Learning Objective:Be able to select and construct solution algorithms for ODEs and PDEs encountered in aerospace and mechanical engineering based on understanding of flow physics and numerical methods. Be able to select and construct solution algorithms for flows that may be modeled as viscous or inviscid, compressible or incompressible, and choose appropriate initial and boundary conditions. Be able to explain consistency, stability, and convergence of numerical methods for PDEs and how they affect the accuracy of numerical solutions. Be able to state and explain factors that affect accuracy of computed solutions generated by research and commercial CFD codes and how errors could be assessed and minimized. Be able to state and explain limitations of CFD analysis because of assumptions invoked and uncertainties in models and inputs.
This course provides an introduction to finite-difference (FD) and finite volume (FV) methods in CFD. The course is divided into three parts. Part 1 reviews the building blocks needed to develop, analyze, and implement CFD, including methods for initial and boundary-value problems, methods for linear and nonlinear algebraic equations, classification and properties of partial differential equations (PDEs), and the equations that govern fluid mechanics, heat transfer, and combustion problems. Part 2 presents FD and FV methods in a step-by-step manner, showing how the building blocks are assembled and their limitations. These include mapping of coordinate systems, grid generation, FD and FV operators, and methods of analysis for consistency, stability, convergence, and errors such as conservation, transportive, dissipation, dispersion, aliasing, and lack of monotonicity and positivity. Part 3 shows how FD and FV methods are applied to the Euler and the Navier-Stokes equations for compressible and incompressible flows with focus on boundary conditions, verification and validation issues, and uncertainty quantification.
Topics Covered:Review of initial and boundary value problems; classification and properties of partial differential equations (PDEs); physical and mathematical properties of the Euler and the Navier-Stokes equations; mapping of PDEs in generalized coordinates and grid generation; methods for deriving discrete operators for finite-difference and finite-volume methods, including central, upwind, flux-vector, flux-difference, and limiters; methods for solving discretized PDEs by explicit (e.g., time-split) and implicit (e.g., ADI, AF, LU, multigrid) methods; methods for compressible and incompressible Navier-Stokes equations; and methods for analyzing consistency, stability, and numerical error.
Prerequisites:Undergraduate courses in thermodynamics, fluid mechanics, and heat transfer. Some background in gas dynamics, numerical methods, and a programming languages such as Fortran or C/C++.
Applied / Theory:50 / 50
Web Content:Syllabus, grades, lecture notes, homework assignments, solutions.
Homework:Several homeworks will be assigned through the semester. These will often involve small programming projects. A larger project, involving a report, will be due near the end of the semester. Late homework may be penalized.
Exams:1 midterms, 1 final exam.
Textbooks:Official textbook information is now listed in the Schedule of Classes. NOTE: Textbook information is subject to be changed at any time at the discretion of the faculty member. If you have questions or concerns please contact the academic department.
Tentative: Recommended textbook: Pletcher, R.H., Tannehill, J.C., and Anderson, D.A., Computational Fluid Mechanics and Heat Transfer, 3rd Edition, Taylor & Francis Book Company, Washington, DC, 2013. ISBN-13: 978-1591690375; ISBN-10: 1591690374.