Hydraulics & Pipe Networks (Hammer, Open Channel, Hydropower) — Engineering Reference

1. At a glance

Hydraulics is the engineering of liquids in conveyance, storage, control, and energy extraction — chiefly water, but also oil, brine, slurry, milk, sewage, and (increasingly in 2026) liquefied hydrogen and green ammonia. It is the applied branch of [[Engineering/fluid-mechanics]] that civil, sanitary, agricultural, and hydropower engineers actually practice. The major subfields are:

• Pipe-flow networks — pressurised conveyance of water, wastewater, irrigation, oil and gas; treats pipes as elements in a graph with junction continuity and loop energy balances.
• Open-channel flow — canals, rivers, drainage ditches, partially-full sewers; free surface at atmospheric pressure, Froude-number governed.
• Hydraulic structures — dams, spillways, weirs, gates, drops, stilling basins, culverts, intakes, outfalls.
• Hydropower and pumped storage — Pelton, Francis, Kaplan turbines; reversible pump-turbines; tidal and wave devices.
• Transient analysis (water hammer + surge) — wave dynamics in elastic pipes after a valve closure, pump trip, or load rejection.
• Groundwater + wells — Darcy flow in porous media, well drawdown (Theis 1935), aquifer recharge.
• Flood hydraulics + routing — 1D and 2D unsteady simulation of overland and channel flow.

The 2026 design context: rising rainfall intensity and sea-level (climate-adapted hydraulics); aging infrastructure (US replaces ~1 % of buried pipe per year, far below the ~100-year service life it implies); drought-driven supply re-engineering; hydrogen and green-ammonia logistics treated with hydraulic-network tools; and the growing role of pumped storage as renewable grids need 4–10 h of dispatchable storage.

Where it sits in the design stack: [[Engineering/fluid-mechanics]] → hydraulics → [[Engineering/pumps-turbomachinery]] / [[Engineering/environmental-engineering]] / [[Engineering/transportation-engineering]] drainage / hydropower → asset management and commissioning.

2. Why it matters

Water and wastewater infrastructure represents over USD 400 billion of installed value globally; the American Society of Civil Engineers’ 2025 infrastructure report card grades US drinking-water systems at C- and stormwater at D. Aging pipe networks lose 10–50 % of supplied water to leakage in older systems. Climate adaptation is rewriting design rainfall and flood return-period charts faster than codes can keep up. Hydropower supplies ~16 % of global electricity — still the largest renewable source — and pumped storage is the dominant grid-scale energy-storage technology by installed energy capacity (USD-per-kWh roughly an order of magnitude below lithium-ion at >4-hour duration).

The hydraulic failures that drive design conservatism are well documented: the Edenville and Sanford dam failures (Michigan, 2020), the Oroville spillway incident (California, 2017), the Mosul dam ongoing seepage risk, and recurring urban surface-flooding losses. On the smaller scale, water-hammer events routinely burst distribution mains and rupture fire-protection piping; under-designed sewer systems back up in storms; mis-set surge protection trips hydropower units; pumped-storage units cavitate during fast mode reversal.

3. First principles

3.1 Energy equation between two points

z₁ + P₁/γ + V₁²/(2g) + h_pump  =  z₂ + P₂/γ + V₂²/(2g) + h_turbine + h_loss

with γ = ρg the specific weight (9.81 kN/m³ for water at 20 °C), and each term in metres of fluid column. The hydraulic grade line (HGL) plots z + P/γ along the pipeline; the energy grade line (EGL) sits V²/(2g) above the HGL. Where the HGL drops below the pipe centreline, the pipe is sub-atmospheric and at risk of air release, column separation, or collapse of thin-walled pipe.

3.2 Major friction losses (Darcy–Weisbach)

h_f = f · (L/D) · V²/(2g)                       (Darcy 1857, Weisbach 1845)

f is the Darcy friction factor, read from a Moody diagram or computed from:

• Colebrook–White (1939) — implicit, iteratively solved:

  1/√f = − 2 log₁₀ [ ε/(3.7D) + 2.51/(Re√f) ]
  

• Swamee–Jain (1976) — explicit, accurate to 1 %:

  f = 0.25 / { log₁₀ [ ε/(3.7D) + 5.74/Re^0.9 ] }²
  

3.3 Hazen–Williams (water-only, US-customary heritage)

SI:  V = 0.849 · C · R_h^0.63 · S^0.54
US:  V = 1.318 · C · R_h^0.63 · S^0.54   (ft/s, ft, ft/ft)

Hazen–Williams (Hazen and Williams, 1905) is the dominant municipal-water sizing formula in North America despite its narrow validity (water at 4–25 °C, turbulent rough-pipe regime). C drops in service — see Table 3.1.

Table 3.1 — Typical Hazen–Williams C values

Pipe (condition)	C (new)	C (10 yr)	C (30 yr)
PVC / HDPE	150	145	140
Cement-lined ductile iron	140	135	130
New ductile iron, unlined	130	100	75
New cast iron, unlined	130	100	65
Welded steel, new (cement-lined)	140	130	120
Concrete (precast)	130	120	110
Riveted steel	110	90	70
Galvanised iron	120	100	80

Tuberculation (the build-up of corrosion products in unlined ferrous pipe) can drop effective C from 130 to 40–60 in 50 years, a 4–6× rise in head loss at the same flow.

