Two Fluid Model Stability Simulation and Chaos 1st Edition by Martín López De Bertodano, William Fullmer, Alejandro Clausse, Victor Ransom 3319449685 9783319449685

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ISBN-10 :  3319449685 

ISBN-13 :  9783319449685

Author:  Martín López De Bertodano, William Fullmer, Alejandro Clausse, Victor H. Ransom 

This book addresses the linear and nonlinear two-phase stability of the one-dimensional Two-Fluid Model (TFM) material waves and the numerical methods used to solve it. The TFM fluid dynamic stability is a problem that remains open since its inception more than forty years ago. The difficulty is formidable because it involves the combined challenges of two-phase topological structure and turbulence, both nonlinear phenomena. The one dimensional approach permits the separation of the former from the latter. The authors first analyze the kinematic and Kelvin-Helmholtz instabilities with the simplified one-dimensional Fixed-Flux Model (FFM). They then analyze the density wave instability with the well-known Drift-Flux Model. They demonstrate that the Fixed-Flux and Drift-Flux assumptions are two complementary TFM simplifications that address two-phase local and global linear instabilities separately. Furthermore, they demonstrate with a well-posed FFM and a DFM two cases of nonlinear two-phase behavior that are chaotic and Lyapunov stable.  On the practical side, they also assess the regularization of an ill-posed one-dimensional TFM industrial code. Furthermore, the one-dimensional stability analyses are applied to obtain well-posed CFD TFMs that are either stable (RANS) or Lyapunov stable (URANS), with the focus on numerical convergence.

Two-Fluid Model Stability, Simulation and Chaos 1st Table of contents:

