MNT-2021-1_Lid-Driven_Cavity

Simulate the flow of a newtonian fluid in a lid-drive cavity with internal obstacles using the explicit and implicit projection methods.

Setup

• Python version: 3.10.7
• Numba version: 0.56.4

Virtual environment

1. To create, run

python -m venv mnt_venv

2. To activate, run

   • On Windows (git bash)

   source mnt_venv/Scripts/activate

   • On Linux

   source mnt_venv/bin/activate

3. To install the dependencies, run

pip install -r requirements.txt

How to use

(mnt-venv)$ python main.py -h
usage: main.py [-h] -re NUM_RE --final_time FINAL_TIME [-i]
               [--grid_size GRID_SIZE] [--dt DT] [--tol TOL] [-v]
               [--num_obs NUM_OBS] [-obs OBSTACLE] [-o OUTPUT]
               [--early_stopping] [--dont_save] [--dont_show]

Simulate the flow of a newtonian fluid in a lid-drive cavity with internal obstacles

options:
  -h, --help            show this help message and exit
  -re NUM_RE, --num_re NUM_RE
                        Reynolds number.
  --final_time FINAL_TIME
                        Final time for the simulation.
  -i, --implicit        Set to use implicit method. (Default: False)
  --grid_size GRID_SIZE
                        Grid discretization. (Default: 100)
  --dt DT               Time increment. (Default: 0.0001)
  --tol TOL             Tolerance of the iteration error. (Default: 1.e-8)
  -v, --validation      Set True for the validation problem. (Default: False)
  --num_obs NUM_OBS     Number of obstacles. (Default: 1)
  -obs OBSTACLE, --obstacle OBSTACLE
                        Obstacle location (i, j) and size as 'i','j','L' for
                        all obstacles. (Default: 40,40,20)
  -o OUTPUT, --output OUTPUT
                        Set the output name. (Default: None)
  --early_stopping      Set True for the early stopping to simulate until the
                        permanent situation or the final time. (Default:
                        False)
  --dont_save           Don't save output plots at the end of the simulation.
                        (Default: False)
  --dont_show           Don't show output plots at the end of the simulation.
                        (Default: False)

Validation

• Explicit method

python main.py --num_re 100 --final_time 60 -v

• Implicit method

python main.py --num_re 100 --final_time 60 -i --dt 0.001 -v

Flow for a square obstacle

• One centered obstacle

python main.py --num_re 100 --final_time 60 -i --dt 0.001 -obs 40,40,20

• One obstacle on arbitrary position

python main.py --num_re 100 --final_time 60 -i --dt 0.001 -obs 20,60,20

• Two obstacles on arbitrary position

python main.py --num_re 100 --final_time 60 -i --dt 0.001 --num_obs 2 -obs 20,60,20,60,20,20

Results

Validation

Maximum value of stream function

• dt = 0.001 and dx = dy = 0.01

Re	This work1	This work2	Kim and Moin (1985)	Ghia et al. (1982)	Marchi et al. (2009)
1	-	0.0996	0.099	-	0.10013
100	0.1033	0.1033	0.103	0.103	0.1035
400	0.1126	0.1126	0.112	0.114	0.1140
1000	0.1149	0.1149	0.116	0.118	0.1189

1 - First order explicit method; 2 - First order implicit method; 3 - In this case Re number was 0.01.

• dt = 0.005 and dx = dy = 0.01

Re	This work2
1	0.0965
100	0.1033
400	0.1125
1000	0.1148

Stream function contour plot

Re = 1	Re = 100
Re = 400	Re = 1000

Velocity u profile for x = 0.5

Re = 100	Re = 1000

Vorticity contour plot

Re = 100	Re = 1000

Flow for a square obstacle

One centered obstacle streamlines plot

Re = 1, L = 20	Re = 100, L = 20
Re = 400, L = 20	Re = 1000, L = 20

One centered obstacle vorticity contour plot

Re = 100, L = 20	Re = 1000, L = 20

One obstacle in various positions streamlines plot

Results for Re = 400.

40x60, L = 20	40x20, L = 20
20x40, L = 20	60x40, L = 20

Two obstacles streamlines plot

Results for Re = 400.

20x60 and 60x20, L = 20	40x20 and 40x60, L = 20
20x20 and 60x60, L = 20	20x40 and 60x40, L = 20

Major references

[1] - Chorin, A.J., 1968. "Numerical solution of the Navier-Stokes equations". Mathematics of computation;

[2] - Ghia, U., Ghia, K. and Shin, C., 1982. "High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method". Journal of Computational Physics;

[3] - Kim, J. and Moin, P., 1985. "Application of a fractional-step method to incompressible Navier-Stokes equations". Journal of Computational Physics;

[4] - Marchi, C.H., Suero, R. and Araki, L.K., 2009. "The lid-driven square cavity flow: Numerical solution with a 1024 x 1024 grid". Journal of the Brazilian Society of Mechanical Science and Engineering;

[5] - Rosa, A.P., 2021. "Roteiro para o Trabalho 5: Resolvendo o Problema do Escoamento de um Fluido em uma Cavidade com o Método de Projeção". Disciplina: Métodos Numéricos em Termofluidos, UnB;

[6] - Temám, R., 1969. "Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II)". Archive for Rational Mechanics and Analysis.