Apply an in-depth understanding of Fluid Mechanics fundamentals - 7013MAA Finite Volume Method for CFD

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Assignment Brief 

Module Title

Computational Fluid Dynamics

Individual

 

Cohort 

September

Module Code

7013MAA

Coursework Title

Finite Volume Method for CFD

Hand out date: 20/09/2021

Lecturer

Due date:

Aula:

Estimated Time (hrs): 20 hrs

Page Limit*: 8 pages

Coursework type: 

Written report

 5 credits

 

Mark and   Feedback date: 01/11/2021

Mark and Feedback method: Individual written feedback, rubrics marking and class discussion

Submission details

  • Reports must be submitted via Aula
  • Submit report as .pdf or .docx files only
  • Equations must be typeset using the native MS Word equation editor (not the Equation Editor add-on)
  • The maximum size of the report is 8 pages (tittle page, abstract, nomenclature, table of contents, list of figures, if included, do not count towards the 8-page size limit)
  • The report must be written using Arial 12pt and all margins must be 1.27 cm (‘narrow’ setting in MS Word). Paragraph text should be justified to the left and right margins
  • Equations must be numbered sequentially using the ‘(N)’ style, where ‘N’ is the equation number ο‚·       No appendices are allowed 
  • Source code files (MATLAB, Octave, Python, C++, etc.) must also be submitted (as separate files). For compiled languages, compilation instructions must be provided
  • References must be included using the APA reference style (more information can be found on the library’s website)
         

Module Learning Outcomes Assessed (in red):

  1. Apply an in-depth understanding of Fluid Mechanics fundamentals.
  2. Demonstrate an understanding of the Finite Volume Methodology as implemented for Computational Fluid Dynamics applications.
  3. Effectively utilise Computational Fluid Dynamics (CFD) software in order to simulate flow configurations relevant to industrial and/or research applications.
  4. Demonstrate the ability to analyse and interpret CFD safely (in this context safely implies that the interpretation of results must be physically valid)
  5. Produce technical report in order to communicate CFD results effectively and accurately through the use of appropriate graphical representation/s and written interpretation of results.

1.0 Introduction

This is an individual coursework. The purpose of this coursework is to investigate the effect that the choice of numerical schemes for convection terms has on the solution of numerical problems using the Finite Volume Method (FVM). A secondary objective is to implement solutions to a simple 1D convection-diffusion problems using computational tools. Essentially, you will be programming a simple CFD code. Finally, the entire process will be critically discussed in writing and it will be presented via a formal technical report. The procedure to complete this coursework is summarised below and further explained in this brief.

  1. To develop a discretised solution for the 1D diffusion-convection equation
  2. To develop a numerical implementation of the diffusion-convection equation using the central difference scheme and the Finite Volume Method (FVM)
  3. To carry out a literature investigation to identify the general conditions and requirements for numerical schemes. From this investigation an upwind-biased scheme will be identified for implementation.
  4. To modify the initial numerical implementation (2) to incorporate an upwind-biased scheme (identified in step 3 above) for convective terms only (central difference should be applied to diffusion terms)
  5. To critically discuss and present findings via a formal technical report. 

2.0 Discretised solution of the convective-diffusion equation using the using the Central Difference scheme and the Finite Volume Method

The first task involved in this course is the development of a general analytical solution for the model 1D convection-diffusion equation:

                                                                                 𝑑                       𝑑        π‘‘πœ™

                                                                                 (πœŒπ‘ˆπœ™) =        (Γ        )

                                                                                𝑑π‘₯                     𝑑π‘₯       𝑑π‘₯

And the flow must also satisfy the 1D continuity equation

𝑑

 (πœŒπ‘ˆ) = 0

𝑑π‘₯

For the development of the discretised general solution, the flow can be assumed to be incompressible, and the velocity field known. The discretised solution must be developed within the Finite Volume Method framework using the Central Difference scheme on all terms. Students are expected to use the classic notation P, E, W, e and w to denote the point of interest, the cellcentre point of the cell to the East, the cell-centre point of the cell to the West, and the east and west cell-face values, respectively.

