Numerical Model
Last Updated: January 22, 2021
*NS2D was developed to solve Navier-Stokes equations. The equations were discretized using a finite-difference approach. The Pressure-Velocity coupling scheme implemented is SIMPLE.
Mass conservation
$$\frac{\partial U}{\partial x}+\frac{\partial V}{\partial y}=0$$
X - Momentum
$$\frac{\partial U}{\partial t}+U\frac{\partial U}{\partial x}+V\frac{\partial U}{\partial y}=-\frac{1}{\rho} \frac{\partial p}{\partial x}+\frac{\mu}{\rho}\left ( \frac{\partial^2 U}{\partial x^2}+\frac{\partial^2 U}{\partial y^2} \right )+g_x$$
Y - Momentum
$$\frac{\partial V}{\partial t}+U\frac{\partial V}{\partial x}+V\frac{\partial V}{\partial y}=-\frac{1}{\rho} \frac{\partial p}{\partial y}+\frac{\mu}{\rho}\left ( \frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2} \right )+g_y$$
Energy
$$\frac{\partial T}{\partial t}+U\frac{\partial T}{\partial x}+V\frac{\partial T}{\partial y}=\frac{k}{\rho c_p}\left ( \frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2} \right )+\frac{\mu}{\rho c_p} \left( 2 \left( \left( \frac{\partial U}{\partial x}\right)^2 + \left( \frac{\partial V}{\partial y} \right)^2 \right) + \left( \frac{\partial V}{\partial x}+\frac{\partial U}{\partial y} \right)^2 \right)$$
