Basic Water Resources Engineering

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This section covers the classification of fluids and the fundamental fluid properties — density, specific weight, viscosity, compressibility, and surface-tension-related phenomena — that underlie all hydraulic analysis.

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Types of Fluids Classification Description Ideal vs. real fluid An ideal fluid is incompressible and has zero viscosity (a theoretical idealization); a real fluid possesses viscosity and, to varying degrees, compressibility Newtonian vs. non- Newtonian fluid A Newtonian fluid has shear stress directly proportional to the rate of shear strain (constant viscosity, e.g. water, air); a non-Newtonian fluid's viscosity varies with the rate of shear (e.g. some slurries, paints) Compressible vs. incompressible fluid A compressible fluid's density changes appreciably with pressure (e.g. gases); an incompressible fluid's density is essentially constant (e.g. liquids under normal conditions), a common simplifying assumption in hydraulics Fluid Properties Property Definition Mass density (ρ) Mass per unit volume of the fluid Specific weight (γ) Weight per unit volume, γ = ρg Specific gravity Ratio of a fluid's density (or specific weight) to that of a reference fluid (water for liquids, air for gases) at standard conditions Specific volume Volume occupied per unit mass, the reciprocal of density Viscosity A fluid's resistance to shear/flow; dynamic viscosity (μ) relates shear stress to velocity gradient, while kinematic viscosity (ν = μ/ρ) is dynamic viscosity divided by density Compressibility The fractional change in volume per unit change in pressure, related to the bulk modulus of elasticity (higher bulk modulus = less compressible) Capillarity The rise or depression of a liquid surface in a narrow tube, caused by the combined effect of surface tension and the adhesive/cohesive forces between liquid and tube material Surface tension The tensile force acting along the surface of a liquid due to intermolecular cohesive forces, causing the surface to behave like a stretched elastic membrane Cavitation & vapour pressure Cavitation is the formation and violent collapse of vapour bubbles in a liquid when local pressure drops to or below the liquid's vapour pressure, which can cause noise, vibration, and pitting damage to hydraulic machinery/surfaces

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This section covers fluid pressure and its relationship to depth, Pascal's law, manometry, pressure force and centre of pressure on submerged surfaces, pressure diagrams, and buoyancy/stability of floating and submerged bodies.

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Pressure, Head & Pascal's Law Pressure at a point in a static fluid is the same in all directions (Pascal's principle for a point);

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Pascal's law states that pressure applied to an enclosed, incompressible fluid is transmitted equally and undiminished to every point of the fluid and the walls of the containing vessel — the working principle behind hydraulic jacks/presses.

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The pressure-depth relationship in a static fluid of constant density is p = γh (or ρgh), where h is the depth below the free surface — pressure increases linearly with depth.

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Head expresses pressure in terms of an equivalent height of a fluid column, h = p/γ.

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Manometers A manometer is a device that measures pressure (or pressure difference) using a column of liquid in a tube, balancing the unknown pressure against a known column of manometric fluid.

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Common types: simple (piezometer) tube (measures gauge pressure directly as a column height, suited only to modest positive pressures of the same fluid), U-tube manometer (uses a heavier manometric liquid, e.g. mercury, to measure larger pressures/pressure differences), and differential manometer (measures the pressure difference between two points in a system).

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Pressure Force & Centre of Pressure on Submerged Bodies The total pressure force on a submerged plane surface equals the pressure at the surface's centroid multiplied by the surface's area:

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F = γhA, where h is the depth of the centroid.

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The centre of pressure (the point of application of the resultant pressure force) lies below the centroid for an inclined/vertical submerged plane surface, since pressure increases with depth, at a location found using the surface's moment of inertia about the centroidal axis.

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For a curved submerged surface, the resultant pressure force is found by separately computing horizontal and vertical force components (the horizontal component equals the force on the vertical projection of the surface; the vertical component equals the weight of the real or virtual fluid volume above/below the surface up to the free surface), then combining them.

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A pressure diagram is a graphical representation of pressure variation (typically triangular for a surface of constant depth-based linear pressure variation) along a submerged surface, useful for visualizing and computing the resultant force and its point of application.

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Buoyancy & Stability of Floating/Submerged Bodies Archimedes' principle (buoyancy): a body wholly or partially submerged in a fluid experiences an upward buoyant force equal to the weight of fluid displaced, acting through the centre of buoyancy (the centroid of the displaced fluid volume).

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Stability of a floating body depends on the relative positions of its centre of gravity (G), centre of buoyancy (B), and metacentre (M) (the point where the line of action of the buoyant force, after a small tilt, intersects the body's original vertical axis): the body is stable if M is above G (positive metacentric height GM), unstable if M is below G, and in neutral equilibrium if M coincides with G.

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For a fully submerged body (e.g. a submarine), stability instead depends directly on the relative positions of G and B: stable if G is below B, since any tilt then creates a righting couple.

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This section covers the classification of fluid flow, the continuity and momentum equations and their applications, Bernoulli's equation, and common flow measurement techniques.

