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Aerodynamics part ii

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Solo Hermelin

The document covers various aspects of aerodynamics, including fundamental laws of fluid dynamics, shock waves, and airflow characteristics in different conditions such as subsonic and supersonic flows. It discusses mathematical models and theories, including the conservation laws, boundary layer theories, and airfoil design techniques. Additionally, the text explores flow phenomena like oblique and normal shock waves, as well as their implications on pressure and temperature changes in supersonic flows.

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1 AERODYNAMICS Part II SOLO HERMELIN http://www.solohermelin.com
2 Table of Content AERODYNAMICS Earth Atmosphere Mathematical Notations SOLO Basic Laws in Fluid Dynamics Conservation of Mass (C.M.) Conservation of Linear Momentum (C.L.M.) Conservation of Moment-of-Momentum (C.M.M.) The First Law of Thermodynamics The Second Law of Thermodynamics and Entropy Production Constitutive Relations for Gases Newtonian Fluid Definitions – Navier–Stokes Equations State Equation Thermally Perfect Gas and Calorically Perfect Gas Boundary Conditions Flow Description Streamlines, Streaklines, and Pathlines AEROD
3 Table of Content (continue – 1) AERODYNAMICS SOLO Circulation Biot-Savart Formula Helmholtz Vortex Theorems 2-D Inviscid Incompressible Flow Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow Aerodynamic Forces and Moments Blasius Theorem Kutta Condition Kutta-Joukovsky Theorem Joukovsky Airfoils Theodorsen Airfoil Design Method Profile Theory by the Method of Singularities Airfoil Design AEROD
4 Table of Content (continue – 2) AERODYNAMICS SOLO Lifting-Line Theory Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞) 3D Lifting-Surface Theory through Vortex Lattice Method (VLM) Incompressible Potential Flow Using Panel Methods Dimensionless Equations Boundary Layer and Reynolds Number Wing Configurations Wing Parameters References AEROD
5 Table of Content (continue – 3) AERODYNAMICS SOLO Shock & Expansion Waves Shock Wave Definition Normal Shock Wave Oblique Shock Wave Prandtl-Meyer Expansion Waves Movement of Shocks with Increasing Mach Number Drag Variation with Mach Number Swept Wings Drag Variation Variation of Aerodynamic Efficiency with Mach Number Analytic Theory and CFD Transonic Area Rule
6 Table of Content (continue – 4) AERODYNAMICS SOLO Linearized Flow Equations Cylindrical Coordinates Small Perturbation Flow Applications: Nonsteady One-Dimensional Flow Applications: Two Dimensional Flow Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack) Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (d) and Lift (l) per Unit Span Computations for Subsonic Flow (M∞ < 1) Prandtl-Glauert Compressibility Correction Computations for Supersonic Flow (M∞ >1) Ackeret Compressibility Correction
7 SOLO Table of Contents (continue – 5) Wings of Finite Span at Supersonic Incident Flow Theoretic Solutions for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) 1. Conical Flow Method 2. Singularity-Distribution Method Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) Arrowhead Wings with Double-Wedge Profile at Zero Incidence Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwise plane [after Lawrence] AERODYNAMICS A E R O D Y N A M I C S P A R T I I I
8 Table of Content (continue – 6) AERODYNAMICS SOLO Aircraft Flight Control References CNα – Slope of the Normal Force Coefficient Computations of Swept Wings Comparison of Experiment and Theory for Lift-Curve Slope of Un-swept Wings Drag Coefficient A E R O D Y N A M I C S P A R T I I I
9 Continue from AERODYNAMICS – Part I AERODYNAMICS
10 SOLO - when the source moves at subsonic velocity V a, it will stay inside the family of spherical sound waves. a V M M =      = − 1 sin 1 µ Disturbances in a fluid propagate by molecular collision, at the sped of sound a, along a spherical surface centered at the disturbances source position. The source of disturbances moves with the velocity V. - when the source moves at supersonic velocity V a, it will stay outside the family of spherical sound waves. These wave fronts form a disturbance envelope given by two lines tangent to the family of spherical sound waves. Those lines are called Mach waves, and form an angle μ with the disturbance source velocity: SHOCK EXPANSION WAVES
11 SOLO SHOCK EXPANSION WAVES M 1 M = 1 M 1 Mach Waves
12 SOLO When a supersonic flow encounters a boundary the following will happen: When a flow encounters a boundary it must satisfy the boundary conditions, meaning that the flow must be parallel to the surface at the boundary. - when the supersonic flow, in order to remain parallel to the boundary surface, must “turn into itself” (see the Concave Corner example) a Oblique Shock will occur. After the shock wave the pressure, temperature and density will increase. The Mach number of the flow will decrease after the shock wave. SHOCK EXPANSION WAVES - when the supersonic flow, in order to remain parallel to the boundary surface, must “turn away from itself” (see the Convex Corner example) an Expansion wave will occur. In this case the pressure, temperature and density will decrease. The Mach number of the flow will increase after the expansion wave. Return to Table of Content
13 SHOCK WAVESSOLO A shock wave occurs when a supersonic flow decelerates in response to a sharp increase in pressure (supersonic compression) or when a supersonic flow encounters a sudden, compressive change in direction (the presence of an obstacle). For the flow conditions where the gas is a continuum, the shock wave is a narrow region (on the order of several molecular mean free paths thick, ~ 6 x 10-6 cm) across which is an almost instantaneous change in the values of the flow parameters. Shock Wave Definition (from John J. Bertin/ Michael L. Smith, “Aerodynamics for Engineers”, Prentice Hall, 1979, pp.254-255) When the shock wave is normal to the streamlines it is called a Normal Shock Wave, otherwise it is an Oblique Shock Wave. The difference between a shock wave and a Mach wave is that: - A Mach wave represents a surface across which some derivative of the flow variables (such as the thermodynamic properties of the fluid and the flow velocity) may be discontinuous while the variables themselves are continuous. For this reason we call it a weak shock. - A shock wave represents a surface across which the thermodynamic properties and the flow velocity are essentially discontinuous. For this reason it is called a strong shock.
14 Normal Shock Wave Over a Blunt Body Normal Shock Wave SHOCK WAVESSOLO Oblique Shock Wave Oblique Shock Wave Return to Table of Content
15 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Conservation of Mass (C.M.) ρ ρ1 1 2 2u u= η ρ ρ = =2 1 1 2 u u Conservation of Linear Momentum (C.L.M.) 2 2 221 2 11 pupu +=+ ρρ ( ) p p u p 2 1 1 2 1 1 1 1= + − ρ η H H h u h u1 2 1 1 2 2 2 21 2 1 2 = → + = + h h u h 2 1 1 2 1 2 1 2 1 1 = + −       η Conservation of Energy (C.E.) Field Equations Constitutive Relations p R T= ρIdeal Gas ( ) ( ) ( ) e e T C Tv= = 1 2 (1) Thermally Perfect Gas (2) Calorically Perfect Gas ργ γ ρρρ γ ρ pp C C C C p R C TC p eh v p vp C C v p v p CCR p TRp p 11 − = − ===+= ≡−== u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2
16 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, First Way h h p p p p p p u h u p 2 1 2 2 1 1 2 1 1 2 2 1 1 2 1 2 1 2 1 1 2 1 1 1 1 2 1 1 1 2 1 1 1 = − − = = = + −       = + − −       γ γ ρ γ γ ρ ρ ρ η η γ γ ρ η or ( ) p p u p u p C L M 2 1 1 2 1 1 1 2 1 1 2 1 1 1 1 1 2 1 1 1 η ρ η η γ γ ρ η = + −           = + − −       ( . . .) after further development we obtain 1 2 1 1 1 1 1 1 2 01 2 1 1 2 1 2 1 1 1 2 1 1 − −       − +           + + −           = γ γ ρ η ρ η γ γ ρ u p u p u p Solving for 1/η , we obtain 1 1 1 2 1 1 1 2 1 1 2 2 1 1 2 1 1 1 2 1 1 2 1 2 1 1 1 2 1 1 η ρ ρ ρ ρ γ γ ρ γ γ ρ γ γ = = = +           − +           − + + −           + u u u p u p u p u p u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2
17 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, We obtain an other relation in the following way: ( ) p p u p p p u p p p p p p p p p p p p p p p 2 1 1 2 1 1 2 2 1 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2 η γ γ ρ η ρ η η γ γ η η γ γ γ γ γ γ η γ γ γ γ γ γ γ γ − = − −       − = −         ⇒ − − = − +       ⇓ − − − −      = + − −       ⇓ = + − − − + + η ρ ρ γ γ γ γ = = = + − − + + − =2 1 1 2 2 1 2 1 2 1 1 2 1 1 1 1 1 u u p p p p p p T T or Rankine-Hugoniot Equation Rankine-Hugoniot Equation (1) William John Macquorn Rankine (1820-1872) u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2 Pierre-Henri Hugoniot (1851 – 1887)
18 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, η ρ ρ γ γ γ γ = = = + − − + + − =2 1 1 2 2 1 2 1 2 1 1 2 1 1 1 1 1 u u p p p p p p T T Rankine-Hugoniot Equation Rankine-Hugoniot Equation (2) p p 2 1 2 1 2 1 1 1 1 1 1 = + − − + − − γ γ ρ ρ γ γ ρ ρ T T p p p p p p p p p p p p 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 = = + + − + − − = + + − + − − = + − − + − − = + − − + − − ρ ρ γ γ γ γ γ γ γ γ γ γ ρ ρ γ γ ρ ρ ρ ρ γ γ ρ ρ γ γ ρ ρ p2 p 1 ρ 2 ρ 1 NormalShockWave Rankine-Hugoniot Isentropic γp2 p 1 ρ 2 ρ 1 ( )= u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2
19 Rankine-Hugoniot Equation (3) u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2 SOLO
20 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Strong Shock Wave Definition: p p u u T T p p R H R H 2 1 2 1 1 2 2 1 2 1 1 1 1 1 → ∞ ⇒ = → + − → − + − −ρ ρ γ γ γ γ Weak Shock Wave Definition: ∆ p p p p p1 2 1 1 1= − ρ ρ ρ2 1 2 1 2 1 = + = + = + ∆ ∆ ∆ p p p h h h For weak shocks u p 1 2 = ∆ ∆ρ ∆ ∆ h u ρ ρ = 1 2 1 u u u u u u2 1 2 1 1 1 1 1 1 1 1 1 1 1 = = + = + ≅ − ρ ρ ρ ρ ρ ρ ρ ρ ρ∆ ∆ ∆ (C.M.) ( ) ( )ρ ρ ρ ρ ρ 1 1 2 1 1 1 2 2 1 1 1 1 1 1u p u u p u u u p p+ = + = −       + + ∆ ∆(C.L.M.)  ordernd uuuhhuuhhuhuh 2 4 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 11 2 1 1 11 2 22 2 11       ∆ + ∆ −+∆+=      ∆ −+∆+=+=+ ρ ρ ρ ρ ρ ρ(C.E.) u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2
21 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Second Way h h u h u0 1 1 2 2 2 21 2 1 2 ≡ + = +Define        − − − =→+ − = − − − =→+ − = 2 10 1 12 1 1 1 0 2 20 2 22 2 2 2 0 11 2 1 1 11 2 1 1 uh p u p h uh p u p h γ γ γ γ ρργ γ γ γ γ γ ρργ γ u u h1 2 0 2 1 1 = − + γ γ Prandtl’s Relation ( )u h u u u p p u p2 0 1 2 1 1 2 2 1 1 2 1 1 2 1 1 1 1 1= − + → = = → = + − γ γ ρ ρ ρη ρ ηFrom this relation, we obtain: Prandtl’s Relation Ludwig Prandtl (1875-1953) u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2 (C.M.) (C.L.M.) ργ γ p h 1− = and use 12 22 2 11 1 2211 2 2 221 2 11 11 uu u p u p uu pupu −=−→    = +=+ ρρρρ ρρ 1221 21 0 2 1 2 1111 uuuu uu h −= − + − −      − − γ γ γ γ γ γ ( )       − −−= −− γ γ γ γ 2 1 1 1 12 21 12 0 uu uu uu h
22 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, (C.M.) Hugoniot Equation ρ ρ ρ ρ 1 1 2 2 2 1 1 2 u u u u= → = ( )ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ 1 1 2 1 2 2 2 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 2 1 2 1 2 2 1 1 2 2 1 2 2 2 2 2 1 2 1 2 2 1 1 2 2 1 1 2 u p u p u p p p u u u p p u p p u u u u + = + =       + → − = −       = − → → = − −       → = − −               = = (C.L.M.) ( )( ) h u h u e p p p e p p p e e p p p p p p p p e e p p h e p 1 1 2 2 2 2 1 1 1 2 1 2 1 2 2 2 2 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2 1 1 2 2 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 + = + → + + − −       = + + − −       → → − = − − −       + − = − − − + − → → − = − + = + ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρρ ρ ρ ( ) ( ) + − = + − − + − → → − = + − +               2 2 2 2 2 2 2 1 2 2 1 2 2 2 2 1 2 1 1 1 2 2 1 2 2 1 2 1 2 1 1 2 1 2 p p p p p p p p e e p p p p ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ (C.E.) e e p p 2 1 1 2 2 1 2 1 1 − = + −       ρ ρ Hugoniot Equation u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2 Pierre-Henri Hugoniot (1851 – 1887)
23 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Fanno’s Line for a Perfect Gas (1) ( )1 1 1 2 2ρ ρu u m A = =  ( ) frictionpupu ++=+ 2 2 221 2 112 ρρ ( )3 1 2 1 2 1 1 2 2 2 2 C T u C T u h C Tp p p+ = + = ( )4 1 1 1 2 2 2 p R T p R T= =ρ ρ ( )5 2 1 2 1 2 1 s s C T T R p p p − = −ln ln (C.M.) (C.L.M.) (C.E.) Ideal Gas ( ) p p T T u u h C T h C T p p T T h C T h C T s s C T T R T T h C T h C T p p p p p p p 2 1 4 2 1 2 1 2 1 1 1 2 3 0 1 0 2 2 1 2 1 0 1 0 2 2 1 2 1 2 1 0 1 0 2 5 =             = = − −        → = − − → − = − − − ( ) ( ) ( ) ln ln ρ ρ ρ ρ Assume that all the conditions of the model are satisfied except the moment equation (2) (a flow with friction) Using , we obtainh C Tp= s s 1 s 2 s max h 1 h 2 h 2 1 s s C h h R h h h h h h p2 1 2 1 2 1 0 1 0 2 − = − − − ln ln Fanno’s Line for a Perfect Gas This is the Adiabatic, Constant Area Flow. u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2 Gino Girolamo Fanno (1888 – 1962)
24 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Fanno’s Line for a Perfect Gas (2) s s 1 s 2 s max h 1 h 2 h 2 1 We have a point of maximum entropy. Let see the significance of this point ρρ dp dh dp dhdsT =→=−= 0 max Gibbs u dud dudu −=→=+ ρ ρ ρρ 0(C.M.) duudh u hd −=→=      + 0 2 2 (C.E.) Therefore )4..( 0 .).( 00 0 EC ds MC dsds ds u du d dpd d dpdp dh =      −      =      == === = ρρ ρ ρρ 0 0 = =       = ds ds d dp u ρ or ds C dT T R dp p ds C dT T R d C C dp p d dp d p dp d p R T p v p v ds ds ds ds p R T = − = = − =        → ≡ = = → = == = = = = max max 0 0 0 0 0 0 ρ ρ γ ρ ρ ρ ρ ρ γ ρ γ ρ We have: u dp d R T a speed of soundds ds = = =       = = =0 0 ρ γ u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2
25 Ideal Gas NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Rayleigh’s Line for a Perfect Gas (1) ( ) A m uu  == 22111 ρρ ( )2 1 1 2 1 2 2 2 2ρ ρu p u p+ = + ( ) QhuTCuTC pp ++=+ 2 22 2 11 2 1 2 1 3 ( )4 1 1 1 2 2 2 p R T p R T= =ρ ρ ( )5 2 1 2 1 2 1 s s C T T R p p p − = −ln ln (C.M.) (C.L.M.) (C.E.) Assume that all the conditions of the model are satisfied except the energy equation (3) (a flow with heating and cooling) Let substitute in (5) , to obtainh C Tp= Rayleigh’s Line for a Perfect Gas This is the Frictionless, Constant Area Flow, with Cooling and Heating. s max s s 1 s 2 h 1 h 2 h M1 M1 Rayleigh2 1 Heating Heating Cooling  m A R T p p m A R T p p x p 1 1 1 2 2 2 1 + = + ( ) 2 1 12 1 1 1 2 12 11 1 2 12 1 2 1 lnln5 p R A m c p TR A m b h C a bbR h h Css p p  =        +=         −+−=− We want to find x p p ≡ 2 1 . Let multiply the result by x p1 x m A R T p b x m A R p c T2 1 1 2 1 1 2 1 21 2 0− +       + =       or x p p b b a T= = + −2 1 1 1 2 1 2 The solution is: John William Strutt Lord Rayleigh (1842-1919) u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2
26 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Rayleigh’s Line for a Perfect Gas (2) We have a point of maximum entropy. Let see the significance of this point u dud dudu −=→=+ ρ ρ ρρ 0(C.M.) (C.L.M.) A Normal Shock Wave must be on both Fanno and Rayleigh Lines, therefore the end points of a Normal Shock Wave must be on the intersection of Fanno and Rayleigh Lines u dp d R T a speed of soundds ds = = =       = = =0 0 ρ γ d p u dp du u+       = → = − 1 2 02 ρ ρ ( )→ = = − −       = dp d dp du du d u u u ρ ρ ρ ρ 2 s s 1 s 2 h 1 h 2 h M1 M1 Rayleigh Fanno 2 1 SHOCK According to the Second Law of Thermodynamics the Entropy must increase. Therefore a Normal Shock Wave from state (1) to state (2) must be such that s2 s1. (from supersonic to subsonic flow only) u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2
27 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Mach Number Relations (1) ( ) ( ) ( )   C M u u C L M u p u p p u p u u u C E a h u a h u a a u a a u a p . . . . . . . ρ ρ ρ ρ ρ ρ γ γ γ γ γ γ γ ρ 1 1 2 2 1 1 2 1 2 2 2 2 1 1 1 2 2 2 2 1 1 2 1 1 2 2 2 2 2 2 1 2 2 1 2 2 2 2 2 2 4 1 1 2 1 1 2 1 2 1 2 1 2 1 2 = + = +    → − = − → − + = − + → = + − − = + − −               = ∗ ∗    − = − a u a u u u1 2 1 2 2 2 2 1 γ γ Field Equations: ( ) γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ + − − − + + − = − ↓ + − + − − = − → + = − − = + ↓ ∗ ∗ ∗ ∗ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 2 1 1 2 2 2 2 1 2 1 1 2 2 2 1 2 1 2 1 2 a u u a u u u u u u u u a u u u u a u u u u a1 2 2 = ∗ u a u a M M1 2 1 21 1∗ ∗ ∗ ∗ = → = Prandtl’s Relation u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2 Ludwig Prandtl (1875-1953)
28 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Mach Number Relations (2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )( ) ( ) M M M M M M M M M 2 2 2 2 1 1 2 1 2 1 2 1 2 1 2 2 1 1 2 1 1 2 1 1 1 2 1 2 1 2 1 1 1 1 1 1 2 = + − − = + − − = + + − + − − = − + + / + − / / + − / + − − ∗ = ∗ ∗ ∗ γ γ γ γ γ γ γ γ γ γ γ γ γ γ or ( ) M M M M M H H A A 2 1 2 1 2 1 2 1 21 2 1 2 1 1 2 1 2 2 1 1 1 2 1 2 1 1 = + − − − = + + − + + − = = γ γ γ γ γ γ γ ( ) ( ) ρ ρ γ γ 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 1 2 = = = = = + − + = ∗ ∗ A A u u u u u u a M M M u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2
29 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Mach Number Relations (3) ( ) ( ) ( ) ( ) ( ) p p u p u u u a M M M M M M M 2 1 1 2 1 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 2 1 1 1 1 2 1 = + −       = + −       = + − − + +       = + / + − / − − + ρ γ ρ ρ γ γ γ γ γ γ γ or (C.L.M.) ( ) p p M2 1 1 2 1 2 1 1= + + − γ γ ( ) ( ) ( ) h h T T p p M M M a a h C T p RTp 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 1 2 1 1 1 2 1 = = = + + −       − + + = = =ρ ρ ρ γ γ γ γ ( ) ( ) ( ) s s R T T p p M M M 2 1 2 1 1 2 1 1 1 2 1 1 1 2 1 2 1 1 2 1 1 1 2 1 − =                       = + + −       − + +                 − − − − ln ln γ γ γ γ γ γ γ γ γ γ ( ) ( ) ( ) ( ) s s R M M M 2 1 1 1 2 1 2 3 2 2 1 2 41 2 2 3 1 1 2 1 1 − ≈ + − − + − + − γ γ γ γ K Shapiro p.125 u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2
30 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Mach Number Relations (4) ( ) p p p p p p p p M M M02 01 02 2 1 01 2 1 2 2 1 2 1 1 2 1 1 2 1 1 2 1 2 1 1= = + − + −           + + −       −γ γ γ γ γ γ ( ) ( ) 1 1 2 1 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 2 1 2 1 1 2 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 + − = + − + − − − = − − + − + −      + + + −       = + + + − γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ M M M M M M M M ( ) ( ) p p M M M02 01 1 2 1 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 1= + + + −             + + −       − − − γ γ γ γ γ γ γ γ u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2
31 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Mach Number Relations (5) ( ) s s R T T p p p p M M M T T 2 1 02 01 1 02 01 1 02 01 1 2 1 2 1 2 02 01 1 1 1 2 1 1 1 1 2 1 1 2 − =                       = −       = − + + −       − − + + −           − − = ln ln ln ln γ γ γ γ γ γ γ γ γ s s 1 s 2 T M1 M1 Rayleigh Fanno 2 1 SHOCK T 2 T 1 T 02 T 01= T 2 T 1=* * p 2 p 1 p 01 p 02 Mollier’s Diagram u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2 John William Strutt Lord Rayleigh )1842-1919( Gino Girolamo Fanno )1888–1062( Return to Table of Content
32 OBLIQUE SHOCK EXPANSION WAVESSOLO →→ →→ += += twnuV twnuV 11 11 222 111   Continuity Eq.: 2211 uu ρρ = ( ) ( ) ( )21222111 ppuuuu +−−=+− ρρ Moment Eq. Tangential Component: ( ) ( ) 0222111 =+− wuwu ρρ Moment Eq. Normal Component: Energy Eq.: 22 2 2 2 2 211 2 1 2 1 1 22 u wu hu wu h ρρ         + +=        + + Continuity Eq.: 2211 uu ρρ = Moment Eq.: 21 ww = 2 222 2 111 upup ρρ +=+ Energy Eq.: 22 2 2 2 2 1 1 u h u h +=+ Summary Calorically Perfect Gas: Tch TRp p= = ρ 6 Equations with 6 Unknowns 222222 ,,,,, hwuTpρ
33 OBLIQUE SHOCK EXPANSION WAVESSOLO For a calorically Perfect Gas ( ) ( ) ( ) ( )[ ] ( )[ ] 2 1 1 2 1 2 2 1 2 12 2 2 1 1 2 2 1 2 1 1 2 11/2 1/2 1 1 2 1 21 1 ρ ρ γγ γ γ γ γ γ ρ ρ p p T T M M M M p p M M n n n n n n = −− −+ = − + += +− + = βsin11 MMn = ( )θβ − = sin 2 2 nM M Now we can compute ( ) ( ) ( ) ( ) ( ) ( )       ⋅+ − = − + −+ === − ⇒          = =− = θββ θβ β θβ βγ βγ ρ ρ β θβ θβ β tantan1tan tantan tan tan sin1 sin12 tan tan tan tan 22 1 22 1 2 1 1 2 12 2 2 1 1 M M u u ww w u w u
34 OBLIQUE SHOCK EXPANSION WAVESSOLO ( )       ++ − = 22cos 1sin cot2tan 2 1 22 1 βγ β βθ M M M,, βθ relation 12 M 12 M .5max =Mforθ β θ 1M 2M Strong Shock Weak Shock θ β We can see that θ = 0 for 1.β = 90° (Normal Shock) 2.sin β = 1/ M1
35 OBLIQUE SHOCK EXPANSION WAVESSOLO 1. For any given M1 there is a maximum deflection angle θmax If the physical geometry is such that θ θmax, then no solution exists for straight oblique shock wave. Instead the shock will be curved and detached.
