Applications in Pipe Networks, Pumps, and Water Distribution Systems

TL;DR

You'll learn how the Navier-Stokes equations collapse into the simple pipe-flow formulas engineers actually use daily. You'll see how pump curves and system curves meet at an operating point, and how Hardy Cross iteration solves flow in looped networks. By the end, you can size a pipe, pick a pump, and balance a small network by hand.

1. The Mental Model

Navier-Stokes is too detailed for a whole city's pipes, so engineers simplify it into one number: head loss per unit length. Every pipe just "costs" energy, and every pump "adds" energy. A network is solved when energy and mass balance everywhere at once. The entire water distribution system is just Navier-Stokes wearing an accountant's hat, tracking energy debits and credits at every junction.

2. The Core Material

2.1 From Navier-Stokes to the Formulas You Actually Use

Start with the full Navier-Stokes equation for incompressible flow. For steady, fully developed, axisymmetric flow in a straight circular pipe (no swirl, gravity absorbed into a modified pressure), the equation collapses dramatically. Convective acceleration vanishes because the velocity profile doesn't change along the pipe. What survives is a balance between the pressure gradient driving the flow and viscous shear resisting it:

$$0 = -\frac{dp}{dz} + \mu\frac{1}{r}\frac{d}{dr}\left(r\frac{du}{dr}\right)$$

Integrate twice with the no-slip condition at the wall ($u=0$ at $r=R$) and a finite centerline velocity, and you get the parabolic Hagen-Poiseuille profile:

$$u(r) = \frac{1}{4\mu}\left(-\frac{dp}{dz}\right)(R^2 - r^2)$$

Integrate that over the cross-section to get volumetric flow rate:

$$Q = \frac{\pi R^4}{8\mu}\left(-\frac{dp}{dz}\right)$$

This is exact for laminar flow. Converting the pressure drop into head loss $h_f = \Delta p /(\rho g)$ gives the laminar head loss formula:

$$h_f = \frac{32\mu L V}{\rho g D^2}$$

Real pipe networks are almost always turbulent, though, where NS has no clean closed-form solution — the convective term no longer disappears and eddies dominate. Engineers handle this by keeping the same structure (head loss proportional to length, inversely to some power of diameter, proportional to some power of velocity) but replacing the exact laminar coefficient with an empirical friction factor $f$. This gives the Darcy-Weisbach equation, the single most-used formula in this whole field:

$$h_f = f\frac{L}{D}\frac{V^2}{2g}$$

$f$ depends on Reynolds number $Re = \rho VD/\mu$ and relative roughness $\varepsilon/D$, found from the Colebrook equation:

$$\frac{1}{\sqrt{f}} = -2\log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}}\right)$$

This is implicit in $f$, so it's solved iteratively or with the Swamee-Jain approximation.

flowchart TD
    A["Full Navier-Stokes<br/>3D, unsteady, viscous"] --> B["Assume: steady, fully developed,<br/>axisymmetric, no swirl"]
    B --> C["Convective terms vanish<br/>pressure grad balances viscous shear"]
    C --> D{"Flow regime?<br/>check Reynolds number"}
    D -->|"Re < 2300<br/>laminar"| E["Exact solution:<br/>Hagen-Poiseuille profile"]
    D -->|"Re > 4000<br/>turbulent"| F["No closed form:<br/>use empirical friction factor f"]
    E --> G["Head loss: h_f = 32*mu*L*V / (rho*g*D^2)"]
    F --> H["Darcy-Weisbach: h_f = f*(L/D)*(V^2/2g)"]
    H --> I["Colebrook equation gives f<br/>from Re and roughness"]

2.2 Pumps: Adding Energy Back Into the System

A pump does the opposite of friction — it adds head instead of removing it. The extended Bernoulli (energy) equation between two points, now including a pump, is:

$$\frac{p_1}{\rho g} + \frac{V_1^2}{2g} + z_1 + H_{pump} = \frac{p_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_f$$

Every real pump has a pump curve: head delivered drops as flow rate rises, typically approximated as

$$H_{pump} = H_0 - B Q^2$$

where $H_0$ is the shutoff head (zero flow) and $B$ is a manufacturer-supplied constant.

The pipe system it's pushing water through has a system curve: the total head the system demands to move flow $Q$, made of static lift plus friction losses:

$$H_{sys} = H_{static} + K Q^2, \qquad K = \left(f\frac{L}{D} + \sum K_{minor}\right)\frac{1}{2gA^2}$$

The pump can only operate where these two curves intersect — that's the operating point, the actual $(Q, H)$ the system settles at. You can't just "pick" a flow rate; the system and the pump negotiate it between them.

2.3 Networks: Continuity Plus Loop Balancing

A distribution network is a graph: pipes are edges, junctions are nodes. Two physical laws must hold everywhere simultaneously — these are just NS's conservation ideas applied at network scale:

1. Continuity at every node: flow in equals flow out.
   $$\sum Q_{in} = \sum Q_{out}$$
2. Energy consistency around every loop: going around any closed loop of pipes, the algebraic sum of head losses must equal zero (you end up back where you started, same energy).
   $$\sum h_f = 0 \text{ around any loop}$$

Because $h_f = KQ^n$ is nonlinear (usually $n=2$ for