Fluid Mechanics Cheatsheet Cheatsheet

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Fluid Properties & Fundamentals

FUNDAMENTAL

Fluid Properties

Density (ρ)mass/volume — kg/m³

Specific weight (γ)ρ·g — N/m³

Specific volume (v)1/ρ — m³/kg

Viscosity (μ)Dynamic viscosity — Pa·s (N·s/m²)

Kinematic visc. (ν)μ/ρ — m²/s

Surface tension (σ)Force per unit length — N/m

Compressibility (β)1/K = fractional vol change per pressure

Bulk modulus (K)−V dP/dV — Pa

Property Values at 20°C

Fluid	ρ (kg/m³)	μ (Pa·s)	ν (m²/s)	K (GPa)
Water	998	1.002×10⁻³	1.004×10⁻⁶	2.15
Air (1 atm)	1.204	1.825×10⁻⁵	1.516×10⁻⁵	—
Mercury	13,546	1.526×10⁻³	1.127×10⁻⁷	25.0
SAE 30 Oil	891	0.29	3.25×10⁻⁴	1.5
Glycerin	1,261	1.412	1.12×10⁻³	4.35
Seawater	1,025	1.08×10⁻³	1.05×10⁻⁶	2.34
Ethanol	789	1.20×10⁻³	1.52×10⁻⁶	1.06
Gasoline	680	2.92×10⁻⁴	4.29×10⁻⁷	1.3

Viscosity & Newton's Law of Viscosity

viscosity.txt

─── Newton's Law of Viscosity ───

  τ = μ × (du/dy)

  τ  = Shear stress (Pa or N/m²)
  μ  = Dynamic viscosity (Pa·s)
  du/dy = Velocity gradient (rate of shear deformation)

Newtonian fluids: μ = constant (independent of shear rate)
  Examples: Water, air, oil, gasoline, mercury

Non-Newtonian fluids: μ varies with shear rate
  • Shear-thinning (pseudoplastic): μ decreases with du/dy
    Examples: Paint, blood, ketchup
  • Shear-thickening (dilatant): μ increases with du/dy
    Examples: Quick-sand, cornstarch suspension
  • Bingham plastic: τ = τ_y + μ_p × (du/dy)
    Examples: Toothpaste, mayonnaise, drilling mud
  • Thixotropic: μ decreases with time under constant shear
  • Rheopectic: μ increases with time under constant shear

No-slip condition: Fluid velocity = 0 at a solid boundary
  (Valid for Newtonian fluids at macroscopic scale)

Surface Tension & Capillarity

surface_tension.txt

─── Surface Tension Effects ───

Capillary rise (wetting liquid, θ < 90°):
  h = 2σ cos θ / (ρ g r)

  σ = surface tension (N/m)
  θ = contact angle between liquid and tube wall
  r = tube radius (m)

  For water-glass: σ ≈ 0.073 N/m, θ ≈ 0°
  Capillary rise: h = 2(0.073)/(998 × 9.81 × r)
    For r = 1 mm: h ≈ 14.9 mm

Capillary depression (non-wetting, θ > 90°):
  Mercury-glass: σ ≈ 0.48 N/m, θ ≈ 140°
  h = 2(0.48)cos(140°)/(13546 × 9.81 × r) → negative (depression)

Pressure inside a droplet (excess pressure):
  ΔP = 2σ / R    (droplet/bubble in liquid)
  ΔP = 4σ / R    (soap bubble, two surfaces)

Compressibility

compressibility.txt

─── Compressibility ───

Bulk modulus:  K = −V (dP/dV)  [Pa]

For liquids (approximately constant K):
  ΔV/V = −ΔP/K

  Water: K ≈ 2.15 GPa
  Mercury: K ≈ 25 GPa
  Oil: K ≈ 1.5 GPa

Speed of sound in a fluid:
  c = √(K/ρ)      [for liquids]
  c = √(γRT)      [ideal gas]
  c = √(γP/ρ)     [ideal gas]

  Water: c ≈ 1480 m/s
  Air at 20°C: c ≈ 343 m/s
  Helium: c ≈ 1007 m/s

Liquids are generally treated as INCOMPRESSIBLE
  (except in water hammer, hydraulic transients)
Gases are COMPRESSIBLE (especially at high Mach).

💡

Ideal vs Real fluid:An ideal (inviscid) fluid has zero viscosity (μ = 0), no friction, and is incompressible. Real fluids have viscosity and exhibit friction. The ideal fluid assumption simplifies analysis (Euler's equation) but cannot model boundary layers, viscous drag, or energy losses in pipes.

