Applications of Bernoulli's equation

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Bernoulli equation is an equation expressing the conservation of mechanical energy of a moving fluid obtained by integrating the equation of motion (i.e. Euler's equation) along the streamline when an ideal barotropic fluid undergoes steady motion under the action of a potential force. It was named after the famous Swiss scientist D. Bernoulli proposed it in 1738. For an incompressible homogeneous fluid in a gravity field, the equation is p+ρgz+(1/2) * ρv^2=C where p, ρ, and v are the pressure, density, and velocity of the fluid respectively. ； z is the vertical height ； g is the acceleration due to gravity. 　　Each term in the above formula represents the pressure energy p, gravitational potential energy ρg z and kinetic energy (1/2) of the unit volume of fluid respectively. * ρv ^2, during the movement along the streamline, the sum remains unchanged, that is, the total energy is conserved. However, the total energy (i.e., the constant value in the above formula) may be different between streamlines. For gases, gravity can be ignored and the equation simplifies to p+(1/2) * ρv ^2 = constant (p0), each term is called static pressure, dynamic pressure and total pressure respectively. Obviously, as the velocity in the flow increases, the pressure decreases ； As the speed decreases, the pressure increases ； When the velocity drops to zero, the pressure reaches its maximum (theoretically it should be equal to the total pressure). The lift force generated by the aircraft wing lies in the fact that the speed of the lower wing surface is low and the pressure is strong, and the speed of the upper wing surface is high and the pressure is small, so the resultant force is upward. According to this equation, the velocity can be obtained by measuring the total pressure and static pressure of the fluid, which becomes the principle of pitot tube speed measurement. In irrotational flow, you can also use irrotational conditions to integrate the Euler equation to obtain the same result but with different meanings. At this time, the constants in the formula remain unchanged in the entire flow field, indicating that the fluid on each streamline has the same total energy. The equation is applicable between any two points in the entire flow field. In viscous flow, viscous friction consumes mechanical energy and generates heat, and mechanical energy is not conserved. When promoting the use of Bernoulli's equation, the mechanical energy loss term should be added. 　　The picture shows an aerodynamic experiment to verify Bernoulli's equation.