CALCULATING MODEL FOR THE FRICTION DRAG OF A SHIP’S HULL

Abstract: This paper presents the contribution of the authors regarding the friction drag calculation of a ship’s hull using the quasi-plane model. The velocity calculation by a dimensional model helps to determine the friction tangential effort. Then, by integration, can be determined the friction drag of a rectangular plane panel. By extension over all the plane plates which compose the ship’s side, results the friction drag for entire ship’s side.


INTRODUCTION
Let us consider T the draft ship full loaded.The careen is reported to a system Oxyz, ensuing that: • the longitudinal plane coincides with the plane xOz; • the plane xOy is the waterline plane full loaded; • the Oz axis is situated in the stem proximity, without crossing it (can be tangent to it, see Figure 1).
The waterline corresponding to the draft T k ; (T 0 -T) is noted by L k ; (k = 0,.., m); the planes which contain this lines are defined by equation:

(
) ( ) Through its marks on the plane xOy, the planes x = x i (i = 1, ..., n-1) has the characteristics below: • the plane x = x 1 coincides with the plane yOz (so x 1 =0); • the plane x = x n-1 is tangent to the stern frame (without crossing it).
The planes z = z k (k = 0,..,m), x = x i (i = 1,...,n-1), the stem and the stern frame establish on the side a succession of curve panels P k,i-1 P k+1,i-1 P k+1,i P ki , (k = 0,...m-1), (i=1,...n).P ki is the notation of the half waterline point k L , on the x i axes (Figure 2).From Figure 2 results that the point P k0 belongs to the stem and the point P kn belong to the stern frame.As regarding the curve panel P k,i-1 P k+1,i-1 P k+1,i P ki , it can be approximate in a rectangular plane panel defined by two consecutive sides: 1) sides P k,i-1 P ki and P k,i-1 P k+1,i-1 ; 2) sides P k,i-1 P ki and P ki P k+1,i ; 3) sides P k+1,i-1 P k+1,i and P k+1,i-1 P k,i-1 ; 4) sides P k+1,i-1 P k+1,i and P k+1,i P ki .
In the situations 1) and 2) the fluid moves on the vector direction and purpose-oriented (so, it is parallel by the respectively plane), of constant velocity , 1 , where and ki v -the velocities corresponding to the points P k,i-1 and P ki (results from the potential flow model), p′the weight which make possible the vectors parallelism Similar, in the situations 3) and 4) the fluid moves on the vector direction (so, it is parallel by the respectively plane), of constant velocity [1]: where: the velocities corresponding to the points P k,i-1 and P ki (results from the potential flow model), p ′ ′ -the weight which make possible the vectors parallelism f F the friction forces developed by the fluid on the plane panels adequate to the situations 1), 2), 3), 4) and 2, 3, 4) projections arithmetic average, noted by , will approximate the ship friction drag component corresponding to the curve panel P k,i-1 P k+1,i-1 P k+1,i P ki .If c f R is the keel friction drag, the ship friction drag can be written as: Double amount applies to all curve panels making up the port side or starboard side of the ship.

THE CALCULATION OF THE FRICTION DRAG COMPONENTS
Be an open polygonal segment  Notations will be used below: As regarding the friction tangential effort, it corresponds to the turbulent flow (characteristic to fluid flow around the ship side), defined by equation [2]: where: ρ -fluid density; υkinematics viscosity of the fluid; v 0velocity module from relations (5) and (6); δboundary layer local thickness developed over the plane panel properly to one of the situations 1), 2), 3), 4) mentioned in the first paragraph; nvalue from the lot (7), (9), (10) chosen depending by the size of the Reynolds number [1], defined by: [ ] is determined from condition:  Using relations (7), ( 9) and (10), the turbulent tangential effort developed over the plane panel along the segments P k,i-1 P ki (in situations 1) and 2)), respectively P k+1,i-1 P k+1,i (in situations 3) and 4)), noted by In the fluid flow case along a rectangular plane panel, of constant velocity parallel with one of the panel segments (for example MQ, Figure 4), the friction tangential effort along the segment MQ has the same distribution along any segment RS parallel to the segment MQ from the parallelogram.I.e. the average value of the friction tangential effort calculated along the segment MQ (or NP) is the same with the average value of the same effort for entire parallelogram surface.The average friction tangential effort τ ′ for entire parallelogram MNPQ surface can be writing as: From these considerations, the friction forces developed by the fluid over the plane panel determined in the situations 1), 2), 3), 4) are: The forces projections i f F (i = 1, 2, 3, 4) over the longitudinal plane can write as: The arithmetic average of the forces i f R (i = 1, 2, 3, 4) represent the ship friction drag component corresponding to the curve panel P k,i-1 P k+1,i-1 P k+1,i P ki : As regarding the c f R term from relation above it is calculated simply: is the plane plate case in a parallel current for constant velocity (the ship regime speed).

CONCLUSIONS
The model presented in this paper is a quasi-plane model, based over a velocity field with two scalar components, x v and y v , defined by equation: It are obtained through the Kármán method of sources [3] arranged in the ship longitudinal plane of variable intensity z; The curve panel approximation of a side in rectangular plane plates (situations 1), 2), 3), 4) from paragraph 1), the consideration that the fluid flow is parallel to the velocities defined by relations (5) and (6) (dependents by z), and the forces average i f R (i = 1, 2 , 3, 4) represents the deviation improve ways from the curve side panels and from velocity variation on the z height.Obviously, the model precision will be better if the calculation is achieved with large number of side panels.

−
which approximates the half waterline k L from Figure 3.If this open polygonal contour is put on the straight line, it is obtained the segment Pkn k P 0 situated on the half axe k Oξ from Figure 3, of the origin O superimposed over a 0 k P point.