# Thin Airfoil Theory. Charles R. O Neill School of Mechanical and Aerospace Engineering Oklahoma State University Stillwater, OK 74078

Save this PDF as:

Size: px
Start display at page:

Download "Thin Airfoil Theory. Charles R. O Neill School of Mechanical and Aerospace Engineering Oklahoma State University Stillwater, OK 74078"

## Transcription

1 13 Thin Airfoil Theory Charles R. O Neill School of Mechanical and Aerospace Engineering Oklahoma State University Stillwater, OK 7478 Project One in MAE 3253 Applied Aerodynamics and Performance March 2

2 Ideal Thin Airfoils 1 ABSTRACT The thin airfoil theory for calculation of section flight properties is reviewed. Lift and moment coefficient and center of pressure calculations are made for cambered and flapped wing sections. Effects of camber and flaps are discussed. The thin wing theory results are compared to experimental airfoil data.

3 Ideal Thin Airfoils 2 INTRODUCTION Aircraft fly by overcoming gravity with a lifting force. This force can be provided by a wing. A wing s cross sectional geometry influences the flow of air. This combined geometry of the wing and the reaction of the air causes any general solution of wing section properties to become too complicated to use or impossible to find. A more simple approach to finding flight properties of wing sections is to assume an inviscid and incompressible flow. A vortex superimposed in a airstream simulates the process of lift generation by a wing section. A vortex distribution placed along the wing will simulate the actual properties of the wing and allow a simple method of calculating wing properties. The thin airfoil theory calculates a distribution of vortices that is compatible with a thin representation of an airfoil. This distribution can be used to find the lift, moment and pressure properties of an airfoil. The objective is to review the thin airfoil theory and to apply the theory to three wing sections. The wing sections will consist of one curved cambered NACA 2412 airfoil, one straight line flapped airfoil and one cambered line segment airfoil. The coefficients of lift and moment and the center of pressure locations will be found. THEORY The thin airfoil theory simulates the aerodynamic properties of an airfoil section with vortex sheets. The vortex sheet consists of a continuous vortex distribution along the chord of the wing. The induced velocity at a distance r caused by a vortex of strength γ is dv = γds 2πr Integrating over the wing chord yields the total downward velocity due to the vortex sheet. v = chord Because fluid cannot flow through the wings surface, the downward velocity at the wing s γds 2πr

4 Ideal Thin Airfoils 3 surface must be zero. Thus a thin wing section at an small angle of attack can be approximated by where U is the flight velocity and dx v(x) U = α dx is the slope of the wing section with respect to the chord line. Substituting for the downward velocity, where r = x x, yields, 1 2πU chord γ(x) dx = α x x dx Additionally, the Kutta condition forces a fluid to flow off a surface tangentailly causing the vortex strength to be zero at the trailing edge. For ease of later calculation, the coordinate system is transformed to a polar system centered at the mid chord by x = c 2 (1 cos(θ)). Thus, 1 2πU γ(θ) sin(θ) dθ = α cos(θ) cos(θ ) dx A general solution of the above equation is unknown and would be in any case too complex to handle. However, an approximation can be found by assuming a distribution of γ, [ 1 + cos(θ) γ(θ) = 2U A + sin(θ) n=1 ] A n sin(nθ) Where values of A through A n are given by (McCormick, 1995) A = α 1 π A n = 2 π dx dθ dx cos(nθ)dθ Applying the Jutta Joukowski expression for lift and moment versus vortex strength yields, C l = 2πA + πa 1 C mle = π 2 (A + A 1 A 2 2 ) The moment at any arbitrary point off the leading edge can be found by adding the contribution due to the lifting force and the distance from the leading edge. At the quarter

5 Ideal Thin Airfoils 4 chord location, the resulting moment does not depend on the angle of attack and is given by (McCormick, 1995), C m.25c = π 4 (A 1 A 2 ) Similarily, the center of pressure is given by x cp = c 4 (1 + π (A 1 A 2 )) C l RESULTS AND DISCUSSION Calculations were performed and are given in Appendix A, B and C. Three profiles were analized as thin wings. These included one cambered wing consisting of constantly changing curvature along a mean chord line. One flapped and one cambered wing both consisting of straight line segments. The NACA 2412 cross section is given in Appendix A. The forward section calculated from the mean camber line is.125 cos(θ).25. The aft section has a of.555 cos(θ).111. The mean camber line equations are switched at θ = 1.369radians. Thus, A = α A 1 = and A 2 = The coefficient of lift is as C l = 2πα and the moment coefficient at the quarter chord is C m.25c =.538. The wing section has zero lift at an angle of attack of 2.7 degrees. The flapped airfoil is given in Appendix B. The forward non-flapped section s dx is and the aft flapped section has a dx of The wing switches from unflapped to flapped segments at θ equals radians. Thus, A = α A 1 = and A 2 equals The effective coefficient of lift is C leff = 2πα and the effective moment coefficient at the quarter chord is C m.25ceff = The ( ) effective center of pressure is at x cp = c πα eff When converted to true chord lengths and true angles of attack, C l = π , C mac = and ( ) x cp =.9849 c πα+.985 The cambered airfoil is given in Appendix C. The forward section s dx =.8 and the aft section has a dx of.8. The wing switches profile lines at θ = 9degrees. A is simply

