Advanced Civil Engineering Handbook › Chapter 7

Chapter 7 of 14 · 40 formulas · 76 one-liners

Fluid Mechanics & Hydraulic Machines — formulas & one-liners

Fluid properties, hydrostatics, flow measurement, pipe and open channel flow, boundary layer, model laws, and turbines and pumps — the core formulas and facts asked in AE/JE civil exams.

7.1 Fluid properties

Newton's law of viscosity

τ = μ \frac{d u}{d y}

τ = shear stress (Pa); μ = dynamic viscosity (Pa·s); du/dy = velocity gradient (1/s)

Fluids obeying this linear law are Newtonian

Kinematic viscosity

ν = \frac{μ}{ρ}

ν = kinematic viscosity (m²/s); ρ = mass density (kg/m³)

Excess pressure inside a drop / bubble / jet

Δ p_{d r o p} = \frac{4 σ}{d}, Δ p_{b u bb l e} = \frac{8 σ}{d}, Δ p_{j e t} = \frac{2 σ}{d}

σ = surface tension (N/m); d = diameter (m)

A soap bubble has two free surfaces, hence 8σ/d

Capillary rise (or fall) in a tube

h = \frac{4 σ cos θ}{ρ g d}

σ = surface tension; θ = contact angle; d = tube diameter

θ ≈ 0° for water–glass (rise); θ > 90° for mercury–glass (depression)

SI unit of dynamic viscosity is Pa·s (N·s/m²); 1 poise = 0.1 Pa·s.
SI unit of kinematic viscosity is m²/s; 1 stoke = 1 cm²/s = 10⁻⁴ m²/s.
Viscosity of liquids decreases with rise in temperature, whereas viscosity of gases increases.
Bingham plastics (e.g. sewage sludge, toothpaste, drilling mud) need a yield stress before they start to flow.
Pseudo-plastic fluids (e.g. blood, paper pulp, polymer solutions) show apparent viscosity decreasing with shear rate; dilatant fluids show it increasing.
Surface tension of water in contact with air at about 20 °C is roughly 0.073 N/m.
Bulk modulus of water is about 2.1–2.2 GPa, so water is treated as incompressible in most hydraulic problems.
Cavitation begins when the local absolute pressure in a liquid falls to its vapour pressure.

7.2 Hydrostatics, manometers & buoyancy

Hydrostatic pressure at depth h

p = ρ g h

p = gauge pressure (Pa); ρ = density; h = depth below free surface (m)

Total pressure force on a submerged plane surface

F = ρ g A \overset{ˉ}{h}

A = area of surface; h̄ = depth of centroid below free surface

Depth of centre of pressure (plane inclined at θ to free surface)

h^{*} = \overset{ˉ}{h} + \frac{I _{G} sin ^{2} θ}{A h ˉ}

I_G = second moment of area about centroidal axis parallel to free surface; θ = 90° for a vertical surface

Differential U-tube manometer (heavier gauge liquid)

h = x (\frac{S _{m}}{S} - 1)

h = piezometric head difference in m of flowing liquid; x = manometer deflection; S_m, S = specific gravity of manometric and flowing liquid

Metacentric height

GM = \frac{I}{V} - B G

I = least moment of inertia of waterline plane; V = volume displaced; BG = distance between centre of buoyancy and centre of gravity

Use BM = I/V; GM = BM − BG when G is above B

Pascal's law: in a fluid at rest, the intensity of pressure at a point is the same in all directions.
Standard atmospheric pressure = 101.325 kPa = 10.33 m of water = 760 mm of mercury.
Absolute pressure = atmospheric pressure + gauge pressure; vacuum pressure is a negative gauge pressure.
Centre of pressure of a submerged plane surface always lies below its centroid, except for a horizontal surface where they coincide.
For a vertical rectangular plate with its top edge at the free surface, the centre of pressure is at two-thirds of the depth.
Horizontal component of hydrostatic force on a curved surface equals the force on its vertical projection; the vertical component equals the weight of liquid vertically above it.
Buoyant force equals the weight of fluid displaced and acts through the centre of buoyancy, the centroid of the displaced volume (Archimedes' principle).
A floating body is stable if the metacentre lies above its centre of gravity, neutral if they coincide and unstable if M lies below G.
A fully submerged body is stable only if its centre of buoyancy lies above its centre of gravity.

