Imperial College London - Dr Valentin Heller

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Model-Prototype SimilarityPictureDr Valentin HellerFluid Mechanics Section, Department of Civil and Environmental Engineering4th CoastLab Teaching School, Wave and Tidal Energy, Porto, 17-20th January 2012Content Introduction Similarities Scale effects Reaching model-prototype similarity Dealing with scale effects ConclusionsLecture follows Heller (2011)21

IntroductionWhy do quantities between model and prototype disagree?Measurement effects: due to non-identical measurement techniquesused for data sampling in the model and prototype (intruding versus nonintruding measurement system etc.).Model effects: due to the incorrect reproduction of prototype featuressuch as geometry (2D modelling, reflections from boundaries), flow orwave generation techniques (turbulence intensity level in approach flow,linear wave approximation) or fluid properties (fresh instead of sea water).Scale effects: due to the inability to keep each relevant force ratioconstant between the scale model and its real-world prototype.3IntroductionExample model effectsReflections from beach or from non-absorbing wave maker (without WEC)Regular water waves in a wave tank: frequency 0.625 Hz, steepness ak 0.05Free water surface [mm]402000255075100-20-40Time [s]Note: Scale effects are due to the scaling. Reflection is not due to the scaling, it istherefore not a scale but a model effect.42

IntroductionExample scale effectsReal-world prototypeMiniature universeJet trajectoryAir concentration1:l 1:30Scale ratio or scale factor l LP/LM with LP a characteristic length in the realworld prototype and LM corresponding length in the model5IntroductionWhy are scale effects relevant? Scale model test results may be misleading if up-scaled to full scale (e.g.incorrect economic prediction of a WEC based on power measurements) Scale effects may be responsible for failures(Sines breakwater, river section is unable todeal with predicted discharge) Whether or not scale effects are significantdepends on the relative importance of theinvolved forces. Understanding which forcesare relevant and which can be neglected is ofkey importance in physical, numerical andmathematical modelling:Does a numerical simulation require an additionalterm to consider surface tension or Coriolis force?Can viscosity be neglected (potential theory)?63

IntroductionRelevance of scale effectsFailure of Sines breakwater in 1978/9 which was strong enough in thescale model investigation (one reason for the failure were scale effectsdue to the incorrect scaling of the structural properties)Sines breakwater failure, 1978/97SimilaritiesPerfect model-prototype similarity: Mechanical similarityA physical scale model satisfying mechanical similarity is completelysimilar to its real-world prototype and involves no scale effects.Mechanical similarity requires three criteria:(i) Geometric similarity:similarity in shape, i.e. all length dimensions in the model are l timesshorter than of its real-world prototype (l LP/LM)(ii) Kinematic similarity:geometric similarity and similarity of motion between model andprototype particles(iii) Dynamic similarity:requires geometric and kinematic similarity and in addition that all forceratios in the two systems are identical84

SimilaritiesMechanical similarity (cont.)Most relevant forces in fluid dynamics are: (rL3)(V2/L)Inertial force mass acceleration rL2V2 rL3gGravitational force mass gravitational accelerationViscous force dynamic viscosity velocity/distance area n(V/L)L2 nVLSurface tension force unit surface tension length sLElastic compression force Young’s modulus area EL2Pressure force unit pressure area pL2r (kg/m3) fluid density L (m) characteristic lengthp (N/m2) pressures (N/m) surface tensionsg (m/s2) gravitational accelerationV (m/s) char. velocityE (N/m2) Young’s modulusn (kg/(ms)) kinematic viscosity9SimilaritiesMechanical similarity (cont.)Relevant force ratios are:Froude number F (inertial force/gravity force)1/2 V/(gL)1/2Reynolds number R inertial force/viscous force LV/nWeber number W inertial force/surface tension force rL2V/sCauchy number C inertial force/elastic force rV2/EEuler number E pressure force/inertial force p/rV2Problem: Only the most relevant force ratio can be identical between model andits prototype, if identical fluid is used, and mechanical similarity is impossible.The most relevant force ratio is selected and the remaining result in scale effects.105

