TEP4280: Introduction to Computational Fluid Dynamics
# Model Equations
## Burger's Equation
$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial ^2 u}{\partial x^2} $$
The inviscid version of the equation is
$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0$$
The linear version of the Burger's Equation is often called the Convection–Diffusion equation
$$\frac{\partial u}{\partial t} + u_0 \frac{\partial u}{\partial x} = \alpha \frac{\partial ^2 u}{\partial x^2} $$
### Numerical Schemes
### Application
## Diffusion Equation
$$\frac{\partial u}{\partial t} = \alpha \frac{\partial ^2 u}{\partial x^2} $$
The diffusjon equation is parabolic.
### Numerical Schemes
### Application
The diffusion equation is applied when a quantity $u$ is being distributed throughout a space or body with out moving the initial source of the quantity.
#### Heat Conduction
$$\frac{\partial T}{\partial t} = \alpha \frac{\partial ^2 T}{\partial x^2} $$
where $T$ is the temperature, and $alpha$ is the heat conduction constant.
#### Flow in porous media
$$\frac{\partial T\u}{\partial t} = c \frac{\partial ^2 u}{\partial x^2} $$
## Poisson Equation
$$\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = f(x, y)$$
Setting $f(x, y) = 0 $ will give the Laplace equation.
### Numerical Schemes
### Application
## Wave Equation
$$\frac{\partial ^2 u}{\partial t^2} = \alpha_0 ^2 \frac{\partial ^2 u}{\partial x^2}$$
The linear version of the eqaution is
$$\frac{\partial u}{\partial t} + \alpha_0 \frac{\partial u}{\partial x} = 0$$
### Numerical Schemes
### Application
# Boundary Conditions
# Numerical Methods
# Termonology