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Computation Of Unsteady Internal Flows: Fundamental Methods with Case Studies

P. G. Tucker

Computation of Unsteady Internal Flows provides an in-depth understanding of unsteady flow modelling algorithms. Also, through Case Studies the practical application of some of these algorithms is illustrated. Case study source codes are included on CD.

Bookcover

Format: Hardcover, English, 355 pages
ISBN: 0792373715
Publisher: Kluwer Academic Publishers
Pub. Date: 2001
Edition: 1

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Publisher Comments

Most Engineering flows are unsteady but due to computational expense, this aspect is often ignored. However, with the ever increasing memory (unsteady simulations require more core memory and, if time histories are required, disc space) and processing speed of computers and their decreasing cost, the use of unsteady simulations is rising dramatically. 'Computation of Unsteady Internal Flows' presents specialized unsteady flow modelling algorithms, their traits, practical tips relating to their use and brief Case Studies. Algorithms, as used for steady flows, are also described along with their performances (and sometimes extensions) when modelling unsteady flows. Both traditional algorithms and future trends are discussed. Algorithms originating from various fields are presented including those intended for geophysical applications.

The focus of the text is viscous internal and Newtonian flows. Initial chapters mostly review previous work and present numerical methods. However, these chapters are connected by common buoyancy driven flow exemplar prediction. This is intended to more immediately illuminate points made at a preliminary elementary level. The initial chapters are followed by brief Case Studies generally considering problems of greater complexity and practical significance. The Case Studies' presentation is harmonized as much as possible so that concepts can be readily connected and algorithm performances compared. Simple to understand and of a training nature source code and set-up files are included, enabling the reader to reproduce, extend some Case Studies and further explore algorithm performances.

Features summary of 'Computation of Unsteady Internal Flows':

  • Specialized unsteady flow modelling algorithms, their traits, and practical tips relating to their use are presented;
  • Case studies considering complex, practically significant problems are given;
  • Source code and set-up files are included. Intended to be of a tutorial nature, these enable the reader to reproduce and extend case studies and to further explore algorithm performances;
  • Mathematical derivations are used in a fashion that illuminates understanding of the physical implications of different numerical schemes. Physically intuitive mathematical concepts are used and
  • New material on adaptive time stepping is included.

Audience:
Researchers in both the academic and industrial areas who wish to gain in-depth knowledge of unsteady flow modelling will find 'Computation of Unsteady Internal Flows' invaluable.


Table of Contents

  PREFACE  
  NOMENCLATURE  
     
1 INTRODUCTION  
1-1 The Importance of Modelling Unsteady Flows  
1-2 The Nature and Causes of Unsteadiness  
1-2 Flow Equations, their Mathematical Nature and Physical Implications  
1-3 Summary of Book Content  
     
2 OVERVIEW OF ELEMENTARY TEMPORAL DISCRETIZATIONS  
2-1 Introduction  
2-2 Profile Assumptions for Variables  
  Dependent Variable Changes with Time  
  Spatial Variation of the Time Derivative  
2-3 The Generic Discretized Equation  
2-4 Two-level Schemes  
  Basic Discretization Approach  
  Simple Accurate Crank-Nicolson Scheme Linearization Example  
  D Form Based on Differential Equation  
  D Form Based in Discretized Equation  
  Stability and Accuracy of Two Level Schemes  
  Two Level Exponential and Modified Crank-Nicolson Schemes  
  General Explicit Methods  
  Lax-Wendroff Scheme  
2-5 Three-level Schemes  
  Multipoint Methods  
  Leapfrog  
  DuFort-Frankel  
2-6 Predictor Corrector Methods  
2-7 Splitting Methods  
2-8 Time Marching and Stiffness  
  Time Marching  
  Stiffness  
2-9 Characterizing Stability and Accuracy of Schemes  
2-10 Boundary Conditions and Solution Initialization  
  Boundary Conditions  
  Solution Initialization  
2-11 Integration of Particle/Droplet Transport Equations  
2-12 Two-level Scheme Exemplar  
  Flow Structure  
2-13 Conclusions  
     
3 TEMPORAL AND SPATIAL DISCRETIZATION RELATIONSHIPS  
3-1 Introduction  
3-2 The Substantial Derivative  
  Making Flows More Steady  
3-3 Convective Term Treatments  
  General Convective Schemes  
  Unsteady Convective Schemes  
  Current Convective Schemes  
3-4 Dissipation, Dispersion and Aliasing  
  The Modified Equation  
  Dissipation and Dispersion  
  Grid Related Effects  
  Stability Analysis  
  Aliasing  
3-5 Moving Boundaries  
  General Modelling Techniques  
  Moving Boundaries with Body Fitted Grids  
  Space Conservation Laws  
  Inertial Coordinate Systems with Moving Boundaries  
  Free Surface Modelling  
  Free Surface Tracking Techniques  
3-6 Time Adaptive Solutions  
  Spatial Grid Adaptation with Time  
3-7 Conclusions  
     
