only need Q2 done
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. 1 The Problem The “chemtrail” [1] conspiracy theory claims that the condensate trails (see Figure 1) left in the air by aircraft are actually comprised of chemical and or biological agents, and that these agents are deliberately distributed by nefarious actors (e.g. the government) for reasons undisclosed to the public (e.g. mind control). Despite a lack of basis in fact, the theory persists in certain sections of the global community [2, 3], to the point that a number of reputable scientific bodies [4, 5] have been compelled to debunk or refute the concept. Figure 1: Condensation trails from an aircraft which are alleged to actually be “chem- trails”. For the purposes of this assignment, pretend that chemtrails are real, in which case the distribution of agents in the atmosphere could be modelled as a combination of diffu- sive and convective processes. 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 towards the ground and manifests as a downwards drift velocity, while prevailing winds would transport the agents horizontally. Consider the two-dimensional scenario shown in Figure 2, which is 10 km wide and includes the first 4 km of the atmosphere. At some point in time, 1000 kg of agent is 2 released into the atmosphere at the location shown in Figure 2. Your task is to predict the propagation of agent in the atmosphere and report on the concentration measured just above the ground at location, A, in the time following its release. 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 agent in air is 0.5 m2/s. Assuming that the diffusion is unbiased (i.e. not influenced by gravity), develop a numerical scheme to predict the change in concentration of agent at location, A, in the eight days after its release. How long does it take for the concentration at location, A, to exceed 0.1 mg/m3? What is the concentration distribution just above the ground and in the air at that time? Assume a unit thickness of 1 m for your numerical model. Generate solutions using explicit, implicit and Crank-Nicolson schemes. [30 marks] 3. Repeat your calculations using a drift velocity of 0.01 m/s, which biases the diffusion of the agent due to the influence of gravity. Choose only one scheme, based on your findings in Part 2. [10 marks] 4. Assuming that the air velocity is horizontal only and can be modelled as, vw (h) = 3 100 ( h 4000 )0.2( 1 + 1 2 sin 2πh 400 ) m/s, repeat your calculations again, after including the effect of the wind. [10 marks] 5. Using more than one grid spacing, determine whether or not your numerical predic- tions for Part 4 are grid independent. It would be ideal if you could show that they were. [5 marks] Figure 2: The location of the plane and observation point at the time of agent release. Location A is just above ground level and the wind velocity profile increases, on average, from ground level. 3 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 PART1(...):), and submitted via a push to the MECH3750 GitHub Classroom. The link to create you repository in the GitHub Classroom is https://classroom.github.com/a/TWlS-l-I. The due date and time applies to both the report and your code (i.e. if either is late, then your submission is late). References [1] https://www.oed.com/view/Entry/318007 [2] https://www.theguardian.com/environment/2017/may/22/ california-conspiracy-theorist-farmers-chemtrails [3] http://www.holmestead.ca/chemtrails/petition-2013.html [4] https://keith.seas.harvard.edu/chemtrails-conspiracy-theory [5] https://www.epa.gov/regulations-emissions-vehicles-and-engines/ information-contrails-aircraft 4 https://classroom.github.com/a/TWlS-l-I https://www.oed.com/view/Entry/318007 https://www.theguardian.com/environment/2017/may/22/california-conspiracy-theorist-farmers-chemtrails https://www.theguardian.com/environment/2017/may/22/california-conspiracy-theorist-farmers-chemtrails http://www.holmestead.ca/chemtrails/petition-2013.html https://keith.seas.harvard.edu/chemtrails-conspiracy-theory https://www.epa.gov/regulations-emissions-vehicles-and-engines/information-contrails-aircraft https://www.epa.gov/regulations-emissions-vehicles-and-engines/information-contrails-aircraft Aim Learning Objectives The Problem The Report Submission