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Aerodynamics part iii

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Solo Hermelin
Solo Hermelin

This document provides a table of contents for a document on aerodynamics. It discusses various topics related to aerodynamics including mathematical notations, basic laws of fluid dynamics, boundary conditions, airfoil design methods, compressible flow, shock waves, and linearized flow equations. Specifically, it summarizes the conical flow method and singularity distribution method for obtaining the theoretical solution for pressure distribution on a finite span wing in supersonic flow. The conical flow method assumes the potential and other flow properties are constant along rays through a common vertex, modeling conical flow patterns seen in supersonic flows.

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1 AERODYNAMICS Part III 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 A E R O D Y N A M I C S P A R T I
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 A E R O D Y N A M I C S P A R T I
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 A E R O D Y N A M I C S P A R T I
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 A E R O D Y N A M I C S P A R T I I
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 A E R O D Y N A M I C S P A R T I I
7 SOLO Table of Contents (continue – 5) AERODYNAMICS 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]
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
9 AERODYNAMICS Continue from AERODYNAMICS – Part II
10 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow The essential physical difference between Subsonic and Supersonic Flow is: - Subsonic Flow: The disturbances of a sound point source propagates in all directions. - Supersonic Flow: The disturbance of a sound point propagates only within a cone that lies downstream of the sound source. This so-called Mach-Cone has the apex semi-angle μ Supersonic V > a a t V t         M 1 sin 1  Sound waves Mach waves 1 1 tan 1 sin 1/: 2      MM aVM 
11 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Mach Cone Wing Leading Edge Mach Cone Wing Leading Edge Mach Cone Wing Leading Edge Mach Cone Wing Leading Edge If the Mach Line lies before the Wing Edge, the component vn of the incident Flow Velocity U∞ normal to the Wing Edge is smaller than the Speed of Sound a∞. Such a Wing Edge is called Subsonic. Conversely, if the Mach Line lies behind the Wing Edge, the component vn of the incident Flow Velocity U∞ normal to the Wing Edge is larger than the Speed of Sound a∞. Such a Wing Edge is called Supersonic. Subsonic Edge vn<a∞ μ>γ m<1 Supersonic Edge vn>a∞ μ<γ m>1
12 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Mach Line Wing Leading Edge Mach :Line Wing Trailing Edge Mach LineWing Leading Edge Mach :Line Wing Trailing Edge Mach Line Wing Leading Edge Mach :Line Wing Trailing Edge Subsonic Leading Edge Subsonic Trailing Edge Subsonic Leading Edge Supersonic Trailing Edge Supersonic Leading Edge Supersonic Trailing Edge Subsonic Leading Edge Flow Subsonic Trailing Edge Flow Supersonic Leading Edge Flow Supersonic Trailing Edge Flow
13 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Mach Line Wing Leading Edge Mach :Line Influence Range of A Wing Trailing Edge Consider a point A’ (x,y,z) on a Wing in a Supersonic Flow (V∞/a∞ > 1). The points on the Wing that, by perturbing the Flow, influence the Flow properties at A’ are only downstream to A’, bounded by the Wing Leading Edges and the Mach Lines (ML) passing through A’ (see Figure). Mach Line Wing Leading Edge Mach :Line Influence Range of A Wing Trailing Edge Subsonic Leading Edge Supersonic Leading Edge Return to Table of Content
14 SOLO Wings in Compressible Flow Theoretic Solutions for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) We present here two solution methods for PDE equation: 1. Conical Flow Method 2. Singularity-Distribution Method     1&1:0 2 2 2 2 2 2 2           MM zyx      This method was proposed by Busemann in 1943 and was extensively used before high speed computers were available. A Conical Flow is defined by velocity, pressure , static temperature, density constant along rays, through a common vertex. The Conical Flow can occur only at Supersonic Speeds. Conical Flow are produced by passing over a conic body, but It can be produced by small supersonic perturbations if the Boundary Conditions satisfy the Conical Conditions. In Supersonic Flow the disturbances are propagated only downstream the Mach Cone. Adolph Busemann (1901 – 1986). This method is similar with that used in Incompressible Flow, but the Singularities are Solutions of Supersonic Hyperbolic PDE. Return to Table of Content
15 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) 1. Conical Flow Method     1&1:0 2 2 2 2 2 2 2           MM zyx      Use for the Conical Flow the potential Start with     x z x y fxzyx   :,: ,:,,                ff f x                                      2 222 2 2 2 2 1111                    ff x f x ff x f x f x f xx Let compute 22 2 22 2 1 , 1 ,                             f xz f z f xy f y     1&1:0/12/1 2 2 2 22 2 2 2 22            MM fff        Mach Cone
16 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) 1. Conical Flow Method Mach Cone     1&1:0/12/1 2 2 2 22 2 2 2 22            MM fff            x z x y fxzyx   :,: ,:,,   Let compute         ,,,,' fzyx x u     The equation of a ray starting at the origin is given by    2121,, cc x z c x y czyxr  We can see that for η = const., ζ = const., we have r (x,y,z) = const.           ., 2 1 ,'. ,' 2, .,' 2 constCUpconst U u C constu pp                 . ,' 1 2,' 1 ,',' const a a T T p p                Isentropic Chain
17 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) 1. Conical Flow Method Regions where Two-Dimensional Flow prevails on Three-Dimensional Wings . Shaded zones signify Two-Dimensional Flow. Because in Supersonic Flows a perturbation is felt only in the Mach cone downstream from the source of disturbance, certain portion of the Wings behave as though they were in the Two – Dimensional Flow.
18 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Inclined Rectangular Wing at Supersonic Flow (a) Planform (b) Pressure disturbance at A-A Section Conical Flow on Rectangular Wings Propagations of Wing Edges (Leading and Side) on the Supersonic Flow propagate over Mach Cones. Looking at the Section A-E-A of the Wing, where E is the intersection of Section A-A with the Mach Line from the Wing Tip, we see that: • Points on A-E (region II) are affected only by the disturbances of the Wing Leading Edge. The Flow is Conical and two dimensional on the Wing, therefore the Pressure Coefficient is given by 22 1 4 2/       MU pp cc plpp   • Points on E-A (region IV) are affected by the disturbances of the Wing Leading Edge and by the Side Edge. The Flow is Conical and two dimensional on the Wing. the Pressure Coefficient is given by   21 2 121cos 1 4       M x y tt M cp  II IV A A E   EdgeLeadingt EdgeSidet M x y tt c c plp p 1 0 121cos 21      y x Area Below Curve = The mean value for is . 1,0t plpp cc 5.0
19 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Mach ConeMach Cone Wing Leading Edge Wing Leading Edge Wing Leading Edge Region II: AMNB Region IV: ADM & BCN Region II: ABE Region IV: ADME & BCNE Region V: MNE Region II: ABE Region IV: AME & BNE Region V: MFNE Conical Flow on Rectangular Wings Propagations of Wing Edges (Leading and Side) on the Supersonic Flow propagate over Mach Cones. Different Regions on the Wing are affected by the Wing Edges. Region II: Flow over points on the Wing in this region are affected only by disturbances of Leading Edge. Region IV: Flow over points on the Wing in this region are affected by disturbances of both Leading Edge and one of Side Edges. Region V: Flow over points on the Wing in this region are affected by disturbances of Leading Edge and both Side Edges. The Flow is not Conical.
20 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Conical Flow on Rectangular Wings Aerodynamic Forces on Inclined Rectangular Wings of various Aspect Ratios at Supersonic Incident Flow (a) Lift Slope (b) Neutral-point Position (c) Drag Coefficient
21 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Conical Flow on Rectangular Wings Pressure Distribution over the Chord and Lift Distribution over the Span for the Inclined Rectangular Plate of Aspect Ratio AR = 2.5 at Supersonic Incident Flow   89.1;41 2   MMARa   13.1; 3 4 1 2   MMARb
n Mm 1 :1tan tan tan 2     1 4 2/ 22       MU pp c plp     tan 1 : : x y t IRange  mtM x y M x y t IIIandIIRange   1tan tan 1 tan 'tan 1 : 22          10';sin11:' 2/ 0 22   EdmmE   Basic Solution for Pressure Distribution of the Inclined Flat Surface in Supersonic Incident Flow (Cone-Symmetric Flow) for Ranges I, II, III and IV SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) 1. Conical Flow Method
SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Conical Flow on Swept-Back Wings Pressure Distribution over in the Wing Chord (schematic) for a section of an Inclined Swept-Back Wing (a) Subsonic Leading and Trailing Edges. (b) Subsonic Leading and Supersonic Trailing Edge. (c) Supersonic Leading and Trailing Edges.
SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Conical Flow on Swept-Back Wings Pressure Distribution over in the Wing Chord and Lift Distribution over the Wing Span of Delta Wings at Supersonic Incident Flow (a) Subsonic Leading Edge, 0 < m < 1. (b) Supersonic Leading Edge, m > 1. n Mm 1 :1tan tan tan 2    
SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Conical Flow on Swept-Back Wings Lift Distribution over the Span of Delta Wings at Supersonic Incident Flow for several values of m: • Subsonic Leading Edge, 0 < m < 1. • Supersonic Leading Edge, m > 1. n Mm 1 :1tan tan tan 2     Return to Table of Content
26 Linearized Flow EquationsSOLO Incompressible Flow (M∞ = 0) Velocity Potential Equations: A Particular Solution is R Q zyx Q   44 222    That can be rewritten as Q – Source Strength Compressible Subsonic Flow (0 < M∞ < 1)     0 11 2 2 2 2 2 2 2 2           zMyMx Potential Equation: A Particular Solution is   2222 14 zyMx Q     That can be rewritten as Q – Subsonic Compressible Source Strength 1 4/4/4/ 222                    Q z Q y Q x Sphere 1 4/ 1 1 4/ 1 14/ 2 2 2 2 2                                         Q M z Q M y Q x Ellipsoid of Revolution 02 2 2 2 2 2          zyx  Elliptic Second Order Linear Partial Differential Equation. Elliptic Second Order Linear Partial Differential Equation. 2. Singularity-Distribution Method
27 Linearized Flow EquationsSOLO Compressible Supersonic Flow (M∞ >1)     1,0 11 2 2 2 2 2 2 2 2           i zMiyMix  Velocity Potential Equation: By analogy with the Subsonic Flow a Particular Solution is   2222 14 zyMx Q     That can be rewritten as Q – Supersonic Compressible Source Strength 1 4/ 1 1 4/ 1 14/ 2 2 2 2 2                                         Q M z Q M y Q x Hyperboloid of Revolution Only the part of the Flow lying downstream Mach Cone is physically significant. Hyperbolic Second Order Linear Partial Differential Equation. 2. Singularity-Distribution Method
28 SOLO Wings in Compressible Flow 2. Singularity-Distribution Method for Supersonic Flow (M∞ >1) Velocity Potential Equation:     1&1:0 2 2 2 2 2 2 2           MM zyx      Flow is Linear even without the assumption of Small Disturbances. This allows to combine Elementary Solutions similar to Subsonic Incompressible Flow (I.e. Source, Sink, Doublet, Vortex, etc.) to obtain General Solution for Supersonic Flow. Those Elementary Solutions are spread on the Aerodynamic Bodies in such a way that satisfy the Boundary Conditions. Example of Supersonic Elementary Solutions are: c S r q   4  Source Doublet c c V r vzq   4  Vortex where         22 1 1 22 2/122 1 22 1 : 1: : zyy xx v M zyyxxr c c        H. Lomax, M.A., Heaslet, F.B., Fuller, “Integrals and Integral Equations in Linearized Wing Theory”, Report 1054, NACA 1951zr zq c D        3 2 4
29 SOLO Wings in Compressible Flow 2. Singularity-Distribution Method for Supersonic Flow (M∞ >1) Four types of problems can be treated by the Singularity Distribution Method: (a) Two Non-lifting Case (Symmetric Wing): 1. Given the Thickness Distribution and the Planform Shape, find the Pressure Distribution on the Wing. 2. Given the Pressure Distribution on a Wing of Symmetrical Section, find the Wing Shape (I.e. the Thickness Distribution and the Planform). (b) Two Lifting Case (Non-Symmetric Wing): 4. A Lifting Surface, find the Pressure Distribution on it. In the Subsonic Case it is necessary to satisfy the Kutta Condition at the Trailing Edge. 3. Given the Pressure Distribution on a Lifting Surface (Zero Thickness) find the Slope of each point on the Surface. Direct Problems: Cases 1 and 3, because they involve Integrals with known Integrands. Indirect Problems: Cases 2 and 4, because the Unknown is inside the Integral Sign. Cases 1 and 2 are more conveniently solved using Source or Doublet Distributions, while Cases 3 and 4 are most often treated using Vortex Distributions. Return to Table of Content
30 SOLO Wings in Compressible Flow Theoretical Solutions for Compressible Supersonic Flow (M∞ >1)   52&8.001 2 2 2 2 2 2 2            MM zyx M  Velocity Potential Equation:     1&1:0 2 2 2 2 2 2 2           MM zyx      By analogy with the Subsonic Flow the influence of the Point Source q located at (ξ’, η’, 0) is given by         2222 ''4 ''0,',' ,, zyx ddq zyxd      The Point Source q must be such that whose boundary are defined by    2222 '' zyx   This is a Mach Cone, with apex at (ξ’, η’, 0) and angle μ = cot-1β             1 2 10 2222 ''4 ''0,',' ,,       zyx ddq zyx     zx zxy zxy       1 222 2 222 1 / /      0'' 2222  zyx      z w y v x u zwUyvxuUu              ',',' 1'1'1'  Elementary Source OfStrength q dξ dη Elementary Source OfStrength q dξ dη Hyperbola (ξ, η) :
