Scaling Issues in Hydraulic Modelling

Introduction

The uncertainties involved in many coastal issues and the lack of complete scientific background in some knowledge fields, especially to evaluate extreme coastal-forcing events, the cumulative environmental evolution and impacts on beaches and coastal structures as well as to confirm design procedures, for instance, lead to the need of using physical modelling. There is also little public awareness of the physics behind several coastal processes and physical modelling can help in describing and illustrating them.

In this paper a brief revue on the importance of physical modelling, its advantages in relation to numerical modelling, some basic aspects of physical modelling, related scaling issues and how to control hydraulic modelling and scale effects as well as three case studies and future challenges will be presented.

The Importance of Physical Modelling

Physical models have played a pivotal role in the growth of coastal engineering as a profession . They have given us insight into the complex hydrodynamic regime of the nearshore region, and they have provided us with reliable and economic design solutions to support man’s activities in the coastal zone. Many of our present-day engineering design techniques were developed using laboratory measurements, and numerous theoretical developments have relied on laboratory experiments for validation.
However, many of us can still list some of the limitations of those design approaches, being in some case considered as empirical formulations. This means further tests and measurements are needed to increase the reliability of those formulations, specially performed at scales closer to the prototype, avoiding scale effects and testing new forcing situations. Especially due to climate change and the demand for bigger structures located at higher depths, more accurate design formulations are needed and this will be the most important role of large laboratories of maritime hydraulics. However, as these large tests in large facilities are more expensive they need to work in close relation with other small/medium facilities for preliminary/cheaper analyses.

Physical and numerical modelling tools have developed enormously during the last years. However several issues need still further developments, namely the physics and modelling of sediment transport, the wave-structure interaction analysis and loads determination, erosion and scour near coastal structures as well as medium to long term accurate simulation tools.

From the management side, for instance, plans should be based on an adequate understanding of coastal dynamics. It is necessary to pursue research on many aspects of coastal dynamics in order to better assess and understand erosion and sedimentation problems, predictions of shoreline positions for various scenarios and time scales of climate variability and direct human influence, the vulnerability of beaches, dunes and coastal structures to storms and other extreme events, the impact of artificial coastal structures and ecological changes.

Physical Modelling versus Numerical Modelling

Numerical models represent the real problem but with some simplifications. Thus, the modeller is forced to make a compromise between the details of the model and the prototype. Several advantages and disadvantages of physical model testing are usually reported.

An incorrectly designed model always provides wrong predictions, independently of the sophistication of the instrumentation and measuring methods. The cost of physical modelling is often more than that of numerical modelling, and less than that of major field experiments, but this depends on the exact nature of the problem being studied. Physical modelling has gathered new perspectives due to the development of new sophisticated equipment, allowing the measurement of variables in complex flows, which was previously impossible. New experimental techniques, automated data acquisition and analysis systems, rapid processing and increased data storage capabilities also provide useful information for the validation of numerical models, Frostick et al. (2011)

Other advantages of physical models are the study of new phenomena, the lower level of simplification, to confirm through measurements theoretical results, to obtain measurements from complex phenomena inaccessible from theory, to test extreme conditions, to test a wide variety of environmental conditions and the immediate visual feedback. Despite all these advantages there are still some problems of physical modelling to solve such as the scale effects, the incomplete modelling, the laboratory effects and the costs of installation and maintenance.

With relation to numerical models it can be said that despite the huge developments made they still exhibit deficiencies and limitations when applied to complex flows and situations like breaking, overtopping, wave structure interaction, etc. However, recent developments such as SPH and in computing capacity have made these tools more powerful than even before, leading to a better description of the complexity of the hydraulic phenomena (physical environment and borders as well as non-linear aspects of the equations used). From another perspective this tool is in general more attractive to researchers and practitioners.

To obtain theoretical solutions, simplifications of the physical environment (especially the boundaries) are needed as well as of the equations that govern the phenomena. As a result of that mathematical solutions may have lower quantitative value, and therefore could be more useful for qualitative or comparative analyses. The geometry can be reproduced with the desired detail but it is not enough to ensure a correct reproduction of the reality in the model as this can generate a behaviour sometimes different from the prototype. So calibration is needed. Physical modelling reproduces both linear and nonlinear aspects of the phenomena, avoiding the simplifications of the numerical modelling that simplifies not only the geometry but also fundamental equations.
Other advantages of physical modelling are intermediate and controllable cost; they represent reality at a certain scale; the involved variables and boundaries can be controlled; measurements are in general easy to perform and the comprehension of the processes is facilitated.

