Hydraulics and Fluid Mechanics is the study of fluids at rest and in motion — from basic fluid properties and pressure measurement, through hydrostatic forces, buoyancy, kinematics and Bernoulli's equation, to viscous and turbulent pipe flow, drag/lift, boundary layer theory, dimensional analysis, weirs, turbines, pumps and open channel flow. Every formula, coefficient value, diagram, solved example and exam-pattern table is included.

Learning Objectives

After studying this chapter you will be able to:

• Apply Newton's law of viscosity, classify Newtonian/non-Newtonian fluids, and compute surface tension and capillarity effects
• Solve manometer problems and compute hydrostatic forces (and centre of pressure) on plane and curved submerged surfaces
• Apply Archimedes' principle and compute the metacentric height to assess the stability of floating bodies
• Classify flow types, apply the continuity equation, and use stream functions/velocity potential for 2D flow analysis
• Apply Bernoulli's equation and the momentum equation to flow-measuring devices (Venturimeter, orifice, Pitot tube) and pipe bends
• Analyse laminar (Hagen-Poiseuille), turbulent, and boundary-layer flow, and compute pipe friction and minor losses (Darcy-Weisbach)
• Apply dimensional analysis (Buckingham π theorem) and similarity laws (Froude, Reynolds, Mach) to model studies
• Design and analyse notches/weirs, impulse/reaction turbines, pumps, and uniform/non-uniform open channel flow including the hydraulic jump

Exam Relevance

Concept Map

Where This Fits

Prerequisite: Foundation Engineering (seepage, dam spillway and drainage design concepts introduced there rely on the flow, pressure and open-channel principles developed here). Leads to: Engineering Hydrology, which applies these flow-measurement and open-channel principles to rainfall-runoff and river hydraulics.

1.1 Definition and Classification

A fluid is a substance that deforms continuously under the application of any shear stress, however small. Unlike a solid (which deforms by a fixed amount), a fluid flows — velocity continues to increase as long as shear stress is applied.

1.2 Newton's Law of Viscosity

Newtonian fluids (water, air, most oils) follow a linear relationship between shear stress and velocity gradient — viscosity μ is constant regardless of the rate of shear. Non-Newtonian fluids do not: the power-law model captures pseudoplastic (shear-thinning) fluids like paint and blood (\(n<1\)) and dilatant (shear-thickening) fluids like cornstarch suspensions and wet sand (\(n>1\)); a Bingham plastic (toothpaste, sewage sludge) additionally requires a minimum yield stress before it will flow at all.

1.3 Viscosity — Temperature Dependence

In liquids, viscosity decreases with rising temperature, since intermolecular cohesive forces (the dominant resistance mechanism) weaken as molecules gain thermal energy (Andrade's equation). In gases, viscosity increases with rising temperature, since the dominant resistance mechanism is momentum transfer between faster-moving molecules, which intensifies with temperature (Sutherland's law) — the opposite trend to liquids, and a frequently tested contrast.

1.4 Compressibility and Bulk Modulus

The bulk modulus \(K\) measures a fluid's resistance to uniform compression; its reciprocal is the compressibility \(\beta\). The speed of sound in a fluid depends directly on \(K\) and inversely on density, giving roughly 1480 m/s in water and 343 m/s in air. The Mach number (velocity relative to the speed of sound) below about 0.3 justifies treating a compressible fluid (like air) as effectively incompressible for most engineering flow calculations.

1.5 Surface Tension and Capillarity

Surface tension creates a pressure jump across any curved liquid interface — a droplet has one curved surface pair (giving \(4\sigma/d\)), a soap bubble has two films (giving \(8\sigma/d\)), and a liquid jet has just one surface (giving \(2\sigma/d\)). Capillary rise or depression in a narrow tube depends on the contact angle: water wets glass (\(\theta\approx0°\), giving a rise), while mercury does not wet glass (\(\theta\approx140°\), giving a depression).

1.6 Cavitation

Cavitation occurs when local pressure in a flowing liquid falls below the vapour pressure, causing vapour bubbles to form; when these bubbles subsequently collapse in a higher-pressure zone, they generate shock waves that cause erosion, noise, and vibration in hydraulic machinery (pump impellers, turbine runners, control valves). Prevention relies on maintaining sufficient Net Positive Suction Head (NPSH) at critical locations such as a pump inlet.

