Similitude and Dimensional Analysis 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, temperature) Units (e.g., m, kg, s, K)
Three independent dimensions of primary interest: • length (L) • mass (M) • time (t)
ML Force: []F = t 2
3 Example of Dimensional Analysis
Assume that the power (P) derived from a tubine depends on flow rate (Q), weight density of the fluid (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), viscosity (m), speed (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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