6 Ideal Thin Airfoils 5 α. A 1 =.1186 and A 2 equals. At 1 degrees angle of attack, the coefficient of lift is C l = and the moment coefficient at the quarter chord is C m.25c =.8. In each case, the coefficient of lift depends on a the angle of attack plus an increment due to the curvature of the wing. For the flapped wing, the increment was 1.58 while the unflapped NACA and cambered wing had an increment of approximately.3. This is consistent with flap increasing the lift at a constant true angle of attack. Abbott and Doenhoff give experimental data for the NACA The experimental coefficient of lift at zero angle of attack is approximately.25 which is close to.228 as given by the thin wing theory. The increase of the coefficient of lift versus angle of attack shown experimentally is approximately.12 per degree. The thin wing gives dc l dα = 2π or.11 per degree. Both experimental and theoretical values of the zero lift angle of attack are approximately 2 degrees. The similarity of the experimental and theoretical values is remarkable considering the assumptions and simple mathematics of the thin wing theory. CONCLUSION The thin airfoil theory is a method of calculating wing section properties. The thin wing theory only requires an expression of the mean chord line and thus can handle flapped and continuous wings. The mathematics are simple and involve only at most integration and differentiation. The results seem to be accurate when compared to experimental data when restricted to moderate angles of attack. The thin wing theory seems to be an adequate method for quick, simple and accurate flight properties of a wing section.

7 Ideal Thin Airfoils 6 REFERENCES McCormick, Barnes W., (1995) Aerodynamics, aeronautics, and flight mechanics, John Wiley & Sons, New York. Abbott, Ira H., Doenhoff, Albert E. (1959) Theory of Wing Sections, Dover Publications, New York.

8 Ideal Thin Airfoils 7 APPENDIX A Calculations: NACA 2412

9 Ideal Thin Airfoils 8 NACA 2412: The NACA 2412 s mean camber line is described piecewise with a switch in equations at x=.4c. z F ront : c =.125(.8x c x 2 ) c z Aft : c =.555(.2 +.8x c x c The derivatives of z with respect to x are, 2 ) f =.25(x.4c)c c 2 =.25 x c +.1 = a.111(x.4c)c c 2 =.111 x c The slopes are transformed into polar coordinates by substituting x = c 2 (1 cos(θ)) into the above derivatives. Thus, f =.25.5(1 cos(θ)) +.1 =.125 cos(θ).25 =.111.5(1 cos(θ)) =.555 cos(θ).111 a Also, the switching point is transformed. x =.4c =.5c(1 cos(θ)). Thus the angle for switching equations is radians. From theory, A = α 1 π dx dθ and A n = 2 π dx cos(nθ)dθ. The integration takes into account the location and distance of the front and aft equation for the section. Thus, A 1 = 2 π A 2 = 2 π A = α 1 π (.125 cos(θ).25)dθ 1 π = α (.23577) = α (.125 cos(θ).25) cos(θ)dθ + 2 π = = (.125 cos(θ).25) cos(2θ)dθ + 2 π (.555 cos(θ).111)dθ (.555 cos(θ).111) cos(θ)dθ (.555 cos(θ).111) cos(2θ)dθ

10 Ideal Thin Airfoils 9 = = From theory, the lift coefficient is C l = 2πA + πa 1 and the moment coefficient about the quarter chord is C m.25c = π 4 (A 1 A 2 ). Thus, the lift and moment coefficients are C l = 2π(α.45165) π = 2πα C m.25c = π ( ) = The section has zero lift when C l equals zero. Solved for alpha, the zero lift angle is α = π =.362radian = 2.7degrees Thus, the angle of attack for zero lift is -2.7 degrees.

