# ITTC School & Reynold Number Theory - Calculation of Total Resistance to Ship Motion

## Introduction

We have been learning about the resistance to ship motion in water, analysis of propeller forces and also learned about a method of calculating this resistance using Froude’s Law of comparison. In this article we will talk about another method of resistance calculation using the ITTC school of thought and Reynold number theory.

Some new coefficients of resistance are included compared to the previous method and these are

- Ct for total resistance
- Cf for frictional resistance
- Cr for the residuary resistance

## The Reynold’s Number

Consider a flat smooth surface such as a plate moving through a viscous fluid. Because of the motion some fluid near the plate travels with it. After some distance from the plate, the fluid has no motion, only the plate along with the layer travels. This layer which is moving along with the plate is called as boundary layer or frictional wake.

The movement of the fluid in the wake takes two forms. One assumption is made here: “The fluid next to the surface has no motion relative to it, but is carried with it.”

The two forms being either laminar or turbulent.

Laminar means the fluid travels in a series of layer in-between the wake without mixing. For example: the movement of oil is laminar.

Next is turbulent which means there will be intermixing of fluid in between the wake because of the eddies. For example: the movement of water is turbulent.

The type of flow depends upon the inertia forces and the viscous forces. The ratio between these two forces is known as the Reynolds number.

Reynolds number Rn = vl/γ, where

v - is the speed in m/s,

l - is the length of the surface in m,

γ - is the co efficient of kinematic viscosity of the fluid in m2 / sec.

Here the value of the γ depends upon the temperature ( deg Celsius ) . The values of the γ are expressed in terms of cst, (ie) centistokes. γ for seawater = 1.190 cst = 1.190 x 10-6 m2 /sec, γ for fresh water = 1.140 cst = 1.140 x 10-6 m2 /sec.

## The co-efficient of frictional resistance

The coefficient of frictional resistance can be given by the formula:

Cf = (0.075)/(logRn - 2)2 where, Rn is the Reynold’s number.

Also Cf = Rf/(0.5 ρ S v2 )

This method has been given by the ITTC (INTERNATIONAL TANK TOWING CONFERENCE).

## The procedure

- Calculate the value of the Reynold’s number Rn by using the formula for the model and take this as Rn ship.
- Calculate the value of the co efficient of frictional resistance using the formula Cf model = (0.075)/(logRn model - 2)2 .
- Now find the value of the total resistance from the tank towing experiment for the model as Rt model
- Now calculate the co efficient of the total resistance Ct model = Rt model /( 0.5 ρ Smodel vmodel2 ). Where Smodel is the wetted surface area of the model and v is the speed of the model in m/s.
- Now we can calculate the coefficient of the residuary resistance by using the formula Cr = Ct model - Cf model (since we have discussed that total resistance is equal to the sum of the two resistances)
- The value of the coefficient of the residuary resistance is the same for the ship and its model. (ie) Cr model = Cr ship.
- Now we have calculated all the co effecients related to the ship and one value related to ship.
- Now calculate the value of the Reynolds number for the ship and take it as Rn ship .
- Calculate the value of the co efficient of frictional resistance using the formula Cf ship = (0.075)/(logRn ship - 2)2 .
- Now calculate the value of the co efficient of the total resistance using the formula
- Ct ship = Cf ship + Cr ship
- The formula for the co efficient of the total resistance is given by: Ct ship = Rt ship /(0.5 ρ Sship vship2 )
- With the above formula the value of the total resistance can be calculated.
- After calculating the total resistance the value of the effective power can be given by the formula:

Effective power = Rt ship X vship

where Rtship is the total resistance offered by the ship in N,

Vship is the speed of the ship in m/s.

In this way we shall determine the resistance of the ship and hence the effective power.