SimilaritiesFroude similarity FM FPFor phenomena where gravity and inertial forces are dominant and effectof remaining forces such as kinematic viscosity are small.Most water phenomena are modeled after Froude, in particular freesurface flows (hydraulic structures, waves, wave energy converters etc.)Anaconda WECHydraulic jump modeled after Froude11SimilaritiesFroude similarity FM FP (cont.)Scale ratios for Froude modelsHow can Froude VM be up-scaled?FM VM/(gMLM)1/2 VP/(gPLP)1/2 FPwith gM gP gLP l LM(not scaled)(geometric similarity)FM VM/(gLM)1/2 VP/(glLM)1/2 FPreduces to VM VP/l1/2VP l1/2VMScale ratio l1/2 is required to upscaleFroude model velocitiesExample: up-scaling model power PM:l 20, power model PM 5 Watts, power prototype PP?12PP l7/2PM 207/25 178885 Watts 0.18 MW!6

SimilaritiesReynolds similarity RM RPFor phenomena where viscous and inertial forces are dominant andeffect of remaining forces such as gravity are small.Not so often applied; examples include vortexes, tidal energy converters,sometimes rivers (water replaced by air to reach high model velocity)Scale ratios for Reynolds modelsVortexes in river modeled with ReynoldsExample: up-scaling Reynolds model velocity vM:l 20, velocity model vM 1 m/s, velocity prototype vP?vP l–1vM 1/20 0.05 m/s vM vP!13Scale effectsGeneralScale effects are due to force ratios which are not identical between modeland its prototype. Consequently, some forces are more dominant in the modelthan in the prototype and distort the results.Four items are relevant, independent of a phenomenon:(i) Physical hydraulic model tests with l 1 always involve scale effects. The relevantquestion is whether or not scale effects can be neglected.(ii)The larger l, the larger are scale effects. However, l alone does not indicate whetheror not scale effects can be neglected.(iii) Each involved parameter requires its own judgement regarding scale effects. If e.g.wave height is not considerably affected by scale effects does not necessarily meanthat e.g. air entrainment is also not affected (relative importance of forces may change).(iv) Scale effects normally have a ‘damping’ effect. Parameters such as relative waveheight or relative discharge are normally smaller in model than in its prototype. A judgement if prediction under or over-estimates prototype value is therefore often possible.147

Scale effectsGeneral (cont.)In a Froude model, scale effects are due to R, W, C and E.In a Reynolds model, scale effects are due to F, W, C and E.Scale effects due to F (in Reynolds models): reduced flow velocity (gravity)Scale effects due to R (in Froude models): larger viscous losses in model, e.g.waves decay faster or energy dissipation is larger, water flows like honeyScale effects due to W (in Froude and Reynolds models): too large air bubblesand faster air detrainment, wave celerity of short wave is affected, reduceddischarge for small water depthsScale effects due to C (in Froude and Reynolds models): structure (WEC) interacting with water behaves too stiff and strong (Sines break water), water andair are too hard in the model (impact phenomena, e.g. wave breaking)Scale effects due to E (in Froude and Reynolds models): cavitation can not beobserved in model if atmospheric pressure is not scaled (reduced)15Scale effectsExamples: jet air entrainment and bottom air concentration onspillway in Froude modelsW0.5limit 140Cb bottom air concentrationfD parameter includingboundary conditions(e.g. slope, F)Pfister and Chanson (2012)168

