Agricultural Engineering · Engineering Sciences · also Hydraulics, Fluid Mechanics
Fluids at rest and in motion, pipe flow and open channels: the base of irrigation, drainage, pumping and conservation structures.
In the exams: GATE AG Section 4 (hydraulics asked in 6 of 8 papers 2019–26); ICAR AIEEA-PG Unit III; ICAR AICE subject 43 Unit 2; NCEES FE 12–18 questions (largest FE area)
Fluid mechanics is the study of liquids and gases at rest and in motion. Hydraulics is its engineering branch for water: how pressure acts on gates and tanks, how water flows through pipes, orifices, weirs and channels, and how much energy it loses on the way. For an agricultural engineer it underlies the design of irrigation mains and pumps, canals and drop structures, flow-measuring weirs and flumes, and scale models of hydraulic structures.
This manual follows the order of a standard second-year course, from fluid properties and hydrostatics through pipe flow, flow measurement and open channels to dimensional analysis. Formulas are in SI units, with notes where US customary constants differ.
The subject is examined heavily everywhere. In India it is part of the Soil and Water Conservation section of GATE Agricultural Engineering, Unit III of the ICAR AIEEA-PG paper and Unit 2 of the ICAR PhD (AICE) Irrigation and Drainage syllabus, and it underlies the state engineering service papers. In the United States it is the largest area of the NCEES FE Other Disciplines exam and recurs in the PE Agricultural and Biological Engineering exam. Most questions are numerical: a candidate who can set up an energy or momentum equation cleanly and keep units straight will score well. Where a coefficient depends on material, age or manufacturer, the manual gives a range and says so.
A fluid is a substance that deforms continuously under any shear stress, however small. Liquids have a free surface and are nearly incompressible; gases fill their container and are compressible.
The SI system is used throughout this manual: length in metres, mass in kilograms, time in seconds, force in newtons (1 N = 1 kg m/s²) and pressure in pascals (1 Pa = 1 N/m²). Standard gravity g is 9.81 m/s² (32.2 ft/s²).
Mass density ρ is mass per unit volume. For fresh water it is 1000 kg/m³ at 4 °C and slightly less at higher temperatures (about 998 kg/m³ at 20 °C); sea water is about 1025 kg/m³ and mercury about 13 600 kg/m³. Specific weight γ = ρg is weight per unit volume: 9.81 kN/m³ for water in routine design. Relative density (specific gravity) S is the ratio of a liquid's density to that of water, so mercury has S ≈ 13.6.
Viscosity is a fluid's resistance to shear deformation. Consider fluid between a fixed plate and a plate moving at velocity U a small distance apart. The fluid touching each plate moves with it (the no-slip condition), and the velocity varies across the gap. Newton's law of viscosity states that the shear stress is proportional to the rate of shear strain, the velocity gradient: τ = μ (du/dy). The constant μ is the dynamic (absolute) viscosity, with units Pa·s or N·s/m².
Kinematic viscosity ν = μ/ρ has units m²/s. The CGS unit is the stoke (1 St = 10⁻⁴ m²/s). Water at 20 °C has ν ≈ 1.0 × 10⁻⁶ m²/s. Kinematic viscosity is the property that enters the Reynolds number, so it is the one used most in pipe and channel calculations.
The viscosity of liquids falls as temperature rises, because the cohesive forces between molecules weaken. The viscosity of gases rises with temperature, because momentum exchange between molecules increases.
Fluids that obey τ = μ du/dy with constant μ are Newtonian: water, air, most light oils. Non-Newtonian fluids do not. A Bingham plastic (such as some slurries and toothpaste-like pastes) needs a yield stress before it flows; a pseudoplastic (shear-thinning) fluid such as many fruit purées and polymer solutions becomes less viscous as shear rate rises; a dilatant (shear-thickening) fluid such as a dense starch suspension becomes more viscous.
At a liquid surface the molecules are pulled inward by their neighbours, and the surface behaves like a stretched membrane. Surface tension σ is the force per unit length along a line in the surface (N/m); for clean water against air at 20 °C it is about 0.073 N/m, and it falls as temperature rises or when detergents are added.
Surface tension raises the pressure inside curved surfaces. Inside a liquid droplet of diameter d the excess pressure is Δp = 4σ/d. A soap bubble has two surfaces, inner and outer, so Δp = 8σ/d. A liquid jet (a cylinder) has Δp = 2σ/d.
Capillarity is the rise or fall of a liquid in a narrow tube. Balancing the vertical component of the surface-tension force around the tube's wetted perimeter against the weight of the raised column gives h = 4σ cos θ/(ρ g d), where θ is the contact angle and d the tube bore. Water wets clean glass (θ ≈ 0), so it rises; mercury does not wet glass (θ is about 130–140°), so cos θ is negative and mercury is depressed. For water in a 1 mm glass tube, h = 4 × 0.073/(1000 × 9.81 × 0.001) ≈ 0.030 m, or about 30 mm. The rise is inversely proportional to bore: halve the diameter and the rise doubles.
Compressibility is measured by the bulk modulus of elasticity K = −dp/(dV/V), the pressure increase needed per unit fractional reduction in volume. For water K is about 2.1–2.2 GPa, so a pressure rise of 1 MPa reduces a volume of water by less than 0.05 per cent. Compressibility matters in water hammer, where a valve is closed quickly and the pressure wave speed depends on K and on the pipe wall's elasticity.
Vapour pressure is the pressure at which a liquid boils at a given temperature: for water it is about 2.3 kPa (absolute) at 20 °C and 101.3 kPa at 100 °C. If the absolute pressure anywhere in a flowing liquid falls to the vapour pressure, vapour bubbles form and then collapse violently when they reach higher pressure. This is cavitation. Standard atmospheric pressure is 101.325 kPa, equal to 10.33 m of water or 760 mm of mercury, so no suction system can lift water more than about 10 m even in theory, and practical suction lifts are much lower once vapour pressure, friction and the pump's net positive suction head requirement are deducted.
