mech

Fluid Dynamics: Bernoulli, Boundary Layer & Pipe Losses

From Bernoulli's equation to Darcy-Weisbach and Navier-Stokes — the head-loss calculation interviewers make you do on the spot.

ThermaxISROTata MotorsBoschL&T Heavy Engineering

Key formulas

p1/ρg + v1²/2g + z1 = p2/ρg + v2²/2g + z2 + h_loss
Bernoulli equation: Head form; add losses to model real flow between two sections.
hf = f·(L/d)·(v²/2g)
Darcy-Weisbach: Major loss in pipes; f from Reynolds number and relative roughness (Moody chart).
Re = ρvd/μ = vd/ν
Reynolds number: Ratio of inertial to viscous forces — the flow-regime decider for every internal flow.

Cheat sheet

▸Bernoulli: p + ½ρv² + ρgz = constant along a streamline for steady, incompressible, inviscid, non-rotating flow — every term is a form of energy per unit volume.
▸Reynolds number Re = ρvd/μ decides laminar (Re < 2300, pipes) vs turbulent; laminar friction factor f = 64/Re, turbulent from the Moody chart / Colebrook equation.
▸Darcy-Weisbach: hf = f·(L/d)·(v²/2g) — the universal head-loss equation; minor losses use K·v²/2g (entrance 0.5, sudden expansion (v1−v2)²/2g, exit 1).
▸Boundary layer: thickness grows with x^½; separation occurs where the wall shear becomes zero under an adverse pressure gradient (dp/dx > 0) — the cause of bluff-body wakes.
▸Navier-Stokes = Newton's second law per unit volume for a viscous fluid; dropping viscosity gives Euler equations, dropping everything gives Bernoulli.
▸Continuity: A1v1 = A2v2 for incompressible flow — the first equation to write in any pipe-network problem.

Laminar vs turbulent pipe flow

Aspect	Laminar	Turbulent
Re (pipes)	< 2300	> 4000
Friction factor	64/Re (independent of roughness)	Moody / Colebrook (roughness matters)
Velocity profile	Parabolic, v_max = 2·v_avg	Flatter, logarithmic, v_max ≈ 1.2·v_avg
Mixing	Diffusion only	Rapid mixing, better heat transfer
Analysis	Exact (Hagen-Poiseuille)	Empirical correlations

Visual

Venturi / Bernoulli application:
   A1 large          A2 throat
  ┌──────\            /──────┐
  │  p1, v1 →   p2, v2 →      │   p2 < p1, v2 > v1 (A2 < A1)
  └──────/            \──────┘
  Pitot tube: stagnation point → v = √(2·(p0 − p)/ρ)

MCQ mastery quiz

1. Bernoulli's equation is valid for…

2. Laminar flow in a pipe occurs below Reynolds number…

3. For laminar pipe flow, the friction factor equals…

4. Head loss in a pipe by Darcy-Weisbach is proportional to…