31 Elementary Source OfStrength q dξ dη Elementary Source OfStrength q dξ dη Hyperbola (ξ, η) : SOLO Wings in Compressible Flow Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -1) Let integrate for all Sources (ξ, η, 0) (on the Wing) that are in the Front Mach Cone with the apex at (x,y,z)             1 2 10 2222 4 0,, ,,       zyx ddq zyx The boundary are defined by     222 2 222 11 /,/, zxyzxyzx   From this we can compute                   1 2 10 2/32222 2 4 0,,,, ,,      zyx ddzq z zyx zyxw We can see that w (x, y, z = 0) is zero everywhere, except at the source x = ξ, y = η where we have a indeterminate value 0/0. This was solved by Puckett in his PhD Thesis at Caltech in 1946 For ϕ (x,y,z), integrate the second integral by parts            222 1 2222 sin 1 4 1 / 4 , zx y vd q ud zyxddv q u                 Note that               12 / / 222 1 ,, 8 1 sin, 4 1 222 12 222 11        qq zx y qvu zxy zxy       
32 SOLO Wings in Compressible Flow                    zxzx d zx yq ddqq             0 222 1 0 12 2 1 sin 4 1 ,, 8 1             1 2 10 2222 4 0,, ,,       zyx ddq zyx we use LEIBNIZ THEOREM from CALCULUS:             )( )( )( )( ),( )),(()),((),(:: tb ta ChangeBoundariesthetodueChange sb sa dx s sxf sd sad ssaf sd sbd ssbfdxsxf sd d LEIBNITZ      To compute                     zyxI zx zyxI zx d zx yq d z dqq zz zyxw ,, 0 222 1 ,, 0 12 2 2 1 1 sin 4 1 ,, 8 1 ,,                                                    zxzx dqq z yzxqdqq z I      0 12 0 121 ,, 8 1 , 4 1 ,, 8 1   yzxyzx  222 12,11 /                                 zxzxy zz d qq zx z yzxqI           0 222 / 1 12 222 8 1 , 4 1 Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -2)
33 SOLO Wings in Compressible Flow We use again LEIBNIZ THEOREM from CALCULUS:             )( )( )( )( ),( )),(()),((),(:: tb ta ChangeBoundariesthetodueChange sb sa dx s sxf sd sad ssaf sd sbd ssbfdxsxf sd d LEIBNITZ      to compute                  22 2 1 21 2 1 2 1 0 222 1 222 1 0 222 1 2 sin 4 1 sinlim 4 1 sin 4 1 I zx I zx zx d zx yq d z d zx yq d zx yq d z I                                                                         0 24 1 lim 24 1 sinlim 4 1 max / 12 max 222 1 21 222 2 1                  yy qq d zx yq I finite zxy zx finite zx               Since we are interested in w (x,y, z=0) (the downwash in the Wing Plane)                          x z yxqd qq zx z yxqzyxI 0 0 2220 1 , 4 1 lim 8 1 , 4 1 0,, 12         Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -3)
34 SOLO Wings in Compressible Flow We use again LEIBNIZ THEOREM from CALCULUS:             )( )( )( )( ),( )),(()),((),(:: tb ta ChangeBoundariesthetodueChange sb sa dx s sxf sd sad ssaf sd sbd ssbfdxsxf sd d LEIBNITZ      to compute                                                                        2 1 12 2 1 2222222 3 0 222 12 0222 12 0222 1 0 22 lim sinlimlimsinlimlimsinlim:                      d zyxzx yzq zx yq zzx yq z d zx yq z Ifrom z zzz                       0sinlimlimlimsinlimlimlim sinlimlimsinlimlim 2/ 222 1 0 0 2220 2/ 222 1 0 0 2220 / 222 12 0222 12 0 1 1 2 2 222 2,1 12                                                                                                                                   zx yq zx z zx yq zx z zx yq zzx yq z zzzz zxy zz           0lim 2 1 2222222 3 0             d zyxzx yzq z     0limlimsinlim 4 1 lim 22 0 21 00 222 1 0 2 0 2 1               IId zx yq d z I zz zx zz         Therefore: Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -4)
35 SOLO Wings in Compressible Flow Finally we obtained:    yxq z zyxw z , 4 1 0,, 0                          1 2 10 2/32222 2 ,,, ,,        zyx ddzU z zyx zyxw Boundary Conditions:                     U xd yxzd U z zyx zyxw S CB z ,,, 0,, .. 0 Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -5) where:       xd yxzd yx S , :,                     zx zxy zxy zyx ddU zyx        1 222 12 222 110 / / 2222 , ,, and: Return to Table of Content
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from P are PB and PD. The Parts of the Wing that influence the Flow at P are located in the Area AEPBA.                                  CPB AEPC AEPBA P yx dd yx ddU yx ddU zyx 222 222 222 0,,             yxBPalong ACBalong     : tan:                 tan , tan tan yxyx B The Limits of Integrations are defined by the points A, E, P, B, C. The Lines of Integrations are
37 SOLO Wings in Compressible Flow                       CPB yx yx y AEPC yxy P yx dd d yx dd d U zyx                                       tan 222 tan tan 2220 0,,                                  yy yx yxy y x d U y x d U yx dd d U 0 1 0 1 0 tan 1 tan 2220 tan cosh1coshcosh                                                                                tan 0 1 0 1tan tan 1 tan 222 tan tan cosh1coshcosh yxyx y yx yx yx y y x d U y x d U yx dd d U  Section aa Mach Cone From P Mach Cone From P                                    32 tan 1 0 1 tan cosh tan cosh0,, I yx y I y P y x d y x d U zyx           Let compute     x zyx zyxu P P    0,, 0,,                       tan tan cosh tan 10,, 0,, 2 1 22 yx yxU x zyx zyxu P P       We obtain Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 1)
38 SOLO Wings in Compressible Flow    tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2                          tn tn n U x zyx zyxuP 1 cosh 1 10,, 0,, 2 1 2    Therefore on the Wing ( t = 0 – Side Edge to t = 1 - Leading Edge) 11 tan tan 2 2 22         n   Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex t =0 (Side Edge), t = 1 (Leading Edge) We found Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 2)                      tan tan cosh tan 10,, 0,, 2 1 22 yx yxU x zyx zyxuP       Section aa Mach Cone From P Mach Cone From P                  tn tn nU zyxu Cp 1 cosh 1 120,, 2 2 1 2  
39 SOLO Wings in Compressible Flow Let find how the disturbances of the Wing on the Flow affect a point N (x,y,0) outside the Wing between the Wing Side-Edge and the Mach Line (see Figure). The Mach Line through N that intersects The Wing between the points L and J Determines the Wing area ALN that affects the Flow at N. Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 3) Section aa Mach Cone From N Mach Cone From N           ALN N yx ddU zyx 222 0,,       yxNLJalong AJalong     : tan:    0,/ tan , tan tan yxL yxyx J                  The Limits of Integrations are defined by the points A, L,J. The Lines of Integrations are                                                                  tan 0 1 0 1tan 0 tan 1 tan 222 tan 0 tan cosh1coshcosh yxyx yx yx yx N y x d U y x d U yx dd d U                       tan 0 1 tan cosh0,, yx N d y xU zyx
40 SOLO Wings in Compressible Flow Let find how the disturbances of the Wing on the Flow affect a point N (x,y,0) Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 4) Section aa Mach Cone From N Mach Cone From N                      tan 0 1 tan cosh0,, yx N d y xU zyx                                                           tan tan cosh tan tan tan tan tan cosh tan 1 tan 1 2 1 tan 0 22 22 22 2 1 22 tan 0 222 yx yxU yx yx U yx d U x u yx yx N N                          tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2       11 tan tan 2 2 22         n   Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex , t = 0 (Side Edge), t =- n (Mach Line)
41 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 5) Section aa Mach Cone From N Mach Cone From N Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex , t = 0 (Side Edge), t =- n (Mach Line)                    tn tn n U x zyx zyxuN 1 cosh 1 10,, 0,, 2 1 2    Therefore between t = 0 (Side Edge ) to t = -n (Mach Line)                  tn tn nU zyxu Cp 1 cosh 1 120,, 2 2 1 2  
42 SOLO Wings in Compressible Flow Let find how the disturbances of the Wing on the Flow affect a point L (x,y,0) outside the Wing between the Wing Leading-Edge and the Mach Line(see Figure). The Mach Line through L that intersects The Wing between the points J and G Determines the Wing area AJG that affects the Flow at L. Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 6) Section aa Mach Cone From A Mach Cone From A           AJG L yx ddU zyx 222 0,,       yxGJNalong AJalong     : tan:    0, tan , tan tan yxG yxyx J                 The Limits of Integrations are defined by the points A, L,J. The Lines of Integrations are                                                           tan 0 1 0 1tan 0 tan 1 tan 222 tan 0 tan cosh1coshcosh yxyx yx yx yx y x d U y x d U yx dd d U                       tan 0 1 tan cosh0,, yx L d y xU zyx
43 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 7) Section aa Mach Cone From A Mach Cone From A                      tan 0 1 tan cosh0,, yx L d y xU zyx Let find how the disturbances of the Wing on the Flow affect a point L (x,y,0)                                                       xy yxU xy yx U yx d U x u yx yx N N tan tan cosh tan tan tan tan tan cosh tan 1 tan 1 2 1 tan 0 22 22 22 2 1 22 tan 0 222                       Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex , t =1(Leading Edge) to t = n (Mach Line)    1 1tan tan tan tan 1tan tan tan tan 2 2 2                          tn tn x y x y x y x y xy yx       11 tan tan 2 2 22         n  
44 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 8) Section aa Mach Cone From A Mach Cone From A Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex , t =1(Leading Edge) to t =+ n (Mach Line)                    1 cosh 1 10,, 0,, 2 1 2 tn tn n U x zyx zyxuL    Therefore between t = 1 (Leading Edge ) to t = +n (Mach Line)                  1 cosh 1 120,, 2 2 1 2 tn tn nU zyxu C L pL  
45 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 9) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Return to Table of Content
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Consider the Point L (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from L are LB and LC. B and C are on the Wing Leading Edge. The Parts of the Wing that influence the Flow at L are located in the Area LBC.           LBC L yx ddU zyx 222 0,,             tan: : :    CBalong yxBLalong yxCLalong                           tan , tan tan tan , tan tan         yxyx C yxyx B The Limits of Integrations are defined by the points C, L and B. The Lines of Integrations are Mach Line                                yx yx y yxy yxL yx dd d U yx dd d U                 tan 222 tan tan 222 tan Section aa
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 1) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Consider the Point L (x,y,z=0) on a Single Wedge Triangular Wing.                                                           y x y x y x y d yx d yx yx yx tan cosh1coshcosh 1 1 0 1 tan 1 tan tan 2222                  y x y x y x y d yx d yx yx yx                                           tan cosh1coshcosh 1 1 0 1 tan 1 tan tan 2222                                 tan tan 1 tan 1 tan 1 tan cosh tan cosh tan cosh                    yx yx yx y y yx L y xU y x d U y x d U                                yx yx y yxy yxL yx dd d U yx dd d U                 tan 222 tan tan 222 tan
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 2) Consider the Point L (x,y,z=0) on a Single Wedge Triangular Wing.                 tan tan 1 tan cosh          yx yx L L d y x x U x u                                       tan tan 222 0 1 1 0 1 1 tan tan tan tan cosh tan 1 tan tan tan cosh tan 1                       yx yx yx dU yx y yx x U yx y yx x U                           tan tan 222 tan         yx yx L yx dU u            tan tan 1 tan cosh           yx yx L d y xU Section aa
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 3) Consider the Point L (x,y,z=0) on a Single Wedge Triangular Wing.         2 22 22 2 22 22 22 22222 tan tan tan tan tan 1 tan tan tan tan 1 tan                                     yx yx yx d yx d                               tan tan 22 22 2 22 1 22 tan tan tan tan tan sin tan 1              yx yx L yx yx U u   2 1 1 1 sin uxd ud xu xd d   use                 2/ 1 22 22 2 22 1 22 2/ 1 22 22 2 22 1 22 tan tan tan tan tan tan sin tan 1 tan tan tan tan tan tan sin tan 1                                                                      yx yxyx U yx yxyx U    22 tan U uL               tan tan 222 tan         yx yx L yx dU u Section aa
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 4) Consider the Point L (x,y,z=0) on a Single Wedge Triangular Wing. 2 222 1:, tan : 1tan             Mn n UU x u L L      Section aa
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β ) (continue – 5) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from P are PB and PD. The Parts of the Wing that influence the Flow at P are located in the Area AEPBA.                                 AEDBPD AEPBA P yx dd yx ddU yx ddU zyx 222222 222 0,,             yxDEPalong DACBalong     : tan:    0, tan , tan tan yxE yxyx D                   The Limits of Integrations are defined by the points A, E, P, B, C. The Lines of Integrations are Mach Line                                         AED yx yx BPD P yx dd dd xU zyx 0 tan tan 22222 tan 0,,            
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 6) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from P are PB and PD.                                         AED yx yx BPD P yx dd dd xU zyx 0 tan tan 22222 tan 0,,                                                                       y x y x y x y d yx d yx yx yx tan cosh1coshcosh 1 1 0 1 tan 1 tan tan 2222                                    AED yx BPD P y x d xU zyx 0 tan 1 22 tan cosh tan 0,,                                                            0 tan 222 0 1 1 22 tan tan tan tan cosh tan 1 tan               yx P P yx d yx y yx x U x u      
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 7) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from P are PB and PD.                           0 tan 22222 tantan          yx P P yx dU x u   0 tan 22 22 2 22 1 22 tan tan tan tan tan sin tan 1                                                      yx L yx yx U u                                                                 2/ 1 22 22 2 22 1 2 1 22 tan tan tan tan tan tan sin tan tan sin tan 1              yx yxyx yx yxU                     tan tan sin 2tan 1 2 1 22 yx yxU uL    