2.1 Pressure and Pascal's Law

Pressure at any point in a static fluid increases linearly with depth, governed by the hydrostatic pressure equation. Pascal's Law states that pressure applied to an enclosed fluid is transmitted equally and undiminished in all directions — the principle behind the hydraulic jack, where a small force on a small piston produces a large force on a large piston in proportion to the area ratio.

2.2 Types of Manometers

2.3 Manometer Calculation Procedure

The standard procedure: start from one point, add \(\rho gh\) for every step going down, and subtract \(\rho gh\) for every step going up; the algebraic sum between two points gives the pressure difference between them. For gas pressure measurement, the weight of the gas column itself is negligible compared to the gauge liquid, so only the mercury (or other gauge fluid) column need be considered.

2.4 Pressure Prism and Hydrostatic Paradox

Pressure at any point depends only on the depth below the free surface and the fluid density — not on the shape or total volume of the container. This is the hydrostatic paradox: a tall narrow tube and a wide, flat, open container filled to the same depth exert identical pressure at the base, despite holding vastly different volumes of fluid.

3.1 Force on a Plane Submerged Surface

The total hydrostatic force on a submerged plane surface depends only on the depth of the centroid below the free surface and the total area — not on the shape's orientation directly. However, the force does not act at the centroid; it acts at the centre of pressure (CP), which always lies below the centroid because pressure increases with depth (the lower portion of the surface experiences higher pressure and therefore a proportionally greater share of the total force). As submergence depth increases, the CP moves progressively closer to the centroid, since the relative variation in pressure across the surface becomes smaller.

3.2 Force on a Curved Submerged Surface

A curved surface is analysed by resolving the resultant hydrostatic force into horizontal and vertical components. The horizontal component equals the force on the vertical projection of the curved surface (computed exactly as for a plane surface). The vertical component equals the weight of the (real or imaginary) volume of fluid directly above the curved surface, and acts through the centroid of that volume. The two components combine vectorially to give the resultant force and its direction.

3.3 Pressure on Dams and Lock Gates

For a rectangular dam retaining water of depth H, the hydrostatic pressure distribution is triangular, so the resultant force acts at \(H/3\) above the base (the centroid of the triangle). This total force, multiplied by its lever arm, gives the overturning moment about the toe, which must be resisted by the moment of the dam's self-weight about the same point, with an appropriate factor of safety.

4.1 Archimedes' Principle

The buoyant force on a submerged or floating body equals the weight of the fluid it displaces. A floating body reaches equilibrium when the buoyant force exactly balances its own weight — the fraction of the body's volume that is submerged equals the ratio of the body's density to the fluid's density. A body whose density exceeds the fluid's density cannot achieve this balance while fully submerged, so it sinks; a body of lower density floats with only a partial fraction submerged.

4.2 Metacentre and Stability of Floating Bodies

When a floating body tilts slightly, its centre of buoyancy (the centroid of the displaced volume) shifts because the submerged shape changes. The metacentre M is the point where a vertical line through the new (shifted) centre of buoyancy intersects the original vertical line through the centre of gravity. If M lies above the centre of gravity G, the resulting couple rights the body back to equilibrium (stable); if M lies below G, the couple instead increases the tilt (unstable, capsizing).

4.3 Metacentric Height

The metacentric height \(GM\) is the distance between the metacentre and the centre of gravity — positive (M above G) means stable equilibrium, with a larger GM giving a "stiffer" (faster-righting, but less comfortable) vessel. \(GM\) can be computed geometrically from the vessel's waterplane geometry (\(BM = I_{WL}/V_{sub}\)), or measured experimentally on an actual vessel by shifting a known weight a known distance and measuring the resulting small angle of tilt.

5.1 Types of Flow

5.2 Flow Lines

A streamline is tangent at every point to the velocity direction at that instant, with no flow crossing it. A pathline is the actual trajectory traced by a single fluid particle over time. A streakline is the locus of all particles that have passed through a fixed point (what dye injection visualises). A streamtube is a bundle of streamlines through which no fluid crosses the walls. For steady flow, all three (streamline, pathline, streakline) coincide exactly; for unsteady flow, they are generally all different.

