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 7.