Bernoulli Equation Calculator – Fluid Dynamics
Solve the Bernoulli equation for fluid flow: compare pressure, velocity, and height at two points along a streamline. Free online calculator with total head, dynamic pressure, and interactive charts for pipe flow, open channels, and aerodynamics.
Position 1
Position 2 (optional)
About This Calculator
The Bernoulli Equation Calculator helps engineers, students, and fluid mechanics professionals analyze steady, incompressible flow along a streamline. Based on Daniel Bernoulli's principle of energy conservation for ideal fluids, this tool computes total head, dynamic pressure, and the Bernoulli constant at one or two points along a flow path. It is ideal for pipe flow analysis, open channel hydraulics, aerodynamics, HVAC duct sizing, and pump system design.
The calculator uses the Bernoulli equation P + ½ρv² + ρgh = constant, where P is static pressure, ρ is fluid density, v is flow velocity, g is gravitational acceleration, and h is elevation. The total head H = P/(ρg) + v²/(2g) + h represents total energy per unit weight in meters. Dynamic pressure ½ρv² represents the kinetic energy per unit volume of the flowing fluid. When two positions are entered, the calculator compares the Bernoulli constants and shows the head difference and pressure difference between the points.
How to Use
Enter the fluid density (default 1000 kg/m³ for water), then fill in pressure, velocity, and height at Position 1. Optionally enter values at Position 2 to compare two points. Leave Position 2 fields blank to analyze a single point. Click Calculate to see results including total head, dynamic pressure, Bernoulli constant, and an optional two-point comparison with bar charts and distribution pie charts.
Regional Notes
Worldwide / SI Units: This calculator uses SI units (pascals, meters per second, meters, kg/m³) as the international standard for fluid mechanics. Pressure can be entered in pascals; for values in bar or psi, convert before entering. Gravitational acceleration is set to 9.80665 m/s² (standard Earth gravity). The results are independent of region and apply globally for any fluid dynamics analysis in engineering, education, and research.
Frequently Asked Questions
What is the Bernoulli equation?
The Bernoulli equation states that along a streamline in steady, inviscid, incompressible flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant: P + ½ρv² + ρgh = constant. It is derived from the conservation of energy principle and is fundamental to fluid dynamics, aerodynamics, and hydraulics.
How does an airplane wing use Bernoulli's principle?
An airplane wing has a curved upper surface and a flatter lower surface. Air traveling over the curved top must travel a longer distance, increasing its speed. According to Bernoulli's principle, higher speed creates lower pressure above the wing while the pressure below remains higher. This pressure difference generates lift, with roughly 70% of lift coming from the pressure differential (the rest from Newton's third law via the downward deflection of air).
What are the assumptions of the Bernoulli equation?
The Bernoulli equation assumes: (1) steady flow — no changes over time, (2) incompressible fluid — constant density, (3) inviscid flow — no friction or viscous losses, (4) flow along a single streamline, and (5) no heat transfer across the fluid boundaries. Real fluids violate some of these assumptions, so Bernoulli provides an idealized approximation that works well for many engineering applications when corrections are applied.
How is Bernoulli used in pipe flow and Venturi meters?
A Venturi meter uses Bernoulli's principle to measure flow rate. As fluid enters a narrow throat section of the pipe, its velocity increases and pressure drops. By measuring the pressure difference between the wide section and the throat, the flow rate can be calculated using the continuity equation (A₁v₁ = A₂v₂) together with Bernoulli's equation. This principle is widely used in industrial flow measurement, HVAC systems, and water distribution networks.
What is total head in fluid systems?
Total head H = P/(ρg) + v²/(2g) + z represents the total energy per unit weight of fluid, measured in meters. It combines pressure head (P/ρg), velocity head (v²/2g), and elevation head (z). Total head is a key concept in selecting pumps, analyzing pipeline systems, and designing water supply networks. Pump performance curves relate flow rate to total head to determine the operating point of a pump.
What is the difference between Bernoulli and Navier-Stokes equations?
Bernoulli's equation is a simplified energy conservation equation for ideal (inviscid, incompressible, steady) flow along a streamline. The Navier-Stokes equations are a complete set of momentum conservation equations that include viscosity, unsteady terms, compressibility effects, and three-dimensional flow patterns. Navier-Stokes are much harder to solve analytically but provide more accurate results for real fluid flows involving boundary layers, turbulence, and separated flows.
Can Bernoulli's principle be applied to both liquids and gases?
Bernoulli's principle applies to both liquids and gases as long as the flow is incompressible (Mach number below 0.3 for gases) and inviscid. For water and most liquids, incompressibility is an excellent assumption. For gases like air at low speeds, the compressibility error is small enough that Bernoulli's equation still gives useful results. For high-speed gas flow above Mach 0.3, the compressible flow equations must be used instead.
What units does this calculator use?
This calculator uses SI units: pressure in pascals (Pa), velocity in meters per second (m/s), height in meters (m), density in kilograms per cubic meter (kg/m³), and gravitational acceleration of 9.80665 m/s². Results show total head in meters, dynamic pressure in pascals, and the Bernoulli constant in pascals. To convert from bar, multiply by 100,000; from psi, multiply by 6,894.76.