DEVELOPMENT OF AN INVERSE DESIGN METHOD FOR PROPELLERS WITH APPLICATION ON LEFT VENTRICULAR ASSIST DEVICES

ENTWICKLUNG EINER INVERSEN AUSLEGUNGSMETHODE FÜR PROPELLER UND DERER ANWENDUNG AUF LINKSVENTRIKULÄRE UNTERSTÜTZUNGSPUMPEN

Der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades

Dr.-Ing.

vorgelegt von

Mihai Bleiziffer geb. Miclea

aus Hermannstadt Rumänien Als Dissertation genehmigt von der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg

Tag der mündlichen Prüfung: 13.07.2017

Vorsitzender des Promotionsorgans: Prof. Dr.-Ing. Reinhard Lerch

Gutachter: Prof. Dr.-Ing. habil. Antonio Delgado Prof. Dr.-Ing. Alexandrina Unt˘aroiu

II This page intentionally left blank

III Copyright Hinweis Die vorliegende Arbeit verwendet die Schriftart Utopia®. Sie ist von Adobe zur Verwendung durch die TEX Users Group (TUG) freigegeben und unterliegt dem Copyright. Der mathematische Zeichensatz wird durch das Paket Fourier-GUTenberg über das Comprehensive TEX Archive Network (CTAN) zur Verfügung gestellt. Copyright 1989, 1991 Adobe Systems Incorporated. All rights reserved. Utopia® Utopia is either a registered trademark or trademark of Adobe Systems Incorporated in the United States and/or other countries. Used under license.

IV To Helen and Sophie

V

Acknowledgments

First, I want to express my deepest gratitude to my mentor Prof. Dr.-Ing. habil. Antonio Delgado for guiding me during the past 9 years, finally adding more value to the present thesis by the suggestions he made during the review. It is due to him that I could go deeper in the fantastic world of fluid me- chanics and turbomachinery while working at the LSTM in Erlangen. He not only guided me all these years but he gave me a platform on which I could freely create and develop myself. I have to thank him for encouraging and making my visiting research stage at the University of Virginia possible. He helped me overcome weaknesses and transform them into strengths and further improve myself.

I want to express my gratitude to Prof. Alexandrina Untaroiu from Virginia Polytechnic Institute for the suggestions she made to the present thesis and for accepting to be a member in the doctoral examination committee. I also want to thank her for inviting me for a research stage at the University of Virginia while I could work on the validation of the ADAP code.

I want to thank Prof. Dr.-Ing. Jovan Jovanovic who not only introduced me to the world of turbulence but also advised me every time I had questions. I also want to thank him for accepting to be the chairman of my doctoral committee.

The former group leader of the turbomachinery group at LSTM, Prof. Dr.-Ing Philipp Epple I want to thank for inviting me to join his group and for sharing with me his fan and blower design experi- ence. We have worked together at many interesting industrial projects, where I was able to extend my knowledge in aerodynamics.

I want to thank Prof. Dr. Ing. Özgür Ertunç for welcoming me in his research group at LSTM and for many advices he gave me over the time. I am also thankful to Prof. Marc Drela from the MIT for his advice on the numerical cascade simulations and for inspiring me with his work.

I express my appreciation to Prof. Dr.-Ing. Jens Peter Majschak from the TU-Dresden who opened me the way to the academic world and offered me the first job as a researcher after finishing my studies.

I am also grateful for the support given by the leaders of the turbomachinery group, Bettina Grashof and Matthias Semel. I want to thank my former colleagues Dr. Frauke Groß, Judith Forstner, Dr. Ana Zbogar-Rasic, Jens Krauß, Dr. Manuel Münch, David Botello-Payro, Balkan Genc, and Dr. Giovanni Luzi for the good time we had at LSTM. I also want to thank my bachelor and master students, es- pecially Patrick Töpfer and Ulrich Schlegel for their work with me and for performing most of the measurements presented in this thesis.

I am thankful to Dipl.-Ing. Klaus Epple from Cardiobridge Gmbh for the good work during the ZIM funded project, and for providing the first version of the MOCK setup as well as the original P 14F pump. I thank him and the company for allowing me to publish pictures and data of the 14F Reitan Catheter Pump. I also want to thank MD PhD. Oyvind Reitan from the Lund University for sharing with me his work and for helping me understand the physiological impact of a blood pump.

Measurements on test rigs would not have been possible without the support given by the LSTM workshop. First I want to thank the head of the LSTM workshop Hermann Lienhart. I want to thank Heinz Hedwig and Herbert Kaiser from the mechanical workshop for building my test rigs. I also want to thank Rolf Zech, Franz Kaschak and especially Horst Weber for helping me build the LDA setup and

VII all electronic devices needed for the measurements. The IT support provided by Sebastian Röhl and Thorsten Bielke is gratefully acknowledged.

For the time I have been at LSTM the secretariat was one of its central points, and its importance grew after I left the institute. I want to thank Mrs. Georgia Bouna, Mrs. Isolina Paulus, Mrs. Anke Lutz and especially Mrs. Franziska Jung for the good cooperation, for organizing my work and helping me keep a good contact to the institute. I also want to thank the administration of LSTM especially to Dr. -Ing. Bernhard Mohr, Sonja Hupfer and Claudia Gerstacker.

The financial support for my research stage at the University of Virginia was granted by the Graduate School for Advanced Optical Technologies (SAOT) in Erlangen. I want to thank Dr. Dubravka Melling, PD. Dr Andreas Bräuer and Joana Stümpfig Barrinho for their support and for the good time I had during the SAOT academies. The support of the Edmund-Bradatsch-Foundation for printing this thesis is also gratefully acknowledged.

I am grateful to my colleagues and friends Charles Comeau, Aleksandar Sekularac and Tim Weiland for the hard work they have done in reviewing the present thesis. I also thank my friend Jaswinder Singh for a first review and for a lot of support in numerical fluid mechanics. My friend Dr. Ionut Georgescu I would like to thank for encouraging me to follow my dream and for offering me his help every time I needed. Without his advice in Matlab a part of this work would not have been possible. I would like to thank Mrs. Renate Krämer, our friend and neighbour in Worms, for the days I could write undisturbed at her home.

I would like to thank my parents for encouraging me to learn and keep persevering in my passion and for the financial support during my studies. I want to thank my grandparents Alice (1925-1989) and Gheorghe Istodorescu (1922-2015) for raising me and teaching me to always be curious and seek for answers. My parents-in-law were a huge source of support for which I am grateful.

For her continuous support and unselfish love I would like to thank my wife Helen. She encouraged me to pursue my passion and has been patient the many nights and weekends in the past 9 years while I was working at this thesis. Her support has extended after the birth of our daughter Sophie for who she cared also in my place so I was able to complete this work. Köszönöm és szeretlek teljes szívemb˝ol!

VIII Contents

List of Figures XI

List of Tables XV

1 Introduction 1 1.1 Short overview of LVAD used in the treatment of cardiogenic shock ...... 2 1.2 Design and analysis methods for VADs ...... 5 1.3 Outline and objectives of the dissertation ...... 5

2 Selected aspects in relevant areas for the design of VADs 7 2.1 Design consideration for VADs ...... 7 2.1.1 Human circulatory system ...... 7 2.1.2 Blood composition and its physical properties ...... 8 2.1.3 Considerations on blood damage for VADs ...... 10 2.1.4 Design requirements (duty point of a LVAD) ...... 11 2.1.5 Head characteristics ...... 12 2.2 Governing equations for fluid dynamics and aerodynamics ...... 13 2.2.1 Governing equations for fluid dynamics ...... 14 2.2.2 RANS turbulence modeling ...... 16 2.2.3 Two-dimensional flows for aero- and hydrodynamics applications ...... 18 2.3 Air- and hydrofoil families: laminar NACA 6-Digit series ...... 19 2.4 Propeller design methods ...... 21 2.4.1 Axial momentum theory for propellers ...... 22 2.4.2 Blade element theory ...... 26 2.4.3 Design theory using the radial loss model proposed by Betz and Prandtl ...... 28 2.4.4 Design theory using the radial loss model proposed by Goldstein ...... 33 2.4.5 Design method correction for moderately loaded propellers ...... 34 2.4.6 Design theory using lifting-line and vortex-lattice theories ...... 34 2.5 Design and construction of a closed loop measurement test rig ...... 36 2.5.1 Test rig set-up and construction ...... 36 2.5.2 Methods and materials ...... 40 2.5.3 Data recording ...... 44 2.5.4 Initialization and characteristics of the LDA flow-rate measurement system . . . . . 48 2.6 Design and construction of a MCL ...... 50

3 Presentation and discussion of the results 53 3.1 Procedure for designing propellers ...... 53 3.2 Development of a propeller design and analysis code ...... 54 3.2.1 Numerical solution for thin airfoil cascades ...... 54 3.2.2 Sensitivity check for the CVL computational model ...... 58 3.2.3 Propeller design framework ...... 68 3.2.4 An iterative method for correcting the lift distribution for propellers with small pitch- to-chord ratios ...... 71

IX Contents

3.2.5 BEM propeller analysis procedure using the Goldstein loss model ...... 73 3.3 Design procedure for multiblade open-water propellers ...... 74 3.3.1 Challenges in designing multiblade propellers ...... 74 3.3.2 CFD setup and simulations ...... 76 3.3.3 Numerical errors of CFD simulations ...... 76 3.3.4 Geometry, grid generation and grid study for the problem ...... 77 3.3.5 Design parameter study ...... 80 3.3.6 Discussion of results ...... 84 3.4 Design procedure for an encased propeller used as a LVAD ...... 90 3.4.1 Experimental and CFD assessment of the 14F RCP pressure-flow performance . . . 90 3.4.2 Analysis of the flow mechanism in the RCP ...... 97 3.4.3 Design and optimization of new VAD propellers ...... 100 3.4.4 Validation of new designs by test-rig measurements ...... 110 3.4.5 Numerical estimations of blood damage ...... 112 3.4.6 Time-dependent CFD simulations of encased propeller VADs ...... 118

4 Conclusion and Outlook 121

Bibliography 123

A Derivation of the scalar shear stress σ from the Navier Stokes equations 131

B Computation of RBC mass-flow 135

C Propeller helping device 137

X List of Figures

1.1.1 The Intra-aortic balloon pump (from [115]) ...... 2 1.1.2 Hemopump®[97] ...... 3 1.1.3 Impella 2.5 ® von Thoratec® and its placement [102] ...... 3 1.1.4 14F Reitan catheter pump (figure courtesy of Cardiobridge Gmbh/Hechingen) . . . . 3 1.1.5 Physiological position of the 14 F RCP in the upper aorta (figure courtesy of Cardio- bridge Gmbh/Hechingen) ...... 4

2.1.1 Human circulatory system [1] ...... 8 2.1.2 Rheological properties of blood ...... 9 2.1.3 Viscosity vs shear rate for different hematocrit levels (adapted from [29]) ...... 9 2.1.4 Shear stress factors contributing to the blood damage ...... 10 2.1.5 Aortic pressure (AoP) distributions for healthy and severe CS - cases ...... 12 2.1.6 Cordier diagram following Lewis [52]showing the ideal and existing design points set foraLVAD...... 13 2.2.1 Local time-average of a fluctuating quantity G ...... 15 2.2.2 Energy spectrum of turbulence as function of the wave number k with the application range of CFD turbulence models (adapted from Hirsch [38, pp.88]) ...... 17 2.2.3 Two-dimensional slice (airfoil) of a blade ...... 18 2.3.1 Sketch of a modern airfoil composed of a mean-line (camber-line) and a thickness distribution ...... 19 2.3.2 Comparison between theoretical and experimental lift slopes of airfoils [72] ...... 20 2.4.1 Stages of propeller design and analysis ...... 21 2.4.2 Exaggerated sketch showing the basis for the actuator disk theory ...... 23 2.4.3 CT,CP and η computed from the axial influence factor a ...... 25 2.4.4 Sketch showing both axial and angular momentum components for a propeller . . . . 25 2.4.5 Blade element ...... 26 2.4.6 Forces and velocities acting on a blade element ...... 27 2.4.7 Propeller wake with helical vortices and the concept of displacement velocity (v’) . . 28 2.5.1 System characteristics of pump indicating different losses inside the measurement rig 36 2.5.2 CAD sketch of the loop test rig concept ...... 38 2.5.3 CFD computed pressure distribution downstream the propeller ...... 39 2.5.4 Bore for pressure measurement according to DIN ...... 39 2.5.5 Ready test rig ...... 40 2.5.6 Rosemount G151 pressure transducer ...... 41 2.5.7 Principle of laser Doppler shift ...... 42 2.5.8 Two-beam LDA setup ...... 42 2.5.9 Laser and optical measurement setup ...... 43 2.5.10 Left hand side: Bandpass filter (LSTM-invent-895-102), right hand side: Philips PM3295A oscilloscope ...... 44 2.5.11 BBC Görtz LSE 01 Doppler signal processor (frequency tracker) ...... 44 2.5.12 NI 9178 USB chassis with 3 NI 9215 BNC input modules ...... 45 2.5.13 Front panel of the measurement program ...... 45

XI List of Figures

2.5.14 Measurement software layout in LabView (adapted from [108]) ...... 45 2.5.15 Computation of the friction velocity (adapted from [108]) ...... 47 2.5.16 Computation of the stationary turbulent velocity profile (adapted from [108]) . . . . 48 2.5.17 Pictures showing the setup for the benchmark measurement from left to right: inflow container (at ~2 m height), flow path in the glass pipe with LDA and throttle ...... 48 2.5.18 comparison of the proposed turbulent profiles ...... 49 2.5.19 Comparison of the volumetric flow-rate computed by using the three turbulent pro- files and the balance measurement ...... 49 2.6.1 Analogy between the human circulatory system (LHS[1]) and the Mock (RHS)) . . . . 50 2.6.2 LabView measurement panel for the mock ...... 51 2.6.3 Validation of the MCL by comparing AoP to literature and human measurements . . . 52

3.1.1 General procedure for designing propellers ...... 53 3.2.1 Sketch of a cylindrical meridian surface through an axial turbomachine and the re- sulting cascade ...... 55 3.2.2 Panel method for thin airfoils (reprinted from Miclea-Bleiziffer et al. [59], with per- mission from Elsevier) ...... 55 3.2.3 Infinite row of vortexes (cascade) (reprinted from Miclea-Bleiziffer et al. [59], with per- mission from Elsevier) ...... 56 3.2.4 Evaluation of the thin flat plate at AOA 5° with points placed at 0.1 c ...... 59 = · 3.2.5 Evaluation of the thin flat plate at AOA 5° with points placed at 0.01 c ...... 59 = · 3.2.6 Evaluation of the thin flat plate at AOA 5° with points placed at 0.001 c ...... 60 = · 3.2.7 CP distribution around a flat plate computed at variable distance from camber . . . . 60 3.2.8 cl results obtained analytically (red) and by the CVL method (green and blue) for cir- cular arcs with different cambers (reprinted from Miclea-Bleiziffer et al. [59], with per- mission from Elsevier) ...... 61 3.2.9 Solution of the flat plate @ AOA 5° with 50 panels ...... 62 = 3.2.10 Solution of the flat plate @ AOA 5° with 500 panels ...... 62 = 3.2.11 Evaluation of a thin 5% circular arc cambered foil at AOA 5° ...... 63 = 3.2.12 Evaluation of a thin 15% circular arc cambered foil at AOA 5° ...... 64 = 3.2.13 Lift ratio k as a function of the solidity σ and stagger angle λ (reprinted from Miclea- Bleiziffer et al. [59], with permission from Elsevier) ...... 64 3.2.14 Figure showing computed streamlines around flat plate cascade with a t/l 1, at = AOA 5° and λ 30°,50°,70° ...... 65 = = 3.2.15 Evaluation of a NACA mean-line (a 0.8 ,c 0.75) at ideal AOA ...... 66 = l = 3.2.16 Evaluation of a NACA mean-line (a 0.8,c 1) at ideal AOA ...... 66 = l = 3.2.17 Evaluation of a NACA mean-line cascade (a 0.8,c 0.75), λ 50°and t/l 1 . . . . 67 = l = = = 3.2.18 Evaluation of a NACA mean-line cascade (a 0.8,c 1), λ 50°and t/l 1 ...... 68 = l = = = 3.2.19 Goldstein factor (G) distribution for propellers with 4 and 6 blades ...... 68 3.2.20 Overview of the design program ADAP (V 0.973) ...... 70 3.2.21 ADAP logo, version and copyright agreements written on the top of every file . . . . . 70 3.2.22 Design framework for propellers with high pitch-to-chord ratio as implemented in ADAP (V 0.973) (reprinted from Miclea-Bleiziffer et al. [59], with permission from El- sevier) ...... 72 3.2.23 Convergence of the ADAP propeller design code ...... 73 3.2.24 Figure 1 and Figure 2 from ADAP showing the stacked airfoils and relevant design data 73 3.2.25 Figure 3 and Figure4 from ADAP showing relevant propeller design data ...... 74 3.2.26 Structure of the BEM analysis method ...... 75 3.3.1 Setup of the studied problem (reprinted from Miclea-Bleiziffer et al. [59], with per- mission from Elsevier) ...... 77 3.3.2 Propeller blade geometry in TurboGrid with figured sections ...... 78 3.3.3 Convergence of solution ...... 79 3.3.4 Details of numerical mesh ...... 79

XII List of Figures

3.3.5 Grid study showing the dependence of K and K at design point V 0.3[m/s](J T Q = = 0.24) upon the number of grid elements (reprinted from Miclea-Bleiziffer et al. [59], with permission from Elsevier) ...... 80 3.3.6 Thickness study results for a 4 bladed propeller (reprinted from Miclea-Bleiziffer et al. [59], with permission from Elsevier) ...... 81 3.3.7 C distribution of both propellers at J 0.7 and 0.7 span ...... 82 p = 3.3.8 C distribution of both propellers at J 0.7 and 0.1 span ...... 83 p = 3.3.9 Influence of ε in the design framework upon K and η at design point V 0.3[m/s] T = (J 0.24)...... 83 = 3.3.10 Results of the present design code are given in red and results of mpvl code [20] are given in blue (reprinted from Miclea-Bleiziffer et al. [59], with permission from Elsevier) 84 3.3.11 Thrust and torque characteristics for propellers with (blue) and without (red) the CVL correction, dashed line results of the BEM method (figure adapted from Miclea-Bleiziffer et al. [59], with permission from Elsevier) ...... 85 3.3.12 Efficiency characteristics for propellers with (blue) and without (red) the CVL correc- tion, dashed line results of the BEM method (figure adapted from Miclea-Bleiziffer et al. [59], with permission from Elsevier) ...... 85 3.3.13 Radial distribution of thrust and torque for design, propeller and propeller with CVL correction (reprinted from Miclea-Bleiziffer et al. [59], with permission from Elsevier) 86 3.3.14 Radial distribution of efficiency at J 2.4 (reprinted from Miclea-Bleiziffer et al. [59], = with permission from Elsevier) ...... 86 3.3.15 C distribution of the 0.1,0.3,0.5 span sections of both investigated propellers at J 2.4 87 P = 3.3.16 C distribution of the 0.7,0.9 span sections of both investigated propellers at J 2.4 . 88 P = 3.3.17 Suction side of the blade with contours of static pressure (J 2.4) (reprinted from = Miclea-Bleiziffer et al. [59], with permission from Elsevier) ...... 88 3.3.18 Pressure side of the blade with contours of static pressure (J 2.4) (reprinted from = Miclea-Bleiziffer et al. [59], with permission from Elsevier) ...... 89 3.3.19 Normalized axial velocity in an aft propeller plane (J 2.4) (reprinted from Miclea- = Bleiziffer et al. [59], with permission from Elsevier) ...... 89 3.4.1 CAD Model of the 14F RCP (courtesy of Cardiobridge GmbH) ...... 90 3.4.2 CAD Model of the reconstructed 14F propeller. Right: stereolitography prototype . . . 91 3.4.3 Measurements of the five 14F RCP configurations ...... 92 3.4.4 CFX setup used for the 14F RCP simulations (lengths are given in mm) ...... 93 3.4.5 Rotor and stator mesh of the 14F setup ...... 93 3.4.6 Output of the CFX mesh statistics ...... 94 3.4.7 Convergence history at duty point (5l/min)...... 95 3.4.8 Plot of the y on the external walls ...... 95 + 3.4.9 Plot of the y on the blade ...... 96 + 3.4.10 Validation of the CFD results by experimental measurements of the 14 RCP running in water ...... 96 3.4.11 Flow recirculation upstream of the propeller at the working point shown by 3D stream- lines clipped by a middle plane ...... 97 3.4.12 Contraction of the core flow under the effect of the swirling back-flow ...... 98 3.4.13 Types of flow in an axial turbomachine depending on the throttle position (adapted from Eck [25]) ...... 98 3.4.14 Performance evaluation of the 14F RCP ...... 99 3.4.15 CFD calculated static efficiency of the P14 propeller-pump ...... 100 3.4.16 Iterative adjustment of the propeller twist (pAoA) shown at two exemplary radii of D25: at hub and at an arbitrary radius ...... 101 3.4.17 Chord distribution and Re of the P14 and of the D25 propeller ...... 102 3.4.18 Grid study results for one of the new designed propellers ...... 102 3.4.19 3D Streamlines of the D25 propellers (0°,10°,20°) at the duty flow-rate ...... 103

XIII List of Figures

3.4.20 Inflow profiles of the axial velocity for the investigated D25 designs anlyzed at the duty point (distance between profiles is not to scale) ...... 103 3.4.21 Surface of evaluation for the through flow-rate ...... 104 3.4.22 Comparison between the three D25 propellers at the duty flow-rate ...... 104 3.4.23 Projected velocity vectors on a middle plane of the D25 p AOA 20 design ...... 105 3.4.24 Comparison of the airfoils between the D25 and D19 designs ...... 105 3.4.25 Radial chord and Re-number distribution for P14, D25 and D19 ...... 106 3.4.26 3D Streamlines of the D25, D19 and D21 propeller run at the duty point ...... 106 3.4.27 Through-flow, thrust and static pressure compared for D25 D19 and D21 at duty flow- rate...... 107 3.4.28 Pressure flow-rate curves of P14, D19,D21 and D25 (20° twist) ...... 107 3.4.29 Comparison of the through-flow, thrust and ∆Pst between 14F RCP and new designed propellers at the duty flow-rate (5l/min)...... 108 3.4.30 Sketch for the momentum theory applied to a propeller in a pipe (adapted from [110]) 108 3.4.31 Pressure flow-rate measurement results with and without helping device ...... 109 3.4.32 Sketch of a propeller slipstream with inlet radius equal to the pipe’s radius ...... 109 3.4.33 Fast product development showing from left to right: CFD model, CAD model and ready to test prototype (D19) ...... 110 3.4.34 Validation of the CFD simulation by measurements on the test rig ...... 111 3.4.35 Computed blood damage index (BDI) for all investigated designs ...... 113 3.4.36 Average BDI for all investigated designs ...... 113 3.4.37 Average exposure time of the investigated pumps ...... 114 3.4.38 Stress (σ) volume distribution for all pump cases ...... 115 3.4.39 Graphical plot of volumes with values of σ 150Pa ...... 115 > 3.4.40 Surface streamlines on the propeller blade ...... 116 3.4.41 Blade surface analysis of stresses ...... 117 3.4.42 Stresses in the range 50 150Pa (medium stresses) ...... 117 − 3.4.43 Transient flow-rate and pressure used as boundary conditions in CFX (LHS: medical units;RHS: SI units) ...... 118 3.4.44 Transient static pressure measured at the outlet of the domain ...... 119

A.0.1 Fluid volume with figured stresses ...... 131

C.0.1 LHS: Graupner rMultiSpeed 280, RHS: Placement of the helping device ...... 137

XIV List of Tables

2.1.1 MCL settings for patients with LFVM [75] ...... 12 2.1.2 Ideal LVAD propeller duty point parameters ...... 12 2.5.1 Measured parameters and their range ...... 38 2.5.2 Components of the test rig shown in figure 2.5.5 ...... 40 2.5.3 Values used for the benchmark ...... 48 2.6.1 Mock components in analogy to the human circulatory system shown in figure 2.6.1 . 51 2.6.2 Average values used to set the mock for ”healthy” condition ...... 52

3.2.1 Summary of the investigated flat plate solution ...... 58 3.2.2 Discretization results over a flat plate airfoil at an AOA of 5° ...... 61 3.2.3 Evaluation error of the cl by using the CVL method for airfoil with different cambers . 63 3.2.4 Comparison of desired and realized cl for different mean-lines ...... 66 3.2.5 Comparison of desired and realized cl for mean-lines stand-alone and in cascades . . 67 3.3.1 Geometric and performance design parameters of the investigated propeller ...... 76 3.3.2 Boundary conditions ...... 78 3.3.3 Simulation parameters ...... 78 3.3.4 Design parameters of the investigated propellers ...... 84 3.4.1 Simulation parameters ...... 94 3.4.2 Boundary conditions ...... 94 3.4.3 Area averaged y ...... 96 + 3.4.4 Properties of the solid phase (RBC) in CFX, mass flow computed according to [119] . . 112 3.4.5 Comparison between the pressure increase in the steady and unsteady simulation . . 119

B.0.1 Computation of RBC mass-flow ...... 135

XV

Nomenclature

Acronyms

Symbol Description

ADAP Advanced Design of Axial Propellers and Pumps Program

AHF Acute Heart Failure

AHFS Acute Heart Failure Syndrome

AOA Angle of Attack

APD Avalanche Photodiode

BDI Blood Damaging Index

BEM Blade Element Method

CFD Computational Fluid Dynamics

CO Cardiac Output

CoD Chord over Diameter, non-dimensional chord distribution CPU Central Processing Unit

CRW Counter Rotating Wall

CS Cardiogenic Shock

CVL Cascade Vortex Lattice

DES Detached Eddy Simulation

DNS Direct Numerical Simulation

EKG Electrocardiogram

FDA Food and Drugs Administration

FFC Failing Fontan Circulation

FFT Fast Fourier Transformation

GUI Guided User Interface

XVII Nomenclature

HF Heart Failure

IABP Intraaortic Balloon Pump

LAP Left Atrium Pressure

LDA Laser Doppler Anemometry

LES Large Eddy Simulation

LSTM Lehrstuhl fur Strömungsmechanik, Universität Erlangen-Nürnberg

LV Left Ventricle

LV AD Left Ventricular Assist Device

LV F M Left Ventricular Failure Model

MAP Mean Aortic Pressure

MCL Mock Circulatory Loop

N AC A National Advisory Committee for Aeronautics

NASA National Aeronautics and Space Administration

NSE Navier-Stokes Equations p AOA prescribed angle of attack

PAP Pulmonary Artery Pressure

PVL Propeller Vortex Lattice

RANS Reynold Averge Navier Stokes

RAP Right Atrium Pressure

RBC Red Blood Cells

RCP Reitan Catheter Pump

SAS Scale Adaptive Simulation

SST Shear Stress Transport turbulence model

VAD Ventricular Assist Device

VLM Vortex Lattice Method

Dimensionless Numbers

Symbol Description Definition

V˙ φ flow coefficient φ = N D3 ·

XVIII Nomenclature

Y ψ head coefficient ψ = N 2 D2 · 2P CP propeller power coefficient CP 3 = ρAP V ∞ µ u ¶2 Cp pressure coefficient Cp 1 = − V ∞ 2T CT propeller thrust coefficient CT 2 = ρAP V ∞ V J propeller advance coefficient J ∞ = nD Q KQ propeller torque coefficient KQ = ρn2D5

T KT propeller thrust coefficient KT = ρn2D4

Ψ1/4 DS specific diameter DS = Φ1/2

Φ1/2 NS specific speed NS = Ψ3/4 Greek Symbols

Symbol Description Dimensions Units

α Angle of Attack °

Cd ² drag-to-lift ratio - ² = Cl

η efficiency - -

λ cascade stagger angle - °

V λ propeller advance ratio - λ ∞ = ΩR d λ pipe friction factor - λ ζ f f = · L

λL laser wavelength L m

µ dynamic viscosity F/A t Pas · ν kinematic viscosity F/A t Pas · Ω rotational speed rot/t rev/s

ω angular velocity rad/t rad/s

φ flow angle on a blade element - °

ρ density W/V kg/m3

XIX Nomenclature

σ cascade solidity - -

τ shear stress F/A Pa

v0 ζ displacement velocity ratio - ζ = V ∞ α,β Heuser and Opitz constants - -

Roman Symbols

Symbol Description Dimensions Units

B number of blades - -

I10 , I20 thrust radial gradients - -

J10 , J20 torque radial gradients - -

R tip radius L m r radial coordinate L m t/l cascade pitch-to-chord ratio - - wn total velocity of the propeller slipstream L/t m/s wt tangential velocity of the propeller slipstream L/t m/s m˙ mass flow rate W/t kg/s

V˙ volumetric flow rate V/t m3/s

∆Hb plasma free hemoglobin W/V mg/L

Hb hemoglobin W/V mg/L

A area L2 m2 a axial interference factor of a propeller - - a0 angular interference factor of a propeller - -

C Heuser and Opitz constant - - c airfoil chord - - cd drag coefficient - - cl lift coefficient - -

2 2 E kinetik energy L W/t− Nm

1 e shear rate 1/t s− f airfoil camber - -

G Glauert factor - -

XX Nomenclature l cascade airfoil chord - -

N turning speed rpm

P pressure P Pa

Q torque FL Nm

S source term for the scalar transport equation - -

T thrust F N t cascade airfoil pinch - -

V velocity L/t m/s v velocity L/t m/s v0 displacement velocity L/t m/s x, y,z spatial coordinates in cartesian coordinate system L m

Y specific work input (turbomachines) FL J

Superscripts

Symbol Description

A test

Subscripts

Symbol Description

infinity ∞ i ideal - used for AOA and lift coefficient s related to propeller’s slipstream st related to a static quantity t turbulent ti p tip tot related to a total quantity

XXI

Abstract

This work addresses the development and optimization of propellers in particular of propeller-pumps used in cardiac support (left ventricular assist devices).

A novel propeller design method is firstly validated for high-Re marine, multiblade propellers and is adapted afterward for designing small propeller-pumps used as left ventricular assist devices. The initial design framework (ADAP) is fully inverse for the case of marine propellers and is coupled with 3D CFD calculations. In addition, for marine propellers, a Blade Element Method (BEM) is devel- oped for predicting off-design performance. The propeller designer can thus engage in a much faster goal-oriented design parameter search without the use of full 3D-CFD calculations. For the cardiac propeller-pump the design is performed both direct and inverse and has to be always accompanied by CFD simulations. The complexity of the flow around an encased propeller prevents the propeller designer from using BEM in this case.

A propeller design framework (ADAP) with different radial momentum loss theories was developed in Matlab r and validated against state of the art vortex-lattice methods. ADAP was programmed to write geometry data to grid generators (e.q. ANSYS TurboGrid r) reducing the time needed from aerodynamic design to CFD simulation. It can also write structured data files which are needed for CAD systems for parametric generation of geometries. In the present work the interface was written for Creo r. Instead of using only predefined NACA camber lines for the radial sections of propellers the framework is improved by implementing a novel method of computing thin airfoil cascades. This method is helpful in the case of designing propellers with more than 4 blades, where blade to blade effects are significant, as for example in the case of propellers used for large container ships. CFD results showed that by using the design code presented in this thesis the thrust of a 6 blades marine propeller was improved by 3.5% and was accompanied by a small propulsive efficiency improvement. Different thickness distributions are implemented in the framework allowing designers to choose the proper one for each application.

ADAP was adapted for small diameter propellers-pumps and together with CFD simulations is shown that it can be used for an iterative improvement of a left ventricular assist device. The work also shows that this type of turbomachine working at the duty point required by the human body is off the optimal design. An iterative optimization with different twist angles for the propeller blades was required so that a relationship between blade twist, pressure increase and propeller through-flow was established. Fast propeller-pump designing was possible in ADAP and relevant aspects of the physics flow were analyzed using CFD simulations.

Stationary CFD simplifies the complex flow physics and therefore experimental confirmation is re- quired. A modular test-rig was developed for the stationary measurement of the LVAD propeller- pump performance. The performance improvement achieved in the CFD was confirmed by mea- surements on the test-rig. A second test rig simulating the instationary conditions in the human body (MOCK) was developed and used for the validation of time-dependent simulations.

The present work has shown that pressure characteristics of propeller-pumps can be improved with- out increasing blood damage. The presented framework was essential in avoiding the prohibitive computational cost of standard CFD simulations.

XXIII

Zusammenfassung

Diese Arbeit befasst sich mit der Entwicklung und Optimierung von Propellern, insbesondere von Propeller-Pumpen, die als linksventrikuläre Unterstützungspumpen verwendet werden.

Ein neuartiges Propellerauslegungsprogramm wurde zunächst für Schiffspropeller validiert. Dieses Auslegungsprogramm wurde danach für die Auslegung kleiner Propellerpumpen, die als linksven- trikuläre Unterstützungspumpen verwendet werden, angepasst. Das ursprüngliche inverse Ausle- gungsprogramm ist mit 3D-CFD-Berechnungen gekoppelt. Darüber hinaus wurde, für die Vorher- sage von Off-Design Performance von Schiffspropellern, eine Blade-Element-Methode (BEM) pro- grammiert. Dies ermöglicht dem Propeller Designer eine wesentlich schnellere zielorientierte De- signparameterstudie ohne Verwendung von vollen 3D-CFD-Berechnungen. Bei der Herzunterstüt- zungspumpe wurde die Auslegung sowohl direkt als auch invers ausgeführt, wobei sie immer von CFD-Simulationen begleitet wurde. Die Komplexität der Strömung um einen umhüllten Propeller hindert den Propellerausleger in diesem Fall die BEM zu verwenden.

Ein Propellerauslegungsprogramm (ADAP) mit unterschiedlichen radialen Impulsverlust-Theorien wurde gegen Stand der Technik Vortex-Lattice-Methoden validiert. ADAP wurde in Matlab program- miert r, um Geometriedaten an Gittergeneratoren (z.B. ANSYS TurboGrid ®) zu schreiben, wodurch die Zeit vom aerodynamischen Design zur CFD-Simulation reduziert wurde. Es kann auch strukturi- erte Datendateien schreiben, die für CAD-Systeme zur parametrischen Erzeugung von Geometrien benötigt werden. In der vorliegenden Arbeit wurde die Schnittstelle für Creo ® erzeugt. Anstelle der Verwendung von vorgegebenen NACA Profilsehnen wurde das Auslegungsprogramm durch die Im- plementierung eines neuartigen Verfahrens zur Berechnung dünner Gitterprofilen verbessert (CVL). Dieses Verfahren ist hilfreich bei der Auslegung von Propellern mit mehr als 4 Schaufeln, wobei die Schaufel-zu-Schaufel-Effekte signifikant sind, wie zum Beispiel bei Propellern, die für große Con- tainerschiffe verwendet werden. CFD Ergebnisse zeigten, dass durch die Verwendung von ADAP mit CVL der Schub eines 6 blättriges Schiffspropellers um 3.5% verbessert wurde. Dazu wurde auch der Wirkungsgrad des Propellers leicht angehoben. Durch die Implementierung von verschiedenen Dick- enverteilungen in ADAP steht Propellerentwickler immer die richtige Wahl für ihre Anwendung zur Verfügung.

ADAP wurde für Propellerpumpen mit kleinem Durchmesser angepasst und zusammen mit CFD- Simulation wurde gezeigt, dass es für die Verbesserung einer linksventrikulärer Unterstützungspumpe verwendet werden kann. In dieser Arbeit wurde gezeigt, dass diese Art von Turbomaschine, die am Arbeitspunkt des menschlichen Kreislaufs angepasst ist, nicht optimal als Propellerpumpe ausgelegt werden kann. Dies führte zu einer iterativen Optimierung mit unterschiedlichen Winkeln für die Pro- pellerschaufel, so dass eine Beziehung zwischen Winkel, Druckerhöhung und Propellerdurchfluss hergestellt wurde. Eine schnelle Propellerpumpenauslegung war in ADAP möglich und durch die Verwendung von CFD-Simulationen wurden Aspekte der Strömungsphysik analysiert.

Stationäres CFD vereinfacht die komplexe Strömungsphysik, so dass eine experimentelle Validierung erforderlich ist. Für die stationäre Messung der LVAD-Propellerpumpe wurde ein modularer Prüfs- tand entwickelt. Die im CFD erzielte Verbesserung wurde durch Messungen am Prüfstand bestätigt. Für die Validierung zeitabhängiger Simulationen wurde ein zweiter Prüfstand entwickelt, der die in- stationären Zustände im menschlichen Körper simuliert (MOCK).

XXV Nomenclature

Die vorliegende Arbeit zeigt, dass die Druckcharakteristik von umhüllten Propellerpumpen durch das neu entwickelte Auslegungsverfahren verbessert werden kann, ohne die Blutschädigung zu beein- trächtigen.

XXVI Chapter 1 Introduction

Today’s technologies enable humans longer life expectation and higher life quality. This is possible due to a rapid development of devices and methods which are proactively involved in daily life, ei- ther as support or replacement of organs. Such devices are, for example, breathing help devices, circulatory support devices, or artificial arms and legs. Circulatory support devices are used more often because of the dramatically increasing number of patients with cardiac problems. According to Nichols et al. [63] over 4 million deaths were caused in 2011 in Europe due to heart or circulatory problems. From this figure almost a half (47%) is accounted by Cardio Vascular Disease (CVD). Heart Failure (HF) is the cause of about 1 million hospitalizations per year in the USA and Europe [70]. It is anticipated that in the future even more patients will need hospital care due to ageing [70]. In re- cent years the usage of cardiac support devices has been established among the therapies used in the treatment of the Acute Heart Failure Syndrome (AHFS). Devices are either Ventricular Assist Devices (VADs) or Left Ventricular Assist Devices (LVADs) and can support the circulatory system alone, in parallel, or in series with the human heart. VADs improve the hemodynamics and can also accel- erate the restoration of the heart after a HF by decreasing the infarct size [102, 90]. In the cases of pulmonary congestion LVADs decrease the Pulmonary Artery Pressure (PAP) or improve the symp- toms of renal deficiency [76]. They are used as a bridge-to-transplant, while the patient awaits a heart transplant, or as a destination therapy for long time support. Although cardiac support devices have been in use for over four decades their design with regard to the biocompatibility, and in particular blood damage could be assessed only in the past 15-20 years. In the context of this thesis biocom- patibility is defined as the ability of a device to be in contact with a living system without affecting it in negative way [112]. A high impact on the design of these devices has the usage of Computational Fluid Dynamics (CFD); both for assessing hemodynamic performances and blood damage computa- tion. Questions regarding the methodology of blood damage prediction as well as the principles used in the design of devices are concerning researchers all over the world. Due to the fact that there are several types of pumps used in the cardiac support, one has to expect that they will behave differently in similar conditions.

