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Lab · Fluid Mechanics & Hydraulics

Flow, pressure & hydraulic machines.

Six working benches in the IIT Virtual Labs format. Pick an experiment, read its aim, theory and procedure, then drive the live apparatus — every slider re-solves the underlying fluid mechanics from continuity, Bernoulli, Reynolds and Darcy-Weisbach — and test yourself with the self-assessment.

Bernoulli & continuity in a varying-area pipe

Experiment 1 · Energy & mass conservation

Converging-diverging pipe with piezometer tubes

Particles advect at the local velocity (continuity: they speed up through the throat). Piezometer column height = static pressure head p / rho g; the throat column drops as the flow accelerates.

Head distribution along the pipe (m)

Energy grade line & hydraulic grade line

To verify the continuity equation and Bernoulli's energy equation for the steady, incompressible flow of a fluid through a converging-diverging pipe, and to demonstrate that where the cross-section narrows the velocity rises and the static pressure falls.

For steady incompressible flow the mass flow rate is the same at every section, so the volume flow Q = A·v is constant: a smaller area must carry a faster velocity. Bernoulli's equation expresses conservation of mechanical energy per unit weight along a streamline — the sum of pressure head, velocity head and elevation head is constant for an ideal (frictionless) fluid.

Continuity:  A1 v1 = A2 v2 = Q
Bernoulli:  p1/(rho g) + v1^2/(2g) + z1 = p2/(rho g) + v2^2/(2g) + z2
Therefore:  p1 - p2 = (rho/2)(v2^2 - v1^2) + rho g (z2 - z1)

At a 2:1 area reduction (A2 = A1/2) the velocity doubles (v2 = 2 v1) and the pressure head falls by (v2^2 - v1^2)/2g = 3 v1^2 / 2g. If the throat pressure approaches the vapour pressure of the liquid, cavitation can occur.

  1. Choose the working fluid and set the inlet and throat diameters.
  2. Set the volume flow rate Q; observe the particles speed up through the throat.
  3. Read the velocities v1 and v2 and confirm A1 v1 = A2 v2 = Q.
  4. Read the throat pressure p2 from Bernoulli and compare the piezometer column heights.
  5. Raise the flow until the cavitation indicator warns that p2 nears the vapour pressure.
  6. Tilt the throat elevation z2 and watch the elevation-head term shift the energy line.
  • F. M. White, Fluid Mechanics, 8th ed., McGraw-Hill, Ch. 3 (Bernoulli equation).
  • B. R. Munson et al., Fundamentals of Fluid Mechanics, Wiley, Ch. 3.
  • IIT Virtual Labs — Fluid Mechanics: Bernoulli's Theorem, vlabs.ac.in.

Reynolds experiment: laminar vs turbulent flow

Experiment 2 · Flow regimes

Glass tube with dye filament (Reynolds apparatus)

A fine dye jet enters the tube. At low Re it stays a straight thread (laminar); near the critical value it wavers (transitional); at high Re it bursts and mixes across the tube (turbulent). The animation samples the live Re.

Velocity profile across the tube

Laminar flow has a parabolic profile (Hagen-Poiseuille); turbulent flow has a flatter, fuller profile from cross-stream momentum mixing. Re = rho v D / mu.

To reproduce Osborne Reynolds' classic experiment — injecting a dye filament into flow through a glass tube — and to identify the laminar, transitional and turbulent regimes, relating the change of regime to the dimensionless Reynolds number.

The Reynolds number is the ratio of inertial forces to viscous forces in a flow. When viscous forces dominate (low Re) disturbances are damped and the flow is orderly and laminar; the dye stays a clean thread. When inertia dominates (high Re) small disturbances grow, the flow becomes chaotic and turbulent, and the dye mixes across the whole tube.

Re = rho v D / mu = v D / nu
Laminar:  Re < 2300 ·  Transitional:  2300 to 4000 ·  Turbulent:  Re > 4000
Critical velocity:  v_c = Re_c · mu / (rho D)

Here rho and mu are taken at the chosen temperature; water's viscosity falls steeply as it warms, so the same velocity gives a higher Re in hot water.

  1. Select the fluid and set the tube diameter.
  2. Start from a low velocity and watch the dye stay a straight filament (laminar).
  3. Raise the velocity until the thread begins to waver near Re = 2300 (transitional).
  4. Increase further until the dye disperses across the tube (turbulent, Re > 4000).
  5. Change the temperature and note how the critical velocity shifts with viscosity.
  6. Compare the laminar parabolic profile with the fuller turbulent profile.
  • O. Reynolds, An Experimental Investigation, Phil. Trans. R. Soc., 1883.
  • F. M. White, Fluid Mechanics, 8th ed., McGraw-Hill, Ch. 6 (Viscous flow in pipes).
  • IIT Virtual Labs — Fluid Mechanics: Reynolds Experiment, vlabs.ac.in.

