## ScalingThe author describes and teaches the art of discovering scaling laws, starting from dimensional analysis and physical similarity, which are here given a modern treatment. He demonstrates the concepts of intermediate asymptotics and the renormalisation group as natural consequences of self-similarity and shows how and when these notions and tools can be used to tackle the task at hand, and when they cannot. Based on courses taught to undergraduate and graduate students, the book can also be used for self-study by biologists, chemists, astronomers, engineers and geoscientists. |

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### Contents

Dimensional analysis and physical similarity | 12 |

12 Dimensional analysis | 22 |

13 Physical similarity | 37 |

Selfsimilarity and intermediate asymptotics | 52 |

the selfsimilar solution | 55 |

23 The intermediate asymptotics | 60 |

very intense groundwater pulse flow the selfsimilar intermediateasymptotic solution | 65 |

Scaling laws and selfsimilar solutions that cannot be obtained by dimensional analysis | 69 |

53 The renormalization group and incomplete similarity | 102 |

Selfsimilar phenomena and travelling waves | 109 |

62 Burgers shock waves steady travelling waves of the first kind | 111 |

steady travelling waves of the second kind Nonlinear eigenvalue problem | 113 |

64 Selfsimilar interpretation of solitons | 119 |

Scaling laws and fractals | 123 |

72 Incomplete similarity of fractals | 129 |

73 Scaling relationship between the breathing rate of animals and their mass Fractality of respiratory organs | 132 |

32 Direct application of dimensional analysis to the modified problem | 71 |

33 Numerical experiment Selfsimilar intermediate asymptotics | 72 |

34 Selfsimilar limiting solution The nonlinear eigenvalue problem | 78 |

Complete and incomplete similarity Selfsimilar solutions of the first and second kind | 82 |

42 Selfsimilar solutions of the first and second kind | 87 |

43 A practical recipe for the application of similarity analysis | 91 |

Scaling and transformation groups Renormalization group | 94 |

the boundary layer on a flat plate in a uniform flow | 96 |

Scaling laws for turbulent wallbounded shear flows at very large Reynolds numbers | 137 |

82 Chorins mathematical example | 140 |

83 Steady shear flows at very large Reynolds numbers The intermediate region in pipe flow | 142 |

84 Modification of IzaksonMillikanvon Mises derivation of the velocity distribution in the intermediate region The vanishingviscosity asymptotics | 150 |

85 Turbulent boundary layers | 154 |

163 | |

170 | |

### Common terms and phrases

absorption applied approximating broken line arbitrary arguments assumed Barenblatt basic bisectrix boundary conditions boundary layer Chapter Chorin and Prostokishin concentrated flooding conservation law considered const constant corresponding density depend determined differential equation dimension function dimensional analysis dimensional parameters dimensionless parameters drag example experimental data Figure fluid flux formulation fractal curves fractal dimension fundamental units G.I. Taylor governing parameters groundwater homogeneity idealized problem incomplete similarity independent dimensions initial condition integral intermediate asymptotics invariant Karman large Reynolds numbers length scale LMT class motion neglected number of governing obtained ordinary differential equation original system parameters with independent phenomena phenomenon pipe pressure properties prototype relation renormalization group scaling law 8.26 segment length self-similar solutions self-similar variables separatrices shear flows shock wave similarity parameters speed of propagation stratum system of units transformation group travelling waves turbulent unit of length universal logarithmic law velocity distribution viscosity Zeldovich