient operator =[ ] Dimensionless Gradient operator =[ ] Vector of velocity components in, directions respectively Dimensionless Vector of velocity components in, respectively directions Fluid Density Fluid pressure Dimensionless fluid pressure Body force per unit volume Fluid dynamic viscosity Fluid characteristic velocity X

11 Characteristic diameter Reynolds Number Penalty Coefficient Fluid Bulk Modulus General variable Time step Artificial viscosity Potential Energy Dimensionless Potential Energy Vertical Deflection Dimensionless Vertical Deflection External load per unit length Dimensionless External load per unit length Young s Modulus of Elasticity Moment of area Weighting function of the SUPG method Continuous weighting function of Galerkin method Discontinuous contribution of the SUPG method The velocity vector calculated at the element center Scalar artificial diffusivity XI

12 Local element co-ordinates Unit vectors in directions Element characteristic lengths in The local velocity components in directions directions Element s Reynolds number based on the local velocities and characteristic lengths Jacobian Matrix The consistent mass matrix The viscous matrix The nonlinear convective vector Vector of gradient operator The transpose of vector The generalized matrix for the pressure term in eqn.( 2.55) The slope at the deflected node of the elastic boundary Element length Cubic shape function Element stiffness matrix for the elastic boundary numerical model Local vector of right hand side of eqn.( 2.67) for the elastic boundary numerical model Global stiffness matrix for the elastic boundary numerical model Global vector of right hand side of eqn.( 2.67) for the elastic boundary numerical model XII

13 Courant Number Lumped mass matrix Cavity length and depth respectively Aspect ratio of the cavity Strouhal Number Volume flow rate Pulsatile Frequency Womersley Number BSUPG Beyond Streamline Upwind Petrov-Galerkin method CFD FEM FDM FOCC Computational fluid dynamics Finite element method Finite difference method Flow Over a Circular Cylinder FSI GLS Fluid-Structure Interaction Galerkin Least Squares method. Index of Fluid-structure Interaction HDAD MPE PBFP PBFEA High-order Derivative Artificial Diffusion method The principle of Minimum Potential Energy Pulsating Blood Flow Problem Pulsating Blood Flow in Elastic Artery XIII

14 PBFRA LDCP SE Pulsating Blood Flow in Rigid Artery Lid-Driven Cavity Problem Strain Energy SUPG S-SUPG WP Streamline Upwind Petrov Galerkin method Simplified SUPG method The work potential XIV

15 Abstract One of the difficulties, faced when simulating fluid flows, is how to account for the two-way interactions with domain boundaries, known as Fluid Structure Interaction (FSI). FSI is the interaction of some deformable or movable structure with an internal or surrounding fluid flow[1]. FSI can be either under steady or pulsating loads. And because one of the most important applications for FSI is the simulation of blood flow in deformable arteries, the topic of this thesis is the simulation of blood flow, with pulsating conditions at its inlet, in elastic tubes. Considering blood as a Newtonian fluid, its flow in the carotid artery is approximated as an unsteady two-dimensional laminar flow. The Navier-Stokes equations are used to model this viscous planar incompressible flow between two infinite rigid parallel plates. Since the continuity equation for incompressible flows becomes a constraint equation for the velocity field, rather than an evolution equation for the density, a special treatment for the continuity equation is required. Hence; an artificial viscosity term is added to the equation in order to yield a Poisson equation for the pressure. The proposed modification overcomes the well-known problem of the incompressibility constraint in primitive variables formulation. A finite element model is developed, using the Streamline Upwind Petrov-Galerkin (SUPG) technique, to numerically solve the three governing coupled non-linear partial differential equations. Two bench mark problems are used for validation of the numerical scheme: i) lid-driven flow in a rectangular cavity, and ii) vortex shedding resulted from uniform flow past a circular cylinder. The obtained numerical results are compared favorably with published experimental works and numerical models with a good degree of qualitative agreement. XV

16 To model blood flows in deformable arteries, the two-way coupling between the fluid flow and the deformability of the boundary needs to be considered.. The deformability of the artery boundary is accounted for by treating it as a simply supported elastic beam under transverse unsteady distributed load, namely the blood pressure. The principle of minimum potential energy (in elasticity in the case of bending for small deflections) is used to model the boundary deflection. The coupling is considered by solving the fluid flow problem, and then the resulting pressure distribution over the boundary is used as the distributed load for the elastic boundary problem. The resulting deformed boundary is used as the new boundary for the fluid flow problem for the following time step. This process continues till the end of the simulation time. A FORTRAN code is developed to implement the proposed dynamical model and the numerical technique to solve the governing equations. The code is tested first, and for purposes of comparison, for blood flows in rigid arteries. Then the targeted FSI problem of blood flow in deformable arteries is fully considered. The fluid velocity field for the pulsating blood flow and the pressure distributions in the deformable artery boundary are plotted in various figures.these model results may be useful in diagnosis because they portray the pulsating blood flow velocity and blood pressure wave for different time points and at different artery axial locations. XVI

17 Chapter 1 : Introduction and Literature Review 1.1 Motivation One of the most important applications of the simulation of pulsatile flow in elastic tube is the blood flow in artery. The number of patients with arterial diseases especially main cardiovascular arteries like carotid continue to rise and consequently the cases of sudden death. The information about flow rate and blood pressure in arteries are necessary in diagnosis. This information can be obtained by imaging techniques and invasive flow and pressure measurements but those techniques are limited and the measurements are restricted by time and physical condition of the patient. So to acquire these important information about flow and pressure, an alternative approach is required which is the numerical simulation. Hence, mathematical modeling and numerical simulation for the pulsatile viscous incompressible flow in elastic tubes is presented in this work. The evolution of the mathematical models and numerical methods relevant to the current study exist in literature is discussed in this chapter. Specifically, the finite element method and its various upwinding techniques are briefly described. Various treatments for the incompressibility constraint are discussed and followed by a survey on different mathematical models used to describe the boundary elasticity. 1.2 Numerical Methods In numerical models of multidimensional problems in fluid dynamics, the frame of reference of the grid in relation to the fluid must be chosen[2]. Classically, the Lagrangian description of a continuum depicts a frame that moves with the continuum i.e. body-fitted grid. The main advantage of this approach is that the momentum equation is linear due to the disappearance of the convective term which is the source of non-linearity. This approach is more suitable for problems in solid mechanics 1

18 since the deformed configuration is usually near the deformed one. On the other hand, the deformations in fluids are much larger and the deformed configuration is largely deviated from the reference configuration. Consequently adopting the Lagrangian grid will result in highly deformed grid in a very small length of time. That is why for fluid mechanics problems, the Eulerian grid is preferred though this results in a nonlinear momentum equation due to the appearance of the convective term. Those are the two main approaches that could be used equally with any numerical technique like the finite difference method, finite element method and finite volume method etc Finite Difference Methods Finite difference methods (FDM's) have been used to model fluid mechanics problems since its first appearance in 1928 by the Courant, Friedrichs and Lewy [3]. The main advantage of the FDM's is their relative simplicity to implement as it is a point-based method. Derivatives in the differential equations are replaced by finite difference approximations at specific points, namely the grid points. FDM's methods lose their simplicity very quickly with complex geometries Finite Element Methods Finite element methods (FEM's) are also numerical techniques that could be used to solve any differential equation and so could be used to model time-dependent hyperbolic problems like fluid flows[4]. FEM is a domain-based method and so doesn t have the disadvantage of the finite difference method with complex geometries. They have been extensively used where irregular moving boundaries are present([5], [6]). One of the advantages of the FEM's is that they are amenable to inhomogeneous boundary conditions. FEM's use the weak form of the flow equations. The most general approach is the method of weighted residuals; in which an integral equation is formulated by the multiplication of the flow equations by a weighting function and then an integration over a prescribed interval called element is carried out [7]. 2

