Fluid Mechanics: Laminar Flow and Turbulence

Fluid Mechanics

Laminar Flow

• Definition: A type of fluid flow where the fluid moves in parallel layers or streams with no turbulence or cross-mixing.
• Characteristics:
  • Smooth, continuous motion
  • No eddies or swirls
  • Velocity profile is parabolic
  • Occurs at low Reynolds numbers (Re < 2000)
• Examples:
  • Flow of oil through a pipe
  • Flow of water through a narrow tube

Turbulence

• Definition: A type of fluid flow characterized by chaotic, irregular motion with eddies and swirls.
• Characteristics:
  • Unpredictable, irregular motion
  • Eddies and swirls are present
  • Velocity profile is flat
  • Occurs at high Reynolds numbers (Re > 4000)
• Effects:
  • Increased energy loss due to friction
  • Enhanced mixing and heat transfer
  • Higher shear stresses on boundaries
• Types:
  • Free turbulence (e.g., ocean currents)
  • Wall-bounded turbulence (e.g., pipe flow)

Bernoulli's Equation

• Statement: The pressure of a fluid decreases as its velocity increases.
• Mathematical Formulation: P + 1/2ρv^2 + ρgy = constant
  • P: pressure
  • ρ: fluid density
  • v: fluid velocity
  • g: acceleration due to gravity
  • y: height above a reference level
• Applications:
  • Design of airplane wings and wind turbines
  • Calculation of pressure drop in pipes
  • Analysis of fluid flow in conduits and orifices

Pipe Flow

• Types:
  • Laminar flow (Re < 2000)
  • Turbulent flow (Re > 4000)
  • Transitional flow (2000 < Re < 4000)
• Characteristics:
  • Laminar flow: parabolic velocity profile, low friction factor
  • Turbulent flow: flat velocity profile, high friction factor
• Pipe Flow Equations:
  • Continuity equation: ∂ρ/∂t + ∇⋅(ρv) = 0
  • Momentum equation: ∂v/∂t + v⋅∇v = -1/ρ ∇P + ν ∇²v
  • Energy equation: ∂e/∂t + ∇⋅(ev) = ρv⋅∇h + ν ∇v⋅∇v
  • Head loss equation: h_L = f \* (L/D) \* (v^2/2g)
    • h_L: head loss
    • f: friction factor
    • L: pipe length
    • D: pipe diameter
    • v: fluid velocity
    • g: acceleration due to gravity

Fluid Mechanics

Laminar Flow

• Smooth, continuous motion with no turbulence or cross-mixing.
• Characterized by:
  • Parabolic velocity profile
  • Occurs at low Reynolds numbers (Re < 2000)
• Examples include:
  • Flow of oil through a pipe
  • Flow of water through a narrow tube

Turbulence

• Chaotic, irregular motion with eddies and swirls.
• Characterized by:
  • Unpredictable, irregular motion
  • Eddies and swirls are present
  • Velocity profile is flat
  • Occurs at high Reynolds numbers (Re > 4000)
• Effects include:
  • Increased energy loss due to friction
  • Enhanced mixing and heat transfer
  • Higher shear stresses on boundaries
• Types of turbulence include:
  • Free turbulence (e.g., ocean currents)
  • Wall-bounded turbulence (e.g., pipe flow)

Bernoulli's Equation

• Pressure decreases as velocity increases.
• Mathematical formulation: P + 1/2ρv^2 + ρgy = constant
• Variables:
  • P: pressure
  • ρ: fluid density
  • v: fluid velocity
  • g: acceleration due to gravity
  • y: height above a reference level
• Applications include:
  • Design of airplane wings and wind turbines
  • Calculation of pressure drop in pipes
  • Analysis of fluid flow in conduits and orifices

Pipe Flow

• Types of flow include:
  • Laminar flow (Re < 2000)
  • Turbulent flow (Re > 4000)
  • Transitional flow (2000 < Re < 4000)
• Characteristics:
  • Laminar flow: parabolic velocity profile, low friction factor
  • Turbulent flow: flat velocity profile, high friction factor
• Important equations:
  • Continuity equation: ∂ρ/∂t + ∇⋅(ρv) = 0
  • Momentum equation: ∂v/∂t + v⋅∇v = -1/ρ ∇P + ν ∇²v
  • Energy equation: ∂e/∂t + ∇⋅(ev) = ρv⋅∇h + ν ∇v⋅∇v
  • Head loss equation: h_L = f \* (L/D) \* (v^2/2g)