Using Euler's equations to model ocean waves

[Signifier]

Hello, I am trying to use Euler's equations to model on a computer the waves that form in the ocean when high-speed winds blow overhead. I'm modeling it in two dimensions, like looking through a camera that is half below and half above the water surface, as winds blow along the water. The lower half of the rectangular slice I'm modeling on has a uniform density of ten times the upper half, initially. This is to represent the upper half as air and the lower half as water. Also, as the wind is blowing and the water is initially static, the initial velocity field is in the upper half the velocity is uniformly to the right (positive, with some magnitude) and in the lower half the velocity is uniformly zero. I've included a simple picture to show what I mean (color indicates density).

This is a complex system to model and I am wondering about a few things. I hope not to reveal my ignorance too nakedly, but so it goes.

1. What state equation should I use to close the system? I am using Euler's equations because I am trying to model the compressible but inviscid flow. I will most likely use the ideal gas equation to close Euler's equations, but I am confused about how to approach the adiabtic index that occurs in the ideal gas equation. I am trying to model two fluids, open air and water. Need I use two adiabatic indexes? Although initially differentiating between wind and water is simple (the top half is wind, the bottom half water), I'm not sure how to make this distinction after the system has evolved the the density, pressure, energy and velocity fields start changing. Or might I use just one adiabatic index for all points in the system at all times?

The ideal gas equation links the pressure, density and internal energy at a point as pressure = (gamma - 1) * density * internal energy with gamma the adiabatic index.

2. I am not sure about how to set up the pressure or internal energy fields initially. The upper half of the rectangular slice is the less dense moving air, the lower half the more dense stationary water (all initially). What sort of pressure distribution (or equivalently, internal energy distribution, because with the equation of state the one determines the other) could correspond to this initial distribution of densities and velocities...? Or do I even need to know this? Is the velocity and density distribution enough to close the system?

I am sorry if this is not the right place to ask these questions, or if these questions are too hefty and not pared down enough. Any help from anyone with more knowledge than I on the subject would be greatly appreciated.

 

[Almsco]

It sounds like you’re on the right track with using Euler’s equations to model the waves in the ocean. As for your questions, it's important to differentiate between the two fluids when modeling this system. Since air is typically more compressible than water, you will likely need to use two different adiabatic indices, one for each fluid. It is also important to consider the pressure and internal energy fields initially. You can use the ideal gas equation to calculate the pressure and energy of each fluid, given the initial density and velocity. It would also be helpful to consult any related literature that has studied similar models of waves in the ocean to better understand how to set up your initial pressure and internal energy fields. Good luck with your project!