# 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

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**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