HF is a clinical syndrome initiated by abnormal function of the heart [79] and which can be explained as the inability of the heart to provide the pump work required to maintain the perfusion of the organs [74]. AHFS’s are one of the most common causes of hospitalization for patients older than 65 years in the USA [17]. According to the same source, the number of hospitalizations will continue to grow in the coming years as a result of an aging population and due to the improved survival after myocardial infarction. The demand for a VAD that can be used for short term support as well as a long-term destination therapy device has increased as well. As the number of patients with AHFS is expected to increase even more in the next years, an increase in the demand of reliable LVAD is also expected.

Cardiogenic shock (CS) is defined as an acute heart failure (AHF) caused by a heart function disorder and has as result the hypoperfusion and hypoxia of organs [58][55]. Recent studies have shown that in the case of an CS the proportion of perfused vessels in micro vascular tissues is decreased due to their inability to dilate as response to hypoxia [46]. CS has an incidence of 1% from all cases of AHFS [17] and according to Michels and Schneider T. [58] between 5 and 8% among the patients with

1 Chapter 1 Introduction myocardial infarction. Killip classifies CS as the most critical stage of a myocardial infarction (Stage IV) [58]. The target of the therapy for patients in CS is the fast improvement of the cardiac pump function [17]. This is required also in order to keep the perfusion and oxygenation of the organs. The treatment of CS can be done either by medication or by using a VAD [58]. In case patients are not responding to the medication the use of an Intraaortic Balloon Pump (IABP) is recommended [58].

1.1 Short overview of LVAD used in the treatment of cardiogenic shock

Intra-aortic Balloon Pump

The working principle of the Intra-aortic Balloon Pump (IABP) is based on counter-pulsation which is achieved by an external pumping chamber (figure 1.1.1). During the diastole the balloon inflates cre- ating a counter pressure and then deflates during the systole. The inflating-deflating of the balloon is controlled by an electrocardiogram (EKG) or blood-pressure signal. The balloon pump was first de- veloped in the early 1960s and is enjoying a wide acceptance today. Because of its counter-pulsating principle this pump can be used only when the heart is still active.

Diastole Systole

Figure 1.1.1: The Intra-aortic balloon pump (from [115])

Hemopump

Another device developed for use in the CS is the Hemopump® (figure 1.1.2). It was developed specif- ically to be used for CS, during minimally invasive coronary bypass surgery, and for pathologies in which the heart must be relieved of approximately 80% of its workload [93]. The main difference com- pared to the IABP is that Hemopump® is an axial continuous flow pumping device whose pumping principle is based on the Archimedes’s screw design. The Hemopump® could be placed inside the ventricle through different blood vessels and achieved a higher Left Ventricle (LV) unloading than the IABP [90]. It has been the first axial blood pump ever used [97]. The Hemopump® is no longer available for clinical use.

Impella

A similar screw design is adopted by Impella (2.5 and 5.0)1; a device currently in clinical use. Like Hemopump, it is an in-ventricle implantable LVAD of very small dimensions 3 mm or 9F , which can be inserted in the aorta through catheterization (figure 1.1.3). The pump has two blades of 0.3 mm

12.5 and 5.0 stand for the maximum cardiac output (http://www.abiomed.com/products/)

2 1.1 Short overview of LVAD used in the treatment of cardiogenic shock

Figure 1.1.2: Hemopump®[97] thickness, which rotate at a speed of 33000 r pm and has a tip gap of only 0.1 mm [93]. The small version (Impella 2.5) is implanted through the femoral artery while Impella 5.0 needs femoral artery surgery. Impella is still associated with high hemolysis which demands an anti-coagulation therapy [4].

Figure 1.1.3: Impella 2.5 ® von Thoratec® and its placement [102]

Reitan Catheter Pump (RCP)

An alternative to the IAPB has been presented in 1999, the Reitan Catheter Pump (RCP), which is an axial propeller-pump (figure 1.1.4)[77]. From the turbomachinery classification this device can be defined as an encased propeller. It has been designed as a short term continuous flow rotating sys- temic circulatory support device. The target patients for this device are myocardial infarction patients in AHF or in CS, and patients having coronary angioplasty. Conceptually, it is an axial two bladed propeller-pump spinning at speeds ranging from 1,000 to 14,000 rpm. The blades have a diameter of 15 mm and the diameter with the protective cage is 21 mm (figure 1.1.4).

Figure 1.1.4: 14F Reitan catheter pump (figure courtesy of Cardiobridge Gmbh/Hechingen)

3 Chapter 1 Introduction

The pump head is foldable like an umbrella resulting in a diameter in folded (closed) position of 4.67[mm] (14)F . This eases the insertion of the pump which is done percutaneous by using an in- ducer in the femoral artery. After insertion it is slowly pushed up until it reaches the upper side of the descending aorta and then unfolded (figure 1.1.5).

Figure 1.1.5: Physiological position of the 14 F RCP in the upper aorta (figure courtesy of Cardiobridge Gmbh/Hechingen)

Figure 1.1.5 shows the open RCP positioned in the upper aorta just after the aortic arch. In this posi- tion the pressure gradient over the pump increases the irrigation of vital organs fed from the main aorta (e.g. kidneys and liver). The rotation of the blades is assured by a flexible shaft inside the catheter which is connected to an external driving unit. Several animal and human clinical studies [74, 77, 91] have proven the benefits of the RCP in recent years, but the device is not yet commercially available2. Similar (axial) propeller-pump concepts have been presented for the treatment of Failing Fontan Circulation (FFC) [104, 103]. They are running at much lower speeds, and there is no evidence that these devices are designed to increase blood pressure at flow-rates between 4 to 5 l/min, which are the target flow-rates in the CS. The benefits of a folding propeller-pump can be briefly summa- rized:

• easy insertion in the femoral arteries, similar to the use of IABP

• easy deployment

• due to a very large tip gap a low hemolysis is expected which was observed in vivo by Smith et al. [91]

• reduced risk of thrombosis by a reduced risk of the activation of platelets by minimal foreign surface exposure (shroud is the blood vessel Throckmorton et al. [104])

• swirling flow at the exit of the propeller inducing transverse velocities on the blood vessels and reducing the risk of stagnation.

The hemodynamic design of such a device influences directly not only the pumping performance such as head or efficiency but also the biocompatibility (hemolysis in this case). Common to many of the rotating pumps used in the cardiac support is the way they were designed: directly, mostly by trial, and error. This method, although it is straight, demands a higher effort in terms of time since there are many iterations needed until the aimed performance is achieved. An alternative to this is the inverse design method, which allows the engineer the lay-out of a device only by knowing the duty points. However, this method requires the knowledge of the device’s physics and deep fluid dynamic understanding.

2as of December 2016

4 1.2 Design and analysis methods for VADs

1.2 Design and analysis methods for VADs

Designing a turbomachine can be a complex task depending on the duty-point, fluid medium, or other conditions imposed. Designing a VAD is challenging because of its dimensions, because of the fluid medium and because of the biocompatibility which has to be guaranteed. These challenges are constrained by material strength and material compatibility with biological tissues. Throckmorton treats in depth the problem of propeller-pump design [104] and suggests the use of empirical equa- tions and CFD simulations for geometry optimization. Clearly this is a good way to design pumps and it is also the method followed by most blood pump designers. In literature concerning turbo- machinery design, it is usually known as the direct design method and basically means, that a pump geometry is created using empirical formulas and then adjusted by trials in order to achieve the de- sired performances. It is a very costly and time intensive process. In contrast, inverse design is based on a set of simplified fluid flow equations linked to the geometry (airfoils, camber-lines, vortex distri- bution) and which deliver a geometry capable of fulfilling the design requirements. The geometry is also verified by CFD means. Both design strategies are available in commercial codes. Several open source codes are available for propeller design: OpenProp [26], Xrotor/Qprop [23] or Qblade [53]. A detailed analysis of existing design methods for propellers is given in section §2.4.

Computational Fluid Dynamics (CFD) is a tool used in today’s engineering practice for the develop- ment of many cardiac support devices. CFD is mainly used as a replacement to experimental mea- surements of pressure-flow curves and it offers, depending on the complexity of simulation, a deeper look in the physics of the flow. This contributes to the understanding of complex phenomena taking place in devices and thus to their enhancement. Hemolysis, the damage of red blood cells or erythro- cytes (RBC), can be evaluated by computing the Blood Damaging Index (BDI) from the results of CFD simulations. This is performed with the formulation of scalar stresses firstly given and validated by Bludszuweit [13],[14]. No deeper details about the numerical schemes of CFD codes are given here since this does not make the object of the present work, but descriptions of methods, algorithms, and their implementation are given in Blazek [12], Hirsch [38] or Ferziger and Peric [30].

Even with today’s advances in computational simulation one can not replace the experimental val- idation. For medical devices the experimental validation has to be carried out directly on animals and lately on humans, in vivo3. The most usual way to asses performances of VADs is to measure the pressure flow-rate curve on a test rig. The construction of a test rig capable of measuring precisely the flow-rate and pressure increase is thus very important. The measurement of a VAD is more complex in conditions similar to in vivo conditions: in a Mock Circulatory Loop (MCL) or simply mock4. This test rig is basically a dynamic replica of the human circulatory system and provides ideal conditions for performance or endurance tests of VADs.

1.3 Outline and objectives of the dissertation

The second chapter of this dissertation reviews the useful literature needed to accomplish this work. It starts with an introduction in the physiologic foundations of the human circulatory system which is complemented by a description of blood and blood damage prediction using CFD methodology. HF and the CS are condensed with the help of numbers and the Cordier diagram in duty points for the future propeller-pump. This is followed by an introduction in the fluid mechanics and a short description of turbulence modeling methodology used in modern CFD codes. Airfoils play an im- portant role in the design of propellers, therefore an introduction is given afterward. This is followed by a section covering all aspects of modern propeller design methods. Validation plays an important

3a study in vivo refers to study in a living organism - www.wikipedia.org, accessed on the 27.12.2016 4mock is synonym of imitation or simulation

5 Chapter 1 Introduction role in the design of turbomachinery and this is accomplished by using test rigs. The design and con- struction of test rigs for VADs both for measuring in steady state condition and pulsating flow (MCL) is shown at the end of the second chapter.

Following the second chapter a method is developed for assessing the performance of airfoils in cas- cades by using potential methods. This is implemented in a propeller design code using a new semi- empirical approach for computing the blade losses by using the Goldstein method. The validation of the cascade code for designing multiblade open-water propellers is shown subsequently. It is shown by CFD simulations that the thrust of propellers can be improved by 3.6% at the propeller design point without penalties in the efficiency by using the new airfoil cascade analysis method. Parameter studies are performed in this chapter for choosing the correct propeller design variables. The vali- dation of the propeller design code delivers the foundation for the design of propeller pumps used as VADs in the next section. Here is shown, however, that the actual challenge is not only the design itself but understanding the flow mechanisms produced by a propeller in a pipe. The overall goal is the performance improvement in form of pressure rise of the existing P 14F RCP without affecting its biocompatibility. The main dimensions (Dti p ,Dhub) and other general characteristics (speed and design flow-rate) of the propeller are fixed. The performance of the reference propeller (14F RCP) is described and analyzed in subsection 2.1.5 and subsequently in subsection 3.4.1 and it is used to cal- ibrate the CFD tools with the help of a measurement data provided by the test stands. As an answer to the direct design approach used in the typical LVAD design, this work proposes a modified propeller inverse design method. Due to the very low Re-number flows in VADs the method is first validated on marine propellers.

Briefly formulated, the specific objectives of the research in this dissertation are :

• increase the pressure rise of the propeller-pump for the same flow-rate

• decrease the shear stresses generated by the propeller

• decrease the resident time spent by the RBC while flowing around the propeller

Most promising designs are built as prototypes and validated on the test rig. Their blood damaging index (BDI) is evaluated afterwards by using state of the art methods exposed in the second chap- ter. Finally, one of the designs, together with the baseline 14 RCP, is evaluated using transient CFD simulation using boundary conditions derived from the MCL setup.

Finally the outcomes of this thesis are presented, both on the design methodology side as well as on the required VAD development itself. Benefits and drawbacks of the propeller design methodology presented in the thesis are reviewed and discussed. The improvements achieved for designing the propeller-pumps are discussed together with results from the literature and an outlook for future work is presented.

6 Chapter 2

Selected aspects in relevant areas for the design of VADs

This chapter summarizes fundamental knowledge needed for the design, simulation and experimen- tal validation of VADs. It contains a literature review about the physiology of the circulatory system and of the heart, figures regarding the desired and real duty point of propeller-pumps, fundamental fluid mechanics and airfoils, propeller design and the experimental test stands used for validation in this thesis. The reader can understand the actual challenges in the development and design of VADs.

2.1 Design consideration for VADs

To understand the clinical meaning of the CS and HF it is necessary to know where and how they take place. In this section the anatomy and physiology of the circulatory system and of the human heart is briefly described. The most important factors contributing to blood damage are presented together with computational prediction methods. The medical facts of the CS are introduced and the duty points of a VAD are computed and then analyzed in the Cordier diagram.

2.1.1 Human circulatory system

One of the essential components of the human body is the circulatory system. The fluid in the system is blood which transports oxygen and the nutrition to body cells and CO2 to the lungs in the reversed direction. In the direction to the cells the circulatory system is formed by arteries and capillaries while on the reverse side the circulatory system is formed out of veins as depicted in figure 2.1.1.

The circulatory system is composed of two subsystems: the pulmonary circulation system and the systemic circulation system. The center of the circulatory system is the heart which pumps the blood with an average of 5 [l/min] 1 through the arteries and veins of the body. To accomplish this effort the heart needs 1W power [67]. The pulmonary circulatory system is powered by the right side of the heart being placed between the right ventricle and left atrium as depicted in figure 2.1.1.

The heart is a muscular organ which pumps the blood in the circulatory system by a cyclical contrac- tion (systole) and dilatation (diastole). Figure 2.1.1 shows that the heart has two main parts: right atrium and ventricle on one side and left atrium and ventricle on the other side. Pumping is realized synchronously by right and left ventricles during the systole. Atria do not have a pumping function.

1average for healthy adult

7 Chapter 2 Selected aspects in relevant areas for the design of VADs

Figure 2.1.1: Human circulatory system [1]

2.1.2 Blood composition and its physical properties

Blood is the fluid in the circulatory system with an average volume of 5,2 l and represents 8% 1% ± from the total body weight [85]. It is composed of a fluid part called blood plasma, which is mixed with suspended solids called blood cells or hematocytes. The three major blood cell types in the blood plasma are [85]:

• red blood cells (RBC- erythrocytes - totalizing about 95% of all cells),

• white blood cells (leukocytes - representing 0.15%),

• platelets (trombocytes - representing less than 5%) .

Blood plasma consists of water (90%) and proteins (7%) and has a viscosity 1.5 higher than that of water. The density is above that of water, 1.035 kg/m3. Erythrocytes have the role of transporting blood gases from and to the cells in the body and have the shape of biconcave discs with a diameter of 7.5 µm [119]. The volumetric fraction of erythrocytes in blood is called hematocrit and has values usually around 45%. Trombocytes are involved in the blood clotting process while leukocytes have the primary role of protecting the body from external organisms.

Fluids shear at a strain rate inversely proportional to the coefficient of viscosity µ [118]. The strain rate can be related to the shear rate (e ∂u/∂y) of the flow [118], so the shear stress for a simplified = case of one-dimensional steady flow and an isotropic and homogeneous fluid can be written as:

∂u τ µ (2.1.1) = ∂y

For a three-dimensional case the relation is given in appendix A. If the relation in equation (2.1.1) is linear one can speak about a Newtonian fluid otherwise the fluid is non-Newtonian. From the rheological point of view the blood plasma can be considered a Newtonian fluid, while the blood with suspended particles may be considered both Newtonian and non- Newtonian. This behavior

8 2.1 Design consideration for VADs

t Plastic m Bingham 103 RBC Pseudo-plastic

102 Dilatant 101 Newtonian 100 e 10-2 10-1 100 101 102 103 e

(a) Shear stresses of different fluids (adapted from [24]) (b) Blood viscosity as function of shear rate (adapted from [67])

Figure 2.1.2: Rheological properties of blood will be explained in this section since it is very important for the blood flow modeling. In figure 2.1.2a are shown various cases of non-Newtonian fluids compared to the Newtonian case. Blood viscosity is inversely proportional to the shear rate as shown in figure 2.1.2b. This is caused due to the orientation of RBC under the effect of shear stresses: when RBC are subjected to high stress in motion, they align along flow direction axis, reducing the resistance in the fluid and thus the viscosity [29]; at small shear rates (no or low flow) RBC’s group together in so called roleaux [29]. Both cases are illustrated simplified in figure 2.1.2b.

Patients with HF might have different levels of hematocrit than usually seen, between 33% and 36% [111] so it is important to know how the rheological properties are changing according to this. The behavior is shown in figure 2.1.3. Normal hematocrit levels for healthy humans are around 45% [29]. It is important to mention that blood viscosity is influenced also by temperature, but its influence will not be discussed in this thesis since temperature will be considered constant for all investigated cases.

1000

oise] 100 H=90%

10 H=45% osity [cP

Visc 1 H=0%

0.1 10-2 10-1 100 101 102 e [s-1]

Figure 2.1.3: Viscosity vs shear rate for different hematocrit levels (adapted from [29])

9 Chapter 2 Selected aspects in relevant areas for the design of VADs

1 Viscosity decreases with decreasing hematocrit level and above a shear rate of 100 [s− ] is constant (figure 2.1.3). These results are also used in recent CFD studies upon VADs where the blood was as- sumed incompressible with constant viscosity [32]. Hence, the blood will be modeled as a Newtonian fluid in this thesis.

2.1.3 Considerations on blood damage for VADs

The action of external factors can lead to the deformation or damage of blood cells. Hemolysis and and thrombosis represent the most important kinds of blood damage. Other types of damage are: platelet activation, emboli, reduced functionality of the white blood cells and destruction of the von Willebrand factor [32].

The damage of RBC under the effect of an external force is the result of traumatic fluid stresses acting with a specific amplitude and a specific frequency for a certain amount of time (the ”triple” action, figure 2.1.4). Early investigations proved the relationship between stress, exposure time and hemol- ysis. Hemolysis reduces the RBC concentration in blood causing a decrease of oxygen transport to the cells. Because of the cell destruction the hemoglobin contained by the RBC is transferred to the plasma. The measurement of hemoglobin concentration in blood quantifies the hemolysis.

Exposure Time

Amplitude Frequency

Figure 2.1.4: Shear stress factors contributing to the blood damage

CFD is an efficient tool for investigating blood hemolysis since it can evaluate stress levels and their exposure times. State of the art models will be presentd and they are used extensively in section §3.4 for the evaluation of the VAD designs proposed in this thesis.

The combination of effects was first investigated by Bludszuweit [13], who was also the first to define the comparative stress level in a fluid. The latter are the equivalent of the von Mises criterion used for mechanical stresses. For solids, the threshold level of stresses is given by the yield stress of each specific material.

The scalar shear stress (σ) is defined ([13]-appendix A) :

r 1 X¡ ¢2 X 2 σ τii τj j τ (2.1.2) = 6 − + i j

Initially a minimal stress threshold, which could cause direct cell damage, has been found experi- mentally: 150Pa [50]. However, as previously stated, stresses alone can not indicate the hemolysis. According to Taskin et al. [98] there are two models for predicting the hemolysis: strain-based models and power-law models. Power-law models differentiate between two fluid analysis approaches: Eule- rian and Lagrangian. The Lagrangian approach is very complex and is based on the evaluation of the stress history of particles which are injected at the inlet of the flow domain. This approach enables the evaluation of hemolysis by all factors: amplitude, time and frequency of stresses. The outcome of

10 2.1 Design consideration for VADs the approach is the blood damage index (BDI) which is the percentage of damaged blood cells. It is based on the evaluation of the scalar shear stresses experienced by a particle during its traveling time in the fluid domain.

This approach is empirical and as a consequence has been best correlated for specific devices. The correlation is performed with the help of exponential coefficients (α and β see equation (2.1.3) ) which are adjusted with the help of regression from experimental data. BDI is then evaluated according to following expression [111], [94]:

∆Hb BDI(%) 100 C ∆T α σβ (2.1.3) = Hb · = · · where ∆Hb represents the plasma free hemoglobin, and Hb the hemoglobin - both are determined experimentally; C,α,β are the model constants (Heuser and Opitz constants [98]) while the exposure time (∆T ) and σ of each particle are obtained from the simulation.

The Lagrangian model was used in the development process of several blood pumps ([111, 105]) and has gained a wide recognition in literature. More precise in the correlation with the experimental data, the model is used for the quantitative comparison of the BDI of the investigated pumps in this thesis. In addition, the σ levels in the fluid volume are analyzed and compared to critical values available in the literature (Eulerian analysis) as for example in [31]. This method enables the VAD designer the chance to analyze visually the stresses produced by a particular pump design. Of course in this way the causes of the possible to high stresses can be evaluated directly from the flow field. In addition to the σ volume analysis Taskin et al. [98] and Fraser et al. [32] proposed more recently an Eulerian approach of calculating the potential blood damage of VADs by using a scalar transport equation:

¡ ¢ d ∆Hb0 ¡ ¢ v ρ ∆Hb0 S (2.1.4) dt + · · ∇ =

1/α where S is the source term and is defined by S ρ ¡Hb C σβ¢ . This model also needs an empirical = · · validation and is therefore also very model-dependent.

Taskin et al. [98] have recently published a benchmark between all up-to-date hemolysis models by using available constant and regression coefficients. It was concluded that the Eulerian model hemol- ysis results do not agree well with experimental results, but because of its well correlated coefficients is very good for relative comparisons. On the other side, the Lagrangian models correlated well with the experimental hemolysis results but the models are highly dependent on the constants.

2.1.4 Design requirements (duty point of a LVAD)

Section 1.1 provided the definition of the CS. In order to define a duty point for a blood pump one needs to analyze the relationship between the flow parameters between a healthy and an ill patient. This will lead to figures which define the duty point of the turbomachine.

CS can be quantified by a systolic arterial pressure below 90mmHg 2 or a Mean Aortic Pressure (MAP) 30mmHg lower than the basal one [39] (in [58] the definition is given as below 70 mmHg). Left ven- tricular failure is more often and, in the case the right ventricle functions normally, it leads to a pul- monary oedema (or backward congestion) with high pressure in the pulmonary circulation system according to [74] and [17]. This is the background of a study upon a left ventricular failure model (LVFM) performed by Reitan et al. [78] where the Left Atrium Pressure (LAP) was increased from val- ues of 5 10mmHg to 25mmHg in order to simulate pulmonary oedema. In this model the outcome − of a VAD was assessed by the decrease of LAP (decongestion).

21 [mm Hg]= 1 [Torr]=133.2 [Pa] - pressure unit intensively used in medicine

11 Chapter 2 Selected aspects in relevant areas for the design of VADs

Systole Diastole 140 18665

120 15999 ]

g 100 13332 P H o u m t a

m 80 10666 o [ r c t i i t c r

60 7999 [ o P a a t ] u o

P 40 5333 P -healthy condition (MOCK) 20 out 2666 P -severe HF - CS (MOCK) out 0 0 0.0 0.3 0.6 0.9 Time [s]

Figure 2.1.5: Aortic pressure (AoP) distributions for healthy and severe CS - cases

The values determined for LFVM can be set in a MCL in order to simulate the effect of the VAD in vitro. Reitan developed a set of parameters describing the hemodynamics of a patient with LFVM based on measurements performs on 10 patients [75], which are depicted in table 2.1.1. Based on the settings for the normal human heart, a set of parameters designed to reproduce the global dynamics in CS or AHFS have been validated on the MCL. With the values provided by Reitan [75] shown in table 2.1.1 a theoretical curve of the aortic pressure for patients in CS can be determined. This curve is measured on the MCL and the results are plotted against the one of measured healthy heart (in section 2.5.4-figure 2.6.3b) in figure 2.1.5.

Table 2.1.1: MCL settings for patients with LFVM [75]

CO AoP MAP LAP RAP Heart rate 1 [l/min] [mmHg] [mmHg] [mmHg] [mmHg] [min− ] 4 70/30 50 15 5 70

2.1.5 Head characteristics

The encased propeller has to be adapted to the adult circulatory system working in series with the human heart at flow-rates between 2 and 10 l/min and increasing pressure with 10-20mmHg . These are values used in the treatment of cardiogenic shock caused by severe left ventricular failure. Ideally, the propeller LVAD should increase the CO from 4 l/min (table 2.1.1) to the normal one (5 l/min) and the AoP pressure from 50mmHg to 90mmHg (which is the minimum limit for AHF). The design input data for the propeller VAD are given in table 2.1.2.

Table 2.1.2: Ideal LVAD propeller duty point parameters

Parameter Value design flow-rate V˙ 5 l/min pressure rise ∆P >40 mmHg ≈ 5328 Pa max propeller speed N 13000 r pm max propeller diameter D 15 mm density ρ 1035kg/m3

12 2.2 Governing equations for fluid dynamics and aerodynamics

The design of a turbomachine is initiated using the Cordier diagram which links the specific speed (NS) to the specific diameter (DS) of a turbomachine. They link the concept of machine type (axial, mixed-flow or radial) to a specific pair of specific speed NS and specific diameter DS.

Figure 2.1.6: Cordier diagram following Lewis [52]showing the ideal and existing design points set for a LVAD

Φ1/2 Ψ1/4 NS and DS (2.1.5) = Ψ3/4 = Φ1/2 where Φ and Ψ are the flow and head coefficients given by:

V˙ Y Φ and Ψ (2.1.6) = N D3 = N 2 D2 · · and Y ∆P/ρ. The pair of Φ,Ψ define the duty point of the turbomachine. = With the data presented in table 2.1.2 one can plot the duty point in the Cordier diagram, here in a ver- sion adapted from Lewis [52]. In figure 2.1.6 the red point shows the pair of NS and DS for an optimum LVAD as computed with the data from table 2.1.2. The blue point represents the actual measured duty point of the existing 14F RCP.None of the points belongs to axial type but to centrifugal respectively to mixed flow pumps. The difference between the two duty points is a result of the pressure rise dif- ference: the 14F RCP has1900Pa ≈ 14mmHg while the ideal pump has 5328 Pa ≈ 40 mmHg at the same flow rate: 5l/min. Since the RCP is an axial propeller-pump but operates as a mixed flow pump it will not operate at its maximum efficiency. The pump will operate at the duty point at off-design condition. These matter will be addressed in detail in the results chapter, in subsection 3.4.

2.2 Governing equations for fluid dynamics and aerodynamics

In this thesis propellers will be designed by using simplified fluid dynamics equations. They are vir- tually tested then in different mediums at high and low Re numbers in different fluids by CFD means. Testing of the propeller pumps occurs in both steady and unsteady flow conditions on different test stands designed to replicate in-vito conditions. In this chapter the fluid mechanics methods em- ployed in the design and simulation of the propellers will be discussed in detail. The versatility of the

13 Chapter 2 Selected aspects in relevant areas for the design of VADs physiological blood flows investigated in this thesis needs a large variety of CFD simulations: steady, unsteady, one and two-phase. All of them have in common turbulence modeling which is explained briefly. On the other side the flow around turbomachinery blades is treated stationary for the duty points. Usually the design of turbomachinery blading is performed for high Re-Numbers. Governing equations will be simplified to inviscid two-dimensional potential flows which allow the design of air-and hydrofoils.

2.2.1 Governing equations for fluid dynamics

Fluid flow is described by the equations of mass, momentum and energy conservation equations [24]. In order to solve the different types of flow, the governing equations of fluid dynamics will be brought in an useful manner. First, the equations of mass and momentum conservation will be arranged in such a way that they can be solved numerically by using turbulence models.

A fluid flowing obeys the mass conservation:

∂ρ ∂(ρUi ) 0 (2.2.1) ∂t + ∂xi = where ρ is the fluid density t is the time and Ui are the velocity components while xi denotes the spatial components (i 1,2,3). For incompressible fluid (ρ constant) equation (2.2.1) simplifies = = to:

∂Ui 0 (2.2.2) ∂xi = which can be written in component form as:

∂u ∂v ∂w 0 (2.2.3) ∂x + ∂y + ∂z =

Fluid motion is described by the generalized momentum equation, which is derived for a continuous fluid using the Newton’s second law. For Newtonian fluids the momentum equation simplifies to the Navier-Stokes equations (NSE). The general form of the momentum equation written in index form reads [24]:

∂ui ∂ui 1 ∂P ∂τik uk ρgi (2.2.4) ∂t + ∂xk = −ρ ∂xi + ∂xk + and for a Newtonian fluid the viscous stress differential term can be written:

2 ∂τik ∂ ui υ (2.2.5) ∂xk = ∂xk ∂xk

In the case of a conservative field, the gravitational term can be dropped so equation (2.2.4) using equation (2.2.5) becomes:

2 ∂ui ∂ui 1 ∂P ∂ ui uk υ (2.2.6) ∂t + ∂xk = −ρ ∂xi + ∂xk ∂xk which are the Navier-Stokes equations (NSE) written in index form. To find P and ui the above system of partial differential equations (PDE) can be solved by using specific boundary conditions and initial conditions. Further details can be found in textbooks like [24],[12] and [30].

14 2.2 Governing equations for fluid dynamics and aerodynamics

In engineering practice flows are almost always turbulent and the transition from a laminar to a tur- bulent state (i.e transition over an airfoil) is very important and is one of the most researched topics in fluid mechanics. Laminar and turbulent flow states are usually defined by the Re (Reynolds) number, named after the British researcher Osborne Reynolds [24]. The Re number defines the ratio between acceleration terms and molecular impulse transport:

u L Re · (2.2.7) = ν where u is the velocity, L is a specific length and ν is the kinematic viscosity of the medium.

To understand the unsteady nature of fluid flow this has to be interpreted in a statistical manner by analyzing the fluctuation of certain quantities around an averaged value. The Reynolds decomposi- tion will be applied to the NSE [24, 40]. Any fluctuating quantity (G) can be decomposed in two main components[40]:

G G g (2.2.8) = + where:

• G is the average part of the quantity (see figure 2.2.1)

• g is the fluctuating part of the quantity (seefigure 2.2.1) where the average si defined by:

T 1 Z G ¡x, y,z¢ lim G ¡x, y,z¢dt (2.2.9) = T T →∞ 0

G

g G

G=G+g

t

Figure 2.2.1: Local time-average of a fluctuating quantity G

Averaging can be applied to the velocity and pressure components leading to:

V ¡x, y,z,t¢ V ¡x, y,z¢ v ¡x, y,z,t¢ (2.2.10a) = +

P ¡x, y,z,t¢ P ¡x, y,z¢ p ¡x, y,z,t¢ (2.2.10b) = + Using the averaged velocities and pressures equation (2.2.6) can be now written:

15 Chapter 2 Selected aspects in relevant areas for the design of VADs

³ ´ ³ ´ ³ ´ ∂ U u ∂ P p ∂2 U u ∂ ³ ´ ³ ´ i + i 1 + i + i U i ui U k uk υ (2.2.11) ∂t + + + ∂xk = −ρ ∂xi + ∂xk ∂xk

Leading to the Reynolds Averaged Navier-Stokes (RANS) equations:

à ! ∂ρU i ∂U i ∂P ∂ ∂U i ρU k ρυ ρUiUk (2.2.12) ∂t + ∂xk = −∂xi + ∂xk ∂xk −

The term ρUiUk is called the Reynolds-Stress-Tensor and represents the closure problem of turbu- lence models. How this term is handled by different turbulence models affects the evaluation of the other terms in the equation, and hence the computed flow-field.

2.2.2 RANS turbulence modeling

CFD programs solve the NSE numerically, in particular the RANS form shown previously in equa- tion (2.2.12). For solving the integro-differential form of the fluid flow equations several methods are commonly used: finite differences, finite element and finite volume methods. Today’s CFD codes pre- fer the finite volume method which directly utilizes the conservation laws [12] which is the integral formulation of the Navier-Stokes equations. This distinguishes the finite volume method significantly from the finite difference method, since the latter discretizes the differential form of the conservation laws. The concept of RANS and Reynold stresses was introduced previously (equation (2.2.12)). In RANS the closure of the transport equations can be achieved either by solving the components of the Reynolds-Stress-Tensor (by so called Reynolds Stress Models) or by modeling the Reynolds-Stress- Tensor using an eddy viscosity, µt . The two-equation models of turbulence base on the Boussinesq assumption, who proposed that the Reynolds-Stresses should be modeled in the same way as normal and shear stresses [66]. With the Boussinesq assumption the Reynolds stresses can be written:

à ! ∂Ui ∂Uk ρUiUk µt (2.2.13) = ∂xk + ∂xi

Apart from the RANS turbulence models, it is today’s practice to include Large Eddy Simulation (LES), Detached Eddy Simulation (DES) or scale adaptive simulation (SAS). They offer a much deeper look in the physics of the fluid problems (see figure 2.2.2) by solving a broad band of time and length scales of the turbulence. The counterpart is that the needed computational effort (time and amount of CPU’s) is much higher than the one needed for a RANS simulations and for many practical applications they are not feasible. However, these methods are still ”modeling” the flow even if only at subgrid scales. In this list should be included also the direct numerical simulation (DNS). This kind of simulation does not model any turbulence scale but solves it accurately in time and space. However, DNS simulations need a very fine computational grid and a very large amount of time.

Referring to RANS modeling the less computing intensive methods are the ones using the model of eddy viscosity. But it is easy to note while looking at figure 2.2.2 that RANS is modeling near-wall flows and solves only the bulk flow. Eddy viscosity models include:

• zero-equation models (or algebraic turbulence models) i.e.: Prandtl mixing-length model, Baldwin- Lomax model, Cebeci-Smith model

• one-equation models i.e.: Spalart-Allmaras

• two-equation models like: k ², k ω, Shear Stress Transport turbulence model SST − −

16 2.2 Governing equations for fluid dynamics and aerodynamics

Large Energy Inertial Viscous scales containing subrange subrange 2 integral scales 10

101 k-5/3 100

E 10-1

10-2

10-3 10-2 10-1 k 100

Computed by RANS Modeled by RANS Computed by Solved by LES LES

Solved by DNS

Figure 2.2.2: Energy spectrum of turbulence as function of the wave number k with the application range of CFD turbulence models (adapted from Hirsch [38, pp.88])

Bardina et al. [7] have investigated the prediction of RANS turbulence models [7] and concluded that the SST model [57] has the best overall prediction results for complex flows. Its unique capability to predict flow separation in steady state simulations was found remarkable by the investigators. This is achieved by switching between two turbulence models in the same model: k ω at boundaries and − k ² in the bulk flow. By combining these two models, the deficits of both are overcome: the excessive − turbulence production at the wall of the k ² model (as described by Menter [56]) and the sensitivity − of the k ω model in free flows. The model has gained a lot of attention since its first presentation − by Menter [57] because it can solve very well flows with adverse pressure gradients (such as airfoils at higher angles of attack), which are found in all aerodynamic and hydrodynamic applications. Hirsch [38] showed that the SST turbulence model has the best predicting capability for a diffuser-case when compared to other turbulence models, while in [9] the k ² model for simulating the performance − characteristics of open-water marine propellers has been used. Later CFD results shown that the ef- ficiency was under-predicted when compared to the experiments. The fact that SST offers reliable results needing reasonable computational time, when compared to other models, lead to the deci- sion to use the SST turbulence model for the CFD simulations in this thesis. Adaptive wall functions are used in this model which verify at every vertex near the walls, if they are yet below or above the boundary layer limit; in the case the grid is fine enough (at least 10 cells grid cells in the layer (y+ <1) ) the transport equation are fully solved otherwise the mixing plane wall boundary law of Prandtl is applied [56]. Details of the used grid as well as the simulation setup are explained in chapter 3 section §3.3 and section §3.4.

In an effort to understand all currently used CFD simulation and methods for blood flows the Food and Drugs Administration (FDA) conducted a complex study evaluating the results of a benchmark flow model at 40 universities around the globe. More than half of the participants in the study were using the SST turbulence model of Menter, while the remaining used mostly the k ² turbulence − model [96]. Previously, Song et al. [95] have compared the results of difference turbulence models

17 Chapter 2 Selected aspects in relevant areas for the design of VADs with results obtained by PIV measurements for a radial type VAD and concluded that the k ω model − was the most suitable for blood pumps simulations. More recently, in [32], the SST turbulence model was used for VAD simulations.

2.2.3 Two-dimensional flows for aero- and hydrodynamics applications

A common practice in designing wing or blade sections is to use low resolution fluid mechanics meth- ods for computing the flow around airfoils. The NSE are simplified to much simpler potential flow equations which are linear and can be solved analytically by using the proper boundary conditions. Not only are the equation simplified but also the flow-field: instead of full three-dimensional simula- tions the flow field reduces to a two-dimensional one. For high Re flows it can be generally assumed, that viscosity effects are very small compared to other effects (further readings in Durst [24]). Viscous terms of the NSE (equation (2.2.4)) can be eliminated leading to the Euler equation of fluid flow:

1 ∂P ∂u j Fi ui (2.2.14) − ρ ∂x j = ∂xi where Fi is the component of the force and P is the pressure in the flow field. All other components are as defined in the NSE. For flows where gradients of velocity and pressure are mostly in two directions it is easy to cut slices perpendicular to the main flow direction as depicted in figure 2.2.3. The resulting slices are easier to analyze using the fluid flow modeling in two-dimensions. Cross-flow effects are considered later on and added to the basic two-dimensional results as it will be shown in section §2.4 of this thesis.

y W x

Figure 2.2.3: Two-dimensional slice (airfoil) of a blade

A strip cut through a wing or a turbomachinery blade is called an airfoil (for liquid flows hydrofoil). If the index style used in equation (2.2.14) is dropped and only two dimensional flow is considered, the Euler equations are:

1 ∂P ∂u ∂u Fx u v (2.2.15a) − ρ ∂x = ∂x + ∂y

1 ∂P ∂v ∂u Fy u v (2.2.15b) − ρ ∂y = ∂x + ∂y

There is still either a direct nor an analytic solution of the above equations, but they are more simple to solve than the NSE. Simplification of the Euler equations by assuming rotation-free flow leads to

18 2.3 Air- and hydrofoil families: laminar NACA 6-Digit series the potential flow. Potential flows have the advantage of linearly combining solutions of basic flows like the one of vortex, source sinks, etc (further readings in [24, 72]).