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7.3 Kinematics, Bernoulli & flow measurement

Continuity equation (incompressible)

Q = A_{1} V_{1} = A_{2} V_{2}

Q = discharge (m³/s); A = area; V = mean velocity

Bernoulli's equation (per unit weight)

\frac{p}{ρ g} + \frac{V ^{2}}{2 g} + z = constant

p/ρg = pressure head; V²/2g = velocity head; z = datum head (all in m)

Steady, incompressible, inviscid flow along a streamline

Discharge through a venturimeter / orifice meter

Q = \frac{C _{d} a _{1} a _{2} 2 g h}{a _{1}^{2} - a _{2}^{2}}

a₁, a₂ = areas at inlet and throat (or orifice); h = piezometric head difference (m of flowing fluid)

C_d of venturimeter ≈ 0.95–0.99; of orifice meter ≈ 0.6–0.65

Velocity by Pitot tube

V = C_{v} 2 g h

h = difference between stagnation and static head (m of flowing fluid); C_v ≈ 0.98

Rectangular notch / sharp-crested weir

Q = \frac{2}{3} C_{d} L 2 g H^{3/2}

L = crest length; H = head over crest (m)

Triangular (V) notch

Q = \frac{8}{15} C_{d} 2 g tan \frac{θ}{2} H^{5/2}

θ = included angle of notch; H = head over vertex

Francis formula for rectangular weir with end contractions

Q = 1.84 (L - 0.1 n H) H^{3/2}

n = number of end contractions; L, H in m; Q in m³/s

Steady flow: fluid properties at a point do not change with time; uniform flow: velocity does not change with position along the flow.
In steady flow, streamlines, pathlines and streaklines coincide.
A stream function exists for any two-dimensional incompressible flow, but a velocity potential exists only for irrotational flow.
Streamlines and equipotential lines intersect at right angles, which is the basis of a flow net.
In a free vortex the product of tangential velocity and radius is constant (Vr = C), while in a forced vortex the velocity varies directly with radius (V = ωr).
Kinetic energy correction factor α is 2.0 and momentum correction factor β is 4/3 for laminar flow in a circular pipe.
In a venturimeter the convergent cone angle is about 20–22° and the divergent cone angle about 5–7°, to limit flow separation.
For a sharp-edged orifice, typical values are C_c ≈ 0.64, C_v ≈ 0.97 and C_d = C_c × C_v ≈ 0.62; the vena contracta lies about d/2 downstream of the orifice.
An external cylindrical mouthpiece has C_d ≈ 0.855, and a Borda (internal) mouthpiece running free has C_c = 0.5.
A 1% error in measuring head gives about 1.5% error in discharge for a rectangular notch and about 2.5% for a triangular notch.
A V-notch is preferred for measuring small discharges because the head is larger for a given flow.
A Cipolletti weir is a trapezoidal weir with side slopes of 1 horizontal to 4 vertical, which offsets the effect of end contractions.

7.4 Flow through pipes

Darcy–Weisbach head loss

h_{f} = \frac{f L V ^{2}}{2 g D}

f = Darcy friction factor; L = pipe length; D = diameter; V = mean velocity

Many Indian texts write h_f = 4f′LV²/(2gD) with f′ = Fanning coefficient = f/4

Friction factor – laminar and smooth turbulent (Blasius)

f_{l am} = \frac{64}{R e}, f_{B l a s i u s} = \frac{0.316}{R e ^{1/4}}

Re = ρVD/μ; Blasius valid for smooth pipes, 4000 < Re < 10⁵

Hagen–Poiseuille equation (laminar pipe flow)

h_{f} = \frac{32 μ V L}{ρ g D ^{2}}

μ = dynamic viscosity; V = mean velocity; L = length; D = diameter

Head loss varies directly with velocity in laminar flow

Loss at sudden enlargement

h_{L} = \frac{( V _{1} - V _{2} ) ^{2}}{2 g}

V₁, V₂ = velocities in smaller and larger pipe

Power transmitted through a pipe

P = ρ g Q (H - h_{f})

H = total head at inlet; h_f = friction loss

Efficiency (H − h_f)/H; maximum power when h_f = H/3, giving η_max = 66.7%

Water hammer pressure rise – rapid (instantaneous) closure

p = ρ V C, C = \frac{K}{ρ}

V = velocity destroyed; C = celerity of pressure wave; K = bulk modulus (rigid pipe)