5.3 Continuity Equation

The continuity equation expresses conservation of mass. In its most general (3D, compressible) form it is a partial differential equation; for steady, incompressible flow it reduces to the divergence of velocity being zero; and for one-dimensional flow through a streamtube it reduces to the simple statement that volumetric flow rate is the same at every cross-section (or mass flow rate, for compressible flow).

5.4 Stream Function and Velocity Potential

The stream function ψ exists for any 2D incompressible flow (it automatically satisfies continuity) — lines of constant ψ are streamlines, and the difference in ψ between two streamlines equals the flow rate between them. The velocity potential φ exists only for irrotational flow, and satisfies Laplace's equation; lines of constant φ (equipotential lines) are always perpendicular to streamlines, forming an orthogonal flow net.

5.5 Vorticity and Circulation

Vorticity is the curl of the velocity field, physically equal to twice the local angular velocity of a fluid element. Circulation is the line integral of velocity around a closed contour, and by Stokes' theorem equals the surface integral of vorticity over any surface bounded by that contour. In irrotational flow, vorticity is zero everywhere, so circulation around any contour not enclosing a singular vortex is also zero.

6.1 Bernoulli's Equation

Euler's equation of motion, integrated along a streamline for steady, incompressible, inviscid, irrotational flow, gives Bernoulli's equation: the sum of pressure head, velocity head and datum (elevation) head remains constant along a streamline — a statement of conservation of mechanical energy per unit weight of fluid. In real (viscous) systems, a modified form accounts for head losses and any pump head added between two points.

6.2 Flow Measuring Devices

A Venturimeter's smooth, gradual contraction and expansion give it a much higher discharge coefficient than the simple orifice meter, at the cost of greater length and expense — this trade-off between accuracy/efficiency and simplicity/cost is a recurring theme across flow-measuring devices.

6.3 Orifice and Vena Contracta

Torricelli's theorem gives the theoretical jet velocity from an orifice under head h. The actual velocity is always slightly less, captured by the coefficient of velocity \(C_v\) (friction losses). The jet also contracts to a minimum cross-section (the vena contracta) downstream of a sharp-edged orifice, captured by the coefficient of contraction \(C_c\). The overall discharge coefficient combines both effects, and its value depends strongly on orifice edge geometry — sharp-edged, bell-mouth, or re-entrant.

6.4 Momentum Equation (Linear Momentum)

The momentum equation (derived from Newton's second law applied to a control volume via the Reynolds Transport Theorem) relates the net force on the fluid to the rate of change of momentum flux across the control surface. Applied to a pipe bend, it gives the force components needed to hold the bend in place — by Newton's third law, the fluid exerts an equal and opposite force on the pipe, which is what anchor blocks and thrust restraints must resist.

7.1 Hagen-Poiseuille Flow (Laminar Pipe Flow)

For fully developed laminar flow in a circular pipe, viscous shear balances the pressure gradient exactly, giving an exact parabolic velocity profile with maximum velocity at the centreline. Integrating this profile over the cross-section gives the Hagen-Poiseuille law for volumetric flow rate, directly proportional to the fourth power of diameter — meaning a small change in pipe diameter has a dramatic effect on capacity. The head loss predicted by this exact solution corresponds to the Darcy-Weisbach friction factor \(f=64/Re\), which is why the Darcy-Weisbach equation applies equally to laminar and turbulent flow.

7.2 Laminar Flow Between Parallel Plates

Flow between two flat plates has two limiting cases superimposed in the general Couette-Poiseuille profile: pure Couette flow (linear velocity profile, driven purely by one plate moving relative to the other, zero pressure gradient) and pure Poiseuille flow (parabolic velocity profile, driven purely by a pressure gradient with both plates fixed). Real bearing and lubrication flows are usually a combination of both mechanisms.

7.3 Reynolds Number and Transition

The Reynolds number is the ratio of inertia forces to viscous forces in a flow, and is the single most important dimensionless parameter for classifying pipe flow. Below the lower critical value, viscous forces dominate and flow remains laminar (smooth, layered); above the upper value, inertia dominates and flow becomes turbulent (chaotic, well-mixed); between the two lies an unstable transition zone that may intermittently switch between the two regimes.

