Fluid Mechanics 101 A Skeleton Guide J. E. Shepherd Aeronautics and Mechanical Engineering California Institute of Technology Pasadena, CA USA 91125 1995-present - Revised December 16, 2007 Foreword This is guide is intended for students in Ae101 “Fluid Mechanics”, the class on the fundamentals of fluid mechanics that all first year graduate students take in Aeronautics and Mechanical Engineering at Caltech. It contains all the essential formulas grouped into sections roughly corresponding to the order in which the material is taught when I give the course (I have done this now five times beginning in 1995). This is not a text book on the subject or even a set of lecture notes. The document is incomplete as description of fluid mechanics and entire subject areas such as free surface flows, buoyancy, turbulent flows, etc., are missing (some of these elements are in Brad Sturtevant’s class notes which cover much of the same ground but are more expository). It is simply a collection of what I view as essential formulas for most of the class. The need for this typeset formulary grew out of my poor chalk board work and the many mistakes that happen when I lecture. Several generations of students have chased the errors out but please bring any that remain to my attention. JES December 16, 2007 Contents 1 Fundamentals 1 1.1 Control Volume Statements ..................................... 1 1.2 Reynolds Transport Theorem .................................... 2 1.3 Integral Equations .......................................... 2 1.3.1 Simple Control Volumes ................................... 3 1.3.2 Steady Momentum Balance ................................. 3 1.4 Vector Calculus ............................................ 3 1.4.1 Vector Identities ....................................... 3 1.4.2 Curvilinear Coordinates ................................... 3 1.4.3 Gauss’ Divergence Theorem ................................. 4 1.4.4 Stokes’ Theorem ....................................... 4 1.4.5 Div, Grad and Curl ..................................... 4 1.4.6 Specific Coordinates ..................................... 5 1.5 Differential Relations ......................................... 5 1.5.1 Conservation form ...................................... 5 1.6 Convective Form ........................................... 6 1.7 Divergence of Viscous Stress ..................................... 7 1.8 Euler Equations ............................................ 7 1.9 Bernoulli Equation .......................................... 7 1.10 Vorticity ................................................ 8 1.11 Dimensional Analysis ......................................... 9 2 Thermodynamics 11 2.1 Thermodynamic potentials and fundamental relations ...................... 11 2.2 Maxwell relations ........................................... 11 2.3 Various defined quantities ...................................... 12 2.4 v(P, s) relation ............................................ 13 2.5 Equation of State Construction ................................... 13 3 Compressible Flow 15 3.1 Steady Flow .............................................. 15 3.1.1 Streamlines and Total Properties .............................. 15 3.2 Quasi-One Dimensional Flow .................................... 15 3.2.1 Isentropic Flow ........................................ 16 3.3 Heat and Friction ........................................... 18 3.3.1 Fanno Flow .......................................... 18 3.3.2 Rayleigh Flow ........................................ 18 3.4 Shock Jump Conditions ....................................... 19 3.4.1 Lab frame (moving shock) versions ............................. 19 3.5 Perfect Gas Results .......................................... 20 3.6 Reflected Shock Waves ........................................ 20 3.7 Detonation Waves .......................................... 22 3.8 Perfect-Gas, 2-γ Model ........................................ 22 3.8.1 2-γ Solution .......................................... 23 3.8.2 High-Explosives ........................................ 23 3.9 Weak shock waves .......................................... 25 3.10 Acoustics ............................................... 26 3.11 Multipole Expansion ......................................... 27 3.12 Baffled (surface) source ....................................... 28 3.13 1-D Unsteady Flow .......................................... 29 3.14 2-D Steady Flow ........................................... 31 3.14.1 Oblique Shock Waves .................................... 31 i 3.14.2 Weak Oblique Waves ..................................... 31 3.14.3 Prandtl-Meyer Expansion .................................. 32 3.14.4 Inviscid Flow ......................................... 