Week 5: Advanced Exercises

Advanced exercises are designed for advanced learners that are ready to take on extra challenges. These exercises are made accessible to all after the class. Note that these exercises typically require more knowledge than what was presented in class.

Adv Exercise: 1D fluid simulation by cellular automata

This week, we’ve learnt that we program cellular automata by specifying rules that will be run by each cell. We will use this property to write a 1D fluid simulation.

Explanation of logic

Use a 1D array to represent the height of water columns in a tank, with each cell representing a column of water in the tank. We will specify rules so that the water will be passed between columns naturally. We will iterate the whole logic over a range of timesteps. At each timestep, we will iterate through each cell and calculate the height of the cell at the next timestep.

The logic is simple to put in words. We check the difference in flow potential for each cell to decide how much fluid flows between the cells in the next timestep. We calculate this by taking the difference between potential energy from height of water, and also the difference in fluid momentum between cells.

Instructions

1. Set N = 15, GRAV = 2, and DAMP = 0.5. N represents grid length, GRAV represents the gravity multiplier, and DAMP represents the damping coefficient for fluid momentum.

2. Initialize four numpy arrays called water, mo_left, mo_right, and dwater, with length set to N=15.

   • Initialize water with the values [100, 100, 100, 80, 70, 60, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20]

   • Initialize the other arrays with all zeros.

   • water will be used to store the height of the water column.

   • mo_left will store the flow of each cell to its neighbor on the left in the previous timestep, representing flow momentum to the left.

   • mo_right will store the flow of each cell to its neighbor on the right in the previous timestep, representing flow momentum to the right.

   • dwater stores the change in water height for the next timestep.

3. Write a for-loop that iterates from 1 to N-1, excluding 0 and N which represents the boundary cells. In this for-loop, calculate for the current cell:

   • fp_left: GRAV * (current cell height - left cell height) + (current cell leftward momentum - left cell rightward momentum)

   • fp_right: GRAV * (current cell height - right cell height) + (current cell rightward momentum - right cell leftward momentum)

   • flow_left and flow_right: fp_left / 4 and fp_right / 4

   • If flow_left is larger than the height of the current cell divided by 4, set it to that.

   • Do the same for flow_right.

4. Still in the same for-loop, assign the flows for the next timestep.

   • If flow_left is positive i.e. larger than zero, update dwater, mo_left and mo_right for the cells:

     • Add flow_left to dwater of the left cell

     • Subtract flow_left from dwater of the current cell

     • Subtract DAMP multiplied by flow_left to mo_left of current cell

     • Add DAMP multiplied by flow_left to mo_right of the left cell.

   • If flow_right is positive i.e. larger than zero, do similar:

     • Add flow_right to dwater of the right cell

     • Subtract flow_right from dwater of the current cell

     • Subtract DAMP multiplied by flow_right to mo_right of current cell

     • Add DAMP multiplied by flow_right to mo_left of the left cell.

5. After this for-loop is done, add code after it for our boundary cells.

   • For the left-most cell, copy-paste the contents of the loop except for everything that has to do with having a cell on the left.

   • For the right-most cell, copy-paste the contents of the loop except for everything that has to do with having a cell on the right.

6. Check momentum of boundary cells reflecting off the walls.

   • if mo_left of leftmost cell is positive, multiply it by -1.

   • if mo_right of rightmost cell is positive, multiply it by -1.

7. Update water by adding to it dwater.

8. Write a bigger for-loop that wraps everything in steps 3 to 7, and iterate over 200 timesteps. Create a variable called water_stored, and initialize it to be an empty list outside this bigger for-loop. At the end of the loop, append water to it. This will serve to store the heights of water over time.

9. If you can get to step 8, ask the instructor for code to visualize this.