Dimensional Analysis of Models and Data Sets
Dimensional analysis of models and data sets
James F. Pricea)
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543
Received 22 May 2002; accepted 4 November 2002 Dimensional analysis is a widely applicable and sometimes very powerful technique that is demonstrated here in a study of the simple, viscous pendulum. The first and crucial step of dimensional analysis is to define a suitably idealized representation of a phenomenon by listing the relevant variables, called the physical model. The second step is to learn the consequences of the physical model and the general principle that complete equations are independent of the choice of units. The calculation that follows yields a basis set of nondimensional variables. The final step is to interpret the nondimensional basis set in the light of observations or existing theory, and if necessary to modify the basis set to maximize its utility. One strategy is to nondimensionalize the dependent variable by a scaling estimate. The remaining nondimensional variables can then be formed in ways that define aspect ratios or that correspond to the ratio of terms in a governing equation. © 2003 American Association of Physics Teachers. DOI: 10.1119/1.1533057
I. ABOUT DIMENSIONAL ANALYSIS Dimensional analysis is a remarkable tool insofar as it can be applied to virtually all quantitative models and data sets. Topics in the recent literature include donuts, dinosaurs,1 and the most fundamental theories of physics.2 In some instances dimensional analysis is very powerful; results include the log-layer profile of a turbulent boundary layer and the spectral slope in the inertial subrange of isotropic turbulence, both landmarks in fluid mechanics.3 More often the result of dimensional analysis is a hint at the form of a solution or a more effective way to display or correlate a large data set. These kinds of results, though seldom complete if taken alone, are an essential element of many investigations. This paper
James F. Pricea)
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543
Received 22 May 2002; accepted 4 November 2002 Dimensional analysis is a widely applicable and sometimes very powerful technique that is demonstrated here in a study of the simple, viscous pendulum. The first and crucial step of dimensional analysis is to define a suitably idealized representation of a phenomenon by listing the relevant variables, called the physical model. The second step is to learn the consequences of the physical model and the general principle that complete equations are independent of the choice of units. The calculation that follows yields a basis set of nondimensional variables. The final step is to interpret the nondimensional basis set in the light of observations or existing theory, and if necessary to modify the basis set to maximize its utility. One strategy is to nondimensionalize the dependent variable by a scaling estimate. The remaining nondimensional variables can then be formed in ways that define aspect ratios or that correspond to the ratio of terms in a governing equation. © 2003 American Association of Physics Teachers. DOI: 10.1119/1.1533057
I. ABOUT DIMENSIONAL ANALYSIS Dimensional analysis is a remarkable tool insofar as it can be applied to virtually all quantitative models and data sets. Topics in the recent literature include donuts, dinosaurs,1 and the most fundamental theories of physics.2 In some instances dimensional analysis is very powerful; results include the log-layer profile of a turbulent boundary layer and the spectral slope in the inertial subrange of isotropic turbulence, both landmarks in fluid mechanics.3 More often the result of dimensional analysis is a hint at the form of a solution or a more effective way to display or correlate a large data set. These kinds of results, though seldom complete if taken alone, are an essential element of many investigations. This paper