Introduction
After no progress has been made in thermal-chemical nonequilibrium for months, I begin to doubt whether my ability can meet the need of basic scientific research work. So I put aside my current job, and spend a day reading a book[^book] about SPH method. Then I try the code provided by the author and gain much fun and confidence.
Here are some method and results.
[^book]: Smoothed Particle Hydrodynamics, a meshfree particle method G.R. Liu, M.B. Liu 2003
Method
The pith and marrow of SPH are fully embodied in that a function and its derivative can be represented in an integral form using dirac function or nascent delta function. It’s called kernel approximation in SPH.
$$
\begin{equation}
\left\langle f (\bf{x}) \right\rangle = \int\limits_\Omega f ({\bf{x’}}) W( {\bf{x}} - {\bf{x’}},h )d{\bf{x’}}
\end{equation}
$$
Then particle approximation is performed to conver the integral representation to the form of discretized particle approximation.
$$
\begin{equation}
\left\langle f\left( {\bf{x}}_i \right) \right\rangle = \sum\limits_{j = 1}^N \frac{m_j}{\rho_j} f\left( {\bf{x}}_j \right) \cdot W_{ij}
\end{equation}
$$
ODEs are produced when kernel approximation and particle approximation are applied to Navier-Stokes equations.
$$
\begin{equation}
\frac{d\rho_i}{dt} = \rho_i \sum\limits_{j = 1}^N \frac{m_j}{\rho_j}v_{ij}^\beta \cdot \frac{\partial W_{ij}}{\partial x_i^\beta }
\end{equation}
$$
$$
\begin{equation}
\frac{dv_i^\alpha }{dt} = \sum\limits_{j = 1}^N {m_j\frac{\sigma_i^{\alpha \beta } + \sigma_j^{\alpha \beta }}{\rho_i \rho_j}} \frac{\partial W_{ij}}{\partial x_i^\beta }
\end{equation}
$$
$$
\begin{equation}
\frac{de_i}{dt} =\frac{1}{2}\sum\limits_{j = 1}^N m_j\frac{p_i + p_j}{\rho_i \rho_j} v_{ij}^\beta \cdot \frac{\partial W_{ij}}{\partial x_i^\beta }+\frac{\mu_i}{2\rho_i}\varepsilon_i^{\alpha \beta} \varepsilon_i^{\alpha \beta}
\end{equation}
$$
Last time integration algorithm is achieved.
Result
I use the code provided by the book[^1] to simulate the shear cavity flow. Here is the result.
Some techniques
Fortran syntax
Write multi files for tecplot with time.
Tecplot
Macro
References
Dynamic text
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