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.064 -0 .475 0.000 -0 .032 160± -2 .148 0.256 0 .000 0 .081 0 .000 0.127 -0 .027 170± -2 .707 0.397 -0 .175 0 .126 0 .000 0.181 0 .000 180± -2 .529 0.376 -0 .174 0 .128 0 .000 0.155 0 .000 Similar polynomials for the lateral force and the horizontal moment are given in the paper by [Isherwood, 1973]. 4.9 Current Loads There are several independent phenomena responsible for the occurrence of current: the ocean circulation system resulting in a steady current, the cyclical change in lunar and solar gravity causing tidal currents, wind and di¤erences in sea water density. The steady wind velocity at the water surface is about 3 per cent of the wind velocity at 10 meters height. Tidal currents are of primary importance in areas of restricted water depth and can attain values up to 10 knots. However, such extreme velocities are rare; a 2-3 knots tidal current speed is common in restricted seas. The prediction of tidal currents is left for the oceanographers. Although surface currents will be the governing ones for ‡oating structures; the current distribution as a function of depth below the surface may also be of importance. For the design of a mooring system of a ‡oating structure, the designer is especially interested in the probability that a particular extreme current velocity will be exceeded during a certain period of time. Observations obtained from current speed measurements are indispensable for this purpose. It may be useful to split up the total measured current in two or more components, for instance in a tidal and a non-tidal component, since the direction of the various components will be di¤erent, in general. The variation in velocity and direction of the current is very slow, and current may therefore be considered as a steady phenomenon. The forces and moment exerted by a current on a ‡oating object is composed of the following parts: ² A viscous part, due to friction between the structure and the ‡uid, and due to pressure drag. For blunt bodies the frictional force may be neglected, since it is small compared to the viscous pressure drag. ² A potential part, with a component due to a circulation around the object, and one from the free water surface wave resistance. In most cases, the latter component is small in comparison with the …rst and will be ignored. 4-30 CHAPTER 4. CONSTANT REAL FLOW PHENOMENA The forces and moments, as given in …gure 4.12, exerted by the current on a ‡oating structure can be calculated from:¯¯¯¯ Xc = 1 2 ½ ¢ V 2c ¢ CXc(®c) ¢ ATS ¯¯¯¯ ¯¯¯¯ Yc = 1 2 ½ ¢ V 2c ¢ CY c(®c) ¢ ALS ¯¯¯¯ ¯¯¯¯ Nc = 1 2 ½ ¢ V 2c ¢ CNc(®c) ¢ ALS ¢ L ¯¯¯¯ (4.56) in which: Xc = steady longitudinal current force (N) Yc = steady lateral current force (N) Nc = steady yaw current moment (Nm) ½ = density of water (kg/m3) Vc = current velocity (m/s) ®c = current direction, from astern is zero (rad) ATS ¼ B ¢ T = submerged transverse projected area (m2) ALS ¼ L ¢ T = submerged lateral projected area (m2) L = length of the ship (m) B = breadth of the ship (m) T = draft of the ship (m) C¡c(®c) = ®c-depending current load coe¢cient (-) Results of model tests are given in the literature for various types of structures and vessels. 4.9.1 Current Loads on Moored Tankers [Remery and van Oortmerssen, 1973] published current loads on several tanker models of di¤erent sizes, tested at MARIN. The coe¢cients CXc, CY c and CNc were calculated from these results. A tanker hull is a rather slender body for a ‡ow in the longitudinal direction and consequently the longitudinal force is mainly frictional. The total longitudinal force was very small for relatively low current speeds and could not be measured accurately. Moreover, extrapolation to full scale dimensions is di¢cult, since the longitudinal force is a¤ected by scale e¤ects. For mooring problems the longitudinal force will hardly be of importance. An estimate of its magnitude can be made by calculating the ‡at plate frictional resistance, according to the ITTC skin friction line as given in equation 4.42:¯¯¯¯ Xc = 0:075 (log10 (Rn) ¡ 2)2 ¢ 1 2 ½Vc 2 ¢ cos®c ¢ j cos®cj ¢ S ¯¯¯¯ (4.57) while: Rn = Vc ¢ j cos®cj ¢ L º (4.58) with: 4.9. CURRENT LOADS 4-31 S ¼ L ¢ (B + 2T) = wetted surface of the ship (m2) L = length of the ship (m) B = breadth of the ship (m) T = draft of the ship (m) Vc = current velocity (m/s) ®c = current direction (-), from astern is zero ½ = density of water (ton/m3) Rn = Reynolds number (-) º = kinematic viscosity of water (m2s) Extrapolation of the transverse force and yaw moment to prototype values is no problem. For ‡ow in the transverse direction a tanker is a blunt body and, since the bilge radius is small, ‡ow separation occurs in the model in the same way as in the prototype. Therefore, the transverse force coe¢cient and the yaw moment coe¢cient are independent of the Reynolds number. The coe¢cients for the transverse force and the yaw moment were expanded by MARIN in a Fourier series, as was done for the wind load coe¢cients as described in a previous section: CY c(®c) = 5X n=1 bn ¢ sin(n ¢ ®c) CNc(®c) = 5X n=1 cn ¢ sin(n ¢ ®c) (4.59) The average values of the coe¢cients bn and cn for the …fth order Fourier series, as published by [Remery and van Oortmerssen, 1973], are given in the table below. n bn 10 ¢ cn 1 0 .908 -0 .252 2 0 .000 -0 .904 3 -0 .116 0 .032 4 0 .000 0 .109 5 -0 .033 0 .011 These results are valid for deep water. For shallow water, the transverse current force and moment coe¢cients have to be multiplied by a coe¢cient, which is given in …gure 4.15. The in‡uence of the free surface is included in the data given on the coe¢cients bn and cn in the previous table. This in‡uence, however, depends on the water depth and on the Froude number, and consequently changes if the current velocity or the tanker dimensions change. For the condition to which these data apply, deep water and a prototype current speed in the order of 3 knots, the e¤ect of the free surface is very small. For the case of a small clearance under the keel and a current direction of 90 degrees, damming up of the water at the weather side and a lowering of the water at the lee side of the ship occurs. 4.9.2 Current Loads on Other Moored Structures Current loads on other types of ‡oating structures are usually estimated in the same way as is used for wind loads. 4-32 CHAPTER 4. CONSTANT REAL FLOW PHENOMENA Figure 4.15: In‡uence of Water Depth on Transverse Current Load on a Tanker 4.9.3 Current Loads on Sailing Ships For sailing ships, generally it is assumed that the ship is moving with the current. So the forward ship speed is not the ground speed but the speed relative to the water. Then there is no current load; the current itself is a navigation problem only. 4.10 Thrust and Propulsion Now that the resistance of a ship or other ‡oating objects moving through still water has been discussed, it is appropriate to approach this phenomena from the other side by discussing the propulsion systems needed to overcome the resistance. This section on that topic is partly based on a text by [Kuiper, 1997] on resistance and propulsion of ships. This text is a good reference for those who wish to know more than is presented here about thrust and propulsion. The basic action of propulsors like propellers is to deliver thrust. In fact, a propulsor is an energy transformer, because torque and rotation, delivered to the propulsor, will be transformed into thrust and translation, delivered by the propulsor. A consequence is that the propulsor also generates water velocities in its wake, which represent a loss of kinetic energy. It is obvious that this will e¤ect the e¢ciency of the propulsor, de…ned by: ¯¯¯¯ ´ = Pout Pin ¯¯¯¯ = PE PD = T ¢ Ve Q ¢ 2¼n (4.60) in which: 4.10.