GATE Agricultural Engineering Soil and Water Conservation Engineering Syllabus

How is Bernoulli's equation modified for real fluids between two sections?

A head-loss term is added: $$\frac{p_{1}}{\rho g} + \frac{V_{1}^{2}}{2g} + z_{1} = \frac{p_{2}}{\rho g} + \frac{V_{2}^{2}}{2g} + z_{2} + h_{L}$$ where $h_{L}$ accounts for frictional and minor energy losses between sections 1 and 2.

What is the Reynolds number and what does it represent?

The Reynolds number is the ratio of inertial to viscous forces: $$Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}$$ For pipe flow it characterizes the flow regime (laminar, transitional, or turbulent).

State the Reynolds number ranges for laminar, transitional, and turbulent flow in a pipe.

Laminar: $Re < 2000$; Transitional: $2000 < Re < 4000$; Turbulent: $Re > 4000$. (The critical value $Re \approx 2000$–$2300$ marks the onset of transition.)

Compare laminar and turbulent pipe flow velocity profiles and friction dependence.

Laminar flow has a parabolic velocity profile with $V_{max} = 2\,V_{avg}$ and friction factor $f = \dfrac{64}{Re}$ (loss $\propto V$). Turbulent flow has a flatter (logarithmic) profile with $V_{max} \approx 1.2\,V_{avg}$ and friction depending on roughness (loss $\propto V^{1.75\text{–}2}$).

State the Hagen-Poiseuille equation for laminar flow in a circular pipe.

The discharge for laminar flow is $$Q = \frac{\pi \Delta p\, D^{4}}{128\, \mu L}$$ and the head loss is $$h_{f} = \frac{32 \mu L V}{\rho g D^{2}}$$ where $D$ is diameter, $L$ length, $V$ mean velocity.

State the Darcy-Weisbach equation for head loss due to friction.

$$h_{f} = \frac{f L V^{2}}{2 g D}$$ where $f$ is the Darcy friction factor, $L$ the pipe length, $V$ the mean velocity, $D$ the pipe diameter, and $g$ gravity.

For laminar flow, how is the Darcy friction factor related to Reynolds number?

$$f = \frac{64}{Re}$$ This applies only in the laminar regime ($Re < 2000$) and shows that friction factor is independent of pipe roughness for laminar flow.

State the Hazen-Williams equation and note where it is used.

The Hazen-Williams formula for velocity in pipes (water) is $$V = 0.849\, C\, R^{0.63} S^{0.54}$$ (SI units) where $C$ is the roughness coefficient, $R$ the hydraulic radius, and $S$ the slope of the energy line. It is empirical and used mainly for water distribution design.

What is the key difference between the Darcy-Weisbach and Hazen-Williams equations?

Darcy-Weisbach is theoretically derived, dimensionally homogeneous, valid for all fluids and all flow regimes (with friction factor from the Moody chart). Hazen-Williams is empirical, restricted to water at ordinary temperatures and turbulent flow, using a fixed roughness coefficient $C$.

What does Moody's diagram plot and what is it used for?

Moody's diagram plots the Darcy friction factor $f$ (y-axis) against Reynolds number $Re$ (x-axis, log scale) for various values of relative roughness $\varepsilon/D$. It is used to determine $f$ for turbulent pipe flow when computing friction head loss.

On the Moody diagram, what is the 'completely turbulent' (rough pipe) zone?

At high Reynolds numbers the curves become horizontal: the friction factor depends only on relative roughness $\varepsilon/D$ and is independent of $Re$. This is the fully rough turbulent zone, described by the Colebrook/von Kármán rough-pipe equation.

State Colebrook's equation for the turbulent friction factor.

$$\frac{1}{\sqrt{f}} = -2 \log_{10}\!\left( \frac{\varepsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}} \right)$$ It is implicit in $f$ and forms the basis of the turbulent region of the Moody chart, covering both smooth and rough pipe behaviour.

Give the discharge equation through a small orifice and define the coefficient of discharge.

$$Q = C_{d}\, a \sqrt{2 g H}$$ where $a$ is the orifice area, $H$ the head over the centre, and $C_{d}$ the coefficient of discharge. Here $C_{d} = C_{c} \times C_{v}$, the product of the coefficients of contraction and velocity (typically $C_d \approx 0.62$).

Define the coefficients of velocity, contraction, and discharge for an orifice.

$C_{v}$ (velocity) = actual velocity ÷ theoretical velocity at the vena contracta ($\approx 0.97$). $C_{c}$ (contraction) = area of vena contracta ÷ orifice area ($\approx 0.64$). $C_{d}$ (discharge) = actual discharge ÷ theoretical discharge $= C_{v} C_{c}$.

State the discharge formula for a rectangular notch/weir.

$$Q = \frac{2}{3} C_{d}\, L\, \sqrt{2g}\; H^{3/2}$$ where $L$ is the crest length and $H$ the head over the crest. Discharge is proportional to $H^{3/2}$.

