DOCSLIB.ORG

Continuity Equation for Compressible Flow

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Continuity Equation for Compressible Flow Continuity Equation for Compressible Flow Velocity potential irrotational steady compressible Momentum (Euler) Equation for Compressible Flow Euler's equation isentropic velocity potential equation for steady, irrotational, isentropic compressible flow Velocity Potential Equation for Compressible Flow nonlinear partial differential equation a can be readily expressed in terms of φ as follows. finite-difference numerical techniques Once φ is known, all the other flow variables can be obtained as: −γ −1 T γ −1 2 −1 p γ −1 ρ γ −1 2 γ − = (1+ M ) = (1+ M 2 )γ −1 = (1+ M ) 1 ρ 2 T0 2 p0 2 0 Velocity Potential Equation for Compressible Flow nonlinear partial differential equation finite-difference numerical techniques Velocity Potential Equation For Incompressible Flow linear partial Laplace’s equation is a second-order linear differential Φ Φ Φ partial differential equation. If 1, 2, 3, equation … , Φn represent n separate solutions of Laplace’s equation, thenΦ =Φ1+Φ2+Φ3+… +Φn is also a solution of Laplace’s Linear algebra equation. Therefore, the solution of a analytical or complex flow are usually in the form of a numerical sum of elementary flow solutions. techniques THE LINEARIZED VELOCITY POTENTIAL EQUATION uˆ <<1 vˆ <<1 (11.12) freestream local uˆ vˆ ~ 0.1<<1; ~ 0.1<<1 V∞ V∞ uˆ 2 vˆ2 <<< <<< 2 ~ 0.01 1; 2 ~ 0.01 1 V∞ V∞ 2 0.32 < 1− M ∞ <1 2 0 < M ∞ < 0.64 for 0 ≤ M ∞ ≤ 0.8 2 0.44 < 1− M ∞ < 24 2 1.44 < M ∞ < 25 for 1.2 ≤ M ∞ ≤ 5 not valid for thick bodies and for not for transonic flow (0.8 < M < 1.2) large angles of attack or hypersonic flow (M> 5). Pressure Coefficient C p Linearized Form of Pressure Coefficient C p For an adiabatic flow of a calorically perfect gas Boundary Conditions θ At infinity At the body surface: The flow-tangency condition holds. Let θ be the angle between the tangent to the surface and the freestream. PRANDTL-GLAUERT COMPRESSIBILITY CORRECTION Compressibility corrections for 0.3<M<0.7 compressibility correction Prandtl-Glauert rule As early as 1922, Prandtl in his lectures at Gottingen, and first formally published by the British aerodynamicist, Hermann Glauert, in 1928. IMPROVED COMPRESSIBILITY CORRECTIONS CRITICAL MACH NUMBER Linearized theory does not apply to the transonic flow regime, 0.8 < M ∞ <1.2. Transonic flow is highly nonlinear. • Consider an airfoil in a low-speed flow, say, with M∞ = 0.3, as sketched in Fig. a. In the expansion over the top surface of the airfoil, the local flow Mach number M increases. Let point A represent the location on the airfoil surface where the pressure is a minimum, hence where M is a maximum. Let us say this maximum is MA = 0.435. • Now assume that we gradually increase the freestream Mach number. As M∞ increases, MA also increases. For example, if M ∞ is increased to M ∞ = 0.5, the maximum local value of M will be 0.772, as shown in Fig. b. CRITICAL MACH NUMBER Linearized theory does not apply to the transonic flow regime, 0.8 < M ∞ <1.2. Transonic flow is highly nonlinear. • Let us continue to increase M∞ until we achieve just the right value such that the local Mach number at the minimum pressure point equals 1, i.e., such that MA = 1.0, as shown in Fig. c. • When this happens, the freestream Mach number M ∞ is called the critical Mach number, denoted by Mcr. By definition, the critical Mach number is that freestream Mach number at which sonic flow is first achieved on the airfoil surface. In Fig. c, Mcr = 0.61. • One of the most important problems in high- speed aerodynamics is the determination of the critical Mach number of a given airfoil, because at values of M ∞ slightly above Mcr the airfoil experiences a dramatic increase in drag coefficient. Estimation of Mcr Cp,A = f (M ∞ , M A ) Estimation of Mcr Thin airfoil Thick airfoil THE SOUND BARRIER C C = d ,0 d 2 1− M ∞ MACH NUMBER DRAG-DIVERGENCE MACH NUMBER According to Prandtl- Glauert rule, Cd becomes infinite at M∞ = 1, C C = d ,0 d 