36 OBLIQUE SHOCK EXPANSION WAVESSOLO 2.For any given θ θmax, there are two values of β predicted by the θ-β-M relation for a given Mach number. WEAKβ STRONGβ ( )       ++ − = 22cos 1sin cot2tan 2 1 22 1 βγ β βθ M M M,, βθ relation - the large value of β is called the strong shock solution In nature the weak shock solution usually occurs. - the small value of β is called the weak shock solution - in the strong shock solution M2 is subsonic (M2 1) - in the weak shock M2 solution is supersonic (M2 1)
37( )       ++ − = 22cos 1sin cot2tan 2 1 22 1 βγ β βθ M M M,, βθ relation SOLO OBLIQUE SHOCK EXPANSION WAVES θ β 4.1=γ θ maxθ θ
38 ( )[ ] ( )[ ] ( )θβ γγ γ β − = −− −+ = = sin 11/2 1/2 sin 2 2 2 1 2 12 2 11 n n n n n M M M M M MM SOLO θ maxθ OBLIQUE SHOCK EXPANSION WAVES Mach Number in Back of Oblique Shock M2 as a Function of the Mach Number in Front of the Shock M , for Different Values of Deflection Angle θ (γ=1.4)
39 ( )1 1 2 1 sin 2 1 1 2 11 − + += = n n M p p MM γ γ β SOLO θ θ OBLIQUE SHOCK EXPANSION WAVES Static Pressure Ratio P2 / P1 as a Function of M1 the Mach Number in Front of an Oblique Shock, for Different Values of Deflection Angle θ (γ=1.4)
40 SOLO θ θ OBLIQUE SHOCK EXPANSION WAVES Stagnation Pressure Ratio P2 0/ P1 0 as a Function of M1 the Mach Number in Front of an Oblique Shock, for Different Values of Deflection Angle θ (γ=1.4)
41 Hodograph Shock Polar SOLO OBLIQUE SHOCK EXPANSION WAVES
42 Hodograph Shock Polar SOLO -For every deflection angle θ the Hodograph gives two solutions, a strong shock (B outside the sonic circle – M21) and a weak shock (D inside the sonic circle – M11) - The line OC tangent to the Hodograph gives the maximum deflection angle θmax. For θ θmax there is no oblique shock wave. - For point E θ=0 and β=π/2, therefore a normal shock. Point A corresponds to the Mach value before the shock M1. - The Shock Angle β corresponding to a given angle θ defined by the points B and D can be found by drawing the line OH normal to line AB. β = angle HOA. OBLIQUE SHOCK EXPANSION WAVES
43 SOLO Family of Hodograph Shock Polars ( γ= 1.4) θ 1 ***1 2 1 ** *** 21 2 1 212 21 2 2 +−      + −       −=      c V c V c V c V c V c V c V c V x x xy γ A. H. Shapiro “The Dynamics and Thermodynamics of Compressible Flow Fluid”,pg.543 45.2 OBLIQUE SHOCK EXPANSION WAVES
44 SOLO θmaxθ maxθ maxθ γ OBLIQUE SHOCK EXPANSION WAVES
45 SOLO OBLIQUE SHOCK EXPANSION WAVES
46 SOLO OBLIQUE SHOCK EXPANSION WAVES
47 SOLO OBLIQUE SHOCK EXPANSION WAVES
48 SOLO OBLIQUE SHOCK EXPANSION WAVES
49 SOLO OBLIQUE SHOCK EXPANSION WAVES Return to Table of Content
50 SOLO OBLIQUE SHOCK EXPANSION WAVES Prandtl-Meyer Expansion Waves Ludwig Prandtl (1875 – 1953) Theodor Meyer (1882 – 1972) The Expansion Fan depicted in Figure was First analysed by Prandtl in 1907 and his student Meyer in 1908. Let start with an Infinitesimal Change across a Mach Wave M ach W ave θd µ µ π − 2 θµ π d−− 2 V VdV + ( ) ( ) θµθµ µ θµπ µπ dddV VdV sinsincoscos cos 2/sin 2/sin − = −− + = + µ θµθ µθ tan / tan1 tan1 1 1 VVd dd dV Vd =⇒+≈ − ≈+ 1 1 tan 1 sin 2 1 − =⇒      = − MM µµ V Vd Md 12 −=θ 1907 - 1908
51 SOLO OBLIQUE SHOCK EXPANSION WAVES Prandtl-Meyer Expansion Waves (continue-1) M ach W ave θd µ µ π − 2 θµ π d−− 2 V VdV + V Vd Md 12 −=θ Integrating this equation gives ∫ −= 2 1 12 M M V Vd Mθ Using the definition of Mach Number: V = M. a a ad M Md V Vd += For a Calorically Perfect Gas 20 2 0 2 1 1 M T T a a − +==      γ MdMM a ad 1 2 2 1 1 2 1 −       − + − −= γγ M Md MV Vd 2 2 1 1 1 − + = γ ∫ − + − = 2 1 2 2 2 1 1 1 M M M Md M M γ θ
52 SOLO OBLIQUE SHOCK EXPANSION WAVES Prandtl-Meyer Expansion Waves (continue-2) The integral ∫ − + − = 2 1 2 2 2 1 1 1 M M M Md M M γ θ ( ) ∫ − + − = M Md M M M 2 2 2 1 1 1 γ ν is called the Prandtl-Meyer Function and is given the symbol ν. Performing the integration we obtain ( ) ( ) ( )1tan1 1 1 tan 1 1 2121 −−− + − − + = −− MMM γ γ γ γ ν Deflection Angle ν and Mach Angle μ as functions of Mach Number       = − M 1 sin 1 µ Finally ( ) ( )12 MM ννθ −= Return to Table of Content
53 Movement of Shocks with Increasing Mach Number Drag rises due to pressure Increase across a Shock Wave •Subsonic Flow - Local airspeed is less than sonic •Transonic Flow - Local airspeed is less than sonic at some points, greater than sonic elsewhere •Supersonic Flow - Local Airspeed is greater than sonic everywhere SOLO AERODYNAMICS
54 Movement of Shocks with Increasing Mach Number ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )87654321 ∞∞∞∞∞∞∞∞ MMMMMMMM SOLO AERODYNAMICS
55 Upper Surface Lower Surface Upper Surface Lower Surface Upper Surface Lower Surface Upper Surface Lower Surface Upper Surface Lower Surface ( c) Shock on upper surface Upper Surface Lower Surface (d ) Shocks on both surfaces Shock Movement of Shocks with Increasing Mach Number SOLO AERODYNAMICS
56 Movement of Shocks with Increasing Mach Number The Mach Number at witch M=1 appears on the Airfoil Upper Surface is called the Critical Mach Number for this Airfoil. The Critical Mach Number can be calculated as follows. Assuming an isentropic flow through the flow-field we have ( )1/ 2 2 2 1 1 2 1 1 − ∞ ∞             − + − + = γγ γ γ A A M M p p p∞, M∞ - Pressure and Mach Number upstream the Airfoil pA, MA- Pressure and Mach Number at a point A on the Airfoil Critical Mach Number The Pressure Coefficient Cp is computed using ( )             −             − + − + =      −= − ∞ ∞∞∞ 1 2 1 1 2 1 1 2 1 2 1/ 2 2 γγ γ γ γγ A A pA M M Mp p M C Definition of Critical Mach Number. Point A is the location of minimum pressure on the top surface of the Airfoil. SOLO AERODYNAMICS
57 Movement of Shocks with Increasing Mach Number Critical Mach Number This relation gives a unique relation between the upstream values of p∞, M∞ and the respective values pA, MA at a point A on the Airfoil. Assume that point A is the point of minimum pressure, therefore maximum velocity, on the Airfoil and that this maximum velocity corresponds to MA = 1. Then by definition M∞ = Mcr . ( )             −             − + − + =      −= − ∞ ∞∞∞ 1 2 1 1 2 1 1 2 1 2 1/ 2 2 γγ γ γ γγ A A pA M M Mp p M C ( )             −             − + − + = − 1 2 1 1 2 1 1 2 1/ 2 γγ γ γ γ cr cr p M M C cr 2 0 1 ∞− = M C C p p ( )             −             − + − + = − 1 2 1 1 2 1 1 2 1/ 2 γγ γ γ γ cr cr p M M C cr 2 0 1 ∞− = M C C p p To find the Mcr we need on other equation describing Cp at subsonic speeds. We can use the Prandtl-Glauert Correction or the Karman-Tsien Rule or Laiton’s Rule SOLO AERODYNAMICS
58 Movement of Shocks with Increasing Mach Number Critical Mach Number AirfoilThickAirfoilMediumAirfoilThin AirfoilThickAirfoilMediumAirfoilThin crcrcr ppp MMM CCC 000 The point of minimum pressure, therefore maximum velocity, does not correspond to the point of maximum thickness of the Airfoil. This is because the point of minimum pressure is defined by the specific shape of the Airfoil and not by a local property. The Critical Mach Number is a function of the thickness of the Airfoil. For the thin Airfoil the Cp0 is smaller in magnitude and because the disturbance in the Flow is smaller. Because of this the Critical Mach Number of the thin Airfoil is greater SOLO AERODYNAMICS
59 Movement of Shocks with Increasing Mach Number Drag Divergence Mach Number The Drag at small Mach number, due to Profile Drag with Induced Drag =0 (αi = 0) is constant (points a, b, and c) until M∞ = Mcr (point c). As the velocity increase above Mcr (point d), a finite region of supersonic flow (Weak Shock boundary)appears on the Airfoil. The Mach Number in this bubble of supersonic flow is slightly above Mach 1, typically 1.02 to 1.05. If M∞ increases more, We encounter a point, e, at which is a sudden increase in Drag. The Value of M∞ at which the sudden increase in Drag starts is defined as the Drag-divergence Mach Number, Mdrag-divergence 1. At this point Shock Waves appear on the Airfoil. The Shock Waves are dissipative phenomena extracting energy (Drag) from the kinetic energy of the Airfoil. In addition the sharp increase of the pressure across the Shock Wave create a strong adverse pressure gradient, causing the Flow to separate From the Airfoil Surface creating Drag increase. Beyond the Drag-divergence Mach Number, the Drag Coefficient becomes very large, increasing by a factor of 10 or more. As M∞ approaches unity (point f) the Flow on both the top and the SOLO AERODYNAMICS
60 Movement of Shocks with Increasing Mach Number Summary of Airfoil Drag The Drag of an Airfoil can be described as the sum of three contributions: wpf DDDD ++= where D – Total Drag of the Airfoil Df – Skin Friction Drag Dp – Pressure Drag due to Flow Separation Dw – Wave Drag (present only at Transonic and Supersonic Speeds; zero for Subsonic Speeds below the Drag-divergence Mach Number) In terms of the Drag Coefficients, we can write: wDpDfDD CCCC ,,, ++= The Sum: pDfD CC ,, + Profile Drag Coefficient SOLO AERODYNAMICS
61 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 SOLO Return to Table of Content AERODYNAMICS
62 AERODYNAMICS Drag Variation with Mach Number SOLO Return to Table of Content
63 AERODYNAMICS Swept Wings Drag Variation Adolf Busemann and Alfred Betz, discovered around 1930 that Drag at Transonic and Supersonic Speeds could be reduced using Swept Back Wings. Assume Mcr for Wing = 0.7 Airfoil Section with Mcr = 0.7 Airfoil Section with Mcr = 0.7 Airfoil ³ sees´ only this component of velocity Mcr for swept wing Adolph Busemann (1901 – 1986) also NACA Colorado U. Albert Betz (1885 – 1968 ), Λ = cos _ cr sweptcr M M From the Figure we see that if Λ is the Swept Angle, than Supersonic L.E. Subsonic L.E. Mach Cone For Supersonic Flow M∞ 1 •If the Leading Edge of Swept Wing is outside the Mach Cone, the component of the Mach Number normal to the Leading Edge is Supersonic. As a result a Strong Oblique Shock Wave will be created on the Wing. •If the Leading Edge of Swept Wing is inside the Mach Cone, the component of the Mach Number normal to the Leading Edge is Subsonic. As a result a Weaker Oblique Shock Wave will be created on the Wing and a Lower Drag will result. SOLO
64 SOLO Wings in Compressible Flow 64 Swept Wings The Swept Wing Theory was first presented by Adolf Busemann at the Fifth Volta Conference in Roma 1935. Busemann made use of so called “Independence Principle”: “The air forces on a sufficient long, narrow Wing Panel are independent of the component of the flight velocity in the direction of the Wing Leading Edge (disregarding friction). The air forces the depend only on the reduced component velocity perpendicular to the Wing Leading Edge” Adolph Busemann (1901 – 1986). The Wing angles relative to Flow Direction are: α – Angle of Attack Λ – Swept Angle The Flow Mach components are: forcesairaffectingnotELtoparallelM forcesairaffecting PlaneWingtheinELtonormalM PlaneWingtonormalM ..sincos ..coscos sin Λ       Λ ∞ ∞ ∞ α α α We have: ( ) ( )[ ] ( ) Λ = Λ = Λ=       Λ =      Λ = Λ−=Λ+= − ∞ ∞− ∞∞∞∞ coscos : cos: cos tan tan coscos sin tan: cossin1coscossin: 11 2/1222/122 τ τ α α α α ααα c t cc M M MMMM e e e e Section A-A Section B-B
65 SOLO Wings in Compressible Flow 65 Swept Wings Section A-A Section B-B ( ) bcM L CL 2 2/ ∞∞ = ργ The Total Lift is: ( ) ( )[ ] ( ) Λ = Λ = Λ=       Λ =      Λ = Λ−=Λ+= − ∞ ∞− ∞∞∞∞ coscos : cos: cos tan tan coscos sin tan: cossin1coscossin: 11 2/1222/122 τ τ α α α α ααα c t cc M M MMMM e e e e Therefore: ( ) ( )α222 cossin1/ Λ−== ∞∞ eLeeLL CMMCC ( ) ( ) ( )ΛΛ == ∞∞∞∞ cos/cos2/2/ 22 bcM L bcM L C eeee eL ργργ and: The Friction Drag is ignored the Tangential Component of Velocity does not contribute to the Drag and the Pressure Drag is normal to the Leading Edge. If D is the Total Pressure Drag the component in the M∞ direction is only D cosΛ. ( ) ( ) ( ) ( )ΛΛ = Λ = ∞∞∞∞ cos/cos2/ ; 2/ cos 22 bcM D C bcM D C e DD ργργ or: ( ) ( )α222 cossin1cos/ Λ−Λ== ∞∞ eDeeDD CMMCC
66 SOLO Wings in Compressible Flow 66 Swept Wings Oblique Wing aircraft, AD-1 was built and flown by NASA.. Oblique Wing concept was developed in the USA by R.T. Jones. Robert Thomas Jones (1910–1999) Oblique Wing Flight Demonstration by the AD-1.
67 AERODYNAMICS Swept Wings Drag Variation SOLO
68 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 AERODYNAMICS Swept Wings Drag Variation SOLO
69 AERODYNAMICS Swept Wings Drag Variation SOLO
70 AERODYNAMICS Swept Wings Drag Variation Comparison of the Transonic Drag Polar for an Unswept Wing with that for a Swept Wing (data from Schlichting) SOLO
71 SOLO Wings in Compressible Flow Profile Drag Coefficients versus Mach Number for an Un-swept and a Swept-back Wing (φ=45°), t/c=0.12, AR=4 Swept Wings
72 AERODYNAMICS Swept Wings Drag Variation SOLO
73 SOLO Return to Table of Content Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
74 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
75 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
76 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
77 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO Return to Table of Content
78 SOLO Return to Table of Content Ray Whitford, “Design for Air Combat”
79 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO Return to Table of Content
80 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane Richard T. Whitcomb (1921 – 2009) SOLO
81 German aerodynamicist named Dr. Adolf Busemann, who had come to work at Langley after World War II, gave a technical symposium on transonic airflows. In a vivid analogy, Busemann described the stream tubes of air flowing over an aircraft at transonic speeds as pipes, meaning that their diameter remained constant. At subsonic speeds, by comparison, the stream tubes of air flowing over a surface would change shape, become narrower as their speed increased. This phenomenon was the converse, in a sense, of a well-known aerodynamic principle called Bernoulli's theorem, which stated that as the area of an airflow was made narrower, the speed of the air would increase. This principle was behind the design of venturis,9 as well as the configuration of Langley's wind tunnels, which were necked down in the test sections to generate higher speeds.10 But at the speed of sound, Busemarm explained, Bernoulli's theorem did not apply. The size of the stream tubes remained constant. In working with this kind of flow, therefore, the Langley engineers had to look at themselves as pipefitters. Busemann's pipefitting metaphor caught the attention of Whitcomb, who was in the symposium audience. Soon after that Whitcomb was, quite literally, sitting with his feet up on his desk one day, contemplating the unusual shock waves he had encountered in the transonic wind tunnel. He thought of Busemann's analogy of pipes flowing over a wing-body shape and suddenly, as he described it later, a light went on. Richard T. Whitcomb (1921 – 2009) Adolph Busemann (1901 – 1986) also NACA Colorado U. Origin of Transonic Area Rule http://history.nasa.gov/SP-4219/Chapter5.html SOLO
82 Richard T. Whitcomb (1921 – 2009) Adolph Busemann (1901 – 1986) also NACA Colorado U. Origin of Transonic Area Rule http://history.nasa.gov/SP-4219/Chapter5.html In practical terms, the area rule concept meant that something had to be done in order to compensate for the dramatic increase in cross- sectional area where the wing joined the fuselage. The simplest solution was to indent the fuselage in that area, creating what engineers of the time described as a Coke bottle or Marilyn Monroe shaped design. The indentation would need to be greatest at the point where the wing was the thickest, and could be gradually reduced as the wing became thinner toward its trailing edge. If narrowing the fuselage was impossible, as was the case in several designs that applied the area rule concept, the fuselage behind or in front of the wing needed to be expanded to make the change in crosssectional area from the nose of the aircraft to its tail less dramatic. Throughout the first quarter of 1952, Whitcomb conducted a series of experiments using various area-rule based wing-body configurations in Langley's 8-Foot High-Speed Tunnel. As he expected, indenting the fuselage in the area of the wing did, indeed, significantly reduce the amount of drag at transonic speeds. In fact, Whitcomb found that indenting the body reduced the drag-rise increments associated with the unswept and delta wings by approximately 60 percent near the speed of sound, virtually eliminating the drag rise created by having to put wings on a smooth, cylindrical shaped body. http://www.youtube.com/watch?v=xZWBVgL8I54 http://www.youtube.com/watch?v=Cn0lSoreB1g SOLO
83 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
84 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
85 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
86 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
87 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
88 SOLO
89 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 AERODYNAMICSSOLO Return to Table of Content
90 AERODYNAMICSSOLO
91 Nguyen X. Vinh, “Flight Mechanics of High Performance Aircraft”, Cambridge University,1993 AERODYNAMICSSOLO
92 Examples of airfoils in nature and within various vehicles Lift and Drag curves for a typical airfoil SOLO
93 SOLO
94 AERODYNAMICSSOLO
95 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 AERODYNAMICSSOLO
96 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 AERODYNAMICSSOLO Return to Table of Content
97 Flow regimes for a slender bodySOLO
98 )A)- Flow field in wing-tail plane, influence of angle of attackSOLO
99 Slender wing-body combinationSOLO
100 ))B)- Flow field in wing-tail plane, influence ofB)- Flow field in wing-tail plane, influence of control deflectioncontrol deflection δδ for pitchfor pitch SOLO
101 Missile controlMissile controlSOLO
102 CHARACTERISTICSCHARACTERISTICS Summary of aerodynamic designSummary of aerodynamic design SOLO
103 ))C)- Flow field in wing-tail plane, influence ofC)- Flow field in wing-tail plane, influence of control deflectioncontrol deflection ξξ for rollfor roll SOLO
104 Types of missile roll control skid-to-turn, bank-to-turnTypes of missile roll control skid-to-turn, bank-to-turnSOLO Return to Table of Content
105 Density Profile Mach 1.2, Color Contours Modified to see Detail on Shock Waves More Fun With CFD – RM-10SOLO
106 Density Profiles, Mach 2.41, simulated altitude of 11,000 ft )Re=76.4x106 ) More Fun With CFD – RM-10SOLO
107 Density Profiles, Mach 2.41 – color contours modified to see detail in shock waves More Fun With CFD – RM-10SOLO
108 Density Profiles, Mach 1.62 – rotated, with plot to show distribution around fins More Fun With CFD – RM-10SOLO
109 The Effect of Leading Edge Slat, Flap, and Trailing Edge Flap Upon Angle of Attack of Basic Wing Darrol Stinton “ The Design of the Aircraft”SOLO
110High Angles of Attack Flows )Development of a High Resolution CFD) SOLO
111High Angles of Attack Flows )Development of a High Resolution CFD) SOLO
112 Three-Element Airfoil Pressure Coefficient and Streamlines at Maximum Lift M=0.2 )Re=4.1x106 ) SALSA Computation AERODYNAMICSSOLO
113Inviscid Transonic Flow Solution Over a 2-D Airfoil at M=0.75 )Re=1000) AERODYNAMICSSOLO
114Inviscid Supersonic Flow Solution Over a 2-D Airfoil at M=1.50 )Re=1000) AERODYNAMICSSOLO
115 AERODYNAMICSSOLO
116 Linearized Flow Equations 1. Irrotational Flow SOLO Assumptions 2. Homentropic 3. Thin bodies ( )0  =×∇ u       = ∂ ∂ =∇ 00..;. t s seieverywhereconsts This implies also inviscid flow ( )~τ = 0 Changes in flow velocities due to body presence are small were - flow velocity as a function of position and time - flow entropy as a function of position and time ( )tzyxu ,,,  ( )tzyxs ,,,