3.4 Manning’s equation (open channel and full-pipe gravity flow)

SI:  V = (1/n) · R_h^(2/3) · S^(1/2)
US:  V = (1.486/n) · R_h^(2/3) · S^(1/2)

R_h = A / P_wet (hydraulic radius). Manning (1889) is the universal open-channel formula and is also used for partially-full sewers.

Table 3.2 — Manning roughness n

Surface	n
Smooth concrete (lined channel, new)	0.012
PVC, fibreglass	0.009–0.011
Concrete pipe (precast)	0.012–0.015
Vitrified clay sewer	0.013
Corrugated metal pipe (CMP)	0.022–0.025
Earth channel, clean straight	0.022
Earth channel, weeds and rocks	0.035
Natural stream, regular	0.030
Natural stream, sluggish + weedy	0.050–0.080
Mountain stream, cobble bed	0.040–0.050
Flood plain, dense brush	0.070–0.160
Asphalt pavement (sheet flow)	0.013

3.5 Froude number, critical depth

Fr = V / √(g · D_h)              D_h = A / T (hydraulic depth, T = top width)

• Fr < 1 → subcritical (tranquil); downstream conditions propagate upstream.
• Fr > 1 → supercritical (rapid); only upstream conditions matter.
• Fr = 1 → critical; section is a hydraulic control.

Critical depth y_c for a rectangular channel: y_c = (q²/g)^(1/3) with q = Q/b unit discharge.

3.6 Wave celerity in a pipe (water hammer)

a = √(K/ρ) / √( 1 + (K/E) · (D/e) · c₁ )

with K = bulk modulus of liquid (2.2 GPa for water at 20 °C), E = Young’s modulus of pipe wall (210 GPa steel; 3 GPa PVC), D pipe ID, e wall thickness, and c₁ a restraint factor (1 for buried pipe with expansion joints, ≈ 1 − ν² for fully anchored). Typical celerity: 1 200–1 450 m/s in steel water pipe; 350–550 m/s in PVC; 800–1 100 m/s in ductile iron.

3.7 Joukowsky head rise

ΔH = a · ΔV / g                  (Joukowsky 1898)

A 2 m/s flow in steel pipe (a = 1 200 m/s) suddenly stopped: ΔH = 1 200 · 2 / 9.81 = 245 m of head = 2.4 MPa = 350 psi — added on top of static pressure.

4. Pipe network analysis

A pressurised distribution network is a graph: nodes (junctions, tanks, reservoirs, pump-inlet and outlet ports, fire hydrants, customer demand points) connected by links (pipes, pumps, valves). The two simultaneous constraints:

• Junction continuity (Kirchhoff I): Σ Q_in − Σ Q_out − Q_demand = 0 at every node.
• Loop head loss (Kirchhoff II): Σ h_f around any closed loop = 0 (sign-aware).

4.1 Hardy Cross method (Hardy Cross, 1936)

Cross’s loop method, devised pre-computer, iteratively corrects assumed loop flows. For Darcy–Weisbach the loop correction is:

ΔQ_loop = − Σ h_f,i / Σ |n · h_f,i / Q_i|       (n = 2 for D–W; n = 1.852 for Hazen–Williams)

Manual implementation is feasible for 3–8-loop networks; convergence to 1 % typically in 4–7 iterations. Still taught — and still useful as a sanity check on a black-box solver.

4.2 Modern solvers — global Newton-Raphson

EPANET, WaterGEMS, InfoWater and KYPIPE all use a gradient (Todini–Pilati 1988) or global gradient Newton-Raphson algorithm that solves all nodes and loops simultaneously. Convergence is quadratic; 5 000-pipe systems solve in seconds. EPANET (Lewis Rossman, US EPA, 1993 → 2.2 in 2020) is the open-source reference engine that many commercial tools embed.

4.3 Network elements

• Junction nodes carry elevation, demand pattern, and optional emitter coefficient (for pressure-driven leakage / fire flow).
• Pipe links carry length, diameter, roughness, minor-loss coefficient, and status (open / closed / CV).
• Pumps are specified by a head curve H(Q), or three-point curve plus speed setting; affinity laws scale for variable speed.
• Tanks introduce time dependence — water level varies with net inflow over the hydraulic timestep.
• Valves include PRV (pressure-reducing), PSV (sustaining), FCV (flow control), PBV (pressure-breaker), TCV (throttle-controlled), GPV (general).

4.4 Operating point: pump curve vs system curve

The system curve is h_static + h_friction(Q), monotone-rising. The pump curve is H_p(Q), monotone-falling for a centrifugal. Their intersection sets the duty point. Multiple pumps:

• Parallel — add flows at common head; lifts Q-axis of combined curve.
• Series — add heads at common flow; lifts H-axis. Used in booster stations and high-head water-supply.

4.5 Variable-speed (VFD) pumping

Affinity laws (covered [[Engineering/pumps-turbomachinery]]): Q ∝ N, H ∝ N², P ∝ N³. A 20 % speed reduction halves the shaft power. VFDs replace throttling-valve control in static-dominant systems with care — minimum-speed limits apply where friction is low because the pump head curve no longer crosses the static lift.

4.6 Skeletonisation, demand allocation, calibration

Real distribution systems have 10⁵+ pipes; skeletonisation prunes the model to manageable size (typically all 6-inch and larger). Demand is allocated to junctions from billing data, area-based methods, or aggregated DMA inflows. Calibration matches modelled pressures and chlorine traces to field measurements (typically AWWA M32 procedure).