2 1− M ∞ • Point c is the critical Mach number. As we very carefully increase M∞ slightly above Mcr, (point d) a finite region of supersonic flow appears on the airfoil. • At point e , the value of M∞ at which this sudden increase in drag starts is defined as the drag-divergence Mach number. Beyond the drag-divergence Mach number, the drag coefficient can become very large, typically increasing by a factor of 10 or more. point a~b point c point d point e drag-divergence Mach number The Bell XS-l-the first rocket-propelled manned aircraft to fly faster than sound, October 14, 1947. Since 1945, research in transonic aerodynamics has focused on reducing the large drag rise. Instead of living with a factor of 10 increase in drag at Mach 1, can we reduce it to a factor of 2 or 3? This is the subject of the remaining sections of this chapter. Reducing Drag at Transonic and Supersonic Flow 1. Thin airfoil section Reducing Drag at Transonic and Supersonic Flow 2. Swept wings Adolf Busemann (1901~1986) The typical swept wing aircraft , F-86 fighter Sweep Wing Obviously, it is desirable to reduce the Mach number of the flow over the airfoil section. A long time ago, it was discovered that the flow could be "fooled" by simply sweeping the wing. Swept Wing Aircrafts F-86 Sabre fighter F-100 Super Sabre fighter Richard Whitcomb (1921~2009) THE WHITCOMB AREA RULE 1950 F-102 fighter Area-Ruled Aircraft YF-102A Area-Ruled Aircraft F-106 Area-Ruled Aircrafts F-104 fighter F-5 fighter B-1B bomber F-16 fighter Reducing Drag at Transonic and Supersonic Flow 3. Supercritical airfoil section Standard NACA 64 series airfoils with different thickness for high speed research in 1949 • Shock wave move downstream as Mach number increases. • A large region of separation flow downstream of shock wave for M=0.94, 0.87 and 0.79. • The separation flow is the primary reason for the increase in the drag near the M=1.0. • Discontinuity pressure increase across a shock wave creates a strong adverse pressure gradient on airfoil surface and this adverse pressure gradient causes flow separation. Computational Fluid Dynamics (CFD) Blended Wing Body (BWB) Aircraft Design Blended Wing Body (BWB) Aircraft Design Area-Ruled Aircraft Computational Fluid Dynamics (CFD) National Transonic Facility (NTF) .
Load more

Recommended publications
  • 7. Transonic Aerodynamics of Airfoils and Wings
    W.H. Mason 7. Transonic Aerodynamics of Airfoils and Wings 7.1 Introduction Transonic flow occurs when there is mixed sub- and supersonic local flow in the same flowfield (typically with freestream Mach numbers from M = 0.6 or 0.7 to 1.2). Usually the supersonic region of the flow is terminated by a shock wave, allowing the flow to slow down to subsonic speeds. This complicates both computations and wind tunnel testing. It also means that there is very little analytic theory available for guidance in designing for transonic flow conditions. Importantly, not only is the outer inviscid portion of the flow governed by nonlinear flow equations, but the nonlinear flow features typically require that viscous effects be included immediately in the flowfield analysis for accurate design and analysis work. Note also that hypersonic vehicles with bow shocks necessarily have a region of subsonic flow behind the shock, so there is an element of transonic flow on those vehicles too. In the days of propeller airplanes the transonic flow limitations on the propeller mostly kept airplanes from flying fast enough to encounter transonic flow over the rest of the airplane. Here the propeller was moving much faster than the airplane, and adverse transonic aerodynamic problems appeared on the prop first, limiting the speed and thus transonic flow problems over the rest of the aircraft. However, WWII fighters could reach transonic speeds in a dive, and major problems often arose. One notable example was the Lockheed P-38 Lightning. Transonic effects prevented the airplane from readily recovering from dives, and during one flight test, Lockheed test pilot Ralph Virden had a fatal accident.