117 SOLO )C.L.M) For an inviscid flow conservation of linear momentum gives:( )~τ = 0 Assume that body forces are conservative and stationary were - flow pressure as a function of position and time( )tzyxp ,,, - flow density as a function of position and time( )tzyx ,,,ρ ( ) Gpuuu t u uu t u tD uD     ρ ∂ ∂ ρ ∂ ∂ ρρ +−∇=      ×∇×−      ∇+=      ∇⋅+= 2 2 1 or ( ) G p uuu t u   + ∇ −=×∇×−      ∇+ ∂ ∂ ρ 2 2 1 Euler’s Equation 0 = ∂ Ψ∂ Ψ−∇= t G  - Body forces as a function of position( )zyxG ,,  Leonhard Euler 1707-1783 Linearized Flow Equations
118 SOLO Let integrate the Euler’s Equation between two points )1) and )2) ( ) ( ) ( ) ∫∫∫∫∫∫ ⋅Ψ∇+ ⋅∇ +×∇⋅×−⋅      ∇+⋅ ∂ ∂ =⋅      Ψ∇+ ∇ +×∇×−      ∇+ ∂ ∂ = 2 1 2 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 0 rd rdp uurdrdurdu t rd p uuuu t    υρ We can chose the path of integration as follows: - along a streamline ) and are collinear; i.e.: )rd  u  0  =×urd - along any path, if the flow is irrotational ( )0  =×∇ u to obtain: ( ) ( ) 0 2 1 =×∇⋅×∫ uurd  Assuming that the flow is irrotational we can define a potential , such that: ( )0  =×∇ u ( )tr ,  Φ Φ∇=u  Let use the identity to obtain: ( ) rdFtrFd constt  ⋅∇== , ( ) 2 1 2 2 1 2 2 1 2 1 0         Ψ+++ ∂ Φ∂ =      Ψ∇++      +Φ ∂ ∂ = ∫∫ ∞ p p pd u t pd udd t ρρ Bernoulli’s Equation for Irrotational and Inviscid Flow Daniel Bernoulli 1700-1782 Linearized Flow Equations
119 SOLO For an isentropic ideal gas we have 2 2 11 a ad T Tdd p pd − = − == γ γ γ γ ρ ρ γ where ρ γ γ ρρ p TR d pdp a s === ∂ ∂ =2 is the square of the speed of sound In this case 2 2 2 1 1 1 2 ad a adppd RTa RTp − = − = = = γργ γ ρ γ ρ and [ ]222 1 1 1 1 2 2 ∞− − = − = ∫∫ ∞∞ aaad pd a a p p γγρ Using the Bernoulli’s Equation we obtain ( ) ( ) ( ) ( )      Ψ−Ψ+−+ ∂ Φ∂ −−=−=− ∞∞∞ ∫ ∞ 2222 2 1 11 Uu t dp aa p p γ ρ γ ( ) 2 1 2 2 1 2 2 1 2 1 0         Ψ+++ ∂ Φ∂ =      Ψ∇++      +Φ ∂ ∂ = ∫∫ ∞ p p pd u t pd udd t ρρ Bernoulli’s Equation for Irrotational and Inviscid Flow Linearized Flow Equations
120 SOLO Let use the conservation of mass )C.M.) equation )C.M.) 0=⋅∇+ u tD D  ρ ρ or tD D u ρ ρ 1 −=⋅∇  Let go back to Bernoulli’s Equation ( ) ( )      Ψ−Ψ+−+ ∂ Φ∂ −= ∞∞∫ ∞ 22 2 1 Uu t pd p p ρ and use the Leibnitz rule of differentiation: ( ) ( )uxFdxuxF xd d x x ,, 0 =∫ to obtain ρρ 1 =∫ ∞ p p pd pd d Now we can compute tD Da tD D d pd tD pD tD pDpd pd dpd tD D p p p p ρ ρ ρ ρρρρρ 2 11 ===         = ∫∫ ∞∞ Therefore ( ) ( )      Ψ−Ψ+−+ ∂ Φ∂ =−=−=⋅∇ ∞∞∫ ∞ 22 22 2 1111 Uu ttD D a pd tD D atD D u p p ρ ρ ρ  Since ( )[ ] 0=Ψ−Ψ= ∞∞ tD D u tD D we have             ∇⋅+ ∂ ∂ ⋅+ ∂ Φ∂ =            ∇⋅+ ∂ Φ∂ ∇⋅+ ∂ ∂ ⋅+ ∂ Φ∂ = =      + ∂ Φ∂       ∇⋅+ ∂ ∂ =      + ∂ Φ∂ =⋅∇ Φ∇= 2 2 1 2 1 2 11 2 11 2 2 2 2 2 2 2 2 2 2 2 2 u u t u u ta u u t u t u u ta u t u ta u ttD D a u u        GOTTFRIED WILHELM von LEIBNIZ 1646-1716 Linearized Flow Equations
121 SOLO             ∇⋅+ ∂ ∂ ⋅+ ∂ Φ∂ =            ∇⋅+ ∂ Φ∂ ∇⋅+ ∂ ∂ ⋅+ ∂ Φ∂ = =      + ∂ Φ∂       ∇⋅+ ∂ ∂ =      + ∂ Φ∂ =⋅∇ Φ∇= 2 2 1 2 1 2 11 2 11 2 2 2 2 2 2 2 2 2 2 2 2 u u t u u ta u u t u t u u ta u t u ta u ttD D a u u        Let substitute Φ∇=u              Φ∇⋅Φ∇∇⋅Φ∇+Φ∇ ∂ ∂ ⋅Φ∇+ ∂ Φ∂ =Φ∇⋅∇ 2 1 2 1 2 2 2 tta ( ) ( ) ( )      Ψ−Ψ+−Φ∇⋅Φ∇+ ∂ Φ∂ −−= ∞∞∞ 222 2 1 1 U t aa γ Special cases 0≈Φ∇⋅∇ Laplace’s equation ∞∞ Ua )subsonic flow) we can approximate the first equation by 1 2 ( ) ( ) 2 2 t uu t uuu ∂ Φ∂ ⋅ ∂ ∂ +⋅∇⋅  we can approximate the first equation by 0 1 2 2 2 = ∂ Φ∂ −Φ∇⋅∇ ta Wave equation Pierre-Simon Laplace 1749-1827 Linearized Flow Equations
122 SOLO Note The equation       + ∂ Φ∂       ∇⋅+ ∂ ∂ =⋅∇ 2 2 2 11 u t u ta u  can be written as Φ=      Φ∇⋅+ ∂ Φ∂       ∇⋅+ ∂ ∂ =      + ∂ Φ∂       ∇⋅+ ∂ ∂ =Φ∇ 2 2 22 2 2 2 11 2 11 tD D a u t u ta u t u ta c c  where the subscript c on and on is intended to indicate that the velocity is treated as a constant during the second application of the operators and . cu  2 2 tD Dc t∂∂/ ( )∇⋅u  This equation is similar to a wave equation. End Note Linearized Flow Equations
123 SOLO Let compute the local pressure coefficient: 2 2 1 : ∞∞ ∞− = U pp C p ρ We have:           −        =           −        =           −      =      −= − ∞∞ =− ∞ ∞ ∞ = − ∞ ∞ ∞       = = ∞ ∞ ∞ ∞ ∞∞∞ − ∞∞ ∞∞∞ 1 2 1 2 1 1 2 1 2 1 2 2 2 /1 2 2 2 2 1 22 2 1 γ γ γ γ γ γ γ ρ γ γ ρ γ γ a a Ma a a U T T U TR p p U p C aUMTRa T T p p TRp p Let use the equation ( ) ( ) ( )      Ψ−Ψ+−Φ∇⋅Φ∇+ ∂ Φ∂ −−= ∞∞∞ 222 2 1 1 U t aa γ to compute ( ) ( ) ( )      Ψ−Ψ+−Φ∇⋅Φ∇+ ∂ Φ∂− −= ∞∞ ∞∞ 2 22 2 2 11 1 U taa a γ Finally we obtain: ( ) ( ) ( )           −             Ψ−Ψ+−Φ∇⋅Φ∇+ ∂ Φ∂− −= − ∞∞ ∞∞ 1 2 11 1 2 1 2 22 γ γ γ γ U taM Cp Linearized Flow Equations
124 SOLO Assuming a stationary flow and neglecting the body forces :      = ∂ ∂ 0 t ( )0=Ψ             Φ∇⋅Φ∇∇⋅Φ∇=Φ∇⋅∇ 2 11 2 a ( ) ( )222 2 1 ∞∞ −Φ∇⋅Φ∇ − −= Uaa γ ( ) ( )           −       −Φ∇⋅Φ∇ − −= − ∞ ∞∞ 1 2 1 1 2 1 2 22 γ γ γ γ U aM Cp Φ∇=u  Linearized Flow Equations
125 SOLO 1 0 332211 323121 =⋅=⋅=⋅ =⋅=⋅=⋅ →→→→→→ →→→→→→ eeeeee eeeeee General Coordinates ( )321 ,, uuu →→→ ∂ Φ∂ + ∂ Φ∂ + ∂ Φ∂ =Φ∇ 3 33 2 22 1 11 111 e uh e uh e uh ( ) ( ) ( )      ∂ ∂ + ∂ ∂ + ∂ ∂ =       ++⋅∇=⋅∇ →→→ 321 3 213 2 132 1321 332211 1 Ahh u Ahh u Ahh uhhh eAeAeAA  Using we obtainΦ∇=:A              ∂ Φ∂ ∂ ∂ +      ∂ Φ∂ ∂ ∂ +      ∂ Φ∂ ∂ ∂ = =Φ∇⋅∇=Φ∇ 33 21 322 13 211 32 1321 2 1 uh hh uuh hh uuh hh uhhh where We have for ( ) ( )321321 ,,,,, uuuAuuu  Φ Linearized Flow Equations
126 SOLO zzyyxx Φ+Φ+Φ=Φ∇=Φ∇⋅∇ 2       Φ+Φ+Φ∇⋅      Φ+Φ+Φ=      Φ∇⋅Φ∇∇⋅Φ∇ →→→ 222 2 1 2 1 2 1 111 2 1 zyxzyx zyx ( ) ( ) ( )=ΦΦ+ΦΦ+ΦΦΦ+ ΦΦ+ΦΦ+ΦΦΦ+ΦΦ+ΦΦ+ΦΦΦ= zzzyzyxzxz yzzyyyxyxyxzzxyyxxxx yzzyxzzxxyyxzzzyyyxxx ΦΦΦ+ΦΦΦ+ΦΦΦ+ΦΦ+ΦΦ+ΦΦ= 222 22             Φ∇⋅Φ∇∇⋅Φ∇=Φ∇⋅∇ 2 11 2 a ( ) ( )222 2 1 ∞∞ −Φ∇⋅Φ∇ − −= Uaa γ ( ) 0 12 2 22111 222 222 2 2 2 2 2 =Φ−ΦΦ+ΦΦ+ΦΦ−Φ ΦΦ − Φ ΦΦ −Φ ΦΦ −Φ        Φ −+Φ         Φ −+Φ        Φ − ttztzytyxtxyz zy xz zx xy yx zz z yy y xx x aaa aaaaa ( ) ( ) ( )      Ψ−Ψ+−Φ+Φ+Φ+ ∂ Φ∂ −−= ∞∞∞ 222222 2 1 1 U t aa zyxγ We finally obtain Cartesian Coordinates ( )zuyuxu === 321 ,, Linearized Flow Equations Return to Table of Content
127 SOLO Cylindrical Coordinates ( )θ=== 321 ,, uruxu →→→→→→ ++=++= zryrxxzzyyxxR 1sin1cos1111 θθ  →→→→→ +−= ∂ ∂ += ∂ ∂ = ∂ ∂ zryr R zy r R x x R 1cos1sin1sin1cos1 θθ θ θθ  r R h r R h x R h = ∂ ∂ == ∂ ∂ == ∂ ∂ = θ  :1:1: 321 →→→→ →→→→→→ =+−= ∂ ∂ ∂ ∂ = =+= ∂ ∂ ∂ ∂ == ∂ ∂ ∂ ∂ = θθθ θ θ θθ 11cos1sin: 11sin1cos:1: 2 21 zy R R e rzy r R r R ex x R x R e       1 0 332211 323121 =⋅=⋅=⋅ =⋅=⋅=⋅ →→→→→→ →→→→→→ eeeeee eeeeee We have Linearized Flow Equations
128 SOLO Cylindrical Coordinates )continue – 1) ( )θ=== 321 ,, uruxu →→→→→→ Φ+Φ+Φ= ∂ Φ∂ + ∂ Φ∂ + ∂ Φ∂ =Φ∇ 321321 11 e r eee r e r e x rx θ θ 2 2 22 1 θΦ+Φ+Φ=Φ∇⋅Φ∇ r rx → →→       ΦΦ+ΦΦ+ΦΦ+       Φ−ΦΦ+ΦΦ+ΦΦ+      ΦΦ+ΦΦ+ΦΦ=       Φ+Φ+Φ∇=      Φ∇⋅Φ∇∇ 322 2 2 3212 2 2 22 11 111 1 2 1 2 1 e rr e rr e r r rrxx rrrrxrxxrxrxxx rx θθθθθ θθθθθ θ θθ θ θθ Φ+Φ+Φ+Φ= ∂ Φ∂ + ∂ Φ∂ + ∂ Φ∂ + ∂ Φ∂ =             ∂ Φ∂ ∂ ∂ +      ∂ Φ∂ ∂ ∂ +      ∂ Φ∂ ∂ ∂ = =Φ∇⋅∇=Φ∇ 22 2 22 2 2 2 2 1111 11 rrrrrrx rr r rx r xr rrrxx Linearized Flow Equations
129 SOLO Cylindrical Coordinates )continue – 2) ( )θ=== 321 ,, uruxu Then equation             Φ∇⋅Φ∇∇⋅Φ∇+Φ∇ ∂ ∂ ⋅Φ∇+ ∂ Φ∂ =Φ∇⋅∇ 2 1 2 1 2 2 2 tta becomes ( ){             ΦΦ+ΦΦ+ΦΦ+       Φ−ΦΦ+ΦΦ+ΦΦ+          ΦΦ+ΦΦ+ΦΦ      Φ+Φ+Φ+ ΦΦ+ΦΦ+ΦΦ+Φ=Φ+Φ+Φ+Φ → → →→→→ 322 2 2 32 12321 22 11 11 11 2 111 e rr e rr e r e r ee arr rrxx rrrrxrx xrxrxxxrx ztzytyxtxttrrrxx θθθθθ θθθ θθθ θθ or ( ) 0 2 112 / 1 1/ 1 1 11 22 222 2 22 2 22 22 2 2 2 =ΦΦ+ΦΦ+ΦΦ− Φ −       ΦΦΦ+ΦΦΦ+ΦΦΦ−         Φ +Φ+Φ        Φ −+Φ        Φ −+Φ        Φ − ztzytyxtx tt rrxxrxrx rrr r xx x aa rra a r ra r raa θθθθ θ θθ θ Linearized Flow Equations
130 SOLO Cylindrical Coordinates )continue – 3) ( )θ=== 321 ,, uruxu becomes ( ) ( ) ( )      Ψ−Ψ+−Φ∇⋅Φ∇+ ∂ Φ∂ −−= ∞∞∞ 222 2 1 1 u t aa γ In cylindrical coordinates, equation ( ) ( )      Ψ−Ψ+      −Φ+Φ+Φ+Φ−−= ∞∞∞ 22 2 2222 1 2 1 1 U r aa rxt θγ Assuming a stationary flow and neglecting body forces      = ∂ ∂ 0 t ( )0=Ψ 0 112 / 1 1/ 1 1 11 222 2 22 2 22 22 2 2 2 =      ΦΦΦ+ΦΦΦ+ΦΦΦ−         Φ +Φ+Φ        Φ −+Φ        Φ −+Φ        Φ − rrxxrxrx rrr r xx x rra a r ra r raa θθθθ θ θθ θ ( )       −Φ+Φ+Φ − −= ∞∞ 22 2 2222 1 2 1 U r aa rx θ γ Linearized Flow Equations Return to Table of Content
131 Linearized Flow EquationsSOLO Boundary Conditions 1. Since the Small Perturbations are not considering the Boundary Layer the Flow must be parallel at the Wing Surface. The Wing Surface S is defined by zU )x,y) – Upper Surface zL )x,y) – Lower Surface 0  =⋅ S un n  - Normal at the Wing Surface 22 1/111       ∂ ∂ +      ∂ ∂ +      + ∂ ∂ − ∂ ∂ −= y z x z zy y z x x z n UUUU U  ( ) ( ) ( ) ( ) zwUyvxuUzwUyvxuUu 1'1'1'1'sin1'1'cos ++++≅++++= ∞∞∞∞ ααα  ( ) ( ) 0,,''' =++ ∂ ∂ − ∂ ∂ +− ∞∞ U UU zyxwU x z v x z uU α For Upper Surface ( ) ( )       − ∂ ∂ ≅ ∂ ∂ + ∂ ∂ += ∞∞ α x z U x z v x z uUzyxw U onPerturbati Small UU U '',,' Therefore ( ) ( ) ( ) Sonyxallfor x z Uzyxw x z Uzyxw L L U U , ,,' ,,'              − ∂ ∂ ≅       − ∂ ∂ ≅ ∞ ∞ α α Section AA (enlarged) Wake region
132 Linearized Flow EquationsSOLO Boundary Conditions )continue -1) 1. Flow must be parallel at the Wing Surface. The Wing Surface S is defined by zU )x,y) – Upper Surface zL )x,y) – Lower Surface Since the Small Perturbation gives Linear Equation we can divide the Airfoil in the Camber Distribution zC )x,y) and the Thickness Distribution zt )x,y) by: ( ) ( ) ( ) Sonyxallfor x z Uyxw x z Uyxw C C t t , 0,,' 0,,'              − ∂ ∂ = ∂ ∂ ±=± ∞ ∞ α ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]   −= += ⇔    −= += 2/,,, 2/,,, ,,, ,,, yxzyxzyxz yxzyxzyxz yxzyxzyxz yxzyxzyxz LUt LUC tCL tCU Because of the Linearity the complete solution can be obtained by summing the Solutions for the following Boundary Conditions Superposition of • Angle of Attack •Camber Distribution •Thickness Distribution Section AA (enlarged) Wake region ( ) ( ) ( ) ( ) ( ) ( ) ( ) Sonyxallfor x z x z Uyxwyxwyxw x z x z Uyxwyxwyxw tC tCL tC tCU , 0,,'0,,'0,,' 0,,'0,,'0,,'              ∂ ∂ −− ∂ ∂ =−+=±       ∂ ∂ +− ∂ ∂ =++=± ∞ ∞ α α
133 Linearized Flow EquationsSOLO Boundary Conditions (continue -2) 2. Disturbances Produced by the Motion must Die Out in all portion of the Field remote from the Wing and its Wake Normally this requirement is met by making ϕ→0 when y→ ±0, z → ±0, x→-∞ Subsonic Leading Edge Flow Subsonic Trailing Edge Flow Supersonic Leading Edge Flow Supersonic Trailing Edge Flow 3. Kutta Condition at the Trailing Edge of a Steady Subsonic Flow There cannot be an infinite change in velocity at the Trailing Edge. If the Trailing Edge has a non-zero angle, the flow velocity there must be zero. At a cusped Trailing Edge, however, the velocity can be non-zero although it must still be identical above and below the airfoil. Another formulation is that the pressure must be continuous at the Trailing Edge. http://nylander.wordpress.com/category/engineering/ Kutta Condition does not apply to Supersonic Flow since the shape and location of the Trailing Edge exert no influence on the flow ahead.
134 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u '2u ∞+Uu '1∞U ( ) ( ) ' ' '2' ' '0 ''2'0 ' 222 1 33 2 11 22 22 11 ρρρ φ += += +≈+= +=Φ += ++=+= += ∞ ∞ ∞∞∞ ∞ ∞∞ ∞ ppp aaaaaa xU uu uuUUuuu uUu O Small Perturbation Assumptions:             ∇⋅+ ∂ ∂ ⋅+ ∂ Φ∂ =⋅∇ 2 2 1 2 2 2 2 u u t u u ta u    (C.M.) +(C.L.M) (C.M.) +(C.L.M) 12 1 12 1 2 2 2 2 − += − ++ ∂ ∂ ∞ ∞ γγ φ a U a u t Bernoulli 121 − ∞ − ∞∞∞       =      =      = γ γ γ γ γ ρ ρ a a T T p p Isentropic Chain Development of the Flow Equations: Flow Equations: ( ) '' 2 1 φφ ∇=+∇⋅∇=⋅∇ ∞ xUu  ( ) 1 1 2 2 1 12 12 2 2 '' ' 1 2 1 x u a U x u uU a u u a ∂ ∂ ≅+ ∂ ∂ +≅      ∇⋅ ∞ ∞ ∞ ∞   ( ) t u UuUU tt u t u u ∂ ∂ =+ ∂ ∂ ≅ ∂ ∂ = ∂ ∂ ⋅ ∞∞∞ ' 2'22 1 1 2 2  ( )  ∞ ∞ ∞ ∞ ∞ ∞ ∞∞ ∞∞ ++ ∂ ∂ =⇒ − += − + +++ ∂ ∂ ρ γ φ γγ φ p a puU t a U aaa uUU t 2 1 2 2 2 1 2 '' ' 0 12 1 1 '2 '2 2 1' ∞∞∞∞∞∞∞∞ − = − ==⇒ − = − == a a T T p p a ad T Tdd p pd ' 1 2' 1 '' 1 2 1 γ γ γ γ ρ ρ γ γ γ γ γ ρ ρ γ Isentropic Chain Bernoulli Linearized Flow Equations
135 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u '2u ∞+Uu '1∞U Small Perturbation Flow Equations: (C.M.) +(C.L.M) 52.18.00 '' 2 '1 ' 2 2 1 1 12 2 2 ≤≤≤≤      ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ ∞∞ ∞ MM tt u U x u U a φ φ ( ) '' ,,,'' 321 φ φφ ∇= = u xxxt  Bernoulli       + ∂ ∂ −= ∞∞ ' ' ' 1uU t p φ ρ ∞∞∞∞ − = − == a a T T p p ' 1 2' 1 '' γ γ γ γ ρ ρ γIsentropic Chain Linearized Flow Equations
136 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Three Dimensional Flow (Thin Wings at Small Angles of Attack) α U Up xd ud θ= L Low xd ud θ−= ∞U x z ( ) 0 ''' 1 2 2 2 2 2 2 2 = ∂ ∂ + ∂ ∂ + ∂ ∂ − ∞ zyx M φφφ (1) ( )zyx ,,'φ(2) z w y v x u ∂ ∂ = ∂ ∂ = ∂ ∂ = ' ', ' ', ' ' φφφ (3) α−=≅ + ∞∞ S xd zd U w uU w ' ' ' (4) x UuUp ∂ ∂ −=−= ∞∞∞∞ ' '' φ ρρ(5) ' 2 1 1 '' 1 2' 1 '' 2 M M M U u M a a T T p p ∞ ∞ ∞ ∞ ∞∞∞∞ − + −=−= − = − == γ γ γ γ γ γ γ ρ ρ γ(6)         ∂ ∂ + ∂∂ ∂ + ∂ ∂ =∇ ∞∞∞ 2 2 2 2 2 2 2 2 '1'2'1 ' tUxtUxM φφφ φ ( ) '' ,,,'' φ φφ ∇= = u zyxt        + ∂ ∂ −= ∞∞ ' ' ' uU t p φ ρ Steady Three Dimensional Flow Small Perturbation Flow Equations: 0 ' 2 2 = ∂ ∂ = ∂ ∂ tt 52.1 8.00 ≤≤ ≤≤ ∞ ∞ M M
137 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Three Dimensional Flow (Thin Wings at Small Angles of Attack) 0 ''' 2 2 2 2 2 2 2 = ∂ ∂ + ∂ ∂ + ∂ ∂ zyx φφφ β(1) Steady Three Dimensional Flow Subsonic Flow M∞ 1 01: 22 −= ∞Mβ ( ) ( ) ( ) ( ) α ξ α φ α ξ α φ −=−= ∂ ∂ = −=−= ∂ ∂ = ∞∞ ∞∞ LowerLower Lower UperUper Upper d zd xd zd zUU w d zd xd zd zUU w '1' '1' 3 4 3 4 Transform of Coordinates ( ) ( )       = = = =−= ∞ ςηξφφ ς η ξβξ ,,,,' 1 2 zyx z y Mx           ∂ ∂ = ∂ ∂ ⇒ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ⇒ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ⇒ ∂ ∂ = ∂ ∂ 2 2 2 2 2 2 2 2 2 2 22 2 '' '' 1'1' ς φφ ς φφ η φφ η φφ ξ φ β φ ξ φ β φ zz yy xx ( ) ( ) SMdcMydycS bb ∞∞ −=−== ∫∫ 2 0 2 0 11 ηη ( ) ( )ηcMyc 2 1 ∞−= ∞∞ − = − == 22 22 11 M AR SM b S b AR 22 1 2 1 12 ∞∞∞∞ − = ∂ ∂ − −= ∂ ∂ −= M C UMxU C p p ξ φφ Section AA (enlarged) Wake region so 02 2 2 2 2 2 = ∂ ∂ + ∂ ∂ + ∂ ∂ ς φ η φ ξ φ Laplace’s Equation like in Incompressible Flow Similarity Rules
138 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack) incpC M 2 1 1 ∞− incLC M 2 1 1 ∞− 22 1 2 1 1 ∞∞ − =      − Md Cd M inc L α α incMC M 2 1 1 ∞− inc0α 4 1 =      inc N c x incMC M 02 1 1 ∞− incLsC M 2 1 1 ∞− incsα LsC sα 0MC c xN MC 0α αd Cd L LC pCPressure Distribution Lift Lift Slope Zero-Lift Angle Pitching Moment Neutral-Point Position Zero Moment Angle of Smooth Leading-Edge Flow Lift Coefficient of Smooth Leading-Edge Flow Aerodynamic Coefficients of a Profile in Subsonic Incident Flow Based on Subsonic Similarity Rules
139 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) α U Up xd ud θ= L Low xd ud θ−= ∞U x y ( ) 0 '' 1 2 2 2 2 2 = ∂ ∂ + ∂ ∂ − ∞ yx M φφ(1) ( )yx,'φ(2) y v x u ∂ ∂ = ∂ ∂ = ' ', ' ' φφ (3) α==≅ + ∞∞ S xd yd U v vU v ' ' ' (4) x UuUp ∂ ∂ −=−= ∞∞∞∞ ' '' φ ρρ(5) ' 2 1 1 '' 1 2' 1 '' 2 M M M U u M a a T T p p ∞ ∞ ∞ ∞ ∞∞∞∞ − + −=−= − = − == γ γ γ γ γ γ γ ρ ρ γ(6)       ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ ∞∞ ∞ 2 2 1 1 12 2 2 '' 2 '1 ' tt u U x u U a φ φ ( ) '' ,,,'' 321 φ φφ ∇= = u xxxt        + ∂ ∂ −= ∞∞ ' ' ' uU t p φ ρ Steady Two Dimensional Flow Small Perturbation Flow Equations: 0 ' 2 2 = ∂ ∂ = ∂ ∂ tt 52.1 8.00 ≤≤ ≤≤ M M Linearized Flow Equations
140 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) 0 '' 2 2 2 2 2 = ∂ ∂ + ∂ ∂ yx φφ β(1) Steady Two Dimensional Flow Subsonic Flow M∞ 1 01: 22 −= ∞Mβ ( ) ( ) ( ) ( ) α φ α φ −= ∂ ∂ = −= ∂ ∂ = ∞∞ ∞∞ Lower Lower Uper Upper xd yd yUU v xd yd yUU v '1' '1' 3 4 3 4 ∞U α Transform of Coordinates ( ) ( )     = = = yx y x ,', φβηξφ βη ξ            ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ =      ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ = ∂ ∂ ∂ ∂ =      ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ = ∂ ∂ 2 2 2 2 2 2 2 2 ' , 1' 11' 111' η φφ ξ φ β φ η φη η φξ ξ φ β φ β φ ξ φ β η η φξ ξ φ β φ β φ yx yyyy xxxx so 02 2 2 2 = ∂ ∂ + ∂ ∂ η φ ξ φ Laplace’s Equation like in Incompressible Flow Linearized Flow Equations
141 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Subsonic Flow M∞ 1 (continue) The Airfoil is defined in (x,y) plane and by (ξ,η) ( ) ( )ξη gxfy AirfoilAirfoil =⇔= The above Transformation relates the Compressible Flow over an Airfoil in (x,y) Space to the Incompressible Flow in (ξ,η) over the same Airfoil. α η φφ −= ∂ ∂ = ∂ ∂ = ∞∞∞ Uper Upper xd yd UyUU v 1'1' α η φφ −= ∂ ∂ = ∂ ∂ = ∞∞∞ Lower Lower xd yd UyUU v 1'1' ( )yx,ρρ = x y η ξ ∞= ρρ Compressible Flow Incompressible Flow α η φ −= ∂ ∂ = ∞∞ Uper Upper xd fd UU v 1' α η φ −= ∂ ∂ = ∞∞ Lower Lower xd fd UU v 1' Linearized Flow Equations
142 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) 0 '1' 2 2 22 2 = ∂ ∂ − ∂ ∂ yx φ β φ (1) ( ) ( ) ( ) ( ) ( ) ( ) yxGyxGyx yxFyxFyx Lower Upper βννβφ βηηβφ +==+= −==−= :,' :,'(7) (8) Steady Two Dimensional Flow Supersonic Flow M∞ 1 01: 22 −= ∞Mβ α U Up xd yd θ= L Low xd yd θ−= ∞U x y 1 1 2 − = ∞Mxd yd 1 1 2 − −= ∞Mxd yd Flow Flow ( ) ( ) ( ) η β α d Fd Uxd yd U v Uper Upper ∞∞ −=−= 1 7 4' ( ) ( ) η φ d Fd xd d u Upper 73 ' ' ==         − − −= ∞ ∞ α Upper Upper xd yd M U u 1 ' 2 ( ) ( ) ( ) ν β α d Gd Uxd yd U v Lower Lower ∞∞ =−= 3 8 4 ' ( ) ( ) ν φ d Gd xd d u Lower 83 ' ' ==         − − = ∞ ∞ α Lower Lower xd yd M U u 1 ' 2         − − =−= ∞ ∞∞ ∞∞ α ρ ρ Upper UpperUpper xd yd M U uUp 1 '' 2 2         − − −=−= ∞ ∞∞ ∞∞ α ρ ρ Lower LowerLower xd yd M U uUp 1 '' 2 2 Linearized Flow Equations
143 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations ( )∫         −−= ∞ S S sd xd yd ppD αsin np α− Upper xd yd ∞U Upper xd yd ∞p∞p α( )∫         −−−= ∞ S S sd xd yd ppL αcos ( )∫         −−≅ ∞ S S sd xd yd ppD α ( )  Γ ∞∞∞ ∫∫ =         −−−≅ SS S sduUsd xd yd ppL 'ρα 1−α Uper xd yd 1−α Uper xd yd Kutta-Joukovsky Define: 2 2 1 : ∞∞ ∞− = U pp Cp ρ ( ) ( ) ∫∫ ∫∫         −−=         − − −≅         −=         − − ≅ ∞∞ ∞∞ ∞ ∞∞ ∞∞ ∞∞ ∞ ∞∞ S S p S S S S p S S sd xd yd CUsd xd yd U pp UL sd xd yd CUsd xd yd U pp UD αρα ρ ρ αρα ρ ρ 2 2 2 2 2 2 2 1 2 12 1 2 1 2 12 1 Linearized Flow Equations
144 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Subsonic Flow (M∞ 1) np α− Upper xd yd ∞U Upper xd yd ∞p∞p α We found: α−= ∞ xd fd U v' α ξ −= ∞ d gd U v ( ) ( )            −= −= = ∞ ∞ yxM yM x ,'1, 1 2 2 φηξφ η ξ ( ) 0 '' 1 2 2 2 2 2 = ∂ ∂ + ∂ ∂ − ∞ yx M φφ 02 2 2 2 = ∂ ∂ + ∂ ∂ η φ ξ φ y v x u ∂ ∂ = ∂ ∂ = ' ', ' ' φφ η φ ξ φ ∂ ∂ = ∂ ∂ = vu , vv M u u = − = ∞ ', 1 ' 2 '' uUp ∞∞−= ρ uUp ∞∞−= ρ xUU u U pp Cp ∂ ∂ −=−= − = ∞∞ ∞∞ ∞ '2'2 2 1 ' : 2 φ ρ ξ φ ρ ∂ ∂ −=−= − = ∞∞ ∞∞ ∞ UU u U pp Cp 22 2 1 : 2 0 2 1 ' ∞− = M p p 2 1 0 ∞− = M C C p p Compressible: Incompressible: Linearized Flow Equations
145 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Subsonic Flow (M∞ 1) np α− Upper xd yd ∞U Upper xd yd ∞p∞p α The Relation: ∫∫ ∫∫       − −=              −−≅               − − =              −≅ ∞ ∞∞ ∞∞ ∞ ∞∞ ∞∞ S p S S p S S p S S p c s dC M U c s d xd yd CUL c s d xd yd C M U c s d xd yd CUD 0 0 2 2 2 2 2 2 1 2 1 2 1 1 2 1 2 1 ρ αρ α ρ αρ 2 1 0 ∞− = M C C p p Prandtl-Glauert Compressibility Correction As earlier in 1922, Prandtl is quoted as stating that the Lift Coefficient increased according to (1-M∞ 2 )-1/2 ; he mentioned this at a Lecture at Göttingen, but without a proof. This result was mentioned 6 years later by Jacob Ackeret, again without proof. The result was finally established by H. Glauert in 1928 based on Linear Small Perturbation. Ludwig Prandtl (1875 – 1953) Hermann Glauert (1892-1934) Linearized Flow Equations Return to Critical Mach Number
SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Several improved formulas where developed: ( )[ ] 2/11/1 0 0 222 p p p CMMM C C ∞∞∞ −++− = Karman-Tsien Rule Linearized Flow Equations ( ) 0 0 2222 12/ 2 1 11 p p p CMMMM C C       −      − ++− = ∞∞∞∞ γ Laitone’s Rule Comparison of several compressibility corrections compared with experimental results for NACA 4412 Airfoil at an angle of attack of α = 1◦ . Return to Table of Content
147 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) 0 '1' 2 2 22 2 = ∂ ∂ − ∂ ∂ zx φ β φ (1) ( ) ( ) ( ) ( ) ( ) ( ) zxFzxGzx zxFzxFzx Lower Upper βννβφ βηηβφ +==+= −==−= :,' :,'(7) (8) Steady Two Dimensional Flow Supersonic Flow M∞ 1 01: 22 −= ∞Mβ α U Up xd zd θ= L Low xd zd θ−= ∞U x z 1 1 2 − = ∞Mxd zd 1 1 2 − −= ∞Mxd zd Flow Flow ( ) ( ) ( ) η β α d Fd Uxd zd U w Upper Upper ∞∞ −=−= 3 7 4' ( ) ( ) η φ d Fd xd d u Upper 73 ' ' == ( ) ( ) ( ) ν β α d Gd Uxd zd U w Lower Lower ∞∞ ==−= 3 8 4 ' ( ) ( ) ν φ d Gd xd d u Lower 83 ' ' ==         − − −= ∞ ∞ α Upper Upper xd zd M U w 1 ' 2         − − = ∞ ∞ α Lower Lower xd zd M U w 1 ' 2         − − =−==− ∞ ∞∞ ∞∞∞ α ρ ρ Upper UpperUpperUpper xd zd M U wUppp 1 '' 2 2         − − −=−==− ∞ ∞∞ ∞∞∞ α ρ ρ Lower LowerLowerLower xd zd M U wUppp 1 '' 2 2 z w x u ∂ ∂ = ∂ ∂ = ' ', ' ' φφ (3) α−=≅ + ∞∞ S xd zd U w uU w ' ' ' (4)
148 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 α U Up xd zd θ= L Low xd zd θ−= ∞U x z 1 1 2 − = ∞Mxd zd 1 1 2 − −= ∞Mxd zd Flow Flow Pressure Distribution and Lift Coefficient         −+ − = − = ∞∞∞ α ρ 2 1 2 2/ '' 22 LowerUpper LowerUpper p xd zd xd zd MU pp C 1 4 2 − = ∞M cL α ( ) ( ) ( ) ( )         −+− − − − =         +      −      − − =      +      −= ∞∞ ∞ ∫∫∫∫      00 22 1 0 1 02 1 0 1 0 00 1 2 1 4 2 1 2 LowerLowerUpperUpper LowerUpper ppL zczzcz MM c x d xd zd c x d xd zd Mc x dC c x dCc LowerUpper α α         − − = ∞ α Upper p xd zd M C Upper 1 2 2         − − −= ∞ α Lower p xd zd M C Lower 1 2 2
149 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 α U Up xd zd θ= L Low xd zd θ−= ∞U x z 1 1 2 − = ∞Mxd zd 1 1 2 − −= ∞Mxd zd Flow Flow Wave Drag Coefficient                         −+              − − =              −−              −= ∫∫∫∫ ∞ 1 0 2 1 0 2 2 1 0 1 0 1 2 c x d xd zd c x d xd zd Mc x d xd zd C c x d xd zd Cc UpperUpperLower p Upper pD LowerUpperW αααα         − − = ∞ α Upper p xd zd M C Upper 1 2 2         − − −= ∞ α Lower p xd zd M C Lower 1 2 2 ( ) ( ) ( ) ( )                             +      −+              +      − − = ∫∫∫∫ =−=− ∞ 1 0 2 00 1 0 2 1 0 2 00 1 0 2 2 22 1 2 c x d xd zd c x d xd zd c x d xd zd c x d xd zd M Lower zcz LowerUpper zcz Upper LowerLowerUpperUpper      αααα ( )22 22 2 1 2 1 4 LowerUpperD MM C W εε α + − + − = ∞∞ ∫ ∫               =               = 1 0 2 2 1 0 2 2 : : c x d xd zd c x d xd zd Lower Lower Upper Upper ε ε
150 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 Wave Drag Coefficient Flat Plate         == 0 LowerUpper xd zd xd zd Double Wedge Airfoil 1 4 2 2 − = ∞M C WD α 022 == LowerUpper εε ( ) ( ) ( )kkc t ck c t k ck c t kc LowerUpper − =       − − +== 14 1 1 14 1 4 11 2 2 2 2 22 2 2 22 εε ( ) ( )       − − =        − − = cxck ck t ckx ck t xd zd cxck ck t ckx ck t xd zd LowerUpper 12 0 2 12 0 2 ( ) ( )kk ct MM C WD −− + − = ∞∞ 1 / 1 1 1 4 2 22 2 α
151 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 Wave Drag Coefficient Biconvex Airfoil ( ) ( )222 2/2/ ctRR +−= The Biconvex Airfoil is obtained by intersection of two Circular Arcs of radius R. c – the chord t – maximum thickness at x = c/2 ( ) ( ) ( )tcttcR tc 4/4/ 222 22 ≈+= θθθθ −≈−=≈= tan,tan LowerUpper xd zd xd zd 2 2 2/2 /2 3 2 1 0 2 1 0 2 2 3 2 34 11 : Lower ct ctUpperUpper Upper c t t c dR c xd xd zd cc x d xd zd ε θ θθε δ δ ==≈≈         =              = + − + −∫∫∫ c t R c xd zd MaxUpper 2 2/ , ≈≈≈         δδ ( ) 2 2 22 2 22 22 2 3 16 1 1 1 4 1 2 1 4 c t MMMM C LowerUpperDW − + − =+ − + − = ∞∞∞∞ α εε α
02/10/15 152 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 Wave Drag Coefficient Parabolic ProfileDesignation Double Wedge Profile Contour Side View Wave Drag ( )kk −13 1 2( )kk −1 1 ( ) ( ) xckck xcxt z 212 22 −+ − ±= ( )       − ± ± = cxckx ck t ckxx ck t z 12 0 2
153 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 Wave Drag Coefficient Wave Drag at Supersonic Incident Flow versus Relative Thickness Position for Double Wedge and Parabolic Profiles k ( )kk −1 1 ( )kk −13 1 2
154 SOLO Wings in Compressible Flow Double Wedge Modified Double Wedge Biconvex τ 2 1 2 122 1 2 ' 2 ==       = c t c t c A τ 3 2 3 2332 1 2 ' 2 == +      = c t c t c t c A
155 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 Pitching Moment Coefficient The Pitching Moment Coefficient about the Leading Edge for any Thin Airfoil is given by xdx xd zd xd zd Mcc x d c x C c x d c x Cc c LowerUpper ppM LowerUpperLE ∫∫∫                 −+         − − −=            +            −=− ∞ 022 1 0 1 0 1 2 αα Thus      + − + − −= ∫∫ ∞∞ xdzxdz McM c c Lower c UpperM LE 00222 1 2 1 2α ( ) ( ) ( ) ( )[ ] xdzxdzczczcxdzzxxdzzxxdx xd zd xd zd c Lower c UpperLowerUpper c Lower cx xLower c Upper cx xUpper c LowerUpper ∫∫∫∫∫ −−−=−+−=         + = = = = 00 0 00000    Using integration by parts Symmetric Airfoil zUpper = -zLower 1 2 2 − −= ∞M cM α The distance of the Airfoil Center of Pressure aft of the Leading Edge is given by cc M M c c c c x L MN 2 1 1/4 1/2 2 2 =⋅ − − =⋅−= ∞ ∞ α α α L ∞U x Return to Table of Content
156 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1)                       −+         − − −=      −≅                       −+         − − =              −≅ ∫∫ ∫∫ ∞ ∞∞ ∞∞ ∞ ∞∞ ∞∞ c x d xd yd xd yd M U c s dCUL c x d xd yd xd yd M U c s d xd yd CUD c LowerUpperS p c LowerUpperS S p 0 2 2 2 0 22 2 2 2 2 1 2 1 2 1 2 1 2 1 2 1 αα ρ ρ αα ρ αρ α U Up xd yd θ= L Low xd yd θ−= ∞U x y 1 1 2 − = ∞Mxd yd 1 1 2 − −= ∞Mxd yd Flow Flow         − − ==− ∞ ∞∞ ∞ α ρ Upper UpperUpper xd yd M U ppp 1 ' 2 2         − − −==− ∞ ∞∞ ∞ α ρ Lower LowerLower xd yd M U ppp 1 ' 2 2 1 2 1 2 2 2 −         − −= −         − = ∞ ∞ M xd yd C M xd yd C Lower p Upper p Lower Upper α α We found: This relation was first derived by Jacob Ackeret in 1925, in a paper “Luftkrafte auf Flugel, die mit groserer als Schall-geschwingigkeit bewegt werden” (“Air Forces on Wings Moving at Supersonic Speeds”), that appeared in Zeitschhrift fur Flugtechnik und Motorluftschiffahrt, vol. 16, 1925, p.72 Jakob Ackeret (1898–1981) Linearized Flow Equations
157 AERODYNAMICS Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1) Supersonic Flow past a Symmetric Double-Edged Airfoil 1 2 3 4 SHOCK LINE SHOCK LINE SHOCK LINE SHOCK LINE EXPANSION EXPANSION Using Ackeret Theory we have ( ) ( ) ( ) ( ) 1 2 , 1 2 1 2 , 1 2 22 22 43 21 − − −= − + −= − + = − − = ∞∞ ∞∞ M C M C M C M C pp pp αδαδ αδαδ ( ) ( ) 1 4 2 1 1 4 2 1 1 4 222 1 2/1 2/1 0 3412 − = − + − =       −+      −=      = ∞∞∞ ∫∫∫ MMM c x dCC c x dCC c s dCC pppp S pX ααα ( ) ( ) ( ) ( ) 1 4 1 4 2 2 22 2 2/ 2 0 2/ 2/ 0 3412 3412 − = − ×=−+−=       −+      −=      = ∞ = ∞ −∫∫∫ MMc t CC c t CC c t c y dCC c y dCC c y dCC ct pppp ct pp ct pp S pX δδ δ XYXYD XYXYL CCCCC CCCCC +≈+= −≈−= ααα ααα α α 1 1 cossin sincos 1 4 1 4 1 4 1 4 2 2 2 21 2 2 2 1 − + − ≈ − − − ≈ ∞∞ ∞∞ MM C MM C D L δα αδα α α
158 AERODYNAMICS Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1)
159 pc− cx / 0.1 pc− cx / 0.1 pc− cx / 0.1 α δα δ ∞M δα ∞M ∞M δα = α ∞M Upper Surface Lover Surface Expansion Shock Shock Expansion Expansion Shock Expansion Shock Shock Expansion Expansion Shock Shock Shock Shock ∞M ∞M ( ) 1 2 2 − − = ∞M cp αδ ( ) 1 2 2 − + = ∞M cp αδ ( ) 1 2 2 − − = ∞M cp αδ ( ) 1 2 2 − + = ∞M cp αδ 1 4 2 − = ∞M cp α 1 4 2 − − = ∞M cp α ( ) 1 2 2 − + −= ∞M cp αδ ( ) 1 2 2 − − −= ∞M cp αδ ( ) 1 2 2 − + −= ∞M cp αδ ( ) 1 2 2 − − −= ∞M cp αδ Supersonic Flow past a Symmetric Biconvex Aerfoil AERODYNAMICS Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1) 2 2 2 2 2 2 2 4 1 1 3 16 3 16 1 4 LD L C M M c t C c tD L Md Cd − + −       =       + = − = ∞ ∞ ∞ α α α
160 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack) Aerodynamic Coefficients of a Profile in Supersonic Incident Flow Based on the Linear Theory Supersonic Rules       − − = ∞ Xd Zd M α 1 1 2 1 4 2 − = ∞M 2 1 = 0DC 0α 0MC c xN αd Cd L pCPressure Distribution Lift Slope Neutral-Point Position Zero Moment Zero-Lift Angle 0= ( ) ∫− −= ∞ 1 02 1 4 XdZ M S Wave Drag L D Cd Cd 1 4 1 2 −−= ∞M ( ) ( ) ∫               +      − −= ∞ 1 0 22 2 1 4 Xd Xd Zd Xd Zd M tS
161 SOLO • Up to point A the flow is Subsonic and it follows Prandtl- Glauert Linear Subsonic Theory. • At point B (M∞=0.81) the flow on the Upper Surface exceeds the Sound Velocity and a Shock Wave occurs. On the Lower Surface the Flow is everywhere Subsonic. • At point C (M∞=0.89) the Flow velocity exceeds the Speed of Sound also on the Lower Surface and a Shock Wave occurs. • At point D (M∞=0.98) the two Shock Waves on the Upper and Lower Surface (weaker than at point C) are located at the Trailing Edge. The Lift is larger than at point C. • At point E (M∞=1.4) pure Supersonic Flow on both Surfaces. Transonic Flow past Airfoils Lift Coefficient of an Airfoil versus Mach Number. Solid Line – Measurement. Dashed Lines - Theory AERODYNAMICS Transonic Flow over an Airfoil at various Mach Numbers; Angle of Attack α=2°. The points A,B, C, D,E correspond to the Lift Coefficients.
162 AERODYNAMICS Return to Table of Content
163 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO AERODYNAMICS
164 AERODYNAMICSSOLO Return to Table of Content Continue to Aerodynamics – Part III
165 I.H. Abbott, A.E. von Doenhoff “Theory of Wing Section”, Dover, 1949, 1959 H.W.Liepmann, A. Roshko “Elements of Gasdynamics”, John Wiley Sons, 1957 Jack Moran, “An Introduction to Theoretical and Computational Aerodynamics” Barnes W. McComick, Jr. “Aerodynamics of V/Stol Flight”, Dover, 1967, 1999 H. Ashley, M. Landhal “Aerodynamics of Wings and Bodies”, 1965 Louis Melveille Milne-Thompson “Theoretical Aerodynamics”, Dover, 1988 E.L. Houghton, P.W. Carpenter “Aerodynamics for Engineering Students”, 5th Ed. Butterworth-Heinemann, 2001 William Tyrrell Thomson “Introduction to Space Dynamics”, Dover References AERODYNAMICSSOLO
166 Holt Ashley “Engineering Analysis of Flight Vehicles”, Addison-Wesley, 1974 J.J. Bertin, M.L. Smith “Aerodynamics for Engineers”, Prentice-Hall, 1979 R.L. Blisplinghoff, H. Ashley, R.L. Halfman “Aeroelasticity”, Addison-Wesley, 1955 Barnes W. McCormick, Jr. “Aerodynamics, Aeronautics, And Flight Mechanics”, W.Z. Stepniewski “Rotary-Wing Aerodynamics”, Dover, 1984 William F. Hughes “Schaum’s Outline of Fluid Dynamics”, McGraw Hill, 1999 Theodore von Karman “Aerodynamics: Selected Topics in the Light of their Historical Development”, Prentice-Hall, 1979 L.J. Clancy “Aerodynamics”, John Wiley Sons, 1975 References (continue – 1) AERODYNAMICSSOLO
167 Frank G. Moore “Approximate Methods for Missile Aerodynamics”, AIAA, 2000 Thomas J. Mueller, Ed. “Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications”, AIAA, 2002 Richard S. Shevell “Fundamentals of Flight”, Prentice Hall, 2nd Ed., 1988 Ascher H. Shapiro “The Dynamics and Thermodynamics of Compressible Fluid Flow”, Wiley, 1953 Bernard Etkin, Lloyd Duff Reid “Dynamics of Flight: Stability and Control”, Wiley 3d Ed., 1995 H. Schlichting, K. Gersten, E. Kraus, K. Mayes “Boundary Layer Theory”, Springer Verlag, 1999 References (continue – 2) AERODYNAMICSSOLO
168 John D. Anderson “Computational Fluid Dynamics”, 1995 John D. Anderson “Fundamentals of Aeodynamics”, 2001 John D. Anderson “Introduction to Flight”, McGraw-Hill, 1978, 2004 John D. Anderson “Introduction to Flight”, 1995 John D. Anderson “A History of Aerodynamics”, 1995 John D. Anderson “Modern Compressible Flow: with Historical erspective”, McGraw-Hill, 1982 References (continue – 3) AERODYNAMICSSOLO Return to Table of Content
February 10, 2015 169 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 –2013 Stanford University 1983 – 1986 PhD AA
170 Ludwig Prandtl (1875 – 1953) University of Göttingen Max Michael Munk (1890—1986)[ also NACA Theodor Meyer (1882 - 1972 Adolph Busemann (1901 – 1986) also NACA Colorado U. Theodore von Kármán (1881 – 1963) also USA Hermann Schlichting (1907-1982) Albert Betz (1885 – 1968 ), Jakob Ackeret (1898–1981) Irmgard Flügge-Lotz (1903 - 1974) also Stanford U. Paul Richard Heinrich Blasius (1883 – 1970)
171 Hermann Glauert (1892-1934) Pierre-Henri Hugoniot (1851 – 1887) Gino Girolamo Fanno (1888 – 1962) Karl Gustaf Patrik de Laval (1845 - 1913) Aurel Boleslav Stodola (1859 -1942) Eastman Nixon Jacobs (1902 –1987) Michael Max Munk (1890 – 1986) Sir Geoffrey Ingram Taylor (1886 – 1975) ENRICO PISTOLESI (1889 - 1968) Antonio Ferri (1912 – 1975) Osborne Reynolds (1842 –1912)
172 Robert Thomas Jones (1910–1999) Gaetano Arturo Crocco (1877 – 1968) Luigi Crocco (1906-1986) MAURICE MARIE ALFRED COUETTE (1858 -1943) Hans Wolfgang Liepmann (1914-2009) Richard Edler von Mises (1883 – 1953) Louis Melville Milne-Thomson (1891-1974) William Frederick Durand (1858 – 1959) Richard T. Whitcomb (1921 – 2009) Ascher H. Shapiro (1916 — 2004)
173 John J. Bertin (1928 – 2008) Barnes W. McCormick (1926 - ) Antonio Filippone John D. Anderson, Jr. Holt Ashley )1923–2006( Milton Denman Van Dyke (1922 – 2010)
174

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Aerodynamics part ii

  • 2. 2 Table of Content AERODYNAMICS Earth Atmosphere Mathematical Notations SOLO Basic Laws in Fluid Dynamics Conservation of Mass (C.M.) Conservation of Linear Momentum (C.L.M.) Conservation of Moment-of-Momentum (C.M.M.) The First Law of Thermodynamics The Second Law of Thermodynamics and Entropy Production Constitutive Relations for Gases Newtonian Fluid Definitions – Navier–Stokes Equations State Equation Thermally Perfect Gas and Calorically Perfect Gas Boundary Conditions Flow Description Streamlines, Streaklines, and Pathlines AEROD
  • 3. 3 Table of Content (continue – 1) AERODYNAMICS SOLO Circulation Biot-Savart Formula Helmholtz Vortex Theorems 2-D Inviscid Incompressible Flow Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow Aerodynamic Forces and Moments Blasius Theorem Kutta Condition Kutta-Joukovsky Theorem Joukovsky Airfoils Theodorsen Airfoil Design Method Profile Theory by the Method of Singularities Airfoil Design AEROD
  • 4. 4 Table of Content (continue – 2) AERODYNAMICS SOLO Lifting-Line Theory Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞) 3D Lifting-Surface Theory through Vortex Lattice Method (VLM) Incompressible Potential Flow Using Panel Methods Dimensionless Equations Boundary Layer and Reynolds Number Wing Configurations Wing Parameters References AEROD
  • 5. 5 Table of Content (continue – 3) AERODYNAMICS SOLO Shock & Expansion Waves Shock Wave Definition Normal Shock Wave Oblique Shock Wave Prandtl-Meyer Expansion Waves Movement of Shocks with Increasing Mach Number Drag Variation with Mach Number Swept Wings Drag Variation Variation of Aerodynamic Efficiency with Mach Number Analytic Theory and CFD Transonic Area Rule
  • 6. 6 Table of Content (continue – 4) AERODYNAMICS SOLO Linearized Flow Equations Cylindrical Coordinates Small Perturbation Flow Applications: Nonsteady One-Dimensional Flow Applications: Two Dimensional Flow Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack) Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (d) and Lift (l) per Unit Span Computations for Subsonic Flow (M∞ < 1) Prandtl-Glauert Compressibility Correction Computations for Supersonic Flow (M∞ >1) Ackeret Compressibility Correction
  • 7. 7 SOLO Table of Contents (continue – 5) Wings of Finite Span at Supersonic Incident Flow Theoretic Solutions for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) 1. Conical Flow Method 2. Singularity-Distribution Method Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) Arrowhead Wings with Double-Wedge Profile at Zero Incidence Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwise plane [after Lawrence] AERODYNAMICS A E R O D Y N A M I C S P A R T I I I
  • 8. 8 Table of Content (continue – 6) AERODYNAMICS SOLO Aircraft Flight Control References CNα – Slope of the Normal Force Coefficient Computations of Swept Wings Comparison of Experiment and Theory for Lift-Curve Slope of Un-swept Wings Drag Coefficient A E R O D Y N A M I C S P A R T I I I
  • 9. 9 Continue from AERODYNAMICS – Part I AERODYNAMICS
  • 10. 10 SOLO - when the source moves at subsonic velocity V a, it will stay inside the family of spherical sound waves. a V M M =      = − 1 sin 1 µ Disturbances in a fluid propagate by molecular collision, at the sped of sound a, along a spherical surface centered at the disturbances source position. The source of disturbances moves with the velocity V. - when the source moves at supersonic velocity V a, it will stay outside the family of spherical sound waves. These wave fronts form a disturbance envelope given by two lines tangent to the family of spherical sound waves. Those lines are called Mach waves, and form an angle μ with the disturbance source velocity: SHOCK EXPANSION WAVES
  • 11. 11 SOLO SHOCK EXPANSION WAVES M 1 M = 1 M 1 Mach Waves
  • 12. 12 SOLO When a supersonic flow encounters a boundary the following will happen: When a flow encounters a boundary it must satisfy the boundary conditions, meaning that the flow must be parallel to the surface at the boundary. - when the supersonic flow, in order to remain parallel to the boundary surface, must “turn into itself” (see the Concave Corner example) a Oblique Shock will occur. After the shock wave the pressure, temperature and density will increase. The Mach number of the flow will decrease after the shock wave. SHOCK EXPANSION WAVES - when the supersonic flow, in order to remain parallel to the boundary surface, must “turn away from itself” (see the Convex Corner example) an Expansion wave will occur. In this case the pressure, temperature and density will decrease. The Mach number of the flow will increase after the expansion wave. Return to Table of Content
  • 13. 13 SHOCK WAVESSOLO A shock wave occurs when a supersonic flow decelerates in response to a sharp increase in pressure (supersonic compression) or when a supersonic flow encounters a sudden, compressive change in direction (the presence of an obstacle). For the flow conditions where the gas is a continuum, the shock wave is a narrow region (on the order of several molecular mean free paths thick, ~ 6 x 10-6 cm) across which is an almost instantaneous change in the values of the flow parameters. Shock Wave Definition (from John J. Bertin/ Michael L. Smith, “Aerodynamics for Engineers”, Prentice Hall, 1979, pp.254-255) When the shock wave is normal to the streamlines it is called a Normal Shock Wave, otherwise it is an Oblique Shock Wave. The difference between a shock wave and a Mach wave is that: - A Mach wave represents a surface across which some derivative of the flow variables (such as the thermodynamic properties of the fluid and the flow velocity) may be discontinuous while the variables themselves are continuous. For this reason we call it a weak shock. - A shock wave represents a surface across which the thermodynamic properties and the flow velocity are essentially discontinuous. For this reason it is called a strong shock.
