Chapter 13 Compressible Thin Airfoil Theory

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CHAPTER 13 COMPRESSIBLE THIN AIRFOIL THEORY

13.1.1 THE FULL POTENTIAL EQUATION In compressible flow, both the lift and drag of a thin airfoil can be determined to a reasonable level of accuracy from an inviscid, irrotational model of the flow. Recall the equations developed in Chapter 6 governing steady, irrotational, homentropic (¢s0= ) flow in the absence of body forces.

¢l()U = 0 UU ¢P ¢£¥------+0------= ¤¦2 l . (13.1) P l a ------= £¥------¤¦ P0 l0 The gradient of the isentropic relation is

2 ¢Pa= ¢l. (13.2) Recall from the development in Chapter 6 that

P a – 1 ¢P ¢£¥--- = £¥------(13.3) ¤¦l ¤¦a l Using (13.3) the momentum equation becomes

a P UU ¢£¥£¥------+ ------= 0 (13.4) ¤¦¤¦a – 1 l 2 Substitute (13.2) into the continuity equation and use (13.3). The continuity equation becomes

2 2 U ¢a + ()a – 1 a ¢ U = 0 . (13.5) Equate the Bernoulli integral to free stream conditions. bjc 13.1 5/31/13

Compressible potential flow

2 2 2 2 a UU a' U' a' a – 1 2 ------+ ------==------+ ------£¥1 + ------M =C T (13.6) a – 1 2 a – 1 2 a – 1¤¦2 ' p t Note that the momentum equation is essentially equivalent to the statement that

the stagnation temperature T t is constant throughout the ﬂow. Using (13.6) we can write

2 a UU £¥------= h – ------(13.7) ¤¦a – 1 t 2 The continuity equation ﬁnally becomes

UU UU ()a – 1 £¥h – ------¢ UU– ¢£¥------= 0 (13.8) ¤¦t 2 ¤¦2 The equations governing compressible, steady, inviscid, irrotational motion reduce to a single equation for the velocity vector U . The irrotationality condition ¢ × U = 0 permits the introduction of a velocity potential.

U = ¢\ (13.9) and (13.8) becomes

¢\ ¢\ 2 ¢\ ¢\ ()a – 1 £¥h – ------¢ \¢\¢– £¥------= 0 . (13.10) ¤¦t 2 ¤¦2

For complex body shapes numerical methods are normally used to solve for \ . However the equation is of relatively limited applicability. If the flow is over a thick airfoil or a bluff body for instance then the equation only applies to the subsonic Mach number regime at Mach numbers below the range where shocks begin to appear on the body. At high subsonic and supersonic Mach numbers where there are shocks then the homentropic assumption (13.2) breaks down. Equation (13.8) also applies to internal flows without shocks such as fully expanded nozzle flow.

13.1.2 THE NONLINEAR SMALL DISTURBANCE APPROXIMATION In the case of a thin airfoil that only slightly disturbs the ﬂow, equation (13.8) can be simpliﬁed using small disturbance theory.

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Consider the ﬂow past a thin 3-D airfoil shown below.

v y=f(x,z) U' + u U'

y x z

The velocity ﬁeld consists of a freestream ﬂow plus a small disturbance

UU= ' + u Vv= (13.11) Ww= where

. (13.12) uU ' « 1 , v U' « 1 , w U' « 1 Similarly the state variables deviate only slightly from freestream values.

PP= ' + P'

TT= ' + T' (13.13)

ll= ' + l' and

. (13.14) aa= ' + a' This decomposition of variables is substituted into equation (13.8). Various terms are

2 2 2 2 UU U' u v w ------= ------uU++++------2 2 ' 2 2 2 (13.15)

¢ U = ux ++vy wz and

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UU ¢£¥------= (u U +++uu vv ww , ¤¦2 x ' x x x (13.16) uyU' +++uuy vvy wwy,

uzU' +++uuz vvz wwz ) as well as

UU ()a – 1 £¥h – ------¢ U = ¤¦t 2

2 2 2 2 £¥£¥U' u v w ()a – 1 ²´ht – ²´------++++uU' ------ux + ¤¦¤¦2 2 2 2 (13.17) 2 2 2 2 £¥£¥U' u v w ()a – 1 ²´ht – ²´------++++uU' ------vy + ¤¦¤¦2 2 2 2

2 2 2 2 £¥£¥U' u v w ()a – 1 ²´ht – ²´------++++uU' ------wz ¤¦¤¦2 2 2 2 and, ﬁnally

