Aerodynamics for Engineering Students

Contents
Preface
	Additional resources
Acknowledgments
1 Basic concepts and definitions
	1.1 Introduction
		1.1.1 Basic concepts
	1.2 Units and dimensions
		1.2.1 Fundamental dimensions and units
		1.2.2 Fractions and multiples
		1.2.3 Units of other physical quantities
		1.2.4 Imperial units
	1.3 Relevant properties
		1.3.1 Forms of matter
		1.3.2 Fluids
		1.3.3 Pressure
			Pressure in fluid at rest
			Pascal’s law
		1.3.4 Temperature
		1.3.5 Density
		1.3.6 Viscosity
			Dynamic viscosity
			Kinematic viscosity
		1.3.7 Speed of sound and bulk elasticity
		1.3.8 Thermodynamic properties
			Specific heat
				Specific heat at constant volume
				Specific heat at constant pressure
				Ratio of specific heats
			Enthalpy
			Entropy
	1.4 Aeronautical definitions
		1.4.1 Airfoil geometry
			Camber
			Thickness distribution
		1.4.2 Wing geometry
			Wingspan
			Chords
			Wing area
			Mean chords
			Aspect ratio
			Wing sweep
			Dihedral angle
			Incidence, twist, wash-out, and wash-in
	1.5 Dimensional analysis
		1.5.1 Fundamental principles
		1.5.2 Dimensional analysis applied to aerodynamic force
	1.6 Basic aerodynamics
		1.6.1 Aerodynamic force and moment
			Lift, L
			Drag, D
			Side force, Y
			Pitching moment, M
			Rolling moment, LR
			Yawing moment, N
		1.6.2 Force and moment coefficients
		1.6.3 Pressure distribution on an airfoil
		1.6.4 Pitching moment
			Aerodynamic center
			Center of pressure
		1.6.5 Types of drag
			Total drag
				Skin-friction drag (or surface-friction drag)
				Pressure drag
			Induced drag (or vortex drag)
			Wave drag
			Form drag (or boundary-layer pressure drag)
			Profile drag (or boundary-layer drag)
			Comparison of drags for various body types
			The wake
		1.6.6 Estimation of lift, drag, and pitching moment coefficients from the pressure distribution
		1.6.7 Induced drag
		1.6.8 Lift-dependent drag
		1.6.9 Airfoil characteristics
			Lift coefficient: incidence
			Effect of aspect ratio on the CL versus α curve
			Effect of Reynolds number on the CL versus α curve
			Drag coefficient versus lift coefficient
			Drag coefficient versus lift coefficient squared
			Pitching moment coefficient
	1.7 Basic flight stability
	1.8 Control-volume analysis
		1.8.1 Froude’s momentum theory of propulsion
		1.8.2 Momentum theory applied to the helicopter rotor
			Actuator disc in hovering flight
			Vertical climbing flight
			Slow, powered descending flight
			Translational helicopter flight
	1.9 Hydrostatics
	1.10 Exercises
2 Equations of motion
	2.1 Introduction
		2.1.1 Selection of reference frame
			Types of flow
		2.1.2 A comparison of steady and unsteady flow
			True unsteady flow
	2.2 One-dimensional flow: the basic equations
		2.2.1 One-dimensional flow: the basic equations of conservation
			Conservation of mass
			Momentum equation
			The conservation of energy
			Equation of state
			Momentum equation for an incompressible fluid
		2.2.2 Comments on the momentum and energy equations
	2.3 Viscous boundary layers
	2.4 Measurement of air speed
		2.4.1 Pitôt-static tube
		2.4.2 Pressure coefficient
		2.4.3 Air-speed indicator: indicated and equivalent air speeds
		2.4.4 Incompressibility assumption
	2.5 Two-dimensional flow
		2.5.1 Component velocities
			Fluid acceleration
		2.5.2 Equation of continuity or conservation of mass
		2.5.3 Equation of continuity in polar coordinates
	2.6 Stream function and streamline
		2.6.1 Stream function ψ
			Sign convention for stream functions
		2.6.2 Streamline
		2.6.3 Velocity components in terms of ψ
	2.7 Momentum equation
		2.7.1 Euler equations
	2.8 Rates of strain, rotational flow, and vorticity
		2.8.1 Distortion of fluid element in flow field
		2.8.2 Rate of shear strain
		2.8.3 Rate of direct strain
		2.8.4 Vorticity
		2.8.5 Vorticity in polar coordinates
		2.8.6 Rotational and irrotational flow
		2.8.7 Circulation
	2.9 Navier-Stokes equations
		2.9.1 Relationship between rates of strain and viscous stresses
		2.9.2 Derivation of the Navier-Stokes equations
	2.10 Properties of the Navier-Stokes equations
	2.11 Exact solutions of the Navier-Stokes equations
		2.11.1 Couette flow: simple shear flow
		2.11.2 Plane Poiseuille flow: pressure-driven channel flow
		2.11.3 Hiemenz flow: two-dimensional stagnation-point flow
	2.12 Exercises
3 Viscous flow and boundary layers
	3.1 Introduction
