MECH3750: Engineering Analysis II
Assignment II: Numerical Modelling of the
Convection-Diffusion Equation
Aim
The aim of this assignment is to synthesise your understanding of parabolic and hy-
perbolic partial differential equations by developing a numerical scheme that models a
physical process where both diffusion and convection are present. Your solutions are to
be implemented in Python and formally documented in a report.
Learning Objectives
This assignment supports the following learning objectives, as listed in the Electronic
Course Profile:
1.1 Understand which types of mathematical model are appropriate for different sys-
tems.
1.2 Model systems using algebraic equations, ordinary differential equations, partial
differential equations and integral equations.
1.3 Construct system models based on rough descriptions of mechanical engineering
situations or problems.
2.4 Compute solutions to partial differential equations using a spectrum of analyt-
ical and numerical methods including separation of variables and finite volume
method.
3.1 Interpret the results of analysis in terms of the behaviour of the physical system
it models.
3.3 Report on the results of analysis in a required format.
4.3 Apply new techniques to engineering applications by implementing them in Python
programs.
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The Problem
COVID-19 has claimed more than 4.7 million lives worldwide [1]. In the early stages
of the pandemic, it was widely believed that airborne transmission of the SARS-CoV-2
virus did not represent a significant risk [2]. However, mounting scientific evidence (e.g.
[3]) pointed to the reality of virus transmission by aerosols (e.g. diameter O (10µm))
and droplets (e.g. diameter O (100µm)) which are ejected from an infectious host via
sneezing or coughing. The behaviour of these particles in the air (see Figure 1) is
heavily dependent on their size, and yet the definitions of both aerosols and droplets
vary in the literature [4].
Figure 1: Computational fluid dynamics simulation of aerosol and droplet transmission
due to a cough, both (below) with and (above) without a mask [5].
With a limited number of assumptions, the distribution of virus-carrying aerosols and
droplets in the air can be modelled as a combination of diffusive and convective pro-
cesses. The unforced, one-dimensional convection and diffusion equations are,
∂u
∂t
+ v
∂u
∂x
= 0,
∂u
∂t
= D
∂2u
∂x2
,
in which u is a function of x and t, v is a velocity that is only dependent on x and t,
and D is a diffusion coefficient. The influence of gravity would act to bias the diffusion
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towards the ground and manifests as a downwards drift velocity, while prevailing cur-
rents would transport the particles horizontally.
Consider the two-dimensional scenario shown in Figure 2, which is 10m wide and 4m
high. The person standing is carrying SARS-CoV-2 and shedding the virus. At some
point in time, that person experiences a fit of coughing which releases 106 particles into
the air. It is assumed here that these particles are initially stationary and all located at
the centre of the domain (i.e. location A), and that they are free to leave the domain
from the sides and top. Your task is to predict the propagation of particles in the
air and report on the particle count at the location of the sitting person’s head (i.e.
location B), in the time following their release.
Figure 2: The location of two people, one standing and one sitting, at the time of the
coughing fit. The air velocity profile increases, on average, from ground level.
1. Define a model equation, including the associated initial and boundary conditions,
that could be used to represent the physical scenario that has been described. [5
marks]
2. The diffusivity of the particles in the air is 0.01m2/s. Assuming that the diffusion is
unbiased (i.e. not influenced by gravity), develop a numerical scheme to predict the
particle count at location B for one minute after their release. Generate contour plots
of the particle distribution in the domain at 15 s, 30 s, 45 s and 60 s after release.
Also create a graph of the particle count at location B with time. Does the count
exceed a threshold value of 10 in this period? If so, when does this occur? Generate
solutions using explicit, implicit, and Crank-Nicolson schemes. [30 marks]
3. Repeat your calculations from Part 2, in order to answer the same questions, using
a drift velocity. This biases the diffusion of the particles due to the influence of
gravity. Undertake this simulation for aerosol particles with a diameter of 10µm
and 20µm, which in accordance with Stokes’ Law have settling velocities of approx-
imately −0.01m/s and −0.04m/s, respectively. Choose only one numerical scheme
for these computations, based on your findings in Part 2. [10 marks]
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4. Assuming that the air velocity is horizontal only and can be modelled as,
vw (h) =
−25
100
(
h
4
)0.2(
1 +
4
5
sin
2pih
4
)
m/s,
repeat your calculations from Part 3, for both the 10µm and 20µm aerosols, after
including the effect of the wind. How do your results change if the diffusion coefficient
is halved to 0.005m2/s? [10 marks]
5. Repeat one calculation (e.g. with d = 10µm and D = 0.01m2/s) from Part 4 for at
least one other choice of grid spacing. Compare your results and comment on the
grid-dependence of your numerical predictions. Hints: Be reasonable, here. If you
try to run with a grid spacing of 1mm you’re most likely going to have a bad time.
Also, include something in your code that tells you how far into your simulation you
are. [5 marks]
The Report
Document your work in a formal report that includes, but is not necessarily limited to,
the following:
i. Introduction: A brief description of the problem you have been asked to solve;
ii. Methodology: The definition of your approach to solving this problem, including
all working and relevant assumptions;
iii. Results: The appropriate presentation of results, which might include figures,
graphs and tables;
iv. Concluding Remarks: A critical discussion of your approach to the problem and
your findings. At a minimum this should include comment on the stability, con-
vergence, accuracy and computational efficiency of your scheme.
Submission
You will be required to submit your report and your code. It is expected that your code
will be neatly structured and well documented so that it can be run and interrogated
during the marking process. A Turnitin submission link for the report will be made
available on Blackboard. Your code should be run from the single main.py Python
script that contains a function for each of Parts 2, 3 and 4 (e.g. def PART2(...):),
and submitted via a push to the MECH3750 GitHub Classroom. The link to create your
repository in the GitHub Classroom is https://classroom.github.com/a/dwHbkTW6.
The due date and time applies to both the report and your code (i.e. if either is late,
then your submission is late).
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References
[1] https://coronavirus.jhu.edu/map.html
[2] https://www.forbes.com/sites/jvchamary/2021/05/04/
who-coronavirus-airborne/?sh=46739df54472
[3] https://www.thelancet.com/article/S0140-6736(21)00869-2/fulltext
[4] Vuorinen et al. (2020) Modelling aerosol transport and virus exposure with nu-
merical simulations in relation to SARS-CoV-2 transmission by inhalation indoors,
Safety Science, Volume 130, 104866.
[5] https://www.ansys.com/covid-19-simulation-insights
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