2.3 Air- and hydrofoil families: laminar NACA 6-Digit series

In [41] are presented several airfoil calculation methods based on the potential flow theory. Although they are not so precise, when compared to simulations performed by using NSE, they are much faster and thus can be used with success in design problems. A new numerical analysis problem for thin airfoil cascades based on the thin airfoil method in [41] is presented in subsection 3.2.1 of this the- sis.

Typically there are two different ways in approaching the aerodynamic design of sections: direct and inverse. By using the direct method the aerodynamic characteristics of a given geometry can be com- puted (cl , cm) while, by using the inverse method, the geometry is computed based on a given aero- dynamic input (cl ). To make this concept clearer, the lift coefficient equation for a cambered circular arc airfoil (thin foil) derived by the Joukowsky transformation is recalled [72]:

µ 2f ¶ c 2π α (2.3.1) l = + c where f is the camber. Let in this equation AOA α 0 and c be known, one obtains following = = l equation for the camber:

c c f · l (2.3.2) = 4π

From the above equation it is obvious that the camber is direct proportional to c . Usually c 1 in l = non-dimensional aerodynamics and the denominator 4π is a constant. If the shape given in this particular case by a thin arc airfoil is searched, the aerodynamicist can compute the required camber straight forward. However, in reality, the airfoils have a specific thickness and this brings a second variable in the problem: the thickness distribution.

y th camber-line

chord th x

Figure 2.3.1: Sketch of a modern airfoil composed of a mean-line (camber-line) and a thickness dis- tribution

Generally a thickness distribution is added to the mean-line [2] since it is generally known that the thickness contributes very little to the force and moment on an airfoil [72]. Figure 2.3.1 depicts the structure of an airfoil composed of a mean camber line to which a thickness distribution was added.

Although mean-lines having circular arc shape are a solution of the inverse airfoil problem they are certainly not the ideal solution given by the thin-airfoil theory. Mostly the aero- and hydrodynamic problem of an airfoil refers to the separation point3 on the suction side which is influenced also by the shape of the camber. Although the viscous flow is not yet taken into consideration there are enough literature references which provide at an early stage the proper direction for the airfoil design. The

3or transition point

19 Chapter 2 Selected aspects in relevant areas for the design of VADs most comprehensive airfoil database which includes also the theoretical backgrounds was provided in [2] and was the result of work at NACA.

Abbott and Doenhoff [2, pp.73] showed, that by combining the zero normal velocity boundary con- dition, setting α 0 and transforming the equation in a similar manner as showed previously for = the Joukovski analytical model, one can obtain the shape of the mean-line by arbitrarily setting the maximum point along the chord (no longer fixed at 0.5 as for cambered arcs). Mean-lines have been computed for uniform load distribution over the camber from x/c 0 to x/c a (a is the position = = in % of chord where load changes) and decreases linearly to x/c 1(trailing edge). They are called = NACA mean-lines and are defined by following function [2]:

½ · ¸ y cl 1 1 ³ x ´2 x 1 ³ x ´2 ³ x ´ 1 ³ x ´2 1 ³ x ´2 i a ln a 1 ln 1 1 a c = 2π(a 1) 1 a 2 − c | − c | − 2 − c − c + 4 − c − 4 − c + − x x x o ln g h (2.3.3) − c c + − c where g and h are given by:

1 ·µ1 1¶ 1¸ g − lna a2 (2.3.4) = 1 a 2 − 4 + 4 −

1 ·1 1 ¸ h (1 a)2 ln(1 a) (1 a)2 g (2.3.5) = 1 a 2 − − − 4 − + − These mean lines have an ideal angle of attack (αi ) which is given by:

cli h αi (2.3.6) = 2π(a 1) + For the NACA mean-line the commonly used thickness distributions are from the NACA 6 series [2, pp.120]. These airfoils have been developed in order to maintain the boundary layer laminar for high Reynolds number over a large area of the airfoil. Because of this the airfoils are called laminar airfoils. The effect of a the large area of laminar boundary layer was a very low drag coefficient at the design lift coefficient. However, if the foil is away from its design point (i.e. at higherAOA) the drag coefficient might increase drasticly [87],[2].

0,13

0,12 63-series

Theoretical 64-series

a / d /

l l 0,11 65-series

dc oil lift slope lift oil 66-series airf 0,10 4 and 5-digit 0,09 0 4 8 12 16 20 24 Thickness in % of chord

Figure 2.3.2: Comparison between theoretical and experimental lift slopes of airfoils [72]

If the performance in terms of lift is compared between the different NACA airfoils it is easy to observe that 6th series are much closer to the theory than the 4 and 5 digits (figure 2.3.2). The figure shows the behavior of the lift curve at different airfoil thicknesses and for different thickness functions. The

20 2.4 Propeller design methods small gradient of the lift supports the previous affirmation that thickness does not have a great con- tribution to lift. Airfoil thicknesses usual do no exceed 20% and one can read from figure 2.3.2 that the 64 and 65 thickness would expect almost no lift increase between 8 and 20%. Kerwin [44] proposed to use NACA 65 series thickness in the design of marine propeller which will be used in this thesis (section §3.3) alongside NACA 4 series thickness.

2.4 Propeller design methods

So far the concept of blade strips (airfoils) has been introduced, so that the reader understands the fundamental aerodynamic concepts. This section presents the fundamentals of propeller design which are connected to the airfoil theory by the blade element theory. Figure 2.4.1 depicts the dif- ferent stages of simplification used in the design and analysis of a propeller. The most simple model is the ”simplified actuator disk model” or the axial momentum theory for propellers. More complex, following this classification, is the blade element theory which is the two-dimensional analysis of pro- peller radial strips; a step which makes use of the airfoil and as it will be shown, of the cascade theory shown in the next chapter.

Axial momentum theory 1-D

Blade element theory 2-D

Prandtl-Betz loss function (1919)

BEM Goldstein (2-D + Radial loss functions) loss function (1929) Design and analysis Vortex-lattice-method Lerbs/Wrench (60‘s)

Generalized actuator disc Actuator line theory Actuator surface theory theory Analysis Full 3-D CFD Analysis (Navier-Stokes)

Figure 2.4.1: Stages of propeller design and analysis

The integration of the radial section results leads to the global propeller performance, but this simple integration does not take into account the tip losses of the propeller in its slipstream. These are given by the radial momentum loss, a theory which was first introduced 1919 by Betz [10] in a form of an approximated tip loss model. This theory was followed in 1929 by the analytical model of Goldstein [36] and latter, in the sixties by the vortex-lattice-methods (VLM). The combination of blade element theory and axial momentum theory is named Blade Element Momentum theory (BEM), which makes the core of the present work and was first published by Glauert [35]4. BEM can be used for both de- signing and analyzing propellers or wind turbines. However, this theory is also based on simplifying assumptions regarding the radial flow and the propeller wake contraction. Models have been devel- oped to overcome its penalties by solving axissymetrical NSE with disk force distributions provided by airfoil data [61]. The generalized actuator disk theory makes the same assumption as the simple actuator disk theory: the flow is axissymetric, there are a infinite blades [61, 64] while the rotor forces are computed using a BEM. It can, for example determine, if the assumptions in the BEM were cor- rect or not but can not be used directly for the blade design. The actuator line theory goes beyond this theory and distributes the force on specific ”lines” which are replacing the blades allowing thus

4first edition from 1926

21 Chapter 2 Selected aspects in relevant areas for the design of VADs to fully analyze the wake [109]. This is however, mostly useful for wind turbine computations were the wake flow is important for wind mill interaction in wind farm design. The advantage is that the viscous flow surrounding the blades is not solved, providing an advantage in terms of computational time and power [64]. But this method can not be used for designing purposes either, because it is just an analysis method. One of the drawbacks is, that the inaccuracy of the 2-D airfoil aerodynam- ics computation is carried through the complete process. However, computers have become more powerful and nowadays it is possible to perform full 3-D CFD simulations of propellers together with their surroundings. This is the most complex way to analyze propellers and can show from the fluid dynamics perspective all details of the flow. Its drawbacks concern wall bounded grid accuracy, tur- bulence modeling, a matter which will be disused in subsection 3.2.2.

In the present research the BEM approach is used to design the propeller, while the validation is performed by both CFD and experimental means. A simplified BEM analysis method is also shown in section §3.3.

2.4.1 Axial momentum theory for propellers

Rankine was the first to propose a very simple theory based on the axial momentum computed on a fluid passing a propeller disk (actuator disk). Similar to the one-dimensional Euler theory for pumps this theory doesn’t take into account the geometry of the propeller. This is concerned later in the more elaborate blade-element theory (subsection 2.4.2). The axial momentum theory gives the upper limit of performances which can be achieved by a propeller. This is very important since it shows the physical limits of every propeller design.

For the actuator disk theory one has to assume the following [19]:

1. The fluid is inviscid and incompressible.

2. The propeller can be replaced by an actuator disk which has an infinite number of blades.

3. Velocity is constant over the disk.

4. The propeller produces thrust without causing rotation in the slipstream5 .

The thrust developed by the propeller is assumed to be uniformly distributed over the disk. Pressure increases suddenly among the propeller disk with the increment ∆p (as depicted in figure 2.4.2). The free stream velocity is denoted by V , while the velocity gradient over propeller is v and at the exit ∞ from the slipstream is v1 which leads to following expressions for the velocities in the sliptream:

V2 V v V3 V v1 (2.4.1) = ∞ + = ∞ + Within this theory is assumed that the pressure increases over the propeller disk from p (upstream) to p p (downstream). By applying the momentum equation between inlet and outlet (figure 2.4.2) +4 the axial force (thrust) reads:

T ρAp v1 (V v) (2.4.2) = ∞ + where A πr 2 represents the area of the propeller disc. p = t After some equalities one obtains:

1 v v1 (2.4.3) = 2

5This is the original Rankine theory which was completed in 1920 by Betz who proposed a momentum theory which includes the effect of the angular momentum [11]

22 2.4 Propeller design methods

V8 V2 V3

1 2 3

Dp p0 p

Figure 2.4.2: Exaggerated sketch showing the basis for the actuator disk theory which shows that half of the acceleration takes place in the front of the propeller and the other half behind it. Inserting equation (2.4.3) in equation (2.4.2) yields to:

T 2ρAp v (V v) (2.4.4) = ∞ +

The kinetic energy balance between the inlet (1) and outlet (3) of the propeller’s slipstream (fig- ure 2.4.2) can be written as:

E T (V v) (2.4.5) = ∞ +

The energy is equal to the total work done by the propeller:

ΩQ T (V v) (2.4.6) = ∞ +

The ratio of the useful work to the total work is given by the efficiency:

TV V η ∞ ∞ (2.4.7) = ΩQ = V v ∞ + If v aV efficiency becomes: = ∞

1 η (2.4.8) = 1 a + This expression represents the ideal efficiency, and is the maximum achievable by a propeller. In this ideal case the only loss considered is the one of kinetic energy of axial velocity in the slipstream and a represents the axial interference factor of a propeller.

Instead of using thrust and power of propellers it is useful to define non-dimensional thrust and power coefficients to replace them. This has the advantage that in some limits the thrust and power characteristics of a certain propeller can be scaled or the analysis completed are more general. The following coefficients are usually found in literature [19]: thrust coefficient:

23 Chapter 2 Selected aspects in relevant areas for the design of VADs

T KT (2.4.9) = ρn2D4 torque coefficient:

Q KQ (2.4.10) = ρn2D5 advance coefficient:

V J ∞ (2.4.11) = nD

Instead of the rotational speed the advance speed V can be used and in that case the thrust coeffi- ∞ cient reads [19]:

2T CT 2 (2.4.12) = ρAP V ∞ and the torque coefficient is replaced by the power coefficient:

2P CP 3 (2.4.13) = ρAP V ∞

CT can be expressed in terms of the axial influence coefficient a by introducing equation (2.4.4) in equation (2.4.12) and simplifying:

C 4a (1 a) (2.4.14) T = + this procedure is applied on equation (2.4.13) by inserting equation (2.4.6):

C 4a (1 a)2 (2.4.15) P = +

Equation 2.4.14 and 2.4.15 are similar to the ones presented by Glauert [35], but he normalized the equations by the square of the propeller diameter (D2 4R2) instead of the propeller area. It is now = easy to develop equations in order to see how the efficiency varies with the thrust or power coefficient. It is though possible to have a look on how the propeller performance coefficients and the efficiency may vary with regard to the axial flow coefficient.

The variation is plotted in figure 2.4.3. For the sake of brevity the plot also shows axial wind tur- bines. In the case of wind turbines the efficiency is computed by using the inverse of the efficiency for propeller: η 1 a [35]. Some interesting propeller characteristics can be observed in figure 2.4.3: = + the propellers designed for low power and thrust coefficients have higher efficiency than the ones designed for higher values. In the case the propeller produces no thrust the efficiency is equal to 1. More interesting is the shape of the C P slope for turbines where the maximum of the power co- 16 efficient equals exactly the Betz-limit: CP Betz 0.593 [35] for which the maximum efficiency − = 27 ' is η 0.67 at a 1/3. For an advance ratio of 1 the turbine has zero efficiency and zero tur bine = = − − power.

24 2.4 Propeller design methods

5

C 4 T C P

η 3 , P

C h , T

C 2

1

0

- 1 - 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 a [ - ]

Figure 2.4.3: CT,CP and η computed from the axial influence factor a

The propeller losses, which have not been mentioned until this moment are very important in the design and analysis of a propeller. The propeller losses are caused by:

1. Blade friction drag

2. Kinetic energy of the rotation of the slipstream

3. Blade tip losses

dr

W

V8 V + v V + v 8

8 1

2w w

Figure 2.4.4: Sketch showing both axial and angular momentum components for a propeller

These additional losses are addressed in the next sections since they are important in achieving high efficient designs. The first step in completing the axial momentum theory is to introduce the terms regarding the angular momentum and thus taking into consideration the influence of the rotation upon the performance (figure 2.4.4). It will be assumed that no induced angular velocity exists before the propeller, and after the propeller the angular velocity is 2ω [35] (as depicted in figure 2.4.4). The change in angular momentum reads:

3 dQ Ω4πr V ρ (1 a)a0dr (2.4.16) = ∞ + where a’ is called the angular interference factor of a propeller.

An expression for the thrust depending on the angular momentum is needed, this can be computed by applying Bernoulli equation for the pressure increase at a blade element r and after some manip- ulation this becomes:

2 3 dT 4ρa0Ω πr (1 a0)dr (2.4.17) = −

25 Chapter 2 Selected aspects in relevant areas for the design of VADs the expression for efficiency of a propeller reads:

V dT η ∞ (2.4.18) = ΩdQ and by inserting the expressions of dT and dQ in the definition of efficiency:

2 3 V 4ρa0Ω πr (1 a0)dr ∞ − η 3 (2.4.19) = Ω4πr V ρ (1 a)a0dr ∞ + which simplifies to:

(1 a0) η − (2.4.20) = (1 a) +

The introduction of angular momentum leads to higher propeller losses and thus lower propeller efficiency shown by comparing equation (2.4.8) with equation (2.4.20).

2.4.2 Blade element theory

The blade element theory models completely differently the action of a propeller by taking into ac- count its geometry and the forces acting on it. In this theory the blade is divided into a large number of elementary strips (figure 3.4.19), similar to the finite wing theory of Prandtl [73]6. The basis of this theory was first introduced by Froude in 1878 [33], but it was not practical to use until Betz published his well known paper ”Screw propeller with minimum energy loss” [10]. Prandtl gave this theory an applicability by publishing a note at the end of the paper in which he showed, how to use the Betz theory for designing a propeller. This will be addressed later in subsection 2.4.3.

dr W

r

W

Figure 2.4.5: Blade element

The blade element depicted in figure 2.4.5 is basically an airfoil subject to a resultant incident velocity W , which similar to a wing airfoil, produces lift and consequently drag. At each radial position of the blade, vortices are shed and move backwards in a helical vortex sheet (see figure 2.4.7). The forces acting on a blade airfoil are depicted in figure 2.4.6a.

6first edition by Prandtl in 1931

26 2.4 Propeller design methods

dL dL

dQ dQ aV8 dT dT W W a’rW

V8 dD b dD b V8 f* f

rW Wr (a) Velocity plan acting on a blade element (b) Induced velocities on a blade element

Figure 2.4.6: Forces and velocities acting on a blade element

It is clear at this moment that the blade element method has its basis in the finite wing theory. β is 7 referred here as the propeller pitch and φ∗ the undisturbed flow angle . However, the forces on a blade element have different meanings, which will be obtained by equating dT and dQ from dL and dD:

1 2 ¡ ¡ ¢ ¡ ¢¢ dT ρ BcW c cos φ∗ c sin φ∗ dr (2.4.21a) = 2 l − d

1 2 ¡ ¡ ¢ ¡ ¢¢ dQ ρ BcW c sin φ∗ c cos φ∗ dr (2.4.21b) = 2 l + d where B and c are the number of blades and the chord length of the airfoil respectively. The results for dT and dQ in equation (2.4.21a) and equation (2.4.21b) can be inserted in the efficiency formulation from equation (2.4.18) which leads to:

¡ ¢ tan φ∗ η (2.4.22) = tan¡φ ε¢ ∗ + where ε represents the drag-to-lift ratio ε c /c [19]. Up to this point there is no relation between = d l the momentum theory presented in the previous section and the blade element theory. Glauert [35] connected both theories for the first time. The velocity plan showed in figure 2.4.6a is changed by the induced axial (aV ) and induced tangential velocity (a0Ω) as depicted in figure 2.4.6b. ∞ The difference between the section’s pitch angle β and the hydrodynamic pitch angle φ is the angle of attack α. From figure 3.4.1 the hydrodynamic angle φ can be written:

V (1 a) tan¡φ¢ + (2.4.23) = Ωr (1 a ) − 0 Consequently the relations 2.4.21a and 2.4.21b become:

1 dT ρ BcW 2 ¡c cos¡φ¢ c sin¡φ¢¢dr (2.4.24a) = 2 l − d

1 dQ ρ BcW 2 ¡c sin¡φ¢ c cos¡φ¢¢dr (2.4.24b) = 2 l + d

7In [44] and [19] hydrodynamic pitch angle is denoted with β and the propeller pitch with θ

27 Chapter 2 Selected aspects in relevant areas for the design of VADs efficiency can be written as:

V dT η ∞ (2.4.25) = ΩdQ which by changing some terms leads finally to: ¡ ¢ ¡ ¢ 1 a0 tan φ η − (2.4.26) = (1 a) tan¡φ ε¢ + +

From equation (2.4.26) it is clear that for inviscid flow ε 0 the result is the same as for the actuator = disk with induced angular momentum in equation (2.4.20).

2.4.3 Design theory using the radial loss model proposed by Betz and Prandtl

It was shown in [73] that optimal span-wise circulation distribution exists for wings of finite length. In subsection 2.4.2 it has been shown that propellers produce lift in the same way as wings resulting in a radial circulation distribution. This is important because propeller blades also have losses at the tips. The causes of the tip vortex have already been reviewed for wings in [73] or in the excellent work of Kerwin [44] and will not be again discussed here.

W

V8

propeller pitch 2p Rtan( b )

w n f w s t axis v’

2’nd vortex sheet

1’st vortex sheet Figure 2.4.7: Propeller wake with helical vortices and the concept of displacement velocity (v’)

Prandtl was the first to find a law for optimum radial circulation written in the article of Betz [10], which was based on a simple idea to be exposed below. For this he introduced the concept of dis- placement velocity denoted here v0. For explaining the concept of displacement velocity a constant helical vortex sheet is considered after the propeller (figure 2.4.7). The helical vortex sheet moves everywhere perpendicular to itself with a velocity w n which is the same with the slipstream velocity

28 2.4 Propeller design methods

8 (figure 2.4.7)[48]. The filament helix angle is φs as depicted in figure 2.4.7 and it is also the angle be- tween the axial component among two consecutive sheets and the filament velocity w n (figure 2.4.7). In this case the displacement velocity (axial apparent velocity between two sheets) can be written as:

wn v0 ¡ ¢ (2.4.27) = cos φs

Betz showed in his paper in 1919 [10] that a propeller with minimum losses requires the vortex sheet to look like a regular screw, which means that r tan¡φ ¢ is constant independent of radius. However, · s this assumption is only valid for propellers with low and medium loads where the contraction of the wake can be neglected9. In this condition one other assumption can be made, the filament helix angle φs can be approximated with the hydraulic pitch angle φ[48]:

φs φ =∼

By recalling equation (2.4.4) for the actuator disk and rearranging the expression:

2 T 2ρAp aV (1 a) (2.4.28) = ∞ +

Differentiating the above expression by radius results in:

dT 4ρπr aV 2 (1 a) (2.4.29) dr = ∞ +

Furthermore the overall radial momentum loss of an infinitely small element can be accounted by a loss factor F [3, 10] resulting in a an expression for the momentum loss: 2V aF dm. Inserting F in ∞ equation (2.4.29) one creates the combined momentum loss expression for the elemental thrust of the actuator disk:

dT 2ρπrV (1 a)(2V aF ) (2.4.30) dr = ∞ + ∞

Following the same argumentation equation (2.4.6) becomes:

dQ 1 2ρπrV (1 a)(2Ωr a0F ) (2.4.31) dr r = ∞ +

Prandtl calculated the blade losses by approximating them with the potential jump computed for a infinitely row of plates [10]. Hence, F has following expression10:

µ B(R r )pv 2 r 2Ω2 ¶ 2 − 0 + F arccos e− 2v0r (2.4.32) = π where B represents the number of blades. There have been fewer attempts to make changes to this theory until Larrabee [48].

He arranged equation (2.4.32) in a more useful way:

8Prandtl denoted the helix angle ε[10] 9this implies also that radial velocities are neglected 10in the original publication F was named Φ

29 Chapter 2 Selected aspects in relevant areas for the design of VADs

2 ³ f ´ F arccos e− (2.4.33) = π where f is given by:

à ! B ³ r ´ pλ2 1 f 1 + (2.4.34) = 2 − R λ

V 11 and here λ represents the advance ratio λ ∞ . = ΩR At a radius r the total lift (of all blades) per radius is given according to the Kutta-Jukowsky law (see [72]):

dL BρW Γ (2.4.35) dr = where Γ represents the circulation around an airfoil; in the propeller wake as a consequence of Helmholtz’s vortex theorems [73] :

BΓ 2πr F w (2.4.36) = t setting the circulations in equation (2.4.35) and equation (2.4.36) equal to each other will lead to a circulation distribution for minimized power losses.

The tangential velocity is given by (2.4.7):

w w sin¡φ¢ (2.4.37) t = n and by replacing wn from equation (2.4.27) it becomes:

¡ ¢ ¡ ¢ w v0 cos φ sin φ (2.4.38) t = by replacing v0 with the displacement velocity ratio ζ v0/V : = ∞

¡ ¢ ¡ ¢ wt V ζcos φ sin φ (2.4.39) = ∞

At this point in [48] is introduced the Glauert or Goldstein factor (G) as an expression of the Prandtl factor (F ):

G F x cos¡φ¢sin¡φ¢ (2.4.40) =

Ωr where x is the non-dimensional speed given by V . Introducing equation (2.4.40) in equation (2.4.36) and rearranging some terms the circulation becomes:∞

G Γ 2πV 2 ζ (2.4.41) = ∞ BΩ

11this is the inverse of the tip speed ratio (TSR) defined for wind turbines

30 2.4 Propeller design methods

The thrust and the torque can be written as a function of the elemental lift and drag:

dT dL dD dL cos(φ) sin(φ) cos(φ)(1 ε) (2.4.42a) dr = dr − dr = dr −

dQ 1 dL dD dL sin(φ) cos(φ) sin(φ)(1 ε) (2.4.42b) dr r = dr + dr = dr +

Larrabee[48] has used equation (2.4.42a) and equation (2.4.42b) and combined them with equa- tion (2.4.30), equation (2.4.31) and equation (2.4.35) in order to compute the displacement velocity 12 v0 . This concludes to optimal design as shown in his publication [48].

Further attention is focused on using some of the expressions derived previously for obtaining an optimal circulation distribution for minimum losses. In [3] was introduced a design scheme by mod- ifying the one of Larrabee and setting boundary conditions for obtaining minimum losses. This will be explained in detail bellow.

First the momentum equations 2.4.30 and 2.4.31 are set equal with 2.4.42a and 2.4.42b for relating the interference factors to the displacement velocity ratio ζ:

ζ a cos2 ¡φ¢¡1 εtan¡φ¢¢ (2.4.43a) = 2 −

µ ¶ ζ ¡ ¢ ε a0 cos φ sin(φ) 1 (2.4.43b) = 2 + tan¡φ¢ where also equation (2.4.40) and equation (2.4.41) have been used for expressing dL/dr . From 2.4.43a and 2.4.43b and by using the velocity plan depicted in 3.4.1 the hydrodynamic pitch angle can be computed as:

³ ζ ´ 1 2 λ tan¡φ¢ + (2.4.44) = ξ where ξ r and is called the non dimensional radius. = R Previously it was defined that the minimum loss condition r tan¡φ ¢ is constant at all radii [10]. · s Equating this in equation (2.4.44) leads to a very important result:

V µ ζ ¶ Cbetz ∞ 1 (2.4.45) = Ω + 2

13 where C betz has been denoted the radial constant for minimum losses .

Equations have been developed until this moment to specify the optimal circulation distribution and to link this to the axial momentum theory. However, in a design scheme a desired thrust is specified. Alternatively when a propeller has to be matched to certain existing engine the power is specified instead of thrust. They will be defined as the constraints or boundary conditions of the problem to solve: design a propeller which achieves a desired thrust and the maximum efficiency at that inflow velocity or advance coefficient.

12Larrabee also gave in his paper expressions for computing the displacement velocity of moderately loaded propeller 13Viscous losses can be accounted or not (by ε)as seen in equation (2.4.43a) and equation (2.4.43b)

31 Chapter 2 Selected aspects in relevant areas for the design of VADs

Expressions of CT and CP (2.4.12 and 2.4.13) were used in [48] to express the thrust and the torque radial gradients in equation (2.4.42a) and equation (2.4.42b) resulting in:

2 C 0 I 0 ζ I 0 ζ (2.4.46a) T = 1 − 2

2 C 0 J 0 ζ J 0 ζ (2.4.46b) P = 1 − 2 where the differentiation of both coefficients has been done with respect to the non-dimensional radius ξ and:

¡ ¡ ¢¢ I 0 4ξG 1 εtan φ (2.4.47a) 1 = −

µ ¶µ ¶ I10 ε ¡ ¢ I 0 λ 1 cos φ sin(φ) (2.4.47b) 2 = 2ξ + tan¡φ¢

µ ε ¶ J 0 4ξG 1 (2.4.47c) 1 = + tan¡φ¢

µ ¶ J10 ¡ ¡ ¢¢ 2 ¡ ¢ J 0 1 εtan φ cos φ (2.4.47d) 2 = 2 −

Recalling equation (2.4.46a) and remembering the Betz condition and the radial constant for mini- mum losses equation (2.4.45), which proves that the speed ratio ζ is independent of radius, and which can be expressed in term of equation (2.4.46a) as:

s µ ¶ µ ¶2 µ ¶ I1 I1 CT ζ (2.4.48) = 2I2 − 2I2 − I2 and the power coefficient is then:

C J ζ J ζ2 (2.4.49) P = 1 + 2 where the integration of C is computed between ξ k (hub) and ξ. If the power is specified then T 0 = solving for ζ equation (2.4.46b) :

s µ ¶ µ ¶2 µ ¶ J1 J1 CP ζ (2.4.50) = − 2J2 + 2J2 − J2 and the thrust becomes:

2 C I ζ I 0 ζ (2.4.51) T = 1 − 2 with this the basis of a design theory has been set. The actual implementation and modification of this theory is shown in sections 3.2.3 and 3.2.4.

32 2.4 Propeller design methods

2.4.4 Design theory using the radial loss model proposed by Goldstein

Although the Prandtl-Betz theory can produce highly efficient propellers, it is limited by the approx- imation Prandtl made: the loss factor F is computed as the potential jump between two plates of an infinite cascade in a cross stream. Goldstein [36] improved the theory ten years after Betz by com- puting the true potential of a three-dimensional vortex sheet. He showed, initially, in his publication, the computation for two propellers: one two bladed propeller and one four bladed propeller. Due to the complexity of the calculus involved in the computation, the ideal circulation distribution for sev- eral propellers having different loadings has been resolved and tabulated by Wrench [120] and more recently by Wald [114]. The ideal circulation distribution according to Goldstein reads:

 ³ 1 ´  2 2 ¡ 1 ¢ λv µ 8 F (µ) 2 µ IB m m 2 Bµ 0 X∞ B,2m 1 0 X∞ + 2 + Γ  + am  (2.4.52) = 2B · 1 µ2 − π2 (2m 1)2 − π 2 ³ 1 ´ m 0 1 µ0 m 0 ¡ 1 ¢ + = + + = IB m m 2 Bµ0 + 2 +

14 where λ,B,v0 were defined previously . µ represents the ratio between displacement velocity mul- tiplied by radius and angular velocity µ v 0r and µ represents the same but for the tip radius only: = Ω 0 µ v 0R [120]. There have been several attempts to solve equation (2.4.52) but the widely used solu- 0 = Ω tions are the one published in [120]. The difficulties, which arise in solving this equation, are posed by the infinite series, which cannot very easily be approximated on one hand, and on solving the am coefficient in the second term of the equation, on the other hand [120]. As pointed out in[120] the terms of the first sum can be transformed in a Lomel function and the ones of the second sum in a n-order modified Bessel function of second kind. This was solved and tabulated for optimal propellers at different advance ratios with 3,4,5 and 6 blades in [120, 106] . Later in [114] the tables in [106] were modified by tabulating the Goldstein factor (G) by the advance ratio (λ) and dimensionless radius (ξ). The transformation made by Theodorsen [100] for the design of highly-loaded propellers were also introduced. For using the tabulated values of G, equation (2.4.41) will be rewritten in terms of G (λ,ξ) and reads:

ΓBΩ G (λ,ξ) (2.4.53) 2πV 2 ζ = ∞ However, if an iterative design approach is employed (for correct wake alignment), as proposed in[3], it will be very complicated because of the values of G are tabulated. These would have to be inserted in a table to be read by the design software in every iteration. The Betz/Prandtl loss factor is an- alytically computed for any propeller configuration and has thus an advantage upon the tabulated Goldstein values. An analytical expression for the Goldstein loss factor, even if approximate, could offer the same advantage for the Goldstein method as for Betz/Prandtl method. Such a fitted func- tion for the Goldstein loss factor can be found in the appendix of the publication of Batten et al. [8] for computing the blade losses of marine water turbines. An exact analytical approach of computing the induced velocity of helical vortices of a lifting line has been presented in [69]. The velocity field is determined analytically by representing the vortex sheets by discrete helical vortex filaments using the method described in [68]. Although the method presented is straight forward and can be used for any propeller and any loading, it implies the discretization of the blade with at least 100 points, mak- ing it slow. However, a comparison with the Goldstein factor results from [106] showed an excellent correlation [69]. The loss coefficient for a marine propeller can be fitted from Goldstein’s charts by following relation [8]:

¡ ¢ 2 µcosh ξ f ¶ κ arccos · (2.4.54) = π cosh¡f ¢

14the original formulation in Goldstein did not include the number of blades

33 Chapter 2 Selected aspects in relevant areas for the design of VADs and f is given by:

B 1 f (2.4.55) = 2ξtan¡φ¢ − 2

κ is inserted in equation (2.4.40) leading to:

G κ x cos¡φ¢sin¡φ¢ (2.4.56) gold = · · which is then inserted into equation (2.4.41):

Ggold Γ 2πV 2 ζ (2.4.57) = ∞ BΩ

2.4.5 Design method correction for moderately loaded propellers

Both the Betz and Goldstein theories were originally proposed for the design of lightly-loaded pro- pellers. Apart from them, propellers can be further classified by the load they produce into moderately- and heavily loaded propellers. The lightly-loaded propellers are different from the moderatly loaded ones by the assumption, which is made in the computation of the ideal circulation distribution (re- placing V by V v0in equation (2.4.41) [114]): ∞ ∞ +

G Γ 2π(V v0)v0 (2.4.58) = ∞ + BΩ

In the assumption of lightly-loaded the speed term is given only by the stream velocity far in front of the propeller, V , while in the moderately loaded formulation this is corrected by adding the dis- ∞ placement velocity v0 of the wake resulting in the term: V v0. An in-detail explanation of this can ∞ + be found in the extensive works [114] citing [100]. For the computation of wake contraction one can for example work on the equations for radial equilibrium for the actuator disk in Lewis[52]. This has been successfully implemented for an axial fan [88].

2.4.6 Design theory using lifting-line and vortex-lattice theories

In [73, 44] it has been shown, how the downwash can be computed for arbitrary wings by using a dis- tribution of vortices over a lifting line. It can be shown, that the same method functions in an inverse way: an elliptic circulation distribution can be obtained for a specific wing geometry. The problem which arises for propellers is that the vortex theory which works for wings cannot be applied for heli- cal vortex sheets, otherwise the lifting line concept can be applied without limitations. Although the solution of Goldstein[36] represents the most accurate flow solution of helical vortices the solution is as mentioned in the previous subsection not very easy to handle.

Since the derivation of the wing lifting line-theory and of its computational method is based on the derivation of the Bio-Savart-Law for a series of vortices this was also the approach for solving the same problem for helical vortices. The solution of the flow potential induced by helical vortices has been solved in [42] followed by [49] and later brought to a much more useful form in [121]. Details of the method can be found in [44].

The basis for this approach has been set by Betz [10] as described in subsection 2.4.2. The Betz con- dition for minimum losses (2.4.2) reads:

r tan¡φ¢ constant =

34 2.4 Propeller design methods from figure 2.4.6b results:

¡ ¢ V aV r tan φ r ∞ + ∞ = Ωr a Ωr − 0 and then:

¡ ¢ V (1 a) r tan φ r ∞ + = Ωr (1 a ) − 0 V recalling figure 2.4.6 and substituting Ωr :

¡ ¢ ¡ ¢ (1 a) r tan φ r tan φ∗ + = (1 a ) − 0 which results in:

¡ ¢ tan φ (1 a) 1 + (2.4.59) tan¡φ ¢ = (1 a ) = η ∗ − 0 equation (2.4.59) shows, that for an optimum propeller the ratio between the hydrodynamic pitch and the hydrodynamic angle has to be constant and is equal to the inverse of efficiency. The result is consistent, since the efficiency for an optimum circulation distribution has to be constant [44, pp 161]. This can be written as:

¡ ¢ tan φ∗ η constant (2.4.60) tan¡φ¢ = =

The constant given by the ratio of the two velocities is only dependent upon the design thrust and advance ratio. This result is however only true for uniform axial inflow. For a non-uniform axial inflow the following was introduced[49]:

¡ ¢ tan φ(r )∗ p Υ 1 wx (r ) (2.4.61) tan¡φ(r )¢ = − where the axial inflow function Va(r ) has the following expression:

Va(r ) [1 wx (r )]V (2.4.62) = − ∞ and Υ is an unknown constant which has to be iteratively determined. Equation 2.4.61 is known as the Lerbs Criterion [19, 44].

The first description of a vortex lattice method (VLM) for designing propellers using the Wrench method [121] belongs to Kerwin [43] who was also the main developer of such programs in the last century. The modern vortex-lattice methods rely mostly on his developments. The computer im- plementation of this theory was done in the PVL code developed at MIT [44]. In an effort to make the code more user friendly and easier to use the code has been reprogrammed from FORTRAN to MATLAB [20]. The most important difference to the original code was the development of a Guided User Interface (GUI). The code was constantly improved15 by including several numerical optimiz- ers, effect of casings and an analysis code, carried out at the MIT’s Department of Ocean Engineering.

15the code was named mpvl and then renamed OpenProp, and is released under GNU General Public License and the current version is 3.3.4[26]

35 Chapter 2 Selected aspects in relevant areas for the design of VADs

The actual implementation can be found in the PhD thesis of Epps [27]. The code’s structure is very flexible allowing a very easy interaction of anyone aware of it to modify or add/remove new methods. For example, Koopmann [45]extended the original OpenProp code for the design of counter-rotating propeller.

Since the solution of the vortex-lattice-method is a numerical approximation for computing the ve- locities induced by a helical vortex sheet, it has to be equal to the solutions given by Goldstein [36] , Tibery and Wrench [106] or the solution of Okulov and Sørensen [69].

2.5 Design and construction of a closed loop measurement test rig

Bridging simulation results to reality is very important in the context of product development. For tur- bomachinery product validation the measurement of characteristic curves (pressure and efficiency) plays a definite role. Blood pumps need a realistic validation because ”in-vivo” validation is hardly possible. Moreover, in the development phase of a VAD, many of iterative steps have to be looped and some of these iterations have to be measured. The test rig presented in this chapter has been de- veloped and validated under the supervision of the author as part of the student works and diploma thesis of Töpfer [107, 108] and Schlegel [81].