Other disadvantages of physical modelling are the time spent and the cost of building alternatives, the particle similarity, the partial control of boundaries and the difficulty in measuring parameters in some model areas. The actual level of research needs common efforts between the various available tools, namely physical and numerical modelling in order to decrease the lack of knowledge in some areas of Maritime Hydraulics. The problems to solve or which are not solved yet are so complex that only this integrated approach is feasible in order to obtain better accurate results not only for researchers but also for designers and practitioners.
There is still a need to design and construct new and more advanced laboratory facilities, develop new reliable measuring instruments and techniques, minimize laboratory effects, and understand the scale effects that arise from incomplete modelling.

Physical and numerical model input conditions can be controlled and systematically varied, whereas field studies have no such control. However, many problems in coastal engineering are not amenable to mathematical analysis because of the nonlinear character of the governing equations of motion, lack of information on wave breaking, turbulence or bottom friction, or numerous connected water channels. In these cases it is often necessary to use physical models for predicting prototype behaviour or observing results not readily examined in nature. The growing use of numerical models in coastal engineering has not stopped the use of physical models and in some cases they made progress in conjunction with each other. Recent trends have included the concept of “hybrid modelling” where results from a physical model of complex region are used as input or boundary conditions for a comprehensive numerical model covering a wider region of interest. Alternatively, numerical model results may be used to provide input conditions at the boundaries of the physical model.

The rationale for continued support of physical modelling in support of project design is that “Theory cannot cover all the complications that are encountered in practice. Consequently, most major hydraulics projects are model tested to optimize design”.
Due to the quantitative deficiencies and limitations of predictive numerical models when applied to complex flows, the need for physical modelling still remains and investments in laboratory facilities, equipment and new techniques are more and more needed, highlighting the need for synergies between the various research tools, physical and numerical modelling included, not only because of the actual complexity of the maritime hydraulics problems, but also to improve some design approaches.

Basic Aspects of Physical Modelling

A physical hydraulic model represents a real prototype and is used to find or confirm solutions for engineering problems. Differences between the model and prototype behaviour and results may be due to scale (similarity laws considered and incomplete reproduction of the forces involved), laboratory (model geometry – 2D or 3D influences, reflections; flow or wave generation techniques – turbulence intensity levels, linear wave theory approach; fluid properties, etc.) or measurement (different equipments used in model and prototype – intrusive or not, probe sizes) effects. The estimation of these effects (qualitatively and quantitatively) affects the results and to know if they can be neglected is a challenge for physical modellers, Heller (2011). They can also justify differences between physical and numerical models (e.g. kinematic viscosity). The examples of the overflow spillway or a wave breaking are classic ones.

Considering the usual geometric scale parameter λ=L_m/L_p or N=L_p/L_m, the required space, time and cost of experiments decrease with λ², λ^1/2 and λ³, but scale effects will increase, and the results cannot be properly extrapolated to the prototype. So a proper selection of λ is an economic and technical issue and if related effects cannot be neglected this must be known and taken into account.

The basic aspects of mechanical similarity (geometric - dimensions, kinematic – time, velocity, acceleration and discharge, and dynamic – forces: inertial, gravitational, viscous, surface tension, elastic, pressure) are well known, leading to the different numbers (similarities or laws) when considering the ratios of all the forces in relation to the inertial one (the most relevant in fluid mechanics) – Froude, Reynolds, Weber, Cauchy, Euler.

It is also known that although a perfect similarity would need the same value of these numbers between the prototype and the model, this is in general not possible as some of these similarities are incompatible (when using the same fluid or considering the same environment – gravity). So the most relevant force(s) present in the prototype must be selected and if compatible the model must be build according to the related similarity.

The relevant forces for most coastal hydrodynamics problems are the gravitational forces, friction, and surface tension (Dalrymple, 1985). Thus, the dimensionless products are combinations of the Froude, Reynolds, and Weber numbers. Neglected are compressibility and elasticity effects. Yet the use of the same fluid on both model and prototype prohibits simultaneously satisfying the Froude, Reynolds and Weber number scaling criteria and thus, most coastal models are run respecting Froude’s similarity only, which implies assuming that gravitational effects are the most significant and that the viscosity and surface tension of water do not play significant roles.

