# An Efficient Strategy to Deliver Understanding of Both Numerical and Practical Aspects When Using Navier-Stokes Equations to Solve Fluid Mechanics Problems

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## Abstract

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## 1. Introduction

## 2. Numerical Elements of Computational Fluid Dynamics (CFD) Teaching

#### 2.1. Introduction

#### 2.2. Navier-Stokes Equations

#### 2.3. Vorticity-Streamfunction Governing Equations

#### 2.4. Boundary Conditions

#### 2.5. Discretization of Transport Equations

#### 2.6. Solution Algorithm

## 3. Practical Elements of CFD Teaching

#### 3.1. Introduction

#### 3.2. Interface Design Features

#### 3.3. Example to illustrate use of the Interface

#### 3.4. Implementation of the CFD Laboratories

- Getting started;
- CFD notation;
- CFD equations-continuity, momentum, energy, concentration of species;
- Finite differencing;
- The finite-volume method;
- Boundary conditions;
- Accounting for the pressure term;
- Closure of averaged equations;
- Time-stepping techniques; and,
- Properties of numerical methods.

## 4. Discussion

#### 4.1. Overall Course Description

- predicting drag forces on bluff, streamline bodies and flat plates;
- analyzing the flow in pipe systems;
- analyzing performance of radial flow pumps and turbines; and,
- matching pumps and turbines for particular applications.

#### 4.2. Use of Mathematica

#### 4.3. Computational Fluid Dynamics methodology

#### 4.4. Individual and Group Projects

#### 4.5. Online Questionnaire

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

- Assanis, D.N.; Heywood, J.B. Development and use of a computer simulation of turbo-compounded Diesel system for engine performance and component heat transfer studies. SAE Trans.
**1986**, 2, 451–476. [Google Scholar] - Devenport, W.J.; Schetz, J.A. Boundary layer codes for students in Java. In Proceedings of the ASME Fluids Engineering Division Summer Meeting, Washington, DC, USA, 21–25 June 1998. [Google Scholar]
- Zheng, H.; Keith, J.M. JAVA-based heat transfer visualization tools. Chem. Eng. Educ.
**2004**, 38, 282–290. [Google Scholar] - Rozza, G.; Huynh, D.B.P.; Nguyen, N.C.; Patera, A.T. Real-time reliable simulation of heat transfer phenomena. In Proceedings of the ASME-American Society of Mechanical Engineers-Heat Transfer Summer Conference, San Franscisco, CA, USA, 19–23 July 2009. [Google Scholar]
- Qian, X.; Tinker, R. Molecular dynamics simulations of chemical reactions for use in education. J. Chem. Educ.
**2006**, 81, 77–90. [Google Scholar] - Pieritz, R.A.; Mendes, R.; Da Silva, R.F.A.F.; Maliska, C.R. CFD: An educational software package for CFD analysis and design. Comput. Appl. Eng. Educ.
**2004**, 12, 20–30. [Google Scholar] [CrossRef] - Stern, F.; Xing, T.; Yarbrough, D.B.; Rothmayer, A.; Rajagopalan, G.; Otta, S.P.; Caughey, D.; Bhaskaran, R.; Smith, S.; Hutchings, B.; et al. Hands-on CFD educational interface for engineering courses and laboratories. J. Eng. Educ.
**2006**, 95, 63–83. [Google Scholar] [CrossRef] - Curtis, J.C. Integration of CFD into undergraduate chemical engineering curriculum. In Proceedings of the Annual AIChE Meeting, Philadelphia, PA, USA, 16–21 November 2008. [Google Scholar]
- Wolfram Research, Inc. Mathematica, Version 12.0; Wolfram Research, Inc.: Champaign, IL, USA, 2019. [Google Scholar]
- Jaeger, M.; Adair, D. Human factors simulation for construction management. Eur. J. Eng. Educ.
**2010**, 35, 299–310. [Google Scholar] [CrossRef] - Jaeger, M.; Adair, D. Communication simulation in construction management simulation: Evaluating the learning effectiveness. Australas. J. Eng. Educ.
**2012**, 18, 1–14. [Google Scholar] [CrossRef] - Patil, A.; Mann, L.; Howard, P.; Martin, F. Assessment of hands-on activities to enhance students’ learning for the first year engineering skills course. In Proceedings of the 20th Australasian Association for Engineering Education Conference, Adelaide, Australia, 6–9 December 2009; pp. 286–292. [Google Scholar]
- Edgar, T.F. Enhancing the undergraduate computing experience. Chem. Eng. Educ.
**2006**, 40, 231–238. [Google Scholar] - Adair, D.; Jaeger, M. Incorporating computational fluid dynamics code development into an undergraduate engineering course. Int. J. Mech. Eng. Educ.
**2015**, 43, 153–167. [Google Scholar] [CrossRef] - Parulekar, S.J. Numerical problem solving using Mathcad in undergraduate reaction engineering. Chem. Educ. Eng.
**2006**, 40, 14–23. [Google Scholar] - Rockstraw, D.A. Aspen Plus in the curriculum-suitable course content and teaching methodology. Chem. Eng. Educ.
**2005**, 39, 68–75. [Google Scholar] - Finlayson, B.A.; Rosendall, B.M. Reactor/transport models for design: How to teach students and practitioners to use the computer wisely. In AIChE Symposium Series; American Institute of Chemical Engineers: New York, NY, USA, 2000; pp. 176–191. [Google Scholar]
- Pomeranz, S. Using a computer algebra system to teach the finite element method. Int. J. Eng. Educ.
**2000**, 16, 362–368. [Google Scholar] - LeVeque, R.J. Finite Difference Methods for Ordinary and Partical Differential Equations: Steady-State and Time-Dependent Problems; Siam: Philadelphia, PA, USA, 2007. [Google Scholar]
- Ferziger, J.H.; Peric, M. Computational Methods for Fluid Dynamics; Springer: New York, NY, USA, 1996. [Google Scholar]
- Bozeman, J.D.; Dalton, C. Numerical study of viscous flow in a cavity. J. Comput. Phys.
**1973**, 12, 348–363. [Google Scholar] [CrossRef] - Adair, D.; Jaeger, M. Developing an understanding of the steps involved in solving Navier-Stokes equations. Math. J.
**2015**, 17, 1–19. [Google Scholar] [CrossRef] - Burggraf, O.R. Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech.
**1966**, 24, 113–151. [Google Scholar] [CrossRef] - Adair, D. Incorporation of Computational Fluid Dynamics into a Fluid Mechanics Curriculum. In Advances in Modeling of Fluid Dynamics; InTech: Berlin, Germany, 2012; ISBN 980-953-307-311. [Google Scholar] [Green Version]
- Gia, U.; Ghia, K.N.; Shin, C.T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys.
**1982**, 48, 387–411. [Google Scholar] [CrossRef] - Spotz, W.F. Accuracy and performance of numerical wall boundary conditions for steady, 2D, incompressible streamfunction vorticity. Int. J. Numer. Methods Fluids
**1998**, 28, 737–757. [Google Scholar] [CrossRef] - Wan, D.C.; Zhou, Y.C.; Wei, G.W. Numerical solution of incompressible flows by discrete singular convolution. Int. J. Numer. Methods Fluids
**2002**, 38, 789–810. [Google Scholar] [CrossRef] - Launder, B.E.; Spalding, B. Mathematical Models of Turbulence; Academic Press: London, UK, 1972. [Google Scholar]
- Agonafer, D.; Liao, D.G.; Spalding, B. The LVEL Turbulence Model for Conjugate Heat Transfer at Low Reynolds Number, Concentration; Heat and Momentum Ltd.: London, UK, 2008. [Google Scholar]
- Gamliel, E.; Davidovitz, L. Online versus traditional teaching evaluation: Mode can mater. Assess. Eval. High. Educ.
**2005**, 30, 581–592. [Google Scholar] [CrossRef] - Nulty, D.D. The adequacy of response rates to outline and paper surveys: What can be done? Assess. Eval. High. Educ.
**2008**, 33, 301–314. [Google Scholar] [CrossRef]