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 8) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from P are PB and PD.                     tan tan sin 2tan 1 2 1 22 yx yxU uL        tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2       2 2 22 1 tan 1tan n           Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex t = 0 (Side Edge), t = n (Leading Edge)                                tn tn n U tn tn n U uL 1 cos 1 1 1 sin 21 1 2 1 2 2 1 2    
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 8) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From N Consider the Point N (x,y,z=0) between the Side Edge of the Triangular and the Mach Lines from A outside the Wing Planform. The f;ow disturbance on N is due to Wing Surface AEC.                tan tan cos tan 1 2 1 22 yx yxU x u N N     Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex t -n (Mach Line), t = 0 (Side Edge)                                tn tn n U tn tn n U uL 1 cos 1 1 1 sin 21 1 2 1 2 2 1 2               ANC N yx ddU zyx 222 0,,       yxCEalong ACalong     : tan:    0, tan , tan tan yxE yxyx C                  The Limits of Integrations are defined by the points A, E, C. The Lines of Integrations are                   tan 0 tan 222 0,,          yx yx N yx dd dd U zyx
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 9) Mach Line Pressure Field for a Semi-Infinite Triangular Wing with a Supersonic Leading Edge Return to Table of Content
57 SOLO Wings in Compressible Flow Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Delta Wing. The Mach Lines from P are PB and PD. The Parts of the Wing that influence the Flow at P are located in the Area ADPBA.                                         CPB AEPC ADE ADPBA yx dd yx dd yx ddU yx ddU zyx 222 222 222 222 0,,              The Limits of Integrations are defined by the points A, D, E, P, B, C. The Lines of Integrations are    yxDEPalong yxBPalong ACBalong ADalong         : : tan: tan:                                   tan , tan tan tan , tan tan yxyx D yxyx B Based on: A.E. Puckett, “Supersonic Wave Drag of Thin Airfoils”, 1949, Caltech PhD Thesis http://thesis.library.caltech.edu/2697/1/Puckett_ae_1949.pdf Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β)
58 SOLO Wings in Compressible Flow Section aa Mach Cone From P Mach Cone From P Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 1)                                    CPB yx yx y AEPC yxy AED yx yx yx dd d yx dd d yx dd d U zyx DE AD                                                      tan 222 tan tan 2220 tan 222 0 tan 0,,                                                0 tan 1 0 1 0 tan tan 1 tan 222 0 tan tan cosh1coshcosh                     yxyx yx yx yx y x d U y x d U yx dd d U                                   yy yx yxy y x d U y x d U yx dd d U 0 1 0 1 0 tan 1 tan 2220 tan cosh1coshcosh                                                                                tan 0 1 0 1tan tan 1 tan 222 tan tan cosh1coshcosh yxyx y yx yx yx y y x d U y x d U yx dd d U 
59 SOLO Wings in Compressible Flow                                                321 tan 1 0 1 0 tan 1 tan cosh tan cosh tan cosh0,, I yx y I y I yx y x d y x d y x d U zyx                We want to compute     x zyx zyxu    0,, 0,,  We use LEIBNIZ THEOREM from CALCULUS:             )( )( )( )( ),( )),(()),((),(:: tb ta ChangeBoundariesthetodueChange sb sa dx s sxf sd sad ssaf sd sbd ssbfdxsxf sd d LEIBNITZ                 yy yx d y x d xd d 0 2220 1 tan 1tan cosh      and   1 1 cosh 2 1   uxd ud xu xd d                                                      tan 222 0 1 1tan 1 tan 1 tan tan tan cosh tan 1tan cosh yx y yx y yx d yx y yx x y x d xd d                                               0 tan 222 0 1 1 0 tan 1 tan 1 tan tan tan cosh tan 1tan cosh               yxyx yx d yx y yx x y x d dx d       Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 2)
60 SOLO Wings in Compressible Flow       2 22 2 22 22 2 222 tan tan tan tan tan tan                              yxyx yx         1 tan tan tan tan tan tan tan tan tan 1 tan 2 22 22 22 2 22 22 22222                                     yx yx yx d yx d           0 tan 22 22 22 2 1 22 0 tan 222 1 tan tan tan tan tan cosh tan 1 tan 1                                                yx yx yx yx yx d x I   1 1 cosh 2 1   uxd ud xu xd d use              0 tan 222 1 tan 1    yx yx d x I EdgeLeadingSubsonictan Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 3) Start with
61 SOLO Wings in Compressible Flow       2 22 2 22 22 2 222 tan tan tan tan tan tan                              yxyx yx         1 tan tan tan tan tan tan tan tan tan 1 tan 2 22 22 22 2 22 22 22222                                      yx yx yx d yx d          y yx d x I 0 222 2 tan 1         y y yx yx yx d x I                                       0 22 22 22 2 1 220 222 2 tan tan tan tan tan cosh tan 1 tan 1   1 1 cosh 2 1   uxd ud xu xd d use Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 4)
62 SOLO Wings in Compressible Flow       2 22 2 22 22 2 222 tan tan tan tan tan tan                              yxyx yx         1 tan tan tan tan tan tan tan tan tan 1 tan 2 22 22 22 2 22 22 22222                                      yx yx yx d yx d                 tan 222 3 tan 1yx y yx d x I                                                        tan 22 22 22 2 1 22 tan 0 222 3 tan tan tan tan tan cosh tan 1 tan 1 yx y yx yx yx yx d x I   1 1 cosh 2 1   uxd ud xu xd d use Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 5)
63 SOLO Wings in Compressible Flow                        x I x I x IU x zyx zyxu 3210,, 0,,                                                                                             tan222 1 0 222 1 0 tan 222 1 22 tan tantan cosh tan tantan cosh tan tantan cosh tan 1 yx y y yx yx yx yx yx yx yxU                                                                     tan 0 222 1 0 tan 222 1 22 tan tantan cosh tan tantan cosh tan 1 yx yx yx yx yx yxU                                                                                        tan tan cosh tan tan tan tan cosh tan tan tan tan cosh tan tan cosh tan 1 2 1 1 222 1 0 1 222 1 2 1 22 yx yx yx yx yx yx yx yx yx yxU                           Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 6)
64 SOLO Wings in Compressible Flow                                          tan tan cosh tan tan cosh tan 10,, 0,, 2 1 2 1 22 yx yx yx yxU x zyx zyxu            tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2          tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2                                            tn tn tn tn n U x zyx zyxu 1 cosh 1 cosh 1 10,, 0,, 2 1 2 1 2    Therefore 11 tan tan 2 2 22         n   Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 7) Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex We found
65 SOLO Wings in Compressible Flow                                      tn tn tn tn n U x zyx zyxu 1 cosh 1 cosh 1 10,, 0,, 2 1 2 1 2    We want to prove that     2 22 1 2 1 2 1 1 cosh2 1 cosh 1 cosh t tn tn tn tn tn                        2 22 1 2 1 cosh 1 120,, 0,, t tn n U x zyx zyxu                tan :&tan:,: n x y t xd zd S Finally we obtain    tn tn tn tn       1 :cosh, 1 :cosh 22  Define Let compute 2 sinh 2 sinh 2 cosh 2 cosh 2 cosh                               tn tnn tn tnn tn tnn tn tnn             12 1 2/1cosh 2 sinh, 12 1 2/1cosh 2 cosh 12 1 2/1cosh 2 sinh, 12 1 2/1cosh 2 cosh         2 22 2 22 2 22 112 1 12 1 2 cosh t tn t tn n n t tn n n           q.e.d. Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 8)
66 SOLO Wings in Compressible Flow     1& 1 cosh 1 120,, 0,, 2 22 1 2          ttn t tn n U x zyx zyxu    Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 9) Since the Pressure and Velocity are constant along t = (y/x) tan Λ, i.e. along rays through the vertex of the Delta Wing, the Solution is of Conical Flows. For t = 1 we get the ray corresponding to the Leading Edge. For t = n=tanΛ/β we get the ray along the “Mach Line” from the vertex of the Delta Wing.   1& 1 cosh 1 140,,2 2 22 1 2          ttn t tn nU zyxu Cp   Pressure Coefficient 1:, tan :&tan:,: 2    Mn x y t xd zd S   
67Theoretical Solution for a Delta Wing (a) Pressure Distribution for a Single-Wedge Delta Wing at α = 0 [From Puckett (1946)] SOLO Wings in Compressible Flow
68 Theoretical Solution for a Delta Wing (b) Thickness Drag of a Double-Wedge Delta Wing with a Supersonic Leading Edge and a Supersonic Line of Maximum Thickness [From Puckett (1946)] SOLO Wings in Compressible Flow
69 Theoretical Solution for a Delta Wing (c) Thickness Drag of a Double-Wedge Delta Wing with a Supersonic Line of Maximum Thickness [From Puckett (1946)] SOLO Wings in Compressible Flow Return to Table of Content
70 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) Consider first a Point P1 (x,y) on the Wing, lying between the Wing Leading Edge and the Mach Line (1 > t > n). This point have a Potential determined only by the Sources lying in region (1) (Defined by Mach Lines P1A1 and P1C1 intersecting only the swept Trailing Edge OA1). But this Potential must be the same as for an Infinite Sweep (Λ) Wing, therefore is given by     2 2121 1:, tan : 1 , 1 1             Mn n U x Pu n xU P P       1 2 The Point P2 (x,y) on the Wing is lying behind the Mach Lines from the Wing Tip. The Mach Line PA intersects the Leading Edge OA and the Mach Line PC intersects the Leading Edge OB. If only the Leading Edge OA exists (no Leading Edge OB) than the Potential at P2 would be the Same as P1. To consider the Leading Edge OB we must subtract the disturbances in the area of region (2) OBC (no sources)       2 )2( 22222 1:, tan :,:, 1                       Mn x z yx ddU n xU P S               2
71 SOLO Wings in Compressible Flow                     ODB y CDB y y yx yx dd d U yx dd d U yx ddU zyx                  0 tan tan 222tan 222 )2( 222 2 2 1 0,,                              tan1 :tan 1 11 11    yx y MLyyxx OALEyx           tan2 :tan 2 22 22    yx y MLyyxx OBLEyx                                                           y x y x y x y d yx d yx yx yx tan cosh1coshcosh 1 1 0 1 tan 1 tan tan 2222                                                              y x y x y x y x y d yx d tan cosh tan coshcosh 1 11 tan tan 1 tan tan 2 tan tan 222 Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 1) 2
72 SOLO Wings in Compressible Flow         )2( 222      yx ddU                       0 111 2 2 1 tan cosh tan cosh tan cosh y y y d y x y xU d y xU                        0 1 0 1 21 tan cosh tan cosh yy d y xU d y xU          The u – velocity associated with this potential is given by                                     2 2 1 1 21 21 0 222 0 222 0 222 0 1 2 212 0 222 0 1 1 111 0 1 0 1 tantan tan tan cosh tan tan cosh tan cosh tan cosh I y I y yy yy yx dU yx dU yx dU yy yx x yU yx dU yy yx x yU d y x x U d y x x U x u                                                                    Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 2) 2
73 SOLO Wings in Compressible Flow         0 2221 1 tan y yx dU I     73       2 22 22 2 2 22 222 tan tan tan tan tan tan                              yxyx yx         2 22 22 22 2 22 22 22222 tan tan tan tan tan 1 tan tan tan tan 1 tan                                     yx yx yx d yx d       0 tan 22 22 22 2 1 22 0 2221 1 1 tan tan tan tan tan sin tan 1 tan                                             yx y y yx yx U yx dU I   2 1 1 1 sin uxd ud xu xd d   use              2/ 1 2 1 22 2 1 22 tan tantan sin tan 1 tan tan sin tan 1                                yx yxyxU yx yxU Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 3)
74 SOLO Wings in Compressible Flow 74       2 22 2 22 2 22 222 tan tan tan tan tan tan                             yxyx yx         2 22 22 2 22 22 22 22222 tan tan tan tan tan 1 tan tan tan tan 1 tan                                     yx yx yx d yx d       0 tan 22 22 2 22 1 22 0 2222 2 2 tan tan tan tan tan sin tan 1 tan                                             yx y y yx yx U yx dU I   2 1 1 1 sin uxd ud xu xd d  use         0 2222 2 tan y yx dU I                    2/ 1 2 1 22 2 1 22 tan tantan sin tan 1 tan tan sin tan 1                                 yx yxyxU yx yxU Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 4)
75 SOLO Wings in Compressible Flow 75 Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 5)             2 2 1 1 0 222 0 222 tantan I y I y yx dU yx dU x u                           21 22 2 1 2222 2 1 22 tan 1 2tan tan sin tan 1 tan 1 2tan tan sin tan 1 II U yx yxUU yx yxU                                                                          S x z yx yx yx yxU : tan tan sin tan tan sin tan 1 2 1 2 1 22    tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2          tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2       2 2 22 1 tan 1tan n           Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex                                 tn tn tn tn n U x u 1 sin 1 sin 1 1 2 1 2 1 2 
76 SOLO Wings in Compressible Flow 76 Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 6)                                 tn tn tn tn n U x u 1 sin 1 sin 1 1 2 1 2 1 2  We want to prove that     2 22 1 2 1 2 1 1 sin2 1 sin 1 sin t tn tn tn tn tn                    2 22 1 2 1 sin 1 12 t tn n U x u                                2 22 1 222 1 sin 2 1 1 1 1 1 2 t tn n U u n U x Pu P      tn tn tn tn       1 :sin, 1 :sin 22 Define Let compute 2 sin2 2 cos 2 sin2 2 cos 2 sin2 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos                                               tn tnn tn tnn tn tnn tn tnn                                     1 1 sin1 2 sin 2 cos, 1 1 sin1 2 sin 2 cos 12 1 sin1 2 sin 2 cos, 1 1 sin1 2 sin 2 cos         2 22 2 22 2 22 1 2 1 1 1 1 2 sin2 t tn t tn n n t tn n n               2 22 1 2 1 2 1 1 sin2 1 sin 1 sin t tn tn tn tn tn                      q.e.d.
