FLUIDS LABORATORY REPORT 3: Modelling and SimilitudeAbstract:A virtual laboratory platform analysing a ball suspended by a jet of air was used to derive a functional relationship between the ball suspension height and fluid, dimensional, and forcing properties. Due to the relatively large number of variable parameters (h–suspended height of ball, ࠵?࠵?–diameter of fluid jet, ࠵?–density of fluid, ࠵?–viscosity of fluid, ࠵?–diameter of ball, ࠵? –weight of ball, Q–volumetric flow rate), dimensionless analysis was employed to determine this relationship while collecting and analysing data from experiments. The dimensionless Pi groups used were ࠵?࠵?࠵?࠵?࠵?,࠵?࠵?࠵?,ℎ࠵?࠵?, and࠵?࠵?࠵?2. The relationship between each of the Pi groups was plotted and analysed by conducting virtual experiments in which the ball diameter, ball weight and fluid flow rate were changed in isolation. The final empirical relationship (using dimensionless groups) was determined to be: ℎ = −338.533314 × ࠵?࠵?× (࠵?࠵?࠵?2) × (࠵?࠵?࠵?࠵?࠵?)2+ 0.44394606 × ࠵?࠵?× (࠵?࠵?࠵?࠵?࠵?)13+ ࠵?࠵?× 3.8023 ×࠵?࠵?࠵?. Or, in simplified functional form, ℎ = −338.533314 × (࠵?࠵?࠵?3࠵?࠵?2) + 0.44394606 × (࠵?࠵?࠵?࠵?2࠵?)13+ 3.8023࠵?Aim: The virtual laboratory experiment was conducted to determine a relationship between the suspended height of a ball in jet of fluid and the many parameters that facilitate this phenomenon. The experiment was conducted to reveal the effectiveness of dimensional analysis, and how it can be employed to analyse real life events. In order to construct the empirical function, dimensional analysis was employed. By studying the dimensions of available parameters and forming dimensionless groups one reduces the variables to be explored, thus imposing limitations on the possible functional form describing the physical system. This way, the amount of experiments and data necessary to define the function is minimised. This is a crucial method to employ when analysing complex multi-variable systems, and attempting to generalise the results. The seven variables were reduced to four dimensionless Pi groups. Experiments were conducted by varying the Pi groups independently from, then in relation to one another, in order to determine the final empirical relationship. Platform and Methodology 1.Determination of Dimensionless Pi Groups To create the dimensionless groups to be tested, the Buckingham Pi theorem was employed. The first step of this procedure is to determine the dimensions of all variables considered. The variables are defined in Figure 1 and their respective dimensions in Table 1.

Table 1:Variables and Dimensions ࠵? − Diameter of Ball Table 2: Dimensionless Groups Secondly, one must determine the number of dimensionless groups that will be present. As there are seven variables and three basic dimensions (M, L,T), there will be four dimensionless groups into which the parameters can be formed. All of these dimensionless groups must include repeating