	3.2 Boundary-layer theory
		3.2.1 Blasius’s solution
		3.2.2 Definitions of boundary-layer thickness
			Displacement thickness
			Momentum thickness
			Kinetic-energy thickness
		3.2.3 Skin-friction drag
		3.2.4 Laminar boundary-layer thickness along a flat plate
		3.2.5 Solving the general case
	3.3 Boundary-layer separation
		3.3.1 Separation bubbles
	3.4 Flow past cylinders and spheres
		3.4.1 Turbulence on spheres
		3.4.2 Golf balls
		3.4.3 Cricket balls
	3.5 The momentum-integral equation
		3.5.1 An approximate velocity profile for the laminar boundary layer
	3.6 Approximate methods for a boundary layer on a flat plate with zero pressure gradient
		3.6.1 Simplified form of the momentum-integral equation
		3.6.2 Rate of growth of a laminar boundary layer on a flat plate
		3.6.3 Drag coefficient for a flat plate of streamwise length L with a wholly laminar boundary layer
		3.6.4 Turbulent velocity profile
		3.6.5 Rate of growth of a turbulent boundary layer on a flat plate
		3.6.6 Drag coefficient for a flat plate with a wholly turbulent boundary layer
		3.6.7 Conditions at transition
		3.6.8 Mixed boundary-layer flow on a flat plate with zero pressure gradient
	3.7 Additional examples of the momentum-integral equation
	3.8 Laminar-turbulent transition
	3.9 The physics of turbulent boundary layers
		3.9.1 Reynolds averaging and turbulent stress
		3.9.2 Boundary-layer equations for turbulent flows
		3.9.3 Eddy viscosity
		3.9.4 Prandtl’s mixing-length theory of turbulence
		3.9.5 Regimes of turbulent wall flow
			Outer boundary layer
		3.9.6 Formulae for local skin-friction coefficient and drag
			Effects of wall roughness
		3.9.7 Distribution of Reynolds stresses and turbulent kinetic energy across the boundary layer
		3.9.8 Turbulence structures in the near-wall region
	3.10 Estimation of profile drag from the velocity profile in a wake
		3.10.1 Momentum-integral expression for the drag of a two-dimensional body
		3.10.2 Jones’s wake traverse method for determining profile drag
		3.10.3 Growth rate of a two-dimensional wake using the general momentum-integral equation
	3.11 Some boundary-layer effects in supersonic flow
		3.11.1 Near-normal shock interaction with the laminar boundary layer
		3.11.2 Shock-wave/boundary-layer interaction in supersonic flow
	3.12 Exercises
4 Compressible flow
	4.1 Introduction
	4.2 Isentropic one-dimensional flow
		4.2.1 Pressure, density, and temperature ratios along a streamline in isentropic flow
		4.2.2 Ratio of areas at different sections of the stream tube in isentropic flow
		4.2.3 Velocity along an isentropic stream tube
		4.2.4 Variation of mass flow with pressure
	4.3 One-dimensional flow: weak waves
		4.3.1 Speed of sound (acoustic speed)
	4.4 One-dimensional flow: plane normal shock waves
		4.4.1 One-dimensional properties of normal shock waves
		4.4.2 Pressure-density relations across the shock
		4.4.3 Static pressure jump across a normal shock
		4.4.4 Density jump across the normal shock
		4.4.5 Temperature rise across the normal shock
		4.4.6 Entropy change across the normal shock
		4.4.7 Mach number change across the normal shock
		4.4.8 Velocity change across the normal shock
		4.4.9 Total pressure change across the normal shock
		4.4.10 Pitôt tube equation
		4.4.11 Converging-diverging nozzle operations
	4.5 Mach waves and shock waves in two-dimensional flow
		4.5.1 Mach waves
		4.5.2 Mach wave reflection
		4.5.3 Mach wave interference
		4.5.4 Shock waves
		4.5.5 Plane oblique shock relations
		4.5.6 Shock polar
		4.5.7 Two-dimensional supersonic flow past a wedge
	4.6 Matlab functions for compressible flow
	4.7 Exercises
5 Potential flow
	5.1 Introduction
		5.1.1 The velocity potential
			Sign convention for velocity potential
		5.1.2 The equipotential
		5.1.3 Velocity components in terms of φ
	5.2 Laplace’s equation
	5.3 Standard flows in terms of ψ and φ
		5.3.1 Two-dimensional flow from a source (or towards a sink)
			To find the stream function ψ of a source
			To find the velocity potential φ of a source
		5.3.2 Line (point) vortex
		5.3.3 Uniform flow
			Flow of constant velocity parallel to Ox axis from left to right
			Flow of constant velocity parallel to Oy axis
			Flow of constant velocity in any direction
		5.3.4 Solid boundaries and image systems
		5.3.5 A source in a uniform horizontal stream
			Method (see Fig. 5.14)
