I am studying jet mixing noise and I am given a velocity field, with $V_x$, $V_y$, $V_z$ in the Cartesian coordinates. I wanted to know if there is a method by which I could find some information relating to the sound level produced from the pressure field.
Since the flow considered represents jet mixing, it is most likely turbulent. Let us assume incompressibility of the flow at the moment for simplicity. Therefore, the Bernoulli's equation cannot be used directly to compute the pressure field given the velocity field. In order to obtain the pressure field, the Navier Stokes equation or the Poisson pressure equation will have to be solved by applying the appropriate boundary conditions.
The specification of a velocity field can help define the flow regimes that can give clues to the underlying physics, but the relevant physics depends greatly on the velocity magnitudes. There are various approaches to sound generation that are dependent upon the flow regimes, and it's impossible to focus on any particular one without knowing the governing flow regimes, such as laminar, turbulent, incompressible, compressible, sub sonic, supersonic, etc. In other words, you need to refine your question and especially, provide the flow conditions in addition to the actual values of the velocity field magnitudes, if you want useful input from us.