Fluid flow in pipes can be laminar or turbulent, depending on the Reynolds number. Laminar flow occurs at low speeds, with fluid moving in parallel layers. Turbulent flow happens at higher speeds, characterized by chaotic mixing and irregular motion.

Understanding pipe flow is crucial for designing efficient systems and analyzing fluid behavior. The Hagen-Poiseuille equation describes laminar flow, while turbulent flow requires more complex analysis using friction factors and velocity profiles.

Laminar and Turbulent Flow in Pipes

Laminar vs turbulent flow

• Reynolds number ($Re$) dimensionless parameter characterizes flow regime in pipes $Re = \frac{\rho VD}{\mu}$, $\rho$ fluid density, $V$ average velocity, $D$ pipe diameter, $\mu$ dynamic viscosity
• Laminar flow occurs at low Reynolds numbers ($Re < 2300$)
  • Fluid particles move in parallel layers without mixing (honey, oil)
  • Velocity profile parabolic, maximum velocity at center, zero at pipe wall
• Turbulent flow occurs at high Reynolds numbers ($Re > 4000$)
  • Fluid particles exhibit irregular and chaotic motion, mixing between layers (fast-flowing rivers, blood flow in arteries)
  • Velocity profile flatter compared to laminar flow, more uniform distribution across pipe cross-section
• Transitional flow occurs between laminar and turbulent regimes ($2300 < Re < 4000$)
  • Flow characteristics less predictable, can exhibit features of both laminar and turbulent flow (slow-moving rivers, blood flow in capillaries)

Hagen-Poiseuille equation applications

• Hagen-Poiseuille equation describes pressure drop ($\Delta P$) in circular pipe for laminar flow $\Delta P = \frac{128\mu LQ}{\pi D^4}$, $L$ pipe length, $Q$ volumetric flow rate
• Rearranging Hagen-Poiseuille equation yields expression for flow rate $Q = \frac{\pi D^4 \Delta P}{128\mu L}$
• Equation assumes steady, fully developed, incompressible, Newtonian fluid flow
• Applications include:
  1. Calculating flow rate in microfluidic devices (lab-on-a-chip systems)
  2. Determining pressure drop in small-diameter pipes (hydraulic lines, capillary tubes)
  3. Analyzing blood flow in small blood vessels (capillaries, arterioles)

Turbulent Flow and Pipe Flow Analysis

Friction factors in turbulent flow

• Darcy-Weisbach equation relates pressure drop to friction factor ($f$) in pipes $\Delta P = f \frac{L}{D} \frac{\rho V^2}{2}$
• Moody diagram graphical representation of relationship between friction factor, Reynolds number, relative roughness ($\varepsilon/D$)
  • Relative roughness ratio of pipe's average roughness height ($\varepsilon$) to its diameter ($D$)
• Colebrook equation implicit formula for calculating friction factor in turbulent flow $\frac{1}{\sqrt{f}} = -2.0 \log_{10} \left(\frac{\varepsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}}\right)$
  • Solving Colebrook equation requires iterative approach or approximations (Haaland equation)
• Friction factor depends on pipe material and surface roughness (smooth pipes: PVC, glass; rough pipes: cast iron, concrete)

Velocity and shear stress profiles

• Velocity profile in laminar flow
  1. Parabolic distribution, maximum velocity at center, zero at pipe wall
  2. Velocity at any radial position ($r$) given by $u(r) = 2V\left[1-\left(\frac{r}{R}\right)^2\right]$, $R$ pipe radius
• Shear stress distribution in laminar flow
  1. Linear variation, maximum shear stress at pipe wall, zero at center
  2. Wall shear stress given by $\tau_w = \frac{4\mu V}{R}$
• Velocity profile in turbulent flow
  1. Flatter compared to laminar flow, more uniform distribution across pipe cross-section
  2. Velocity profile approximated using power-law relationship, $\frac{u}{u_{\max}} = \left(\frac{r}{R}\right)^{1/n}$, $n$ depends on Reynolds number
• Shear stress distribution in turbulent flow
  1. Non-linear variation, higher shear stress near pipe wall, lower values towards center
  2. Wall shear stress given by $\tau_w = \frac{1}{8}f\rho V^2$, $f$ friction factor
• Understanding velocity and shear stress profiles crucial for:
  • Designing piping systems (minimizing pressure drop, optimizing flow distribution)
  • Analyzing heat transfer in pipes (convective heat transfer depends on velocity and shear stress)
  • Studying erosion and corrosion in pipes (high shear stress can lead to material damage)