Fluid Dynamics & Bernoulli Calculator
Calculate pipe flow velocity, dynamic pressure, pressure drop, and total head using Bernoulli’s principle and continuity equation. Features Venturi/Pitot flow meters, Torricelli tank discharge/drainage time. Ideal-fluid study model; no cavitation diagnosis.
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Key Formula & Conditions
Learning modelComputes p + ½ρv² + ρgh = const along streamlines for steady, incompressible, inviscid fluid flow. Solves Venturi static pressure drops from cross-sectional area ratios and Torricelli exit velocity v = √(2gh).
An ideal-fluid model for learning. Use one pressure reference throughout (gauge or absolute). The low-pressure flag is not a cavitation diagnosis: vapor pressure at the fluid temperature, pipe losses and equipment specifications need separate checks.
Fluid & Pipe Inputs
p + ½ρv² + ρgh = constDone
Fluid Dynamics & Bernoulli Calculator
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Worked Examples & Verified Values
Torricelli exit velocity
Venturi pressure drop
Dynamic pressure (air)
Common Pitfalls & Calculation Rules
- •The Bernoulli equation assumes inviscid flow. For long pipes where viscous friction causes substantial energy loss, Darcy-Weisbach head loss equations must be used.
- •Do not mix gauge and absolute pressure values; apply consistent definitions across both measuring stations. Standard atmospheric pressure is 101.325 kPa.
- •When gas velocities exceed Mach 0.3, density changes significantly, rendering the incompressible Bernoulli equation invalid.
Verification scope and citation information
Use worked examples within their stated inputs, units and assumptions. Tests cover specific cases at a specific time; they do not certify every input, expert review or standards compliance. Check the references and tool-specific verification notes, and enter your own access date when citing.
@misc{tools_hub_bernoulli_calculator_2026,
author = {{Tools Hub}},
title = {Fluid Dynamics & Bernoulli Calculator},
year = {2026},
url = {https://the-tools-hub.net/en/tools/bernoulli-calculator/},
note = {Accessed: YYYY-MM-DD}
}How to use
- 1
Select calculation mode
Choose Pipe Flow & Bernoulli (flow velocity & pressure drop), Venturi & Pitot Tubes (differential flow meters), or Torricelli Orifice & Tank Drainage.
- 2
Enter fluid and flow parameters
Select fluid preset (water, air, seawater, ethanol, oil, gasoline, or custom density), diameters D₁/D₂, heights h₁/h₂, inlet pressure p₁, differential pressure Δp, or tank liquid head H.
- 3
Inspect velocity, pressure, heads & animation
Review velocities v₁/v₂, outlet pressure p₂, dynamic/total heads, discharge rate and tank empty time, with interactive pipe flow simulation.
Features
- Coupled solution of continuity equation A₁v₁ = A₂v₂ and Bernoulli’s equation p₁ + ½ρv₁² + ρgh₁ = p₂ + ½ρv₂² + ρgh₂
- Breakdown of static pressure, dynamic pressure, elevation pressure, and total head components
- Presets for water, air, seawater, ethanol, motor oil, gasoline, plus custom fluid densities
- Venturi flow meter discharge solver with discharge coefficient Cd and diameter ratio β correction
- Pitot-static airspeed calculation with output in m/s, km/h, knots, and Mach number estimation
- Torricelli’s orifice velocity v = √(2gH) and instantaneous discharge rate calculation
- Tank draining time simulation t_empty = (2A / Cd A₀)√(H/2g) for cylindrical tanks
- Low-pressure notice; does not diagnose cavitation from temperature and vapor pressure
- Real-time SVG animation depicting pipe narrowing, velocity vectors, and manometer liquid heights
- Practical presets: pipe constriction, elevation rise, Venturi meter, aircraft Pitot, and tank draining
- One-click Markdown laboratory report export formatted for engineering and physics coursework
Use cases
Fluid mechanics & engineering coursework
Verify pipe flow continuity, pressure changes, dynamic heads, and energy balances without unit conversion slips.
Venturi meter & Pitot tube laboratory reports
Convert manometer liquid differential heights into flow rates and airspeeds with discharge coefficient adjustments.
Piping design, tank emptying & hydraulic estimates
Check ideal-model pressure changes and tank drainage times for study. Real equipment losses and safety require separate analysis.
Details
Bernoulli’s principle represents the conservation of mechanical energy for steady, incompressible, frictionless fluid flow along a streamline: p + ½ρv² + ρgh = constant. In head form, pressure head p/(ρg), velocity head v²/(2g), and elevation head h sum to a constant total head H_total. When a conduit constricts, the continuity equation A₁v₁ = A₂v₂ dictates an increase in velocity, converting static pressure into dynamic pressure and causing a drop in static pressure.
A Venturi meter utilizes this pressure drop across a throat constriction (β = D₂/D₁) to measure volumetric flow rate via Q = C_d A₂ √[2Δp / (ρ(1 - β⁴))], where C_d is the discharge coefficient (typically 0.96–0.99 for smooth Venturi tubes). Similarly, Pitot-static tubes measure airspeed by sampling total (stagnation) pressure and static pressure, yielding flow velocity v = √[2Δp / ρ].
