Scale effects in physical hydraulic engineering models

Abstract

Beside analytical approaches, physical modelling represents probably the oldest design tool in hydraulic engineering. It is thus a pleasure to see this Forum Paper in JHR. The Discussers focus on one aspect of the publication, thereby specifying the information of the Forum Paper. Free surface flows are typically scaled with the Froude similitude keeping identical F = V /(gh) 0.5 both in the model and in the prototype. The air transport in models is affected by scale effects because the internal flow turbulence, represented by the Reynolds number R = Vh/ν, is underestimated, while surface tension, represented by the Weber number W = (ρV 2 h)/σ , is overestimated (Chanson 2009), with V = flow velocity, g = gravity constant, h = flow depth, ρ = water density, σ = water surface tension, and ν = water kinematic viscosity. Because a strict dynamic similitude exists only at a full-scale, the underestimation of the air transport is minimized if limitations in terms of W or R are respected. The Forum Paper overlooks a number of aspects and probably recommends too optimistic limitations. As stated in Table D1, the literature mentions limitations around W 0.5 = 110-170 and R = 1.0-2.5 × 10 5. These values focus on air entrainment at hydraulic jumps, general chute air entrainment and aerated stepped spill-way flows, as well as the air entrainment coefficient β and the streamwise bottom air concentration C b generated by chute aer-ators. Pfister and Hager (2010a, b) identified an underestimation up to one magnitude in terms of C b if W 0.5 < 140 (Fig. D1). There, the abscissa corresponds to the streamwise normalization given by these authors, and the trend lines correspond to the best fit of all C b curves from tests with W 0.5 ≥ 140, i.e. without significant scale effects. As can be noted from Table D1, two criteria are often applied relating to the herein discussed scale effects, i.e. limiting values for W 0.5 and R for a range of air-water flow parameters. This results in an over-determined system, as the two numbers depend on each other, besides F and the Morton number M. The latter characterizes the shape of bubbles or drops moving in a surrounding medium, solely as a function of the fluid properties and the gravity constant (Wood 1991, Chanson 1997). With a negligible inner bubble density, as is typical for air-water flows, the Morton number is with μ = dynamic water viscosity M = gμ 4 σ 3 ρ = W 3 F 2 R 4 (D1) For air-water two-phase flows M = 3.89 × 10 −11. If using the Froude similitude: (1) M = constant, and (2) F is similar in the model and the prototype. Isolating these two numbers results in MF 2 = W 3 R 4 (D2) For a given F, the right-hand side of Eq. (D2) thus has to be identical both in the model and in the prototype flows. The theoretical function MF 2 versus F is shown in Fig. D2(a). The theoretical MF 2 values (curve) are identical with the experimentally derived W 3 /R 4 values (symbols; Pfister and Hager 2010a, 2010b), as expected from Eq. (D2). To visualize the limitations, Fig. D2(b) shows the measured...

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