Closure is 'rapid' if valve closing time t < 2L/C

Flow in a pipe is generally laminar for Re < 2000 and turbulent for Re > 4000.
In laminar pipe flow the velocity profile is parabolic and the maximum (centre-line) velocity is twice the mean velocity.
In laminar pipe flow shear stress varies linearly from zero at the centre to a maximum at the wall.
For turbulent pipe flow following the 1/7th power law, the mean velocity is about 0.82 times the centre-line velocity.
In the fully rough zone of the Moody chart the friction factor depends only on relative roughness and not on Reynolds number.
A pipe surface is hydraulically smooth when the roughness projections are buried within the laminar sub-layer.
Loss at pipe entrance is about 0.5 V²/2g and loss at exit into a reservoir is V²/2g.
For pipes in series the discharge is the same and the head losses add; for pipes in parallel the head loss is the same and the discharges add.
Celerity of a pressure wave in water in a rigid pipe is about 1440 m/s; pipe elasticity reduces it.
Surge tanks are provided on long pipelines and penstocks to absorb water hammer pressure and to store or supply water during load changes.

7.5 Boundary layer, drag & dimensional analysis

Laminar boundary-layer thickness on a flat plate (Blasius)

δ = \frac{5 x}{R e _{x}}

x = distance from leading edge; Re_x = Ux/ν

Exact Blasius constant 4.91; 5.0 commonly used

Drag force on a body

F_{D} = C_{D} A \frac{ρ U ^{2}}{2}

C_D = drag coefficient; A = projected area normal to flow; U = approach velocity

Stokes' law for a sphere (creeping flow)

F_{D} = 3 π μ D V, C_{D} = \frac{24}{R e}

D = sphere diameter; V = velocity; valid for Re < 1

Froude model law – scale ratios

V_{r} = L_{r}, T_{r} = L_{r}, Q_{r} = L_{r}^{5/2}, P_{r} = L_{r}^{7/2}

L_r = length scale ratio (prototype/model); V, T, Q, P = velocity, time, discharge, power

Force ratio F_r = L_r³ (same fluid)

Boundary-layer thickness is the distance from the wall at which the velocity reaches 99% of the free-stream velocity.
On a flat plate the laminar boundary layer becomes turbulent at a critical Reynolds number Re_x of about 5 × 10⁵.
Laminar boundary-layer thickness grows as x^(1/2), whereas turbulent boundary-layer thickness grows as x^(4/5).
Shape factor H = δ*/θ is about 2.6 for a laminar (Blasius) boundary layer and about 1.3–1.4 for a turbulent one.
Boundary-layer separation occurs under an adverse pressure gradient (dp/dx > 0), at the point where the velocity gradient at the wall becomes zero.
For streamlined bodies skin-friction drag dominates, whereas for bluff bodies pressure (form) drag dominates.
Buckingham π theorem: n variables with m fundamental dimensions form (n − m) independent dimensionless groups.
Reynolds number = inertia/viscous force; Froude number = √(inertia/gravity force); Weber number relates inertia to surface tension; Mach number relates inertia to elastic force.
Froude's law governs models of free-surface flows such as spillways, weirs, open channels and ship waves.
Reynolds' law governs models of fully enclosed flows such as pipes, submarines and low-speed aircraft.
Under Reynolds' law with the same fluid in model and prototype, the velocity ratio is the inverse of the length ratio (V_r = 1/L_r).

7.6 Open channel flow

Chezy formula

V = C R S

C = Chezy coefficient (m^1/2/s); R = hydraulic radius A/P (m); S = bed slope

Manning formula

V = \frac{1}{n} R^{2/3} S^{1/2}

n = Manning roughness coefficient; R in m; V in m/s

Relation with Chezy: C = R^1/6/n

Specific energy

E = y + \frac{V ^{2}}{2 g} = y + \frac{q ^{2}}{2 g y ^{2}}

y = depth of flow; q = discharge per unit width (rectangular channel)

Critical depth in a rectangular channel

y_{c} = (\frac{q ^{2}}{g})^{1/3}, E_{min} = \frac{3}{2} y_{c}

q = discharge per unit width (m²/s)

Froude number of channel flow

F_{r} = \frac{V}{g D}

D = hydraulic depth = A/T (T = top width); D = y for a rectangle

Conjugate (sequent) depths of a hydraulic jump – rectangular channel

\frac{y _{2}}{y _{1}} = \frac{1}{2} (1 + 8 F_{1}^{2} - 1)

