Lecture on Bernoulli's Equation

Introduction

• Fundamental aerodynamic equation: Bernoulli's equation
• Used to calculate pressures and velocities in incompressible flows.

Background

• Derived from the Euler equation for fluid flow based on Newton's second law (force = mass x acceleration).
• Euler equation: dp = -rhoVdV
• Assumptions: Steady flow, no gravity, no viscosity forces.

Derivation of Bernoulli's Equation

• Integrate Euler equation along a streamline between two points (1 and 2):
  • Integral of dp from p1 to p2 plus integral from v1 to v2 of rhoVdV = 0.
  • Develops into: p2 - p1 + rho(1/2 V2^2 - 1/2 V1^2) = 0.
  • Rearranged: p1 + 1/2 rhoV1^2 = p2 + 1/2 rhoV2^2.
• Along a streamline: p + 1/2 rhoV^2 = constant.
• Terminology:
  • Total Pressure (pt): Constant along streamline.
  • Static Pressure (p): Pressure at a point.
  • Dynamic Pressure (1/2 rho V^2).
• Known as Bernoulli's principle or Bernoulli's equation.

Historical Context

• Daniel Bernoulli:

  • Born in January 1700, Groningen, Netherlands.
  • Family from Basel, Switzerland, moved back when Daniel was 5.
  • Influenced by father Johann Bernoulli, a renowned mathematician.
  • Initially studied philosophy, logic, and medicine.
  • Academic career: Chair of mathematics at University of St. Petersburg.

• Significant Works:

  • Published Hydrodynamica in 1738, basis for Bernoulli’s equation.
  • Relationship with Euler who contributed to the derivation.
  • Awarded prestigious Prize of the Paris Academy of Sciences 10 times.

Practical Applications

• Pressure Distribution on Airfoils:

  • Pressure constant perpendicular to airfoil surface.
  • Pressure distribution helps derive velocity around the airfoil.

• Flight Speed Measurement:

  • Use of a Pitot Tube to measure total pressure.
  • Developed by Henri Pitot in 1732.
  • Static Pressure measured by a port in the fuselage.
  • Pitot-static tube measures dynamic pressure, derives flight speed V = sqrt (2q/rho).

Conclusion

• Bernoulli's equation applicable in incompressible flows, neglecting viscosity.
• Next lecture will explore compressible flows.

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Note: These points provide a high-level summary and reference to the key concepts and historical context presented in the lecture on Bernoulli's equation, including its derivation, historical significance, and practical applications.