Similitude and II

Hydromechanics VVR090

Scale Models

Installation of offshore substructure 1:45 scale model

Model subjected to extreme waves

1 Model Study of Dolos I

Model Study of Dolos II

2 Dimensional Analysis

Built on the principle of dimensional homogeneity: An equation expressing a physical relationship between quantities must be dimensionally homogeneous.

Useful method to extract a maximum of information from a minimum of experiment.

Dimensionless groups are formed in a systematic manner.

Joseph Fourier

Metric System

Dimensions (e.g., length, mass, time, ) Units (e.g., m, kg, s, K)

Three independent dimensions of primary interest: • length (L) • mass (M) • time (t)

ML : []F = t 2

3 Example of Dimensional Analysis

Assume that the power (P) derived from a tubine depends on flow rate (Q), weight of the (γ = ρg), and energy extracted from the fluid flowing through the

turbine (ET):

PfQE=γ{ ,, T }

Relationship must be a product of the variables to some power:

ab c PCQE=γT

C is a dimensionless constant

Write the equation in terms of dimensions:

23a b ML⎛⎞ L⎛⎞ M c 322= ⎜⎟⎜⎟()L ttLt⎝⎠⎝⎠

Nm kgm2 ML []P == = (example) ss33 t

Homogeneous dimensions: 1 = b a =1 23=−+abc 2 b =1 −=−−32ab c =1

Equation: PCQE=γT

4 Buckingham’s P-Theorem

Three fundamental dimensions exist (M, L, and t) in most of our problems Æ Three equations only (still useful method)

Buckingham provided a systematic approach to dimensional analysis through his theorem expressed as:

1. If n variables are involved in the problem, then k equations of their exponents can be written 2. In most cases k is the number of independent dimensions (e.g., M, L, t) 3. The functional relationship may be expressed in terms of n- k distinct dimensionless groups

Example of Dimensional Analysis

Drag force (D) on a ship. Assume that D is related to length (l), density (ρ), (m), (V), and acceleration due to gravity (g):

fDl{ ,,ρμ , , Vg ,} = 0

Problem involves n = 6 variables and k = 3 fundamental dimensions Æ k - n = 6 – 3 = 3 dimensionless groups can be formed:

f ',,{ΠΠΠ123} = 0

Many different ways to combine the variables into dimensionless groups – rational approach needed.

5 Method for Deriving Dimensionless Groups

1. Find the largest number variables which do not form a dimensionless P-group 2. Determine the number of P-groups to be formed 3. Combine sequentially the variables in 1. with the remaining variables to form P-groups.

Present example: select ρ, V, and l and combine with remaining variables:

Π=11f {DVl,, ρ ,}

Π=22f {} μρ,,Vl ,

Π=33f {}gVl,, ρ ,

First P-group:

ab cd Π=1 DVl ρ

Analyze dimensions:

abc 000 ⎛⎞⎛⎞⎛⎞ML M L d M Lt= ⎜⎟⎜⎟⎜⎟23 () L ⎝⎠⎝⎠⎝⎠tLt

Mab:0=+ ba= − Labcd:0=− 3 ++ ca= −2 tac:0=+ 2 da= −2

a ⎛⎞D Result: Π=1 ⎜⎟22 ⎝⎠ρlV

6 Analysis in the same manner for P2 and P3 yields:

V Π= =Fr 2 gl Vlρ Π= =Re 3 μ

Thus, the following relationship is obtained:

⎧⎫D f ',Re,Fr0⎨⎬22 = ⎩⎭ρlV

D = f ''{} Re, Fr ρlV22

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