Torricelli’s law is an immediate consequence of Bernoulli’s theorem for liquid draining from an open tank orifice under gravity: the discharge velocity equals that of free fall from depth H, v = √(2gH). Because head diminishes during drainage, the total time to empty a uniform tank of cross-sectional area A through an orifice of area A₀ is given by t_empty = [2A / (C_d A₀)] √(H / 2g), exactly twice the duration calculated assuming a constant initial discharge rate.
This tool calculates velocities, pressures, heads, low-pressure notices, Venturi discharge rates, and tank drainage times with interactive fluid vector diagrams, manometer height displays, and one-click report formatting.
Heat Engine, Gas Law & Flow Analysis
Analyze thermal cycle work/efficiency, working fluid PV=nRT state changes, and pipe flow velocity/Venturi losses in sequence.
FAQ
Why does static pressure drop when a fluid enters a constricted pipe?
By mass continuity, fluid must speed up through narrower sections. To accelerate, a forward pressure gradient is required, meaning pressure upstream must be higher than downstream. Thermodynamically, mechanical energy is conserved: an increase in kinetic energy (dynamic pressure ½ρv²) comes at the direct expense of static pressure energy p.
What are the fundamental assumptions of Bernoulli’s equation?
Bernoulli’s equation strictly applies to: (1) Incompressible flow (constant density ρ; liquids and gases below Mach 0.3), (2) Inviscid flow (negligible frictional viscosity), (3) Steady flow (parameters invariant with time), and (4) Flow along a single streamline. Real-world long pipelines experience additional head losses due to viscous wall friction.
What is the discharge coefficient (Cd) in a Venturi flow meter?
The discharge coefficient Cd is an empirical correction factor representing the ratio of actual flow rate to theoretical ideal flow rate. It accounts for fluid viscosity, boundary layer growth, and vena contracta effects. High-accuracy standard Venturi meters typically achieve Cd ≈ 0.96–0.99, whereas sharp-edged orifice plates exhibit Cd ≈ 0.60–0.65.
What causes cavitation in constricted fluid lines?
When local flow velocity accelerates dramatically, static pressure drops. If it plunges below the liquid’s saturation vapor pressure (~2.3–3.2 kPa for ambient water), the liquid boils at room temperature, spawning micro vapor bubbles. As these bubbles travel into higher pressure regions, they violently implode, generating microjets and shock waves that pit pipe walls and impellers. This calculator has no fluid-temperature, vapor-pressure or pressure-reference diagnostic model; its low-pressure notice cannot determine cavitation.
Why does draining a tank take twice as long as constant-rate discharge?
Under Torricelli’s law v = √(2gH), discharge velocity decays continuously as liquid depth drops. Integrating -A dH/dt = Cd A₀ √(2gH) yields t_empty = [2A / (Cd A₀)] √(H₀ / 2g). Because the average draining velocity across the process is exactly half the initial peak velocity v₀, draining takes precisely twice the time compared to discharge sustained constantly at v₀.
What is Torricelli’s Law?
A theorem stating that the speed of efflux of a fluid through a sharp orifice at depth h below the free surface equals the speed acquired by a body falling freely from height h, given by v = √(2gh).
Why does static pressure drop in the throat of a Venturi tube?
By the continuity equation (A₁v₁ = A₂v₂), velocity increases as cross-sectional area narrows. Because total mechanical energy is conserved along a streamline, an increase in dynamic pressure (½ρv²) requires a corresponding drop in static pressure (p).
Related guide
GuideBernoulli Equation: Velocity, Pressure, Flow and Limits
Reproduce pipe-flow, Venturi, Pitot and tank-drainage examples with stated inputs and units. Check pressure reference, the 183.16 kPa pipe result and the limits of the ideal-fluid model.
GuideCircular Motion & Centripetal Force: Formulas, Centrifuge RCF, Bank Angles & Loops
Master uniform circular motion calculations: angular velocity ω, period T, rpm, speed v, centripetal acceleration a, and force F. Understand centripetal vs centrifugal force, laboratory centrifuge RCF (×g) conversion, highway bank angles, and vertical loop clearance physics.
GuideIdeal Gas Law vs Combined Gas Law: When to Use Each
Choose between PV = nRT and P₁V₁/T₁ = P₂V₂/T₂ by checking whether the amount of gas changes. Includes unit conversion (C to K, mL to L, mmHg to Pa), the 22.4 L molar volume at STP, gas density and molar mass, common mistakes, and a free calculator for checking answers.
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Verified: Checked October 15, 2026: density 998.2 kg/m³, internal diameters 100/50 mm, v1=1.5 m/s, p1=200 kPa, equal heights yield v2=6.0000 m/s and p2=183.16 kPa; h2=0.5 m gives 178.26 kPa. Venturi Δp=12 kPa and Cd=0.98 gives 6.24 L/s. Tank H=2.5 m, outlet 25 mm, tank 1.2 m and Cd=0.62 gives about 2653 s. Independently recalculated and checked in Chrome in both languages; not measurements, compliance certification or cavitation diagnosis.
Did you know?
At the same volume flow rate, halving a circular pipe’s internal diameter quarters its area and multiplies speed by four. At unchanged density, dynamic pressure ρv²/2 becomes sixteen times larger. Check elevation, losses and the pressure reference when comparing static pressures.