11 Ideal Thin Airfoils 1 APPENDIX B Calculations: Flapped airfoil

12 Ideal Thin Airfoils 11 Flapped airfoil: The thin airfoil shown above has a 2 percent flap deflected 25 degrees. The effective chord length is found from the law of cosines, c 2 = A 2 + B 2 2ABcos(c) = cos(18 25) which yields an effective chord length of.9849 of the original chord. From the law of sines, the angle between the front section and the chord line is found. sin(β).2 = sin(155).9849 Thus, β = 4.923degrees and δ = 2.77degrees. The flap width projected along the effective chord line is.2c sin(2.77) =.1878c or.19726c eff when in terms of the effective chord length. So, the effective chord length of the unflapped section of the wing is.8927c eff. The slope of the lines forming the wing and flap are dx f = tan(4.923) = = tan(2.77) =.3655 dx a The switch from the forward to the aft equation is at x = =.5(1 cos(θ)) which is at radians in the polar coordinate system. From theory, the A coefficients are, A = α 1 π dθ 1 π dθ A 1 = 2 π A 2 = 2 π = α = α cos(θ)dθ + 2 π = = cos(2θ)dθ + 2 π = = cos(θ)dθ.3655 cos(2θ)dθ

13 Ideal Thin Airfoils 12 The effective lift and moment coefficients are C leff = 2π(α ) π = 2πα C m.25ceff = π ( ( )) = The location of the center of pressure with respect to the effective chord length is x cp = c 4 (1 + π C l (A 1 A 2 )) = c 4 ( ) 2πα eff The lift and moment coefficients and the center of pressure can be converted to the true chord length and angle of attack. α = α eff c = c eff.9849 Substituting the true values for the effective values and then simplifying yields, C l = C leff ( c eff c ) =.9849 (2π(α ) +.985) = π C mac = C m.25ceff ( c eff c )2 =.27858(.9849) 2 = ( ) 1 + x cp = x cpeff ( c eff c ) =.9849 c πα +.985

14 Ideal Thin Airfoils 13 APPENDIX C Calculations: Thin Cambered Airfoil

15 Ideal Thin Airfoils 14 Thin Cambered Airfoil: The slope of the front piece of the airfoil is found from the thickness and half chord length. dx =.4.5 =.8 The aft piece has the negative magnitude of the front slope. So, dx =.4.5 =.8 Transforming the coordinates yields 9 degrees ( π 2 location to switch slope equations. Thus, radians) as the A = α 1 π.5π.8dθ 1 π.5π.8dθ A 1 = 2 π A 2 = 2 π.5π = α.4 (.4) = α.8 cos(θ)dθ + 2 π.5π = = π The coefficient of lift is The quarter chord moment coefficient is.8 cos(2θ)dθ + 2 π = + =.5π C l = 2πα π = 2πα +.32 C m.25c = π ( ) = cos(θ)dθ.8 cos(2θ)dθ At 1 degrees (.1745 radians) angle of attack as shown, the lift and moment coefficients are, C l = 2π = C m.25c =.8

### NACA Nomenclature NACA 2421. NACA Airfoils. Definitions: Airfoil Geometry

0.40 m 0.21 m 0.02 m NACA Airfoils 6-Feb-08 AE 315 Lesson 10: Airfoil nomenclature and properties 1 Definitions: Airfoil Geometry z Mean camber line Chord line x Chord x=0 x=c Leading edge Trailing edge

### Lift and Drag on an Airfoil ME 123: Mechanical Engineering Laboratory II: Fluids

Lift and Drag on an Airfoil ME 123: Mechanical Engineering Laboratory II: Fluids Dr. J. M. Meyers Dr. D. G. Fletcher Dr. Y. Dubief 1. Introduction In this lab the characteristics of airfoil lift, drag,

### AOE 3134 Complete Aircraft Equations

AOE 3134 Complete Aircraft Equations The requirements for balance and stability that we found for the flying wing carry over directly to a complete aircraft. In particular we require the zero-lift pitch

### NACA 2415- FINDING LIFT COEFFICIENT USING CFD, THEORETICAL AND JAVAFOIL

NACA 2415- FINDING LIFT COEFFICIENT USING CFD, THEORETICAL AND JAVAFOIL Sarfaraj Nawaz Shaha 1, M. Sadiq A. Pachapuri 2 1 P.G. Student, MTech Design Engineering, KLE Dr. M S Sheshgiri College of Engineering

### The aerodynamic center

The aerodynamic center In this chapter, we re going to focus on the aerodynamic center, and its effect on the moment coefficient C m. 1 Force and moment coefficients 1.1 Aerodynamic forces Let s investigate

### Lengths in Polar Coordinates

Lengths in Polar Coordinates Given a polar curve r = f (θ), we can use the relationship between Cartesian coordinates and Polar coordinates to write parametric equations which describe the curve using