77 SOLO Wings in Compressible Flow 77 Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 7)   2 22 1 22 22 1 222 1 cos 1 12 1 sin 21 12 1 1 2 t tn n U t tn n U u n U x Pu P                                1:,1 tan :&tan:,: 2    Mn x y t xd zd S    For an Un-swept Wing (Flat Surface) (Λ = 0) we have t = 0 & n = 0                        SP x z M UU x Pu 122 2     2 22 1 22 22 1 2 2 2 1 cos 1 14 1 sin 2 1 1 1 2 2 t tn nt tn nU Pu PCp                       
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow Pressure Field for a Semi-Infinite Triangular Wing with a Supersonic Leading Edge Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Mach Line Summary
SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow Inclined Delta Wing with Subsonic Leading Edge (0 < m < 1) (a) Wing Planform (Triangular Wing) (b) Pressure Distribution on a Section Normal to the Flow Direction, m = 0.6.             22 2 1 2 1 cos 1 tm t m m             22 2 1 2 1 cosh 1 tm t m m nMm /11tan tan tan : 2     mtM x y M x y t   1tan tan 1 1: 22   Inclined Delta Wing with Supersonic Leading Edge ( m > 1) (a) Wing Planform (Triangular Wing) (b) Pressure Distribution on a Section Normal to the Flow Direction, m = 1.5. Summary
80 SOLO Wings in Compressible Flow
81 Delta wing vortices Delta wing pressure distribution (suction effect at the tip) SOLO Wings in Compressible Flow
82 (A)- Flow field in wing-tail plane, influence of angle of attack SOLO Wings in Compressible Flow
83 (B)- Flow field in wing-tail plane, influence of control deflection  for pitch SOLO Wings in Compressible Flow
84 (C)- Flow field in wing-tail plane, influence of control deflection  for roll SOLO Wings in Compressible Flow Return to Table of Content
85 SOLO Wings in Compressible Flow Arrowhead Wings with Double-Wedge Profile at Zero Incidence [after Puckett and Steward] Nomenclature
86 SOLO Wings in Compressible Flow Arrowhead Wings with Double-Wedge Profile at Zero Incidence [after Puckett and Steward] Thickness Drag for e = 0
87 SOLO Wings in Compressible Flow Arrowhead Wings with Double-Wedge Profile at Zero Incidence [after Puckett and Steward] Thickness Drag for e = 0.5
88 SOLO Wings in Compressible Flow Arrowhead Wings with Double-Wedge Profile at Zero Incidence [after Puckett and Steward] Thickness Drag for b = 0.2 Return to Table of Content
89 SOLO Wings in Compressible Flow Arrowhead Wings with constant Chord, Biconvex Profile, and Subsonic Leading Edge [after Jones] (a) Platform (b) Pressure Distribution at various Spanwise Stations (c) Thickness Drag Coefficient at various Spanwise Stations Return to Table of Content
90 SOLO Wings in Compressible Flow 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] (a) Nomenclature and Geometrical Relationships. Note that e is negative if C lies aft of B.
91 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
92 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
93 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
94 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
95 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
96 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
97 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leanding and Trailing Edges and the same dimensionless profile in all chordwise plane [after Lawrence] (h) Wings with Biconvex Parabolic Arc Profile Return to Table of Content
98 SOLO Wings in Compressible Flow λ – Taper Ratio, 12  M CNα – Slope of the Normal Force Coefficient Computations of Swept Wings ΛLE – Leading Edge Swept Angle Wing Planform S – Wing Area     2 1 2 1 2 b c b c c c b ccS r r t rtr        AR – Aspect Ratio        1 12 2 1 22 r r c b b c b S b AR LE b c  tan 2                     2/ 0 2 322 2/ 0 2 2 22 2/ 0 2 2/3 1 2/ 1 2/1 2 2/ 1 2/ 121 2/1 2 2 1 b r b r r b b y b y y b c yd b y b y c bc ydyc S c                                   2 0 2/ 11 2/ b y b y c b y cccyc rrtr                                        1 1 3 2 12 3 1 11 1 2 1 62 1 22/1 2 2 22 rrr ccbbb b c  2 22 1 14 1 1 3 2           b Sc c r
99 SOLO Wings in Compressible Flow λ – Taper Ratio, 12  M ΛLE – Leading Edge Swept Angle CNα – Slope Computation is done as follows: 1. Compute s = β/tan ΛLE. If s<1 use the abscissa on the left side of the chart. If s>1 use the right side of the chart with the abscissa tanΛLE/β. 2. Pick the chart corresponding to the Taper Ratio λ. If λ is between the values of the given charts interpolation is needed. 3. Calculate AR tanΛLE for the given Airfoil. This is the parameter in the charts. If λ is between curves in the chart interpolation is needed. 4. Read the corresponding value from the ordinate; this value will correspond to tanΛLE (CNα) if the left side of the chart is used, and it will correspond to β(CNα) if the right side of the charts is used. 5. Extract CNα by dividing the left ordinate by tanΛLE , or by dividing the right ordinate by β, as the case may be. “USAF Stability and Control DATCOM Handbook” , Air Force Flight Dynamics Lab. Wright-Patterson AFB, Ohio, 1965 CNα – Slope of the Normal Force Coefficient Computations of Swept Wings
100 SOLO Wings in Compressible Flow λ = 0 – Taper Ratio CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M
101 SOLO Wings in Compressible Flow CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M λ = 1/5 – Taper Ratio
102 SOLO Wings in Compressible Flow CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M λ = 1/4 – Taper Ratio
103 SOLO Wings in Compressible Flow CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M λ = 1/3 – Taper Ratio
104 SOLO Wings in Compressible Flow CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M λ = 1/2 – Taper Ratio
105 SOLO Wings in Compressible Flow CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M λ = 1 – Taper Ratio Return to Table of Content
106 Comparison of Experiment and Theory for Lift-Curve Slope of Un-swept Wings (after Vincenti, 1950) SOLO Wings in Compressible Flow
107 Comparison of Experiment and Theory for Lift-Curve Slope of Swept Wings (after Vincenti, 1950) SOLO Wings in Compressible Flow
108 Thickness plus Skin-Friction Drag as a function of Sweep Angle (after Vincenti, 1950) SOLO Wings in Compressible Flow
109 Thickness plus Skin-Friction Drag as a function of position of Maximum Thickness (after Vincenti, 1950) SOLO Wings in Compressible Flow
110 Effect of radius of Subsonic Leading Edge on Pressure-Drag Ratio due to Lift (after Vincenti, 1950) SOLO Wings in Compressible Flow
111 Effect of radius of Subsonic Leading Edge on Lift-to-Drag Ratio (after Vincenti, 1950) SOLO Wings in Compressible Flow
112 SOLO Wings in Compressible Flow Lift Slope of Swept-Back Wings (taper λ = 1) at Supersonic Incident Flow, 0 < m <1; Subsonic Leading Edge; Supersonic Leading Edge.
113 SOLO Wings in Compressible Flow Drag Coefficient due to Lift versus Mach Number for a Trapezoidal, a Swept-Back, and a Delta Wing of Aspect Ratio Λ = 3. Dashed curve: with suction force. Solid curve: without suction force.
114 SOLO Wings in Compressible Flow Drag Coefficient (Wave Drag) at Zero Lift for Delta Wing (Triangular Wing) versus Mach Number. Profile I: Double Wedge profile. Profile II: Parabolic Profile, 0 < m < 1: Subsonic Leading Edge, m > 1: Supersonic Leading Edge Return to Table of Content
115 SOLO Wings in Compressible Flow Drag Coefficient (Wave Drag) at Zero Lift of Swept-Back Wings (tape λ = 1) at Supersonic Incident Flow. 0 < m < 1: Subsonic Leading Edge, m > 1: Supersonic Leading Edge
116 SOLO Wings in Compressible Flow Lift Slope versus Mach Number for a Trapezoidal, a Swept-Back, and a Delta Wing of Aspect Ratio Λ = 3. Total Drag Coefficient (Wave Drag + Friction Drag) versus Mach Number for a Trapezoidal, a Swept-Back, and a Delta Wing of Aspect Ratio Λ = 3. Double-Wedge profile t/c = 0.05, x c = 0.50
117 Lifting Properties of Three Planforms (after Jones, 1946) SOLO Wings in Compressible Flow Induced Drag of Three Planforms (after Jones, 1946) Return to Table of Content
118 Aircraft Flight ControlSOLO
119 centre stickailerons elevators rudder Aircraft Flight Control Generally, the primary cockpit flight controls are arranged as follows: a control yoke (also known as a control column), centre stick or side-stick (the latter two also colloquially known as a control or B joystick), governs the aircraft's roll and pitch by moving the A ailerons (or activating wing warping on some very early aircraft designs) when turned or deflected left and right, and moves the C elevators when moved backwards or forwards rudder pedals, or the earlier, pre-1919 "rudder bar", to control yaw, which move the D rudder; left foot forward will move the rudder left for instance. throttle controls to control engine speed or thrust for powered aircraft. SOLO
120 Stick Stick Rudder Pedals Aircraft Flight Control An aircraft 'rolling', or 'banking', with its ailerons Rudder Animation SOLO
121 Stick Stick Rudder Pedals Aircraft Flight ControlSOLO
122 Stick Stick Rudder Pedals Aircraft Flight ControlSOLO
123 Aircraft Flight ControlSOLO
124 Differential ailerons Aircraft Flight ControlSOLO
125 Aircraft Flight ControlSOLO
126 Aircraft Flight ControlSOLO
127 Aircraft Flight ControlSOLO
128 Aircraft Flight ControlSOLO
129 Aircraft Flight ControlSOLO
130 Aircraft Flight ControlSOLO
131 The effect of left rudder pressure Four common types of flaps Leading edge high lift devices The stabilator is a one-piece horizontal tail surface that pivots up and down about a central hinge point. Aircraft Flight ControlSOLO
SOLO 132 Flight Control Aircraft Flight Control
SOLO 133 Aerodynamics of Flight Aircraft Flight Control
SOLO 134 Aircraft Flight Control Specific Stabilizer/Tail Configurations Tailplane Fuselage mounted Cruciform T-tail Flying tailplane The tailplane comprises the tail-mounted fixed horizontal stabiliser and movable elevator. Besides its planform, it is characterised by: • Number of tailplanes - from 0 (tailless or canard) to 3 (Roe triplane) • Location of tailplane - mounted high, mid or low on the fuselage, fin or tail booms. • Fixed stabilizer and movable elevator surfaces, or a single combined stabilator or (all) flying tail.[1] (General Dynamics F-111) Some locations have been given special names: • Cruciform: mid-mounted on the fin (Hawker Sea Hawk, Sud Aviation Caravelle) • T-tail: high-mounted on the fin (Gloster Javelin, Boeing 727) Sud Aviation Caravelle Gloster Javelin
SOLO 135 Aircraft Flight Control Specific Stabilizer/Tail Configurations Tailplane Some locations have been given special names: • V-tail: (sometimes called a Butterfly tail) • Twin tail: specific type of vertical stabilizer arrangement found on the empennage of some aircraft. • Twin-boom tail: has two longitudinal booms fixed to the main wing on either side of the center line. The V-tail of a Belgian Air Force Fouga Magister de Havilland Vampire T11, Twin-Boom Tail A twin-tailed B-25 Mitchell
SOLO 136 Aircraft Avionics Aerodynamics of Flight Return to Table of Content
SOLO 137 Aircraft Avionics Aerodynamics of Flight
SOLO 138 Control Surfaces Aircraft Flight Control Return to Table of Content
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Aerodynamics part iii

  • 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 A E R O D Y N A M I C S P A R T I
  • 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 A E R O D Y N A M I C S P A R T I
  • 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 A E R O D Y N A M I C S P A R T I
  • 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 A E R O D Y N A M I C S P A R T I I
  • 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 A E R O D Y N A M I C S P A R T I I
  • 7. 7 SOLO Table of Contents (continue – 5) AERODYNAMICS 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]
  • 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
  • 10. 10 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow The essential physical difference between Subsonic and Supersonic Flow is: - Subsonic Flow: The disturbances of a sound point source propagates in all directions. - Supersonic Flow: The disturbance of a sound point propagates only within a cone that lies downstream of the sound source. This so-called Mach-Cone has the apex semi-angle μ Supersonic V > a a t V t         M 1 sin 1  Sound waves Mach waves 1 1 tan 1 sin 1/: 2      MM aVM 
  • 11. 11 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Mach Cone Wing Leading Edge Mach Cone Wing Leading Edge Mach Cone Wing Leading Edge Mach Cone Wing Leading Edge If the Mach Line lies before the Wing Edge, the component vn of the incident Flow Velocity U∞ normal to the Wing Edge is smaller than the Speed of Sound a∞. Such a Wing Edge is called Subsonic. Conversely, if the Mach Line lies behind the Wing Edge, the component vn of the incident Flow Velocity U∞ normal to the Wing Edge is larger than the Speed of Sound a∞. Such a Wing Edge is called Supersonic. Subsonic Edge vn<a∞ μ>γ m<1 Supersonic Edge vn>a∞ μ<γ m>1
  • 12. 12 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Mach Line Wing Leading Edge Mach :Line Wing Trailing Edge Mach LineWing Leading Edge Mach :Line Wing Trailing Edge Mach Line Wing Leading Edge Mach :Line Wing Trailing Edge Subsonic Leading Edge Subsonic Trailing Edge Subsonic Leading Edge Supersonic Trailing Edge Supersonic Leading Edge Supersonic Trailing Edge Subsonic Leading Edge Flow Subsonic Trailing Edge Flow Supersonic Leading Edge Flow Supersonic Trailing Edge Flow
