Fluid Mechanics and Hydraulic Machines — ESE Mechanical

Weightage: Fluid mechanics and turbomachines form a major block of the Mechanical papers, with a mix of conceptual and numerical questions. Pelton and Francis turbines, pump laws and dimensionless numbers are asked in almost every cycle.

1. Fluid properties

Newton's law of viscosity says shear stress is proportional to velocity gradient: $\tau =\mu du/dy$. Viscosity of liquids falls with temperature, and that of gases rises with temperature. Surface tension gives a capillary rise $h=4\sigma \cos \theta /(\rho gd)$ in a tube of diameter $d$, and the excess pressure inside a droplet is $4\sigma /d$ (for a bubble in a soap film, $8\sigma /d$ because there are two surfaces).

2. Momentum equation and jets

The linear momentum equation gives the force on a control volume: $F=\rho Q(V_{out}-V_{in})$. Standard cases:

• A jet striking a fixed flat plate at right angles exerts $F=\rho a{V}^2$.
• On a moving plate at speed $u$ in the jet direction, $F=\rho a(V-u{)}^2$.
• On a series of vanes (a wheel), the mass flow is the full jet, so $F=\rho aV(V-u)$.
• The force on a pipe bend combines the momentum change and the pressure forces.

For a curved vane that deflects the jet through an angle, the work per second on a moving vane series is $P=\rho Qu(V-u)(1+k\cos \varphi )$, with $\varphi$ the deflection angle's supplement and $k$ a friction factor.

3. Dimensional analysis and similitude

The Buckingham $\pi$ theorem says $n$ variables involving $m$ fundamental dimensions form $n-m$ independent dimensionless groups. The groups to know:

Dynamic similarity needs the dominant group equal in model and prototype. For a river or dam model use Froude scaling, and for a pipe or submerged body use Reynolds scaling.

4. Boundary layer and drag

A boundary layer is the thin region near a surface where viscosity matters. For a flat plate, the laminar thickness grows as $\delta =5x/\sqrt{R{e}_x}$, and the skin-friction coefficient averaged over length is $C_f=1.328/\sqrt{R{e}_L}$. Transition to turbulence occurs near $R{e}_x\approx 5\times 10^5$.

A boundary layer separates where the pressure gradient is adverse, giving a wake and pressure drag. A streamlined body reduces separation, and a rough surface such as a golf ball delays separation. Stokes' law gives the drag on a small sphere, $F=3\pi \mu DV$, for $Re<1$, equivalent to $C_D=24/Re$.

5. Compressible flow

The speed of sound in a perfect gas is $a=\sqrt{\gamma RT}$. The Mach number $M=V/a$. Isentropic stagnation relations are:

$\frac{T_0}{T}=1+\frac{\gamma -1}{2}{M}^2,\frac{p_0}{p}={\left(\frac{T_0}{T}\right)}^{\gamma /(\gamma -1)}$

For air at $M=2$, $T_0/T=1.8$ and $p_0/p=7.82$.

In a converging-diverging nozzle, flow accelerates to sonic at the throat only if the back pressure is low enough, after which the throat is choked and the mass flow cannot rise further. Beyond the throat, a diverging section speeds up supersonic flow, and a normal shock produces a sudden rise in pressure and temperature with a fall to subsonic speed. Across a shock, stagnation temperature is unchanged and stagnation pressure falls.

6. Impulse turbines: the Pelton wheel

A Pelton wheel converts the kinetic energy of a high-velocity jet in an open-air wheel. It suits high head and low flow, with a specific speed from about 10 to 35 in SI-style units. The jet speed is $V=C_v\sqrt{2gH}$.

The ideal bucket deflects the jet through 180 degrees. Maximum hydraulic efficiency occurs when the bucket speed is half the jet speed, $u=V/2$. In practice buckets turn the jet about 165 degrees, so a little jet energy leaves with the water.

Worked example. A jet with $V=40$ m/s and $Q=0.08{\text{m}}^3/\text{s}$ strikes a wheel with $u=20$ m/s. With full deflection, $P=\rho Qu(V-u)\times 2=1000\times 0.08\times 20\times 20\times 2=64$ kW. The jet power is $\frac{1}{2}\rho Q{V}^2=64$ kW as well, so the ideal efficiency is 100 percent. Real wheels reach about 85 to 90 percent.

7. Reaction turbines

A Francis turbine is a radial-inward (mixed) flow reaction turbine for medium head and flow. A Kaplan is an axial-flow turbine with adjustable runner blades for low head and large flow, keeping high efficiency at part load. Reaction turbines run full of water, so a draft tube is used to convert exit kinetic energy to pressure recovery and to let the runner sit above tail-water level.

Specific speed $N_s=N\sqrt{P}/H^{5/4}$ for turbines, and $N_s=N\sqrt{Q}/H^{3/4}$ for pumps. Governing is done by a spear valve in a Pelton wheel and by guide vanes in a Francis turbine. Cavitation in a reaction turbine occurs when local pressure falls to vapour pressure, causing pitting, noise and loss of efficiency. The Thoma cavitation factor compares the available suction head with the head, and a plant needs a larger value than the critical one.

8. Centrifugal pumps

A centrifugal pump adds energy to liquid through an impeller. The theoretical head is from the Euler equation, $H=(u_2{V}_{w2}-u_1{V}_{w1})/g$. With a radial entry, $V_{w1}=0$.

Backward-curved vanes are normally used because they are efficient and stable. Real head is lower than the Euler head because of slip and losses.

The affinity laws at constant impeller diameter:

$Q\propto N,H\propto N^2,P\propto N^3$

Worked example. A pump gives 30 L/s at 20 m head at 1000 rpm. At 1500 rpm: $Q=45$ L/s, $H=20\times 2.25=45$ m, and power rises 3.375 times.

NPSH (net positive suction head) is the total head at the suction above vapour pressure. To avoid cavitation, $NPS{H}_{available}>NPS{H}_{required}$. Priming is needed because a centrifugal pump cannot lift air. Reciprocating pumps deliver pulsating flow and suit high heads and small discharges, and an air vessel smooths the flow. Pumps in series add heads and pumps in parallel add discharges.

Common traps

• Using the wrong scaling law. Free-surface models need Froude, not Reynolds.
• Forgetting that $u=V/2$ is an ideal-wheel result for full deflection.
• Applying affinity laws across different diameters without the diameter terms.
• Treating a choked nozzle as still speeding up with lower back pressure.
• Mixing the two specific speeds. One is for turbines and one for pumps.

Memory aids

• "Pelton high, Kaplan low": head ranges.
• "Q, N, N squared, N cubed": pump laws.
• "Froude for free surface": similitude choice.

Summary

The momentum equation gives jet forces and underlies turbine theory, while dimensionless groups decide which model law applies. Boundary layers explain friction and drag, and compressible flow adds Mach-number relations and choking.

Pelton, Francis and Kaplan turbines are matched to head and flow by specific speed, and centrifugal pump behaviour follows the Euler head, affinity laws and NPSH.

Exam protocol

• Identify the dominant force before choosing a similitude law.
• Write the control volume before applying momentum.
• Check NPSH and cavitation for every pump or reaction turbine question.
• Use consistent SI units for all turbine power calculations.