3.0 Development of a numerical solution and its implementation using the Central Difference scheme and the Finite Volume Method

Once a general discretised solution for the diffusion-convection equation in 1D has been developed. A numerical algorithm to solve the equation (preferably using MatLab or Octave, however, other programming languages will also be accepted) will be developed using the Finite Volume Method and the Central Difference Scheme (for all equation terms). This code will be developed to solve the following 1D convection-diffusion problem:

 

                                                                                                                                         π‘₯=0                 π‘₯=𝐿

                                                                                                                            Figure 1 - System domain and boundary conditions

Where πœ™π΄ and πœ™π΅ are the boundary conditions for the transported property πœ™. Finally, the velocity, U, can be assumed to be known and constant throughout the domain. The analytical solution to this scenario is given by:

 πœŒπ‘ˆπ‘₯

                                                                                    πœ™ − πœ™0                𝑒 Γ − 1

                                                                                = 

                                                                                   πœ™πΏ − πœ™0           π‘’πœŒπ‘ˆπΏΓ − 1

The specific boundary conditions and problem variables are individualised based on student names. They are provided in the table below: (FIRST LETTER OF MY NAME IS ‘F’ , FIRST LETTER OF MY SURNAME IS ‘E’.

First letter of name or surname

Based on First letter of your name

Based on the first letter of your surname

A – D

𝜌 = 1 π‘˜π‘”/π‘š3

𝐿 = 1 π‘š

Γ = 0.1kg/m/s

πœ™π΄ = 1

πœ™π΅ = 0

π‘ˆ = 0.1m/s and 2.5 m/s

E – G

𝜌 = 1.1 π‘˜π‘”/π‘š3

𝐿 = 1.1 π‘š

Γ = 0.2kg/m/s

πœ™π΄ = 2

πœ™π΅ = 1

π‘ˆ = 0.2m/s and 3 m/s

H – L

𝜌 = 1.2 π‘˜π‘”/π‘š3

𝐿 = 1.2 π‘š

Γ = 0.3kg/m/s

πœ™π΄ = 3

πœ™π΅ = 2

π‘ˆ = 0.3m/s and 3.5 m/s

M – S

𝜌 = 1.3 π‘˜π‘”/π‘š3

𝐿 = 1.3 π‘š

Γ = 0.3kg/m/s

πœ™π΄ = 1

πœ™π΅ = 0

π‘ˆ = 0.4m/s and 4 m/s

T – Z

𝜌 = 1.4 π‘˜π‘”/π‘š3

𝐿 = 1.4 π‘š

Γ = 0.4kg/m/s

πœ™π΄ = 2

πœ™π΅ = 1

π‘ˆ = 0.35m/s and 3.25 m/s

It is recommended to start developing the numerical solution with a small number of cells (say 10 cells) and then increase the number of cells. Also, for simplicity make sure that the cells are equally spaced so that the implementation of the code is not complicated by having to include the capability to deal with various cell sizes. Also, notice that the code is to be tested at two different velocity conditions. Students are also asked to solve the 1D convection-diffusion on several meshes by increasing the total number of cells (a minimum of 3 grid densities are to be included). Apply an in-depth understanding of Fluid Mechanics fundamentals - 7013MAA Finite Volume Method for CFD Finally, the solutions obtained with your code must be compared with the analytical solution provided above.