### A. Hyll and V. Horák * Department of Mechanical Engineering, Faculty of Military Technology, University of Defence, Brno, Czech Republic

AiMT Advances in Military Technology Vol. 8, No. 1, June 2013 Aerodynamic Characteristics of Multi-Element Iced Airfoil CFD Simulation A. Hyll and V. Horák * Department of Mechanical Engineering, Faculty

### PROBLEM SET. Practice Problems for Exam #1. Math 2350, Fall 2004. Sept. 30, 2004 ANSWERS

PROBLEM SET Practice Problems for Exam #1 Math 350, Fall 004 Sept. 30, 004 ANSWERS i Problem 1. The position vector of a particle is given by Rt) = t, t, t 3 ). Find the velocity and acceleration vectors

### NACA airfoil geometrical construction

The NACA airfoil series The early NACA airfoil series, the 4-digit, 5-digit, and modified 4-/5-digit, were generated using analytical equations that describe the camber (curvature) of the mean-line (geometric

### High-Lift Systems. High Lift Systems -- Introduction. Flap Geometry. Outline of this Chapter

High-Lift Systems Outline of this Chapter The chapter is divided into four sections. The introduction describes the motivation for high lift systems, and the basic concepts underlying flap and slat systems.

### Flap Optimization for Take-off and Landing

Proceedings of the 10th Brazilian Congress of Thermal Sciences and Engineering -- ENCIT 2004 Braz. Soc. of Mechanical Sciences and Engineering -- ABCM, Rio de Janeiro, Brazil, Nov. 29 -- Dec. 03, 2004

### Computational Aerodynamic Analysis on Store Separation from Aircraft using Pylon

International Journal of Engineering Science Invention (IJESI) ISSN (Online): 2319 6734, ISSN (Print): 2319 6726 www.ijesi.org ǁ PP.27-31 Computational Aerodynamic Analysis on Store Separation from Aircraft

### Development and Design of a Form- Adaptive Trailing Edge for Wind Turbine Blades

Development and Design of a Form- Adaptive Trailing Edge for Wind Turbine Blades Master Thesis Alireza Taheri REMENA Batch 5 Matriculation Number - 33105532 Institut Thermische Energietechnik 10.02.2015

### EXPERIMENTAL ANALYSIS OR AIRFOIL FOR HIGH ANGLE OF ATTACK

UNIVERSIDADE FEDERAL DO PARÁ MECHANICAL ENGINEERING DEPARTMENT TURBOMACHINERY GROUP SCIENTIFIC INICIATION MAGAZINE PROJECT IDENTIFICATION Professor: André Luiz Amarante Mesquita Title: Doctor Student:

### AE 430 - Stability and Control of Aerospace Vehicles

AE 430 - Stability and Control of Aerospace Vehicles Atmospheric Flight Mechanics 1 Atmospheric Flight Mechanics Performance Performance characteristics (range, endurance, rate of climb, takeoff and landing

### Application of CFD Simulation in the Design of a Parabolic Winglet on NACA 2412

, July 2-4, 2014, London, U.K. Application of CFD Simulation in the Design of a Parabolic Winglet on NACA 2412 Arvind Prabhakar, Ayush Ohri Abstract Winglets are angled extensions or vertical projections

### Performance. 13. Climbing Flight

Performance 13. Climbing Flight In order to increase altitude, we must add energy to the aircraft. We can do this by increasing the thrust or power available. If we do that, one of three things can happen:

### Wing Design: Major Decisions. Wing Area / Wing Loading Span / Aspect Ratio Planform Shape Airfoils Flaps and Other High Lift Devices Twist

Wing Design: Major Decisions Wing Area / Wing Loading Span / Aspect Ratio Planform Shape Airfoils Flaps and Other High Lift Devices Twist Wing Design Parameters First Level Span Area Thickness Detail Design

### Deflections. Question: What are Structural Deflections?

Question: What are Structural Deflections? Answer: The deformations or movements of a structure and its components, such as beams and trusses, from their original positions. It is as important for the

### I think that a smaller radius of curvature will produce more lift using the Coanda Effect.

The Coanda Conundrum Mackenzie Carter Background My experiment was constructed to test the Coanda effect which occurs when a fluid that is traveling parallel to a nearby surface adheres to that surface.

### XFlow CFD results for the 1st AIAA High Lift Prediction Workshop

XFlow CFD results for the 1st AIAA High Lift Prediction Workshop David M. Holman, Dr. Monica Mier-Torrecilla, Ruddy Brionnaud Next Limit Technologies, Spain THEME Computational Fluid Dynamics KEYWORDS

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### Experiment #1, Analyze Data using Excel, Calculator and Graphs.