UU 2 U ¢£¥------= u U +++uu U vv U ww U + ¤¦2 x ' x ' x ' x ' 2 uu U +++u u uvv uww + x ' x x x . (13.18) 2 vuyU' +++vuuy v vy vwwy + 2 wuzU' +++wuuz wvvz w wz Neglect terms in (13.17) and (13.18) that are of third order in the disturbance velocities. Now

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UU UU ()a – 1 £¥h – ------¢ UU– ¢£¥------ ¤¦t 2 ¤¦2 2 £¥£¥U' ()a – 1 ²´ht – ²´------+ uU' ux + ¤¦¤¦2 2 £¥£¥U' (13.19) ()a – 1 ²´ht – ²´------+ uU' vy + ¤¦¤¦2 2 £¥£¥U' 2 ()a – 1 ²´ht – ²´------+ uU' wz – ()uxU' +++uuxU' vvxU' wwxU' – ¤¦¤¦2

uuxU' – vuyU' – wuzU' or

UU UU ()a – 1 £¥h – ------¢ UU– ¢£¥------ ¤¦t 2 ¤¦2 2 (13.20) ()a – 1 ()h' – uU' ()ux ++vy wz – ()uxU' +++uuxU' vvxU' wwxU'

–uuxU' – vuyU' – wuzU'

2 Recall that ()a – 1 h' = a' . Equation (13.20) can be rearranged to read

UU UU ()a – 1 £¥h – ------¢ UU– ¢£¥------ ¤¦t 2 ¤¦2 2 2 (13.21) a'()ux ++vy wz – U'ux – ()a + 1 uuxU' – () vvxU' + wwxU'

()a – 1 ()uvyU' + uwzU' – vuyU' – wuzU'

2 Divide through by a' .

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UU UU ()a – 1 £¥h – ------¢ UU– ¢£¥------ ¤¦t 2 ¤¦2

2 ()a + 1 M' ()1M– ' ux ++vy wz – ------uux – (13.22) a'

M' ------()()a – 1 ()uvy + uwz ++vuy wuz ++vvx wwx a' This equation contains both linear and quadratic terms in the velocity disturbances and one might expect to be able to neglect the quadratic terms. But note that the

ﬁrst term becomes very small near M' = 1 . Thus in order to maintain the small disturbance approximation at transonic Mach numbers the uux term must be retained. The remaining quadratic terms are small at all Mach numbers and can be dropped. Finally the small disturbance equation is

2 ()a + 1 M' . (13.23) ()1M– ' ux ++vy wz – ------uux = 0 a' The velocity potential is written in terms of a freestream potential and a disturbance potential

. (13.24) \ = U'x + q()xyz,, The small disturbance equation in terms of the disturbance potential becomes

2 M' (13.25) ()1M– ' qxx ++qyy qzz = ()a + 1 ------qxqxx a' Equation (13.25) is valid over the whole range of subsonic, transonic and supersonic Mach numbers.

13.1.3 LINEARIZED POTENTIAL FLOW If we restrict our attention to subsonic and supersonic ﬂow, staying away from Mach numbers close to one, the nonlinear term on the right side of (13.25) can be dropped and the small disturbance potential equation reduces to the linear wave equation.

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2 .` qxx – ()qyy + qzz = 0 (13.26)

2 where ` = M' – 1 . In two dimensions

2 ` qxx – qyy = 0 . (13.27) The general solution of (13.27) can be expressed as a sum of two arbitrary functions

q()xy, = Fx()– `y + Gx()+ `y . (13.28)

2 Note that if M' < 1 the 2-D linearized potential equation (13.27) is an elliptic equation that can be rescaled to form Laplace’s equation and (13.28) expresses the solution in terms of conjugate complex variables. In this case the subsonic ﬂow can be analyzed using the methods of complex analysis. Presently we will restrict our attention to the supersonic case. The subsonic case is treated later in the chapter.

2 If M' > 1 then (13.27) is the 2-D wave equation and has solutions of hyperbolic type. Supersonic flow is analyzed using the fact that the properties of the flow are constant along the characteristic lines x ± `y = constan t . The figure below illustrates supersonic flow past a thin airfoil with several characteristics shown. Notice that in the linear approximation the characteristics are all parallel to one another and lie at the Mach angle µ' of the free stream. Information about the flow is carried in the value of the potential assigned to a given characteristic and in the spacing between characteristics for a given flow change. Right-leaning characteristics carry the information about the flow on the upper surface of the wing and left-leaning characteristics carry information about the flow on the lower surface.