19 Several variants have been developed out of this approach e.g. least-squares method, collocation method etc. The most well-known variant is the standard Galerkin method. FEM's are spatial discretization; therefore, the Galerkin method must be coupled with standard time-stepping algorithms (Crank-Nicolson, backward Euler, Lax-Wendroff or Runge-Kutta, for example) to address time-dependent advection problems. Donea[8] notes several disadvantages to this coupled approach: it suffers from a rapid fall off in accuracy as the time step is increased, and it has a reduced range of stability as compared to a corresponding FDM. Deficiencies in the coupling of the classical Galerkin method with standard integrators for advection dominated problems motivated the development of generalized Galerkin methods([8], [9]). In the generalized approach, the weights are integrated differently than the velocity and pressure. Also, most of these methods more properly account for the link between space and time via the direct or indirect use of characteristics. Upwinding is a major issue when dealing with hyperbolic equations, the Petrov- Galerkin method is an early upwind FEM used to model the one-dimensional linear wave equation where special weighting functions are used in order to control the direction in which diffusion is added [10]. Streamline upwinding has been added to Petrov-Galerkin to modify the weighting function[11]. This yields a consistent formulation without the familiar spurious wiggles of the pure method[11]. Taylor-Galerkin methods are different in that the discretization in time is addressed before the spatial discretization in an attempt to achieve greater accuracy[12]. They are based on Taylor series expansions in the time step and do not account for characteristic behavior. Least-squares Galerkin methods add least-squares forms of residuals to the standard Galerkin approach in order to improve stability and maintain accuracy. Stabilized space-time FEM's have been introduced to overcome limitations in the classical Galerkin approach when modeling the incompressible Navier-Stokes equations by FEM's([4], [13] [15]). There is no need for the FDM's in these methods as the unsteady terms are discretized by a finite element approximation. 3

20 In fluid flows or convective heat transfer, the matrix associated with the convective term is nonsymmetric, and as a result, the 'best approximation' property is lost. In practice, solutions are often corrupted by spurious node-to-node oscillations or 'wiggles'. Wiggles are most likely to appear in convection dominated cases (high Peclet or Reynolds number) when a downstream boundary condition forces a rapid change in the solution[11]. It is well known that the Galerkin finite element method gives rise to central-difference type approximations of differential operators. It is thus not surprising that 'wiggles' have also afflicted central-difference finite difference solutions. In fact, obtaining solutions at all has been problematic due to the reliance on iterative solvers in finite differences coupled with the jack of diagonal dominance of central-difference equations[11]. It was discovered, however, that wiggle-free solutions could be obtained by the use of 'upwind' differencing on the convective term. Upwind differencing amounts to approximating the convective derivatives with solution values at the upstream and central nodes of a three-node stencil. The drawback is that upwind differences are only first-order accurate (central differences are second-order accurate). The loss of accuracy is manifested as overly diffuse solutions. It is well known that the upwinded convective term can be constructed simply by adding artificial diffusion to a central difference treatment. This artificial diffusion interpretation has been the basis for extensive criticism of upwind methods[11]. There are several upwind techniques, they can be listed as([16], [17]): 1- High-order Derivative Artificial Diffusion (HDAD) method. 2- Streamline Upwind Petrov Galerkin (SUPG) method. 3- Beyond Streamline Upwind Petrov-Galerkin (BSUPG) 4- Simplified SUPG (S-SUPG) procedure. 4

21 5- Galerkin Least Squares (GLS) method. 6- Positive Coefficient Upwinding Procedure. Hendriana [16] attempted to identify the optimum upwinding scheme used in the quadrilateral fournode finite element. She achieved this by comparing several previously published upwinding techniques via testing their stability and accuracy in two numerical convection-diffusion problems. Her experiment shows that the SUPG and BSUPG upwinding procedures are easy to implement, however the results still contain small oscillations near shock fronts and boundary layers. Hendriana and Bathe [17]studied the performance of various upwind techniques implemented in parabolic finite element discretization for incompressible high Reynolds number flow. The characteristics of an ideal upwind procedure are first discussed. Then the streamline upwind Petrov- Galerkin method, a simplified version thereof, the Galerkin least squares technique and a high-order derivative artificial diffusion method are evaluated on test problems. They concluded that none of the methods displays the desired solution characteristics and that for very large Reynolds number all techniques failed to converge after certain iteration except HDAD method. Many researchers have studied the stability of the SUPG method([18] [21]) which makes it a wellknown method. The SUPG upwinding technique is used in this study for many reasons, the most important of which are: it s easier to implement and the range of Reynolds number for flows solved does not reach the range which causes the technique to fail. One more thing is the facility of stabilization of the SPUG technique. In regards to the time marching scheme, an explicit time marching is used because of its easiness and facility of stabilization using courant condition (courant number). Also it saves computer memory since there is no need to solve a system of coupled algebraic equations which is the case with the implicit time marching schemes. 5

22 1.3 The Incompressibility Constraint Assuming incompressible fluid implies that the speed of sound is infinity which means that pressure signals are felt the same instant they happen which is not physically correct and consequently has numerical implications. One of the implications is that the continuity equation becomes a constraint equation for the velocity field rather than an evolution equation for the density as the case in the compressible assumption. In other words, there is no evolution equation for the pressure unlike the velocities. Throughout last decades researchers aimed to find a method to overcome this problem by adding a pressure term or terms to the continuity equation multiplied by a small number. The nature of those terms is very affecting the discretization method of the equations and the numerical integration rule. Penalty formulation is one of the most known methods used in solving Navier-stokes equations[22]. The formulation defines that the pressure is equal to the continuity equation multiplied by the penalty coefficient. Then substituting the penalty formulation in the momentum equations to eliminate the pressure as an unknown, hence the system is just two equations in two unknowns; the velocity components. This method leads to the simplest effective finite element implementation of the incompressibility but it restricts the shape of pressure element and numerical integration rule for penalty coefficient terms. The pressure element must be less by one order from the velocity element and the integration rule order must be lowered which is not an easy issue while programing. The penalty function formulation leads to unphysical results in some problems due to potential difficulties[11]. An alternative formulation may be used to deal with this difficulties is the slightly compressible formulation[11]. The pressure is defined to be a function of the density multiplied by the bulk modulus of the fluid. The bulk modulus is assumed to be large so that the density is almost constant. This assumption is implemented in the full continuity equation and the modified continuity 6

23 equation is obtained. Finally this equation is linked to the momentum equations and pressure is calculated using time-stepping algorithm. Another treatment used in [11] which is a combination of the penalty and slightly compressible formulations. This combination results an auxiliary equation for calculating pressure which must be treated implicitly. However it has the same restrict on element shape and integration rule, it shows a great success in solving the flow over circular cylinder with shedding vortex problem. A new Petrov-Galerkin method has been presented in[23] for steady state problems, the formulation is equivalent to using local time-stepping with a centered in time/least-squares in space approximation. It used a modification that minimizes the squared-residual of the momentum equations with respect to the pressure parameters. That produces a strong coupling between the pressure field and the incompressibility condition, and equal order elements for velocity and pressure becomes possible. A certain stabilized formulations is proposed in [24] with bilinear and linear equal-order interpolation velocity-pressure elements for the computation of steady and unsteady incompressible flows. The stabilization procedure involves a slightly modified Galerkin/least-squares formulation of the steadystate equations. These elements were implemented using the one-step and multi-step formulations of the Navier-Stokes equations. Three numerical examples solved: the standing vortex problem, the liddriven cavity flow, and flow past a circular cylinder. A different approach is developed by [25], in which the continuity equation is modified by adding an artificial viscosity in terms of the Laplacian of the pressure to stabilize the solution algorithm. This formulation allows equal order of interpolation for pressure and velocity, and does not impose any restrictions on the integration rules like other approaches. Flow in sudden expansion geometry is solved and the results show that the value of the artificial viscosity coefficient affect the solution except when the pressure and velocity are explicitly coupled. 7

