сопствено дело

Brief description of the numerical method

The following code leverages some numerical methods to simulate the solution of the 2-dimensional Navier-Stokes equation.

We choose the simplified incompressible flow Navier-Stokes Equation as follows:

${\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=\mu \nabla ^{2}\mathbf {v} .}$

The iterations here are based on the velocity change rate, which is given by

${\displaystyle {\frac {\partial \mathbf {v} }{\partial t}}={\frac {\mu }{\rho }}\nabla ^{2}\mathbf {v} -\mathbf {v} \cdot \nabla \mathbf {v} .}$

Or in X coordinates:

${\displaystyle {\frac {\partial v_{x}}{\partial t}}={\frac {\mu }{\rho }}({\frac {\partial ^{2}v_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial y^{2}}})-v_{x}{\frac {\partial v_{x}}{\partial x}}-v_{y}{\frac {\partial v_{x}}{\partial y}}.}$