    [Show full text]
  • Section 7, Lecture 3: Effects of Wing Sweep
    Section 7, Lecture 3: Effects of Wing Sweep • All modern high-speed aircraft have swept wings: WHY? 1 MAE 5420 - Compressible Fluid Flow • Not in Anderson Supersonic Airfoils (revisited) • Normal Shock wave formed off the front of a blunt leading g=1.1 causes significant drag Detached shock waveg=1.3 Localized normal shock wave Credit: Selkirk College Professional Aviation Program 2 MAE 5420 - Compressible Fluid Flow Supersonic Airfoils (revisited, 2) • To eliminate this leading edge drag caused by detached bow wave Supersonic wings are typically quite sharp atg=1.1 the leading edge • Design feature allows oblique wave to attachg=1.3 to the leading edge eliminating the area of high pressure ahead of the wing. • Double wedge or “diamond” Airfoil section Credit: Selkirk College Professional Aviation Program 3 MAE 5420 - Compressible Fluid Flow Wing Design 101 • Subsonic Wing in Subsonic Flow • Subsonic Wing in Supersonic Flow • Supersonic Wing in Subsonic Flow A conundrum! • Supersonic Wing in Supersonic Flow • Wings that work well sub-sonically generally don’t work well supersonically, and vice-versa à Leading edge Wing-sweep can overcome problem with poor performance of sharp leading edge wing in subsonic flight. 4 MAE 5420 - Compressible Fluid Flow Wing Design 101 (2) • Compromise High-Sweep Delta design generates lift at low speeds • Highly-Swept Delta-Wing design … by increasing the angle-of-attack, works “pretty well” in both flow regimes but also has sufficient sweepback and slenderness to perform very Supersonic Subsonic efficiently at high speeds. • On a traditional aircraft wing a trailing vortex is formed only at the wing tips.
    [Show full text]
  • Naca Research Memorandum
    RM No. L8A28c NACA RESEARCH MEMORANDUM EFFECTS OF SWEEP ON CONTROLS By John G. Lowry, John A. Axelson, and Harold I. Johnson Langley Memorial Aeronautical La 00 ratory Langley Field, Va. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WASHINGTON June 3, 1948 Declassified February 10, 1956 NACA PM No. L8A28c NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS RESEARCH MEMORANDUM EFFECTS OF SWEEP ON CONTROLS By John G. Lowry., John A. Axelson, and Harold I. Johnson SUMMARY An analysis of principal results of recent control-surface research pertinent to transonic flight has -been made. Available experimental data on control surfaces of both unswept and sweptback configurations at transonic speeds are used to indiäate the control-surface characteristics in the transonic speed range. A design procedure for controls on swept- back wings based on low-speed experimental data is also discussed. The results indicated, that no serious problems resulting from compressibility effects would be encountered as long as the speeds are kept below the critical speed of the wing and the trailing-edge angle is kept small. Above critical speed, however, the behavior of the' controls depended to a large extent on the wing sweep angle. The design procedures presented for controls on swept wings, although of a preliminary nature, appear to offer a method of estimating the effec- tiveness of flap-type controls on swept wings of normal aspect ratio and taper ratio. INTRODUCTION The design of controls for unswept wings that fly at low speed has been discussed in several papers (references 1 to 7). The design procedures set forth in these papers are adequate to allow for the prediction of control characteristics within small limits.