24 In the current work the artificial viscosity presented by [25] will be used because of its simplicity, effectiveness and the ability to use equal order interpolation functions for the pressure and velocity. 1.4 Mathematical Models of Elastic Boundaries Modeling the movement of the boundaries in fluid-structure interaction problems is one of the most challenging research points for the time being. Fluid structure interaction has to be taken into account in analyzing many engineering applications. The relevant application to the current work is Hemodynamics (dynamics of blood flow in human body) especially blood flow in arteries and tissues. A linearized kinematics approach is the simplest computational framework. This approach assumes that the deformation of the tube wall is small enough to use the same Eulerian frame as in the fluid equations([26] [28]). In this framework, the interface between the fluid and the tube walls is fixed but the nodes can have non zero velocities. Transpiration condition formulations use this linearized kinematics approach. [26]proposed a modified fixed-point algorithm with a transpiration formulation to reduce computational time. [27]also used a linearization principle with a reduced linear structure to solve problems of fluid-structure interaction. [28]developed the coupled momentum method by adopting the linearized kinematics formulation for the tube walls. This method couples the elastodynamic equations of the tube walls to the Navier-Stokes equations using a shear-enhanced membrane model for the vessel walls. The second computational framework is the immersed boundary method([29], [30]). This computational approach is implemented in the areas where the mass of the solid is insignificant compared to the mass of the fluid and can be ignored. This approach is used in many studies describing the flow across heart valves and inside the heart itself. The third computational framework is the Arbitrary Lagrangian-Eulerian (ALE) formulation for fluidsolid interaction problems([5], [31]). In this framework, the flow domain is no longer constant, but on 8

25 a moving grid. The wall deformation problem is solved in Lagrangian formulation, and the time evolution of the grid of the domains should be evaluated each time step. ALE formulations are computationally expensive when considering large models of the vasculature and less robust than linearized kinematics methods since they necessitate the continual updating of the geometry of the fluid and structural elements. A new approach is developed in this work that can model flows in simple elastic tubes. The deformability of the boundary is accounted for by considering the boundary as a simply supported beam under transverse unsteady distributed load, namely the fluid pressure. The principal of minimum potential energy in elasticity in the case of bending in the regime of small deflections is used to model the boundary deflection. The proposed approach is explained in details in chapter 2 and is tested by simulating a typical pulsatile blood flow in a deformable artery. 1.5 Scope of the Current Work The purpose of the current work could be summarized in the flowing points: 1- Presenting a mathematical model that describes the two-way coupling of pulsating viscous incompressible fluid flowing in elastic tubes using. 2- Developing a new technique to overcome the incompressibility constraint problem for unsteady flows. 3- Using the SUPG technique of the FEM to numerically solve the proposed mathematical model. 4- Using the proposed model to simulate pulsating blood flow in elastic arteries. 9

26 Chapter 2 : Mathematical Modeling and Numerical Technique 2.1 Introduction The mathematical model describing the fluid flow and the elastic boundary deformability is presented in this chapter. Also the numerical technique and discretization method are explained in details. 2.2 Fluid Mathematical Model Figure 2-1: Problem definition. The physical phenomena that a numerical simulation could capture depend on the mathematical model used to describe the phenomenon. In the present study the two-dimensional, unsteady, incompressible and viscous Navier-Stokes equations are used. In this model the pressure waves are travelling at infinite speed since the continuity equation behaves as a constraint equation for the velocity field rather than an evolution equation for the density field 10

27 2.2.1 Governing Equations The following assumptions are adopted: 1. The flow is laminar and incompressible. 2. The flow domain is planar (the variation in the third direction is neglected). 3. The body forces are neglected. 4. The fluid is Newtonian. 5. The flow is unsteady So the flow of an unsteady viscous and incompressible fluid is governed by the continuity and the momentum equations (Navier-Stokes) in the form: ( 2.1) ( 2.2) Where the symbols of equations 2.1 and 2.2 stand for: [ ] is the gradient operator w.r.t. and (the Cartesian space coordinates defined in Figure 2-1), denotes time, is the total time differentiation operator, is the vector of fluid particle velocity in, directions respectively, is the fluid velocity, is the thermodynamic pressure, is the dynamic viscosity of the fluid. It s convenient to use the dimensionless form of the equations, so a dimensionaliztion scheme is used. The following dimensionless variables are defined ( 2.3) 11

28 Where D and are the problem characteristic diameter and velocity respectively. Then equations ( 2.1) and ( 2.2) become: ( 2.4) ( 2.5) Where is the Reynolds number that is defined as: So the Cartesian form of the equations is: ( 2.6) ( ) ( 2.7) ( ) The Incompressibility Constraint As mentioned in the first chapter that the main problem in the numerical treatment of the Navier- Stokes equations is how to deal with the incompressibility constraint as the continuity equation becomes a constraint equation for the velocity field. On the contrary, in the case of the compressible assumption an evolution equation for the density exists. In other words, an evolution equation for the pressure like the velocities is a must. Many solutions exist in the literature, most of them deal with the shape interpolation functions, numerical integration rules and the pressure evolution equation used. The treatment done by Hughes [22] is defining the pressure as: 12

29 ( 2.8) Where stands for a penalty coefficient. This restricts the pressure interpolation functions to be less by one order than the velocity and the same for the numerical integration rule for the pressure terms which represents a programming difficulty. Brooks et al [11] add a time dependent term to the equation (2.8) and is referred to as a slightly compressible formulation to be combined with the penalty formulation. So the final modified equation will be in the form: ( 2.9) Where is the fluid s bulk modulus. This modified equation requires the same treatment regarding the interpolation functions and numerical integration rules. Habashi et al [25]developed a new algorithm to overcome the compressibility problem. For steady incompressible viscous flow, they redefined the velocities as: ( 2.10) Where is an artificial viscosity. Then the continuity equation ( 2.6) is modified to be: ( ) ( 2.11) And then the momentum equation will be affected and for example the x-direction equation will be: ( ) ( 2.12) This formulation allows equal order of interpolation for pressure and velocity and does not restrict the choice of element[25]. But this algorithm works only for the steady flow. 13

30 Another technique introduced by Chung[32] is to multiply the momentum equation ( 2.5) in vector form by neglecting the viscous and time dependent terms resulting in a Poisson equation for the pressure: ( 2.13) Equation ( 2.13) is used to get the pressure distribution but its major disadvantage is the nonlinearity appearing on the left hand side. A theoretical study is carried out by Guaily and Epstein[33] regarding the time integration effects on the modified equation. This analysis is used to justify the used technique regarding treating the incompressibility constraint. The analysis is repeated here for convenience; consider one dimensional nonlinear partial differential equation: ( 2.14) Using Backward Euler scheme for the unsteady term: ( 2.15) And the Taylor expansion series: ( ) ( 2.16) Doing some algebraic work the modified equation is: ( * ( 2.17) Comparing the modified ( 2.17)with the original ( 2.14) ; one concludes the following[33]: 1. The original advection speed changed by the amount { }. 2. An artificial dissipation is introduced { } 14

31 3. The mathematical nature of the original PDE changed from being a first order PDE eqn.( 2.14) to be a second order PDE eqn. ( 2.17). The same procedure is applied to one of the conservation laws[33]. The conservation of energy for an isothermal perfect gas as well as it represents the compressible continuity equation by replacing the pressure by the density and for γ =1: ( 2.18) The modified PDE shows that the second order dissipation terms appeared not for the pressure only but also for the other variables u and v[33]. The first observation from this analysis is that the same procedure when implemented on ( 2.9) that introduced by Brooks et al[11] adds implicitly second order terms of the other variables which is a defect of this treatment. The second observation; this formulation satisfies also the momentum equations as the time dependent velocity terms are always discretized in the same way, so second order dissipation terms of pressure will automatically appear in the momentum equations. The analysis of [33] shows that the solution suggested by [11]using equation( 2.9) is actually adding the second order pressure term which is equivalent to the term added by Habashi in eqn.( 2.11), so we intend in this study to use eqn.( 2.11) directly without the redefinition of velocities in eqn.( 2.10). This assumption overcomes all difficulties found in the previous works. For the ease of notation the artificial viscosity coefficient will be renamed, so equation ( 2.11) is modified to be: ( ) ( 2.19) 15

