# Lorena Barba

## ME 702 - CFD

1. Introduction to CFD
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### ME 702 - Computational Fluid Dynamics - Video Lesson 1

by bu 20,265 views

Introduction to CFD, starting from recalling the Navier-Stokes equation and the meaning of each term: unsteady term, convective term, viscous term. Why is CFD needed? Some reasons: when a system is difficult to study experimentally; to quickly test many scenarios; when it's faster or easier than experiment; fluid simulations for entertainment. The basic ingredients of CFD: (i) the mathematical model; (ii) the discretization method; (iii) analysis of the numerical scheme; (iv) solving a resulting algebraic system of equations, and; (v) post-processing and visualization.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

2. Derivation of the continuity equation
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### ME 702 - Computational Fluid Dynamics (Lecture "zero", part 1)

by bu 27,939 views

Pencast explaining the derivation of the equation of conservation of mass (a.k.a., "continuity") in differential form.

More ME 702 videos on iTunes U:
http://itunes.apple.com/itunes-u/computational-fluid-dynamic­s/id452560554

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

3. Derivation of the momentum equation
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### ME 702 - Computational Fluid Dynamics (Lecture "zero", part 2)

by bu 12,120 views

Pencast explaining the derivation of the equation of conservation of momentum in differential form.

More ME 702 videos on iTunes U:
http://itunes.apple.com/itunes-u/computational-fluid-dynamic­s/id452560554

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

4. Derivation of Navier-Stokes equation
4

### ME 702 - Computational Fluid Dynamics (Lecture "zero", part 3)

by bu 8,102 views

The Navier-Stokes equations for a Newtonian fluid (assumes a linear relationship between stresses and rates of deformation).

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

5. Introduction to finite differences
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### ME 702 - Computational Fluid Dynamics - Video Lesson 2

by bu 9,478 views

Presents the Finite Difference (FD) method, the basis of which is the approximation of derivatives using nearby points on a spatial grid. The formulas for backward, forward and central difference are shown, and the accuracy of these FD approximations is determined via Taylor series.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

6. Order of accuracy, formula for mid-point, model equations
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### ME 702 - Computational Fluid Dynamics - Video Lesson 3

by bu 6,635 views

Review of the FD method. Definition of "order of accuracy" and "big-O notation". FD formulas for the mid-point: gain an order of accuracy. Simplified model equations: numerical discretizations must be appropriate to the physical properties of the model.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

7. Explains the midpoint scheme
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### ME 702 - Computational Fluid Dynamics - Explains the midpoint scheme

by bu 1,625 views

A one-sided first order FD formula can be considered as a central difference with respect to the midpoint. Graphical explanation.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

8. First four steps of practical module "12 steps to Navier-Stokes"
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### ME 702 - Computational Fluid Dynamics - Video Lesson 4

by bu 9,118 views

Practical module: "12 steps to the Navier-Stokes equations". (Step 1) 1D linear convection. (Step 2) 1D nonlinear convection—can generate shocks. (Step 3) 1D diffusion or heat equation.

When pseudocode is shown —i.e., something that looks like code, but is not code— remember that it is not meant to be syntactically correct. It is written in a Matlab-esque way, but it is not Matlab code! (This course is taught using Python since 2010.)

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

9. Explicit vs. implicit schemes
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### ME 702 - Computational Fluid Dynamics - Video Lesson 5

by bu 6,438 views

Explicit vs Implicit FD schemes. Using "stencils" for numerical schemes. Crank-Nicolson method.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

10. Steps 5 to 8 of practical module "12 steps to Navier-Stokes"
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### ME 702 - Computational Fluid Dynamics - Video Lesson 6

by bu 5,199 views

Practical module: "12 steps to the Navier-Stokes equations". (Step 5) 2D linear convection. (Step 6) 2D convection. (Step 7) 2D diffusion (Step 8) Burgers' equation.

When pseudocode is shown —i.e., something that looks like code, but is not code— remember that it is not meant to be syntactically correct. It is written in a Matlab-esque way, but it is not Matlab code! (This course is taught using Python since 2010.)

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

11. Analysis of numerical schemes I
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### ME 702 - Computational Fluid Dynamics - Video Lesson 7

by bu 4,372 views

Analysis of numerical schemes—Setting the scene: stability. In the assignment several solutions blew up. Why? Many different schemes: upwind schemes, implicit schemes, 2nd-order schemes, leapfrog scheme ... all have different behavior. Demonstration with advection equation.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

12. Analysis of numerical schemes II
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### ME 702 - Computational Fluid Dynamics - Video Lesson 8

by bu 3,814 views

Analysis of numerical schemes—Definitions of consistency, stability, convergence. Different definitions of error. Equivalence Theorem of Lax. The modified differential equation.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