    [Show full text]
  • Conceptual Design for a Supersonic Advanced Military Trainer
    Conceptual Design for a Supersonic Advanced Military Trainer A project presented to The Faculty of the Department of Aerospace Engineering San José State University In partial fulfillment of the requirements for the degree Master of Science in Aerospace Engineering by Royd A. Johansen May 2019 approved by Dr. Nikos Mourtos Faculty Advisor © 2019 Royd A. Johansen ALL RIGHTS RESERVED ABSTRACT CONCEPTUAL DESIGN FOR A SUPERSONIC ADVANCED MILITARY TRAINER by Royd A. Johansen The conceptual aircraft design project is based off the T-X program requirements for an advanced military trainer (AMT). The design process focused on a top-level design aspect, that followed the classic aircraft design process developed by J. Roskam’s Airplane Design. The design process covered: configuration selection, weight sizing, performance sizing, fuselage design, wing design, empennage design, landing-gear design, Class I weight and balance, static longitudinal and directional stability, subsonic drag polars, supersonic area rule applied to supersonic drag polars, V-n diagrams, Class II weight and balance, moments and products of inertia, and cost estimation. Throughout the process other materials and references are consulted to verify or develop a better understanding of the concepts in the Airplane Design series. ACKNOWLEDGEMENTS I would like to express my deep gratitude to Dr. Nikos Mourtos for his support throughout this project and his invaluable academic advisement throughout my undergraduate and graduate studies. I would also like to thank Professor Gonzalo Mendoza for introducing Aircraft Design to me as an undergraduate and for providing additional insights throughout my graduate studies. My thanks are also extended to Professor Jeanine Hunter for helping me develop my theoretical foundation in Dynamics, Flight Mechanics, and Aircraft Stability and Control.
    [Show full text]
  • Ac 61-107A - Operations of Aircraft at Altitudes Above 25,000 Feet Msl And/Or Mach Numbers (Mmo) Greater Than .75
    AC 61-107A - OPERATIONS OF AIRCRAFT AT ALTITUDES ABOVE 25,000 FEET MSL AND/OR MACH NUMBERS (MMO) GREATER THAN .75 {New-2003-09} Department of Transportation Federal Aviation Administration 1/2/03 Initiated by: AFS-820 1. PURPOSE. This advisory circular (AC) is issued to alert pilots who are transitioning from aircraft with less performance capability to complex, high-performance aircraft that are capable of operating at high altitudes and high airspeeds, of the need to be knowledgeable about the special physiological and aerodynamic considerations involved in these kinds of operations. 2. CANCELLATION. AC 61-107, Operations of Aircraft at Altitudes Above 25,000 Feet MSL and/or Mach Numbers (Mmo) Greater Than .75, dated January 23, 1991, is cancelled. 3. DEFINITIONS. a. Aspect Ratio is the relationship between the wing chord and the wingspan. A short wingspan and wide wing chord equal a low aspect ratio. b. Aileron buzz is a very rapid oscillation of an aileron, at certain critical air speeds of some aircraft, which does not usually reach large magnitudes nor become dangerous. It is often caused by shock- induced separation of the boundary layer. c. Drag Divergence is a phenomenon that occurs when an airfoil's drag increases sharply and requires substantial increases in power (thrust) to produce further increases in speed. This is not to be confused with MACH crit. The drag increase is due to the unstable formation of shock waves that transform a large amount of energy into heat and into pressure pulses that act to consume a major portion of the available propulsive energy.