Pipe friction & head loss (Darcy-Weisbach)

Experiment 3 · Major losses

Moody chart · friction factor vs Reynolds number

Log-log Moody diagram. The cross marks your operating point. Laminar uses f = 64/Re; turbulent uses the Swamee-Jain explicit form of the Colebrook equation for the chosen relative roughness.

Head loss vs flow rate

hf = f (L/D)(v^2 / 2g) ·  laminar f = 64/Re ·  turbulent 1/sqrt(f) = -2 log10(e/3.7D + 2.51/(Re sqrt(f))) (Colebrook).

To measure the major head loss due to friction in a uniform circular pipe as a function of flow rate, to determine the Darcy friction factor from the Reynolds number and relative roughness, and to locate the operating point on the Moody chart.

As a real fluid flows through a pipe, wall shear dissipates mechanical energy; the lost energy per unit weight is the friction head loss. The Darcy-Weisbach equation relates this loss to the velocity head through a dimensionless friction factor f, which itself depends on the flow regime. In laminar flow f follows directly from the Hagen-Poiseuille solution; in turbulent flow it depends on both Reynolds number and the pipe's relative roughness, captured by the Colebrook equation (plotted as the Moody chart).

Darcy-Weisbach:  hf = f (L/D)(v^2 / 2g)
Laminar (Re < 2300):  f = 64 / Re
Turbulent (Swamee-Jain):  f = 0.25 / [log10(e/3.7D + 5.74/Re^0.9)]^2
Pressure drop:  dp = rho g hf

Head loss grows roughly with the square of the velocity (and hence flow rate) in turbulent flow, so doubling the flow quadruples the loss.

  1. Choose the fluid and set the pipe diameter, length and wall roughness.
  2. Set the flow rate Q; read the mean velocity and Reynolds number.
  3. Note whether the flow is laminar or turbulent and which friction-factor law applies.
  4. Read the friction factor and the resulting Darcy-Weisbach head loss.
  5. Locate the operating point on the Moody chart and follow it as Q changes.
  6. Increase the roughness and observe f rise toward its fully-rough plateau.
  • L. F. Moody, Friction Factors for Pipe Flow, Trans. ASME, 1944.
  • P. K. Swamee & A. K. Jain, Explicit equations for pipe-flow problems, J. Hydraulics Div., 1976.
  • IIT Virtual Labs — Fluid Mechanics: Friction in Pipes, vlabs.ac.in.

Venturi / orifice flow meter

Experiment 4 · Differential-pressure metering

Flow meter with differential manometer

The constriction accelerates the flow and lowers its pressure; the U-tube manometer reads the head difference. Discharge is recovered from the deflection with the meter's discharge coefficient.

Calibration: discharge vs square-root of head

Q = Cd A2 sqrt( 2 (p1-p2) / (rho (1 - (A2/A1)^2)) ) ·  p1-p2 = (SG_m - 1) rho g h (mercury over water).

To measure volume flow rate with a venturi or orifice-plate meter by reading the differential pressure across the constriction, to apply the meter equation, and to understand the role of the discharge coefficient that corrects the ideal Bernoulli flow for real losses.

A flow meter creates a known constriction. Combining continuity with Bernoulli between the upstream pipe (1) and the throat (2) gives the ideal discharge in terms of the pressure difference. Real flow separates slightly and loses some energy, so the measured discharge is the ideal value multiplied by a discharge coefficient Cd. A venturi recovers most of its pressure and has Cd near 0.97-0.98; a sharp-edged orifice plate has stronger contraction (the vena contracta) and Cd near 0.6-0.65.

From Bernoulli + continuity:  Q_ideal = A2 sqrt( 2 dp / (rho (1 - beta^4)) ),  beta = D2/D1
Real discharge:  Q = Cd · Q_ideal
Manometer:  dp = (SG_m - 1) rho_water g h

Because Q is proportional to the square root of the head difference, a calibration plot of Q against sqrt(h) is a straight line through the origin.

  1. Select the meter type (venturi or orifice plate) and set the pipe and throat diameters.
  2. Choose the manometer fluid (mercury for large flows, lighter fluids for small).
  3. Set the manometer deflection h, or switch to drive the actual flow Q directly.
  4. Read the pressure difference, the ideal and actual discharge, and the throat velocity.
  5. Compare the discharge coefficient of the venturi with that of the orifice plate.
  6. Confirm that Q rises with the square root of the head, giving a linear calibration.
  • B. R. Munson et al., Fundamentals of Fluid Mechanics, Wiley, Ch. 8 (Flow measurement).
  • ISO 5167 — Measurement of fluid flow by pressure differential devices.
  • IIT Virtual Labs — Fluid Mechanics: Venturimeter & Orifice meter, vlabs.ac.in.