			The position of the stagnation point
			The local velocity
		5.3.6 Source-sink pair
		5.3.7 A source set upstream of an equal sink in a uniform stream
		5.3.8 Doublet
		5.3.9 Flow around a circular cylinder given by a doublet in a uniform horizontal flow
			The pressure distribution around a cylinder
		5.3.10 A spinning cylinder in a uniform flow
			The normal force on a spinning circular cylinder in a uniform stream
			The flow pattern around a spinning cylinder
		5.3.11 Bernoulli’s equation for rotational flow
	5.4 Axisymmetric flows (inviscid and incompressible flows)
		5.4.1 Cylindrical coordinate system
		5.4.2 Spherical coordinates
		5.4.3 Axisymmetric flow from a point source (or towards a point sink)
		5.4.4 Point source and sink in a uniform axisymmetric flow
		5.4.5 The point doublet and the potential flow around a sphere
		5.4.6 Flow around slender bodies
	5.5 Computational (panel) methods
	5.6 A computational routine in Matlab®
	5.7 Exercises
6 Thin airfoil theory
	6.1 Introduction
		6.1.1 The Kutta condition
		6.1.2 Circulation and vorticity
		6.1.3 Circulation and lift (the Kutta–Zhukovsky theorem)
	6.2 The development of airfoil theory
	6.3 General thin-airfoil theory
	6.4 Solution to the general equation
		6.4.1 Thin symmetrical flat-plate airfoil
			Aerodynamic coefficients for a flat plate
		6.4.2 General thin-airfoil section
			Lift and moment coefficients for a general thin airfoil
	6.5 The flapped airfoil
		6.5.1 Hinge moment coefficient
	6.6 The jet flap
	6.7 Normal force and pitching moment derivatives due to pitching
		6.7.1 (Zq)(Mq) wing contributions
	6.8 Particular camber lines
		6.8.1 Cubic camber lines
		6.8.2 NACA four-digit wing sections
	6.9 The thickness problem for thin-airfoil theory
		6.9.1 Thickness problem for thin airfoils
	6.10 Computational (panel) methods for two-dimensional lifting flows
	6.11 A “new theory of lift”?
		6.11.1 Why search for a new theory?
		6.11.2 What is different in the new theory?
	6.12 Exercises
7 Wing theory
	7.1 The vortex system
		7.1.1 Starting vortex
		7.1.2 Trailing vortex system
		7.1.3 Bound vortex system
		7.1.4 Horseshoe vortex
	7.2 Laws of vortex motion
		7.2.1 Helmholtz’s theorems
		7.2.2 The Biot-Savart law
			Special cases of the Biot-Savart law
		7.2.3 Variation of velocity in vortex flow
	7.3 The wing as a simplified horseshoe vortex
		7.3.1 Influence of downwash on the tailplane
		7.3.2 Ground effects
	7.4 Vortex sheets
		7.4.1 Use of vortex sheets to model the lifting effects of a wing
			Lifting effect
	7.5 Relationship between spanwise loading and trailing vorticity
		7.5.1 Induced velocity (downwash)
		7.5.2 The consequences of downwash—trailing vortex drag
		7.5.3 Characteristics of simple symmetric loading—elliptic distribution
			Lift for elliptic distribution
			Downwash for elliptic distribution
			Induced drag (vortex drag) for elliptic distribution
		7.5.4 General (series) distribution of lift
