## Dimensional Analysis

**Introduction: **Similitude refers to the formulation of a description for physical behavior that is general and independent of the individual dimensions, physical properties, forces, etc. In this subsection the treatment of similitude is restricted to dimensional analysis; for a more general treatment see Zlokarnik (1991). The full power and utility of dimensional analysis is often underestimated and underutilized by engineers. This technique may be applied to a complete mathematical model or to a simple listing of the variables that deﬁne the behavior. Only the latter application is described here. For a description of the application of dimensional analysis to a mathematical model see Hellums and Churchill (1964).

**General Principles: **Dimensional analysis is based on the principle that all additive or equated terms of a complete relationshipbetween the variables must have the same net dimensions. The analysis starts with the preparation of alist of the individual dimensional variables (dependent, independent, and parametric) that are presumedto deﬁne the behavior of interest. The performance of dimensional analysis in this context is reasonablysimple and straightforward; the principal difﬁculty and uncertainty arise from the identiﬁcation of thevariables to be included or excluded. If one or more important variables are inadvertently omitted, thereduced description achieved by dimensional analysis will be incomplete and inadequate as a guide forthe correlation of a full range of experimental data or theoretical values.

The familiar band of plottedvalues in many graphical correlations is more often a consequence of the omission of one or morevariables than of inaccurate measurements. If, on the other hand, one or more irrelevant or unimportantvariables are included in the listing, the consequently reduced description achieved by dimensionalanalysis will result in one or more unessential dimensionless groupings. Such excessive dimensionlessgroupings are generally less troublesome than missing ones because the redundancy will ordinarily berevealed by the process of correlation. Excessive groups may, however, suggest unnecessary experimentalwork or computations, or result in misleading correlations. For example, real experimental scatter mayinadvertently and incorrectly be correlated in all or in part with the variance of the excessive grouping.

**Fully Developed Flow of Water through a Smooth Round Pipe**: Choice of Variables. The shear stress two on the wall of the pipe may be postulated to be a function of the density r and the dynamic viscosity m of the water, the inside diameter D of the pipe, and the space mean of the time-mean velocity um. The limitation to fully developed ﬂow is equivalent to a postulate of independence from distance x in the direction of ﬂow, and the speciﬁcation of a smooth pipe is equivalent to the postulate of independence from the roughness e of the wall. The choice of two rather than the pressure drop per unit length –dP/dx avoids the need to include the acceleration due to gravity g and the elevation z as variables. The choice of um rather than the volumetric rate of ﬂow V, the mass rate of ﬂow w, or the mass rate of ﬂow per unit area G is arbitrary but has some important consequences as noted below. The postulated dependence may be expressed functionally as f {tw, r, m, D, um} = 0 or tw = f{r, m, D, um}.

**Correlation of Experimental Data and Theoretical Values:** Correlations of experimental data are generally developed in terms of dimensionless groups rather than in terms of the separate dimensional variables in the interests of compactness and in the hope of greater generality. For example, a complete set of graphical correlations for the heat transfer coefﬁcient h of**.**