I am relatively new to studying in-depth fluid mechanics, so do bear with me. My skills in math are quite lacking, so I've been spending considerable time trying to hone my skills with DEs and Calculus. In the isentropic, unsteady potential flow Bernoulli equation, I'm trying to mathematically understand how to apply measured values to this equation. An image is included in this link that shows the equation I'm working with.
The first term I believe to be frame velocity potential, of which I am unsure. The second I understand it to be synonymous with V^2/2, instead this time with the dot product of two gradients of velocity potentials. If that's correct, what would that look like, mathematically? Additionally, my flow scenario should generally be simple, as it's a compressible flow through a circular straight tube, ideally with the total gradient flow in one direction. The third term I know to be potential, which should be zero as there is no change in height. The fourth I know is basically heat enthalpy of a compressible isentropic gas.
What I'm trying to solve for in my system is generally what the velocity profile in this tube would look like over time given that we know the initial and final pressures and densities? How would I apply such an equation to actual engineering design and basic simulation?