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Dimensional Analysis, Similitude and Physical Experiments at Laboratory Scale

  • Kolumban HutterEmail author
  • Yongqi Wang
Chapter
  • 1.7k Downloads
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

Abstract

This chapter is devoted to the subjects ‘Dimensional analysis, similitude and physical experimentation at laboratory scale’, topics often not systematically taught at higher technical education. However, no insider would deny the usefulness of these specialties. Books treating these subjects separately and in sufficient detail have appeared since the mid 20th century. We give an account of dimensional analysis, define dimensional homogeneity of functions of mathematical physics, the properties of which culminate in Buckingham’s theorem (which is proved in an appendix to the chapter); its use is illustrated by a diversity of problems from general fluid dynamics, gas dynamics and thermal sciences, e.g., propagation of a shock from a point source, rising gas bubbles, RayleighBénard instability, etc. The theory of physical models develops rules, how to down- or up-scale physical processes from the size of a prototype to the size of the model. The theory shows that in general such scaling transformations are practically never exactly possible, so that scale effects enter in these cases, which distort the model results in comparison to those in the prototype. In hydraulic applications, this leads to the so-called Froude and Reynolds models, in which the Froude or Reynolds numbers, respectively, remain mapping invariants but not the other. Application on sediment transport in rivers, heat transfer in forced convection, etc. illustrate the difficulties. The chapter ends with the characterization of dimensional homogeneity of the equations describing physical processes by their governing differential equations. The NavierStokesFourierFick fluid equations serve as illustration.

Keywords

Similitude and model experiments Dimensional homogeneity Buckingham’s theorem Viscosity in a kinetic gas Motion of a shock front after an explosion RayleighBénard instability Sediment transport in a shallow lake Model theory and differential equations NavierStokesFourierFick fluid 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.c/o Versuchsanstalt für Wasserbau, Hydrologie und GlaziologieETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany

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