  • 13. 13 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Mach Line Wing Leading Edge Mach :Line Influence Range of A Wing Trailing Edge Consider a point A’ (x,y,z) on a Wing in a Supersonic Flow (V∞/a∞ > 1). The points on the Wing that, by perturbing the Flow, influence the Flow properties at A’ are only downstream to A’, bounded by the Wing Leading Edges and the Mach Lines (ML) passing through A’ (see Figure). Mach Line Wing Leading Edge Mach :Line Influence Range of A Wing Trailing Edge Subsonic Leading Edge Supersonic Leading Edge Return to Table of Content
  • 14. 14 SOLO Wings in Compressible Flow Theoretic Solutions for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) We present here two solution methods for PDE equation: 1. Conical Flow Method 2. Singularity-Distribution Method     1&1:0 2 2 2 2 2 2 2           MM zyx      This method was proposed by Busemann in 1943 and was extensively used before high speed computers were available. A Conical Flow is defined by velocity, pressure , static temperature, density constant along rays, through a common vertex. The Conical Flow can occur only at Supersonic Speeds. Conical Flow are produced by passing over a conic body, but It can be produced by small supersonic perturbations if the Boundary Conditions satisfy the Conical Conditions. In Supersonic Flow the disturbances are propagated only downstream the Mach Cone. Adolph Busemann (1901 – 1986). This method is similar with that used in Incompressible Flow, but the Singularities are Solutions of Supersonic Hyperbolic PDE. Return to Table of Content
  • 15. 15 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) 1. Conical Flow Method     1&1:0 2 2 2 2 2 2 2           MM zyx      Use for the Conical Flow the potential Start with     x z x y fxzyx   :,: ,:,,                ff f x                                      2 222 2 2 2 2 1111                    ff x f x ff x f x f x f xx Let compute 22 2 22 2 1 , 1 ,                             f xz f z f xy f y     1&1:0/12/1 2 2 2 22 2 2 2 22            MM fff        Mach Cone
  • 16. 16 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) 1. Conical Flow Method Mach Cone     1&1:0/12/1 2 2 2 22 2 2 2 22            MM fff            x z x y fxzyx   :,: ,:,,   Let compute         ,,,,' fzyx x u     The equation of a ray starting at the origin is given by    2121,, cc x z c x y czyxr  We can see that for η = const., ζ = const., we have r (x,y,z) = const.           ., 2 1 ,'. ,' 2, .,' 2 constCUpconst U u C constu pp                 . ,' 1 2,' 1 ,',' const a a T T p p                Isentropic Chain
  • 17. 17 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) 1. Conical Flow Method Regions where Two-Dimensional Flow prevails on Three-Dimensional Wings . Shaded zones signify Two-Dimensional Flow. Because in Supersonic Flows a perturbation is felt only in the Mach cone downstream from the source of disturbance, certain portion of the Wings behave as though they were in the Two – Dimensional Flow.
  • 18. 18 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Inclined Rectangular Wing at Supersonic Flow (a) Planform (b) Pressure disturbance at A-A Section Conical Flow on Rectangular Wings Propagations of Wing Edges (Leading and Side) on the Supersonic Flow propagate over Mach Cones. Looking at the Section A-E-A of the Wing, where E is the intersection of Section A-A with the Mach Line from the Wing Tip, we see that: • Points on A-E (region II) are affected only by the disturbances of the Wing Leading Edge. The Flow is Conical and two dimensional on the Wing, therefore the Pressure Coefficient is given by 22 1 4 2/       MU pp cc plpp   • Points on E-A (region IV) are affected by the disturbances of the Wing Leading Edge and by the Side Edge. The Flow is Conical and two dimensional on the Wing. the Pressure Coefficient is given by   21 2 121cos 1 4       M x y tt M cp  II IV A A E   EdgeLeadingt EdgeSidet M x y tt c c plp p 1 0 121cos 21      y x Area Below Curve = The mean value for is . 1,0t plpp cc 5.0
  • 19. 19 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Mach ConeMach Cone Wing Leading Edge Wing Leading Edge Wing Leading Edge Region II: AMNB Region IV: ADM & BCN Region II: ABE Region IV: ADME & BCNE Region V: MNE Region II: ABE Region IV: AME & BNE Region V: MFNE Conical Flow on Rectangular Wings Propagations of Wing Edges (Leading and Side) on the Supersonic Flow propagate over Mach Cones. Different Regions on the Wing are affected by the Wing Edges. Region II: Flow over points on the Wing in this region are affected only by disturbances of Leading Edge. Region IV: Flow over points on the Wing in this region are affected by disturbances of both Leading Edge and one of Side Edges. Region V: Flow over points on the Wing in this region are affected by disturbances of Leading Edge and both Side Edges. The Flow is not Conical.
  • 20. 20 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Conical Flow on Rectangular Wings Aerodynamic Forces on Inclined Rectangular Wings of various Aspect Ratios at Supersonic Incident Flow (a) Lift Slope (b) Neutral-point Position (c) Drag Coefficient
  • 21. 21 SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Conical Flow on Rectangular Wings Pressure Distribution over the Chord and Lift Distribution over the Span for the Inclined Rectangular Plate of Aspect Ratio AR = 2.5 at Supersonic Incident Flow   89.1;41 2   MMARa   13.1; 3 4 1 2   MMARb
  • 22. n Mm 1 :1tan tan tan 2     1 4 2/ 22       MU pp c plp     tan 1 : : x y t IRange  mtM x y M x y t IIIandIIRange   1tan tan 1 tan 'tan 1 : 22          10';sin11:' 2/ 0 22   EdmmE   Basic Solution for Pressure Distribution of the Inclined Flat Surface in Supersonic Incident Flow (Cone-Symmetric Flow) for Ranges I, II, III and IV SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Finite Span Wing in a Supersonic Flow (M∞ > 1) 1. Conical Flow Method
  • 23. SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Conical Flow on Swept-Back Wings Pressure Distribution over in the Wing Chord (schematic) for a section of an Inclined Swept-Back Wing (a) Subsonic Leading and Trailing Edges. (b) Subsonic Leading and Supersonic Trailing Edge. (c) Supersonic Leading and Trailing Edges.
  • 24. SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Conical Flow on Swept-Back Wings Pressure Distribution over in the Wing Chord and Lift Distribution over the Wing Span of Delta Wings at Supersonic Incident Flow (a) Subsonic Leading Edge, 0 < m < 1. (b) Supersonic Leading Edge, m > 1. n Mm 1 :1tan tan tan 2    
  • 25. SOLO Wings in Compressible Flow Wings of Finite Span at Supersonic Incident Flow Conical Flow on Swept-Back Wings Lift Distribution over the Span of Delta Wings at Supersonic Incident Flow for several values of m: • Subsonic Leading Edge, 0 < m < 1. • Supersonic Leading Edge, m > 1. n Mm 1 :1tan tan tan 2     Return to Table of Content
  • 26. 26 Linearized Flow EquationsSOLO Incompressible Flow (M∞ = 0) Velocity Potential Equations: A Particular Solution is R Q zyx Q   44 222    That can be rewritten as Q – Source Strength Compressible Subsonic Flow (0 < M∞ < 1)     0 11 2 2 2 2 2 2 2 2           zMyMx Potential Equation: A Particular Solution is   2222 14 zyMx Q     That can be rewritten as Q – Subsonic Compressible Source Strength 1 4/4/4/ 222                    Q z Q y Q x Sphere 1 4/ 1 1 4/ 1 14/ 2 2 2 2 2                                         Q M z Q M y Q x Ellipsoid of Revolution 02 2 2 2 2 2          zyx  Elliptic Second Order Linear Partial Differential Equation. Elliptic Second Order Linear Partial Differential Equation. 2. Singularity-Distribution Method
  • 27. 27 Linearized Flow EquationsSOLO Compressible Supersonic Flow (M∞ >1)     1,0 11 2 2 2 2 2 2 2 2           i zMiyMix  Velocity Potential Equation: By analogy with the Subsonic Flow a Particular Solution is   2222 14 zyMx Q     That can be rewritten as Q – Supersonic Compressible Source Strength 1 4/ 1 1 4/ 1 14/ 2 2 2 2 2                                         Q M z Q M y Q x Hyperboloid of Revolution Only the part of the Flow lying downstream Mach Cone is physically significant. Hyperbolic Second Order Linear Partial Differential Equation. 2. Singularity-Distribution Method
  • 28. 28 SOLO Wings in Compressible Flow 2. Singularity-Distribution Method for Supersonic Flow (M∞ >1) Velocity Potential Equation:     1&1:0 2 2 2 2 2 2 2           MM zyx      Flow is Linear even without the assumption of Small Disturbances. This allows to combine Elementary Solutions similar to Subsonic Incompressible Flow (I.e. Source, Sink, Doublet, Vortex, etc.) to obtain General Solution for Supersonic Flow. Those Elementary Solutions are spread on the Aerodynamic Bodies in such a way that satisfy the Boundary Conditions. Example of Supersonic Elementary Solutions are: c S r q   4  Source Doublet c c V r vzq   4  Vortex where         22 1 1 22 2/122 1 22 1 : 1: : zyy xx v M zyyxxr c c        H. Lomax, M.A., Heaslet, F.B., Fuller, “Integrals and Integral Equations in Linearized Wing Theory”, Report 1054, NACA 1951zr zq c D        3 2 4
  • 29. 29 SOLO Wings in Compressible Flow 2. Singularity-Distribution Method for Supersonic Flow (M∞ >1) Four types of problems can be treated by the Singularity Distribution Method: (a) Two Non-lifting Case (Symmetric Wing): 1. Given the Thickness Distribution and the Planform Shape, find the Pressure Distribution on the Wing. 2. Given the Pressure Distribution on a Wing of Symmetrical Section, find the Wing Shape (I.e. the Thickness Distribution and the Planform). (b) Two Lifting Case (Non-Symmetric Wing): 4. A Lifting Surface, find the Pressure Distribution on it. In the Subsonic Case it is necessary to satisfy the Kutta Condition at the Trailing Edge. 3. Given the Pressure Distribution on a Lifting Surface (Zero Thickness) find the Slope of each point on the Surface. Direct Problems: Cases 1 and 3, because they involve Integrals with known Integrands. Indirect Problems: Cases 2 and 4, because the Unknown is inside the Integral Sign. Cases 1 and 2 are more conveniently solved using Source or Doublet Distributions, while Cases 3 and 4 are most often treated using Vortex Distributions. Return to Table of Content
  • 30. 30 SOLO Wings in Compressible Flow Theoretical Solutions for Compressible Supersonic Flow (M∞ >1)   52&8.001 2 2 2 2 2 2 2            MM zyx M  Velocity Potential Equation:     1&1:0 2 2 2 2 2 2 2           MM zyx      By analogy with the Subsonic Flow the influence of the Point Source q located at (ξ’, η’, 0) is given by         2222 ''4 ''0,',' ,, zyx ddq zyxd      The Point Source q must be such that whose boundary are defined by    2222 '' zyx   This is a Mach Cone, with apex at (ξ’, η’, 0) and angle μ = cot-1β             1 2 10 2222 ''4 ''0,',' ,,       zyx ddq zyx     zx zxy zxy       1 222 2 222 1 / /      0'' 2222  zyx      z w y v x u zwUyvxuUu              ',',' 1'1'1'  Elementary Source OfStrength q dξ dη Elementary Source OfStrength q dξ dη Hyperbola (ξ, η) :
  • 31. 31 Elementary Source OfStrength q dξ dη Elementary Source OfStrength q dξ dη Hyperbola (ξ, η) : SOLO Wings in Compressible Flow Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -1) Let integrate for all Sources (ξ, η, 0) (on the Wing) that are in the Front Mach Cone with the apex at (x,y,z)             1 2 10 2222 4 0,, ,,       zyx ddq zyx The boundary are defined by     222 2 222 11 /,/, zxyzxyzx   From this we can compute                   1 2 10 2/32222 2 4 0,,,, ,,      zyx ddzq z zyx zyxw We can see that w (x, y, z = 0) is zero everywhere, except at the source x = ξ, y = η where we have a indeterminate value 0/0. This was solved by Puckett in his PhD Thesis at Caltech in 1946 For ϕ (x,y,z), integrate the second integral by parts            222 1 2222 sin 1 4 1 / 4 , zx y vd q ud zyxddv q u                 Note that               12 / / 222 1 ,, 8 1 sin, 4 1 222 12 222 11        qq zx y qvu zxy zxy       
  • 32. 32 SOLO Wings in Compressible Flow                    zxzx d zx yq ddqq             0 222 1 0 12 2 1 sin 4 1 ,, 8 1             1 2 10 2222 4 0,, ,,       zyx ddq zyx we use LEIBNIZ THEOREM from CALCULUS:             )( )( )( )( ),( )),(()),((),(:: tb ta ChangeBoundariesthetodueChange sb sa dx s sxf sd sad ssaf sd sbd ssbfdxsxf sd d LEIBNITZ      To compute                     zyxI zx zyxI zx d zx yq d z dqq zz zyxw ,, 0 222 1 ,, 0 12 2 2 1 1 sin 4 1 ,, 8 1 ,,                                                    zxzx dqq z yzxqdqq z I      0 12 0 121 ,, 8 1 , 4 1 ,, 8 1   yzxyzx  222 12,11 /                                 zxzxy zz d qq zx z yzxqI           0 222 / 1 12 222 8 1 , 4 1 Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -2)