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x – `y = constant y

µ' x

M' > 1

µ -1 ' = Sin (1/M') x +const`y = tan

All properties of the flow, velocity, pressure, temperature, etc. are constant along the characteristics. Since disturbances only propagate along downstream running characteristics we can write the velocity potential for the upper and lower surfaces as q()xy, = Fx()– `y y>0 . (13.29) q()xy, = Gx()+ `y y<0 Let yfx= () define the coordinates of the upper surface of the wing and ygx= () define the lower surface. The full nonlinear boundary condition on the upper surface is v df ---- = ------. (13.30) U dx yf= In the spirit of the thin airfoil approximation this boundary condition can be approximated by the linearized form v df ------= ------(13.31) U' dx y0= which we can write as

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,q()xy, df ------= U £¥------(13.32) ,y '¤¦dx y0= or

U' df F'()x = –------£¥------. (13.33) ` ¤¦dx On the lower surface the boundary condition is

U' dg G'()x = ------£¥------. (13.34) ` ¤¦dx In the thin airfoil approximation the airfoil itself is, in effect, collapsed to a line along the x-axis, the velocity potential is extended to the line y0= and the surface boundary condition is applied at y0= instead of at the physical airfoil surface. The entire effect of the airfoil on the flow is accounted for by the vertical velocity perturbation generated by the local slope of the wing. The linearized boundary condition is valid on 2-D thin wings and on 3-D wings of that are of “thin planar form”. Recall that (13.26) is only valid for subsonic and supersonic flow and not for tran- 2 sonic flow where ()1M– ' « 1 .

13.1.4 THE PRESSURE COEFFICIENT Let’s work out the linearized pressure coefﬁcient. The pressure coefﬁcient is

PP– ' 2 P C ==------£¥------– 1 . (13.35) P 1 2 2 ¤¦P ---l U a M ' 2 ' ' ' The stagnation enthalpy is constant throughout the ﬂow thus

T 1 2 2 2 2 . (13.36) ------= 1 + ------()U' – ()U ++v w T ' 2CpT ' Similarly the entropy is constant and thus the pressure and temperature are related by

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a ------P 1 2 2 2 2 a – 1 ------= £¥ 1 + ------()U – ()U ++v w (13.37) ¤¦' P' 2CPT ' and the pressure coefﬁcient is

a ¨¬------2 ««1 2 2 2 2 a – 1 C = ------£¥1 + ------()U – ()U ++v w – 1 . (13.38) P 2 ©¤¦2C T ' a M ««p ' 'ª® The velocity term in (13.37) is small

2 2 2 2 2 2 2 . (13.39) U' – ()U ++v w = –()2uU' +++u v w

n 2 Now use the binomial expansion ()1 – ¡ 1n– ¡ + nn()– 1¡  2 to expand the term in parentheses in (13.39). Note that the expansion has to be carried out to second order. The pressure coefﬁcient is approximately

2 2 2 £¥2u 2 u v + w C – ------++() 1– M ------. (13.40) P ²´' 2 2 ¤¦U' U' U' Equation (13.40) is a valid approximation for small perturbations in subsonic or supersonic flow. For 2-D flows over planar bodies it is sufficient to retain only the first term in (13.40) and we use the expression u CP –2------. (13.41) U' For 3-D flows over slender approximately axisymmetric bodies we must retain the last term and so

2 2 £¥2u v + w C – ------+ ------. (13.42) P ²´2 ¤¦U' U'

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As was discussed above, if the airfoil is a 2-D shape deﬁned by the function yfx= () the boundary condition at the surface is df v ------==------tane (13.43) dx U' + u where e is the angle between the airfoil surface and the horizontal. For a thin airfoil this is accurately approximated by

df qy ------------e . (13.44) dx U' For a thin airfoil in supersonic ﬂow the wall pressure coefﬁcient is

2 df C = ------£¥------. (13.45) Pwall 2 12 ¤¦dx ()M' – 1

2 2 2 12 where dU  U =M–£¥2M ()– 1 de , which is valid for * 1 , has ¤¦' ' been used. In the thin airfoil approximation in supersonic flow the local pressure coefficient is determined by the local slope of the wing. The figure below shows the wall pressure coefficient on a thin, symmetric biconvex wing.

(-) x (+)

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This airfoil will have shock waves at the leading and trailing edges and at ﬁrst sight this would seem to violate the isentropic assumption. But for small disturbances the shocks are weak and the entropy changes are negligible.

