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Chapter 6 Pipe Flow and Dimensional Analysis

# Chapter 6 Pipe Flow and Dimensional Analysis - Chapter 6...

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Chapter 6 PIPE FLOW AND DIMENSIONAL ANALYSIS 1.1 Introduction Fluid mechanics problems often depend on a large number of variables, and determining the analytic relation between those variables is usually impossible. Consider the problem of evaluating the pressure drop for turbulent flow in a pipe. After nearly 100 years of effort, this basic problem with obvious engineering relevance has not yet been solved completely. Flow prediction for more complicated geometric configurations is no better. Hence, fluid mechanics relies heavily on experimentation; much more so than, say, structural mechanics or electronic circuit analysis. This dependence on experimentation explains the extensive use in fluid mechanics of a technique called “dimensional analysis” [1]. Dimensional analysis significantly reduces the number of relevant variables involved in a problem. The decrease in variables reduces the number of tests required and clarifies data presentation. Non-dimensional data are also more general. For example, non-dimensionless information on the speed at which a marble falls in water can also be used to predict the speed at which a balloon rises in air. The basic principles of dimensional analysis are fairly straightforward. The Buckingham Pi theorem [2] forms the basis of dimensional analysis. “If it is known that a physical process is governed by a dimensionally homogeneous relation involving n dimensional parameters, such as ( 29 n x x x f x , , 3 2 1 = , where the x’s are dimensional variables, then there exists an equivalent relation involving a smaller number, (n-k), of dimensionless parameters, such as: ( 29 k n F - Π Π Π = Π , , 3 2 1 , where the Π’s same situation as above are dimensionless groups constructed from the x’s. The reduction, k, is usually equal to, but never more than, the number of fundamental dimensions involved in the x’s. Length, mass, time, charge, and luminosity are all examples of fundamental dimensions.” For a more physical explanation of the Buckingham Pi theorem, see the Appendix at the end of this chapter consisting of an extract from Reference [3]. Note that the Pi theorem tells us that we can organize variables efficiently and suggests possible ways that the organization can be carried out; however, it does not lead to unique results. The method of dimensional analysis involves a measure of art and experience. Also note that Chapter 8 in your fluids text ( A Physical Introduction to Fluid Mechanics by Alexander Smits) MAE 224: Introduction to Fluid Mechanics Laboratory 1

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discusses dimensional analysis but uses a slightly different method (using a matrix of dimensions) for determining the non-dimensional groups. This method will be introduced in lecture. Here we will use the example of fully developed turbulent pipe flow to illustrate one method of dimensional analysis. The flow in a constant area pipe or duct is said to be fully developed if the shape of the velocity profile is the same at all cross sections. We
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