The compressible flow project is based on numerical analysis work and is done in groups of up to four students. All numerical work in the project is done in the commercial CFD software STARCCM+. The project is a pass/failtype of course element and an approved project is needed to get the final course grade. The project can result in total seven bonus points for the written exam at the end of the course. However, the maximum number of bonus points will only be rewarded to groups that complete all parts of the project on time and pass the assessment without revision. If you do not hand in your deliverables on time, you will not get any bonus points. If revision is needed, you will still get bonus points but depending on the severity of the revision the total number may be modified. A detailed scheme for the project assessment and criteria for getting bonus points can be found below.
There are in total seven cases to choose from for the compressible Flow project. Each of these cases is connected to two project group numbers according to the specification in the casedescription list below.
Project Cases  
Case 1 (project groups 1 & 8) Inviscid flow through engine intake: The flow field inside a supersonic engine intake (SR71 Blackbird) will be calculated both analytically and numerically. The flow is assumed to be steady and twodimensional. 

Case 2 (project groups 2 & 9) Unsteady and steady inviscid compressible flow around airfoil: The unsteady flow developing around an airfoil after an impulsive start will be simulated numerically. The solution should (after some time) converge towards a steadystate flow condition. The flow is assumed to be twodimensional. In addition, a supercritical airfoil is investigated and results are compared. 

Case 3 (project groups 3 & 10) Inviscid compressible underexpanded nozzle flow: The steady state flow in and outside of a convergent nozzle including possible shocks downstream of the nozzle will be calculated numerically. The flow is assumed to be steady, twodimensional, and axisymmetric. 

Case 4 (project groups 4 & 11) Inviscid compressible overexpanded nozzle flow: The steady state flow in and outside of a convergentdivergent nozzle including possible shocks inside and downstream of the nozzle will be calculated numerically. The flow is assumed to be steady, twodimensional, and axisymmetric. 

Case 5 (project groups 5 & 12) Supersonic flow over a biconvex airfoil: The supersonic flow over a biconvex airfoil is investigated both numerically and analytically. The numerical predictions are compared with the analytical solution and available experimental data. 

Case 6 (project groups 6 & 13) Inviscid supersonic flow over a 2D compression/expansion ramp: The supersonic flow over a compression/expansion ramp is investigated both numerically and analytically. The numerical predictions are compared with the analytical solution. The detached shock limit is investigated. 

