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Chapter 7: Problem 78

A 1: 50 -scale model of a submarine is to be tested in a towing tank under two conditions: motion at the free surface and motion far below the surface. The tests are performed in freshwater. On the surface, the submarine cruises at 24 knots. At what speed should the model be towed to ensure dynamic similarity? Far below the surface, the sub cruises at 0.35 knot. At what speed should the model be towed to ensure dynamic similarity? What must the drag of the model be multiplied by under each condition to give the drag of the full-scale submarine?

Short Answer

Expert verified
The model should be towed at a speed of \(3.396\) knots when at the surface and \(0.0495\) knots for far below the surface to maintain dynamic similarity. To determine the drag of the full-scale submarine, the model's drag should be multiplied by \(125,000\) (which is \(50^3\)).

Step by step solution

01

Calculate the speed for motion at free surface

We will use Froude's law of dynamic similarity to calculate the speed. The Froude number is maintained constant, given by the formula: Fr1=Fr2, where v1 and v2 are velocities of the actual submarine and the model, L1 and L2 are their lengths. Therefore, \(v2 = v1 \sqrt{\frac {L1}{L2}}\). Here, the scale model is 1:50 and surface speed of the submarine is 24 knots. Substituting these values, we find the speed of the model on the surface.
02

Calculate the speed for motion far below the surface

Following similar steps as in Step 1, we use Froude's law of dynamic similarity to calculate the speed far below the surface. The only difference here is the speed of the actual submarine that cruises at 0.35 knot. Substituting this value we calculate the speed of the model submarine for motion far below the surface.
03

Calculation of drag on the model

Remember that forces scale with the cube of the lengths. Therefore, the drag of the model has to be multiplied by the scale factor cubed to obtain the full-scale submarine drag. The scale factor is 50, so calculate the cube of the scale factor, which is the multiplication factor for the model's drag.

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