Fluid Mechanics: Laminar Flow and Turbulence

Study Notes

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

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)

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

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

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

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)

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

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)