Math Review in Fluid Mechanics

Transcription

1 Math Review in Fluid Mechanics 2017 Ohio-PKAL Annual Conference Brett Batson, Ph.D. Mechanical & Aerospace Engineering Trine University, Angola, Indiana 1

2 Math Review in Fluid Mechanics?! Brett Batson, Ph.D. Mechanical & Aerospace Engineering Trine University, Angola, Indiana 2

3 Math Review in Fluid Mechanics Relabeling: Integration of Effective Mathematics Teaching review and reinforcement across the STEM Curriculum Supporting, not replacing, the Math Department Bridging the gap between just math and applications Just math Math review Applications 3

4 The Problem The minute we need a thing, we begin paying for it whether we buy it or not. Considering the lack of knowledge in, and skill with, algebra, calculus, and differential equations: Students begin paying by an impaired ability to grasp concepts in fluid mechanics (and other subjects). They continue paying through lost credit on assignments. Hines, S. W. ed Laura Ingalls Wilder Farm Journalist. University of Missouri Press, Columbia, MO. 4

5 The Problem Defined Fluid Mechanics is probably the most math-intensive course in the core curriculum. Students prove they are poorly prepared, mathematically. Fall 2010 o Curriculum change o Single fluids course Fluid Mechanics is very full. 5

6 The Problem Defined A prerequisite for this Fluid Dynamics class is Differential Equations o In order to take Differential Equations, students must have earned at least a C in Calculus I and Calculus II, and passed Calculus III. o These students should be well prepared to learn Fluid Dynamics. 6

7 Math Review The Need ~48 Students ~17 pts Exams Before the Math Review Exam II is math intensive 0 Exam 1 I Exam 2 II Exam 3 III Final 4 7

8 8 Fluid Mechanics Cover a lot of material Properties Pressure fields Bernoulli equation Acceleration fields Kinematics Integral analysis Conservation of mass Newton s second law 1 st law of thermodynamics Differential analysis Continuity Navier-Stokes equations Potential flow Stream function Velocity potential Pipe flow Laminar Turbulent External flow Lift, drag

9 Class Inversion In the conventionally-taught (lecture) class, there was inadequate time to review math. Outside Class View lecture video In Class Take quizzes Work Practice Problems Outside Class Finish Practice Problems 9

10 Inverting Fluid Mechanics Advantages Frequent quizzes Helps keep students on track Shorter lectures ~30 minute average per lecture for a 50 minute class Now I have time to review mathematics! Math practice problems included Math problems fair game on quizzes and exams 10

11 Math Review The Timing A math review may be done all at once and at the start of the mathematical-related course. By the time these skills are needed for the course, students usually have forgotten them, again. Importantly, the plurality of review topics should be covered immediately before they are needed in the mathematical-related course. 11

12 Math Review Topics Properties Pressure fields Taylor series Gradient ( ), logarithms, 1 st order ODEs Bernoulli equation Reynolds Transport Acceleration fields Kinematics Partial derivatives, total derivatives Evaluating determinants, curl 12

13 Math Review Topics Differential analysis Conservation of mass o Stream function Newton s 2 nd law o Exact solutions Divergence Total differential Exact differential equations 2 nd order differential equations Method of undetermined coefficients Surface integrals 13

14 Math Review The Results Exams ~48 Students ~144 Students Exam 1 I Exam 2 II Exam 3 III Final 4 Before math review After math review 14

15 Math Review Student Comments Anonymous Course Evaluations: Math review was also very helpful getting the concepts back into our minds and thinking about math again. Dr. Batson has done a wonderful job with the math review and I found it to be very helpful. Personal conversations (unprompted) revealed the same feeling. 15

16 Math Review Important Points If your course is too full to review math, consider flipping. Integrate math review when it will be needed. Put teeth into the review. o Quizzes Don t be afraid to review math not necessarily needed for your class: o very helpful getting the concepts back into our minds and thinking about math again. The results could be astonishing! 16

17 Math Review Questions??? 17