Physics 182 - Fall 2014 - Experiment #1 1 Experiment #1, Analyze Data using Excel, Calculator and Graphs. 1 Purpose (5 Points, Including Title. Points apply to your lab report.) Before we start measuring

### Introduction to Aircraft Stability and Control Course Notes for M&AE 5070

Introduction to Aircraft Stability and Control Course Notes for M&AE 57 David A. Caughey Sibley School of Mechanical & Aerospace Engineering Cornell University Ithaca, New York 14853-751 211 2 Contents

### Aerodynamic Department Institute of Aviation. Adam Dziubiński CFD group FLUENT

Adam Dziubiński CFD group IoA FLUENT Content Fluent CFD software 1. Short description of main features of Fluent 2. Examples of usage in CESAR Analysis of flow around an airfoil with a flap: VZLU + ILL4xx

### ESTIMATING R/C MODEL AERODYNAMICS AND PERFORMANCE

ESTIMATING R/C MODEL AERODYNAMICS AND PERFORMANCE Adapted from Dr. Leland M. Nicolai s Write-up (Technical Fellow, Lockheed Martin Aeronautical Company) by Dr. Murat Vural (Illinois Institute of Technology)

### Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving

### Chapter 6 Lateral static stability and control - 3 Lecture 21 Topics

Chapter 6 Lateral static stability and control - 3 Lecture 21 Topics 6.11 General discussions on control surface 6.11.1 Aerodynamic balancing 6.11.2 Set back hinge or over hang balance 6.11.3 Horn balanace

### Principles of Flight. Chapter 3. Introduction. Structure of the Atmosphere

Chapter 3 Principles of Flight Introduction This chapter examines the fundamental physical laws governing the forces acting on an aircraft in flight, and what effect these natural laws and forces have

### We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model

CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant

### Exam 1 Sample Question SOLUTIONS. y = 2x

Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

### Aerodynamic Design Optimization Discussion Group Case 4: Single- and multi-point optimization problems based on the CRM wing

Aerodynamic Design Optimization Discussion Group Case 4: Single- and multi-point optimization problems based on the CRM wing Lana Osusky, Howard Buckley, and David W. Zingg University of Toronto Institute

### APPENDIX 3-B Airplane Upset Recovery Briefing. Briefing. Figure 3-B.1

APPENDIX 3-B Airplane Upset Recovery Briefing Industry Solutions for Large Swept-Wing Turbofan Airplanes Typically Seating More Than 100 Passengers Briefing Figure 3-B.1 Revision 1_August 2004 Airplane

### 13.021 Marine Hydrodynamics Lecture 24B Lifting Surfaces. Introduction What are the characteristics of a lifting surface?

13.021 Marine Hydrodynamics Lecture 24B Lifting Surfaces Introduction What are the characteristics of a lifting surface? Lifting surfaces in marine hydrodynamics typically have many applications such as

### Design and Structural Analysis of the Ribs and Spars of Swept Back Wing

Design and Structural Analysis of the Ribs and Spars of Swept Back Wing Mohamed Hamdan A 1, Nithiyakalyani S 2 1,2 Assistant Professor, Aeronautical Engineering & Srinivasan Engineering College, Perambalur,

### Principles of glider flight

Principles of glider flight [ Lift, drag & glide performance ] Richard Lancaster R.Lancaster@carrotworks.com ASK-21 illustrations Copyright 1983 Alexander Schleicher GmbH & Co. All other content Copyright

### The Influence of Aerodynamics on the Design of High-Performance Road Vehicles

The Influence of Aerodynamics on the Design of High-Performance Road Vehicles Guido Buresti Department of Aerospace Engineering University of Pisa (Italy) 1 CONTENTS ELEMENTS OF AERODYNAMICS AERODYNAMICS

### Lab 8 Notes Basic Aircraft Design Rules 6 Apr 06

Lab 8 Notes Basic Aircraft Design Rules 6 Apr 06 Nomenclature x, y longitudinal, spanwise positions S reference area (wing area) b wing span c average wing chord ( = S/b ) AR wing aspect ratio C L lift

### PARAMETRIC INVESTIGATION OF A HIGH-LIFT AIRFOIL AT HIGH REYNOLDS NUMBERS. John C. Lin* NASA Langley Research Center, Hampton, VA 23681-0001.