2.5.1 Test rig set-up and construction

The aim of a test rig is to allow the measurement of specified quantities, which define the perfor- mance of turbomachinery. Specifically in the case of pumps the pressure flow-rate characteristics and the hydraulic efficiency are measured. In the case of VADs the efficiency is not always an im- portant issue. In the case of blood pumps the flow-rates are very low (5 l/min is the flow-rate of the human heart) and in the specific case of the RCP pump the maximum pressures are also very re- duced, since, as already mentioned, the pump just helps the heart work. In this case the design of a test rig capable of measuring the pressure flow-rate curve is a challenge because the pressure rise of the pump is low, so all the losses in the system have to be kept small. If losses are too high only a small part of the pressure curve can be measured (the green part shown in figure 2.5.1), while for small losses the curve follows the path until point 3 in figure 2.5.1.

1 Pump slope with high system losses 2 Pump slope with medium system losses DP 1 3 Pump slope with low system losses 2

3

. V

Figure 2.5.1: System characteristics of pump indicating different losses inside the measurement rig

For this reason the construction of the test rig was focused on keeping the head losses as low as pos- sible. The diameter of the pipe was first set at φ 25[mm], a dimension which represents the normal = size of an adult aorta (23 - 25 mm according to [101]). The head losses are given by:

w¯2 ∆P ζ ρ (2.5.1) ζ = · · 2

36 2.5 Design and construction of a closed loop measurement test rig where ζ represents the head-loss parameter, ρ the density and w¯ the average speed in the pipe. Fur- ther on, one can write ζ:

L ζ λ (2.5.2) = f · d where λf represents the friction factor and L and d the length and pipe diameter, respectively. λf depends on both Re-number and roughness of the pipe and can be found in the Moody (or Cole- brook) chart, for example in [118, pp. 349]. One look at equation (2.5.2) reflects the good choice of the major diameter: the head-loss parameter is smaller. Simultaneously λf is increased because of the decreased Re-number:

v d Re · (2.5.3) = ν

If the expression is evaluated for specific figures and λf is computed for a smooth pipe of 2 m length and for a flow rate of 5 l/min with the help of the Colebrook formula [118, eq.6.64 on pp.348], the head losses are ~40% smaller for the pipe with the 25 mm diameter when compared to the ones computed for a 23 mm pipe. The chosen diameter for the pipes of the test rig is 25 mm.

The loop was designed as a rectangle with two long and two short sides. The length of the long side was determined by the need for a fully developed flow profile. Since laminar as well as turbulent flow regimes were expected in the pipe, the condition for both of them has to be investigated relative to the pipe length. For a fully developed laminar profile the recommended length is given by Wagner [113]:

L 0.06 Re d (2.5.4) l aminar = · · which in the case of a Re 2300 and a 0.025 m pipe results in a length of 3.5 m. In the case of turbulent = flow the length is given by [113]:

L 5.3 Re0.12 d (2.5.5) tur bulent = · · in which case for 5 l/min and 0.025 m pipe diameter results a length of 0.35 m. The measurements were supposed to take place predominantly in the turbulent flow regime (over 2.8 l/min) so in this case the length is quite small. For the laminar regime the length is hard to be handled in lab condi- tions. The length of the straight pipe was extended from 0.35 m to 0.75 m and a stator was added after the bends. One straight segment was made of calibrated glass having a diameter of 0.024 mm (did not comply the DIN standard DN 25~0.025 mm which was used for the steel components). This was needed for the LDA measurements.

The test rig was constructed as modular concept (figure 2.5.2), which means it can be easily disas- sembled and if necessary changed to the desired shape. The concept is shown in figure 2.5.2. The structure is based on 4 steel pipe bends having a curvature radius of 150 mm. This dimension is con- strained by the depth of the optical bank on which the set up had to be built. The 180° pipe bend has ideally the smallest losses when it is constructed in one piece [89], but because other compo- nents had to be placed in-between the solution with two 90° pieces was chosen. On the opposite side of the glass pipe was placed the candidate propeller prototype on a shaft, which was centered with the help of a stator (figure 2.5.2). In order to visualize the prototype, an adapter was milled from Plexiglas (polymethylmethacrylat-PMMA). This was connected to the bend by two straight steel pipe segments. The small segments contain on one side a manual valve for controlling the flow-rate in the system and on the other side an opened reservoir to fill the system with fluid. This also helped with the aeration of the system. All pipes were connected by PVC milled connectors which were sealed

37 Chapter 2 Selected aspects in relevant areas for the design of VADs with rubber O-rings. The measured pump characteristics were: relative pressure difference, flow-rate and rotational speed.

Table 2.5.1: Measured parameters and their range

Parameter Range (units) Converted range (SI units) flow-rate 0-30 l/min 0-0.68 m/s (average speed) pressure difference 0-15 mmHg 0-2500 Pa rotational speed 1000-14000 r pm

The range of the measured quantities is presented in table 2.5.1. They have been inspired by the original measurements of the 14F RCP pump shown by its originator in [74].

Regulating Glass pipe valve

Steel pipes Laser source

Photodiode Connectors(PVC)

connection to motor

de-swirler Shaft Transparent window

Figure 2.5.2: CAD sketch of the loop test rig concept

The tested propeller prototype is installed in front of a de-swirler (stator) in the middle16 of a long pipe in order to provide a correct measurement of the static pressure. Initial CFD simulations (figure 2.5.3) have shown that the flow after the propeller is highly turbulent and the pressure distribution is not homogeneous in the section where the pressure is measured. The left picture in figure 2.5.3 presents the pressure distribution downstream the propeller showing that this is almost double at the walls (where the pressure taps are mounted) than at its minimum close to the middle of the pipe.

Measuring the pressure at the walls is erroneous since this is not the real pressure acting on the sur- face of the section but only the local wall pressure where this is measured. The actual pressure in the pipe would have to be tested by a sensor in every point of the measurement surface, but this is impossible for such dimensions and this would be also a very costly and time consuming procedure. Instead, a de-swirler, or a stator was placed after the propeller in order to homogenize the fluid field downstream the propeller. At the beginning a honeycomb was used, but this had high pressure losses so it was replaced by a stator with 4 ”wings” which also centered the propeller shaft.

16in figure 2.5.1 is not figured in the middle

38 2.5 Design and construction of a closed loop measurement test rig

Figure 2.5.3: CFD computed pressure distribution downstream the propeller

d‘B

pipe wall

lB

dB

Figure 2.5.4: Bore for pressure measurement according to DIN

The bores for the pressure measurement (figure 2.5.4) were designed according to the DIN ISO 5198 [21] in order to avoid any malfunctions in the pressure measuring system. This specifies the diam- eter of the bore (dB ), the length inside the pipe wall(lB ) as well as their connection to the pressure sensors. Four bores placed at an angle of 90° around the pipe were used for the measurement of the pressure once down- and once upstream the propeller. The distance between propeller and each measurement section was set at 10 pipe diameters.

The final test rig was layed on a optical bench (figure 2.5.5) with the help of aluminum profiles. This eases the adjustment between different components which are shifted along them. On the profiles are mounted the pipe holders which can be fixed flexibly as requested by the setup. They can be adjusted in the height by screws in order to provide a perfect plane for the pipes. The driving motor is a MAXONN brushless and it is controlled by a computer through a ST Micro controller board via the USB port. The motor is fixed on an adjustable motor bench on a profile.

The motor is connected through a cardan universal joint to a long shaft which reaches the transparent window in the middle of the long side of the test rig. The shaft is centered by two friction bearing holders which are inside the pipe. At the other end of the shaft the tested propeller is mounted.

39 Chapter 2 Selected aspects in relevant areas for the design of VADs

8 9 10 11 12 13 14 15

7

6

1 5 4 3 2

Figure 2.5.5: Ready test rig

Table 2.5.2: Components of the test rig shown in figure 2.5.5

1 Laser beam 2 Diaphragm 3 Avalanche photo-diode 4 Glass pipe 5 PVC connector 6 Pipe holder 7 Throttle 8 Propeller drive motor 9 Transmision shaft 10 Upstream pressure measuring section 11 Pressure sensor 12 Prototype PMMA window 13 Downstream pressure measuring section 14 Flexible tube 15 Fluid reservoir

2.5.2 Methods and materials

The relative pressure difference is read by a Rosemount G1151 pressure transducer. This is produced by Emerson Electric Co and it can read relative or absolute pressure for both liquids and gases. The pressure gauge is read in front and behind the propeller-pump at a distance of 345mm. The trans- ducer was calibrated at LSTM for a range between 0 and 25mbar (0 2500Pa). The output of the − gauge is analog; its signal is converted to a voltage between 0 and 10V with a special in-house con- verter. The signal is sent via a coaxial cable to a data acquisition card.

40 2.5 Design and construction of a closed loop measurement test rig

Figure 2.5.6: Rosemount G151 pressure transducer

The voltage signal is converted to pressure according to the sensor range (linearly) in a LabViewr program who’s structure and functionality is explained in this subsection.

For the measurement of the flow-rate a LDA-based system was preferred. This complex measurement system has important benefits when compared to other flow- or mass-measurement systems:

• LDA flow-rate measurement is non-invasive so the head losses in the system are minimal

• LDA is an absolute measurement method (only one calibration is required)

• it is independent of the flow state (laminar or turbulent)

• it is independent of the fluid (can be either water or water-glycerin mixture)

• LDA can measure precisely very small flow-rates (below 1.667 l/min which is the limit for typical magnetic inductive or ultrasound devices [99, 16])

• no additional costs for a sensor were needed, the components were available in-house.

The original LDA flow rate measurement system was developed at LSTM at the beginning of the 1990’s by Teufel [99]. The fundamentals of the method are explained also in [18] and more detailed in the book of Ruck [80]. The physical background used in Laser Doppler Anemometry (LDA) is the Doppler effect. The movements of the scattering particles are measurable by changes of the laser light fre- quency. These frequency changes are measured and related to the movement of the particles. This implies also that for the measurement of velocity suitable light-scattering particles must be present in the driving fluid. These can be either natural or they can be added separately.

An incoming beam from a laser light source with the beam direction li is intersected by a particle (figure 2.5.7). This scatters the light in all directions, including A and B with the vectors ki and mi respectively. The movement of the particle at the speed ui scatters light that has a Doppler shift of the frequency. At the same time the particle acts like a sender of the shifted frequency. However, direct measurement of the scatter spectrum is not possible because the frequencies are in the range of 1015 Hz. In contrast, the frequencies of the Doppler shift (about 2 105Hz) can be measured. Thus, in order · to determine the laser light frequency caused by Doppler shift, two laser signals have to interfere (figure 2.5.9).

41 Chapter 2 Selected aspects in relevant areas for the design of VADs

Direction of velocity vector Receiver A

}i Particle {k

{li} Ray scattering from {m i } source {Ui} Instantaneous velocity of the particle

ReceiverB

Figure 2.5.7: Principle of laser Doppler shift

Focal length

Lens Lens

Laser l light 1 j

l2 Aperture Beam splitter

Interfering fringes

Figure 2.5.8: Two-beam LDA setup

According to [24] in the case of interfered laser beams interference fringes occur in the overlapped area. These are divided into successive maxima and minima of intensity. Hence, the fringe spacing ∆x depends on the laser wavelength λL and the on the angle φ:

λL ∆x (2.5.6) = 2 sin¡φ¢ ·

The angle φ can be determined geometrically:

µ BS ¶ φ arctan (2.5.7) = 2 f · where BS represents the distance between the two beams and f represents the focal length (as de- picted in figure 2.5.9). The signal frequency observed by the detector (not shown in figure 2.5.9) is given by:

¡ ¢ 1 U 2 sin φ f s ⊥ · · (2.5.8) = ∆t = λL

42 2.5 Design and construction of a closed loop measurement test rig where ∆t is:

∆x ∆t (2.5.9) = U ⊥ and represents the time needed by a particle to pass a single fringe from the interfered surface.

As a result the velocity read by the LDA system is:

λL f U · ¡ ¢ (2.5.10) ⊥ = 2 sin φ · In hardware the system consists of a He-Ne-Laser (Mellis Griot ) with λ 630nm which is posi- r L = tioned parallel to the flow direction (shown on the left side of figure 2.5.9).

APD

Figure 2.5.9: Laser and optical measurement setup

After leaving the source, the beam is split in two beams of equal intensity at a distance BS 50mm. = They are then refracted by a bi-convex lens having a focal length of 300mm. The interfered area of the beams is set exactly in the middle of the glass pipe. An aperture and another lens focuses the light into avalanche photodiode (APD), which is a combination of both photomultiplier and a photodiode and which amplifies the optical signal in the present case with 40dB. After the photodiode the signal is amplified again (38dB) and filtered by a bandpass filter (LHS of figure 2.5.10). This is set manually according to the signal which is visualized with the help of an oscilloscope as shown on the RHS of figure 2.5.10.

The oscilloscope is not detecting the frequency, but the filtered and triggered signal coming from the APD can be visualized on the monitor. The frequency can be computed by setting the proper time scale on the monitor and counting the divisions of an oscillation.

The Doppler frequency is detected by a frequency tracker which is shown in figure 2.5.11. Basically, LDA trackers are modulators of the trailing frequency [80]. They use the filtered signals of the Doppler frequency and compare them to the frequency of an internal oscillator. If the Doppler frequency differs from the one of the oscillator, this one is adjusted in such a way that both frequencies are equal. The frequency of the oscillator is known so this is the searched frequency. The tracker needs a high

43 Chapter 2 Selected aspects in relevant areas for the design of VADs

Figure 2.5.10: Left hand side: Bandpass filter (LSTM-invent-895-102), right hand side: Philips PM3295A oscilloscope

Figure 2.5.11: BBC Görtz LSE 01 Doppler signal processor (frequency tracker) concentration of particles in the measured fluid because it requires a continuous signal. One of its advantages, which has also been used in the present setup, is that it has an analog output (0 10V ). − This can be calibrated according to the frequency range and directly read by the data acquisition card.

The data from the pressure sensor and from the tracker was acquired by a National Instruments (NI) r data acquisition card. This has a NI chassis (NI 7128) with 8 slots with data input modules (NI 9125 with BNC connectors17), as depicted in figure 2.5.12. The chassis has an USB 2.0 connection to the PC. The system can acquire data at a frequency of 100kS/s.

Signals coming from all sensors can be read simultaneously from all channels, they are then triggered and computed in the LabView software from NI. This is presented in the next subsection.

2.5.3 Data recording

As stated previously the signals (from pressure measuring tap and tracker) are forwarded to the eval- uation software on the PC via data acquisition card. This software converts the voltage directly into volumetric flow rates or pressures, respectively. The measurement panel is shown on figure 2.5.13.

As stated in the previous subsection the pressure will be noted in SI pressure units Pascal as well as in Torr. The panel allows to set user-defined conversion factors for volt into millibar, since there

17BNC connector (Bayonet Neill–Concelman) is a miniature quick connect/disconnect RF connector used for coaxial cable

44 2.5 Design and construction of a closed loop measurement test rig

Figure 2.5.12: NI 9178 USB chassis with 3 NI 9215 BNC input modules

Figure 2.5.13: Front panel of the measurement program are various measuring field of different pressure sensors. One can also insert a deviation in percent needed in the computation of the volumetric flow-rates which has been determined during several preliminary tests at a value of 0.94%. The complete lay-out of the software is shown in figure 2.5.14.

Figure 2.5.14: Measurement software layout in LabView (adapted from [108])

The incoming signal is received from the USB data acquisition card by the DAQ-assistant, shown on the left hand side in figure 2.5.14. For the pressure computation only some simple arithmetical pro- cedures are needed since here the voltage is linearly converted to units. Further on the bottom course of VI’s shows the conversion of the voltage signal received from the tracker into volumetric flow rate,

45 Chapter 2 Selected aspects in relevant areas for the design of VADs

first by transforming the Volt signal into a frequency succeeded in Sub-VI (”Teufel” in figure 2.5.14) by a mathematical computation of the maximum velocity. Finally the signal is transformed into the corresponding volumetric flow rate in an additional Sub-VI. It is essential to mention that the cal- culation to determine the average velocity is made in two steps followed by their summing up first the boundary layer and then the one from the boundary layer to the symmetry axis. The basis of the method is exposed in Teufel [99] and will be not extended in this work. However, the basics, as well as some necessary ideas about how it works, will be given at this point. Basically, after the maximum velocity in the center of the pipe is known (by computing it from the frequency with equation (2.5.8)) a velocity profile has to be assumed and integrated over the pipe in order to find the flow-rate. For the laminar case the velocity profile is known and the integration leads to the flow-rate:

π R2 V˙ · umax (2.5.11) = 2 · where R is the pipe radius, and umax is the maximum velocity in center of the pipe.

The turbulent stationary case is a bit more complicated since there are different levels of approxima- tion of the profile; the most simple approximation being the power law profile:

1 ³ r ´ n u umax (2.5.12) = R where u is the local velocity corresponding to the r position in the radius and n is the power coefficient depending on the Re number (values of this can be found for example in [15]). The average velocity − is given in this case by:

2 n2 u¯ umax · (2.5.13) = · (n 1)(2n 1) + + and from this the volumetric flow-rate is computed. A function, which is founded theoretically and validated experimentally, is given by Prandtl et al. [Prandtl 73] and it is based on the Prandtl mixing length concept. The average velocity is given in this case by:

µ 1 ¶ u¯ uτ ln y+ C (2.5.14) = κ +

However, these functions are simply approximating empirically the complicated mechanism in the boundary layer. This is composed from 3 components (from wall to center): the laminar layer (0 < y+ 15), the overlap layer (or turbulent viscous layer)(y+ 15) and the wake layer. Every layer has < > its own analytical expression and this was combined by Neubert and Walz [62]. It was improved and simplified in [99] where the velocity equation for the whole profile is given by:

½µ 1 ¶ ¾ 1 P ³ y ´ a y+ ¡ ¢ u+ 1 C a y+ C e− · ln y+ 1 C W (2.5.15) = − κ − · · − · + κ + + + κ · · δ where

³ y ´ ³ ³ y ´´ W 0.5 1 cos π (2.5.16) · δ = · − δ and represents the ”wake layer”.

46 2.5 Design and construction of a closed loop measurement test rig

The constants have following values:

P 0.55 = C 5.1 = κ 0.4 = a 0.3 = and C and κ(von Karman constant) are the constants of the logarithmic wall function.

The ”Teufel” function is much more complex than the power-law profile which was presented previ- ously. In order to see the differences between the different turbulent velocity profiles and their influ- ence upon the aimed volumetric flow-rate, they have been compared against a balance measurement of the mass flow-rate. These were performed for different flow-rates as shown in subsection 2.5.4. The function in Teufel [99] (equation (2.5.15)) has been chosen. This is programmed in the sub-VI called ”Teufel” shown in figure 2.5.14.

The steps followed in the computation of the velocity profile are:

Computation of the friction velocity (uτ) by using a Newton-Raphson method on a modified equa- tion (2.5.15) which is shown in figure 2.5.15. This is needed because it is directly related to the non- dimensional velocity by:

u¯max u+ (2.5.17) = uτ umax is computed from the measured frequency by equation (2.5.10). The variables are transferred to an array, which is converted in a for-loop into a "fractional string"(figure 2.5.15).

Figure 2.5.15: Computation of the friction velocity (adapted from [108])

The boundary layer thickness and the wall shear stress is computed in a sub-VI by[113]:

2.9 63.1 δ · 2 rpi pe (2.5.18) = Re0.875 · ·

The yet known friction velocity is introduced in equation (2.5.15), which is then integrated over the pipe to compute the flow rate as depicted in figure 2.5.16.

In the next section the methods of computing the volumetric flow-rate of turbulent flows are bench- marked in order to determine which of them is more suitable for the present case.

47 Chapter 2 Selected aspects in relevant areas for the design of VADs

Figure 2.5.16: Computation of the stationary turbulent velocity profile (adapted from [108])

2.5.4 Initialization and characteristics of the LDA flow-rate measurement system

This section will present the benchmark used for comparing the 3 presented turbulent flow profiles for computing the turbulent flow rate in a pipe. This way will choose one method for all future mea- surements.

The benchmark are measurements of the mass-flow-rate performed by using an electronic balance on which a container is filled with water. The time is counted at prescribed intervals by using an electronic time counter. This is performed several times for four positions of the throttle valve. Water flows from a container placed at 2 m height (left hand side of figure 2.5.17) and flows through a setup composed from the glass pipe with LDA system and the throttle (middle picture of figure 2.5.17) into the container placed on the balance (not shown).

Figure 2.5.17: Pictures showing the setup for the benchmark measurement from left to right: inflow container (at ~2 m height), flow path in the glass pipe with LDA and throttle

The values used for the benchmark are presented in table 2.5.3. They are the average of several mea- surements (4-10) performed for each throttle position.

Table 2.5.3: Values used for the benchmark

units throttle 1 throttle 2 throttle 3 throttle 4 V˙ [l/min] 21.235 21.034 19.276 11.294 Doppler frequency [kHz] 234.755 240.064 223.095 134.309

Next, a graphic comparison between the three proposed turbulent profiles will be presented. For the comparison a Doppler frequency of 300[kHz] has been chosen.

In figure 2.5.18a the profiles are plotted from the wall to the pipe axis. It is visible that the ”Teufel” profile is different from thef Prantdl log-law and from the power-law. Their shape is tough very sim-

48 2.5 Design and construction of a closed loop measurement test rig

1.257 0.880 1.132 0.792 1.006 0.704 s] s] 0.880 0.616 0.754 0.528 0.629 0.440 0.503 0.352 elocity [m/ elocity [m/ V V 0.377 Teufel 0.264 Teufel 0.251 Prandtl Prandtl Power-law 0.176 Power-law 0.126 0.088

0 0.0012 0.0048 0.0084 0.012 0 1.05x10-5 3.15x10-5 5.25x10-5 Radial distance [m] Radial distance [m] (a) Flow profiles from wall to pipe axis (b) Flow profiles from wall to δ

Figure 2.5.18: comparison of the proposed turbulent profiles ilar. In order to see the differences, the profiles are plotted from wall to the computed boundary layer thickness (which is showed in dimensional length) in figure 2.5.18b. The difference between the profiles are more evident in this region. The shape of the ”Teufel” solution is different than the ones of the power-law and Prandtl log-law and is very similar to the shape of the laminar boundary layer solution of Blasius which can be found for example in [118]. The next diagrams show the dif- ferences between the computed volumetric flow-rate obtained by the use of all three profiles and the benchmark described previously.

100

Balance Teufel 98 Log-law Power-law

96

94

92

90 Difference to volumetric-flow-rate of the balance [%] balance of the volumetric-flow-rate to Difference

88 Throttle 1 Throttle 2 Throttle 3 Throttle 4

Figure 2.5.19: Comparison of the volumetric flow-rate computed by using the three turbulent profiles and the balance measurement

The comparison reveals the performance of the Teufel profile which delivers in three from four cases the closest value to the balance measurement. Another interesting trend can be seen from the dia- gram: the precision of all methods improves towards low flow-rates. In the case of the lowest flow-rate (throttle 4) the simple log-law profile computes the flow-rate with the highest accuracy. An explana- tion for this can be that the boundary layer is turbulent and in this case the log-law as well as the power-law predict better the shape of the profile close to the wall (as depicted in figure 2.5.18b). However, the precision of the measurement is very good when using the ”Teufel” formulation and the error stays in the investigated flow-rate windows between 1 and 6%.

49 Chapter 2 Selected aspects in relevant areas for the design of VADs

2.6 Design and construction of a MCL

In order to assess the VAD performance in close to reality conditions it is required to build an exper- imental setup which reproduces the condition of the human body. Such setups are called MCL or simply mock and are used for the research and development of blood pumps. A mock has been con- structed and improved in order to assess the performances of the investigated pumps and to provide realistic boundary conditions for the instationary CFD simulations. The work has been performed as part of a student work under the supervision of the author by Schlegel [81]. The setup and some specific measurements will be presented very briefly in this section.

The MCL is built analog to the human circulatory system as depicted in figure 2.6.1. In table 2.6.1 are listed the elements of the mock. This MCL system is based on the two component windkessel model, described in [117]. The two ventricles are driven by FESTO pneumatic actuators which provide air for the ventricles. Ventricles are half filled with Glycerol mixture from the hydraulic system and the other half is filled with air which is pulled or suctioned by actuators. The airflow is controlled by elec- tromagnetic valves (FESTO CPE18-M1H-5J-1/4 ) which are driven by a rotating magnet. Two sensors 1 placed on it give the pull signal for 3 of the cycle duration (systole) and then suction signal for the 2 other 3 of it (diastole) replicating in this way the pulsation of the heart.

Components Measurement locations

Figure 2.6.1: Analogy between the human circulatory system (LHS[1]) and the Mock (RHS))

The fluid is then pulled through a flexible pipe in the direction of the aortic arch. After the ventricle a check valve is placed in order to prevent the back-flow when the air is suctioned back by the actuator. Further two windkessels are placed to attenuate the pressure peaks produced by the ventricle. In the human body this function is taken by the elasticity of the blood vessels. Downstream the aortic windkessel (3) a transparent replica of the aortic arch18 is placed, which continues with a straight plastic pipe simulating the descending aorta in the human body. An adapter for introducing the test VAD is placed at the lower end of the pipe(4). On the way to the capillaries of the lower body region a flow-rate sensor (V) is placed. The sensor is a magnetic-inductive type (Promag 50P15 Endress & Hauser). It is followed by an adjustable resistance (5) which replicates the capillaries in the human body. Downstream comes the right atrium and then the right ventricle. The ventricle is followed by a windkessel and then a resistance, which replicates the lungs. After the lungs is the left atrium and afterward the left ventricle. The heart valves are modeled by check valves.

18provided by Cardiobridge GmbH

50 2.6 Design and construction of a MCL

Table 2.6.1: Mock components in analogy to the human circulatory system shown in figure 2.6.1

human circulatory system Mock Description 4 10 left atrium 5 1 left ventricle 2 windkessel / ventricle attenuation 6 3 windkessel / aortic attenuation 3 aorta (aortic arch) 4 VAD adapter 8 5 capillaries 11 6 right atrium 1 7 right ventricle 8 right windkessel 3 9 lungs

The pressure of the liquid is monitored at five locations (P1-P5) and the air pressure in the ventricles. Five pressure transducers of type 23Y (Keller AG) were used for the pressure measurement of the liquid. They were calibrated (by the manufacturer) for a pressure range from 0 to 500 mbar (relative pressure). Four circumferential bores connected by a ring channel to the pressure transducer were used at each location on the pipe for the pressure measurement. In the ventricles the air pressure was measured by 2 26 PC0250 D6A pressure transducers build by Sensortechnics.

Pressure

Averaged values

P3 filtered

Air pressure Flow-rate filtered

Figure 2.6.2: LabView measurement panel for the mock

It was mentioned earlier that flow-rate was measured by a magnetic-inductive sensor (Promag 50) which can measure time dependent the signal. The sampling frequency was rough (ca.18 samples / s) so this had to be filtered and smoothed in the data measurement program, which is shown in figure 2.6.2. All other sensors can measure the signal at higher frequency of 1000 samples / s. The data acquisition was achieved by using the same equipment as for the loop rig and consisted of the same NI 9178 USB chassis with 3 NI 9215 BNC input modules as used for the loop test rig (subsection2.5.2). The signal was received as a voltage from sensors and was calibrated for each of them. An offset can be set by the user for each of the sensors. The front panel of the program is shown on figure 2.6.2 and contains on the left side two windows showing the real time signal of all pressure transducers (liquid flow on the upper side and air in ventricles on the lower side). On the right upper side are shown time averaged values of both pressure and flow rate. In middle of the right side are placed the real time filtered P3 pressure signal and the filtered flow rate signal while on the lower side is a NI built in Fast

51 Chapter 2 Selected aspects in relevant areas for the design of VADs

Fourier Transformation (FFT) analysis for the P3 signal. The FFT analysis was used to detect the main frequency and for finding eventual interference frequencies.

Target output parameters of the mock were initially set for average values of healthy humans. Typical average values for the mock settings are presented in table 2.6.2 and were provided by Reitan [75].

Table 2.6.2: Average values used to set the mock for ”healthy” condition

CO AoP MAP LAP RAP Heart rate 1 [l/min] [mmHg] [mmHg] [mmHg] [mmHg] [min− ] 5 80/120 100 10 5 70

Time dependent pressure and flow-rate curves can be found in physiology text books i.e. [84]. Figure 2.6.3 depicts comparisons of the time-dependent MCL measured AoP with values taken from [84] and with values obtained by using similar setups (figure 2.6.3a).

1 3 0 1 7 3 3 2 1 3 0 6 . 0

1 2 0 1 5 9 9 9 1 2 0 5 . 5 ]

P g ] o g H u F t ( H l m a 1 1 0 1 4 6 6 5 1 1 0 5 . 0 o m o w m r [ - m t ) r [ i ) a c c i c t ) t 1 0 0 1 3 3 3 2 i 1 0 0 4 . 5 e [ t r P r

[ o a o l a / ] a ( m t ( t u 9 0 1 1 9 9 9 9 0 4 . 0 P ( l i t ) u i o o o u t P ( l i t . ) n

P o u t P ] P ( p r e s e n t ) P ( M O C K ) 8 0 o u t 1 0 6 6 6 8 0 o u t 3 . 5 P ( Q U T ) o u t F l o w - r a t e ( M O C K ) 7 0 9 3 3 3 7 0 3 . 0 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 T i m e [ s ] T i m e [ s ]

(a) Comparison of the MCL AoP (Pout or P3) to human aortic (b) MCL vs human AoP and flow-rate pressure [83] and to the MCL AoP results of QUT [37]

Figure 2.6.3: Validation of the MCL by comparing AoP to literature and human measurements

The matching of the MCL AoP measurement to the measured human AoP shown in figure 2.6.3b is very good. Moreover, when comparing it also to values found in literature [37] the MCL results are closer to the physiological values. The larger systolic pressure of the QUT shown in figure 2.6.3a is caused by a smaller pulsating frequency of (60 pulses/min) which was used in the measurements. The isovolumic relaxation which follows the ejection from the ventricle (the small buckle after the systole in figure 2.6.3b) is matched only by the present MCL. This is a result of using two serial windkessels on the left side of the heart.

In this chapter were introduced considerations about human circulatory system, heart and blood damage. These explained the impact of a HF on the hemodynamics of the circulatory system. Con- sequently the duty point of a LVAD was derived ant then plotted in the Cordier diagram. This shows that propeller-pumps are not the ideal turbomachine to be used as LVAD for HF.Governing equations of fluid flow were introduced with a short survey on turbulence modeling and airfoil families. These were followed by a review of state of the art methods for airfoil and propeller analysis and design, and last by a description of the experimental test-rigs and of the methodology used for the validation of the designs presented in this thesis.

52 Chapter 3 Presentation and discussion of the results

This chapter is divided in four sections and presents the general procedure for designing propellers, its application and validation on propeller design. The first section presents the general procedure used for the design of propellers while the second section presents the implementation of a design method into a propeller design code. This is extended to the design of multiblade propellers by using an airfoil cascade method which is developed and validated. A further extension is the propeller analysis code based on the BEM method. The application of the design code is presented in the next section for designing open water marine propellers. Last section concentrates on the application of the code on the analysis, design and optimization of encased propellers used as LVAD.

3.1 Procedure for designing propellers

The concept of optimum propeller has been introduced in the last chapter. There have also been presented nearby all up-to-date design methods for optimum propellers. One of the challenges of this thesis is to validate the proposed design frameworks by full 3-D-CFD simulations and experimental measurements.

Figure 3.1.1: General procedure for designing propellers

Figure 3.1.1 shows the general propeller design procedure, which is used in this thesis. It is based on the propeller design method presented in the previous chapter, which is combined with full 3-D CFD simulations used for the validation of the propeller designs. The importance of CFD simulation in the validation of propeller designs is mentioned for example in the works of Benini [9] or in Carlton [19]. Using CFD simulations in the design phase significantly reduces the development time and costs of

53 Chapter 3 Presentation and discussion of the results a propeller and can present very detailed pictures of the flow. In this way expensive prototypes are produced only for a few chosen designs for the final validation.

This chapter is structured in three main parts: development of a propeller design and analysis code by using the CVL method, design procedure for multiblade propellers and design procedure for encased propeller pumps used as LVADs. During the development of the propeller design code it become clear, that in the propeller theory existed a gap in the propeller design methodology, where the blade interference plays a major role. Since the propeller code needed a validation for propellers running at ideal conditions (i.e. high Re), it was decided that it should be used for validating the CVL method for marine propeller design. For the cases studied the analysis code was validated. After these steps the code was used for designing LVADs. CFD was used also here for validating and analyzing the designs and the best ones were experimentally tested. Finally the most promising of them was simulated by CFD time-dependent while using information provided by the MCL to show the improvements in conditions close to the ones in the human body.

3.2 Development of a propeller design and analysis code

This section is divided in six subsections and covers the development and validation of a new nu- merical solution of potential flows around airfoils cascades followed by its implementation in a new propeller design code called ADAP.The potential code is presented in subsection 3.2.1 and 3.2.2 while data flow in this propeller code is presented in subsection 3.2.4.

3.2.1 Numerical solution for thin airfoil cascades

Assuming that there is no radial flow and having computed an optimal circulation distribution in ra- dial direction, the design process demands the two-dimensional sections to fulfill the required circu- lation (lift). There are two ways of fulfilling this request: an existing hydrofoil is chosen from a catalog based on its aerodynamic characteristics (CL,CD ) or a hydrofoil is designed to fulfill the demands. NACA investigated both methods in the last century, as summarized in [2]. The inverse design of a hydrofoil is performed by generating analytically a camber-line (mean line) which delivers a certain lift [2], presented shortly in section §2.3. As already mentioned the thickness contributes very little to the force and moment on an airfoil [72]. Generating a proper camber-line is therefore the most im- portant step in designing a hydrofoil. A vortex-based analysis method for thin airfoils was presented in [41] where the numerical solution for arbitrary thin-cambered airfoils is named the Vortex Lumped Method.

A numerical method of computing the incompressible two-dimensional flows in turbomachinery cascades (figure 3.2.1) is described in [52] and has originally been developed by [54]. Lewis [52] solves the complex potential problem for two-dimensional cascade numerically, by using a vortex panel dis- tribution. This method has been developed for thick airfoils which means it solves both the camber and the thickness distribution problem. Previously to Martensen, Numachi followed by Weinig have solved analytically the complex potential problem of two-dimensional flat plates cascades [116, 65]. Later, Weinig [116] developed a method of solving the exact potential flow through the cascades of infinitely thin circular arcs. However, none of these last methods could solve the problem for camber- line cascades of arbitrary shape. Schlichting and Scholz [82] have developed a method for computing the vortex distribution for compressor cascades by using the classic Glauert Fourier series method. Several other frameworks for thick profile cascade analysis have been developed, probably the most used being (M)ISES, an MIT code developed by Drela [22]. This, however, is a two-dimensional CFD code which solves the inviscid Euler equation. It also computes the transition point and the boundary layer.

54 3.2 Development of a propeller design and analysis code

However, since none of the present methods enables a turbomachinery designer the generation of a proper camber-line by taking in account the influences of the blades (cascades) a numerical method based on the Vortex Lumped (or Lattice) Method is developed and validated in this section. Its usage on the design of propellers is shown on a test case validated by 3D CFD in section §3.3.

The complex potential of a singular vortex reads:

Γ F (z) i lnz (3.2.1) = −2π and this has following velocity components:

Γ y Γ x u v (3.2.2) = 2π x2 y2 = −2π x2 y2 + +

Figure 3.2.1: Sketch of a cylindrical meridian surface through an axial turbomachine and the resulting cascade

This basic solution can be used in solving the complex problem of a thin airfoil: a distribution of vortices will be placed on the line and by using boundary conditions (zero flow normal to the panel) the value of each vortex will be found. The Kutta condition is satisfied automatically by the present method since it has a singular tangential velocity component on the flat plate (panel) as shown in figure 3.2.2b [41].To the RHS of the system will be added the velocities of the free stream. The air- foils are divided in panels with a vortex placed at 1/4 l and a collocation point, where the boundary · conditions are placed at 3/4 l, also shown in figure 3.2.2b. ·

y y (xi=3 , y i=3 ) (xj=3 , y j=3 ) n = (sinb , cos b ) n n i i i 3 4 h(x) G j n2 n5 G G 4 n 3 G5 6 b b n1 ti = (cos i , - sin i ) G2 G6 n7

G7 b G1 i

V 8 b a V U x x (a) (b)

Figure 3.2.2: Panel method for thin airfoils (reprinted from Miclea-Bleiziffer et al. [59], with permis- sion from Elsevier)

55 Chapter 3 Presentation and discussion of the results

If panels are placed arbitrary on the camber line as shown in figure 3.2.2a, the velocity influence coefficients are given by [41]:

¡ ¢ ¡ ¢ 1 y y sin β − 0 · i ax 2 = −2π (x x )2 ¡y y ¢ − 0 + − 0 ¡ ¢ 1 (x x ) cos β − 0 · i ay 2 (3.2.3) = −2π (x x )2 ¡y y ¢ − 0 + − 0 with the RHS:

¡ ¡ ¢ ¡ ¢ ¢ RHS V cos βi sin(α) sin βi cos(α) (3.2.4) = − ∞ +

To solve the unknown vortex strength the influence coefficients (one for each direction) are summed ¡a ax ay¢, resulting in the following system of linear equations in matrix form: = +      a11 a12 ... a1N Γ1 RHS1       a21 a22 ... a2N  Γ2   RHS2   . . .  .   .  (3.2.5)  . . .. .  .  =  .   . . . .  .   .  aN1 aN2 ... aNN ΓN RHSN which can be solved by usual linear algebra methods.

y y

G G G

8 v v8 2t 2t t v

u

g(s)ds G

l

x x G

Figure 3.2.3: Infinite row of vortexes (cascade) (reprinted from Miclea-Bleiziffer et al. [59], with per- mission from Elsevier)

The complex potential of an infinite vortex array (figure 3.2.3) reads [51] in the transformed Z-plane:

iΓ iΓ F ln (Z 1) ln (Z ) (3.2.6) = 2π − − 4π which leads to following velocity potential in the z-plane:

iΓ ³ z ´ u i v cosh (3.2.7) − = 2t 2 this is comfortably written in normalized coordinates:

56 3.2 Development of a propeller design and analysis code

 ³ 2πy ´ sinh¡ 2πx ¢ i sin iΓ t − t u i v  ³ ´  (3.2.8) − = 2t cosh¡ 2πx ¢ cos 2πy t − t where the velocity vector can be split into its components:

¡ 2πx ¢ Γ sinh t u ³ ´ = 2t cosh¡ 2πx ¢ cos 2πy t − t

¡ 2πx ¢ Γ sin t v ³ ´ (3.2.9) = −2t cosh¡ 2πx ¢ cos 2πy t − t If this is now applied to a infinite vortex array formed by thin airfoils, each of them as shown in figure 3.2.2a, it is possible to find the unknown vortex strength in equation (3.2.9). Basically equa- tion (3.2.9) is solved by applying the Vortex Lumped Method in which the velocities are separated from vortices into the influence coefficients, which are then solved for each panel on the airfoil:

 2π ¡ ¢  sin t y y0 1 l − ¡ ¢ axc  cos βi λ = 2t cosh 2π (x x ) cos 2π ¡y y ¢ + t − 0 − t − 0

à 2π ! 1 sinh t (x x0) ¡ ¢ ayc − sin βi λ (3.2.10) = 2t cosh 2π (x x ) cos 2π ¡y y ¢ + t − 0 − t − 0 Because the airfoils are placed at an angle λ relative to the ordinate axis (figure 3.2.3), this has to be added to both tangential and normal velocity components at the panel (to bring it to the original ordinate). Therfore λ is added to the influence coefficients (as shown in equation (3.2.10)) and also to the RHS:

¡ ¡ ¢ ¡ ¢ ¢ RHS V cos βi λ sin(α λ) sin βi λ cos(α λ) (3.2.11) = − ∞ + + + + +

The final influence coefficient is summed to give: ac axc ayc and its RHS is similar to equa- = + tion (3.2.11). The resulting equation system is solved similarly to that in equation (3.2.5).

