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Terminology

Scales and Scaling

The purpose of scaling usually is to inter- or extrapolate quantities a along a scale or a reference quantity A = a_r . Scales may be basic quantities like meter m , kilogram kg , or second s , then they are frequently referred to as units. However, in the following scales are rather natural or even composed quantities of several constants which are characteristic for the observed phenomenon. In any case scales allow to introduce dimensionless quantities denominated wit a star in the following: $$ a = A a^\ast $$

The actual law how-to scale might be found by observation and intuition. These observations provide an empirical "law" which relates an observed, dependent quantity to the varied, independent quantity. Thus predictions for certain relevant quantities for non tested domains might be provided. If a phenomenon scales, then the dependent quantity correlates to the independent quantity and the pairs of values plotted in a double log diagram should will lie on a straight line.

The chances for success, i.e. getting a more universal relation, are usually furthered if the dependent and independent quantities are transformed with a sub-set of scales to dimensionless quantities.

Scale factors

The scale factor \hat{a} relates the two corresponding scales in the model A^' = {a_r}^' and in the prototype A = a_r : $$ \hat{a} = A^'/A $$ If there is a homologous scaling corresponding values of this quantity are scaled from the model to the prototype with this scale factor. Of course the scaling factor for the different quantities like length (classical size scale), time, temperature, electrical current, force or mass might differ. These scale factors are called primary scale factors as they described the scaling of the basic units, whereas all derived quantities are scaled by secondary scale factors. For instance, the secondary scale factor for the velocities \hat{v} can be easily derived by dividing the scale factor for the lengths \hat{l} by the scale factor for the time \hat{t} : $$ \hat{v} =\hat{l} / \hat{t} $$

In many engineering applications observations are made with a size-scale model, which is a geometrical model of the prototype. For some applications not all of the dimensions of the model have to be scaled equally, instead the known laws require special rules to preserve a consistent scaling of forces, for instance. One example is the scale modelling of airfoils, sometimes labelled as parametric scaling.

Similitude, Similarity, Model, Prototype

Similitude and similarity are synonyms in this context and define the possibility to transfer observations from a model to a prototype.

The methods, described in the following, allow to get a conceptional picture of a complicated phenomena, which might be not directly accessible in a prototype. Reasons which make a direct study impossible are the real size of the prototype, too big or too small to handle the prototype easily, the general possibility to conduct experiments in our usual environment (e.g. gravity) and simply the implied costs of the prototype which might be lost or damaged in an experiment. Another motivation might be to exclude unwanted qualities of a scenario to be studied, e.g. to use a non-combustible gas for studying the dispersion of a burning gas like hydrogen.

Similarity methods include the Parameter Approach, the Law and the Equation Approach. The Parameter Approach requires experience, the Law Approach the identification of basic laws involved, and the Equation Approach needs a valid theoretical model for the whole coupled phenomena, usually defined by a set of differential equations.

Dimensional Analysis

The dimensional analysis provides a methodology to identify suitable combinations of parameters influencing the problem on the basis of the fundamental dimensions of a problem and some pure mathematical recipes. With the help of dimensional analysis the minimum number of independent dimensionless parameters, including the observed and varied quantity might be identified in a unique way. Based on this set of dimensionless parameters other observations might be easily translated in this usually unique systems of variables. However, this methodology requires a sound understanding of the relevant physical phenomena and a proper pre-selection of the parameters entering the mathematical scheme.

All methods allow to derive the dimensionless parameters and in some cases even the relative importance of single effects. For similarity all dimensionless parameters have to be the same in the model and in the prototype.

Relaxation

At least for complex phenomena a perfect scaling, similarity respectively, with identical dimensionless parameters in the model and prototype cannot be fulfilled or only fulfilled if model and prototype are identical, what would make the whole effort obsolete. Then one has to relax the model requirements. One way is to identify less important influences or partial phenomena and to omit their modelling. Alternatively a temporal or spatial separation might help, when certain phenomena prevail in a certain time span or area only.

The following chapters first introduce the established methodology and then show applications to a few hydrogen safety relevant dispersion problems, the SBEPV3 Garage problem and a horizontal buoyant jet.