In a fluid at rest there is no shear stress, so the only surface force is pressure, acting normal to every surface. Pascal's law states that pressure at a point in a fluid at rest is the same in all directions.
For a fluid at rest under gravity, a force balance on a thin vertical element gives the hydrostatic law dp/dz = −ρg, where z is measured upward. For a liquid of constant density this integrates to p = p₀ + ρgh, where h is depth below the free surface and p₀ the pressure on the surface. Pressure depends only on depth, not on the shape of the container: the bottoms of a wide tank and a narrow tube filled to the same depth carry the same pressure (the hydrostatic paradox). All points at the same level in the same continuous body of liquid at rest have the same pressure; this rule is the basis of every manometer calculation.
Pressure can be expressed as a head, h = p/(ρg), the height of a column of a stated liquid that would produce that pressure. Converting a head of one liquid to another uses relative density: h_water = S × h_liquid.
Absolute pressure is measured from a perfect vacuum; gauge pressure is measured from local atmospheric pressure. Absolute = atmospheric + gauge. A negative gauge pressure is called a vacuum or suction. Most hydraulic instruments read gauge pressure.
A piezometer is a vertical open tube tapped into a pipe; the liquid rises to a height h such that p = ρgh. It is simple and accurate but cannot measure large pressures (the tube becomes impractically tall), negative pressures (air would be drawn in), or gas pressures.
A U-tube manometer holds a heavier, immiscible manometric liquid, usually mercury, in a U-shaped tube. One limb connects to the pipe; the other is open to air. To solve any manometer, start at a point of known pressure, move through the tube adding ρgh when going down and subtracting ρgh when going up, and set the result equal to the pressure at the far end. For a pipe carrying water at the level of the top of the water column, with mercury deflection h₂ above the lower meniscus and the pipe centre h₁ above the same meniscus, p_A = ρ_Hg g h₂ − ρ_w g h₁ (gauge).
A single-column (well-type) manometer has one limb of large cross-section, so the level in the well barely changes and only one reading is taken. An inclined manometer tilts the reading limb at a small angle θ, so a vertical rise h appears as a length L = h/sin θ along the tube, magnifying small pressure differences in air ducts and grain-drying fans.
A differential manometer measures the difference in pressure between two points, for instance across a venturimeter, an orifice plate or a length of pipe. When both points are in pipes carrying the same liquid of relative density S₁ and the manometer holds a heavier liquid S₂ with deflection x, the pressure-head difference in metres of the flowing liquid is h = x (S₂/S₁ − 1). For water over mercury this is 12.6x: a 100 mm mercury deflection means a 1.26 m difference in piezometric head of water. The expression gives the difference in piezometric head (p/ρg + z), not in pressure alone; if the two tappings are at different heights, the height difference must be added back to get the true pressure difference. Forgetting the '−1', and so using 13.6x, is the most common slip in exam answers.
When the manometric liquid is lighter than the flowing liquid, the manometer is inverted: the U is upside down and the top holds a light liquid such as air or oil. The difference in piezometric head is then h = x (1 − S_light/S₁).
The resultant hydrostatic force on a plane surface of area A submerged in a liquid, with atmospheric pressure on both the free surface and the back of the plate, is F = ρ g A h̄, where h̄ is the vertical depth of the centroid of the area below the free surface.
The force acts at the centre of pressure, which is always below the centroid (except for a horizontal surface, where they coincide) because pressure increases with depth. For a vertical surface, h_cp = h̄ + I_G/(A h̄). For a surface inclined at θ to the horizontal, h_cp = h̄ + I_G sin²θ/(A h̄), with depths measured vertically. I_G is the second moment of area about the horizontal axis through the centroid: bd³/12 for a rectangle of width b and height d, πd⁴/64 for a circle of diameter d, and bh³/36 for a triangle of base b and height h.
Example: a vertical rectangular sluice gate 2 m wide and 3 m high has its top edge at the water surface. Then h̄ = 1.5 m and A = 6 m², so F = 9810 × 6 × 1.5 = 88.3 kN. I_G = 2 × 3³/12 = 4.5 m⁴, so h_cp = 1.5 + 4.5/(6 × 1.5) = 2.0 m, which is two-thirds of the depth, as expected for a triangular pressure distribution.
On a curved surface the elementary forces change direction, so the resultant is found from components. The horizontal component equals the force on the vertical projection of the curved surface, and acts through the centre of pressure of that projection. The vertical component equals the weight of liquid vertically above the curved surface up to the free surface, real or imaginary, and acts through the centroid of that volume. If the liquid is below the surface, the vertical component acts upward and equals the weight of the imaginary liquid that would fill the space above. The resultant is √(F_H² + F_V²), inclined at tan⁻¹(F_V/F_H) to the horizontal. For a circular surface every elementary force passes through the centre, so the resultant does too. Radial (Tainter) gates on canal regulators exploit this: the thrust passes through the trunnion pin, so the hoist lifts almost only the gate's weight and friction.
A body wholly or partly immersed in a fluid experiences an upward force equal to the weight of fluid it displaces (Archimedes' principle). This buoyant force acts through the centre of buoyancy B, the centroid of the displaced volume. A floating body displaces its own weight of liquid. So a timber block of relative density 0.6 floats in fresh water with 60 per cent of its volume submerged, and a body's apparent weight in water is its weight minus the weight of the displaced water.
A submerged body is stable only if its centre of gravity G lies below B. A floating body can be stable even with G above B, because when it tilts the shape of the displaced volume changes and B moves sideways.
When a floating body is given a small angle of heel, the new line of action of the buoyant force cuts the original vertical centreline at the metacentre M. The distance GM is the metacentric height. If M lies above G, the weight and buoyancy form a restoring couple and the body is stable; if M coincides with G it is in neutral equilibrium; if M is below G it is unstable and capsizes.
