Question

The Buckingham \(\pi\) theorem is used for which purpose in fluid mechanics?

• A. Determining the velocity profile in turbulent flow
• B. Formulating dimensionless groups from dimensional variables
• C. Predicting the onset of turbulence
• D. Calculating pressure losses in pipes

Hint:

Buckingham Pi Theorem. A key tool in dimensional analysis. Used to determine the number of independent dimensionless groups (\(\pi\) terms) required to describe a physical relationship involving dimensional variables. Reduces complexity and aids scaling.

Answer 1

The Correct Option is B

The Buckingham \(\pi\) theorem is a fundamental principle in dimensional analysis.
It states that if a physical phenomenon involves \(n\) dimensional variables and these variables can be described using \(k\) fundamental dimensions (like Mass, Length, Time), then the relationship between the variables can be expressed in terms of \(n-k\) independent dimensionless groups (called \(\pi\) terms).
In fluid mechanics and other fields, this theorem is used to reduce the number of variables needed to describe a system by grouping them into meaningful dimensionless parameters (like Reynolds number, Froude number, Mach number), which simplifies experimentation and allows for scaling results.
Velocity profiles, onset of turbulence, and pressure losses are related to these dimensionless groups, but the theorem's primary purpose is the formulation of the groups themselves.