Text provided under a Creative Commons Attribution license, CC-BY. All code is made available under the FSF-approved BSD-3 license. (c) Lorena A. Barba, Gilbert F. Forsyth 2017. Thanks to NSF for support via CAREER award #1149784.

12 steps to Navier–Stokes


This Jupyter notebook continues the presentation of the 12 steps to Navier–Stokes, the practical module taught in the interactive CFD class of Prof. Lorena Barba. You should have completed Step 1 before continuing, having written your own Python script or notebook and having experimented with varying the parameters of the discretization and observing what happens.

Step 2: Nonlinear Convection


Now we're going to implement nonlinear convection using the same methods as in step 1. The 1D convection equation is:

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0$$

Instead of a constant factor $c$ multiplying the second term, now we have the solution $u$ multiplying it. Thus, the second term of the equation is now nonlinear. We're going to use the same discretization as in Step 1 — forward difference in time and backward difference in space. Here is the discretized equation.

$$\frac{u_i^{n+1}-u_i^n}{\Delta t} + u_i^n \frac{u_i^n-u_{i-1}^n}{\Delta x} = 0$$

Solving for the only unknown term, $u_i^{n+1}$, yields:

$$u_i^{n+1} = u_i^n - u_i^n \frac{\Delta t}{\Delta x} (u_i^n - u_{i-1}^n)$$

As before, the Python code starts by loading the necessary libraries. Then, we declare some variables that determine the discretization in space and time (you should experiment by changing these parameters to see what happens). Then, we create the initial condition $u_0$ by initializing the array for the solution using $u = 2\ @\ 0.5 \leq x \leq 1$ and $u = 1$ everywhere else in $(0,2)$ (i.e., a hat function).

In [1]:
import numpy                 # we're importing numpy 
from matplotlib import pyplot    # and our 2D plotting library
%matplotlib inline


nx = 41
dx = 2 / (nx - 1)
nt = 20    #nt is the number of timesteps we want to calculate
dt = .025  #dt is the amount of time each timestep covers (delta t)

u = numpy.ones(nx)      #as before, we initialize u with every value equal to 1.
u[int(.5 / dx) : int(1 / dx + 1)] = 2  #then set u = 2 between 0.5 and 1 as per our I.C.s

un = numpy.ones(nx) #initialize our placeholder array un, to hold the time-stepped solution

The code snippet below is unfinished. We have copied over the line from Step 1 that executes the time-stepping update. Can you edit this code to execute the nonlinear convection instead?

In [ ]:
for n in range(nt):  #iterate through time
    un = u.copy() ##copy the existing values of u into un
    for i in range(1, nx):  ##now we'll iterate through the u array
    
     ###This is the line from Step 1, copied exactly.  Edit it for our new equation.
     ###then uncomment it and run the cell to evaluate Step 2   
      
           ###u[i] = un[i] - c * dt / dx * (un[i] - un[i-1]) 

        
pyplot.plot(numpy.linspace(0, 2, nx), u) ##Plot the results

What do you observe about the evolution of the hat function under the nonlinear convection equation? What happens when you change the numerical parameters and run again?

Learn More

For a careful walk-through of the discretization of the convection equation with finite differences (and all steps from 1 to 4), watch Video Lesson 4 by Prof. Barba on YouTube.

In [2]:
from IPython.display import YouTubeVideo
YouTubeVideo('y2WaK7_iMRI')
Out[2]:
In [3]:
from IPython.core.display import HTML
def css_styling():
    styles = open("../styles/custom.css", "r").read()
    return HTML(styles)
css_styling()
Out[3]:

(The cell above executes the style for this notebook.)