Bernoulli's Equation: Pressure and Velocity
In 1738, Daniel Bernoulli published his masterwork Hydrodynamica, establishing the fundamental relationship between fluid velocity and pressure. Along a streamline in an ideal fluid: P + ½ρv² + ρgh = constant. This elegantly simple equation says that when fluid speeds up, its pressure must drop — and when it slows down, pressure rises. This inverse relationship between velocity and pressure is the heart of how wings generate lift.
How an Airfoil Creates Lift
An airfoil (wing cross-section) is shaped so that air travels a longer path over the curved upper surface than the flatter lower surface. Combined with the angle of attack, this forces air over the top to accelerate. By Bernoulli's principle, faster air means lower pressure. The resulting pressure difference — low above, high below — creates a net upward force: lift. The lift equation L = ½ρv²SC_L quantifies this, where C_L depends on the airfoil shape and angle of attack.
Understanding the Simulation
This visualization shows streamlines flowing around an airfoil shape. Notice how streamlines are compressed (closer together) above the wing, indicating faster flow and lower pressure (shown in blue/cyan). Below the wing, streamlines are spread apart — slower flow, higher pressure (shown in red). The green lift arrow shows the net upward force. Increase the angle of attack to see more deflection and higher lift — until you approach stall, where the flow separates from the upper surface.
Beyond Simple Bernoulli
While Bernoulli's principle captures an essential truth about lift, the complete picture involves circulation theory (Kutta condition) and the fact that the wing deflects air downward (Newton's third law). Race car wings are inverted airfoils generating downforce to improve traction. At speeds above Mach 0.3, compressibility effects mean Bernoulli's incompressible equation needs modification — this is where transonic and supersonic aerodynamics begin.