# Aeronautical Engineering (MEng)

## Aircraft Aerodynamics

### Module aims

This course covers the essential aircraft aerodynamics over a range of Reynolds numbers & Mach numbers. The course is divided in to two parts, one part focussing on incompressible aerodynamics, the second exploring compressible aerodynamics.

The aims for the incompressible aerodynamics part of the course are:

• Convey the essential fluid dynamics associated with aircraft flight at low Mach number.  For the simple case of a thin-wing in incompressible flight, and by the use of a small-perturbation analysis, the Navier-Stokes equations can be reduced to the linear Laplace’s equation, which permits the linear superposition of simple solutions.  Thus the effects of incidence, thickness and camber can be examined separately, and we confine our attention to thin wings of varying planforms, but making the important distinction between large and small aspect ratios.  Large aspect-ratio wings are examined principally by using the distinction of bound and trailing vorticity, and in particular, by use of lifting-line theory – a “lumped-vortex” model.  This is then modified to account for the effects of sweep.  Slender wings are examined in a similar fashion, although there are some fundamental approximations.
• Viscous effects are accounted for by use of both laminar and turbulent boundary layer theory in two dimensions.  Simple boundary layer solutions are sought, as well as solutions to the lifting-line equation.  The physical aspects of separated flow on swept and slender wings are also examined.

The aims for the compressible aerodynamics part of the course are:

• To provide an introduction to available analytical and computational methods for the aerodynamic calculation of flow about aerofoils and wings based on the inviscid compressible fluid model. The non-linear character of flow discontinuities such as shocks is emphasised.  The shock-expansion theory is applied to the calculation of aerofoils in supersonic flows. Effects of friction and heat-transfer on compressible flow are emphasised.
• The theory of characteristics (TOC) is then introduced to gain a mathematical insight onto the behaviour of a compressible inviscid flow.  The TOC provides a numerical method for calculating unsteady 1-D isentropic flow and 2-D supersonic flow.  In this last case the TOC leads to an elegant method for wind tunnel nozzle design.
• Concentrating in the aerodynamic aspects of compressible inviscid flow, the behaviour of the flow about aerofoils and wings is characterised in terms of the Mach number and the previous theory is used to discuss the feasibility and applicability of linear theories at the different regimes.  The linearised theory of compressible flow is described and particular attention is given to the 2-D and 3-D subsonic similarity rules.  The critical Mach number is used as a limit of the validity of the linearised theory of subsonic flow.

### Learning outcomes

Knowledge and understanding:
On successfully completing this course unit, students will be able to:

• Understand the fundamental processes at work on thin wings, in terms of bound and trailing vorticity.
• Understand the effects of wing camber, sweep and taper, and the rationale for their use in design.
• Understand the effect of compressibility on the behaviour of an inviscid fluid in terms of wave propagation and its dependence on the Mach number.
• Understand the effects of friction and heat-transfer on compressible flow.
• Understand the variation of the aerodynamic coefficients of aerofoils in steady flow as a function of the Mach number.
• Understand the area rule for wing-body configurations and its application to aeroplane design.

Skills and other attributes:
On successfully completing this course unit, students will be able to design wings for incompressible and compressible flow:

Intellectual skills

• Perform analyses of wing sections for a range of thickness, camber and incidence, including the effects of boundary layer behaviour.
• Obtain simple solutions to the lifting-line equation for simple swept and unswept wings of large aspect ratio.
• Apply the TOC for the numerical and graphical solution of 1-D unsteady flows and 2-D supersonic flows
• Use the TOC for 2-D nozzle design.
• Justify the classification of the flow into subsonic, transonic, supersonic and hypersonic through the description of the main physical phenomena involved at the different regimes and to explain their effect in the aerodynamic coefficients.
• Calculate solutions for aerofoils and wings in subsonic flow given an incompressible solution by using similarity transformations.
• Use the critical Mach number as a criterion for determining the range of validity of the subsonic similarity rules for aerofoils and wings.
• Carry out experiments in a wind tunnel, utilising appropriate measurement equipment

Practical skills

• Use fundamental knowledge to investigate and assess new and emerging technologies
• Apply mathematical and computer-based models for solving aerodynamic fluid-flow problems in engineering

