Page 9 - Aerodynamics for Engineering Students

vi Contents

2 Governing equations of fluid mechanics 52
Preamble 52
2.1 Introduction 52
2.1.1 Air flow 53
2.1.2 A comparison of steady and unsteady flow 54
2.2 One-dimensional flow: the basic equations 56
2.2.1 One-dimensional flow: the basic equations of conservation 56
2.2.2 Comments on the momentum and energy equations 62
2.3 The measurement of air speed 62
2.3.1 The Pit&-static tube 62
2.3.2 The pressure coefficient 64
2.3.3 The air-speed indicator: indicated and equivalent air speeds 64
2.3.4 The incompressibility assumption 66
2.4 Two-dimensional flow 68
2.4.1 Component velocities 68
2.4.2 The equation of continuity or conservation of mass 71
2.4.3 The equation of continuity in polar coordinates 72
2.5 The stream function and streamline 73
2.5.1 The stream function 11, 73
2.5.2 The streamline 75
2.5.3 Velocity components in terms of 11, 76
2.6 The momentum equation 78
2.6.1 The Euler equations 83
2.7 Rates of strain, rotational flow and vorticity 83
2.7.1 Distortion of fluid element in flow field 83
2.7.2 Rate of shear strain 84
2.7.3 Rate of direct strain 85
2.1.4 Vorticity 86
2.7.5 Vorticity in polar coordinates 86
2.7.6 Rotational and irrotational flow 87
2.7.7 Circulation 87
2.8 The Navier-Stokes equations 89
2.8.1 Relationship between rates of strain and viscous stresses 89
2.8.2 The derivation of the Navier-Stokes equations 91
2.9 Properties of the Navier-Stokes equations 91
2.10 Exact solutions of the Navier-Stokes equations 95
2.10.1 Couette flow - simple shear flow 95
2.10.2 Plane Poiseuille flow - pressure-driven channel flow 96
2.10.3 Hiemenz flow - two-dimensional stagnation-point flow 97
Exercises 101
3 Potentialflow 104
Preamble 104
3.1 Introduction 104
3.1.1 The velocity potential 105
3.1.2 The equipotential 106
3.1.3 Velocity components in terms of q5 107
3.2 Laplace’s equation 109
3.3 Standard flows in terms of 11, and q5 110
3.3.1 Two-dimensional flow from a source (or towards a sink) 110
3.3.2 Line (point) vortex 112