¯Rt ship = 1
2½V2¢ Ct ship¢ S
¯¯¯
¯ (4.47)
and the resistance coe¢cient, Cts, has to be determined.
A number of methods to determine the still water resistance coe¢cients of ships, based on (systematic series of) model test data, are given in the literature. A very well known method, developed at MARIN, is described by [Holtrop, 1977], [Holtrop and Mennen, 1982]
and [Holtrop, 1984]. The method is based on the results of resistance tests carried out by MARIN during a large number of years and is available in a computerized format. The reader is referred to these reports for a detailed description of this method, often indicated by the ”Holtrop and Mennen” method. An example for a tug of the correlation be-tween the results of this ”Holtrop and Mennen” method and model test results is given in
…gure 4.11.
4.8 Wind Loads
Like all environmental phenomena, wind has a stochastic nature which greatly depends on time and location. It is usually characterized by fairly large ‡uctuations in velocity and
Figure 4.11: Comparison of Resistance Prediction with Model Test Results
direction. It is common meteorological practice to give the wind velocity in terms of the average over a certain interval of time, varying from 1 to 60 minutes or more.
Local winds are generally de…ned in terms of the average velocity and average direction at a standard height of 10 meters above the still water level. A number of empirical and theoretical formulas are available in the literature to determine the wind velocity at other elevations. An adequate vertical distribution of the true wind speed z meters above sea level is represented by:
Vtw(z)
Vtw(10) =³ z 10
´0:11
(at sea) (4.48)
in which:
Vtw(z) = true wind speed at z meters height above the water surface Vtw(10) = true wind speed at 10 meters height above the water surface
Equation 4.48 is for sea conditions and results from the fact that the sea is surprisingly smooth from an aerodynamic point of view - about like a well mowed soccer …eld.
On land, equation 4.48 has a di¤erent exponent:
Vtw(z)
Vtw(10) =³ z 10
´0:16
(on land) (4.49)
At sea, the variation in the mean wind velocity is small compared to the wave period.
The ‡uctuations around the mean wind speed will impose dynamic forces on an o¤shore structure, but in general these aerodynamic forces may be neglected in comparison with the hydrodynamic forces, when considering the structures dynamic behavior. The wind will be considered as steady, both in magnitude and direction, resulting in constant forces and a constant moment on a …xed ‡oating or a sailing body.
The wind plays two roles in the behavior of a ‡oating body:
² Its …rst is a direct role, where the wind exerts a force on the part of the structure exposed to the air. Wind forces are exerted due to the ‡ow of air around the various parts. Only local winds are needed for the determination of these forces.
² The second is an indirect role. Winds generate waves and currents and through these in‡uence a ship indirectly too. To determine these wind e¤ects, one needs information about the wind and storm conditions in a much larger area. Wave and current generation is a topic for oceanographers; the e¤ects of waves and currents on
‡oating bodies will be dealt with separately in later chapters.
Only the direct in‡uence of the winds will be discussed here.
Forces and moments will be caused by the speed of the wind relative to the (moving) body.
The forces and moments which the wind exerts on a structure can therefore be computed by:
Xw = steady longitudinal wind force (N) Yw = steady lateral wind force (N)
Nw = steady horizontal wind moment (Nm)
½air¼ ½water=800 = density of air (kg/m3) Vrw = relative wind velocity (m/s)
®rw = relative wind direction (-), from astern is zero AT = transverse projected wind area (m2)
AL = lateral projected wind area (m2)
L = length of the ship (m)
C¤w(®rw) = ®rw-dependent wind load coe¢cient (-)
Note that it is a ”normal” convention to refer to the true wind direction as the direction from which the wind comes, while waves and currents are usually referred to in terms of where they are going. A North-West wind will cause South-East waves, therefore!
