Lid Driven Cavity Flow Simulation using Finite Differences Method

Overview

Implementation of a lid driven cavity flow simulator using regular finite differences method.

Example: Periodic drive

The following image shows the simulation results for the following non-default configuration parameters:

• Simulation time: $120\ sec$
• Reynolds number: $2000$
• Lid velocity : $v_{top}(t)=sin(t/3)$

Example: Constant drive

The following image shows the simulation results for the following (default) configuration parameters:

• Simulation time: $60\ sec$
• Reynolds number: $500$
• Lid velocity : $v_{top}(t)=1.0$

Installation

This project uses setuptools for packaging. To install the package simply run the following command in the project directory

pip3 install -e .

Execution

The simulation can then be executed by running

python3 src/main.py

Governing equations in streamfunction-vorticity formulation

Streamline function

$$ \frac{\partial^2 \psi}{\partial\ x^2} + \frac{\partial^2 \psi}{\partial\ y^2} = -w $$

Vorticity

$$ \frac{\partial w}{\partial t} + \frac{\partial \psi}{\partial y} \frac{\partial w}{\partial x} + \frac{\partial \psi}{\partial x} \frac{\partial w}{\partial y} = \frac{1}{Re} \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right) $$

Transport equation

$$ \frac{\partial C}{\partial t} + \frac{\partial \psi}{\partial y} \frac{\partial C}{\partial x} + \frac{\partial \psi}{\partial x} \frac{\partial C}{\partial y} = \frac{1}{Re \cdot Pr} \left( \frac{\partial^2 C}{\partial x^2} + \frac{\partial^2 C}{\partial y^2} \right) $$

For readability we introduce:

$$ \frac{\partial \psi}{\partial x} =: u_x $$ $$ \frac{\partial \psi}{\partial y} =: u_y $$

Finite Differences Method formulation

Streamline function update

$$ \psi_{i,j}^{n+1} = \frac{1}{4} \left( \psi_{i+1,j}^n + \psi_{i-1,j}^n + \psi_{i,j+1}^n + \psi_{i,j-1}^n + h^2 w_{i,j}^n \right) $$

Vorticity update

$$ \frac{\partial w}{\partial t} = \frac{1}{Re} \left( \frac{w_{i+1, j}^n +w_{i, j+1}^n + w_{i-1, j}^n + w_{i,j-1}^n - 4 w_{i,j}^n }{h^2} \right) - u^n_{x|i,j}\ \left( \frac{w_{i+1,j}^n - w_{i-1,j}^n}{2h}\right) - u^n_{y|i,j}\left(\frac{w_{i,j+1}^n - w_{i,j-1}^n}{2h} \right) := f_{i,j}^n $$

Using Tayler-approximation of $w(t_{n+1})$ we find the update rule

$$ w_{i,j}^{n+1} = w_{i,j}^n + \Delta t\ f_{i,j}^n $$

Velocity update

$$ u^n_{x|i,j} = \frac{\left( \psi^n_{i,j+1} - \psi^n_{i,j-1}\right)}{2h} $$ $$ u^n_{y|i,j} = \frac{\left( \psi^n_{i+1,j} - \psi^n_{i-1,j}\right)}{2h} $$

Transport equation update

$$ C_{i,j}^{n+1} = C_{i,j}^n - \hat{Cu}_x - \hat{Cu}_y $$

with the following flow-direction dependent terms to accomodate the upwind-scheme:

$$ \hat{Cu}_x := \left\{ \begin{array}{ll} u^n_{x|i,j}\ \frac{\Delta t}{\Delta x}\ \left( C_{i,j}^{n} - C_{i-1,j}^{n} \right) & u^n_{x|i,j} \geq 0 \\ \\\ u^n_{x|i,j}\ \frac{\Delta t}{\Delta x}\ \left( C_{i+1,j}^{n} - C_{i,j}^{n} \right) & u^n_{x|i,j} < 0 \\\ \end{array} \right. $$ $$ \hat{Cu}_y := \left\{\begin{array}{ll} u^n_{y|i,j}\ \frac{\Delta t}{\Delta x}\ \left( C_{i,j}^{n} - C_{i,j-1}^{n} \right) & u^n_{y|i,j} \geq 0 \\ \\\ u^n_{y|i,j}\ \frac{\Delta t}{\Delta x}\ \left( C_{i,j+1}^{n} - C_{i,j}^{n} \right) & u^n_{y|i,j} < 0 \\\ \end{array} \right. $$