  • 14. 14 Normal Shock Wave Over a Blunt Body Normal Shock Wave SHOCK WAVESSOLO Oblique Shock Wave Oblique Shock Wave Return to Table of Content
  • 15. 15 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Conservation of Mass (C.M.) ρ ρ1 1 2 2u u= η ρ ρ = =2 1 1 2 u u Conservation of Linear Momentum (C.L.M.) 2 2 221 2 11 pupu +=+ ρρ ( ) p p u p 2 1 1 2 1 1 1 1= + − ρ η H H h u h u1 2 1 1 2 2 2 21 2 1 2 = → + = + h h u h 2 1 1 2 1 2 1 2 1 1 = + −       η Conservation of Energy (C.E.) Field Equations Constitutive Relations p R T= ρIdeal Gas ( ) ( ) ( ) e e T C Tv= = 1 2 (1) Thermally Perfect Gas (2) Calorically Perfect Gas ργ γ ρρρ γ ρ pp C C C C p R C TC p eh v p vp C C v p v p CCR p TRp p 11 − = − ===+= ≡−== u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2
  • 16. 16 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, First Way h h p p p p p p u h u p 2 1 2 2 1 1 2 1 1 2 2 1 1 2 1 2 1 2 1 1 2 1 1 1 1 2 1 1 1 2 1 1 1 = − − = = = + −       = + − −       γ γ ρ γ γ ρ ρ ρ η η γ γ ρ η or ( ) p p u p u p C L M 2 1 1 2 1 1 1 2 1 1 2 1 1 1 1 1 2 1 1 1 η ρ η η γ γ ρ η = + −           = + − −       ( . . .) after further development we obtain 1 2 1 1 1 1 1 1 2 01 2 1 1 2 1 2 1 1 1 2 1 1 − −       − +           + + −           = γ γ ρ η ρ η γ γ ρ u p u p u p Solving for 1/η , we obtain 1 1 1 2 1 1 1 2 1 1 2 2 1 1 2 1 1 1 2 1 1 2 1 2 1 1 1 2 1 1 η ρ ρ ρ ρ γ γ ρ γ γ ρ γ γ = = = +           − +           − + + −           + u u u p u p u p u p u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2
  • 17. 17 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, We obtain an other relation in the following way: ( ) p p u p p p u p p p p p p p p p p p p p p p 2 1 1 2 1 1 2 2 1 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2 η γ γ ρ η ρ η η γ γ η η γ γ γ γ γ γ η γ γ γ γ γ γ γ γ − = − −       − = −         ⇒ − − = − +       ⇓ − − − −      = + − −       ⇓ = + − − − + + η ρ ρ γ γ γ γ = = = + − − + + − =2 1 1 2 2 1 2 1 2 1 1 2 1 1 1 1 1 u u p p p p p p T T or Rankine-Hugoniot Equation Rankine-Hugoniot Equation (1) William John Macquorn Rankine (1820-1872) u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2 Pierre-Henri Hugoniot (1851 – 1887)
  • 18. 18 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, η ρ ρ γ γ γ γ = = = + − − + + − =2 1 1 2 2 1 2 1 2 1 1 2 1 1 1 1 1 u u p p p p p p T T Rankine-Hugoniot Equation Rankine-Hugoniot Equation (2) p p 2 1 2 1 2 1 1 1 1 1 1 = + − − + − − γ γ ρ ρ γ γ ρ ρ T T p p p p p p p p p p p p 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 = = + + − + − − = + + − + − − = + − − + − − = + − − + − − ρ ρ γ γ γ γ γ γ γ γ γ γ ρ ρ γ γ ρ ρ ρ ρ γ γ ρ ρ γ γ ρ ρ p2 p 1 ρ 2 ρ 1 NormalShockWave Rankine-Hugoniot Isentropic γp2 p 1 ρ 2 ρ 1 ( )= u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2
  • 20. 20 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Strong Shock Wave Definition: p p u u T T p p R H R H 2 1 2 1 1 2 2 1 2 1 1 1 1 1 → ∞ ⇒ = → + − → − + − −ρ ρ γ γ γ γ Weak Shock Wave Definition: ∆ p p p p p1 2 1 1 1= − ρ ρ ρ2 1 2 1 2 1 = + = + = + ∆ ∆ ∆ p p p h h h For weak shocks u p 1 2 = ∆ ∆ρ ∆ ∆ h u ρ ρ = 1 2 1 u u u u u u2 1 2 1 1 1 1 1 1 1 1 1 1 1 = = + = + ≅ − ρ ρ ρ ρ ρ ρ ρ ρ ρ∆ ∆ ∆ (C.M.) ( ) ( )ρ ρ ρ ρ ρ 1 1 2 1 1 1 2 2 1 1 1 1 1 1u p u u p u u u p p+ = + = −       + + ∆ ∆(C.L.M.)  ordernd uuuhhuuhhuhuh 2 4 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 11 2 1 1 11 2 22 2 11       ∆ + ∆ −+∆+=      ∆ −+∆+=+=+ ρ ρ ρ ρ ρ ρ(C.E.) u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2
  • 21. 21 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Second Way h h u h u0 1 1 2 2 2 21 2 1 2 ≡ + = +Define        − − − =→+ − = − − − =→+ − = 2 10 1 12 1 1 1 0 2 20 2 22 2 2 2 0 11 2 1 1 11 2 1 1 uh p u p h uh p u p h γ γ γ γ ρργ γ γ γ γ γ ρργ γ u u h1 2 0 2 1 1 = − + γ γ Prandtl’s Relation ( )u h u u u p p u p2 0 1 2 1 1 2 2 1 1 2 1 1 2 1 1 1 1 1= − + → = = → = + − γ γ ρ ρ ρη ρ ηFrom this relation, we obtain: Prandtl’s Relation Ludwig Prandtl (1875-1953) u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2 (C.M.) (C.L.M.) ργ γ p h 1− = and use 12 22 2 11 1 2211 2 2 221 2 11 11 uu u p u p uu pupu −=−→    = +=+ ρρρρ ρρ 1221 21 0 2 1 2 1111 uuuu uu h −= − + − −      − − γ γ γ γ γ γ ( )       − −−= −− γ γ γ γ 2 1 1 1 12 21 12 0 uu uu uu h
  • 22. 22 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, (C.M.) Hugoniot Equation ρ ρ ρ ρ 1 1 2 2 2 1 1 2 u u u u= → = ( )ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ 1 1 2 1 2 2 2 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 2 1 2 1 2 2 1 1 2 2 1 2 2 2 2 2 1 2 1 2 2 1 1 2 2 1 1 2 u p u p u p p p u u u p p u p p u u u u + = + =       + → − = −       = − → → = − −       → = − −               = = (C.L.M.) ( )( ) h u h u e p p p e p p p e e p p p p p p p p e e p p h e p 1 1 2 2 2 2 1 1 1 2 1 2 1 2 2 2 2 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2 1 1 2 2 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 + = + → + + − −       = + + − −       → → − = − − −       + − = − − − + − → → − = − + = + ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρρ ρ ρ ( ) ( ) + − = + − − + − → → − = + − +               2 2 2 2 2 2 2 1 2 2 1 2 2 2 2 1 2 1 1 1 2 2 1 2 2 1 2 1 2 1 1 2 1 2 p p p p p p p p e e p p p p ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ (C.E.) e e p p 2 1 1 2 2 1 2 1 1 − = + −       ρ ρ Hugoniot Equation u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2 Pierre-Henri Hugoniot (1851 – 1887)
  • 23. 23 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Fanno’s Line for a Perfect Gas (1) ( )1 1 1 2 2ρ ρu u m A = =  ( ) frictionpupu ++=+ 2 2 221 2 112 ρρ ( )3 1 2 1 2 1 1 2 2 2 2 C T u C T u h C Tp p p+ = + = ( )4 1 1 1 2 2 2 p R T p R T= =ρ ρ ( )5 2 1 2 1 2 1 s s C T T R p p p − = −ln ln (C.M.) (C.L.M.) (C.E.) Ideal Gas ( ) p p T T u u h C T h C T p p T T h C T h C T s s C T T R T T h C T h C T p p p p p p p 2 1 4 2 1 2 1 2 1 1 1 2 3 0 1 0 2 2 1 2 1 0 1 0 2 2 1 2 1 2 1 0 1 0 2 5 =             = = − −        → = − − → − = − − − ( ) ( ) ( ) ln ln ρ ρ ρ ρ Assume that all the conditions of the model are satisfied except the moment equation (2) (a flow with friction) Using , we obtainh C Tp= s s 1 s 2 s max h 1 h 2 h 2 1 s s C h h R h h h h h h p2 1 2 1 2 1 0 1 0 2 − = − − − ln ln Fanno’s Line for a Perfect Gas This is the Adiabatic, Constant Area Flow. u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2 Gino Girolamo Fanno (1888 – 1962)
  • 24. 24 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Fanno’s Line for a Perfect Gas (2) s s 1 s 2 s max h 1 h 2 h 2 1 We have a point of maximum entropy. Let see the significance of this point ρρ dp dh dp dhdsT =→=−= 0 max Gibbs u dud dudu −=→=+ ρ ρ ρρ 0(C.M.) duudh u hd −=→=      + 0 2 2 (C.E.) Therefore )4..( 0 .).( 00 0 EC ds MC dsds ds u du d dpd d dpdp dh =      −      =      == === = ρρ ρ ρρ 0 0 = =       = ds ds d dp u ρ or ds C dT T R dp p ds C dT T R d C C dp p d dp d p dp d p R T p v p v ds ds ds ds p R T = − = = − =        → ≡ = = → = == = = = = max max 0 0 0 0 0 0 ρ ρ γ ρ ρ ρ ρ ρ γ ρ γ ρ We have: u dp d R T a speed of soundds ds = = =       = = =0 0 ρ γ u p ρ T e u p ρ T e τ 11 q 1 1 1 1 1 2 2 2 2 2 1 2
  • 25. 25 Ideal Gas NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Rayleigh’s Line for a Perfect Gas (1) ( ) A m uu  == 22111 ρρ ( )2 1 1 2 1 2 2 2 2ρ ρu p u p+ = + ( ) QhuTCuTC pp ++=+ 2 22 2 11 2 1 2 1 3 ( )4 1 1 1 2 2 2 p R T p R T= =ρ ρ ( )5 2 1 2 1 2 1 s s C T T R p p p − = −ln ln (C.M.) (C.L.M.) (C.E.) Assume that all the conditions of the model are satisfied except the energy equation (3) (a flow with heating and cooling) Let substitute in (5) , to obtainh C Tp= Rayleigh’s Line for a Perfect Gas This is the Frictionless, Constant Area Flow, with Cooling and Heating. s max s s 1 s 2 h 1 h 2 h M1 M1 Rayleigh2 1 Heating Heating Cooling  m A R T p p m A R T p p x p 1 1 1 2 2 2 1 + = + ( ) 2 1 12 1 1 1 2 12 11 1 2 12 1 2 1 lnln5 p R A m c p TR A m b h C a bbR h h Css p p  =        +=         −+−=− We want to find x p p ≡ 2 1 . Let multiply the result by x p1 x m A R T p b x m A R p c T2 1 1 2 1 1 2 1 21 2 0− +       + =       or x p p b b a T= = + −2 1 1 1 2 1 2 The solution is: John William Strutt Lord Rayleigh (1842-1919) u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2
  • 26. 26 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Rayleigh’s Line for a Perfect Gas (2) We have a point of maximum entropy. Let see the significance of this point u dud dudu −=→=+ ρ ρ ρρ 0(C.M.) (C.L.M.) A Normal Shock Wave must be on both Fanno and Rayleigh Lines, therefore the end points of a Normal Shock Wave must be on the intersection of Fanno and Rayleigh Lines u dp d R T a speed of soundds ds = = =       = = =0 0 ρ γ d p u dp du u+       = → = − 1 2 02 ρ ρ ( )→ = = − −       = dp d dp du du d u u u ρ ρ ρ ρ 2 s s 1 s 2 h 1 h 2 h M1 M1 Rayleigh Fanno 2 1 SHOCK According to the Second Law of Thermodynamics the Entropy must increase. Therefore a Normal Shock Wave from state (1) to state (2) must be such that s2 s1. (from supersonic to subsonic flow only) u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2
  • 27. 27 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Mach Number Relations (1) ( ) ( ) ( )   C M u u C L M u p u p p u p u u u C E a h u a h u a a u a a u a p . . . . . . . ρ ρ ρ ρ ρ ρ γ γ γ γ γ γ γ ρ 1 1 2 2 1 1 2 1 2 2 2 2 1 1 1 2 2 2 2 1 1 2 1 1 2 2 2 2 2 2 1 2 2 1 2 2 2 2 2 2 4 1 1 2 1 1 2 1 2 1 2 1 2 1 2 = + = +    → − = − → − + = − + → = + − − = + − −               = ∗ ∗    − = − a u a u u u1 2 1 2 2 2 2 1 γ γ Field Equations: ( ) γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ + − − − + + − = − ↓ + − + − − = − → + = − − = + ↓ ∗ ∗ ∗ ∗ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 2 1 1 2 2 2 2 1 2 1 1 2 2 2 1 2 1 2 1 2 a u u a u u u u u u u u a u u u u a u u u u a1 2 2 = ∗ u a u a M M1 2 1 21 1∗ ∗ ∗ ∗ = → = Prandtl’s Relation u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2 Ludwig Prandtl (1875-1953)
  • 28. 28 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Mach Number Relations (2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )( ) ( ) M M M M M M M M M 2 2 2 2 1 1 2 1 2 1 2 1 2 1 2 2 1 1 2 1 1 2 1 1 1 2 1 2 1 2 1 1 1 1 1 1 2 = + − − = + − − = + + − + − − = − + + / + − / / + − / + − − ∗ = ∗ ∗ ∗ γ γ γ γ γ γ γ γ γ γ γ γ γ γ or ( ) M M M M M H H A A 2 1 2 1 2 1 2 1 21 2 1 2 1 1 2 1 2 2 1 1 1 2 1 2 1 1 = + − − − = + + − + + − = = γ γ γ γ γ γ γ ( ) ( ) ρ ρ γ γ 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 1 2 = = = = = + − + = ∗ ∗ A A u u u u u u a M M M u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2
  • 29. 29 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Mach Number Relations (3) ( ) ( ) ( ) ( ) ( ) p p u p u u u a M M M M M M M 2 1 1 2 1 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 2 1 1 1 1 2 1 = + −       = + −       = + − − + +       = + / + − / − − + ρ γ ρ ρ γ γ γ γ γ γ γ or (C.L.M.) ( ) p p M2 1 1 2 1 2 1 1= + + − γ γ ( ) ( ) ( ) h h T T p p M M M a a h C T p RTp 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 1 2 1 1 1 2 1 = = = + + −       − + + = = =ρ ρ ρ γ γ γ γ ( ) ( ) ( ) s s R T T p p M M M 2 1 2 1 1 2 1 1 1 2 1 1 1 2 1 2 1 1 2 1 1 1 2 1 − =                       = + + −       − + +                 − − − − ln ln γ γ γ γ γ γ γ γ γ γ ( ) ( ) ( ) ( ) s s R M M M 2 1 1 1 2 1 2 3 2 2 1 2 41 2 2 3 1 1 2 1 1 − ≈ + − − + − + − γ γ γ γ K Shapiro p.125 u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2
  • 30. 30 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Mach Number Relations (4) ( ) p p p p p p p p M M M02 01 02 2 1 01 2 1 2 2 1 2 1 1 2 1 1 2 1 1 2 1 2 1 1= = + − + −           + + −       −γ γ γ γ γ γ ( ) ( ) 1 1 2 1 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 2 1 2 1 1 2 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 + − = + − + − − − = − − + − + −      + + + −       = + + + − γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ M M M M M M M M ( ) ( ) p p M M M02 01 1 2 1 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 1= + + + −             + + −       − − − γ γ γ γ γ γ γ γ u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2
  • 31. 31 NORMAL SHOCK WAVESSOLO Normal Shock Wave ( Adiabatic), Perfect Gas   G Q= =0 0, Mach Number Relations (5) ( ) s s R T T p p p p M M M T T 2 1 02 01 1 02 01 1 02 01 1 2 1 2 1 2 02 01 1 1 1 2 1 1 1 1 2 1 1 2 − =                       = −       = − + + −       − − + + −           − − = ln ln ln ln γ γ γ γ γ γ γ γ γ s s 1 s 2 T M1 M1 Rayleigh Fanno 2 1 SHOCK T 2 T 1 T 02 T 01= T 2 T 1=* * p 2 p 1 p 01 p 02 Mollier’s Diagram u p ρ T e u p ρ T e τ 11 q Q 1 1 1 1 1 2 2 2 2 2 1 2 John William Strutt Lord Rayleigh )1842-1919( Gino Girolamo Fanno )1888–1062( Return to Table of Content
  • 32. 32 OBLIQUE SHOCK EXPANSION WAVESSOLO →→ →→ += += twnuV twnuV 11 11 222 111   Continuity Eq.: 2211 uu ρρ = ( ) ( ) ( )21222111 ppuuuu +−−=+− ρρ Moment Eq. Tangential Component: ( ) ( ) 0222111 =+− wuwu ρρ Moment Eq. Normal Component: Energy Eq.: 22 2 2 2 2 211 2 1 2 1 1 22 u wu hu wu h ρρ         + +=        + + Continuity Eq.: 2211 uu ρρ = Moment Eq.: 21 ww = 2 222 2 111 upup ρρ +=+ Energy Eq.: 22 2 2 2 2 1 1 u h u h +=+ Summary Calorically Perfect Gas: Tch TRp p= = ρ 6 Equations with 6 Unknowns 222222 ,,,,, hwuTpρ
  • 33. 33 OBLIQUE SHOCK EXPANSION WAVESSOLO For a calorically Perfect Gas ( ) ( ) ( ) ( )[ ] ( )[ ] 2 1 1 2 1 2 2 1 2 12 2 2 1 1 2 2 1 2 1 1 2 11/2 1/2 1 1 2 1 21 1 ρ ρ γγ γ γ γ γ γ ρ ρ p p T T M M M M p p M M n n n n n n = −− −+ = − + += +− + = βsin11 MMn = ( )θβ − = sin 2 2 nM M Now we can compute ( ) ( ) ( ) ( ) ( ) ( )       ⋅+ − = − + −+ === − ⇒          = =− = θββ θβ β θβ βγ βγ ρ ρ β θβ θβ β tantan1tan tantan tan tan sin1 sin12 tan tan tan tan 22 1 22 1 2 1 1 2 12 2 2 1 1 M M u u ww w u w u
  • 34. 34 OBLIQUE SHOCK EXPANSION WAVESSOLO ( )       ++ − = 22cos 1sin cot2tan 2 1 22 1 βγ β βθ M M M,, βθ relation 12 M 12 M .5max =Mforθ β θ 1M 2M Strong Shock Weak Shock θ β We can see that θ = 0 for 1.β = 90° (Normal Shock) 2.sin β = 1/ M1
  • 35. 35 OBLIQUE SHOCK EXPANSION WAVESSOLO 1. For any given M1 there is a maximum deflection angle θmax If the physical geometry is such that θ θmax, then no solution exists for straight oblique shock wave. Instead the shock will be curved and detached.