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Fluid Statics

STATIC

Pascal's Law & Pressure at a Depth

pressure_depth.txt

─── Pressure at a Point in a Static Fluid ───

  P = P₀ + ρgh

  P₀ = Pressure at the free surface (usually P_atm)
  ρ  = Fluid density (kg/m³)
  g  = 9.81 m/s²
  h  = Depth below free surface (m)

Pascal's Law:
  Pressure applied to a confined fluid is transmitted
  equally and undiminished in ALL directions.

Hydraulic press (Pascal's law application):
  F₁/A₁ = F₂/A₂ = P (same pressure)
  F₂ = F₁ × (A₂/A₁)
  Mechanical advantage = A₂/A₁

Gauge pressure:  P_gauge = P − P_atm  (can be negative)
Absolute pressure: P_abs = P_gauge + P_atm
Vacuum: P_gauge < 0 (absolute pressure below atmospheric)

Standard atmospheric pressure:
  P_atm = 101.325 kPa = 1 atm = 760 mm Hg
         = 10.33 m water = 1.01325 bar

Manometry

manometry.txt

─── Simple Manometer ───

P_A = P_B + ρ_manometer × g × h

  h = difference in manometer levels (m)

U-tube differential manometer:
  P₁ − P₂ = g × h × (ρ_m − ρ_f)

  ρ_m = manometer fluid density
  ρ_f = flowing fluid density
  h   = manometer reading difference

Inverted U-tube:
  P₁ − P₂ = g × h × (ρ_f − ρ_m)
  Used when ρ_m < ρ_f (lighter manometer fluid)

Inclined manometer (improved sensitivity):
  h_vertical = L × sin(θ)
  Where L = reading along tube, θ = inclination angle
  Sensitivity = 1/sin(θ) → greater for small angles

Hydrostatic Forces on Surfaces

hydrostatic_force.txt

─── Force on a Submerged Plane Surface ───

Total hydrostatic force:
  F = ρg × h̄ × A = P̄ × A

  h̄ = depth of centroid of area from free surface
  A = area of the surface

Center of pressure (where resultant force acts):
  h* = h̄ + I_G / (A × h̄)

  I_G = Second moment of area about centroidal axis
  (parallel to the surface)

  h* is always BELOW h̄ (pressure increases with depth)

For vertical rectangular surface (b × D):
  I_G = bD³/12
  h̄ = D/2 (centroid at half depth)

Force on curved surface:
  F_H = ρg × h̄ × A_v  (horizontal component)
  F_V = ρg × V_below  (vertical component = weight of liquid above)

  F = √(F_H² + F_V²)
  tan(θ) = F_V / F_H

hydrostatic_calc.py

import math

# Hydrostatic force on a vertical rectangular gate
b = 2.0    # m (width)
D = 3.0    # m (depth of gate, water surface at top)
rho = 998  # kg/m³
g = 9.81   # m/s²

# Centroid depth
h_bar = D / 2  # 1.5 m

# Force
F = rho * g * h_bar * (b * D)
# F = 998 × 9.81 × 1.5 × 6 = 88,083 N = 88.08 kN

# Center of pressure
I_G = b * D**3 / 12  # 4.5 m⁴
h_star = h_bar + I_G / (b * D * h_bar)
# h* = 1.5 + 4.5 / (6 × 1.5) = 1.5 + 0.5 = 2.0 m

print(f"Force = {F/1e3:.2f} kN, CP depth = {h_star:.2f} m")

Buoyancy & Archimedes' Principle

buoyancy.txt

─── Archimedes' Principle ───

Buoyant force = Weight of fluid displaced
  F_B = ρ_fluid × g × V_displaced

Conditions for stability of floating bodies:
  Metacentric height (GM):
    GM = BM − BG = I/V_submerged − BG

    BM = I / V_sub  (distance from center of buoyancy
                       to metacenter)
    I = Second moment of waterplane area
    V_sub = Volume submerged
    BG = Distance between center of gravity
          and center of buoyancy

  Stable:    GM > 0  (M above G)
  Neutral:   GM = 0
  Unstable:  GM < 0  (M below G)

  Metacentric height for common sections:
  Rectangular: BM = b²/(12d)   (b=width, d=draft)
  Circular:    BM = r²/(4d)    (r=radius, d=draft)

  Recommended: GM > 0.2 m for ships

⚠️

Key distinction: Pressure in a static fluid acts equally in all directions(Pascal's law) andincreases linearly with depth. The pressure at any point depends only on depth, not on the shape of the container. A gauge reads the difference between local pressure and atmospheric pressure.

