But then I understand turbulence, and I love Applied Math. So maybe I can suggest something that might interest you.
You can change the density of a liquid mixture by mixing liquids of different densities. Fresh water and sea water. Water and oil. Cooking oil and Petroleum oil. Honey and molasses. Water, mercury and liquid gold. Etc.
However, since they are incompressible, you change the mixture, you change the volume of the mixture. So your matrix of equal squares may have to be modified to changing sizes of squares too. Better for more chaotic presentation. Or, if you want to maintain the equal squares, then change the height/thickness of the typical square in a changed mixture. Three-Dimension! Better still for more chaotic/turbulent effect.
The change in volume doesn't have to be due to a mixture of different liquids. Increasing or decreasing the volume of the same, single liquid will do.
Instead of a matrix of squares, why not a combination of rows of flows? The rows may have the same width (like the wood flooring in a basketball court) or varying widths. Each row is a different sinusoidal wave (sine or cosine waves). The rows are of varying amplitudes, periods, and/or vertical shifts. The rows may be of squares of varying heights---the heights following the sinusoidal wave in their respective rows.
You may want to create a laminar flow to be transitioned into a turbulent flow here. In the laminar section, use almost similar waves of longer periods and lower amplitudes--or flatter waves. And in the turbulence, use various waves of very short periods and higher amplitudes---for higher and closer peaks and dips.