5.7 Normal shock waves 5.7.1 1D flow A shock wave is a pressure front which travels at speed through a gas. Shock waves cause an increase in pressure, temperature, density and entropy and a decrease in normal velocity. Equations of state and equations of conser- vation applied to a unit area of shock wave give (see Figure 5.10): State p 1 / 1 T 1 = p 2 / 2 T 2 Mass flow m = 1 u 1 = 2 u 2 Basic fluid mechanics 91 uu 1 p 1 1 p 2 2 Shock wave travels into area of stationary gas Fig. 5.10(a) 1-D shock waves uu 1 p 1 1 p 2 2 Shock wave becomes a stationary discontinuity Fig. 5.10(b) Aircraft shock waves 92 Aeronautical Engineer’s Data Book 2 Momentum p 1 + p 1 u 1 2 = p 2 + 2 u 2 2 2 u 1 u 2 Energy c T 1 + ᎏ = c p T 2 + ᎏ = c p T 0p 2 2 Pressure and density relationships across the shock are given by the Rankine-Hugoniot equations: ␥ + 1 2 ᎏᎏ ᎏᎏ – 1 p ␥ –1 1 2 ᎏᎏ = ᎏᎏ p ␥ + 1 2 1 ᎏᎏ – ᎏᎏ ␥ –1 1 ( ␥ + 1)p ᎏᎏ 2 + 1 2 ( ␥ –1)p 1 ᎏᎏ = ᎏᎏ 1 ␥ + 1 p 2 ᎏᎏ + ᎏᎏ ␥ –1 p 1 Static pressure ratio across the shock is given by: p 1 2 ␥ M 2 2 – ( ␥ – 1) ᎏ = ᎏᎏ p 2 ␥ + 1 Temperature ratio across the shock is given by: T 2 p 2 2 ᎏ = ᎏ ᎏ T 1 p 1 / 1 T 2 2 ␥ M 2 1 – ( ␥ + 1) 2 + ( ␥ – 1)M 2 1 ᎏ = ᎏᎏ ᎏᎏ T 1 ␥ + 1 ( ␥ + 1)M 2 1 Velocity ratio across the shock is given by: From continuity: u 2 /u 1 = 1 / 2 u 2 2 + ( ␥ – 1)M 2 so: ᎏ = ᎏᎏ 1 u 1 ( ␥ + 1)M 2 1 In axisymmetric flow the variables are indepen- dent of so the continuity equation can be expressed as: 1 ∂(R 2 q R ) 1 ∂(sin q ) ᎏ ᎏ + ᎏ ᎏᎏ = 0 R 2 ∂R R sin ∂ Similarly in terms of stream function : ᎏᎏ 93 Basic fluid mechanics 1 ∂ q R = ᎏ ᎏ R 2 sin ∂ 1 ∂ q = ᎏ ᎏ R sin ∂R Additional shock wave data is given in Appen- dix 5. Figure 5.10(b) shows the practical effect of shock waves as they form around a super- sonic aircraft. 5.7.2 The pitot tube equation An important criterion is the Rayleigh super- sonic pitot tube equation (see Figure 5.11). M ᎏ ␥ /( ␥ – 1) 2 1 ␥ + 1 p 02 2 Pressure ratio: ᎏ = p 1 2 M 1 1 p 1 u 1 p 2 p 02 M 2 Fig. 5.11 Pitot tube relations 2 ␥ M 2 1 – ( ␥ – 1) ␥ + 1 5.8 Axisymmetric flows Axisymmetric potential flows occur when bodies such as cones and spheres are aligned ᎏ 94 Aeronautical Engineer’s Data Book y x z R r q R q θ q ϕ θ ϕ ᎏ ∂ Fig. 5.12 Spherical co-ordinates for axisymmetric flows ᎏ ∂R into a fluid flow. Figure 5.12 shows the layout of spherical co-ordinates used to analyse these types of flow. Relationships between the velocity compo- nents and potential are given by: ∂ ∂ R sin 1 ᎏ ∂ 1 ᎏ r ∂ q R = q = q = 5.9 Drag coefficients Figures 5.13(a) and (b) show drag types and ‘rule of thumb’ coefficient values. U U U U Shape Pressure drag Friction drag D P (%) D f (%) 0 100 ≈ 10 ≈ 90 ≈ 90 ≈ 10 100 0 Fig. 5.13(a) Relationship between pressure and fraction drag: ‘rule of thumb’ 95 Basic fluid mechanics d l d d d ␣ l U U U U U Cylinder (flow direction) Shape Dimensional ratio Datum area, A Approximate drag coefficient, C D Cylinder (right angles to flow) Hemisphere (bottomless) Cone d I I/d = 1 0.91 2 0.85 4 0.87 7 0.99 I/d = 1 0.63 2 0.68 5 0.74 10 0.82 40 0.98 ∞ 1.20 I 0.34 II 1.33 a = 60˚ 0.51 a = 30˚ 0.34 1.2 π – d 2 4 π – d 2 4 dl π – d 2 4 π – d 2 4 Bluff bodies Rough Sphere ( Re = 10 6 ) 0.40 Smooth Sphere ( Re = 10 6 ) 0.10 Hollow semi-sphere opposite stream 1.42 Hollow