5. Water hammer and transient analysis

5.1 Joukowsky surge and critical time

A pressure surge ΔH = a · ΔV / g is generated by any rapid change in velocity. The critical (or pipe-period) time is:

T_c = 2 L / a

Any change accomplished in less than T_c is “rapid” and yields the full Joukowsky head; slower changes yield a reduced surge that can be estimated by Allievi’s interlocking equations (Allievi 1903 — the second great pioneer of transient analysis).

5.2 Method of Characteristics (MOC)

The 1-D unsteady flow PDEs in an elastic pipe are:

∂H/∂t + (a²/g) · ∂V/∂x = 0           (continuity)
∂V/∂t + g · ∂H/∂x + f · V|V|/(2D) = 0 (momentum, with quasi-steady friction)

These convert along characteristic lines dx/dt = ±a into four ODEs (C⁺ and C⁻ characteristics) that march in (x, t). MOC is the dominant numerical method in commercial surge software because it natively handles the wave physics and easily incorporates boundary-condition devices (valves, pumps, surge tanks, accumulators, check valves).

Earlier graphical surge analysis: Bergeron / Wood (1937), the precursor to MOC, is occasionally still used for sanity checks and teaching.

5.3 Vacuum and column separation

When the local pressure falls below the liquid vapour pressure (2.34 kPa absolute for water at 20 °C), a vapour pocket forms — column separation. When the vapour later collapses (downstream pressure rises faster than the column refills), the columns rejoin with a celerity-driven impact. The rejoinder surge can be 2–3 × the original Joukowsky head — the deadliest mode in pump-trip events with check-valve slam.

5.4 Mitigation hierarchy

Table 5.1 — Transient mitigation methods

Method	Acts on	Cost	Notes
Slow valve closure	T_c control	Free	Closure time > 2 T_c is the cheapest fix
Pump flywheel (added inertia)	Pump rundown	Low–Med	Extends V(t) ramp from seconds to ~10 s
Surge tank (open)	Mass storage	High	Civil works; common at hydropower headworks
Air chamber (closed)	Compressible buffer	Med	Pressure vessel + air-charge maintenance
Air-inlet / air-release valve	Vacuum protection	Low–Med	Allows air in to prevent column collapse
Pressure-relief valve	Over-pressure cap	Low	Discharges to atmosphere; sized for max V
Surge anticipator valve	Pre-empts return surge	Med	Opens on detected pump trip
By-pass with check valve	Re-feeds suction	Low	Centrifugal pumps with positive suction

5.5 Software for transients

• Bentley HAMMER (CONNECT Edition) — leader for municipal water and raw-water transient.
• KYPIPE Surge — embedded in KYPIPE network suite.
• AFT Impulse — popular in process industries and oil-and-gas.
• Flowmaster (Mentor Graphics / Siemens) — 1D thermo-fluid systems including transients.
• Wanda (Deltares) — open-source-friendly, strong in dredging-industry and large-scale water.
• TSnet / Allievi (free / academic) — research and teaching.

6. Open-channel flow

6.1 Steady uniform flow

For a channel of constant cross-section and slope at terminal velocity, Manning’s equation directly yields V (Section 3.4). The corresponding normal depth y_n is found by iterating Manning’s equation for the unknown depth at given Q.

6.2 Steady non-uniform — gradually-varied flow (GVF)

When channel slope, roughness, or section changes, depth varies along x. The GVF equation:

dy/dx = ( S_0 − S_f ) / ( 1 − Fr² )

with S_0 the bed slope and S_f the energy-grade-line slope. Twelve classical profiles (M1/M2/M3 mild-slope, S1/S2/S3 steep-slope, C1/C3 critical, H2/H3 horizontal, A2/A3 adverse) describe every backwater curve. Numerical integration is by the standard-step (channel-control) or direct-step (rectangular) method. HEC-RAS is the industry-standard implementation, used for FEMA flood-insurance studies in the US.

6.3 Rapidly-varied flow (RVF)

Local discontinuities — hydraulic jumps, weirs, gates, falls — change depth in a length of order the depth itself. The energy equation alone fails because of significant turbulence dissipation; momentum is the controlling balance.

Hydraulic jump (sequent-depth relation):

y₂ / y₁ = 0.5 · ( √(1 + 8 Fr₁²) − 1 )
ΔE = (y₂ − y₁)³ / (4 y₁ y₂)             (energy loss in the jump)

The jump is the workhorse energy dissipator below spillways and chutes. USBR classifies stilling-basin Types I–IV by Fr₁ (II for Fr₁ 4.5–8 with chute blocks and a dentated end sill; III for high-velocity small basins; IV for Fr₁ 2.5–4.5 with diagonal sills).

6.4 Weirs and gates

Sharp-crested rectangular weir (Kindsvater–Carter):

Q = (2/3) · C_d · √(2g) · L_e · H_e^(3/2)

with C_d ≈ 0.602 + 0.075 (H/P) for fully-aerated nappe.

V-notch (Cone–Thomson):

Q = (8/15) · C_d · tan(θ/2) · √(2g) · H^(5/2)       (θ = notch angle)

Broad-crested:

Q = C_d · L · (2/3)^(3/2) · √g · H^(3/2)             C_d ≈ 0.85–0.95

Ogee spillway (USBR / Creager profile):

Q = C · L · H^(3/2)

with C ≈ 2.0–2.2 (SI; 3.6–4.0 US) for an unsubmerged design head. The crest is shaped so the pressure on the surface is near atmospheric at the design head — anything significantly above design head produces sub-atmospheric pressures and cavitation risk on the spillway face (Oroville 2017).