13. The truncation error, modified differential equation
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### ME 702 - Computational Fluid Dynamics - Video Lesson 9

by bu 3,819 views

The truncation error: the difference between the numerical scheme and the differential equation. Modified Differential Equation: satisfied by the exact solution of the numerical scheme. Demonstrations using convection with central difference in space or 1st-order upwind. CFL condition.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

14. Von Neumann stability analysis
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### ME 702 - Computational Fluid Dynamics - Video Lesson 10

by bu 4,969 views

Von Neumann stability analysis; the amplification factor. Examples: (1) convection with CD in space, (2) CD in space, implicit in time, (3) 1st order upwind, (4) implicit upwind (5) diffusion eqn. with CD in space.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

15. Poisson equation for pressure
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### ME 702 - Computational Fluid Dynamics - Video Lesson 11

by bu 4,065 views

Computation of the Navier-Stokes equations for incompressible flow: there is no obvious way to couple velocity and pressure; take the divergence of the momentum and use continuity to get a Poisson equation for pressure.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

16. Steps 9 to 12 of practical module "12 steps to Navier-Stokes"
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### ME 702 - Computational Fluid Dynamics - Video Lesson 12

by bu 5,124 views

Steps 9 to 12 of the Navier-Stokes programming assignment. (Step 9) Laplace equation, (Step 10) Poisson equation, (Step 11) Navier-Stokes case 1, "cavity"; (Step 12) Navier-Stokes case 2, "channel".

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

17. New schemes for convection
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### ME 702 - Computational Fluid Dynamics - Video Lesson 13

by bu 3,540 views

New schemes for convection (including stencil and von Neumann stability): (1) Leapfrog (2) Lax-Friedrichs (3) Lax-Wendroff

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

18. Multi-step methods
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### ME 702 - Computational Fluid Dynamics - Video Lesson 14

by bu 3,165 views

Some schemes presented for linear equations are not well-suited to the solution of non-linear problems. Multi-step methods work well in nonlinear hyperbolic equations—these are FD schemes at split time levels; also called "predictor-corrector" methods. This video lecture presents the following schemes: (1) Richtmyer/Lax-Wendroff (2) Mac-Cormack, as applied to the linear equation. In the final 8 minutes, an exercise dealing with inviscid Burgers' equation is presented.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

19. Spectral analysis I
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### ME 702 - Computational Fluid Dynamics - Video Lesson 15

by bu 2,399 views

Spectral analysis of numerical schemes — Recall von Neumann analysis: introducing a Fourier decomposition of the solution, define an amplification factor, G. Then the stability condition is given by G less than 1. But what else can we know about the errors? Amplitude errors: numerical diffusion / error on phase of the solution: numerical dispersion.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

20. Spectral analysis II
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### ME 702 - Computational Fluid Dynamics - Video Lesson 16

by bu 3,493 views

Spectral analysis of numerical schemes II — Error analysis for hyperbolic problems: 1D linear advection. "Leading error": numerical convection faster than physical; "lagging error": numerical convection slower than physical.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

21. Spectral analysis III
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### ME 702 - Computational Fluid Dynamics - Video Lesson 17

by bu 2,782 views

Spectral analysis of numerical schemes III — Analysis of Lax-Friedrichs: amplitudes strongly damped for smaller values of CFL number; no diffusion error at phase angle=180º explains odd-even decoupling (error oscillation of wavelength 2 mesh widths); dispersion error always greater than 1 (leading phase error). Analysis Lax-Wendroff: large range of phase angles with very low diffusion error; dispersion error mostly lagging type. Analysis of leapfrog: it has no diffusion errors (i.e., leapfrog scheme is particularly useful in long-term simulations, e.g., long weather forecasts); dispersion error is of lagging type; neutral stability can cause problems, as high-frequency errors will not be damped. Leaprog is not recommended for high-speed flows where shocks can occur! Presentation of the explicit Beam-Warming method, using second-order backward-difference (upwind) scheme. Analysis of Beam-Warming: wide region of accuracy with respect to diffusion error; dispersion error greater than one (oscillations move ahead of solution). Final note: a key quantity is the number of mesh points per wavelength.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

22. Nonlinear convection: inviscid Burgers' equation with Lax-Friedrichs and Lax-Wendroff schemes
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### ME 702 - Computational Fluid Dynamics - Video Lesson 18

by bu 2,604 views

Nonlinear convection—Inviscid Burgers' equation in conservative form, and physical interpretation. Discretization of the Burgers' equation with (i) Lax-Friedrichs scheme, and stability consideratins, and (ii) Lax-Wendroff scheme.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

23. Nonlinear convection: inviscid Burgers' equation with MacCormack and Beam-Warming implicit
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### ME 702 - Computational Fluid Dynamics - Video Lesson 19

by bu 2,214 views

Nonlinear convection—Inviscid Burgers' equation, discretized with (iii) MacCormack scheme, (iv) Beam & Warming implicit method, and (v) Beam & Warming with artificial damping.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

