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Mathematical Challenges Of Teaching A Graduate Fluids Course From Both The Classical And Numerical Standpoint

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Conference

2006 Annual Conference & Exposition

Location

Chicago, Illinois

Publication Date

June 18, 2006

Start Date

June 18, 2006

End Date

June 21, 2006

ISSN

2153-5965

Conference Session

Integrating Math, Science, & Engineering

Tagged Division

Mathematics

Page Count

13

Page Numbers

11.914.1 - 11.914.13

DOI

10.18260/1-2--666

Permanent URL

https://peer.asee.org/666

Download Count

61

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Paper Authors

biography

Phillip Smith New Mexico State University

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Phillip Smith is currently an Emeritus Professor of Mechanical Engineering at New Mexico State University in Las Cruces, New Mexico. He received his B.S. in Mechanical Engineering and his M.S. in Aeronautical Engineering from Purdue University in West Lafayette, Indiana. In 1966 he received a Ph.D. in Engineering Mechanics from the University of Kansas. Dr. Smith has been actively involved in teaching and research in fluid mechanics, applied mathematics, and computational methods since joining the NMSU faculty in 1964.

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Abstract
NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

Mathematical Challenges of Teaching a Graduate Fluids Course from Both the Classical and Numerical Standpoint

Introduction

We are very fortunate today to have computer software packages available for teaching subjects that are very difficult from the standpoint of the mathematical models that represent the fundamental equations of those subjects. The Navier-Stokes equations, which are the equations of motion for fluids, along with the energy equation and the continuity equation (the equation representing the conservation of mass) are a case in point. Often, however, when we teach a course in fluid mechanics using a commercial software package, we wonder if the students are grasping a basic physical understanding of the fluid mechanics, whether they have an appreciation for the mathematical techniques being applied within the software, or have an understanding of the limitations of the software. In an effort to assure ourselves that these three areas were being properly treated, a pilot master’s level fluid mechanics course was taught within the Mechanical Engineering Department at New Mexico State University treating the subject from both the classical mathematical and the numerical perspective. The material for this course was obtained by combining the material from a traditional Master’s level theoretical fluid mechanics course, taught for several years within the department, and a computational fluids mechanics course which we taught during an eight month period over the internet to Master’s level students at Boeing Aircraft Company1 . It was hoped that this would give the students (future theoretical and experimental researchers in the fluids area) a sound understanding of (1) the derivation and limitations of the fluids equations, (2) the classical linear and non-linear mathematical methods for solving the fluids equations, and (3) various numerical methods for solving the fluids equations. The mathematical challenges faced by the students included learning both classical mathematical techniques and numerical techniques for solving linear and non-linear time dependent partial differential equations in various orthogonal coordinate systems.

In this paper, examples will be given of student solutions to various advanced fluids prob- lems using both classical and numerical methods. The perceived success of teaching the course in this manner will also be discussed.

Brief Overview of the Course Content

Because the subject of solution techniques for the fluid mechanics equations is very broad, the topics covered in a one-semester course in this area of necessity must be very selective. Hence, we limited the topics to the following: Derivations of the fluid mechanics equations, presentation of a commercial computer software package for student use, Introduction of three numerical methods common to software packages, grid generation, and a few classical mathe- matical methods for solving linear and non-linear partial differential equations. No text book was required for the course, but class notes were provided to the students and several reference text were made available in the university library (see the Bibliography at the end of this paper).

Derivation of the Fluids Equations

The fluid mechanics equations were derived in class using basic concepts in the manner of Schlichting2 , White3 , and Warsi4 . Fundamental assumptions and limitations of the equations

Smith, P. (2006, June), Mathematical Challenges Of Teaching A Graduate Fluids Course From Both The Classical And Numerical Standpoint Paper presented at 2006 Annual Conference & Exposition, Chicago, Illinois. 10.18260/1-2--666

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