8.1 Darcy-Weisbach Equation

The Darcy-Weisbach equation is the universal formula for friction (major) head loss in pipe flow, applicable to both laminar and turbulent flow provided the correct friction factor \(f\) is used. For laminar flow, \(f=64/Re\) exactly; for turbulent flow, \(f\) depends on both Reynolds number and relative pipe roughness, given approximately by the Blasius formula for smooth pipes at moderate Re, or more generally (and implicitly) by the Colebrook-White equation across the full turbulent range, visualised graphically in the Moody diagram.

8.2 Minor Losses

Minor losses occur at pipe fittings, bends, valves, and sudden changes in cross-section — despite the name, they can be significant in short pipe systems with many fittings. Each loss is expressed as a multiple \(K\) of the velocity head; sudden expansion losses can be derived theoretically (Borda-Carnot), while most other loss coefficients are determined experimentally.

8.3 Pipes in Series and Parallel

Pipes in series carry the same flow rate through every pipe, and head losses add up along the path (just like resistors in series). Pipes in parallel share the same head loss across each parallel branch, and the total flow rate is the sum of the flow through each branch (just like resistors in parallel, but for flow rather than current). An equivalent single pipe, matching the same total head loss for the same total flow, can be computed for either arrangement.

8.4 Pipe Network Analysis (Hardy-Cross)

Real distribution networks (water supply grids) form closed loops rather than simple series/parallel combinations. The Hardy-Cross method solves such networks iteratively: an initial guess of flow in each pipe is corrected loop-by-loop until both continuity at every junction and zero net head loss around every loop are satisfied simultaneously.

8.5 Water Hammer

Sudden valve closure in a pipeline abruptly decelerates the flow, converting kinetic energy into a pressure surge that propagates as a wave at the speed of sound in the fluid-pipe system (the wave speed is reduced from the value in an infinite fluid by the elasticity of the pipe wall). If the valve closes faster than the time for the pressure wave to travel to the reservoir and back (the critical closure time), the full Joukowsky pressure rise develops; slower closure allows partial pressure relief, reducing the severity of the surge.

9.1 Drag and Lift Definitions

Drag is the force component parallel to the flow direction (resisting motion), while lift is the force component perpendicular to it. Both are expressed via empirical or theoretical dimensionless coefficients \(C_D\) and \(C_L\), which depend on body shape, orientation, and Reynolds number, and are typically obtained from wind-tunnel or towing-tank experiments.

9.2 Types of Drag

• Pressure (form) drag: due to the pressure difference between the high-pressure front and low-pressure wake of a body; dominant for bluff (non-streamlined) bodies
• Friction (skin) drag: due to viscous shear stress on the surface; dominant for streamlined bodies where flow separation is minimal
• Wave drag: free-surface (ships) or compressibility (supersonic aircraft) effects
• Induced drag: arising from vortices shed at the tips of a finite-span wing

9.3 Drag Coefficients for Standard Bodies

9.4 Stokes' Law (Creeping Flow)

For very low Reynolds number (Re < 1), viscous forces completely dominate inertia, and Stokes derived an exact expression for the drag force on a sphere. Setting this drag equal to the net gravity/buoyancy force on a settling particle gives the terminal (settling) velocity — the constant velocity a particle reaches once these forces balance. This principle underlies the design of sedimentation tanks, centrifuges, and particle-size analysis by sedimentation.

10.1 Boundary Layer Concept (Prandtl)

Prandtl (1904) introduced the boundary layer concept: for high-Reynolds-number flows over a body, viscous effects are confined to a thin layer near the surface — the boundary layer. Outside this layer, the flow is essentially inviscid (potential flow), which is why potential flow theory (accurate everywhere except this thin layer) can still predict pressure distributions well, even though it completely ignores viscosity.

10.2 Laminar Boundary Layer (Blasius Solution)

Blasius solved the boundary layer equations exactly for laminar flow over a flat plate with zero pressure gradient, giving closed-form expressions for boundary layer thickness, displacement thickness, momentum thickness, and skin friction, all as functions of the local Reynolds number \(Re_x\) (based on distance from the leading edge). Transition to turbulence on a smooth flat plate typically occurs around \(Re_x \approx 5\times10^5\).

10.3 Turbulent Boundary Layer

Once the boundary layer transitions to turbulence, it grows faster (a 1/5 power law rather than the laminar 1/2 power law) and has higher skin friction, since turbulent mixing continuously replenishes momentum near the wall. The turbulent velocity profile near a wall follows the universal law of the wall: a linear viscous sublayer very close to the surface, transitioning to a logarithmic profile further out.

