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Usefulness of Dimensional Analysis with Particular Reference to Scale Model Testing Introduction Dimensional analysis is a widely used technique within the fields of engineering and physical sciences. This technique is applied for achieving reduced number of variables about any particular subject matter before experiments. There are two strong uses of dimensional analysis which include reduction of dimensions before experiments and scalability of results (Albrecht et al ., 2013). Typically, a dimension is characterized by geometric expression of a dimension and is usually expressed in one-, two-, or three-dimensional space but it also defines the physical expression of a value (Kowalewski et al ., 2017). The classic example which can be illustrated here to further explain the reduction of variables is given in this equation (Sonin, 2001): d = f (V, p, D, E,^ γ) In this equation which explains the deformation of elastic balls, d denotes the diameter of circular mark left after the ball impacted the wall, V denotes the velocity, ρ denotes the material density, D denotes the diameter, and E represents the elasticity modulus of the ball while γ denotes Poisson’s ratio. There are five independent variables in this case which on applying dimensional analysis can be reduced to two through the Buckingham Π-Theorem and is given in the following equation (Buckingham, 1914): π0 = h ( π1,^ π2) In this equation, the above model has been characterized with dimensionless response where π 0 = d / D ; π 1 = E /(ρ V^2 ); and π 2 = γ. Here h is referred as dimensional analysis (DA) model. Given the fact that the dimensional analysis results in dimensionless variables, the scalability of the results is major advantage. A dimensional analysis process undergoes four steps to develop a DA model. This includes; 1) identification of dependent and independent variables, 2) identification of complete and dimensionally independent subset, 3) identification of the dimensionless forms
of the variables not in the basis set, and 4) application of Buckingham’s Π-Theorem for developing DA model (Albrecht et al ., 2013). In this report, two case studies will be discussed in length to explain the usefulness of dimensional analysis with particular reference to scale model testing. One case study will be focused on its application in marine engineering while the second case study will explore the usefulness of dimensional analysis in geotechnical engineering. A conclusion will highlight the important aspects of this report. Case Study1: Physical Hydraulic Engineering Model Within the field of experimental fluid mechanics, dimensional analysis is considered as the most effective method for determining the implied criteria used to identify dynamic similarity (Kobus, 1980). On development of DA model, the dimensionless variables are helpful to represent results without any requirement of upscale them. Any problem which has n independent variables q 1 ,^ q2 …, qn can be reduced to^ n – r^ dimensionless variables. Due to reduction of variables from n to n – r , the quantity of relevant tests is also reduced in number (Heller, 2011). In this case study, the dimension analysis of impulse wave generation is presented along with illustration of a definition sketch. There were seven independent variables which were found during dimensional analysis of impulse wave generation. These seven independent variables included; “the still water depth h [L], slide impact velocity V (^) s [LT21 ], slide thickness^ s [L], bulk slide volume^ −V^ s [L3], bulk slide density ρ (^) s [ML -3^ ], grain (subscript g) diameter dg [L] and the slide impact angle α [ο]” (Heller,
2011, p.298). In addition, the water density ρ^ [ML^ -3^ ], gravitational acceleration g [LT^ -2], horizontal distance x [L] from the coordinate origin and time t [T] are found to impact the unknown wave features which involve the maximum wave amplitude or water-surface displacement (Heller et al. 2008).