4 SOLVER ALGORITHMS AND GENERAL SOLUTION PROCEDURES  
4-1 Introduction  
4-2 Simultaneous Equation Solvers  
  Diagonal Dominance Issues  
  TDMA Solver  
  Convergence Criteria  
  The Multilevel Algorithm  
  Specific Multilevel Convergence Temporal Applications  
  The Newton Method  
4-3 Parallel Processing  
  Parallel Processing with Multilevel Convergence  
  Domain Decompositions in Time  
4-4 Evaluation of the Pressure Field  
  Streamfunction Vorticity Method  
  Basic SIMPLE type Methods - Including PISO  
  SIMPLE2 and SIMPLE*  
  AVPI Method  
  Consistent 'Time-Step' variant of SIMPLE  
  Fractional Step Methods  
  Artificial Compressibility Methods  
  Penalty Function Method  
  Implicit Continuous-fluid Eulerian (ICE) Method  
  Fully Coupled Solvers  
  Reduced Pressures  
  Pressure/velocity Locations  
  Pressure Subcycling  
4-5 Review of Commercial Programs and their Unsteady Flow Capabilities  
  Codes Considered  
  General Code Features  
  Capabilities Specific to Unsteady Flows  
4-6 Features of Codes used for Case Studies  
4-7 Conclusions  
     
5 SOLUTION ADAPTED TIME STEPS  
5-1 Introduction  
5-2 Numerical Methods  
  Crude Time Step Adaptation with no Direct Error Estimate  
  Time Step Adaptation with Altered Scheme Estimate  
  Time Step Adaptation with Altered Step Error Estimate  
  Time Step Adaptation with Altered Order Error Estimate  
  Evaluating a Representative Solution Error  
  Relating Solution Error to New Time Steps  
  Solution Correction  
  Specific Implementation Details for Schemes  
  Novel Sub-cycling with Adaptive Time Stepping  
  Novel Forcing Function Element Adaptation  
  Diagonal Dominance  
5-3 Discussion of Results  
  Case (A), Transient Wall Driven Square Box Flow  
  Case (B), Transient Buoyancy Driven Flow between Cylinders  
  Case (C), Transient Wall Driven Free Disc Flow  
5-4 Conclusions  
     
6 TURBULENCE MODELLING  
6-1 Introduction  
6-2 VLES/URANS Modelling  
6-3 LES Modelling  
6-4 Zonal LES/DES  
6-5 URANS Models used in Case Studies  
  Turbulence Averaging  
  URANS Equations  
  Turbulence and Scalar Transport Equations  
  Mixing Length Models  
  l-v Models  
  k-l Models  
  k-e Model  
  Zonal Models  
  Non-linear eddy viscosity models  
  Stagnation Modification  
  Calculation of Normal Wall Distances  
  Particle Trajectories in Turbulent Flows  
6-6 Conclusions  
     
7 CYCLIC ANNULAR CAVITY FLOWS  
7-1 Introduction  
7-2 Numerical Details  
  Substantial Derivative  
  Solution Initialization  
  Grid Structure and Numerical Parameter Settings  
  Convergence Criteria  
7-3 Discussion of Results  
  Experimental Conditions and Fluid Properties  
  Comparisons with Experimental Data  
7-4 Conclusions  
     
8 CYCLIC AERO ENGINE MOTIVATED CAVITY FLOWS  
8-1 Introduction  
8-2 Numerical Details  
  Solution Initialization Boundary Conditions  
  Convergence Criteria  
  Numerical Parameter Settings  
8-3 Lower Reynolds Number Results  
  Flow Structure Features  
  Comparison between Numerical Predictions and Experimental Flow Visualization  
  The Flow Drift  
  The Substantial Derivative  
  Performances of Different Numerical Schemes  
8-4 Higher Reynolds Number Results  
  Computation of Bubble Vortex Breakdown  
  URANS/VLES Predictions  
8-5 Adaptive Time Stepping  
8-6 Conclusions  
     
9 CYCLIC MOVING BOUNDARY FLOWS  
9-1 Introduction  
9-2 Numerical Details  
  Boundary Conditions  
  Dynamic Cavitation Modelling  
  General Case Details  
  The Substantial Derivative  
9-3 Discussion of Results for Journal Bearing  
  Analytical Isothermal Case  
  Experimental Isothermal Comparison  
9-4 Discussion of Results for Magnetic Bearing  
9-5 Conclusions  
     
10 CYCLIC COMPLEX SYSTEM FLOW  
10-1 Introduction  
10-2 Numerical Details  
  Wall Distance Algorithm  
  Boundary Conditions and Solution Initialization  
  Structure of Grids  
  General Case Details  
10-3 Discussion of Results for Fluid Flow  
  Spatial Velocity Variations  
  Temporal Velocity Variations  
  Spatial Turbulence Intensity Variations  
10-4 Adaptive Time Stepping  
10-5 Discussion of Results for Scalar and Particle Transport  
  Scalar Transport  
  Particle Transport  
10-6 Conclusions  
     
11 TRANSIENT COMPLEX SYSTEM FLOW  
11-1 Introduction  
11-2 Numerical Details  
  Boundary Conditions and Solution Initialization  
11-3 Discussion of Results  
  Steady Flow Comparisons  
  Transient Flow Comparisons  
  Adaptive Time Stepping  
11-4 Conclusions  
     
12 OVERALL CONCLUSIONS WITH FUTURE PERSPECTIVES  

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