Computation of Viscous Incompressible Flows

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1 Computation of Viscous Incompressible Flows

2 Scientific Computation Editorial Board J.-J. Chattot, Davis, CA, USA P. Colella, Berkeley, CA, USA R. Glowinski, Houston, TX, USA Y. Hussaini, Tallahassee, FL, USA P. Joly, Le Chesnay, France J.E. Marsden, Pasadena, CA, USA D.I. Meiron, Pasadena, CA, USA O. Pironneau, Paris, France A. Quarteroni, Lausanne, Switzerland and Politecnico of Milan, Milan, Italy J. Rappaz, Lausanne, Switzerland R.Rosner,Chicago,IL,USA P. Sagaut, Paris, France J.H. Seinfeld, Pasadena, CA, USA A. Szepessy, Stockholm, Sweden E. Weinan, Princeton, NJ, USA M.F. Wheeler, Austin, TX, USA For further volumes:

3 Dochan Kwak Cetin C. Kiris Computation of Viscous Incompressible Flows 123

4 Dochan Kwak NASA Ames Research Center NASA Advanced Supercomputing Division Mail Stop Bldg Moffet Field USA Cetin C. Kiris NASA Ames Research Center Applied Modeling & Simulations Branch Mail Stop Bldg Moffett Field USA ISSN ISBN e-isbn DOI / Springer Dordrecht Heidelberg London New York 2011 to the complete printed work by Springer, except as noted. The U.S. Government retains a nonexclusive and nontransferable license to all exclusive rights provided by copyright. Any opinions, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of their respective employers. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 To my family: Soonup Kwak Sally, Nancy & Brian, Lawrence and our granddaughter Subin

7 To my family: Cahide Kiris Eren

9 Foreword Numerical Simulation of incompressible flows has become an essential tool for studying many important problems in science and engineering, thanks to advances both in numerical methods and computer technology. In life and earth sciences, complicated flow phenomena can be simulated today because of such sophisticated tools. For example, in biology the circulation of blood in the human heart and brain, air in the lungs, and urine in kidneys is the subject of many extensive studies. Similarly, simulations of oil well and oil field flows provide critical information to geologists. Ocean circulation and weather prediction are among the fields that have become dependent on computer simulations. Engineering applications of internal and external incompressible flows are plentiful, including laminar and turbulent flows of pipes, pumps and turbines, hydrofoils, and flow around ships and submarines. With faster and more powerful computers available every year, scientists and engineers are running numerical simulations of highly sophisticated problems and developing efficient numerical methods. To handle complex geometry, overset grids have proven to be of practical use. Higher order upwinding schemes are used for high Reynolds number flows, and approximate (LU) factorization methods and/or relaxation schemes can be used for both structured and unstructured grids. With these advances, together with enhanced turbulence modeling (algebraic, oneand two-equation models), commercial software today is being applied to a wide spectrum of flow simulation problems. Historically, numerical simulations of compressible and incompressible flows were based on two different mathematical formulations. For compressible flows, the density and velocity components are updated using the continuity and momentum equations, respectively, and the pressure is calculated from the energy equation together with the equation of state. On the other hand, incompressible flow calculations, where density is constant, are usually based on artificial compressibility or pressure correction methods. In the first approach, the continuity equation is augmented by an artificial, time-dependent term of the pressure, while in the second approach, a Poisson s equation for the pressure is derived by taking the divergence of the momentum equations with the constraint of mass conservation. In this book, NASA computational fluid dynamics researchers Dochan Kwak and Cetin C. Kiris discuss and analyze these two approaches in detail. Moreover, ix

10 x Foreword they introduce a unified approach that is validated for both compressible and incompressible flows using standard benchmark cases. The authors present many applications, for both laminar and turbulent flows, with an emphasis on practical applications that is clear throughout the book. Three separate chapters are devoted to simulations of liquid propellant rocket engine subsystems, turbopumps, and hemodynamics related to simulation of blood circulation in the human brain and in mechanical heart assist devices. All calculations presented are based on finite differences or finite volumes, using structured grids. For complex geometries, overset grids are used. In order to obtain steady-state solutions in an efficient manner, several methods of convergence acceleration are included using parallel computations. Unlike other books on incompressible flow simulations (in particular those based on finite elements), no abstract mathematics, such as functional analysis or tensors, are used in the presentation. The authors appeal to more physical approaches. Based on papers and reports written by the authors and colleagues at NASA and elsewhere over the last two decades, this collection of material is very useful for both researchers and graduate students. The book is easy to read and understand. The only mathematical prerequisites are first-level courses on linear algebra, calculus, and differential equations. This book is a valuable contribution to the subject of incompressible flow simulations, and I am proud to have collaborated with the authors on numerous projects in this area. Davis, CA March 2010 Mohamed Hafez