10.4 Boundary Layer Separation

Separation occurs when the wall shear stress falls to zero, caused by an adverse pressure gradient (pressure increasing in the flow direction) that decelerates the near-wall flow faster than the free stream. Common triggers include bluff bodies, sharp corners, diverging channels, and airfoils at high angle of attack; separation causes a large wake, increased pressure drag, and (for a wing) loss of lift known as stall. Prevention strategies include streamlining the body, boundary-layer suction, deliberately tripping the boundary layer turbulent (since a turbulent BL resists separation better), or vortex generators.

10.5 von Kármán Momentum Integral Equation

The von Kármán momentum integral equation is an approximate, integrated form of the boundary layer equations, relating wall shear stress to the growth rate of momentum thickness along the surface. It underlies most approximate (non-exact) boundary layer solution methods, including flows with a non-zero pressure gradient where an exact solution (like Blasius') is not available.

11.1 Nature of Turbulence

Turbulent flow is characterised by chaotic, irregular velocity fluctuations in space and time. It involves mixing, vortex stretching, an energy cascade from large eddies down to progressively smaller eddies, and ultimate dissipation as heat at the smallest (Kolmogorov) scale. Reynolds decomposition splits the instantaneous velocity into a time-averaged mean component and a fluctuating component, which by definition averages to zero over time but still transports momentum — giving rise to an additional apparent stress (Reynolds stress) beyond ordinary viscous shear.

11.2 Velocity Distribution in Turbulent Pipe Flow

Prandtl's mixing length theory models turbulent shear stress using an empirical length scale that grows linearly with distance from the wall. The resulting velocity profile in a turbulent pipe is much flatter than the parabolic laminar profile, commonly approximated by a power law whose exponent depends on Reynolds number; the ratio of mean to maximum velocity is consequently much closer to 1.0 than the laminar value of 0.5, reflecting the more uniform velocity distribution that turbulent mixing produces.

11.3 Turbulent Flow in Pipes

Because the velocity profile is flatter in turbulent flow, the kinetic energy and momentum correction factors (α and β) are much closer to 1.0 than their laminar counterparts (2.0 and 4/3). The Moody diagram summarises the Darcy friction factor as a function of both Reynolds number and relative pipe roughness across the entire laminar-to-fully-rough-turbulent range, with the Blasius formula and Colebrook-White equation as its two key underlying correlations.

12.1 Buckingham π Theorem

The Buckingham π theorem states that if a physical phenomenon can be described by \(n\) variables involving \(m\) fundamental dimensions, it can be reduced to a relationship between \((n-m)\) independent dimensionless groups (π groups). This dramatically reduces the number of experiments needed to characterise a phenomenon — instead of varying every variable independently, only the dimensionless groups need be varied.

12.2 Important Dimensionless Numbers

12.3 Model Analysis and Similarity Laws

Complete similarity between a model and its prototype requires geometric similarity (same shape), kinematic similarity (same velocity pattern, scaled), and dynamic similarity (same force ratios). Since it is generally impossible to match every dimensionless number simultaneously, the dominant force ratio for the phenomenon being studied is matched: Froude similarity for free-surface flows (open channels, ships, spillways, where gravity dominates), Reynolds similarity for fully-enclosed viscous flows (pipes, submarines), and Mach similarity for compressible/high-speed aerodynamic flows.

Froude and Reynolds similarity generally cannot both be satisfied in the same model using the same fluid — this is the fundamental model-scale problem in hydraulic engineering, and is why ship model tests (which must satisfy Froude similarity for wave-making) accept some Reynolds-number mismatch, correcting for it separately.

13.1 Rectangular Notch / Weir

A rectangular sharp-crested weir gives discharge proportional to \(H^{3/2}\), derived by integrating the velocity (from Torricelli's theorem) over the flow depth above the crest. Real weirs deviate from this theoretical value through the discharge coefficient. When the weir does not span the full channel width, end contractions reduce the effective crest length (the Francis formula), and when the approach velocity is significant, a velocity-of-approach correction adds the upstream kinetic energy head to the effective head.