32 2.2.3 Boundary Conditions Computational fluid dynamics (CFD) deals with the well-known boundary conditions; Neumann boundary condition, Dirichlet boundary condition and mixed condition[32]. For the velocity field results from eqn.( 2.7) the following set of boundary conditions are applied[32]: A. For fixed walls: (no slip condition). B. For symmetry line: = 0 and. C. For entry, exit and far field condition: and are usually specified at inlet but it will be treated separately in each problem as shown in chapter 3. Boundary conditions on the modified continuity equation( 2.19) are as follow[34]: A. Fixed walls: a homogeneous Neumann boundary condition is applied = 0. B. Entry and exit: Pressure is usually specified at exits but it depends on the problem examined and it will be discussed in each problem. 2.3 Elastic Boundary Potential Energy The deformation of the elastic boundary containing the fluid is accounted for by considering the boundary as a simply supported beam (see Figure 2-2) under transverse unsteady distributed load namely, the pressure. This deformation is calculated using the principle of Minimum Potential Energy so the potential energy should be defined. The principle of Minimum Potential Energy (MPE)[35], given below, is used to reach the local matrix of the FEM for a beam element. It states that: For conservative structural systems, of all the kinematically admissible deformations, those corresponding to the equilibrium state extremize (i.e., minimize or maximize) the total potential energy. If the extremum is a minimum, the equilibrium state is stable. 16

33 For further explanation, a constrained structural system, i.e., a supported structure points is deforming when subjected to external loads. Deformation of a structural system refers to the incremental change to the new deformed state from the original undeformed state. The Potential energy (PE) of a structural system is defined as the sum of the strain energy (SE) and the work potential (WP)[35]. The strain energy is the elastic energy stored in deformed structure and is defined as : ( 2.20) ( 2.21) And the strain energy is given by, ( 2.22) The work potential WP, is the negative of the work done by the external forces acting on the structure. Work done by the external forces is simply the forces multiplied by the displacements at the points of application of forces. Applying this principle to the case of bending of a symmetric beam of a regime of small deflection, the functional of the total PE is defined by [36]: [ ( ) ] ( 2.23) Where is the vertical deflection, is the external load (assuming the load positive downwards), is the modulus of elasticity and I is the beam moment of area. Considering the boundary as the beam and that the equilibrium state is achieved every time step the deflection of the boundary is calculated. The non-dimensional variables we define: 17

34 ,,, ( 2.24) Using ( 2.24) in ( 2.23) results; [ ( ) ] ( 2.25) The dimensionless form of the potential energy of a simply supported beam will be: [ ( ) ] ( 2.26) This non-dimensionaliztion procedure is applied using fluid s characteristic diameter, characteristic velocity and density to ensure unity of the problem as this model is coupled with the fluid model to form the main problem in this study. A dimensionless number resulted from non-dimensionaliztion: ( 2.27) is the Index of Fluid-Structure Interaction. is the ratio of structure (boundary) flexural rigidity and the fluid momentum flux. is usually large for small deflections and it becomes infinite for rigid boundary Equation ( 2.26) is a one dimensional ODE in one variable, namely, the deflection, which makes the boundary condition very simple. It s assumed that the boundary is simply supported at ends, so the only boundary condition is zero deflection at these ends (see Figure 2-2). 18

35 Figure 2-2: Simply supported beam 19

36 2.4 Numerical Technique and Discretization Method Both of the fluid and solid models are solved using FEM. For the fluid model; two-dimensional system of PDEs are solved for three variables, then the resulting pressure distribution at the lower boundary is used as an input for the deflection problem. Every model uses a different approach of the FEM and discretization method Finite Element Model of The Fluid Flow The most general approach in FEM is the method of weighted residuals; in which the weak form is formulated by multiplying the flow equations by a weighting function and then an integration is carried out over the element area[7]: ( 2.28) The well-known standard Galerkin weighted residual formulation considers a continuous weighting function over interelement boundaries but this consideration fails in the convection dominated problems as spurious wiggles often appear[11].this leads to the development of upwinding techniques. Petrov-Galerkin upwinding technique modifies the weighting function of a typical node that is to weight the element upwind of the node more heavily than the downwind element as shown in Figure 2-3. Unfortunately, when Petrov- Galerkin upwind applied to more complicated situations like multidimensional problems, generalizations of this method often gives results that are much worse than those obtained by the standard Galerkin s method due to the crosswind diffusion [11]. The Streamline upwind method is introduced by Brooks and Hughes in[11] using the artificial diffusion concept to the multidimensional advection-diffusion equations. In this method, the artificial diffusion operator is set to affect only the direction of flow eliminating any possibility of crosswind diffusion. 20

37 Figure 2-3: Galerkin and Petrov-Galerkin weighting functions. Though, the streamline upwind method overcomes the crosswind problem, however it still suffers several deficiencies; e.g. the upwinding of the convection term without the centrally weighted and transient terms resulted in excessively diffuse solution[11]. Clearly, applying the upwinding to all terms of the equation is needed which gives the desired effect and defines a consistent Petrov- Galerkin formulations. This technique is presented as streamline upwind Petrov-Galerkin (SUPG) method [11]. The SUPG formulation requires discontinuous weighting function of the form: ( 2.29) Where is a continuous weighting function and is the discontinuous streamline upwind contribution (see Figure 2-4). Figure 2-4: SUPG and Galerkin weighting functions for node A. 21

38 The perturbation part added to the weighting function is defined as[11]: ( * ( 2.30) Where is a scalar artificial diffusivity, is the velocity vector calculated at the element center and ( 2.31) The value of depends on the element shape and local coordinates used for the numerical integration rule. In this work, the bilinear isoparametric quadrilateral four node element is used with local axes as described in Figure 2-5: Figure 2-5: Quadrilateral element in Cartesian and local coordinates Defining the unit vectors in the local element coordinates: ( 2.32) 22

39 ( 2.33) Where and are the unit vectors in the Cartesian coordinates and. And element characteristic lengths ( 2.34) ( 2.35) Figure 2-6 explains the element s geometry, unit vectors and characteristic lengths: Figure 2-6: Typical four-node element Now we introduce the local velocity components ( 2.36) And element s Reynolds number based on the local velocities and characteristic lengths as: Hence the artificial diffusivity is calculated [11]: ( 2.37) 23

40 ( ) ( 2.38) where ( 2.39) Using equations ( 2.28), ( 2.29) and ( 2.30) we can write the finite element integral equations according to SUPG technique as: ( ( )+ ( 2.40) *( ( *) ( ( )++ ( 2.41) *( ( *) ( ( )++ ( 2.42) The integral equations are calculated over the element s surface area. The weighting function in eqn. ( 2.40) does not contain the upwind contribution as the continuity equation represents the conservation of mass which is independent on the flow direction Discretization of Finite Element Equations of Fluid Finite Element discretization is implemented by approximating both problem geometry and variables using interpolation shape functions. As a result of the pressure equation developed in section 2.2.2, equal order interpolation functions for the pressure and velocity are assumed. For an isoparametric quadrilateral bilinear element the local shape functions are: ( 2.43) 24

41 Where are the local coordinates of the local corner node i of any quadrilateral element, while is the element local axes as illustrated in Figure 2-5. The integer index i takes values from 1 to 4 according to the corresponding node. The dependent variables are approximated by ( 2.44) Where and are the nodal values. The local coordinates are related to the Cartesian coordinates by ( 2.45) Partial derivatives w.r.t. and are related to partial derivatives w.r.t. and by [ ] * + * + * + ( 2.46) is the Jacobian matrix. Area integrals are transformed according to: is the Jacobian determinant. ( 2.47) Equations ( 2.40)-( 2.42) along with eqn. ( 2.44) completes the finite element integral equations. The nonlinear terms in ( 2.41) and ( 2.42) represent a numerical problem. Hughes et al suggested in [22] taking average of the velocity and velocity gradient over the element. We define this average values as: ( 2.48) 25

42 ( 2.49) ( 2.50) and for ease of notation we define: ( 2.51) The discretized integral equations in index notation are: [ ( ) ] ( 2.52) ( ) * ( ) + ( 2.53) ( ) * ( ) + ( 2.54) The viscous terms and the pressure gradient terms are not largely affected by the upwind contribution except the case of higher order elements[11], so it can usually be neglected. Usually the second order terms are integrated by parts using Green theorem resulting in contour integrals. But Habashi et al[25] mention that, integration by parts must be held on all terms of equations ( 2.52)-( 2.54) not the second order terms only which is done also by Brooks et al [11]. This means that the first order terms will be affected resulting in other contour integrals but those integrals are neglected since the Dirichlet boundary conditions are imposed directly not via the contour integral. Doing so, the equations are: 26

