Fluid Pressure and Flow Simulator

Understanding Fluid Pressure and Flow: A Comprehensive Guide

When I first started teaching fluid mechanics, students would glaze over at equations. Then I discovered something: show them a garden hose nozzle, and suddenly Bernoulli makes sense. This simulator brings that intuition to five fundamental fluid phenomena.

Fluid pressure and flow govern everything from your morning shower to aircraft flight. The equations are elegant--Bernoulli, Torricelli, Pascal, Poiseuille--but they become powerful when you see them animate before your eyes. After spending years troubleshooting hydraulic systems and analyzing flow problems, I've distilled the core concepts into interactive demonstrations that actually stick [1].

The Big Picture: Pressure-Velocity Tradeoff

Here's the fundamental insight that unlocks fluid mechanics: in a moving fluid, pressure and velocity trade off. Speed up the flow, pressure drops. Slow it down, pressure rises. This single principle explains airplane lift, carburetors, atomizers, and why your shower curtain gets sucked inward [2].

How to Use This Simulation

Picture the streamlines as you explore each demo mode. The flow behaves differently depending on geometry, velocity, and fluid properties, and this simulator lets you visualize those relationships through color-coded pressure fields and animated particle tracers.

Selecting a Demo Mode

Start by clicking one of the five mode buttons at the top:

Input Parameters by Mode

Each mode exposes different sliders. The flow behaves differently depending on these variables:

Venturi Tube:

Airplane Wing:

Hydraulic Press:

Water Tower:

Blood Vessel:

Understanding the Visualization

The canvas displays several layers of information:

1. Pressure Field (toggle with checkbox): Red = high pressure, blue = low pressure. Watch how colors shift as flow accelerates or decelerates.

2. Particles (toggle with checkbox): White dots trace the flow. In laminar conditions, they follow smooth paths. At high Reynolds numbers, you might see more chaotic motion.

3. Velocity Arrows (toggle with checkbox): Longer arrows = faster flow. Watch them stretch through constrictions and shrink in expansions.

4. Animation (toggle to pause): Freeze the display to examine static patterns, or let it run to see dynamic behavior.

Reading the Results

Each mode displays different outputs, but two primary values always appear:

The stats panel below shows additional parameters relevant to each demo. Reynolds number appears in modes where flow regime matters (venturi, blood vessel).

Tips for Exploration

Regime transitions: In the Venturi and Blood Vessel modes, watch the Reynolds number. The flow behaves differently depending on whether Re is below 2000 (laminar) or above 4000 (turbulent). At high Reynolds numbers, mixing increases and pressure losses deviate from simple theory.

Pressure-velocity tradeoff: Start with the Venturi mode. Increase inlet velocity and watch the throat turn bluer (lower pressure) while particles accelerate. This is Bernoulli in action: kinetic energy gained comes from pressure energy lost.

r^4 sensitivity: In Blood Vessel mode, start with 0% stenosis, then increase in 25% increments. The flow rate plummets faster than you expect because Poiseuille's law has radius to the fourth power. A 50% diameter reduction means flow drops to 1/16 of baseline, not 1/2. This explains why even moderate arterial plaque causes serious problems.

Force multiplication limits: In Hydraulic Press mode, maximize the diameter ratio. Notice that output force can be enormous (mechanical advantage > 100x), but remember the tradeoff: the output piston moves that many times slower. Conservation of energy wins every time.

Stall exploration: In Airplane Wing mode, gradually increase angle of attack beyond 12 degrees. The lift curve starts to flatten, then eventually the simple thin-airfoil model shows its limits. Real wings would experience flow separation and stall, which this simplified model cannot fully capture.

The Five Demo Modes Explained

1. Venturi Tube: Bernoulli in Action

The Venturi tube is the textbook example of Bernoulli's principle, and for good reason. As fluid enters the narrowed throat section, continuity demands it speed up (A1V1 = A2V2). That velocity increase comes at the cost of pressure--the throat region shows distinctly lower pressure (blue) compared to the inlet and outlet (red) [3].

What to observe:

• Watch particles accelerate through the throat
• Note the pressure color gradient
• Adjust throat diameter and see dramatic velocity changes

Real application: Venturi meters measure flow rate in industrial piping by measuring this pressure difference. The same principle powers paint sprayers, carburetors, and laboratory aspirators.

2. Airplane Wing: Lift Generation

This demonstration shows why aircraft fly, and it's not what most people think. The popular "equal transit time" explanation is actually wrong [4]. Lift occurs because the wing shape and angle of attack deflect air downward, and by Newton's third law, the air pushes the wing up.