11 Acknowledgements Many colleagues contributed to the material presented in this monograph. We especially thank those who have worked with us at Ames over various periods since the 1980s; Stuart Rogers, Moshe Rosenfeld, Marcel Vinokur, Jeff Housman, Changsung Kim, Jennifer Dacles-Mariani, Seokkwan Yoon, Jong-Youb Sa, and William Chan; a number of researchers from Rocketdyne during the Space Shuttle engine redesign period, including James L.C. Chang, Steve Barson, and Gary Belie to name just a few; researchers from NASA Marshall Space Flight Center throughout the 1980s and 1990s: Luke Schutzenhofer, Paul McConnaughey, Robert Garcia, Robert Williams, and others. We are very grateful to Prof. Mohamed Hafez for reviewing the entire draft and writing the Foreword for this monograph. We would like to acknowledge our many years of collaboration with him in developing computational methods for incompressible flow. There are many others not explicitly listed here you know who you are whom we have interacted with during the course of algorithm and applications procedure development. We truly appreciate their cooperation and encouragement throughout the course of our efforts in performing NASA mission-related tasks. We hope this monograph passes some of their ideas on numerical simulation of incompressible flow to the next generation of scientists and engineers working on real-world problems. We would also like to thank Jill Dunbar for editing multiple drafts and Marco Librero for preparing the final manuscript. Most notably, DK would like to thank his wife Soonup for her continuous support and encouragement throughout his CFD career over the past several decades. Without her support, completion of this monograph would not have been possible. Moffett Field, CA Moffett Field, CA May 2010 Dochan Kwak Cetin C. Kiris xi

13 Contents 1 Introduction Flow Physics History of Computational Approaches Scope of this Monograph Methods for Solving Viscous Incompressible Flow Problems Overview Mathematical Models Formulation for General Geometry Overview of Solution Approaches Pressure-Based Method Artificial Compressibility Method Methods Based on Derived Variables Pressure Projection Method in Generalized Coordinates Overview Formulation in Integral Form Discretization Geometric Quantities Mass Conservation Equation Momentum Conservation Equation Solution Procedure Fractional-Step Procedure Solution of Momentum Equations Using an Upwind Scheme Pressure Poisson Solver Validation of the Solution Procedure Artificial Compressibility Method Artificial Compressibility Formulation and Physical Characteristics Characteristics of Pseudo Waves Wave-Vorticity Interaction Rate of Convergence Limit of Incompressibility xiii

14 xiv Contents 4.2 Steady-State Formulation Steady-State Algorithm Difference Equations Approximate Factorization Scheme LU-SGS Scheme Line Relaxation Scheme Numerical Dissipation or Smoothing Boundary Conditions Time-Accurate Procedure Time-Accurate Algorithm Using Upwind Differencing Upwind Differencing Scheme Implicit Scheme Boundary Conditions for Upwind Scheme Validation of Solution Procedure Two-Dimensional (2-D) Channel Flow Flow over a Backward-Facing Step Unified Formulation Time-Derivative Preconditioning Method Numerical Results Flow Solvers and Validation Scope of Validation Artificial Compressibility Codes Pressure Projection Code Selection of Codes for Engineering Applications Steady Internal Flow: Curved Duct with Square Cross Section Time-Dependent Flow Flow Over a Circular Cylinder Impulsively Started Flat Plate at Pulsatile Flow Through A Constricted 2-D Channel Flapping Foil in a Duct External and Juncture Flow Cylinder on a Flat Plate Wing-Body Junction Wingtip Vortex Flow Simulation of a Liquid-Propellant Rocket Engine Subsystem Historical Background Flow Analysis in the Space Shuttle Main Engine (SSME) Flow Analysis Task and Computational Model for the SSME Powerhead Computational Model Description Multiple-Zone Computation Grid and Geometry Effects

15 Contents xv 6.4 Turbulence Modeling Issues Selection of Turbulence Model for Internal Flow Turbulence Modeling Issues Involving Strong Streamwise Curvature Analysis of the Original Three-Circular-Duct HGM Configuration Development of New Two Elliptic-Duct HGM Configuration From Redesign to Flight Turbopumps Historical Background Turbopumps in Liquid-Propellant Rocket Engines Mathematical Formulation for a Steady Rotating Frame of Reference Validation of Simulation Procedures Using a Steadily Rotating Inducer Application to Impeller Simulation SSME Impeller Advanced Impeller Simulation of a Complete Pump Geometry Geometry and Computational Grid Issues Related to Large-Scale Computations Issues Related to Flange-to-Flange Simulation High-Fidelity Unsteady Flow Application to SSME Flowliners Description of the Flow Simulation Task Computational Model and Grid System Computed Results Some Aspects of a Parallel Implementation Hemodynamics Issues in Computational Hemodynamics for Humans Geometry of the Human Vascular System Modeling Non-Newtonian or Stress-Supporting Flow Turbulence Model Geometry and Morphology Arterial Wall Model Boundary Conditions Cardiovascular Model Brain Model Model Equations for Blood Flow Simulation Blood Flow Model Deformable Wall Model Vascular Bed Model Arteriolar Auto-Regulation Model Validation of the Simulation Procedure Carotid Bifurcation Circular Tube with 90 Bend

16 xvi Contents Effect of Arterial Wall Distensibility Effects of Altered Gravity on Blood Circulation Blood Circulation in the Human Brain Collateral Circulation Under Auto-Regulation Extraction of Geometry Data from Anatomical Picture Effects of Gravitational Variations Simulations of Blood Flow in Mechanical Devices Artificial Heart Valves Ventricular Assist Devices Closing Remarks References Index

Contents. I Introduction 1. Preface. xiii

Contents. I Introduction 1. Preface. xiii Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics.........................