How does a river freeze when the water keeps moving? It's cold outside right now, and the biggest river in the country has frozen over. We're talking about a minimum of 500m in width, and I've no idea how deep (but some pretty big ships can sail there).
And this got me wondering - how does a big river like that freeze over? When water freezes, it starts off small - thin membranes of ice, tiny grains, etc. But the water is constantly moving. Any paper-thin layer ice that could form would be broken apart immediately. And yet, given the right conditions, it can somehow freeze over thickly enough that a car can be safely driven over.
So how does this process happen? How can a large, moving, undulating river just freeze over without the ice breaking apart as it does so?
 A: You know that ice is less dense than water. Then, water that freezes will stay at the surface. Also, take in mind that water will only freeze on the surface. Then as you said, any ice that forms will break apart. But there are places on the river where these tiny pieces of ice can accumulate, forming mushy ice. There are many studies of mushy ice and ice formation on the north pole if you want to check. If the temperatures stay under the fusion point of the river water, the mushy ice will eventually become layers of ice. Once the layers form in the low-velocity stream places they will start growing and expanding. Because ice is solid, water will doge the ice, then, if the layer is thick enough water won't break it and will eventually manage to cover the whole river. Only the surface will freeze. Beneath the ice water still flows normally. You'll be surprised at how thin can the ice layer be without breaking.
To simulate this, I'll start with the Navier-stokes equations for the water motion. Then, you have to add some advection-diffusion heat transport equations. To couple them to the fluid just make the density of water a function of temperature. You could try to make viscosity also a function of temperature but that can result in some numerical instabilities. For the phase change, the enthalpy method is quite simple to implement. You know that most of the water on the river will be almost at the fusion point but won't lose enough energy to become solid. Just the water on top will. Then, you have to take into account that you have 3 mediums. Ice, water, and air. All those have different densities and heat capacitances that are important for the simulation. You'll notice that ice will become some sort of heat insulator preventing the underneath water to freeze after a certain thickness. Then the difficult part. You want the ice to move and stay on top.
So, you can have 2 approaches.
The first one is to use some boundary tracking method to separate ice from water. Then, calculate what the drag force would be and the buoyancy so the ice floats and apply that to the boundary to move it around. As you probably noticed, the problem with this approach is that the boundary has to move, and that can be tricky.
The second way is using some Brinkmann penalization for the fluid but you'll need an extra auxiliary field for the ice, and somehow, you have to calculate the forces from the fluid velocity field and pass them over to the auxiliary field. At last, you have to implement a fracture model for the ice. This model won't be that hard because ice is very well studied and there are plenty already proposed.
I hope you got a general idea of the physical phenomena. If you have any questions I'm happy to answer them.
A: The river does not freeze in its moving parts. It begins to freeze from the banks in places where the water is motionless or nearly so. Then the frozen area gradually grows towards the center of the river. Also, pieces of the frozen bank break off, then eventually pile up, dam and create stationary areas where the water freezes further. This process can be clearly seen if you examine photographs of rivers over the course of a freezing winter:

In ideal channels that have constantly moving water, such as water pipes, then even if the temperature of the pipe is well below freezing the water will not freeze, so long as the pipe has a continuous flow.
A: A video like this is probably the most direct answer possible.
My interpretation: The river carries patches of floating ice, which eventually jam up and form a semi-closed surface. This significantly slows the water flow at the top, which allows the remaining liquid parts on the surface to freeze as well.
