Prerequisites: CE380S (Environmental Fluid Mechanics) or equivalent graduate course in fluid mechanics, and knowledge of any programming language (Fortran, C++, Matlab, Python etc.).

Objectives: Learn the basics of computational methods and their application to fluid mechanics problems in Civil, Environmental, Ocean or other types of Engineering.

• Why should I take this course? The field of Computational Fluid Dynamics (CFD) has matured so much in the last decades that its use has become common place in the industry and academia. The availability of cheap powerful multiple-CPU personal computers (that can do what supercomputers could do two decades ago) has rendered CFD an indispensable tool for the design of (internal or external) complex flow devices. CFD has cut down the number of required expensive and time-consuming experiments (which often suffer from scaling and/or confined flow effects) drastically. The course will be addressing APPLICATIONS, through examples in class, homework assignments, and term projects. NOTE: The term project can be a part of a student's research, thesis, or dissertation.
• Why do I need this course, given that there are so many commercial CFD software available? Actually, this course not only will teach you the basics behind any commercial software, i.e. the brains inside the blackbox, but, most importantly, will provide you with valuable expertise on how to avoid "pitfalls", which can often lead to erroneous answers.
• Who should take it? Graduate EWRE, Ocean, Civil or other types of Engineering students who need expertise on computational fluid mechanics in their research, thesis, and future professional careers as engineers.
• What will the course include?

1. Review of fluid mechanics equations
2. Review of basic numerical methods (e.g., matrix inversion, finite difference methods as applied to ordinary or partial differential equations, etc.)
3. Finite volume methods; theory and applications (e.g. Euler, Reynolds-Averaged Navier-Stokes, and advection-diffusion equations) 

• Would the students have access to commercial CFD software? Yes, ANSYS_FLUENT, OpenFOAM, or Star-CCM+,UT's VISVE to be used in their term projects

• Review of fluid mechanics equations for an incompressible fluid - Proof of symmetry for shear stresses
• Constitutive equations for Newtonian Fluids - Proof of continuity equation for compressible fluids
• Useful identities from vector calculus and a complementary explanation of the gradient of a scalar
• HW#1
• Definition of elementary mass rate coming out of a control volume and the Reynolds Transport Theorem
• Definition of linear operator and examples
• Vorticity and its physical meaning in 2D
  • Vorticity plots for various cases of 2D flow
  • Simulation (mp4 file) of flow around cylinder, using UT/OEG's VISVE, a method which solves the 2D vorticity equation (Note how the vorticity travels downstream with the flow, and at the same time, it diffuses the farther it travels downstream)
  • Physical meaning of the "vortex-stretching" term of the vorticity equation in 3-D (NOT covered in class, only for those who wish to read more)
• Summary of Navier-Stokes, Vorticity, and Potential flow equations
• Navier-Stokes equations with respect to rotating system of coordinates
• Common Orthogonal Curvilinear Coordinate Systems
• Advection-Diffusion Equation for a dispersed substance and the Heat Equation for temperature
• HW#2
• Basic vector and tensor products (NEW!) - Explanation of symbols used in CFD textbook, 2020 edition
• NOT COVERED IN CLASS: Notes on energy equation - Definition of tractions

• Useful Finite Differences for derivatives (excerpt from the textbook with notes by SAK)
• Notes on grids with expansion ratio, and on the associated truncation error
• Section from textbook on TDMA (TriDiagonal Method Algorithm), with added information by SAK
• Proof of the TDMA (TriDiagonal Method Algorithm) - Special case of the TDMA (with N=3 intervals) - TDMA in reverse order
• HW#3
• TDMA involving Neumann condition at one end
• The "ghost" node for the application of Neumann boundary condition, and its treatment in TDMA
• A note on Richardson extrapolation - A note on norms of error
• Advection-Diffusion equation for Pe=0
• Results from solution of steady 1D advection-diffusion equation (excerpt from textbook with SAK's notes)
• Proof of condition for local Pe<2 to avoid oscillatory solutions
• Fig. 3.11 from the textbook with SAK's annotations
• HW#4

• Heuristic Stability Analysis
• HW#5
• von Neumann Stability Analysis - Basic properties of complex numbers
• von Neumann Stability Analysis (old version) & Application to FTCS Scheme for the Advection-Diffusion Equation
• von Neumann Stability Analysis for UDS in the Advection-Diffusion Equation
• Excel Tools for stability analysis:
  • For FTCS and UDS
  • For Leapfrog and DuFort-Frankel
• Comparison of three implicit methods
• Figs 6.3 and 6.4 from the textbook with SAK's annotations
• 1-D wave equation, Lax and Lax-Wendroff methods (NEW!)
• HW#6
• Application of various numerical schemes to the first order 1-D wave equation CORRECTION: On p.2 under equ. (4): Change: "Replacing (2a), (3) & (4) in (2)" to "Replacing (3) and (4) in (2a)"

• Numerical Evaluation of Integrals in FVM
• Proof that FVM is the same as FDM for cartesian grids
• Numerical Diffusivity for UDS in FVM (NOTE: We take u_e>0 in the shown example)
• FVM for the steady and unsteady advection-diffusion equation in 2-D
• Some notes on Gauss elimination and LU decomposition ...with my notes (includes ADI)
• A note on the conditions for the Gauss-Seidel method to converge and a related excel tool
• Notes on Additive ADI ... with my notes - ADI for Poisson equation
• Solution of the Navier-Stokes equations and the equation for the pressure (in the case of Euler explicit scheme in time)
• Boundary conditions for the Navier-Stokes equations - Example of results with no-slip and symmetry conditions
• Solution of Navier-Stokes equations using Euler implicit in time
• Supplement on the pressure correction methods
• >>>> HW#7 posted on Canvas; DUE: Tuesday, December 14, at 11:59pm on Canvas - Submit BEFORE you take the Test!

• The artificial compressibility method and treatment of general geometry cells
• Evaluation of derivatives at the cell centroids, and of cell volume, using Gauss theorem and evaluation of derivatives at the cell centroids, using least squares method
• Overview of the k-epsilon and the k-omega turbulence models
• Wall functions (from "Turbulence Modeling in CFD" by D.C. Wilcox) - Laminar Sub-layer
• VISVE - a method developed at UT/OEG, based on the solution of viscous vorticity equation - movie of computed vorticity patterns at Re=40,000 (turbulent), using UT/OEG's VISVE