Computational Fluid Dynamics (CFD)

Discretization methods for approximating the Partial Differential Equations (PDEs):

Finite Difference (FD) $\checkmark$
Finite Elements (FE)
Finite Volume (FV)

Finite Difference (FD)

Taylor’s polynomial

$f(x_0 + h) = f(x_0) + \frac{f'(x_0)}{1!}h + \frac{f^{(2)}(x_0)}{2!}h^2 + \cdots + \frac{f^{(n)}(x_0)}{n!}h^n + R_n(x)$

$f'(a)\approx {f(a+h)-f(a)\over h}.$

$f'(a)\approx{f(a+h)+f(a-h)-2f(h)\over \Delta h^2$

Incompressible 2D $\frac{\partial \rho}{\partial t}=0$

$\frac{\partial \Vec{V}}{\partial t} + (\Vec{V}\cdot \nabla)\Vec{V}=- \frac{1}{\rho}\nabla P + \nu \nabla^2 \vec{V}^2$
$\nabla \vec{V}=0$

3 equations $propto$ $u$ , $v$ , $P$

We obtain $\vec{V}$ by transposing the equation and we obtain the explicit form of the equation:

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

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

$P$ is obtained by multiplying the Navier Stokes equations by $\nabla$ , then we sum them:

$-\frac{1}{\rho}(\frac{\partial^2 P}{\partial x^2}+\frac{\partial^2 P}{\partial y^2})=(\frac{\partial u}{\partial x})^2+(\frac{\partial v}{\partial y})^2+2\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}$

We have a Poisson equation:

$\nabla^2 P=-f$

An iterative method can solve this equation:

$P_{ij}=\frac{\Delta x^2(P^n_{i,j+1}+P^n_{i,j-1}) + \Delta y^2(P^n_{i+1,j}+P^n_{i-1,j}) + \Delta x^2\Delta y^2 f_{ij}}{2(\Delta x^2+\Delta y^2)}$

There are several algorithms that were designed to solve Poisson equation, it can be discretised as a tridiagonal block matrix:

Thomas algorithm $\mathcal{O}(n^{2.5})$
Successive overrelaxation $\mathcal{O}(n^{1.5})$
Fast Furier transforms $\mathcal{O}(n \log(n))$
multigrid methods $\mathcal{O}(n)$

Computational Fluid Dynamics (CFD)

Finite Difference (FD)

Incompressible 2D $\frac{\partial \rho}{\partial t}=0$

#1 Cavity Flow

#2 Cavity Flow + Obstacle

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