The project focuses on developing a solver combining surface-to-surface radiation and conduction. Consider a square cavity containing a transparent gas such as air. The walls of the cavity are cooled on the outside by convection.The gas itself is quiescent, and heat transfer through it is by conduction alone. The walls, however, are hot enough that they radiate to each other through the gas. Develop a finite volume solver to compute enclosure radiation heat transfer by writing the radiosity equations for all wall control surfaces, finding view factors, and solving the resulting linear system to find the wall temperatures, which are then used as boundary conditions for the conduction problem in the gas. Explore the influence of enclosure geometry and Planck number on the ratio of conduction to radiation heat transfer from the walls.

2. Algebraic Multigrid Scheme for Structured and/or Unstructured Meshes

In class, we will develop an algebraic multigrid scheme for unstructured meshes. This project involves implementing the multigrid idea either using a Gauss-Seidel relaxation sweep or, for structured meshes, an LBL-TDMA, and tying this to a multigrid cycle. Explore the performance of the scheme for problems with large anisotropies in coefficients due to properties, mesh aspect ratios and other complexities.

3. Phonon Boltzmann Transport Equation Solver

In recent years there has been a great deal of interest in solving the
phonon Boltzmann trasnport equation to model micro-scale conduction heat
transfer. In this project, take the convection schemes we developed in class and
apply them to solve the phonon BTE. Show that you can recover Fourier conduction
in macro domains, and explore the numerical properties of your solver over
a range of acoustic thicknesses.

4. Solver for Chemically Reacting Systems

Given a flow field, write a solver to solve for species transport and
chemical reaction in a reacting system, taking care to properly linearize and
couple your species transport equations. Explore the performance of your
numerical algorithm for a range of governing parameters.

5. Solver for Radiative Transfer Equation for Participating Radiation

Develop a solver for coupled thermal transport and participating
radiation. Participating radiation is described by the radiative transfer
equation (RTE), which is amenable to solution using the ideas about
convection-diffusion equations developed in class. The RTE is coupled to the
energy equation through radiative source terms. Develop a solver for the coupled
system, and test your solver against a variety of published solutions.

6. Combined Lagrangian-Eulerian Solver for Particle Transport Through a
Gas

Solvers for spray combustion and particle transport through gases and
liquids sometimes employ a coupled Lagrangian-Eulerian method for low
solid/droplet volume fraction. In this approach, droplets or particles
traversing the fluid are tracked individually in a Lagrangian frame of
reference. Particle tracks are then located in a background mesh on which the
gas phase governing equations are solved. In general, the momentum, mass and
energy loss by the particles is that gained by the gas. The project involves
implementing a coupled Lagrangian-Eulerian solver within the framework of a
finite volume scheme.

7. SIMPLE Solver for Flow in a Driven Cavity

Implement the SIMPLE algorithm for sequential solution of the
incompressible continuity and momentum equations within a finite volume
framework. Establish that your code works against a variety of published
solutions and examine the convergence properties of your scheme using the driven
cavity problem as a test problem. This is a substantial project and will require
good programming skills, but you will learn a great deal of CFD by doing
it.

8. Two-Temperature Model for Porous Media

Thermal transport in porous media is sometimes treated using a "two
temperature" formulation. If the thermal properties of the solid and fluid media
are very different from each other, the two media cannot be assumed to be in
thermal equilibrium. One approach to modeling this type of situation is to
pretend that each point in the medium is described by two temperatures, one for
the solid and one for the fluid, and write two separate energy equations,
accounting for the volume occupied by each medium, and coupling them through
energy exchange terms. The objective of the project is to write a solver for
this class of problem using a finite volume framework and testing the properties
of the scheme against analytical and published solutions.

9. Effective Conductivity of Foams

The objective of this project is to develop a solver for computing the
effective thermal conductivity of metallic foams impregnated with either a solid
such as paraffin, or immersed in a fluid such as air or water. For the purposes
of this project, you may assume that the fluid is stationary, and further,
assume that the foam can be represented by a regular structure that can be
modeled in Cartesian coordinates. Develop a solver to address this coupled
problem and present the effective thermal conductivity of the medium as a
function of geometry and metal-substrate conductivity ratio.

10. Effective Conductivity of Particle Composites

The objective of this project is to develop a solver for computing the
effective thermal conductivity of particulate composites, where particles are
embedded in a substrate such as polymer. The substrate is stationary.
Assume that the bed can be represented by a regular or random matrix of cuboidal
particles that can be modeled in Cartesian coordinates. Develop a solver to
address this coupled problem and present the effective thermal conductivity of
the medium as a function of geometry and particle-substrate conductivity ratio.

11. Convection-Diffusion Solver Using Control Volume-Based Finite
Element Method

In this class, we have looked mainly at cell-based schemes. An
alternative is to develop node-based schemes. During the 1980's, a variety of
node-based schemes using the so-called control-volume finite element methods
(CVFEM) were developed which sought to combine the conservative property of
finite volume schemes with the geometric flexibility of finite element schemes.
The project involves developing a CVFEM solver for the
convection-diffusion equation and to examine its properties.

**12. Unstructured Mesh Solver for the Convection Diffusion Equation**

The objective of this project is to implement the scheme that we have
developed in this class, working out all the issues with respect to data
structures and solution algorithms. Test your solver against published solutions
and examine the numerical properties of the underlying schemes.