## Model Similitude

The principle of dynamically similar fluid motion forms the basis for the design and operation of hydraulic models and the interpretation of experimental data. The basic concept of dynamic similarity is that two systems with geometrically similar boundaries have similar flow patterns at corresponding instants of time.11,12 To achieve this, all individual forces acting on corresponding fluid elements must have the same ratios in the two geometrically similar systems. The condition required for complete similitude may be developed from Newton's second law of motion:

where Ft = inertial force, defined as mass m times acceleration a

Fp = pressure force connected with or resulting from the motion Fg = gravitational force Fv = viscous force Ft = surface tension force

Additional forces, such as fluid compression, magnetic, or Coriolis forces, may be relevant under special circumstances, but generally these forces have little influence and are, therefore, not considered in the following development.

Equation 1 can be made dimensionless by dividing all the terms by Ft. Two systems of different size that are geometrically similar are dynamically similar if both satisfy the same dimensionless form of Equation 1. We may write each of the forces of Equation 1 as

Fp = net pressure difference X area = a1\pL2 Fg = specific weight X volume = a2gL3 Fv = shear stress X area = (aymAa/Ay) (area) = a3muL Ft = surface tension X length = a4sL

Fi = density X volume X acceleration = a5pL3u2/L = a5pu2L2

where

 a1, a2, and so on = proportionality factors Ap = net pressure difference

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