13.2 Triangular (V-Notch) Weir

A V-notch gives discharge proportional to \(H^{5/2}\) — a steeper power than the rectangular weir's \(H^{3/2}\) — because both the head and the effective flow width increase together as the water rises in the V shape. This makes the V-notch far more sensitive at low flows (a small change in Q produces a larger, more measurable change in H) and self-cleaning (no stagnant zone at low flow, unlike a rectangular notch), which is why it is preferred for measuring small or highly variable discharges.

13.3 Trapezoidal Weir (Cipolletti)

The Cipolletti weir uses side slopes of 1:4 (horizontal:vertical), chosen specifically so that the increase in flow area from the sloping sides exactly compensates for the reduction in discharge caused by end contractions — allowing the same simple formula as a suppressed (no end contraction) rectangular weir to be used directly, without a separate contraction correction.

13.4 Broad-Crested and Ogee Weirs

A broad-crested weir is wide enough (relative to head) that the flow becomes critical on the crest itself, giving a discharge formula derived from critical-flow theory rather than the sharp-crested weir's orifice-like formula. An ogee (overflow spillway) profile is shaped to exactly follow the underside of the nappe from a sharp-crested weir at its design head, minimising negative pressure on the spillway face and making it the most hydraulically efficient shape for large dam spillways.

13.5 Submerged Weir

When the downstream (tailwater) level rises above the weir crest, the discharge is reduced compared to free (unsubmerged) flow — the Villemonte equation quantifies this reduction as a function of the submergence ratio (downstream head over upstream head). Only mild submergence (ratio below about 0.67) causes a small reduction; near-complete submergence causes a severe reduction in discharge capacity for the same upstream head.

14.1 Impact of Jets on Stationary and Moving Vanes

A jet striking a stationary flat plate perpendicular to its path exerts a force but does no work (the plate doesn't move, so power = force × 0 = 0). A jet striking a series of curved vanes moving at velocity \(u\) does useful work, since momentum is exchanged between the fluid and the moving vane — this is the fundamental principle behind impulse turbines. Work done is maximised at a specific vane-to-jet velocity ratio, and the theoretical maximum efficiency approaches 100% only for a semicircular vane deflecting the jet through a full 180° with no friction.

14.2 Pelton Wheel (Impulse Turbine)

A Pelton wheel is a pure impulse turbine: a high-velocity jet (accelerated through a nozzle from a high head) strikes a series of curved buckets on a wheel, converting the jet's kinetic energy into mechanical work at essentially atmospheric pressure throughout the runner. Maximum hydraulic efficiency occurs when the bucket speed is exactly half the jet velocity — beyond this point, the bucket "runs away" from the jet too fast for effective energy transfer; below it, the bucket doesn't move fast enough to fully utilise the jet's relative velocity.

14.3 Francis Turbine (Mixed Flow, Reaction)

Unlike the Pelton wheel, a Francis turbine is a reaction turbine — pressure drops continuously as water flows through the runner, not just in a nozzle beforehand. Work done is captured by the Euler turbine equation, relating the change in whirl (tangential) velocity component to the work extracted. A draft tube below the runner allows the turbine to be positioned above tailwater level (easing maintenance access) while still recovering the kinetic energy leaving the runner as useful pressure head, rather than wasting it.

14.4 Kaplan / Propeller Turbine

A Kaplan turbine is an axial-flow reaction turbine suited to low head, high flow-rate sites (run-of-river hydropower); its adjustable blades (vs. the fixed blades of a simple propeller turbine) let it maintain high efficiency across a wide range of flow conditions, which is essential where river flow varies substantially with season.

15.1 Classification of Pumps

15.2 Centrifugal Pump — Performance

Euler's pump equation (the mirror image of the turbine equation) gives the theoretical head developed by a centrifugal pump impeller in terms of the change in whirl velocity across it — for the common case of purely radial entry (no pre-swirl), the equation simplifies considerably. Net Positive Suction Head (NPSH) is the margin between the actual pressure at the pump inlet and the fluid's vapour pressure; if the available NPSH falls below the pump's required NPSH (a design characteristic), local pressure drops below vapour pressure somewhere inside the pump and cavitation occurs, which is why pump suction lift is practically limited to about 6–8 m for water.

15.3 Specific Speed and Affinity Laws

Specific speed is a dimensionless (or semi-dimensionless, depending on unit convention) parameter that characterises the "shape" of a pump's performance and is the standard basis for pump classification. The affinity laws predict how a pump's performance changes with rotational speed (for the same pump) or with impeller diameter (for geometrically similar pumps): flow scales linearly, head scales with the square, and power scales with the cube — the cubic power relationship means even small speed increases substantially raise motor power requirements.

