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The Physical Basis of DIMENSIONAL ANALYSIS

1 The Physical Basis ofDIMENSIONAL ANALYSISAin A. SoninSecond Edition2 Copyright 2001 by Ain A. SoninDepartment of Mechanical EngineeringMITC ambridge, MA 02139 First Edition published 1997. Versions of this material have been distributed in Fluid Mechanics and other courses at MIT since picture by Pat Keck (Untitled, 1992)3 Contents1. Introduction12. Physical Quantities and quantities and base Unit and numerical Derived quantities, dimension, and dimensionless quantities Physical equations, DIMENSIONAL homogeneity, andphysical constants Derived quantities of the second of units Recapitulation 273.

Step 2: Dimensional considerations 30 Step 3: Dimensional variables 32 Step 4: The end game and Buckingham’s Π -theorem 32 3.2 Example: Deformation of an elastic sphere striking a wall 33 Step 1: The independent variables 33 Step 2: Dimensional considerations 35 Step 3: Dimensionless similarity parameters 36 Step 4: The end game 37

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Transcription of The Physical Basis of DIMENSIONAL ANALYSIS

1 1 The Physical Basis ofDIMENSIONAL ANALYSISAin A. SoninSecond Edition2 Copyright 2001 by Ain A. SoninDepartment of Mechanical EngineeringMITC ambridge, MA 02139 First Edition published 1997. Versions of this material have been distributed in Fluid Mechanics and other courses at MIT since picture by Pat Keck (Untitled, 1992)3 Contents1. Introduction12. Physical Quantities and quantities and base Unit and numerical Derived quantities, dimension, and dimensionless quantities Physical equations, DIMENSIONAL homogeneity, andphysical constants Derived quantities of the second of units Recapitulation 273.

2 DIMENSIONAL ANALYSIS steps of DIMENSIONAL ANALYSIS and Buckingham sPi-Theorem29 Step 1: The independent variables29 Step 2: DIMENSIONAL considerations30 Step 3: DIMENSIONAL variables32 Step 4: The end game and Buckingham s Example: Deformation of an elastic sphere striking a wall33 Step 1: The independent variables33 Step 2: DIMENSIONAL considerations35 Step 3: Dimensionless similarity parameters36 Step 4: The end the utility of DIMENSIONAL ANALYSIS and some difficultiesand questions that arise in its application 37 Similarity 37 Out-of-scale modeling38 DIMENSIONAL ANALYSIS reduces the number of variablesand minimizes incomplete set of independent quantities maydestroy the analysis40 Superfluous independent quantities complicate the resultunnecessarily40On the importance of simplifying assumptions41On choosing a complete set of independent variables42 The result is independent of how one chooses a dimensionallyindependent subset43 The result

3 Is independent of the type of system of ANALYSIS in Problems Where Some IndependentQuantities Have Fixed Values45 Cited References49 Other Selected References51 AcknowledgementsMy thanks to Mark Bathe, who volunteered to perform thecomputation for the elastic ball. This work was begun withsupport from the Gordon Bacon (1561-1628)1: I found that I was fitted for nothing so well as the study ofTruth; as having a nimble mind and versatile enough tocatch the resemblance of things (which is the chief point),and at the same time steady enough to fix and distinguishtheir subtle Think things, not words.

4 Albert Einstein (1879-1955)2: .. all knowledge starts from experience and ends in arrived at by purely logical means arecompletely empty as regards reality."Percy W. Bridgman (1882-1961)3: ..what a man means by a term is to be found by observingwhat he does with it, not by what he says about it. 1 Catherine Drinker Bowen, 19632 Einstein, 19333 Bridgman, 195061. IntroductionDimensional ANALYSIS offers a method for reducing complex physicalproblems to the simplest (that is, most economical) form prior to obtaininga quantitative answer.

5 Bridgman (1969) explains it thus: "The principaluse of DIMENSIONAL ANALYSIS is to deduce from a study of the dimensions ofthe variables in any Physical system certain limitations on the form of anypossible relationship between those variables. The method is of greatgenerality and mathematical simplicity".At the heart of DIMENSIONAL ANALYSIS is the concept of similarity. Inphysical terms, similarity refers to some equivalence between two thingsor phenomena that are actually different. For example, under some veryparticular conditions there is a direct relationship between the forcesacting on a full-size aircraft and those on a small-scale model of it.

6 Thequestion is, what are those conditions, and what is the relationshipbetween the forces? Mathematically, similarity refers to a transformationof variables that leads to a reduction in the number of independentvariables that specify the problem. Here the question is, what kind oftransformation works? DIMENSIONAL ANALYSIS addresses both thesequestions. Its main utility derives from its ability to contract, or makemore succinct, the functional form of Physical relationships. A problemthat at first looks formidable may sometimes be solved with little effortafter DIMENSIONAL problems so well understood that one can write down inmathematical form all the governing laws and boundary conditions, andonly the solution is lacking, similarity can also be inferred by normalizingall the equations and boundary conditions in terms of quantities thatspecify the problem and identifying the dimensionless groups that appearin the resulting dimensionless equations.

7 This is an inspectional form ofsimilarity ANALYSIS . Since inspectional ANALYSIS can take advantage of theproblem's full mathematical specification, it may reveal a higher degree ofsimilarity than a blind (less informed) DIMENSIONAL ANALYSIS and in thatsense prove more powerful. DIMENSIONAL ANALYSIS is, however, the onlyoption in problems where the equations and boundary conditions are notcompletely articulated, and always useful because it is simple to apply andquick to give insight. Some of the basic ideas of similarity and DIMENSIONAL ANALYSIS hadalready surfaced in Fourier's work in the nineteenth century's first quarter,7but the subject received more methodical attention only toward the closeof that century, notably in the works of Lord Rayleigh, Reynolds,Maxwell, and Froude in England, and Carvallo, Vaschy and a number ofother scientists and engineers in France (Macagno, 1971)4.

8 By the 1920'sthe principles were essentially in place: Buckingham's now ubiquitous theorem had appeared (Buckingham, 1914), and Bridgman hadpublished the monograph which still remains the classic in the field(Bridgman, 1922, 1931). Since then, the literature has grown now include aerodynamics, hydraulics, ship design,propulsion, heat and mass transfer, combustion, mechanics of elastic andplastic structures, fluid-structure interactions, electromagnetic theory,radiation, astrophysics, underwater and underground explosions, nuclearblasts, impact dynamics, and chemical reactions and processing (see forexample Sedov, 1959, Baker et al, 1973, Kurth, 1972, Lokarnik, 1991)

9 ,and also biology (McMahon & Bonner, 1983) and even economics (deJong, 1967).Most applications of DIMENSIONAL ANALYSIS are not in question, nodoubt because they are well supported by experimental facts. The debateover the method's theoretical-philosophical underpinnings, on the otherhand, has never quite stopped festering ( Palacios, 1964).Mathematicians tend to find in the basic arguments a lack of rigor and aretempted to redefine the subject in their own terms ( Brand, 1957),while physicists and engineers often find themselves uncertain about thephysical meanings of the words in terms of which the ANALYSIS cast.

10 Theproblem is that DIMENSIONAL ANALYSIS is based on ideas that originate atsuch a substratal point in science that most scientists and engineers havelost touch with them. To understand its principles, we must return to someof the very fundamental concepts in ANALYSIS is rooted in the nature of the artifices weconstruct in order to describe the Physical world and explain itsfunctioning in quantitative terms. Einstein (1933) has said, "Pure logicalthinking cannot yield us any knowledge of the empirical world; allknowledge starts from experience and ends in it.


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