The numerical method will be named Cascade Vortex Lattice, or simply CVL and is programmed in the MATLAB©environment.

The assessment of pressure at the airfoil surface indicates the airfoil’s performance. The analysis is performed by analyzing the pressure coefficient,Cp which represents the local pressure coefficient defined at the surface of the airfoil by:

µ v ¶2 Cp 1 (3.2.12) = − V ∞

57 Chapter 3 Presentation and discussion of the results

3.2.2 Sensitivity check for the CVL computational model

Before using the CVL code one has to proof its stability and limits as well as the convergence to analytical solutions which will define its accuracy in this case. The validation is accomplished by comparing results of the numerical method to results of analytical solutions or results available in the literature.

Because of the method’s specific, following studies have been carried out:

1. Optimal distance to read out the velocities needed for computing the local CP . The velocities cannot be read at the panel’s surface since they would be exactly in the singularity (vortex ori- gin). The result would have no physical meaning, so an optimal distance has to be found.

2. Spatial discretization. The solution (and thus both CP distribution and cl ) depends on the num- ber of panels in which the airfoil is divided. An optimum number has to be found since too many panels have an negative influence on the speed of the computation.

3. Maximum camber. The present theory is based on the small angles assumptions. This means that for a very high camber no reasonable comparison is possible between Joukowski analytical solution and the numerical framework. Here is searched a maximum camber for which results do not diverge more than 5% from the analytical solution.

For the validation will be used the cl and CP distribution solved by using the Joukowski transforma- tion. To simulate an isolated airfoil the pitch-to-chord-ratio (t/l) is set at a value of t 100 in the = CVL code. The validation of the cascade flow is achieved by solving the cascade flow of flat plates for different stagger angles and comparing it to the ”Weinig diagram” Weinig [116] .

Optimal distance for CP evaluation

As mentioned previously the numerical method implies a distribution of singularities over the airfoil surface. The evaluation of the local velocity vector needed for the CP computation (equation (3.2.12)) is not possible at the vortex origin1. The distance from the singularity (vortex origin) where this can be evaluated is searched here. A flat plate plate is simulated at an angle of attack (AOA) of 5°. Compu- tations of CP are performed at three distances as summarized in table 3.2.1.

Table 3.2.1: Summary of the investigated flat plate solution

Distance [-] Solution Lift coefficient cl Figure 0.1 c numerical 0.54757 3.2.4 · 0.01 c numerical 0.54757 3.2.5 · 0.001 c numerical 0.54757 3.2.6 · - analytical 0.54762 all

All simulation were performed by using 250 discretization panels over the chord. Diagrams of CP dis- tribution will be presented in the aerodynamics formalism by setting the suction side results (upper side of the airfoil) with negative numbers on the ”positive” side of the graphs.

Figure 3.2.4a shows the comparison between the CP distribution obtained by evaluating the velocities far away from the vortex (figure 3.2.4b). Although the distribution above 0.25 c is quantitatively very · close to the one of the analytical solution, at the leading edge as well as in the first quarter-chord this differs completely. The fluid acceleration present on the suction side is not visible, both sides having an almost equal CP . Approaching the LE shows huge differences which reach over 400. This is caused by the very high distance from the vortex to the evaluation points and so the effect of high vortex strength at LE is wasted. This is not acceptable for a simulation and cannot be used.

1the evaluation will show here a zero velocity which has no physical meaning

58 3.2 Development of a propeller design and analysis code

- 2 . 0 0.5 C J o u k o w s k i P - 1 . 5 C C V L P 0.25 - 1 . 0

- 0 . 5

p 0 C 0 . 0 −0.25 0 . 5 Y−Coordinate [−]

1 . 0 −0.5 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 0 0.25 0.5 0.75 1 X−Coordinate [−] x [ - ] (b) Flat plate with figured evaluation points (a) CP distribution

Figure 3.2.4: Evaluation of the thin flat plate at AOA 5° with points placed at 0.1 c = ·

The next step is according to table 3.2.1 a distance 10 times lower than the previous one, respectively 0.01 c. Results of the simulation are shown in figure 3.2.5a. Differences between analytical and nu- · merical solution are small and the curves have the same shape. Nearby the stagnation point there is a region where a relative difference between 5% and20% (at peak) is still present (first 5% of chord). It can be concluded that the 0.01 c position for the evaluation points is relatively precise. ·

- 2 . 0 0.5 C J o u k o w s k i P C C V L - 1 . 5 P 0.25 - 1 . 0

- 0 . 5

p 0 C 0 . 0 −0.25 0 . 5 Y−Coordinate [−]

1 . 0 −0.5 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 0 0.25 0.5 0.75 1 X−Coordinate [−] x [ - ] (b) Flat plate with figured evaluation points (a) CP distribution

Figure 3.2.5: Evaluation of the thin flat plate at AOA 5° with points placed at 0.01 c = ·

In the next step, the distance is once again decreased 10 times to the value of 0.001 c (depicted in · figure 3.2.6b). The resulting CP distribution is shown in figure 3.2.6a. Again is the region of interest around the stagnation point. Results show a very good agreement between numeric and analytic methods. Unfortunately the distribution downstream the stagnation point is up to 10% below of the one of the analytical solution. The differences appear at chord ratios between 10% and 70 which makes the evaluation in this region useless.

The first obvious conclusion is that the evaluation at 0.1 c from the camber-line delivers the most · unrealistic results not catching the physics of the problem at all (e.q. stagnation point). The closer the points the better seems to be the description of the physics around the stagnation point. On the other side the middle distance of 0.01 c agrees perfectly elsewhere than at and around the stagnation ·

59 Chapter 3 Presentation and discussion of the results

- 2 . 0 0.5 C J o u k o w s k i P - 1 . 5 C C V L P 0.25 - 1 . 0

- 0 . 5

p 0 C 0 . 0 −0.25 0 . 5 Y−Coordinate [−]

1 . 0 −0.5 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 0 0.25 0.5 0.75 1 X−Coordinate [−] x [ - ] (b) Flat plate with figured evaluation points (a) CP distribution

Figure 3.2.6: Evaluation of the thin flat plate at AOA 5° with points placed at 0.001 c = · point with the analytic result. A compromise solution is chosen to satisfy all the regions of the airfoil: the analysis before the stagnation point up to the same distance downstream from it is analyzed at a small distance (0.001 c) while from here downstream it is analyzed at 0.01 c. Results showing this · · excellent agreement are shown in figure 3.2.7.

- 2 . 0

- 1 . 5 C J o u k o w s k i p - 1 . 0 C C V L p - 0 . 5 P C 0 . 0

0 . 5

1 . 0 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 x [ - ]

Figure 3.2.7: CP distribution around a flat plate computed at variable distance from camber

Sensitivity to the spatial discretization

The accuracy of the solution depends on the discretization. The number of singularities distributed over the chord influences, among others, the accuracy of the solution. In the previous study 250 panels were considered distributed on the camber. The present investigation studies how the CP distribution behaves by using more ore less panels. However, it is shown in table 3.2.2 that for all con- sidered panel distributionscl remains constant. The same airfoil used in the previous investigation is used: flat plate airfoil at an AOA of 5°.

A more detailed analysis of the code limitations is shown in figure 3.2.8. The problem depicted is more complex: the variation of both camber and panels is investigated. Figure 3.2.8 shows clearly that by increasing the number of panels the difference between analytically computed and numerically computed CL decreases. However, the error rises while increasing the camber. An acceptable error

60 3.2 Development of a propeller design and analysis code

Table 3.2.2: Discretization results over a flat plate airfoil at an AOA of 5°

Panels[-] method Lift coefficient cl [-] C p distribution 50 numeric 0.54757 figure 3.2.8 100 numeric 0.54757 - 250 numeric 0.54757 figure 3.2.7 500 numeric 0.54757 figure 3.2.9 1000 numeric 0.54757 - - analytic 0.54762 - of 5% is fulfilled for a range of foils with cambers up to 10%. For cambers up to 15% the error rises at 10%. Figure 3.2.8 depicts in the graphical way the results presented in table 3.2.2 which express briefly that the solution is more accurate for a low camber airfoil than for a high camber. In fact for the flat plate the number of panels doesn’t influence at all the cl . Instead, for airfoils cambered until 10% the solution remains in an acceptable error of 4-5%. Although the number is increased drastic (from 50 to 250) the improvement of the solution is rather small. Circular arc airfoils cambered at 10% of the chord length have already at an AOA of 5° a c 1.23. This value is already high enough for all l = usual propeller and turbomachinery applications. The computation for a foil with 250 panels takes about 1.25 sec on an usual PC.

3

c J o u k o w s k i l c C V L 2 l 2 5 0 p a n e l s c C V L l 5 0 p a n e l s ] - [

l c 1

0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 C a m b e r [ % o f c h o r d ]

Figure 3.2.8: cl results obtained analytically (red) and by the CVL method (green and blue) for circular arcs with different cambers (reprinted from Miclea-Bleiziffer et al. [59], with permission from Elsevier)

Going back to the C p sensitivity investigation, figure 3.2.9 depicts the result of a simulation performed with 50 panels. The C p distribution follows the same path as the one computed by the Joukowski method. The poor discretization around the stagnation point has as result the poor agreement in this area between the numerical and analytical computation. Compared with the results computed by using 250 panels depicted in figure 3.2.7 the quality can be described as poor.

The result of increasing the number of panels to 500 is depicted in figure 3.2.10. In this case the agree- ment to the analytical solution very good with no quantifiable differences. However, there is no sig- nificant difference between the 250 and 500 panels solutions. The solution with 50 panels shows 20% difference in Cp to the analytical solution (figure 3.2.9) and is, as a result, not so accurate. Though, 50 panels are not recommended for such a simulation.

The conclusion of this investigation is that for using the CVL method properly one should always use in computations at least 250 panels. On the other hand this is only needed if the C p distribution is

61 Chapter 3 Presentation and discussion of the results

- 2 . 0

- 1 . 5 C J o u k o w s k i p C C V L - 1 . 0 p 5 0 - P a n e l s

- 0 . 5 p C 0 . 0

0 . 5

1 . 0 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 x [ - ]

Figure 3.2.9: Solution of the flat plate @ AOA 5° with 50 panels =

- 2 . 0

- 1 . 5 C J o u k o w s k i P C C V L P 5 0 0 P a n e l s - 1 . 0

- 0 . 5 p C 0 . 0

0 . 5

1 . 0 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 x [ - ]

Figure 3.2.10: Solution of the flat plate @ AOA 5° with 500 panels = important for the results. If only the cl is important the simulation can be performed also by using less than 250 panels.

Maximum camber of an airfoil which can be evaluated by the CVL method

Last analysis has shown that while computing the flat plate by different panel distributions the lift coefficient cl has not changed (table 3.2.2). However, while analyzing figure 3.2.8 has been clear that if the camber of an airfoil is raised the difference between the analytic and numerical solution raises. This can be a problem since the camber is very important for the zero lift angle 2 and also a problem because most of the airfoils are cambered. In order to find out what is the maximum camber at which the CVL method still has an acceptable error a series of simulations was performed for airfoils having different cambers. Details of the investigated foils as well as their cl results for both analytical and numerical simulation is shown in table 3.2.3.

An acceptable error below 5% is found generally for airfoils having a camber smaller than10% of the chord length as depicted in the last column of table 3.2.3. In fact the error for an airfoil having a camber below 5% is negligible. However, similar to the last study the C p distribution given by the

2its importance is exposed later in section §3.3

62 3.2 Development of a propeller design and analysis code

Table 3.2.3: Evaluation error of the cl by using the CVL method for airfoil with different cambers

airfoil camber [%] cl analytic cl CVL (250 panels) Error [%] C p distribution 0 0.54762 0.54757 0.0091 figure 3.2.7 5 1.1815 1.1679 1.15 figure 3.2.11a 10 1.8524 1.7713 4.4 - 15 2.5932 2.3467 10.5 figure 3.2.12a 20 3.4400 2.8898 16 - numerical method will be compared to the one given by Joukowski. Since in this study differences are visible already for the integral quantities like cl , major differences in C p distribution are expected.

The investigation of the flat plate does not need to be presented anymore since this has been investi- gated in the previous subsections, the C p distribution is depicted in figure 3.2.7.

- 2 . 0 0.5

- 1 . 5 C 5 % a r c c a m b e r e d J o u k o w s k i f o i l P C 5 % a r c c a m b e r e d J o u k o w s k i f o i l - C V L 0.25 - 1 . 0 P ]

- - 0 . 5 [ 0

p C 0 . 0 −0.25 0 . 5 Y−Coordinate [−]

1 . 0 −0.5 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 0 0.25 0.5 0.75 1 X - C o o d i n a t e [ - ] X−Coordinate [−] (b) 5% thin cambered foil with figured eval- (a) C distribution for a 5% thin cambered foil P uation points

Figure 3.2.11: Evaluation of a thin 5% circular arc cambered foil at AOA 5° =

Figure 3.2.11a depicts the result of the numerical and analytical solution of a flow past a 5% cambered arc airfoil at an AOA 5◦. Deviations of the numerical method are again found around the stagnation = point but the error is around 20 at Cp peak. As discussed previously a better refinement of the panels in this area could improve the results.

By further raising the camber to 15% the agreement between the C p distribution given by the analyti- cal Joukowski solution and the numerical solution becomes poor. This is shown in figure 3.2.12a. One can not longer speak about zonal disagreements because the differences pass 50 which is definitely due to inappropriate modeling. It is just a confirmation of the results computed for the cl which were presented earlier in table 3.2.3.

The comparison between numerics and analytics of both C p distribution and cl leads to the conclu- sion that the present method can be efficiently and with a sufficient small error used for thin cam- bered airfoils having cambers up to 10% of the chord length.

Validation of the cascade solution

The Joukowski transformation can not be applied for cascades of airfoils since it was developed to transform a circle into a singular airfoil. However, there have been developed analytical methods ca- pable of transforming a circle to an airfoil cascade (or a cascade of infinitely thin airfoils). The first

63 Chapter 3 Presentation and discussion of the results

- 3 . 0 0.5 C 1 5 % a r c c a m b e r e d J o u k o w s k i f o i l P - 2 . 5 C 1 5 % a r c c a m b e r e d J o u k o w s k i f o i l - C V L P - 2 . 0 0.25 - 1 . 5 ]

- - 1 . 0 [ 0

p

C - 0 . 5 0 . 0 −0.25 Y−Coordinate [−] 0 . 5 1 . 0 −0.5 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 0 0.25 0.5 0.75 1 X - C o o d i n a t e [ - ] X−Coordinate [−] (b) 15% thin cambered foil with figured (a) C distribution for a 15% thin cambered foil P evaluation points

Figure 3.2.12: Evaluation of a thin 15% circular arc cambered foil at AOA 5° = method mentioned in the literature was published by Numachi [65] where the influence of the stagger angle (λ,0 90°) and pitch-to-chord ratio (t/l) upon the lift ratio k of flat plate cascades was investi- − gated. The Weinig-diagram published a few years later by Weinig [116] depicts the same comparison and has remained in the turbomachinery literature as the reference for analytical cascade flows. To validate the code for cascade flows, data of the Weinig-diagram in the version published in [86] are ex- tracted and plotted against the values computed using the CVL. The version in [86] shows the values of k plotted against the cascade solidity σ which is the inverse of the pitch-to-chord ratio. The results of the validation are presented in figure 3.2.13. Basically, k tends to 1 as the solidity is decreased, which means the aerodynamic effect of the cascade is decreased as the cascade pitch is increased.

l=0° l=20° l=30° 3.0 l=40° l=50° l=60° l=70° 2.5 l=80° l=88° l=0° -CVL l=20° -CVL l=30° -CVL t 2.0 l=40° -CVL l=50° -CVL l=60° -CVL l=70° -CVL

l k 1.5 l=80° -CVL l=88° -CVL

1.0 l a 0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s

Figure 3.2.13: Lift ratio k as a function of the solidity σ and stagger angle λ (reprinted from Miclea- Bleiziffer et al. [59], with permission from Elsevier)

In the figure above one can see that for stagger angles (λ) up to 70◦ the curves obtained by the nu- merical simulation using the CVL method are identical to the ones read from the diagram computed analytically. The 80◦ curve shows differences for a σ 2. The computation for an angle λ 90◦ is not > = possible because of the singularity in equation (3.2.10). Instead, a computation with a value of 88◦

64 3.2 Development of a propeller design and analysis code was performed. The results do not match the analythics in this case. Thus, a cascade of flat plates having λ 90◦ is an extreme case not used in practice. =

For all the investigated solidities and stagger angles up to 80◦ the results computed by CVL show excellent agreement with the analytical ones (from [86]). This agreement to analytical data presented in the literature validates the presented CVL method for cascade flows and opens the doors to its use for both inverse design, optimization and computation of axial turbomachinery blades.

Figure 3.2.14: Figure showing computed streamlines around flat plate cascade with a t/l 1, at = AOA 5° and λ 30°,50°,70° = =

To show graphically what the effect of cascade upon an incoming flow is, the streamlines of cascade flow have been computed for different stagger angles. The AOA has been set at a value of 5° in order to enhance the view of the cascade effect while the pitch-to-chord ratio (t/l) has been set at a value where the effects are good visible (t/l 1 ; σ 1). The results of these simulations are presented in = = figure 3.2.14. It is visible from the geometry of the streamlines that the higher the stagger is, the more deflection is added to the flow. Or, it means the lift force is also higher.

The new CVL method for computing the flow of thin airfoil cascades has been validated in this sub- section. The modeling of various validation cases with extreme values has shown also the limitations of the code. However, all cases presented in this thesis are found in-between these limits. Moreover, almost all of the aerodynamic flows in the axial turbomachinery are covered by the working field of the present code. Computation of cascades with staggers above 70° are not needed as will be shown in the design cases presented later in section §3.3.

After the mean-line is generated a first analysis using the CVL framework can be performed in order to verify the C p and lift distribution. Proceeding in this way one can check very fast if the desired cl is reached by the mean-line (this is shown for a few cases depicted in table 3.2.4). Typically the mean-line is computed by equation (2.3.3) to which a thickness distribution is added.

It was shown in section §2.3 that NACA airfoils can be generated by using the inverse method by setting a desired cl . To see if the prescribed cl is realized by the mean-line 4 airfoil cases were tested by using the CVL framework. The test matrix is presented in table 3.2.4.

Two cases from table 3.2.4 have been chosen for a more detailed analysis. The mean-lines with a 0.8 = have been chosen since, as it will been shown in subsection 3.2.4, these will be the mean-lines used for propeller design. More interesting for the present thesis are the values of the maximum camber which lay between 5 and 7.5%. This confirms the statement made in the previous section that the usual

65 Chapter 3 Presentation and discussion of the results

Table 3.2.4: Comparison of desired and realized cl for different mean-lines

a desired cl computed cl difference [%] max camber [%] Figures 0.6 0.75 0.74940 0.08 5.4 - 0.6 1 0.99862 0.138 7.4 - 0.8 0.75 0.74813 0.2493 5.1 figure 3.2.15afigure 3.2.15b 0.8 1 0.99509 0.491 6.8 figure 3.2.16afigure 3.2.16b range of maximum camber is 10%. It was shown previously that the CVL framework can compute such mean-lines with good accuracy.

- 2 . 0 0.4 C N A C A 6 m e a n - l i n e c = 0 . 7 5 P l - 1 . 5 0.3

0.2 - 1 . 0 0.1 ]

- - 0 . 5 [

0 p C 0 . 0 −0.1

0 . 5 −0.2 −0.3 1 . 0 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 X - C o o d i n a t e [ - ] (b) NACA mean-line for cl 0.75 (a) C distribution of NACA mean-line for c 0.75 = P l = Figure 3.2.15: Evaluation of a NACA mean-line (a 0.8 ,c 0.75) at ideal AOA = l =

In figure 3.2.15a is depicted the C p distribution over a NACA mean-line designed for c 0.75. The l = shape of the distribution is exactly as predicted by the theory: from 0 up to a it is constant and after it decays linearly to 0. In table 3.2.4 is shown that the difference between the computed cl and the desired one is very small (below 1%).

- 2 . 0 0.4 C N A C A 6 m e a n - l i n e c = 1 P l - 1 . 5 0.3

0.2 - 1 . 0 0.1 ]

- - 0 . 5 [

0 p C 0 . 0 −0.1

0 . 5 −0.2

−0.3 1 . 0 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 X - C o o d i n a t e [ - ] (b) NACA mean-line for cl 1 (a) C distribution of NACA mean-line for c 1 = P l = Figure 3.2.16: Evaluation of a NACA mean-line (a 0.8,c 1) at ideal AOA = l =

For the mean-line with desired c 1 results show the same pattern of agreement as in the previous l = case. The higher load is observed in the higher C p distribution which is depicted in figure 3.2.16a.

66 3.2 Development of a propeller design and analysis code

The difference between the desired in computed cl is in this case slightly higher but still below 1%.

These results show how powerful the inverse airfoil design method is an it can be concluded that this is a good way to start the design the sections of a propeller. For the sake of brevity it has to be noted that these methods exist for over 60 years, but they still are the state of the art in starting the design of propellers as many present thesis and publication show: [28, 27, 6, 45].

It is now legitimate to ask how the C p distribution looks and what happens with the cl if the foils are in a cascade. To see how they change the investigated mean-lines are computed in a cascade having t/l 1/σ 1 and a stagger angle λ 50°. The main differences are shown in table bellow. = = =

Table 3.2.5: Comparison of desired and realized cl for mean-lines stand-alone and in cascades

a desired cl computed cl diff. [%] k camber [%] t/l λ[°] Figures 0.8 0.75 0.74813 0.2493 1 5.1 - figure 3.2.15a ∞ 0.8 0.75 0.65919 12.1 0.88 5.1 1 50 figure 3.2.17a 0.8 1 0.99509 0.491 1 6.8 - figure 3.2.16a ∞ 0.8 1 0.85311 14.7 0.85 6.8 1 50 figure 3.2.18a

Looking at the differences in the fourth column of the table the data found in the Weinig diagram comes to mind (3.2.13). It is impressive how close the results computed for NACA mean-lines are to the ones for flat plates (at λ=50° stagger). However, if for the same stagger and solidity the camber increases for a NACA mean-line, the cascade influence factor k sinks (column 5 in table 3.2.5). This means that the more lift is prescribed for an airfoil the less will be realized if the airfoil is in cascade.

CVL solution for lambda:50 [°] - 2 . 0 1 C NACA 6 mean-line c =0.75, l =50°, t/l=1 P l 0.8 - 1 . 5 0.6 - 1 . 0 0.4 0.2 ]

- - 0 . 5

[ 0

p

C −0.2 0 . 0 −0.4

0 . 5 −0.6

−0.8 1 . 0 −1 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 −0.5 0 0.5 1 1.5 X - C o o d i n a t e [ - ] (b) Streamlines plot around the cascade at αi (a) CP distribution of NACA mean-line cascade

Figure 3.2.17: Evaluation of a NACA mean-line cascade (a 0.8,c 0.75), λ 50°and t/l 1 = l = = =

In 3.2.17a is depicted the shape of the C p distribution over the mean-line designed for c 75. The l = discordance is obvious when compared with the case of an isolated airfoil (3.2.15a). The pressure side has a radical change; instead of a constant distribution the C p slips beyond the leading edge to negative values sinking in this way the lift. The load slides to the rear part of the foil heading towards the trailing edge. The streamlines in 3.2.17b show perfectly, how the flow is deflected starting with the leading edge of the foil. The cascade computation results of the foil designed for for a c 1 l = show even better the change of the C p shape (3.2.18a) when compared with the same isolated profile (figure 3.2.16a) . In this case is also visible how the suction side also changes its shape. In the front part the values are decreasing and in the rear part they are decreasing. The loading increases toward the trailing edge.

67 Chapter 3 Presentation and discussion of the results

CVL solution for lambda:50 [°] - 2 . 0 1 C NACA 6 mean-line c =1, l =50°, t/l=1 P l 0.8 - 1 . 5 0.6 - 1 . 0 0.4 0.2 ]

- - 0 . 5

[ 0

p

C −0.2 0 . 0 −0.4

0 . 5 −0.6

−0.8

1 . 0 −1 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 −0.5 0 0.5 1 1.5 X - C o o d i n a t e [ - ] (b) Streamlines plot around the cascade at αi (a) CP distribution of NACA mean-line cascade

Figure 3.2.18: Evaluation of a NACA mean-line cascade (a 0.8,c 1), λ 50°and t/l 1 = l = = =

3.2.3 Propeller design framework

The aim of this work is to develop improved propellers of different sizes and for different purposes. Because the available codes do not have the flexibility to design both large and small propellers a new code was developed. The purpose of the new code was to allow quick parametric studies as well as to have different export possibilities for example to the Ansys Turbogrid©and Creo©3 softwares. Tur- bogrid©is part of the Ansys©CFD suite and builds blade channels of turbomachines from radial dis- tributed airfoils bounded by hub and shroud curves with the purpose of creating a three-dimensional hexahedral grid used in CFD simulations. Once a turbomachine geometry can be exported to Turbo- grid©parameter studies are very easy to run since the mesh setup boundary conditions are always the same. The propeller design program is called ADAP which stands for Advanced Design for Ax- ial Propellers and pumps. In the first part of this subsection is described the implementation fol- lowing the method shown in section 2.4.2 and 2.4.3, last with the radial loss function of Goldstein (equation (2.4.54) and equation (2.4.55)). The Goldstein function using the empirical formulation presented in subsection 2.5.4 is first proved against the analytical presented in Wald [114].

1 . 0 0 . 5 4 B G o l d s t e i n 4 B c o m p u t e d 0 . 8 0 . 4 6 B G o l d s t e i n 6 B c o m p u t e d 0 . 6 0 . 3 ] ] - - [ [

G G 0 . 4 4 B G o l d s t e i n 0 . 2 4 B c o m p u t e d 0 . 2 6 B G o l d s t e i n 0 . 1 6 B c o m p u t e d 0 . 0 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 ξ [ - ] ξ [ - ] (a) Propellers designed for J 0.63 (b) Propellers designed for J 2.1 = = Figure 3.2.19: Goldstein factor (G) distribution for propellers with 4 and 6 blades

3formely known as Wildfire ProE

68 3.2 Development of a propeller design and analysis code

The steps in subsection 2.4.2 following equation (2.4.41) can be used for completing the Goldstein theory without using any other modification. In order to prove the validity of the empirical method presented in [8] and shown in equation (2.4.54) and equation (2.4.55) G is computed for several pro- pellers having a different number of blades and advance ratios and plotted against the analytical val- ues presented in [114]. The results for two propellers with 4 and 6 blades computed for two different advance ratios are depicted bellow. The main dimensions used for this computations are the one of the propellers investigated in section §3.4 (table 3.3.1). The higher advance ratio (J 2.1) was chosen = to match the design advance ratio of the propellers in section §3.4 while the small one was chosen to show the performance at low advance ratios (J 0.63). = The results for propellers designed at an advance ratio J 0.63, depicted in figure 3.2.19a, show very = good agreement between the fitted Goldstein function and the results tabulated in [114]. No differ- ence can be observed between the fitted function and the theoretical values. For J 2.1 shown in = figure 3.2.19b small differences arise between the fitted Goldstein function and the analytical results. The differences appear only at radii between 20% and 70%. In the case of 4 blades the differences are higher and reach a maximum of 30% at 20% radius and then decrease to 0 at 70% radius. 6 blades de- signs show a smaller difference of 20% at 20% radius and then dropping to below 10% between 40% and 80%. This method is very comfortable because the values of the Goldstein factor are instantly computed. If a high resolution is desired, the best way is to use the method of Okulov and Sørensen [69] which agrees very well with the results of Goldstein.

Both Prandtl/Betz and Goldstein loss functions are implemented in the propeller design and analysis code. Brief description of the main scripts, functions and variables is presented in these sections. This should help the reader to understand the data-flow within the framework and how this can be changed, modified or improved if this is desired. A propeller design framework was developed based on the blade element momentum method described in subsection 2.5.3. For computing the correct wake alignment and the evaluation of the displacement velocity (v0) an iterative design approach is employed in the design framework as suggested in [3]. The structure of the program and the data flow are presented in 3.2.20.

The input is followed by a user specification of the wished propeller geometry: radial chord distribu- tion (non-dimensional CoD), air or hydrofoil class or type and the point in percent of chord where the airfoils are stacked. Also in the input section is specified the ideal AOA for the used airfoils. First computations with this code have used the NACA formulation in section §2.3 but later this has been changed to a constant value of 1.54° which is found also in OpenProp[26]. The input can be changed by editing the variables, while the options can be chosen by setting a variable equal to one or zero (one means the option shown is valid, zero means the default is chosen).

If no chord distribution is specified an optimum chord distribution according to the Prandtl-Betz framework is computed. The ideal circulation is found by calling the circulation function (Adkins_BP figure 3.2.22) which delivers the main program an optimum circulation and, hence the lift distribu- tion. The variables needed and the ones resulted by running the function are transferred from the main code implicitly by the function. The iterative scheme used in the computation of the correct wake alignment is straightforward and uses for each new iteration the value obtained in the last iter- ation. This needs usually only a few iterations for convergence.

To fulfill the desired lift distribution a NACA 6 mean-line is computed for each radial section accord- ing to equation (2.3.3). A thickness distribution is then added to the mean-line. In the current version of the program (V 0.973 - Oct 2016) following thickness distribution are available: NACA 65A010 or a NACA 4 thickness distribution. In the final step the geometry is scaled and the sections are exported in a text file as ASCII format. The user can choose to export the data to the CAD program (PTC Creo©) and/or to the CFD mesh generator ANSYS TurboGrid©. For Creo©files are exported in .ibl format while the TurboGrid©files are exported in .curve files.

69 Chapter 3 Presentation and discussion of the results

Propeller Input Data turning speed N, diameter D, number of blades B, thrust T, ship speed V

Propeller geometry chord to diameter ratio (CoD), stack rotation point (CR) hydrofoil class

Startz=0 ( v/V8 )

f l z f Compute tan(t )= (1+ /2) and then (r)

b a naca6.m Compute f, F, G, a, a’, , Cl, ideal

Adkins_BP.m Compute I1’, I2’, J1’, J2’, Integrate to I1, I2, J1, J2

z = ComputeNEW I1/(2 I2)-sqrt((I1/(2 I2))^2-CT /I2)

NO z z ? NEW=

YES

G(r)

Generation of NACA 6 Meanline (Cl) naca6.m

Generation of thickness distribution (NACA 4) naca4.m

Compute 3D Coordinates of radial sections blade_coordinate.m

Export data (ASCII) for ProEngineer Export data (ASCII) for TurboGrid (Rapid Prototyping) (automatic grid generation)

Figure 3.2.20: Overview of the design program ADAP (V 0.973)

Figure 3.2.21: ADAP logo, version and copyright agreements written on the top of every file

70 3.2 Development of a propeller design and analysis code

3.2.4 An iterative method for correcting the lift distribution for propellers with small pitch-to-chord ratios

Previously was mentioned that blade interferences arise in all rotating machinery when the value of ¡ t ¢ cascade pitch-to-chord ratio l is less then 3. Lifting line theory and the vortex methods presented in section §2.4 essentially do not take account for this interferences. A good example of this hypothesis, showing two propellers having the same lifting-line loading but different profile lengths is presented in [44]. Carlton [19] refers to the blade interference problem by comparing the effect of the cascade for airfoils and advices the reader about the problems which can arise in such configurations.

However, up to now there has not been presented any method for the propeller design which over- comes the aerodynamic deficiencies produced by the cascade. In order to consider the interference between blades, especially at the design stage, an extension of the design framework presented in the previous subsection is proposed here, and it was first published in [59]. The new propeller de- sign framework is presented in figure 3.2.22. The red square marks the actual blade element design method like the one in figure 3.2.20. Alternatively, for validation purposes only, an interface was writ- ten for using the vortex-lattice propeller design method presented in subsection 2.4.6 implemented in the MIT mpvl code [20] (subsection 2.4.6 ).

The circulation function (red square in Figure 3.2.22) delivers the main program an optimum circula- tion and hence lift distribution. In the classic approach (figure 3.2.20) this lift distribution is directly used to the generation of mean lines. At this point an iterative procedure involving the CVL method (presented in subsection 3.2.1) was implemented in a function (light blue square). The code is basi- cally the same as the one used for designing an usual propeller but if the option ”correction” is spec- ified at the beginning of the main code the optimal circulation data is delivered to a function called ”vector_sub”. This data contain the desired cl , the velocity, the tip radius (rt ) and the pitch angle. In the code hierarchy below the vector_sub function, which is just an intermediate function, there are two other functions for the iterative computation: an airfoil generation function with the cl correction procedure and below this one the CVL code which computes the cl of the cascade. This code is based on the theory presented in section §3.3 and it was validated for the Quasi-3D computation of axial fans in [60]. For the desired cl at every section, a NACA mean line is generated (orange square inside) and with the stagger angle, solidity values and angle of attack an analysis of the section is performed. The cascade’s lift coefficientcl NEW is compared to the desired cl . For high pitch-to-chord ratios the − values are exactly same, while for σ 1 the values differ. In this case, for correction, a small value δc ≥ l is added to the original cl and with this one a new analysis is started. The loop runs until cl NEW cl . − = The corrected value cl NEW is then used for designing the section in the next step. Finally, a thickness − distribution is added to the mean-line. The current implemented thickness distribution investigated are: NACA 6 (NACA 65A010 as in [44]) or a NACA 4 thickness distribution. The final step has the same functions as in the standard framework to write, scale and export the coordinates. However, the user of the propeller design code (aero-designer) is interested on how the parameters influence the final design result. In order to visualize this, plots of all design relevant quantities within the code were programed. The plots are grouped in the current code version in five windows as depicted in fig- ure 3.2.23, figure 3.2.24 and figure 3.2.25. The convergence of the wake alignment method is shown in figure 3.2.23. Error is computed based on the difference between the ζ - values of the current and the previous iteration and the convergence is achieved when the error is lower than 0.0001%. As depicted in figure 3.2.23 the wake alignment method needs only 6 iterations for a design. Typical designs which are shown in this thesis needed maximum 20 iterations. One of the factors increasing he number of the iterations is the reduced number of blades which increases the blade load. A too higher load leads to no convergence and thus to no design.

In figure 3.2.24 are shown the plots of the first two figures in ADAP: figure 1 shows five stacked airfoils of the propeller (every second from the exported is shown) and figure 2 depicts the pitch-to-chord ratio distribution, the undisturbed flow angle (φ ), the flow angle (φ) (see figure 2.4.6) and their ratio. ∗ In this way one can figure out how the airfoils look like, if the camber is not too high or if the chord is

71 Chapter 3 Presentation and discussion of the results CVL NEW L C L C NEW L C Export data (ASCII) for TurboGrid (ASCII) for Export data grid generation) (automatic ? YES NEW l s a LL ,,, C =C L L C Compute CCompute cascade( for ) Generation of NACA Meanline ( of NACA Generation ) Generation of thickness distribution Generation d NEW Compute 3D Coordinates of radial sections of radial 3D Coordinates Compute Generation of NACA Meanline ( of NACA Generation ) LL C + C NO L C = Export data (ASCII) for ProEngineer (ASCII) for Export data (Rapid Prototyping) NO T L ideal b a 8 YES (r) = L C NEW t z z ? f z f z=0 ( Start v/V ) Propeller Input Data Propeller Propeller geometry Propeller NEW z = Compute f, F, G, a, a’, F, f, Compute , C , Compute tan(Compute )=SR(1+ /2) and then (r) Compute I1’, I2’, J1’, J2’, Integrate to I1, I2, J1, J2 to Integrate J2’, J1’, I2’, I1’, Compute Compute I1/(2 I2)-sqrt((I1/(2 I2))^2-C /I2) chord distribution , stack rotation point (CR) point rotation , stack distribution chord class, hydrofoil rpm, diameter D, number of blades B, desired thrust T, thrust desired number of blades B, D, rpm, diameter ship speed V

Figure 3.2.22: Design framework for propellers with high pitch-to-chord ratio as implemented in ADAP (V 0.973) (reprinted from Miclea-Bleiziffer et al. [59], with permission from Else- vier)

72 3.2 Development of a propeller design and analysis code Error

Figure 3.2.23: Convergence of the ADAP propeller design code correct. Furthermore with the help of the pitch-to-chord ratio one is able to determine if the design needs to be updated with the help of the CVL method. The angle ratio is constant, as is the efficiency distribution which has been shown previously in subsection 2.4.6: 2.4.59 and 2.4.60.