Gates — vertical slide / sluice (rectangular, simple, lower capacity), radial / Tainter (curved face with horizontal trunnions; allows much wider span with lower hoist load — dominant on large spillways), roller and stoplog for emergency closure.

6.5 Culverts (FHWA HDS-5)

Culverts cross roads and railways under flow. The hydraulic control is either:

• Inlet control — entrance geometry chokes flow; depth in barrel can be supercritical; raise inlet headwall for capacity.
• Outlet control — barrel friction or tailwater controls; flow is subcritical, possibly submerged.

Design rule: compute both and use the higher headwater. FHWA HY-8 is the standard public-domain culvert sizing tool.

6.6 Flood routing

• Kinematic-wave — channel slope and friction balance only; one-dimensional; explicit; used for overland flow and steep streams.
• Muskingum / Muskingum–Cunge — storage routing; lumped reach approach; standard in hydrology for reach-by-reach attenuation.
• Dynamic-wave (full St. Venant) — both inertia terms retained; required where backwater, tidal influence, or rapid hydrograph dominate. HEC-RAS Unsteady and MIKE 11 implement this; 2D variants (HEC-RAS 2D, MIKE 21, TUFLOW) solve over a mesh for floodplain inundation.

7. Hydropower

7.1 Power equation and head definitions

P = ρ · g · Q · H_net · η         η_overall = η_hydraulic · η_mechanical · η_generator

Gross head H_g is the difference in reservoir-to-tailrace elevations. Net head H_net = H_g − h_loss (intake, penstock, draft-tube residual). Typical η_overall for modern units = 80–92 % (Kaplan and large Francis hit the top end).

7.2 Turbine selection (head + flow)

Table 7.1 — Hydraulic turbine selection

Turbine	Head range (m)	Flow range (m³/s)	N_s (SI)	Mode	Notes
Pelton	200 – 2 000	0.5 – 50	5 – 70	Impulse	Free jets onto buckets; high head
Turgo	50 – 250	0.5 – 10	30 – 80	Impulse	Inclined-jet variant of Pelton
Crossflow (Banki)	5 – 200	0.05 – 5	20 – 80	Impulse-ish	Simple; small / pico applications
Francis	30 – 700	1 – 700	60 – 400	Reaction	Most installed capacity worldwide
Kaplan (adjustable blades)	5 – 70	50 – 800	350 – 1100	Reaction	Low head, very high flow; double regulation
Propeller (fixed blades)	2 – 40	1 – 200	350 – 1000	Reaction	Run-of-river micro; cheaper than Kaplan
Bulb / tubular	2 – 25	10 – 600	600 – 1200	Reaction	Tidal and low-head river; horizontal axis
Reversible pump-turbine	50 – 700	1 – 500	80 – 400	Both	Pumped-storage workhorse

7.3 Pumped storage

A pumped-storage hydropower (PSH) plant has an upper and lower reservoir and a reversible Francis pump-turbine (or separate pump and turbine for very high head). Round-trip efficiency 75–82 %. Cycle: pump at off-peak / surplus-renewable cost, generate at peak. PSH dominates global grid-scale storage energy by an order of magnitude in MWh — Snowy 2.0 (Australia, 2 GW / 350 GWh, commissioning 2027), Fengning (China, 3.6 GW), Bath County (US, 3 GW / 24 GWh) are the marquee plants. Closed-loop PSH (no natural inflows) is the project type with the cleanest permitting outlook in 2026.

7.4 Run-of-river

Minimal storage; turbine flow follows river discharge. Low ecological impact, predictable diurnal capacity, common at small dams and run-of-river weirs. Often combined with environmental flow releases.

7.5 Tidal and wave

• Tidal barrage — La Rance (France, 240 MW, 1966) and Sihwa (S Korea, 254 MW, 2011) are the only utility-scale barrages worldwide; bulb turbines on a barrier across an estuary. Ecological footprint is the limiting issue.
• Tidal-stream — open-flow turbines (Atlantis-AR1500, ORPC) in tidal currents. MeyGen (Pentland Firth, UK, 6 MW Phase 1A, building to 86 MW) is the bellwether.
• Wave energy — emerging; Eco Wave Power (Gibraltar pilot), CETO (Carnegie Clean Energy). Pelamis (Scotland) went insolvent in 2014; the sector remains pre-commercial.

7.6 Mini, micro, and pico

• Small hydro = < 10 MW (definition varies by jurisdiction).
• Mini ≈ 100 kW – 1 MW.
• Micro ≈ 5 – 100 kW (rural electrification).
• Pico < 5 kW (off-grid household / village).

Crossflow (Banki–Michell) and Pelton dominate the micro/pico segment; turbine cost per kW rises sharply below 100 kW, so civil works simplicity (low-head, gravity-fed) is key to economics.

8. Worked examples

Example A — Hardy-Cross loop iteration

Problem. A single closed loop has four pipes (1, 2, 3, 4) with assumed initial flows Q₁ = +30 L/s, Q₂ = +20 L/s, Q₃ = −15 L/s, Q₄ = −25 L/s (sign convention: clockwise positive). Pipes are 200 mm SDR-17 PVC (C = 150). Pipe lengths L = 300, 200, 250, 350 m respectively. Demand allocations are consistent with the assumed flows. Find the corrected loop flows after one Hardy-Cross step (Hazen-Williams form).

Step 1 — Hazen-Williams head loss per pipe. SI: h_f = 10.67 · L · Q^1.852 / (C^1.852 · D^4.87), with Q in m³/s.