  • 33. 33 SOLO Wings in Compressible Flow We use again LEIBNIZ THEOREM from CALCULUS:             )( )( )( )( ),( )),(()),((),(:: tb ta ChangeBoundariesthetodueChange sb sa dx s sxf sd sad ssaf sd sbd ssbfdxsxf sd d LEIBNITZ      to compute                  22 2 1 21 2 1 2 1 0 222 1 222 1 0 222 1 2 sin 4 1 sinlim 4 1 sin 4 1 I zx I zx zx d zx yq d z d zx yq d zx yq d z I                                                                         0 24 1 lim 24 1 sinlim 4 1 max / 12 max 222 1 21 222 2 1                  yy qq d zx yq I finite zxy zx finite zx               Since we are interested in w (x,y, z=0) (the downwash in the Wing Plane)                          x z yxqd qq zx z yxqzyxI 0 0 2220 1 , 4 1 lim 8 1 , 4 1 0,, 12         Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -3)
  • 34. 34 SOLO Wings in Compressible Flow We use again LEIBNIZ THEOREM from CALCULUS:             )( )( )( )( ),( )),(()),((),(:: tb ta ChangeBoundariesthetodueChange sb sa dx s sxf sd sad ssaf sd sbd ssbfdxsxf sd d LEIBNITZ      to compute                                                                        2 1 12 2 1 2222222 3 0 222 12 0222 12 0222 1 0 22 lim sinlimlimsinlimlimsinlim:                      d zyxzx yzq zx yq zzx yq z d zx yq z Ifrom z zzz                       0sinlimlimlimsinlimlimlim sinlimlimsinlimlim 2/ 222 1 0 0 2220 2/ 222 1 0 0 2220 / 222 12 0222 12 0 1 1 2 2 222 2,1 12                                                                                                                                   zx yq zx z zx yq zx z zx yq zzx yq z zzzz zxy zz           0lim 2 1 2222222 3 0             d zyxzx yzq z     0limlimsinlim 4 1 lim 22 0 21 00 222 1 0 2 0 2 1               IId zx yq d z I zz zx zz         Therefore: Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -4)
  • 35. 35 SOLO Wings in Compressible Flow Finally we obtained:    yxq z zyxw z , 4 1 0,, 0                          1 2 10 2/32222 2 ,,, ,,        zyx ddzU z zyx zyxw Boundary Conditions:                     U xd yxzd U z zyx zyxw S CB z ,,, 0,, .. 0 Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -5) where:       xd yxzd yx S , :,                     zx zxy zxy zyx ddU zyx        1 222 12 222 110 / / 2222 , ,, and: Return to Table of Content
  • 36. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from P are PB and PD. The Parts of the Wing that influence the Flow at P are located in the Area AEPBA.                                  CPB AEPC AEPBA P yx dd yx ddU yx ddU zyx 222 222 222 0,,             yxBPalong ACBalong     : tan:                 tan , tan tan yxyx B The Limits of Integrations are defined by the points A, E, P, B, C. The Lines of Integrations are
  • 37. 37 SOLO Wings in Compressible Flow                       CPB yx yx y AEPC yxy P yx dd d yx dd d U zyx                                       tan 222 tan tan 2220 0,,                                  yy yx yxy y x d U y x d U yx dd d U 0 1 0 1 0 tan 1 tan 2220 tan cosh1coshcosh                                                                                tan 0 1 0 1tan tan 1 tan 222 tan tan cosh1coshcosh yxyx y yx yx yx y y x d U y x d U yx dd d U  Section aa Mach Cone From P Mach Cone From P                                    32 tan 1 0 1 tan cosh tan cosh0,, I yx y I y P y x d y x d U zyx           Let compute     x zyx zyxu P P    0,, 0,,                       tan tan cosh tan 10,, 0,, 2 1 22 yx yxU x zyx zyxu P P       We obtain Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 1)
  • 38. 38 SOLO Wings in Compressible Flow    tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2                          tn tn n U x zyx zyxuP 1 cosh 1 10,, 0,, 2 1 2    Therefore on the Wing ( t = 0 – Side Edge to t = 1 - Leading Edge) 11 tan tan 2 2 22         n   Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex t =0 (Side Edge), t = 1 (Leading Edge) We found Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 2)                      tan tan cosh tan 10,, 0,, 2 1 22 yx yxU x zyx zyxuP       Section aa Mach Cone From P Mach Cone From P                  tn tn nU zyxu Cp 1 cosh 1 120,, 2 2 1 2  
  • 39. 39 SOLO Wings in Compressible Flow Let find how the disturbances of the Wing on the Flow affect a point N (x,y,0) outside the Wing between the Wing Side-Edge and the Mach Line (see Figure). The Mach Line through N that intersects The Wing between the points L and J Determines the Wing area ALN that affects the Flow at N. Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 3) Section aa Mach Cone From N Mach Cone From N           ALN N yx ddU zyx 222 0,,       yxNLJalong AJalong     : tan:    0,/ tan , tan tan yxL yxyx J                  The Limits of Integrations are defined by the points A, L,J. The Lines of Integrations are                                                                  tan 0 1 0 1tan 0 tan 1 tan 222 tan 0 tan cosh1coshcosh yxyx yx yx yx N y x d U y x d U yx dd d U                       tan 0 1 tan cosh0,, yx N d y xU zyx
  • 40. 40 SOLO Wings in Compressible Flow Let find how the disturbances of the Wing on the Flow affect a point N (x,y,0) Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 4) Section aa Mach Cone From N Mach Cone From N                      tan 0 1 tan cosh0,, yx N d y xU zyx                                                           tan tan cosh tan tan tan tan tan cosh tan 1 tan 1 2 1 tan 0 22 22 22 2 1 22 tan 0 222 yx yxU yx yx U yx d U x u yx yx N N                          tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2       11 tan tan 2 2 22         n   Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex , t = 0 (Side Edge), t =- n (Mach Line)
  • 41. 41 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 5) Section aa Mach Cone From N Mach Cone From N Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex , t = 0 (Side Edge), t =- n (Mach Line)                    tn tn n U x zyx zyxuN 1 cosh 1 10,, 0,, 2 1 2    Therefore between t = 0 (Side Edge ) to t = -n (Mach Line)                  tn tn nU zyxu Cp 1 cosh 1 120,, 2 2 1 2  
  • 42. 42 SOLO Wings in Compressible Flow Let find how the disturbances of the Wing on the Flow affect a point L (x,y,0) outside the Wing between the Wing Leading-Edge and the Mach Line(see Figure). The Mach Line through L that intersects The Wing between the points J and G Determines the Wing area AJG that affects the Flow at L. Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 6) Section aa Mach Cone From A Mach Cone From A           AJG L yx ddU zyx 222 0,,       yxGJNalong AJalong     : tan:    0, tan , tan tan yxG yxyx J                 The Limits of Integrations are defined by the points A, L,J. The Lines of Integrations are                                                           tan 0 1 0 1tan 0 tan 1 tan 222 tan 0 tan cosh1coshcosh yxyx yx yx yx y x d U y x d U yx dd d U                       tan 0 1 tan cosh0,, yx L d y xU zyx
  • 43. 43 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 7) Section aa Mach Cone From A Mach Cone From A                      tan 0 1 tan cosh0,, yx L d y xU zyx Let find how the disturbances of the Wing on the Flow affect a point L (x,y,0)                                                       xy yxU xy yx U yx d U x u yx yx N N tan tan cosh tan tan tan tan tan cosh tan 1 tan 1 2 1 tan 0 22 22 22 2 1 22 tan 0 222                       Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex , t =1(Leading Edge) to t = n (Mach Line)    1 1tan tan tan tan 1tan tan tan tan 2 2 2                          tn tn x y x y x y x y xy yx       11 tan tan 2 2 22         n  
  • 44. 44 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 8) Section aa Mach Cone From A Mach Cone From A Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex , t =1(Leading Edge) to t =+ n (Mach Line)                    1 cosh 1 10,, 0,, 2 1 2 tn tn n U x zyx zyxuL    Therefore between t = 1 (Leading Edge ) to t = +n (Mach Line)                  1 cosh 1 120,, 2 2 1 2 tn tn nU zyxu C L pL  
  • 45. 45 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 9) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Return to Table of Content
  • 46. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Consider the Point L (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from L are LB and LC. B and C are on the Wing Leading Edge. The Parts of the Wing that influence the Flow at L are located in the Area LBC.           LBC L yx ddU zyx 222 0,,             tan: : :    CBalong yxBLalong yxCLalong                           tan , tan tan tan , tan tan         yxyx C yxyx B The Limits of Integrations are defined by the points C, L and B. The Lines of Integrations are Mach Line                                yx yx y yxy yxL yx dd d U yx dd d U                 tan 222 tan tan 222 tan Section aa
  • 47. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 1) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Consider the Point L (x,y,z=0) on a Single Wedge Triangular Wing.                                                           y x y x y x y d yx d yx yx yx tan cosh1coshcosh 1 1 0 1 tan 1 tan tan 2222                  y x y x y x y d yx d yx yx yx                                           tan cosh1coshcosh 1 1 0 1 tan 1 tan tan 2222                                 tan tan 1 tan 1 tan 1 tan cosh tan cosh tan cosh                    yx yx yx y y yx L y xU y x d U y x d U                                yx yx y yxy yxL yx dd d U yx dd d U                 tan 222 tan tan 222 tan
  • 48. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 2) Consider the Point L (x,y,z=0) on a Single Wedge Triangular Wing.                 tan tan 1 tan cosh          yx yx L L d y x x U x u                                       tan tan 222 0 1 1 0 1 1 tan tan tan tan cosh tan 1 tan tan tan cosh tan 1                       yx yx yx dU yx y yx x U yx y yx x U                           tan tan 222 tan         yx yx L yx dU u            tan tan 1 tan cosh           yx yx L d y xU Section aa
  • 49. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 3) Consider the Point L (x,y,z=0) on a Single Wedge Triangular Wing.         2 22 22 2 22 22 22 22222 tan tan tan tan tan 1 tan tan tan tan 1 tan                                     yx yx yx d yx d                               tan tan 22 22 2 22 1 22 tan tan tan tan tan sin tan 1              yx yx L yx yx U u   2 1 1 1 sin uxd ud xu xd d   use                 2/ 1 22 22 2 22 1 22 2/ 1 22 22 2 22 1 22 tan tan tan tan tan tan sin tan 1 tan tan tan tan tan tan sin tan 1                                                                      yx yxyx U yx yxyx U    22 tan U uL               tan tan 222 tan         yx yx L yx dU u Section aa
  • 50. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 4) Consider the Point L (x,y,z=0) on a Single Wedge Triangular Wing. 2 222 1:, tan : 1tan             Mn n UU x u L L      Section aa
  • 51. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β ) (continue – 5) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from P are PB and PD. The Parts of the Wing that influence the Flow at P are located in the Area AEPBA.                                 AEDBPD AEPBA P yx dd yx ddU yx ddU zyx 222222 222 0,,             yxDEPalong DACBalong     : tan:    0, tan , tan tan yxE yxyx D                   The Limits of Integrations are defined by the points A, E, P, B, C. The Lines of Integrations are Mach Line                                         AED yx yx BPD P yx dd dd xU zyx 0 tan tan 22222 tan 0,,            
  • 52. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 6) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from P are PB and PD.                                         AED yx yx BPD P yx dd dd xU zyx 0 tan tan 22222 tan 0,,                                                                       y x y x y x y d yx d yx yx yx tan cosh1coshcosh 1 1 0 1 tan 1 tan tan 2222                                    AED yx BPD P y x d xU zyx 0 tan 1 22 tan cosh tan 0,,                                                            0 tan 222 0 1 1 22 tan tan tan tan cosh tan 1 tan               yx P P yx d yx y yx x U x u      
  • 53. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 7) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from P are PB and PD.                           0 tan 22222 tantan          yx P P yx dU x u   0 tan 22 22 2 22 1 22 tan tan tan tan tan sin tan 1                                                      yx L yx yx U u                                                                 2/ 1 22 22 2 22 1 2 1 22 tan tan tan tan tan tan sin tan tan sin tan 1              yx yxyx yx yxU                     tan tan sin 2tan 1 2 1 22 yx yxU uL    
  • 54. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 8) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Triangular Wing. The Mach Lines from P are PB and PD.                     tan tan sin 2tan 1 2 1 22 yx yxU uL        tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2       2 2 22 1 tan 1tan n           Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex t = 0 (Side Edge), t = n (Leading Edge)                                tn tn n U tn tn n U uL 1 cos 1 1 1 sin 21 1 2 1 2 2 1 2    
  • 55. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 8) Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Section aa Mach Cone From P Mach Cone From N Consider the Point N (x,y,z=0) between the Side Edge of the Triangular and the Mach Lines from A outside the Wing Planform. The f;ow disturbance on N is due to Wing Surface AEC.                tan tan cos tan 1 2 1 22 yx yxU x u N N     Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex t -n (Mach Line), t = 0 (Side Edge)                                tn tn n U tn tn n U uL 1 cos 1 1 1 sin 21 1 2 1 2 2 1 2               ANC N yx ddU zyx 222 0,,       yxCEalong ACalong     : tan:    0, tan , tan tan yxE yxyx C                  The Limits of Integrations are defined by the points A, E, C. The Lines of Integrations are                   tan 0 tan 222 0,,          yx yx N yx dd dd U zyx