y₁, F₁ = depth and Froude number before the jump; y₂ = depth after

Energy loss in a hydraulic jump

Δ E = \frac{( y _{2} - y _{1} ) ^{3}}{4 y _{1} y _{2}}

y₁, y₂ = sequent depths

Gradually varied flow equation

\frac{d y}{d x} = \frac{S _{0} - S _{f}}{1 - F _{r}^{2}}

S₀ = bed slope; S_f = energy (friction) slope

Most economical rectangular channel section has width equal to twice the depth (b = 2y) and R = y/2.
Most economical trapezoidal section is half a regular hexagon, with sides at 60° to horizontal and R = y/2.
Most economical triangular channel has side slopes of 1:1 (45° to vertical).
In a circular channel, maximum velocity occurs at a depth of about 0.81D and maximum discharge (Chezy) at about 0.95D.
Flow is subcritical (tranquil) when F_r < 1, critical when F_r = 1 and supercritical (rapid or shooting) when F_r > 1.
For a given discharge, specific energy is minimum at critical depth; for a given specific energy, discharge is maximum at critical depth.
At critical flow in a rectangular channel the velocity head equals half the critical depth.
A hydraulic jump forms only when flow changes from supercritical to subcritical.
Jumps are classified by F₁: undular 1–1.7, weak 1.7–2.5, oscillating 2.5–4.5, steady 4.5–9 and strong above 9.
The length of a hydraulic jump is commonly taken as about 5 to 7 times its height (y₂ − y₁).
The M1 profile (backwater curve) forms upstream of a dam or weir on a mild slope; M2 forms near a free overfall; M3 forms downstream of a sluice gate.
Mild slope means normal depth exceeds critical depth; steep slope means normal depth is less than critical depth.
Mean velocity in a stream is approximately the velocity at 0.6 depth below the surface, or the average of velocities at 0.2 and 0.8 depth.

7.7 Impact of jets, turbines & pumps

Force of a jet on a flat plate normal to the jet

F_{f i x e d} = ρ a V^{2}, F_{m o v in g} = ρ a (V - u)^{2}

a = jet area; V = jet velocity; u = plate velocity in jet direction

Euler's head (work done per unit weight)

H_{e} = \frac{V _{w 1} u _{1} \pm V _{w 2} u _{2}}{g}

V_w = whirl velocity; u = blade velocity; 1 = inlet, 2 = outlet

For a centrifugal pump with radial entry, H_e = V_w2u₂/g

Specific speed of a turbine

N_{s} = \frac{N P}{H ^{5/4}}

N = speed (rpm); P = shaft power (kW); H = net head (m)

Specific speed of a pump

N_{s} = \frac{N Q}{H ^{3/4}}

N = speed (rpm); Q = discharge (m³/s); H = manometric head (m)

Overall efficiency

η_{o} = η_{h} \times η_{m}

η_h = hydraulic efficiency; η_m = mechanical efficiency

Turbine: η_o = shaft power/(ρgQH)

Thoma's cavitation factor

σ = \frac{H _{a} - H _{v} - H _{s}}{H}

H_a = atmospheric head; H_v = vapour pressure head; H_s = suction head (height of runner above tailrace); H = net head

Cavitation occurs if σ falls below the critical value σ_c

For a jet striking a series of flat vanes on a wheel, the maximum efficiency is 50%, reached when vane velocity is half the jet velocity.
Pelton wheel is a tangential-flow impulse turbine for high heads and small discharges.
Francis turbine is a mixed (inward radial) flow reaction turbine for medium heads; Kaplan is an axial-flow reaction turbine with adjustable runner blades for low heads and large discharges.
Specific speed ranges (metric, kW): Pelton about 8.5–30 (single jet), Francis about 51–255, Kaplan/propeller about 255–860.
Maximum hydraulic efficiency of a Pelton wheel occurs theoretically when bucket speed is half the jet speed; in practice the speed ratio is about 0.46.
A draft tube is used with reaction turbines to recover kinetic energy at exit and to allow the runner to be set above tail-water level; it is not used with a Pelton wheel.
In reaction turbines cavitation is most likely at the runner outlet and draft-tube inlet; in centrifugal pumps it is most likely at the impeller eye (inlet).
A centrifugal pump must be primed before starting because the pressure it develops is proportional to the density of the fluid in the impeller.
Backward-curved vanes are generally used in centrifugal pump impellers because they give a stable, non-overloading characteristic.
Pumps are connected in series to increase the head and in parallel to increase the discharge.
For a given pump, discharge varies as N, head as N² and power as N³ (affinity laws).
NPSH available must exceed NPSH required to avoid cavitation in a pump.
Air vessels on a reciprocating pump reduce acceleration head and save about 84.8% of the work against friction for a single-acting pump.

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Values follow the current Indian Standards and IRC / Indian Railways documents named in each item. Always check the latest amendment of a code before using a value in design.