Question 4 (Part A, Question A4): Full-Similarity Water-Tunnel Testing of a Suspension-Bridge Deck: 22-Mec-B6 Advanced Fluid Mechanics, May 2014, Q4

Question 4 (Part A, Question A4): Full-Similarity Water-Tunnel Testing of a Suspension-Bridge Deck (14 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Prototype (air)	Model (water)
Characteristic length	deck length 50 m, width 5 m	1/20 scale: 2.5 m, 0.25 m
Density \(\rho\)	1.2 kg/m³	1000 kg/m³
Dynamic viscosity \(\mu\)	18×10⁻⁶ Pa·s	0.001 Pa·s
Kinematic viscosity \(\nu=\mu/\rho\)	15×10⁻⁶ m²/s	1×10⁻⁶ m²/s
Speed	30 m/s (given)	to be found
Measured lift / drag	to be predicted	1.0 MN / 0.1 MN
Oscillation frequency	to be predicted	5 Hz

Find. (a) the water-tunnel speed that gives full similarity, (b) the full-scale lift and drag implied by the measured model forces, and (c) the full-scale oscillation frequency.

Figure 4.1 — Model-to-prototype mapping. Reynolds similarity fixes the tunnel speed; force and Strouhal coefficients then transfer the measured loads and frequency.

Approach. "Full similarity" for a bluff body in a single-phase flow means matching the Reynolds number, which fixes the model speed; the force coefficient \(F/(\rho V^2 L^2)\) and the Strouhal number \(fL/V\) are then equal between model and prototype and transfer the measurements.

Part (a) — impose Reynolds similarity. Setting \(Re_m=Re_p\) with \(\nu=\mu/\rho\), \[\frac{V_m L_m}{\nu_m}=\frac{V_p L_p}{\nu_p} \quad\Rightarrow\quad V_m=V_p\left(\frac{L_p}{L_m}\right)\left(\frac{\nu_m}{\nu_p}\right).\] The scale ratio is \(L_p/L_m=20\) and the viscosity ratio is \(\nu_{\text{water}}/\nu_{\text{air}}=1\times10^{-6}/15\times10^{-6}=1/15\), so \[V_m=(30)(20)\left(\tfrac{1}{15}\right)\] \[\boxed{V_m=40\ \text{m/s}}\] As a check, both Reynolds numbers on the deck length are \(Re=(30)(50)/(15\times10^{-6})=(40)(2.5)/(1\times10^{-6})=1.0\times10^{8}\). Water is chosen precisely because its low kinematic viscosity lets a 1/20 model reach full-scale \(Re\) at a manageable speed; the same test in air would need 600 m/s and be hopelessly compressible.
Part (b) — transfer the forces through the force coefficient. Similarity of \(C_F=F/(\rho V^2 L^2)\) gives \[\begin{aligned}\frac{F_p}{F_m}&=\left(\frac{\rho_p}{\rho_m}\right)\left(\frac{V_p}{V_m}\right)^{2}\left(\frac{L_p}{L_m}\right)^{2}\\&=\left(\frac{1.2}{1000}\right)\left(\frac{30}{40}\right)^{2}(20)^{2}\\&=(1.2\times10^{-3})(0.5625)(400)=0.270 .\end{aligned}\] The prototype loads are therefore smaller than the model loads, because the density ratio of 1/833 overwhelms the 400-fold area ratio.
Evaluate the two forces. Applying the same factor to each measurement, \[\begin{aligned}\boxed{L_p=(0.270)(1.0\ \text{MN})=0.270\ \text{MN}=270\ \text{kN}}\\\boxed{D_p=(0.270)(0.1\ \text{MN})=0.0270\ \text{MN}=27\ \text{kN}}\end{aligned}\] The lift-to-drag ratio of 10 is of course preserved, since both forces scale identically.
Part (c) — transfer the frequency through the Strouhal number. Matching \(St=fL/V\), \[f_p=f_m\left(\frac{V_p}{V_m}\right)\left(\frac{L_m}{L_p}\right)=(5)\left(\frac{30}{40}\right)\left(\frac{1}{20}\right)\] \[\boxed{f_p=0.1875\ \text{Hz}\quad(\text{period }5.33\ \text{s})}\] The common Strouhal number is \(St=(5)(2.5)/40=(0.1875)(50)/30=0.3125\). A period of five seconds is exactly the order of the torsional oscillation that destroyed the Tacoma Narrows bridge, which is why this frequency — not the force — is usually the governing result of such a test.

