# 伯努利方程

流体动力学和压强-速度关系的交互式可视化

> 标准页面: https://elysiatools.com/zh/visualizations/bernoulli-equation

- **分类:** Physics

## 概述

Interactive visualization of Bernoulli's equation demonstrating fluid dynamics, pressure-velocity relationship, and Venturi effect. Features the fundamental equations: P + ½ρv² + ρgh = constant (Bernoulli equation showing energy conservation per unit volume), A₁v₁ = A₂v₂ (continuity equation for incompressible flow), v₂ = (A₁/A₂)v₁ = (D₁/D₂)²v₁ (velocity relation from continuity), P₂ = P₁ + ½ρ(v₁² - v₂²) (pressure relation from Bernoulli), Q = A₁v₁ (volume flow rate), ṁ = ρA₁v₁ (mass flow rate). Real-time visualization includes: (1) Venturi tube animation showing pipe constriction (inlet → throat → outlet) with smooth radius transitions, animated flow particles with velocity-dependent speed (faster in throat section), color-coded particles (blue=slow, red=fast), pressure gradient visualization showing high pressure (red) at inlet, low pressure (blue) at throat, pressure recovery at outlet; (2) Manometer tubes displaying liquid heights proportional to local pressure, real-time pressure readings P₁, P₂, P₃ with proper units (Pa), dynamic liquid level changes responding to parameter adjustments; (3) Streamline visualization showing smooth flow paths through varying cross-section, velocity vectors with magnitude indication, laminar flow patterns with particle tracking; (4) Pressure distribution chart P(x) showing pressure drop in constriction and recovery, velocity distribution chart v(x) showing velocity increase in throat, both with real-time updates and key point markers. Interactive parameters: fluid type (water ρ=1000 kg/m³, air ρ=1.225 kg/m³, oil ρ=900 kg/m³, gasoline ρ=720 kg/m³), inlet velocity v₁ (1-20 m/s), inlet pressure P₁ (50,000-200,000 Pa), inlet diameter D₁ (5-20 cm), throat diameter D₂ (2-15 cm), tube length (50-200 cm). Display options: show/hide flow particles, streamlines, pressure gradient color map, velocity vectors, manometer tubes. Presets include: Water Flow (standard water parameters), Air Flow (low density, high velocity), High Velocity (enhanced visualization), Extreme Contraction (maximum pressure drop). Educational content covers Bernoulli equation derivation (energy conservation along streamline, work-energy principle for fluids), key concepts (pressure energy P, kinetic energy ½ρv², potential energy ρgh, speed-pressure trade-off), Venturi effect physics (constriction accelerates flow, pressure drops in throat, applications in flow measurement), real-world applications (airplane wings lift, carburetors, perfume atomizers, chimney draft, sailing), limitations (inviscid assumption, incompressible flow, steady flow, laminar vs turbulent), and historical context (Daniel Bernoulli 1738, Euler refinement, Venturi's 1797 work on pipe flow). Multi-language support (zh, en, de, fr, es, pt, ru).

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