Resistance - Dimensional Analysis

From MarineWiki

1 Introduction
2 Ship resistance analysis
3 Froude’s law of comparison
4 Model Experiment
5 References
6 External links
7 Other Lessons

Introduction

When one has a partial knowledge of a problem, dimensional analysis is useful to help us understand the problem mathematically. The basic principle is that every equation that expresses relationship must be dimensionally homogenous.

The three basic quantities that can be used to express all the physical quantities are the mass [M], length [L], time [T].

For example, speed is distance traveled in unit time i.e, Distance/Time = L/T = LT-1

Similarly, acceleration is rate of change of speed i.e, Speed/Time = LT-1/T = LT-2

Force is product of mass and acceleration i.e, mass x acceleration = MLT-2

Ship resistance analysis

In case of the resistance of ship, the variable that influence it are: Speed (V), size of the body (L), mass density of fluid (ρ), viscosity of fluid (μ), acceleration due to gravity (g), and pressure (p).

Speed	V	LT-1
Size of body	L	L
Mass density of fluid	ρ	ML-3
viscosity of fluid	μ	ML-1T-1
Acceleration due to gravity	g	LT-2
Pressure	P	MLT-2

…..Eq.(1)

Substituting the dimensions of these terms we get,

…..Eq.(2)

Equating coefficient of M gives, 1 = a + d + f …..Eq.(3)

Similarly equating that of L gives, 1 = -3a + b + c –d + e – f …..Eq.(4)

And that of T gives, -2 = -b – d – 2e – 2f …..Eq.(5)

From Eq.(3), a = 1 – d – f

And from Eq.(5), b = 2 – d – 2e – 2f

Substituting the above two in Eq.(4) gives, c = 2 – d + e

That gives,

…..Eq.(6)

…..Eq.(7)

This is what one gets by dimensional analysis. The three groups in the brackets are dimensionless quantities.

One can conclude that Resistance in a function of the following dimensionless quantities, with a modification that kinematic viscosity,ν = μ / g

…..Eq.(8)

is called as drag coefficient and is generally written as CT(coefficient of total resistance)

: Reynold’s number (Rn) in honor of Osborne Reynolds

: Froude number (Fn) in honor of William Froude

: Pressure coefficient

This gives,

…..Eq.(9)

In case of Froude number, it was a practice to represent the coefficient, also known as speed-length coefficient as with V in knots and L in feet. The disadvantage of this terms is that it is not non-dimensionless.

The conversion from non-dimensional Froude number to this is:

…..Eq.(10)

or inversely, …..Eq.(11)

William Froude, about 150 yrs ago, formulated a postulate to split the total resistance in to two components;

…..Eq.(12)

where, RF is frictional resistance and RR is residuary resistance.

Froude’s law of comparison

The residuary resistance of geometrically similar ships is in the ratio of the cube of their linear dimension if their speeds are in the ratio of square root of their linear dimension.

If - Scale ratio

and the speeds are in ratio (Corresponding speed)

Then, …..Eq.(13)

This is the basic principle of a model test.

The above corresponding speed means that or , ie. the two ships have the same Froude number.

…..Eq.(14)

Substituting for CR1 and λ, we get

This implies that if two ships are geometrically similar and they are moving at same Froude number, then their residuary resistance coefficient is constant.

In the above equation (from dimensional analysis),

we can see that if the fluid is non-viscous fluid then there is no frictional resistance and the total resistance coefficient is equal to residuary resistance coefficent and is a function of Froude number and pressure coefficient.

The p is the total pressure, which is sum of atmospheric pressure, static water head and dynamic water pressure. In a model test it is difficult to make this pressure head equal to that of the real ship, as the atmospheric pressure is same in both cases.

In normal cases (luckily) the atmospheric pressure is not affecting the total resistance and only the difference is important. This difference is proportional to the dimension of the body.

So, will be constant as p is in ratio of λ and V is in ratio of

This gives, for a non-viscous fluid.

Please note that atmospheric pressure will become important only when the pressure is low and in comparable to atmospheric pressure, i.e., when it is close to cavitating phenomenon. This means that the above assumption on neglecting the pressure coefficient do not hold in case of cavitation.

Since water is not viscous we cannot apply the above.

For a submerged body, the wave making resistance will vanish, ie., CR = 0,

Therefore CT = CF = f(Rn)

In a general case, where the fluid is neither non-viscous and not fully submerged, we get,

So, we can conclude that if we can find the CF for a ship and move the model in same Froude number to get the and find the total ship resistance.

Similar to the above analysis, if we try to make the Reynolds number of the ship and model same to get same frictional resistance, we get

or , which means

ie., to get the same Reynolds number the model must have the velocity increased in the ratio of the length. So, if we scale the model 5 times then the speed has to scaled up 5 times. This means that we cannot attain the Reynolds similarity in practice.

Frictional resistance can be estimated by other means of statistical analysis on model test on 2-D and 3-D bodies.