PARAMETRIC INVESTIGATION OF A HIGH-LIFT AIRFOIL AT HIGH REYNOLDS NUMBERS John C. Lin* NASA Langley Research Center, Hampton, VA 23681-0001 and Chet J. Dominik McDonnell Douglas Aerospace, Long Beach, CA

### (1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its

(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:

### A NUMERICAL METHOD TO PREDICT THE LIFT OF AIRCRAFT WINGS AT STALL CONDITIONS

Braz. Soc. of Mechanical Sciences and Engineering -- ABCM, Rio de Janeiro, Brazil, Nov. 29 -- Dec. 3, 24 A NUMERICAL METHOD TO PREDICT THE LIFT OF AIRCRAFT WINGS AT STALL CONDITIONS Marcos A. Ortega ITA

### Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE

1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object

### MATH 118, LECTURES 14 & 15: POLAR AREAS

MATH 118, LECTURES 1 & 15: POLAR AREAS 1 Polar Areas We recall from Cartesian coordinates that we could calculate the area under the curve b taking Riemann sums. We divided the region into subregions,

### We can use more sectors (i.e., decrease the sector s angle θ) to get a better approximation:

Section 1.4 Areas of Polar Curves In this section we will find a formula for determining the area of regions bounded by polar curves. To do this, wee again make use of the idea of approximating a region

### Mathematical Modeling and Engineering Problem Solving

Mathematical Modeling and Engineering Problem Solving Berlin Chen Department of Computer Science & Information Engineering National Taiwan Normal University Reference: 1. Applied Numerical Methods with

### Lecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.8-4.12, second half of section 4.7

Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.8-4.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal

### Aerodynamics B Summary

Aerodynamics B ummary 1. Basic Concepts 1.1 Flow types If there is friction, thermal conduction or diffusion in a flow, it is termed viscous. If none of these things is present, the flow is inviscid. Inviscid

### FLUID MECHANICS, EULER AND BERNOULLI EQUATIONS

FLUID MECHANIC, EULER AND BERNOULLI EQUATION M. Ragheb 4//03 INTRODUCTION The early part of the 8 th century saw the burgeoning of the field of theoretical fluid mechanics pioneered by Leonhard Euler and

### How do sails work? Article by Paul Bogataj

How do sails work? Article by Paul Bogataj Sails are wings that use the wind to generate a force to move a boat. The following explanation of how this occurs can help understand how to maximize the performance

### Aircraft Design. Lecture 2: Aerodynamics. G. Dimitriadis. Introduction to Aircraft Design

Aircraft Design Lecture 2: Aerodynamics G. Dimitriadis Introduction! Aerodynamics is the study of the loads exerted by the flow of air over an aircraft (there are other applications but they are boring)!

### 9210-228 Level 7 Post Graduate Diploma in Mechanical Engineering Aerospace engineering

9210-228 Level 7 Post Graduate Diploma in Mechanical Engineering Aerospace engineering You should have the following for this examination one answer book non-programmable calculator pen, pencil, drawing

### When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.

Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs

### A COMPARISON BETWEEN THE AERODYNAMIC PRESSURE FACTORING AND THE AERODYNAMIC DERIVATIVES FACTORING METHODS FOR THE DOUBLET LATTICE PROGRAM

A COMPARISON BETWEEN THE AERODYNAMIC PRESSURE FACTORING AND THE AERODYNAMIC DERIVATIVES FACTORING METHODS FOR THE DOUBLET LATTICE PROGRAM Paper Reference No. 1-1 by Emil Suciu* esuciu@yahoo.com ABSTRACT

### Core Maths C2. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

### The Influence of Aerodynamics on the Design of High-Performance Road Vehicles

The Influence of Aerodynamics on the Design of High-Performance Road Vehicles Guido Buresti Department of Aerospace Engineering University of Pisa (Italy) 1 CONTENTS ELEMENTS OF AERODYNAMICS AERODYNAMICS

### Lab 1a Wind Tunnel Testing Principles & Lift and Drag Coefficients on an Airfoil

Lab 1a Wind Tunnel Testing Principles & Lift and Drag Coefficients on an Airfoil OBJECTIVES - Calibrate the RPM/wind speed relation of the wind tunnel. - Measure the drag and lift coefficients of an airfoil

### DESIGN OF THE MODERN FAMILY OF HELICOPTER AIRFOILS 51

DESIGN OF THE MODERN FAMILY OF HELICOPTER AIRFOILS Wojciech KANIA, Wieńczysław STALEWSKI, Bogumiła ZWIERZCHOWSKA Institute of Aviation Summary The paper presents results of numerical design and experimental

### Integration Involving Trigonometric Functions and Trigonometric Substitution

Integration Involving Trigonometric Functions and Trigonometric Substitution Dr. Philippe B. Laval Kennesaw State University September 7, 005 Abstract This handout describes techniques of integration involving

### Mark Howell Gonzaga High School, Washington, D.C.

Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,

### 13.04 LECTURE NOTES HYDROFOILS AND PROPELLERS

13.04 LECTURE NOTES HYDROFOILS AND PROPELLERS Justin E. Kerwin January 2001 Contents 1 TWO DIMENSIONAL FOIL THEORY 1 1.1 Introduction.................................. 2 1.2 Foil Geometry.................................