43 [ ( ) ] ( 2.55) [( ) ( )( ) ( ) ] ( 2.56) [( ) ( )( ) ( ) ] ( 2.57) In matrix form as following: ( 2.58) [ ] ( 2.59) Where is the consistent mass matrix, is the viscous matrix, is nonlinear convective vector, is the gradient operator, is the generalized matrix for the pressure term in eqn. ( 2.55). The vectors, are the nodal values for the velocity time derivative, velocity component respectively ( ) and is the nodal values of pressure. The matrices in equations ( 2.58) and ( 2.59) are in the global form which means that they are calculated in across every element then an assembly procedure is applied to result their global form. All the numerical integrals are calculated using Gauss-Legendre quadrature rule with three points in each direction. Calculations of the contour integrals in equations ( 2.55)-( 2.57) are explained briefly in the Appendix. 27

44 2.4.3 Finite Element Model of The Elastic Boundary The variational approach is one of the FEM approaches used for problems that are governed by a variational principle. The mathematical model for the elastic boundary is derived by minimization of the potential energy defined in section 2.3 meets the variational method requirement. Equation ( 2.26) could be rewritten as: ( 2.60) The FEM solution to the problem consists of determining the values of at the nodes that extremize the functional [37]. To achieve that, the total variation of the functional is carried out as: ( 2.61) Where n is the number of nodes, equation ( 2.61) results in a system of equations for the nodal values to form the local element equation and then assembled to the global equation Discretization of Finite Element Equation of The Elastic Boundary Using 1-D element with two nodes, define the nodal unknowns in vector form: [ ] ( 2.62) Where. This means that the unknowns are not just the nodal deflection but also the slope at each node. Using the following finite element approximation for the nodal unknowns: ( 2.63) 28

45 Where is a cubic shape function[36]. From equation ( 2.60) the functional U is broken down as the sum of its values over the individual elements: ( 2.64) is the number of elements over the lower boundary of the two-dimensional domain. Using ( 2.61) we can find that[37]: ( 2.65) One should note that each will only contain the nodal values of W associated with that element. Therefore in each element we have: ( 2.66) Using equation ( 2.66) to form the local element equation as: ( 2.67) where ( 2.68) [ ] is the element stiffness matrix[38], is the element length and [ ] ( 2.69) is the local right-hand side. Equation ( 2.67) after assembly forms the global matrix equation as: 29

46 ( 2.70) Where is global matrix, is the global right-hand side and is the vector of all nodal unknowns Time Marching Scheme Time marching scheme is one of the fundamentals of the numerical simulation of time dependent problems. There are two types of time marching; explicit and implicit. An implicit scheme is usually more stable and allows for a larger time step but requires more computer memory and usually used when steady state solution is the target. On the other hand; an explicit scheme requires smaller time steps for stability reasons but needs less computer memory. In the current study, a first order explicit scheme is used since we are interested in a time accurate solution and consequently a small time step is used. The stability of the explicit scheme is achieved by applying Courant condition[11]. This condition depends on a dimensionless number called Courant number and is defined as: ( ) ( 2.71) The stability conditions for the SUPG method is[11]: ( 2.72) Brooks et al[11] show that the mass matrix in equation ( 2.58) is affected by the time marching scheme and consequently its definition is determined according to which part of the system of equations is treated implicitly or explicitly. They replaced the mass matrix by a generalized mass matrix. Hence for the case at hand, in which we treat everything explicitly: 30

47 ( 2.73) Where is the lumped mass matrix. Lumped mass matrix is a member in a family of techniques that have been employed for further computational simplifications[22]. Lumping means diagonalization which can be done by several methods. Hughes et al[22] show that the only diagonalization method which circumvent all difficulties is the row sum method; e.g. for an element in the M* matrix : ( 2.74) Where k is the size of the matrix Solution Algorithm At the end of this chapter we conclude all the work done by listing the steps of the solution algorithm which represents the main guidelines in programing. Equations ( 2.58), ( 2.59) and ( 2.70) are coupled together to simulate the whole problem. The solution algorithm is: Step1. Set initial condition for and i.e. at. Step2. Form the matrices, and Step3. Calculate at and. Step4. Solve for Step5. Calculate Step6. Solve at for Step7. If boundary deflection is required go to Step 8, if not skip to Step 11. Step8. Form the matrix. Step9. Solve for. Step10. Update boundary position by the deflection. Step11. Update time increment. Step12. If additional time steps are required go to Step 2. If not, stop. A flow chart of the solution algorithm is presented in Figure

48 Figure 2-7: Flow chart of the solution algorithm. 32

49 Chapter 3 : Validation Cases 3.1 Introduction In this chapter, the proposed model and numerical technique are validated via two bench-mark problems. The first one is the lid-driven cavity for different Reynolds numbers while the second one is the unsteady flow over a cylinder with vortex shedding. Then the proposed model and numerical technique are applied to an important engineering application which is the pulsating blood flow in elastic artery. 3.2 Lid-Driven Cavity Problem (LDCP) The LDCP is the recirculation of a fluid bounded by a rectangular geometry. The fluid flow is initiated by moving one side of the rigid boundaries at a constant velocity as described Figure Problem Definition As shown in Figure 3-2, the cavity flow consists of a powerful shear driven flow besides the upper moving plate, highly curved streamlines of flow at the four corners and the core-vortex flow in the center of the cavity. The main affecting parameters on the cavity flow are; Reynolds number and aspect ratio. Reynolds number is based on the cavity width such that: ( 3.1) 33

50 Figure 3-1: LDCP-Problem description. Figure 3-2: LDCP -Effect of the aspect ratio and Reynolds number.([39]) 34

51 And the aspect ratio as defined based on the cavity depth and as: As shown in Figure 3-2, for and a symmetric core-vortex formed is in the upper part ( 3.2) of the cavity near the moving boundary. Increasing the only translates the center-vortex to downstream as the center of rotation moves. Also the corner eddies gets larger till a critical Reynolds number is reached after which an instability state occurs[39]. The cavity width affects the characteristic residence time as it represents the interval at which the fluid element experiencing the shearing motion close to the moving plate[39]. Furthermore, the characteristic shear rate can be generally estimated using the cavity depth for shallow cavities where the aspect ratio is. But in the case of deep cavities the shear rate does not strikingly affected by the cavity depth[39]. Also for deep cavities the core vortex is divided into two vertices circulating in opposite directions as shown in Figure Boundary and Initial Conditions Boundary conditions are set as discussed in section 0, lower, left and right boundary are considered fixed walls (no slip conditions);. For the upper moving boundary;. According to the pressure boundary condition, at the four boundaries [40]. The initial condition used in this problem is zero velocity and pressure Results Using bilinear elements mesh (see Figure 3-3), non-dimensional artificial viscosity and ; results are obtained for and, both at. To reach a steadystate, it takes about 1000 time steps for and time steps for. 35

52 Figure 3-3: LDCP-Computational domain. Figure 3-4: LDCP-Pressure isocontours for. 36

53 Figure 3-5: LDCP-Streamlines for. Figure 3-6: LDCP-Image of the cavity flow at taken from Pakdel et al[39]. 37

54 Figure 3-7: LDCP- the -component velocity at =0.5 for. Solution at is compared with the experimental work done by Pakdel et al[39]. The streamlines shown in Figure 3-5 when compared with the experimental work of Pakdel et al[39] shown in Figure 3-6 assures the success of the current model in capturing, qualitatively, the basic flow field features, specifically the location of the center of the main vortex which is calculated to be ( = 0.50, = 0.780) while it is ( = 0.48, = 0.788) using the experimental work of Pakdel et al. For further verification, as shown in Figure 3-7, the -component velocity profile at is plotted and compared with the experimental work of Pakdel et al. [39] that is acquired using two apparatuses; digital particle image velocimetry (1) and laser doppler velocimetry (2). The comparison again assures the success of proposed model. A higher Reynolds number, namely, solution is compared with the numerical work of Hirsch [40] and Chung et al[32].figure 3-8 shows the solution at different dimensionless times; at t = 38