Reaching model-prototype similarity4 available methodsInspectional analysis: similarity criteria between model and prototype arefound with set of equations describing a hydrodynamic phenomenon,which have to be identical between model and prototype.Dimensional analysis: a method to transform dimensional in dimensionless parameters. Those dimensionless parameters have to be identicalbetween model and prototype.Calibration: calibration and validation of model tests with real-world data(discharge in river, run-up height of tsunami). The model is then appliedwith some confidence to other scenarios.Scale series: a method comparing results of models of different sizes(different scale effects) to quantify scale effects.17Reaching model-prototype similarityExample: Landslide-tsunamis (Froude model)Dimensional test parametersStill water depth hSlide impact velocity VsBulk slide volume VsSlide thickness sBulk slide density rsSlide impact angle aGrain diameter dg189

Reaching model-prototype similarityExample: Dimensional analysisA physical problem with n independent parameters q1, ., qn can be reduced toa product of n – r independent, dimensionless parameters P1, ., Pn–r with r asthe minimum number of reference dimensions required to describe the dimensions of these n parameters. Each of Pn–r have to be identical between modeland prototype.n 9 independent par. qn: h [L], Vs [LT 1], Vs [L3], s [L], rs [ML 3], a [-], dg [L], r [ML 3], g [ML 2]r 3 reference dimensions: [L], [T], [M]n – r 6 dimensionless parameters: P1, ., P6r 3 selected reference parameters: h, g, r (include different combinations of ref. dim.)Example Vs: P1 Vshbggrdor[-] [LT 1][L]b[LT 2]g[ML 3]d[L] : 0 1 1b 1g 3d[T] : 0 1 0b 2g 0d[M] : 0 0 0b 0g 1d b –1/2, g –1/2 and d 0P1 Vs/(gh)1/2, the Froude number F19Reaching model-prototype similarityExample: Dimensional analysisAll dimensionless parameters P1, ., P6Slide Froude numberRelative slide thicknessRelative grain diameterRelative slide densityRelative slide volumeSlide impact angleP1 P2 P3 P4 P5 P6 F Vs /(gh)1/2S s/hDg dg /hD rs /rwV Vs /(bh2)a2010

Reaching model-prototype similarityExample: CalibrationLituya Bay 1958 caseRun-up height observed in natureR 524 mRun-up height measured in study of Fritz et al. (2001) at scale 1:675R 526 m21Reaching model-prototype similarityScale series: Results from tests conducted at three scales are comparedQuantification (schematic)Level full-scale Prototype?Scaleeffects2211

Reaching model-prototype similarityExample: Scale seriesWave generationF 2.5W 5350R 290000F 2.5W 1340R 103400F 2.5W 5350R 290000F 2.5W 1340R 10340023Reaching model-prototype similarityExample: Scale seriesS4/1: h 0.400 mS4/2: h 0.200 mS4/3: h 0.100 mScale effects relative to aM are negligible ( 2%) if:Reynolds number: R 300000Weber number:W 5000R g1/2h3/2/nwW rwgh2/swchar. velocity V (gh)1/2char. length L h2412

Dealing with scale effectsAvoidance: With rules of thumbSatisfy limiting criteriaIn Froude models: R Rlimit, W W limit etc.In practice rules of thumb are often applied. Some examples: Linear wave propagation is affected less than 1% by surface tension ifT 0.35 s (corresponding to L 0.17 m, Hughes 1993) Free surface water flows should be 5 cm to avoid significant surfacetension scale effects (e.g. Heller et al. 2005) Wave height to measure wave force on slope during wave breakingshould be larger than 0.50 m (Skladnev and Popov 1969) Free surface air-water flows should be conducted at W 0.5 140 (Pfisterand Chanson 2012)25Dealing with scale effectsAvoidance: Replacement of fluidWater replaced by water-isopropyl alcohol (reduced surface tension, increased W)Stagonas et al. (2011)Water replaced by air (reduced kinematic viscosity, increased R)2613

