CFD_Julia Python and C++ Implementation

This repo is the Python implementation of CFD_Julia.

The corrigendum of its paper, CFD Julia: A Learning Module Structuring an Introductory Course on Computational Fluid Dynamics, 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

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.

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, $F=$

$$F^L,\qquad if\ S_L \geq 0$$

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

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

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

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CFD_Julia - Python and C++ Implementation

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