The metacentric height is GM = BM − BG, where BM = I/V. Here I is the second moment of area of the waterline plane about the longitudinal axis of tilting (for a rectangular pontoon of breadth b and length L, I = Lb³/12) and V is the displaced volume. BG is positive when G is above B. GM can be measured by an inclining experiment, moving a known weight w a distance x across the deck and measuring the angle of heel θ: GM = wx/(W tan θ).
Flow is steady if conditions at a point do not change with time, and unsteady if they do. It is uniform if velocity does not change from point to point along a streamline at a given instant, and non-uniform if it does.
A streamline is everywhere tangent to the velocity; in steady flow streamlines, pathlines and streaklines coincide. Rotational flow has fluid particles spinning about their own axes; irrotational flow does not, and it allows a velocity potential. The acceleration of a fluid particle has a local part (change with time at a point, zero in steady flow) and a convective part (change due to moving to a place where velocity differs, non-zero in a tapering pipe even at steady discharge).
Continuity is conservation of mass. For steady flow through a stream tube or pipe, the mass flow rate is the same at every section: ρ₁A₁V₁ = ρ₂A₂V₂. For an incompressible liquid ρ is constant, so the volumetric discharge Q = A V is constant: A₁V₁ = A₂V₂. Here V is the mean velocity over the section. Velocity is inversely proportional to area, and for circular pipes to the square of diameter: halving the diameter quadruples the velocity.
At a junction, the sum of inflows equals the sum of outflows. In differential form, for steady incompressible three-dimensional flow, ∂u/∂x + ∂v/∂y + ∂w/∂z = 0; a proposed velocity field that does not satisfy this is not a possible incompressible flow, and GATE papers test this by giving u and v and asking whether continuity holds or for the missing component.
Euler's equation of motion along a streamline, integrated for steady, incompressible, frictionless flow, gives Bernoulli's equation: p/(ρg) + V²/(2g) + z = constant along a streamline. The three terms are the pressure head, velocity head and elevation (datum) head, each energy per unit weight in metres. The line joining total heads along a pipe is the energy grade line (EGL); the line of p/(ρg) + z, the piezometric head, is the hydraulic grade line (HGL), which lies one velocity head below the EGL.
The assumptions are: steady flow, incompressible fluid, no friction (ideal fluid), flow along a streamline, and no shaft work or heat transfer between the sections. For real flow between sections 1 and 2 the equation becomes p₁/ρg + V₁²/2g + z₁ + h_p = p₂/ρg + V₂²/2g + z₂ + h_L + h_t, where h_L is head loss, h_p head added by a pump and h_t head taken by a turbine. Because the velocity varies across a real section, the velocity head is multiplied by the kinetic energy correction factor α: α = 2 for laminar flow in a pipe and about 1.03–1.1 for turbulent flow, often taken as 1.
Free jets are at atmospheric pressure. Water leaving a small orifice at depth H below a free surface has ideal velocity √(2gH) (Torricelli's theorem).
A venturimeter is a converging cone, a short throat and a long diverging cone installed in a pipe. Applying Bernoulli and continuity between the inlet (area a₁) and throat (area a₂), with h the difference in piezometric head, gives Q = C_d a₁a₂ √(2gh)/√(a₁² − a₂²). The coefficient of discharge C_d is typically about 0.95–0.99 because the gradual cones cause little loss. The divergent cone is longer than the convergent one (a small divergence angle, about 5–7°) to avoid flow separation. Example: a 200 mm × 100 mm venturi with a mercury-water manometer reading 250 mm gives h = 0.25 × 12.6 = 3.15 m of water; with C_d = 0.98, a₁ = 0.0314 m² and a₂ = 0.00785 m², Q ≈ 0.0625 m³/s, or 62.5 L/s. The reading does not depend on whether the venturi is horizontal, inclined or vertical, because the manometer measures piezometric head difference directly.
An orifice meter is a thin plate with a sharp-edged hole clamped between flanges. It uses the same equation but has C_d of roughly 0.6–0.65, because the jet contracts to a vena contracta downstream and the sudden expansion afterwards wastes energy. It is cheap and compact but causes much larger permanent head loss than a venturi.
A pitot tube is a bent tube facing upstream. Fluid stops at its mouth, the stagnation point, where the pressure rises by ρV²/2. With a static tapping, the difference is the dynamic head h, and V = C_v √(2gh), with C_v close to 0.98–1.0. A pitot-static tube combines both tappings. Pitot tubes measure point velocity, so discharge in a pipe or channel requires a traverse.
Newton's second law applied to a fixed control volume in steady flow states that the net external force on the fluid in the control volume equals the rate of momentum leaving minus the rate of momentum entering: ΣF = ρQ(V₂ − V₁), a vector equation applied separately in the x, y and z directions. The force the fluid exerts on the structure is equal and opposite to the wall force on the fluid.
Unlike Bernoulli, the momentum equation needs no knowledge of internal losses, so it is the tool for problems where losses are unknown or large: hydraulic jumps, sudden expansions, forces on bends, nozzles and vanes. Where velocity varies across the section, the momentum flux is multiplied by the momentum correction factor β, which is 4/3 for laminar pipe flow and about 1.01–1.05 for turbulent flow.
At a pipe bend deflecting flow through an angle θ, with pressures p₁ and p₂ acting on areas A₁ and A₂, the x-component of the force exerted by the fluid on the bend is F_x = p₁A₁ − p₂A₂ cos θ + ρQ(V₁ − V₂ cos θ), and the y-component is F_y = ρQ V₂ sin θ + p₂A₂ sin θ (signs depend on the chosen axes). These forces must be carried by thrust blocks at bends, tees, reducers and dead ends of buried irrigation mains, especially in PVC pipelines whose push-fit joints have no tensile resistance.
At a nozzle on the end of a hose or a sprinkler, the fluid accelerates and the pressure drops to atmospheric at the exit. The force on the nozzle joint is found from p₁A₁ − F = ρQ(V₂ − V₁), with V₁ and V₂ from continuity and p₁ from Bernoulli. The reaction of the jet on the operator or on a travelling gun is ρQV.
