## 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. |

### 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 |

References | 163 |

170 | |

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### Common terms and phrases

according appears applied approximating arguments assumed asymptotics Barenblatt basic boundary broken line Chapter complete consideration considered constant corresponding curve definition density depend derivation determined dimensional analysis dimensionless dimensions distance distribution drag equal equation example exists experimental experiments explosion expressed fact factor Figure finite flow fluid formulation fractal front function fundamental gives governing parameters head idealized important incomplete similarity independent dimensions initial integral intense intermediate asymptotics layer length limit mass mathematical means measurement motion natural neglected obtained original phenomenon physical pipe possible presented pressure problem propagation properties quantities region relation remains represented Reynolds number scaling law segment self-similar solutions side similarity similarity parameters system of units transformation turbulent units universal values variables velocity viscosity wall wave zero