Flow Visualization

Standard Bernoulli Equation

P + 1/2ρv2 + ρgh = constant

The fundamental energy conservation equation for incompressible, inviscid flow along a streamline. Sum of pressure, kinetic, and potential energy per unit volume remains constant.

Static Pressure

Pa (Pascals)

Fluid Density

kg/m3

Flow Velocity

m/s

Gravity

9.81 m/s2

Energy Terms

Conservation

Term	Description
Static Pressure (P)	Thermodynamic pressure of fluid
Dynamic Pressure (1/2ρv2)	Kinetic energy per unit volume
Hydrostatic (ρgh)	Gravitational potential energy

Practical Examples

• Pipe flow analysis: Calculate pressure drop from velocity change
• Tank discharge: Torricelli's theorem (v = √2gh)
• Siphon design: Maximum height limited by vapor pressure

Flow Visualization

Head Form (Energy per Weight)

P/γ + v2/2g + h = h_total

Bernoulli equation expressed as head (length units). Commonly used in hydraulics and pipe flow analysis. Each term represents energy per unit weight of fluid.

P/γ

Pressure Head

m (meters)

v2/2g

Velocity Head

m (meters)

Elevation Head

m (meters)

Specific Weight

N/m3 (ρg)

Energy Terms

Conservation

Term	Description
Pressure Head	Height of fluid column for given pressure
Velocity Head	Height fluid would fall to reach velocity v
Elevation Head	Geometric height above datum

Practical Examples

• Pump selection: Total dynamic head (TDH) calculations
• Pipe network analysis: Hardy-Cross method
• Open channel flow: Specific energy diagrams

Flow Visualization

Engineering Applications

P1 + 1/2ρv12 = P2 + 1/2ρv22

Practical applications where elevation change is negligible. Pressure difference relates directly to velocity change. Foundation for flow measurement and aerodynamic lift.

ΔP

Pressure Difference

v1, v2

Velocities at points 1, 2

m/s

Fluid Density

kg/m3

Energy Terms

Conservation

Term	Description
Venturi Effect	Pressure drop in constricted flow
Pitot Tube	Velocity measurement from stagnation pressure
Airfoil Lift	Pressure difference from velocity difference

Practical Examples

• Venturi meter: Q = A2√[2(P1-P2)/ρ(1-(A2/A1)2)]
• Pitot-static tube: Aircraft airspeed measurement
• Atomizers/spray bottles: Low pressure draws fluid upward
• Baseball curveball: Magnus effect pressure differential

Flow Visualization

Limitations & Assumptions

Assumptions must hold

Bernoulli equation applies only under specific conditions. Real flows with viscosity, turbulence, or compressibility require modifications or alternative approaches.

Steady Flow

∂/∂t = 0

Incompressible

ρ = constant

Inviscid

μ ≈ 0

Along Streamline

Same streamline

Energy Terms

Conservation

Term	Description
No Pumps/Turbines	No external work addition/extraction
No Friction Losses	Negligible viscous dissipation
Single Phase	No cavitation or phase change

Practical Examples

• Boundary layers: Viscous effects near solid surfaces
• Turbulent flow: Energy losses require Darcy-Weisbach
• Compressible flow: Use isentropic relations (M > 0.3)
• Pumps/turbines: Add head terms (h_pump, h_turbine)