semi-sphere facing stream 0.38 Hollow semi-cylinder opposite stream 1.20 Hollow semi-cylinder facing stream 2.30 Squared flat plate at 90° 1.17 Long flat plate at 90° 1.98 Open wheel, rotating, h / D = 0.28 0.58 Streamlined bodies Laminar flat plate ( Re = 10 6 ) 0.001 Re = 10 6 ) 0.005 0.006 0.025 0.025 0.05 0.05 0.16 0.005 0.09 n.a. Aircraft -general 0.012 M = 2.5 0.016 Airship 0.020–0.025 Helicopter download 0.4–1.2 II Turbulent flat plate ( Airfoil section, minimum Airfoil section, at stall 2-element airfoil 4-element airfoil Subsonic aircraft wing, minimum Subsonic aircraft wing, at stall Subsonic aircraft wing, minimum Subsonic aircraft wing, at stall Aircraft wing (supersonic) Subsonic transport aircraft Supersonic fighter, Fig. 5.13(b) Drag coefficients for standard shapes Section 6 Basic aerodynamics 6.1 General airfoil theory When an airfoil is located in an airstream, the flow divides at the leading edge, the stagna- tion point. The camber of the airfoil section means that the air passing over the top surface has further to travel to reach the trail- ing edge than that travelling along the lower surface. In accordance with Bernoulli’s equation the higher velocity along the upper airfoil surface results in a lower pressure, producing a lift force. The net result of the velocity differences produces an effect equiv- alent to that of a parallel air stream and a rotational velocity (‘vortex’) see Figures 6.1 and 6.2. For the case of a theoretical finite airfoil section, the pressure on the upper and lower surface tries to equalize by flowing round the tips. This rotation persists downstream of the wing resulting in a long U-shaped vortex (see Figure 6.1). The generation of these vortices needs the input of a continuous supply of energy; the net result being to increase the drag of the wing, i.e. by the addition of so-called induced drag. 6.2 Airfoil coefficients Lift, drag and moment (L, D, M) acting on an aircraft wing are expressed by the equations: U 2 Lift (L) per unit width = C L l 2 ᎏ 2 97 Basic aerodynamics An effective rotational velocity (vortex) superimposed on the parallel airstream + + + + + – – – – – – ( a) Pressures equalize by flows ( b) around the tip – – – – – – – – + + + + + + + + Tip Midspan Tip Core of vortex ( c) Finite airfoil ‘Horse-shoe’ vortex persists downstream Fig. 6.1 Flows around a finite 3-D airfoil Camber line edge Chord Camber Thickness Leading edge l L D a U General airfoil section Trailing Profile of an asymmetrical airfoil section Centre line Chord line x t c Fig. 6.2 Airfoil sections: general layout 98 Aeronautical Engineer’s Data Book U 2 Drag (D) per unit width = C D l 2 ᎏ 2 Moment (M) about LE or U 2 1/4 chord = C M l 2 ᎏ 2 per unit width. C L , C D and C M are the lift, drag and moment coefficients, respectively. Figure 6.3 shows typical values plotted against the angle of attack, or incidence, ( ␣ ). The value of C D is small so a value of 10 C D is often used for the characteristic curve. C L rises towards stall point and then falls off dramatically, as the wing enters the stalled condition. C D rises gradually, increasing dramatically after the stall point. Other general relationships are: • As a rule of thumb, a Reynolds number of Re Х 10 6 is considered a general flight