13.1.5 DRAG COEFFICIENT OF A THIN SYMMETRIC AIRFOIL A thin, 2-D, symmetric airfoil is situated in a supersonic stream at Mach number M' and zero angle of attack. The y-coordinate of the upper surface of the airfoil is given by the function /x yx()= A Sin£¥------(13.46) ¤¦C where C is the airfoil chord. The airfoil thickness to chord ratio is small, 2A C « 1 . Determine the drag coefﬁcient of the airfoil. Solution The drag integral is C (13.47) D2= () PP– ' Sin()_ dx 00 where the factor of 2 accounts for the drag of both the upper and lower surfaces and _ is the local angle formed by the upper surface tangent to the airfoil and the x-axis. Since the airfoil is thin the angle is small and we can write the drag coefﬁcient as

1 £¥ D PP– ' x C ==------2 ²´------()_ d£¥---- (13.48) D 1 2 ²´1 2 ¤¦C ---l U C 00 ¤¦---l U 2 ' ' 2 ' ' The local tangent is determined by the local slope of the airfoil therefore Tan()_ = dy dx and for small angles _ = dy dx . Now the drag coefﬁ- cient is

1 £¥ D PP– ' dy x C ==------2 ²´------£¥------d£¥---- (13.49) D 1 2 ²´1 2 ¤¦dx ¤¦C ---l U C 00 ¤¦---l U 2 ' ' 2 ' ' The pressure coefﬁcient on the airfoil is given by thin airfoil theory (13.45) as

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PP– ' 2 dy C ==------£¥------(13.50) P 1 2 2 ¤¦dx ---l U M – 1 2 ' ' ' and so the drag coefﬁcient becomes

2 2 2 2 4 1 £¥dy £¥x 4A / / 2£¥/x £¥/x 2A / CD ==------d ------Cos ------d ------=------(13.51) 2 00 ¤¦dx ¤¦C 2 2 00 ¤¦C ¤¦C 2 2 M' – 1 C M' – 1 C M' – 1 The drag coefﬁcient is proportional to the square of the wing thickness-to-chord ratio.

13.1.6 THIN AIRFOIL WITH LIFT AND CAMBER AT A SMALL ANGLE OF ATTACK A thin, cambered, 2-D, airfoil is situated in a supersonic stream at Mach number M' and a small angle of attack as shown below. y

S' x y =

The y-coordinate of the upper surface of the airfoil is given by the function

x x x bx f £¥---- = Ao£¥---- + Bm£¥---- – ------(13.52) ¤¦C ¤¦C ¤¦C C and the y-coordinate of the lower surface is

x x x bx g£¥---- = –Ao£¥---- + Bm£¥---- – ------(13.53) ¤¦C ¤¦C ¤¦C C where 2A C « 1 and b  C « 1 . The airfoil surface is deﬁned by a dimensionless thickness function

x o£¥---- ; o()0 ==o()1 0 , (13.54) ¤¦C a dimensionless camber function

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x m£¥---- ; m()0 ==m()1 0 (13.55) ¤¦C and the angle of attack

b Tan()_ = –---- (13.56) C Determine the lift and drag coefﬁcients of the airfoil. Let

x j = ---- (13.57) C Solution The lift integral is

C C LP= ()– P Cos()_ dx – ()P – P Cos()_ dx (13.58) 00 lower ' lower 00 upper ' upper where _ is the local angle formed by the tangent to the surface of the airfoil and the x-axis. Since the angle is everywhere small we can use Cos()_ 1 . The lift coefﬁcient is

D 1 1 C ==------Cdj – C dj (13.59) L 1 2 Plower Pupper ---l U C 00 00 2 ' ' The pressure coefﬁcient on the upper surface is given by thin airfoil theory as

2 £¥df 2 £¥do dm CP ==------A------+ B------– b (13.60) upper 2 ¤¦dj 2 ¤¦dj dj CM' – 1 CM' – 1 and on the lower surface the pressure coefﬁcient is

2 £¥dg 2 £¥do dm CP ==–------–------–A------+ B------– b . (13.61) lower 2 ¤¦dj 2 ¤¦dj dj CM' – 1 CM' – 1 Substitute into the lift integral

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–2 £¥1 df 1 dg –2 £¥–b –b CL ==------dj + ------dj ------df+ dg (13.62) 2 ¤¦00 dj 00 dj 2 ¤¦00 00 CM' – 1 CM' – 1 In the thin airfoil approximation the lift is independent of the airfoil thickness.

