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2020 | Book

Mechanics of Flow Similarities

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About this book

The mechanics of similarity encompasses the analysis of dimensions, performed by various procedures, the gasdynamic similarity and the model technology. The analysis of dimensions delivers the dimensionless numbers by which specific physical challenges can be described with a reduced number of variables. Thereby the assessment of physical problems is facilitated. For fluid dynamics and all sorts of heat transfer the discipline of the mechanics of similarity was so important in the past, that the historical background is highlighted of all the persons who have contributed to the development of this discipline.

The goal of the classical gasdynamic similarity was to find rules, which enables the aerodynamic engineer to perform transformations from existing flow fields to others, which meet geometrical and other specific flow field parameters. Most of these rules and findings do no longer play a role today, because a lot of potent experimental and theoretical/numerical methods are now available. This problem is addressed in the book.

A recent investigation regarding the longitudinal aerodynamics of space vehicles has revealed, that there exist other astonishing similarities for hypersonic and supersonic flight Mach numbers. It seems, that obviously most of the longitudinal aerodynamics is independent from the geometrical configurations of the space vehicle considered, if a simple transformation is applied. A section of this book is devoted to these new findings.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
Over a large number of decades, after Leonhard Euler (1707–1783) had formulated the system of equations for the description of convective processes in three-dimensional flow fields and further after its extension to flows with friction by Navier (1822), Poisson (1831), Saint-Vernant (1843) and Stokes (1845), a solution of such kind of systems of equations seemed to be impossible.
Claus Weiland
Chapter 2. Dimensional Analysis—Buckingham’s Theorem
Abstract
Edgar Buckingham was born on July 8, 1867 in Philadelphia, Pennsylvania, USA and died on April 25, 1940 in Washington, DC. He had his studies in Harvard, USA, at the University of Strasbourg, France and the University of Leipzig, Germany, where he received his PhD in 1893. From 1902 to 1906 he worked as a soil physicist at the US Bureau of Soils and afterwards accepted (1907) a post at the National Bureau of Standards, where he remained until his retirement in 1937. He performed a lot of work in the discipline of soil physics and published the results in some widely acknowledged papers, [1, 2]. In 1914 he presented the paper [3], in which he summed up the theory of the \(\varPi \) theorem with its fundamental significance for the analysis of dimensions, see also [4].
Claus Weiland
Chapter 3. The Fractional Analysis Method
Abstract
The Fractional Analysis Method is a heuristic method and was developed in the second part of the 19th century by J. W. Strutt, third Baron Rayleigh, 1842–1919, [1], the man who received the Nobel Prize in 1904, see also Sect. 6.​16, [2]. It encompasses the relationships of single forces and of single energy parts, which are compared under physical aspects. Of course, in the final outcome there is nothing different to the method of differential equations, Chap. 4, or the dimensional analysis, Chap. 2. Nevertheless it makes sense to have a view on this method in particular from the physical, but also from the historical standpoint. Physically, the reader gets a feeling about the quantities involved. Historically, it shows which interests the scientists had at that time.
Claus Weiland
Chapter 4. Method of Differential Equations
Abstract
First we state that the governing equations, which describe general three dimensional flow fields, can be found in more or less all fluiddynamic textbooks. Some of them give the complete mathematical derivations of the equations both in integral and differential form, [1–6].
Claus Weiland
Chapter 5. Classification of Dimensionless Numbers—Similarity Parameters
Abstract
This chapter gives an overview over the dimensionless numbers treated in this book, whereat the details of these quantities are investigated in Chap. 6. The basic equations for the aerodynamic, aerothermodynamic and all sorts of heat transfer problems are the Navier–Stokes equations with the appropriate initial and boundary conditions and if necessary expanded by the relations for the thermodynamic equilibrium state and in particular by the equations for the thermodynamic non-equilibrium state (species continuity, excitation of molecular vibrations, electron modes).
Claus Weiland
Chapter 6. Dimensionless Numbers—Similarity Parameters: A Look at the Name Holders
Abstract
In Chaps. 2, 3 and 4 we presented various methods with which a count of dimensional numbers, depending on different methods, can be derived. Obviously, Buckingham’s \(\varPi \)theorem has, there is no doubt, the capacity with the greatest possible extent. Historically many of the similarity quantities for the first time were formulated as a single event, in particular those known from the 18th and the first part of the 19th century. Therefore, the appearance of the dimensionless numbers obviously was an evolutionary process. Most of the power products, later noted as dimensionless numbers or similarity parameters, were established before the mathematical calculus of the analysis of dimension, Buckingham’s \(\varPi \)theorem, was developed.
Claus Weiland
Chapter 7. Gasdynamic Similarity
Abstract
As we have already discussed in Chaps. 2, 3 and 4 the calculus and the aim of the procedure and the determination of dimensionless numbers is to reduce the amount of variables, which describe fluid dynamical and thermal processes.
Claus Weiland
Chapter 8. Model Test Entity
Abstract
When humans have built large technical devices like bridges, ships, buildings, embankment dams, airplanes, space planes, reactors for chemical purposes, hydraulic engines (pumps, compressors, turbines) etc. there was a need to fabricate geometrically similar models of these devices in order to investigate and test their physical behavior in ground based test facilities. The main question from the physical point of view in this regard is under what conditions are the experimental test results, received with geometrically similar models in test facilities, transferable to the situation of the original devices? We define the three similarity rules, which must be kept for a suitable testing of the physics of originals in test facilities, [1, 2].
Claus Weiland
Backmatter
Metadata
Title
Mechanics of Flow Similarities
Author
Dr. Claus Weiland
Copyright Year
2020
Electronic ISBN
978-3-030-42930-0
Print ISBN
978-3-030-42929-4
DOI
https://doi.org/10.1007/978-3-030-42930-0

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