Fluids In Motion

Suppose you wanted to study current patterns in a tidal basin or the effect of gale winds on a tall building. One approach would be to construct scale models and run water through one and blow air on the other. Right? Well, it's not really quite that simple.

In trying to relate flow patterns around a model with what would be observed in the real thing, there are fully eight parameters that must be taken into consideration. In most applications, the most important of these are the relative size of the model, and the viscosity, speed and density of whatever is flowing past it.

Some surprises can pop up. For instance, if engineers are testing scale models of automobiles in a wind tunnel, they must use a higher wind velocity than the highway speed they are trying to duplicate, If, on the other hand, a ship model is being tested in a tank, it must be towed at a lower velocity than the "full-size" speed it is trying to simulate.

In the case of the car and the wind tunnel, a factor called the "Reynolds number" dictates that the dimensions of the model must vary in inverse proportion to the wind velocity. For example, to duplicate the airflow pattern of a full-size car around a 1/10 scale model, a wind velocity 10 times as great as the actual highway speed must be used.

In the case of the ship model, however, the dominant factor is that the model is riding on a "free surface" (unlike the car, which is completely surrounded by the air through which it is passing). In this situation, a factor called the "Froude number" outweighs the Reynolds number, and the scaling factor depends largely on the weight of the water that must be moved. To maintain a constant Froude number between model and full-size ship (which would result in the same wave pattern from both), the dimensions of the model must vary in proportion to the square of the velocity ratio. This means, for example, that a 1/10 scale model would have to be towed through the water at a speed a little less than one-third (1 over the square root of 10) of that for which the real ship is being tested.

These are very simplistic examples of very complex problems. In practice, both the Reynolds and Froude numbers should be satisfied simultaneously (which isn't always possible). In addition, other factors must be taken into consideration, such as the Mach number (when speeds approach that of sound), and equivalent roughnesses of the model and prototype.

Nature, however, has no difficulty in providing us with examples illustrating the mechanics of fluids in motion, even if we don't always recognize them. For example, the equations of motion for aurora-producing plasma in the earth's magnetic field are essentially the same as those governing liquid particle motion in ocean turbidity currents or cream in a cup of coffee.