Incompressible Flow

AERE-3100

Previous offerings

Course Objectives

Introduction to fluid mechanics and aerodynamics. Fluid properties and kinematics. Conservation equations in differential and integral form. Bernoulli’s equation. Basic potential flow concepts and solutions. Boundary layer concept. Incompressible flow over airfoils and wings. Examples of numerical methods. Applications of multi-variable calculus to fluid mechanics and aerodynamics.

Outcomes

On completion of the course the attentive student will understand:

• Incompressible flow equations
• Dimensional analysis and similitude for fluid mechanics
• Potential flow analysis: elementary flows and superposition
• Airfoil theory
• Wing (lifting line) theory

Syllabus

• Concepts of pressure, etc.
• Similitude: Buckingham-Pi theorem
• Equations (conservation laws) governing fluid flow
  • Euler equations
  • Bernoulli’s equation
  • Local, convective, and material derivatives
• Potential flow:
  • Streamlines, streaklines, and path lines
  • Velocity measurement: Venturi meter, Pitot-static tube
  • Stream function and velocity potential
  • Elementary flows: uniform, source/sink, doublet, vortex
  • Linear superposition of elementary flows
  • Conformal mapping and Jukouwski transformation
• Flow over airfoils:
  • Circulation, Kelvin’s theorem, lift generation
  • Kutta condition
  • Thin airfoil theory
  • Panel methods (source/vortex)
• Flow over finite wings
  • Biot-Savart’s law
  • Downwash and induced drag
  • Prandtl’s lifting-line theory
• Viscous flows (overview)
  • Governing equations
  • Non-dimensionalization and order-of-magnitude analysis
  • Laminar & turbulent boundary layers

Textbook and reading mtl.

The suggested reading materials include the following.

• Anderson, J. D. (2016). Fundamentals of Aerodynamics (6th ed.). McGraw Hill. (textbook)
• White, F. M., & others. (2015). Fluid mechanics. SI Units.
• Bertin, J. J., & Cummings, R. M. (2021). Aerodynamics for engineers. Cambridge University Press.
• Katz, J., & Plotkin, A. (2001). Low-speed aerodynamics (Vol. 13). Cambridge university press.

Lecture videos

• Week 1
  • Lecture 1: Introduction, logistics; what’s a fluid?
  • Lecture 2: Molecular vs. continuum description; Lagrangian and Eulerian frames; fluid properties.
  • Lecture 3: Forces and moments on an airfoil; center of pressure.
• Week 2
  • Lecture 4: Aerodynamic center; dimensional analysis - principle of dimensional homogeneity.
  • Lecture 5: Dimensional analysis - reference/scaling variables; Buckingham-pi theorem; example.
• Week 3
  • Lecture 6: Buckingham-Pi theorem: examples; Tensors - scalars, vectors, and tensors; coordinate systems
  • Lecture 7: Partial derivatives; chain rule; gradient and divergence operators.
  • Lecture 8: Gradient Operator (transformation From Cartesian To Polar Coordinates), Curl Operator And Fluid Rotation.
• Week 4
  • Lecture 9: Integral operators; Stokes, Divergence, Gradient theorems (short lecture).
  • Lecture 10: Concept of flux; mass conservation - integral and differential forms; example problem.
  • Lecture 11: Mass conservation examples; momentum balance; drag on a 2-D body.
• Week 5
  • Lecture 12: Mass/momentum conservation examples; Energy conservation; Reynolds Transport theorem.
  • Lecture 13 (redacted): Material derviative example.
  • Lecture 14: Pathlines, streamlines, and streaklines; examples; angular velocity, rotation, and strain.
• Week 6
  • Lecture 15: Stream/Path/Streak lines, vorticity, strain, stream function.
  • Lecture 16: Strain/rotation-rate tensors, Stream functions and potential functions.
  • Lecture 17: Example problems (vorticity; line integral V.ds).
• Week 7
  • Lecture 18: Example problem (given velocity potential, find velocity field, stream function, etc.).
  • Lecture 19: Example problem; Bernoulli’s equation.
• Week 8
  • Lecture 20: Review before midterm
  • Midterm exam
  • Midterm exam review
• Week 9
  • Lecture 23: Bernoulli’s eq w/ example; Venturi meter; static & stagnation pressures.
  • Lecture 24: Static pressure calculation; Pitot-static tube with examples.
  • Lecture 25: Pitot-static tube example; coeff. of pressure, governing eqs.: $\nabla^2\phi=0$
• Week 10
  • Lecture 26: Elementary flows: uniform and source/sink flows.
  • Lecture 27: Rankine half-body, Rankine oval, Doublet, flow over a cylinder.
  • Lecture 28: Non-lifting flow over a cylinder.
• Week 11
  • Lecture 29: Vortex flow & lifting flow over a cylinder.
  • Lecture 30: Velocity/pressure distributions over a lifting cylinder, complex potential and conformal transformations
  • Lecture 31: Source and vortex panel methods; method of images (qualitative treatment).
• Week 12
  • Lecture 32: Airfoil nomenclature and characteristics; the concept of vortex sheet.
  • Lecture 33: Vortex sheet; thin airfoil theory - application to symmetric airfoils.
  • Lecture 34: Thin airfoil theory: symmetric and cambered airfoils; aerodynamic center.
• Week 13
  • Lecture 35: Circulation theorem; vortex panel method; lumped vortex model; high-lift devices.
  • Lecture 36: Viscous flow over airfoils; Finite-wing aerodynamics: downwash, induced alpha/drag, Prandtl’s lifting line theory overview.
  • Lecture 37: Prandtl’s lifting line theory.
• Week 14
  • Lecture 38: Cancelled due to weather
  • Lecture 39: Elliptic and general lift distribution, numerical lifting-line theory, Assignment #6 discussion.
  • Lecture 40: Lifting surface theory; introduction to viscous flows.
• Week 15
  • Lecture 41: Introduction to viscous flows.
  • Lecture 42: Boundary layer equations: non-dimensionalization and order-of-magnitude analysis. Review of topics covered in the class.
  • Lecture 43: Review of select topics