Nondimensionalization of Physical Problems

Most of the physical quantities considered up to now have dimensions. That is, one must specify a unit of measure for the physical quantity. For example, the length of an object is measured in units of meters or millimeters. These fixed measuring units are man-made and do not depend on the specific problem at hand. It may be advantageous to express the dimensional physical quantities in terms of units that are natural to the problem that is, the unit of measure depends on the characteristics of the problem such as boundary and initial conditions and physical constants. 

Nondimensionalization of a dimensional physical quantity x usually involves specifying a reference xref and a scale xscaie. The nondimensional quantity x is then expressed as 

The reference and the scale have the same units as x. A nonzero reference indicates that the difference between x and xref is important to the specific problem at hand rather than the absolute value of x itself. xref is determined from the problem and is typically some value of x at the boundary or some initial time. The scale is some combination of other dimensional quantities that are relevant to the physical problem, such as boundary and initial conditions and physical constants. If chosen properly, the scale provides a measure of the range of values that the variable x — xref will take for the particular physical problem. The ideal choice of the reference and scale quantities for a given problem will result in order-unity values for the nondimensional quantities. 

To illustrate the nature of nondimensionalization of a variable, take the example of pressure, p. For high-speed compressible flows, an appropriate nondimensional pressure 

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