Dimensional analysis of models and data sets

American Journal of Physics

Volumn 71, Issue 5, 2003, Pages 437-447

  Price, James F ^a,b  

^a WOODS HOLE OCEANOGRAPHIC INSTITUTION   (United States)

^b OFFICE OF NAVAL RESEARCH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038393253     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1533057     Document Type: Article

References (28)

1. E. Thurairajasingam, E. Shayan, and S. Masood,'"Modeling of a continuous food pressing process by dimensional analysis," Comput. Ind. Eng. (in press); J. R. Hutchinson and M. Garcia, "Tyrannosaurus was not a fast runner," Nature (London) 415, 1018-1022 (2002).
2. E. Thurairajasingam, E. Shayan, and S. Masood,'"Modeling of a continuous food pressing process by dimensional analysis," Comput. Ind. Eng. (in press); J. R. Hutchinson and M. Garcia, "Tyrannosaurus was not a fast runner," Nature (London) 415, 1018-1022 (2002).
3. F. Wilczek, "Getting its from bits," Nature (London) 397, 303-306 (1999).
4. This essay builds upon the introduction to dimensional analysis that can be found in most comprehensive fluid mechanics textbooks. Recent examples include P. K. Kundu and I. C. Cohen, Fluid Mechanics (Academic, New York, 2001); B. R. Munson, D. F. Young, and T. H. Okiishi, Fundamentals of Fluid Mechanics (Wiley, New York, 1998), 3rd ed.; D. C. Wilcox, Basic Fluid Mechanics (DCW Industries, La Canada, CA, 2000); F. M. White, Fluid Mechanics (McGraw-Hill, New York, 1994), 3rd ed. An older but very useful reference is by H. Rouse, Elementary Mechanics of Fluids (Dover, New York, 1946). A particularly good discussion of the relationship between dimensional analysis and other analysis methods is by C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (MacMillan, New York, 1974).
5. This essay builds upon the introduction to dimensional analysis that can be found in most comprehensive fluid mechanics textbooks. Recent examples include P. K. Kundu and I. C. Cohen, Fluid Mechanics (Academic, New York, 2001); B. R. Munson, D. F. Young, and T. H. Okiishi, Fundamentals of Fluid Mechanics (Wiley, New York, 1998), 3rd ed.; D. C. Wilcox, Basic Fluid Mechanics (DCW Industries, La Canada, CA, 2000); F. M. White, Fluid Mechanics (McGraw-Hill, New York, 1994), 3rd ed. An older but very useful reference is by H. Rouse, Elementary Mechanics of Fluids (Dover, New York, 1946). A particularly good discussion of the relationship between dimensional analysis and other analysis methods is by C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (MacMillan, New York, 1974).
6. This essay builds upon the introduction to dimensional analysis that can be found in most comprehensive fluid mechanics textbooks. Recent examples include P. K. Kundu and I. C. Cohen, Fluid Mechanics (Academic, New York, 2001); B. R. Munson, D. F. Young, and T. H. Okiishi, Fundamentals of Fluid Mechanics (Wiley, New York, 1998), 3rd ed.; D. C. Wilcox, Basic Fluid Mechanics (DCW Industries, La Canada, CA, 2000); F. M. White, Fluid Mechanics (McGraw-Hill, New York, 1994), 3rd ed. An older but very useful reference is by H. Rouse, Elementary Mechanics of Fluids (Dover, New York, 1946). A particularly good discussion of the relationship between dimensional analysis and other analysis methods is by C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (MacMillan, New York, 1974).
7. This essay builds upon the introduction to dimensional analysis that can be found in most comprehensive fluid mechanics textbooks. Recent examples include P. K. Kundu and I. C. Cohen, Fluid Mechanics (Academic, New York, 2001); B. R. Munson, D. F. Young, and T. H. Okiishi, Fundamentals of Fluid Mechanics (Wiley, New York, 1998), 3rd ed.; D. C. Wilcox, Basic Fluid Mechanics (DCW Industries, La Canada, CA, 2000); F. M. White, Fluid Mechanics (McGraw-Hill, New York, 1994), 3rd ed. An older but very useful reference is by H. Rouse, Elementary Mechanics of Fluids (Dover, New York, 1946). A particularly good discussion of the relationship between dimensional analysis and other analysis methods is by C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (MacMillan, New York, 1974).
