Well, firstly you need to decide how accurate you want your wind modeling to be, but generally, the less complicated an equation you can use for your desired level of accuracy, the better. It'll end up being a vector field, but I would probably make simple rules for how the wind interacts with objects in a given tile and the tiles around it, rather than trying to model fluid dynamics precisely.

Another approach might be a ray-tracing style approach, where the rays can bounce off object geometry and lose some momentum, objects can respond to ray impacts and gain momentum, and you do some vector math to figure out what happens when rays and moving objects intersect.

The exact approach depends on the specifics of your game, and the "right" answer might be neither of these.

Air is a compressible fluid, so ultimately you'll need Navier-Stokes, but actually figuring out how to translate those equations into code is infamously annoying, so definitely check out an existing solution instead of trying to implement from the equations. Foster & Metaxas is an old a grid-based solution. Theirs is designed for incompressible fluids (water), but I think it should be fine if you raise β₀.

There's a lot of math in the paper, but it's easier to implement than it looks (each step, run [2] on every cell once, then run [6] [7] [8] on every cell ~6 times), and it's mostly just transcribing equations. It's super fiddly though, just incredibly fiddly, expect to have to spend a lot of time debugging.

There are probably suitable simpler solutions too though, so definitely spend some time searching for them (or just copyable implementations of Foster & Metaxas) before diving into that.

I forgot most of my simulation subject, but we did a 2D particle simulator and yes, you will need Navier-Stokes if you want the effect to be physically correct. You will need to solve a numerical problem each frame to compute turbulences. Wind is a force with a source that will propagate depending on its parameters (like viscosity)

If you don't implement that, you won't be able to implement things like walls that block wind and such, which I guess is something you might had in mind. Also, having a changing environment is probably pretty hard, so prepare yourself.

I just took a look at the source code of The Powder Toy, and the class Air (src/simulation/Air.hpp/cpp) seems to implement what you need. Sadly it is not very well documented, so you'd have to work out the relation between code and physical formulation.

Sometimes physically incorrect simulations are easier to implement, have lighter computations, give the designer more control of the game, and makes the player able to better predict how the game will behave and have more fun.