GATE Civil Engineering Water Resources Engineering Syllabus

Define the Reynolds number and give the critical values distinguishing laminar and turbulent flow in a pipe.

$$Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}$$ In pipes: laminar for $Re < 2000$, transitional for $2000 < Re < 4000$, and turbulent for $Re > 4000$.

For fully developed laminar flow in a circular pipe (Hagen-Poiseuille), give the velocity profile shape, the mean-to-max velocity ratio, and the friction factor.

The velocity profile is parabolic. The mean velocity is half the maximum: $V_{avg} = \dfrac{1}{2}V_{max}$. The Darcy friction factor is $f = \dfrac{64}{Re}$.

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

$$Q = \frac{\pi \Delta p\, R^{4}}{8 \mu L} = \frac{\pi \Delta p\, D^{4}}{128 \mu L}$$

Compare laminar and turbulent flow with respect to velocity profile and head loss dependence on velocity.

Laminar: smooth, ordered layers, parabolic profile, head loss $\propto V$. Turbulent: chaotic mixing with eddies, flatter (logarithmic) profile, head loss $\propto V^{1.75\text{ to }2}$.

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

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

Give the formula for head loss due to sudden expansion in a pipe.

$$h_L = \frac{(V_1 - V_2)^{2}}{2g}$$ where $V_1$ and $V_2$ are the velocities before and after the expansion (Borda-Carnot loss).

State the three governing conditions used to analyze pipe networks by the Hardy Cross method.

(1) Continuity: net flow into each junction is zero; (2) Energy: algebraic sum of head losses around each closed loop is zero; (3) Each head loss follows a relation $h_f = r Q^{n}$ (e.g., $n = 2$ for Darcy-Weisbach).

Give the Hardy Cross flow correction formula for one loop iteration.

$$\Delta Q = -\frac{\sum r Q^{n}|Q|^{\,0}}{\sum n\, r |Q|^{n-1}} = -\frac{\sum h_f}{n \sum \left(\dfrac{h_f}{Q}\right)}$$

Define equivalent pipe and state the rule for combining pipes in series and parallel.

An equivalent pipe carries the same discharge with the same total head loss as the system it replaces. In series: discharge same, head losses add. In parallel: head loss same across branches, discharges add.

Define the hydraulic boundary layer and the boundary layer thickness $\delta$.

The boundary layer is the thin region near a solid surface where viscous effects are significant and velocity rises from zero (no-slip) to the free-stream value. The thickness $\delta$ is the distance from the surface at which the velocity reaches $0.99 U$ (the free-stream velocity).

Define displacement thickness and momentum thickness of a boundary layer.

Displacement thickness: $\delta^{*} = \displaystyle\int_0^{\delta}\left(1 - \frac{u}{U}\right)dy$ (deficit in mass flow). Momentum thickness: $\theta = \displaystyle\int_0^{\delta}\frac{u}{U}\left(1 - \frac{u}{U}\right)dy$ (deficit in momentum flow).

For a flat plate, give the Reynolds number criterion for transition from laminar to turbulent boundary layer.

Transition typically occurs around $Re_x = \dfrac{U x}{\nu} \approx 5 \times 10^{5}$, where $x$ is the distance from the leading edge.

Define boundary layer separation and state the condition at the wall when it begins.

Separation is the detachment of the boundary layer from the surface, occurring under an adverse pressure gradient ($\dfrac{dp}{dx} > 0$). It begins where the wall velocity gradient vanishes: $\left(\dfrac{\partial u}{\partial y}\right)_{y=0} = 0$.

Define drag force and lift force on an immersed body and give their formulas.

Drag is the force component parallel to flow: $F_D = C_D \cdot \dfrac{1}{2}\rho A V^{2}$. Lift is the component perpendicular to flow: $F_L = C_L \cdot \dfrac{1}{2}\rho A V^{2}$, where $C_D, C_L$ are drag and lift coefficients and $A$ a reference area.

Distinguish skin friction drag from pressure (form) drag.

Skin friction drag arises from viscous shear stress acting tangentially over the body surface. Pressure (form) drag arises from the pressure difference between the front and rear, largely due to boundary-layer separation and the resulting wake.

What is a streamlined body versus a bluff body in terms of drag?

