Surrogate models for incompressible fluid dynamics in periodic regime Open Access

Abstract

Computational fluid dynamics (CFD) plays an important role in modeling the system of Left Ventricle Assist Device (LVAD), which is now a main solution for the patients who reach to an end-stage heart failure. However, because the post-surgery conditions create an abnormal hemodynamics that may eventually lead to long-term complications and diseases in aorta. CFD can be used to understand the interplay between post-op morphology, hemodynamics and clinical outcomes. However, a main fallback of using CFD is the high computational cost. In order to reach to accurate solutions with an inexpensive cost, a surrogate model for the unsteady Navier-Stokes equation is preferred. Acknowledging that heart has a periodic behavior in beats, we try to use the solution for steady Navier-Stokes equation to approximate the time-average solution for unsteady Navier-Stokes equation in periodic regime. Because of the existence of the non-linear term in the Navier-Stokes equation, this approximation will present some differences between those two solutions. By using FreeFem++ to run the numerical simulations on different geometries for those two problems, this paper will discuss how the differences between those two solutions will be affected by various of factors including the amplitude of boundary conditions, number of time steps at which the unsteady problem is solved, the quality of meshes and different geometries. From the results obtained from the numerical tests, it is concluded that the geometry will have a dominated effect on the differences between those two solutions. When the geometry is regular and indicates a non- linear term approaching to 0, the amplitude of the boundary condition will have a polynomial- like relationship with that difference. The quality of messes and number time steps will only affect the computational cost but the differences between those two solutions.

Table of Contents

Contents

1 Introduction 9 1.1 Motivations......................................... 9

1.2 Outline ........................................... 11

2 The Stokes and the Navier-Stokes Equations 13

2.1 TheStokesflowproblem.................................. 13

2.2 TheNavier-Stokesproblem ................................ 15

2.3 Numerical approximation of the (Navier)-Stokes equations . . . . . . . . . . . . . . . 15

3 Surrogates for periodic equations 17

3.1 Thelinearcase(Stokes) .................................. 17

3.2 Thenonlinearcase(Navier-Stokes)............................ 18

4 The numerical code 21

4.1 TheFreeFem++environment............................... 21

4.2 Numericaltest1:flowinacylinder............................ 21

4.2.1 Themeshesandboundaryconditions ...................... 21

4.2.2 Thealgorithmtosolvethesteadyproblem ................... 22

4.2.3 Thealgorithmtosolvetheunsteadyproblem . . . . . . . . . . . . . . . . . . 24

4.2.4 Testthecode’scorrectnessincylinder ...................... 24

4.2.5 Variablessetinthecodefortesting ....................... 24

4.3 Numericaltest2:flowinaroom ............................. 24

4.3.1 Themeshesandboundaryconditions ...................... 24

4.3.2 Thealgorithmtosolvetheproblem ....................... 25

5 Numerical results 27

5.1 Testcase1:Flowinacylinder .............................. 27

5.1.1 Thelinearcase:codetesting ........................... 27

5.1.2 Thenonlinearcase................................. 27

5.2 Testcase2: Flowinaroom(Navier-StokesProblem). . . . . . . . . . . . . . . . . . 33

5.2.1 Changingthequalityofthemeshes........................ 33

5.2.2 Changingtheinflowfactoroftheboundarycondition . . . . . . . . . . . . . 34

5.3 Discussion.......................................... 35

5.3.1 Forthenonlineartermintheequation...................... 35

5.3.2 Fortheirregulargeometry............................. 35

5.3.3 Future directions.................................. 36

References.................................. 39

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	LVAD CFD Navier Stokes Equation
Committee Chair / Thesis Advisor	Alessandro Veneziani, Emory University
Committee Members	Manuela Manetta, Emory University Steve La Fleur, Emory University

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