CE319F (©Kinnas) FALL 1998
Some notes on similitude

The shown (elementary) force vectors are those acting on the elementary fluid cubes (fluid particles), also shown on the graphs. The pressure force vector (superscript p) is due to the pressures applying on all sides of the fluid cube, the viscous force vector (superscript v) is due to the shear stresses acting on all sides of the fluid cube, the gravity force vector (superscript g) is the weight of the cube, and the inertial force vector (superscript i) is the product of the mass of the fluid cube times the fluid acceleration vector.

Note 1: The shown fluid cubes are at the same relative location with respect to the body (e.g. building) and are also scaled by the same scale ratio.

Note 2: As shown in the graph, due to Newton's law applying on each fluid cube, the vector sum of the pressure, the viscous, and the gravity forces, equals the inertial force vector. In other words, the pressure, the viscous, the gravity, and the inertial force vectors form a closed polygon!

In order to have similarity of the flow patterns between the model and the prototype (also called Kinematic Similarity) we must have Dynamic Similarity (or similarity of the shown force polygons):

where the subscript ``p'' stands for prototype and ``m'' for model. The superscript ``v'' stands for viscous, ``p'' for pressure, ``g'' for gravity, and ``i'' for inertial. The inertial force is defined as:

where M is the mass of the fluid particle and a the acceleration.

Due to similarity the following proportionality relations can be written:

where (Delta p) the difference of pressure from the ambient pressure

Using the above equations we can get:

• from equating the first fraction with the last fraction in equation (1)

  

  or

  

  where Re is the Reynolds number

• from equating the third fraction with the last fraction in equation (1)

  

  or

  

  where Fr is the Froude number

• from equating the second fraction with the last fraction in equation (1)

  

  or

  

  where CP is the pressure coefficient

As mentioned in class it is not feasible to enforce equality of the Reynolds and the Froude numbers at the same time. Thus, depending on the application we can only force equality of the Reynolds number (internal flow or external flow in unbounded fluid) or of the Froude number (flow of body through or under free surface, flow over a spill-way). The equality of the pressure coefficients results AUTOMATICALLY from the equality of Re or Fr numbers, and thus does not need to be enforced.

So, once the dynamic similarity has been enforced we can:

• relate the pressure in the prototype to that (measured) in the model via the equality of the pressure coefficients (see Ex. 8.6 of the textbook).
• relate the total force on the prototype (Fp) to the total force (measured) on the model (Fm) by using the dynamic similarity relations, e.g. in a case where only the Reynolds number is involved:

  

  You can use any of the fractions in the above equation to determine Fp/Fm. In Ex. 8.7 the textbook uses (together with the fact that CPm=CPp)

  

  In the solution of Ex. 8.7 which was presented in class I used:

  

  We could have also used:

  

  All of the above choices will give us the same answer for Fp/Fm, due to dynamic similarity (which has been enforced via equality of Reynolds numbers in the case of Ex. 8.7).