Computational Aerodynamics Assignment 1 — Euler Equations
The first assignment consists of deriving from basic principles the mass, momentum, and energy transport equations commonly used in CFD.
 06.27.16
 Question #1
Starting from the principle of conservation of mass, show that the mass conservation equation for a fluid corresponds to: $$\frac{\partial \rho}{\partial t} + \frac{\partial \rho v_x}{\partial x} + \frac{\partial \rho v_y}{\partial y} + \frac{\partial \rho v_z}{\partial z} =0$$ with $\rho$ the mass density, and $v_x,~v_y,~v_z$ the $x,y,z$ components of the velocity vector.
 Question #2
Starting from Newton's law $\vec{F}_y=m\frac{dv_y}{dt}$ and the mass conservation equation show that the $y$-component of the momentum transport equation for a fluid corresponds to: $$\frac{\partial \rho v_y}{\partial t} + \frac{\partial \rho v_x v_y}{\partial x} + \frac{\partial \rho v_y^2}{\partial y} + \frac{\partial \rho v_z v_y}{\partial z} = -\frac{\partial P}{\partial y}$$ with $P$ the pressure.
 Question #3
Starting from the 1st law of thermo $${\rm d}(mh)-V {\rm d}P=\delta Q-\delta W$$ the $x$ and $y$ momentum equations in 2D: $$\rho \frac{\partial v_x}{\partial t} + \rho v_x \frac{\partial v_x}{\partial x} + \rho v_y \frac{\partial v_x}{\partial y}=-\frac{\partial P}{\partial x}$$ $$\rho \frac{\partial v_y}{\partial t} + \rho v_x \frac{\partial v_y}{\partial x} + \rho v_y \frac{\partial v_y}{\partial y}=-\frac{\partial P}{\partial y}$$ show that the total energy transport equation for a fluid corresponds to: $$\frac{\partial \rho E}{\partial t} + \frac{\partial \rho v_x H}{\partial x} + \frac{\partial \rho v_y H}{\partial y} = 0$$ with the total energy $E\equiv e+\frac{1}{2}q^2$, the total enthalpy $H\equiv h +\frac{1}{2}q^2$, $q$ the speed of the flow, and $T$ the temperature.
 02.24.17
 Due on Thursday March 21st at 16:30. Do Questions #2 and #3 only.
 03.12.19
 $\pi$