# In actuator disk theory, why does changing the inertial reference frame work for Bernoulli's equation?

I was reading this website: https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node86.html, which goes over actuator disk theory, and it discusses using Bernoulli's equation in a frame of reference that moves with the actuator disk. I don't understand how this works mathematically. Let's say we're in a stationary reference frame and P0=10, u0=5, and P1 = 15. If we ignore the 1/2 and rho coefficients, we can find u1: 10 + 5^2 = 15 + u1^2 -> u1=4.47. Now let's do the same with a moving reference frame that is moving with a speed of 4 toward the oncoming fluid. Now P0=10, u0=9, and P1=15, so 10 + 9^2 = 15 + u1^2 -> u1 = 8.72. But, if we add the reference frame velocity to the original answer, we get 8.47. I'm clearly missing a nuance here, can someone help me to understand?

• You just wrote energy conservation between 1 and 2 with two different conditions (inlet velocity change from 5 to 9 m/s and same Delta p). This is not really surprising to find different Delta v since there is no linearity. In you equation (not consistent, please add at least density): u1 = sqrt(u0^2 + p0 - p1) – Acsed. Sep 4 '20 at 12:11