State the discharge formula for a triangular (V-notch) weir.

$$Q = \frac{8}{15} C_{d}\, \tan\!\left(\frac{\theta}{2}\right) \sqrt{2g}\; H^{5/2}$$ where $\theta$ is the notch angle and $H$ the head. Discharge varies as $H^{5/2}$, making V-notches accurate for small flows.

Why is a triangular notch preferred over a rectangular notch for measuring low discharges?

For a V-notch a single sharp-edged opening handles a wide range of flows, only one head measurement is needed, and ventilation is unnecessary. Its $H^{5/2}$ relationship gives greater head change per unit flow, so small discharges are measured more accurately than with a rectangular notch ($H^{3/2}$).

State the discharge equation for a Cipolletti weir.

A Cipolletti weir is trapezoidal with side slopes $1$ horizontal to $4$ vertical, designed so the end contractions compensate the sloping sides. Discharge follows the simple rectangular form: $$Q = \frac{2}{3} C_{d}\, L\, \sqrt{2g}\; H^{3/2}$$ without an end-contraction correction.

Define specific energy in open channel flow.

Specific energy $E$ is the energy per unit weight measured relative to the channel bed: $$E = y + \frac{V^{2}}{2g} = y + \frac{Q^{2}}{2 g A^{2}}$$ where $y$ is the flow depth. It is minimum at critical flow.

What is the Froude number and how does it classify open channel flow?

$$Fr = \frac{V}{\sqrt{g\, D}}$$ where $D$ is hydraulic depth. $Fr < 1$ = subcritical (tranquil) flow; $Fr = 1$ = critical flow; $Fr > 1$ = supercritical (rapid/shooting) flow.

State Manning's equation for uniform open channel flow.

$$V = \frac{1}{n} R^{2/3} S^{1/2}$$ where $n$ is Manning's roughness coefficient, $R = A/P$ the hydraulic radius, and $S$ the bed slope. Discharge is $Q = AV = \dfrac{1}{n} A R^{2/3} S^{1/2}$.

What defines critical depth in an open channel and the critical condition?

Critical depth is the flow depth at which specific energy is minimum for a given discharge, where $Fr = 1$. The general condition is $$\frac{Q^{2} T}{g A^{3}} = 1$$ where $T$ is the top width. For a rectangular channel, $y_{c} = \left(\dfrac{q^{2}}{g}\right)^{1/3}$.

What is a hydraulic jump and what does it dissipate?

A hydraulic jump is a sudden transition from supercritical ($Fr>1$) to subcritical ($Fr<1$) flow, marked by an abrupt rise in depth and turbulence. It dissipates excess kinetic energy as heat, and is used downstream of spillways/structures for energy dissipation.

State Buckingham's $\pi$-theorem.

If a physical problem involves $n$ variables expressed in $m$ fundamental dimensions, the relationship can be reduced to $(n - m)$ independent dimensionless $\pi$ terms. Each $\pi$ group is formed by combining repeating variables with one of the remaining variables.

Give the definition and physical meaning of the Weber number.

$$We = \frac{\rho V^{2} L}{\sigma}$$ It is the ratio of inertial force to surface-tension force. It is significant where surface tension dominates, such as in capillary flows, droplet formation, and thin sheets of liquid.

Define the Euler number and the Mach number.

Euler number: $Eu = \dfrac{V}{\sqrt{p/\rho}}$ (or $\dfrac{\Delta p}{\rho V^{2}}$), the ratio of inertial to pressure force. Mach number: $M = \dfrac{V}{C}$, the ratio of fluid velocity to the speed of sound $C$, indicating compressibility effects ($M>1$ = supersonic).

State the three-phase relationship void ratio (e) versus porosity (n).

Void ratio $e = \dfrac{V_{v}}{V_{s}}$ (voids to solids); porosity $n = \dfrac{V_{v}}{V}$ (voids to total). They relate as $$n = \frac{e}{1+e}, \qquad e = \frac{n}{1-n}.$$

Define degree of saturation and the basic phase relationship $S e = w G_s$.

Degree of saturation $S = \dfrac{V_{w}}{V_{v}}$ (fraction of voids filled with water). The fundamental identity linking the phases is $$S\, e = w\, G_{s}$$ where $w$ is water content, $G_{s}$ specific gravity of solids, and $e$ void ratio.

Give the formula relating bulk (moist) unit weight to specific gravity, void ratio, and saturation.

$$\gamma = \frac{(G_{s} + S e)\,\gamma_{w}}{1 + e}$$ For saturated soil ($S=1$): $\gamma_{sat} = \dfrac{(G_{s}+e)\gamma_{w}}{1+e}$; for dry soil ($S=0$): $\gamma_{d} = \dfrac{G_{s}\gamma_{w}}{1+e}$.

What are the Atterberg limits and their associated indices?

The Atterberg limits are the liquid limit ($w_L$), plastic limit ($w_P$), and shrinkage limit ($w_S$), marking water-content boundaries between liquid, plastic, semi-solid, and solid states. Plasticity index $I_{P} = w_{L} - w_{P}$, the range of water content over which soil is plastic.

Define consistency index and liquidity index.

