Similitude and Dimensional Analysis
Experiments are usually necessary to solve scientific and technologic problems because they can provide theoretical basis and criterion. The theoretical fundamentals for experimental design and evaluation are similitude and dimensional analysis. This chap
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Similitude and Dimensional Analysis
Abstract
Experiments are usually necessary to solve scientific and technologic problems because they can provide theoretical basis and criterion. The theoretical fundamentals for experimental design and evaluation are similitude and dimensional analysis. This chapter includes similitude, dimensional analysis, and its application. The learning goals are as follows. First, we should understand the concepts of mechanics similitude and similarity criterion. Then we have to master the applications of approximate similitude, the p theorem and dimensional analysis. Finally, three important approximate models should be grasped including Froude model, Reynolds model, and Euler model. Keywords
Mechanics similitude similitude Pi theorem
8.1
Froude number
Euler number
Approximate
Similitude
8.1.1 Basic Concepts of Mechanics Similitude The experiments of engineering fluid mechanics mainly have two categories. One is engineering model experiment, aiming at hydraulic prediction in large machineries and upcoming constructed structures. The other is observation experiment, aiming at figuring out unknown flow characteristics. When investigating the inner mechanics and physical nature of fluid motion, all steps have to be based on scientific experiments, including proposing research method, developing fluid mechanics theory and solving engineering problems. © Metallurgical Industry Press, Beijing and Springer Nature Singapore Pte Ltd. 2018 H. Song, Engineering Fluid Mechanics, https://doi.org/10.1007/978-981-13-0173-5_8
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8 Similitude and Dimensional Analysis
To represent actual flow phenomena and predict flow properties utilizing physical model under actual conditions, it is necessary that prototype and model have mechanics similitude [1, 2]. Mechanics similitude indicates that all model variables at corresponding points are proportional to its prototype, and includes three aspects: geometric, kinematic and dynamic similitude. (1) Geometric similitude It means that the model and its prototype have identical shape, and differ only in scale. Set the subscript p denote the prototype, and m denote the model. The length scale ratio kl (also named linear scale ratio) is defined as kl ¼
lp lm
ð8:1Þ
The area scale ratio kA and the volume scale ratio kV can be expressed as kA ¼
l2p Ap ¼ 2 ¼ k2l Am l m
ð8:2Þ
kV ¼
l3p Vp ¼ 3 ¼ k3l Vm l m
ð8:3Þ
The length scale ratio kl is the first basic scale ratio of mechanics similitude, which can be used to derive the area scale ratio kA and the volume scale ratio kV . The dimensions of length l, area A and volume V are L, L2, and L3, respectively. (2) Kinematic similitude It means that in addition to geometric similitude, velocities at all corresponding points are in the same ratio. The velocity scale ratio is kv ¼
vp vm
ð8:4Þ
It is the second basic scale ratio of mechanics similitude, and other kinematic scale ratios can be deduced with kl and kv according to the definition and dimension. The time scale ratio is kt ¼
tp lp =vp kl
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