  • 36. 36 OBLIQUE SHOCK EXPANSION WAVESSOLO 2.For any given θ θmax, there are two values of β predicted by the θ-β-M relation for a given Mach number. WEAKβ STRONGβ ( )       ++ − = 22cos 1sin cot2tan 2 1 22 1 βγ β βθ M M M,, βθ relation - the large value of β is called the strong shock solution In nature the weak shock solution usually occurs. - the small value of β is called the weak shock solution - in the strong shock solution M2 is subsonic (M2 1) - in the weak shock M2 solution is supersonic (M2 1)
  • 37. 37( )       ++ − = 22cos 1sin cot2tan 2 1 22 1 βγ β βθ M M M,, βθ relation SOLO OBLIQUE SHOCK EXPANSION WAVES θ β 4.1=γ θ maxθ θ
  • 38. 38 ( )[ ] ( )[ ] ( )θβ γγ γ β − = −− −+ = = sin 11/2 1/2 sin 2 2 2 1 2 12 2 11 n n n n n M M M M M MM SOLO θ maxθ OBLIQUE SHOCK EXPANSION WAVES Mach Number in Back of Oblique Shock M2 as a Function of the Mach Number in Front of the Shock M , for Different Values of Deflection Angle θ (γ=1.4)
  • 39. 39 ( )1 1 2 1 sin 2 1 1 2 11 − + += = n n M p p MM γ γ β SOLO θ θ OBLIQUE SHOCK EXPANSION WAVES Static Pressure Ratio P2 / P1 as a Function of M1 the Mach Number in Front of an Oblique Shock, for Different Values of Deflection Angle θ (γ=1.4)
  • 40. 40 SOLO θ θ OBLIQUE SHOCK EXPANSION WAVES Stagnation Pressure Ratio P2 0/ P1 0 as a Function of M1 the Mach Number in Front of an Oblique Shock, for Different Values of Deflection Angle θ (γ=1.4)
  • 41. 41 Hodograph Shock Polar SOLO OBLIQUE SHOCK EXPANSION WAVES
  • 42. 42 Hodograph Shock Polar SOLO -For every deflection angle θ the Hodograph gives two solutions, a strong shock (B outside the sonic circle – M21) and a weak shock (D inside the sonic circle – M11) - The line OC tangent to the Hodograph gives the maximum deflection angle θmax. For θ θmax there is no oblique shock wave. - For point E θ=0 and β=π/2, therefore a normal shock. Point A corresponds to the Mach value before the shock M1. - The Shock Angle β corresponding to a given angle θ defined by the points B and D can be found by drawing the line OH normal to line AB. β = angle HOA. OBLIQUE SHOCK EXPANSION WAVES
  • 43. 43 SOLO Family of Hodograph Shock Polars ( γ= 1.4) θ 1 ***1 2 1 ** *** 21 2 1 212 21 2 2 +−      + −       −=      c V c V c V c V c V c V c V c V x x xy γ A. H. Shapiro “The Dynamics and Thermodynamics of Compressible Flow Fluid”,pg.543 45.2 OBLIQUE SHOCK EXPANSION WAVES
  • 45. 45 SOLO OBLIQUE SHOCK EXPANSION WAVES
  • 46. 46 SOLO OBLIQUE SHOCK EXPANSION WAVES
  • 47. 47 SOLO OBLIQUE SHOCK EXPANSION WAVES
  • 48. 48 SOLO OBLIQUE SHOCK EXPANSION WAVES
  • 49. 49 SOLO OBLIQUE SHOCK EXPANSION WAVES Return to Table of Content
  • 50. 50 SOLO OBLIQUE SHOCK EXPANSION WAVES Prandtl-Meyer Expansion Waves Ludwig Prandtl (1875 – 1953) Theodor Meyer (1882 – 1972) The Expansion Fan depicted in Figure was First analysed by Prandtl in 1907 and his student Meyer in 1908. Let start with an Infinitesimal Change across a Mach Wave M ach W ave θd µ µ π − 2 θµ π d−− 2 V VdV + ( ) ( ) θµθµ µ θµπ µπ dddV VdV sinsincoscos cos 2/sin 2/sin − = −− + = + µ θµθ µθ tan / tan1 tan1 1 1 VVd dd dV Vd =⇒+≈ − ≈+ 1 1 tan 1 sin 2 1 − =⇒      = − MM µµ V Vd Md 12 −=θ 1907 - 1908
  • 51. 51 SOLO OBLIQUE SHOCK EXPANSION WAVES Prandtl-Meyer Expansion Waves (continue-1) M ach W ave θd µ µ π − 2 θµ π d−− 2 V VdV + V Vd Md 12 −=θ Integrating this equation gives ∫ −= 2 1 12 M M V Vd Mθ Using the definition of Mach Number: V = M. a a ad M Md V Vd += For a Calorically Perfect Gas 20 2 0 2 1 1 M T T a a − +==      γ MdMM a ad 1 2 2 1 1 2 1 −       − + − −= γγ M Md MV Vd 2 2 1 1 1 − + = γ ∫ − + − = 2 1 2 2 2 1 1 1 M M M Md M M γ θ
  • 52. 52 SOLO OBLIQUE SHOCK EXPANSION WAVES Prandtl-Meyer Expansion Waves (continue-2) The integral ∫ − + − = 2 1 2 2 2 1 1 1 M M M Md M M γ θ ( ) ∫ − + − = M Md M M M 2 2 2 1 1 1 γ ν is called the Prandtl-Meyer Function and is given the symbol ν. Performing the integration we obtain ( ) ( ) ( )1tan1 1 1 tan 1 1 2121 −−− + − − + = −− MMM γ γ γ γ ν Deflection Angle ν and Mach Angle μ as functions of Mach Number       = − M 1 sin 1 µ Finally ( ) ( )12 MM ννθ −= Return to Table of Content
  • 53. 53 Movement of Shocks with Increasing Mach Number Drag rises due to pressure Increase across a Shock Wave •Subsonic Flow - Local airspeed is less than sonic •Transonic Flow - Local airspeed is less than sonic at some points, greater than sonic elsewhere •Supersonic Flow - Local Airspeed is greater than sonic everywhere SOLO AERODYNAMICS
  • 54. 54 Movement of Shocks with Increasing Mach Number ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )87654321 ∞∞∞∞∞∞∞∞ MMMMMMMM SOLO AERODYNAMICS
  • 55. 55 Upper Surface Lower Surface Upper Surface Lower Surface Upper Surface Lower Surface Upper Surface Lower Surface Upper Surface Lower Surface ( c) Shock on upper surface Upper Surface Lower Surface (d ) Shocks on both surfaces Shock Movement of Shocks with Increasing Mach Number SOLO AERODYNAMICS
  • 56. 56 Movement of Shocks with Increasing Mach Number The Mach Number at witch M=1 appears on the Airfoil Upper Surface is called the Critical Mach Number for this Airfoil. The Critical Mach Number can be calculated as follows. Assuming an isentropic flow through the flow-field we have ( )1/ 2 2 2 1 1 2 1 1 − ∞ ∞             − + − + = γγ γ γ A A M M p p p∞, M∞ - Pressure and Mach Number upstream the Airfoil pA, MA- Pressure and Mach Number at a point A on the Airfoil Critical Mach Number The Pressure Coefficient Cp is computed using ( )             −             − + − + =      −= − ∞ ∞∞∞ 1 2 1 1 2 1 1 2 1 2 1/ 2 2 γγ γ γ γγ A A pA M M Mp p M C Definition of Critical Mach Number. Point A is the location of minimum pressure on the top surface of the Airfoil. SOLO AERODYNAMICS
  • 57. 57 Movement of Shocks with Increasing Mach Number Critical Mach Number This relation gives a unique relation between the upstream values of p∞, M∞ and the respective values pA, MA at a point A on the Airfoil. Assume that point A is the point of minimum pressure, therefore maximum velocity, on the Airfoil and that this maximum velocity corresponds to MA = 1. Then by definition M∞ = Mcr . ( )             −             − + − + =      −= − ∞ ∞∞∞ 1 2 1 1 2 1 1 2 1 2 1/ 2 2 γγ γ γ γγ A A pA M M Mp p M C ( )             −             − + − + = − 1 2 1 1 2 1 1 2 1/ 2 γγ γ γ γ cr cr p M M C cr 2 0 1 ∞− = M C C p p ( )             −             − + − + = − 1 2 1 1 2 1 1 2 1/ 2 γγ γ γ γ cr cr p M M C cr 2 0 1 ∞− = M C C p p To find the Mcr we need on other equation describing Cp at subsonic speeds. We can use the Prandtl-Glauert Correction or the Karman-Tsien Rule or Laiton’s Rule SOLO AERODYNAMICS
  • 58. 58 Movement of Shocks with Increasing Mach Number Critical Mach Number AirfoilThickAirfoilMediumAirfoilThin AirfoilThickAirfoilMediumAirfoilThin crcrcr ppp MMM CCC 000 The point of minimum pressure, therefore maximum velocity, does not correspond to the point of maximum thickness of the Airfoil. This is because the point of minimum pressure is defined by the specific shape of the Airfoil and not by a local property. The Critical Mach Number is a function of the thickness of the Airfoil. For the thin Airfoil the Cp0 is smaller in magnitude and because the disturbance in the Flow is smaller. Because of this the Critical Mach Number of the thin Airfoil is greater SOLO AERODYNAMICS
  • 59. 59 Movement of Shocks with Increasing Mach Number Drag Divergence Mach Number The Drag at small Mach number, due to Profile Drag with Induced Drag =0 (αi = 0) is constant (points a, b, and c) until M∞ = Mcr (point c). As the velocity increase above Mcr (point d), a finite region of supersonic flow (Weak Shock boundary)appears on the Airfoil. The Mach Number in this bubble of supersonic flow is slightly above Mach 1, typically 1.02 to 1.05. If M∞ increases more, We encounter a point, e, at which is a sudden increase in Drag. The Value of M∞ at which the sudden increase in Drag starts is defined as the Drag-divergence Mach Number, Mdrag-divergence 1. At this point Shock Waves appear on the Airfoil. The Shock Waves are dissipative phenomena extracting energy (Drag) from the kinetic energy of the Airfoil. In addition the sharp increase of the pressure across the Shock Wave create a strong adverse pressure gradient, causing the Flow to separate From the Airfoil Surface creating Drag increase. Beyond the Drag-divergence Mach Number, the Drag Coefficient becomes very large, increasing by a factor of 10 or more. As M∞ approaches unity (point f) the Flow on both the top and the SOLO AERODYNAMICS
  • 60. 60 Movement of Shocks with Increasing Mach Number Summary of Airfoil Drag The Drag of an Airfoil can be described as the sum of three contributions: wpf DDDD ++= where D – Total Drag of the Airfoil Df – Skin Friction Drag Dp – Pressure Drag due to Flow Separation Dw – Wave Drag (present only at Transonic and Supersonic Speeds; zero for Subsonic Speeds below the Drag-divergence Mach Number) In terms of the Drag Coefficients, we can write: wDpDfDD CCCC ,,, ++= The Sum: pDfD CC ,, + Profile Drag Coefficient SOLO AERODYNAMICS
  • 61. 61 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 SOLO Return to Table of Content AERODYNAMICS
  • 62. 62 AERODYNAMICS Drag Variation with Mach Number SOLO Return to Table of Content
  • 63. 63 AERODYNAMICS Swept Wings Drag Variation Adolf Busemann and Alfred Betz, discovered around 1930 that Drag at Transonic and Supersonic Speeds could be reduced using Swept Back Wings. Assume Mcr for Wing = 0.7 Airfoil Section with Mcr = 0.7 Airfoil Section with Mcr = 0.7 Airfoil ³ sees´ only this component of velocity Mcr for swept wing Adolph Busemann (1901 – 1986) also NACA Colorado U. Albert Betz (1885 – 1968 ), Λ = cos _ cr sweptcr M M From the Figure we see that if Λ is the Swept Angle, than Supersonic L.E. Subsonic L.E. Mach Cone For Supersonic Flow M∞ 1 •If the Leading Edge of Swept Wing is outside the Mach Cone, the component of the Mach Number normal to the Leading Edge is Supersonic. As a result a Strong Oblique Shock Wave will be created on the Wing. •If the Leading Edge of Swept Wing is inside the Mach Cone, the component of the Mach Number normal to the Leading Edge is Subsonic. As a result a Weaker Oblique Shock Wave will be created on the Wing and a Lower Drag will result. SOLO
  • 64. 64 SOLO Wings in Compressible Flow 64 Swept Wings The Swept Wing Theory was first presented by Adolf Busemann at the Fifth Volta Conference in Roma 1935. Busemann made use of so called “Independence Principle”: “The air forces on a sufficient long, narrow Wing Panel are independent of the component of the flight velocity in the direction of the Wing Leading Edge (disregarding friction). The air forces the depend only on the reduced component velocity perpendicular to the Wing Leading Edge” Adolph Busemann (1901 – 1986). The Wing angles relative to Flow Direction are: α – Angle of Attack Λ – Swept Angle The Flow Mach components are: forcesairaffectingnotELtoparallelM forcesairaffecting PlaneWingtheinELtonormalM PlaneWingtonormalM ..sincos ..coscos sin Λ       Λ ∞ ∞ ∞ α α α We have: ( ) ( )[ ] ( ) Λ = Λ = Λ=       Λ =      Λ = Λ−=Λ+= − ∞ ∞− ∞∞∞∞ coscos : cos: cos tan tan coscos sin tan: cossin1coscossin: 11 2/1222/122 τ τ α α α α ααα c t cc M M MMMM e e e e Section A-A Section B-B
  • 65. 65 SOLO Wings in Compressible Flow 65 Swept Wings Section A-A Section B-B ( ) bcM L CL 2 2/ ∞∞ = ργ The Total Lift is: ( ) ( )[ ] ( ) Λ = Λ = Λ=       Λ =      Λ = Λ−=Λ+= − ∞ ∞− ∞∞∞∞ coscos : cos: cos tan tan coscos sin tan: cossin1coscossin: 11 2/1222/122 τ τ α α α α ααα c t cc M M MMMM e e e e Therefore: ( ) ( )α222 cossin1/ Λ−== ∞∞ eLeeLL CMMCC ( ) ( ) ( )ΛΛ == ∞∞∞∞ cos/cos2/2/ 22 bcM L bcM L C eeee eL ργργ and: The Friction Drag is ignored the Tangential Component of Velocity does not contribute to the Drag and the Pressure Drag is normal to the Leading Edge. If D is the Total Pressure Drag the component in the M∞ direction is only D cosΛ. ( ) ( ) ( ) ( )ΛΛ = Λ = ∞∞∞∞ cos/cos2/ ; 2/ cos 22 bcM D C bcM D C e DD ργργ or: ( ) ( )α222 cossin1cos/ Λ−Λ== ∞∞ eDeeDD CMMCC
  • 66. 66 SOLO Wings in Compressible Flow 66 Swept Wings Oblique Wing aircraft, AD-1 was built and flown by NASA.. Oblique Wing concept was developed in the USA by R.T. Jones. Robert Thomas Jones (1910–1999) Oblique Wing Flight Demonstration by the AD-1.
  • 68. 68 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 AERODYNAMICS Swept Wings Drag Variation SOLO
  • 70. 70 AERODYNAMICS Swept Wings Drag Variation Comparison of the Transonic Drag Polar for an Unswept Wing with that for a Swept Wing (data from Schlichting) SOLO
  • 71. 71 SOLO Wings in Compressible Flow Profile Drag Coefficients versus Mach Number for an Un-swept and a Swept-back Wing (φ=45°), t/c=0.12, AR=4 Swept Wings
  • 73. 73 SOLO Return to Table of Content Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
  • 74. 74 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
  • 75. 75 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
  • 76. 76 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
  • 77. 77 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO Return to Table of Content
  • 78. 78 SOLO Return to Table of Content Ray Whitford, “Design for Air Combat”
  • 79. 79 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO Return to Table of Content
  • 80. 80 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane Richard T. Whitcomb (1921 – 2009) SOLO
  • 81. 81 German aerodynamicist named Dr. Adolf Busemann, who had come to work at Langley after World War II, gave a technical symposium on transonic airflows. In a vivid analogy, Busemann described the stream tubes of air flowing over an aircraft at transonic speeds as pipes, meaning that their diameter remained constant. At subsonic speeds, by comparison, the stream tubes of air flowing over a surface would change shape, become narrower as their speed increased. This phenomenon was the converse, in a sense, of a well-known aerodynamic principle called Bernoulli's theorem, which stated that as the area of an airflow was made narrower, the speed of the air would increase. This principle was behind the design of venturis,9 as well as the configuration of Langley's wind tunnels, which were necked down in the test sections to generate higher speeds.10 But at the speed of sound, Busemarm explained, Bernoulli's theorem did not apply. The size of the stream tubes remained constant. In working with this kind of flow, therefore, the Langley engineers had to look at themselves as pipefitters. Busemann's pipefitting metaphor caught the attention of Whitcomb, who was in the symposium audience. Soon after that Whitcomb was, quite literally, sitting with his feet up on his desk one day, contemplating the unusual shock waves he had encountered in the transonic wind tunnel. He thought of Busemann's analogy of pipes flowing over a wing-body shape and suddenly, as he described it later, a light went on. Richard T. Whitcomb (1921 – 2009) Adolph Busemann (1901 – 1986) also NACA Colorado U. Origin of Transonic Area Rule http://history.nasa.gov/SP-4219/Chapter5.html SOLO
  • 82. 82 Richard T. Whitcomb (1921 – 2009) Adolph Busemann (1901 – 1986) also NACA Colorado U. Origin of Transonic Area Rule http://history.nasa.gov/SP-4219/Chapter5.html In practical terms, the area rule concept meant that something had to be done in order to compensate for the dramatic increase in cross- sectional area where the wing joined the fuselage. The simplest solution was to indent the fuselage in that area, creating what engineers of the time described as a Coke bottle or Marilyn Monroe shaped design. The indentation would need to be greatest at the point where the wing was the thickest, and could be gradually reduced as the wing became thinner toward its trailing edge. If narrowing the fuselage was impossible, as was the case in several designs that applied the area rule concept, the fuselage behind or in front of the wing needed to be expanded to make the change in crosssectional area from the nose of the aircraft to its tail less dramatic. Throughout the first quarter of 1952, Whitcomb conducted a series of experiments using various area-rule based wing-body configurations in Langley's 8-Foot High-Speed Tunnel. As he expected, indenting the fuselage in the area of the wing did, indeed, significantly reduce the amount of drag at transonic speeds. In fact, Whitcomb found that indenting the body reduced the drag-rise increments associated with the unswept and delta wings by approximately 60 percent near the speed of sound, virtually eliminating the drag rise created by having to put wings on a smooth, cylindrical shaped body. http://www.youtube.com/watch?v=xZWBVgL8I54 http://www.youtube.com/watch?v=Cn0lSoreB1g SOLO
  • 83. 83 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
  • 84. 84 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
  • 85. 85 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
  • 86. 86 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
  • 87. 87 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO
  • 89. 89 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 AERODYNAMICSSOLO Return to Table of Content
  • 91. 91 Nguyen X. Vinh, “Flight Mechanics of High Performance Aircraft”, Cambridge University,1993 AERODYNAMICSSOLO
  • 92. 92 Examples of airfoils in nature and within various vehicles Lift and Drag curves for a typical airfoil SOLO
  • 95. 95 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 AERODYNAMICSSOLO
  • 96. 96 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 AERODYNAMICSSOLO Return to Table of Content
  • 97. 97 Flow regimes for a slender bodySOLO
  • 98. 98 )A)- Flow field in wing-tail plane, influence of angle of attackSOLO
  • 100. 100 ))B)- Flow field in wing-tail plane, influence ofB)- Flow field in wing-tail plane, influence of control deflectioncontrol deflection δδ for pitchfor pitch SOLO
  • 102. 102 CHARACTERISTICSCHARACTERISTICS Summary of aerodynamic designSummary of aerodynamic design SOLO
  • 103. 103 ))C)- Flow field in wing-tail plane, influence ofC)- Flow field in wing-tail plane, influence of control deflectioncontrol deflection ξξ for rollfor roll SOLO
  • 104. 104 Types of missile roll control skid-to-turn, bank-to-turnTypes of missile roll control skid-to-turn, bank-to-turnSOLO Return to Table of Content