For coastal sediment models, another set of scale relationships governing the initiation of motion, the transport mode, and the transport rate have to be introduced into the model, again with inevitable scale effects. The coastal mobile bed sediment transport and morphology model is perhaps the most difficult of all physical hydraulic models (Kamphuis, 2009); yet despite the shortcomings it is, in many cases, the most important available instrument to bring about improvements with respect to sediment transport, and erosion.

Scaling Issues in Hydraulic Modelling

As was referred before the differences between the model and the prototype behaviour are due to several reasons, however:

• Scale effects are always present. When can they be neglected?
• As smaller is λ larger will be the scale effects. Can a given value of λ indicate if scale effects can be neglected?
• Scale effects do not affect in the same way all phenomenon/parameter under investigation – qualitatively they may be differently reproduced between model and prototype, but quantitatively they can be properly scaled (discharge vs air entrainment). How to evaluate these differences?
• In general some parameters are smaller in the model than in prototype – relative wave height, relative discharge, transported relative volume of sand, etc. How small is this damping effect?

Froude similarity is normally considered in open-channel hydraulics, where friction effects are negligible (deep-water wave propagation) or highly turbulent phenomena, since the energy dissipation depends mainly on the turbulent shear stress terms which are statistically correctly scaled even though the turbulent fine structures and the average velocity distribution differ between the model and prototype flows (Hughes, 1993). The gravitational acceleration is not scaled as well as the other numbers. Note: for the most important scaling ratios see Hughes (1993)/Martin and Pohl (2000). To avoid scale effects the influence of the gravity force on fluid flow should be negligible in a Reynolds model.

Reynolds similarity is normally considered in air models, laminar boundary layer problems, intake structures, seepage flows, creeping around immersed bodies or head losses elements (for lower values of Re). The other force ratios (numbers) are not properly modelled. A serious disadvantage is the scale velocity which will lead, for the same fluid, to higher velocities rather than in prototype. For that the use of air models is more appropriate. Hydraulic Froude models should be run under the same hydraulic rough regime as in nature in order to have the same losses’ level and not a faster decay as append sometimes with waves in a model.

It is not normal to build a model according to Weber similarity, so this can lead to some scale effects. Surface tension is negligible in most of hydraulic prototypes, but is important in scale models for air entrainment (wave breaking), small water depths, small wave heights and periods (capillary waves) or large fluid paths. Since many dimensions are smaller in the model this can lead to a dominant surface tension – larger relative air bubbles’ sizes, faster air detrainment and smaller volume fractions of air. Phenomenon that includes air flow (that depends also on the atmospheric pressure) needs a relative large scale.

Cauchy number is related with Young’s modulus, E, which in the case of water is not scaled, so its behaviour in impact phenomena, like wave-structure interaction, should be analysed with precaution. In fact a distensible structure (with Est) must be scaled without changing the Poisson number. In a Froude model, Est scales linearly with 1/λ, so if same material as the prototype is used the model structure will be λ too stiff which lead to erroneous resistance obtained in the model. This is also the case of air-water mixtures, for instance during the interaction of waves with a vertical quay. To model fenders is another example, where sometimes its Est is modelled with a spring system that includes the non-linear behaviour.

The Euler number considers pressure forces as relevant especially for high pressures (pipes, cavitation of turbines, pumps or hydraulic structures). If the cavitation number is not correctly modelled in a Froude model, the cavitation phenomena will be not observed.

Control of Hydraulic Modelling and Scale Effects

When trying to control hydraulic modelling some “tools/procedures” are available:

• If a certain physical phenomenon can be described mathematically (Navier-Stokes equations) both the model and prototype should follow the related equations (inspectional analysis);
• A physical problem can be reduced to a product of dimensionless parameters (geometrical and forces ratios), that must have the same values both in the model and in the prototype (dimensional analysis);
• Perform tests of known prototype situations, from which there is data available, allowing testing of other unknown situations (calibration);
• Performing tests with at least three λ being the smallest considered as the prototype; deviations of the dimensionless results of the smaller compared with largest one are due to scale effects; this requires large experimental efforts and does not inform about the scale effects of the larger model (scale series).

When trying to control scale effects some “tools/procedures” are available:

Avoidance: The best way to avoid scale effects would be to satisfy the force ratios limits for which there are no differences between the model and the prototype. There are also some practical rules/guidelines resulting from many tests and theoretical analysis if such a relation model/prototype is to be made. For limiting criteria and reference model scales with “reasonable size” and moderate scale effects (not necessary negligible) see Heller, 2011.