Common multiplier M_HW = 10.67 / (150^1.852 · 0.200^4.87).

• 150^1.852 ≈ 10 590.
• 0.200^4.87 ≈ 4.07 × 10⁻⁴.
• M_HW = 10.67 / (10 590 · 4.07 × 10⁻⁴) = 10.67 / 4.310 = 2.475 m / (m of pipe · (m³/s)^1.852).

For Q in L/s convert: Q[m³/s] = Q[L/s] · 10⁻³, so Q^1.852 picks up a factor 10⁻³·¹·⁸⁵² ≈ 1.405 × 10⁻⁶.

Working in absolute |Q|, signed h_f = sign(Q) · M_HW · 1.405 × 10⁻⁶ · L · |Q[L/s]|^1.852.

| Pipe | L (m) | Q (L/s) | |Q|^1.852 | h_f (m, signed) | |------|-------|---------|-----------|------------------| | 1 | 300 | +30 | 521 | +0.544 | | 2 | 200 | +20 | 246 | +0.171 | | 3 | 250 | −15 | 145 | −0.126 | | 4 | 350 | −25 | 372 | −0.452 |

Σ h_f = +0.544 + 0.171 − 0.126 − 0.452 = +0.137 m (loop residual; should be 0).

Step 2 — Loop derivative. Σ |n · h_f / Q| with n = 1.852, Q in L/s:

• 1.852 · 0.544 / 30 = 0.0336
• 1.852 · 0.171 / 20 = 0.0158
• 1.852 · 0.126 / 15 = 0.0156
• 1.852 · 0.452 / 25 = 0.0335
• Sum = 0.0985 m / (L/s).

Step 3 — Correction.

ΔQ = − Σ h_f / Σ |n h_f / Q| = − 0.137 / 0.0985 = − 1.39 L/s

Step 4 — Corrected flows. All flows decrease by 1.39 L/s (loop convention):

• Q₁ = 30 − 1.39 = 28.61 L/s
• Q₂ = 20 − 1.39 = 18.61 L/s
• Q₃ = −15 − 1.39 = −16.39 L/s (|Q| rises since direction is “anti-loop”)
• Q₄ = −25 − 1.39 = −26.39 L/s

A second iteration reduces |Σ h_f| typically by an order of magnitude; 4–6 iterations to < 1 % convergence is normal for a single-loop problem. EPANET’s gradient method converges the same problem in 2–3 global steps because it solves all loops jointly.

Example B — Joukowsky surge and the case for slow closure

Problem. A 1 000 m steel transmission main, D_i = 500 mm, wall thickness e = 10 mm. Steady water flow at V = 1.5 m/s. A downstream butterfly valve closes. Compute the wave celerity, critical closure time, and full-Joukowsky surge. Then estimate the maximum allowable closure time so that ΔP remains below 0.5 MPa (≈ 73 psi) over static.

Step 1 — Wave celerity. K = 2.20 GPa (water 20 °C), ρ = 998 kg/m³, E = 210 GPa (steel), D/e = 50.

a² = (K/ρ) / [ 1 + (K/E)·(D/e) ]
   = (2.20 × 10⁹ / 998) / [ 1 + (2.20/210)·50 ]
   = 2.204 × 10⁶ / [ 1 + 0.524 ]
   = 2.204 × 10⁶ / 1.524
   = 1.446 × 10⁶ m²/s²
a = √1.446 × 10⁶ = 1 203 m/s.

Step 2 — Critical (pipe-period) closure time.

T_c = 2L/a = 2 · 1 000 / 1 203 = 1.66 s.

Any closure in < 1.66 s yields the full Joukowsky head.

Step 3 — Full Joukowsky surge.

ΔH = a · ΔV / g = 1 203 · 1.5 / 9.81 = 184 m of water.
ΔP = ρ · g · ΔH = 998 · 9.81 · 184 = 1.80 MPa ≈ 261 psi.

A 0.5 s valve closure produces this on top of static pressure — easily enough to burst a Class 150 ANSI flange (rated 1.97 MPa at room temperature, but margins must include water-hammer per ASME B31.4 / B31.8).

Step 4 — Allowable closure time for ΔP < 0.5 MPa. Allievi’s interlocking equation for slow closure (linear closure law) gives ΔH_max ≈ ΔH_J · (T_c / T_close) when T_close > T_c. For ΔP_target = 0.5 MPa = 51 m head:

T_close ≈ T_c · ΔH_J / ΔH_target = 1.66 · 184 / 51 = 6.0 s.

Specifying ≥ 6 s linear closure (or a two-speed closure that decelerates the final 10 % more slowly) keeps the transient inside Class 150 envelope without an accumulator. The valve actuator and stop-block design follow from this.

Example C — Pelton turbine sizing

Problem. Hydro project: H_gross = 420 m, penstock losses 5 %, Q = 5.0 m³/s, generator coupled at N = 600 rpm. Estimate net head, hydraulic power, plant power at η = 0.90, and confirm Pelton is appropriate.

Step 1 — Net head. H_net = H_gross · (1 − 0.05) = 420 · 0.95 = 399 m.

Step 2 — Hydraulic and shaft power.

P_hyd = ρ g Q H_net = 998 · 9.81 · 5.0 · 399 = 1.953 × 10⁷ W ≈ 19.5 MW.
P_shaft = η · P_hyd = 0.90 · 19.53 = 17.58 MW.

