Scale effects 
Introduction 
Scale effects arise due to forces which are more dominant in the model than in its prototype. This results in deviations between the upscaled model and prototype observations. Figure 1 shows how scale effects affect the jet over an overflow spillway at scale 1:l (delta) = 1:30 (l (delta) = length_{Prototype}/length_{Model}). While the jet trajectory between model and prototype agree well, the air concentration in the jet differs considerably and would be misleading if upscaled from the model to its prototype. Scale effects can potentially result in an inappropriate design and failure of the prototype (cf. Sines breakwater in 1978/9). Dr Heller has investigated scale effects in various phenomena, including landslidetsunamis and ski jumps, and he wrote several review articles about scale effects in general (Heller 2007, 2011, 2017). 


Personal research website of Dr Valentin Heller 
Review articles 
Fig. 1. Overflow spillway of Gebidem Dam, Valais, Switzerland: (a) model at scale 1 : 30, (b) prototype in 1967; the air entrainment of free jet differs considerably due to scale effects (Heller 2011) 
Landslidetsunamis/impulse waves 
Selected publications 
Journals Kesseler, M., Heller, V., Turnbull, B. (2018). A laboratorynumerical approach for modelling scale effects in dry granular slides. Landslides (under review). Heller, V. (2017). Selfsimilarity and Reynolds number invariance in Froude modelling. Journal of Hydraulic Research 55(3):293309 (http://dx.doi.org/10.1080/00221686.2016.1250832). Heller, V. (2017). Two Replies to Discussions of „Selfsimilarity and Reynolds number invariance in Froude modelling“ by Ettema, R., and SokorayVarga, B., and Vladimir, N. Journal of Hydraulic Research (in press). Heller, V. (2012). Three Replies to Discussions of „Scale effects in physical hydraulic engineering models“ by Pfister, M., and Chanson, H.; Tafarojnoruz, A., and Gaudio; Echávez, G., R. Journal of Hydraulic Research 50(2):246; 248; 249250. Heller, V. (2011). Scale effects in physical hydraulic engineering models. Journal of Hydraulic Research 49(3):293306 (http://dx.doi.org/10.1080/00221686.2011.578914). Heller, V., Hager, W.H., Minor, H.E. (2008). Scale effects in subaerial landslide generated impulse waves. Experiments in Fluids 44(5):691703 (http://dx.doi.org/10.1007/s0034800704277). Heller, V. (2007). Massstabseffekte im hydraulischen Modell. Wasser Energie Luft 99(4):153159 (in German). 
The aim of the review articles Heller (2007, 2011) was to generically describe scale effects. The number of such generic rules applicable for all phenomena is limited. Scale effects are complex and depend on the involved forces and their relative importance changing from phenomenon to phenomenon. However, generally speaking: (i) physical hydraulic model tests with l (delta) ≠ 1 always involve scale effects; (ii) the larger l (delta), the larger are scale effects; each involved parameter requires its own judgement regarding scale effects (if the jet trajectory in Fig. 1(a) is not considerably affected, this does not exclude significant scale effects for air entrainment); (iii) and scale effects normally have a ‘damping’ effect, i.e. parameters such as relative wave height are normally smaller in the model than in its prototype. 
Fig. 4. Impulse waves scale series: test (a,c) conducted at half scale in test (b,d); the Froude number (inertial force/gravity force)^{1/2} is F = 2.5 for both scales while the Reynolds number R (inertial force/viscous force) and Weber number W (inertial force/surface tension force) differ between (a,c) R = 289,329, W = 5,345 and (b,d) R = 103,415, W = 1,336 resulting in different air entrainment and detrainment (Heller et al. 2008) 
Four approaches help to obtain modelprototype similarity, to quantify scale effects, to investigate how they affect the parameters, and to establish limiting criteria where they can be neglected: 
The scale series approach was applied in Heller et al. (2008) to define limiting criteria for scale effects in subaerial landslidegenerated tsunamis/impulse wave. Tests involving granular materials were conducted at different scales as shown in Fig. 4. The employed wave flume was 0.50 m wide and the water depths were in the range 0.075 ≤ h ≤ 0.600 m. All parameters within a scale series were scaled according to Froude scaling, i.e. length scales were linearly reduced by the scale factor l (delta). Plotting the wave profiles for a scale series in dimensionless from (Fig. 5) results in a mismatch due to scale effects. With a total of seven scale series, it was possible to define that scale effects relative to the maximum wave amplitude are negligible (<2%) if: 
Fig. 5. Impulse waves scale series: normalized water surface displacement h/h (eta/h) versus relative time t(g/h)^{1/2 }for three scales (Heller et al. 2008) 
In practice, scale effects are commonly avoided (with relevant rules of thumb, replacement of fluid, Fig. 2), compensated or corrected for. Some rules of thumb are the following (for a more comprehensive list with references see Heller 2011): 
· Wave propagation is affected less than 1% by surface tension if period > 0.35 s · Free surface water flows should be > 5 cm to avoid significant scale effects · Wave height for wave force on a slope during wave breaking should be > 0.50 m · Free surface airwater flows should be conducted at Weber number > 140 
Fig. 2. Replacement of fluid: similar morphologies in sand caused by fluid (a) water: ripples in hydraulic model to investigate bridge pier scours, (b) air: ripples on sand dune (Heller 2011) 
· Inspectional analysis: modelprototype similarity criteria are found with an analytical set of equations describing the phenomenon. · Dimensional analysis: a method to transform dimensional into dimensionless parameters, which have to be identical between model and prototype. · Calibration: calibration and validation of model tests with realworld data (discharge, runup height of tsunami). · Scale series: a method comparing results of models of different sizes (and different amounts of scale effects) in order to quantify scale effects. 
The parameter g is the gravitational acceleration, n_{w} (nu) the kinematic viscosity, s_{w} (sigma) the surface tension and r_{w} (rho) the water density. The corresponding rule of thumb is h ≥ 0.200 m. 
Reynolds number R = g^{1/2}h^{3/2}/n_{w} ≥ 300,000 Weber number W = r_{w}gh^{2}/s_{w} ≥ 5,000 
This part of Dr Heller’s research aims to exclude Reynolds number R scale effects in Froude models. Two concepts were identified for these purposes: (i) selfsimilarity and (ii) Reynolds number invariance. Figure 3 shows a simple example of selfsimilarity namely a Romanesco broccoli with a geometry identical to the geometry of a small fraction of itself. Similar, the selfsimilarity may also apply to fluid flows where the properties (e.g. velocity distribution) at small scale are identical to the properties at full scale, such that no scale effects are observed. Concept (ii) (R invariance) makes use of the fact that many fluid features at large R become independent of R. As R tends to increases with the model size, these independent fluid features are also observed at full scale (again no R scale effects are observed). 
Selfsimilarity and Reynolds number invariance in Froude Modelling 
The two concepts (i) and (ii) were reviewed and illustrated in Heller (2017) with a wide range of examples: (i) irrotational vortices, wakes, jets and plumes, sheardriven entrainment, highvelocity open channel ﬂows, sediment transport and homogeneous isotropic turbulence; and (ii) tidal energy converters, complete mixing in contact tanks and gravity currents. The content of the forum paper Heller (2017) is summarised in a presentation available in the Downloads section. 
Fig. 3. Example of a selfsimilar geometry in nature: Romanesco broccoli (Heller 2017) 
Last modified: 15.01.2018 