Hydraulics & Fluid Mechanics Formulas

🧪 Fluid Properties

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Density Fundamental

ρ = m / V

Mass per unit volume. Fundamental fluid property.

ρ = Density (kg/m³)

m = Mass

V = Volume

Water: ρ ≈ 1000 kg/m³ | Air (20°C): ρ ≈ 1.204 kg/m³

Specific Weight

γ = ρ × g

Weight per unit volume. Also called unit weight.

γ = Specific weight (N/m³)

g = Acceleration due to gravity (9.81 m/s²)

Water: γ ≈ 9810 N/m³ = 9.81 kN/m³

Dynamic Viscosity (Newton's Law)

τ = μ × (du/dy)

Shear stress proportional to velocity gradient. Newtonian fluids obey this linearly.

τ = Shear stress

μ = Dynamic (absolute) viscosity (Pa·s)

du/dy = Velocity gradient

Kinematic Viscosity

ν = μ / ρ

Ratio of dynamic viscosity to density. Governs flow behavior independent of density.

ν = Kinematic viscosity (m²/s)

Water (20°C): ν ≈ 10⁻⁶ m²/s | Air (20°C): ν ≈ 1.5 × 10⁻⁵ m²/s

Bulk Modulus of Elasticity

K = −ΔP / (ΔV/V)

Resistance of fluid to compression. Inverse of compressibility.

K = Bulk modulus (Pa)

ΔP = Pressure change

ΔV/V = Volumetric strain

Water: K ≈ 2.2 × 10⁹ Pa (effectively incompressible for most engineering)

Surface Tension

σ = F / L

Force per unit length acting at the interface between two fluids.

σ = Surface tension (N/m)

F = Force

L = Length

Water-air (20°C): σ ≈ 0.0728 N/m. Capillary rise: h = (4σ cosθ)/(γd)

🌊 Hydrostatics & Pressure

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Hydrostatic Pressure Fundamental

P = ρ × g × h = γ × h

Pressure increases linearly with depth in a static fluid.

P = Gauge pressure (Pa)

h = Depth below free surface

Absolute Pressure

P_abs = P_atm + P_gauge

Sum of atmospheric and gauge pressures.

P_atm = Atmospheric pressure (≈ 101.325 kPa)

Hydrostatic Force on Plane Surface

F = γ × h̄ × A

Total force on a submerged plane area. Acts at centre of pressure.

h̄ = Depth of centroid of the area

A = Area of surface

Centre of Pressure

h* = h̄ + I_G / (A × h̄)

Point where resultant hydrostatic force acts on a submerged surface.

h* = Depth of centre of pressure

I_G = MOI of area about its own centroidal axis

Centre of pressure is always below the centroid

Buoyancy Force (Archimedes)

F_B = γ_f × V_disp

Upward force equal to weight of displaced fluid.

γ_f = Specific weight of fluid

V_disp = Volume of fluid displaced

Metacentric Height (GM)

GM = BM − BG (for floating body)

Stability criterion. Positive GM = stable. Negative = unstable.

BM = I_WP / V_disp

I_WP = MOI of waterplane area

BG = Distance between centre of buoyancy and centre of gravity

🔀 Flow Kinematics & Dynamics

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Continuity Equation Conservation of Mass

A₁V₁ = A₂V₂ = Q

For incompressible steady flow, discharge is constant across sections.

A = Cross-sectional area

V = Mean velocity

Q = Discharge (m³/s)

Reynolds Number Flow Regime

R_e = (ρ × V × D) / μ = (V × D) / ν

Dimensionless number indicating laminar vs turbulent flow regime.

D = Characteristic length (pipe diameter)

Pipe: R_e < 2000 Laminar | R_e > 4000 Turbulent | 2000–4000 Transitional

Bernoulli's Equation Energy

P/γ + V²/(2g) + z = Constant

Sum of pressure, kinetic, and potential energy heads remains constant along a streamline (ideal, incompressible, steady flow).

P/γ = Pressure head

V²/(2g) = Velocity head

z = Elevation head

Modified Bernoulli (Real Fluid)

P₁/γ + V₁²/(2g) + z₁ = P₂/γ + V₂²/(2g) + z₂ + h_L

Includes head loss term h_L to account for viscous friction.

h_L = Total head loss between sections 1 and 2

Euler's Equation of Motion

(1/ρ)(∂P/∂s) + V(∂V/∂s) + g(∂z/∂s) = 0

Differential form of Bernoulli along a streamline for inviscid flow.

s = Direction along streamline

🔧 Pipe Flow & Friction Losses

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Darcy-Weisbach Equation Major Loss

h_f = (f × L × V²) / (2 × g × D)

Head loss due to friction in a pipe. The most general and accurate formula.

f = Darcy friction factor (from Moody chart)

L = Length of pipe

D = Diameter

V = Mean velocity

Friction Factor (Laminar)

f = 64 / R_e

Darcy friction factor for laminar flow (R_e < 2000). Exact solution from Hagen-Poiseuille.

Hazen-Williams Equation

V = 0.85 × C_HW × R^0.63 × S^0.54

Empirical formula for water supply design. Easier than Darcy-Weisbach but less accurate.

C_HW = Hazen-Williams coefficient (material dependent)

R = Hydraulic radius = D/4 for pipes

S = Slope of energy line = h_f/L

New CI pipe: C ≈ 130 | Old corroded: C ≈ 60–80

Minor Losses (Fittings)

h_m = K × V² / (2g)

Head loss through valves, bends, expansions, contractions.