Chapter 1: Introduction
1.1 Summary
1.2 Outline of Contents
References
Part I: Horizontal and Near Horizontal Wavy Flow
Chapter 2: Fixed-Flux Model
2.1 Introduction
2.2 Compressible Two-Fluid Model
2.2.1 One-Dimensional Model Equations
2.2.2 Characteristics
2.3 Incompressible Two-Fluid Model
2.3.1 One-Dimensional Model Equations
2.3.2 Derivation of the Fixed-Flux Model
2.4 Linear Stability
2.4.1 Dispersion Relation for the Kelvin-Helmholtz Instability ()
2.4.2 Dispersion Relation for the SWT Instability ()
2.4.3 Sheltering Effect
2.5 Numerical Stability
2.5.1 Obtaining a Well-Posed Numerical Model
2.5.2 First-Order Semi-Implicit Scheme (Inviscid)
2.5.3 First-Order Semi-Implicit Scheme (with Viscous Terms)
2.5.4 First-Order Fully Implicit Scheme (with Viscous Terms)
2.5.5 Second-Order Semi-Implicit Scheme
2.6 Verification
2.6.1 Kreiss-Yström Equations
2.6.2 Characteristic Analysis
2.6.3 Dispersion Relation
2.6.4 Method of Manufactured Solutions
2.6.5 Water Faucet Problem
2.7 Kelvin-Helmholtz Instability
2.8 Summary and Discussion
References
Chapter 3: Two-Fluid Model
3.1 Introduction
3.2 Incompressible Two-Fluid Model
3.3 Linear Stability
3.3.1 Characteristics
3.3.2 Dispersion Analysis
3.3.3 KH Instability
3.4 Numerical Stability
3.4.1 TFIT Two-Fluid Model
3.4.2 Staggered Cell Structure
3.4.3 First-Order Semi-Implicit Scheme
3.4.4 Implicit Pressure Poisson Equation
3.4.5 von Neumann Analysis
3.4.6 Numerical Regularization
3.4.7 Second-Order Semi-implicit Scheme
3.5 Verification
3.5.1 Sine Wave
3.5.2 Water Faucet Problem
3.5.3 Modified Water Faucet Problem
3.5.4 Convergence
3.6 Nonlinear Simulations
3.6.1 Thorpe Experiment
3.6.2 Viscous Stresses
3.6.3 Wall Shear
3.6.4 Interfacial Shear
3.6.5 Single Nonlinear Wave
3.6.6 Thorpe Experiment Validation
3.6.7 Convergence
3.7 Summary and Discussion
References
Chapter 4: Fixed-Flux Model Chaos
4.1 Introduction
4.2 Chaos and the Kreiss and Yström Equations
4.2.1 Nonlinear Simulations
4.2.2 Sensitivity to Initial Conditions
4.2.3 Lyapunov Exponent
4.2.4 Fractal Dimension
4.2.5 The Route to Chaos
4.2.6 Numerical Convergence
4.3 Fixed-Flux Model Chaos
4.3.1 Nonlinear Simulations with the FFM
4.3.2 Extension of Thorpe Experiment into Chaos
4.3.3 Fixed-Flux Model for Fully Developed Laminar Flow in a Pipe
4.3.4 Kelvin-Helmholtz Instability
4.3.5 Nonlinear Simulations
4.3.6 Lyapunov Exponent
4.3.7 Numerical Convergence
4.3.8 Fractal Dimension
4.4 Summary and Discussion
References
Part II: Vertical Bubbly Flow
Chapter 5: Fixed-Flux Model
5.1 Introduction
5.2 Compressible Two-Fluid Model
5.2.1 Compressible Model Equations
5.2.2 Virtual Mass Force
5.3 Incompressible Two-Fluid Model
5.3.1 Interfacial Pressure
5.3.2 Fixed-Flux Model Derivation
5.4 Linear Stability
5.4.1 Characteristic Analysis
5.4.2 Collision Force
5.4.3 Dispersion Relation: Kinematic Instability
5.4.4 Drag Force
5.4.4.1 Laminar Regime (Stokes Flow)
5.4.4.2 Turbulent Regime (Distorted Bubbles)
5.5 Nonlinear Simulations
5.5.1 Stable Wave Evolution
5.5.2 Kinematically Unstable Waves in Guinness
5.6 Summary and Discussion
References
Chapter 6: Drift-Flux Model
6.1 Introduction
6.2 Void Propagation Equation
6.3 Applications of Void Propagation Equation
6.3.1 Level Swell
6.3.2 Drainage
6.3.3 Propagation of Material Shocks
6.4 Dynamic Drift-Flux Model
6.4.1 Mixture Momentum Equation
6.4.2 Integral Momentum Equation
6.5 Delay Drift-Flux Model
6.6 Flow Excursion
6.6.1 Homogeneous Equilibrium Model
6.6.2 Drift-Flux Model
6.7 Density Wave Instability
6.7.1 Homogeneous Equilibrium Model
6.7.2 Transfer Function
6.7.3 Drift-Flux Model
6.8 Summary and Discussion
References
Chapter 7: Drift-Flux Model Nonlinear Dynamics and Chaos
7.1 Introduction
7.2 Nonlinear Mapping of the Boiling Channel Dynamics
7.3 Model of a Boiling Channel with Moving Nodes
7.4 Dynamics of a Boiling Channel with an Adiabatic Riser
7.4.1 Summary of MNM Equations for the Channel-Riser System
7.4.2 Low Power Oscillations at Low Fr Numbers in a Heated Channel with Adiabatic Riser
7.4.3 Experimental Validation of Quasi-periodic Oscillations
7.5 Summary and Discussion
References
Chapter 8: RELAP5 Two-Fluid Model
8.1 Introduction
8.2 Material Waves
8.2.1 RELAP5 Adiabatic Two-Fluid Model
8.2.2 Characteristics
8.2.3 Bernier´s Experiment
8.3 Low Pass Filter Regularization of the TFM
8.3.1 Dispersion Analysis
8.3.2 Numerical Viscosity
8.3.3 Artificial Viscosity Model
8.3.4 Water Faucet Problem
8.4 Summary and Discussion
References
Chapter 9: Two-Fluid Model CFD
9.1 Introduction
9.2 Incompressible Multidimensional TFM
9.2.1 Model Equations
9.2.2 Interfacial Momentum Transfer
9.2.3 Drag Force
9.2.4 Lift Force
9.2.5 Wall Force
9.2.6 Laminar Pipe Flow
9.3 RANS Two-Fluid Model
9.3.1 Reynolds Stress Stabilization
9.3.2 Single-Phase k-epsi Model
9.3.3 Two-Phase k-epsi Model
9.3.4 Decay of Grid Generated Turbulence
9.3.4.1 Single Time-Constant Model
9.3.4.2 Two Time-Constant Model
9.3.5 Turbulent Pipe Flow
9.3.6 Turbulent Diffusion Force
9.3.7 Bubbly Jet
9.4 Near-Wall Two-Fluid Model
9.4.1 Wall Boundary Conditions
9.4.2 Two-Phase Logarithmic Law of the Wall of Marie et al. (1997)
9.4.3 Near-Wall Averaging
9.4.4 Laminar Pipe Flow Revisited
9.4.5 Turbulent Bubbly Boundary Layer
9.4.6 Turbulent Pipe Flow Revisited
9.5 URANS Two-Fluid Model
9.5.1 Stability
9.5.2 Constitutive Relations
9.5.3 Plane Bubble Plume
9.6 Summary and Discussion

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Tags: Model Stability, Simulation, Chaos, Martín López De Bertodano, William Fullmer, Alejandro Clausse, Victor Ransom