Liquidity index $I_{L} = \dfrac{w - w_{P}}{I_{P}}$ and consistency index $I_{C} = \dfrac{w_{L} - w}{I_{P}}$, where $w$ is natural water content. Note $I_{L} + I_{C} = 1$. High $I_{L}$ (near 1) means soft soil near liquid limit; high $I_{C}$ means stiff soil.

State Darcy's law for flow through soils.

$$q = k\, i\, A, \qquad v = k\, i$$ where $q$ is discharge, $k$ the coefficient of permeability, $i = h/L$ the hydraulic gradient, and $A$ the cross-sectional area. $v$ is the discharge (superficial) velocity. Valid for laminar flow.

How is the coefficient of permeability of layered soil computed for flow parallel and perpendicular to bedding?

Parallel to layers (equivalent horizontal): $$k_{H} = \frac{\sum k_{i} H_{i}}{\sum H_{i}}$$ Perpendicular to layers (equivalent vertical): $$k_{V} = \frac{\sum H_{i}}{\sum (H_{i}/k_{i})}$$ Always $k_{H} \geq k_{V}$.

What is the seepage discharge from a flow net, and what is its formula?

From a flow net, $$q = k\, H\, \frac{N_{f}}{N_{d}}$$ where $H$ is the total head loss, $N_{f}$ the number of flow channels, and $N_{d}$ the number of equipotential drops. $N_f/N_d$ is the shape factor of the net.

Define the critical hydraulic gradient and the quicksand condition.

The critical hydraulic gradient is $$i_{c} = \frac{G_{s} - 1}{1 + e} = \frac{\gamma'}{\gamma_{w}}$$ Quicksand (boiling) occurs when the upward seepage gradient reaches $i_c$, making effective stress zero so the soil loses shear strength.

State Terzaghi's principle of effective stress.

Total stress equals effective stress plus pore water pressure: $$\sigma = \sigma' + u$$ Effective stress $\sigma'$ (carried by the soil skeleton) governs shear strength and volume change; $u$ is the pore water pressure.

State the Mohr-Coulomb shear strength equation.

$$\tau_{f} = c + \sigma' \tan\phi$$ where $\tau_f$ is shear strength on the failure plane, $c$ the cohesion, $\sigma'$ the effective normal stress, and $\phi$ the angle of internal friction. For sands $c \approx 0$; for saturated clays (undrained) $\phi_u \approx 0$.

On a Mohr's circle of stress, what do the centre and radius represent?

For principal stresses $\sigma_{1}$ and $\sigma_{3}$, the centre lies at $\left(\dfrac{\sigma_{1}+\sigma_{3}}{2}, 0\right)$ and the radius is $\dfrac{\sigma_{1}-\sigma_{3}}{2}$, which equals the maximum shear stress $\tau_{max}$.

Give the normal and shear stress on a plane inclined at angle $\theta$ from Mohr's circle.

$$\sigma_{\theta} = \frac{\sigma_{1}+\sigma_{3}}{2} + \frac{\sigma_{1}-\sigma_{3}}{2}\cos 2\theta$$ $$\tau_{\theta} = \frac{\sigma_{1}-\sigma_{3}}{2}\sin 2\theta$$ where $\theta$ is measured from the major principal plane.

What is the orientation of the failure plane in terms of the friction angle $\phi$?

The failure plane makes an angle $$\theta_{f} = 45^{\circ} + \frac{\phi}{2}$$ with the major principal plane (the plane on which $\sigma_1$ acts), where $\phi$ is the angle of internal friction.

State the relationship between major and minor principal stresses at failure (Mohr-Coulomb).

$$\sigma_{1} = \sigma_{3} \tan^{2}\!\left(45^{\circ}+\frac{\phi}{2}\right) + 2c\,\tan\!\left(45^{\circ}+\frac{\phi}{2}\right)$$ Using $N_{\phi} = \tan^{2}(45^{\circ}+\phi/2)$: $\sigma_{1} = \sigma_{3} N_{\phi} + 2c\sqrt{N_{\phi}}$.

What is the uniformity coefficient and coefficient of curvature in soil grading?

$$C_{u} = \frac{D_{60}}{D_{10}}, \qquad C_{c} = \frac{(D_{30})^{2}}{D_{60}\, D_{10}}$$ where $D_{10}, D_{30}, D_{60}$ are particle sizes at 10%, 30%, 60% passing. A soil is well graded if $C_{u}$ is high and $1 < C_{c} < 3$.

Differentiate the discharge velocity and seepage velocity in soil.

Discharge (superficial) velocity $v = ki$ is based on total cross-sectional area. Seepage velocity is the actual velocity through the voids: $$v_{s} = \frac{v}{n} = \frac{ki}{n}$$ where $n$ is porosity. Since $n<1$, $v_{s} > v$.

What are the three common laboratory tests for shear strength of soil and the drainage conditions they represent?

(1) Unconsolidated-Undrained (UU) test — no drainage during consolidation or shear. (2) Consolidated-Undrained (CU) test — drainage during consolidation only. (3) Consolidated-Drained (CD) test — full drainage throughout. They are run in a triaxial cell (or direct shear/vane for field).