8. This essay builds upon the introduction to dimensional analysis that can be found in most comprehensive fluid mechanics textbooks. Recent examples include P. K. Kundu and I. C. Cohen, Fluid Mechanics (Academic, New York, 2001); B. R. Munson, D. F. Young, and T. H. Okiishi, Fundamentals of Fluid Mechanics (Wiley, New York, 1998), 3rd ed.; D. C. Wilcox, Basic Fluid Mechanics (DCW Industries, La Canada, CA, 2000); F. M. White, Fluid Mechanics (McGraw-Hill, New York, 1994), 3rd ed. An older but very useful reference is by H. Rouse, Elementary Mechanics of Fluids (Dover, New York, 1946). A particularly good discussion of the relationship between dimensional analysis and other analysis methods is by C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (MacMillan, New York, 1974).
9. This essay builds upon the introduction to dimensional analysis that can be found in most comprehensive fluid mechanics textbooks. Recent examples include P. K. Kundu and I. C. Cohen, Fluid Mechanics (Academic, New York, 2001); B. R. Munson, D. F. Young, and T. H. Okiishi, Fundamentals of Fluid Mechanics (Wiley, New York, 1998), 3rd ed.; D. C. Wilcox, Basic Fluid Mechanics (DCW Industries, La Canada, CA, 2000); F. M. White, Fluid Mechanics (McGraw-Hill, New York, 1994), 3rd ed. An older but very useful reference is by H. Rouse, Elementary Mechanics of Fluids (Dover, New York, 1946). A particularly good discussion of the relationship between dimensional analysis and other analysis methods is by C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (MacMillan, New York, 1974).
10. An algorithm for computing nondimensional variables has been implemented in MATLAB and can be downloaded from the author's web page, 〈http://www.whoi.edu/science/PO/people/jprice/misc/Danalysis.m〉 or from the MATLAB archive (the file name is Danalysis.m). Also available is a manuscript that treats aspects of dimensional analysis that are not touched on here, 〈http://www.whoi.edu/science/PO/people/jprice/class/ ND.pdf〉.
11. The simple pendulum is the starting point for most discussion of dimensional analysis including the classic text by P. W. Bridgman, Dimensional Analysis (Yale U.P., New Haven, CT, 1937), 2nd ed., which is an excellent introduction to the topic, and the more advanced treatment by L. I. Sedov, Similarity and Dimensional Methods in Mechanics (Academic, New York, 1959). Still more advanced is G. I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics (Cambridge U.P., Cambridge, 1996).
12. The simple pendulum is the starting point for most discussion of dimensional analysis including the classic text by P. W. Bridgman, Dimensional Analysis (Yale U.P., New Haven, CT, 1937), 2nd ed., which is an excellent introduction to the topic, and the more advanced treatment by L. I. Sedov, Similarity and Dimensional Methods in Mechanics (Academic, New York, 1959). Still more advanced is G. I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics (Cambridge U.P., Cambridge, 1996).
13. The simple pendulum is the starting point for most discussion of dimensional analysis including the classic text by P. W. Bridgman, Dimensional Analysis (Yale U.P., New Haven, CT, 1937), 2nd ed., which is an excellent introduction to the topic, and the more advanced treatment by L. I. Sedov, Similarity and Dimensional Methods in Mechanics (Academic, New York, 1959). Still more advanced is G. I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics (Cambridge U.P., Cambridge, 1996).
14. An excellent discussion of physical measurement and much else that is relevant to dimensional analysis is given by A. A. Sonin, "The physical basis of dimensional analysis," 2001. This unpublished manuscript is available from 〈http://me.mit.edu/people/sonin/html〉.
15. -1 is the rotation rate of the earth and R is the distance normal to the rotation axis).
16. T. Szirtes, Applied Dimensional Analysis and Modeling (McGraw-Hill, Englewood Cliffs, NJ, 1997).
17. S. Brückner and the University of Stuttgart Pi-Group 〈http:// www.pigroup.de/〉, is an excellent resource for advanced applications of dimensional analysis.
