๐ Reynolds Number & Entry Length
Compute Reynolds number, classify flow regimes, find hydraulic diameter, and estimate entry lengths for pipes, flat plates, annuli, and rectangular ducts.
Fluid Mechanics
๐ Configuration
Re = ฯยทVยทD_h / ฮผ
D_h = D (pipe), L (plate)
D_h = D_oโD_i (annulus)
D_h = 4WH/[2(W+H)] (duct)
Laminar: Re < 2300, Turbulent: Re > 4000
๐ Results
Configure inputs and click Compute to view results.
๐ Methodology
Reynolds Number
Re = ฯVD_h/ฮผ is the ratio of inertial to viscous forces. It determines the flow regime: laminar (Re < 2300 for internal flow, Re < 5ร10โต for external), transition, or turbulent.
Hydraulic Diameter
For non-circular cross-sections, the hydraulic diameter D_h = 4A/P (4ร cross-sectional area / wetted perimeter) is used. For an annulus D_h = D_o โ D_i, for a rectangular duct D_h = 2WH/(W+H).
Entry Length
The hydrodynamic entry length is the distance for the velocity profile to fully develop. Laminar: L_e โ 0.05ยทReยทD. Turbulent: L_e โ 4.4ยทRe^(1/6)ยทD (shorter relative to pipe diameter).
๐ Calculation Methodology: Reynolds Number & Flow Regime Classification
Mathematical Model & Theory
The Reynolds number $Re$ represents the fundamental ratio of inertial forces to viscous forces in a fluid flow, determining the boundary between laminar, transitional, and turbulent regimes:
Assumptions
- Newtonian fluid with constant dynamic viscosity $\mu$.
- Characteristic length $D$ defined as internal pipe diameter or hydraulic diameter $D_h = 4A/P$.
Academic References
- Reynolds, O. (1883): An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, Phil. Trans. Roy. Soc.
- White, F. M.: Fluid Mechanics, Ch. 6.
Worked Engineering Example
Fuel oil ($\rho = 890\text{ kg/m}^3$, $\mu = 0.080\text{ Pa}\cdot\text{s}$) flows through a pipeline of diameter $D = 150\text{ mm}$ at flow rate $Q = 45\text{ m}^3/\text{h}$ ($0.0125\text{ m}^3/\text{s}$). Determine the flow regime.
Step-by-step Solution:
1. Calculate mean velocity: $V = 4 Q / (\pi D^2) = 4(0.0125) / (\pi \times 0.15^2) = 0.050 / 0.07069 \approx 0.7073\text{ m/s}$.
2. Compute Reynolds number: $Re = (890 \times 0.7073 \times 0.15) / 0.080 = 94.42 / 0.080 \approx 1180$.
3. Classify regime: $Re = 1180 < 2300 \implies$ **Laminar Flow**.
Final Result:
$Re = \mathbf{1180}$ (Strictly Laminar Flow with parabolic velocity profile).