Step 3 — Specific speed (SI).

N_s = N · √P_shaft / H_net^1.25
    = 600 · √(17 580 [kW]) / 399^1.25
    = 600 · 132.6 / 1 783
    = 44.6

N_s ≈ 45 is squarely in the Pelton range (5–70 SI). ✓

Step 4 — Jet diameter (single-jet preliminary). Jet velocity c = √(2g H_net) = √(2 · 9.81 · 399) = 88.5 m/s. Jet discharge q = (π/4) d_j² · c. For a single jet handling all 5 m³/s: d_j = √(4 · 5.0 / (π · 88.5)) = √0.0719 = 0.268 m. Bucket pitch diameter D_pcd typically 12–15 × d_j → D_pcd ≈ 3.3–4.0 m, large for a single jet.

Step 5 — Number of jets. Splitting the discharge across 4 jets reduces d_j to 0.134 m and D_pcd to ~ 1.8 m, much more buildable. The bucket pitch tangential speed U_opt = 0.46 · c = 40.7 m/s, so D_pcd = 2 U / ω = 2 · 40.7 / (2π · 600/60) = 1.30 m for fine-tuned design. Four-jet Pelton at 600 rpm with D_pcd ≈ 1.3 m is the design starting point.

9. Hydraulic structures and dams

9.1 Dam types

Table 9.1 — Dam types

Type	Material	Typical height (m)	Foundation needs	Notable example
Gravity concrete	Mass concrete	Up to 285	Strong rock	Grand Coulee (USA, 168 m)
Arch (single / double)	Reinforced concrete	Up to 305	Strong canyon walls	Hoover (USA, 221 m); Jinping-I (CN, 305 m)
Buttress	Concrete + steel	Up to 180	Moderate rock	Daniel-Johnson (CA, 214 m, multi-arch buttress)
Embankment, earthfill	Compacted soil	Up to 300	Wide range	Nurek (TJ, 300 m)
Embankment, rockfill (CFRD)	Concrete-face rockfill	Up to 233	Moderate to weak	Shuibuya (CN, 233 m)
Roller-compacted concrete (RCC)	Lean concrete	Up to 180	Rock	Gibe III (Ethiopia, 250 m)
Tailings (mining)	Mine waste	Up to 300	Highly variable	Brumadinho (BR) failure 2019

9.2 Spillways

Function: pass the design flood without over-topping the dam. Capacity is set to the Probable Maximum Flood (PMF) for high-hazard dams (loss-of-life consequence) and to a fraction of PMF (typically 0.5 PMF or 10 000-year flood) for significant-hazard. Layouts:

• Ogee (overflow) — the dominant high-capacity spillway; Creager profile fitted to design head.
• Chute — concrete trough; energy dissipated in stilling basin at toe.
• Side-channel — entrance perpendicular to discharge axis, used where canyon width is limited.
• Tunnel and shaft (morning glory) — vertical shaft into a tunnel; common at arch-dam sites.
• Ski-jump (flip bucket) — throws flow into the air and into a plunge pool downstream.
• Stepped (CFRD and RCC) — successive drops dissipate energy on the dam face; popular post-2000.
• Labyrinth and piano-key weir (PKW) — geometrically increased crest length; used to raise capacity of existing dams without raising HW level.

9.3 Stilling basins and energy dissipators

Standard USBR Types I (sloping apron, low Fr), II (Fr 4.5–8 with chute blocks, baffle piers, dentated sill), III (high-velocity small basins), IV (Fr 2.5–4.5 with diagonal sills). SAF basin (St Anthony Falls), USBR/USACE Type-VI impact basin, and plunge pools (ski-jump) are the alternatives.

9.4 Outlet works and dam break

• Outlet works — bottom outlets for reservoir draw-down, irrigation release, low-flow augmentation. Sized for ~ 0.05 PMF and emergency drawdown.
• Dam-break analysis — required for emergency-action plans (EAP). Failure modes: overtopping (largest by count), piping (internal erosion), structural failure, foundation. Tools: HEC-RAS 2D (USACE), DAMBRK / FLDWAV (NWS legacy), NWS HEC-RAS Unsteady, FLO-2D. Outputs feed inundation maps with arrival times and depths for downstream warning.

9.5 Reservoir sedimentation

Trap efficiency η_t (Brune curve) is a function of capacity-to-inflow ratio. Large reservoirs (C/I > 1) trap 95+% of sediment, reducing capacity over decades. Counter-measures: sluicing (pass sediment-laden flow through low-level outlets), flushing (drawdown release), dredging, upstream check dams + reforestation. Three Gorges has built in 22 km³ of sediment storage by design.

9.6 Fish passage

• Fish ladders / fishways (pool-and-weir, vertical-slot, Denil) for upstream migration.
• Fish lifts for very high dams.
• Bypass channels that simulate natural streams.
• Downstream screens and surface bypass at intakes (Bonneville, Snake River salmon).

9.7 Concrete dam stress and uplift

Gravity method analyses sliding (Σ H ≤ μ Σ V + cohesion · contact area), overturning (resultant within middle third), and crushing (compressive stress at heel/toe). Uplift pressure is critical — controlled by grout curtain near upstream face and drainage gallery. Modern FE method (3D + thermal + seismic + uplift) supersedes hand-method on critical structures but the gravity method remains the screening tool.

10. Tools / software

Pressurised water distribution.