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Fluid Kinematics

KINEMATICS

Flow Types & Descriptions

Classification	Criteria	Types	Examples
By viscosity effect	Reynolds number	Viscous (real) / Inviscid (ideal)	Pipe flow (viscous) vs flow far from body (inviscid)
By compressibility	Mach number	Incompressible / Compressible	Water (incompressible) vs air at high speed
By time	Dependence on time	Steady / Unsteady	Constant pump (steady) vs surge (unsteady)
By space	Dependence on position	Uniform / Non-uniform	Constant pipe section (uniform) vs nozzle (non-uniform)
By rotationality	Vorticity ω	Irrotational (ω=0) / Rotational	Free vortex (irrotational), forced vortex (rotational)
By randomness	Orderly vs chaotic	Laminar / Turbulent	Re < 2300 (laminar), Re > 4000 (turbulent)

Continuity Equation

continuity.txt

─── Continuity Equation (Conservation of Mass) ───

General form (control volume):
  ∂ρ/∂t + ∇·(ρV) = 0

For incompressible, steady flow:
  A₁V₁ = A₂V₂ = Q = constant

  Q = Volume flow rate (m³/s)
  A = Cross-sectional area (m²)
  V = Average velocity (m/s)

Mass flow rate:
  ṁ = ρ × A × V = ρ × Q  [kg/s]

For compressible flow:
  ρ₁A₁V₁ = ρ₂A₂V₂

Differential form (2D Cartesian):
  ∂u/∂x + ∂v/∂y = 0  (incompressible)

  u = x-velocity component
  v = y-velocity component

Stream Function & Velocity Potential

stream_potential.txt

─── Stream Function (ψ) ───

  u = ∂ψ/∂y,   v = −∂ψ/∂x

  • ψ = constant along a streamline
  • Difference between two streamlines = flow rate
  • Automatically satisfies continuity for 2D
    incompressible flow
  • Exists for both rotational and irrotational flow

─── Velocity Potential (φ) ───

  u = −∂φ/∂x,   v = −∂φ/∂y

  • φ = constant along equipotential lines
  • Streamlines ⊥ equipotential lines
  • Exists ONLY for irrotational flow
  • Automatically satisfies irrotationality

  Laplace equation: ∇²φ = 0 (irrotational + incompressible)

Cauchy-Riemann equations:
  ∂φ/∂x = ∂ψ/∂y  and  ∂φ/∂y = −∂ψ/∂x

Reynolds Number & Flow Regimes

reynolds_number.py

import math

# Reynolds number
# Re = ρVD/μ = VD/ν

# Example: Water flow in pipe
V = 2.0       # m/s (velocity)
D = 0.05      # m (50 mm pipe)
rho = 998     # kg/m³ (water)
mu = 1.002e-3  # Pa·s (water at 20°C)
nu = mu / rho  # kinematic viscosity

Re = rho * V * D / mu
# Re = 998 × 2.0 × 0.05 / 0.001002 = 99,600

if Re < 2300:
    regime = "Laminar"
elif Re < 4000:
    regime = "Transitional"
else:
    regime = "Turbulent"

print(f"Re = {Re:.0f} → {regime}")

# Critical velocities
Re_crit = 2300
V_lam = Re_crit * nu / D   # max laminar velocity
# V_lam = 2300 × 1.004e-6 / 0.05 = 0.0462 m/s

Flow Parameter	Laminar (Re < 2300)	Turbulent (Re > 4000)
Velocity profile	Parabolic (u = umax[1−(r/R)²])	Flatter (logarithmic/power law)
u_avg / u_max	0.5	≈ 0.8 (varies with Re)
Head loss	h_f ∝ V (linear)	h_f ∝ V^(1.75–2.0)
Mixing	Minimal cross-sectional mixing	Intense mixing
Entry length	L_e ≈ 0.05 × Re × D	L_e ≈ 4.4 × Re^(1/6) × D
Friction factor	f = 64/Re (exact)	f depends on Re and ε/D (Moody)

💡

Stream tube: A stream tube is a tube bounded by streamlines. No fluid crosses the walls of a stream tube. The cross-section of a stream tube is called a stream filament. For an incompressible flow, Q = A₁V₁ = A₂V₂ through any stream tube — a narrower section means higher velocity (and vice versa).

