Lesson 5.7: Dimensional Analysis & Similitude

PSU & GATE Mechanical Engineering Master Course

Lesson 5.7: Dimensional Analysis & Similitude

Dimensional Analysis and Similitude are important for model testing, scaling experiments, and simplifying complex fluid problems. GATE and PSU exams often test dimensionless numbers and scaling laws.

🔹 1. Introduction

Definition: Dimensional analysis studies physical quantities in terms of fundamental dimensions (M, L, T).
Applications:
- Predicting prototype behavior from model tests
- Scaling in hydraulic machines
- Correlating experimental data using dimensionless numbers
Key Concepts:
1. Fundamental dimensions: Mass (M), Length (L), Time (T), Temperature (θ)
2. Derived quantities: Force, Pressure, Velocity

🔹 2. Buckingham π Theorem

Purpose: Reduces number of variables in a physical problem
Steps:
1. Identify n variables affecting the system
2. Determine r fundamental dimensions involved
3. Number of dimensionless groups = n − r
Example: Flow through pipe depends on Q, ρ, μ, D, ΔP → n = 5, r = 3 → 2 π-groups (e.g., Reynolds number, friction factor)

🔹 3. Common Dimensionless Numbers

Number	Symbol	Formula	Significance
Reynolds	Re	$ρVDμ\frac{\rho V D}{\mu}$	Laminar/turbulent flow
Froude	Fr	$VgL\frac{V}{\sqrt{gL}}$	Gravity effects in open channel/ship model
Euler	Eu	$ΔPρV2\frac{\Delta P}{\rho V^2}$	Pressure-velocity relation
Mach	Ma	$Va\frac{V}{a}$	Compressible flow (speed of sound a)
Prandtl	Pr	$να\frac{\nu}{\alpha}$	Heat transfer in fluids

🔹 4. Similitude

Definition: Condition where model and prototype behave similarly
Types:
1. Geometric similarity: Model and prototype have same shape ratios
2. Kinematic similarity: Velocity ratios are maintained
3. Dynamic similarity: All forces are proportional → ensures dimensionless numbers equal
Applications: Hydraulic model tests, wind tunnel testing, aircraft models

🔹 5. Solved Examples (PYQ Style)

Example 1 (GATE ME 2017):
Determine Reynolds number for flow in pipe: D = 0.1 m, V = 2 m/s, ρ = 1000 kg/m³, μ = 0.001 Pa·s → Re = 2×10⁵ → turbulent.

Example 2 (PSU Exam):
Geometric similarity: Model ship length = 5 m, prototype = 50 m → Fr same → velocity of model = √(scale) × velocity of prototype.

Example 3:
Identify dimensionless groups for forced convection heat transfer: variables = q, k, L, ΔT, ρ, Cp, μ → n − r = 7 − 3 = 4 π-groups.

🔹 6. Practice Exercises

Use Buckingham π theorem to find dimensionless groups for pipe flow and open channel flow.
Compute Reynolds and Froude numbers for given pipe and model scales.
Verify dynamic similarity between model and prototype for hydraulic machines.
Explain geometric, kinematic, and dynamic similarity with examples.
Solve problems involving scaling laws in laboratory experiments.

🔹 7. Summary

Dimensional Analysis: Express quantities in terms of fundamental dimensions
Buckingham π Theorem: Reduces variables, forms dimensionless groups
Similitude: Ensures similarity between model and prototype (geometric, kinematic, dynamic)
Applications: Model testing, ship design, hydraulic and aerodynamic experiments