15.4 Pumps in Series and Parallel

Pumps in series pass the same flow through each pump, with heads adding — used when a single pump cannot generate sufficient head. Pumps in parallel operate at the same head with flows adding — used when a single pump cannot deliver sufficient flow. The actual operating point of any pump-system combination is the intersection of the pump's head-flow characteristic curve with the system's resistance curve (static lift plus friction losses, which grow with the square of flow rate).

15.5 Reciprocating Pump

A reciprocating pump displaces a fixed volume per stroke (piston area × stroke length), giving theoretical discharge directly proportional to speed — a double-acting design (pumping on both the forward and return stroke) roughly doubles output for the same piston. Slip is the shortfall between theoretical and actual discharge (due to leakage past valves and the piston); an air vessel near the pump smooths the inherently pulsating flow of a reciprocating pump and reduces the acceleration head that would otherwise stress the suction/delivery pipework.

16.1 Types and Basic Equations

Open channel flow (a free surface exposed to atmosphere) is governed by Chezy's and Manning's formulas, both relating mean velocity to the hydraulic radius (area over wetted perimeter) and the channel's energy slope. Manning's formula is by far the most widely used in practice, since Manning's roughness coefficient \(n\) is well-tabulated for a huge range of natural and constructed channel surfaces.

16.2 Manning's n Values

16.3 Most Efficient Channel Section

For a given cross-sectional area, slope, and roughness, discharge is maximised by minimising the wetted perimeter (which maximises the hydraulic radius) — this defines the "most efficient" or "best hydraulic" section for each shape. Among all possible cross-sectional shapes, the semicircle is theoretically the most efficient of all; among practically constructible shapes, the best rectangle has width equal to twice the depth, and the best trapezoid has 60° side walls (half of an equilateral triangle).

16.4 Specific Energy and Critical Flow

Specific energy is the total mechanical energy measured relative to the channel bed (rather than an absolute horizontal datum) — the sum of flow depth and velocity head. For a given discharge, specific energy has a minimum value at a particular depth, the critical depth, at which the Froude number equals exactly 1. Below critical depth, flow is supercritical (fast, shallow, "shooting", controlled from upstream); above critical depth, flow is subcritical (slow, deep, "tranquil", controlled from downstream) — this upstream/downstream control distinction is fundamental to understanding how disturbances propagate in open channels.

16.5 Hydraulic Jump

A hydraulic jump is the abrupt, turbulent transition from supercritical to subcritical flow, dissipating substantial kinetic energy as turbulence and heat (rather than smoothly, as in a gradual transition) — it is deliberately used at the base of spillways and stilling basins to dissipate excess energy before water re-enters a natural channel, protecting the channel bed from erosion. The ratio of downstream to upstream depth (the sequent or conjugate depth ratio) depends only on the upstream Froude number, and jumps are classified by strength (and energy dissipation effectiveness) according to that Froude number.

16.6 Gradually Varied Flow (GVF)

When the channel bed slope, roughness, or cross-section changes gradually (rather than abruptly, as in a hydraulic jump), the flow depth adjusts gradually along the channel length according to the GVF equation, relating the rate of change of depth to the mismatch between the actual bed slope and the friction slope required to sustain uniform flow. Depending on how the actual depth compares to the normal depth (uniform-flow depth) and critical depth, twelve standard water-surface profile types are classified (M1-M3 for mild slopes, S1-S3 for steep slopes, and further categories for critical, horizontal and adverse slopes).

16.7 Uniform Flow — Non-Circular Sections

Trapezoidal channels (the most common constructed cross-section) and partially-filled circular pipes (sewers, culverts) both have closed-form expressions for area, wetted perimeter and hydraulic radius as functions of depth, allowing direct application of Manning's formula. For a circular section, maximum discharge and maximum velocity occur at slightly different partial-depth ratios (94% and 81% of full diameter respectively) — a subtlety in sewer design that is frequently tested.

A. Essential Formulae at a Glance

B. Key Number Summary

C. Exam-Angle Summary

D. Mnemonics & Memory Aids

Practice Questions

Related Topics

Mini Quiz — Full Chapter Recap