• EPANET 2.2 (US EPA, open-source) — dominant academic / municipal engine.
• Bentley WaterCAD / WaterGEMS — commercial GIS-integrated.
• Innovyze (Autodesk) InfoWater Pro — heavy in large-utility EU and Asia.
• KYPIPE — long-running US-civil tool.
• Aquis / MIKE URBAN+ (DHI) — strong in Europe.

Wastewater and stormwater.

• SWMM 5 (US EPA, open-source) — dominant municipal.
• Bentley SewerCAD / StormCAD — commercial.
• InfoWorks ICM (Autodesk / Innovyze) — flagship 1D/2D urban drainage.
• MIKE+ (DHI) — full urban + flood.
• PCSWMM (CHI) — popular SWMM-with-tools front-end.

Transient / surge. Bentley HAMMER, KYPIPE Surge, AFT Impulse, Flowmaster, Wanda, TSnet, Allievi.

Open-channel + flood routing.

• HEC-RAS / HEC-HMS (USACE, free) — US standard for floodplain, FEMA studies.
• MIKE 11 / MIKE 21 / MIKE FLOOD (DHI) — strong in Europe and Asia.
• TUFLOW — Australian-origin, dominant in many AU/UK markets.
• InfoWorks ICM (River) — combined network + 2D.
• FLO-2D — debris flow and mudflow specialty.
• Iber — open-source 2D (Spain).
• Delft3D / D-Flow FM (Deltares) — large-scale coastal, estuarine.

Hydraulic structures and dam-break. HEC-RAS Unsteady + 2D, DAMBRK / FLDWAV (NWS legacy), BREACH (NWS), HEC-LifeSim (downstream consequence).

CFD specialty for hydraulics.

• FLOW-3D Hydro (Flow Science) — the favourite for spillways, intakes, fishways, locks.
• ANSYS Fluent, Siemens Star-CCM+, OpenFOAM — general CFD.
• TELEMAC-MASCARET (EDF, open-source) — coastal and river.

GIS integration. ArcGIS Pro + Spatial Analyst + ArcHydro toolkit, QGIS + SAGA + GRASS — terrain pre-processing, watershed delineation, model-to-GIS coupling.

Time-series and field data. USGS NWIS (US streamflow + stage), EPA STORET (water quality), NOAA water-data and tide stations; HEC-DSS time-series format underpins HEC-RAS / HEC-HMS / HEC-ResSim.

11. Edge cases / gotchas

1. Aging pipe tuberculation turns 130-C ductile iron into 50-C in 50 years; size networks with explicit service-life C, not new-pipe C.
2. Distribution-system leakage of 10–50 % is unrecorded “non-revenue water”; pressure management (PRVs, DMAs) and acoustic-survey programmes are usually cheaper than mains replacement.
3. Water-hammer in HVAC and steam systems is widespread because building services design too rarely runs a transient analysis; isolation valves, condensate-induced hammer in steam, and chilled-water riser slam are the typical failure modes.
4. Pump-trip + check-valve slam + column rejoin produces a return surge of 2–3× Joukowsky; the deadliest transient mode and a primary driver of in-line air chambers.
5. Cavitation at partly-open valves during transient — cavitation can damage seats and trim that survive steady-state duty; throttling valves are commonly the wrong choice at the surge-pressure location.
6. Dead-ends and stagnant manifolds create water-quality issues (chlorine residual loss, biofilm, nitrification in chloraminated systems); fold loop-feeding into the network plan, not after-the-fact.
7. Thermal expansion in buried mains (steel and ductile iron) governs joint design; mechanical-joint and restrained-joint specs by AWWA C600 / C111.
8. Cathodic protection on buried steel and ductile iron — sacrificial anode or impressed-current; AWWA C105 polyethylene encasement on ductile iron is the modern alternative.
9. Trenchless replacement — HDD (horizontal directional drilling) and CIPP (cured-in-place pipe) reduce surface restoration cost by 60–80 %; CIPP UV-cure replacing steam-cure (lower volatile emissions, 2026 norm in EU).
10. Climate-adapted design rainfall — current IDF curves (intensity-duration-frequency) underestimate 2050 storm; jurisdictions adopt + 10–20 % uplift factors (NYC, Toronto, London).
11. Reservoir sedimentation is a 30–70 yr slow loss of capacity; sluicing strategies are now a basic design feature on new dams (Three Gorges, Itaipu).
12. California state-water and Colorado River allocations — engineering hydraulics is reshaped by curtailment rules; 2026 designs build in scenario flexibility.
13. Minimum environmental flow for hydropower licensing — FERC and EU Water Framework Directive impose habitat-protective releases that reduce nameplate energy by 5–15 %.
14. Pumped-storage GHG accounting — energy used to pump is greenhouse-neutral if from surplus renewable, fossil if from thermal grid; lifecycle CO₂-eq depends on dispatch model, not nameplate.
15. Karst geology and leakage — limestone foundations leak around dam abutments; grout curtains and monitoring are essential (Kemano, Boruca-Naranjo).
16. Vacuum and pipe collapse — HGL below pipe centreline + insufficient air-inlet valves collapses thin-wall HDPE and large-diameter steel mains in seconds during pump trip.
17. Cold-region pipe freeze and frost depth — bury below frost line (AWWA M-22 region-specific); insulated above-grade where geology forbids burial.
18. Sluicing of sediment without full drawdown — flow-through bottom-outlet operations with monitoring of downstream turbidity to respect fish-habitat regulations.
19. Spillway pressures below atmospheric at off-design heads — cavitation on ogee crest (Oroville 2017, Karun-III 2014); aeration grooves and step roughening are the mitigations.
20. Fish-passage retrofit of existing dams — vertical-slot fishways and ladders perform poorly for shad and lamprey; species-specific design is necessary.

