ME 310 Chap.8 Dimensional Analysis and Similitude
Dimensional Analysis and Similitude
Why the course emphasises it
The course description lists learning to use dimensional analysis to dramatically simplify complex parameter spaces as one of its aims, and the content outline sums it up as nature caring not about metres and seconds.
Four lectures spanning late October and early November are devoted to it, and it feeds directly into pipe flow and drag.
The Pi theorem
If a quantity depends on a list of variables, and those variables involve a certain number of independent dimensions, the relationship can be rewritten using fewer variables: the number of dimensionless groups equals the number of variables minus the number of reference dimensions.
The method of repeating variables finds the groups systematically by combining each remaining variable with powers of a density, a velocity and a length chosen to cancel all dimensions.
Groups worth recognising
The Reynolds number compares inertia with viscous forces; the Froude number, inertia with gravity; the Mach number, flow speed with sound speed; the Weber number, inertia with surface tension; and pressure and force coefficients compare pressure or force with dynamic pressure.
Knowing which ratio a group represents tells you when it matters.
Models and similarity
A model predicts full-scale behaviour when it is geometrically similar and every independent group matches. Then every dependent group, such as a drag coefficient, matches too, and measurements can be scaled.
When two groups cannot both be matched, as with Froude and Reynolds numbers for a ship model tested in water, engineers match the dominant one and correct for the other.
Early practice
The first homework sets already practise the skill: finding the dimensions of the drag coefficient in the wind-load formula, and checking whether a slow pipe-flow formula with a numerical constant is valid in any unit system.
Choosing repeating variables
Pick variables that are easy to control in an experiment and that together contain mass, length and time, typically density, a characteristic speed and a characteristic length.
Never choose the dependent variable as a repeating one, or it will appear in every group.
What this chapter covers
- 01
Dimensional homogeneity and reference dimensions
- 02
Buckingham Pi theorem
- 03
Method of repeating variables
- 04
Common groups and the force ratios they represent
- 05
Geometric, kinematic and dynamic similarity
- 06
Scaling model measurements to prototypes
- 07
Conflicting similarity requirements
Worked example · free
Groups for drag on a smooth sphere
- 2Count: five variables (F, D, V, ρ, μ) and three dimensions (M, L, T), so two groups.
- 2Repeating variables ρ, V, D. First group: F / (ρV²D²).
- 1Second group: μ / (ρVD), the inverse of the Reynolds number.
- 1Result: F / (ρV²D²) = φ(ρVD/μ), a drag coefficient that depends only on Reynolds number.
Key terms
- Buckingham Pi Theorem
- The Buckingham Pi theorem states that a relation among k variables with r reference dimensions can be written among k minus r dimensionless groups.
- Reynolds Number
- The Reynolds number is the ratio of inertial to viscous forces, density times speed times length divided by viscosity.
- Froude Number
- The Froude number is the ratio of flow speed to the speed of gravity waves, important for free-surface flows.
- Dynamic Similarity
- Dynamic similarity is the condition in which all independent dimensionless groups have the same values in a model and its prototype.
Dimensional Analysis and Similitude FAQ
How many dimensionless groups should I expect?
Subtract the number of independent reference dimensions, usually mass, length and time, from the number of variables, including the dependent one. Five variables in three dimensions give two groups.
Why can a ship model not match both Froude and Reynolds numbers?
Matching Froude number requires the model to move more slowly than the ship, while matching Reynolds number in the same water requires it to move faster. Both cannot hold at once, so the dominant effect is matched and the other corrected.
Does dimensional analysis give the actual equation?
No. It tells you which dimensionless combinations the answer can depend on, reducing the number of experiments or calculations needed. The shape of the function between those groups still has to come from data or from solving the governing equations.
Exam move
For each group you meet, say aloud which two forces it compares; that single sentence decides whether the group can be ignored in a new problem.
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