Case 7 (project groups 7 & 14) Supersonic flow over a gradual 2D compression ramp: The supersonic flow over a gradual compression ramp is analysed numerically. Both inviscid and viscous simulations are done and results are compared. 
Intended Learning Outcomes
The learning outcomes for the course Compressible Flow TME085 are listed below. Learning outcomes for the Compressible Flow course not directly addressed in The Compressible Flow Project are greyed.
 Define the concept of compressibility for flows
 Explain how to find out if a given flow is subject to significant compressibility effects
 Describe typical engineering flow situations in which compressibility effects are more or less predominant (e.g. Mach number regimes for steadystate flows)
 Present at least two different formulations of the governing equations for compressible flows and explain what basic conservation principles they are based on
 Explain how thermodynamic relations enter into the flow equations
 Define the special cases of calorically perfect gas, thermally perfect gas and real gas and explain the implication of each of these special cases
 Explain why entropy is important for flow discontinuities
 Derive ( marked) and apply (all) of the presented mathematical formulae for classical gas dynamics
 1D isentropic flow
 Normal shocks
 1D flow with heat addition
 1D flow with friction
 Oblique shocks in 2D
 Shock reflection at solid walls
 Contact discontinuities
 PrandtlMeyer expansion fans in 2D
 Detached blunt body shocks, nozzle flows
 Unsteady waves and discontinuities in 1D
 Basic acoustics
 Solve engineering problems involving the abovementioned phenomena (8.a  8.k)
 Explain how the incompressible flow equations are derived as a limiting case of the compressible flow equations
 Explain how the equations for aeroacoustics and classical acoustics are derived as limiting cases of the compressible flow equations
 Explain the main principles behind a modern Finite Volume CFD code and such concepts as explicit/implicit time stepping, CFL number, conservation, handling of compression shocks, and boundary conditions
 Apply a given CFD code to a particular compressible flow problem
 Analyse and verify the quality of the numerical solution
 Explain the limitations in fluid flow simulation software
 Report numerical analysis work in form of a technical report
 Describe a numerical analysis with details such that it is possible to redo the work based on the provided information
 Write a technical report (structure, language)
 Search for literature relevant for a specific physical problem and summarize the main ideas and concepts found
 Present engineering work in the form of oral presentations
General Instructions
In order to avoid conflicts related to the project work and the efforts put down by each individual group member, it is recommended that each group agrees on the level of ambition before starting to work on the project  set up a contract if needed.
When finished, the project reports should be uploaded on Canvas (only pdf format accepted). The reports should be named as follows:
 groupXX_literature_survey.pdf
 groupXX_preliminary_report.pdf
 groupXX_final_report.pdf
where XX should be replaced with your group number.
When uploaded, the reports will be checked for plagiarism using URKUND.
The final report should contain a section where the contribution of each individual group member is described in some detail.
In the end of the course there will be an oral presentation session where each group will present the case that they have studied and the results. Each group will have 10 minutes at their disposal for the presentation and additionally 5 minutes will be used for questions. Detailed assessment criteria for the presentation session can be below.
Project Outline
In essence the project work is divided in four parts (all compulsory) described in detail below. Assessment criteria for each of the four project parts can be found here.
 Literature survey (week 14):
The first part of the project is to do a literature survey for the chosen case. The literature survey should be summarized in a report that is to be handed in week 4.
You are expected to find theory and background information relevent for your specific case and sumarize your findings in a 23 page report (see Assessment Criteria and Writing Instructions for more details).
Note: The literature report will later be reused in the writing of the introduction/background section of the project report.  Numerical analysis work (week 37):
It is recommended to start with the numerical analysis work first after completing Assignment 2 since that assignment is an introduction to STARCCM+, which is the software to be used for the numerical analysis part of this project.
Note: In the preliminary project report (to be handed in week 5) you should describe the methodology that you have applied: mesh strategy, numerical scheme and solver settings, boundary conditions, etc. If you have obtained preliminary results, those can be included as well. The purpose of the preliminary report is to review your project work at an early stage and thus make sure to avoid unnecessary mistakes, i.e. you are not expected to present the final version of your simulations in this report.  Project report (week 58):
After the numerical analysis part of the project is completed, the work should be presented in the form of a technical report (no more than ten pages). A suggested structure for the report is given here. The project report shall be handed in week 8.  Oral presentation (week 8):
The last part of the project is to make an oral presentation of the project and the obtained results. The presentations are done groupwise and it is important that all group members are active in the presentation.
The figure below gives a schematic overview of project activities over the eightweek course period. The indicated deliverables are: D0 group identification and case selection, D1 literature survey report, D2 preliminary project report, D3 oral presentation, and D4 final project report. The indicated feedback are: F1 feedback on the literature survey report, F2 feedback on the preliminary project report, F3 feedback on the oral presentation, and F4 feedback on the final project report. In the numerical analysis activity, C13 are scheduled consultations.