55 1 the solution exhibits a transient stage and then reaches the fully developed state at t =10. For comparison purposes solution of Hirsch[40] with an apparent degree of success at the same times is presented in Figure 3-9. The location of the center of the main vortex which is calculated to be ( = 0.61, = 0.748) while it ( = 0.61, = 0.75) for Hirsch[40]. Again plotting the -component velocity profile and comparing with Hirsch[40] ( 41x41 Mesh using FDM and vorticity transport formulation ) and Chung et al[32] ( 20x20 Mesh using FDM and vorticity transport formulation) in Figure 3-10 which again assures the success of the present work. Figure 3-8: LDCP-Streamlines at different dimensionless times: (a) t =1, (b) t =10 for 39

56 Figure 3-9: LDCP-Hirsch s solution at different times :(a) t =1, (b) t =10 for. [40] Figure 3-10: LDCP- the -component velocity at =0.5 and = 10 for. 40

57 Figure 3-11: LDCP- Mesh-independent solution test using the -component velocity at =0.5 for Re 0. Reaching a mesh independent solution is an important criterion of the numerical solver. So the solution at different finite element meshes is obtained and compared with the experimental results of Pakdel et al[39]. Figure 3-11 shows the x-component velocity profile plotted for 40 40, and meshes (at ) which prove the solution independency of mesh. One of the important issues regarding the cavity flow, is the instability estimated by bifurcation [41]. Bifurcation means a local loss of stability and in our case depends on a critical at which the solution becomes unstable and periodic. Auteri et al[41] found that the critical Reynolds number is The present model is used to obtain solutions at = 500, 1000, 5000 and Figure 3-12 shows the solutions at t=2. A stable vortex is formed for cases =500 and =

58 but for the case of = 5000 and =8000 flow becomes turbulent where complete unstable solution is observed. Figure 3-12: LDCP-Solution at t=2 for different Re :(a)500, (b)1000, (c)5000,(d)

59 3.3 Flow Over a Circular Cylinder (FOCC) Flow over a cylinder is a shear as well as pressure driven flow and adding its practical significance makes it a bench-mark problem. Enormous implementations are relevant to this problem, e.g. the submarine case. Due to its importance, a lot of experiments as well as numerical simulations exist for this problem Problem Definition The problem consists of a circular cylinder surrounded by a rectangular domain, with downstream length 29 times the cylinder radius, upstream length 9 times the radius, domain width 18 times the radius and the flow is moving from left to right (see Figure 3-13). Sumer and Fredsbe[42] found, experimentally, that the main parameter affecting the flow behavior is the Reynolds number ( ). Their observations shows that the flow experiences asymmetric behavior at low and with the increase of the fluid particles begins to separate gradually forming a symmetric wakes behind the cylinder then for higher periodic vortex shedding is achieved. The flow regimes behind the cylinder for different values of Reynolds number are presented in Figure 3-14 taken from Sumer and Fredsbe [42]. Figure 3-13: FOCC-Physical domain description. 43

60 Figure 3-14: FOCC-Different flow regimes(taken from[42]). 44

61 As seen from Figure 3-14, the flow experiences a periodic vortex shedding starting from a relatively low Reynolds number namely = 40. The main parameter controlling the periodic shedding is the Strouhal number,, which is defined as: ( 3.3) Where is the shedding period. An experimental data for different values are collected by Roshko[43];e.g. for =500, = Boundary and Initial Conditions McMullen et al [44] mentioned that a symmetric initial uniform flow results in a symmetric solution with steady symmetric eddies regardless of the Reynolds number value. So, to start the vortex shedding phenomenon, the solution should be interfered by adding a perturbation to the flow field. McMullen et al [44] use an initial condition that already contains a vortex shedding to converge to the periodic vortex[44]. While Brooks et al [11] perturb the flow by adding nodal forces to the boundary layer nodes for number of time steps.. Another technique is suggested by Heshmat [45] in which he uses an upward vertical flow =1) as an initial condition then the periodic shedding is achieved. The initial condition used by Heshmat [45] is adopted in the current work. On the other hand, the set of boundary conditions are; inlet uniform flow, stress-free at exit, for the cylinder surface and for the upper and lower boundaries. There is no boundary condition on the pressure at inlet, so the contour integral in ( 2.55) is calculated Results A clustered finite element mesh of 1100 elements and 1200 nodes (shown in Figure 3-15) is used. Figure 3-16 shown a zoom-in around the cylinder surface to better capture the flow details. The 45

62 artificial viscosity ( ) equals to and the used time step ( ) is The results are presented for = 500; first with symmetric initial condition to get the steady symmetric vortex and then using the upward vertical ( =1) condition to start the shedding. For the symmetric solution it takes 5000 time steps but to get the fully developed vortex shedding it takes time steps. The streamlines and velocity vectors shown in Figure 3-17 and Figure 3-18 respectively present how the separation of the fluid particles occur and form the symmetric eddies behind the cylinder. The full symmetry is captured with an apparent degree of success showing a high degree of robustness and stability of the current model and numerical technique. Pressure isocontours are shown in Figure The broken pressure contour line is due to the use of clustered grid around the cylinder surface resulting in a coarse grid away from it. Figure 3-15: FOCC- Computational domain and the grid 46

63 Figure 3-16: FOCC- Zoom on the mesh around the cylinder. Figure 3-17: FOCC-Velocity vectors of steady symmetric vortices for =

64 Figure 3-18: FOCC-Steady symmetric vortices streamlines for =500. Figure 3-19: FOCC-Pressure isocontours of symmetric vortices for =

65 Figure 3-20: FOCC-Streamlines of shedding vortex for =500: (a) at =0, (b) at =. 49

66 Figure 3-21: FOCC-Pressure contours of shedding vortex for =500: (a) at =0, (b) at =. 50

67 Figure 3-22 FOCC-Streamlines of Heshmat s solution for =500: (a) at =0, (b) at = [45]. Streamlines are shown in Figure 3-20 at the beginning of the period (a) and at half of the period (b). The corresponding pressure isocontours are shown in Figure 3-21 (a) and (b) respectively. The results shows a good agreement with the experimental observations of Sumer and Fredsbe [42] and the numerical solutions of Brooks et al[11] and Heshmat[45] shown in Figure This agreement enforces the numerical solver proposed here and adds another validation success to it. The results show how elegantly we treat the incompressibility-constraint problem known for incompressible flow simulations using the primitive variables formulation. 51

68 Chapter 4 : Pulsating Blood Flow (PBF) 4.1 Introduction The pulsating blood flow problem (PBF) in rigid and elastic tubes is considered in this chapter. A pulsatile flow is one of the most important types of unsteady flows in which the inlet and/or outlet flow boundary conditions are modeled as a wave propagating in or out of the domain of interest. The most crucial application to this type of flows is the study of the dynamics of blood flow in elastic arteries which is the main subject of the Hemodynamics discipline. Flow in an artery as along tube is best described by cylindrical coordinates. The blood flow in an artery is two dimensional axisymmetric. The governing equations would have taken the form: ( 4.1) ( ( * ) ( ( * ) ( 4.2) Where and are the radial and longitudinal directions respectively, and are the velocity components in the same directions respectively. However, in this study the flow in elastic tube will be approximated by flow between two infinite deformable plates, so equations ( 2.19) and ( 2.7) can therefore describe this flow[46]. The constitutive equation describing the response of blood to external stimuli depends on many factors, the most important of which are: i) the relative size of the domain on interest, in which the blood is flowing, to the size of blood constituents e.g. platelets, white blood cells, etc. ii) the physical phenomena we need to include in the simulation e.g. shear-thinning behavior, clotting, fluid 52

69 fading memory etc. In general, blood is considered as a thixotropic viscoelastic incompressible fluid [47]. The size of the arteries is varying considerably and consequently affects the choice of the blood constitutive model. In the case of large arteries like carotid the blood is exposed to high shear rates and so could be considered as a Newtonian fluid[46] [48]. On the other hand, in small arteries and tissues other models should be considered. The scope of the current work is the study of the hemodynamics in the carotid artery in which the Newtonian assumption could be safely adopted. 4.2 Problem Definition The physical domain of interest is shown in Figure 4-1 in which a carotid artery is modeled with a 0.3 cm radius, 12.6 cm length and 0.03 thickness. the artery wall properties are; Young s Modulus of MPa and 1000 kg/m 3 density [28]. While the Newtonian properties of the blood are: density of 1060 kg/m 3 and Pa dynamic viscosity. The computational domain and the grid used in the simulation are presented in Figure 4-2 in which 25 x 75 bilinear quadrilateral elements are used with a total number of nodes of The inlet pulsating flow rate is calculated by an approximate analytical formula, by Nichols et al [48] in which they assume the pressure gradient as a harmonic function resulting in the following analytical form for the blood volumetric flow rate: ( 4.3) Where is the volume flow rate, number of harmonics, is the th frequency, is the tube radius, are constants depending on the value of the dimensionless number which is known as Womersley number. This number is the main parameter affecting pulsatile flows and is defined as[48]: ( 4.4) 53