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Fluid Dynamics — Euler & Bernoulli

DYNAMICS

Euler's Equation of Motion

euler_equation.txt

─── Euler's Equation (along a streamline) ───

For inviscid, incompressible flow:

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

Along a streamline (s-direction):
  dp/ρ + V·dV + g·dz = 0

Integrating (Euler → Bernoulli):
  P/ρ + V²/2 + gz = constant

Or per unit weight (total head):
  P/(ρg) + V²/(2g) + z = H = constant

  P/(ρg) = Pressure head [m]
  V²/(2g) = Velocity head [m]
  z       = Datum head / Potential head [m]

Bernoulli's Equation

bernoulli.txt

─── Bernoulli's Equation ───

For steady, incompressible, inviscid flow along a streamline:

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

  OR (pressure form):
  P₁ + ½ρV₁² + ρgz₁ = P₂ + ½ρV₂² + ρgz₂

Assumptions:
  1. Steady flow
  2. Incompressible fluid
  3. Inviscid (frictionless)
  4. Flow along a streamline
  5. No energy addition/removal (no pump/turbine between points)

With pump/turbine:
  P₁/ρg + V₁²/2g + z₁ + h_pump = P₂/ρg + V₂²/2g + z₂ + h_turbine + h_loss

  h_pump = head added by pump [m]
  h_turbine = head extracted by turbine [m]
  h_loss = head lost to friction [m]

Bernoulli's Applications

bernoulli_applications.txt

─── Venturimeter (Flow rate measurement) ───

  Q_theoretical = (A₁A₂/√(A₁²−A₂²)) × √(2gh)

  where h = (P₁−P₂)/(ρg) = manometer reading

  Q_actual = C_d × Q_theoretical
  C_d = coefficient of discharge ≈ 0.95–0.99

─── Orifice Meter ───

  Q = C_d × A₀ × √(2ΔP/ρ)

  C_d ≈ 0.60–0.65 (orifice plate)
  A₀ = area of orifice opening

─── Pitot Tube (velocity measurement) ───

  V = √(2(P_stagnation − P_static)/ρ) = √(2gh)

  Measures velocity at a point.
  Stagnation point: V = 0, P = P + ½ρV²

Bernoulli — Numerical Example

bernoulli_example.py

import math

# Venturimeter: water flows through a pipe
D1 = 0.10  # m (pipe diameter)
D2 = 0.06  # m (throat diameter)
h  = 0.15  # m (manometer reading, water)
Cd = 0.98  # discharge coefficient
rho = 998  # kg/m³
g = 9.81

# Areas
A1 = math.pi * D1**2 / 4  # 0.00785 m²
A2 = math.pi * D2**2 / 4  # 0.00283 m²

# Theoretical flow rate
Q_th = (A1 * A2 / math.sqrt(A1**2 - A2**2)) * math.sqrt(2*g*h)
# Q_th = 0.00785×0.00283/√(6.16e-5 − 8.0e-6) × √(2.943)
#      = 2.21e-5 / 0.00732 × 1.7155 = 5.186 L/s

# Actual flow rate
Q_act = Cd * Q_th
V1 = Q_act / A1

print(f"Q = {Q_act*1e3:.3f} L/s")
print(f"V_pipe = {V1:.2f} m/s")
print(f"V_throat = {Q_act/A2:.2f} m/s")

🚫

Modified Bernoulli (with losses): Real fluid flow always involves energy losses due to friction. Add h_L to the downstream side: P₁/ρg + V₁²/2g + z₁ = P₂/ρg + V₂²/2g + z₂ + h_L. For minor losses: h_L = K × V²/2g where K is the loss coefficient (elbow, valve, etc.).

🔧

Pipe Flow & Head Losses

PIPES

Darcy-Weisbach Equation (Major Losses)

darcy_weisbach.txt

─── Darcy-Weisbach Equation ───

  h_f = f × (L/D) × (V²/2g)

  h_f = Head loss due to friction [m]
  f   = Darcy friction factor (dimensionless)
  L   = Pipe length [m]
  D   = Pipe diameter [m]
  V   = Average flow velocity [m/s]
  g   = 9.81 m/s²

Power lost to friction:
  P_loss = ρ × g × Q × h_f  [Watts]

  = ρ × g × A × V × f × (L/D) × V²/2g
  = f × ρ × A × V³ × L / (2D)

Friction Factor — Laminar & Turbulent

friction_factor.txt

─── Laminar Flow (Re < 2300) ───

  f = 64 / Re     (exact, no roughness dependence)

─── Turbulent Flow (Re > 4000) ───

Colebrook-White equation (implicit):
  1/√f = −2 log₁₀[(ε/D)/3.7 + 2.51/(Re√f)]

  ε = pipe roughness (m)
  ε/D = relative roughness

Moody approximation (explicit, Haaland):
  1/√f ≈ −1.8 log₁₀[(ε/D/3.7)¹·¹¹ + 6.9/Re]

Fully rough zone (very high Re):
  1/√f = −2 log₁₀[(ε/D)/3.7]
  (f depends only on relative roughness)