		7.5.5 Aerodynamic characteristics for symmetrical general loading
			Lift on the wing
			Downwash
			Induced drag (vortex drag)
			Minimum induced drag condition
	7.6 Determination of load distribution on a given wing
		7.6.1 General theory for wings of high aspect ratio
		7.6.2 General solution to Prandtl’s integral equation
		7.6.3 Load distribution for minimum drag
	7.7 Swept and delta wings
		7.7.1 Yawed wings of infinite span
		7.7.2 Swept wings of finite span
		7.7.3 Wings of small aspect ratio
	7.8 Slope of the lift curve for wings
	7.9 Computational (panel) methods for wings
	7.10 Exercises
8 Airfoils and wings in compressible flow
	8.1 Wings in compressible flow
		8.1.1 Transonic flow: the critical Mach number
		8.1.2 Subcritical flow: the small-perturbation theory (Prandtl-Glauert rule)
			The equations of motion of a compressible fluid
			Small disturbances
			Prandtl-Glauert rule: the application of linearized theories of subsonic flow
			Constant chordwise ordinates
			Constant normal ordinates
			Critical pressure coefficient
			Application to swept wings
		8.1.3 Supersonic linearized theory (Ackeret’s rule)
			Symmetrical double wedge airfoil in supersonic flow
			Supersonic biconvex circular arc airfoil in supersonic flow
			General airfoil section
			Airfoil section made up of unequal circular arcs
			Double-wedge airfoil section
		8.1.4 Other aspects of supersonic wings
			The shock-expansion approximation
			Wings of finite span
			Computational methods
	8.2 Exercises
9 Computational fluid dynamics
	9.1 Computational methods
		9.1.1 Methods based on the momentum-integral equation
		9.1.2 Transition prediction
		9.1.3 Computational solution for the laminar boundary-layer equations
		9.1.4 Computational solution for turbulent boundary layers
		9.1.5 Zero-equation methods
			Cebeci-Smith method
		9.1.6 k−ε: a typical two-equation method
		9.1.7 Large-eddy simulation
			Two common choices of filter function
			Subgrid scale modeling
10 Flow control and wing design
	10.1 Introduction
	10.2 Maximizing lift for single-element airfoils
	10.3 Multi-element airfoils
		10.3.1 The slat effect
		10.3.2 The flap effect
		10.3.3 Off-the-surface recovery
		10.3.4 Fresh boundary-layer effect
		10.3.5 The Gurney flap
		10.3.6 Movable flaps: artificial bird feathers
	10.4 Boundary layer control prevention to separation
		10.4.1 Boundary-layer suction
		10.4.2 Control by tangential blowing
		10.4.3 Other methods of separation control
	10.5 Reduction of skin-friction drag
		10.5.1 Laminar flow control by boundary-layer suction
		10.5.2 Compliant walls: artificial dolphin skins
		10.5.3 Riblets
	10.6 Reduction of form drag
	10.7 Reduction of induced drag
	10.8 Reduction of wave drag
A Symbols and notation
	Subscripts
	Primes and superscripts
B The international standard atmosphere
C A solution of integrals of the type of Glauert’s integral
D Conversion of Imperial units to Systéme International (SI) units
Bibliography
Index