condition. • Maximum C L increases steadily for Reynolds numbers between 10 5 and 10 7 . • C D decreases rapidly up to Reynolds numbers of about 10 6 , beyond which the rate of change reduces. • Thickness and camber both affect the maximum C L that can be achieved. As a general rule, C L increases with thickness and then reduces again as the airfoil becomes even thicker. C L generally increases as camber increases. The minimum C D achievable increases fairly steadily with section thickness. 6.3 Pressure distributions The pressure distribution across an airfoil section varies with the angle of attack ( ␣ ). Figure 6.4 shows the effect as ␣ increases, and the notation used. The pressure coefficient C p reduces towards the trailing edge. 99 Basic aerodynamics Characteristics for an asymmetrical ‘infinite-span 2D airfoil’ 75 50 25 0 –25 1.5 1.0 0.5 0 –0.5 –5˚ 20˚15˚ 10 C D 10˚5˚ α L/D L/D C L C L and 10 C D C L = 0 at the no-lift angle (–α) Stall point Characteristic curves of a practical wing 2.0 0.20 1.6 0.16 C L C D C M 1/4 1.2 0.12 0.8 0.08 0.4 0.04 C L C M 1/4 C D 0 0 –0.4 –0.04 –0.08 –0.12 –8˚ –4˚ 0˚ 4˚ 8˚ 12˚ 16˚ 20˚ α Fig. 6.3 Airfoil coefficients 100 Aeronautical Engineer’s Data Book Arrow length represents the magnitude of pressure coefficient C p P ∞ = upstream pressure S Stagnation point (S) moves backwards on the airfoil lower surface ( p – p ∞ ) α Ӎ 5˚ S Pressure coefficient C = p 1 V 2 α Ӎ 12˚ 2 Fig. 6.4 Airfoil pressure coefficient (Cp) 6.4 Aerodynamic centre The aerodynamic centre (AC) is defined as the point in the section about which the pitching moment coefficient (C M ) is constant, i.e. does not vary with lift coefficient (C L ). Its theoreti- cal positions are indicated in Table 6.1. Table 6.1 Position of aerodynamic centre Condition Theoretical positon of the AC ␣ < 10° At approx. 1/4 chord somewhere near the chord line. Section with high At 50% chord. aspect ratio Flat or curved plate: At approx. 1/4 chord. inviscid, incompressible flow [...]... 0.4 0 .6 0.8 1.0 x/c Fig 6. 6 Variation of pressure deterioration (2-D airfoil) 6. 7 Wing loading: semi-ellipse assumption The simplest general loading condition assump tion for symmetric flight is that of the semiellipse The equivalent equations for lift, downwash and induced drag become: For lift: VK0πs L=ᎏ 2 replacing L by CL 1/2V 2S gives: CLVS K0 = ᎏ πs 104 Aeronautical Engineer s Data Book AR... ratio λ = Ct/CR = 1.0 2.0 6 1.8 5 1 .6 4 1.4 Xa.c 1.2 3 1.0 Cr 2 0.8 1 0 .6 0.4 Subsonic 0.2 Supersonic AR tanΛLE Ct/CR = 0.5 1.4 6 E 1.2 Xa.c Cr 5 Sonic T 1.0 4 3 0.8 2 0 .6 1 Unswept T.E 0.4 0.2 Subsonic 0 Supersonic AR tanΛLE Ct/CR = 0.25 Cr Sonic 1.0 Xa.c 6 5 T.E 1.2 0.8 0 .6 4 3 2 Unswept T.E 1 0.4 0.2 Subsonic 0 0 1 tanΛLE β β tanΛLE Supersonic 0 β tanΛLE 1 0 tanΛLE β Fig 6. 7 Wing aerodynamic centre... CL The CP is conventionally shown at distance kCP back from the section leading edge (see Figure 6. 5) Using Lift and drag only cut at the CP C Lift MAC MLE Drag xAC Aerodynamic centre Lift M Drag kCP Centre of pressure (CP) Fig 6. 5 Aerodynamic centre and centre of pressure 102 Aeronautical Engineer s Data Book the principle of moments the following expres sion can be derived for kCP: xAC CMAC kCP =... of M2 – 1 the wing 6. 6.3 Supersonic effects on aerodynamic centre Figure 6. 