  • 56. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 9) Mach Line Pressure Field for a Semi-Infinite Triangular Wing with a Supersonic Leading Edge Return to Table of Content
  • 57. 57 SOLO Wings in Compressible Flow Section aa Mach Cone From P Mach Cone From P Consider the Point P (x,y,z=0) on a Single Wedge Delta Wing. The Mach Lines from P are PB and PD. The Parts of the Wing that influence the Flow at P are located in the Area ADPBA.                                         CPB AEPC ADE ADPBA yx dd yx dd yx ddU yx ddU zyx 222 222 222 222 0,,              The Limits of Integrations are defined by the points A, D, E, P, B, C. The Lines of Integrations are    yxDEPalong yxBPalong ACBalong ADalong         : : tan: tan:                                   tan , tan tan tan , tan tan yxyx D yxyx B Based on: A.E. Puckett, “Supersonic Wave Drag of Thin Airfoils”, 1949, Caltech PhD Thesis http://thesis.library.caltech.edu/2697/1/Puckett_ae_1949.pdf Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β)
  • 58. 58 SOLO Wings in Compressible Flow Section aa Mach Cone From P Mach Cone From P Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 1)                                    CPB yx yx y AEPC yxy AED yx yx yx dd d yx dd d yx dd d U zyx DE AD                                                      tan 222 tan tan 2220 tan 222 0 tan 0,,                                                0 tan 1 0 1 0 tan tan 1 tan 222 0 tan tan cosh1coshcosh                     yxyx yx yx yx y x d U y x d U yx dd d U                                   yy yx yxy y x d U y x d U yx dd d U 0 1 0 1 0 tan 1 tan 2220 tan cosh1coshcosh                                                                                tan 0 1 0 1tan tan 1 tan 222 tan tan cosh1coshcosh yxyx y yx yx yx y y x d U y x d U yx dd d U 
  • 59. 59 SOLO Wings in Compressible Flow                                                321 tan 1 0 1 0 tan 1 tan cosh tan cosh tan cosh0,, I yx y I y I yx y x d y x d y x d U zyx                We want to compute     x zyx zyxu    0,, 0,,  We use LEIBNIZ THEOREM from CALCULUS:             )( )( )( )( ),( )),(()),((),(:: tb ta ChangeBoundariesthetodueChange sb sa dx s sxf sd sad ssaf sd sbd ssbfdxsxf sd d LEIBNITZ                 yy yx d y x d xd d 0 2220 1 tan 1tan cosh      and   1 1 cosh 2 1   uxd ud xu xd d                                                      tan 222 0 1 1tan 1 tan 1 tan tan tan cosh tan 1tan cosh yx y yx y yx d yx y yx x y x d xd d                                               0 tan 222 0 1 1 0 tan 1 tan 1 tan tan tan cosh tan 1tan cosh               yxyx yx d yx y yx x y x d dx d       Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 2)
  • 60. 60 SOLO Wings in Compressible Flow       2 22 2 22 22 2 222 tan tan tan tan tan tan                              yxyx yx         1 tan tan tan tan tan tan tan tan tan 1 tan 2 22 22 22 2 22 22 22222                                     yx yx yx d yx d           0 tan 22 22 22 2 1 22 0 tan 222 1 tan tan tan tan tan cosh tan 1 tan 1                                                yx yx yx yx yx d x I   1 1 cosh 2 1   uxd ud xu xd d use              0 tan 222 1 tan 1    yx yx d x I EdgeLeadingSubsonictan Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 3) Start with
  • 61. 61 SOLO Wings in Compressible Flow       2 22 2 22 22 2 222 tan tan tan tan tan tan                              yxyx yx         1 tan tan tan tan tan tan tan tan tan 1 tan 2 22 22 22 2 22 22 22222                                      yx yx yx d yx d          y yx d x I 0 222 2 tan 1         y y yx yx yx d x I                                       0 22 22 22 2 1 220 222 2 tan tan tan tan tan cosh tan 1 tan 1   1 1 cosh 2 1   uxd ud xu xd d use Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 4)
  • 62. 62 SOLO Wings in Compressible Flow       2 22 2 22 22 2 222 tan tan tan tan tan tan                              yxyx yx         1 tan tan tan tan tan tan tan tan tan 1 tan 2 22 22 22 2 22 22 22222                                      yx yx yx d yx d                 tan 222 3 tan 1yx y yx d x I                                                        tan 22 22 22 2 1 22 tan 0 222 3 tan tan tan tan tan cosh tan 1 tan 1 yx y yx yx yx yx d x I   1 1 cosh 2 1   uxd ud xu xd d use Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 5)
  • 63. 63 SOLO Wings in Compressible Flow                        x I x I x IU x zyx zyxu 3210,, 0,,                                                                                             tan222 1 0 222 1 0 tan 222 1 22 tan tantan cosh tan tantan cosh tan tantan cosh tan 1 yx y y yx yx yx yx yx yx yxU                                                                     tan 0 222 1 0 tan 222 1 22 tan tantan cosh tan tantan cosh tan 1 yx yx yx yx yx yxU                                                                                        tan tan cosh tan tan tan tan cosh tan tan tan tan cosh tan tan cosh tan 1 2 1 1 222 1 0 1 222 1 2 1 22 yx yx yx yx yx yx yx yx yx yxU                           Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 6)
  • 64. 64 SOLO Wings in Compressible Flow                                          tan tan cosh tan tan cosh tan 10,, 0,, 2 1 2 1 22 yx yx yx yxU x zyx zyxu            tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2          tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2                                            tn tn tn tn n U x zyx zyxu 1 cosh 1 cosh 1 10,, 0,, 2 1 2 1 2    Therefore 11 tan tan 2 2 22         n   Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 7) Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex We found
  • 65. 65 SOLO Wings in Compressible Flow                                      tn tn tn tn n U x zyx zyxu 1 cosh 1 cosh 1 10,, 0,, 2 1 2 1 2    We want to prove that     2 22 1 2 1 2 1 1 cosh2 1 cosh 1 cosh t tn tn tn tn tn                        2 22 1 2 1 cosh 1 120,, 0,, t tn n U x zyx zyxu                tan :&tan:,: n x y t xd zd S Finally we obtain    tn tn tn tn       1 :cosh, 1 :cosh 22  Define Let compute 2 sinh 2 sinh 2 cosh 2 cosh 2 cosh                               tn tnn tn tnn tn tnn tn tnn             12 1 2/1cosh 2 sinh, 12 1 2/1cosh 2 cosh 12 1 2/1cosh 2 sinh, 12 1 2/1cosh 2 cosh         2 22 2 22 2 22 112 1 12 1 2 cosh t tn t tn n n t tn n n           q.e.d. Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 8)
  • 66. 66 SOLO Wings in Compressible Flow     1& 1 cosh 1 120,, 0,, 2 22 1 2          ttn t tn n U x zyx zyxu    Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 9) Since the Pressure and Velocity are constant along t = (y/x) tan Λ, i.e. along rays through the vertex of the Delta Wing, the Solution is of Conical Flows. For t = 1 we get the ray corresponding to the Leading Edge. For t = n=tanΛ/β we get the ray along the “Mach Line” from the vertex of the Delta Wing.   1& 1 cosh 1 140,,2 2 22 1 2          ttn t tn nU zyxu Cp   Pressure Coefficient 1:, tan :&tan:,: 2    Mn x y t xd zd S   
  • 67. 67Theoretical Solution for a Delta Wing (a) Pressure Distribution for a Single-Wedge Delta Wing at α = 0 [From Puckett (1946)] SOLO Wings in Compressible Flow
  • 68. 68 Theoretical Solution for a Delta Wing (b) Thickness Drag of a Double-Wedge Delta Wing with a Supersonic Leading Edge and a Supersonic Line of Maximum Thickness [From Puckett (1946)] SOLO Wings in Compressible Flow
  • 69. 69 Theoretical Solution for a Delta Wing (c) Thickness Drag of a Double-Wedge Delta Wing with a Supersonic Line of Maximum Thickness [From Puckett (1946)] SOLO Wings in Compressible Flow Return to Table of Content
  • 70. 70 SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) Consider first a Point P1 (x,y) on the Wing, lying between the Wing Leading Edge and the Mach Line (1 > t > n). This point have a Potential determined only by the Sources lying in region (1) (Defined by Mach Lines P1A1 and P1C1 intersecting only the swept Trailing Edge OA1). But this Potential must be the same as for an Infinite Sweep (Λ) Wing, therefore is given by     2 2121 1:, tan : 1 , 1 1             Mn n U x Pu n xU P P       1 2 The Point P2 (x,y) on the Wing is lying behind the Mach Lines from the Wing Tip. The Mach Line PA intersects the Leading Edge OA and the Mach Line PC intersects the Leading Edge OB. If only the Leading Edge OA exists (no Leading Edge OB) than the Potential at P2 would be the Same as P1. To consider the Leading Edge OB we must subtract the disturbances in the area of region (2) OBC (no sources)       2 )2( 22222 1:, tan :,:, 1                       Mn x z yx ddU n xU P S               2
  • 71. 71 SOLO Wings in Compressible Flow                     ODB y CDB y y yx yx dd d U yx dd d U yx ddU zyx                  0 tan tan 222tan 222 )2( 222 2 2 1 0,,                              tan1 :tan 1 11 11    yx y MLyyxx OALEyx           tan2 :tan 2 22 22    yx y MLyyxx OBLEyx                                                           y x y x y x y d yx d yx yx yx tan cosh1coshcosh 1 1 0 1 tan 1 tan tan 2222                                                              y x y x y x y x y d yx d tan cosh tan coshcosh 1 11 tan tan 1 tan tan 2 tan tan 222 Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 1) 2
  • 72. 72 SOLO Wings in Compressible Flow         )2( 222      yx ddU                       0 111 2 2 1 tan cosh tan cosh tan cosh y y y d y x y xU d y xU                        0 1 0 1 21 tan cosh tan cosh yy d y xU d y xU          The u – velocity associated with this potential is given by                                     2 2 1 1 21 21 0 222 0 222 0 222 0 1 2 212 0 222 0 1 1 111 0 1 0 1 tantan tan tan cosh tan tan cosh tan cosh tan cosh I y I y yy yy yx dU yx dU yx dU yy yx x yU yx dU yy yx x yU d y x x U d y x x U x u                                                                    Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 2) 2
  • 73. 73 SOLO Wings in Compressible Flow         0 2221 1 tan y yx dU I     73       2 22 22 2 2 22 222 tan tan tan tan tan tan                              yxyx yx         2 22 22 22 2 22 22 22222 tan tan tan tan tan 1 tan tan tan tan 1 tan                                     yx yx yx d yx d       0 tan 22 22 22 2 1 22 0 2221 1 1 tan tan tan tan tan sin tan 1 tan                                             yx y y yx yx U yx dU I   2 1 1 1 sin uxd ud xu xd d   use              2/ 1 2 1 22 2 1 22 tan tantan sin tan 1 tan tan sin tan 1                                yx yxyxU yx yxU Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 3)
  • 74. 74 SOLO Wings in Compressible Flow 74       2 22 2 22 2 22 222 tan tan tan tan tan tan                             yxyx yx         2 22 22 2 22 22 22 22222 tan tan tan tan tan 1 tan tan tan tan 1 tan                                     yx yx yx d yx d       0 tan 22 22 2 22 1 22 0 2222 2 2 tan tan tan tan tan sin tan 1 tan                                             yx y y yx yx U yx dU I   2 1 1 1 sin uxd ud xu xd d  use         0 2222 2 tan y yx dU I                    2/ 1 2 1 22 2 1 22 tan tantan sin tan 1 tan tan sin tan 1                                 yx yxyxU yx yxU Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 4)
  • 75. 75 SOLO Wings in Compressible Flow 75 Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 5)             2 2 1 1 0 222 0 222 tantan I y I y yx dU yx dU x u                           21 22 2 1 2222 2 1 22 tan 1 2tan tan sin tan 1 tan 1 2tan tan sin tan 1 II U yx yxUU yx yxU                                                                          S x z yx yx yx yxU : tan tan sin tan tan sin tan 1 2 1 2 1 22    tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2          tn tn x y x y x y x y yx yx                          1 tan1 tan tan tan tan1 tan tan tan 2 2 2       2 2 22 1 tan 1tan n           Define 1:1 tan :&tan: 2    Mn x y t   y/x=t/tanΛ is the equation of a ray starting from Wing apex                                 tn tn tn tn n U x u 1 sin 1 sin 1 1 2 1 2 1 2 
  • 76. 76 SOLO Wings in Compressible Flow 76 Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 6)                                 tn tn tn tn n U x u 1 sin 1 sin 1 1 2 1 2 1 2  We want to prove that     2 22 1 2 1 2 1 1 sin2 1 sin 1 sin t tn tn tn tn tn                    2 22 1 2 1 sin 1 12 t tn n U x u                                2 22 1 222 1 sin 2 1 1 1 1 1 2 t tn n U u n U x Pu P      tn tn tn tn       1 :sin, 1 :sin 22 Define Let compute 2 sin2 2 cos 2 sin2 2 cos 2 sin2 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos                                               tn tnn tn tnn tn tnn tn tnn                                     1 1 sin1 2 sin 2 cos, 1 1 sin1 2 sin 2 cos 12 1 sin1 2 sin 2 cos, 1 1 sin1 2 sin 2 cos         2 22 2 22 2 22 1 2 1 1 1 1 2 sin2 t tn t tn n n t tn n n               2 22 1 2 1 2 1 1 sin2 1 sin 1 sin t tn tn tn tn tn                      q.e.d.
