example is rather extreme it nevertheless hints at the necessity to consider

Example is rather extreme it nevertheless hints at

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example is rather extreme, it nevertheless hints at the necessity to consider some not-so-obvious details. These are stated in the following definition. Definition 3.2 Two objects are said to be geometrically similar if all linear length scales of one object are a fixed ratio of all corresponding length scales of the second object.
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84 CHAPTER 3. THE EQUATIONS OF FLUID MOTION Here, “linear” length scale simply means any length that can be associated with a straight line extending from a chosen coordinate origin to an appropriate part of the object being considered. The definition immediately implies that the two objects are of the same general shape, for otherwise there could be no “corresponding” linear length scales. To clarify this idea we present the following example. EXAMPLE 3.6 In Fig. 3.12 we show an axisymmetric ogive that represents a typical shape for missile nose cones. The external tank of the space shuttle, for example, has a nose cone of this shape. ( ) S θ L (b) θ L (a) R R Figure 3.12: Missile nose cone ogive (a) physical 3-D figure, and (b) cross section indicating linear lengths. In this simple example there is actually only one linear length scale: the distance S ( θ ) from the base of the ogive to the ogive surface in any fixed plane through the nose and center of the base. We can see this because S ( θ ) = R when θ = 0, and S ( θ ) = L when θ = π/ 2. So both of the obvious scales are covered by the distance to the surface, S ( θ ). The reader may wish to consider how many independent linear length scales are needed to describe a rectangular parallelopiped with length, width and height such that L negationslash = W negationslash = H . Now suppose we wish to design a wind tunnel model of this nose cone in order to understand details of the flow field around the actual prototype. We will require that the wind tunnel model have a base radius r with r R in order for it to fit into the wind tunnel. Then we have r/R = α with α 1, and by the definition of geometric similarity we must require that s ( θ ) S ( θ ) = α for all θ [0 , π/ 2] . In particular, ℓ/L = α , where is the axial length of the model nose cone; s ( θ ) in the above equation is distance from the center of the base to the surface of the model. It should be clear that the surface areas and volume ratios of the model to the actual object will scale as α 2 and α 3 , respectively. For example, consider the area of any cross section. For the prototype nose cone the area of the base is πR 2 , and for the scale model it is πr 2 = π ( αR ) 2 = α 2 πR 2 . Hence, the ratio is α 2 , as indicated.
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3.6. SCALING AND DIMENSIONAL ANALYSIS 85 Dynamic Similarity We begin our treatment of dynamic similarity with a formal definition. Definition 3.3 Two geometrically similar objects are said to be dynamically similar if the forces acting at corresponding locations on the two objects are everywhere in the same ratio.
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