Figure 3.2.24: Figure 1 and Figure 2 from ADAP showing the stacked airfoils and relevant design data

In figure 3.2.25 are shown the next two plotted figures. Here are plotted Re-number, Glauert’s factor, radial loss function or the ideal AOA (which is not constant when computed for the NACA 6 mean- lines with equation (2.3.6)). On the other figure panel are plotted the non-dimensional circulation as well as the lift and thrust distribution and the lift coefficient distribution. Latter is an indication for the sections airfoil camber and is important since a too high camber leads to flow separation.

3.2.5 BEM propeller analysis procedure using the Goldstein loss model

Once a propeller is designed the designer is tempted to know how the propeller might work at off design points. It was shown at the beginning of section §2.4 that performances of propellers can be precisely assessed by means of 3-D CFD. Latter is still expensive in terms of computational time and hardware resources and the BEM offers a fast, cheap possibility to asses the performances of propeller at off-design. The obvious drawback of this method is that the simulations are based on 2- D data obtained either from measurements or from simulations which are then integrated radially by using a loss function. However, at the initial design stage of the propeller, rough parameters studies might be performed by such methods. They have the advantage of being very fast, so fine setups of the design might be done later on by using the 3-D CFD methods. This was done for example for axial fans in [60]. The data flow in the code is opposite to the one of the design method starting in this case with the geometry of the propeller, as received from ADAP. Basically the code follows the

73 Chapter 3 Presentation and discussion of the results

Figure 3.2.25: Figure 3 and Figure4 from ADAP showing relevant propeller design data same theory as the design code with the difference that here the sections at different positions are known. The function of the code is very similar to other propeller analysis codes: the geometry is read at every radial section, followed by a computation of the section’s aerodynamic characteristics (or an interpolation from a previously measured / computed) for the specified operating conditions (free stream velocity). In the present version of the code only NACA mean-lines can be analyzed, but with a few modifications this can be adapted to arbitrary airfoil analysis (i.e. the Martensen method which was newly used in [92]). Viscous effects are accounted by using the data of a NACA 65 airfoil [2, pp.662]. The drag coefficient values have been interpolated to a function (equation (3.2.13)) so for every computed Cl (subsection 3.2.1 ) the code calculates a Cd . Alternatively, the value of Cd can be fixed, but this offers poor correlation to reality. The Cd as a function of Cl reads:

C 0.0002 C 0.0001 C 0.009 (3.2.13) d = · l − · l +

Similarly to ADAP the BEM assumes at the beginning a zero displacement velocity (figure 3.2.26) [3]. Then, a computation of the aerodynamic coefficients (cl , cd ) and of radial loss momentum is per- formed (with the Goldstein radial momentum loss equation (2.4.54), equation (2.4.55)) for each ra- dial section. From these values the flow angle φ is computed again and its value is compared with the initial one. If they are not equal within a tolerance of 0.05% the code starts to iterate the proce- dure, including the computation of loads, until the tolerance is fulfilled. For the final value of φ the axial and tangential momentum induced factors are the correct ones. The usage of the code and a comparison to 3D CFD are given in section §3.3.

3.3 Design procedure for multiblade open-water propellers

This section proposes the validation of the method for correcting the lift coefficient of two-dimensional hydrofoils in cascades by an iterative procedure developed for arbitrary camber-lines. The method was presented in subsection 3.2.4 and combines the optimum propeller design theory with the itera- tive cascade correcture procedure which uses the CVL method. For the validation of the theory a case study with several propeller designs is simulated by using steady-state RANS CFD simulations. Parts of the results in this section were presented in [59].

3.3.1 Challenges in designing multiblade propellers

As stated already an optimum (marine) propeller is a propeller having maximum efficiency and sat- isfying the required thrust for a given advance coefficient (J). Blade interference effects arise in all

74 3.3 Design procedure for multiblade open-water propellers

Simulation basic data rpm, working velocity range,fluid density

Propeller geometry chord type/geometry , twist angle, chord length, diameter, number of blades, hub-to-tip ratio

for r:rh ...r t

Startz=0 ( v/V8 )

Compute tan(f( r ) )=(1+ z /2)V8 /( W r )

Compute f, F

a=b-f t

G Compute cl, CVL.m

Compute cd

Compute a, a‘

f ( )8 W Compute tan(new r )=(1+a)V /( r(1-a‘))

NO f ( ) tan(new r ) =tan(f( r ) )

YES Adkins_solver_goldstein.m Compute T, Q, h

Figure 3.2.26: Structure of the BEM analysis method rotating machinery when the value of cascade pitch-to-chord ratio ¡ t ¢ is less then 3. In English lan- ³ ´ l guage literature is more often used the blade solidity σ l which is the inverse of the pitch-to-chord = t ratio. A typical rotating machinery cascade is depicted in 3.2.1. It was shown in subsection 3.2.2 that in the case of cascades with t 3 the lift coefficient of an airfoil C has a smaller value (for cascade l ≤ LC stagger angles λ 50°) than the lift coefficient C of the same airfoil without interference. This is de- ≤ L scribed by the lift ratio k which has been plotted against σ and λ was in figure 3.2.14. A discussion about the behavior of CP distribution of stand alone airfoils and in cascades has been already shown in subsection 3.2.3 of section §3.3.

For generating the radial sections of every propeller it was shown that the presented design frame- work uses an inverse airfoil design method to generate them. But airfoils can be also generated based on experimental data. However, there exist only limited aerodynamic data for airfoils in cascade and

75 Chapter 3 Presentation and discussion of the results there are no known inverse design frameworks for airfoil cambers in cascades. Propellers with high solidity will have a lift penalty and thus a theoretical thrust penalty if they are designed with data or methods for non-cascade airfoils. As a result the propeller will not reach the desired thrust. This af- fects the entire design cycles by demanding more time and effort from the designer for reaching the desired thrust at the highest possible efficiency. For validating the improvements achieved by using the CVL correction a real propeller was designed by using Goldstein- and mpvl (implemented in [20]) optimal circulation distribution. These sections of the propellers were designed using a NACA mean- line with different thickness distributions by using or not the correction, and were finally simulated by 3D CFD means.

3.3.2 CFD setup and simulations

Simulation were performed as a stationary flow case by using the commercial solver Ansys CFX® (V14.0). The geometric and performance design parameters of the investigated propeller are pre- sented in table 3.3.1. These dimensions were chosen because another PhD thesis focused on the ex- perimental propeller investigation is ongoing at LSTM. In particular propellers having the parameters in table 3.3.1 are used in wastewater management. The duty point represents the one of wastewater clearing plant. These data were used for the research presented in this sections of this thesis.

Table 3.3.1: Geometric and performance design parameters of the investigated propeller

Parameter Name Value Units outer diameter D 0.25 [m] hub diameter DHUB 0.03 [m] blades B 4 [-] rotation speed r pm 300 [rev/min] design thrust T 21 [N] free stream speed V 0.3 [m/s]

In order to correctly evaluate the effect of the blade-to-blade interference several other parameters involved in the design process of a propeller were investigated: drag-to-lift ratio (ε) and the thickness distribution. A summary of these parameter studies is presented in the next subsections. In the CFD simulations one stationary and a rotating domain were used: a surrounding domain (cylinder) and the propeller, which are shown in figure 3.3.1a and figure 3.3.1b. Between the rotational and station- ary domain the GGI4 frozen rotor domain interface has been used.

The dimensions of the outer domain were chosen in such a way that external walls do not interfere with the propeller. A rotational speed of 300 rpm was applied to the propeller domain. For gener- ating the propeller grid the Ansys TurboGrid software was used. The outer domain surrounding the propeller (cylinder) was block-meshed with Ansys ICEM. The dimensions of the domain relative to the propeller diameter are shown in figure 3.3.4a. Numerical simulations were performed on high performance computers (HPC) using ANSYS CFX® 14.0. As mentioned previously the Shear Stress Turbulence model (SST) of Menter was chosen for the simulations.

3.3.3 Numerical errors of CFD simulations

Numerical simulation has different erros sources, which have to be known in order to understand correctly their results. The accuracy of the solution depends on more factors amonge them [30]: mod- eling errors, discretization errors, iteration errors (in the present case is used an iterative method).

4General Grid Interface

76 3.3 Design procedure for multiblade open-water propellers

Dprop 5xDprop

5xDprop

(a) Propeller and cylindric domain as used in the (b) 3-D View of the assembly CFD simulation

Figure 3.3.1: Setup of the studied problem (reprinted from Miclea-Bleiziffer et al. [59], with permis- sion from Elsevier)

Modeling errors include simplification and assumptions made in the flow modeling i.e turbulence modeling or how good the geometry is rendered by the grid. It should be noticed that these are not always known and in order to eliminate them one should compare their results to experimental mea- surements or to more precise models (e.g. DNS)[30].

The discretization errors are caused by the grid refinement. In order to eliminate them a grid inde- pendence study should be performed. The study ultimately shows the grid dimension from which no change in the solution is expected.

Finally, the iteration errors are caused by insufficiently converged solution, which dramatically differs from the one for each more iteration would be used. The monotony of the solution is important for the convergence of the solution.

The influence of these factors upon the solution of the simulations shown in this work is addressed in the next subsections for both marine propellers and LVAD.

3.3.4 Geometry, grid generation and grid study for the problem

The rotor grid was generated by using the turbo-machine dedicated grid generator from ANSYS, Tur- boGrid figure 3.3.4a. In this software the geometry of a single blade channel is automatically created based on a set of radial-spaced-airfoils, hub and shroud curves (shown in figure 3.3.2). The shape of the airfoils and of the hub and shroud curves has been computed by ADAP (section §3.2) while using the settings presented in the next pages.

ADAP writes the sections coordinates (axial, traversal and radial) in a ASCII text file of type ”.curve”. Coordinates arrays of the sections are separated by two empty lines. In TurboGrid the surface of the blade is generated by span-wise lofting with B spline curves and surfaces. Leading and trailing edge are longer than the blade span as it is visible in figure 3.3.2 in the tip region. This is required for the correct construction of the blade tip, the surface of the blade having the correct design span. Shroud and hub curves are defined in this case by straight lines called ”hub.curve” and ”shroud.curve” defined by axial and radial coordinates.

Different grid generation options are available in the program including a full automatic grid genera- tion by specifying the target number of elements in the blade channel. For the geometries presented in this study the grid generation was performed automatically using the ATM option. In [30] and [38] are given in-detail descriptions about the correct grid treatment for computations of internal flows

77 Chapter 3 Presentation and discussion of the results

Trailing Edge(TE) Leading Edge(TE) Shroud

2D-sections

ADAP output

Hub

Figure 3.3.2: Propeller blade geometry in TurboGrid with figured sections

Table 3.3.2: Boundary conditions

BC Parameter Value Units Turbulence inlet V 0.2-1 [m/s] medium intensity (5%) ∞ outlet relative static pressure 0 [Pa] medium intensity (5%) rotation nrot 300 [r pm] - walls no-slip walls - - - using two-component turbulence models. Their recommendations are followed in this work. Con- 5 vergence criteria was set for the RMS residuals of mass and momentum at 1 10− . × Table 3.3.3: Simulation parameters

Simulation parameter Definition Value Unit density 998 [kg/m3] temperature 25 [◦C] physical time scale 0.05 [s] fluid model incompressible (no cavitation) - - fluid type water - - advection scheme high resolution - -

Simulations were performed with the boundary conditions shown in table 3.3.2. The turbulence was set according to the Ansys CFX best practice guide [5] which recommends a medium intensity turbu- lence in case the real turbulence is not known. Monitors set in the solver for the thrust and torque were fully converged. Actually, worth to be mentioned that the simulations were fully converged af- ter 45 iterations since neither RMS residuals nor the monitored quantities showed any change. The summary of the simulation parameters is presented in table 3.3.3. Here has to be mentioned that the physical time scale is computed according to the best practice guide for turbomachinery [5] as 1/Ω 0.032[s], which is increased to 0.5[s] because of an improved and faster convergence. In order = to ensure the convergence of the solution and the optimal grid parameters of the studied geometries a grid study was performed for one chosen propeller. The number of elements and their distribution at interfaces was set to be equal on both sides, this improves accuracy and convergence (3.3.4a).

78 3.3 Design procedure for multiblade open-water propellers

0 . 0 1

R M S P - M a s s 1 0 1 E - 3 R M S U - M o m s

R M S V - M o m t s l n i a

R M S W - M o m o u P d

i r s 1 E - 4 o e t i R

n S 1 o M M R 1 E - 5 M o n i t o r P o i n t η [ - ] M o n i t o r P o i n t t h r u s t [ N ] M o n i t o r P o i n t t o r q u e [ N m ] 1 E - 6 0 3 0 6 0 9 0 0 3 0 6 0 9 0 I t e r a t i o n c o u n t I t e r a t i o n c o u n t (a) Convergence of mass and momentum in for an exemplary (b) Convergence of user monitor points propeller simulation

Figure 3.3.3: Convergence of solution

A propeller having the parameters depicted in table 3.3.1 was designed for the grid study. A value of 0.02 was chosen for ε and a NACA 65A010 thickness distribution was used for the initial design. The grid independence has been investigated by using 4 numerical grids which were generated for propeller and cylinder.

(a) Detail of the interface at the propeller inlet (b) Detail of the propeller mesh (blue are lines of the propeller, black of the cylinder)

Figure 3.3.4: Details of numerical mesh

The dependence of thrust coefficient KT and the torque coefficient KQ on grid density were inves- tigated. The initial propeller domain mesh had around 1 106 elements alone and the number was × increased to 2 , 3 and 4 106 elements while the total amount of grid cells has been doubled from × × × the coarse to the fine grid. This broad range offers the possibility to see how the monitored quan- tities may vary. The thrust coefficient K shows a very small variation between above 7 106 and T × 8 106(1%). This is caused by change in the blade y which decreases from an average of 17.2 to 12.8. × + This means that in the fine mesh there are many regions which are not modeled by the wall func- tion but are solved, when compared to the previous meshes. However, the torque coefficient KQ still shows a very small gradient as it can be seen in figure 3.3.5. KQ is much sensitive at grid settings be- cause it depends more on the cd (equation (2.4.21b)), and this is more sensitive to grid settings nearby the wall. Since for grids having more then 7 106 (propeller having 3 106 ) the changes found in the × × monitored quantities were very small, this grid was used for all further simulations in this study.

79 Chapter 3 Presentation and discussion of the results

0 . 2 4 0 . 2 4

0 . 2 3 0 . 2 3 K T ]

1 0 x K -

Q [

] Q - [

0 . 2 2 0 . 2 2 K

T x K

0 1 0 . 2 1 0 . 2 1

0 . 2 0 0 . 2 0 4 x 1 0 6 5 x 1 0 6 6 x 1 0 6 7 x 1 0 6 8 x 1 0 6 G r i d P o i n t s [ - ]

Figure 3.3.5: Grid study showing the dependence of K and K at design point V 0.3[m/s](J T Q = = 0.24) upon the number of grid elements (reprinted from Miclea-Bleiziffer et al. [59], with permission from Elsevier)

3.3.5 Design parameter study

During the design process one has to choose among different parameters for designing the optimal propeller. Since, for some of them there is no recommendation which values should be used it is wise to make a pre-study to determine them. The results of this study should give the recommendation for the values to be used. The parameters investigated in this section are the thickness distribution and the lift-to-drag ratio (ε). Complementary to to this a comparison of the prescribed lift distribution resulted from two design methods (Goldstein loss factor) and the mpvl method [20] is performed. This ultimately shows how good the present design method performs if compared to state-of-the-art methods using vortex-lattice methods.

Airfoil thickness distribution study

In subsection 3.2.3 has been shown how an air- and hydrofoil fulfilling the load demands is created. The procedure used a NACA mean-line to which a 10% thickness distribution is applied at all radial sections of the propeller. In subsection 3.2.3 was also mentioned that NACA mean-lines were devel- oped to be used with NACA 6 series thickness distributions. Though, it is interesting for the designer to see what happens if other distributions are applied. For example in OpenProp [20, 28] a parabolic distribution can be chosen while for low pressure axial fans a constant thickness distribution was in [60]. Besides the standard NACA 65A010 thickness distribution a propeller with NACA 4 thickness distribution was investigated by using CFD computations. Both propeller designs were simulated for all advance coefficients. Results are depicted in figure 3.3.6. The open water characteristics of both designs show almost no difference up to the J 0.6. Above this advance ratio the propeller designed = with NACA 4 thickness performs better in terms of efficiency (figure 3.3.6b).

This improvement in efficiency leads to the conclusion that the airfoils with 10% NACA 4 distribu- tion perform better at high angles of attack than airfoil with NACA 65A010 thickness. This positive behavior at off-design-point advance ratios of the propeller having NACA 4 airfoils is not known from any literature. For assessing the better performance a deeper investigation is performed for both pro- pellers at the advance ratio J 0.7. Here the difference between them is visible and so, some effects = should be visible in the fluid analysis. For this purpose were analyzed the Cp distributions of the air- foils at different blade spans. When plotting the Cp distribution for 30% span (figure 3.3.7 left above) it is visible that the stagnation point of both propellers is moved away from its normal position to the suction side. This can be viewed in the pictures on the left of the figure. The peak of the suction pres- sure is also moved on the pressure side, depicted by the peak of the C , and also clearly visible in − p the pictures on the left side. They indicate that the hydrofoil is subject to a high incidence angle and

80 3.3 Design procedure for multiblade open-water propellers

0 . 2 5 0 . 7 N A C A 6 5 A T h i c k n e s s N A C A 4 T h i c k n e s s 0 . 6 0 . 2 0 0 . 5

0 . 1 5 0 . 4 ] ] - - [ [

T 0 . 3 η K 0 . 1 0 N A C A 6 5 A T h i c k n e s s 0 . 2 N A C A 4 T h i c k n e s s 0 . 0 5 0 . 1

0 . 0 0 0 . 0 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 J [ - ] J [ - ]

(a) KT (b) Propeller efficiency

Figure 3.3.6: Thickness study results for a 4 bladed propeller (reprinted from Miclea-Bleiziffer et al. [59], with permission from Elsevier)

therefore it might not work optimal. From the plot one can read that the pressure difference between suction and pressure side of the foil is higher for the NACA 4 foil. This means the NACA 4 section has a higher lift. Actually, the larger radius of the NACA 4 section should damp high Cp variations across the leading edge. This is however, not the case but the slope of the Cp is smoother over the NACA 4 foil indicating lower losses. The higher losses are also suggested by the none-uniform load distribution of the NACA 6 sections up to 0.7 span. However, in the region above 0.7 span the tip losses might affect more the proper foil loading. The Cp distribution of NACA 6 foils shows a peak on the suction side which is always moved much behind the one of the NACA 4 foil. This is more effective at the ideal AOA when the distribution looks similar to the one of the camber (as shown for example by Abbott and Doenhoff [2]). Figure 3.3.8 presents the plot of the airfoil Cp at 10% span where, on the NACA 6 foil, a disturbance is visible on the suction side. Although the pressure difference of the NACA 6 foil suggests a higher lift, the peak at the leading edge suggests higher losses. The separation of the flow starts at the hub (marked with the blue circle) and grows through the span up to 50-60% in the case of the NACA 6 propeller and 40% in the case of NACA 4 propeller. Scott [87] notes that the NACA 6 airfoils present higher drag when subject to AOA’s different than the optimum, which is confirmed by the present results. However, the Cp distribution of the NACA 6 propeller improves through the span resulting in a higher lift then on the NACA 4 propeller, especially above 70% span. No difference can be observed, however, at the design point (J 0.24). Because of the efficiency improvement at higher = advance ratios, the NACA 4 Digit thickness distribution was used for further designs in this study.

Sensitivity to the drag-to-lift ratio (²)

In order to find an optimum design drag-to-lift ratio coefficient ε (necessary in the computation of blade forces from equation (2.4.42a) and equation (2.4.42b)) a study was performed for the investi- gated propeller. Initially is assumed that ε 0 (i.e. frictionless flow) which is then incremented to = 0.02 and 0.04. Three propellers having the specifications depicted in table 3.3.1 were designed for this values and simulated at the design advance ratio: J 0.24. The results of this study are presented = in figure 3.3.7. No notable differences are seen for the propellers having different values of ε, which are shown in 3.3.7. Based on these findings, a value of ε 0.02 was used for further studies. =

81 Chapter 3 Presentation and discussion of the results

-2.0

-1.5 Cp NACA 4 Thickness @ 0.3 Span NACA 4 Cp NACA 65A Thickness @ 0.3 Span -1.0

-0.5 p C

0.0 Span 0.3 NACA 65A 0.5

1.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Coordinate -1.0

NACA 4 -0.5

0.0 p C Span 0.5 Cp NACA 4 Thickness @ 0.5 Span NACA 65A 0.5 Cp NACA 65A Thickness @ 0.5 Span

1.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Coordinate -1.0 Cp NACA 4 Thickness @ 0.7 Span Cp NACA 65A Thickness @ 0.7 Span -0.5 NACA 4

0.0 p C Span 0.7

0.5 NACA 65A

1.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Coordinate -1.0

-0.5 NACA 4

0.0 p C

Cp NACA 4 Thickness @ 0.9 Span Span 0.9 0.5 Cp NACA 65A Thickness @ 0.9 Span NACA 65A

1.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Coordinate

Figure 3.3.7: C distribution of both propellers at J 0.7 and 0.7 span p =

82 3.3 Design procedure for multiblade open-water propellers

-3.0

-2.5 Cp NACA 4 Thickness @ 0.1 Span Cp NACA 65A Thickness @ 0.1 Span NACA 4 -2.0 -1.5 -1.0 p C

-0.5 Span 0.1

0.0 NACA 65A 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Coordinate

Figure 3.3.8: C distribution of both propellers at J 0.7 and 0.1 span p =

0 . 2 1 2 0 . 3 5 2

0 . 3 5 0 0 . 2 0 8 0 . 3 4 8 ] η - [

T K 0 . 3 4 6 0 . 2 0 4 0 . 3 4 4

0 . 2 0 0 0 . 3 4 2 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 0 0 . 0 2 0 . 0 4 ε ε

(a) KT variation with ² (b) η variation with ²

Figure 3.3.9: Influence of ε in the design framework upon K and η at design point V 0.3[m/s] T = (J 0.24) =

Optimal circulation study

In the previous studies the effects of the most important design parameters were isolated, and by performing parameter studies an optimum was found for each of them. The actual study aimed in this chapter is the influence of the blade-to-blade correction. It is more useful for the application of this theory to change the duty parameters of the propeller to some more close to the ones found in marine applications. Some rough reference of duty parameters of marine propellers can be found for example in [20]. Starting from this work the duty of the ”high advance ratio propeller” was shifted to higher advance ratios5 (depicted in Table 3.3.4).

The geometry of the propeller was defined by using the chord distribution of a N4148 propeller [20]. This had to be modified at the tip region (ξ 0.9 1) because of the grid quality demands. The stan- = − dard radial chord distribution lead to very high skew angles and thus negative elements in this re- gion.

Figure 3.3.10a depicts the non-dimensional circulation distribution obtained by using the present code with the Goldstein circulation function (in red) and for comparison the results of the mpvl code (blue). There is no considerable difference between the two methods except the near hub part, where the mpvl code prescribes zero circulation (hub load correction). In Figure 3.3.10b are presented the

5This was done in order to enhance the effects

83 Chapter 3 Presentation and discussion of the results

Table 3.3.4: Design parameters of the investigated propellers

Parameter Value

KT 0.20 J 2.4 mean-line NACA a 0.8 = design ε 0.02 thickness distribution NACA 4 Digit chord distribution N4148 (modified) hub ratio 0.2

0 . 0 0 9 0 . 3 c L - G o l d s t e i n Γ c G o l d s t e i n L - G o l d s t e i n - C V L Γ c m p v l L - m p v l c 0 . 0 0 6 0 . 2 L - m p v l - C V L ] ] - [ -

[ L

c Γ 0 . 0 0 3 0 . 1

0 . 0 0 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 ξ [ - ] ξ [ - ]

(a) Non-dimensional circulation (b) Radial CL distribution: standard (squares) and corrected (inverse triangles)

Figure 3.3.10: Results of the present design code are given in red and results of mpvl code [20] are given in blue (reprinted from Miclea-Bleiziffer et al. [59], with permission from Else- vier) differences between the standard approach and the use of CVL iterative method in the design of a 6 bladed propeller. The Goldstein momentum loss implementation is shown in red and the results computed using the mpvl are presented in blue. Independent of the radial loss method differences are observed in the prescribed cl increase while performing the CVL correction loop (Figure 3.3.10b). Two propellers have been designed adopting the present design code by making or not use of the CVL correction. Results are presented in next section.

3.3.6 Discussion of results

This section shows the results of the CFD simulation. Nearby the CFD simulation a BEM analysis has been performed. The BEM ”simulation” program was described in subsection 3.2.5. The BEM method although not exact, has the advantage of beeing very fast6. Basically the BEM method was val- idated by the test cases presented in this section. Figure 3.3.11a presents the CFD results of the thrust coefficient against the advance ratio. A violet point marks the desired thrust coefficient (K 0.2). T = The main difference observed for all advance ratios is that the thrust of the CVL propeller increased with a δK 0.007 in both CFD and BEM simulations, at the design advance ratio. This is a result of T = the lift increase applied by the CVL method, that is clearly shown in figure 3.3.1b. Also the efficiency, although well predicted for low advance ratios, is not so well matched for the higher ones. Benini [9] relates the poor result matching to a combination of effects as the stream-tube contraction, radial equilibrium and tip vortex flows. Worth mentioning that he used for the BEM simulations a higher

6an analysis of a propeller over the advance ratio takes only about 90s

84 3.3 Design procedure for multiblade open-water propellers order panel software Xfoil 7,which can fairly predict the boundary layer and the viscous drag. That is one of the reasons in his study the thrust and torque were not over-predicted at low advance ratios.

0 . 6 2 . 8 C F D K C F D 1 0 x K T Q 0 . 5 2 . 4 C F D K C F D 1 0 x K T - C V L Q - C V L B E M K 2 . 0 B E M 1 0 x K 0 . 4 T Q

B E M K ]

T - C V L - 1 . 6 B E M 1 0 x K

[ Q - C V L ] Q - 0 . 3 [ K

T T a r g e t x K

0 1 . 2 0 . 2 1 0 . 8

0 . 1 0 . 4

0 . 0 0 . 0 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 2 . 8 3 . 0 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 2 . 8 3 . 0 J [ - ] J [ - ]

(a) Thrust coefficient KT (b) Torque coefficient KQ

Figure 3.3.11: Thrust and torque characteristics for propellers with (blue) and without (red) the CVL correction, dashed line results of the BEM method (figure adapted from Miclea- Bleiziffer et al. [59], with permission from Elsevier)

0 . 9 0 . 8 0 . 7 0 . 6 ]

- 0 . 5 η [ C F D T η 0 . 4 C F D η T - C V L 0 . 3 B E M η 0 . 2 T B E M η 0 . 1 T - C V L T a r g e t η 0 . 0 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 2 . 8 3 . 0 J [ - ]

Figure 3.3.12: Efficiency characteristics for propellers with (blue) and without (red) the CVL correc- tion, dashed line results of the BEM method (figure adapted from Miclea-Bleiziffer et al. [59], with permission from Elsevier)

However, the relative KT increase of the CVL design is captured entirely also by the BEM method, visible in the curves of figure 3.3.11a. The relative increase obtained by the CVL design, as shown by the CFD results in figure 3.3.11a is about 3.6% (which means an absolute increase of 3.1%) at the design advance ratio. There is also a small increase in torque as shown in figure 3.3.11b (both CFD and BEM method). As a result the propeller efficiency is the same for both CVL and classic designs (figure 3.3.12). The total performance characteristics are though better for the CVL method because a higher thrust is reached for the same efficiency and advance ratio. The thrust developed by the propeller with corrected cl represents 84.6 % from the required design thrust while the one of the propeller design by the classical approach represents 81.5 %. For advance ratios higher than the de- sign ratio a significant increase of the efficiency can be noted in the case of the CVL method. This secondary effect is also well captured by the BEM code.

7http://web.mit.edu/drela/Public/web/xfoil/

85 Chapter 3 Presentation and discussion of the results

Through the integral propeller characteristics was shown that thrust improvements can be achieved by using the CVL correction method. Hence, it is important to understand the mechanism of this improvement and it can be seen by investigating the hydrofoil dynamics in the CFD simulations. First, the span-wise propeller blade loading is investigated. This is done by comparing the prescribed blade thrust and torque with the results delivered by the RANS CFD simulation at the design advance ratio.

0 . 4 0 . 0 5 T h r u s t d e s i g n T o r q u e d e s i g n T h r u s t C F D ( n o C V L ) T o r q u e C F D ( n o C V L ) T h r u s t C F D ( C V L ) 0 . 0 4 T o r q u e C F D ( C V L ) 0 . 3

] 0 . 0 3 m ] 0 . 2 N N [ [

T Q 0 . 0 2

0 . 1 0 . 0 1

0 . 0 0 . 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 ξ [ - ] ξ [ - ] (a) Radial thrust distribution at J 2.4 (b) Radial torque distribution at J 2.4 = = Figure 3.3.13: Radial distribution of thrust and torque for design, propeller and propeller with CVL correction (reprinted from Miclea-Bleiziffer et al. [59], with permission from Elsevier)

0 . 9

0 . 8

] 0 . 7 - [

η η η ( n o C V L ) 0 . 6 η ( C V L )

0 . 5 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 ξ [ - ]

Figure 3.3.14: Radial distribution of efficiency at J 2.4 (reprinted from Miclea-Bleiziffer et al. [59], = with permission from Elsevier)

Figure 3.3.13a presents the span-wise thrust distribution of a blade for both investigated designs and the one prescribed in the design program. The thrust read from the CFD simulations is, as expected, below the prescribed thrust. The difference between the thrust of the blade design by correcting the cl (CVL) and the baseline blade is constant over the radius. In the same way the torque distribution is higher for the CVL corrected propeller. This is expected since a higher lift should be produced by the propeller’s sections. However, in the span-wise distribution of efficiency small advantages of the CVL corrected design can be observed in the hub region.

In figure 3.3.15 and figure 3.3.16 are depicted the CP distribution of both propellers at key blade spans for the simulations run at the design advance ratio:J 2.4. In the comparison of the blade loads = shown in figure 3.3.13 the value of the span loads of both designs were very close. This is also visible in the CP analysis were only at the hub near spans (0.1 and 0.3 ) one can see a very slight difference between the profiles. This are also the only regions where blade-to-blade interferences are significant, the pcr values are in the high interference region shown in figure 3.2.13: pcr 0.9 for 0.1 span and =

86 3.3 Design procedure for multiblade open-water propellers

-0.8 C classic design @ 0.1 Span -0.6 p C CVL @ 0.1 Span p -0.4 Classic -0.2 0.0 p

C 0.2

0.4 Span 0.1 0.6 CVL 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Coordinate -0.8 C classic design @ 0.3 Span -0.6 p C CVL @ 0.3 Span -0.4 p Classic -0.2 0.0 p

C 0.2

0.4 Span 0.3 0.6 CVL 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Coordinate -0.8

-0.6 C classic design @ 0.5 Span p C CVL @ 0.5 Span -0.4 p Classic -0.2 0.0 p

C 0.2 0.4 Span 0.5 0.6 CVL 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Coordinate

Figure 3.3.15: C distribution of the 0.1,0.3,0.5 span sections of both investigated propellers at J 2.4 P = pcr 1.3 for 0.3 span and stagger angles (λ) of about 18° and 33° respectively. It can be also observed = very well that the hydrofoils do not show any separation point and the slopes are very gentle.

The CP distributions computed for the hydrofoils of the outer spans (0.7 and 0.9) are shown in fig- ure 3.3.16. The pcr for the 0.7 span has the value of 1.6 and for the 0.9 span 3. Thus, blade-to-blade interferences do not exist at this spans as also confirmed by both graphs and contour plots in fig- ure 3.3.16.

It is important to notice how changes in hydrofoil geometry influence the flow field around the blade. To understand the radial blade load, which has already been shown in figure 3.3.13, the pressure distribution was analyzed on the blades of both cases at the design advance ratio (J 2.4), as depicted = in figure 3.3.17 and figure 3.3.18.

87 Chapter 3 Presentation and discussion of the results

-0.8 C classic design @ 0.7 Span -0.6 p C CVL @ 0.7 Span p -0.4 Classic -0.2 0.0 p C 0.2 Span 0.7 0.4 0.6 CVL 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Coordinate -0.8

-0.6 C classic design @ 0.9 Span p C CVL @ 0.9 Span -0.4 p Classic -0.2 0.0 p

C 0.2

0.4 Span 0.9 0.6 CVL 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Coordinate

Figure 3.3.16: C distribution of the 0.7,0.9 span sections of both investigated propellers at J 2.4 P =

No-CVL CVL

1 1

2 2

3 3

Figure 3.3.17: Suction side of the blade with contours of static pressure (J 2.4) (reprinted from = Miclea-Bleiziffer et al. [59], with permission from Elsevier)

The major differences between the static pressure on the suction side surface of the investigated de- signs appear in regions close to the leading edge. They can be summed up in three regions (denoted 1, 2, 3) as depicted in figure 3.3.17. The contours of the CVL design (right) have been indicated with a dashed dark line and are overlapped by those of the design without CVL which are marked in white

88 3.3 Design procedure for multiblade open-water propellers

(left). In all three labeled regions, the contours of the design without CVL have a sharper shape than those of the CVL designs. As a consequence, the pressure distribution is more uniform in the case of the CVL design. This can enhance performances by inducing less losses.

No-CVL CVL

Figure 3.3.18: Pressure side of the blade with contours of static pressure (J 2.4) (reprinted from = Miclea-Bleiziffer et al. [59], with permission from Elsevier)

A similar pattern can be observed on the pressure side (figure 3.3.18). For better visualization, the contours are marked in the CVL design plot by a black dashed line which are overlapped on the pres- sure contour without CVL. The minimum pressure contour of the CVL design is marked with a dashed dark line (right) and then overlapped on the blade design without CVL. Here, the contour of same magnitude is marked with a white line. This contour is shifted very little in the direction of the trail- ing edge (difference between the two islands black and white on the left side of figure 3.3.18). The next contour is marked at the leading edge, and this contour is also very little shifted .

No-CVL CVL

Normalized axial velocity

Figure 3.3.19: Normalized axial velocity in an aft propeller plane (J 2.4) (reprinted from Miclea- = Bleiziffer et al. [59], with permission from Elsevier)

The axial velocity fields at a 0.5 D aft plane are almost identical, as depicted in figure 3.3.19. A dif- · ference appears between 40% and 75% radial distance, where the velocity is no longer uniform on the CVL blade (marked with white on the right side of figure 3.3.19) when compared with the contour of the non-CVL design shown on the left (contours marked with a dashed dark line). The contours of the CVL blade are also shifted circumferential, as shown by the differences between the white and black lines on the right. In the lower radial side the CVL blade shows a shifted island of high velocity. This could be connected to the differences in pressure distribution at low radii on figure 3.3.17. This marked high velocity contour is smaller in the case of the CVL design.

89 Chapter 3 Presentation and discussion of the results

In this subchapter has been investigated the influence of the blade-to-blade interference influence on the design of open water propellers. Also the influence of other important parameters used in the propeller design process has been investigated and their optimum has been found. The results of the investigation has shown that blade-to-blade interference plays an important role in the design of open water propellers. It is possible that the blade-to-blade interferences plays more vital role in the design of encased propellers which are theoretically more closer to axial fans Lewis [52]. Moreover, the propeller design program ADAP developed in this thesis has been validated for the design of open water propellers. It can now be used for the design of propellers used as VAD.

3.4 Design procedure for an encased propeller used as a LVAD

The Reitan Cathether Pump (RCP) concept has been presented first in 1999 as a promising option in the treatment of the AHFS and cardiogenic shock. This concept, as already mentioned in section 1.1 is available for clinical testing, but not yet commercially. At the starting moment of this work the only available version was the 14 F. In a common project with the company producing the RCP, Cardiobridge Gmbh, the 10F version was developed, and a part of this research is shown in the present section.

Cage Blade Driving shaft Catheter 4.6 [mm]

15 [mm]

6.5 [mm] 31.6 [mm] 13.8 [mm]

Figure 3.4.1: CAD Model of the 14F RCP (courtesy of Cardiobridge GmbH)

The RCP pump is composed of following main parts: a protective cage, two foldable blades attached to the driving shaft and a catheter (figure 3.4.1). The main dimensions of the propeller are depicted in figure 3.4.1. On the inflow side the pump has a rounded head which improves the inflow through the propeller. For the study presented in this thesis the most important parameter is the outer diam- eter of the blades which is fixed. The number of blades is also fixed ensuring the propeller remains foldable.

In contrast with the case presented in section §3.3 the inverse design of the axial-propeller-pumps has been performed under the constrains of dimensions and number of blades. For the optimization iterations the existing pump geometry has been used as reference. The design process will follow the path shown in figure 3.1.1: the baseline design, 14F RCP, is measured and simulated to provide the design input data. The original geometry shown in figure 3.4.1 is simplified allowing the designer greater and faster performance assessment.

3.4.1 Experimental and CFD assessment of the 14F RCP pressure-flow performance

The motivation of the first experimental work was to offer a validation base for the CFD simulation used in the development phase. Although different measurements are available from the literature

90 3.4 Design procedure for an encased propeller used as a LVAD

[77, 74] the actual model has been modified through different development stages. Because the pump assembly is very complex (as shown in figure 3.4.1) information about the effect of different compo- nents on the performance were needed i.e: what are the pressure losses due to the cage or what is the influence of the propeller foot on the flow8. Through successive measurement and simplifications the influences of all components on the overall performance of the pump are isolated. This is needed be- cause the design method applies only to the rotor. Measurements were performed for two fluids: a 40 % glycerol mixture composed of 40% glycerin and 60% distillate water (as proposed in [74] and usu- ally found in literature) or water. The glycerol mixture has a viscosity close to blood: 3.9mPa s and a · density of 1.106g/ml, values which have been measured by the author at LSTM. The propellers were measured in a closed measurement loop as it was presented in section §2.5 starting with the original 14F RCP (unmodified). The same geometry with a downstream stator was measured. A stator or a de-swirler is needed in order to eliminate the swirl and with this the inhomogeneous pressure distri- bution (this matter is discusses in detail in section 2.5.4). With removed cage another measurement was performed. Finally the original propeller geometry was reconstructed in the CAD and instead of the ”foot” the blade surface was connected directly to a hub. Nevertheless this configuration al- lows the stereolitography prototyping, which is a very fast and cheap method. The transformation of the original CAD is shown in figure 3.4.2. Because of the high spinning speed the structural stresses were too high for the material used, so the connection of the blade to the hub had to be reinforced by adding some material and filleting to the connection with the hub (marked with a red circle in figure 3.4.2). This led to a geometry which was stable enough to perform measurements.