  • 105. 105 Density Profile Mach 1.2, Color Contours Modified to see Detail on Shock Waves More Fun With CFD – RM-10SOLO
  • 106. 106 Density Profiles, Mach 2.41, simulated altitude of 11,000 ft )Re=76.4x106 ) More Fun With CFD – RM-10SOLO
  • 107. 107 Density Profiles, Mach 2.41 – color contours modified to see detail in shock waves More Fun With CFD – RM-10SOLO
  • 108. 108 Density Profiles, Mach 1.62 – rotated, with plot to show distribution around fins More Fun With CFD – RM-10SOLO
  • 109. 109 The Effect of Leading Edge Slat, Flap, and Trailing Edge Flap Upon Angle of Attack of Basic Wing Darrol Stinton “ The Design of the Aircraft”SOLO
  • 110. 110High Angles of Attack Flows )Development of a High Resolution CFD) SOLO
  • 111. 111High Angles of Attack Flows )Development of a High Resolution CFD) SOLO
  • 112. 112 Three-Element Airfoil Pressure Coefficient and Streamlines at Maximum Lift M=0.2 )Re=4.1x106 ) SALSA Computation AERODYNAMICSSOLO
  • 113. 113Inviscid Transonic Flow Solution Over a 2-D Airfoil at M=0.75 )Re=1000) AERODYNAMICSSOLO
  • 114. 114Inviscid Supersonic Flow Solution Over a 2-D Airfoil at M=1.50 )Re=1000) AERODYNAMICSSOLO
  • 116. 116 Linearized Flow Equations 1. Irrotational Flow SOLO Assumptions 2. Homentropic 3. Thin bodies ( )0  =×∇ u       = ∂ ∂ =∇ 00..;. t s seieverywhereconsts This implies also inviscid flow ( )~τ = 0 Changes in flow velocities due to body presence are small were - flow velocity as a function of position and time - flow entropy as a function of position and time ( )tzyxu ,,,  ( )tzyxs ,,,
  • 117. 117 SOLO )C.L.M) For an inviscid flow conservation of linear momentum gives:( )~τ = 0 Assume that body forces are conservative and stationary were - flow pressure as a function of position and time( )tzyxp ,,, - flow density as a function of position and time( )tzyx ,,,ρ ( ) Gpuuu t u uu t u tD uD     ρ ∂ ∂ ρ ∂ ∂ ρρ +−∇=      ×∇×−      ∇+=      ∇⋅+= 2 2 1 or ( ) G p uuu t u   + ∇ −=×∇×−      ∇+ ∂ ∂ ρ 2 2 1 Euler’s Equation 0 = ∂ Ψ∂ Ψ−∇= t G  - Body forces as a function of position( )zyxG ,,  Leonhard Euler 1707-1783 Linearized Flow Equations
  • 118. 118 SOLO Let integrate the Euler’s Equation between two points )1) and )2) ( ) ( ) ( ) ∫∫∫∫∫∫ ⋅Ψ∇+ ⋅∇ +×∇⋅×−⋅      ∇+⋅ ∂ ∂ =⋅      Ψ∇+ ∇ +×∇×−      ∇+ ∂ ∂ = 2 1 2 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 0 rd rdp uurdrdurdu t rd p uuuu t    υρ We can chose the path of integration as follows: - along a streamline ) and are collinear; i.e.: )rd  u  0  =×urd - along any path, if the flow is irrotational ( )0  =×∇ u to obtain: ( ) ( ) 0 2 1 =×∇⋅×∫ uurd  Assuming that the flow is irrotational we can define a potential , such that: ( )0  =×∇ u ( )tr ,  Φ Φ∇=u  Let use the identity to obtain: ( ) rdFtrFd constt  ⋅∇== , ( ) 2 1 2 2 1 2 2 1 2 1 0         Ψ+++ ∂ Φ∂ =      Ψ∇++      +Φ ∂ ∂ = ∫∫ ∞ p p pd u t pd udd t ρρ Bernoulli’s Equation for Irrotational and Inviscid Flow Daniel Bernoulli 1700-1782 Linearized Flow Equations
  • 119. 119 SOLO For an isentropic ideal gas we have 2 2 11 a ad T Tdd p pd − = − == γ γ γ γ ρ ρ γ where ρ γ γ ρρ p TR d pdp a s === ∂ ∂ =2 is the square of the speed of sound In this case 2 2 2 1 1 1 2 ad a adppd RTa RTp − = − = = = γργ γ ρ γ ρ and [ ]222 1 1 1 1 2 2 ∞− − = − = ∫∫ ∞∞ aaad pd a a p p γγρ Using the Bernoulli’s Equation we obtain ( ) ( ) ( ) ( )      Ψ−Ψ+−+ ∂ Φ∂ −−=−=− ∞∞∞ ∫ ∞ 2222 2 1 11 Uu t dp aa p p γ ρ γ ( ) 2 1 2 2 1 2 2 1 2 1 0         Ψ+++ ∂ Φ∂ =      Ψ∇++      +Φ ∂ ∂ = ∫∫ ∞ p p pd u t pd udd t ρρ Bernoulli’s Equation for Irrotational and Inviscid Flow Linearized Flow Equations
  • 120. 120 SOLO Let use the conservation of mass )C.M.) equation )C.M.) 0=⋅∇+ u tD D  ρ ρ or tD D u ρ ρ 1 −=⋅∇  Let go back to Bernoulli’s Equation ( ) ( )      Ψ−Ψ+−+ ∂ Φ∂ −= ∞∞∫ ∞ 22 2 1 Uu t pd p p ρ and use the Leibnitz rule of differentiation: ( ) ( )uxFdxuxF xd d x x ,, 0 =∫ to obtain ρρ 1 =∫ ∞ p p pd pd d Now we can compute tD Da tD D d pd tD pD tD pDpd pd dpd tD D p p p p ρ ρ ρ ρρρρρ 2 11 ===         = ∫∫ ∞∞ Therefore ( ) ( )      Ψ−Ψ+−+ ∂ Φ∂ =−=−=⋅∇ ∞∞∫ ∞ 22 22 2 1111 Uu ttD D a pd tD D atD D u p p ρ ρ ρ  Since ( )[ ] 0=Ψ−Ψ= ∞∞ tD D u tD D we have             ∇⋅+ ∂ ∂ ⋅+ ∂ Φ∂ =            ∇⋅+ ∂ Φ∂ ∇⋅+ ∂ ∂ ⋅+ ∂ Φ∂ = =      + ∂ Φ∂       ∇⋅+ ∂ ∂ =      + ∂ Φ∂ =⋅∇ Φ∇= 2 2 1 2 1 2 11 2 11 2 2 2 2 2 2 2 2 2 2 2 2 u u t u u ta u u t u t u u ta u t u ta u ttD D a u u        GOTTFRIED WILHELM von LEIBNIZ 1646-1716 Linearized Flow Equations
  • 121. 121 SOLO             ∇⋅+ ∂ ∂ ⋅+ ∂ Φ∂ =            ∇⋅+ ∂ Φ∂ ∇⋅+ ∂ ∂ ⋅+ ∂ Φ∂ = =      + ∂ Φ∂       ∇⋅+ ∂ ∂ =      + ∂ Φ∂ =⋅∇ Φ∇= 2 2 1 2 1 2 11 2 11 2 2 2 2 2 2 2 2 2 2 2 2 u u t u u ta u u t u t u u ta u t u ta u ttD D a u u        Let substitute Φ∇=u              Φ∇⋅Φ∇∇⋅Φ∇+Φ∇ ∂ ∂ ⋅Φ∇+ ∂ Φ∂ =Φ∇⋅∇ 2 1 2 1 2 2 2 tta ( ) ( ) ( )      Ψ−Ψ+−Φ∇⋅Φ∇+ ∂ Φ∂ −−= ∞∞∞ 222 2 1 1 U t aa γ Special cases 0≈Φ∇⋅∇ Laplace’s equation ∞∞ Ua )subsonic flow) we can approximate the first equation by 1 2 ( ) ( ) 2 2 t uu t uuu ∂ Φ∂ ⋅ ∂ ∂ +⋅∇⋅  we can approximate the first equation by 0 1 2 2 2 = ∂ Φ∂ −Φ∇⋅∇ ta Wave equation Pierre-Simon Laplace 1749-1827 Linearized Flow Equations
  • 122. 122 SOLO Note The equation       + ∂ Φ∂       ∇⋅+ ∂ ∂ =⋅∇ 2 2 2 11 u t u ta u  can be written as Φ=      Φ∇⋅+ ∂ Φ∂       ∇⋅+ ∂ ∂ =      + ∂ Φ∂       ∇⋅+ ∂ ∂ =Φ∇ 2 2 22 2 2 2 11 2 11 tD D a u t u ta u t u ta c c  where the subscript c on and on is intended to indicate that the velocity is treated as a constant during the second application of the operators and . cu  2 2 tD Dc t∂∂/ ( )∇⋅u  This equation is similar to a wave equation. End Note Linearized Flow Equations
  • 123. 123 SOLO Let compute the local pressure coefficient: 2 2 1 : ∞∞ ∞− = U pp C p ρ We have:           −        =           −        =           −      =      −= − ∞∞ =− ∞ ∞ ∞ = − ∞ ∞ ∞       = = ∞ ∞ ∞ ∞ ∞∞∞ − ∞∞ ∞∞∞ 1 2 1 2 1 1 2 1 2 1 2 2 2 /1 2 2 2 2 1 22 2 1 γ γ γ γ γ γ γ ρ γ γ ρ γ γ a a Ma a a U T T U TR p p U p C aUMTRa T T p p TRp p Let use the equation ( ) ( ) ( )      Ψ−Ψ+−Φ∇⋅Φ∇+ ∂ Φ∂ −−= ∞∞∞ 222 2 1 1 U t aa γ to compute ( ) ( ) ( )      Ψ−Ψ+−Φ∇⋅Φ∇+ ∂ Φ∂− −= ∞∞ ∞∞ 2 22 2 2 11 1 U taa a γ Finally we obtain: ( ) ( ) ( )           −             Ψ−Ψ+−Φ∇⋅Φ∇+ ∂ Φ∂− −= − ∞∞ ∞∞ 1 2 11 1 2 1 2 22 γ γ γ γ U taM Cp Linearized Flow Equations
  • 124. 124 SOLO Assuming a stationary flow and neglecting the body forces :      = ∂ ∂ 0 t ( )0=Ψ             Φ∇⋅Φ∇∇⋅Φ∇=Φ∇⋅∇ 2 11 2 a ( ) ( )222 2 1 ∞∞ −Φ∇⋅Φ∇ − −= Uaa γ ( ) ( )           −       −Φ∇⋅Φ∇ − −= − ∞ ∞∞ 1 2 1 1 2 1 2 22 γ γ γ γ U aM Cp Φ∇=u  Linearized Flow Equations
  • 125. 125 SOLO 1 0 332211 323121 =⋅=⋅=⋅ =⋅=⋅=⋅ →→→→→→ →→→→→→ eeeeee eeeeee General Coordinates ( )321 ,, uuu →→→ ∂ Φ∂ + ∂ Φ∂ + ∂ Φ∂ =Φ∇ 3 33 2 22 1 11 111 e uh e uh e uh ( ) ( ) ( )      ∂ ∂ + ∂ ∂ + ∂ ∂ =       ++⋅∇=⋅∇ →→→ 321 3 213 2 132 1321 332211 1 Ahh u Ahh u Ahh uhhh eAeAeAA  Using we obtainΦ∇=:A              ∂ Φ∂ ∂ ∂ +      ∂ Φ∂ ∂ ∂ +      ∂ Φ∂ ∂ ∂ = =Φ∇⋅∇=Φ∇ 33 21 322 13 211 32 1321 2 1 uh hh uuh hh uuh hh uhhh where We have for ( ) ( )321321 ,,,,, uuuAuuu  Φ Linearized Flow Equations
  • 126. 126 SOLO zzyyxx Φ+Φ+Φ=Φ∇=Φ∇⋅∇ 2       Φ+Φ+Φ∇⋅      Φ+Φ+Φ=      Φ∇⋅Φ∇∇⋅Φ∇ →→→ 222 2 1 2 1 2 1 111 2 1 zyxzyx zyx ( ) ( ) ( )=ΦΦ+ΦΦ+ΦΦΦ+ ΦΦ+ΦΦ+ΦΦΦ+ΦΦ+ΦΦ+ΦΦΦ= zzzyzyxzxz yzzyyyxyxyxzzxyyxxxx yzzyxzzxxyyxzzzyyyxxx ΦΦΦ+ΦΦΦ+ΦΦΦ+ΦΦ+ΦΦ+ΦΦ= 222 22             Φ∇⋅Φ∇∇⋅Φ∇=Φ∇⋅∇ 2 11 2 a ( ) ( )222 2 1 ∞∞ −Φ∇⋅Φ∇ − −= Uaa γ ( ) 0 12 2 22111 222 222 2 2 2 2 2 =Φ−ΦΦ+ΦΦ+ΦΦ−Φ ΦΦ − Φ ΦΦ −Φ ΦΦ −Φ        Φ −+Φ         Φ −+Φ        Φ − ttztzytyxtxyz zy xz zx xy yx zz z yy y xx x aaa aaaaa ( ) ( ) ( )      Ψ−Ψ+−Φ+Φ+Φ+ ∂ Φ∂ −−= ∞∞∞ 222222 2 1 1 U t aa zyxγ We finally obtain Cartesian Coordinates ( )zuyuxu === 321 ,, Linearized Flow Equations Return to Table of Content
  • 127. 127 SOLO Cylindrical Coordinates ( )θ=== 321 ,, uruxu →→→→→→ ++=++= zryrxxzzyyxxR 1sin1cos1111 θθ  →→→→→ +−= ∂ ∂ += ∂ ∂ = ∂ ∂ zryr R zy r R x x R 1cos1sin1sin1cos1 θθ θ θθ  r R h r R h x R h = ∂ ∂ == ∂ ∂ == ∂ ∂ = θ  :1:1: 321 →→→→ →→→→→→ =+−= ∂ ∂ ∂ ∂ = =+= ∂ ∂ ∂ ∂ == ∂ ∂ ∂ ∂ = θθθ θ θ θθ 11cos1sin: 11sin1cos:1: 2 21 zy R R e rzy r R r R ex x R x R e       1 0 332211 323121 =⋅=⋅=⋅ =⋅=⋅=⋅ →→→→→→ →→→→→→ eeeeee eeeeee We have Linearized Flow Equations
  • 128. 128 SOLO Cylindrical Coordinates )continue – 1) ( )θ=== 321 ,, uruxu →→→→→→ Φ+Φ+Φ= ∂ Φ∂ + ∂ Φ∂ + ∂ Φ∂ =Φ∇ 321321 11 e r eee r e r e x rx θ θ 2 2 22 1 θΦ+Φ+Φ=Φ∇⋅Φ∇ r rx → →→       ΦΦ+ΦΦ+ΦΦ+       Φ−ΦΦ+ΦΦ+ΦΦ+      ΦΦ+ΦΦ+ΦΦ=       Φ+Φ+Φ∇=      Φ∇⋅Φ∇∇ 322 2 2 3212 2 2 22 11 111 1 2 1 2 1 e rr e rr e r r rrxx rrrrxrxxrxrxxx rx θθθθθ θθθθθ θ θθ θ θθ Φ+Φ+Φ+Φ= ∂ Φ∂ + ∂ Φ∂ + ∂ Φ∂ + ∂ Φ∂ =             ∂ Φ∂ ∂ ∂ +      ∂ Φ∂ ∂ ∂ +      ∂ Φ∂ ∂ ∂ = =Φ∇⋅∇=Φ∇ 22 2 22 2 2 2 2 1111 11 rrrrrrx rr r rx r xr rrrxx Linearized Flow Equations
  • 129. 129 SOLO Cylindrical Coordinates )continue – 2) ( )θ=== 321 ,, uruxu Then equation             Φ∇⋅Φ∇∇⋅Φ∇+Φ∇ ∂ ∂ ⋅Φ∇+ ∂ Φ∂ =Φ∇⋅∇ 2 1 2 1 2 2 2 tta becomes ( ){             ΦΦ+ΦΦ+ΦΦ+       Φ−ΦΦ+ΦΦ+ΦΦ+          ΦΦ+ΦΦ+ΦΦ      Φ+Φ+Φ+ ΦΦ+ΦΦ+ΦΦ+Φ=Φ+Φ+Φ+Φ → → →→→→ 322 2 2 32 12321 22 11 11 11 2 111 e rr e rr e r e r ee arr rrxx rrrrxrx xrxrxxxrx ztzytyxtxttrrrxx θθθθθ θθθ θθθ θθ or ( ) 0 2 112 / 1 1/ 1 1 11 22 222 2 22 2 22 22 2 2 2 =ΦΦ+ΦΦ+ΦΦ− Φ −       ΦΦΦ+ΦΦΦ+ΦΦΦ−         Φ +Φ+Φ        Φ −+Φ        Φ −+Φ        Φ − ztzytyxtx tt rrxxrxrx rrr r xx x aa rra a r ra r raa θθθθ θ θθ θ Linearized Flow Equations
  • 130. 130 SOLO Cylindrical Coordinates )continue – 3) ( )θ=== 321 ,, uruxu becomes ( ) ( ) ( )      Ψ−Ψ+−Φ∇⋅Φ∇+ ∂ Φ∂ −−= ∞∞∞ 222 2 1 1 u t aa γ In cylindrical coordinates, equation ( ) ( )      Ψ−Ψ+      −Φ+Φ+Φ+Φ−−= ∞∞∞ 22 2 2222 1 2 1 1 U r aa rxt θγ Assuming a stationary flow and neglecting body forces      = ∂ ∂ 0 t ( )0=Ψ 0 112 / 1 1/ 1 1 11 222 2 22 2 22 22 2 2 2 =      ΦΦΦ+ΦΦΦ+ΦΦΦ−         Φ +Φ+Φ        Φ −+Φ        Φ −+Φ        Φ − rrxxrxrx rrr r xx x rra a r ra r raa θθθθ θ θθ θ ( )       −Φ+Φ+Φ − −= ∞∞ 22 2 2222 1 2 1 U r aa rx θ γ Linearized Flow Equations Return to Table of Content
  • 131. 131 Linearized Flow EquationsSOLO Boundary Conditions 1. Since the Small Perturbations are not considering the Boundary Layer the Flow must be parallel at the Wing Surface. The Wing Surface S is defined by zU )x,y) – Upper Surface zL )x,y) – Lower Surface 0  =⋅ S un n  - Normal at the Wing Surface 22 1/111       ∂ ∂ +      ∂ ∂ +      + ∂ ∂ − ∂ ∂ −= y z x z zy y z x x z n UUUU U  ( ) ( ) ( ) ( ) zwUyvxuUzwUyvxuUu 1'1'1'1'sin1'1'cos ++++≅++++= ∞∞∞∞ ααα  ( ) ( ) 0,,''' =++ ∂ ∂ − ∂ ∂ +− ∞∞ U UU zyxwU x z v x z uU α For Upper Surface ( ) ( )       − ∂ ∂ ≅ ∂ ∂ + ∂ ∂ += ∞∞ α x z U x z v x z uUzyxw U onPerturbati Small UU U '',,' Therefore ( ) ( ) ( ) Sonyxallfor x z Uzyxw x z Uzyxw L L U U , ,,' ,,'              − ∂ ∂ ≅       − ∂ ∂ ≅ ∞ ∞ α α Section AA (enlarged) Wake region
  • 132. 132 Linearized Flow EquationsSOLO Boundary Conditions )continue -1) 1. Flow must be parallel at the Wing Surface. The Wing Surface S is defined by zU )x,y) – Upper Surface zL )x,y) – Lower Surface Since the Small Perturbation gives Linear Equation we can divide the Airfoil in the Camber Distribution zC )x,y) and the Thickness Distribution zt )x,y) by: ( ) ( ) ( ) Sonyxallfor x z Uyxw x z Uyxw C C t t , 0,,' 0,,'              − ∂ ∂ = ∂ ∂ ±=± ∞ ∞ α ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]   −= += ⇔    −= += 2/,,, 2/,,, ,,, ,,, yxzyxzyxz yxzyxzyxz yxzyxzyxz yxzyxzyxz LUt LUC tCL tCU Because of the Linearity the complete solution can be obtained by summing the Solutions for the following Boundary Conditions Superposition of • Angle of Attack •Camber Distribution •Thickness Distribution Section AA (enlarged) Wake region ( ) ( ) ( ) ( ) ( ) ( ) ( ) Sonyxallfor x z x z Uyxwyxwyxw x z x z Uyxwyxwyxw tC tCL tC tCU , 0,,'0,,'0,,' 0,,'0,,'0,,'              ∂ ∂ −− ∂ ∂ =−+=±       ∂ ∂ +− ∂ ∂ =++=± ∞ ∞ α α
  • 133. 133 Linearized Flow EquationsSOLO Boundary Conditions (continue -2) 2. Disturbances Produced by the Motion must Die Out in all portion of the Field remote from the Wing and its Wake Normally this requirement is met by making ϕ→0 when y→ ±0, z → ±0, x→-∞ Subsonic Leading Edge Flow Subsonic Trailing Edge Flow Supersonic Leading Edge Flow Supersonic Trailing Edge Flow 3. Kutta Condition at the Trailing Edge of a Steady Subsonic Flow There cannot be an infinite change in velocity at the Trailing Edge. If the Trailing Edge has a non-zero angle, the flow velocity there must be zero. At a cusped Trailing Edge, however, the velocity can be non-zero although it must still be identical above and below the airfoil. Another formulation is that the pressure must be continuous at the Trailing Edge. http://nylander.wordpress.com/category/engineering/ Kutta Condition does not apply to Supersonic Flow since the shape and location of the Trailing Edge exert no influence on the flow ahead.
  • 134. 134 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u '2u ∞+Uu '1∞U ( ) ( ) ' ' '2' ' '0 ''2'0 ' 222 1 33 2 11 22 22 11 ρρρ φ += += +≈+= +=Φ += ++=+= += ∞ ∞ ∞∞∞ ∞ ∞∞ ∞ ppp aaaaaa xU uu uuUUuuu uUu O Small Perturbation Assumptions:             ∇⋅+ ∂ ∂ ⋅+ ∂ Φ∂ =⋅∇ 2 2 1 2 2 2 2 u u t u u ta u    (C.M.) +(C.L.M) (C.M.) +(C.L.M) 12 1 12 1 2 2 2 2 − += − ++ ∂ ∂ ∞ ∞ γγ φ a U a u t Bernoulli 121 − ∞ − ∞∞∞       =      =      = γ γ γ γ γ ρ ρ a a T T p p Isentropic Chain Development of the Flow Equations: Flow Equations: ( ) '' 2 1 φφ ∇=+∇⋅∇=⋅∇ ∞ xUu  ( ) 1 1 2 2 1 12 12 2 2 '' ' 1 2 1 x u a U x u uU a u u a ∂ ∂ ≅+ ∂ ∂ +≅      ∇⋅ ∞ ∞ ∞ ∞   ( ) t u UuUU tt u t u u ∂ ∂ =+ ∂ ∂ ≅ ∂ ∂ = ∂ ∂ ⋅ ∞∞∞ ' 2'22 1 1 2 2  ( )  ∞ ∞ ∞ ∞ ∞ ∞ ∞∞ ∞∞ ++ ∂ ∂ =⇒ − += − + +++ ∂ ∂ ρ γ φ γγ φ p a puU t a U aaa uUU t 2 1 2 2 2 1 2 '' ' 0 12 1 1 '2 '2 2 1' ∞∞∞∞∞∞∞∞ − = − ==⇒ − = − == a a T T p p a ad T Tdd p pd ' 1 2' 1 '' 1 2 1 γ γ γ γ ρ ρ γ γ γ γ γ ρ ρ γ Isentropic Chain Bernoulli Linearized Flow Equations
  • 135. 135 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u '2u ∞+Uu '1∞U Small Perturbation Flow Equations: (C.M.) +(C.L.M) 52.18.00 '' 2 '1 ' 2 2 1 1 12 2 2 ≤≤≤≤      ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ ∞∞ ∞ MM tt u U x u U a φ φ ( ) '' ,,,'' 321 φ φφ ∇= = u xxxt  Bernoulli       + ∂ ∂ −= ∞∞ ' ' ' 1uU t p φ ρ ∞∞∞∞ − = − == a a T T p p ' 1 2' 1 '' γ γ γ γ ρ ρ γIsentropic Chain Linearized Flow Equations
  • 136. 136 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Three Dimensional Flow (Thin Wings at Small Angles of Attack) α U Up xd ud θ= L Low xd ud θ−= ∞U x z ( ) 0 ''' 1 2 2 2 2 2 2 2 = ∂ ∂ + ∂ ∂ + ∂ ∂ − ∞ zyx M φφφ (1) ( )zyx ,,'φ(2) z w y v x u ∂ ∂ = ∂ ∂ = ∂ ∂ = ' ', ' ', ' ' φφφ (3) α−=≅ + ∞∞ S xd zd U w uU w ' ' ' (4) x UuUp ∂ ∂ −=−= ∞∞∞∞ ' '' φ ρρ(5) ' 2 1 1 '' 1 2' 1 '' 2 M M M U u M a a T T p p ∞ ∞ ∞ ∞ ∞∞∞∞ − + −=−= − = − == γ γ γ γ γ γ γ ρ ρ γ(6)         ∂ ∂ + ∂∂ ∂ + ∂ ∂ =∇ ∞∞∞ 2 2 2 2 2 2 2 2 '1'2'1 ' tUxtUxM φφφ φ ( ) '' ,,,'' φ φφ ∇= = u zyxt        + ∂ ∂ −= ∞∞ ' ' ' uU t p φ ρ Steady Three Dimensional Flow Small Perturbation Flow Equations: 0 ' 2 2 = ∂ ∂ = ∂ ∂ tt 52.1 8.00 ≤≤ ≤≤ ∞ ∞ M M
  • 137. 137 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Three Dimensional Flow (Thin Wings at Small Angles of Attack) 0 ''' 2 2 2 2 2 2 2 = ∂ ∂ + ∂ ∂ + ∂ ∂ zyx φφφ β(1) Steady Three Dimensional Flow Subsonic Flow M∞ 1 01: 22 −= ∞Mβ ( ) ( ) ( ) ( ) α ξ α φ α ξ α φ −=−= ∂ ∂ = −=−= ∂ ∂ = ∞∞ ∞∞ LowerLower Lower UperUper Upper d zd xd zd zUU w d zd xd zd zUU w '1' '1' 3 4 3 4 Transform of Coordinates ( ) ( )       = = = =−= ∞ ςηξφφ ς η ξβξ ,,,,' 1 2 zyx z y Mx           ∂ ∂ = ∂ ∂ ⇒ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ⇒ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ⇒ ∂ ∂ = ∂ ∂ 2 2 2 2 2 2 2 2 2 2 22 2 '' '' 1'1' ς φφ ς φφ η φφ η φφ ξ φ β φ ξ φ β φ zz yy xx ( ) ( ) SMdcMydycS bb ∞∞ −=−== ∫∫ 2 0 2 0 11 ηη ( ) ( )ηcMyc 2 1 ∞−= ∞∞ − = − == 22 22 11 M AR SM b S b AR 22 1 2 1 12 ∞∞∞∞ − = ∂ ∂ − −= ∂ ∂ −= M C UMxU C p p ξ φφ Section AA (enlarged) Wake region so 02 2 2 2 2 2 = ∂ ∂ + ∂ ∂ + ∂ ∂ ς φ η φ ξ φ Laplace’s Equation like in Incompressible Flow Similarity Rules
  • 138. 138 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack) incpC M 2 1 1 ∞− incLC M 2 1 1 ∞− 22 1 2 1 1 ∞∞ − =      − Md Cd M inc L α α incMC M 2 1 1 ∞− inc0α 4 1 =      inc N c x incMC M 02 1 1 ∞− incLsC M 2 1 1 ∞− incsα LsC sα 0MC c xN MC 0α αd Cd L LC pCPressure Distribution Lift Lift Slope Zero-Lift Angle Pitching Moment Neutral-Point Position Zero Moment Angle of Smooth Leading-Edge Flow Lift Coefficient of Smooth Leading-Edge Flow Aerodynamic Coefficients of a Profile in Subsonic Incident Flow Based on Subsonic Similarity Rules
  • 139. 139 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) α U Up xd ud θ= L Low xd ud θ−= ∞U x y ( ) 0 '' 1 2 2 2 2 2 = ∂ ∂ + ∂ ∂ − ∞ yx M φφ(1) ( )yx,'φ(2) y v x u ∂ ∂ = ∂ ∂ = ' ', ' ' φφ (3) α==≅ + ∞∞ S xd yd U v vU v ' ' ' (4) x UuUp ∂ ∂ −=−= ∞∞∞∞ ' '' φ ρρ(5) ' 2 1 1 '' 1 2' 1 '' 2 M M M U u M a a T T p p ∞ ∞ ∞ ∞ ∞∞∞∞ − + −=−= − = − == γ γ γ γ γ γ γ ρ ρ γ(6)       ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ ∞∞ ∞ 2 2 1 1 12 2 2 '' 2 '1 ' tt u U x u U a φ φ ( ) '' ,,,'' 321 φ φφ ∇= = u xxxt        + ∂ ∂ −= ∞∞ ' ' ' uU t p φ ρ Steady Two Dimensional Flow Small Perturbation Flow Equations: 0 ' 2 2 = ∂ ∂ = ∂ ∂ tt 52.1 8.00 ≤≤ ≤≤ M M Linearized Flow Equations
  • 140. 140 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) 0 '' 2 2 2 2 2 = ∂ ∂ + ∂ ∂ yx φφ β(1) Steady Two Dimensional Flow Subsonic Flow M∞ 1 01: 22 −= ∞Mβ ( ) ( ) ( ) ( ) α φ α φ −= ∂ ∂ = −= ∂ ∂ = ∞∞ ∞∞ Lower Lower Uper Upper xd yd yUU v xd yd yUU v '1' '1' 3 4 3 4 ∞U α Transform of Coordinates ( ) ( )     = = = yx y x ,', φβηξφ βη ξ            ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ =      ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ = ∂ ∂ ∂ ∂ =      ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ = ∂ ∂ 2 2 2 2 2 2 2 2 ' , 1' 11' 111' η φφ ξ φ β φ η φη η φξ ξ φ β φ β φ ξ φ β η η φξ ξ φ β φ β φ yx yyyy xxxx so 02 2 2 2 = ∂ ∂ + ∂ ∂ η φ ξ φ Laplace’s Equation like in Incompressible Flow Linearized Flow Equations
  • 141. 141 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Subsonic Flow M∞ 1 (continue) The Airfoil is defined in (x,y) plane and by (ξ,η) ( ) ( )ξη gxfy AirfoilAirfoil =⇔= The above Transformation relates the Compressible Flow over an Airfoil in (x,y) Space to the Incompressible Flow in (ξ,η) over the same Airfoil. α η φφ −= ∂ ∂ = ∂ ∂ = ∞∞∞ Uper Upper xd yd UyUU v 1'1' α η φφ −= ∂ ∂ = ∂ ∂ = ∞∞∞ Lower Lower xd yd UyUU v 1'1' ( )yx,ρρ = x y η ξ ∞= ρρ Compressible Flow Incompressible Flow α η φ −= ∂ ∂ = ∞∞ Uper Upper xd fd UU v 1' α η φ −= ∂ ∂ = ∞∞ Lower Lower xd fd UU v 1' Linearized Flow Equations