24. Burgers' equation debrief (i)
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### ME 702 - Computational Fluid Dynamics - Video Lesson 20

by bu 2,045 views

Burgers' equation debrief—Review of the discretization in conservative form. For hyperbolic equations, we have 'strong solutions' and 'weak solutions'; the conservarive form lets us obtain weak solutions (e.g., shocks). What happens with the Lax-Friedrichs scheme? It is a first-order scheme and exhibits numerical diffusion; stability requires CFL less than 1 (defining CFL number with the maximum value of velocity)f. With different CFL numbers, we see more numerical diffusion as the CFL number is decreased and the quality of the solution deteriorates. Note that there is cancellation of errors with CFL=2 and the solution is exact! (similarly to the linear convection example with CFL=1.) With the Lax-Wendroff scheme, which is second order, we see dispersion errors; with different CFL numbers, we see more oscillations as CFL decreases.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

25. Burgers' equation debrief (ii)
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### ME 702 - Computational Fluid Dynamics - Video Lesson 21

by bu 2,889 views

Burgers' equation debrief—What happens with MacCormack method? Watch out for the possibility of over-shoots destroying stability! As you decrease CFL number, there are more dispersive oscillations. Among these schemes, which is best? Lax-Friedrichs, Lax-Wendroff, MacCormack? When solving Burgers' equation with the implicit Beam & Warming method, you use the Thomas algorithm to solve a tri-diagonal system. If you use a source like Wikipedia for the Thomas algorithm, watch out! We have boundary conditions ...

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

26. 26

### ME 702 - Computational Fluid Dynamics - Video Lesson 22

by bu 2,918 views

Euler equations, in conservation form: continuity equation, momentum equation, energy equation. "Short form" of the full Euler equations using vector notation, and the analogy with the 1D convection equation that allows us to use the same schemes that we have been studying thus far.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

27. Euler equation & the shock-tube problem
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### ME 702 - Computational Fluid Dynamics - Video Lesson 23

by bu 2,258 views

Euler equations and the shock-tube problem—The Riemann problem consists of a conservation law, and piecewise constant initial data with a single jump discontinuity. It has an analytical solution for the Euler equations, which is very useful to test numerical schemes. The Riemann problem itself appears as an integral part of the formulation of many CFD approaches, called "wave capturing methods". Classic example: "The shock-tube problem": generates a one-dimensional unsteady flow consisting of a right-traveling shock and a left-traveling expansion wave.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

28. Discretizing Euler equations
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### ME 702 - Computational Fluid Dynamics - Video Lesson 24

by bu 2,105 views

Discretizing the Euler equations, using (i) Lax-Friedrichs scheme, (ii) Lax-Wendroff scheme, (iii) Richtmyer method, (iv) MacCormack method. "Sod's test problems — initial conditions and parameters to use. Overview of results with different schemes, compared with the analytical condition.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

29. The finite volume method
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### ME 702 - Computational Fluid Dynamics - Video Lesson 25

by bu 2,289 views

Introduction to the Finite Volume Method. The FVM is the most widely used method in CFD, and one of the reasons is that it is very general and flexible, and it allows the use of any unstructured mesh. It is characterized by using the integral formulation of the conservation laws, and the discretization is applied over infinitesimal control volumes. One of the most important features of the FVM is that it automatically results in a conservative discretization. A simple demonstration of this feature appears at the end of this video, where one domain is divided in three sub-domains, and the internal fluxes are seen to cancel out.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

30. 30

### ME 702 - Computational Fluid Dynamics - Video Lesson 26

by bu 2,140 views

The conservative discretization in the FVM — A simple example considering a 1D conservation law: using a central difference scheme shows the cancellation of internal fluxes. For the same example, when using the non-conservative form of the equation, all flux terms do not cancel out, resulting in additional internal source terms. Conservative and non-conservative form are mathematically equivalent—but their numerical implementation is not equivalent. The extra source terms are of the same order as the truncation error, but their magnitude depends on the value of derivatives. For discontinuous flows (e.g., shocks), the numerical source terms can become important.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

31. 31

### ME 702 - Computational Fluid Dynamics - Video Lesson 27

by bu 3,441 views

The finite volume method discretization— Cell-centered vs. cell-vertex schemes, and various types of mesh; requirements for a consistent scheme. Applying the integral conservation law to each control volume; cell-averaged quantities.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com

32. 32

### ME 702 - Computational Fluid Dynamics - Video Lesson 28

by bu 3,908 views

Finite volume formulas in two dimensions— If FVM is applied on a Cartesian grid, we recover FD formulas! Finite volume equation for a general quadrilateral cell. The evaluation of fluxes is the crucial feature that differentiates the various schemes; examples: central scheme & cell-centered FVM / central scheme & cell-vertex FVM.

This video is part of the open courseware prepared by Prof. Lorena A. Barba.

For more, visit: http://lorenabarba.com