June 18, 2006
June 18, 2006
June 21, 2006
11.914.1 - 11.914.13
Mathematical Challenges of Teaching a Graduate Fluids Course from Both the Classical and Numerical Standpoint
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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