Mathematics for Engineers: The Capstone Course

The Hong Kong University of Science and Technology

Mathematics for Engineers: The Capstone Course

This course is part of Mathematics for Engineers Specialization

Instructor: Jeffrey R. Chasnov

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What you'll learn

Computational Fluid Dynamics
Scientific Computing

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Assessments

1 assignment

Taught in English

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This course is part of the Mathematics for Engineers Specialization

When you enroll in this course, you'll also be enrolled in this Specialization.

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There are 3 modules in this course

Mathematics for Engineers: The Capstone Course provides a capstone project for students who are completing the Mathematics for Engineers specialization. Students will first learn some basic concepts in computational fluid dynamics, and then apply these concepts to compute the fluid flow around a cylinder. Access to MATLAB online and the MATLAB grader is given to all students who enroll.

Before enrolling, students should have already taken courses in matrix algebra, differential equations, vector calculus and numerical methods, and be able to program in MATLAB. The course contains 22 short video lectures and a full set of lecture notes. After each lecture, there are problems to solve, and at the end of the second and third weeks, there is a substantial MATLAB programming assignment. Download the lecture notes from the link https://www.math.hkust.edu.hk/~machas/flow-around-a-cylinder.pdf Watch the promotional video from the link https://youtu.be/FlM1de9Sxh0

We learn the governing equations for the flow around a cylinder. We discuss the Navier-Stokes equations and the continuity equation, and derive a pair of coupled equations for the stream function and scalar vorticity. We nondimensionalize these equations so that they contain only a single dimensionless parameter called the Reynolds number. We then simplify the nondimensional governing equations using log-polar coordinates.

What's included

10 videos16 readings1 assignment1 plugin

10 videosTotal 67 minutes

Course Overview2 minutesPreview module
Week One Introduction1 minute
Navier-Stokes Equations | Lecture 18 minutes
Vorticity Equation | Lecture 27 minutes
Geometry of the Flow | Lecture 37 minutes
Two-dimensional Flow | Lecture 48 minutes
Stream Function | Lecture 57 minutes
Streamlines | Lecture 66 minutes
Reynolds Number | Lecture 710 minutes
Log-polar Coordinates | Lecture 86 minutes

16 readingsTotal 147 minutes

Welcome and Course Information2 minutes
MATLAB Online5 minutes
Plane Couette Flow10 minutes
Channel Flow10 minutes
Pipe Flow10 minutes
Vector Identities10 minutes
Flow Boundaries10 minutes
Curl of the Vorticity-Velocity Cross Product10 minutes
Scalar Vorticity in Cartesian Coordinates10 minutes
Scalar Vorticity Equation in Cartesian Coordinates10 minutes
Stream Function in Cartesian Coordinates10 minutes
Scalar Vorticity in terms of the Stream Function in Cartesian Coordinates10 minutes
Stream Function as a Vector Potential10 minutes
Streamlines in Cartesian Coordinates10 minutes
Stream Function Equation when Re = 010 minutes
Steady Flow Equations10 minutes

1 assignmentTotal 30 minutes

Week One Assessment30 minutes

1 pluginTotal 17 minutes

Deep Dive into the Flow Around a Cylinder17 minutes

We formulate the computational fluid dynamics problem of the steady flow around a cylinder. We introduce the finite difference method and derive iteration equations. We derive boundary conditions and discuss the outline of a MATLAB program. Students will write a MATLAB code to compute the stream function at a Reynolds number of ten.

What's included

8 videos6 readings1 app item

8 videosTotal 62 minutes

Week Two Introduction 1 minutePreview module
Finite Difference Method | Lecture 99 minutes
Iteration Equations | Lecture 1010 minutes
Free-stream Boundary Conditions | Lecture 116 minutes
Cylinder Boundary Conditions | Lecture 1214 minutes
Summary of the Boundary Conditions | Lecture 135 minutes
MATLAB Program (Steady) (Part A) | Lecture 146 minutes
MATLAB Program (Steady) (Part B) | Lecture 157 minutes

6 readingsTotal 37 minutes

Even and Odd Functions5 minutes
SOR Equations10 minutes
Vorticity Free-Stream Boundary Condition10 minutes
Test the Cylinder Boundary Condition10 minutes
Starting SOR with zero vorticity1 minute
Reference solution to "Steady Flow at Re = 10"1 minute

1 app itemTotal 60 minutes

Steady Flow at Re = 1060 minutes

We formulate the computational fluid dynamics problem of the unsteady flow around a cylinder. We introduce periodic boundary conditions in the polar angle, and show how to solve for the stream function using matrix methods. We show how to use a MATLAB ODE integrator to solve for the scalar vorticity. Students will write a MATLAB code to compute the time-dependent scalar vorticity at a Reynolds number of sixty.