Notes on MatLab & Octave

Notice that the coefficients for each cell can be stored in a 2-dimensional array of any size (only limited by your computer’s RAM memory). This process can be automated, for example, by using for loops and embedded loop. For example, in pseudo-code

                                                            For i = length(mesh variable)

                                                    A(i) = value to be assigned

                             End

Also, the system of equation can be conveniently written in matrix form

                               [A][πœ™]=[B]

These linear systems of equations can be readily solved using Matlab’s inverse division command ‘’ and, therefore, their solution does not need to be coded (i.e. using low level code within your script/s)

4.0 Literature investigation on convective discretisation schemes

When solving the 1D convection-diffusion problem you will notice, that the solution using the central difference scheme can become unstable or oscillatory in comparison to the analytical solution. This is the result of using this scheme for convective terms. As a result, you are also tasked with carrying out a literature survey and discuss 3 upwind-biased schemes. You will then choose one of these schemes and you will extend/re-write your code to include your chosen scheme to discretise convective terms only (diffusive terms should again be discretised using central difference).

5.0 Reporting the findings

The final step for this coursework is to present your findings via a written formal report. This report should succinctly discuss the various steps followed in order to develop your numerical code. It must also discuss the lessons learnt in regards to the effects of grid density and choice of discretisation scheme for convection-diffusion problems. Make sure to include high-quality graphs comparing your computed results with the analytical solution provided above, making sure to discuss any discrepancies. Apply an in-depth understanding of Fluid Mechanics fundamentals - 7013MAA Finite Volume Method for CFD. For information on how to export MatLab figures into other graphic formats please refer to MatLab’s user guide (the plot, print and hold commands will be particularly useful)

Notes:

  1. You are expected to use the APAReference Style. For support and advice on this students can contact Centre for Academic Writing (CAW).
  2. Please notify your registry course support team and module leader for disability support.
  3. Any student requiring an extension or deferral should follow the university process as outlined here. 
  4. The University cannot take responsibility for any coursework lost or corrupted on disks, laptops or personal computer. Students should therefore regularly back-up any work and are advised to save it on the University system.
  5. If there are technical or performance issues that prevent students submitting coursework through the online coursework submission system on the day of a coursework deadline, an appropriate extension to the coursework submission deadline will be agreed. This extension will normally be 24 hours or the next working day if the deadline falls on a Friday or over the weekend period.

This will be communicated via your Module Leader.

  1. Assignments that are more than 10% over the page limit will result in a deduction of 10% of the mark i.e. a mark of 60% will lead to a reduction of 6% to 54%. The word limit includes quotations, but excludes the bibliography, reference list and tables.
  2. You are encouraged to check the originality of your work by using the draft Turnitin links on your Moodle Web.
  3. Collusion between students (where sections of your work are similar to the work submitted by other students in this or previous module cohorts) is taken extremely seriously and will be reported to the academic conduct panel. This applies to both courseworks and exam answers.
  4. A marked difference between your writing style, knowledge and skill level demonstrated in class discussion, any test conditions and that demonstrated in a coursework assignment may result in you having to undertake a Viva Voce in order to prove the coursework assignment is entirely your own work.
  5. If you make use of the services of a proof reader in your work you must keep your original version and make it available as a demonstration of your written efforts. 
  6. You must not submit work for assessment that you have already submitted (partially or in full), either for your current course or for another qualification of this university, unless this is specifically provided for in your assignment brief or specific course or module information. Where earlier work by you is citable, i.e. it has already been published/submitted, you must reference it clearly. Identical pieces of work submitted concurrently will also be considered to be selfplagiarism.

Mark allocation guidelines to students (to be edited by staff per assessment)

Report

 

 

  

Discretised solution (general form)

 

15

Code implementing solution to discretised equations

 

20

Literature review of 3 upwind-biased discretisation schemes

 

15

Code adaptation/expansion to include upwind-biased scheme

 

5

Discussion of results – to include the effect of grid spacing and choice of discretisation scheme on numerical solutions using the FVM

 

20

Plots comparing computational results and analytical solution

 

15

Presentation and structure – including typesetting of equations

 

5

Referencing (APA)

 

5

Total

 



 100%

 Apply an in-depth understanding of Fluid Mechanics fundamentals - 7013MAA Finite Volume Method for CFD


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