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Physical similarity and dimensional analysis
Published in Bernard S. Massey, John Ward-Smith, Mechanics of Fluids, 2018
Bernard S. Massey, John Ward-Smith
Kinematic similarity is similarity of motion. This implies similarity of lengths (i.e. geometric similarity) and, in addition, similarity of time intervals. Then, since corresponding lengths in the two systems are in a fixed ratio and corresponding time intervals are also in a fixed ratio, the velocities of corresponding particles must be in a fixed ratio of magnitude at corresponding times. Moreover, accelerations of corresponding particles must be similar. If the ratio of corresponding lengths is rl and the ratio of corresponding time intervals is rt, then the magnitudes of corresponding velocities are in the ratio rl/rt and the magnitudes of corresponding accelerations in the ratio rl/rt2.
Preliminary Concepts
Published in Hillel Rubin, Joseph Atkinson, Environmental Fluid Mechanics, 2001
The result of dimensional analysis is a definition of a relationship between the appropriate dimensionless variables resulting from grouping the parameters of the problem. The specific form of the relationship is not revealed using dimensional analysis — physical experiments must be performed to provide additional information. For example, dimensional analysis can be used to show that a dimensionless group incorporating the drag on a sphere moving at constant velocity through a fluid should depend on Re. However, the actual form of the relationship is determined from experimental results. One other important application of dimensional analysis is in providing a means of scaling the results of a model study to prototype conditions. This is necessary, for instance, in extrapolating results from laboratory physical modeling studies to field conditions. In order to do this, conditions of similarity must be satisfied. There are three kinds of similarity. Intuitively, a model or experiment should be geometrically similar to the field situation, which means that the ratio of all length scales is the same between the model and the prototype. Kinematic similarity incorporates similarity of length and time quantities. Dynamic similarity also must be satisfied in order to properly scale results concerning forces and stresses. Kinematic and dynamic similarity are obtained when appropriate dimensionless parameters are the same in the model and in the prototype. Dynamic similarity is equivalent to saying the ratios of relevant forces are the same.
Modeling of Thermal Systems
Published in Yogesh Jaluria, Design and Optimization of Thermal Systems, 2019
where λ2 is the scale factor and the subscripts p and m again indicate the prototype and the model. For kinematic similarity, the model and the prototype must both have the same length-scale ratio and the same time-scale ratio. Consequently, derived quantities such as acceleration and volume flow rate also have a constant scale factor. For a given value of the magnitude of the gravitational acceleration g, the Froude number Fr (Table 3.1) represents the scaling for velocity and length. Therefore, this kinematic parameter is used for scaling wave motion in water bodies.
Investigations on the dynamics of container stack and securing system under rolling motion using a scaled model test
Published in Ships and Offshore Structures, 2022
Jiaqi Liu, Chuntong Li, Deyu Wang, Zhonghua Cai
This study employed 4 scaled models of the 20-ft ISO freight container with a 1/10 geometric similitude (λ=10). The basic geometric features of the scaled models, such as length, breadth and height follow a strict numerical relation with the full-scaled container. The similarity of velocity and acceleration, which is kinematic similarity, can be guaranteed based on Froude scaling laws. Dynamic similarity is presented when the ratio between forces acting on the system is a fixed ratio. The scaled model is established based on two dimensionless parameters, respectively, including the gravity force/inertia force ratio, and the elastic force/inertia force ratio. The scaled container model follows the scaling laws (Vassalos 1998; Bursi and Wagg 2009), which is summarised in Table 1.
Mobile-bed similitude evaluation of hydraulic sediment response models
Published in Journal of Applied Water Engineering and Research, 2019
Muhammed T. Mustafa, Amanda L. Cox, Robert D. Davinroy, Bradley J. Krischel, Ivan H. Nguyen
Similitude, or similarity, is the existence of a relationship between a model and prototype such that hydraulic conditions in a model at a specific location and time are proportionally related to hydraulic conditions in the associated prototype at the corresponding location and time. Perfect similitude requires that the model and prototype be geometrically, kinematically, and dynamically similar. Geometric similarity is the similarity between length ratios, and kinematic similarity is the similarity between motions. Dynamic similarity is achieved when similarity between forces exist while maintaining geometric and kinematic similarity. For physical hydraulic modelling, the relevant forces include inertial, gravitational, viscous, pressure, and surface-tension forces (Heller 2011).
Criterion of vehicle instability in floodwaters: past, present and future
Published in International Journal of River Basin Management, 2021
Syed Muzzamil Hussain Shah, Zahiraniza Mustaffa, Eduardo Martinez-Gomariz, Do Kyun Kim, Khamaruzaman Wan Yusof
A variation of the Shu et al. (2011) formula was proposed by Xia et al. (2013). The experimental runs were conducted in a flume of the Experimental Hall for Sediment and Flood Control Engineering, Wuhan University, China. The horizontal flume was 1.2 wide, 60 m long and with the bed covered by a thin cement layer and two glass sides. To obtain the conditions of water depth and corresponding velocity at the threshold of vehicle instability, three orientation angles (0°, 180° and 90°), as shown Figure 15, and two ground slopes (1:50 and 1:100) were selected. Two sets of different die-cast-model vehicles were selected (Honda Accord and Audi Q7) at the scale ratio of 1:14 (larger die-cast-model) and 1:24 (small die-cast-model) strictly following the laws of similarity. The wheels of the vehicle were locked; thus, only the incipient motion under sliding was considered. The physical models were meant to evaluate how the vehicle’s size, kerb weight and design shape could affect the threshold of vehicle instability in floodwaters. In free-surface flows, the effects of gravity are predominant and the model-prototype similarity was generally satisfied following the scaling criterion of Froude similarity. According to the requirements for kinematic similarity, the scale ratio of the inertia force to the gravity force gives the relationship between the scale ratios of velocity and length, thus satisfying the Froude similarity. Dynamic similarity implies that the ratios of the prototype forces to model forces are equal if the density of a model vehicle is equal to the value of a prototype vehicle (Xia et al.2013).