Blasius Equation and Boundary Layers

Chapter 11: Boundary Layer Equations

• Covers boundary layer equations in fluid dynamics

11.1 Introduction

Governing Equations

• Continuity equation: ∂u/∂x + ∂v/∂y = 0
• Momentum equation: u(∂u/∂x) + v(∂u/∂y) = -(1/ρ)(dp/dx) + ν(∂²u/∂y²)

Boundary Conditions

• At the surface (y=0): u(x, 0) = 0, v(x, 0) = 0
• At infinity (far from the surface): u(x, ∞) = U(x)

Similarity Solution Notes

• A similarity solution is sought using the similarity variable η = y√(U/(νx))
• Stream function (ψ) is defined such that u = ∂ψ/∂y and v = -∂ψ/∂x
• Stream function is expressed as ψ(x, y) = √(νxU)f(η)

Velocity Components

• u = Uf'(η)
• v = (1/2)√(νU/x) (ηf' - f)

Velocity Derivatives

• ∂u/∂x = U'f' - (UU'/2)ηf''
• ∂u/∂y = (U^(3/2) / √(νx)) f''
• ∂²u/∂y² = (U²/νx)f'''

Blasius Equation

• Substituting into the momentum equation (with dp/dx = 0) yields: f''' + (1/2)ff'' = 0
• Boundary conditions for the Blasius equation: f(0) = f'(0) = 0, f'(∞) = 1

11.2 Blasius Solution

Numerical Solution Steps

• Convert to a first-order system:
  • f' = g
  • g' = h
  • h' = -(1/2)fh
• Initial conditions: f(0) = 0, g(0) = 0, h(0) = ?
• Shoot for g(∞) = 1, iterate on h(0)

Key Results

• f''(0) ≈ 0.332
• Boundary layer thickness (δ_99) ≈ 5√(νx/U)
• Displacement thickness (δ*) ≈ 1.72√(νx/U), where δ* = ∫₀^∞ (1 - u/U)dy
• Momentum thickness (θ) ≈ 0.664√(νx/U), where θ = ∫₀^∞ (u/U)(1 - u/U)dy
• Skin friction coefficient (C_f) = 0.664Re_x^(-1/2), where C_f = (τ_w) / (1/2 ρU²) = (2μ ∂u/∂y|_(y=0)) / (ρU²)

11.3 Falkner-Skan Solutions

Pressure Gradient Considerations

• Considers non-zero pressure gradient flows
• Velocity form: U(x) = U₀x^m, where U₀ and m are constants
• Pressure gradient: dp/dx = -ρU(dU/dx) = -ρU₀²x^(2m-1)m
• Favorable pressure gradient: m > 0
• Adverse pressure gradient: m < 0

Similarity Variable Definitions

• Similarity variable: η = y√(U/(νx))
• Stream function: ψ = √(νxU)f(η)

Velocity Components

• u = Uf'
• v = (1/2)√(ν(U/x))[(m+1)ηf' - (m+1)f]

Governing Equation

• Substituting into the momentum equation results in: f''' + ((m+1)/2)ff'' + m(1 - f'²) = 0
• Boundary conditions: f(0) = f'(0) = 0, f'(∞) = 1
• Note: m = 0 recovers the Blasius equation