A shallow water simulator in a rectangular tub that uses the Navier-Stokes Equations to create an animation of how the surface of the water will move over time when given the initial state of the surface of the water.

• 1 - Explanation of Inputs
• 2 - Animations
  • (i) Water droplet as an input
  • (ii) Water wall as an input
• Improvements to Make

The calculations for the simulation as well as the function that does the calculations is contained in the 'shallow_water_simulation.py' file. The function that performs the calculation is shallow_water_simulation. It takes in the parameters (Lx, Ly, T, eps, dx, dy, dt, init_cond, H), where

• Lx: The width of the rectangular tub in meters.
• Ly: The length of the rectangular tub in meters.
• T: The total duration of the simulation in seconds (i.e. the simulation will show how the surface of the water behaves from t=0 to t=T).
• eps: The smoothing parameter. This determines how smoothed out the waves are after every calculation. This makes it so that the waves in the animation look smooth rather than jagged. Note that eps is short for epsilon.
• dx: Step size for the horizontal spatial dimension. Splits the grid into N = Lx/dx number of cells horizontally. Program calculates surface height at every grid point.
• dy: Step size for the vertical spatial dimension. Splits the grid into M = Ly/dy number of cells vertically.
• dt: Step size for the time dimension. Program calculates the state of the surface of the water after every time step from t=0 to t=T.
• init_cond: A numpy array containing the initial state of the surface of the water. It is an array containing 3 arrays in the form init_cond = [u_0, v_0, eta_init]. Each of the following arrays has M rows and N columns so that there are N elements horizontally and M elements vertically:
  • u_0: Contains the horizontal component of the velocity of the water at each grid point.
  • v_0: Contains the vertical component of the velocity of the water at each grid point.
  • eta_init: Contains the height of the water at each grid point.
• H: The bathymetry function. I.e., this is a function that returns the height (and thus the shape) of the surface of the bottom of the tub.

To input the initial state and create the animation, use the Animation code and results.py file.

The output of the simulation with a still water droplet (i.e., Gaussian peak) at some position in the tub initially:

The same output with a different surface colouring:

The output of the simulation with a still water "wall" at some position in the tub initially:

The same output with a different surface colouring:

• Work on animations (fill in under the plot, reduce height of initial water droplet/wall, and make waves easier to see).
• Combine all plot and animation code into one "create_animation" function.
• Create a user interface where changing the input of the simulation is a lot easier/more accesible to someone not familiar with programming OR offer a range of inputs to click on.
• Add in math & physics section for more details on how or why the values are calculated the way they are and what to expect when testing.