# Detailed Outline, M E 521: Foundations of Fluid Mechanics I

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1 Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Second-order tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic Fluid Mechanics C. Chapter 2: Cartesian Tensors 1. Definitions a. scalar b. vector c. second-order tensor d. axes rotation rules for Cartesian tensors 2. Tensor notation a. definitions (1) free index (2) dummy index b. rules in tensor notation (1) the words "for i = 1, 2, or 3" are implied when i is a free index (2) summation from 1 to 3 is implied by a dummy index (3) in an equation, two terms should not have different free indices (4) an index should not appear more than twice in any given term c. consequences of tensor notation rules (1) the number of free indices determines the order of an expression (2) any letter can be used for a dummy index (3) a dummy index letter can be changed to another dummy letter d. other definitions in tensor notation (1) Kronecker delta function (2) dot product (3) permutation symbol (4) the epsilon-delta relation (5) cross product (6) short-hand comma notation (7) contraction of a tensor (8) symmetric and antisymmetric tensors (9) eigenvalues and principal axes (10) tensor invariants (11) Gauss and Stokes theorems in tensor notation e. relationship between principal axes and eigenvalues II. Kinematics 1. Lagrangian description 2. Eulerian description 3. Material derivative B. Motion of Fluid Particles 1. Rate of translation 2. Linear strain rate a. strain rate tensor b. volumetric strain rate 3. Shear strain rate a. definition Prepared by Professor J. M. Cimbala, Penn State University Latest revision, 02 January 2008

2 b. strain rate tensor 4. Rate of rotation, vorticity a. definition b. vorticity C. Circular Flows 1. Solid body rotation 2. Line vortex D. Miscellaneous Topics in Chapter 3 1. Stream function 2. Polar coordinates a. plane polar coordinates b. cylindrical polar coordinates c. spherical polar coordinates III. Conservation Laws B. Reynolds Transport Theorem 1. Derivation in Kundu's book a. 1-D Leibnitz theorem b. extension of Leibnitz theorem to a volume integral c. fixed control volume, Reynolds transport theorem d. moving control volume C. Conservation of Mass 1. Integral form 2. Differential form D. The Linear Momentum Equation 1. Lagrangian (system) law a. following a fluid particle b. following a material volume 2. Control volume (integral) form 3. Differential form a. conservative form b. non-conservative form 4. Constitutive equation for Newtonian fluid a. definition b. Newtonian fluids (Stokes s assumptions) c. deviatoric stress tensor 5. The Navier-Stokes equation a. compressible b. incompressible 6. Example problems a. flat plate boundary layer (control volume solution) b. incompressible Couette flow (differential equation solution) E. Conservation of Energy 1. Notation 2. Mechanical energy equation a. introduction b. derivation c. deformation work d. viscous dissipation 3. Thermal energy equation (heat equation) a. derivation (1) first law of thermodynamics (for a material volume) (2) transformation to control volume form (using RTT) (3) transformation into differential form (using Gauss) (4) alternate form (using the mechanical energy equation)

3 b. heat equation in terms of primary unknowns F. Bernoulli Equation 1. Incompressible Newtonian flow a. incompressible, unsteady, and irrotational b. incompressible, steady, and irrotational (the beloved form) c. incompressible, steady, and inviscid 2. Compressible, inviscid flow a. compressible, inviscid, steady, isentropic, and irrotational b. compressible but barotropic, and inviscid (1) barotropic, inviscid, and steady (2) barotropic, inviscid, steady, and irrotational (3) barotropic, inviscid, unsteady, and irrotational G. Boussinesq Approximation IV. Vorticity Dynamics 1. Viscosity and rotationality 2. Comparison of solid body rotation and line vortex B. Kelvin's Circulation Theorem 1. Statement of the theorem 2. Physical significance of the theorem C. Helmholtz Vortex Theorems 1. Vortex lines move with the fluid 2. Strength of a vortex tube is constant along its length 3. Vortex cannot "end" in a fluid 4. Strength of a vortex tube remains constant in time D. Mutual Interaction of Vortices 2. Examples a. self-propelled vortex pair b. orbiting vortex pairs c. two arbitrary vortices 3. Time-marching schemes a. forward time-difference scheme b. backward time-difference scheme c. central time-difference scheme d. predictor-corrector time marching scheme e. Runge-Kutta time marching scheme E. The Vorticity Equation 1. Derivation 2. Physical meaning of vorticity equation a. rate of change of vorticity following a fluid particle b. vortex stretching and tilting term c. viscous diffusion term 3. Usefulness of vorticity equation a. vorticity is independent of frame of reference b. sometimes easier computationally to work with vorticity c. further simplification for creeping flow F. Production of Vorticity 2. Wall production of vorticity 3. The heat flux - vorticity flux analogy 4. How does vorticity get produced at a wall? a. equations of motion at the wall b. physical significance c. e.g. 2-D airfoil