4 b C = ------£¥---- (13.63) L 2 ¤¦C M' – 1 The drag integral is

C C DP= ()– P Sin()_ dx + ()P – P Sin()–_ dx (13.64) 00 upper ' upper 00 lower ' lower Since the airfoil is thin the angle is small and we can write the drag coefﬁcient as

(13.65) £¥ £¥ D 1 Pupper – P' 1 Plower – P' C ==------²´------()_ dj + ²´------()–_ dj D 1 2 00 ²´1 2 upper 00 ²´1 2 lower ---l U C ¤¦---l U ¤¦---l U 2 ' ' 2 ' ' 2 ' ' The local tangent is determined by the local slope of the airfoil therefore

Tan()_ = dy dx and for small angles _ = dy dx . Now the drag coefﬁcient is

D 1 1 dyupper 1 1 dylower C ==------C £¥------dj + ---- C £¥–------dj (13.66) D 1 2 C00 Pupper¤¦dj C00 Plower¤¦dj ---l U C 2 ' ' The drag coefﬁcient becomes

2 £¥1 do dm 2 1 do dm 2 C = ------²´£¥A------+ B------– b dj + £¥–A------+ B------– b dj (13.67) D 2 2 ¤¦00 ¤¦dj dj 00 ¤¦dj dj C M – 1 ' Note that most of the cross terms cancel. The drag coefﬁcient breaks into several terms.

4 £¥A 2 1 do 2 B 2 1 dm 2 B b 1 dm b 2 1 C = ------²´£¥---- £¥------dj ++£¥---- £¥------dj–£¥------£¥------dj £¥---- dj (13.68) D 2 ¤¦¤¦C 00 ¤¦dj ¤¦C 00 ¤¦dj ¤¦C C00 ¤¦dj ¤¦C 00 M – 1 '

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The third term in (13.68) is

1 1 B b dm B b B b £¥------£¥------dj ==£¥------dm £¥------[]m()1 – m()0 =0 (13.69) ¤¦ ¤¦ ¤¦ ¤¦ C C00 dj C C00 C C Finally

4 A 2 1 do 2 B 2 1 dm 2 b 2 C = ------£¥£¥---- £¥------dj ++£¥---- £¥------dj £¥---- . (13.70) D 2 ¤¦¤¦C 00 ¤¦dj ¤¦C 00 ¤¦dj ¤¦C M – 1 ' In the thin airfoil approximation, the drag is a sum of the drag due to thickness, drag due to camber and drag due to lift.

13.2 SIMILARITY RULES FOR HIGH SPEED FLIGHT The figure below shows the flow past a thin symmetric airfoil at zero angle of attack. The fluid is assumed to be inviscid and the flow Mach number U  a , '1 ' 2 where a' = a P'  l' is the speed of sound, is assumed to be much less than

one. The airfoil chord is ct and the maximum thickness is 1 . The subscript one is applied in anticipation of the fact that we will shortly scale the airfoil to a new shape with subscript two and the same chord.

c U '1 t1

–C p 0.2 4x 1 ------2 -1 1 2 c -0.2 -0.4 -0.6 -0.8 -1

Figure 13.1 Pressure variation over a thin symmetric airfoil in low speed ﬂow.

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The surface pressure distribution is shown below the wing, expressed in terms of the pressure coefﬁcient.

Ps – P' C = ------. (13.71) P1 1 2 ---l U 2 ' '1 The pressure and ﬂow speed throughout the ﬂow satisfy the Bernoulli relation, in particular, near the airfoil surface,

1 2 1 2 P +P---l U = + ---l U . (13.72) ' 2 ' '1 s1 2 ' s1 The pressure is high at the leading edge where the flow stagnates, then as the flow accelerates about the body, the pressure falls rapidly at first, then more slowly reaching a minimum at the point of maximum thickness. From there the surface velocity decreases and the pressure increases continuously to the trailing edge. In the absence of viscosity, the flow is irrotational.

¢ × u1 = 0 (13.73) this permits the velocity to be described by a potential function.

u1 = ¢q1 (13.74) when this is combined with the condition of incompressibility, ¢ u1 = 0 , the result is Laplace’s equation.

2 2 , q1 , q1 ------+0------= . (13.75) 2 2 , x1 , y1

Let the shape of the airfoil surface in ()x1, y1 be given by

y1 ----- = o gx[] c (13.76) c 1 1 where o1 = t1  c is the thickness to chord ratio of the airfoil. The boundary conditions that the velocity potential must satisfy are,

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£¥,q1 £¥dy1 dg[] x1  c ²´------==U ²´------U o ------y x1 '1 dx ' 1 dx c ¤¦, 1 y = co g£¥----- ¤¦1 ()1  1 1 ¤¦c body . (13.77)

q1 = U'1 x1 A ' Any number of methods of solving for the velocity potential are available includ- ing the use of complex variables. In the following we are going to restrict the airfoil to be thin, o1 « 1 . In this context we will take the velocity potential to be a perturbation potential so that.

