CE 1501 CE 150 Fluid Mechanics G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico

CE 1502 Dimensional Analysis and Modeling Reading: Munson, et al., Chapter 7

CE 1503 Introduction The solutions to most fluid mechanics problems involving real fluids require both analysis and experimental data In this section, we look at the techniques used in designing experiments and correlating data Specifically, we will learn how laboratory experiments (or models) can be used to describe similar phenomena outside the laboratory

CE 1504 Dimensional Analysis Example Consider an experiment that investigates the pressure drop in an incompressible Newtonian fluid flowing through a long, smooth- walled circular pipe Based upon our “experience”, the pressure drop per unit length is

CE 1505 Dimensional Analysis Example To determine the nature of this function, an experiment could be designed which isolates and measures the effect of each variable:

CE 1506 Buckingham Pi Theorem Isolating the variables and performing this experiment would be difficult and time-consuming The number of independent variables can be reduced by a dimensional analysis technique known as the Buckingham pi theorem: A dimensionally homogeneous equation with k variables can be reduced to k - r dimensionless products, where r is the minimum no. of reference dimensions needed to describe the variables

CE 1507 Determining the Pi Terms Method of Repeating Variables: 1) List all variables in the problem 2) Express each variable in terms of basic dimensions 3) Determine the number of pi terms 4) Select a number of repeating variables that equals the no. of reference dimensions 5) Form a pi term for each non-repeating variable such that the combination is dimensionless 6) Write an expression as a relationship of pi terms and consider its meaning

CE 1508 Comments on Dimensional Analysis Selection of variables –no simple procedure –requires understanding of the phenomena and physical laws –variables can be categorized by geometry, material properties, and external effects Basic dimensions –usually use MLT or FLT –all three not always required –occasionally, the no. of reference dimensions is less than no. of basic dimensions required

CE 1509 Comments on Dimensional Analysis Repeating variables –number must equal the no. of reference dimensions –must include all basic dimensions contained in variables –must be dimensionally independent of each other Pi terms –number of terms is unique but form of each term is not unique, since selection of repeating variables is somewhat arbitrary (unless there is only one pi term)

CE Dimensionless Groups in Fluid Mechanics Common groups given in Table 7.1 –Reynolds number –Froude number –Euler number –Mach number –Strouhal number –Weber number

CE Correlation of Experimental Data Dimensional analysis and experimental data can be used together to determine the specific relationship between pi terms Problems with one pi term: –relationship is determined by dimensional analysis but constant must be determined by experiment

CE Correlation of Experimental Data Problems with two or more pi terms: –typically requires an experiment where one variable in each pi term is varied and its effect on the dependent variable is measured –an empirical mathematical relationship can be developed if the data shows good correlation –the mathematical relationship, or correlation equation, is only valid for the range of pi values tested

CE Modeling & Similitude A model is a representation of a physical system used to predict the behavior of the system –mathematical model –computer model –physical model, usually of different size and operating conditions Similitude refers to ensuring that the results of the model study are similar to the actual physical system

CE Theory of Models For a given physical system (i.e., the prototype): For a model of this system: –where the formof the function  will be the same as long as the physical phenomena are the same

CE Theory of Models If the model is designed and operated such that –Thus, if all pi terms are equal, then the measured value of  1m for the model will be equal to the corresponding  1 for the prototype. These equations provide the modeling laws that will ensure similarity between prototype and model.