On further explanation, it was observed that it was necessary to find a similar value for each of the dimensionless parameters π (^) 1, π2… π 8 (Heller, 2011). As far as scaling effects were concerned, Heller et al. (2008) computed the scale effects using seven scale series. While applying the Froude scale ratios, the scaling of seven governing variables along with other geometric parameters was performed. This included reduction of space between wave gauges from 1.00m to 0.50m standardized with scale ratio of λ = 2 and to 0.25m with scale ratio of λ = 4. The purpose of these reductions was to ensure consistency of dimensionless parameters within defined limits of scale ratios. However, three different approaches were used to minimize the scale effects and these three approaches included avoidance, compensation, and correction. In avoidance, the standard rule is to achieve limiting values for force ratios. During compensation, the prototype similarity of an enhanced model is preferred over geometric similarity modeling. However, it is implicated that dimensional analysis in hydraulics can be considered for prototype modeling without satisfying the scaling ratios necessarily (Heller, 2011). This case study is an illustration of how the dimensional analysis can be used in developing DA model with particular reference to scale modeling. The scale effects, however, have varying effects on prototype modeling and should be determined through applying equations as mentioned above. This is only one aspect of the usefulness of dimensional analysis in marine engineering and cannot be applied in broad terms. Case Study2: Three-Dimensional Finite Element Analysis in Geotechnical Engineering Due to significance of three-dimensional finite element analyses in geotechnical engineering practice originating from several decades ago, it is primarily applied in building excavation. There is sufficient empirical literature that investigated the role of three-dimensional analyses in building excavations. Back in 2004 when the collapse of the Circle Line of the
Singapore Mass Rapid Transit was reported on 20th^ April, the team of researchers conducted
multiple three-dimensional finite element analyses to determine the three-dimensional impacts on the collapse. These three-dimensional analyses involved evaluation of the soil profile, overview, sequence of construction and comparison with field measurements (Lee et al., 2010).
Figure SEQ Figure * ARABIC 2: Overall view of Collapse Zone (Lee et al., 2010) Upon soil profiling, it was found that the soils of the Kallang Formation were detected under the site of collapse. The identified soil profiles were mathematically inserted and modeled derived from the data based upon 22 boreholes. Each element was allocated to its respective soil type which was performed through dimensional analysis. This is further illustrated in the following figures (Lee et al., 2010):
Figure SEQ Figure * ARABIC 5: Finite element mesh a) before excavation b) after excavation level 10 (Lee et al.,2010)
During analysis of collapse behavior, it was particularly determined that the wall deflection could demonstrate relatively smaller impact till the excavation level 10. However, this phenomenon was particularly obvious after the excavation level 10. This was also similar to the accounts narrated by the witnesses of the collapse (Lee et al., 2010). The usefulness of three-dimensional analyses has always been in debates but it is an established fact that these analyses are helpful to describe complex situation in a precise manner. However, it has largely been argued that these analyses in particular reference to collapse of buildings have been effective in post-collapse. Also, there is a recurring debate whether these analyses can be useful in preparation attempts for any larger incidents. This is somehow answered in a classical case of its application in urban underground construction works. In the past, these analyses had met with challenges due to hardware and software lacking. But since
now there is an enormous advancement in technology, these analyses can be performed using robust technologies which will further enhance their precision and accuracy. Recent developments in algorithms have further strengthened the scope of dimensional analysis whether this is one-, two-, or three-dimensional in nature. The computation systems in line with dimensional analysis requirements are already developed in multiple fields including engineering. However, the adoption and implementation of these technology-driven dimensional analyses remain still a question (Lee et al., 2010). The future research may be helpful to determine the usefulness of dimensional analysis equipped with technology, such as 3D technology. Conclusion Within the professional practice of engineering and physical sciences, dimensional analysis is a common approach to derive dimensionless variables which make the process of analysis easy and precise. With this technique, the reduced number of variables about any particular subject matter is achieved before experiments, which is useful for experimentation. The dimensional analysis offers two major advantages in any field and these advantages include reduction of dimensions before experiments and scalability of results. A dimension is usually described in geometric expressions of a dimension and is usually expressed in one-, two-, or three-dimensional space. The dimension represents the physical expression of a value as well. There are four steps involved in a dimensional analysis process that ultimately leads to a DA model. These four steps include identification of dependent and independent variables, identification of complete and dimensionally independent subset, identification of the dimensionless forms of the variables not in the basis set, and application of Buckingham’s Π- Theorem for developing DA model.
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