(13.78) u1 ==U'1 +vu1' ; v1 1' or

,q1 ,q1 . (13.79) u1 ==U'1 + ------; v1 ------,x1 ,y1 The boundary conditions on the perturbation potential in the thin airfoil approximation are,

¬ £¥,q1 £¥dy1 dg[] x1  c ²´------==U ²´------U o ------« ,y '1 dx ' 1 dx() c « ¤¦1 y1 = 0 ¤¦1 body 1 . (13.80) « q = 0 1x A ' « 1 ® The surface pressure coefﬁcient in the thin airfoil approximation is,

2 £¥,q1 C = –------²´------. (13.81) P1 U ,x '1¤¦1 y1 = 0 Note that the boundary condition on the vertical velocity is now applied on the line y1 = 0 . In effect the airfoil has been replaced with a line of volume sources whose strength is proportional to the local slope of the actual airfoil. This sort of approximation is really unnecessary in the low Mach number limit but it is essen-

5/31/13 13.18 bjc Similarity rules for high speed flight tial when the Mach number is increased and compressibility effects come in to play. Equally, it is essential in this example where we will map a compressible ﬂow to the incompressible case.

13.2.1 SUBSONIC FLOW M' < 1 Now imagine a second ﬂow at a free stream velocity, U in a new space ()x , y '2 2 2 over a new airfoil of the same shape (deﬁned by the function gx[] c ) but with a new thickness ratio o2 = t2  c « 1 . Part of what we need to do is to determine how o1 and o2 are related to one another. The boundary conditions that the new perturbation velocity potential must satisfy are,

£¥,q2 £¥dy2 dg[] x1  c ²´------==U ²´------U o ------,y '2 dx '2 2 dx() c ¤¦2 y2 = 0 ¤¦2 body 1 . (13.82)

q2 = 0 x2 A ' In this second flow the Mach number has been increased to the point where compressibility effects begin to occur: the density begins to vary significantly and the pressure distribution begins to deviate from the incompressible case. As long as the Mach number is not too large and shock waves do not form, the flow will be nearly isentropic. In this instance the 2-D steady compressible flow equations are,

,u2 ,u2 1 ,P u2------++v2------= 0 ,x2 ,y2 l,x2

,v2 ,v2 1 ,P u2------++v2------= 0 ,x2 ,y2 l,y2 . (13.83) ,l ,l ,u2 ,v2 u2------+++v2------l------l------= 0 ,x2 ,y2 ,x2 ,y2 P l a ------= £¥------¤¦ P' l' Let

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(13.84) u2 ====U'2 +vu'2 ; v2 '2 ; ll' +Pl' ; P ' + P' where the primed quantities are assumed to be small compared to the free stream conditions. When quadratic terms in the equations of motion are neglected the equations (13.83) reduce to,

2 £¥l U ' '2 ,u'2 ,v'2 ²´1 – ------+0------= . (13.85) ²´a P ,x ,y ¤¦' 2 2 Introduce the perturbation velocity potential

,q2 ,q2 u'2 ==------; v'2 ------. (13.86) ,x2 ,y2 The equation governing the disturbance ﬂow becomes,

2 2 2 , q2 , q2 ()1M– ------+0------= . (13.87) '2 2 2 ,x2 ,y2 Notice that (13.87) is valid for both sub and supersonic ﬂow in the thin airfoil approximation. Since the ﬂow is isentropic, the pressure and velocity disturbances are related to lowest order by,

(13.88) P' +0l'U'2u2' = and the surface pressure coefﬁcient retains the same basic form as in the incompressible case,

2 £¥,q2 C = –------²´------. (13.89) P2 U ,x '2¤¦2 y2 = 0 Since we are at a ﬁnite Mach number, this last relation is valid only within the thin airfoil, small disturbance approximation and therefore may be expected to be invalid near the leading edge of the airfoil where the velocity change is of the order of the free stream velocity. For example for the “thin” airfoil depicted in Figure

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13.1 which is actually not all that thin, the pressure coefﬁcient is within

–0.2< CP < 0.2 except over a very narrow portion of the chord near the leading edge. Equation (13.87) can be transformed to Laplace’s equation, (13.75) using the following change of variables.