18. The calculation of a null space basis is, in effect, what all computational methods accomplish, and was noted by E. A. Bender, An Introduction to Mathematical Modelling (Dover, New York, 1977). We delegate the calculation to the computer, and emphasize those properties of the null space basis that are essential for the present purpose.
19. 5.
20. G. Strang, Introduction to Linear Algebra (Wellesley-Cambridge Press, Wellesley, MA, 1998).
21. 0=π. Is dimensional analysis of any further use for explaining this? Consider energy conservation.
22. Detailed treatment of damping processes are by P. T. Squire, "Pendulum damping," Am. J. Phys. 54, 984-991 (1986) and R. A. Nelson and M. G. Olsson, "The pendulum: Rich physics from a simple system," ibid. 54, 112-121 (1985).
23. Detailed treatment of damping processes are by P. T. Squire, "Pendulum damping," Am. J. Phys. 54, 984-991 (1986) and R. A. Nelson and M. G. Olsson, "The pendulum: Rich physics from a simple system," ibid. 54, 112-121 (1985).
24. 1.
25. More recent textbooks (Ref. 3), like this article, show only the curve that runs through the middle of a tight cloud of data points that have accumulated from many laboratory experiments, see for example, Rouse (Ref. 3). What is most important, but not evident from this kind of presentation, is that drag coefficients inferred from experiments made using a very wide range of spheres and cylinders moving at widely differing speeds and through many different viscous fluids (Newtonian fluids) do indeed collapse to a well-defined function of Reynolds number alone, just as dimensional analysis had indicated. This is a result, characteristic of dimensional analysis generally, that is at once profound and trivial. One might say trivial because, after all, dimensional analysis told us that the drag coefficient must depend upon Re alone. From this perspective, an effective collapse of the data verifies that carefully controlled laboratory conditions can indeed approximate the idealized physical model. It is profound in that dimensional analysis has shown the way to a useful result (Fig. 5), where there would otherwise have been be an unwieldy mass of highly specific data (as in going from Fig. 1 to Fig. 2). An open question of considerable practical importance is whether the steady drag laws are robust in the sense of giving useful estimates in practical problems, say our pendulum, in which the idealized conditions are inevitably violated. Other data sets have been developed to define the effects of idealized surface roughness, for example, but our pendulum has a long list of violations - time-dependence, a nearby solid boundary (the floor), slight surface roughness, etc., all present at once, so that we are on our own. About all that can be said is that it is important to understand the full set of assumptions under which a similarity law has been defined, and to be skeptical of applications outside of those bounds.
26. 5 due to changes in the viscous boundary layer and the width of the wake behind a moving sphere. This is the Re range of a well-hit golf ball or tennis ball, and is part of the reason that aerodynamic drag on these objects has a surprising sensitivity to surface roughness or spin. For much more detail on these phenomenon see S. Vogel, Life in Moving Fluids (Princeton U.P., New York, 1994), and P. Timmerman and J. P. van der Weele, "On the rise and fall of a ball with linear and quadratic drag," Am. J. Phys. 67, 538-546 (1999).
27. 5 due to changes in the viscous boundary layer and the width of the wake behind a moving sphere. This is the Re range of a well-hit golf ball or tennis ball, and is part of the reason that aerodynamic drag on these objects has a surprising sensitivity to surface roughness or spin. For much more detail on these phenomenon see S. Vogel, Life in Moving Fluids (Princeton U.P., New York, 1994), and P. Timmerman and J. P. van der Weele, "On the rise and fall of a ball with linear and quadratic drag," Am. J. Phys. 67, 538-546 (1999).
28. 2/s at a temperature = 40°C). Given these results, can you think of a name more apt than "viscous" pendulum?