简介
In January 1937, Nobel laureate in Physics Subrahmanyan Chandrasekhar was recruited to the University of Chicago. He was to remain there for his entire career, becoming Morton D. Hull Distinguished Service Professor of Theoretical Astrophysics in 1952 and attaining emeritus status in 1985. This is where his then student Ed Spiegel met him during the summer of 1954, attended his lectures on turbulence and jotted down the notes in hand. His lectures had a twofold purpose: they not only provided a very elementary introduction to some aspects of the subject for novices, they also allowed Chandra to organize his thoughts in preparation to formulating his attack on the statistical problem of homogeneous turbulence. After each lecture Ed Spiegel transcribed the notes and filled in the details of the derivations that Chandrasekhar had not included, trying to preserve the spirit of his presentation and even adding some of his side remarks. The lectures were rather impromptu and the notes as presented here are as they were set down originally in 1954. Now they are being made generally available for Chandrasekhar鈥檚 centennial.
目录
Prologue 6
Contents 13
The Turbulence Problem 16
The Meaning of `Turbulence' 16
Two Fundamental Aspects of Turbulence 17
The Net Energy Balance 18
Interchange of Energy Between States of Motion 22
Some Remarks 25
On the Harmonic Analysis 25
On the Concept of Isotropy 25
On the Possibility of a Universal Theory 26
The Spectrum of Turbulent Energy 27
The Spectrum 27
An Equation for the Spectrum 28
Some Preliminaries to the Development of a Theory of Turbulence 32
Heisenberg's Theory of Turbulence 34
The Fundamental Equation of the Theory 34
Chandrasekhar's Solution of (7.17) for the Case of Stationary Turbulence 36
Other Derivations of the k-5/3 Law 41
Fermi's Approach 41
Kolmogorov's Theory 42
The Method of von Neumann 43
Conclusion 45
An Alternate Approach: Correlations 46
The Equations of Isotropic Turbulence 48
The Concept of Isotropy 48
Qij as an Isotropic Tensor 49
Two More Examples 50
The Isotropic Vector 50
The Isotropic Tensor of Third Order 50
Solenoidal Isotropic Tensors 51
Isotropic Vectors Li 51
Isotropic Second Order Tensor Qij 52
Further Manipulations 53
The Isotropic Third Order Tensor, Tijk 55
The Karman-Howarth Equations 60
The Meanings of the Defining Scalars 62
Some Results from the Karman-Howarth Equation 65
The Taylor Microscale 65
The Study of the Decay of Turbulence 66
The Connection Between the Karman-Howarth Equation and the Kolmogorov Theory 67
The Double Correlation 67
The Triple Correlation 68
The Relation Between the Fourth and Second Order Correlations When the Velocity Follows a Gaussian Distribution 70
Some Properties of the Gaussian Distribution 70
One-Dimensional Gaussian Distribution 70
Characteristic Function 71
The Importance of the Characteristic Function 72
n-Dimensional Gaussian Function 73
Characteristic Function 74
Two-Dimensional Gaussian Function 76
Addition Theorem for Gaussian Distributions 76
Proof of (14.2) 78
Chandrasekhar's Theory of Turbulence 80
A More Subjective Approach to the Derivation of Chandrasekhar's Equation 89
The Dimensionless Form of Chandrasekhar's Equation 91
Some Aspects and Advantages of the New Theory 92
A Mathematical Justification of the Assumptionsof the Heisenberg Theory 92
Compatibility with the Kolmogorov Theory 93
The Problem of Introducing the Boundary Conditions 95
Discussion of the Case of Negligible Inertial Term 96
The Case in Which Viscosity Is Neglected 100
Solution of the Non-Viscous Case Near r=0 103
Solution of the Heat Equation 105
Solution of the Quasi-Wave Equation 106
The Introduction of Boundary Conditions 113
Epilogue 116
Contents 13
The Turbulence Problem 16
The Meaning of `Turbulence' 16
Two Fundamental Aspects of Turbulence 17
The Net Energy Balance 18
Interchange of Energy Between States of Motion 22
Some Remarks 25
On the Harmonic Analysis 25
On the Concept of Isotropy 25
On the Possibility of a Universal Theory 26
The Spectrum of Turbulent Energy 27
The Spectrum 27
An Equation for the Spectrum 28
Some Preliminaries to the Development of a Theory of Turbulence 32
Heisenberg's Theory of Turbulence 34
The Fundamental Equation of the Theory 34
Chandrasekhar's Solution of (7.17) for the Case of Stationary Turbulence 36
Other Derivations of the k-5/3 Law 41
Fermi's Approach 41
Kolmogorov's Theory 42
The Method of von Neumann 43
Conclusion 45
An Alternate Approach: Correlations 46
The Equations of Isotropic Turbulence 48
The Concept of Isotropy 48
Qij as an Isotropic Tensor 49
Two More Examples 50
The Isotropic Vector 50
The Isotropic Tensor of Third Order 50
Solenoidal Isotropic Tensors 51
Isotropic Vectors Li 51
Isotropic Second Order Tensor Qij 52
Further Manipulations 53
The Isotropic Third Order Tensor, Tijk 55
The Karman-Howarth Equations 60
The Meanings of the Defining Scalars 62
Some Results from the Karman-Howarth Equation 65
The Taylor Microscale 65
The Study of the Decay of Turbulence 66
The Connection Between the Karman-Howarth Equation and the Kolmogorov Theory 67
The Double Correlation 67
The Triple Correlation 68
The Relation Between the Fourth and Second Order Correlations When the Velocity Follows a Gaussian Distribution 70
Some Properties of the Gaussian Distribution 70
One-Dimensional Gaussian Distribution 70
Characteristic Function 71
The Importance of the Characteristic Function 72
n-Dimensional Gaussian Function 73
Characteristic Function 74
Two-Dimensional Gaussian Function 76
Addition Theorem for Gaussian Distributions 76
Proof of (14.2) 78
Chandrasekhar's Theory of Turbulence 80
A More Subjective Approach to the Derivation of Chandrasekhar's Equation 89
The Dimensionless Form of Chandrasekhar's Equation 91
Some Aspects and Advantages of the New Theory 92
A Mathematical Justification of the Assumptionsof the Heisenberg Theory 92
Compatibility with the Kolmogorov Theory 93
The Problem of Introducing the Boundary Conditions 95
Discussion of the Case of Negligible Inertial Term 96
The Case in Which Viscosity Is Neglected 100
Solution of the Non-Viscous Case Near r=0 103
Solution of the Heat Equation 105
Solution of the Quasi-Wave Equation 106
The Introduction of Boundary Conditions 113
Epilogue 116
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