    [Show full text]
  • Sound Propagation of Sonic Booms Through Real Atmospheres Emitted from a New Supersonic Business Aircraft
    SOUND PROPAGATION OF SONIC BOOMS THROUGH REAL ATMOSPHERES EMITTED FROM A NEW SUPERSONIC BUSINESS AIRCRAFT Martina Kästner and Dietrich Heimann Institute of Atmospheric Physics, German Aerospace Center, Oberpfaffenhofen, Germany E-mail: [email protected] Abstract: Supersonic air transport generates sonic booms that can affect people, animals and structures. There is a renewed interest in high-speed transport for civilian application. The widely used standard atmosphere for sonic boom prediction is neither capable of describing topographical variation, nor temporal variations, nor the impact of wind on the propagation of a sonic boom. However, within the frame of the 6th FP EU-project HISAC (HIgh Speed AirCraft), our ray tracing studies are performed through real atmospheres from emissions of a supersonic aircraft that can impact people in case of overland flights. Obviously, the mountains are affected more often by the noise carpet than the lowlands. The used ERA-40 data over the domain of Europe are appropriate to define flight tracks for supersonic aircrafts that avoid the noise carpet of sonic booms at the ground depending on the actual state of the atmosphere. A key role plays the atmospheric refraction due to temperature and wind gradients and the atmospheric absorption due to the humidity. Primary use of improved prediction possibility is expected to operational route planning, especially as supersonic activity increases. Recently, flight demonstration of shaped sonic boom that generates low overpressure N-waves show the potential for quiet overland supersonic flights. Keywords: supersonic aircraft, sound propagation, mountainous area INTRODUCTION The human response to aircraft noise is complicated because of the multitude of factors involved.
    [Show full text]
  • The Aerodynamic Performance of an Aerofoil in Tri-Sonic Ground Effect
    Proc. of the 32nd International Symposium on Shock Waves (ISSW32) doi:10.3850/978-981-11-2730-4_0376-cd Copyright c 2019 ISSW32. All rights reserved. The Aerodynamic Performance of an Aerofoil in Tri-sonic Ground Effect S. R. Morrow, H. Roohani, B. W. Skews, I. Gledhill Flow Research Unit, University of the Witwatersrand, 1 Jan Smuts Avenue, Braamfontein, Johannesburg, 2000, South Africa B. J. Evans College of Engineering, Swansea University, Bay Campus, Fabian Way, Skewen, Swansea, SA1 8EN, United Kingdom Sean Morrow: [email protected] Abstract Inspired by the world land speed record vehicles, the Thrust and Bloodhound Supersonic Cars (SSC); this investigation considers the influence of ground effect and acceleration on the aerodynamic performance of an aerofoil accelerating in ground effect from subsonic to supersonic speeds. Using Fluent as the computational fluid dynamic solver, a steady state analysis of the RAE 2822 aerofoil at two different ground clearances was used to evaluate the influence of ground effect. The effects of acceleration were considered by comparing the steady state results against transient models with extreme accelerations of 175g. The aerodynamic performance and shock wave development were analysed from Mach 0.20 to Mach 2.00. The analysis revealed that both ground effect and acceleration impact on the shock wave development and propagation through the flow. An increase in ground effect proved to dramatically improve the supersonic lift performance of the aerofoil and alter the shock system that forms below the aerofoil. The extreme acceleration revealed that transient flow lags behind the steady state flow and the lift and drag performance of the aerofoil are impacted differently by the flow history and inertia of the fluid.
    [Show full text]
  • Ntsb/Aar-82-04
    .. PB82-910404 AIRCRAFT ACCIDENT REPORT .SKY TRAIN AIR, INC. GATES LEARJET 24, N44CJ FELT, OKLAHOMA OCTOBER 1, 1981 2' 1 NTSB-AAR-82-4 0, I UNITED STATES GOVERNMENT TECHNICAL REPORT DOCUMENTATION PAGE . Report No. 2.Government Accession No. 3.Recipient's Catalog No. NTSB-AAR-82-4 PB82-910404 . Tit le and Subtitle Aviation Accident Report--Sky 5.Report Date Train Air, Inc., Gates Learjet 24, N44CJ, Felt, May 18,1982 Oklahoma, October 1, 1981. 6.Performing- Organization- Code , . Author(s) 8.Performing Organization ~ Report No. 88 National Transportation Safety Board 3435A Bureau of Accident Investigation 11.Contract or Grant No. Washington, D.C. 20594 13.Type of Report and Period Covered 2.Sponsoring Agency Name and Address Aircraft Accident Report October, 1, 1981 NATIONAL TRANSPORTATION SAFETY BOARD - Washington, 0. C. 20594 14.Sponsoring Agency Code 5.Supplcmentrry Notes l6.AbstractAbout 1502 c.d.t., on October 1, 1981, a Sky Train Air, Inc., Learjet 24, N44CJ, crashed 2.5 miles southwest of Felt, Oklahoma. The flightcrew and one passenger, the only occupants on board, were killed. At 1449:39, while in cruise flight at Flight Level 450, en route to McAllen, Texas, from Casper, Wyoming, the flightcrew made initial contact with the Albuquerque Air Route Traffic Control Center. About 1 minute later, the flightcrew failed to respond to a frequency change instruction and the aircraft's transponder beacon code was lost. The controller made several attempts to contact the aircraft but to no avail. Witnesses at Felt heard an aircraft overhead at a very high speed; one witness who saw the aircraft momentarily, stated it was in I a descent angle of about 45'before it struck the ground.