The pressure field visualization shows lower pressure above the wing (faster flow) and higher pressure below. At positive angles of attack, this asymmetry creates lift.

Key parameters to explore:

• Angle of attack: Too high causes stall (the thin-airfoil approximation breaks down above ~12 degrees)
• Airspeed: Lift scales with velocity squared--double the speed, quadruple the lift
• Air density: Explains why aircraft struggle at high altitude

3. Hydraulic Press: Pascal's Multiplication

Pascal's principle feels like magic until you understand it: pressure applied to a confined fluid transmits equally in all directions. The hydraulic press exploits this by using different piston areas to multiply force [5].

With a 20mm input piston and 100mm output piston, the area ratio is 25:1. Apply 100 N, get 2500 N out. But here's the catch--conservation of energy means the output piston moves 25x slower. There's no free lunch, just force-distance tradeoff.

Practical insight: Every car brake system uses this principle. Your foot applies modest force to a small master cylinder; the larger wheel cylinders multiply that into the braking force needed to stop a 2-ton vehicle.

4. Water Tower: Torricelli's Theorem

Torricelli showed that fluid exiting a tank behaves like an object falling the same height: V = sqrt(2gh). A water tower 20 meters tall produces exit velocity of about 20 m/s--same as dropping a ball from that height [6].

The simulation lets you vary:

• Tower height (determines potential energy)
• Water level (effective driving head)
• Outlet diameter (affects flow rate but not velocity)

Municipal water systems use this principle. Those elevated tanks aren't just storage; the height provides the pressure head that pushes water through miles of pipes to your faucet without pumping.

5. Blood Vessel: Poiseuille Flow

Here's where fluid mechanics gets medical. Poiseuille's equation describes viscous flow through tubes, and the key insight is devastating: flow rate depends on radius to the fourth power [7].

A 50% stenosis (narrowing) doesn't cut flow in half--it cuts it by about 94%. This explains why even moderate arterial plaque causes serious problems. The simulation shows how stenosis creates:

• Accelerated flow through the narrowing (blue = low pressure)
• Increased wall shear stress (damages endothelium)
• Dramatically reduced overall flow

Clinical relevance: Cardiologists use pressure-velocity relationships to assess stenosis severity. The simulator's Reynolds number output indicates turbulence risk--healthy arteries maintain laminar flow (Re < 2000).

Key Equations Reference

Typical Fluid Properties

Learning Objectives

After working through this simulator, you should be able to:

1. Explain the inverse relationship between fluid velocity and pressure using Bernoulli's principle
2. Calculate velocity changes through varying cross-sections using the continuity equation
3. Apply Pascal's principle to determine force multiplication in hydraulic systems
4. Predict exit velocities from tanks using Torricelli's theorem
5. Analyze the dramatic effect of vessel diameter on viscous flow using Poiseuille's equation
6. Interpret color-coded pressure fields to identify high and low pressure regions

Exploration Activities

Activity 1: Venturi Flow Measurement

1. Set inlet diameter to 50mm, throat to 25mm, velocity to 2 m/s
2. Record the pressure drop shown
3. Use Bernoulli's equation to calculate expected pressure drop manually
4. Compare your calculation to the simulator output
5. Vary throat diameter and observe how pressure drop changes with area ratio squared

Activity 2: Aircraft Lift Investigation

1. Start with 0 degree angle of attack--note zero lift
2. Increase angle to 5 degrees, record lift force
3. Double the airspeed from 50 to 100 m/s
4. Verify lift increased by factor of 4 (velocity squared relationship)
5. Find the angle where the thin-airfoil approximation becomes inaccurate (lift stops increasing linearly)

Activity 3: Hydraulic Advantage Verification

1. Set small piston to 20mm, large to 100mm
2. Calculate mechanical advantage (area ratio = diameter ratio squared = 25)
3. Apply 100N input force
4. Verify output shows 2500N (25x multiplication)
5. Consider: what's the tradeoff for this force gain?

Activity 4: Blood Flow and Stenosis

1. Start with 0% stenosis, record flow rate
2. Increase stenosis to 25%, 50%, 75%
3. Calculate the expected flow reduction using the r^4 relationship
4. At what stenosis level does turbulence become likely (Re > 2000)?
5. Explain why even moderate stenosis causes significant symptoms

Challenge Questions

Basic:

1. If inlet velocity is 2 m/s and area ratio is 4:1, what is throat velocity?
2. A hydraulic jack has piston areas of 10 cm2 and 100 cm2. What force does 50 N produce?