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### Lecture L5 - Other Coordinate Systems

S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates

### CFD results for TU-154M in landing configuration for an asymmetrical loss in wing length.

length. PAGE [1] CFD results for TU-154M in landing configuration for an asymmetrical loss in wing length. Summary: In CFD work produced by G. Kowaleczko (GK) and sent to the author of this report in 2013

### CFD ANALYSIS OF RAE 2822 SUPERCRITICAL AIRFOIL AT TRANSONIC MACH SPEEDS

CFD ANALYSIS OF RAE 2822 SUPERCRITICAL AIRFOIL AT TRANSONIC MACH SPEEDS K.Harish Kumar 1, CH.Kiran Kumar 2, T.Naveen Kumar 3 1 M.Tech Thermal Engineering, Sanketika Institute of Technology & Management,

### THE CFD SIMULATION OF THE FLOW AROUND THE AIRCRAFT USING OPENFOAM AND ANSA

THE CFD SIMULATION OF THE FLOW AROUND THE AIRCRAFT USING OPENFOAM AND ANSA Adam Kosík Evektor s.r.o., Czech Republic KEYWORDS CFD simulation, mesh generation, OpenFOAM, ANSA ABSTRACT In this paper we describe

### Aerodynamics of Flight

Chapter 2 Aerodynamics of Flight Introduction This chapter presents aerodynamic fundamentals and principles as they apply to helicopters. The content relates to flight operations and performance of normal

### Parametric Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015

Parametric Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point

### MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 BASIC INTEGRATION

MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 ASIC INTEGRATION This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning

### Theory of turbo machinery / Turbomaskinernas teori. Chapter 3

Theory of turbo machinery / Turbomaskinernas teori Chapter 3 D cascades Let us first understand the facts and then we may seek the causes. (Aristotle) D cascades High hub-tip ratio (of radii) negligible

CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

### 1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### Mathematical Model and Simulation for a Helicopter with Tail Rotor

Mathematical Model and Simulation for a Helicopter with Tail Rotor TULIO SALAZAR School of Mechanical Engineering and Automation Beijing University of Aeronautics and Astronautics XueYuan Road No.37, HaiDian

### the points are called control points approximating curve

Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.

### Recitation Week 4 Chapter 5

Recitation Week 4 Chapter 5 Problem 5.5. A bag of cement whose weight is hangs in equilibrium from three wires shown in igure P5.4. wo of the wires make angles θ = 60.0 and θ = 40.0 with the horizontal.

### AN INVESTIGATION ON THE AERODYNAMIC PERFORMANCE OF A VERTICAL AXIS WIND TURBINE ETESH VAISHNAV

AN INVESTIGATION ON THE AERODYNAMIC PERFORMANCE OF A VERTICAL AXIS WIND TURBINE By ETESH VAISHNAV Bachelor of Science in Mechanical Engineering Bhilai Institute of Technology Durg, India 2007 Submitted

### A LAMINAR FLOW ELEMENT WITH A LINEAR PRESSURE DROP VERSUS VOLUMETRIC FLOW. 1998 ASME Fluids Engineering Division Summer Meeting

TELEDYNE HASTINGS TECHNICAL PAPERS INSTRUMENTS A LAMINAR FLOW ELEMENT WITH A LINEAR PRESSURE DROP VERSUS VOLUMETRIC FLOW Proceedings of FEDSM 98: June -5, 998, Washington, DC FEDSM98 49 ABSTRACT The pressure

### FRICTION, WORK, AND THE INCLINED PLANE

FRICTION, WORK, AND THE INCLINED PLANE Objective: To measure the coefficient of static and inetic friction between a bloc and an inclined plane and to examine the relationship between the plane s angle

### HOW BOARDS AND RUDDERS WORK

HOW BOARDS AND RUDDERS WORK Lift theory By B. Kohler Copyright 2006 by B. Kohler K-designs HOW BOARDS & RUDDERS WORK Our boats become faster and faster and a better understanding of how a sail, rudder,

### A Simple Model for Ski Jump Flight Mechanics Used as a Tool for Teaching Aircraft Gliding Flight

e-περιοδικό Επιστήμης & Τεχνολογίας 33 A Simple Model for Ski Jump Flight Mechanics Used as a Tool for Teaching Aircraft Gliding Flight Vassilios McInnes Spathopoulos Department of Aircraft Technology

### Triple integrals in Cartesian coordinates (Sect. 15.4) Review: Triple integrals in arbitrary domains.

Triple integrals in Cartesian coordinates (Sect. 5.4 Review: Triple integrals in arbitrar domains. s: Changing the order of integration. The average value of a function in a region in space. Triple integrals

### Behavioral Animation Simulation of Flocking Birds

Behavioral Animation Simulation of Flocking Birds Autonomous characters determine their actions Simulating the paths of individuals in: flocks of birds, schools of fish, herds of animals crowd scenes 1.