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Classification of Fluid Flow Basis Types Time dependence Steady flow (flow properties at a point do not change with time) vs. unsteady flow (flow properties change with time) Spatial variation Uniform flow (velocity does not change along the flow direction at a given instant) vs. non-uniform flow (velocity varies along the flow direction) Viscous behaviour Laminar flow (smooth, orderly, parallel layers, dominant viscous forces, low Reynolds number) vs. turbulent flow (chaotic, mixing, dominant inertial forces, high Reynolds number) Rotationality Rotational flow (fluid particles rotate about their own axes as they move) vs. irrotational flow (fluid particles translate without net rotation) Continuity Equation & Momentum Equation The continuity equation expresses conservation of mass: for steady, incompressible flow through a stream tube/pipe, A1V1 = A2V2 (=Q, the constant volumetric flow rate), where A is cross-sectional area and V is average velocity.

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The momentum equation, derived from Newton's second law applied to a control volume, relates the net force acting on a fluid mass to the rate of change of momentum — used to determine forces exerted by flowing fluid on pipe bends, nozzles, vanes, and other flow-altering boundaries.

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Bernoulli's Equation Bernoulli's equation, derived from the principle of conservation of energy for steady, incompressible, inviscid flow along a streamline, states that the sum of pressure head, velocity head, and elevation head is constant: p/γ + V²/2g + z = constant.

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Applications include flow measurement (venturi meter, orifice meter, pitot tube), analysis of flow through orifices/nozzles, and siphon design; practical application requires correction for real-fluid energy losses (friction) between sections, typically accounted for via a head-loss term or an empirical coefficient of discharge/velocity.

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Flow Measurement Device Principle Venturi meter A gradually converging-then-diverging pipe section; the pressure drop at the throat (smaller area, higher velocity per continuity) is related to flow rate via Bernoulli's equation, offering low head loss due to the gradual diverging section Orifice meter A plate with a sharp-edged opening inserted in a pipe; simpler and cheaper than a venturi meter but causes greater head loss due to the abrupt contraction and expansion of flow Pitot tube Measures local flow velocity by sensing the difference between stagnation (total) pressure and static pressure at a point, related to velocity via Bernoulli's equation

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This section covers the types and governing equations of pipe flow, major and minor head losses, hydraulic and energy grade lines, pipe network analysis, and unsteady flow phenomena in pipes.

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Types & Governing Equations Pipe flow is classified (as in Section 3.3) by regime — laminar (low Reynolds number, Re < ~2000, parabolic velocity profile, head loss ∝ velocity) or turbulent (Re > ~4000, flatter velocity profile, head loss approximately ∝ velocity²) — governing which head-loss formula and friction-factor relationship applies.

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The Reynolds number, Re = ρVD/μ (V = mean velocity, D = pipe diameter), is the key dimensionless parameter distinguishing laminar from turbulent flow in a pipe.

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Major & Minor Head Losses Loss Type Description Major (friction) loss Continuous energy loss along the pipe length due to wall friction, computed using the Darcy– Weisbach equation: hf = fLV²/(2gD), where f is the friction factor (function of Reynolds number and relative pipe roughness, e.g. via the Moody chart), L is pipe length, D is diameter Minor (local) losses Losses at fittings, bends, valves, entrances, exits, and sudden area changes, generally expressed as hL = KV²/2g, where K is an empirical loss coefficient specific to the fitting type HGL and TEL The Hydraulic Grade Line (HGL) represents the sum of pressure head and elevation head (p/γ + z) along the pipe, and physically corresponds to the level water would rise to in a piezometer tube at each point.

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The Total Energy Line (TEL) represents the sum of pressure head, velocity head, and elevation head (total energy per unit weight) along the pipe;

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TEL is always above HGL by the velocity head (V²/2g), and TEL continuously drops in the direction of flow due to head losses.

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Design, Pipe Networks & Unsteady Flow Pipe design selects diameter, material, and layout to deliver a required flow rate at acceptable head loss/pressure, balancing construction cost against pumping/operating cost.

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Pipe network problems (interconnected pipes forming loops, as in water distribution systems) are commonly solved using the Hardy Cross method, an iterative technique that balances assumed flows in each loop until head loss around every loop sums to (approximately) zero.

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Unsteady flow in pipes (water hammer) occurs when flow velocity changes rapidly (e.g. sudden valve closure or pump trip), generating a pressure surge (transient) that propagates through the pipe as a wave, potentially causing pipe damage/rupture if not controlled.

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Relief devices — such as surge tanks, air vessels, and pressure relief/surge-anticipating valves — are used to absorb or dissipate the pressure surge from water hammer and protect the pipeline system.

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This section covers the geometric properties of open channels, types of flow, energy and momentum principles, gradually varied flow profiles, hydraulic jump, and flow in mobile boundary (erodible) channels.