Intermediate: 3. Water exits a hole 5 m below the surface. What is the exit velocity? 4. An artery narrows from 4mm to 2mm diameter. By what factor does flow velocity increase?

Advanced: 5. Design a Venturi meter to measure flow rates from 1-10 L/s with measurable (but not excessive) pressure drops 6. Calculate the lift generated by a 10m wing span at 250 km/h with CL = 0.5

View Answers

1. 8 m/s (continuity: V2 = V1 * A1/A2)
2. 500 N (Pascal: F2 = F1 * A2/A1)
3. 9.9 m/s (Torricelli: V = sqrt(2 * 9.81 * 5))
4. 4x increase (area ratio is 4:1, so velocity increases 4x by continuity)
5. Throat area ~1/4 of inlet area gives manageable pressure drops. For 10 L/s at 2 m/s inlet, need inlet diameter ~80mm, throat ~40mm
6. L = 0.5 * 1.225 * (69.4)^2 * 10 * 0.5 = 14,755 N or about 1500 kg of lift

Common Mistakes to Avoid

1. Confusing pressure and velocity direction: Higher velocity means lower pressure--counterintuitive but fundamental
2. Forgetting the area squared relationship: When diameter doubles, area quadruples, so hydraulic advantage is diameter ratio squared
3. Ignoring the r^4 dependence: In viscous flow, small diameter changes cause huge flow changes
4. Misapplying Bernoulli: Only valid for incompressible, inviscid, steady flow along a streamline
5. Equal transit time fallacy: Airplane lift is NOT because air must "meet" at the trailing edge--this is physically incorrect

Frequently Asked Questions

Q: Why doesn't the Bernoulli equation include viscosity? A: Bernoulli assumes ideal (inviscid) flow. For viscous effects in pipes and narrow vessels, use Poiseuille's equation or add head loss terms [8].

Q: How do airplanes fly upside down if lift depends on wing shape? A: Angle of attack matters more than shape. Inverted aircraft use negative angle of attack (relative to normal flight) to maintain lift downward relative to the aircraft [4].

Q: Why is blood flow so sensitive to vessel diameter? A: Poiseuille's equation shows flow proportional to r^4. This fourth-power dependence means a 10% diameter reduction cuts flow by 34% [7].

Q: Can I use Torricelli's theorem for gases? A: Only approximately. Gases compress, so the full analysis requires thermodynamics. For small pressure differences, it's reasonable [6].

Q: What's the difference between gauge and absolute pressure in these calculations? A: Bernoulli works with either, but you must be consistent. The pressure difference is what matters for flow, not absolute values [3].

Real-World Applications

About the Data

Property values in this simulator come from standard engineering references. Water properties follow NIST Chemistry WebBook data [9]. Air properties use ISA standard atmosphere values. Blood viscosity represents typical adult values; actual viscosity varies with hematocrit and temperature [7].

How to Cite This Simulator

For academic work, cite as:

Simulations4All. (2025). Fluid Pressure and Flow Simulator [Interactive web simulation]. Retrieved from https://simulations4all.com/simulations/fluid-pressure-flow

References

1. Munson, B.R., Young, D.F., & Okiishi, T.H. (2012). Fundamentals of Fluid Mechanics (7th ed.). Wiley.
2. White, F.M. (2015). Fluid Mechanics (8th ed.). McGraw-Hill. Chapter 3: Integral Relations.
3. Cengel, Y.A. & Cimbala, J.M. (2017). Fluid Mechanics: Fundamentals and Applications (4th ed.). McGraw-Hill.
4. NASA Glenn Research Center. (2021). "Incorrect Lift Theory." https://www.grc.nasa.gov/www/k-12/airplane/wrong1.html
5. Meriam, J.L. & Kraige, L.G. (2012). Engineering Mechanics: Statics (7th ed.). Wiley.
6. Torricelli, E. (1644). Opera Geometrica. Historical derivation of efflux theorem.
7. Fung, Y.C. (1996). Biomechanics: Circulation (2nd ed.). Springer. Chapter 3: Blood Rheology.
8. Bird, R.B., Stewart, W.E., & Lightfoot, E.N. (2006). Transport Phenomena (2nd ed.). Wiley.
9. NIST Chemistry WebBook. Thermophysical Properties of Fluid Systems. https://webbook.nist.gov/chemistry/fluid/

Verification Log