# Similarity Parameters

Any physical problem can be analyzed in the manner just illustrated. It always happens that dimensional reasoning alone cannot show in detail how each parameter (e. g., the viscosity or speed of sound) enters the mathematical expression governing the problem. However, the dimensionless groupings that appear have enormous significance. Hundreds of these groups have been identified as important in engineering applications; they usually are assigned the name of the individual who first recognized its importance. For example, in the aerodynamic force expression derived in Eq. 2.14, the two groups that appear are as follows:

Mach number |
M = V, a |
(2.16) |

Reynolds number |
*,=pVd. e b |
(2.17) |

The first group was identified as a key parameter in describing the effects of compressibility in aerodynamic flows. It was introduced by the Austrian physicist Ernst Mach in the 1870s. The second is named for Osborne Reynolds, who showed in the 1880s that this group governs the role of viscous forces in the flow of liquids or gases. In particular, it controls the transition between laminar and turbulent flow.

Inertia force |
pV2L? |
~ Mass X acceleration |

Pressure force |
ApL2 |
~ Pressure increment X area |

Viscous force |
pVL |
~ Shear stress X area |

Gravity force |
PgL3 |
~ Mass X gravitational acceleration |

Surface-tension force |
cL |
~ Surface tension X length |

Compressibility force |
p dpL2 dp |
~ Pressure required to change density |

Table 2.3. Basic fluid forces |

Each dimensionless group can be identified as the ratio of sets of forces governing the fluid flow. Clearly, the ratio of any two such forces is a dimensionless number like those we found. Some key forces are:

where the dimensional form is expressed by means of the notation introduced in Table 2.2. Thus, the ratio:

inertia force = [pV2!*2] viscous force " WL]

is the Reynolds number. All dimensionless groups can be expressed similarly as ratios of the forces governing the fluid motion. Table 2.3 lists a set of forces and their dimensions that may occur in fluid-dynamics problems. Of these, only the viscous force, pressure force, and force related to compressibility are likely to be encountered in aerodynamics problems.

It is important to demonstrate the power of the concept of similarity parameters. Using the Reynolds number to illustrate, consider what the difference would be in a flow with a high Re compared to one with a low Re value. For example, the flow over an airplane wing in low-speed flight typically has a Re number (based on the flight speed and average width of the wing) of two million or three million. From Eq. 2.20, it is clear that this means that the inertia forces—those due to acceleration or deceleration of gas particles moving over the wing—are enormous compared to the viscous forces due to the shearing stresses along the surface. For a second example, consider the flow of ketchup from an upturned bottle. Here, the Re is small, indicating the overwhelming importance of the viscous forces in controlling the flow.

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