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Geometric Properties & Types of Flow Key geometric properties of an open channel section: depth of flow, top width, wetted perimeter (length of channel boundary in contact with water), hydraulic radius R = (flow area)/(wetted perimeter), and hydraulic depth = (flow area)/(top width).

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Open channel flow is further classified as uniform (depth/velocity constant along the channel) vs. varied (non-uniform) flow (depth/velocity changes along the channel — gradually varied or rapidly varied), and as subcritical, critical, or supercritical based on the Froude number (Fr = V/√(gDh), where Dh is hydraulic depth):

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Fr < 1 (subcritical, tranquil flow, controlled from downstream), Fr = 1 (critical), Fr > 1 (supercritical, rapid flow, controlled from upstream).

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Energy & Momentum Principles Specific energy (E) is the energy per unit weight of flow measured relative to the channel bed:

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E = y + V²/2g, where y is flow depth; for a given discharge, a specific-energy curve shows that a given E (above the minimum) corresponds to two possible depths (subcritical and supercritical, called alternate depths), with minimum specific energy occurring at critical depth.

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Specific force (momentum function, M) is the sum of the momentum flux and hydrostatic pressure force per unit weight at a channel section, used (analogous to specific energy) to analyze situations such as the hydraulic jump where energy is not conserved but momentum is: for a given specific force, two depths (called conjugate/sequent depths) can satisfy the same M.

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Gradually Varied Flow Profiles & Hydraulic Jump Gradually varied flow (GVF) profiles describe the gradual change in water surface depth along a channel (e.g.

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M1/M2/M3 profiles for mild slope, S1/S2/S3 for steep slope, etc.), classified by channel slope type and whether the actual depth is above, between, or below the normal and critical depths.

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A hydraulic jump is an abrupt transition from supercritical to subcritical flow, accompanied by significant energy dissipation (turbulence) and a sudden rise in depth; for a horizontal, rectangular channel, jump theory relates the sequent depths before/after the jump via the momentum equation (not energy, since energy is dissipated in the jump), and the jump is characterized by type (undular, weak, oscillating, steady, strong) depending on the upstream Froude number.

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Flow in Mobile Boundary Channels A mobile boundary (alluvial/erodible) channel has a bed/banks composed of erodible material (sediment) that can be scoured or deposited depending on flow conditions, requiring design that considers sediment transport, not just hydraulic capacity.

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The inception of motion (threshold at which sediment particles begin to move under flowing water) is commonly assessed using the Shield diagram, which relates a dimensionless shear stress parameter (Shields parameter) to a particle Reynolds number to determine whether a given flow condition will initiate sediment movement for a given particle size.

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This final section of the chapter covers the hydrologic cycle, streamflow measurement and hydrograph analysis, rainfall-runoff analysis, flood hydrology, and groundwater hydrology.

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Hydrologic Cycle & Water Balance The hydrologic cycle is the continuous circulation of water between the atmosphere, land surface, and oceans, through processes including evaporation, transpiration, condensation, precipitation, infiltration, runoff, and groundwater flow.

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A water balance for a catchment/region is an accounting of inflows and outflows over a period, generally:

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Precipitation = Runoff + Evapotranspiration + Infiltration/Groundwater recharge ± Change in storage — used for water resources planning and management.

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Flow Measurement, Rating Curves & Hydrograph Analysis Streamflow (discharge) is measured using methods such as the velocity-area method (current meter), weirs/flumes, or dilution gauging; a rating curve is a graph relating measured river stage (water level) to discharge at a gauging station, allowing continuous discharge estimation from stage readings alone once established.

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A hydrograph is a plot of discharge versus time at a point in a stream, typically showing a rising limb, peak, and falling/recession limb in response to a rainfall event; hydrograph analysis separates the baseflow (groundwater contribution) from the direct runoff component.

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A unit hydrograph is the direct runoff hydrograph resulting from one unit depth of effective rainfall occurring uniformly over a catchment in a unit time; a synthetic unit hydrograph is derived (using catchment characteristics such as area, length, slope) for catchments lacking sufficient observed rainfall- runoff data to derive an observed unit hydrograph directly.

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Rainfall-Runoff Analysis & Flood Hydrology Rainfall-runoff analysis relates catchment rainfall to the resulting runoff, using methods ranging from simple empirical formulas and the rational method (for small catchments) to more sophisticated hydrological models and the unit hydrograph approach.

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Flood frequency analysis uses statistical methods (fitting a probability distribution, e.g.

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Gumbel or Log- Pearson Type III, to historical annual maximum flood series) to estimate the flood magnitude associated with a given return period (recurrence interval), used to determine the design flood for hydraulic structures (spillways, bridges, culverts) sized to an appropriate risk/return period.

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Groundwater Hydrology Groundwater hydrology covers the occurrence and movement of water in the subsurface (aquifers), including aquifer types (unconfined, confined, perched), aquifer properties (porosity, permeability/hydraulic conductivity, storage coefficient), groundwater flow governed by Darcy's law, and well hydraulics (pumping tests, drawdown, cone of depression).