# cfd_practice

**Repository Path**: wt1900/cfd_practice

## Basic Information

- **Project Name**: cfd_practice
- **Description**: CFD_Julia - Python and C++ Implementation
- **Primary Language**: Unknown
- **License**: Not specified
- **Default Branch**: main
- **Homepage**: None
- **GVP Project**: No

## Statistics

- **Stars**: 0
- **Forks**: 1
- **Created**: 2023-07-16
- **Last Updated**: 2023-07-16

## Categories & Tags

**Categories**: Uncategorized

**Tags**: None

## README

# CFD_Julia Python and C++ Implementation

This repo is the Python implementation of [CFD_Julia](https://github.com/surajp92/CFD_Julia).

The corrigendum of its paper, [CFD Julia: A Learning Module Structuring an Introductory Course on Computational Fluid Dynamics](https://www.mdpi.com/2311-5521/4/3/159), is also provided.

Personally, I am more interested in compressible flow solvers. Currently, the tutorials untill Rusanov solver have been implemented. Incompressible flow solvers may be implemented in the future.

Note:
 - most recent updates can be found in branch/basic.
 - GitHub supports LaTeX mathematical expressions, finally! However, it doesn't work as expected. Please refer to corresponding jupyter notebook for Corrigendum.

## Content
- [01. 1D heat equation: Forward time central space (FTCS) scheme](./jupyter/01_Heat_Equation_FTCS/ftcs.ipynb)
- [02. 1D heat equation: Third-order Runge-Kutta (RK3) scheme](./jupyter/02_Heat_Equation_RK3/rk3.ipynb)
- [03. 1D heat equation: Crank-Nicolson (CN) scheme](./jupyter/03_Heat_Equation_CN/cn.ipynb)
- [04. 1D heat equation: Implicit compact Pade (ICP) scheme](./jupyter/04_Heat_Equation_ICP/icp.ipynb)
- [05. 1D inviscid Burgers equation: WENO5 with Dirichlet and periodic boundary condition](./jupyter/05_Inviscid_Burgers_WENO/weno5.ipynb)
- [06. 1D inviscid Burgers equation: CRWENO5 with Dirichlet and periodic boundary conditions](./jupyter/06_Inviscid_Burgers_CRWENO/crweno5.ipynb)
- [07. 1D inviscid Burgers equation: Flux-splitting approach with WENO5](./jupyter/07_Inviscid_Burgers_Flux_Splitting/flux_splitting_periodic.ipynb)
- [08. 1D inviscid Burgers equation: Riemann solver approach with WENO5 using Rusanov solver](./jupyter/08_Inviscid_Burgers_Rieman/rusanov_riemann_periodic.ipynb)
- [09. 1D Euler equations: Roe solver, WENO5, RK3 for time integration](./jupyter/09_Euler_1D_Roe/roe.ipynb)
- [10. 1D Euler equations: HLLC solver, WENO5, RK3 for time integration](./jupyter/10_Euler_1D_HLLC/hllc.ipynb)
- [11. 1D Euler equations: Rusanov solver, WENO5, RK3 for time integration](./jupyter/11_Euler_1D_Rusanov/rusanov.ipynb)
- [12. 2D Poisson equation: Finite difference fast Fourier transform (FFT) based direct solver](./jupyter/12_Poisson_Solver_FFT/fft.ipynb)
- [13. 2D Poisson equation: Spectral fast Fourier transform (FFT) based direct solver](./jupyter/13_Poisson_Solver_FFT_Spectral/fft_spectral.ipynb)

## Compile C++ test code (Linux)
```
    git clone https://github.com/fengyiqi/cfd_practice.git
    cd cfd_practice
    mkdir build; cd build
    cmake ..; make
    ./CFD_Practice
```
Simulation results are stored in **build/*.csv** that can be posteprocessed by [plot_cpp_results](./plotting/plot_cpp_results.ipynb). 