1 1 £¥U'2 x2 ==x1 ; y2 ------y1 ; q2 =--- ²´------q1 (13.90) 2 A¤¦U'1 1M– '2 where, at the moment, A is an arbitrary constant. The velocity potentials are related by,

1 £¥U'2 q2[]x2, y2 = --- ²´------q1[]x1, y1 (13.91) A¤¦U'1 or,

£¥U'1 1 q1[]x1, y1 = A²´------q2 x1, ------y1 (13.92) ¤¦U'2 2 1M– '2 and the boundary conditions transform as,

£¥ £¥,q1 Ao2 dg[] x1  c ¬ ²´------= U ²´------« ,y '1²´2 dx() c « ¤¦1 y1 = 0 1 ¤¦1M– '2 . (13.93) « « q1 ==q2 0 x1 A ' x2 A ' ® The transformation between ﬂows one and two is completed by the correspondence,

Ao2 o1 = ------(13.94) 2 1M– '2 or

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t1 A t2 £¥---- = ------£¥---- . (13.95) ¤¦c 2 ¤¦c 1M– '2 Finally the transformed pressure coefﬁcient is

CP1 = ACP2. (13.96) These results may be stated as follows. The solution for incompressible ﬂow over a thin airfoil with shape gx[]1  c and thickness ratio, t1  c at velocity U1 is iden- tical to the subsonic compressible ﬂow at velocity, U2 and Mach number M2 over an airfoil with a similar shape but with the thickness ratio,

2 t2 1M– '2 t1 ---- = ------£¥---- . (13.97) c A ¤¦c The pressure coefﬁcient for the compressible case is derived by adjusting the incompressible coefﬁcient using CP2 = CP1  A . This result comprises several different similarity rules that can be found in the aeronautical literature depending on the choice of the free constant, A .

M= 0.5 –C M= 0.75 P

4x ------2 -1 1 2 c

-0.5

M0= -1

-1.5

Figure 13.2 Pressure coefﬁcient over the airfoil in Figure 13.1 at several Mach numbers as estimated using the Prandtl-Glauert rule (13.99)

Perhaps the one of greatest interest is the so-called Prandtl-Glauert rule that describes the variation of pressure coefﬁcient with Mach number for a body of a given shape and thickness ratio. In this case we select,

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2 (13.98) A1M= – '2 so that the two bodies being compared in (13.97) have the same shape and thickness ratio. The pressure coefﬁcient for the compressible ﬂow is,

CP1 CP2 = ------. (13.99) 2 1M– '2 Several scaled proﬁles are shown in Figure 13.2. Keep in mind the lack of validity of (13.99) near the leading edge where the pressure coefﬁcient is scaled to inac- curate values.

13.2.2 SUPERSONIC SIMILARITY M' > 1 All the theory developed in the previous section can be extended to the supersonic 2 2 case by simply replacing 1M– ' with M' – 1 . In this instance the mapping is between the equation,

2 2 2 , q2 , q2 ()M – 1 ------– ------= 0 (13.100) '2 2 2 ,x2 ,y2 and the simple wave equation

2 2 , q1 , q1 ------– ------= 0 (13.101) 2 2 ,x1 ,y1 A generalized form of the pressure coefﬁcient valid for subsonic and supersonic ﬂow is,

C p o ------F= ------(13.102) A 2 A1M– '

2 where A is taken to be a function of 1M– ' .

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13.2.3 TRANSONIC SIMILARITY, M' 1 When the Mach number is close to one, the simple linearization used to obtain (13.87) from (13.83) loses accuracy. In this case the equations (13.83) reduce to the nonlinear equation

2 2 ,u'1 ,v'1 ()a 1 + 1 M'1 ,u'1 . (13.103) ()1M– '1 ------+0------– ------u'1------= ,x1 ,y1 U'1 ,x1 In terms of the perturbation potential,

2 2 2 2 2 , q1 , q1 ()a 1 + 1 M'1,q1, q1 ()1M– ------+0------– ------= . (13.104) '1 2 2 2 U'1 ,x1 ,x1 ,y1 ,x1 This equation is invariant under the change of variables,

2 1M– '1 1 £¥U'2 x2 ==x1 ; y2 ------y1 ; q2 = --- ²´------q1 (13.105) 2 A¤¦U'1 1M– '2 where

£¥2 £¥2 £¥1 + a 2 1M– '1 M'2 A = ------²´------²´------. (13.106) ²´²´2 ²´2 ¤¦1 + a 1 ¤¦1M– '2 ¤¦M'1 Notice that, due to the nonlinearity of the transonic equation (13.104), the constant A is no longer arbitrary. The pressure coefﬁcient becomes,

£¥2 £¥2 £¥1 + a 2 1M– '1 M'2 C = ------²´------²´------C (13.107) P1 ²´²´2 ²´2 p2 ¤¦1 + a 1 ¤¦1M– '2 ¤¦M'1 and the thickness ratios are related by,