Orifice discharge & tank draining (Torricelli)

Experiment 5 · Free jet & falling head

Draining tank with free jet (side orifice)

The jet leaves at v = sqrt(2 g h) and follows a projectile path; as the head falls the jet slows and its range shortens. The water level drops at dh/dt = -Cd a sqrt(2 g h) / A_t.

Head vs time (draining curve)

v = sqrt(2 g h) (Torricelli) ·  Q = Cd a sqrt(2 g h) ·  time to empty t = (2 A_t / (Cd a sqrt(2g))) sqrt(H0).

To study the discharge of a liquid through a sharp-edged orifice under a gravity head, to verify Torricelli's theorem for the jet velocity, to find the coefficient of discharge, and to compute the time taken to empty a draining tank.

Applying Bernoulli between the free surface and the orifice (both at atmospheric pressure) gives the ideal efflux velocity Torricelli's theorem. Real jets contract at the vena contracta and lose a little energy, so the actual discharge is reduced by the coefficient of discharge Cd. As the tank drains, the head falls, the velocity decreases, and the level descends ever more slowly — the draining time follows by integrating the unsteady mass balance.

Torricelli:  v = sqrt(2 g h)
Discharge:  Q = Cd · a · sqrt(2 g h),  a = orifice area
Unsteady drain:  A_t dh/dt = -Q →  t_empty = (2 A_t / (Cd a sqrt(2g))) sqrt(H0)
Jet range (orifice at height y):  x = 2 sqrt(h · y)

The simulation integrates dh/dt numerically each frame, so the live draining curve reproduces the closed-form sqrt(H0) emptying time.

  1. Set the tank cross-section, the initial head and the orifice diameter.
  2. Set the coefficient of discharge (about 0.62 for a sharp-edged orifice).
  3. Press Open valve and watch the jet issue and the head fall.
  4. Observe the jet velocity following sqrt(2 g h) and its range shrinking as h drops.
  5. Read the time to empty and compare it with the theoretical sqrt(H0) law.
  6. Double the orifice area and confirm the emptying time roughly halves.
  • F. M. White, Fluid Mechanics, 8th ed., McGraw-Hill, Ch. 3 (Frictionless flow).
  • R. K. Bansal, Fluid Mechanics and Hydraulic Machines, Laxmi, Ch. 8 (Orifices).
  • IIT Virtual Labs — Fluid Mechanics: Flow through an Orifice, vlabs.ac.in.

Hydrostatic pressure & U-tube manometer

Experiment 6 · Fluid statics

Apparatus

In depth mode the gauge measures p = p0 + rho g h, rising linearly with depth. In U-tube mode the heavy manometer fluid is displaced by the applied pressure; the deflection reads the head difference.

Pressure variation with depth

p = p0 + rho g h (hydrostatic) ·  U-tube: p = (rho_m - rho_c) g h_def + rho_c g a ·  head = p / rho g.

To study the linear variation of hydrostatic pressure with depth in a static liquid, to express pressures as equivalent fluid heads, and to read a gauge pressure difference with a U-tube manometer using a heavier indicating fluid.

In a fluid at rest the pressure increases linearly with depth because each layer must support the weight of the column above it. The hydrostatic law follows from a force balance on a fluid element. A manometer turns this law into a measuring instrument: the unknown pressure displaces a column of indicating fluid, and the deflection — multiplied by the difference in specific weights — gives the pressure.

Hydrostatic law:  p = p0 + rho g h
Pressure head:  h_p = p / (rho g)
U-tube (heavy fluid):  p = (rho_m - rho_c) g h_def + rho_c g a
Equivalent water column:  h_water = p / (rho_water g)

Mercury (SG 13.6) gives a small, readable deflection for large pressures; lighter oils give a magnified deflection for small pressures.

  1. In depth mode, choose the liquid and move the depth probe; read the gauge pressure.
  2. Confirm the pressure varies linearly with depth from the curve.
  3. Express the pressure as an equivalent head and an equivalent water column.
  4. Switch to U-tube mode and apply a gauge pressure.
  5. Watch the manometer columns displace and read the deflection.
  6. Change the manometer fluid SG and see the deflection scale inversely with its weight.
  • F. M. White, Fluid Mechanics, 8th ed., McGraw-Hill, Ch. 2 (Pressure distribution).
  • B. R. Munson et al., Fundamentals of Fluid Mechanics, Wiley, Ch. 2 (Fluid statics).
  • IIT Virtual Labs — Fluid Mechanics: Manometry, vlabs.ac.in.