Lecture Notes on Boundary Layer Theory

Introduction to Boundary Layer Concept

• Developed by Ludwig Prandtl in 1904.
• Unites two separate groups in fluid mechanics:
  • Theoreticians: Focused on theoretical hydrodynamics (e.g., potential and inviscid flow).
  • Experimentalists: Conducted experiments to understand viscous flow.

Importance of Boundary Layer Theory

• Acknowledges that viscosity is significant only near the wall, in the boundary layer.
• Outside this region, fluid can be treated as inviscid.
• Coupling between potential flow and viscous flow was established.

Initial Applications of Boundary Layer Theory

• Applied to the simplest flow scenario: Flat plate with zero pressure gradient.
• Key Features:
  • Leading edge at x = 0, where there is no velocity along the wall (zero slip condition).
  • As flow moves downstream, viscous diffusion occurs, leading to a velocity deficit.
  • Boundary layer thickness increases as fluid flows over the plate.

Types of Boundary Layers

1. Laminar Boundary Layer: Starts off smooth, can transition to turbulent.
2. Turbulent Boundary Layer: Higher growth rate and different velocity characteristics compared to laminar flow.
   • Transition may be induced by disturbances (e.g., sandpaper or wire).

Reynolds Number in Boundary Layer

• Characterizes boundary layer flow (
  • Re_X = (Density * Velocity * X) / Dynamic Viscosity
  • X denotes the spatial dimension along the plate.
• Reynolds number increases along the plate from the leading edge.

Transition from Laminar to Turbulent Flow

• Critical Reynolds Number: approximate value for transition is 5 x 10^5 for flat plates.
• Previous examples noted critical Reynolds numbers for flow (e.g., 2000-2300 for other cases).

Future Topics of Study

• Analyze early theoretical developments on boundary layer growth and friction.
• Discuss simplified approaches developed by Theodore von Karman.
• Move towards more sophisticated analysis using Navier-Stokes equations (not covered in this course).