  • 142. 142 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) 0 '1' 2 2 22 2 = ∂ ∂ − ∂ ∂ yx φ β φ (1) ( ) ( ) ( ) ( ) ( ) ( ) yxGyxGyx yxFyxFyx Lower Upper βννβφ βηηβφ +==+= −==−= :,' :,'(7) (8) Steady Two Dimensional Flow Supersonic Flow M∞ 1 01: 22 −= ∞Mβ α U Up xd yd θ= L Low xd yd θ−= ∞U x y 1 1 2 − = ∞Mxd yd 1 1 2 − −= ∞Mxd yd Flow Flow ( ) ( ) ( ) η β α d Fd Uxd yd U v Uper Upper ∞∞ −=−= 1 7 4' ( ) ( ) η φ d Fd xd d u Upper 73 ' ' ==         − − −= ∞ ∞ α Upper Upper xd yd M U u 1 ' 2 ( ) ( ) ( ) ν β α d Gd Uxd yd U v Lower Lower ∞∞ =−= 3 8 4 ' ( ) ( ) ν φ d Gd xd d u Lower 83 ' ' ==         − − = ∞ ∞ α Lower Lower xd yd M U u 1 ' 2         − − =−= ∞ ∞∞ ∞∞ α ρ ρ Upper UpperUpper xd yd M U uUp 1 '' 2 2         − − −=−= ∞ ∞∞ ∞∞ α ρ ρ Lower LowerLower xd yd M U uUp 1 '' 2 2 Linearized Flow Equations
  • 143. 143 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations ( )∫         −−= ∞ S S sd xd yd ppD αsin np α− Upper xd yd ∞U Upper xd yd ∞p∞p α( )∫         −−−= ∞ S S sd xd yd ppL αcos ( )∫         −−≅ ∞ S S sd xd yd ppD α ( )  Γ ∞∞∞ ∫∫ =         −−−≅ SS S sduUsd xd yd ppL 'ρα 1−α Uper xd yd 1−α Uper xd yd Kutta-Joukovsky Define: 2 2 1 : ∞∞ ∞− = U pp Cp ρ ( ) ( ) ∫∫ ∫∫         −−=         − − −≅         −=         − − ≅ ∞∞ ∞∞ ∞ ∞∞ ∞∞ ∞∞ ∞ ∞∞ S S p S S S S p S S sd xd yd CUsd xd yd U pp UL sd xd yd CUsd xd yd U pp UD αρα ρ ρ αρα ρ ρ 2 2 2 2 2 2 2 1 2 12 1 2 1 2 12 1 Linearized Flow Equations
  • 144. 144 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Subsonic Flow (M∞ 1) np α− Upper xd yd ∞U Upper xd yd ∞p∞p α We found: α−= ∞ xd fd U v' α ξ −= ∞ d gd U v ( ) ( )            −= −= = ∞ ∞ yxM yM x ,'1, 1 2 2 φηξφ η ξ ( ) 0 '' 1 2 2 2 2 2 = ∂ ∂ + ∂ ∂ − ∞ yx M φφ 02 2 2 2 = ∂ ∂ + ∂ ∂ η φ ξ φ y v x u ∂ ∂ = ∂ ∂ = ' ', ' ' φφ η φ ξ φ ∂ ∂ = ∂ ∂ = vu , vv M u u = − = ∞ ', 1 ' 2 '' uUp ∞∞−= ρ uUp ∞∞−= ρ xUU u U pp Cp ∂ ∂ −=−= − = ∞∞ ∞∞ ∞ '2'2 2 1 ' : 2 φ ρ ξ φ ρ ∂ ∂ −=−= − = ∞∞ ∞∞ ∞ UU u U pp Cp 22 2 1 : 2 0 2 1 ' ∞− = M p p 2 1 0 ∞− = M C C p p Compressible: Incompressible: Linearized Flow Equations
  • 145. 145 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Subsonic Flow (M∞ 1) np α− Upper xd yd ∞U Upper xd yd ∞p∞p α The Relation: ∫∫ ∫∫       − −=              −−≅               − − =              −≅ ∞ ∞∞ ∞∞ ∞ ∞∞ ∞∞ S p S S p S S p S S p c s dC M U c s d xd yd CUL c s d xd yd C M U c s d xd yd CUD 0 0 2 2 2 2 2 2 1 2 1 2 1 1 2 1 2 1 ρ αρ α ρ αρ 2 1 0 ∞− = M C C p p Prandtl-Glauert Compressibility Correction As earlier in 1922, Prandtl is quoted as stating that the Lift Coefficient increased according to (1-M∞ 2 )-1/2 ; he mentioned this at a Lecture at Göttingen, but without a proof. This result was mentioned 6 years later by Jacob Ackeret, again without proof. The result was finally established by H. Glauert in 1928 based on Linear Small Perturbation. Ludwig Prandtl (1875 – 1953) Hermann Glauert (1892-1934) Linearized Flow Equations Return to Critical Mach Number
  • 146. SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Several improved formulas where developed: ( )[ ] 2/11/1 0 0 222 p p p CMMM C C ∞∞∞ −++− = Karman-Tsien Rule Linearized Flow Equations ( ) 0 0 2222 12/ 2 1 11 p p p CMMMM C C       −      − ++− = ∞∞∞∞ γ Laitone’s Rule Comparison of several compressibility corrections compared with experimental results for NACA 4412 Airfoil at an angle of attack of α = 1◦ . Return to Table of Content
  • 147. 147 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) 0 '1' 2 2 22 2 = ∂ ∂ − ∂ ∂ zx φ β φ (1) ( ) ( ) ( ) ( ) ( ) ( ) zxFzxGzx zxFzxFzx Lower Upper βννβφ βηηβφ +==+= −==−= :,' :,'(7) (8) Steady Two Dimensional Flow Supersonic Flow M∞ 1 01: 22 −= ∞Mβ α U Up xd zd θ= L Low xd zd θ−= ∞U x z 1 1 2 − = ∞Mxd zd 1 1 2 − −= ∞Mxd zd Flow Flow ( ) ( ) ( ) η β α d Fd Uxd zd U w Upper Upper ∞∞ −=−= 3 7 4' ( ) ( ) η φ d Fd xd d u Upper 73 ' ' == ( ) ( ) ( ) ν β α d Gd Uxd zd U w Lower Lower ∞∞ ==−= 3 8 4 ' ( ) ( ) ν φ d Gd xd d u Lower 83 ' ' ==         − − −= ∞ ∞ α Upper Upper xd zd M U w 1 ' 2         − − = ∞ ∞ α Lower Lower xd zd M U w 1 ' 2         − − =−==− ∞ ∞∞ ∞∞∞ α ρ ρ Upper UpperUpperUpper xd zd M U wUppp 1 '' 2 2         − − −=−==− ∞ ∞∞ ∞∞∞ α ρ ρ Lower LowerLowerLower xd zd M U wUppp 1 '' 2 2 z w x u ∂ ∂ = ∂ ∂ = ' ', ' ' φφ (3) α−=≅ + ∞∞ S xd zd U w uU w ' ' ' (4)
  • 148. 148 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 α U Up xd zd θ= L Low xd zd θ−= ∞U x z 1 1 2 − = ∞Mxd zd 1 1 2 − −= ∞Mxd zd Flow Flow Pressure Distribution and Lift Coefficient         −+ − = − = ∞∞∞ α ρ 2 1 2 2/ '' 22 LowerUpper LowerUpper p xd zd xd zd MU pp C 1 4 2 − = ∞M cL α ( ) ( ) ( ) ( )         −+− − − − =         +      −      − − =      +      −= ∞∞ ∞ ∫∫∫∫      00 22 1 0 1 02 1 0 1 0 00 1 2 1 4 2 1 2 LowerLowerUpperUpper LowerUpper ppL zczzcz MM c x d xd zd c x d xd zd Mc x dC c x dCc LowerUpper α α         − − = ∞ α Upper p xd zd M C Upper 1 2 2         − − −= ∞ α Lower p xd zd M C Lower 1 2 2
  • 149. 149 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 α U Up xd zd θ= L Low xd zd θ−= ∞U x z 1 1 2 − = ∞Mxd zd 1 1 2 − −= ∞Mxd zd Flow Flow Wave Drag Coefficient                         −+              − − =              −−              −= ∫∫∫∫ ∞ 1 0 2 1 0 2 2 1 0 1 0 1 2 c x d xd zd c x d xd zd Mc x d xd zd C c x d xd zd Cc UpperUpperLower p Upper pD LowerUpperW αααα         − − = ∞ α Upper p xd zd M C Upper 1 2 2         − − −= ∞ α Lower p xd zd M C Lower 1 2 2 ( ) ( ) ( ) ( )                             +      −+              +      − − = ∫∫∫∫ =−=− ∞ 1 0 2 00 1 0 2 1 0 2 00 1 0 2 2 22 1 2 c x d xd zd c x d xd zd c x d xd zd c x d xd zd M Lower zcz LowerUpper zcz Upper LowerLowerUpperUpper      αααα ( )22 22 2 1 2 1 4 LowerUpperD MM C W εε α + − + − = ∞∞ ∫ ∫               =               = 1 0 2 2 1 0 2 2 : : c x d xd zd c x d xd zd Lower Lower Upper Upper ε ε
  • 150. 150 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 Wave Drag Coefficient Flat Plate         == 0 LowerUpper xd zd xd zd Double Wedge Airfoil 1 4 2 2 − = ∞M C WD α 022 == LowerUpper εε ( ) ( ) ( )kkc t ck c t k ck c t kc LowerUpper − =       − − +== 14 1 1 14 1 4 11 2 2 2 2 22 2 2 22 εε ( ) ( )       − − =        − − = cxck ck t ckx ck t xd zd cxck ck t ckx ck t xd zd LowerUpper 12 0 2 12 0 2 ( ) ( )kk ct MM C WD −− + − = ∞∞ 1 / 1 1 1 4 2 22 2 α
  • 151. 151 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 Wave Drag Coefficient Biconvex Airfoil ( ) ( )222 2/2/ ctRR +−= The Biconvex Airfoil is obtained by intersection of two Circular Arcs of radius R. c – the chord t – maximum thickness at x = c/2 ( ) ( ) ( )tcttcR tc 4/4/ 222 22 ≈+= θθθθ −≈−=≈= tan,tan LowerUpper xd zd xd zd 2 2 2/2 /2 3 2 1 0 2 1 0 2 2 3 2 34 11 : Lower ct ctUpperUpper Upper c t t c dR c xd xd zd cc x d xd zd ε θ θθε δ δ ==≈≈         =              = + − + −∫∫∫ c t R c xd zd MaxUpper 2 2/ , ≈≈≈         δδ ( ) 2 2 22 2 22 22 2 3 16 1 1 1 4 1 2 1 4 c t MMMM C LowerUpperDW − + − =+ − + − = ∞∞∞∞ α εε α
  • 152. 02/10/15 152 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 Wave Drag Coefficient Parabolic ProfileDesignation Double Wedge Profile Contour Side View Wave Drag ( )kk −13 1 2( )kk −1 1 ( ) ( ) xckck xcxt z 212 22 −+ − ±= ( )       − ± ± = cxckx ck t ckxx ck t z 12 0 2
  • 153. 153 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 Wave Drag Coefficient Wave Drag at Supersonic Incident Flow versus Relative Thickness Position for Double Wedge and Parabolic Profiles k ( )kk −1 1 ( )kk −13 1 2
  • 154. 154 SOLO Wings in Compressible Flow Double Wedge Modified Double Wedge Biconvex τ 2 1 2 122 1 2 ' 2 ==       = c t c t c A τ 3 2 3 2332 1 2 ' 2 == +      = c t c t c t c A
  • 155. 155 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Steady Two Dimensional Flow Supersonic Flow M∞ 1 Pitching Moment Coefficient The Pitching Moment Coefficient about the Leading Edge for any Thin Airfoil is given by xdx xd zd xd zd Mcc x d c x C c x d c x Cc c LowerUpper ppM LowerUpperLE ∫∫∫                 −+         − − −=            +            −=− ∞ 022 1 0 1 0 1 2 αα Thus      + − + − −= ∫∫ ∞∞ xdzxdz McM c c Lower c UpperM LE 00222 1 2 1 2α ( ) ( ) ( ) ( )[ ] xdzxdzczczcxdzzxxdzzxxdx xd zd xd zd c Lower c UpperLowerUpper c Lower cx xLower c Upper cx xUpper c LowerUpper ∫∫∫∫∫ −−−=−+−=         + = = = = 00 0 00000    Using integration by parts Symmetric Airfoil zUpper = -zLower 1 2 2 − −= ∞M cM α The distance of the Airfoil Center of Pressure aft of the Leading Edge is given by cc M M c c c c x L MN 2 1 1/4 1/2 2 2 =⋅ − − =⋅−= ∞ ∞ α α α L ∞U x Return to Table of Content
  • 156. 156 SOLO Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1)                       −+         − − −=      −≅                       −+         − − =              −≅ ∫∫ ∫∫ ∞ ∞∞ ∞∞ ∞ ∞∞ ∞∞ c x d xd yd xd yd M U c s dCUL c x d xd yd xd yd M U c s d xd yd CUD c LowerUpperS p c LowerUpperS S p 0 2 2 2 0 22 2 2 2 2 1 2 1 2 1 2 1 2 1 2 1 αα ρ ρ αα ρ αρ α U Up xd yd θ= L Low xd yd θ−= ∞U x y 1 1 2 − = ∞Mxd yd 1 1 2 − −= ∞Mxd yd Flow Flow         − − ==− ∞ ∞∞ ∞ α ρ Upper UpperUpper xd yd M U ppp 1 ' 2 2         − − −==− ∞ ∞∞ ∞ α ρ Lower LowerLower xd yd M U ppp 1 ' 2 2 1 2 1 2 2 2 −         − −= −         − = ∞ ∞ M xd yd C M xd yd C Lower p Upper p Lower Upper α α We found: This relation was first derived by Jacob Ackeret in 1925, in a paper “Luftkrafte auf Flugel, die mit groserer als Schall-geschwingigkeit bewegt werden” (“Air Forces on Wings Moving at Supersonic Speeds”), that appeared in Zeitschhrift fur Flugtechnik und Motorluftschiffahrt, vol. 16, 1925, p.72 Jakob Ackeret (1898–1981) Linearized Flow Equations
  • 157. 157 AERODYNAMICS Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1) Supersonic Flow past a Symmetric Double-Edged Airfoil 1 2 3 4 SHOCK LINE SHOCK LINE SHOCK LINE SHOCK LINE EXPANSION EXPANSION Using Ackeret Theory we have ( ) ( ) ( ) ( ) 1 2 , 1 2 1 2 , 1 2 22 22 43 21 − − −= − + −= − + = − − = ∞∞ ∞∞ M C M C M C M C pp pp αδαδ αδαδ ( ) ( ) 1 4 2 1 1 4 2 1 1 4 222 1 2/1 2/1 0 3412 − = − + − =       −+      −=      = ∞∞∞ ∫∫∫ MMM c x dCC c x dCC c s dCC pppp S pX ααα ( ) ( ) ( ) ( ) 1 4 1 4 2 2 22 2 2/ 2 0 2/ 2/ 0 3412 3412 − = − ×=−+−=       −+      −=      = ∞ = ∞ −∫∫∫ MMc t CC c t CC c t c y dCC c y dCC c y dCC ct pppp ct pp ct pp S pX δδ δ XYXYD XYXYL CCCCC CCCCC +≈+= −≈−= ααα ααα α α 1 1 cossin sincos 1 4 1 4 1 4 1 4 2 2 2 21 2 2 2 1 − + − ≈ − − − ≈ ∞∞ ∞∞ MM C MM C D L δα αδα α α
  • 158. 158 AERODYNAMICS Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1)
  • 159. 159 pc− cx / 0.1 pc− cx / 0.1 pc− cx / 0.1 α δα δ ∞M δα ∞M ∞M δα = α ∞M Upper Surface Lover Surface Expansion Shock Shock Expansion Expansion Shock Expansion Shock Shock Expansion Expansion Shock Shock Shock Shock ∞M ∞M ( ) 1 2 2 − − = ∞M cp αδ ( ) 1 2 2 − + = ∞M cp αδ ( ) 1 2 2 − − = ∞M cp αδ ( ) 1 2 2 − + = ∞M cp αδ 1 4 2 − = ∞M cp α 1 4 2 − − = ∞M cp α ( ) 1 2 2 − + −= ∞M cp αδ ( ) 1 2 2 − − −= ∞M cp αδ ( ) 1 2 2 − + −= ∞M cp αδ ( ) 1 2 2 − − −= ∞M cp αδ Supersonic Flow past a Symmetric Biconvex Aerfoil AERODYNAMICS Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack) Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1) 2 2 2 2 2 2 2 4 1 1 3 16 3 16 1 4 LD L C M M c t C c tD L Md Cd − + −       =       + = − = ∞ ∞ ∞ α α α
  • 160. 160 SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0  =×∇ u Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack) Aerodynamic Coefficients of a Profile in Supersonic Incident Flow Based on the Linear Theory Supersonic Rules       − − = ∞ Xd Zd M α 1 1 2 1 4 2 − = ∞M 2 1 = 0DC 0α 0MC c xN αd Cd L pCPressure Distribution Lift Slope Neutral-Point Position Zero Moment Zero-Lift Angle 0= ( ) ∫− −= ∞ 1 02 1 4 XdZ M S Wave Drag L D Cd Cd 1 4 1 2 −−= ∞M ( ) ( ) ∫               +      − −= ∞ 1 0 22 2 1 4 Xd Xd Zd Xd Zd M tS
  • 161. 161 SOLO • Up to point A the flow is Subsonic and it follows Prandtl- Glauert Linear Subsonic Theory. • At point B (M∞=0.81) the flow on the Upper Surface exceeds the Sound Velocity and a Shock Wave occurs. On the Lower Surface the Flow is everywhere Subsonic. • At point C (M∞=0.89) the Flow velocity exceeds the Speed of Sound also on the Lower Surface and a Shock Wave occurs. • At point D (M∞=0.98) the two Shock Waves on the Upper and Lower Surface (weaker than at point C) are located at the Trailing Edge. The Lift is larger than at point C. • At point E (M∞=1.4) pure Supersonic Flow on both Surfaces. Transonic Flow past Airfoils Lift Coefficient of an Airfoil versus Mach Number. Solid Line – Measurement. Dashed Lines - Theory AERODYNAMICS Transonic Flow over an Airfoil at various Mach Numbers; Angle of Attack α=2°. The points A,B, C, D,E correspond to the Lift Coefficients.
  • 163. 163 Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane SOLO AERODYNAMICS
  • 164. 164 AERODYNAMICSSOLO Return to Table of Content Continue to Aerodynamics – Part III
  • 165. 165 I.H. Abbott, A.E. von Doenhoff “Theory of Wing Section”, Dover, 1949, 1959 H.W.Liepmann, A. Roshko “Elements of Gasdynamics”, John Wiley Sons, 1957 Jack Moran, “An Introduction to Theoretical and Computational Aerodynamics” Barnes W. McComick, Jr. “Aerodynamics of V/Stol Flight”, Dover, 1967, 1999 H. Ashley, M. Landhal “Aerodynamics of Wings and Bodies”, 1965 Louis Melveille Milne-Thompson “Theoretical Aerodynamics”, Dover, 1988 E.L. Houghton, P.W. Carpenter “Aerodynamics for Engineering Students”, 5th Ed. Butterworth-Heinemann, 2001 William Tyrrell Thomson “Introduction to Space Dynamics”, Dover References AERODYNAMICSSOLO
  • 166. 166 Holt Ashley “Engineering Analysis of Flight Vehicles”, Addison-Wesley, 1974 J.J. Bertin, M.L. Smith “Aerodynamics for Engineers”, Prentice-Hall, 1979 R.L. Blisplinghoff, H. Ashley, R.L. Halfman “Aeroelasticity”, Addison-Wesley, 1955 Barnes W. McCormick, Jr. “Aerodynamics, Aeronautics, And Flight Mechanics”, W.Z. Stepniewski “Rotary-Wing Aerodynamics”, Dover, 1984 William F. Hughes “Schaum’s Outline of Fluid Dynamics”, McGraw Hill, 1999 Theodore von Karman “Aerodynamics: Selected Topics in the Light of their Historical Development”, Prentice-Hall, 1979 L.J. Clancy “Aerodynamics”, John Wiley Sons, 1975 References (continue – 1) AERODYNAMICSSOLO
  • 167. 167 Frank G. Moore “Approximate Methods for Missile Aerodynamics”, AIAA, 2000 Thomas J. Mueller, Ed. “Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications”, AIAA, 2002 Richard S. Shevell “Fundamentals of Flight”, Prentice Hall, 2nd Ed., 1988 Ascher H. Shapiro “The Dynamics and Thermodynamics of Compressible Fluid Flow”, Wiley, 1953 Bernard Etkin, Lloyd Duff Reid “Dynamics of Flight: Stability and Control”, Wiley 3d Ed., 1995 H. Schlichting, K. Gersten, E. Kraus, K. Mayes “Boundary Layer Theory”, Springer Verlag, 1999 References (continue – 2) AERODYNAMICSSOLO
  • 168. 168 John D. Anderson “Computational Fluid Dynamics”, 1995 John D. Anderson “Fundamentals of Aeodynamics”, 2001 John D. Anderson “Introduction to Flight”, McGraw-Hill, 1978, 2004 John D. Anderson “Introduction to Flight”, 1995 John D. Anderson “A History of Aerodynamics”, 1995 John D. Anderson “Modern Compressible Flow: with Historical erspective”, McGraw-Hill, 1982 References (continue – 3) AERODYNAMICSSOLO Return to Table of Content
  • 169. February 10, 2015 169 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 –2013 Stanford University 1983 – 1986 PhD AA
  • 170. 170 Ludwig Prandtl (1875 – 1953) University of Göttingen Max Michael Munk (1890—1986)[ also NACA Theodor Meyer (1882 - 1972 Adolph Busemann (1901 – 1986) also NACA Colorado U. Theodore von Kármán (1881 – 1963) also USA Hermann Schlichting (1907-1982) Albert Betz (1885 – 1968 ), Jakob Ackeret (1898–1981) Irmgard Flügge-Lotz (1903 - 1974) also Stanford U. Paul Richard Heinrich Blasius (1883 – 1970)
  • 171. 171 Hermann Glauert (1892-1934) Pierre-Henri Hugoniot (1851 – 1887) Gino Girolamo Fanno (1888 – 1962) Karl Gustaf Patrik de Laval (1845 - 1913) Aurel Boleslav Stodola (1859 -1942) Eastman Nixon Jacobs (1902 –1987) Michael Max Munk (1890 – 1986) Sir Geoffrey Ingram Taylor (1886 – 1975) ENRICO PISTOLESI (1889 - 1968) Antonio Ferri (1912 – 1975) Osborne Reynolds (1842 –1912)
  • 172. 172 Robert Thomas Jones (1910–1999) Gaetano Arturo Crocco (1877 – 1968) Luigi Crocco (1906-1986) MAURICE MARIE ALFRED COUETTE (1858 -1943) Hans Wolfgang Liepmann (1914-2009) Richard Edler von Mises (1883 – 1953) Louis Melville Milne-Thomson (1891-1974) William Frederick Durand (1858 – 1959) Richard T. Whitcomb (1921 – 2009) Ascher H. Shapiro (1916 — 2004)
  • 173. 173 John J. Bertin (1928 – 2008) Barnes W. McCormick (1926 - ) Antonio Filippone John D. Anderson, Jr. Holt Ashley )1923–2006( Milton Denman Van Dyke (1922 – 2010)
  • 174. 174

Editor's Notes

  • #54: “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367
  • #56: L.J. Clancy, “Aerodynamics”, Pitman International, 1975, pg.287
  • #57: John D._Anderson, “Fundamentals_of_Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  • #58: John D._Anderson, “Fundamentals_of_Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  • #59: John D._Anderson, “Fundamentals_of_Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  • #60: John D._Anderson, “Fundamentals_of_Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001, pp.612-619
  • #61: John D._Anderson, “Fundamentals_of_Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001, pp.612-619
  • #63: “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367, Talay, 1975 J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
  • #64: J.J. Bertin, M.L. Smith, “Aerodynamics for Engineers”,Prentice-Hall, 1979 J.D. Anderson, Jr., “Introduction to Flight”, McGraw Hill, 1978
  • #65: A.H. Shapiro, “The Dynamics and Thermodynamics of Compressible Flow”, Ronald Press, Vol II, 1954, pg. 710 J.J. Bertin, M.L. Smith, “Aerodynamics for Engineers”, Prentice-Hall, 1979, pg. 343 R.T. Jones, “Wing Theory” Princeton University Press, 1990, pp. 90-104
  • #66: A.H. Shapiro, “The Dynamics and Thermodynamics of Compressible Flow”, Ronald Press, Vol II, 1954, pg. 710 J.J. Bertin, M.L. Smith, “Aerodynamics for Engineers”, Prentice-Hall, 1979, pg. 343 R.T. Jones, “Wing Theory” Princeton University Press, 1990, pp. 90-104
  • #67: R.T. Jones, “Wing Theory” Princeton University Press, 1990, pg. 94
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