Energy conversion of water hammer

Water hammer refers to the shock wave that occurs when liquid is flowing in a pipe, as the kinetic energy of the flow is converted into pressure energy due to reasons such as the sudden closure of a valve or a change in flow velocity. This process can be described by the law of conservation of energy. Assume that at the initial moment, the flow velocity of the liquid in the pipe is v1, its density is ρ1, and its pressure is P1 ； After the water hammer occurs, the fluid velocity becomes v2, the density becomes ρ2, and the pressure becomes P2. According to the law of conservation of energy, in the absence of external forces and friction, the total energy of a system remains constant. Thus, we have: Initial total energy = Final total energy, that is: (1/2)ρ1v1^2 + P1 + ρ1gh1 = (1/2)ρ2v2^2 + P2 + ρ2gh2. Here, (1/2)ρ1v1^2 and (1/2)ρ2v2^2 represent the kinetic energy of the liquid, while ρ1, ρ2, h1, and h2 represent the density of the liquid and the height differences, respectively; P1 and P2 represent the pressures of the liquid. This is the energy conversion formula for water hammer, which describes the process by which the kinetic energy of fluid flow is converted into static pressure energy. .

Hard work. I thought about it further based on your reasoning: since v2 ends up being 0, p2 increases. But this is an endpoint value; p2 represents the pressure when the valve is closed. In fact, due to the pressure buildup caused by the closing of the valve, p2 ends up increasing. My reasoning is as follows: a = Δv/Δt (where Δt is the time it takes to close the valve). When the valve is closed instantly, the acceleration a is very large. Using the formula F=ma, the pressure at the valve closure point can be calculated as F = pressure before the valve * cross-sectional area + v/time to close the valve * mass of liquid in the pipe before the valve. From this equation, it can be seen that two factors influence the intensity of water hammer: ① the shorter the time it takes to close the valve, and ② the greater the mass of liquid in the pipe before the valve

I think it’s not accurate to use Bernoulli’s equation for water hammer. Bernoulli's equation is used for fluids with constant flow. The actual flow rate changes before and after water hammer. Here are the diagrams I drew. This question isn’t one to answer by oneself; after posting it yesterday, I spent the whole afternoon working on it