A jet of area a and velocity V striking a fixed flat plate normally is turned through 90°, losing all its momentum in the original direction: F = ρaV². If the plate is inclined so the jet meets it at angle θ to the plate surface, the normal force is ρaV² sin θ. A jet striking a fixed curved vane symmetrically and deflected through 180° (turned back on itself) exerts 2ρaV², twice the flat-plate force, because the momentum change is doubled; this is the principle of the Pelton wheel bucket.
If a single flat plate moves away from the jet at speed u, the relative velocity is V − u and the mass striking per second is ρa(V − u), so F = ρa(V − u)². The work done per second is Fu, and the efficiency, work done over the jet's kinetic energy, is maximum when u = V/3, giving 8/27, about 29.6 per cent. For a series of flat plates mounted on a wheel, all the jet's mass is used, F = ρaV(V − u), and efficiency is maximum at u = V/2 with a value of 50 per cent. For a series of curved buckets deflecting the jet nearly 180°, as in a Pelton wheel, the theoretical maximum efficiency approaches 100 per cent at u = V/2.
Osborne Reynolds injected dye into water flowing through a glass tube. At low velocity the dye stayed as a straight thread (laminar flow); at higher velocity it wavered and then mixed across the whole tube (turbulent flow). The transition depends on the dimensionless Reynolds number Re = ρVD/μ = VD/ν, the ratio of inertial to viscous forces. For pipe flow, Re below about 2000 is laminar, above about 4000 is turbulent, and between them is a transitional zone; some texts use 2300 as the lower critical value. Laminar flow occurs with viscous oils (hydraulic oil at low temperature), in soil pores and in micro-tubes at very low flow.
In open channels the Reynolds number is based on hydraulic radius R, Re = VR/ν, and the critical value is about 500.
For steady laminar flow in a circular pipe, a force balance on a cylindrical element gives a shear stress that varies linearly from zero at the centre to a maximum at the wall, τ = (−dp/dx) r/2, and a parabolic velocity profile u = (−dp/dx)(R² − r²)/(4μ). The maximum velocity at the centre is twice the mean velocity. Integrating gives the Hagen-Poiseuille equation for head loss over length L: h_f = 32μVL/(ρgD²). Head loss is proportional to the first power of velocity, and inversely to D². Comparing with the Darcy-Weisbach form shows that the friction factor for laminar flow is f = 64/Re, independent of roughness. The kinetic energy correction factor for laminar pipe flow is 2 and the momentum correction factor 4/3.
In turbulent flow, eddies transfer momentum across the pipe, giving a much flatter velocity profile and much larger wall shear. Next to the wall a thin viscous sublayer remains. If the wall roughness projections are buried inside this sublayer the pipe behaves as hydraulically smooth; if they project well through it, the pipe is hydraulically rough and viscosity no longer matters.
Friction head loss in any steady pipe flow, laminar or turbulent, is given by the Darcy-Weisbach equation h_f = f (L/D) V²/(2g), where f is the dimensionless Darcy friction factor. In terms of discharge, h_f = 8fLQ²/(π²gD⁵), which shows the powerful effect of diameter: at constant Q and f, head loss varies as 1/D⁵, so a 10 per cent increase in diameter cuts friction loss by about 38 per cent.
Many Indian textbooks write h_f = 4fLV²/(2gD), where their f is the Fanning coefficient of friction, one quarter of the Darcy factor. Always check which f a question uses: f = 0.005 (Fanning) is the same as f = 0.02 (Darcy).
Example: 30 L/s of water in 300 m of 150 mm pipe with Darcy f = 0.02. The velocity is 0.030/(π × 0.15²/4) = 1.70 m/s, and h_f = 0.02 × (300/0.15) × 1.70²/19.62 ≈ 5.88 m.
The Moody diagram plots f against Re on log scales for curves of relative roughness ε/D, where ε is the equivalent sand-grain roughness of the pipe material. It has four zones: the laminar line f = 64/Re; the critical zone where f is uncertain; the transition zone where f depends on both Re and ε/D; and the fully rough (complete turbulence) zone to the right of a dashed line, where the curves are horizontal and f depends only on ε/D.
The curves in the turbulent region follow the Colebrook-White equation, 1/√f = −2 log₁₀[(ε/D)/3.7 + 2.51/(Re√f)], which is implicit in f. The Swamee-Jain explicit approximation, f = 0.25/[log₁₀(ε/(3.7D) + 5.74/Re^0.9)]², is accurate to within a few per cent over the practical range and is convenient for calculators. For smooth pipes at Re up to about 10⁵, the Blasius formula f = 0.316/Re^0.25 is widely used; it gives head loss proportional to V^1.75.
Three classic problem types use these tools: find h_f from Q and D (direct); find Q from h_f and D (guess f, compute V and Re, update f, iterate); and find D from Q and allowable h_f (iterate on D).
The Hazen-Williams equation is an empirical formula for turbulent flow of water at ordinary temperatures in pipes of practical size. In SI units, V = 0.849 C R^0.63 S^0.54, where V is mean velocity (m/s), R the hydraulic radius (D/4 for a full circular pipe, in m), S the friction slope h_f/L, and C the Hazen-Williams coefficient. In US customary units the constant is 1.318 instead of 0.849. Rearranged for head loss in a full circular pipe in SI: h_f = 10.67 L Q^1.852/(C^1.852 D^4.87), with Q in m³/s and D in m.
C is a property of the pipe material and condition, not dimensionless and not related to the Darcy f in a simple way. Higher C means a smoother pipe. Typical design values quoted in irrigation and water-supply handbooks are about 140–150 for PVC and polyethylene, around 130 for new cast or ductile iron, and 100 or lower for old, tuberculated iron; they vary by source, manufacturer and age, so the value used should come from the relevant national standard or supplier data. Head loss varies as Q^1.852 rather than Q², so doubling the flow multiplies friction loss by about 3.6, not 4.
