Course objectives

Introduction.
The student will be able to...
1. explain what a fluid is,
2. list its important properties and the laws that govern its motion,
3. write down and explain Newton's Law of Viscosity and Fouriers Law of Heat Conduction,
4. explain the meaning of Reynolds number and Mach number,
5. quantitatively analyze pressure and velocity variations in a flow using Newton's second law,
6. explain the meaning of basic flow terms (e.g. boundary layer, no-slip condition, separation, turbulence etc.). 

Vector algebra. The student will be able to...
1. explain and give an example of the intrinsic property of direction,
2. explain the concept of polar, axial, unit vectors,
3. demonstrate use of the right hand rule,
4. make use of and physically interpret vector addition, multipication and triple products,
5. use Cartesian, cylindrical, and spherical coordinate systems, and express vectors (including the position vector) and vector functions in terms of their unit vectors. 

Vector calculus. The student will be able to... 
1. make appropriate use the formulae for differential changes in unit vectors in when differentiating vector functions with respect to time or space 
2. perform the various types of integration w.r.t. space in three-dimensions, in particular the flowrate and circulation integrals, and explain their significance 
3. define and, if necessary, evaluate a derivative w.r.t. 3D space from first principles (i.e. in integral form), and explain in principal how a differential form is obtained 
4. explain the physical significance of div, grad, curl and the convective operator particularly when applied to fields such as the velocity or pressure of a flow 
5. apply the gradient, divergence, curl and Stokes' theorems, to regions with multiple boundaries (by virtue of knowing where they come from) 
6. manipulate div, grad and curl, the convective operator, associated second order operators and the integral theorems to simplify a vector differential or integral expression 
7. explain what solenoidal and irrotational fields are, and use their special properties. 

Derivation of the equations of motion. The student will be able to...
1. explain what the Eulerian and Lagrangian perspectives are and how they relate to the laws of motion, the fluid, and the governing equations, 
2. derive the both integral and differential forms of the Eulerian equation for the transport of a property, given its governing law, 
3. convert integral to differential forms, and vice versa, 
4. derive of the Eulerian expression for the substantial derivative, and derive the Reynolds transport theorem, 
5. explain the origin/meaning of the terms in the governing equations, 
6. explain what constitutive relations, a Newtonian fluid, an isotropic fluid, Stokes hypothesis, the second coefficient of viscosity are and how these concepts are used, 
7. explain Cauchy Stokes decomposition and use it to extract expressions for divergence, curl and rates of strain in cartesian and non-cartesian systems, 
8. accurately list the assumptions implied by the continuity, Navier Stokes and viscous flow energy equations. 

Fluid statics and kinematics. The student will be able to...
1. determine the pressure distribution in a static fluid given a body force field, determine when a fluid may not remain static
2. write down and explain the properties of streamlines/surfaces/tubes, fluid lines, particle paths, vortex lines/surfaces/tubes
3. determine the equations of streamlines and streamfunctions for flows, given a suitable velocity field,
4. determine the equations of vortex lines, given a velocity field, and whether the vortex lines convect with the fluid lines,
5. state Helmholtz's and Kelvin's theorems and the conditions under which they hold. 

Fundamentals of Ideal Flow. The student will be able to...
1. derive, distinguish and explain the assumptions behind the two forms of Bernoulli's equation.
2. list the ideal flow assumptions, above and beyond the assumptions of the Navier Stokes
3. show why gravity can be ignored in many aero/hydro problems
4. write down the ideal flow boundary condition for velocity of potential at an arbitrary moving solid surface
5. use the various topological concepts to describe flow spaces and, with Stokes' theorem, identify the number of circulation values that must be prescribed in an ideal flow.
6. set up the derivation for force on a body in arbitrary motion through an ideal fluid (step 1), and explain the remainder of the derivation.
7. interpret the results of this derivation and use them to explain/derive d'Alemberts paradox, the Kutta Joukowski Theorem, and added mass.