Part	Quantity	Value
(a)	Water-tunnel speed \(V_m\)	40 m/s
(a)	Common Reynolds number (deck length)	1.0×10⁸
(b)	Force scale \(F_p/F_m\)	0.270
(b)	Full-scale lift	0.270 MN = 270 kN
(b)	Full-scale drag	0.0270 MN = 27 kN
(c)	Common Strouhal number	0.3125
(c)	Full-scale oscillation frequency	0.1875 Hz (5.33 s period)

Check: what "full similarity" can and cannot deliver. Geometric similarity is assumed at 20:1 in every dimension, including deck details and cable diameters, and the model is assumed rigid enough that its measured forces are not contaminated by its own structural response. Reynolds matching alone is used because the flow is single-phase and low-speed; if free-surface effects or gravity waves mattered, Froude similarity would also be required and could not be satisfied simultaneously with Reynolds in the same fluid. Compressibility is ignored in both streams (\(M_{\text{air}}=0.09\)) and cavitation in the water tunnel at 40 m/s is assumed to be suppressed by adequate tunnel pressure.

Topic. Dimensional analysis and model testing: Reynolds, force-coefficient and Strouhal similarity between a water-tunnel model and a full-scale structure in wind.

Key relations. \(Re=\rho VL/\mu=VL/\nu\); \(C_F=F/(\tfrac12\rho V^2 A)\Rightarrow F_p/F_m=(\rho_p/\rho_m)(V_p/V_m)^2(L_p/L_m)^2\); \(St=fL/V\); scale ratio \(\lambda=L_m/L_p=1/20\).

Why this works. Buckingham's theorem says the dimensionless force on a bluff body in incompressible flow depends only on geometry and Reynolds number. Match the geometry and \(Re\) and the dimensionless force coefficient must be identical, whatever the fluid, so a single measured model force converts to full scale by pure algebra. The same argument applied to a shedding or oscillating flow produces the Strouhal number: the dimensionless frequency depends only on geometry and \(Re\), so matching \(Re\) transfers the frequency too. The scale factors combine in a way that often surprises: forces here shrink by a factor of 3.7 going to full scale because the working fluid is 833 times lighter, and frequencies shrink by a factor of 27 because the prototype is both larger and slower.

Common pitfall. The dominant error is inverting the viscosity ratio and obtaining 22.5 m/s instead of 40 m/s; write the Reynolds equality out symbolically before substituting. The second is forgetting the density ratio in the force transfer, which leaves the full-scale loads 833 times too large. The third is scaling frequency by the length ratio only (giving 0.25 Hz) instead of the full Strouhal group. Finally, note that \(\nu\) must be computed from \(\mu/\rho\) for each fluid; using \(\mu\) directly in \(Re=VL/\mu\) is dimensionally wrong.

Source. F. M. White, Fluid Mechanics, 8th ed., Ch. 5.3–5.4 (modelling and similarity, incomplete similarity); R. W. Fox, A. T. McDonald and J. W. Mitchell, Introduction to Fluid Mechanics, 10th ed., Ch. 7.5–7.7 (flow similarity and model studies).

Question 4 of 7: Question 4 (Part A, Question A4): Full-Similarity Water-Tunnel Testing of a Suspension-Bridge Deck

Question 4 (Part A, Question A4): Full-Similarity Water-Tunnel Testing of a Suspension-Bridge Deck (14 marks)