Example: 20 L/s through 500 m of 150 mm pipe with C = 130 gives h_f = 10.67 × 500 × 0.02^1.852/(130^1.852 × 0.15^4.87) ≈ 4.8 m.
It applies only to water at ordinary temperatures in turbulent flow; for micro-tubes, drip laterals and viscous fluids, Darcy-Weisbach with an appropriate f (often Blasius for small smooth plastic pipe) is preferred.
Minor (local) losses occur at fittings, valves, bends, entrances, exits and changes of section, where the flow separates and eddies dissipate energy. They are expressed as h_m = K V²/(2g), where K is a loss coefficient found by experiment and V is normally the velocity in the smaller pipe (or the pipe the fitting is attached to). An alternative is the equivalent-length method: the fitting is replaced by a length of straight pipe L_e = KD/f that would cause the same loss.
K values for valves and fittings depend on type, size, opening and manufacturer, so design uses published tables.
Sudden expansion. Applying the momentum equation across a sudden enlargement from velocity V₁ to V₂, assuming the pressure on the annular face equals the upstream pressure, gives the Borda-Carnot result h_e = (V₁ − V₂)²/(2g).
Exit loss. When a pipe discharges into a large reservoir, V₂ ≈ 0 and the loss is V₁²/(2g), that is K = 1: the entire velocity head is lost.
Sudden contraction. The flow contracts to a vena contracta smaller than the downstream pipe and then re-expands. The loss is h_c = (1/C_c − 1)² V₂²/(2g), where C_c is the contraction coefficient and V₂ the velocity in the smaller pipe. With C_c = 0.62 this gives K ≈ 0.375, and many texts round the sudden-contraction loss to about 0.5 V₂²/2g when C_c is not given.
Entrance loss. From a reservoir into a sharp-edged (square) pipe entrance, K ≈ 0.5; a well-rounded bell-mouth entrance reduces K to about 0.05 or less; a re-entrant (projecting) pipe has K ≈ 0.8–1.0.
In an exam, list every loss from the upstream free surface to the downstream free surface, write the energy equation, and express all velocity heads in terms of one unknown using continuity. Many candidates forget the exit loss, which is often the largest minor loss in a short pipe connecting two tanks.
Pipes of different diameters and lengths joined end to end carry the same discharge, and the total head loss is the sum of the friction losses in each pipe plus the minor losses at the joints, entrance and exit: H = Σ h_fi + Σ h_m. If minor losses are neglected and all pipes have the same f, the system can be replaced by a single equivalent pipe of diameter D and length L that gives the same loss at the same discharge. Because h_f ∝ fLQ²/D⁵, the condition is L/D⁵ = L₁/D₁⁵ + L₂/D₂⁵ + L₃/D₃⁵ + …, known as Dupuit's equation.
Example: 300 m of 300 mm pipe followed by 200 m of 200 mm pipe, replaced by a single 250 mm pipe. The equivalent length is L = 0.25⁵ (300/0.3⁵ + 200/0.2⁵) ≈ 731 m.
When a pipeline divides into two or more branches that rejoin, each branch runs between the same two junctions and so carries the same head loss, while the total discharge is the sum of the branch discharges: Q = Q₁ + Q₂ + …, h_f1 = h_f2 = …. For two parallel branches with the same f, Q ∝ √(D⁵/L) in each, so the split of flow follows from Q₁/Q₂ = √(D₁⁵L₂/(D₂⁵L₁)).
A classic result: a single pipe of diameter D replaced by two parallel pipes of equal length and diameter d carrying the same total flow at the same head loss needs d = D/2^0.4 ≈ 0.758D. If instead a second pipe identical to the first is laid beside it, then at the same head loss each pipe carries the original discharge, so capacity doubles; at the same total discharge each pipe carries half the flow and, since h_f ∝ Q², the head loss falls to one quarter.
Larger networks, such as village water-supply loops or a farm's buried irrigation mains, are solved by the Hardy Cross method: assume flows satisfying continuity at every node, compute the head-loss imbalance around each loop, correct each loop's flows by ΔQ = −Σh_f/(n Σ h_f/Q), where n is the flow exponent (2 for Darcy, 1.852 for Hazen-Williams), and repeat.
A siphon is a pipe that rises above the hydraulic grade line of its supply before falling to a lower outlet. The pressure at the summit is below atmospheric; flow continues only if the absolute pressure there stays above the vapour pressure. The theoretical limit on summit height above the upstream water surface is therefore somewhat less than 10.3 m of water, and practical siphons are designed well below it.
When a pipeline carries water from a reservoir to a turbine or nozzle, the power delivered is ρgQ(H − h_f). For a given pipe and supply head H this is maximum when h_f = H/3, and the transmission efficiency at maximum power is then two-thirds (66.7 per cent). For a nozzle at the end of a pipe, the nozzle diameter that gives maximum jet power satisfies d = (D⁵/(2fL))^(1/4) with the Darcy f; the same result is written d = (D⁵/(8fL))^(1/4) in texts that use the Fanning coefficient.
An orifice is an opening in the wall or base of a tank, with a closed perimeter, through which liquid flows. For a small sharp-edged orifice under head H, the jet contracts to the vena contracta about half a diameter downstream. Three coefficients describe the real jet. The coefficient of contraction C_c is the ratio of jet area at the vena contracta to orifice area, about 0.61–0.69 and often taken as 0.64 for a sharp edge. The coefficient of velocity C_v is the ratio of actual to ideal jet velocity, about 0.95–0.99, often 0.97–0.98. The coefficient of discharge C_d = C_c × C_v, about 0.60–0.64, often 0.62. Discharge is Q = C_d a √(2gH).
C_v can be found from the trajectory of the jet: if the jet travels a horizontal distance x while falling a vertical distance y, then C_v = x/√(4yH).