    [Show full text]
  • Analysis of Supercritical Airfoils for Increasing Critical Mach Number for Subsonic Aircrafts
    IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org ISSN (e): 2250-3021, ISSN (p): 2278-8719 PP 01-05 Analysis of Supercritical Airfoils for Increasing Critical Mach number for Subsonic Aircrafts K.Soundarya1,K.Sriram2,M.Divya3,G.Muralidharan4 1(Department of Aeronautical Engineering,Surya Group of Institutions/Anna university,India) 2(Department of Aeronautical Engineering,Surya Group of Institutions/Anna university,India) 3(Department of Aeronautical Engineering,Surya Group of Institutions/Anna university,India) 4(Department of Aeronautical Engineering,Surya Group of Institutions/Anna university,India) Abstract: The aerodynamic characteristics of an airfoil plays a vital role in terms of design aspects and experimental valuation This led to increasing of critical Mach Number with the help of comparing various performance characteristics of supercritical airfoils at cruising conditions .The main objective of this project is to carry out design and analysis of various supercritical airfoils with same cruising local velocity along with the study of aerodynamic characteristics such as drag co-efficient and critical Mach number.The objective is to improve the Mach number of a subsonic aircraft at transonic Mach speeds. The design profile chosen is based upon existing airfoils used in Airbus A380-800 . Keywords – Supercritical airfoils, Critical Mach number, Co-efficient of drag,Cruising Speed Velocity I. Introduction A flow over a supercritical airfoils is very from the flow over an cambered airfoils Airbus A380-800 fly at maximum speed of 0.83 Mach number at the local velocity reaches sonic velocity somewhere over the wing and compressibility effects start of two grow up the free stream Mach number at which local sonic velocities develop is called critical Mach number.
    [Show full text]
  • Transonic Aerodynamics
    Transonic Aerodynamics • Critical Pressure Coefficient and Critical Mach Number • Drag Divergence • Mitigating the Drag Problem – The supercritical airfoil – Sweep – The Area Rule Limit of the Compressibility Correction NACA 0012 AT 8 DEGREES M∞ -10 -9 C M=0 -16 -8 p1 0.1 C p -14 Cp -7 2 0.2 min 1 M 0.3 -12 -6 040.4 -10 -5 0.5 0.6 -8 -4 0.7 -Cp -6 -3 M∞ 080.8 0.9 -4 -2 -2 -1 0 0 00.51 1 M∞ 0.0 0.2 0.4 0.6 0.8x/c 1.0 Critical Pressure Coefficient PRESSURE COEFFIENT WHERE THE LOCAL MACH NUMBER IS 1 Critical Pressure Coefficient PRESSURE COEFFIENT WHERE THE LOCAL MACH NUMBER IS 1 1 1 1 ( 1)M 2 1 Cp 2 1 CR 1 2 1 2 M 2 ( 1) -16 -14 CpCR -12 -10 -8 -6 -4 -2 0 00.20.40.60.81 M∞ Critical Mach Number Cp1min Cpmin 1 M 2 1 1 1 ( 1)M 2 1 Cp 2 1 CR 1 2 1 2 M 2 ( 1) M∞ -16 -14 -12 -10 -8 MCR -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 M∞ Critical Mach Number MCR Cp1min Cpmin 1 M 2 1 1 1 ( 1)M 2 1 Cp 2 1 CR 1 2 1 2 M 2 ( 1) -16 -14 -12 0012 at 8 deg. -10 -8 MCR MCR 0012 at 4 deg. -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 M∞ M=0.5 Drag Divergence INVISCID CALCULATION M065M=0.65 M=0.
