《METHODS IN COMPUTATIONAL PHYSICS Volume 3 Fundamental Methods in Hydrodynamics》

TWO-DIMENSIONAL LAGRANGIAN HYDRODYNAMIC DIFFERENCE EQUATIONS＆William D．Schulz1

Ⅰ．Introduction1

Ⅱ．Definitions,Notation,and Transformation Relations2

Ⅲ．Conservation Equations6

Ⅳ．Tensor Artificial Viscosity8

Ⅴ．Stability of the Differential Equations15

Ⅳ．Hydrodynamic Difference Equations21

Ⅶ．Dt Control32

Ⅷ．Boundary Treatment39

Ⅸ．Sample Problem40

Reference45

MIXED EULERIAN-LAGRANGIAN METHOD＆R．M．Frank and R．B．Lazarus47

Ⅰ．Introduction47

Ⅱ．Cell Equations48

Ⅲ．Euler-Lagrange Mesh49

Ⅳ．Choice of Variables50

Ⅴ．Space Differencing51

Ⅵ．Time Differencing55

Ⅶ．Artificial Forces55

Ⅷ．Courant Stability56

Ⅸ．Test Calculations57

Ⅹ．Details of the Machine Program58

Ⅺ．Heat Flow66

THE STRIP CODE AND THE JETTING OF GAS BETWEEN PLATES＆John G．Trulio69

Ⅰ．Introduction69

Ⅱ．Simple Theory for the Jetting of Gas between Two Plates71

Ⅲ． The Strip Code75

Ⅳ． Comments on the Convergence of Numerical Methods in Hydrodynamics87

Ⅴ． Application of the Strip Code to the Problem of the Jetting of Gas92

Ⅵ． Conclusion100

Ⅶ． Appendices103

References115

CEL:A TIME-DEPENDENT,TWO-SPACE-DIMENSIONAL,COUPLED EULERIAN-LAGRANGE CODE＆W．F．Noh117

Ⅰ．Introduction117

Ⅱ．A General Discussion of Hydrodynamical Calculations121

Ⅲ．Description of the CEL Code125

Ⅳ．Generalization of the Difference Approximation for Partial Derivatives over an Arbitrary Region130

Ⅴ．Eulerian Difference Equations for Interior Cells of the Mesh136

Ⅵ．The Conservation Form of the Hydrodynamical Equations When the Space Variables x and y May Have an Arbitrary Velocity Relative to the Fluid Velocity140

Ⅶ．General Considerations for the Boundary Zone Calculations143

Ⅷ．Boundary Zones and Boundary Sequences144

Ⅸ．Boundary Zone Difference Equations for Variables p,e,etc．,Which Are Centered in Boundary Zones Cn(k,l)148

Ⅹ．Difference Equations for the Components of Momentum(m=pu,n=pv) Which Are Centered in the Boundary Zones Cn(k+1/2,l+1/2)151

Ⅺ．Lagrange Difference Equations153

Ⅻ．Discussion and Graphical Results of Several CEL Calculations156

References179

THE TENSOR CODE＆G．Maenchen and S．Sack181

Ⅰ．Introduction181

Ⅱ．Definitions and Notation184

Ⅲ．Momentum Equation187

Ⅳ．Boundary Conditions191

Ⅴ．Volume and Strain Rate192

Ⅵ．Plastic Yield196

Ⅶ．Fracture198

Ⅷ．Damping202

Ⅸ．Energy Equation and Stability Control205

Ⅹ．Examples206

References210

CALCULATION OF ELASTIC-PLASTIC FLOW＆Mark L．Wilkins211

Ⅰ．Introduction211

Ⅱ．Equations of State212

Ⅲ．One-Dimensional Elastic-Plastic Flow223

Ⅳ．Two-Dimensional Elastic-Plastic Flow226

Appendix Ⅰ．Finite Difference Equations for the Equations of Section Ⅲ241

Appendix Ⅱ．Finite Difference Equations for the Equations of Section Ⅳ244

References262

SOLUTION BY CHARACTERISTICS OF THE EQUATIONS OF ONE-DIMENSIONAL UNSTEADY FLOW＆N．E．Hoskin265

Ⅰ．Introduction265

Ⅱ．Equations of Motion268

Ⅲ．Structure of a Computer Program274

Ⅳ．Calculation by Characteristics at Fixed Time Intervals288

Ⅴ．Conclusions293

References293

THE SOLUTION OF TWO-DIMENSIONAL HYDRODYNAMIC EQUATIONS BY THE METHOD OF CHARACTERISTICS＆D．J．Richardson295

Ⅰ．Introduction295

Ⅱ．The Equations of Hydrodynamic Flow296

Ⅲ．Characteristic Surfaces and Bi-Characteristic Curves296

Ⅳ．Relations Holding along Bi-Characteristics298

Ⅴ．The General Bi-Characteristic Method300

Ⅵ．Shock Surfaces302

Ⅶ．Computational Aspects of the Two-Dimensional Characteristics Method303

Ⅷ．Topological Aspects of the Computer Program306

Ⅸ．Appendices310

References318

THE PARTICLE-IN-CELL COMPUTING METHOD FOR FLUID DYNAMICS＆Francis H．Harlow319

Ⅰ．Introduction319

Ⅱ．The Basic Computing Method322

Ⅲ．Modified Forms of the Method331

Ⅳ．Properties of the PIC Method333

Ⅴ．Applicability and Limitations of the PIC Method339

Ⅵ．Application to a Specific Problem342

References343

THE TIME DEPENDENT FLOW OF AN INCOMPRESSIBLE VISCOUS FLUID＆Jacob Fromm345

Ⅰ．Introduction346

Ⅱ．The Numerical Method349

Ⅲ．Initial and Boundary Conditions355

Ⅳ．Modification of the Vorticity Equation362

Ⅴ．Functionals of the Motion;A Test of the Method371

Ⅵ．Applications376

References382

AUTHOR INDEX383

SUBJECT INDEX384