Dimensional Analysis and Similitude¶

Lesson Outline¶

topic
topic

Background¶

Fluid mechanics design problems rely upon experimental data, either collected directly or inferred from prior published works. In most cases empirical results do not directly apply to general situations so engineers need to interpret the original data in normal design practice.

Such data are found in handbooks, journals, and other authorative sources.

Some examples are friction loss coefficients for pipes, valves, and other fittings; drag coefficients; fluid properties.

Dimensional Analysis¶

Is a tool used to guide expermental design and practice

Similitude¶

Is a tool used to scale laboratory results to full-scale situations. Many practical problems have unusual enough geometry that studies on a replica at small scale are used to predict flow patterns, pressure variation, frictional losses.

When such tests are performed:

Replica == Model
Full Size == Prototype

Note

Physical modeling is common in fluid mechanics studies even today (circa 2022). Scale models are used to study and estimate behavior at full scale (aka “prototype”), although CFD methods are steadily replacing physical modeling as a primary tool.

Example 1:¶

# computational thinning
import math
q = 0.1 #discharge
v_jet = 50.0
omega = 500*2*math.pi/60
radius = 0.5
rho = 1000.0
power = rho*q*v_jet*radius*omega - rho*q*omega**2*radius**2
print("Power ",round(power,2)," Newton-meters/sec    ")

Power  62360.77  Newton-meters/sec

Example 2:¶

Readings¶

CE-3305-2022-1 Syllabus. http://54.243.252.9/ce-3305-webroot/0-Syllabus/ce-3305-2022-1-syllabus.html
Hibbeler, R.C, Fluid Mechanics, 2ed. Prentice Hall, 2018. ISBN: 9780134655413 pp. 444-469
DF Elger, BC Williams, Crowe, CT and JA Roberson, Engineering Fluid Mechanics 10th edition, John Wiley & Sons, Inc., 2013. (placeholder file to get links working). http://54.243.252.9/ce-3305-webroot/3-Readings/EFM-10.pdf
Cleveland, T. G. (2014) Fluid Mechanics Notes to Accompany CE 3305 at Jade-Holshule (TTU Study Abroad 2015-2019), Department of Civil, Environmental, and Construction Engineering, Whitacre College of Engineering. http://54.243.252.9/ce-3305-webroot/3-Readings/ce3305-lecture11.pdf
Cleveland, T. G. (2009) Error Analysis in Instructor Notes to Accompany CE 5333 Studies in Dimensional Analysis and Similitude, Department of Civil, Environmental, and Construction Engineering, Whitacre College of Engineering.
Cleveland, T. G. (2009) Propagation of Error in Instructor Notes to Accompany CE 5333 Studies in Dimensional Analysis and Similitude, Department of Civil, Environmental, and Construction Engineering, Whitacre College of Engineering.
Cleveland, T. G. (2009) Dimensional Analysis in Instructor Notes to Accompany CE 5333 Studies in Dimensional Analysis and Similitude, Department of Civil, Environmental, and Construction Engineering, Whitacre College of Engineering.
Doebelin, E. O. (1990). Measurement Systems (4th ed.). New York, NY: McGraw-Hill.
Holman, J. P. (1989). Analysis of Experimental Data in Experimental Methods for Engineers, 5th Edition. New York, NY: McGraw-Hill.
Kline, S. J. and F. A. McClintock (1953). Describing uncertainties in single-sample experiments. Mechanical Engineering (No. 75), 3–9.
Buckingham, E. (1915). Model experiments and the forms of empirical equations. Transactions of American Society of Mechancial Engineers (No. 37), 263.
Hwang, N. H. C. and C. E. Hita (1987). Dimensional Analysis in Fundamentals of Hydraulic Engineering Systems. San Diego, CA: Prentice Hall.
White, F. M. (1979). Fluid Mechanics. New York, NY: McGraw-Hill
Dimensional Analysis Part 1
Dimensional Analysis Part 2
Dimensional Analysis Part 3
Dimensional Analysis Part 4

Engineering Fluid Mechanics