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Bernoulli Flow Network

This example simulates a fluid dynamics network using Causal Monads to chain state updates across pipe segments.

How to Run

cargo run -p physics_examples --example bernoulli_flow_network

Expected Output

=== Bernoulli Flow Network Simulation ===

Fluid: Water (1000 kg/m³)
Flow Rate: 0.10 m³/s

[0] Reservoir       | P=200000.0 Pa | v=0.00 m/s | h=10.0 m
[1]       Main Pipe | P=194933.9 Pa | v=3.18 m/s | h=10.0 m
[2]         Venturi | P=118943.1 Pa | v=12.73 m/s | h=10.0 m  (Venturi: pressure drops as velocity increases)
[3]   Ground Outlet | P=293000.4 Pa | v=3.18 m/s | h=0.0 m   (Drop: potential energy → pressure)

=== Simulation Complete ===
Final State: Ground Outlet at P=293000.4 Pa, v=3.18 m/s, h=0.0 m

Physics Overview

The simulation tracks a fluid parcel moving through a pipe network, applying Bernoulli's Principle at each segment transition:

$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$$

This equation expresses conservation of energy along a streamline in an ideal (inviscid, incompressible) fluid.

Network Segments

Segment Description Diameter Height Key Physics
0 Reservoir Large (10 m) 10 m Starting point with high pressure potential
1 Main Pipe 0.2 m 10 m Velocity increases as flow enters narrower pipe
2 Venturi 0.1 m 10 m Venturi Effect: pressure drops as velocity increases in constriction
3 Ground Outlet 0.2 m 0 m Hydrostatic Gain: potential energy (height) converts to pressure

Key Principles Demonstrated

  1. Continuity Equation: $A_1 v_1 = A_2 v_2 = Q$ (constant volumetric flow rate)

    • As pipe area decreases, velocity must increase to maintain constant flow

  2. Venturi Effect (Segment 1→2):

    • Diameter halves (0.2m → 0.1m), area decreases by factor of 4
    • Velocity quadruples (3.18 → 12.73 m/s)
    • Pressure drops significantly (195 kPa → 119 kPa)

  3. Hydrostatic Pressure (Segment 2→3):

    • Height drops from 10m to 0m
    • Potential energy $\rho g h = 1000 \times 9.8 \times 10 ≈ 98$ kPa converts to pressure
    • Final pressure is highest despite starting lower

Simulation Configuration

Parameter Value Description
Fluid Density 1000 kg/m³ Water at STP
Flow Rate 0.1 m³/s 100 L/s volumetric flow
Initial Pressure 200 kPa ~2 atm
Initial Height 10 m Reservoir elevation

Causal Chain Architecture

The simulation uses CausalEffectPropagationProcess to model fluid state propagation:

Reservoir → Main Pipe → Venturi → Ground Outlet
    ↓           ↓          ↓          ↓
FluidState  FluidState  FluidState  FluidState

FluidState Structure

Each state tracks:

  • Pressure ($P$): Current static pressure in Pascals
  • Velocity ($v$): Current flow velocity in m/s
  • Height ($h$): Current elevation in meters
  • Diameter ($D$): Pipe diameter at this segment

Key Design Patterns

  1. First Bind: Uses the state parameter (initial FluidState from with_state)
  2. Subsequent Binds: Use the first closure parameter (prev_state.into_value()) which is the propagating FluidState from the previous segment
  3. Captured Variables: density and flow_rate_volumetric are captured from outer scope

Key APIs Used

API Purpose
deep_causality_physics::bernoulli_pressure Computes new pressure using Bernoulli equation
deep_causality_physics::Pressure Type-safe pressure values (Pa)
deep_causality_physics::Speed Type-safe velocity values (m/s)
deep_causality_physics::Length Type-safe distance values (m)
deep_causality_physics::Density Type-safe density values (kg/m³)
deep_causality_core::CausalEffectPropagationProcess Monadic effect propagation

Physical Verification

The simulation results can be verified:

  1. Continuity: At d=0.2m, A=π(0.1)²≈0.0314 m², so v=Q/A=0.1/0.0314≈3.18 m/s ✓
  2. Venturi: At d=0.1m, A=π(0.05)²≈0.00785 m², so v=0.1/0.00785≈12.73 m/s ✓
  3. Energy Conservation: Pressure gain at outlet ≈ ρgh = 1000×9.8×10 = 98 kPa

The final pressure (~293 kPa) is higher than initial (~200 kPa) because the elevation drop converts potential energy into pressure.