Similarity Laws in Naval Architecture and Marine Engineering

Training TSL_Nov2010

Nguyễn Anh TuấnEngineer of Naval Architecture and Marine Engineering http://sites.google.com/site/tuanshiplandtuanshipland@gmail.com(+84) (0) 944 113 787

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1. Geometric Similitude (Đồng dạng hình học)

2. Kinematic Similitude (Đồng dạng động học)

3. Dynamic Similitude (Đồng dạng động lực học)

4. Similitude in Ship Model Test (Đồng dạng trong kiểm tra mô hình tàu)

Main Sections

1. Geometric Similitude

Geometric Simulitude states that the ratio of full-scale model’s geometric dimension (length, width, height and so on) to model-scale’s geometric dimension is contant.The scale ratio of the length

The full - scale length (Chiều dài tàu thật) [L]The model length (Chiều dài tàu mô hình) [L]

For corresponding area A[] and volume [] :

Geometric Similitude

2. Kinematic Similitude

Kinematic Similitude

Kinematic Similitude states that the ratio of full-scale times to model-scale times is contant. The scale ratio of the time

The full - scale time (Thời gian thử trên tàu thật) [T]The model time (Thời gian thử trên mô hình) [T]

For velocity V and acceleration a:

3. Dynamic Similitude

Dynamic Similitude

Dynamic similitude states that forces act on the full-scale ship and forces act on the model are similar:

the dynamic ratio (or the dynamic model scale)Forces act the ship such as inertial forces, gravity, frictional forces, buoyancy forces and etc.

Force F = mass m x acceleration a (Newton’s Law)Mass of ship m = density of water x volume of displacement

Hence the Newton’s law of similitude:

Hydrodynamic forces c coeffient (e.g. lift coe. whether F is lift force or drag whether F is drag force)V reference speed (e.g. speed of ship) []A reference area [ (e.g. wetted surface in calm water)

Hence with :

With contant for both ship and model

4. Similitude in Ship Model Test

Gravity forces of full-scale ship Corresponding gravity forces of model-scale ship The ratio of gravity of full-scale and model-scale

For dynamic similitude,

Applied Geometric and Kinemetic Similitude,

In Ship Hydrodynamics, the velocity divides square root of ship’s length and square root of acceleration of gravity yields the Froude Number:

For testing resistance, propulsion, seakeeping, and manoeuvring of ship, Froude number is an enssential criteria. As a result, the scaling ratios are for speeds, forces and power:

Similitude in Ship Model Test“The same Froude number in model and full scale ensures dynamical similarity only if inertial and gravity forces are present (Froude’s Law). For the same Froude number, the wave pattern in model and full-scale are geometrically similar. This is only true for waves of small amplitude where gravity is the only relevant physical mechanism. Breakingwaves and splashes involve another physical mechanism(e.g. surface tension) and do not scale so easily.” (Volker Bertram, 2000)

Frictional forces

dynamic viscosity (material constant)A area is acted by the frictional stresses (because two layers of fluid have friction forces) velocity gradient (normal to the flow direction)The scale ratio of friction

Demanding the ratio of frictional forces and inertial forces are similar , thus:

In addition, the kinematic viscosity , thus:

Hence, The Reynolds number is an essential non-dimentional speed parameter in viscous flows

Newton’s Similar Law

⟹

“The same Reynolds number in model and full-scale ensures dynamic similarity if only internial and frictional forces are present (Reynolds’Law). This is some what simplified as viscous flows are complicated by transition from laminar to turbulent flows, microscope viscosity effects such as surface roughness, flow separation etc.”(Volker Bertram, 2000)

Similitude in Ship Model Test

The relationship between Froude number and Reynolds number is:

Similitude of Froude number can easibily satisfy in model test. Otherwise, Similitude of Reynolds number is hard to obtain for contant kinematic viscosity. Hence, forces can not scale down with contant viscosity in scale test.

Ship moves in free surface, therefore it has gravity forces (wave) and frictional forces. Finally, similitude of ship model should follow Froude’s law and Reynolds’ law:

“Such fluids do not exist or at least are not cheap and easy to handle for usual model scales. However, sometimes the test water is heated to improve the viscosity ratio and reduce the scaling errors for viscous effects” (Volker Bertram, 2000)

Cauchy number

E the modulus of elasticity (mô dun đàn hồi), I the moment of innertia (moment quán tính), T time and L length

The similitude of Caushy and Froude number states that is similar to density and the modulus of elasticity.Similitude of vibrations are according to Caushy number, so that the Caushy number is similar in model and full-scale.

Model scaleViscosity ratio of the test fluid

Similitude in Ship Model Test

[1] Volker Bertram, 2000, Practical Ship Hydrodynamics, Butterworth Heinemann, Great Britain.[2] Trần Công Nghị, 2004, LT Tàu 2 - Sức Cản Vỏ Tàu và Thiết Bị Đẩy Tàu, NXB ĐH QG tp.HCM, Vietnam.

References

Thank you!Nguyễn Anh Tuấn

Naval Architecture and Marine Engineering

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