## Corrigendum 

### 2.2 Runge-Kutta Numerical Scheme

- The third-order Runge-Kutta scheme is given as 

    $$
    \begin{align*}
    u_i^{(1)} =&\ u_i^{(n)} + \frac{\alpha \Delta t}{\Delta x^2} \left( u_{i+1}^{(n)} - 2u_i^{(n)} + u_{i-1}^{(n)} \right)\\
    u_i^{(2)} =&\ \frac{3}{4}u_i^{(n)} + \frac{1}{4}u_i^{(1)} + \frac{\alpha \Delta t}{4\Delta x^2}\left( u_{i+1}^{(1)} - 2u_i^{(1)} + u_{i-1}^{(1)} \right)\\
    u_i^{(n+1)} =&\ \frac{1}{3}u_i^{(n)} + \frac{2}{3}u_i^{(2)} + \frac{2\alpha \Delta t}{3\Delta x^2}\left( u_{i+1}^{(2)} - 2u_i^{(2)} + u_{i-1}^{(2)} \right)
    \end{align*}
    $$

### 2.3 Crank-Nicolson Scheme
  
- In Listing 3

    ```
    beta: (alpha*dt)/(2*dx*dx)
    ```

### 2.4 Implicit Compact Pade (ICP) Scheme

- Equation 22:

    $$\frac{1}{\alpha}\left( \frac{1}{12}\frac{\partial u}{\partial t}\Bigr|_{i-1} + \frac{10}{12}\frac{\partial u}{\partial t}\Bigr|_{i} + \frac{1}{12}\frac{\partial u}{\partial t}\Bigr|_{i+1} \right) = \frac{u_{i-1} - 2u_i + u_{i+1}}{\Delta x^2}$$

- Equation 23:

    $$\frac{1}{\alpha}\left( \frac{u_{i-1}^{n+1} - u_{i-1}^{n}}{12\Delta t} + 10\frac{u_{i}^{n+1} - u_{i}^{n}}{12\Delta t} + \frac{u_{i+1}^{n+1} - u_{i+1}^{n}}{12\Delta t} \right) = \frac{u_{i-1}^{n+1} - 2u_{i}^{n+1} + u_{i+1}^{n+1} + u_{i-1}^{n} - 2u_{i}^{n} + u_{i+1}^{n}}{2\Delta x^2}$$

- Equation 28:

    $$r_i = \frac{-2}{\alpha\Delta t}\left( u_{i-1}^{(n)} + 10u_{i}^{(n)} + u_{i+1}^{(n)} \right) - \frac{12}{\Delta x^2}\left( u_{i-1}^{(n)} - 2u_{i}^{(n)} + u_{i+1}^{(n)} \right)$$

### 3.1 WENO-5 Scheme

- Equition 40
  
    $$\beta_2 = \frac{13}{12}(u_i-2u_{i+1}+u_{i+2})^2 + \frac{1}{4}(3u_i - 4u_{i+1}+u_{i+2})^2$$

### 4.2 Riemann Solver: Rusanov Scheme

- Equation 59:

    $$f_{1+1/2} = \frac{1}{2}\left( f_{i+1/2}^L + f_{1+1/2}^R \right) - \frac{c_{1+1/2}}{2}\left( u_{1+1/2}^R - u_{i+1/2}^L \right)$$

- Equation 60:

    $$c_{i+1/2} = max(|u_i|, |u_{i+1}|)$$

### 5 One-Dimensional Euler Solver

- In Equation 62, an important step that derives $F$ with respect to $q$ is missed.

    $$F = 
    \begin{Bmatrix}
    q_2 \\
    \frac{q_2^2}{q_1} + p \\
    \frac{q_2 q_3}{q_1} + p \frac{q_2}{q_1}
    \end{Bmatrix}
    $$

    and

    $$p = (\gamma-1)(q_3 - \frac{q_2^2}{2q_1})$$


### 5.2 HLLC Riemann Solver (can not render last two expressions, see tutorial for full version)

- Equation 81, $F=$
    
    $$F^L,\qquad if\ S_L \geq 0$$

    $$F^R,\qquad if\ S_R \leq 0$$

    $$\frac{S_*(S_L u_L - F^L) + S_L P_{LR} D_*}{S_L - S_*},\qquad if\ S_L \leq 0\ and\ S_* \geq 0$$
    
    $$\frac{S_*(S_R u_R - F^R) + S_R P_{LR} D_*}{S_R - S_*},\qquad if\ S_R \geq 0\ and\ S_* \leq 0$$