Replacement of Fluid: In order to avoid scale effects due to kinematic viscosity, sometimes air is used in the model instead of water, also because of the more advanced measurement techniques (turbulence e.g.). Although inertial and viscous forces are correctly modelled, however gravity, free-surface and cavitation effects are not reproduced. Example: ripple forms (water and sediments) were simulated with coal dust in glycerine with a surfactant to lower surface tension effects; a mixture of distilled water with isopropyl alcohol solution changed the breaker’s shape and evolution, air entrainment and energy dissipation.

Compensation: The model geometry is distorted in order to obtain a better similarity when compared to the prototype: roughness, vertical distortion and grain diameter.

Roughness: many models are built with a fixed bed (Froude similarity) leading to important scale effects due to surface tension and viscosity; as the Reynolds number is lower in the model the boundary roughness coefficient is higher and as geometric roughness similarity is ignored same friction coefficient can be achieved, resulting in a compensation of scale effects and a similarity of the water surface and energy gradients (only for models not too small and for a certain range of Re); the boundary friction is also relevant for the drag force in small-scale ship models – artificial roughness elements are included on the hull of the ship to shift boundary layer roughness to the turbulent rough regime leading to an appropriate prototype roughness modelling.

Vertical distortion: more used in fluvial hydraulics increases water depth, decreases scale effects since Re and We will be higher, higher flow velocities, a shift from hydraulic smooth to the rough flow regime, improvement of relative measurement accuracy and the duration of a run is reduced; as negative effects, 2D and 3D flow processes are incorrectly modelled and the model roughness must be calibrated.

Movable bed: Among many factors that influence sediment transport (Fr*, Re*, Energy line slope - SE, etc) the grain density and diameter may be the most critical ones as they cannot follow any scale without being affected by cohesion or changing in its transport mode (bed load to suspended load) in the model; This is only possible satisfying various similarity criteria that need to be fulfil simultaneously and for that only with a distorted model; the sediment density may be reduced and a larger grain diameter can be used, leading to an unknown time scale that can only be evaluated through prototype data; if λ is used to scale d₅₀ the corresponding value in the model will be smaller than 0,22 mm that would change the flow-grain interactions and related sediment transport. Limit model d₅₀ references state values of 0.5, 0.8 or 1.0 mm.

Correction: Sometimes the model is small for economic, limited space or time reasons and for that scale effects are expected; however some model results can be corrected afterwards for phenomena where enough information on the quantitative influence of scale effects is available; ex: correction factors for: solitary waves decay – boundary layer effects and fluid viscosity; coefficients for the stability of rubble mound breakwaters; equilibrium scour-depth estimates in cylinders; slopes of movable surf zone beach profiles; wave impact pressures and air leakage in the impact zone.

The Future of Physical Modelling

Similitude theory is quite old, new modern laboratory techniques and instrumentation exists and better field data is known allowing better calibrations. However several questions can be raised:

• What’s the role of un-scaled atmospheric pressure in air entrainment?
• How the scaling of g would affect the results?
• How numerical simulations can be used to improve physical ones (composite modelling) specially for phenomena that are difficult to reproduce in models – turbidity currents, distensible structures, turbulent flows and bridge pier scour?
• Can numerical models used for a certain scale confirm some model scale effects as the effect of kinematic viscosity;

Another important issue is related with the development of harbours and related challenges. For several reasons the volume of goods actually transported by ships is increasing, and in order to optimize this, the size of ships has also been increasing. These bigger ships need bigger water depths for their pathways that can only be obtained, in general, further away the coast. This leads also, in general, to higher wave design conditions and other safety problems related with the foundations, currents, scour, breaking, etc. Many new harbour facilities are reaching dimensions never reached before. Taking in mind all this it will result in new interesting problems for hydraulic engineers namely for the development of design tools for these new sea conditions and harbour structures’ dimensions.

Finally, as it was referred physical hydraulic modelling is an important field within civil engineering and others scientific areas that can use several tools to resolve complex problems. How to avoid completely duplication of work and how to stimulate the exchange of information as each laboratory has their particular method for conducting model studies that sometimes makes difficult the comparison of results and data transfer will be a future challenge.

See also

• Modelling coastal hydrodynamics
• Modelling Review (all generic types)
• Dynamics of mud transport
• Manual Sediment Transport Measurements in Rivers, Estuaries and Coastal Seas

References

The main author of this article is Taveira Pinto, Francisco With contributions by: Paulo Rosa-Santos, Luciana das Neves, Raquel Silva For other articles by this author see Category:Articles by Taveira Pinto, Francisco