A streamlined body has gradual contours that delay separation, so it is dominated by friction drag and has low total drag. A bluff body causes early separation and a large wake, so it is dominated by pressure (form) drag and has high total drag.

State the principle and discharge equation of a venturimeter.

A venturimeter uses a converging-throat-diverging section; the pressure drop at the throat (via Bernoulli + continuity) gives discharge: $$Q = C_d \frac{A_1 A_2}{\sqrt{A_1^{2} - A_2^{2}}}\sqrt{2g h}$$ where $h$ is the differential head.

Compare a venturimeter and an orifice meter in terms of coefficient of discharge and head loss.

Venturimeter: high $C_d$ (about $0.95$-$0.98$), low permanent head loss, but bulky and costly. Orifice meter: lower $C_d$ (about $0.6$-$0.65$), higher permanent head loss, but compact and cheap.

Give the discharge equation for a rectangular notch/weir and a triangular (V-notch) weir.

Rectangular: $Q = \dfrac{2}{3} C_d \, L \sqrt{2g}\; H^{3/2}$. Triangular (V-notch of angle $\theta$): $Q = \dfrac{8}{15} C_d \sqrt{2g}\,\tan\!\dfrac{\theta}{2}\; H^{5/2}$.

Define the Froude number and explain its physical meaning.

$$Fr = \frac{V}{\sqrt{g L}}$$ It is the ratio of inertia force to gravity force, governing free-surface (open-channel) flow. For channels, $L$ is the hydraulic depth.

State Buckingham's $\pi$-theorem.

If a physical problem involves $n$ variables expressible in $m$ fundamental dimensions, the relationship can be reduced to $(n - m)$ independent dimensionless $\pi$ groups.

Name the three types of similarity required for complete dynamic similitude between model and prototype.

Geometric similarity (same shape / scale ratios), kinematic similarity (similar flow patterns / velocity ratios), and dynamic similarity (same ratios of corresponding forces).

List the main dimensionless numbers and the dominant force each represents.

Reynolds (inertia/viscous), Froude (inertia/gravity), Euler (inertia/pressure), Weber (inertia/surface tension), and Mach (inertia/elastic/compressibility).

Define specific energy in open-channel flow and write its expression.

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

Define critical depth and give the general condition for critical flow.

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

For a rectangular channel, relate critical depth to minimum specific energy.

At critical flow in a rectangular channel, the velocity head equals half the critical depth, so $$E_{min} = \frac{3}{2} y_c$$

Classify open-channel flow as subcritical, critical, or supercritical using the Froude number and depth.

Subcritical (tranquil): $Fr < 1$, $y > y_c$. Critical: $Fr = 1$, $y = y_c$. Supercritical (rapid/shooting): $Fr > 1$, $y < y_c$.

Define a hydraulic jump and state the sequent (conjugate) depth relationship for a rectangular channel.

A hydraulic jump is an abrupt transition from supercritical to subcritical flow with energy dissipation. The conjugate depths relate as $$\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8 Fr_1^{2}} - 1\right)$$

Give the energy loss in a hydraulic jump in a rectangular channel.

$$\Delta E = \frac{(y_2 - y_1)^{3}}{4\, y_1 y_2}$$

Write Manning's equation for uniform flow in an open channel and define hydraulic radius.

$$V = \frac{1}{n} R^{2/3} S^{1/2}$$ where $n$ is Manning's roughness, $S$ the bed slope, and the hydraulic radius $R = \dfrac{A}{P}$ (area divided by wetted perimeter). Uniform flow occurs at normal depth where slope of energy line equals bed slope.

State the dynamic equation of gradually varied flow (GVF) and name the surface profiles classified by it.

$$\frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^{2}}$$ where $S_0$ is bed slope and $S_f$ the friction slope. Water surface profiles are classified by bed slope and depth zone as M (mild), S (steep), C (critical), H (horizontal), and A (adverse) curves.

List the main components/processes of the hydrologic cycle.

The continuous circulation of water through: evaporation and transpiration (evapotranspiration), condensation, precipitation, interception, infiltration, surface runoff, percolation to groundwater, and return flow to oceans/atmosphere. It is a closed cycle driven by solar energy and gravity.