CIVL2611 Chap.7 Dimensional Analysis and Similitude
Dimensional Analysis and Similitude
Dimensional analysis reorganises a physically chosen variable list into fewer independent nondimensional groups. This chapter applies Buckingham pi, explains repeating-variable selection and connects matched groups to model testing. The final examples distinguish geometric similarity from Reynolds or Froude similarity and show how operating scales affect transferred forces.
The method begins with physical judgement because omitted mechanisms cannot reappear through algebra and redundant descriptors add no independent information. With k variables and r independent basic dimensions, k minus r groups should result. Repeating variables must span those dimensions, remain independent and avoid forming a nondimensional product by themselves.
Exponent equations then construct each group, followed by an explicit dimensionless check. A drag derivation produces a response coefficient related to Reynolds number, while a wave-speed example produces a Froude-type ratio. Equivalent reciprocals or constant multiples are acceptable when the groups remain independent. Model testing adds geometric, kinematic and dynamic similarity as separate requirements.
Matching Reynolds number in the same fluid can force a smaller model to run faster, whereas Froude matching makes a smaller gravity-wave model slower. That opposite direction exposes why the governing force competition must be named before scaling.
The chapter transfers measured response through density, speed and length ratios and requires any unmatched parameter or impractical facility condition to be reported as a scale effect.
What this chapter covers
- 01
Relevant-variable selection
- 02
Base dimensions and group count
- 03
Repeating-variable criteria
- 04
Buckingham pi exponent equations
- 05
Reynolds and Froude similarity
- 06
Model to prototype response scaling
Worked example · free
Drag relation from Buckingham pi
- +1Count five variables and three base dimensions, giving two groups.
- +1Choose density, speed and diameter as dimensionally independent repeating variables.
- +3Construct the force group and solve exponents to obtain force divided by density times speed squared times diameter squared.
- +2Construct the viscosity group and recognise its reciprocal as Reynolds number.
- +1Verify both groups are dimensionless and write their functional relation.
Key terms
- Repeating variables
- A dimensionally independent set used in every Buckingham-pi product to span the basic dimensions.
- Geometric similarity
- A condition in which corresponding model and prototype lengths share constant scale ratios.
- Scale effect
- A discrepancy caused when a model cannot match every nondimensional parameter controlling the prototype.
Dimensional Analysis and Similitude FAQ
How many pi groups should I obtain?
For k dimensional variables involving r independent base dimensions, Buckingham's theorem gives k minus r independent groups. Use that count to detect missing or redundant groups before interpreting the result.
How do I choose repeating variables?
Choose enough variables to span all base dimensions, make sure they are dimensionally independent and avoid forming a dimensionless group among themselves. In the usual construction, do not select the dependent response as a repeating variable.
Can a scale model match every similarity parameter?
Often it cannot. A free-surface model may need Froude similarity while viscous effects call for Reynolds similarity, and one speed scale in one fluid may not satisfy both. Match the dominant physics and disclose the remaining scale effect.
Exam move
Write variable dimensions in a compact table and count the expected groups before selecting repetitions. Begin by stating the dependent response and defending every independent variable physically. Remove duplicate geometric descriptions such as radius and diameter unless both genuinely represent different features. After identifying the basic dimensions, compute k minus r before constructing any group.
Choose repeating variables that collectively span mass, length and time, remain dimensionally independent and do not already form a dimensionless product. For each nonrepeating variable, write a product with unknown exponents and solve the mass, length and time equations in separate rows. Verify the resulting group explicitly rather than assuming the algebra succeeded.
Compare your form with common alternatives: a reciprocal or constant multiple can express the same independent relationship. Practise deriving a drag coefficient and Reynolds number, then perform a second derivation for pipe pressure drop including relative length.
Plotting language matters too: identify which nondimensional response belongs on the vertical axis and which controlling groups belong on the horizontal axis or as parameters. For similitude questions, state geometric, kinematic and dynamic similarity separately. Name the force competition represented by Reynolds or Froude number before matching it.
Derive the model-to-prototype speed ratio from the chosen parameter so a Reynolds rule is not accidentally used for a free-surface Froude problem. Build a scale table for length, speed, density, viscosity and response force; carry exponents visibly. Test whether the required facility speed or fluid is practical and identify any nondimensional parameter that remains unmatched. A scale effect should be reported, not hidden.
Finish by checking a limiting case in which model and prototype properties are identical: every scale ratio should return one. In timed work, show the variable list, dimension table, group count, repeating-variable justification, exponent solution and dimensionless verification. Those stages provide a markable method even when the final conventional group is written in an equivalent form.
Compare the answer with its reciprocal and explain why both encode the same similarity condition despite their different appearance. Verify independence before interpreting groups.