70 In the current work, four sine harmonics are considered to calculate the volumetric flow rate using eqn. ( 4.3) with the following Womersley numbers and other constants taken from Nichols et al [48]: : = 3.34, = 4.7, = 5.78, = The corresponding constants are ( = 0.78, = 1.32, = -0.74, = -0.41), ( = 0.64, = 0.74, = 0.78, = 0.81), ( =10.65 o, = o, = 26.5 o, = o ) and ( =31.5 o, = 20.7 o, = 16.2 o, = 13.7 o ) [48]. 54

71 Figure 4-1: PBF- Carotid artery model. Figure 4-2: PBF-Finite element mesh of the Carotid artery model. 55

72 Figure 4-3: PBF - The pulsating volumetric flow rate versus time. The resulting volumetric flow rate is shown in Figure 4-3. As shown in the figure, the periodic time is = 1.37 sec and maximum volumetric flow rate = cc/sec. The characteristic velocity used in the definition of non-dimensional numbers is the maximum flow velocity ( ). The calculated is Boundary and initial conditions The no-slip condition is imposed on the upper and lower walls and for the pressure ( ). The pulsating flow rate wave shown in Figure 4-3 used as an inlet boundary condition and for the outlet the pressure is set to 85 mmhg. The initial condition is zero for the velocity and for the pressure as well. In the following two subsections, two cases are considered; the first is the blood flow in rigid artery while in the second the artery elasticity is taken into account. 56

73 4.4 Pulsating Blood Flow in a Rigid Artery (PBFRA) The used numerical values are artificial viscosity ( ) = and time step ( ). The following results are for a total time of two periods which corresponds to 6900 time steps. The evolution of pressure and volumetric flow rate is traced in two cross-sections; S1 and S2 (shown in Figure 4-1) S1 location is just one node away from the inlet boundary node at the centerline and S2 is the same but at exit. The inlet pressure is also traced. Figure 4-4 shows the pressure as well as the volumetric flow rate distribution. The time delay between the leading pressure wave and the volumetric flow rate wave is approximately 0.15 sec. The flow rate wave is plotted in Figure 4-5 at both locations S1 and S2 as well as the pressure in Figure 4-6. The results show that the mean flow value at S2 is less than the mean flow at S1. The velocity profiles and velocity magnitude distribution at different axial locations (A, B, C, D, E and F) for different time points over the flow wave period is presented in Figure 4-7 and Figure 4-8 respectively. These results are consistent with the conclusion reached by Hyun [46] in that the backflow occurs around point D shown in Figure

74 Figure 4-4: PBFRA- Pulsating inlet flow rate and pressure. Figure 4-5: PBFRA-Volumetric flow rate at both S1 and S2. 58

75 Figure 4-6: PBFRA - Pressure distribution at both locations S1 and S2. 59

76 Figure 4-7: PBFRA -Velocity profiles at different axial locations at different time points. 60

77 Figure 4-8: PBFRA -Velocity distribution at different time points. 61

78 4.5 Pulsating Blood Flow in an Elastic Artery (PBFEA) In this section the deformability of the boundary is taken into account. The elastic boundary model in section is used considering the pressure as the load and the artery boundary a simply supported beam. Due to the problem symmetry, the deflection of the upper boundary is calculated only. Using the same numerical values for the parameters ( ) the presented solution is obtained after 6900 time steps. Deflection calculations code is activated after a reasonable number of time steps after which the pressure distribution inside the domain becomes harmonic. In Figure 4-9, the deflection variation and the pressure wave are plotted at five sections along the artery (B1, B2, B3, B4 and B5). B1 and B5 are at the same axial location of S1 and S2 respectively but on the upper wall. The other three axial locations B2, B3 and B4 are at =3.36, = 6.3 and =8.3 cm respectively. For further details the deflection of the whole boundary is shown in Figure 4-10 and Figure 4-11 at different time points (A, B, C and D) over the inlet pressure wave period and the deflection flooded by the pressure field and velocity field respectively. Finally the pressure gradient along the artery wall at the same time points (in Figure 4-10) is plotted in Figure These results are consistent with the conclusion reached by Figueroa et al [28] in that the pressure and deflection waves are in-phase. Also the leading time of the pressure wave is 0.15 while it for Figueroa et al [28] The maximum deflection is cm and for Figueroa et al [28] is cm. To show the importance of taking the wall elasticity into account; the pressure fluctuation at B1 and the wall shear stress at B3 is plotted in Figure 4-13 and Figure 4-14 respectively for the case of rigid wall and deformable wall at two numbers. The results show that decreasing the number is highly affecting the pressure distribution and the wall shear stress. 62

79 Figure 4-9: PBFEA - Boundary deflection and boundary pressure at different axial locations. 63

80 Figure 4-10: PBFEA Pressure field in the boundary layer blood flow. 64

81 Figure 4-11: PBFEA Velocity field in the boundary layer blood flow. 65

82 Figure 4-12: PBFEA-Pressure gradient along the boundary at different time points of the pressure wave. Figure 4-13: PBFEA Pressure wave at B1 for rigid and deformable wall. 66

83 Figure 4-14: PBFEA Wall shear stress at B3 for rigid and deformable wall at different numbers. 67

84 Chapter 5 : Summary and Conclusions 5.1 Summary In chapter one, A quick review is given for the two well-known numerical techniques namely the finite difference method and the finite element method with more emphasis on the upwinding techniques specially the SUPG method which is used in current work. Then the main problem regarding the numerical simulation of incompressible flow is discussed in details, namely the incompressibility constraint of the continuity equation being not an evolution equation like the case for compressible flows. Finally a survey regarding the two-way coupling of the boundary deformability and the flow field is presented. In chapter two, the mathematical model describing the fluid flow and the boundary deformability is presented. The theoretical reasoning for modifying the continuity equation to overcome the incompressibility constraint is presented. Also the numerical technique and discretization method are explained in details for both the fluid and the elastic boundary. At the end of the chapter the time marching algorithm is presented followed by the solution algorithm. Finally in chapter three, the mathematical model and the numerical technique proposed in this work is validated via two bench-mark problems. The first is the lid-driven cavity flow and the second is the flow over a circular cylinder with vortex shedding. The results are compared against both numerical and experimental published work. Then the proposed model and the numerical technique are used to approach an important engineering application which is a pulsating blood flow in rigid and elastic large arteries. An idealized model of common carotid artery is used. For the rigid artery case, results of pressure variation versus the flow rate and velocity profiles at different locations along the artery are obtained and compared with published works. The same work is repeated taking into account the 68

85 artery deformability, the two-way coupling between the wall artery deflection and the pressure is simulated and results are compared with published work. 5.2 Conclusion A new model for simulating pulsatile incompressible flow in elastic tube is successfully developed and implemented. The incompressibility constraint problem is treated by the artificial viscosity technique developed by Habashi et al[25] for steady-state flows and shown to be equivalent to other techniques developed for unsteady flows. The Newtonian constitutive equation assumption is shown to be accurate enough to represent the blood behavior in large arteries. The streamline/upwind Petrove-Galerkin finite element technique is shown to be highly effective and robust in simulating flows accounting with deformable boundaries. The two-way coupling between the flow field and the boundary elasticity is considered and the wall deformability effect is found to be dependent on the IFSI number. 5.3 Recommendation for Future Work When it comes to the physics of the fluid under consideration, it is well known that the blood belongs to the category of complex fluids. So, to account for this property, at least the viscosity-sheardependency if not other complex properties should be considered. In regards to the boundary conditions, different types of boundary condition on the artery exit could be used e.g. impedance and resistance boundary condition[49]. Finally, the proposed model could be extended to be three-dimensional axi-symmytric in space in order to be able to simulate arteries including their branches. 69