Common roughness values (ε):
  ┌─────────────────────┬──────────┐
  │ Pipe material       │ ε (mm)   │
  ├─────────────────────┼──────────┤
  │ Drawn copper/brass  │ 0.0015   │
  │ PVC, Plastic        │ 0.0015   │
  │ Commercial steel    │ 0.045    │
  │ Galvanized iron     │ 0.15     │
  │ Cast iron           │ 0.26     │
  │ Riveted steel       │ 0.9–9.0  │
  │ Concrete            │ 0.3–3.0  │
  └─────────────────────┴──────────┘

Minor Losses

Fitting / Loss	K (loss coefficient)	Formula
Sudden enlargement	K = (1−A₁/A₂)²	h_L = K × V₁²/2g
Sudden contraction	K ≈ 0.5(1−A₂/A₁)	h_L = K × V₂²/2g
Gradual expansion (diffuser)	K = 0.2–0.8	Depends on angle (best < 8°)
90° elbow (standard)	K ≈ 0.9	h_L = K × V²/2g
45° elbow	K ≈ 0.4	h_L = K × V²/2g
Globe valve (fully open)	K ≈ 10	High loss, flow control
Gate valve (fully open)	K ≈ 0.2	Low loss
Swing check valve	K ≈ 2.5	One-way flow
Tee (branch flow)	K ≈ 1.8	h_L = K × V_branch²/2g
Pipe entrance (sharp)	K ≈ 0.5	h_L = K × V²/2g
Pipe exit	K ≈ 1.0	All KE lost

pipe_loss_calcs.py

import math

# Pipe flow problem — total head loss
L = 100      # m (pipe length)
D = 0.05     # m (50 mm diameter)
Q = 5e-3     # m³/s (5 L/s)
rho = 998
g = 9.81
eps = 0.045e-3  # m (commercial steel)

A = math.pi * D**2 / 4
V = Q / A  # 2.546 m/s
Re = rho * V * D / 1.002e-3  # Re ≈ 127,000

# Turbulent: use Haaland equation for initial f estimate
eps_D = eps / D
f_est = 1 / (-1.8 * math.log10((eps_D/3.7)**1.11 + 6.9/Re))**2
# f ≈ 0.0217

# Major loss (Darcy-Weisbach)
h_major = f_est * (L/D) * V**2 / (2*g)

# Minor losses: 2 elbows (K=0.9 each) + 1 gate valve (K=0.2)
K_total = 2*0.9 + 0.2 + 0.5 + 1.0  # elbows + gate + entrance + exit
h_minor = K_total * V**2 / (2*g)

h_total = h_major + h_minor
P_loss = rho * g * Q * h_total

print(f"Re = {Re:.0f}, f = {f_est:.4f}")
print(f"h_major = {h_major:.2f} m, h_minor = {h_minor:.2f} m")
print(f"h_total = {h_total:.2f} m")
print(f"Power loss = {P_loss:.1f} W")

⚠️

Moody chart: Plot of f vs Re for various ε/D values. Laminar region: f = 64/Re (straight line). Transition: f jumps around. Turbulent smooth: Blasius f = 0.316/Re^0.25. Fully rough: horizontal lines. Always read the Darcy friction factor (not Fanning). Darcy f = 4 × Fanning f.

⚡

Hydraulic Turbines

TURBINES

Turbine	Type	Head Range	Flow	Specific Speed (Ns)	Application
Pelton	Impulse	H > 300 m (high head)	Low Q	10–70	Mountain hydropower
Francis	Reaction	H = 30–300 m (medium)	Medium Q	60–300	Most common, dams
Kaplan	Reaction (axial)	H < 30 m (low head)	High Q	300–1000	Run-of-river, tidal
Propeller	Reaction (axial)	H < 30 m	High Q	300–1000	Fixed blades (lower efficiency)
Bulb turbine	Reaction	H < 10 m	Very high Q	—	Low-head river/tidal

Pelton Wheel (Impulse Turbine)

pelton_turbine.txt

─── Pelton Wheel (Impulse Turbine) ───

Jet velocity: V₁ = C_v × √(2gH)
  C_v = velocity coefficient ≈ 0.98

Bucket velocity (optimum): u = V₁/2 = √(gH)/2

Velocity at bucket outlet: V₂ = V₁ − u
  (water leaves bucket with reversed relative velocity)

Power: P = ρ × Q × (V₁ − u) × u × (1 + k cos β)
  k = friction factor for bucket ≈ 0.85–0.95
  β = bucket angle at outlet (typically 165°–170°)

Maximum efficiency occurs at: u/V₁ = 0.5 (or cos(α/2))
  η_max ≈ (1 + k cos β)/2 × C_v²

