Bernoulli’s Principle Simulator

This simulator demonstrates Bernoulli's Principle — the fundamental relationship P + ½ρv² + ρgh = constant along a streamline in steady, incompressible flow. Through an interactive Venturi tube visualisation, you can observe how fluid velocity increases and pressure decreases when a pipe narrows, and vice versa. The continuity equation (A₁v₁ = A₂v₂) ensures mass conservation, while Bernoulli's equation relates the resulting flow velocity to pressure changes.

Designed for mechanical and aerospace engineering students, fluid mechanics trainees, and physics learners, this tool covers the Venturi effect, dynamic pressure, flow rate calculation, and practical applications like Pitot tubes, spray guns, and airplane lift. Four interactive modes let you simulate, explore concepts, practise calculations, and test your knowledge.

The simulator opens in Simulate mode showing a Venturi tube with animated streamlines. The default configuration has an inlet velocity of 3.0 m/s, wide diameter d₁ = 100 mm, and throat diameter d₂ = 50 mm. Animated streamlines show the flow accelerating through the constriction, and pressure bars visually compare the pressures at the wide section and the throat.

Each parameter has both a slider (drag for quick adjustment) and a [−] number [+] stepper (click for precise steps or type an exact value). The readout cards instantly show v₁, v₂, P₁, P₂, ΔP, Q, speed ratio, area A₁, Reynolds number, and flow regime. Try the Presets — Water Pipe, Venturi Meter, Fire Nozzle, Garden Hose, Wind Tunnel, Oil Riser — to load realistic engineering configurations in one click. Click the 🧮 Show Calculations button on the canvas to open a full step-by-step Bernoulli walkthrough for the current simulation state.

In Simulate mode the canvas renders an animated Venturi tube. Streamlines are colour-coded — faster regions show brighter, faster-moving dots. Pressure bars above the pipe compare static pressures at the wide and narrow sections.

• Fluid selector — Switch between Water (ρ = 1000 kg/m³), Oil (ρ = 850), Air (ρ = 1.225), and Mercury (ρ = 13 600). Each changes the Reynolds number and flow behaviour.
• Δh slider — Set a height difference (−2 to +2 m) between the wide section and the throat. This activates the gravitational term ρgΔh in Bernoulli’s equation. Use the Oil Riser preset as an example.
• Show Manometer checkbox — Overlays a U-tube manometer connecting the wide section and throat, showing the pressure difference as a fluid column height.
• Cavitation warning — A red banner appears if throat pressure P₂ drops below zero, indicating cavitation risk. Reduce velocity or increase the throat diameter to clear it.
• SI / Imperial toggle — Switch all readouts between SI (m/s, mm, kPa, L/s) and Imperial (ft/s, in, psi, gal/s). Stepper inputs also update to the selected unit system.
• 🧮 Show Calculations button — Click this button on the canvas (bottom-right) to open a detailed step-by-step Bernoulli walkthrough based on the current simulation values.

Keyboard shortcuts: Right-click the canvas to access Export PNG, Export CSV, and Reset Defaults.

Switch to Explore mode to browse concept cards across three categories: Fundamentals, Applications, and Calculations. Fundamentals covers Bernoulli's equation derivation, the continuity equation for incompressible flow, static versus dynamic pressure, and the assumptions behind the equation (steady, inviscid, incompressible flow along a streamline).

The Applications category explains airplane lift (faster air over the wing creates lower pressure), Pitot tubes for airspeed measurement, Venturi meters for pipe flow measurement, carburettors, atomisers, and spray nozzles. The Calculations category provides worked examples showing how to compute throat velocity from the continuity equation and pressure drops from Bernoulli's equation. Each card includes formulas, diagrams, and numerical examples to reinforce your understanding of flow velocity and pressure relationships.

Practice mode generates random fluid mechanics problems involving Bernoulli's equation and the continuity equation. You might be asked to find the throat velocity given inlet conditions and diameter ratio, calculate the pressure drop across a constriction, or determine the flow rate from measured pressures. Enter your answer and click Check for immediate feedback. Your running score tracks your progress.

Quiz mode presents five questions per session, combining conceptual and numerical problems. Topics include the Venturi effect, the relationship between flow area and velocity, pressure-velocity trade-offs, and real-world applications. After completing the quiz, review your results to identify areas for further study. This is valuable preparation for fluid mechanics examinations covering Bernoulli's equation and pipe flow analysis.

• Remember that Bernoulli's equation applies along a streamline in steady, inviscid, incompressible flow. Real flows with viscosity, turbulence, or compressibility require corrections.
• The continuity equation A₁v₁ = A₂v₂ means velocity scales with the inverse of area. Since area scales with diameter squared, halving the diameter quadruples the velocity.
• Watch the pressure bars — they give immediate visual feedback about where pressure is high versus low. Faster flow always means lower static pressure.
• Use the Presets to quickly compare different engineering scenarios. The Fire Nozzle preset shows extreme acceleration; the Oil Riser preset demonstrates the gravitational term ρgΔh in action.
• Use the SI / Imperial toggle to practice with the unit system used in your course or workplace. All readout cards and stepper inputs switch together.
• Right-click on the canvas at any time to export a PNG snapshot or a CSV file with all calculated values in both SI and Imperial units.
• If a red cavitation warning appears, the throat pressure has gone negative — this is physically impossible in a real liquid. Reduce inlet velocity or increase d₂ to bring P₂ back above zero.
• Pair this tool with the Fluid Flow Simulator to study Reynolds number, friction losses, and turbulent flow — effects that Bernoulli's ideal equation does not account for.