86 Appendix Contour Integral Calculations For the following contour integral in equation ( 2.55): (1) Where n is the outward unit normal to the boundary. In Cartesian two-dimensional coordinates, the differential element length is: Since and are functions of the local coordinates then: (2) (3) So the differential element length could be rewritten as, ( * ( * (4) Using the four node quadrilateral finite element geometry shown in Figure A-1. The contour integral is: (5) where ( * ( * (6) 70

87 Figure A-1: Four node element geometry. ( * ( * (7) To calculate the unit normal vectors we define the following points: (8) Then, the boundary tangent vectors are:: (9) So, And the corresponding unit tangent vectors are: 71

88 (10) Finally the boundary normal vectors are: (5) Where is the unit vector in the -direction. Then the contour integral will be calculated on each side of the element as follows: (( ) ( * ( *, (6) (( ) ( * ( *, (7) (( ) ( * ( *, (8) (( ) ( * ( *, (9) 72

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93 ممخص الرسالة أحد الصعوبات التى تواجه جهود بناء نماذج لمحاكاة سريان الموائع هى كيفية اخذ التفاعل الثنائى بين الموائع و الجد ارن الصمبة فى االعتبار. و هى مشكمة تواجهنا سواء كان المائع يسرى داخل او حول جد ارن صمبة و سواء كان السريان نبضيا او ثابتا مع الزمن. و ألن تدف الدم فى الش اريين المرنة هو احد اهم تطبياات هذ المسألة كان موضوع هذ الرسالة هو محاكاة التدف النبضى لمدم فى الش اريين المرنة. باعتبار الدم مائع نيوتونى يمكن تاريب تدف الدم فى الش اريين الواسعة كسريان طباى متغير مع الزمن فى بعدين. و لهذا تستخدم معادالت نافيير ستوكس لنمذجه السريان الال منضغط فى مستوى بين لوحين مستويين متوازيين. نحتاج لمعاممة خاصة لمعادلة االستم ارر لمكتمه وذلك النها تصمح فى حالة السريان الال منضغط كمعادلة قيد عمى السرعات و ليست معادلة لمتغير الزمنى لمكثافة. و لهذا الغرض يضاف معامل لزوجة مصطنعة لتحويمها الى معادلة بواسون فى ضغط السائل. لتصبح المعادالت الحاكمه ثالث معادالت تفاضمية مرتبطة فى ثالثة متغي ارت. تم تطوير نموذج لمحل العددى لهذ المعادالت التفاضمية الال خطية المرتبطة معتمد عمى اسموب (SUPG) فى طرياة العنصر المحدود و وضعه عمى هيئة مخطط لمحل. لمتحا من دقة النموذج تمت ماارنة نتائجه بنتائج حمول منشورة لمسألتين شهيرتين السريان فى فجوة مستطيمة مدفوعا بحركة غطاءها الدوامات الناتجة من سريان سائل لزج حول اسطوانة دائرية -1-2 و اظهرت الماارنة تااربا كيفيا حا دقة النموذج. من اجل نمذجه سريان الدم فى الش اريين ذات الجد ارن المرنة البد من اعتبار التفاعل بين الدم و الجد ارن. يتم ذلك باعتبار الجد ارن كمرة مرنة بسيطه معرضة لحمل عرضى منتشر وموزع و متغير مع الزمن هو ضغط الدم. يستخدم مبدأ الحد األدنى لطاقة الوضع) لالجسام المرنة التى تكون تشوهات شكمها صغيرة( لحساب هذا التفاعل. تحل مسألة تدف المائع اوال ويحسب توزيع ضغطه عمى الجد ارن ثم يستخدم هذا الضغط كحمل موزع عمى الكمرة الممثمة لمجد ارن لحساب انحناءها. ثم تحل مسألة تدف المائع فى المحظة التالية عمى الجد ارن المنحنية. و هكذا يتم تك ارر هذ العممية حتى نهاية زمن المحاكاة ( و هو زمن نبضة كاممة ). أ

94 استخدمت لغة (FORTRAN) فى عمل كود لتنفيذ النموذج العددى لحل المعادالت التفاضمية الحاكمه تم تجربة الكود اوال باعتبار الجد ارن صمبة ثم تم تطبياه عمى الحالة المرجوة لمجد ارن المرنة. هذا وقد قدمت النتائج فى اشكا ل بيانيه ورسومات عديدة لتوزيع السرعات و ضغط الدم و انحناءات جد ارن الش اريين عند لحظات مختمفةتاليه لبدء النبضه و مواضع متباينة عمى الشريان وهى معمومات تفيد فى تشخيص االم ارض. ب

95 مهندس : تاريخ الميالد: الجنسية: تاريخ التسجيل: تاريخ المنح: القسم: الدرجة: المشرفون: أسامة عمي عبد المنعم البنهاوي 1188 \ 8 \ 14 مصري 2112 \ 3 \ \...\... الرياضيات والفيزياا الهندسية ماجستير العموم ا.د. محمد سمير طوسون د. عمرو جمال جويمي الممتحنون: أ.د. مصطفى صابر مصطفى ابودينه )أستاذ باسم الرياضيات- كمية العموم- جامعة الااهرة( )الممتحن الخارجي( أ.د. ممدوح عبد الحميد فهمي )الممتحن الداخمي( أ.د. محمد سمير طوسون )المشرف الرئيسي( د. عمرو جمال جويمي )مشرف( عنوان الرسالة: محاكاة عددية لتدفق نبضي لسائل لزج غير قابل للنضغاط في أنابيب مرنة. الكممات الدالة: سائل لزج غير قابل النضغاط طرياة العنصر المحدود و أسموب (SUPG) ديناميكا الموائع الحسابية تفاعل المائع - الصمب ممخص الرسالة: يستخدم أسموب (SUPG) لد ارسة تصرف سائل لزج غير قابل النضغاط في االنبايب المرنة. تعتمد المحاكاة عمى حل الصغية المتغيرة مع الزمن و الثنائية االبعاد لمعادالت نافيير ستوكس مع معادلة االتصال. يضاف لمعادلة االتصال معامل لزوجة مصطنعة وتصبح معادلة معدلة لحساب الضغط لمتغمب عمى مشكمة معادلة االستم ارر كونها تمثل قيد في حل المعادالت. مرونة الحدود ستؤخذ في االعتبار كانها قضيب مثبت تثبيت بسيط يتعرض لحمل متغير مع الزمن في االتجا العرضي ويعتبر هذا الحمل هو ضغط المائع. تعتمد نمذجة انح ارف الحدود عمى مبدأ الحد األدنى لطاقة الوضع في حالة التاوس لنظام االنح ارفات الصغيرة. تم تطبي هذا النموذج لد ارسة سريان الدم في الش اريين المرنة بدرجة مناسبة من النجاح.

96 محاكاة عدد ة لتدفق نبض لسائل لزج غ ر قابل لإلنضغاط ف أناب ب مرنة مهندس/ إعداد أسامة عل عبد المنعم البنهاوي رسالة مقدمة إلى كل ة الهندسة - جامعة القاهرة كجزء من متطلبات الحصول على درجة ماجست ر العلوم ف الم كان كا الهندس ة تحت اشراف أ.د. محمد سمير طوسون أستاذ الم كان كا الهندس ة قسم الر اض ات والف ز قا الهندس ة كل ة الهندسة جامعة القاهرة د. عمرو جمال جويلي مدرس قسم الر اض ات والف ز قا الهندس ة كل ة الهندسة جامعة القاهرة كل ة الهندسة - جامعة القاهرة الج زة - جمهور ة مصر العرب ة 5102

97 محاكاة عدد ة لتدفق نبض لسائل لزج غ ر قابل إعداد لإلنضغاط ف أناب ب مرنة مهندس/ أسامة عل عبد المنعم البنهاوي رسالة مقدمة إلى كل ة الهندسة - جامعة القاهرة كجزء من متطلبات الحصول على درجة ماجست ر العلوم ف الم كان كا الهندس ة كل ة الهندسة - جامعة القاهرة الج زة - جمهور ة مصر العرب ة 5102