Problem Description

Governing Equations

The mnomentum equation in vector for a velocity field $\vec{V}$

$$\frac{\partial \vec{V}}{\partial t}+(\vec{V}\cdot\nabla)\vec{V}=-\frac{1}{\rho}\nabla p + \nu \nabla^2\vec{V}$$

This represents three scalar equations, one for each velocity component $(u,v,w)$. We are solving it in 2-D, so there will be two scalar equations:

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} = -\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu \left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} \right) $$$$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y} = -\frac{1}{\rho}\frac{\partial p}{\partial y}+\nu\left(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}\right) $$

We need one ore more equation, which is the POisson equation to link pressure and velocity:

$$\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2} = -\rho\left(\frac{\partial u}{\partial x}\frac{\partial u}{\partial x}+2\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}+\frac{\partial v}{\partial y}\frac{\partial v}{\partial y} \right)$$

Problem Formulation

Input Data:

  • niter = 51 - maximum number of iterations
  • nx = 21 - number of x spatial points
  • ny = 11 - number of y spatial points
  • xmax = 2
  • ymax = 1
  • infinity = 100 - a large number

ICs

  • $\forall{(n,x,y)}$: $n=0 \rightarrow p =0$

BCs

  • Dirichlet:

    • $\forall{(n,y)}$: $x=0, x=2, y=0, y=1\rightarrow p=0$

  • Neumann:

    • $\forall{(n,x)}$ if $y=0 \rightarrow \frac{\partial{p}}{\partial{y}}=0$
    • $\forall{(n,x)}$ if $y=1 \rightarrow \frac{\partial{p}}{\partial{y}}=0$

Output Data:

  • $\forall{(x, y, n)}p = ?$
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