  • 77. 77 SOLO Wings in Compressible Flow 77 Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 7)   2 22 1 22 22 1 222 1 cos 1 12 1 sin 21 12 1 1 2 t tn n U t tn n U u n U x Pu P                                1:,1 tan :&tan:,: 2    Mn x y t xd zd S    For an Un-swept Wing (Flat Surface) (Λ = 0) we have t = 0 & n = 0                        SP x z M UU x Pu 122 2     2 22 1 22 22 1 2 2 2 1 cos 1 14 1 sin 2 1 1 1 2 2 t tn nt tn nU Pu PCp                       
  • 78. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a Supersonic Flow Pressure Field for a Semi-Infinite Triangular Wing with a Supersonic Leading Edge Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge Mach Line Summary
  • 79. SOLO Wings in Compressible Flow Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow Inclined Delta Wing with Subsonic Leading Edge (0 < m < 1) (a) Wing Planform (Triangular Wing) (b) Pressure Distribution on a Section Normal to the Flow Direction, m = 0.6.             22 2 1 2 1 cos 1 tm t m m             22 2 1 2 1 cosh 1 tm t m m nMm /11tan tan tan : 2     mtM x y M x y t   1tan tan 1 1: 22   Inclined Delta Wing with Supersonic Leading Edge ( m > 1) (a) Wing Planform (Triangular Wing) (b) Pressure Distribution on a Section Normal to the Flow Direction, m = 1.5. Summary
  • 80. 80 SOLO Wings in Compressible Flow
  • 81. 81 Delta wing vortices Delta wing pressure distribution (suction effect at the tip) SOLO Wings in Compressible Flow
  • 82. 82 (A)- Flow field in wing-tail plane, influence of angle of attack SOLO Wings in Compressible Flow
  • 83. 83 (B)- Flow field in wing-tail plane, influence of control deflection  for pitch SOLO Wings in Compressible Flow
  • 84. 84 (C)- Flow field in wing-tail plane, influence of control deflection  for roll SOLO Wings in Compressible Flow Return to Table of Content
  • 85. 85 SOLO Wings in Compressible Flow Arrowhead Wings with Double-Wedge Profile at Zero Incidence [after Puckett and Steward] Nomenclature
  • 86. 86 SOLO Wings in Compressible Flow Arrowhead Wings with Double-Wedge Profile at Zero Incidence [after Puckett and Steward] Thickness Drag for e = 0
  • 87. 87 SOLO Wings in Compressible Flow Arrowhead Wings with Double-Wedge Profile at Zero Incidence [after Puckett and Steward] Thickness Drag for e = 0.5
  • 88. 88 SOLO Wings in Compressible Flow Arrowhead Wings with Double-Wedge Profile at Zero Incidence [after Puckett and Steward] Thickness Drag for b = 0.2 Return to Table of Content
  • 89. 89 SOLO Wings in Compressible Flow Arrowhead Wings with constant Chord, Biconvex Profile, and Subsonic Leading Edge [after Jones] (a) Platform (b) Pressure Distribution at various Spanwise Stations (c) Thickness Drag Coefficient at various Spanwise Stations Return to Table of Content
  • 90. 90 SOLO Wings in Compressible Flow 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] (a) Nomenclature and Geometrical Relationships. Note that e is negative if C lies aft of B.
  • 91. 91 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
  • 92. 92 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
  • 93. 93 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
  • 94. 94 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
  • 95. 95 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
  • 96. 96 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence] (b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles
  • 97. 97 SOLO Wings in Compressible Flow Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leanding and Trailing Edges and the same dimensionless profile in all chordwise plane [after Lawrence] (h) Wings with Biconvex Parabolic Arc Profile Return to Table of Content
  • 98. 98 SOLO Wings in Compressible Flow λ – Taper Ratio, 12  M CNα – Slope of the Normal Force Coefficient Computations of Swept Wings ΛLE – Leading Edge Swept Angle Wing Planform S – Wing Area     2 1 2 1 2 b c b c c c b ccS r r t rtr        AR – Aspect Ratio        1 12 2 1 22 r r c b b c b S b AR LE b c  tan 2                     2/ 0 2 322 2/ 0 2 2 22 2/ 0 2 2/3 1 2/ 1 2/1 2 2/ 1 2/ 121 2/1 2 2 1 b r b r r b b y b y y b c yd b y b y c bc ydyc S c                                   2 0 2/ 11 2/ b y b y c b y cccyc rrtr                                        1 1 3 2 12 3 1 11 1 2 1 62 1 22/1 2 2 22 rrr ccbbb b c  2 22 1 14 1 1 3 2           b Sc c r
  • 99. 99 SOLO Wings in Compressible Flow λ – Taper Ratio, 12  M ΛLE – Leading Edge Swept Angle CNα – Slope Computation is done as follows: 1. Compute s = β/tan ΛLE. If s<1 use the abscissa on the left side of the chart. If s>1 use the right side of the chart with the abscissa tanΛLE/β. 2. Pick the chart corresponding to the Taper Ratio λ. If λ is between the values of the given charts interpolation is needed. 3. Calculate AR tanΛLE for the given Airfoil. This is the parameter in the charts. If λ is between curves in the chart interpolation is needed. 4. Read the corresponding value from the ordinate; this value will correspond to tanΛLE (CNα) if the left side of the chart is used, and it will correspond to β(CNα) if the right side of the charts is used. 5. Extract CNα by dividing the left ordinate by tanΛLE , or by dividing the right ordinate by β, as the case may be. “USAF Stability and Control DATCOM Handbook” , Air Force Flight Dynamics Lab. Wright-Patterson AFB, Ohio, 1965 CNα – Slope of the Normal Force Coefficient Computations of Swept Wings
  • 100. 100 SOLO Wings in Compressible Flow λ = 0 – Taper Ratio CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M
  • 101. 101 SOLO Wings in Compressible Flow CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M λ = 1/5 – Taper Ratio
  • 102. 102 SOLO Wings in Compressible Flow CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M λ = 1/4 – Taper Ratio
  • 103. 103 SOLO Wings in Compressible Flow CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M λ = 1/3 – Taper Ratio
  • 104. 104 SOLO Wings in Compressible Flow CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M λ = 1/2 – Taper Ratio
  • 105. 105 SOLO Wings in Compressible Flow CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle λ – Taper Ratio, 12  M λ = 1 – Taper Ratio Return to Table of Content
  • 106. 106 Comparison of Experiment and Theory for Lift-Curve Slope of Un-swept Wings (after Vincenti, 1950) SOLO Wings in Compressible Flow
  • 107. 107 Comparison of Experiment and Theory for Lift-Curve Slope of Swept Wings (after Vincenti, 1950) SOLO Wings in Compressible Flow
  • 108. 108 Thickness plus Skin-Friction Drag as a function of Sweep Angle (after Vincenti, 1950) SOLO Wings in Compressible Flow
  • 109. 109 Thickness plus Skin-Friction Drag as a function of position of Maximum Thickness (after Vincenti, 1950) SOLO Wings in Compressible Flow
  • 110. 110 Effect of radius of Subsonic Leading Edge on Pressure-Drag Ratio due to Lift (after Vincenti, 1950) SOLO Wings in Compressible Flow
  • 111. 111 Effect of radius of Subsonic Leading Edge on Lift-to-Drag Ratio (after Vincenti, 1950) SOLO Wings in Compressible Flow
  • 112. 112 SOLO Wings in Compressible Flow Lift Slope of Swept-Back Wings (taper λ = 1) at Supersonic Incident Flow, 0 < m <1; Subsonic Leading Edge; Supersonic Leading Edge.
  • 113. 113 SOLO Wings in Compressible Flow Drag Coefficient due to Lift versus Mach Number for a Trapezoidal, a Swept-Back, and a Delta Wing of Aspect Ratio Λ = 3. Dashed curve: with suction force. Solid curve: without suction force.
  • 114. 114 SOLO Wings in Compressible Flow Drag Coefficient (Wave Drag) at Zero Lift for Delta Wing (Triangular Wing) versus Mach Number. Profile I: Double Wedge profile. Profile II: Parabolic Profile, 0 < m < 1: Subsonic Leading Edge, m > 1: Supersonic Leading Edge Return to Table of Content
  • 115. 115 SOLO Wings in Compressible Flow Drag Coefficient (Wave Drag) at Zero Lift of Swept-Back Wings (tape λ = 1) at Supersonic Incident Flow. 0 < m < 1: Subsonic Leading Edge, m > 1: Supersonic Leading Edge
  • 116. 116 SOLO Wings in Compressible Flow Lift Slope versus Mach Number for a Trapezoidal, a Swept-Back, and a Delta Wing of Aspect Ratio Λ = 3. Total Drag Coefficient (Wave Drag + Friction Drag) versus Mach Number for a Trapezoidal, a Swept-Back, and a Delta Wing of Aspect Ratio Λ = 3. Double-Wedge profile t/c = 0.05, x c = 0.50
  • 117. 117 Lifting Properties of Three Planforms (after Jones, 1946) SOLO Wings in Compressible Flow Induced Drag of Three Planforms (after Jones, 1946) Return to Table of Content
  • 119. 119 centre stickailerons elevators rudder Aircraft Flight Control Generally, the primary cockpit flight controls are arranged as follows: a control yoke (also known as a control column), centre stick or side-stick (the latter two also colloquially known as a control or B joystick), governs the aircraft's roll and pitch by moving the A ailerons (or activating wing warping on some very early aircraft designs) when turned or deflected left and right, and moves the C elevators when moved backwards or forwards rudder pedals, or the earlier, pre-1919 "rudder bar", to control yaw, which move the D rudder; left foot forward will move the rudder left for instance. throttle controls to control engine speed or thrust for powered aircraft. SOLO
  • 120. 120 Stick Stick Rudder Pedals Aircraft Flight Control An aircraft 'rolling', or 'banking', with its ailerons Rudder Animation SOLO
  • 131. 131 The effect of left rudder pressure Four common types of flaps Leading edge high lift devices The stabilator is a one-piece horizontal tail surface that pivots up and down about a central hinge point. Aircraft Flight ControlSOLO
  • 134. SOLO 134 Aircraft Flight Control Specific Stabilizer/Tail Configurations Tailplane Fuselage mounted Cruciform T-tail Flying tailplane The tailplane comprises the tail-mounted fixed horizontal stabiliser and movable elevator. Besides its planform, it is characterised by: • Number of tailplanes - from 0 (tailless or canard) to 3 (Roe triplane) • Location of tailplane - mounted high, mid or low on the fuselage, fin or tail booms. • Fixed stabilizer and movable elevator surfaces, or a single combined stabilator or (all) flying tail.[1] (General Dynamics F-111) Some locations have been given special names: • Cruciform: mid-mounted on the fin (Hawker Sea Hawk, Sud Aviation Caravelle) • T-tail: high-mounted on the fin (Gloster Javelin, Boeing 727) Sud Aviation Caravelle Gloster Javelin
  • 135. SOLO 135 Aircraft Flight Control Specific Stabilizer/Tail Configurations Tailplane Some locations have been given special names: • V-tail: (sometimes called a Butterfly tail) • Twin tail: specific type of vertical stabilizer arrangement found on the empennage of some aircraft. • Twin-boom tail: has two longitudinal booms fixed to the main wing on either side of the center line. The V-tail of a Belgian Air Force Fouga Magister de Havilland Vampire T11, Twin-Boom Tail A twin-tailed B-25 Mitchell
  • 136. SOLO 136 Aircraft Avionics Aerodynamics of Flight Return to Table of Content
  • 138. SOLO 138 Control Surfaces Aircraft Flight Control Return to Table of Content