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3 --- £¥2 2£¥2 t2 £¥1 + a 1 1M– '2 M'1 t1 ---- = ------²´------²´------. (13.108) ²´²´2 ²´2 c ¤¦1 + a 2 c ¤¦1M– '1 ¤¦M'2 In the transonic case, it is not possible to compare the same body at different Mach numbers or bodies with different thickness ratios at the same Mach number except by selecting gases with different a . For a given gas it is only possible to map the pressure distribution for one airfoil to an airfoil with a different thickness ratio at a different Mach number. A generalized form of (13.102) valid from subsonic to sonic to supersonic Mach numbers is,

2 13 2 CP()()a + 1 M' 1M– ' ------= F ------. (13.109) 23 23 o 2 ()oa()+ 1 M' Prior to the advent of supercomputers capable of solving the equations of high speed ﬂow, similarity methods and wind tunnel correlations were the only tools available to the aircraft designer and these methods played a key role in the early development of transonic and supersonic ﬂight.

13.3 THE EFFECT OF SWEEP Return to the continuity equation (13.8) written as

1 UU ¢ U – ----- U ¢£¥------= 0 (13.110) 2 ¤¦ a 2

Using the irrotationality condition ¢ × U = 0 , (13.110) can expressed as (13.111) 2 2 2 £¥U ,U £¥V ,V £¥W ,W UV,U WV,V UW,U 1 – ------++1 – ------1 – ------– 2£¥------++------= 0 ²´2 ²´2 ²´2 ¤¦2 2 2 ¤¦a ,x ¤¦a ,y ¤¦a ,z a ,y a ,z a ,z

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The figure below shows a plan view of of a long slender wing with the free stream flow approaching at a sweep angle ` . The component of the free stream flow parallel to the long axis of the wing is assumed to play little role in determining the pressure variation over the wing except at the wing tips.

Figure 13.3 Slender wing with free stream approaching at sweep angle `

Therefore the ﬂow ﬁeld in the ()xy, plane normal to the long axis of the wing will be nearly two-dimensional so that ,W  ,z and ,U  ,z can be assumed to be very small. Now

2 2 £¥U ,U £¥V ,V V U ,U W ,V ²´1 – ------+ ²´1 – ------– 2----£¥------+ ------= 0 (13.112) 2 x 2 y a ¤¦a y a z ¤¦a , ¤¦a , , , Assume small disturbances.

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UU= 'Cos()` + u Vv= (13.113) WU= 'Sin()` + w

aa= ' + a' Neglect terms that are cubic in the small disturbances. Equation (13.112) becomes (13.114) £¥2 2 U'Cos ()` ,U ,V V £¥U'Cos()` ,U U'Sin()` ,V ²´1 – ------+ ------– 2------+ ------= 0 ²´2 ²´ ,x ,y a'¤¦a' ,y a' ,z ¤¦a'

The quantity Va ' is very small and we can neglect terms in (13.114) that are quadratic in the disturbances. Introduce the disturbance potential. Equation (13.111) reduces to

2 2 2 2 , q , q ()1M– Cos ()` ------+ ------= 0 (13.115) ' 2 2 ,x ,y For the swept wing the disturbance potential is governed by the component of the free stream velocity normal to the leading edge of the wing U'Cos()` . If 2 2 M'Cos ()` > 1 then the normal ﬂow is supersonic and we would use supersonic theory developed above to relate the pressure coefﬁcient to the slope of the airfoil surface in the ()xy, plane.

2 2 If M'Cos ()` < 1 then the normal flow is subsonic and we can use the mapping (13.90) to transform (13.115) to Laplace’s equation. The methods of subsonic thin airfoil theory can be used to determine the M0= disturbance potential q1 . The pressure coefficient of the M0= solution is used with the Prandtl-Glauert rule (13.99) to determine the pressure coefficient on the swept wing.

CP1 C = ------(13.116) PM Cos()` ' 2 2 1M– 'Cos ()`

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The best discussion of this topic that I know of appears in NACA Technical Report 863 by R. T. Jones published in 1945. The rest of the discussion of this topic comprises images taken directly from this report. Where Jones refers to equations (4) and (6) he is referring to Laplaces equation and the wave equation.

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5/31/13 13.36 bjc Problems

13.4 PROBLEMS Problem 1 - A thin, 2-D, airfoil is situated in a supersonic stream at Mach number M' and a small angle of attack as shown below. y

S' x y =

The y-coordinate of the upper surface of the airfoil is given by the function

x x b fx()= A----£¥1 – ---- – ----x C¤¦C C and the y-coordinate of the lower surface is

x x b gx()= –A----£¥1 – ---- – ----x C¤¦C C where2A C « 1 and b  C « 1 . Determine the lift and drag coefﬁcients of the airfoil.

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