Project Assessment
The assessment of the numerical analysis project will be done using the criteria presented below. Note that the numbers within parenthesis at the end of each description indicate which of the learning outcomes that are addressed. Criteria marked with a , do not have to be fulfilled in order to pass this course element. However, it is those criteria that can result in bonus points for the final exam (the number of bonus points for each of these criteria is indicated in the list). In addition, there are specific tasks for each case for which instructions are given in the case description document. To get the maximum number of bonus points it is necessary to meet all the marked criteria, hand in reports on time, participate in the oral presentation session and pass the compressible flow project course element without a major revision, i.e. projects that needs a major revision will not be awarded with bonus points even if the marked criteria are fulfilled. A minor revision may lead to a withdrawal of 13 bonus points depending on the severity of the mistakes made.
Literature survey (project deliverable D1)
 Relevant literature for the problem found (17)
 At least five relevant references used for the literature survey report (1p)
 The literature survey report gives a summary of ideas and concepts relevant for the chosen case (16, 17)
Preliminar numerical results (project deliverable D2)
 This report is just a project status check and will not be assessed. It is a way to identify misstakes (if any) at an early stage.
Oral presentation (project deliverable D3)
 Wellstructured presentation (18)
 Case description (2, 3, 12, 13, 16, 18)
 Geometry described
 Boundary conditions are described
 Numerical approach described (12, 13, 14, 15, 16, 18)
 Mesh strategy and mesh quality is discussed
 Numerical scheme
 Convergence criteria (if steadystate) and chosen CFLnumber are specified
 Findings from numerical accuracy are presented (if applicable)
 Results and discussion
 Discussion on physical properties of the flow (2, 3, 18)
 References to literature (if applicable) (15, 16, 17)
 Presentation finished on time (10 minutes per group + 5 minutes for questions) (18)
 All members in the presenting group should be able to answer basic questions on the flow studied and the numerical method applied (mesh quality, mesh strategy, numerical scheme, etc.) (2, 3, 12, 13, 15, 16)
Project final report (project deliverable D4)
General project structure
 The report follows the suggested structure or similar (no more than ten pages) (16)
 The report describes the case studied and related compressible effects (2, 3)
 It is possible to redo the analysis using the information in the report (boundary conditions, mesh strategy, solver settings, etc.) (12, 13, 16)
 The report should present details about the analysis work such as convergence criteria (for steadystate cases), chosen CFL number, chosen numerical scheme, etc. (12, 13, 14, 15, 16)
 Software limitations are addressed (15)
 The literature survey material is used to support the conclusions drawn from the results obtained in the numerical analysis work (3, 16, 17) (2p)
Numerical analysis work
 A sound mesh strategy is used (13, 14, 15)
 High resolution is used where needed (9, 15)
 The mesh is not unnecessarily fine in flow regions where only large scales needs to be resolved
 Mesh stretching is used in a constructive way (In the report, figures showing the mesh for the whole domain and near the investigated object should be included)
 Numerical accuracy is adressed (12, 13, 14, 15)
 At least three meshes with significant difference in cell count and/or meshing strategy were evaluated (2p)
 At least two numerical schemes have been used (study the influence of numerical scheme on the solution for one case and one grid  order of accuracy, flux type) (2p)
Suggested Report Structure
1. Introduction
Introduction to the studied problem:
 Description of the application
 Relevance of compressible flow effects for the application
 References to published work related to your case (these publications can report work done on for example methods related to your work, investigations of related applications (numerical and/or experimental), investigations of related flow phenomena, etc.)
2. Method description
Describe your simulation approach in some detail:
 Mesh strategy (if a mesh sensitivity study has been done, the strategy for the study should be given here but the differences in the obtained results should be presented and discussed in the results and discussion sections)
 Chosen Numerical scheme (time and space. If a investigation of the effects on the predicted flow field of different numerical schemes has been done that should be reported here but the corresponding affect on the obtained results should be presented and discussed in the results and discussion sections)
 Turbulence model (described chosen approach if applicable)
 Convergence criteria (if steadystate)
 Chosen CFL number
 Software limitations
3. Case description
Describe details of your case in terms of:
 Geometrical definition (schematic view of the case geometry, main features, difficulties, etc.)
 Boundary conditions (in/outflow conditions, wall treatment, etc)
 Hand calculations (if applicable)
4. Results
Present results from the simulations:
 Discuss numerical accuracy (if applicable)
 Mesh dependency study (discuss differences in results as a consequence of mesh quality)
 Compare results obtained using different numerical schemes (discuss the effects of numerical scheme on the flow predictions for your specific case)
 What was the main findings when it comes to choice of mesh quality and numerical schemes
 Present results for best (believed to be most accurate) configuration (mesh, scheme)
 Discuss your results in physical terms (main flow features)
5. Discussion
 Summarize your study
 Did you expect to get the result that you got (based on what you have learned about compressible flows in the course and based on literature related to your specific case)
6. Conclusions
 Short overview of study
 Main findings
 Reflection on results
7. Division of Work (if more than one group member)
 Description of the contribution to the project from each of the group members