**Figure 1.**Summary of boundary conditions for cavity with moving lid. (Reproduced with permission from [22]).

**Figure 2.**Typical Cartesian mesh used for the lid-cavity flow. (Reproduced with permission from [21]).

**Figure 3.**Summary of Computational Fluid Dynamics (CFD) interface. (Reproduced with permission from [24]).

**Figure 4.**Geometry and contents of a 2D computational domain with steady turbulent flow of air and heat exchange.

**Figure 7.**Contour plot of streamfunction for lid-cavity problem. (Reproduced with permission from [22]).

Geometry | Solid and Other Fluid Boundaries, Special Shapes of Objects. |
---|---|

Physics | Incompressible/compressible fluid, which quantities to be solved for, closure of equations, materials within the computational domain, initial and boundary conditions |

Mesh | The choice here is Cartesian meshing or orthogonal meshing. The Cartesian mesh can be automatically generated or built by hand, and can be refined close to solid objects or in areas of high velocity gradient. Fine-grid embedding can also be used when appropriate. |

Numerics | Convergence monitoring, selection of numerical scheme, maximum number of iterations, convergence criteria for each variable. |

Post-processing | Flow visualization, analysis, validation using published experimental data or analytical calculation data. |

No. | Question/Statement |
---|---|

1 | I have used CFD modelling before. |

2 | I found the use of vorticity and streamfunction to be instructive. |

3 | I could relate vorticity and streamfunction to the primitive variables easily. |

4 | I found the use of collaboration with other students useful. |

5 | I preferred working on my own to working in groups. |

6 | Do you think the balance between CFD theory and practice was correct? |

7 | This CFD course enhances my understanding of fluid mechanics theory. |

8 | This CFD course is a useful addition to the fluid mechanics laboratories. |

9 | The ‘hands-on’ aspects of this CFD course has taught me extra skills. |

10 | Do you feel you can continue to model basic flows without much help? |

11 | Does this course give you confidence that you can model more difficult flows? |

12 | By using CFD I have learned things that could not be taught through traditional theory or experiments. |

13 | I have now a sound knowledge of CFD good practice. |

14 | I have enjoyed this course. |

15 | I would recommend this CFD course to others. |

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**MDPI and ACS Style**

Adair, D.; Jaeger, M.
An Efficient Strategy to Deliver Understanding of Both Numerical and Practical Aspects When Using Navier-Stokes Equations to Solve Fluid Mechanics Problems. *Fluids* **2019**, *4*, 178.
https://doi.org/10.3390/fluids4040178

**AMA Style**

Adair D, Jaeger M.
An Efficient Strategy to Deliver Understanding of Both Numerical and Practical Aspects When Using Navier-Stokes Equations to Solve Fluid Mechanics Problems. *Fluids*. 2019; 4(4):178.
https://doi.org/10.3390/fluids4040178

**Chicago/Turabian Style**

Adair, Desmond, and Martin Jaeger.
2019. "An Efficient Strategy to Deliver Understanding of Both Numerical and Practical Aspects When Using Navier-Stokes Equations to Solve Fluid Mechanics Problems" *Fluids* 4, no. 4: 178.
https://doi.org/10.3390/fluids4040178