    [Show full text]
  • Swept Wing Configuration
    * Presented by: S.H.Hoseini When aircraft enters the transonic speeds just below the speed of sound, an effect known as wave drag starts. * Near the speed of sound oblique shock wave is generated. *Shock waves require energy to form. *The energy reduces airplanes power. * *Anglo-Irish engineer John William Dunne was experimenting along similar lines, obsessed with achieving innate stability in flight. He was able to successfully employ severely swept wings in his tailless aircraft as a means of creating positive longitudinal static stability.[2] For a low- speed aircraft, swept wings are useful to avoid problems with the *1-center of gravity, to move the wing spar into a more convenient location, *2- to improve the sideways view from the pilot's position * * *One of the simplest and best explanations of how the swept wing works was offered by Robert T. Jones: "Suppose a cylindrical wing (constant chord, incidence, etc.) is placed in an airstream at an angle of yaw - ie., it is swept back. Now, even if the local speed of the air on the upper surface of the wing becomes supersonic, a shock wave cannot form there because it would have to be a sweptback shock - swept at the same angle as the wing - ie., it would be an oblique shock. Such an oblique shock cannot form until the velocity component normal to it becomes supersonic." * Mcrit : ‘Mcrit’ in aerodynamics, is in fact the ‘critical Mach Number’ of an aircraft. It is the lowest Mach number at which the airflow over some point on the aircraft reaches the speed of sound.
    [Show full text]
  • Induced Drag and High-Speed Aerodynamics
    Induced Drag and High-Speed Aerodynamics Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 • Drag-due-to-lift and effects of wing planform • Effect of angle of attack on lift and drag coefficients • Mach number (i.e., air compressibility) effects on aerodynamics • Newtonian approximation for lift and drag Reading: Flight Dynamics Aerodynamic Coefficients, 85-96 Airplane Stability and Control Chapter 1 Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html http://www.princeton.edu/~stengel/FlightDynamics.html 1 Induced Drag 2 1 Aerodynamic Drag 1 1 Drag = C ρV 2S ≈ C + εC 2 ρV 2S D 2 ( D0 L ) 2 2 1 ≈ ⎡C + ε C + C α ⎤ ρV 2S ⎣⎢ D0 ( Lo Lα ) ⎦⎥ 2 3 2 Induced Drag of a Wing, εCL § Lift produces downwash (angle proportional to lift) § Downwash rotates local velocity vector clockwise in figure § Lift is perpendicular to velocity vector § Axial component of rotated lift induces drag § But what is the proportionality factor, ε? 4 2 Induced Angle of Attack C C sin , where Di = L α i α i = CL πeAR, Induced angle of attack C = C sin C πeAR C 2 πeAR Di L ( L ) ! L 5 Three Expressions for Induced Drag of a Wing C 2 C 2 1+δ C L L ( ) C 2 Di = ! ! ε L πeAR π AR e = Oswald efficiency factor = 1 for elliptical distribution δ = departure from ideal elliptical lift distribution 1 (1+δ ) ε = = πeAR π AR 6 3 Spanwise Lift Distribution of Elliptical and Trapezoidal Wings Straight Wings (@ 1/4 chord), McCormick Spitfire TR = taper ratio, λ P-51D Mustang For some taper ratio between 0.35 and 1, trapezoidal lift distribution is nearly elliptical 7 Induced Drag Factor, δ 2 CL (1+ δ ) C = Di π AR • Graph for δ (McCormick, p.
    [Show full text]