Dealing with scale effectsCompensationCompensation is achieved by distorting a model geometry by giving upexact geometric similarity of some parameters in favour of an improvedmodel-prototype similarity.Examples: Distorted models: the length lL scale factor of a model (say a river) issmaller than the height and width scale factor l to compensateincreased friction effects with a larger flow velocity The grain diameter dg in sediment transport can often not be scaledwith the same scale factor l as the model main dimensions since it mayresult in dg 0.22 mm for which the flow-grain interaction characteristics changes. Zarn (1992) proposes a method to modify the model grainsize distribution accordingly.27Dealing with scale effectsCorrectionEconomic considerations, limited space or time may be reasons to intentionally build a small model where significant scale effects are expected. Themodel results may afterwards be corrected for phenomena where enoughinformation on the quantitative influence of scale effects is available.Examples: Solitary waves decay too fast in small scale physical models, which canbe corrected with an analytical relation from Keulegan (1950) Correction factors for wave impact pressures from small-scale Froudemodels are included by Cuomo et al. (2010) Correction coefficients for the stability of rubble mound breakwatermodel tests were presented by Oumeraci (1984)2814

Conclusions Similarity theory between physical model and real-world prototype wasreviewed including mechanical, Froude and Reynolds similarities A model with l 1 always results in scale effects (with identical fluid)since only one relevant force ratio can be satisfied. The relevant questionto ask is whether or not scale effects are negligible For each phenomenon or parameter in a model, the relative importance ofthe involved forces may vary and limitations should be defined relative tospecific parameters and prototype features Inspectional analysis, dimensional analysis, calibration and scale seriesare available to obtain model-prototype similarity, to quantify scale effects,to investigate how they affect the parameters and to establish limitingcriteria where they can be neglected Scale effects can be minimised with three methods namely avoidance,compensation and correction Similarity theory (scale effects) is not an exact science and requiresengineering judgement for each particular problem29ReferencesCuomo, G., Allsop, W., Takahashi, S. (2010). Scaling wave impact pressures on vertical walls.Coastal Engineering 57(6), 604-609.Fritz, H.M., Hager, W.H., Minor, H.-E. (2001). Lituya bay case: Rockslide impact and wave run-up.Science of Tsunami Hazards 19(1), 3-22.Heller, V. (2011). Scale effects in physical hydraulic engineering models. Journal of HydraulicResearch 49(3), 293-306.Hughes, S.A. (1993). Advanced series on ocean engineering 7. Physical models and laboratorytechniques in coastal engineering. World Scientific, London.Keulegan, G.H. (1950). Wave motion. Engineering hydraulics, 711-768. H. Rouse, ed. Wiley, NewYork.Le Méhauté, B. (1990). Similitude. Ocean engineering science, the sea. B. Le Méhauté, D.M.Hanes, eds. Wiley, New York, 955-980.Oumeraci, H. (1984). Scale effects in coastal hydraulic models. Symp. Scale effects in modellinghydraulic structures 7(10), 1-7. H. Kobus ed. Technische Akademie, Esslingen.Pfister, M. and Chanson, H. (2012). Discussion of Discussion of “Scale effects in physical hydraulicengineering models”. Journal of Hydraulic Research.Skladnev, M.F., Popov, I.Y. (1969). Studies of wave loads on concrete slope protections of earthdams. Symp. Research on wave action 2(7), 1-11 Delft Hydraulics Laboratory, Delft NL.Stagonas, D., Warbrick, D., Muller, G., and Magnaga, D. (2011). Surface tension effects on energydissipation by small scale, experimental breaking waves. Coastal Engineering 58: 826-836.Zarn, B., (1992). Lokale Gerinneaufweitung: Eine Massnahme zur Sohlenstabilisierung der Emme beiUtzenstorf (Local river expansion: A measure to stabilise the bed of Emme River at Utzendorf).VAW Mitteilung 118. D. Vischer ed. ETH Zurich, Zürich [in German].3015

(ii) Kinematic similarity: geometric similarity and similarity of motion between model and prototype particles (iii) Dynamic similarity: requires geometric and kinematic similarity and in addition that all force ratios in the two systems are identical Perfect model-prototype similarity: Mechanical similarity 8 . 5 Similarities Most relevant forces in fluid dynamics are: Inertial force mass .

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