Prologue |
The Turbulence Problem / 1: |
The Meaning of 'Turbulence' / 1.1: |
Two Fundamental Aspects of Turbulence / 1.2: |
The Net Energy Balance / 2: |
Interchange of Energy Between States of Motion / 3: |
Some Remarks / 4: |
On the Harmonic Analysis / 4.1: |
On the Concept of Isotropy / 4.2: |
On the Possiblity of a Universal Theory / 4.3: |
The Spectrum of Turbulent Energy / 5: |
The Spectrum / 5.1: |
An Equation for the Spectrum / 5.2: |
Some Preliminaries to the Development of a Theory of Turbulence / 6: |
Heisenberg's Theory of Turbulence / 7: |
The Fundamental Equation of the Theory / 7.1: |
Chandrasekhar's Solution of (7.17) for the Case of Stationary Turbulence / 7.2: |
Fermi's Approach / 8: |
Kolmogorov's Theory / 8.2: |
The Method of von Neumann / 8.3: |
Conclusion / 8.4: |
An Alternate Approach: Correlations / 9: |
The Equations of Isotropic Turbulence / 10: |
The Concept of Isotropy / 10.1: |
Two More Examples / 10.2: |
Solenoidal Isotropic Tensors / 10.3: |
Further Manipulations / 10.3.1: |
The Karman-Howarth Equations / 10.3.3: |
The Meanings of the Defining Scalars / 12: |
Some Results from the Karman-Howarth Equation / 13: |
The Taylor Microscale / 13.1: |
The Study of the Decay of Turbulence / 13.2: |
The Connection Between the Karman-Howarth Equation and the Kolmogorov Theory / 13.3: |
The Double Correlation / 13.3.1: |
The Triple Correlation / 13.3.2: |
The Relation Between the Fourth and Second Order Correlations When the Velocity Follows a Gaussian Distribution / 14: |
Some Properties of the Gaussian Distribution / 14.1: |
One-Dimensional Gaussian Distribution / 14.1.1: |
n-Dimensional Gaussian Function / 14.1.2: |
Two-Dimensional Gaussian Function / 14.1.3: |
Addition Theorem for Gaussian Distributions / 14.2: |
Proof of (14.2) / 14.3: |
Chandrasekhar's Theory of Turbulence / 15: |
A More Subjective Approach to the Derivation of Chandrasekhar's Equation / 16: |
The Dimensionless Form of Chandrasekhar's Equation / 17: |
Some Aspects and Advantages of the New Theory / 18: |
A Mathematical Justification of the Assumptions of the Heisenberg Theory / 18.1: |
Compatibility with the Kolmogorov Theory / 18.2: |
The Problem of Introducing the Boundary Conditions / 19: |
Discussion of the Case of Negligible Inertial Term / 20: |
The Case in Which Viscosity Is Neglected / 21: |
Solution of the Non-Viscous Case Near r = 0 / 22: |
Solution of the Heat Equation / 23: |
Solution of the Quasi-Wave Equation / 24: |
The Introduction of Boundary Conditions / 25: |
Epilogue |
Prologue |
The Turbulence Problem / 1: |
The Meaning of 'Turbulence' / 1.1: |
Two Fundamental Aspects of Turbulence / 1.2: |
The Net Energy Balance / 2: |
Interchange of Energy Between States of Motion / 3: |