  For β = 165°, k = 0.9: η_max ≈ 0.967 × C_v² ≈ 93%

Specific speed: Ns = N√P / H^(5/4)
  For Pelton: Ns = 10–70 (typically designed with Ns < 30)

Number of jets: n = D/(2.5 × d) + 1
  D = runner pitch diameter, d = jet diameter

Francis Turbine (Reaction)

francis_turbine.txt

─── Francis Turbine (Reaction Turbine) ───

Euler turbine equation:
  P = ρ × Q × (u₁V_w1 − u₂V_w2)

  u₁, u₂ = tangential velocity at inlet/outlet
  V_w1, V_w2 = whirl component of velocity

For Francis turbine:
  V_w2 = 0 (designed for zero whirl at exit)
  → P = ρQ × u₁V_w1

Degree of reaction: R = (P_static_change) / P_total
  Francis: R ≈ 0.5–0.8 (significant pressure drop through runner)

Flow: Radial inward, exits axially
Guide vanes (wicket gates): Control flow & angle

Efficiency: η_max ≈ 90–95%
  Hydraulic η ≈ 93%
  Mechanical η ≈ 98%
  Overall η = η_h × η_m × η_v

Draft tube: Converts KE to pressure energy
  Recovered head = V₂²/(2g) × (1 − (A₂/A₁)²)

Kaplan Turbine & Cavitation

kaplan_cavitation.txt

─── Kaplan Turbine (Axial Flow Reaction) ───

• Adjustable runner blades (pitch control)
• Adjustable guide vanes
• Axial flow — water flows parallel to shaft
• High specific speed: Ns = 300–1000
• Best for low head, high flow
• Efficiency: 88–93%

─── Cavitation ───

Cavitation occurs when local pressure drops below
vapor pressure → bubbles form → collapse violently.

Thoma's cavitation parameter:
  σ = (P_atm − P_v − H_s) / H

  P_v = vapor pressure at water temperature
  H_s = suction head (positive if below tailwater)
  H = net head on turbine

Critical sigma (σ_c): Minimum sigma to avoid cavitation
  From model tests or empirical charts.

To prevent cavitation: σ > σ_c
  → H_s < (P_atm − P_v)/ρg − σ_c × H

  OR set runner deeper below tailwater.

Effects of cavitation:
  • Pitting and erosion of blades
  • Vibration and noise
  • Loss of efficiency
  • eventual failure

turbine_calcs.py

import math

# Francis turbine: calculate power and specific speed
H = 150      # m (net head)
Q = 10       # m³/s (flow rate)
N = 600      # RPM
rho = 998    # kg/m³
g = 9.81
eta = 0.92   # overall efficiency

# Power output
P = rho * g * Q * H * eta / 1e6  # MW
# P = 998 × 9.81 × 10 × 150 × 0.92 / 1e6 = 13.53 MW

# Specific speed (metric: P in kW)
P_kW = P * 1000
Ns = N * math.sqrt(P_kW) / H**(5/4)
# Ns = 600 × √13530 / 150^1.25
#    = 600 × 116.3 / 498.7 = 139.8

print(f"Power = {P:.2f} MW")
print(f"Specific speed Ns = {Ns:.1f}")
# Ns = 139.8 → Francis turbine (in range 60-300) ✓

💡

Turbine selection: Use specific speed (Ns)as the primary selection criterion. Ns < 30 → Pelton. Ns 30–300 → Francis. Ns > 300 → Kaplan. Also consider head, flow availability, and site conditions. Francis is the most versatile and widely used turbine type worldwide, accounting for ~60% of global hydro capacity.

🔄

Pumps

PUMPS

Pump Type	Mechanism	Head Range	Flow	Application
Centrifugal	Rotating impeller throws fluid outward by centrifugal force	Low–Medium	High	Water supply, HVAC, irrigation
Reciprocating	Piston/cylinder displaces fluid	Very High	Low	High-pressure cleaning, oil drilling
Gear pump	Meshing gears trap and move fluid	Medium	Low–Medium	Hydraulics, lubrication, fuel
Vane pump	Sliding vanes in eccentric rotor	Medium	Medium	Hydraulics, fuel, vacuum
Axial flow	Propeller pushes fluid along axis	Very Low	Very High	Drainage, flood control
Submersible	Sealed motor + pump, submerged	Medium	Medium–High	Deep wells, sewage
Jet pump	Venturi creates suction + ejects	Low–Medium	Low–Medium	Shallow wells, domestic water

Centrifugal Pump — Key Parameters

centrifugal_pump.txt

─── Centrifugal Pump Characteristics ───

Manometric (pump) head:
  H_m = (P₂−P₁)/(ρg) + (V₂²−V₁²)/(2g) + (z₂−z₁)