Figure 3.4.2: CAD Model of the reconstructed 14F propeller. Right: stereolitography prototype

Next, the CAD reconstructed P14 RCP propeller was build by stereolitography. This last configuration has been measured both in water and glycerol mixture. Measurements in water need less preparing time and they were preferred for testing the new prototypes.

Measured data were recorded by a digital data acquisition system from National Instruments with the help of a acquisition and recording software written and implemented in the LabView r soft- ware. The development and validation of the test rig together with details regarding the measurement system are presented in sub-subsection 2.5.3. Each measurement of the pressure slope has been re- peated two more times and then an average of the measurements was computed. The average is plotted with error bars showing the standard deviation of each measurement.

The five 14F measurements are resumed in figure 3.4.3. They were measured around the working range of VADs situated between 0 and 10l/min . At the duty point of the pump (5l/min) the mea- surement of the original 14F pump shows a pressure increase of 12.3mmHg. This value agrees with the results shown by the RCP developer Dr. Reitan in his PhD work [74]. The pressure curve is very flat in the flow-rate range on the graphic. The same was observed also by Reitan in his work [74]

8this matter is actually very important because the new pump prototypes were build with blades directly attached to the hub

91 Chapter 3 Presentation and discussion of the results

Flow-rate [dm3/s] 0.00 0.03 0.07 0.10 0.13 0.17 15 2000 14F RCP Original (Glycerol)

14F RCP with stator (Glycerol)

12 1600 D 14F RCP with stator w/o cage (Glycerol) ] P g s t H [ P

m 14F RCP with stator w/o cage (H2O) a ] m [ t s 9 1200 P 14F RCP with modified hub (H2O) D

14F RCP with modified hub (Glycerol) 6 800 0 2 4 6 8 10 Flow-rate [l/min]

Figure 3.4.3: Measurements of the five 14F RCP configurations when comparing the performance of the pump by changing different pipe diameters. By adding a stator immediately after the pump the measured pressure difference decreases as it is visible in fig- ure 3.4.3. The curve modifies its shape from almost constant to more inclined, a more usual pump characteristics. At the design point the pressure decreases with about 2mmHg, representing 85 % from the previous measured value. As a result cage produces about 15% losses for this configuration at the duty point. For this configuration the performances are similar with the ones of the measure- ment without stator (red and blue lines in figure 3.4.2). By changing the fluid from glycerol-mixture to water the pressure reduces (violet and dark red slopes in figure 3.4.2). The glycerol mixture has a density of 1106kg/m3 compared to the 998kg/m3 of water. This causes the differences between the measurements since the pressure produced by a rotating machine is proportional to the density of the fluid. However, the viscosity is also different for both fluids, the mixture having an increased viscosity when compared to water (3.9mPas to 1.002mPas-dynamic viscosity) . This changes the local Reynold number of airfoils, which affects the propeller performances. In the case of the recon- structed propeller the pressure increases at lower-flow-rates and decreases at higher flow-rates. Gen- erally speaking the pressure curve of the reconstructed propeller is similar to the one of the original propeller running in water and without cage.

Once the evaluation of different modification stages of the 14F RCP has been closed the last geometry is prepared for the CFD simulation. The performance characteristics of the 14F RCP with the modified hub shown in figure 3.4.2 (dark red) represent the base for the validation of the CFD simulation.

The simulations setup (figure 3.4.4) is built according to the measurement setup shown in figure 2.5.5. However, only parts relevant to the pressure measurement are used for the simulation setup: the pipe in the front of the propeller (inlet-pipe), propeller, de-swirler and outlet pipe. The main dimen- sions correspond to the ones described in section §2.5. The simulation setup divided in four domains (shown in figure 3.4.4), for each of them a grid is generated and they are connected by the GGI in- terface. For connecting the rotor (propeller) to the non-rotating domains (inlet pipe and de-swirler) the frozen rotor interface is used. The rotor and de-swirler grids were generated by using the Ansys Turbo-Grid r software. In contrast to the new generated designs the rotor geometry has to be first read in Ansys Design Modeler r , which is part of the Ansys Workbench r. This reads the airfoil pro- files at chosen radial stations which are then exported in a format readable by Turbo-Grid. The blade geometry is finally constructed automatically from this curves and a mesh is generated (as described for marine propellers in subsection 3.3.4). The rotor grid is generated with a very large tip gap (40% of the nominal pipe diameter) in order to minimize the number of used interfaces (figure 3.4.5)9. Trials have shown that dividing the computational space with a interface of the rotating domain parallel to

9instead of meshing a volume just arround the propeller and use an additional radial interface which is not

92 3.4 Design procedure for an encased propeller used as a LVAD

Outlet-pipe

De-swirler Propeller

160

Inlet-pipe

30 9.5

170

Figure 3.4.4: CFX setup used for the 14F RCP simulations (lengths are given in mm) the rotating axis did not lead to correct results. However, the kind of grid used in here is not usual nor standard in turbomachinery simulation, where tip gaps are in the order of 1 2% of the nominal − diameter.

de-swirler domain interface rotor-stator propeller domain

interface to outlet

interface to inlet

Figure 3.4.5: Rotor and stator mesh of the 14F setup

Additional profiles are generated in the radial direction, above the actual tip radius so that an imagi- nary propeller surface is generated up to the pipe walls. This surface is sliced at the profile represent- ing the propeller tip. The interface surface up- and downstream of the propeller geometry has been set at a distance of about 5mm (one chord length). Iterative trials have shown that this distance has to be kept as large as possible in order to avoid any convergence or error issues.

The setup parameters used in Ansys CFX-Pre are depicted in table 3.4.1. For simulations with glycerol mixture the density as well as the viscosity are changed with the values measured at LSTM. Boundary conditions are set similarly to the ones in the marine propeller study presented in section §3.3. They are however, adapted to the present case which has some particularities, presented in the following.

93 Chapter 3 Presentation and discussion of the results

Table 3.4.1: Simulation parameters

Simulation parameter Definition Value Units fluid water - - density rho 998 kg/m3 reference pressure pref 100000 Pa temperature temp 21 °C fluid model incompressible (no cavitation) - - advection type high resolution - - fluid timescale control physical timescale 0.005 s

In the case of the RCP the runs are started in a specific sequence: the first run is started at a rotational speed of 3000r pm by using the upwind advection scheme (first order) which is followed by a run at nominal speed (13000r pm). Last is again used as initialization for the runs with high resolution advection scheme (second order) which are run at the nominal speed. First simulation is run at the duty point flow-rate of 5l/min, while all other simulations have this one as initial definition file. Hence, their convergence is much faster.

Table 3.4.2: Boundary conditions

BC Parameters Value Units Turbulence inlet volumetric flow-rate 2 10 l/s low intensity − outlet opening pressure (relative) 0 Pa medium intensity interface frozen-rotor / GGI - - - rotation - 3000 13000 rev/min - − walls no slip - - - CRW surface: propeller shroud - - -

The convergence history is shown graphically in figure 3.4.7. For domain partitioning was used the 10 RCB method , which is available in the Ansys CFX r.

Figure 3.4.6: Output of the CFX mesh statistics

The CFX-solver has a mesh quality check routine which plots its results in the ”out” file of the simu- lation. In the case of the 14F RCP pump all meshes show a quality above the minimum requirements as depicted in figure 3.4.6. This ensured a good convergence on one side and a good quality of results (compared to the measurements) on the other side.

10RCB Recursive Coord Bisection

94 3.4 Design procedure for an encased propeller used as a LVAD

0 . 0 1 R M S P - M a s s 1 0 0 0 ∆ R M S U - m o m 1 0 0 M o n i t o r P o i n t P R M S V - m o m M o n i t o r P o i n t t h r u s t [ N ] 1 0 R M S W - m o m s M o n i t o r P o i n t t o r q u e [ N m ] t

s 1 E - 3 l n i a 1 o u P d

i r s o e 0 . 1 t R i

n S

o 0 . 0 1 M 1 E - 4 M R 1 E - 3 1 E - 4 1 E - 5 1 E - 5 0 4 0 0 8 0 0 1 2 0 0 1 6 0 0 2 0 0 0 0 4 0 0 8 0 0 1 2 0 0 1 6 0 0 2 0 0 0 I t e r a t i o n c o u n t I t e r a t i o n c o u n t (a) Mass and momentum convergence history at duty (b) User monitor points convergence history at duty point point (5l/min) (5/min )

Figure 3.4.7: Convergence history at duty point (5l/min)

Convergence of the solution is stable as shown by the history of mass and momentum RMS resid- uals (figure 3.4.7a). The monitor points set by the user tracked following quantities: ∆Pst in-out (static pressure difference between the inlet and outlet of the whole setup), flow rate at outlet, pro- peller axial force and propeller torque. They are defined as expressions in CFX-Pre. Simulations are stopped when both the convergence of monitor points and the one of mass and momentum RMS 4 residuals below 1.0− are achieved. For achieving the convergence, the simulations (only the high resolution@13000r pm) needed about ~400 (interval above 1600 from figure 3.4.7b).

Figure 3.4.8: Plot of the y on the external walls + In order to decide on mesh accuracy, in particular in the wall bounded regions, the values of the y for the simulation run at the design flow-rate (5l/min) are plotted. The Ansys solver manual + [5] suggests values below an y of 1 only for high accurate simulations. In any case the solver uses + the automatically scalable wall function which decides if a logarithmic function (similar to the one presented in equation (2.5.14)) is used in the wall region or if the full momentum equations are solved. In figure 3.4.8 is shown that the actual local y never exceeds the value of 40 and the region with + values above 30 is actually very restricted only in the region of the propeller (both on the pipe walls). The average y on the walls has values up to 32.3 (table 3.4.3). + On the other hand when looking at the values on blades, which are actually more important from the point of view of the actual study, the y has values between 0 and 10 on the lifting surface, with +

95 Chapter 3 Presentation and discussion of the results

PS SS

Figure 3.4.9: Plot of the y on the blade +

some locally higher values at the tip region. However, the lifting surfaces are the ones responsible for both pressure increase and high stresses, as it will be shown in subsection 3.4.3. The y area average + including the tip region has a value of 5.7 (table 3.4.3) which is an indicator for the good accuracy of the results. High values of y are normal for tip regions due to the high shear. +

Table 3.4.3: Area averaged y + Surface y (area average ) + inlet wall 9.9 propeller blades 5.7 shroud propeller 32.3 shroud stator 12.8 stat blades 8.9 outlet wall 2.9

Simulations were performed for flow-rates between 0 10l/min to be compared with the experi- − mental results in figure 3.4.10. The evaluation of the pressure was made between the inlet and outlet of the simulation, which corresponded to the measurement positions (10 and 13) of the test rig (fig- ure 2.5.5).

F l o w - r a t e [ d m 3 / s ] 0 . 0 3 0 . 0 7 0 . 1 0 0 . 1 3 0 . 1 7 1 6 2 1 3 3

1 4 1 8 6 7

1 2 1 6 0 0 ] g ∆ H P

s

1 0 1 3 3 3 t m

[ P m [ a

t ] s 8 ∆P E X P - 1 4 F R C P w i t h m o d i f i e d h u b ( H O ) 1 0 6 7

P s t 2

∆ ∆P C F D - 1 4 F R C P w i t h m o d i f i e d h u b ( H O ) s t 2 ∆P E X P - 1 4 F R C P w i t h m o d i f i e d h u b ( G l y c ) 6 s t 8 0 0 ∆P C F D - 1 4 F R C P w i t h m o d i f i e d h u b ( G l y c ) s t 4 5 3 3 2 4 6 8 1 0 F l o w - r a t e [ l / m i n ]

Figure 3.4.10: Validation of the CFD results by experimental measurements of the 14 RCP running in water

96 3.4 Design procedure for an encased propeller used as a LVAD

Accuracy of CFD prediction is usually assessed by a comparison with experimental measurements. For the propeller setup chosen here (RCP with modified hub) the pressure flow-rate characteristics for runs in water and glycerol are shown on figure 3.4.10. The agreement between CFD and mea- surements is good, especially at the duty flow-rate (5l/min). Here the difference between simulation and experiment is only 2% for the water measurement and 5% for the Glycerol measurement (always referenced to the mean value of the respective measurement). The difference between measurement and CFD is growing for both cases reaching 9% for the water case and 6% for the Glycerol case. This is due to the flow becoming more unstable as the flow-rate decreases, caused by the increased flow recirculation, which is discussed in the next subsection.

In this subsection it was shown how the original impeller geometry was simplified in order to isolate the effects of different components on performances. Results of measurements performed for dif- ferent propeller configurations were presented and commented. The performance of the pump was also assessed by using CFD simulations. The results of CFD simulations show a good agreement to the experimental values.

3.4.2 Analysis of the flow mechanism in the RCP

It has been already shown that the RCP is a propeller encased in a pipe with a relatively high tip gap (or a pump with high tip gap). This unique arrangement leads to a behavior different than the one of normal axial pump and also different than the one of a typical propeller. As a result of this behaviour usual design methods can not be used in the case of RCP.To develop a proper design framework one has first to understand the flow physics of such devices. This is investigated in the present subsec- tion.

Figure 3.4.11: Flow recirculation upstream of the propeller at the working point shown by 3D stream- lines clipped by a middle plane

Results of the CFD simulations are investigated deeper so that the flow physics shown in figure 3.4.11 understood. First, the attention is focused on the duty point (5l/min) of the glycerol mixture simu- lations. A streamline of relative velocity starting at the inlet of the simulation domain is depicted in figure 3.4.11.

The flow upstream of the propeller changes its expected trajectory, from the outlet direction and re- verses in the direction of the inlet. Actually the arrows in the figure are planar but the flow has already a swirl velocity, as illustrated by the streamlines going in the opposite direction of the core flow. The flow can be defined by two major directions: the core flow following the main direction (from inlet to outlet - bordered by the blue lines) and the secondary flow following the direction from downstream of the rotor to the inlet (illustrated by the red arrows). The separation of the two flow regions is easily

97 Chapter 3 Presentation and discussion of the results

Figure 3.4.12: Contraction of the core flow under the effect of the swirling back-flow observed in a figure depicting the upstream pipe in its full length. Here it is clear how the swirling back-flow contracts the core flow (figure 3.4.12). Extrapolated from the axial fans and pump theory, this can be explained by the back-flow appearing at the tip of the blades at off-design conditions. This is pointed out in figure 3.4.13 by the red square and the red line from the point on the pressure curve to the ordinate (symbolizing the flow-rate). This flow is caused by the positive pressure difference between propeller pressure- and suction side.

Figure 3.4.13: Types of flow in an axial turbomachine depending on the throttle position (adapted from Eck [25])

As explained before the gap of the RCP is much larger than the one of an usual axial turbomachine (40% instead of 1 2% in case of a fan) so the flow will recirculate easier through this gap due to the − pressure difference. However, this recirculation should not be confused with the tip vortex. For a turbomachine the duty point is defined as the ”best efficiency point” found over the flow-rate char- acteristics, e.g. the point c in figure 3.4.13. To support this idea an analysis of the computed propeller efficiency has been performed. A measurement of the efficiency was not possible by the present test-

98 3.4 Design procedure for an encased propeller used as a LVAD rig because of the high losses produced by the bearings (as described in subsection 2.5.2).

The hydraulic static efficiency is defined in this case by:

∆Pst V˙ ηst · (3.4.1) = ω M · where the evaluated P is the one shown in figure 3.4.10 computed between the inlet and outlet of 4 st the flow domain, while ω and M are evaluated at the rotor. Looking at the typical characteristic of the axial fan shown in figure 3.4.13, figure 3.4.14 and figure 3.4.15 one can conclude that the propeller- pump is working off duty at the desired duty flow-rate. This is mostly visible by the slope of the static efficiency which on the depicted flow-rate range has not yet reached its maximum (figure 3.4.15). The efficiency is also very small, an indicator for the poor matching to the requirements11. Poor machine matching to the duty point as well as high recirculation zones mean highly turbulent flow zones with mixing and shearing. Such phenomena are not wished in the operation of a VAD. The question, which arises is, if there is any way to understand the machine performance and if it is possible with this new earned knowledge to improve the performance of the 14F RCP.It has been shown in previous sections that the propeller design method proposed in this work can design a propeller exactly at the desired working point. It is intended to further analyse the present design and to provide a new framework for designing encased propeller.

F l o w - r a t e [ d m 3 / s ] F l o w - r a t e [ d m 3 / s ] 0 . 0 3 0 . 0 7 0 . 1 0 0 . 1 3 0 . 0 3 0 . 0 7 0 . 1 0 0 . 1 3 0 . 7 8 1 5 2 0 0 0 0 . 7 6 T h r u s t 1 4 F R C P 1 4 ∆P 1 4 R C P 1 8 6 6 0 . 7 4 s t ] ]

g 0 . 7 2 ∆

N 1 3 1 7 3 3 H P [

s m t t

s 0 . 7 0 [ m P u [

r 1 2 1 6 0 0

a t s h 0 . 6 8 ] P T 0 . 6 6 ∆ 1 1 1 4 6 7 0 . 6 4 1 0 1 3 3 3 0 . 6 2 0 . 6 0 9 1 2 0 0 2 4 6 8 2 4 6 8 F l o w - r a t e [ l / m i n ] F l o w - r a t e [ l / m i n ]

(a) Thrust of the 14F RCP (b) ∆Pst for the 14F RCP

Figure 3.4.14: Performance evaluation of the 14F RCP

The back-flow produced by the propeller, shown in figure 3.4.11 and figure 3.4.12, is explained by the shift from optimum duty point of the propeller pump as it is depicted in figure 3.4.13 and figure 3.4.14. However, the flow-path along the inlet pipe (figure 3.4.12) shows a contraction of the inlet flow under the influence of the back-flow. This means that a mass fraction from propeller downstream moves to the front of the propeller. Figure 3.4.12 suggests that this mass fraction is relative high when com- pared with the initial one coming from the pipe. This matter will be explained in detail in the next subsections. By looking again at figure 3.4.14a and figure 3.4.14b one can see that both the thrust and the pressure of the propeller have the same behavior which is correct, since in a propeller the pres- T sure difference and trust are related by ∆Pst . The pumping efficiency of the 14 RCP based on = AP the static head depicted in 3.4.14b is shown in 3.4.15. It can be noted that the ηst has very low values, below 10% and at the duty point is down to 4% (figure 3.4.15).

11for this duty point a mixed flow machine is required see figure 2.1.6

99 Chapter 3 Presentation and discussion of the results

F l o w - r a t e [ d m 3 / s ] 0 . 0 3 0 . 0 7 0 . 1 0 0 . 1 3 0 . 0 6

0 . 0 5 η 1 4 R C P s t

] 0 . 0 4 % [ t s η 0 . 0 3

0 . 0 2

0 . 0 1 2 4 6 8 F l o w - r a t e [ l / m i n ]

Figure 3.4.15: CFD calculated static efficiency of the P14 propeller-pump

3.4.3 Design and optimization of new VAD propellers

Up to now it has been shown that the flow about the RCP is presenting a high recirculating flow at the duty point. Hence, the design problem can be formulated as follows: the designer has no exact information about the correlation between the actual working point (flow-rate) of the encased pro- peller and the analytical one (set as the free stream velocity in front of the propeller-as mentioned in subsection 2.4.3). As consequence the design can not be performed fully inverse, like the case of marine propellers presented in section §3.3, a relationship needs to be established first between the propeller characteristics and the recirculating mass.

Some iterations are needed in order to find a correlation between the inverse design method and the actual working point and the real performance of the encased propeller. In the first design iteration a propeller shape is computed with the design software ADAP with following input:

• Free stream velocity V is assumed from a volumetric flow-rate 17l/min for the point set as ∞ desired inlet flow-rate (5l/min). The velocity is computed by taking the reference area of the pipe.

• The desired thrust at the working point is set at 0.75N. This thrust value is an extrapolation of the results from CFD simulation results in figure 3.4.14a (0.68N) multiplied by a ”loss” factor (1.1) which represents the approximate difference between analytical design and CFD results for the marine propellers discussed in subsection 3.3.6.

• Dimensions: D 15mm (like the 14F RCP figure 3.4.1); D /D 0.3 (like the re-engineered Ti p = Hub Ti p = 14F RCP figure 3.4.2) ; Chord distribution CoD CoD 0.8 . Airfoils: camber-line: = N4148,modi f ied · NACA 6 with a=0.8, thickness distribution type: NACA 4, radial thickness distribution: modified N4148.

• Spinning speed: 13000rev/min (as the original propeller, see table 3.4.3) . The glycerol mixture is considered for the fluid properties: density: ρ 1106kg/m3 and the dynamic viscosity: gl yc = µ 3.9mPa s. gl yc = · • Three prescribed twist angles (or prescribed angles of attack p Ao A) for the blades: 0, 10, 20[°]. The geometry arrangement for the 0° angle is the direct result of running ADAP for the mentioned input, while 10° and 20° represent an angle at which the blade is rotated (or all sections as shown in figure 3.4.16) at 30% chord length (close to the mass center given by the thin airfoil theory of 25% [72]). However, this twist angle (or p Ao A) may not directly represent the real angle of attack seen by the fluid, since the interaction between the propeller thrust and the recirculated mass-flow (which can also change the incidence) is not established at this point.

100 3.4 Design procedure for an encased propeller used as a LVAD

0° 10° 10° 0° 20° 20°

Whub

W

Figure 3.4.16: Iterative adjustment of the propeller twist (pAoA) shown at two exemplary radii of D25: at hub and at an arbitrary radius

Once the input parameters are given in the ADAP design program, it computes the optimal blade shape as described in section §3.4. The output of the design program is used to analyze the different settings. In the case of VADs an important parameter is the Re-number because of their dimensions. The only parameter which can adjust the Re-number of the sections is the chord distribution since the velocity at the duty point is fixed. Figure 3.4.17a illustrates the chord distribution of the original P14 propeller (actually constant) compared to the one of the new designed D25. The chord distribution is always given in the program by a non-dimensional distribution, normalized by the max diameter (CoD-ratio see figure 3.2.20). As depicted in figure 3.4.17 the modified N4148 chord distribution is set to sink to 0 at the tip, helping to reduce the load and thus the tip losses. This should also be visible when analyzing the tip vortex of both propellers. How Re-number is influenced by the chord distribution is shown in figure 3.4.17b: at the hub it is almost equal to the one of P14 while rising quickly up to the maximum (approximately at R 0.006m or ξ 0.7) and afterwards sinking to 0 at = = the tip. Also shown in the figure is an assumed laminar-turbulent transition at a value of Re 2300. = The chord distribution of the D25 propeller has been chosen in such a way that the chord length corresponds to the one of P14 (see figure 3.4.17 a). However, it has been shown that for marine pro- pellers (figure 3.3.11a) the most thrust is produced by the propeller at 0.6 ξ 0.9. At this range the > < new chord distribution reaches its peak so the Re-number is higher, sinking towards zero at the tip. The old chord distribution allowed the P14 a turbulent Re-number at all radii except the hub (~2200) while with the new distribution more sections are running in the laminar region (up to ξ 0.35 and = after ξ 0.99). = While the sections at higher radius might not be critical, the ones at the hub could lead to perfor- mance penalties. Very small Re-numbers, even if the flow is still turbulent, have a negative impact on the general aerodynamic performance of airfoils as it has already been shown in a PhD work in- vestigating the design of miniature helicopter rotors (Kunz [47]). His CFD simulations showed that by decreasing the Re-number cl sinks and cd grows which means that generally the aerodynamic efficiency of the foil drops.

The D25 propellers (0, 10, 20[°]) were generated in Ansys TurboGrid r in the same manner as shown in subsection 3.3.4 and taking in consideration the specific problems of mesh generation which were discussed for the 14F RCP in subsection (3.4.1). The grid settings for the rotor were the same with some adjustments needed by the new shape. All other grids were re-used so no further modifications were needed.

101 Chapter 3 Presentation and discussion of the results

0 . 2 0 8 0 0 0 R e P 1 4 R C P R e D 2 5 0 . 1 5 6 0 0 0 R e T u r b u l e n t ] - [ ]

- [ D

0 . 1 0 4 0 0 0 o e C R

0 . 0 5 C o D P 1 4 R C P 2 0 0 0 C o D D 2 5

0 . 0 0 0 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 ξ [ - ] ξ [ - ] (a) Non-dimensional chord distribution (CoD) (b) Re distribution over the radius

Figure 3.4.17: Chord distribution and Re of the P14 and of the D25 propeller

1 9 0 0 0 . 0 0 4 0 0 ∆P 1 8 5 0 T o r q u e 0 . 0 0 3 7 5 T o r q ] u a 1 8 0 0 0 . 0 0 3 5 0 P e [

[ N P ∆ m 1 7 5 0 0 . 0 0 3 2 5 ]

1 7 0 0 0 . 0 0 3 0 0 1 . 5 x 1 0 6 2 . 0 x 1 0 6 2 . 5 x 1 0 6 3 . 0 x 1 0 6 3 . 5 x 1 0 6 4 . 0 x 1 0 6 G r i d P o i n t s [ - ]

Figure 3.4.18: Grid study results for one of the new designed propellers

A grid independence study based on the procedure shown in subsection 3.3.4 is performed for the new designs. Three different grids ranging from 1.75 mil. to 3.75 mil elements are tested (figure 3.4.18). First, the number of elements in the propeller passage is modified followed, if necessary, by an ad- justment of the elements number and distribution in the other domains. This allows equal element distributions on both faces. Pressure difference between inlet and outlet and the torque of the pro- peller are monitored in the grid convergence study. The pressure is shown in the SI units (Pa). Be- tween the highest and the lowest point, which are the result of the coarse and fine grids respectively, is a difference of 41Pa. Despite the fact that such difference is hardly measurable (see section §2.5) it represents only 2% of the pressure computed for the fine grid case. Such a difference is insignificant compared to the increase in mesh size: double elements of the initial grid. The evaluated difference in pressure between the course and medium grids is 30Pa while the difference between the fine and medium grid is only 10Pa. Hence, one can assume a converged grid study. The same tendency is observed for the torque, which is a direct result of the flow over blade. Same grid settings were used in the P14 RCP simulation shown previously in subsection3.4.1 and therefore results are comparable.

The influence of the blade’s twist angle on the flow is shown in figure 3.4.19: top: 0°, middle: 10° and bottom: 20°. The flow-paths are changing from top to bottom: the recirculation region becomes larger with increasing pAoA. Streamlines depicted for the 20° twisted case suggest a multiple recircu- lation, meaning that a particle travels several times the way back from the propeller upstream. From this comparison it can be also assumed that a connection between the propeller thrust and the recir- culated mass flow may exist.

102 3.4 Design procedure for an encased propeller used as a LVAD

AOA 0°

AOA 10°

AOA 20°

Figure 3.4.19: 3D Streamlines of the D25 propellers (0°,10°,20°) at the duty flow-rate

AOA 0 0.014 -0.5 0.012 -1 0.01 -2 0.008 -3 0.006 -4 0.004 -5 0.002 -6 0 -7 -25 -20 -15 -10 -5 0 5

AOA 10 0.014 0.012 -0.5 -1 0.01 -2 0.008 -3 0.006 -4 0.004 -5 0.002 -6

0 -7 -25 -20 -15 -10 -5 0 5

AOA 20 0.014 0.012 -0.5 0.01 -1 0.008 -2 0.006 -3 0.004 -4 0.002 -5 0 -6 -25 -20 -15 -10 -5 0 5

Figure 3.4.20: Inflow profiles of the axial velocity for the investigated D25 designs anlyzed at the duty point (distance between profiles is not to scale)

To quantify the back-flow shown by the streamlines in figure 3.4.19 an analysis of the axial flow pro- files is performed. These are shown in figure 3.4.20, which depicts the flow profiles analyzed starting from a distance equal to 9 D upstream up to 0.5 D upstream (just before the shaft). · pr opeller · pr opeller In the figure only one half of the pipe is shown, for each of the presented profiles the bottom side rep- resents the pipe axis while the top side represents the pipe wall. A red line drawn through all graphs shows the separation between the back-flow and the core flow. One can see that the recirculation reaches far more axial length in the twisted cases than in the case without. This was also visible in the streamlines shown in figure 3.4.19. The first flow profiles near the propeller are different for each

103 Chapter 3 Presentation and discussion of the results of the cases, the designs with pAoA have a very high back-flow. The results reflect one tendency: the higher the twist, the higher the back-flow. But in order to confirm this some more analysis have to be performed. The easiest way to do that is to analyze the flow through the propeller. Obviously, the difference between the flow-rate on the propeller disk and the inflow specified flow-rate is the recirculated flow.

Figure 3.4.21: Surface of evaluation for the through flow-rate

CFD is used to analyze the actual flow-rate through the propeller. To evaluate the flow-rate a circular disk is created in the middle of the propeller in CFD Post r (figure 3.4.21) which is bounded by the maximum propeller radius. The average of the axial flow velocity is evaluated over the shown disk. It is computed using the function massflowAveAbs which is integrated in CFD Post and is recommended for computing averages for recirculating flows. The velocity is multiplied by the disk area to compute the through disk flow-rate.

20 0.33 12 1600 F ] l

n 15 0.25 9 1200 o i w ] m D g - / r l P H a [ s t t m e e [ t P

10 0.17 m 6 800 [ a a [ d r - t ] m s w P 3 o / D l s F 5 0.08 ] 3 400

0 0.00 0 0 D25_AOA_0 D25_AOA_10 D25_AOA_20 D25_AOA_0 D25_AOA_10 D25_AOA_20

Figure 3.4.22: Comparison between the three D25 propellers at the duty flow-rate

The flow-rate induced by the 0° propeller is the highest of all, followed by the 20°. Last produces the highest pressure rise ∆Pst (depicted in figure 3.4.22) and the highest thrust. Actually, the thrust and pressure rise are growing with the p AOA angle. The through flow-rate would be expected to grow likely since it it dependent on the pressure difference between suction- and pressure side, but it does not.

The flow field of the D25 p AOA 20 propeller running at duty point flow-rate is depicted in figure 3.4.23. The complete velocity field around the propeller is shown on the LHS of the figure. There are basi- cally three main flow regions: one near the hub recirculating, the principal one in the flow-direction through the propeller disk and the third one between propeller tip and the tube wall where flow is recirculated. Velocity is projected on the main flow axis so that their magnitude reflects the flow-rate. Regions with high momentum are visible at the tip and in the upper half of the propeller shown on the RHS of figure 3.4.23. The radii where ξ 0.7 are expected to have a higher flow, since the highest > propeller loading is found here. The high recirculation nearby hub (region 1) impacts the blood dam- age in a negative way, as it creates high shears in the flow which can increase the blood damage and it

104 3.4 Design procedure for an encased propeller used as a LVAD

Region 3

Region 1 Region 2

All vectors Suction side Pressure side

Figure 3.4.23: Projected velocity vectors on a middle plane of the D25 p AOA 20 design can impact the structure in a negative way. At this point it is worth to notice that the aimed pressure increase is almost reached only in the case with 20° twist (11.82mmHg compared to 12mmHgof the P14 RCP propeller) and the thrust is below the one of the original P14: 0.63Nto 0.68N.

The second design run is based on the results of the first design iteration and improves some of the disadvantages found there. Two new propellers have been designed (D19 and D21) in this iteration by setting following into ADAP:

• Like in the first design study, the free stream velocity V is computed from an assumed flow- ∞ rate of 17l/min.

• The desired thrust is increased at the duty point is set at 0.9N for D19 and 0.8N for D21.

• Dimensions: D 15mm] (as of the original P14 figure 3.4.1); D /D 0.3 (as of the Ti p = Hub Ti p = reconstructed P14 RCPfigure 3.4.2) ; Chord distribution CoD CoD 2.8 . Air- = N4148,modi f ied · foils: camber-line: NACA 6 with a=0.8, thickness distribution type: NACA 4, radial thickness distribution: N4148.

• Spinning speed: 13000rev/min]. The glycerol mixture is considered for the fluid properties: density: ρ 1106kg/m3 and the dynamic viscosity: µ 3.9mPa s. gl yc = gl yc = · • Prescribed angle of attack (p Ao A) for the blades: 30 ◦.

D25 Airfoils D19 Airfoils

2 2.5

2 1.5

1.5 1 1

0.5 0.5

0 0

-0.5 -0.5

-1

-1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Figure 3.4.24: Comparison of the airfoils between the D25 and D19 designs

Each of the new settings is explained in detail in the following: in the first place the chord length has been increased in order to rise the local Re, as depicted in figure 3.4.25. However, the increase of the chord length creates a problem for the propeller folding. At its maximum the chord length of the D19 is almost double than the one of P14. As a result the Re of the sections shown in figure 3.4.25 exceeds the transition value at the hub (Re 3461) and over some 30% of the radius it reached values over = 10000 (figure 3.4.25). Hence, a higher lift and a decreased drag over the airfoils is insured.

105 Chapter 3 Presentation and discussion of the results

0 . 5 0 1 4 0 0 0 R e P 1 4 R C P 0 . 4 5 C o D P 1 4 R C P 1 2 0 0 0 R e D 2 5 0 . 4 0 C o D D 2 5 R e D 1 9 0 . 3 5 C o D D 1 9 1 0 0 0 0 R e T u r b u l e n t ]

- 0 . 3 0

[ 8 0 0 0

] - D 0 . 2 5 [

o e

C 6 0 0 0 0 . 2 0 R 0 . 1 5 4 0 0 0 0 . 1 0 2 0 0 0 0 . 0 5 0 . 0 0 0 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 ξ [ - ] ξ [ - ] (a) Non-dimensional chord distribution (CoD) (b) Re distribution over the radius

Figure 3.4.25: Radial chord and Re-number distribution for P14, D25 and D19

The increased chord length decreases the cl value because the thrust is proportional to both cl and c (as shown by equation (2.4.24a)), as at equal thrust the product c c is constant. The decreased · l cl demands less cambered airfoils (see equation (2.3.2) and equation (2.3.3)), shown in figure 3.4.24. Because of this they have lower tendency for flow separation, thus improving the hydrodynamic per- formance. The pAoA is further increased in order to further increase the performance which was just above the one of the reference P14 RCP.Also the prescribed thrust is increased from 0.75 to 0.9 (D19) and 0.8 (D21). Actually, the only difference between the two new designs is the designed thrust. Pro- pellers were built in Ansys TurboGrid r and the grids were generated using the medium grid settings (figure 3.4.18). Simulations were performed for the flow range 2 8[l/min]. −

D25 AOA 20°

D19

D21

Figure 3.4.26: 3D Streamlines of the D25, D19 and D21 propeller run at the duty point

The influence of the new propeller designs on the flow is shown in figure 3.4.26: top: D25 AOA 20°, middle: D19 and bottom: D21. The flow-paths are changing from top to bottom: the recirculation region are different from one design to the other and the streamlines of D25 AOA 20° and D19 suggest

106 3.4 Design procedure for an encased propeller used as a LVAD a larger recirculation. Once again the propeller through-flow is analyzed in order to show how much flow is recirculated and how the flow in front of the propeller is swirled (figure 3.4.26). The flow in front of the propeller is pre-swirled as also indicated by the flow-paths in 3.4.26. This improves the performances of the rotor at low flow-rates because it reduces the angle of attack relative to the blade ([71]). Stability of the flow is improved by the pre-swirl as the inlet relative velocity drops, leading on the other hand to a head decrease. Actually the propeller through flow-rate is the one ”seen” by the machine and not the one given as boundary condition. On one hand, the designer has to start the development process of every propeller by approximating the real mass flowing through the propeller, in case of this research 17l/min. On the other hand, the recirculation in the pipe increases the risk of hemolysis, which is favored by a longer resident time discussed extensively in subsection 3.4.5.

25 0.42 15 2000

20 0.33 12 1600 F ] l o n ] i w g D

15 0.25 - 9 1200 m r H / P a l [ s m t t e [ e m P t [ [ t a d a 10 0.17 s 6 800 r ] m P - D 3 w / o s l ] F 5 0.08 3 400

0 0.00 0 0 D25_AOA_20 D19 D21 D25_AOA_20 D19 D21

Figure 3.4.27: Through-flow, thrust and static pressure compared for D25 D19 and D21 at duty flow- rate

The initial analysis is performed for the same parameters as for the D25 propeller study: through-flow, thrust and static pressure (figure 3.4.27). The desired effect of pressure and thrust rise is achieved within this design iteration, the static pressure increase is higher, as a result of the higher thrust. D19 shows the highest pressure increase (13.6mmHg) and a through flow-rate equal to the one of D21 which has 13% less pressure increase. Generally, recirculation rises proportional to the pressure (and thrust), and this is evident while looking at figure 3.4.5. Both D19 and D21 designs have the potential to replace the P14 RCP from the point of view of pressure rise. However, an equal or an even more important role plays the blood damage, since no pump with high hemolysis risk can be used for humans.

F l o w - r a t e [ d m 3 / s ] 0 . 0 3 0 . 0 5 0 . 0 7 0 . 0 8 0 . 1 0 0 . 1 2 0 . 1 3 1 8 2 4 0 0

1 5 2 0 0 0 ] g

∆ H P m s t

m [ [ P

t 1 2 1 6 0 0 a s ] P ∆ P 1 4 R C P 9 D 2 5 1 2 0 0 D 1 9 D 2 1 6 8 0 0 2 3 4 5 6 7 8 F l o w - r a t e [ l / m i n ]

Figure 3.4.28: Pressure flow-rate curves of P14, D19,D21 and D25 (20° twist)

The analysis of the CFD computed ∆Pst is depicted in figure 3.4.28. The previous analysis at the duty flow-rate is confirmed by the simulations over the flow-range: D19 and D21 propeller designs pro- duce more pressure than the P14 RCP.There is an even more positive aspect: the differences between the new designs and the P14 is constant over the whole flow-rate range. In contrast, D25 shows a con-

107 Chapter 3 Presentation and discussion of the results stant pressure increase between 2 and 6 l/min. However, no convergent solution could be achieved after this flow-rate for the D25 design, which might indicate that this design is very unstable and consequently it is unsuitable for the present application.

