# High speed winds blowing over the ocean

## Homework Statement

Hello. This is not a homework question (my school will never offer a fluid dynamics course), but I figure this would be the best place to ask it.

I am interested in writing a computer model of high speed winds blowing over the ocean - or any body of water. I am wondering how I might do this. I have only a... how should I put it, "MCAT style" introduction to fluid dynamics, and so it may seem a bit silly of me to be trying something so complex. Nonetheless...

I thought I'd model the wind and the water in 2-dimensions, like looking at a rectangular slice of the water-wind intersection... like looking through a camera half in and half out of water. On this rectangular slice, I'd initially have the upper half have some uniform density rho and the lower half have some uniform density 10 * rho, to model the different densities of wind and ocean water.

What I'm not sure about exactly are two things: 1) how to model the wind and 2) which fluid equations to use. For 1), I thought that I'd just have the velocity field of the upper half of the slice (wind) be uniformly to the right (positive) and the bottom half (water) be uniformly zero. But I wasn't sure if keeping the velocity -constant- throughout the simulation would work or be sensible... might I just initially have the velocity field set up in some way, and then solve the equations based on that? If I keep the velocity field constant, then a lot of the terms in various fluid equations disappear...

For 2), I think I'll use:

## Homework Equations

Euler's equations of inviscid gas dynamics!

## The Attempt at a Solution

I think Euler's equations apply because the wind and the water are Newtonian and I can model them as inviscid. The fluid is definitely compressible because the Mach number would be around 1 or above, etc. But I'm not positive that Euler's equations would be best. A few things about Euler's equations from wikipedia made me wonder, for instance, the energy equation was confusing (is e, the internal energy, a dynamical variable I have to monitor as well as pressure, density, and velocity?).

Also, for the wind-water system I'm trying to simulate, would the ideal gas law be a good equation of state? If so, what would the adiabatic index (gamma) be?

Sorry if this is rambling. Any help, criticisms, ideas etc. would be awesome.

## The Attempt at a Solution

I dont know much myself on the subject, but since high speed winds will be blowing, you will have to use bernoulli's equation as there will be a pressure gradient...

You will also have to take viscosity into account. The contact layers of both fluids will be slower than the other layers above them, and this will lead to the formation of waves as faster layers come in contact and the slower ones are left behind... like one big circle...

As the layers pick up speed, they will encounter resistance from the water in front of them. I think you can apply conservation of linear momentum in the vertical direction here as initially, the system was in equilibrium and no net vertical force is applied here...

Dunno if that helps or not...

If I assume the contact region to be without shear (slip surface), by perturbing the density at the contact region (with a wave) I can generate waves.

I'm mainly looking to simulate the complex shapes formed by Kelvin-Helmholtz instabilities beyond the simple sinusoidal beginnings. I don't think I need to consider viscosity to do this, even though in the real ocean it is obviously viscosity that even leads to waves starting up! You are right.

As I understand it Bernoulli's equation is only for steady flow. I've realized by now that keeping the velocity fields constant would be ridiculous, so dv/dt is not zero everywhere and furthermore rot v is not necessarily zero (anywhere) either. So Bernoulli's equation cuts out too much...

I've been thinking more about this, and now I'm wondering about something else. I know what the initial setups of the velocity field and density field will be, but I do not have any idea right now of what the pressure field and energy field will be initially. I think that using Euler's equations I'd have a total of 4 dynamic fields to take care of, three scalars (density, pressure, internal energy) and one vector (velocity). Will knowing the initial conditions for two of these fields (density and velocity) be enough to let me simulate the evolution of the system? Or need I also know about the initial setup of the other two fields?

How would I put together these initial setups? For density and velocity here it seems straightforward to me: water is more dense than the wind, uniformly; the wind is uniformly moving, the water is not. But I'm not sure about pressure or internal energy initially.

Thank you!