K = Loss coefficient (empirical, depends on fitting type)

90° elbow: K ≈ 0.9 | Gate valve (open): K ≈ 0.2 | Sudden expansion: K = (1 − A₁/A₂)²

Hardy Cross Method

ΔQ = −Σh / (n × Σ(h/Q))

Iterative method for balancing flows in pipe networks. Corrects assumed flow ΔQ per loop.

h = Head loss in each pipe (h = K × Qⁿ)

n = Exponent (≈ 2 for Darcy-Weisbach)

🏞️ Open Channel Flow

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Manning's Equation Uniform Flow

V = (1/n) × R^2/3 × S^1/2

Most widely used empirical equation for steady uniform flow in open channels.

V = Mean velocity (m/s)

n = Manning's roughness coefficient

R = Hydraulic radius = A/P

S = Bed slope (m/m)

Concrete: n ≈ 0.013 | Earth channel: n ≈ 0.025 | Mountain stream: n ≈ 0.05

Discharge (Manning)

Q = (1/n) × A × R^2/3 × S^1/2

Volume flow rate using Manning's equation.

A = Cross-sectional area of flow

Froude Number Flow Type

F_r = V / √(g × D)

Ratio of inertial to gravitational forces. Determines subcritical vs supercritical flow.

D = Hydraulic depth = A / T (top width)

F_r < 1 Subcritical (tranquil) | F_r = 1 Critical | F_r > 1 Supercritical (rapid)

Specific Energy

E = y + V² / (2g) = y + Q² / (2g × A²)

Energy per unit weight measured relative to the channel bed.

y = Depth of flow

Critical depth (minimum energy): E_min = (3/2)y_c

Critical Depth (Rectangular)

y_c = (q² / g)^1/3

Depth at which specific energy is minimum for a given discharge in rectangular channel.

q = Discharge per unit width = Q/b

Chezy's Formula

V = C × √(R × S)

Earlier empirical formula for open channel velocity. C depends on roughness and hydraulic radius.

C = Chezy coefficient

Relation: C = (1/n) × R^1/6 (connecting Chezy and Manning)

💧 Orifices, Notches & Weirs

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Orifice Discharge (Tank)

Q = C_d × A × √(2gH)

Discharge through a sharp-edged orifice under head H.

C_d = Coefficient of discharge (≈ 0.62 for sharp orifice)

A = Area of orifice

H = Head above centre of orifice

Time to Empty Tank (Orifice)

T = (2A_t / C_dA_o√(2g)) × √H₁

Time for a tank of area A_t to empty from initial head H₁ through orifice of area A_o.

A_t = Area of tank (constant cross-section)

A_o = Area of orifice

Rectangular Weir (Sharp Crested)

Q = (2/3) × C_d × L × √(2g) × H^3/2

Discharge over a rectangular weir of length L under head H.

L = Length of weir crest

H = Head over the crest

Francis formula: Q = 1.84 × (L − 0.2H) × H^3/2 (with end contractions)

Triangular (V-Notch) Weir

Q = (8/15) × C_d × tan(θ/2) × √(2g) × H^5/2

Discharge over a V-notch. More sensitive at low flows than rectangular weirs.

θ = Vertex angle of notch

H = Head above vertex

For 90° V-notch: Q ≈ 1.38 × H^5/2

Broad-Crested Weir

Q = 1.705 × C_d × L × H^3/2

Discharge over a broad-crested weir where critical flow occurs on the crest.

C_d ≈ 0.85 (typical)

⚙️ Hydraulic Machines

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Pump Power

P = γ × Q × H_m / η

Shaft power required to drive a pump delivering discharge Q at total head H_m.

H_m = Manometric head (total head developed)

η = Overall efficiency

Specific Speed (Pump)

N_s = (N × √Q) / H^3/4

Dimensionless (or dimensional) number for pump selection. Determines impeller type.

N = Speed (rpm)

Q = Discharge (m³/s)

H = Head per stage (m)

Centrifugal: N_s ≈ 10–80 | Mixed flow: 80–160 | Axial: 160–400+

Turbine Power

P = η × γ × Q × H

Power output from a hydraulic turbine under head H with discharge Q.

η = Overall efficiency

H = Net available head

Specific Speed (Turbine)

N_s = (N × √P) / H^5/4

Classifies turbine type. P in kW (metric).

P = Output power (kW)

Pelton: N_s ≈ 8–30 | Francis: 50–300 | Kaplan: 300–800

Affinity Laws (Pump)

Q ∝ N | H ∝ N² | P ∝ N³

Effect of speed change on pump performance. Same pump, different speeds.

N = Rotational speed

Also for diameter: Q ∝ D, H ∝ D², P ∝ D³ (same speed, different impeller)

📐 Dimensional Analysis & Similitude

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Froude Number (Model Scaling)

(F_r)_m = (F_r)_p

For free-surface flow models (spillways, weirs, channels). Gravity forces dominate.

Scale ratios: V_r = √L_r | Q_r = L_r^5/2 | t_r = √L_r

Reynolds Number (Model Scaling)

(R_e)_m = (R_e)_p

For enclosed flow models (pipes, valves, aircraft). Viscous forces dominate.

Scale ratios: V_r = L_r⁻¹ × (ν_r) | Q_r = L_r × ν_r

Conflict: Cannot simultaneously satisfy both Froude and Reynolds similarity in the same model (unless same fluid at L_r=1)

Weber Number

W_e = ρVL / σ

Ratio of inertial to surface tension forces. Important for thin jets, droplets, sprays.

σ = Surface tension

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Hydraulics & Fluid Mechanics
Formula Reference

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Formula Reference