25 0.42 15 2000

20 0.33 12 1600 F ] l o n i w ] g

15 0.25 9 1200 D m - r / H P l a [ s t m t e e [ t P m [ [ a d 10 0.17 6 800 a r t ] m s - P w 3 / D o s l ] F 5 0.08 3 400

0 0.00 0 0 14F RCP D19 D21 D25 14F RCP D19 D21 D25

Figure 3.4.29: Comparison of the through-flow, thrust and ∆Pst between 14F RCP and new designed propellers at the duty flow-rate (5l/min)

A detailed analysis of all new designs and the original propeller is depicted in figure 3.4.29. As already mentioned, only the thrust of the D25 design is lower than the one of the 14F RCP. ∆Pst is also higher in the case of D19 and D21 than the reference propeller. The flow-rate through propeller resembles the thrust tendency shown in the middle of the figure except the flow of the 14F propeller, which seems to be higher as expected. Again this behavior can be the result of a local flow-reversal or some other flow disturbance in the P 14F RCP case.

Several propeller design have been analyzed so far and in all cases a difference has been shown be- tween the through-propeller disk flow-rate and the pipe flow-rate. ADAP considers only the case of a free stream propeller for the design. Such effects can only be described by experimental and CFD flow analysis. However, the wall interference on propeller performance has been investigated from the very beginning, for example by Glauert [35] who offered a first physical model of the propeller flow in wind tunnel effect. The case of the propeller in wind tunnel effect is treated in the same way (figure 3.4.30) in the fluid mechanics text book of Truckenbrodt [110].

rpipe

rp

v8 v2 v3

Figure 3.4.30: Sketch for the momentum theory applied to a propeller in a pipe (adapted from [110])

As depicted in figure 3.4.30 the flow in the pipe is smooth and free of recirculation. However this is the case for high flow-rates where the flow in the pipe is much higher (similar to flows at high advance ratios J - see section §3.3) . This might be the case when another device generates in-flow in series with the propeller so the high advance ratio corresponds to the case of a free stream flow of an airplane propeller in flight.

Measurements have been performed to find the full operation range of the 14F RCP (depicted in fig- ure 3.4.31). While the operating range without any help device ends for 14F RCP at 17l/min the maximum flow is 31l/min for zero pressure. The flow-rates above 17l/min are only possible with the help of an extra propeller (figure C.0.1) placed in the test rig after the LDA measurement point (see appendix C). In the flow regions above 17l/min the propeller behaves like a usual propulsion

108 3.4 Design procedure for an encased propeller used as a LVAD

F l o w - r a t e [ d m 3 / s ] 0 . 0 0 0 . 0 8 0 . 1 7 0 . 2 5 0 . 3 3 0 . 4 2 0 . 5 0 0 . 5 8 1 0 1 3 3 3

8 1 0 6 7 ] g H ∆ P m 6 8 0 0 s t m [ [ P t s a P ]

∆ 4 5 3 3

2 2 6 7 ∆P P 1 4 R C P w i t h f l o w s t r a i g h t e n e r s t ∆P P 1 4 R C P w i t h f l o w s t r a i g h t e n e r a n d h e l p p u m p s t 0 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 F l o w - r a t e [ l / m i n ]

Figure 3.4.31: Pressure flow-rate measurement results with and without helping device propeller while in the range below 17l/min it is a pumping device thus increasing the pressure from up-to-downstream. This pressure difference leads to a mass flow. In the actuator disk theory one can reverse the problem and instead of considering the thrust T as unknown, it considers it as known. Thus one can compute for the known thrust (T ) and a free stream velocity (v ) the required inlet ∞ diameter of the slipstream or for a given diameter the required free stream velocity.

rpipe

rp

v8 v2 v3

Figure 3.4.32: Sketch of a propeller slipstream with inlet radius equal to the pipe’s radius

The propeller slipstream for such a theoretical case is depicted in figure 3.4.32. In order to determine the slipstream’s inlet velocity v one first considers the continuity equation applied between the pro- ∞ peller’s disk surface and the inlet surface which is given by the pipe’s diameter:

v Api pe (v v) Ap (3.4.2) ∞ · = ∞ + · and now equating for v:

2 ·µrpi pe ¶ ¸ v v 1 (3.4.3) = ∞ · rp − which can be introduced in the expression of thrust (equation (2.4.2)) leading to:

·µ ¶2 ¸ µ ·µ ¶2 ¸¶ 2 rpi pe rpi pe T 2 ρ Ap v 1 1 1 (3.4.4) = · · · ∞ · rp − · + rp −

For the P14 propeller case the values for thrust and flow-rate read T 0.68N respectively 5l/min and = the inlet velocity v 0.17m/s (figure 3.4.14a). For this velocity and thrust the slipstream inlet can ∞ =

109 Chapter 3 Presentation and discussion of the results be computed by rearranging the terms in equation (3.4.4) resulting a radius r 0.024[m] which is in = almost double than then pipe radius r 0.0125m. Computing the flow-rate required for this case p = by multiplying the area with the velocity results in V˙ 18.402l/min, which is more than three times = higher than the original flow-rate. The CFD result for the through disk flow-rate (figure 3.4.29) is V˙ 24l/min and that is more than four times higher than the initial flow in the pipe. If the CFD = pipe diameter is considered as a boundary condition and a velocity is calculated based on the latter value, than this is v 0.815m/s. This is roughly five times higher than the original velocity given by ∞ = the flow-rate and it impacts the design of the propeller because according to figure 2.4.6 this directly affects the flow kinematics of the blade sections.

Instead of investigating different designs (or for example different pAOA) it is possible to roughly es- timate the through-propeller disk flow by using the above described method (equation (3.4.4)). This method is not considering all the flow physics as the CFD method, but it offers the possibility to find a closer estimation of the through flow rate without spending too much computational time.

3.4.4 Validation of new designs by test-rig measurements

The validation of the propeller designs by experimental measurements is presented in this subsec- tion. Prototypes were build by using the stereolitography procedure described in subsection 3.4.1 by the company VISIOTECH GmbH12. The first step in building the propellers was to design them in the parametric design software Pro-Engineer-Wildfire 3.0. The coordinates of each radial section are written (exported) from the design software directly as .ibl files to be readable by the design software (figure 3.2.22).

CFD Model CAD Model Prototype

Figure 3.4.33: Fast product development showing from left to right: CFD model, CAD model and ready to test prototype (D19)

Airfoils are generated by connecting the points imported from the .ibl files in splines, which are then used to generate the surfaces of the blade. Surfaces are merged in a closed surface which is filled to a solid. The result is a volume 3D-model (middle of figure 3.4.33) to which a hub is added. This model is exported to a .stl file, which is needed for the stereolitography manufacturing process. In the present study the validation has been performed for the last designs: D19 and D21.

Measurements are performed under the same conditions as in the case of the re-engineered P14 RCP (in subsection 3.4.1) on the loop test rig described in section §2.5 by using the 40% glycerol-water mixture. Three measurements for each propeller are performed and an average is computed. Com- parison to the CFD results is depicted in figure 3.4.34 and shows an acceptable agreement between

12http://www.visiotech-gmbh.de

110 3.4 Design procedure for an encased propeller used as a LVAD them. D19 presents a difference of 12% at the working point and is plotted with y-error bars sym- bolizing the standard deviation. The absolute difference is 1.6mmHg and rises to 2.2mmHg at the minimum flow-rate. Similar studies performed with axial-propeller pumps in same conditions show differences up to 20% (in absolute values 3 to 10mmHg, [103]). The second propeller, D21 shows a difference of 17% at the working point (1.95mmHg), which, the same as in the case of D19, increases as the flow-rate is decreased. CFD predicts a linear pressure increase along the complete flow-rate line which does not correspond well with the experiments. The results obtained for D19 are well re- produced for high flow-rates but become worst at low ones. Although D21 conserves the pressure curve shape from the simulation, it delivers even lower performance than the baseline. D19 however, shows in the experiments the potential of producing more pressure than the original design. At the design flow-rate the propeller produces more than 0.5 mmHg compared to the baseline design, which means approximately 5% more pressure. D21 is 0.2mmHg below the baseline. However, in its case the measurement shows a decreased spreading of results as shown by the error bars in figure 3.4.34.

F l o w - r a t e [ d m 3 / s ] 0 . 0 3 0 . 0 5 0 . 0 7 0 . 0 8 0 . 1 0 0 . 1 2 0 . 1 3 1 8 2 4 0 0 ∆P C F D P 1 4 R C P s t ∆P C F D D 1 9 s t ∆P C F D D 2 1 s t

∆ ∆ ] 1 5 P E X P P 1 4 R C P 2 0 0 0 s t g ∆P E X P D 1 9 P s H s t t

∆ [

m P E X P D 2 1 s t P a m [ 1 2 1 6 0 0 ]

t s P ∆ 9 1 2 0 0

6 8 0 0 2 3 4 5 6 7 8 F l o w - r a t e [ l / m i n ]

Figure 3.4.34: Validation of the CFD simulation by measurements on the test rig

The stereolitography manufacturing process might have led to differences between the CAD model and the prototype, in particular in the case of high curved geometries of the new designed propellers. Differences arising for the D19 and D21 designs are most probably caused by the bending of the pro- peller. While structural weakness lead to the addition of a fillet at the blade’s foot (figure 3.4.2) of the P14 RCP for the new designs this was not added. High bending of all prototypes was noticed during experiments which was also caused by a decrease in the section thicknesses. Thus the stiffness of the blade is decreased and without the fillet it leads to deformation and ultimately to rupture. On the other side the 14F RCP has constant chord and constant thickness.

The CFD results over-predict, as expected, the experimental measured performance. In the duty- point the relative differences between CFD measurement was 8% in the D19 case 3.4% in the P14 case and 11% in the D21 case. There are though two points which should be discussed here: the dif- ferences, which are not equal between the CFD of the designs and the experiment, and the relatively large deviation shown for the P14 RCP case. If, for example the lower values of the P 14F RCP mea- surements are considered the difference between CFD and experiment is 6%. This value is closer to the 8% in the case of the D19. Also in this case the value would be very close to the one of the D21 measurement. The deviations in the case of the new designs show a very reduced deviation (2% - 5%) compared to 10% in the case of the P 14F RCP. One reason for the high deviation in the case of the P 14F RCP could be the instability of the flow induced. This would require a much deeper ex- perimental investigation which would exceed the frame of the present thesis. On the other hand the prototype models could be improved by adding fillets to increase their structural stiffness and thus the measurement reliability.

111 Chapter 3 Presentation and discussion of the results

3.4.5 Numerical estimations of blood damage

The concept of hemolysis evaluation by using Eularian CFD methods was introduced in chapter 2. In previous subsection new propeller models were validated by experimental measurements and they can be used for simulations which will include the evaluation of hemolysis. Simulations setup is changed by removing the stator domain since this influences the flow and particularly the stresses. All other grids are preserved and the outlet grid is translated and attached to the propeller domain. In order to simulate the blood cells and to evaluate the exposure time and the resulting blood damaging index, the setup is changed from one phase simulation to a two phase simulation. First phase is the already defined liquid phase (glycerol mixture) while the second phase is defined by solid particles injected at the inlet boundary. The main settings are summarized in table 3.4.4. Taskin et al. [98] and Fraser et al. [32] did not use particle tracking in the Lagrange analysis, but preferred to track path- lines of fluid and simulated only one phase in contrast to [94, 105, 111]. Last set a constant number of particles at the inlet of the domain.

Table 3.4.4: Properties of the solid phase (RBC) in CFX, mass flow computed according to [119]

Property Value Material Solid Molar mass 67000g/mol Density 1094.1kg/m3 Reference temperature 21◦ (C) Mass and Momentum of particles zero slip velocity Particle mass flow 0.01275kg/s Particle diameter 0.0075mm Multiphase particle model Euler-Euler

In the present study a particle mass flow was specified at the inlet of the domain (3.4.4). This has been computed from the average RBC concentration in blood (table B.0.1), which was used to deter- mine their concentration. Then, by knowing the flow-rate of blood, the mass-flow of particles can be computed by assuming an uniform particle distribution in the fluid (see table B.0.1 ).

The fluid phase used the same properties of the glycerol-water mixture from the previous simulations. Steady state simulation were performed by using Ansys CFXr. Results were post-processed by CFD Post r. BDI is computed as desribed in section §2.4 in equation (2.1.3) from σ in equation (2.1.2). In the evaluation of equation (2.1.2) only the viscous stresses are evaluated, because if accounting also Reynolds stresses one would introduce a fictive shear force in the evaluation. The Reynolds stresses are artificially created while statistically decomposing the NSE, as it was shown in subsection 3.2.1 and equation (2.2.12). That means they would overestimate the hemolysis as already shown in [34] and recently used in [98]. The particles are tracked on their path-lines through the flow field from the inlet to the outlet. Lagrangian track is used to account for the stress exposure of the particles. The computation of the BDI has been performed by an excel macro which reads the data exported by CFX-Post in a .csv data file.

In figure 3.4.35 the results of the BDI computations for all investigated pumps are presented. Results are distributed in index groups for each model. Most of the particles show the lowest BDI (up to 0.025%) while the maximum one is reached by the D25 propeller design at 0.125%.

However, it is clear that the reference pump (14F-RCP) shows the lowest BDI while D25 shows the highest one. This is shown by both distributed BDI (figure 3.4.35) and averaged BDI (shown in fig- ure 3.4.36). The order of magnitude of the results is the same as the one computed for centrifugal blood pumps (CentriMag pumps) in n Taskin et al. [98] using the same BDI formulation and coeffi- cients.

112 3.4 Design procedure for an encased propeller used as a LVAD

100% 90% P14

80% D19 70% D21 60% D25 50% 40%

Percentage of particles of Percentage 30% 20% 10% 0% 0.00000 0.00025 0.00050 0.00075 0.00100 0.00125 Blood damage index, BDI(%) ,

Figure 3.4.35: Computed blood damage index (BDI) for all investigated designs

0.00025

0.00020

0.00015

0.00010 Average BDI BDI [%] Average

0.00005

0.00000 P14 D19 D21 D25

Figure 3.4.36: Average BDI for all investigated designs

Untaroiu [111] found a value of about 0.0015% for axial VADs13 and Fraser et al. [32], who used the Eu- lerian implementation from [98] found for similar ”screw” axial pumps (14) values between 0.0004% and 0.000062%. BDI values computed in the present research are in-between the values mentioned previously in [32]. However, in the mentioned study the coefficients of the power law model have been corrected with the help of measurements results. In [98] was shown that BDI calculated by using the Eularian model correlates very well with the one of the Lagrangian one while using the HO constants. In this way it is possible to compare the results obtained in this thesis by Lagrangian methods with the ones in [32] which uses the Eularian method. The tendency shown in figure 3.4.35 is reflected

13here are meant axial levitated VADs - screw type 14model is not specified in [32]

113 Chapter 3 Presentation and discussion of the results by the average BDI results: P14 shows the lowest BDI followed by the D21 design and then by D19 and D25. Now it is useful to disseminate further these results in order to create links between the hydrodynamic design and blood damage.

0.8

0.7

0.6

0.5

0.4

0.3 Average exposure time [s] exposure Average 0.2

0.1

0 P14 RCP D19 D21 D25

Figure 3.4.37: Average exposure time of the investigated pumps

By comparing the flow-rates evaluated through the propeller disk (figure 3.4.29) and averaged expo- sure time values (figure 3.4.37) one can recognize that the differences between the designs show the same tendency. This means that particles in a higher recirculated flow travel longer than the ones in a lower recirculated flow. Since the exposure time is part of BDI this is the first link between pro- peller design, resulting hydrodynamics and the hemolysis. It was shown in the previous subsection that high pressure mostly induces higher through-propeller-flow (and recirculation). Optimal design should therefore provide for low stress that a particle is experiencing.

The average exposure time (shown in figure 3.4.37) of the P14 RCP model is higher than the one of the other models, which contradicts the results in figure 3.4.36 indicating this pump has the lowest BDI. In the same way the other investigated pumps show higher BDIs but lower exposure times. An answer to this apparent contradiction is searched for in order to formulate a conclusion about the blood damage caused by the axial propeller-pumps. An analysis of the stress distribution in the simulated domain has to be performed with a method also used by Fraser et al. [32] to explain the results in the respective study. Stresses are divided in 5 major groups, each of them being named according to the level of stresses (from small to very high according to Fraser et al. [32]- figure 3.4.38). The ”small” stresses are the ones considered to have a small influence in the hemolysis so this is why their ”amount” has to be as high as possible. The very high stresses (over 150[Pa]) are responsible for the so called ”fast hemolysis”. The reference pump, P 14F RCP produces the highest amount of ”small” stresses (63%) as depicted in figure 3.4.38, while D25 produces the smallest amount of small stresses.

The difference is filled up by the high, higher and very high stresses as it can be seen also from the figure. Although D25 has the smallest amount of volume at the very high stresses from all investigated pumps (0.003% compared to the 0.005% of P14) the difference for the medium stresses is much higher and this is decisive for hemolysis. The same is valid for D19 and D21.

Actually, from the results presented in figure 3.4.38 for the medium stresses, one deduces that their influence on the final BDI is the highest from all other stress categories. The tendency of the medium stresses corresponds again exactly to the one of the average BDI shown in figure 3.4.36. A critical high

114 3.4 Design procedure for an encased propeller used as a LVAD

70%

60% P14

50% D19 40% D21 D25

30% Fluid volume [%] volume Fluid 20%

10%

0% <1 Pa (Small) 1-9 Pa 9-50 Pa (high) 50-150 Pa >150 Pa (very (medium) (higher) high)

Figure 3.4.38: Stress (σ) volume distribution for all pump cases value for the σ was set in [31] at a value of 150Pa (very high stresses). Other studies [104] suggest a critical σ value of 250Pa. For relative comparisons, as the ones in the present study the value does not play such an important role as the differences arising between the different designs.

P14 RCP D19

D21 D25

Figure 3.4.39: Graphical plot of volumes with values of σ 150Pa >

Critical σ of all investigated propellers are plotted in figure 3.4.39. First one notices that in all studied cases the critical stresses appear near the surface of the propeller. This is natural since stresses are coupled to the velocity gradients, which are higher in the vicinity of the rotating body. However, the way the flow is directed along the blade surface plays an important role in the stress distribution, as it appears from the plot of the different analyzed propellers. P 14F RCP seems to have the thickest layer of very high stresses around the blade which is confirmed by the numerical values and which is caused by flow separation shown in figure 3.4.40. The flow separates just after the leading edge and does not re-attach to the blade. As a consequence the velocity field is strongly turbulent and

115 Chapter 3 Presentation and discussion of the results thus fluctuates resulting in high gradients and in high stresses. The surface streams demonstrate actually very good, within the known limitations of numerical accuracy, the correlation between flow separation and high stresses. The surface stresses which are plotted in figure 3.4.41 show that the stresses caused by the separation directly increase the stress on the blade.

P14 RCP D19

D21 D25

Figure 3.4.40: Surface streamlines on the propeller blade

The original P 14F RCP propeller experiences the highest surface stresses when compared to the new designs as depicted in figure 3.4.41. In fact, the results correlate with the total volume of high stresses depicted in figure 3.4.38. The high stresses at the leading edge of P14 are a result of its thickness; the new propellers have much thinner airfoils which are better suited to ”cut” the flow. However, on the new blades the flow fails to re-attach. Figure 3.4.41 shows that stresses on the new designs are lower in both intensity and distribution than on the P 14F RCP.Actually, the high stresses on the surface of the blade might not play such an important role as in the pumps where the tip gap is very small.

In figure 3.4.42 the stresses of the fluid in the range between 50 and 150Pa are shown for all investi- gated propellers. The diagram in the middle shows the distribution of volume in percent of the total volume. The D25 design produces a volume almost double than the one of the P14 RCP while the other two designs are in-between. In the pictures of the flow one can also identify the higher volume around the D25 design.

The distribution shows the shape of the tip vortex in all cases, which is recognizable by high stress volumes. The tip vortex is a region where two relative different energy layers meet: suction and pres- sure side, and is in consequence a region where high levels of energy dissipation are expected. As a consequence to this, the shear is high and thus the stresses. On the other hand the investigation of the new designs shows a high trailing edge vorticity which has its origin in the separation on the blade. If this can be improved the levels of BDI can be reduced dramatically.

This subsection has investigated the numerical blood damage produced by encased propeller pumps. Moreover, with the help of statistical analysis and Eularian investigation of the flow it was possible to connect high level stresses to flow phenomena like separation, transition and reattachment. BDI

116 3.4 Design procedure for an encased propeller used as a LVAD

P14 RCP D19

D21 D25

Figure 3.4.41: Blade surface analysis of stresses

P14 RCP D19

0.050% P14 D19

0.040% D21 D25

0.030%

D21 0.020% D25

0.010%

0.000% 50-150 Pa (higher)

Figure 3.4.42: Stresses in the range 50 150Pa (medium stresses) − could be connected, based on CFD simulations results, to the recirculated flow. Finally all phenom- ena are linked the geometry of the blade. Thus, the result of this investigation is exactly this link between the geometry, stresses and BDI. This can be resumed to the following statements:

117 Chapter 3 Presentation and discussion of the results

• blade areas with high σ do not necessary imply high BDI

• highly cambered airfoils of the propeller might lead to high σ and a result to high BDI

• thin airfoils avoid big surfaces having high stresses

• thick airfoils (with thick LE) avoid better the separation as thin airfoils and the stress over that surface

• propellers with high thrust (and pressure) lead to high recirculation which means higher resi- dent time and thus higher σ

3.4.6 Time-dependent CFD simulations of encased propeller VADs

It was shown up to this section that performance of encased propeller pumps can be assessed using CFD. These simulations have been performed by assuming stationary flow conditions. However, the blood flow in the human body is not stationary, but time dependent, as induced by the human heart. Therefore the working point of the pump is permanently switched from the duty point to off-design operating points. To evaluate the performances in this case, the boundary conditions have to be changed from stationary to transient. It has to be mentioned that transient simulations are more expensive in terms of CPU time and storage capacity than steady ones and as a result they are used only in the last stages of product development. The simulations have been performed for the original P14 RCP pump and the new D19 design.

Boundary conditions require knowledge of both pressure and flow-rate, as for the stationary case (table 3.4.2). They were derived from measurements performed on the MCL (section §2.6) which was set for patient conditions with AHFS [75]. Pressure measurements were always in the format presented in figure 2.6.2 .

Average flow-rate for AHFS was set in the MCL at 4l/min and the AoP at 50mm Hg according to table 2.1.1. Time dependent signals were recorded with the help of the recording software described in section §2.6 on the MCL test rig. They were transformed into Fourier series with 3 coefficients for the flow-rate and 10 coefficients for the pressure, as depicted in figure 3.4.43. An advantage of the Fourier-fitted signals is that they are automatically filtered depending on the amount of coefficients chosen. The time period of one pulsation was T 0.85s. Flow-rate was varying between 4.46l/min = and 3.57l/min and the pressure between 68.88mm Hg and 31.178mm Hg (figure 3.4.43).

80 5.0 10667 0.08 P in V 60 4.5 8000 out 0.07 V o u t V ] ] [ a o d g u P m t H [ [

n 5333

40 i 0.07

4.0 l 3 m / / P m s m ] [ i n n i ] P 20 P 3.5 2667 0.06 in V out 0 3.0 0 0.05 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 t [s] t [s]

Figure 3.4.43: Transient flow-rate and pressure used as boundary conditions in CFX (LHS: medical units;RHS: SI units)

The Fourier series were programmed in CFX as a CCL-user-expression dependent on the simulation time. Simulation’s time step was set according to the time period of the blood pulsation: 1/40 T . How- · ever, the time-step is very large compared to the rotational speed of the pump: 216.7rev/s, which means that in every time-step of the simulation the pump rotates 4.6 times. From this point of view

118 3.4 Design procedure for an encased propeller used as a LVAD the simulation is ”quasi-transient” since the whole transient effect of the pump is not caught. A sim- ulation, which would catch all transient effects of the pump, would need a time-step representing at least 1/120 1rev/min resulting in a very small time-step and it is not necessary for assessing the · macroscopic pump performance. In the simulation the interface between rotating and steady part was changed from ”Frozen Rotor” to the ”Transient Rotor Stator”. A total time of 4 T was set for · the simulation with a steady state starting solution from which only the last two periods were evalu- ated. At the inlet was set the time-dependent pressure boundary condition and at outlet the flow-rate boundary condition (figure 3.4.43). Results of the simulations are presented in figure 3.4.44. The shown pressure curves are measured at the outlet of the domain and correspond to p3 in the mea- surements. It should, however, be noticed that in the measurements that include the propeller the pressure increase has to be computed as: ∆Ppr op p3 p2 or to be converted only in pressure at p3; ¡ ¢ = − p3 p3pr op p2no pr op p2pr op . This has to be done in order to add to the pressure increase of the = + − − propeller (from the pressure side) the pressure decrease generated on the suction side of the propeller. In the simulation the pressure on the suction side is fixed so the complete work done by the propeller is seen only on the pressure side. Independent of the flow-rate, the pumps running at 13000rev/m show a constant increase of pressure. This is about 19mmHg at the peak and about 18mmHg at the lowest point. The average pressure is increased at 61.1mmHg by P14RCP and 64.8mmHg by D19. In the case without pump the average computed pressure at outlet is 47.4mmHg. The average pressure increase by using the P14RCP is 13.7mmHg respectively 17.4mmHg by using D19. This values are much higher than the ones shown in figure 3.4.28 for the steady state case.

1 0 0 1 3 3 3 3 9 0 1 2 0 0 0 8 0 1 0 6 6 7 7 0 9 3 3 3 ]

g 6 0 8 0 0 0 P H o u

5 0 6 6 6 7 t m

[ P m [ 4 0 5 3 3 3 a

t ] u o 3 0 4 0 0 0 P P o u t w / o p u m p ( C F D ) 2 0 P o u t 1 4 R C P ( C F D ) 2 6 6 7 1 0 P o u t D 1 9 ( C F D ) 1 3 3 3 P o u t 1 4 R C P ( E X P ) 0 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 T i m e [ s ]

Figure 3.4.44: Transient static pressure measured at the outlet of the domain

Figure 3.4.44 depicts also the comparison to the time-dependent measurement of the original P 14F RCP prototype ran with cage in the MCL. The agreement between the measurement and simulation is very good even if the geometries are not exactly identical and in the experiment the pump had the protection cage.

Table 3.4.5: Comparison between the pressure increase in the steady and unsteady simulation

Steady state pressure increase Unsteady average pressure increase P14RCP 12.88mmHg 13.7mmHg D19 14.6mmHg 17.4mmHg ∆(D19 P14) 1.72mmHg 3.7mmHg − Values from the steady state simulations (figure 3.4.28-4l/min) are presented in comparison with average values of the unsteady simulation in table 3.4.5. The difference between the pressure increase of the two pumps is almost double in the unsteady simulation than in the steady state simulation. It might be assumed that the pumps are running at a much lower flow-rate within the pulsation and so they produce a higher pressure increase.

119 Chapter 3 Presentation and discussion of the results

This chapter has presented the application of the propeller design program ADAP on two cases: a multiblade propeller and an encased propeller pump used as VAD. In the first case the developed CVL method has been used to increase the thrust of a marine propeller at the same or greater efficiency. The VAD propeller-pump has used the experience gained for designing the marine propellers and extended the theory for small Reynolds number devices. This however, have particularities because of the way they are placed inside the human body. A simplified model for analysis of the back-flow has been developed and successfully validated for VAD propeller-pumps. Further on, it has been shown, that correlations exist between geometry and the predicted blood damage, and a first step has been done in order to establish them. Time-dependent simulations confirmed the steady state ones. They have shown that propeller with improved pressure rise can perform even better in-vivo.

120 Chapter 4

Conclusion and Outlook

The scope of this works was the development and optimization of propellers used as LVAD. An inverse propeller design method was applied for the first time for the design of LVAD. The design frame- work works in combination with 3D CFD being ultimately a combined direct-inverse design method. As a result the D19 design showed improvements in the pressure rise at all investigated flow-rates (2 8l/min) when compared to the baseline P14F RCP. This benefit was validated experimentally − on the loop test rig developed for this purpose at LSTM. Time-dependent simulations confirmed the pressure increase of the pump which exceeded the stationary measurements. BDI of D19 is double when compared to the original P14F RCP,but still remaining between literature values while the ex- posure time of the particles is decreased.

For the first time were optimized open water low-solidity propellers by using a novel method for com- puting the aerodynamics of airfoils cascades. This method applied on the design of propellers fills a gap in their design methodology and has shown a thrust improvement of 3.6% at a higher efficiency for the same advance ratio. The method is based on the cascade vortex-lattice numerical method for the computation of thin airfoil cascades, which was validated against the Joukowsky thin airfoil and later against the Weinig-diagram for flat plate cascades. Calculations showed that high cambered airfoil show higher differences between the lift values alone and in cascades. This is an important result for the design of low-solidity propellers as well as for axial fans or compressors. CVL was im- plemented in the ADAP framework for the design and analysis of propellers and was validated on a high advance propeller case in section §3.3. The design method for propellers is based on the classi- cal Betz/Prandtl method which is improved by an iterative wake alignment method and by using the radial momentum loss of Goldstein. Its implementation, based on an empirical formulation found in the literature was successfully validated against analytical values. A comparison to the results pro- vided by state-of-the-art vortex-lattice methods has also been shown to further prove the validity of the code. The validation of the design code was performed on a typical case of open water marine propellers having sufficient large dimensions so that the Re-number of the sections was higher than 100000. It was shown, that for the same mean-line different thickness distributions can lead to dif- ferent results. NACA 4 thickness was preferred in this case for propeller design because it has shown better stability at higher advance ratios than the NACA 65A thickness.

The investigation of LVAD propeller has shown that they cannot be designed fully-inverse in a straight- forward manner as the open water ones. This is caused mainly by the reversed flow coming from the propeller downstream to upstream. It was shown that the flow downstream transfers the angular momentum to the unswirled flow, thus creating a pre-swirl in front of the propeller. This reduces the angle-of-attack of the respective section, which in consequence reduces the risk of flow sepa- ration helping the propeller to perform stable over a larger range of flow-rates. A step forward in accelerating the design process of LVAD - propellers was accomplished by roughly approximating the through-flow of the encased propeller in subsection 3.4.3.

It was shown for the first time, how exposure time and blade stresses can be related to the design characteristics of propellers and also how the performance in terms of pressure rise correlate to the

121 Chapter 4 Conclusion and Outlook latter. The exposure time is proportional to the recirculated flow and thus to the thrust of the pro- peller, while the peak of the stresses are not always correlating with the highest BDI. Basically both the residence time and shear stresses have a similar relevance on the BDI. The analysis of several designs has shown that design producing low stress levels in the fluid have also the lowest BDI risk. Keeping a large volume of stresses at the lowest level is one of the key factors in designing low BDI devices. Results of the BDI for the investigated propellers show this is connected through the shear stresses to the airfoil camber and chord length. Higher camber as well as short chord increase the potential of flow separation and thus the risk of high shear stresses. In the design of D19 was shown that by combining an optimal design thrust with an appropriate chord and camber, pressure rise can be improved without affecting the blood damage properties.

Fast design and analysis of propellers is achieved by using the newly developed ADAP code. However, its analysis accuracy can be improved by using a thick airfoil analysis method like the one presented in [52]. By adopting a boundary layer prediction method as presented in the Open Source codes Xfoil or Panda the accuracy of the cascade method could be further improved while further maintaining the speed advantage over full-3D-CFD methods. An extension of the code for the design of axial fans and compressors is possible as it has been shown in Miclea-Bleiziffer et al. [60]. By using conformal methods [51] CVL can be adapted to simulate radial cascades which would allow its use for the inverse design of radial fans and compressors.

Further improvements for the design of LVAD propellers can be achieved by implementing a reliable prediction method for the propeller through-flow by using a more detailed axial and angular momen- tum computation. In this work the flow physics for encased propellers was calculated by stationary CFD and validated only by integral measurements. Future studies should concentrate on experimen- tal investigation of the flow for encased propellers and also more detailed instationary CFD simu- lations. These can be used for validating analytical theories developed for the computation of the propeller through-flow.

Accelerated hydrodynamic improvement of LVAD propeller designs can be achieved by implement- ing ADAP in a genetic algorithm optimization framework such as for example ModeFrontier. Such frameworks enable the multiple object optimization needed in the design of blood pumps: low BDI, low exposure time for RBC and high pressure increase.

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129

Appendix A

Derivation of the scalar shear stress σ from the Navier Stokes equations

Continuing from equation (2.2.4) the viscous stress-tensor can be resumed:

  τxx τx y τxz τ τyx τy y τyz  (A.0.1) = τzx τzy τzz

The components of the tensor’s trace τii (main diagonal) are known as normal stresses, the other ones are called the shear stresses. The shear stresses are usually denoted in a general form as τi j which means by convention that the particular stress affects a plane perpendicular to the i-axis, in the direction of j-axis (Blazek [12]). This is better viewed in figure A.0.1.

Z t zz

t zy t zx

t yz

t yy t Y yx

t xz

t xy t xx X

Figure A.0.1: Fluid volume with figured stresses

The fluid stresses are dependent on the dynamical properties of the medium. For general fluids like air or water, Newton stated that the shear-stresses are proportional to the velocity gradient. This is stated in equation (2.2.5) Therefore these fluids are called Newtonian fluids. Hence, the components of the stress-tensor (Durst [24], Blazek [12]) can be written:

Normal stresses:

µ∂u ∂v ∂w ¶ ∂u τxx λ 2µ (A.0.2a) = ∂x + ∂y + ∂z + ∂x

131 Appendix A Derivation of the scalar shear stress σ from the Navier Stokes equations

µ∂u ∂v ∂w ¶ ∂v τy y λ 2µ (A.0.2b) = ∂x + ∂y + ∂z + ∂y

µ∂u ∂v ∂w ¶ ∂w τzz λ 2µ (A.0.2c) = ∂x + ∂y + ∂z + ∂z

Shear stresses:

µ∂u ∂v ¶ τx y τyx µ (A.0.3a) = = ∂y + ∂x

µ∂u ∂w ¶ τxz τzx µ (A.0.3b) = = ∂z + ∂x

µ∂v ∂w ¶ τyz τzy µ (A.0.3c) = = ∂z + ∂y

In which l represents the second viscosity coefficient and µ represents the dynamic viscosity which is related to the kinematic viscosity by:

µ υ (A.0.4) = ρ

For closing the expressions of the normal stresses, Stokes introduced the hypothesis:

2 λ µ 0 (A.0.5) + 3 =

However this is only needed for the closing the expressions for compressible flow of gases. In the incompressible case, this leads to:

∂u ∂u τxx λ(di v ~v) 2µ 2µ (A.0.6a) = + ∂x = ∂x

∂v ∂v τy y λ(di v ~v) 2µ 2µ (A.0.6b) = + ∂y = ∂y

∂w ∂w τzz λ(di v ~v) 2µ 2µ (A.0.6c) = + ∂z = ∂z

Eq.A.0.1And hence the stress tensor:

 ∂u 1 ³ ∂u ∂v ´ 1 ³ ∂u ∂w ´ ∂x 2 ∂y ∂x 2 ∂z ∂x  ³ ´ + ³ + ´ τ 2µ 1 ∂u ∂v ∂v 1 ∂v ∂w  (A.0.7) =  2 ∂y + ∂x ∂y 2 ∂z + ∂y   1 ³ ∂u ∂w ´ 1 ³ ∂v ∂w ´ ∂w  2 ∂z + ∂x 2 ∂z + ∂y ∂z or:

τ 2µD (A.0.8) =

132 In which D represents the deformation tensor. The shear terms τi j can be expressed upon the rate of strain tensor (ei j ):

τ 2µe (A.0.9) i j = i j where the rate of strain tensor is given by:

µ ¶ 1 ∂ui ∂u j ei j (A.0.10) = 2 ∂x j + ∂xi

The deformation tensor is therefore a sum of the normal-stress tensor and the rate of strain tensor (or invariant tensor). This is a very important consideration for the later use of these stresses in the interpretation of the blood damage disseminated in section 3.4.3 and the effect of each of them upon it.

The comparative shear stress are evaluated in a scalar function (or scalar shear stresses -[13]):

· ¸ 1 1 X¡ ¢2 X 2 2 σ τii τj j τ (A.0.11) = 6 − + i j which in cartesian components can be decomposed and written:

1 ·1 ³ ´ ³ ´¸ 2 σ 2 τ2 τ2 τ2 2τ2 τ2 2τ2 τ2 2τ2 τ2 2 τ2 τ2 τ2 (A.0.12) = 6 · · xx + y y + zz − xx y y − xx zz − y y zz + · x y + xz + yz

The stresses in the above equation are programmed as an expression in CFD Post r and evaluated in the computation of blood damage.

133

Appendix B Computation of RBC mass-flow

One RBC weights about 3E-11g. The concentration of RBC in blood ist between 4800000 and 5400000 per µl. From here one can compute the RBC mass per litre, and further by using the average blood flow-rate in human body (5l/min) the RBC mass per minute can be computed. This is averaged from the two values.

Table B.0.1: Computation of RBC mass-flow

min max Units Weight of one RBC - 3E 11 g − Concentration 4800000 5400000 RBC/µl Mass per l 144 162 g/l Mass per minute 720 810 g/min Mass per second 12 13.5 g/s 0.012 0.135 kg/s Average - 0.01275 kg/s

135

Appendix C Propeller helping device

In figure C.0.1 is shown the helping device used in the loop test rig. This was a standard brush motor for ship models, Graupner r MULTISpeed 280. It was installed in the loop after the glass pipe in the corner, as shown in figure C.0.1 .

Figure C.0.1: LHS: Graupner rMultiSpeed 280, RHS: Placement of the helping device

137