12. Cross-references

• [[Engineering/fluid-mechanics]] — foundational continuity / momentum / Bernoulli framework.
• [[Engineering/pumps-turbomachinery]] — pump and turbine curves, NPSH, affinity laws, BEP.
• [[Engineering/environmental-engineering]] — water + wastewater + stormwater process overlap.
• [[Engineering/transportation-engineering]] — culverts, road drainage, bridge hydraulics.
• [[Engineering/structural-analysis]] — dam stress and stability.
• [[Engineering/soil-mechanics]] — embankment slope stability, seepage, foundation.
• [[Engineering/cfd-deep]] — hydraulic CFD for spillways, intakes, locks.
• [[Engineering/electric-motors]] — pump and turbine drives and generators.
• [[Engineering/electromagnetics-engineering]] — hydroelectric synchronous generators.
• [[Engineering/vibration-dynamics]] — flow-induced vibration in penstocks, draft tubes, gates.
• [[Languages/Tier3/geospatial]] — terrain, watershed, and flood-map data formats.

13. Citations

1. Chaudhry, M. H. Applied Hydraulic Transients, 3rd ed. Springer, 2014. ISBN 978-1461485377. The canonical transient-analysis text.
2. Wylie, E. B.; Streeter, V. L. Fluid Transients in Systems, Prentice Hall, 1993. ISBN 978-0139344237.
3. Chow, V. T. Open-Channel Hydraulics, McGraw-Hill, 1959. The classic open-channel reference.
4. Roberson, J. A.; Cassidy, J. J.; Chaudhry, M. H. Hydraulic Engineering, 2nd ed. Wiley, 1998.
5. Mays, L. W. Water Resources Engineering, 3rd ed. Wiley, 2019. ISBN 978-1119490562.
6. Featherstone, R. E.; Nalluri, C. Civil Engineering Hydraulics, 5th ed. Wiley-Blackwell, 2009.
7. Walski, T. M.; Chase, D. V.; Savic, D. A.; Grayman, W.; Beckwith, S.; Koelle, E. Advanced Water Distribution Modeling and Management, Haestad / Bentley, 2003.
8. Rossman, L. EPANET 2.2 Users Manual, US EPA / EPA/600/R-20/133, 2020.
9. USBR Design of Small Dams, 3rd ed. Bureau of Reclamation, 1987.
10. USBR Hydraulic Design of Stilling Basins and Energy Dissipators, Engineering Monograph 25, 1978.
11. USACE Hydraulic Design of Spillways, EM 1110-2-1603, 1990.
12. USACE HEC-RAS River Analysis System — Hydraulic Reference Manual, Version 6.5, 2024.
13. USACE HEC-HMS Technical Reference Manual, 2024.
14. Hazen, A.; Williams, G. S. Hydraulic Tables, Wiley, 1905. The eponymous formula.
15. Manning, R. “On the flow of water in open channels and pipes.” Transactions of the Institution of Civil Engineers of Ireland, vol. 20, 1889.
16. Darcy, H. Recherches expérimentales relatives au mouvement de l’eau dans les tuyaux, 1857.
17. Weisbach, J. Lehrbuch der Ingenieur- und Maschinen-Mechanik, 1845.
18. Colebrook, C. F. “Turbulent Flow in Pipes, with Particular Reference to the Transition Region.” Journal of the ICE, vol. 11(4), 1939, pp. 133–156.
19. Joukowsky, N. E. “Über den hydraulischen Stoss in Wasserleitungsröhren.” Mémoires de l’Académie Impériale des Sciences de St-Pétersbourg, 1898 (Russian original 1900). Water-hammer foundation.
20. Allievi, L. Teoria generale del moto perturbato dell’acqua nei tubi in pressione, Milan, 1903. Slow-closure (Allievi interlocking) equations.
21. Cross, H. Analysis of Flow in Networks of Conduits or Conductors, University of Illinois Engineering Experiment Station, Bulletin No. 286, 1936.
22. Swamee, P. K.; Jain, A. K. “Explicit Equations for Pipe-Flow Problems.” J. Hydraulics Div. (ASCE), vol. 102(HY5), 1976.
23. Todini, E.; Pilati, S. “A gradient method for the analysis of pipe networks.” International Conference on Computer Applications for Water Supply and Distribution, 1988. Basis for EPANET solver.
24. ASCE Manual of Practice No. 37 Design and Construction of Sanitary and Storm Sewers, 1969 (and successive ASCE / WEF reprints).
25. AWWA M-Series — M11 (steel pipe), M22 (sizing water mains), M23 (PVC), M32 (calibration), M41 (ductile iron), M55 (PE), M60 (drought planning).
26. IEC 60193:2019 — Hydraulic turbines, storage pumps and pump-turbines — Model acceptance tests.
27. IEC 62256:2017 — Hydraulic machines — Rehabilitation and performance improvement.
28. FEMA P-94 Selecting Analytic Tools for Concrete Dams; FEMA P-919 Federal Guidelines for Dam Safety.
29. FHWA HDS-5 Hydraulic Design of Highway Culverts, 3rd ed., 2012.
30. Hydraulic Institute (HI) Pump Standards — ANSI/HI 14.6 (rotodynamic pump tests), 9.6.1 (NPSH), 9.8 (intake design).