SIMPLE algorithm based CFD solver in Rust

This project is a computational fluid dynamics (CFD) solver written in Rust and post-processed with matplotlib, using the SIMPLE algorithm with a collocated grid to integrate the 2D incompressible steady Navier-Stokes equations. This project is based on the lectures by Dr. Sandip Mazumder. It can solve three cases: the lid-driven cavity flow, the pipe flow with a velocity inlet/gauge pressure outlet, and the backward facing step flow.

Usage

To run this project, you need to have Rust, Python 3 and matplotlib installed on your system. To run the solver for a specific case, use the following command:

cargo run --release -- -c <case>

where <case> can be one of lid_driven_cavity, pipe_flow, backward_facing_step. The solver uses a uniform mesh of size $n_x \times n_y$, which can be specified by the user. To get information about what variables you can change in the solver, you can use the --help or -h flag.

Lid-driven cavity flow

The lid-driven cavity flow is a classic benchmark problem for CFD. It consists of a square domain with all the boundaries being solid walls. The top wall moves in the x-direction at a constant speed while the other walls are stationary. The flow is governed by the incompressible Navier-Stokes equations:

$$ \nabla \cdot \mathbf{u} = 0 $$

$$ (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{1}{\rho} \nabla p + \nu \nabla^2 \mathbf{u} $$

where $\mathbf{u} = (u,v)$ is the velocity vector, $p$ is the pressure, $\rho$ is the density, and $\nu$ is the kinematic viscosity.

The boundary conditions are:

$$ u = 1,\ v = 0 \quad \text{at} \quad y = 1 $$

$$ u = 0,\ v = 0 \quad \text{at} \quad x = 0,\ x = 1,\ y = 0 $$

To validate the results of the lid-driven cavity flow solver, we compare them with the experimental data of Ghia et al. ¹ for Reynolds number 100.

Results

Pipe flow

The pipe flow is another benchmark problem for CFD. It consists of a rectangular domain with a velocity inlet at the left boundary and a gauge pressure outlet at the right boundary. The flow is governed by the same incompressible Navier-Stokes equations as the lid-driven cavity flow.

The boundary conditions are:

$$ u = U_{in},\ v = 0 \quad \text{at} \quad x = 0 $$

$$ p = p_{out} \quad \text{at} \quad x = L $$

$$ u = 0,\ v = 0 \quad \text{at} \quad y = 0,\ y = H $$

where $U_{in}$ is the inlet velocity, $p_{out}$ is the outlet pressure, $L$ is the length of the domain, and $H$ is the height of the domain.

The Reynolds number based on the inlet velocity and the height is defined as:

$$ Re = \frac{U_{in} H}{\nu} $$

To validate the results of the pipe flow solver, they're compared with the analytical solution of the Poiseuille flow, in which the u velocity profile and the pressure drop are given by:

$$ u(y) = 1.5 U_{in}(1-y^2/h^2) $$

$$ \frac{\partial p}{\partial x} = -3 \frac{\mu}{h^2}U_{in} $$

where $\mu$ is the viscosity and $h$ is $H/2$. The figures show a good agreement between the numerical and analytical solutions, confirming the accuracy of the solver.

Results

Backward facing step flow

The backward facing step flow is a more complex problem for CFD. It consists of a rectangular domain with a step at the bottom wall.

The boundary conditions are:

$$ u = U_{in},\ v = 0 \quad \text{at} \quad x = 0 $$

$$ p = p_{out} \quad \text{at} \quad x = L $$

$$ u = 0,\ v = 0 \quad \text{at} \quad y = H_1,\ 0 < x < L_1 $$

$$ u = 0,\ v = 0 \quad \text{at} \quad y = 0,\ L_1 < x < L $$

where $U_{in}$ is the inlet velocity, $p_{out}$ is the outlet pressure, $L$ is the length of the domain, $H$ is the height of the domain, $L_1$ is the length of the step, and $H_1$ is the height of the step.

The Reynolds number based on the inlet velocity and the total height is defined as:

$$ Re = \frac{U_{in} H}{\nu} $$

Results

Footnotes

1. U. Ghia, K. N. Ghia, and C. T. Shin, "High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method," Journal of Computational Physics, vol. 48, no. 3, pp. 387-411, 1982. ↩