### CHAPTER 7 CLIMB PERFORMANCE

CHAPTER 7 CLIMB PERFORMANCE 7 CHAPTER 7 CLIMB PERFORMANCE PAGE 7.1 INTRODUCTION 7.1 7.2 PURPOSE OF TEST 7.1 7.3 THEORY 7.2 7.3.1 SAWTOOTH CLIMBS 7.2 7.3.2 STEADY STATE APPROACH TO CLIMB PERFORMANCE 7.4

### PRECISION AERIAL DELIVERY SEMINAR RAM-AIR PARACHUTE DESIGN

13th AIAA Aerodynamic Decelerator Systems Technology Conference Clearwater Beach, May 1995 PRECISION AERIAL DELIVERY SEMINAR RAM-AIR PARACHUTE DESIGN J.Stephen Lingard Martin-Baker Aircraft Co. Ltd. Higher

### CRANFIELD UNIVERSITY F MORTEL CRANFIELD TEAM F1: THE FRONT WING SCHOOL OF ENGINEERING CRANFIELD COLLEGE OF AERONAUTICS. MSc THESIS

CRANFIELD UNIVERSITY F MORTEL CRANFIELD TEAM F1: THE FRONT WING SCHOOL OF ENGINEERING CRANFIELD COLLEGE OF AERONAUTICS MSc THESIS CRANFIELD UNIVERSITY SCHOOL OF ENGINEERING CRANFIELD COLLEGE OF AERONAUTICS

### Practice Problems on Boundary Layers. Answer(s): D = 107 N D = 152 N. C. Wassgren, Purdue University Page 1 of 17 Last Updated: 2010 Nov 22

BL_01 A thin flat plate 55 by 110 cm is immersed in a 6 m/s stream of SAE 10 oil at 20 C. Compute the total skin friction drag if the stream is parallel to (a) the long side and (b) the short side. D =

### JET ENGINE PERFORMANCE. Charles Robert O Neill. School of Mechanical and Aerospace Engineering. Oklahoma State University. Stillwater, OK 74078

JET ENGINE PERFORMANCE Charles Robert O Neill School of Mechanical and Aerospace Engineering Oklahoma State University Stillwater, OK 74078 Honors Project in ENGSC 3233 Fluid Mechanics December 1998 JET

### Lift Distributions on Low Aspect Ratio Wings at Low Reynolds Numbers

Lift Distributions on Low Aspect Ratio Wings at Low Reynolds Numbers by Sagar Sanjeev Sathaye A Thesis Submitted to the Faculty of WORCESTER POLYTECHNIC INSTITUTE in partial fulfillment of the requirements

### THEORETICAL MECHANICS

PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents

### Chapter 24 Physical Pendulum

Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...

### MECHANICAL PRINCIPLES OUTCOME 4 MECHANICAL POWER TRANSMISSION TUTORIAL 2 BELT DRIVES

MECHANICAL PRINCIPLES OUTCOME 4 MECHANICAL POWER TRANSMISSION TUTORIAL BELT DRIVES Simple machines: lifting devices e.g. lever systems, inclined plane, screw jack, pulley blocks, Weston differential pulley

### Orbits of the Lennard-Jones Potential

Orbits of the Lennard-Jones Potential Prashanth S. Venkataram July 28, 2012 1 Introduction The Lennard-Jones potential describes weak interactions between neutral atoms and molecules. Unlike the potentials

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### An Application of Analytic Geometry to Designing Machine Parts--and Dresses

Electronic Proceedings of Undergraduate Mathematics Day, Vol. 3 (008), No. 5 An Application of Analytic Geometry to Designing Machine Parts--and Dresses Karl Hess Sinclair Community College Dayton, OH

### The Verification and Validation of Preliminary CFD Results for the Construction of an A Priori Aerodynamic Model.

Master of Science Thesis The Verification and Validation of Preliminary CFD Results for the Construction of an A Priori Aerodynamic Model. W.G.M. (Wim) Vos February 2006 Verification and Validation of