Last Updated: 7/10/2018

University of Virginia
School of Engineering & Applied Science
Department of Mechanical & Aerospace Engineering

Computational Fluid Dynamics & Heat Transfer I -- Projects

Spring 2010 Projects


Much of the course will focus on six or seven programs which students will be expected to develop and submit. Generally a subroutine or section of the code will be provided as a start, but by and large the student will be independently writing the majority of the program. All students will emerge from this course as good scientific/technical programmers. Projects may those described below.    Click on the title for the writeup.   



 Superposition of Elementary Solutions




Students are given functions for one or two of the four elementary functions and will write the others. The streamfield due to the superposition of several combinations of the elementary solutions will be computed on a rectangular mesh and a plot of the computed values will be made using a contouring package. A major goal of this assignment is to familiarize students with whatever software package they plan to use for the rest of the semester.


Heat Transfer in a Self-Similar Boundary Layer




 Students will be given a VBA program for a 4th order Runge-Kutta solution of the Blasius equation. With the velocity distribution from that solution, they will solve the energy equation in the boundary layer both with dissipation (aerodynamic heating) and without. Another Excel/VBA program demonstrating the direct solution of the resulting tridiagonal system of linear equations will be provided.





Panel Method for Ideal Flow Over an Airfoil.



Students will implement the vortex panel method for a 2-D airfoil. Some coding of geometric quantities will be supplied. An Excel workbook that generates and plots the coordinates for a particular airfoil will be provided or the student may design his or her own shape. An online source of coordinate data for over a thousand airfoils is also available. The system of linear equations may be solved using subroutines from the Linpack or Lapack collection. The students will plot the computed pressure coefficient versus position on the airfoil for several angles of attack and may want to plot the streamlines as well.  A highly developed version of this 2-D panel method calculation created by a former student and which includes the effects of boundary layers, a separation criterion and capability for the user to design an airfoil to meet his or her performance specifications may be downloaded from his website.






Solution of Elliptic Equations



Students will solve for the velocity profile for Hagen-Poiseuille flow in an oddly-shaped duct and determine the resulting friction factor. This problem may then be extended to handle non-Newtonian fluids. A simple pointwise iterative method will be used initially, and students will be encouraged to try one of the more advanced techniques for solving the system of linear equations resulting from the discretization of elliptic equations; sample implementations of several will be provided.



Grid Generation for Odd Geometries



Students will generate a computational grid for an airfoil or other odd geometry using elliptic methods. Some of the coding will be provided, and the elliptic equations will be solved using an iterative method. The boundary-fitted grid will be plotted.  A DRAFT version of a highly interactive workbook that uses elliptic methods (both Laplace and Poisson) to generate a grid for a symmetric nozzle may be downloaded here.






Thermal Entry Length Problem



Assuming a fully developed (parabolic) laminar velocity profile in a cylindrical pipe, students will solve for the temperature distribution resulting from a sudden change in the wall thermal boundary condition. This parabolic problem results in a tridiagonal system of equations at each axial location along the pipe, which may be solved using an algorithm provided. The local Nusselt Number (non-dimensional convective heat transfer coefficient) will be computed as a function of the axial coordinate. Thorough implementation instructions (for both laminar and turbulent flows) and a complete VB program implementing this project are included in Heat Transfer Today.




Driven Cavity Problem


This standard test problem of viscous flow in a rectangular region driven by a sliding upper surface will be solved using a time-marching, primitive variable (i.e., pressure, velocity components) method. This problem involves both parabolic and elliptic equations, and the algorithm is typical of methods commonly used for low speed flows. The resulting velocity field will be plotted, either as velocity vectors or as streamlines.





Two-D Incompressible, Viscous, Buoyancy-Induced Flow in a Saturated Porous Matrix


This final project involves a coupled system, including one non-linear partial differential equation. The primitive variable algorithm we will implement is typical of those embedded in commercial CFD software. Thorough implementation instructions and a complete VB program implementing this project are included in Heat Transfer Today. Equally complete instructions for completing the same project using the alternative vorticity-streamfunction formulation and for the closely-related problem of Rayleigh-Benard convection are included as appendices on the HTT CD-Rom.





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