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 \( r \)

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 \( \Delta x = \Delta y \)

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

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

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

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