A large orifice, whose height is not small relative to the head, must be integrated strip by strip: for a rectangular orifice of width b between heads H₁ and H₂, Q = (2/3) C_d b √(2g)(H₂^1.5 − H₁^1.5). A drowned (submerged) orifice discharges into liquid on the downstream side; H is then the difference between upstream and downstream water levels.
A mouthpiece is a short pipe, two to three diameters long, fitted to an orifice. In an external cylindrical mouthpiece the jet contracts inside the tube and then expands to fill it, so the issuing jet has no contraction; the pressure at the vena contracta inside is below atmospheric, which draws more flow. C_d is about 0.855, higher than a plain orifice, and the loss is mainly the sudden expansion inside. The absolute pressure at the vena contracta limits the head at which the mouthpiece runs full; beyond it air enters and flow reverts to orifice behaviour.
A convergent-divergent mouthpiece, shaped to follow the jet and expand gradually, can have a theoretical C_d of 1.0 based on the throat area. An internal (re-entrant or Borda) mouthpiece projects inward: running free, the jet does not touch the tube walls and C_c = 0.5, giving C_d = 0.5; running full, C_d is about 0.707.
When a tank drains through an orifice, the head falls with time, so the flow is unsteady; it is solved by treating each instant as quasi-steady. For a tank of constant plan area A draining through an orifice of area a with coefficient C_d, the continuity statement −A dh = C_d a √(2gh) dt integrates to T = 2A(√H₁ − √H₂)/(C_d a √(2g)) to fall from H₁ to H₂. To empty completely, set H₂ = 0. Note that the second half of the depth takes longer than the first: falling from H to H/2 takes about 29 per cent of the total emptying time, and the remaining half depth takes about 71 per cent.
Example: a tank with plan area 1 m² and water 2 m deep drains through a 25 mm orifice with C_d = 0.62. a = 4.91 × 10⁻⁴ m², so T = 2 × 1 × √2/(0.62 × 4.91 × 10⁻⁴ × 4.43) ≈ 2100 s, about 35 minutes.
For tanks whose area varies with depth (hemispherical bowls, circular horizontal cylinders, conical hoppers), A is written as a function of h before integrating.
A notch is an opening in the side of a small tank or channel with its top above the water surface; a weir is the same idea built across a larger channel or river. Both measure discharge from the head H above the crest, measured far enough upstream (about four or more times the maximum head) to avoid the drawdown curve.
Rectangular sharp-crested weir or notch of crest length L: Q = (2/3) C_d L √(2g) H^1.5. With C_d = 0.62 this is about 1.84 L H^1.5 in SI. The Francis formula allows for end contractions: Q = 1.84 (L − 0.1 n H) H^1.5, where n is the number of end contractions (2 for a notch narrower than its channel, 0 for a suppressed weir spanning the channel). Velocity of approach is allowed for by adding h_a = V_a²/2g to H.
Triangular (V) notch of included angle θ: Q = (8/15) C_d √(2g) tan(θ/2) H^2.5. For a 90° notch with C_d = 0.6 this is about 1.417 H^2.5 in SI. The V-notch is preferred for small flows, as in measuring well yield or drip-system supply, because the head is large relative to flow and C_d stays nearly constant.
Trapezoidal (Cipolletti) weir: side slopes of 1 horizontal to 4 vertical exactly compensate for the loss of flow due to end contractions, so Q ≈ 1.86 L H^1.5 in SI with no correction. It is common on farm channels.
Broad-crested weir: flow passes through critical depth on the crest; the maximum discharge occurs when the depth on the crest is two-thirds of the upstream head, giving Q = 1.705 C_d L H^1.5 in SI.
Error sensitivity: since Q ∝ H^n, a small error in head gives dQ/Q = n dH/H. A 1 per cent error in head gives a 1.5 per cent error in discharge over a rectangular weir and 2.5 per cent over a V-notch. Ventilation matters too: the space under the nappe of a suppressed weir must be vented to the atmosphere, otherwise air is carried away, the nappe clings and the discharge rises above the rated value.
An open channel has a free surface at atmospheric pressure; the driving force is the component of gravity along the slope, not a pressure difference. Flow is classified by time (steady or unsteady) and by space (uniform, where depth is the same at every section, or varied).
The geometric elements are: depth y; flow area A; wetted perimeter P, the length of the boundary in contact with water (the free surface is not included); hydraulic radius R = A/P; top width T; and hydraulic depth D = A/T. For a rectangular channel of bed width b: A = by, P = b + 2y. For a trapezoidal channel with side slope z horizontal to 1 vertical: A = (b + zy)y, P = b + 2y√(1 + z²), T = b + 2zy. For a very wide channel, R ≈ y.
In uniform flow the depth (called normal depth y_n), area and velocity are constant along the reach, so the water surface, bed and energy line are all parallel: the friction slope equals the bed slope S₀. The gravity force component along the channel is balanced by boundary shear, which gives the average bed shear stress τ₀ = ρgRS₀. This tractive force is the basis of design for erodible, unlined channels: the shear must stay below the permissible value for the soil or lining.
The Chezy equation, V = C√(RS), follows from the uniform-flow force balance if boundary shear is proportional to V². The Chezy coefficient C has units of m^0.5/s and is not constant; it depends on roughness and hydraulic radius.
The Manning equation is the most widely used uniform-flow formula: V = (1/n) R^(2/3) S^(1/2) in SI units, so Q = (A/n) R^(2/3) S^(1/2). In US customary units the constant 1.49 replaces 1: V = (1.49/n) R^(2/3) S^(1/2), with R in ft. Comparing the two gives C = R^(1/6)/n. Manning's n depends on the lining material, vegetation, channel irregularity and alignment; for concrete linings it is of the order of 0.012–0.017, for clean earth channels roughly 0.02–0.03, and for weedy or vegetated channels considerably higher.
Example: a trapezoidal earth channel with bed width 2 m, depth 1 m, side slopes 1.5H:1V, n = 0.025 and S = 0.0005. A = (2 + 1.5)(1) = 3.5 m², P = 2 + 2√(1 + 2.25) = 5.61 m, R = 0.624 m, V = (1/0.025)(0.624^0.667)(0.0005^0.5) = 0.653 m/s, and Q = 2.29 m³/s.
