Hydraulics, Hydrology and Irrigation Engineering — ESE Civil

Weightage: Fluid mechanics, hydrology and irrigation together make up a major block of the Civil Prelims Paper II and the Mains water-resources paper. The questions are formula-driven, and many reuse the same four ideas: energy, momentum, continuity and probability.

1. Hydrostatics and Bernoulli

The force on a submerged plane surface is $F=\gamma \bar{h}A$, with $\bar{h}$ the depth of the centroid. It acts at the centre of pressure, below the centroid by $I_G{\sin }^2\theta /(A\bar{h})$ along a plane inclined at $\theta$ to the free surface, and for a vertical surface by $I_G/(A\bar{y})$ along the plane.

A body floats stably if its metacentre lies above its centre of gravity.

Bernoulli's equation for steady, incompressible, frictionless flow along a streamline is:

$\frac{p}{\gamma }+\frac{V^2}{2g}+z=\text{constant}$

Add head loss $h_L$ between sections for a real fluid. Continuity gives $Q=AV$.

2. Pipe flow

Reynolds number $Re=\rho VD/\mu$ separates flow regimes: laminar below about 2000 and turbulent above about 4000 in pipes.

The Darcy-Weisbach equation gives friction loss:

$h_f=\frac{fL{V}^2}{2gD}$

For laminar flow $f=64/Re$, and the Hagen-Poiseuille result gives a parabolic velocity profile with a mean velocity half the maximum. Minor losses are $K{V}^2/2g$, with a sudden expansion losing $(V_1-V_2{)}^2/2g$.

For pipes in series the head losses add and the discharge is common. For pipes in parallel the head loss is common and the discharges add. Maximum power transmission through a pipeline occurs when friction loss is one-third of the supply head.

3. Open-channel flow

Manning's equation is:

$V=\frac{1}{n}{R}^{2/3}{S}^{1/2},R=\frac{A}{P}$

The most efficient cross-section for a given area has the least wetted perimeter. For a rectangle it is $b=2y$, and for a trapezoid it is half of a regular hexagon.

Specific energy is $E=y+V^2/2g$. For a rectangular channel with discharge $q$ per unit width, critical depth is:

$y_c={\left(\frac{q^2}{g}\right)}^{1/3},E_{min}=\frac{3}{2}{y}_c$

The Froude number $Fr=V/\sqrt{gy}$ is 1 at critical flow, below 1 subcritical and above 1 supercritical.

A hydraulic jump forms when supercritical flow changes to subcritical. The sequent depth is:

$\frac{y_2}{y_1}=\frac{1}{2}\left(\sqrt{1+8F{r}_1^2}-1\right)$

and the energy loss is $(y_2-y_1{)}^3/(4y_1{y}_2)$.

Worked example. A rectangular channel carries $q=4{\text{m}}^2/\text{s}$. Then $y_c=(16/9.81{)}^{1/3}=1.17$ m and $E_{min}=1.75$ m.

For a sharp-crested rectangular weir, $Q=\frac{2}{3}{C}_d\sqrt{2g}L{H}^{3/2}$, and for a V-notch $Q=\frac{8}{15}{C}_d\sqrt{2g}\tan (\theta /2)H^{5/2}$.

4. Hydrology

The hydrologic cycle moves water through precipitation, infiltration, runoff and evapotranspiration. Rainfall is measured by gauges, and the average over a catchment uses the arithmetic mean, Thiessen polygon or isohyetal method (the last is the most accurate in hilly areas).

Runoff. The rational method estimates peak flow from small catchments:

$Q=\frac{CiA}{360}$

with $Q$ in ${\text{m}}^3/\text{s}$, rainfall intensity $i$ in mm/h and area $A$ in hectares. The intensity is for a storm lasting the time of concentration.

A unit hydrograph is the direct-runoff hydrograph from one unit of effective rainfall over a stated duration. By linearity and superposition it generates the response to any storm.

Flood frequency. The return period is $T=1/p$. The probability that a flood of return period $T$ occurs at least once in $n$ years is:

$1-{\left(1-\frac{1}{T}\right)}^n$

For a 100-year flood in a 50-year design life this is $1-0.99^{50}\approx 0.39$, which is why designers do not treat a 100-year flood as a once-in-a-century event.

5. Irrigation: duty, delta and crop water

Duty $D$ is the area irrigated per unit discharge, in hectares per cumec, and delta $\Delta$ is the depth of water a crop needs over its base period $B$ in days. They are linked by:

$\Delta =\frac{8.64B}{D}$

with $\Delta$ in metres. Crop water requirement is evapotranspiration plus losses, and irrigation requirement subtracts effective rainfall. Irrigation efficiency compares water used by the crop with water delivered. Methods include surface, sprinkler and drip irrigation, and drip suits scarce water.

6. Canals and silt theories

Kennedy's theory says a non-silting, non-scouring velocity is $V_0=0.55m{y}^{0.64}$, with $y$ the depth and $m$ a critical velocity ratio. Lacey's regime theory describes an alluvial channel in dynamic balance. Its main results are:

$V={\left(\frac{Q{f}^2}{140}\right)}^{1/6},P=4.75\sqrt{Q},f=1.76\sqrt{d_{mm}}$

with $f$ the silt factor and $P$ the wetted perimeter. A channel in regime is stable, with no net silting or scouring over a year.

7. Dams and spillways

A gravity dam resists water pressure by its own weight. Stability checks are:

• Overturning: factor of safety at least 1.5.
• Sliding: a shear-friction factor is used.
• Tension: no tension at the heel, so the resultant should lie within the middle third of the base.
• Crushing: the toe stress must not exceed the permissible value.

The base pressures are $p=\frac{\sum V}{B}\left(1\pm \frac{6e}{B}\right)$, and the uplift reduces effective weight. An ogee spillway is shaped to the underside of a nappe, and a stilling basin dissipates energy by forcing a hydraulic jump.

8. Headworks on permeable foundations

For a weir on pervious soil, water seeps under the floor. Bligh's creep theory needs a minimum creep length $L=CH$. Khosla's theory uses flow nets and checks the exit gradient, which must stay below the critical value, typically with a safe limit near one-fifth to one-sixth. Cut-off piles at the upstream and downstream ends lengthen the seepage path and reduce uplift.

Common traps

• Using the diameter in Reynolds number for open channels. Use hydraulic radius.
• Treating $E_{min}=1.5y_c$ as general. It is for rectangular sections.
• Quoting the rational formula without checking units. Hectares and mm/h give the 360.
• Reading a 100-year flood as once per century. The risk is annual.
• Using duty in the wrong direction. Higher duty means less water per hectare.

Memory aids

• "8.64 B over D": delta.
• "Q over 360 with C, i, A": rational method.
• "Middle third": no tension in a gravity dam.

Summary

Hydraulics rests on continuity, energy and momentum. Pipe problems use Darcy-Weisbach, open channels use Manning, specific energy and the jump relation.

Hydrology treats rainfall and runoff as probabilistic, using return periods and the rational method. Irrigation links crop needs to canals, dams and headworks through duty, regime theory and stability checks.

Exam protocol

• Check units: head in metres of fluid, area in hectares for the rational method.
• Decide sub- or supercritical with the Froude number.
• Use the middle-third rule to check dam bases.
• Compute return-period risk with the complement.