32 3.14.5 Potential Flow ........................................ 32 3.14.6 Natural Coordinates ..................................... 33 3.14.7 Method of Characteristics .................................. 33 4 Incompressible, Inviscid Flow 34 4.1 Velocity Field Decomposition .................................... 34 4.2 Solutions of Laplace’s Equation ................................... 34 4.3 Boundary Conditions ......................................... 35 4.4 Streamfunction ............................................ 35 4.4.1 2-D Cartesian Flows ..................................... 36 4.4.2 Cylindrical Polar Coordinates ................................ 37 4.4.3 Spherical Polar Coordinates ................................. 38 4.5 Simple Flows ............................................. 38 4.6 Vorticity ................................................ 40 4.7 Key Ideas about Vorticity ...................................... 41 4.8 Unsteady Potential Flow ....................................... 42 4.9 Complex Variable Methods ..................................... 42 4.9.1 Mapping Methods ...................................... 44 4.10 Airfoil Theory ............................................. 44 4.11 Thin-Wing Theory .......................................... 45 4.11.1 Thickness Solution ...................................... 46 4.11.2 Camber Case ......................................... 48 4.12 Axisymmetric Slender Bodies .................................... 49 4.13 Wing Theory ............................................. 50 5 Viscous Flow 52 5.1 Scaling ................................................. 52 5.2 Two-Dimensional Flow ........................................ 53 5.3 Parallel Flow ............................................. 53 5.3.1 Steady Flows ......................................... 54 5.3.2 Poiseuille Flow ........................................ 56 5.3.3 Rayleigh Problem ...................................... 57 5.4 Boundary Layers ........................................... 57 5.4.1 Blasius Flow ......................................... 59 5.4.2 Falkner-Skan Flow ...................................... 59 5.5 K´arm´anIntegral Relations ...................................... 60 5.6 Thwaites’ Method .......................................... 60 5.7 Laminar Separation ......................................... 61 5.8 Compressible Boundary Layers ................................... 61 5.8.1 Transformations and Approximations ........................... 61 5.8.2 Energy Equation ....................................... 62 5.8.3 Moving Shock Waves ..................................... 64 5.8.4 Weak Shock Wave Structure ................................ 64 5.9 Creeping Flow ............................................ 66 A Famous Numbers 69 B Books on Fluid Mechanics 71 ii 1 FUNDAMENTALS 1 1 Fundamentals 1.1 Control Volume Statements Ω is a material volume, V is an arbitrary control volume, ∂Ω indicates the surface of the volume. mass conservation: d Z ρ dV = 0 (1) dt Ω Momentum conservation: d Z ρu dV = F (2) dt Ω Forces: Z Z F = ρG dV + T dA (3) Ω ∂Ω Surface traction forces T = −P nˆ + τ · nˆ = T · nˆ (4) Stress tensor T T = −P I + τ or Tik = −P δik + τik (5) where I is the unit tensor, which in cartesian coordinates is I = δik (6) Viscous stress tensor, shear viscosity µ, bulk viscosity µv 1 τ = 2µ D − δ D + µ δ D implicit sum on j (7) ik ik 3 ik jj v ik jj Deformation tensor 1 ∂ui ∂uk 1 T Dik = + or ∇u + ∇u (8) 2 ∂xk ∂xi 2 Energy conservation: d Z |u|2 ρ e + dV = Q˙ + W˙ (9) dt Ω 2 Work: Z Z W˙ = ρG · u dV + T · u dA (10) Ω ∂Ω Heat: Z Q˙ = − q · nˆ dA (11) ∂Ω heat flux q, thermal conductivity k and thermal radiation qr q = −k∇T + qr (12) Entropy inequality (2nd Law of Thermodynamics): d Z Z q · nˆ ρs dV ≥ − dA (13) dt Ω ∂Ω T 1 FUNDAMENTALS 2 1.2 Reynolds Transport Theorem The multi-dimensional analog of Leibniz’s theorem: d Z Z ∂φ Z φ(x, t) dV = dV + φuV · nˆ dA (14) dt V (t) V (t) ∂t ∂V The transport theorem proper. Material volume Ω, arbitrary volume V . d Z d Z Z φ dV = φ dV + φ(u − uV ) · nˆ dA (15) dt Ω dt V ∂V 1.3 Integral Equations The equations of motions can be rewritten with Reynolds Transport Theorem to apply to an (almost) arbi- trary moving control volume. Beware of noninertial reference frames and the apparent forces or accelerations that such systems will introduce. Moving control volume: d Z Z ρdV + ρ (u − uV ) · nˆ dA = 0 (16) dt V ∂V d Z Z Z Z ρudV + ρu (u − uV ) · nˆ dA = ρG dV + T dA (17) dt V ∂V V ∂V d Z |u|2 Z |u|2 ρ e + dV + ρ e + (u − uV ) · nˆ dA = dt V 2 ∂V 2 Z Z Z ρG · u dV + T · u dA − q · nˆ dA (18) V ∂V ∂V d Z Z Z q ρsdV + ρs (u − uV ) · nˆ dA + · nˆ dA ≥ 0 (19) dt V ∂V ∂V T Stationary control volume: d Z Z ρdV + ρu · nˆ dA = 0 (20) dt V ∂V d Z Z Z Z ρudV + ρuu · nˆ dA