Theoretical head (Euler):
  H_th = u₂V_w2/g  (for zero inlet whirl: V_w1 = 0)

  u₂ = πD₂N/60 (tip speed)
  V_w2 = whirl component at impeller exit

Affinity Laws (speed change, N₁ → N₂):
  Q₂/Q₁ = N₂/N₁
  H₂/H₁ = (N₂/N₁)²
  P₂/P₁ = (N₂/N₁)³

System curve:
  H_system = H_static + K × Q²
  H_static = elevation + pressure difference

Operating point: Where pump curve intersects system curve

NPSH (Net Positive Suction Head):
  NPSH_available = P_atm/(ρg) − P_v/(ρg) − h_s − h_f_suction
  
  To avoid cavitation: NPSH_available > NPSH_required

Pump Efficiency & Power

pump_power.py

import math

# Centrifugal pump calculation
Q = 0.05    # m³/s (50 L/s)
H = 30      # m (head)
rho = 998   # kg/m³
g = 9.81
eta = 0.75  # pump efficiency (75%)

# Hydraulic power (fluid power)
P_hydraulic = rho * g * Q * H  # Watts
# P_h = 998 × 9.81 × 0.05 × 30 = 14,688 W = 14.69 kW

# Shaft power (motor power needed)
P_shaft = P_hydraulic / eta
# P_shaft = 14688 / 0.75 = 19,584 W = 19.58 kW

# Motor selection (add 15-20% safety factor)
P_motor = P_shaft * 1.20
# P_motor = 23.5 kW → use 22 kW or 30 kW motor

# Specific speed of pump
N = 1450  # RPM
Ns = N * math.sqrt(Q) / H**(3/4)
# Ns = 1450 × √0.05 / 30^0.75
print(f"P_hydraulic = {P_hydraulic/1e3:.2f} kW")
print(f"P_shaft = {P_shaft/1e3:.2f} kW")
print(f"P_motor = {P_motor/1e3:.1f} kW")
print(f"Ns = {Ns:.1f}")

Pumps in Series & Parallel

pumps_series_parallel.txt

─── Pumps in Series ───
  Same flow Q, heads add up.

  H_total = H₁ + H₂
  Q = Q₁ = Q₂

  Use when: High head needed, moderate flow.
  The system curve is steep (high resistance).

─── Pumps in Parallel ───
  Same head H, flows add up.

  Q_total = Q₁ + Q₂
  H = H₁ = H₂

  Use when: Large flow needed, moderate head.
  The system curve is flat (low resistance).

Note: Pumps must have similar characteristics
for effective series/parallel operation.

⚠️

NPSH explained:NPSH_available depends on the suction system (atmospheric pressure, elevation, pipe friction). NPSH_required is a property of the pump (provided by manufacturer). If NPSH_available < NPSH_required, cavitation occurs. Solutions: reduce suction lift, increase pipe diameter, cool the fluid, or use a pump with lower NPSH_requirement.

💨

Compressible Flow

COMPRESSIBLE

Mach Number & Speed of Sound

mach_number.txt

─── Speed of Sound ───

  c = √(γRT) = √(γP/ρ)

  c_air at 20°C = √(1.4 × 287 × 293) = 343.2 m/s
  c_air at 0°C  = √(1.4 × 287 × 273) = 331.3 m/s
  c_helium      = √(1.66 × 2077 × 293) = 1007 m/s

─── Mach Number ───

  Ma = V/c

  Ma < 0.3  → Incompressible (ρ ≈ constant)
  0.3 < Ma < 0.8 → Subsonic (compressible effects begin)
  0.8 < Ma < 1.2 → Transonic (mixed sub/supersonic regions)
  1.0 < Ma < 5   → Supersonic
  Ma > 5         → Hypersonic

Stagnation properties (isentropic flow):
  T₀ = T + V²/(2cp)   = T(1 + (γ−1)/2 × Ma²)
  P₀ = P(1 + (γ−1)/2 × Ma²)^(γ/(γ−1))
  ρ₀ = ρ(1 + (γ−1)/2 × Ma²)^(1/(γ−1))

Isentropic Flow Relations

isentropic_flow.txt

─── Isentropic Flow Through Nozzles ───

Critical (throat) conditions at Ma = 1:
  T* = T₀ × 2/(γ+1)     = T₀ × 0.833 (air)
  P* = P₀ × (2/(γ+1))^(γ/(γ−1)) = P₀ × 0.528 (air)
  ρ* = ρ₀ × (2/(γ+1))^(1/(γ−1)) = ρ₀ × 0.634 (air)