4 d. e.g. the laminar flat plate boundary layer V. Irrotational Flow B. Equations of Motion 1. Velocity potential 2. Stream function C. Solution Techniques 1. Straightforward approach using real variables (computational) 2. Superposition using real variables (inverse approach) 3. Solutions using complex variables D. Complex Variables and Irrotational Flow 1. Review of complex variables a. complex conjugate b. magnitude of a complex variable c. miscellaneous equations d. separating a complex function into real and imaginary parts e. derivatives of complex functions 2. Application to 2-D incompressible irrotational flow a. complex potential b. analytic function c. Cauchy-Riemann conditions d. complex velocity (1) magnitude and direction of complex velocity (2) complex velocity in cylindrical coordinates 3. Elementary planar irrotational flows in complex variables a. uniform stream in x-direction b. uniform stream in arbitrary direction c. line source at origin d. line vortex at origin e. source or line vortex at arbitrary z location f. doublet g. power function 4. Superposition of complex potentials a. flow past a 2-D half-body b. flow over a circular cylinder c. D'Alembert's paradox d. flow over a circular cylinder with circulation e. physical significance (1) do these flows really model anything in real life? (2) lift force on 2-D bodies - the Kutta-Zhukhovski lift theorem 5. Conformal mapping (Zhukhovski transformation) E. Axisymmetric Irrotational Flow 2. Equations for axisymmetric irrotational flow 3. Similarities and differences between 2-D and axisymmetric irrotational flows 4. Simple axisymmetric irrotational flows a. uniform flow b. point source c. axisymmetric doublet 5. Superposition of axisymmetric flows a. Rankine nose shape b. flow over a sphere c. torpedo-shaped bodies F. Three-Dimensional Irrotational Flows

5 2. Lift and drag on 3-D bodies in irrotational flow 3. Induced drag a. introduction b. simple 2-D wing of finite span (1) kinetic energy arguments (2) momentum arguments (3) downwash arguments c. consequences of tip vortices (1) unwanted drag on airplanes (2) problems for airplane in wake of another airplane (3) tip vortices can be used to advantage, however. d. how to reduce induced drag (1) make wing with very long span (2) vary wing geometry (3) add winglets at the tips of the wing 4. Propulsion of fish and birds a. fish b. birds 5. Forces on a sailboat VI. Laminar Flow Solutions 1. Review 2. The "dynamic pressure" 3. General solution techniques for laminar flows B. Examples of Exact Laminar Solutions 1. Steady flow between infinite parallel plates 2. Steady pipe flow 3. Flow between concentric cylinders 4. Axial flow in an annulus between concentric cylinders 5. Cylinder rotating in an infinite fluid 6. Falling film flow 7. Others C. Similarity Solutions 1. Stokes's first problem a. introduction b. equations of motion c. similarity solution d. vorticity in Stokes s first problem 2. Diffusion of a vortex sheet 3. Viscous decay of a line vortex 4. 2-D stagnation point flow D. Creeping Flow a. nondimensionalization of the equations of motion b. Stokes equations 2. Observations about Stokes's equations a. density does not appear in the equations b. time does not appear in the equations c. Stokes equations cannot be applied to 2-D flow d. Stokes equations are linear e. creeping flow over a symmetric body has a symmetric velocity field 3. Creeping flow over a sphere a. summary of solution b. drag (1) pressure drag

6 (2) viscous drag (3) total drag c. observations about Stokes flow over a sphere (1) pressure is antisymmetric fore and aft (2) viscous drag is twice the pressure drag (3) streamlines are symmetric fore and aft (4) velocity does not depend on viscosity (5) flow everywhere near sphere is slower than freestream d. comparison of potential flow to creeping flow over a sphere 4. Oseen's improvement E. Compressible Flow 2. E.g. compressible Couette flow a. problem set up b. solution c. qualitative result d. energy equation VII. Laminar Boundary Layers 1. Definition 2. E.g. flow over an airfoil 3. Boundary layer approximation B. Differential Equations of Motion 1. Assumptions 2. BL coordinate system 3. Nondimensionalization and normalization 4. 2-D incompressible BL equations 5. General procedure for BL problems 6. Limitations of the BL technique 7. Character of the BL equations C. Blasius Flat Plate BL 2. Solution (similarity solution) 3. Results of Blasius solution a. boundary layer thickness b. displacement thickness c. momentum thickness d. shape factor e. wall shear stress f. drag coefficient g. velocity normal to the plate D. Pressure Gradients in BL's 2. Equations at the wall 3. Examples 4. Separation E. Falkner-Skan Wedge Flows F. 2-D Laminar Free Shear Layers 2. 2-D laminar jet. to be continued in M E 522

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