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situation. 2.3.9 Inclining Experiment Much of the data used in stability calculations depends only on the geometry of the struc- ture. The total mass of the structure m follows from the displacement volume, r. The longitudinal position of the center of gravity, G; follows simply from the longitudinal posi- tion of the center of buoyancy, B, which can be determined from the under water geometry of the structure. The vertical position of the center of gravity must be known before the stability can be completely assessed for a given loading condition. Since the vertical po- sition of the center of gravity, KG, sometimes can be 10 times greater than the initial metacentric height GM , KG must be known very accurately if the metacentric height is to be assessed with reasonable accuracy. KG can be calculated for a variety of loading conditions, provided it is accurately known for one precisely speci…ed loading condition. The displacement volume, r can be calculated, given the geometry of the structure, in combination with measured drafts fore and aft (at both ends) in the upright condition (Á = Á0 ¼ 0±). With this displacement and the measured angle of heel (Á = Á1) after shifting a known mass p on the structure over a distance c, the vertical position of the center of gravity can be found easily by writing equation 2.51 in another form:¯¯¯¯ GM = p ¢ c ½r ¢ tan (Á1¡ Á0) ¯¯¯¯ with: Á0 ¿ Á1 and Á1 < §10± (2.52) The known underwater geometry of the structure and this ”measured” GM value yields the vertical position of the center of gravity G. An experiment to determine the metacentric height in this way is called an inclining experiment. The purposes of this experiment are to determine the displacement, r; and the position of the center of gravity of the structure under a precisely known set of conditions. It is usually carried out when construction is being completed (a‡oat) before leaving the construction yard. An additional inclining experiment may be carried out following an extensive modernization or a major re…t. Keep in mind that r and G have to be corrected when the mass p is removed from the structure, again. 2.3. STATIC FLOATING STABILITY 2-25 2.3.10 Free Surface Correction Free surfaces of liquids inside a ‡oating structure can have a large in‡uence on its static stability; they reduce the righting moment or stability lever arm. Figure 2.19: Floating Structure with a Tank Partially Filled with Liquid When the structure in …gure 2.19 heels as a result of an external moment MH , the surface of the ‡uid in the tank remains horizontal. This means that this free surface heels relative to the structure itself, so that the center of gravity of the structure (including liquid) shifts. This shift is analogous to the principle of the shift of the center of buoyancy due to the emerged and immersed wedges, as discussed above. Of course, the under water geometry of the entire structure and the boundaries of the wedges at the water plane as well as in the tank play a role. They determine, together with the angle of heel, the amount by which the centers of buoyancy and of gravity now shift. In case of a vertical wedge boundary, (a wall-sided tank) as given in …gure 2.20, this shift can be calculated easily with the …rst moment of volumes with triangular cross sections. In this …gure, ½0 is the density of the liquid in the tank, v is the volume of the liquid in the Figure 2.20: E¤ect of a Liquid Free Surface in a Wall Sided Tank tank and i is the transverse moment of inertia (second moment of areas) of the surface of 2-26 CHAPTER 2. HYDROSTATICS the ‡uid in the tank (Á = 0). The center of gravities are marked here by b and not by g, to avoid confusions with the acceleration of gravity. The in‡uence of the movement of the mass ½0v from b to bÁ on the position of the over- all center of buoyancy can also be determined using the …rst moment of volumes with triangular cross sections: shift of shift of direction center of gravity center of gravity of ‡uid of total structure horizontal iv ¢ tanÁ ½ 0i ½r ¢ tanÁ vertical iv ¢ 12 tan2 Á ½ 0i ½r ¢ 12 tan2 Á The amount of ‡uid in the tank, v, does not matters, only the transverse moment of inertia of the surface of the ‡uid in the tank, i, counts. This has been made more clear in …gure 2.21. Figure 2.21: Metacentric Height Reduction Caused by Free Surfaces in Wall Sided Tanks This means that the righting stability lever arm will be reduced by GG00 ¢ sinÁ, with: GG00 = ½0i ½r ¢ µ 1 + 1 2 tan2 Á ¶ (2.53) 2.3. STATIC FLOATING STABILITY 2-27 The magnitude GG00 is called the free surface correction or the reduction of the metacentric height. For small angles of heel for which 12tan 2 Á is small relative to 1:0, one can write for the reduction of the metacentric height: GG00 ¼ GG0 = ½ 0i ½r (2.54) For more free surfaces (more tanks) the reduction of the metacentric height becomes:¯¯¯¯ GG00 = Pf½0ig ½r ¢ µ 1 + 1 2 tan2 Á ¶¯¯¯¯ (2.55) Figure 2.22: GZ-Curve, Corrected for Free Surface Note that the e¤ect of the free surface on the stability lever arm (see …gure 2.22) is inde- pendent of the position of the tank in the ‡oating structure; the tank can be at any height in the structure and at any position in its length or breadth. The e¤ect is also independent of the amount of liquid in the tank, provided that the moment of inertia (second moment of areas) of the free surface remains substantially unchanged when inclined. The moment of inertia of the free surface of a tank which is almost empty or almost full can change signi…cantly during an inclining experiment. This is why tanks which cannot be completely full or empty during an inclining experiment are speci…ed to be half full. It is common practice to treat the free surface e¤ect as being equivalent to a virtual rise of the center of gravity of the structure. One should appreciate that this is merely a convention which has no factual basis in reality. A similar free surface e¤ect can arise from movements of granular cargo such as grain which is stowed in bulk. Here, however, there is no simple relation between the angle of inclination of the ship and the slope of the surface of the grain. At small angles of heel the grain is not likely to shift, so there is no in‡uence on initial stability. When the heel 2-28 CHAPTER 2. HYDROSTATICS angle becomes great enough, a signi…cant shift can take place very abruptly; its e¤ect can be equally abrupt! 2.3. STATIC FLOATING STABILITY 2-29 . 2-30 CHAPTER 2. HYDROSTATICS Chapter 3 CONSTANT POTENTIAL FLOW PHENOMENA 3.1 Introduction This chapter introduces hydrodynamics by treating the simplest of ‡ows: constant, in- compressible, potential ‡ows; time is therefore unimportant. These ‡ows obey reasonably simpli…ed laws of ‡uid mechanics and mathematics as will be explained in this chapter. In spite of its limitations, potential ‡ow is widely used in ‡uid mechanics. It often gives at least a qualitative impression of ‡ow phenomena. Discrepancies between real and idealized ‡ows are often accommodated via experimentally determined coe¢cients. The concepts introduced in this chapter will be extended and applied in later chapters for a wide variety of solutions including those involving waves. Indeed, potential theory can very easily be extended to time dependent ‡ows as well. This chapter starts by de…ning a potential ‡ow …eld and its properties in a very general way. The power of the method will become more obvious as it is applied to constant ‡ow situations later in this chapter. Much of the information in this chapter may be well-known to the reader; it can be seen as a quick review. In any case, this chapter introduces the notation used in many of the remaining chapters as well. 3.2 Basis Flow Properties All ‡ows are treated here