I have been looking on Euler's equations for a while and can't grasp one thing.
Suppose we have initial system state with volumes of fluid "hanging" in air (time is frozen and equal to zero), each of them has its initial (x, y) coordinates and velocity vector (Vx, Vy). The space is divided in stationary volumes in which measurement takes place (Euler measurement model).
As I understand Euler's equations impose mutual restrictions between functions of velocity ( u(x, y, t) ), mass density ( m(x, y, t) ), and pressure ( p(x, y, t) ) at any instance of time for stationary volume of space with coordinates (x, y) that contains volume of fluid (in that instance).
So why each function has time parameter when equations must be solved for any instance of time (iteration process) ? We must have new set of these functions for any moment of time that satisfy equations. And it's unclear whether volume of space contains volume of fluid or not, how do we take this in consideration ?
And the second question is how state of each volume of fluid (its coordinates and velocity vector) is related to these functions ? I need a system that has some initial state (array of fluid volumes with their coordinates and velocities) and uses Euler's equations to compute next state, which becomes initial for next iteration and so on. Something like this:
I don't need math details, I just want to grasp basic idea in context of computer simulation (how method of simulation can be bounded to standard analytic equations). Most papers are very complicated, with additional physical factors and user interaction, and they don't cover this "link". I want to consider the easiest case and how to start iteration process (just idea and how it is usually done). Sorry if some things sound ridiculous, please adjust my statements if they are wrong. If my model of understanding is completely wrong just point me in right direction, thanks.