Area-Mach relation:
  A/A* = (1/Ma)[(2/(γ+1))(1 + (γ−1)/2 × Ma²)]
        ^((γ+1)/(2(γ−1)))

Converging nozzle:
  • P_back > P*: Subsonic throughout (no choking)
  • P_back = P*: Choked flow (Ma=1 at exit)
  • P_back < P*: Choked; expansion waves outside nozzle

Converging-DIVERGING (C-D) nozzle (de Laval):
  • P_exit > P*: Subsonic throughout
  • P_exit = P*: Choked at throat, subsonic in diverging
  • P_exit < P*: Isentropic supersonic in diverging section
    (shock-free, design condition)

Normal Shock Waves

normal_shock.txt

─── Normal Shock Relations (air, γ=1.4) ───

Upstream (1) → Shock → Downstream (2)

Mach number:
  Ma₂² = (1 + (γ−1)/2 × Ma₁²) / (γ×Ma₁² − (γ−1)/2)

Pressure ratio:
  P₂/P₁ = 1 + 2γ/(γ+1) × (Ma₁² − 1)

Temperature ratio:
  T₂/T₁ = [1 + 2γ/(γ+1)(Ma₁²−1)] × [2+(γ−1)Ma₁²] / [(γ+1)Ma₁²]

Density ratio:
  ρ₂/ρ₁ = V₁/V₂ = (γ+1)Ma₁² / [2+(γ−1)Ma₁²]

Stagnation pressure ratio (entropy increase):
  P₀₂/P₀₁ = [(γ+1)Ma₁²/(2+(γ−1)Ma₁²)]^(γ/(γ−1))
             × [(2γMa₁²−(γ−1))/(γ+1)]^(-1/(γ−1))

Key: Normal shock is IRREVERSIBLE (entropy increases).
  Ma₁ > 1, Ma₂ < 1 always.
  Stagnation temperature T₀ is constant across shock.

compressible_flow.py

import math

# Isentropic flow — converging-diverging nozzle
T0 = 500    # K (stagnation temperature)
P0 = 500    # kPa (stagnation pressure)
gamma = 1.4
R = 287     # J/(kg·K)

# Critical conditions (Ma=1 at throat)
T_star = T0 * 2 / (gamma + 1)
P_star = P0 * (2/(gamma+1))**(gamma/(gamma-1))
rho_star_ratio = (2/(gamma+1))**(1/(gamma-1))

# Sound speed at throat
c_star = math.sqrt(gamma * R * T_star)
# c* = √(1.4 × 287 × 416.7) = 409.3 m/s

# If fully expanded to Ma = 2.0 in diverging section
Ma = 2.0
T_exit = T0 / (1 + (gamma-1)/2 * Ma**2)
P_exit = P0 * (1 + (gamma-1)/2 * Ma**2)**(-gamma/(gamma-1))
V_exit = Ma * math.sqrt(gamma * R * T_exit)

print(f"Throat: T* = {T_star:.1f} K, P* = {P_star:.1f} kPa")
print(f"Exit (Ma=2): T = {T_exit:.1f} K, P = {P_exit:.1f} kPa")
print(f"Exit velocity = {V_exit:.1f} m/s")

# Normal shock at Ma=2.0
Ma2_sq = (1 + (gamma-1)/2 * Ma**2) / (gamma * Ma**2 - (gamma-1)/2)
Ma2 = math.sqrt(Ma2_sq)  # Ma2 = 0.577
P2_P1 = 1 + 2*gamma/(gamma+1) * (Ma**2 - 1)  # = 4.5
print(f"After shock: Ma = {Ma2:.3f}, P₂/P₁ = {P2_P1:.2f}")

Flow Regimes Summary

Regime	Mach Number	Flow Behavior	Key Equations
Incompressible	Ma < 0.3	Constant density, Bernoulli applies	P + ½ρV² = const
Subsonic	0.3 < Ma < 1	Disturbances propagate in all directions	Isentropic relations, A/V variation
Sonic (choking)	Ma = 1	Flow velocity = sound speed at throat	A = A* (minimum area), mass flow limit
Supersonic	Ma > 1	Disturbances confined within Mach cone	Mach angle: μ = arcsin(1/Ma)
Hypersonic	Ma > 5	Extreme heating, real gas effects, dissociation	Modified relations needed

🚫

Choked flow:In a converging nozzle, the maximum mass flow rate is reached when Ma = 1 at the exit (throat). Beyond this, reducing the back pressure does NOT increase the flow rate. The mass flow is "choked" at: ṁ_max = A* × P₀ × √(γ/(RT₀)) × (2/(γ+1))^((γ+1)/(2(γ−1))). This is critical for safety valves, rocket nozzles, and jet engines.

⏳

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