7 shows the location of wing aerody namic centre for several values of tip chord/root chord ratio (␥) These are empirically based results which can be used as a ‘rule of thumb’ Basic aerodynamics 103 M1 (local) M∞ > Mcrit M∞ > Mcrit –1.2 –0.8 –0.4 –Cp –Cp 0 0.4 0.8 1.2 0 0.2 0.4 0 .6 0.8 1.0 x/c 0 0.2 0.4 0 .6 0.8 1.0 x/c M∞ >... location on the airfoil surface Figure 6. 6 shows approximate forms of the pressure distribution on a two-dimensional airfoil around the critical region Owing to the complex non-linear form of the equations of motion which describe high speed flow, two popular simplifica tions are used: the small perturbation approxima tion and the so-called exact approximation 6. 6.2 Supersonic effects on drag In the... Х1 and CD sin ␣ Х 0 gives: xAC CMAC kCP Х ᎏ – ᎏ c CL 6. 6 Supersonic conditions As an aircraft is accelerated to approach super sonic speed the equations of motion which describe the flow change in character In order to predict the behaviour of airfoil sections in upper subsonic and supersonic regions, compressible flow equations are required 6. 6.1 Basic definitions M Mach number M∞ Free stream Mach... the Oxaxis orientated parallel to the velocity vector V0 (see Figure 7.3) 7.2.4 Motion variables The important motion and ‘perturbation’ variables are force, moment, linear velocity, 108 Aeronautical Engineer s Data Book O xb xw yb,.yw V0 zw zb Conventional body axis system O xb is parallel to the ‘fuselage horizontal’ datum O zb is ‘vertically downwards’ Conventional wind (or‘stability’) axis system:... и Roll rate p = – sin where , , и cos are attitude Pitch rate q = и rates with + sin cos respect to и cos cos Yaw rate r = datum axes и – sin �� � �� � � 110 Aeronautical Engineer s Data Book Inverting gives: и 1 sin tan cos tan и –sin = 0 cos и 0 sin sec cos sec �� � �� � p q r 7.3 The generalized force equations The equations of motions for a... approximations, the following equations can be derived: 9 d xAC ᎏ = ᎏ – ᎏ (CMa) c c dCL where CMa = pitching moment coefficient at distance a back from LE xAC = position of AC back from LE c = chord length 6. 5 Centre of pressure The centre of pressure (CP) is defined as the point in the section about which there is no pitching moment, i.e the aerodynamic forces on the entire section can be represented by... Unswept T.E 1 0.4 0.2 Subsonic 0 0 1 tanΛLE β β tanΛLE Supersonic 0 β tanΛLE 1 0 tanΛLE β Fig 6. 7 Wing aerodynamic centre location: subsonic/ supersonic flight Originally published in The AIAA Aerospace Engineers Design Guide, 4th Edition Copyright © 1998 by The American Institute of Aeronautics and Astronautics Inc Reprinted with permission Basic aerodynamics 105 For downwash velocity (w): K0 w = ᎏ . ᎏ πs 104 Aeronautical Engineer s Data Book 2.0 1.8 1 .6 1.4 X a.c. 1.2 C r 1.0 0.8 0 .6 0.4 0.2 1.4 1.2 1.0 X a.c. 0.8 C r 0 .6 0.4 0.2 0 1.2 1.0 X a.c. 0.8 C r 0 .6 0.4. asymmetrical airfoil section Centre line Chord line x t c Fig. 6. 2 Airfoil sections: general layout 98 Aeronautical Engineer s Data Book U 2 Drag (D) per unit width = C D l 2 ᎏ 2 Moment. 0.20 1 .6 0. 16 C L C D C M 1/4 1.2 0.12 0.8 0.08 0.4 0.04 C L C M 1/4 C D 0 0 –0.4 –0.04 –0.08 –0.12 –8˚ –4˚ 0˚ 4˚ 8˚ 12˚ 16 20˚ α Fig. 6. 3 Airfoil coefficients 100 Aeronautical