I understand that the constant in the Courant-Friedrichs-Lewy condition is defined as $\mathrm{CFL} = \frac{u \Delta t}{\Delta x}$, where $u$ is the principal coefficient. I came across this post: Estimating the Courant number ..., which basically suggests that the CFL number can be used to decide on the time step size to ensure stability and convergence. I have seen mentioned in many papers that the results were obtained with for instance $\mathrm{CFL} = 0.5$, which would suggest that the CFL is a constant.
But how is the CFL number defined if the coefficient $u$ changes with time? I am particularly interested in the solution of hyperbolic inviscid Euler equations, where $u = u$, $u+a$ or $u-a$, where the velocity $u$ varies with time and $a$ is the speed of sound.