2D Elementary Ideal Flow. The student will be able to...
1. explain and use all the variables involved in 2D ideal flow and the relations between them 
2. use the relation between streamfunction and volumetric flow rate and that relating velocity potential to a partial circulation integral in analyzing flow solutions.
3. explain the definition of an analytic function and know that any analytic function is a solution to Laplace's equation. 
4. use complex numbers, variables and functions and their properties in the solution of a flow problem 
5. explain the complex potential, complex velocity (polar and cartesian) and the relations between them 
6. explain and demonstrate the character of the various elementary flows
7. explain the method of images, the rules about singularity convection, Bernoulli's equation in terms of pressure coefficient, the equivalence between solid surfaces and streamlines, the Kutta Joukowski theorem and the Milne-Thompson circle theorem
8. to use the above items to construct, analyze the streamlines, stagnation points, convective motion velocity fields, pressure distributions of a range of elementary flows. 

Conformal Mapping. The student will be able to...
1. state and explain the justification and need for the Kutta condition, and explain and use the implications of the Kutta condition in the case of cusped and finite trailing edges
2. use and explain the Blasius theorem to compute forces and moments in simple potential flows, and explain why Kutta Joukowski theorem cannot always be used
3. explain and demonstrate what is meant by 'conformal mapping' and 'critical point' 
4. map a flow and determine the properties of the mapped field, identify the critical points in a mapping, identify distinct branches and when they occur
5. recall and use the properties of the basic conformal mappings (translation, rotation, power, log) and be able to use them to obtain a desired result
6. recall and explain the Joukowski transformation, its effects on the space, its critical points and their effects and characteristics and how the final flow depends on the positioning of the circle and how the Kutta condition is applied. 
7. recall and explain how the zero-lift angle of attack and circulation are determined in the zeta (unmapped) plane. 
8. define the aerodynamic center and the center of pressure and calculate forces, moments, aerodynamic center, zero-lift angle of attack for a Joukowski airfoil.

2D Panel Methods.The student will be able to...
1. analyze and explain the flowfield generated by distributed singularities such as a source or vortex panel, and how they are used in a panel method.
2. explain what a control point is, where it should be located, and what needs to be satisfied there
3. explain what an influence coefficient matrix is, why its a matrix, and how it is used in obtaining a panel method solution and then using that solution
4. explain how circulation can be introduced in a panel method and how a Kutta condition is satisfied
5. write a simple panel method of their own and/or modify and enhance an existing panel method to include, for example, different types of panels, other bodies, different boundary conditions or  multiple bodies.
6. plot, integrate or analyze features of a solved flow.

Thin airfoil theory. The student will be able to...
1. explain the motivations behind thin airfoil theory
2. explain the assumptions, and how they simplify the pressure coefficient, boundary conditions and lead to the simplification of the problem 
3. analyze a vortex+source sheets, explain the limiting velocity distribution on the sheet, and how it is used to satisfy the boundary conditions and the Kutta condition
4. explain the extent to which airfoil thickness and camber contribute to the lift, moment and pressure coefficient distribution on the airfoil surface,
5. calculate the above using the relations given on the handouts. 

3D ideal flow for non-lifting bodies. The student will be able to...
1. use point sources and doublets and vortex filaments to model simple 3D flow fields.
2. explain the properties of vortex filaments, use the Biot Savart law for 3D flows, and explain its origins. 
3. explain panels in 3D flow, in particular the doublet panel, and its relation to a vortex ring.
4. explain how a 3D panel method for a non-lifting body works
5. use and interpret a 3D panel method code to analyze flow around 3D non-lifting objects. 

3D ideal flow for wings. The student will be able to...
1. explain the flow around an wing, in particular the circulation distribution, downwash distribution, wake and relationship between them.
2. explain the origin of induced drag
3. explain the Kutta condition for a 3D wing
4. Know the nomenclature for a wing
5. explain how a panel method for a 3D wing may work
6. use and interpret a 3D panel method code for wings
7. explain the motivations and assumption associated with lifting line theory. 
8. use Prandtl's lifting line theory to calculate lift and drag, and compare and assess the lift and drag of different wings
9. explain the properties of an eliptical wing