4.8.1 Wind Loads on Moored Ships
For moored ships, only the true wind speed and direction determine the longitudinal and lateral forces and the yaw moment on the ship, as given in …gure 4.12. Because of the absence of a steady velocity of the structure, the relative wind is similar to the true wind:
jVrw= Vtwj and j®rw = ®twj (4.51) The total force and moment experienced by an object exposed to the wind is partly of viscous origin (pressure drag) and partly due to potential e¤ects (lift force). For blunt
Figure 4.12: De…nitions Used here for Forces and Moments
bodies, the wind force is regarded as independent of the Reynolds number and proportional to the square of the wind velocity.
[Remery and van Oortmerssen, 1973] collected the wind data on 11 various tanker hulls.
Their wind force and moment coe¢cients were expanded in Fourier series as a function of the angle of incidence. From the harmonic analysis, it was found that a …fth order representation of the wind data is su¢ciently accurate, at least for preliminary design purposes:
with wind coe¢cients as listed below.
Ta nke r N o . 1 2 3 4 5 6 7 8 9 1 0 11
Figure 4.13 shows, as an example, the measured wind forces and moment together with their Fourier approximation, for one of the tankers.
Figure 4.13: Example of Wind Load Coe¢cients
4.8.2 Wind Loads on Other Moored Structures
The wind forces on other types of structures, as for instance semi-submersible platforms, can be approximated by dividing the structure into a number of components, all with a more or less elementary geometry, and estimating the wind force on each element.
Drag coe¢cients are given in the literature for a lot of simple geometrical forms, such as spheres, ‡at plates and cylinders of various cross sectional shapes. [Hoerner, 1965] and [Delany and Sorensen, 1970] are good sources of this information. The total wind load on the structure is found by adding the contributions of all the individual component parts.
The fact that one element may in‡uence the wind …eld of another element is neglected in this analysis.
4.8.3 Wind Loads on Sailing Ships
For sailing merchant ships and tankers, only the longitudinal wind resistance, Xw = Rw, is of importance for determining a sustained sea speed.
The relative wind speed, Vrw, and the relative wind direction, ®rw, have to be determined from this true wind speed, Vtw, and the true wind direction, ®tw, together with the forward ship speed, Vs, and the heading of the ship, see …gure 4.14.
As opposed to the hydromechanical notation, for seamen head wind has a direction equal to zero. The e¤ect of the lateral force and the yaw moment on the ship will be corrected via a small course correction. Then the relative wind speed and the relative wind direction (so head wind has a direction equal to zero) follows from:
¯¯¯¯Vrw= q
Vs2+ Vtw2 + 2 ¢ Vs ¢ Vtw¯¯
¯¯
¯¯¯¯®rw = arctan
µ Vtw¢ sin ®tw Vs+ Vtw¢ cos®tw
¶¯¯¯¯ (4.53)
Figure 4.14: Relative Wind
[Isherwood, 1973] published a reliable method for estimating the wind resistance. He ana-lyzed the results of wind resistance experiments carried out at di¤erent laboratories with models covering a wide range of merchant ships. He determined empirical formulas for the two horizontal components of the wind force and the wind-induced horizontal moment on any merchant ship form for a wind coming from any direction.
The longitudinal wind resistance is de…ned by:
¯¯¯¯Xw = 1
2½airVrw2 ¢ CXw ¢ AT¯¯
¯¯ (4.54)
with for the longitudinal wind resistance coe¢cient CXw:
CXw = A0+ A1¢ 2AL
L2oa + A2¢ 2AT
B2 + A3¢ Loa
B + A4¢ S Loa
+ A5¢ C Loa
+ A6¢ M (4.55)
in which:
Loa = length over all (m)
B = beam (m)
S = length of perimeter of lateral projection excluding water line and slender bodies such as masts and ventilators (m)
C = distance from bow of centroid of lateral projected area (m) AL = lateral projected wind area (m2)
AT = transverse projected wind area (m2)
M = number of distinct groups of masts or kingposts seen in lateral projection (-); kingposts close against the bridge are not included
The coe¢cients for equation 4.55 are listed below.
®rw A0 A1=10 A2 A3 A4 A5=10 A6
Similar polynomials for the lateral force and the horizontal moment are given in the paper by [Isherwood, 1973].