Finding discharge for a known depth is direct. Finding normal depth for a given discharge requires trial or iteration, because A and R both depend on y: compute the section factor AR^(2/3) = nQ/√S and find the y that gives it. In design, the velocity must lie between a minimum non-silting velocity and a maximum non-scouring velocity; freeboard is added above the design water surface.
For a given area, slope and roughness, Manning's equation shows that discharge is maximum when the wetted perimeter is minimum, because Q ∝ A^(5/3)/P^(2/3). Such a section is called the most economical or best hydraulic section; it needs the least lining and excavation per unit discharge.
Rectangular section: the bed width is twice the depth, b = 2y, and R = y/2.
Trapezoidal section: for a given side slope, the best section has the sloping side length equal to half the top width, and R = y/2. The three sides are tangent to a semicircle centred on the water surface. If the side slope is also free, the best trapezoid is half of a regular hexagon, with sides at 60° to the horizontal (z = 1/√3) and all three sides equal.
Triangular section: the best section has a 90° vertex angle, that is side slopes of 1:1.
Circular section (partly full): maximum velocity occurs at about 0.81 of the diameter, and maximum discharge at about 0.94 (0.938) of the diameter, both by Manning; a pipe flowing just under full carries slightly more than a full pipe.
In practice the best hydraulic section is not always the cheapest overall: side slopes are set by soil stability, depth by water table, and bed width by construction machinery.
Specific energy is the energy per unit weight measured with the channel bed as datum: E = y + V²/(2g). For a rectangular channel carrying discharge per unit width q = Q/b, E = y + q²/(2gy²). Plotting E against y for fixed q gives the specific energy curve. It has two limbs: an upper limb where depth is large and velocity small (subcritical, tranquil flow) and a lower limb where depth is small and velocity large (supercritical, rapid or shooting flow). For any E above the minimum there are two possible depths, called alternate depths, with the same specific energy.
The minimum specific energy occurs at the critical depth y_c. At critical depth, for a given E the discharge is maximum, and for a given q the specific energy is minimum.
Setting dE/dy = 0 for a rectangular channel gives y_c = (q²/g)^(1/3). At that depth V_c = √(g y_c), the velocity head is y_c/2 and the minimum specific energy is E_min = 1.5 y_c, so y_c = (2/3)E_min. For example, q = 2 m²/s gives y_c = (4/9.81)^(1/3) = 0.742 m and E_min = 1.112 m.
For a channel of any shape the critical condition is Q²T/(gA³) = 1, where T is top width. For a triangular channel the minimum specific energy is 1.25 y_c.
The Froude number Fr = V/√(gD), with D = A/T the hydraulic depth (equal to y for a rectangle), is the ratio of flow velocity to the speed of a small surface wave in still water. Fr < 1 is subcritical: surface waves can travel upstream, and the flow is controlled from downstream. Fr > 1 is supercritical: disturbances cannot travel upstream, and control is from upstream. Fr = 1 is critical. Critical flow is unstable, with an undulating surface, so canals are designed to avoid depths near critical, but control structures deliberately force critical depth because it gives a unique depth-discharge relationship: that is how broad-crested weirs, free overfalls and flumes measure flow. At a free overfall the brink depth is about 0.715 y_c.
A channel slope is called mild if normal depth exceeds critical depth, steep if normal depth is less than critical, and critical if they are equal. The slope that makes uniform flow critical is the critical slope.
Specific energy explains flow over a raised bed (hump) or through a contraction, neglecting losses. In subcritical flow over a hump of height Δz, the specific energy on the hump is E₁ − Δz; on the upper limb, lower E means lower depth, so the water surface drops over the hump. In supercritical flow the depth increases over the hump. If the hump is high enough that E₁ − Δz falls to E_min, the flow on the hump becomes critical; any higher hump chokes the flow, and the upstream depth rises (afflux) until the energy is enough to pass the discharge. Contractions in width act the same way, because q rises as width falls.
A hydraulic jump is an abrupt rise of the water surface where supercritical flow meets subcritical flow, for example below a sluice gate, at the foot of a spillway or a chute, or where a steep channel meets a mild one. Intense turbulence and a surface roller dissipate a large part of the energy. Because the loss is unknown, the jump is analysed with the momentum equation, not Bernoulli.
For a horizontal rectangular channel, equating the hydrostatic forces and momentum fluxes on either side gives the Bélanger equation for the sequent (conjugate) depths: y₂/y₁ = ½[√(1 + 8Fr₁²) − 1], where Fr₁ = V₁/√(gy₁). The energy loss is ΔE = (y₂ − y₁)³/(4y₁y₂). The two depths are also related by q² = g y₁ y₂ (y₁ + y₂)/2.
Example: y₁ = 0.3 m and V₁ = 6 m/s. Fr₁ = 6/√(9.81 × 0.3) = 3.50; y₂ = 0.15 × [√(1 + 8 × 3.50²) − 1] = 1.34 m; ΔE = (1.04)³/(4 × 0.3 × 1.34) = 0.70 m.
The US Bureau of Reclamation classifies jumps by Fr₁: about 1–1.7 undular; 1.7–2.5 weak; 2.5–4.5 oscillating, which sends waves downstream and is avoided in stilling-basin design; 4.5–9 steady, the well-balanced jump preferred for stilling basins; above 9 strong, which dissipates up to about 85 per cent of energy but is rough. The length of a jump on a horizontal floor is roughly 6 times the sequent depth y₂ for steady jumps; Indian textbooks often quote 5 to 7 times the jump height (y₂ − y₁). In a stilling basin the tailwater depth must match y₂ for the jump to form within the basin.