Transferable skills

• Communicate the importance of flight to a general audience

### Module syllabus

• Introduction: The need for numerical methods.  Equations of motion: derivation of 3-D, incompressible, Navier Stokes equations.  Reynolds stresses.  Review of small perturbation theory: hierarchy of small disturbances.  Numerical solution of incompressible, potential flow.  Effects of thickness and camber.
• Surface singularity methods.  Surface source method (A.M.O. Smith).
• Introduction to boundary layers: the thin-shear-layer approximation. Blasius solutions Laminar: Thwaites’ approximate method.  Turbulent boundary layers, Reynolds stresses, the concept of eddy-viscosity, the law of the wall.
• Lifting line theory: wings of large aspect ratio: basis of theory for wings of finite span.  Downwash and induced drag.  Prandtl’s theory.  Use of lifting line theory: solution method.  Elliptic loading.  Solutions for general planforms – collocation and iteration methods.
• Swept wings in incompressible flow. Comparison with/without sweep: the use of taper.
• Separated flow and stall.
• Compressible flow: governing equations.  Waves and speed of sound.  Validity of the incompressible assumption.  Normal and oblique shock waves.  Prandtl-Meyer expansion waves.  Shock expansion theory.
• Rayleigh flow, Fanno flow.
• Method of characteristics.  Introduction: 1-D unsteady inviscid flow.  Unsteady jump conditions.  Unsteady motion in a constant area duct.  Characteristic equations and compatibility conditions.  Examples: simple and non-simple regions.  Shock tube problem.
• Steady 2-D irrotational isentropic flow.  Governing equations.  Characteristic lines.  Compatibility conditions.  Example: nozzle design.
• Aerodynamic analysis methods for compressible flow.  Introduction: Characteristics of flight at different regimes.
• Small perturbation analysis for isentropic irrotational flows.  Governing equations.  Boundary conditions.  Linearised pressure coefficient.
• Similarity rules for 2-D aerofoils (infinite wing).  Prandtl-Glauert rule.  Critical Mach number.  Other similarity relations:  Karman-Tsien, Laitonde.  Aerofoils in supersonic flow: Ackeret’s rule.  Spreiter’s rule for transonic flow.  Similarity rules for 3-D wings (finite wing).  Gothert’s rule (subsonic/supersonic).
• Critical Mach number for swept back wings.  Transonic wings (geometry only). Area rule for wing/body combinations.
• An introduction to elements of hypersonic flow, including thin shock layers, Newtonian theory, entropy layers and heat transfer.

### Pre-requisites

AERO40001 Aerodynamics 1
AERO50001 Aerodynamics 2

### Teaching methods

• Lectures (with gapped lecture notes and practical examples)
• Six tutorial sheets
• Surgeries to focus on specific areas of difficulty.
• Laboratory sessions for low speed and supersonic wind tunnel testing

### Assessments

Examined Assessment
3-hour written examination in the Summer term (80%).
Low Speed Flow Past a High Aspect Ratio Wing Lab (10%)
Compressible Flow Lab (10%)
Non-Examined Assessment
Peer-reviewed class questionnaire

### Core

• #### Modern compressible flow : with historical perspective /

Anderson, John David.

3rd ed., McGraw-Hill

• #### Modern compressible flow : with historical perspective

Anderson, John D.,

Fourth edition.; International student edition., McGraw-Hill

• #### Chapter 7 : "Unsteady Wave Motion". in Modern compressible flow : with historical perspective

John D. Anderson, Jr.

• #### Section 3.3-3.5. in Modern compressible flow : with historical perspective

John D. Anderson, Jr.

• #### Section 3.8-3.9. in Modern compressible flow : with historical perspective

John D. Anderson, Jr.

• #### Fundamentals of aerodynamics /

Anderson, John David ,

Fifth edition in SI Units, McGraw-Hill,

• #### Fundamentals of aerodynamics (SI units)

Anderson, John D.

5th,

### Supplementary

• #### An Introduction to Theoretical and Computational Aerodynamics.

Moran, Jack.

New York : Dover Publications

Moran, Jack

Wiley

• #### Elements of Gas Dynamics.

Liepmann, H. W.

New York : Dover Publications

• #### Elements of gas dynamics /

Liepmann, H. W.

Wiley

• #### Low-speed aerodynamics / Joseph Katz, Allen Plotkin.

Katz, Joseph.

2nd ed., Cambridge University Press

• #### Low-Speed Aerodynamics / Joseph Katz, Allen Plotkin.

Katz, Joseph,

2nd ed., Cambridge University Press,

• #### Boundary-layer theory /

Schlichting, Hermann,

9th edition., Springer,

• #### Boundary-layer theory /

Schlichting, Hermann,

Ninth edition., Springer,