Dimensional Analysis

We have a bunch of problems in physics that we want to non-dimensionalize so they are scale independent and we can model things in an easier way.

Summary

We want to find groups of variables that do not have units that describe our problem

Notes

• Relationship between dimensionless groups does not need to be linear
• Buckingham Pi Theorem tells us what these groups are for a given problem
• We have seen some of these groups before (e.g. Reynolds number)
• Dimensional analysis is a very universal tool (used across disciplines)
• Dimensional analysis (or scale analysis) are used in
• Experimental design
• Performance analysis

Reduction in Number of Variables

• Say we have 5 dimensional variables for our problem, 𝑛 = 5
  • Velocity, diameter…
  • Say these are defined by three primary dimensions, 𝑗 = 3
• Mass [M], length [L], time [T]
• We can then find 𝑘 = 𝑛 − 𝑗 = 2 dimensionless groups
  • ${\pi }_1$, ${\pi }_2$

Buckingham Pi Theorem / Recipe

• Step 1: List and count the n variables (with dimensions) that are in the problem
• Step 2: Express units of each variable in terms of primary dimensions/units
• Step 3: Determine, j, the number of repeating variables (usually the number of primary units)
• Step 4: Choose j variables that must contain the primary dimensions and must be dimensionally independent
• Step 5: Non-dimensionalize the remaining variables

Similarity

• Similarity is used to design a prototype to model a physical phenomena of interest
• We often seek dynamic similarity (match non-dimensional parameters)

The main principle…

• If model is an exact geometric model of real problem and relevant 𝜋 parameters that establish the flow are the same, then the 𝜋 performance parameters are the same
• 𝜋 parameters that establish the flow, $R_e$, $F_r$, $M_a$
• 𝜋 performance parameters, $C_L$, $C_D$ Common non-dimensional parameters include…
• Re: $\frac{\rho vD}{\mu }$ = inertia over viscosity
• Froude Number: $\frac{V^2}{gL}$ inertia over gravity
• Mach Number: $\frac{u}{c}$ viscosity over the speed of sound