How do you relate prototype and model properties using dimensionless groups?

Conditions

• Applied to the prototype-model pair written on the board.
• The method separates the first group, used for prediction, from the remaining groups, used as similarity requirements.
• The underlying physics governing the flow must be identical for both the prototype and the model.

Reasoning, step by step

1. Start from the prototype relation and the model relation written on the board.
2. Equate corresponding dimensionless groups between prototype and model.
3. Apply the matching rule to all non-first groups; these are grouped under the label "Similarity requirements".
4. Apply the same matching rule to the first group; this equality is singled out as the "prediction equation".

Example

The lecturer announces, "here's how I'm going to relate the prototype and model properties. I will equate the corresponding pi terms to each other." He then writes $\Pi_1^{(p)} = \Pi_1^{(m)}$, $\Pi_2^{(p)} = \Pi_2^{(m)}$, $\Pi_3^{(p)} = \Pi_3^{(m)}$, and $\Pi_{N-1}^{(p)} = \Pi_{N-1}^{(m)}$.

Common misconceptions

• Do not conflate the prediction equation with the similarity requirements: treating every matched $\Pi$-equality as if it had the same role.
• Assuming that the same functional form is not assumed for prototype and model beyond the displayed notation.

Watch the explanation

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