All courses offered by CMTL require some form of report writing or presentation. All reports and presentations must be properly formatted, as would be expected for publication or conference presentation.

This course is the equivalent of a Computational Fluid Dynamics (CFD) course. After covering the fundamentals of numerical solution, three mainstream methods, namely Finite Difference, Finite Volume, and Finite Element are discussed.

Prerequisite(s): CS 108 and ME 421 or consent of instructor.

We consider the time evolution of a flow in a parallel plate channel. The fluid is considered initially be at rest, and then evolves to its steady state solution.

The governing Navier-Stokes equations for this flow may be simplifed to: \[ \frac{\partial u}{\partial t} = - \frac{1}{\rho} \frac{dp}{dx} + \nu \frac{\partial^2 u}{\partial y^2}, \] where \( \frac{dt}{dx} \) is a constant pressure gradient.

For this project, the student is expected to:

- State the boundary conditions for the problem
- Non-dimensionalize the boundary conditions and governing equation
- Mathematically determine the analytical solution
- Write a numerical finite-difference code in either Fortran or C/C++ using:
- An explicit method
- An implicit method
- 4th-order Runge-Kutta method

- Investigate the following:
- Accuracy considering the effects \( \Delta t \), \( \Delta y \), and \( r=\Delta t / \left (\Delta y \right)^2 \)
- Stability considering the effects of \( \Delta t \), \( \Delta y \), and \( r \)

The video depicts the solution to this problem using an forward in time, central in space (FTCS), finite-difference scheme

The project handout may be downloaded here:

In this project, the student will develop a two-dimensional incompressible Navier-Stokes solver. The numerical method is based on the Marker and Cell approach by Harlow and Welch.

The code will then be used to solve a benchmark problem, the lid driven cavity, in order to verify its accuracy.

For this project, the student is expected to:

- State the boundary conditions for the problem
- Non-dimensionalize the boundary conditions and governing equations
- Write an explicit MAC method code in either Fortran or C/C++
- Compare with previously published results for the one-sided lid-driven cavity
- Investigate:
- The effect of Reynolds number
- Stability considering the effects of \( \Delta t \) and \( \Delta x = \Delta y \)

Here we see an example of the flow in a single-lid driven cavity at Reynolds number equal to 100. The solution is shown with velocity vectors and vorticiy. Note that the solution is laminar and steady.

Validation data may be downloaded here:

Here we see an example of the flow in a single-lid driven cavity at Reynolds number equal to 10,000. Note that the velocity vectors are not steady, and a quasi-steady behaviour emerges.

The project handout may be downloaded here:

This course is focused primarily on supersonic flows. It covers fundamental mathematical descriptions of benchmark linear and nonlinear problems, leading to the numerical solution of the Riemann Problem for Euler equations.

Prerequisite(s): CS 108 and ME 428 or consent of instructor.

This project deals with the exact solution of the Riemann Problem for 1D Euler equations, as discussed in Chapter 4 of Toro’s book. Start from the source code E1RPEX.F in the library NUMERICA that is available online.

For this project, the student is expected to:

- Run the code for tests 1 through 5
- Modify the subroutine GUESSP to enforce the coice of initial guess values
- Modify the boundary conditions for tests 1 through 5 and compare to the initial case

Exact solution for density, velocity, pressure and specific internal energy for \(\rho_L = 1.0 \) at \( t = 0.15 \) units and for \( \rho_L = 0.2 \) at \( t = 0.1 \) units.

The project handout may be downloaded here:

This project deals with the solution of the 1D inviscid Burgers equation using the Godunov method described in Chapter 5 of Toro’s book. Start from the source code BUGOD.F in the library NUMERICA that is available online.

For this project, the student is expected to:

- Run the code for Test 3 in Chapter 5 to validate the code
- Modify subroutine BCONDI to implement Dirichlet type boundary conditions
- Obtain numerical results for the 1D Burgers equation with periodic boundary conditions
- Determine the effect of \( \Delta t \) on stability
- Determine the effect of \( \Delta x \) on numerical diffusion

Exact solution for velocity with transmissive boundary conditions.

The project handout may be downloaded here:

This project deals with the solution of the 1D Euler equations using the Godunov method described in Chapter 6 of Toro’s book. Start from the source code E1GODS.F in the library NUMERICA that is available online.

For this project, the student is expected to:

- Run the code for Tests 1 to 5 in Chapter 6 and validate the results
- Run Tests 1 to 5 for REFLECTIVE boundary conditions and compare the results with the results from part 1
- Study the effect of INTERCELL FLUX for Tests 1 to 5

Test 2 at \( t=0.15 \) units.

The project handout may be downloaded here:

This course discusses various analytical methods for solving heat conduction problems. The main focus is on the method of Separation of Variables and its various modifications in different coordinate systems. Numerical solution is also discussed using the Finite Element Method.

Prerequisite(s): CS 108 and ME 421 or consent of instructor.

For this project, the student should select any 2-D conduction problem for which an analytical solution is available. It may be in either cartesian or cylindrical coordinates.

You will then perform a numerical analysis to determine the transient solution for the problem by modifying the provided FEM code. Finally, write a report summarizing your work.

For this project, the student is expected to:

- Perform a resolution study by comparing various mesh sizes and time-step sizes
- Compare the analytical solution to the numerical solution to validate your results
- Use grid streching/compression to show you are able to achieve higher accuracy with fewer elements than a uniform mesh
- Select different initial conditions to show they all give the same steady-state solution
- Change boundary conditions and compare results

Consider a rectangular plate with initial temperature of 300 and a uniform heat generation with a magnitude of 10000.

The project handout and sample code/materials may be downloaded here: