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Similitude is a concept used in the testing of engineering models. A model is said to have similitude with the real application if the two share Geometric similarity, Kinematic similarity and Dynamic similarity.
Similarity and Similitude are interchangeable in this context. The term Dynamic Similitude is often used because it implies that geometric and kinematic similitude have already been met. The model, and the conditions under which it is tested, are designed to insure Dynamic Similitude with the application.
Similitude's main application is in hydraulic and aerospace engineering to test fluid flow conditions with scaled models. It is also the primary theory behind many textbook formulas in fluid mechanics.
Engineering models are used to study complex fluid dynamics problems where calculations and computer simulations aren't reliable. Models are usually smaller than the final design, but not always. Scale models allow testing of a design prior to building, and in many cases are a critical step in the development process.
Construction of a scale model, however, must be accompanied with an analysis to determine what conditions it is tested under. While the geometry may be simply scaled, other parameters, such as pressure, temperature or the velocity and type of fluid may need to be altered. Similitude is achieved when testing conditions are created such that the test results are applicable to the real design.
The following criteria are required to achieve similitude;
To satisfy the above conditions the application is analyzed;
It is often impossible to achieve strict similitude during a model test. The greater the departure from the application's operating conditions, the more difficult achieving similitude is. In these cases some aspects of similitude may be neglected, focusing on only the most important parameters.
Consider a submarine modeled at 1/40th scale. The application operates in sea water at .5 ° C, moving at 5 meters/sec. The model will be tested in fresh water at 20 ° C. Find the power required for the submarine to operate at the stated speed.
A free body diagram is constructed and the relevent relationships of force and velocity are formulated using techniques from continuum mechanics. The variables which describe the system are;
| Variable | Application | Scaled model | Units |
|---|---|---|---|
| L (dimension of submarine) | 1 | 1/40 | (unit length) |
| T (Temperature) | .5 | 20 | ° C |
| V (Velocity) | 5 | Calculate | (m/s) |
| (Density) | 1028 | 998 | (Kg/m3) |
| (Dynamic Viscosity) | 1.88 | 1.00 | (N s/m2) |
| F (force) | Calculate | To be measured | (N) |
The Buckingham Pi theoremThe Buckingham π theorem is a key theorem in dimensional analysis. The theorem states that the functional dependence between a certain number (e. n of variables can be reduced by the number (e. k of independent dimensions occurring in those variables t is invoked to show that the system can be modeled with one independent variable (F) and two dimensionless numbers (in this case the Reynolds numberThe Reynolds number is the most important dimensionless number in fluid dynamics and provides a criterion for dynamic similarity; in other words when two similar objects of possibly different sizes in perhaps different fluids with different flowrates will () and Pressure coefficientThe pressure coefficient is a dimensionless number used in aerodynamics. Most often in the design and analysis of an airfoil. The relationship between the coefficient and the dimensional number is: where is the free stream pressure is the fluid density (s ()). The dimsionless numbers account for all the variables listed above except F. Since the dimensionless parameters will stay constant for both the test and the real application, they will be used to formulate scaling laws for the test.
Scaling Laws;
This gives a required test velocity of;
The force measured from the model at that velocity is then scaled to find the force that can be expected for the real application;
The power required by the submarine is then;
Note that even though the model is scaled smaller, the water velocity needs to be increased for testing. This remarkable result shows how similitude in nature is often counterintuitive.