So what is the tiniest possible fluid simulation? Can we get anything interesting from a single cell? For this we'll be using a standard MAC grid, which means we'll represent horizontal velocities (red) on the vertical edges and vertical velocities (green) on the horizontal edges of each cell. - ThreadSky

vassvik.bsky.social • 87 days ago

So what is the tiniest possible fluid simulation?

Can we get anything interesting from a single cell?

For this we'll be using a standard MAC grid, which means we'll represent horizontal velocities (red) on the vertical edges and vertical velocities (green) on the horizontal edges of each cell.

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vassvik.bsky.social•87 days ago

Most real fluids can be reasonably approximated as incompressible fluids at standard atmospheric conditions, including gases and liquids.

This "incompressibility constraint" is typically modelled as requiring that the net flow (flux) through all the faces (edges in 2D) of each fluid cell is zero.

vassvik.bsky.social•87 days ago

The velocity field varies continuously in space, but in practice we treat the velocity as a fixed value for the each face of our cell. By placing the perpendicular velocity component on each respective cell face we just so happen to have the right ingredients to approximate the Divergence through it

vassvik.bsky.social•87 days ago

So our first step is to discretize the divergence, and estimate it in each cell (i, j) by using the horizontal (u) and vertical (v) velocity components on each adjacent face (which is half a cell size apart).

vassvik.bsky.social•87 days ago

We can assume our grid size is 1 without loss of generality (https://en.wikipedia.org/wiki/Without_loss_of_generality), to simplify the discretized Incompressibility constraint.

And then what?

vassvik.bsky.social•87 days ago

So let's say we explicitly specify the horizontal velocity on the leftmost vertical face, and all the other face components are zero.

In our case here we're interpolating the velocity field, where the velocity is defined to be zero out of bounds as well

Is this field (discretely) divergence free?

vassvik.bsky.social•87 days ago

Hopefully it's clear that it's not. There's no vertical velocity on the top/bottom faces, and the velocity on the rightmost face is zero, so the net flux cannot be zero. In fact the net flux is negative (local contraction). Anything that flows into the cell cannot flow out again.

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