Every physical quantity can be expressed in fundamental dimensions: mass M, length L and time T (with temperature Θ where heat is involved). Velocity is LT⁻¹, acceleration LT⁻², force MLT⁻², pressure and stress ML⁻¹T⁻², dynamic viscosity ML⁻¹T⁻¹, kinematic viscosity L²T⁻¹, surface tension MT⁻², power ML²T⁻³ and discharge L³T⁻¹. A physically correct equation must be dimensionally homogeneous: every term has the same dimensions. Empirical formulae such as Manning's and Hazen-Williams' are not homogeneous, which is why their constants change with the unit system.
Dimensional analysis groups the variables of a problem into dimensionless groups. It reduces the number of variables to be tested experimentally, gives the form of relationships, and provides the rules for building and interpreting scale models.
Rayleigh's method writes the dependent variable as a product of powers of the independent variables, equates exponents of M, L and T, and solves. It suits problems with few variables (up to about four).
The Buckingham π theorem states that if a physical problem involves n variables containing m fundamental dimensions, the variables can be arranged into (n − m) independent dimensionless π groups. The procedure: list the n variables; choose m repeating variables that together contain all fundamental dimensions but do not themselves form a dimensionless group (typically one geometric, one flow or kinematic, and one fluid property, such as D, V and ρ); combine each remaining variable with the repeating variables to form a π term; solve each for the exponents. For the pressure drop in a pipe, depending on D, L, V, ρ, μ and ε (seven variables including Δp, three dimensions), there are four π groups: Δp/(ρV²), L/D, ρVD/μ and ε/D. The Darcy friction factor and Moody diagram are exactly this relationship displayed.
Each standard dimensionless number is a ratio of the inertial force to another force. Reynolds number Re = ρVL/μ is inertia over viscous force, governing pipe flow, flow around submerged bodies and settling of fine particles. Froude number Fr = V/√(gL) is the square root of inertia over gravity, governing free-surface flows: open channels, weirs, spillways, waves and ship resistance. Weber number We = ρV²L/σ involves surface tension (droplets, sprays). Euler number Eu = V/√(Δp/ρ) (some texts use Δp/(ρV²)) relates pressure and inertia and governs cavitation and flow meters. Mach number Ma = V/c, where c is the speed of sound, governs compressible flow and water hammer.
Hydraulic models of spillways, canal falls, intakes and pump sumps are built at reduced scale. Complete similarity requires geometric similarity (all length ratios equal, the scale ratio L_r), kinematic similarity (velocity ratios equal at corresponding points) and dynamic similarity (ratios of all forces equal). In practice it is impossible to keep both Re and Fr equal in a model using water, so the dominant force is chosen.
Froude model law, for free-surface flows where gravity dominates: Fr_model = Fr_prototype. With the same g, velocity ratio V_r = L_r^0.5, time ratio T_r = L_r^0.5, discharge ratio Q_r = L_r^2.5, force ratio F_r = L_r³ (same fluid), and power ratio P_r = L_r^3.5. For a 1:25 model, prototype discharge is 25^2.5 = 3125 times model discharge, and prototype velocity is 5 times model velocity.
Reynolds model law, for closed-conduit flow and fully submerged bodies where viscosity dominates: Re_model = Re_prototype, so V_r = ν_r/L_r. With the same fluid, the model velocity must be higher than the prototype velocity by the scale factor, which can be impractically high, so air models or pressurised models are sometimes used.
Learn to draw a control volume or a pressure diagram before writing any equation, and to write the energy equation from one free surface to another listing every loss. Practise unit conversions until they are automatic: litres per second to cubic metres per second, millimetres of mercury to metres of water, and SI to US units for the FE exam. Memorise the few results that recur (f = 64/Re, 2/3 depth centre of pressure, y_c = (q²/g)^(1/3), sequent-depth equation, Froude scaling) and derive everything else. Use a laboratory venturi, notch and Reynolds apparatus at least once so the coefficients mean something.
In practice this subject shows up as design checks: friction loss in a PVC main by Hazen-Williams to size the pump; the thrust on a buried bend to size a thrust block; the head over a V-notch or Parshall flume to measure a tubewell's yield; Manning's equation to size a lined field channel and check its velocity against silting and scouring limits; the stilling-basin length below a drop structure. Keep a small set of spreadsheet tools, state the coefficient and its source in every calculation, and check field results against design: a flow meter reading well below design usually means a clogged screen, partly closed valve or worn impeller.
At senior level the job is judgement about coefficients and margins: choosing Hazen-Williams C for pipe that will age over thirty years; choosing Manning's n for a canal that will weed up between cleanings; deciding whether a structure needs a physical model or whether standard designs suffice; reviewing a consultant's surge and cavitation checks. Senior engineers also standardise measurement, deciding where flumes or weirs are installed for water accounting in a command area and how they are calibrated and maintained, because volumetric water charging and allocation disputes depend on those numbers.
Research questions sit where the textbook coefficients are weakest: roughness of vegetated and eroding channels, flow resistance in micro-irrigation laterals and emitters where flow is transitional, hydraulic behaviour of non-standard farm structures, and sediment transport and scour. Computational fluid dynamics and physical models are both used and must be validated against each other; dimensional analysis sets the model laws and exposes scale effects. A senior researcher also judges which empirical formula a field dataset supports and publishes calibration data so practitioners can use better coefficients than generic handbook ranges.
This manual does not yet cover water hammer and surge analysis in detail, gradually varied flow profiles (M, S, C, H and A curves) and their computation, unsteady open-channel flow, boundary-layer theory and drag on immersed bodies, compressible flow, pump characteristics and cavitation numbers (treated in the groundwater, wells and pumps subject), sediment transport and scour, or computational fluid dynamics. Coefficient values (Hazen-Williams C, Manning's n, fitting K) are given as ranges because they vary with material, age, condition and source; design values must come from the national standard or supplier data. There are no diagrams yet: the Moody diagram and specific-energy curve are described in words.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Every numerical answer was computed by script; coefficients such as C_d, Hazen-Williams C and Manning's n are stated in each question because real values vary with the installation.
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