Pipe flow refers to the movement of a fluid (liquid or gas) through a closed conduit, such as a pipe or tube, under the influence of pressure, gravity, or external forces.
\[ Re = \frac{\rho v D}{\mu} \]
where:
| Classification | Flow Type | Key Characteristics |
|---|---|---|
| Flow Regime | Laminar | Re < 2000, smooth layers |
| Turbulent | Re > 4000, chaotic mixing | |
| Transitional | 2000 < Re < 4000 | |
| Time Dependency | Steady/Unsteady | Time-invariant vs. time-varying |
| Compressibility | Incompressible/Compressible | Constant vs. variable density |
Pipe flow and open channel flow represent two fundamental types of fluid flow systems in engineering, each with distinct characteristics and governing principles.
| Parameter | Pipe Flow | Open Channel Flow |
|---|---|---|
| Definition | Flow through completely filled, enclosed conduits | Flow with a free surface exposed to atmospheric pressure |
| Driving Force | Pressure gradient | Gravity (slope of channel) |
| Cross-Section | Fixed (circular, rectangular, etc.) | Variable (can change with flow depth) |
| Pressure Distribution | Hydraulic pressure throughout | Atmospheric pressure at free surface |
| Flow Control | Valves, pumps | Weirs, gates, channel slope |
| Velocity Profile | Developed by viscous forces | Influenced by surface roughness and channel geometry |
| Typical Applications | Water supply systems, oil pipelines | Rivers, canals, drainage systems |
Pipe Flow occurs when a fluid completely fills the conduit, creating a closed system where the primary driving force is pressure difference. The entire cross-section is wetted, and flow characteristics are determined by the pipe's material, diameter, and the fluid's viscosity.
Open Channel Flow features a free surface that's exposed to atmospheric pressure, with gravity serving as the main driving force. The flow depth can vary, making the flow cross-section changeable. Surface waves and varying roughness significantly affect the flow behavior.
In pipe flow, the Darcy-Weisbach equation is commonly used to calculate head loss due to friction:
hf = f(L/D)(v²/2g)
For open channel flow, the Manning equation is frequently employed to determine flow velocity:
v = (1/n)R2/3S1/2
where n is Manning's roughness coefficient, R is hydraulic radius, and S is channel slope.
Pipe systems are generally more expensive to construct but offer better control and efficiency for pressurized systems. Open channels are typically more economical for large-scale water transport but are subject to environmental factors like evaporation and contamination.
Osborne Reynolds' seminal experiment (1883) demonstrated the transition between flow regimes using a glass pipe with dye injection:
The dimensionless Reynolds number determines flow regime:
\[ Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu} \]Where:
| Characteristic | Laminar Flow (Re < 2000) | Turbulent Flow (Re > 4000) |
|---|---|---|
| Flow Structure | Ordered parallel layers | Chaotic eddies and mixing |
| Velocity Profile | Parabolic (Poiseuille) | Flattened logarithmic profile |
| Energy Loss | \( h_f \propto V \) | \( h_f \propto V^n \) (1.75 ≤ n ≤ 2) |
| Mixing | Molecular diffusion only | Strong convective mixing |
For fully developed laminar flow in circular pipes:
Velocity distribution:
\[ u(r) = \frac{1}{4\mu}\left(-\frac{dP}{dx}\right)(R^2 - r^2) \]Volumetric flow rate (Poiseuille's Law):
\[ Q = \frac{\pi R^4}{8\mu}\left(-\frac{dP}{dx}\right) = \frac{\pi D^4 \Delta P}{128 \mu L} \]Where \( \Delta P \) is the pressure drop over length \( L \). This shows the fourth-power dependence on diameter, making pipe sizing critical.
For infinite parallel plates separated by distance \( h \):
Velocity profile:
\[ u(y) = \frac{1}{2\mu}\left(-\frac{dP}{dx}\right)(hy - y^2) \]Volumetric flow rate per unit width:
\[ q = \frac{h^3}{12\mu}\left(-\frac{dP}{dx}\right) \]Maximum velocity occurs at midplane:
\[ u_{max} = \frac{h^2}{8\mu}\left(-\frac{dP}{dx}\right) \]From dimensional analysis and experimental data, the head loss equation:
\[ h_f = f \frac{L}{D} \frac{V^2}{2g} \]Where \( f \) is the Darcy friction factor. For laminar flow:
\[ f = \frac{64}{Re} \]For turbulent flow, the Colebrook-White equation (1939) provides an implicit relationship:
\[ \frac{1}{\sqrt{f}} = -2 \log_{10}\left(\frac{\epsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}}\right) \]Where \( \epsilon \) is the equivalent sand-grain roughness height. This requires iterative solution, leading to the Moody chart development.
In fully developed pipe flow, the shear stress varies linearly:
\[ \tau(r) = \frac{r}{R} \tau_w \]Where wall shear stress \( \tau_w \) is:
\[ \tau_w = \frac{D}{4}\left(-\frac{dP}{dx}\right) = \frac{f}{4} \rho V^2 \]The shear stress is maximum at the wall and zero at the centerline.
Total head loss is the sum of individual losses:
\[ h_{f,total} = \sum_{i=1}^n h_{f,i} \]Dupuit equation for equivalent pipe:
\[ \frac{L_{eq}}{D_{eq}^5} = \sum \frac{L_i}{D_i^5} \]Head loss is equal across all branches:
\[ h_{f,1} = h_{f,2} = \cdots = h_{f,n} \]Total flow is the sum of branch flows:
\[ Q_{total} = \sum_{i=1}^n Q_i \]Applying Bernoulli's equation between reservoirs:
\[ \frac{P_{atm}}{\rho g} + z_1 = \frac{P_{vap}}{\rho g} + z_2 + h_f + \frac{V^2}{2g} \]Where \( P_{vap} \) is the vapor pressure limiting the maximum siphon height.