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Chapter 14

Thin airfoil theory

14.1 Compressible potential flow

14.1.1 The full potential equation

In compressible flow, both the lift and drag of a thin airfoil can be determined to a reason-able level of accuracy from an inviscid, irrotational model of the flow. Recall the equationsdeveloped in Chapter 6 governing steady, irrotational, homentropic (rs = 0) flow in theabsence of body forces.

r�⇢U�= 0

r✓U · U2

◆+

rP

⇢= 0

P

P0

=

✓⇢

⇢0

◆�

(14.1)

The gradient of the isentropic relation is

rP = a2r⇢. (14.2)

Recall from the development in Chapter 6 that

r✓P

⇢

◆=

✓� � 1

�

◆rP

⇢. (14.3)

14-1

CHAPTER 14. THIN AIRFOIL THEORY 14-2

Using (14.3) the momentum equation becomes

r✓✓

�

� � 1

◆P

⇢+

U · U2

◆= 0. (14.4)

Substitute (14.2) into the continuity equation and use (14.3). The continuity equationbecomes

U ·ra2 + (� � 1) a2r · U = 0. (14.5)

Equate the Bernoulli integral to free stream conditions.

a2

� � 1+

U · U2

=a12

� � 1+

U12

2=

a12

� � 1

✓1 +

� � 1

2M1

2

◆= CpTt (14.6)

Note that the momentum equation is essentially equivalent to the statement that thestagnation temperature Tt is constant throughout the flow. Using (14.6) we can write

a2

� � 1= ht �

U · U2

. (14.7)

The continuity equation finally becomes

(� � 1)

✓ht �

U · U2

◆r · U � U ·r

✓U · U2

◆= 0. (14.8)

The equations governing compressible, steady, inviscid, irrotational motion reduce to asingle equation for the velocity vector U. The irrotationality condition r⇥ U = 0 permitsthe introduction of a velocity potential.

U = r� (14.9)

and (14.8) becomes

(� � 1)

✓ht �

r� ·r�

2

◆r2��r� ·r

✓r� ·r�

2

◆= 0. (14.10)

For complex body shapes numerical methods are normally used to solve for � . Howeverthe equation is of relatively limited applicability. If the flow is over a thick airfoil or a blu↵


body for instance then the equation only applies to the subsonic Mach number regime atMach numbers below the range where shocks begin to appear on the body. At high subsonicand supersonic Mach numbers where there are shocks then the homentropic assumption(14.2) breaks down. Equation (14.8) also applies to internal flows without shocks such asfully expanded nozzle flow.

14.1.2 The nonlinear small disturbance approximation

In the case of a thin airfoil that only slightly disturbs the flow, equation (14.8) can besimplified using small disturbance theory. Consider the flow past a thin 3-D airfoil shownin Figure 14.1.

Figure 14.1: Flow past a thin 3-D airfoil

The velocity field consists of a free-stream flow plus a small disturbance

U = U1 + u

V = v

W = w

(14.11)

where

u

U1<< 1

v

U1<< 1

w

U1<< 1.

(14.12)

Similarly the state variables deviate only slightly from free-stream values


P = P1 + P 0

T = T1 + T 0

⇢ = ⇢1 + ⇢0

(14.13)

and

a = a1 + a0. (14.14)

This decomposition of variables is substituted into equation (??)(eq:original014008). Var-ious terms are

U · U2

=U1

2

2+ uU1 +

u2

2+

v2

2+

w2

2

r · U = ux + vy + wz

(14.15)

and

r✓U · U2

◆= (uxU1 + uux + vvx + wwx, uyU1 + uuy + vvy + wwy, uzU1 + uuz + vvz + wwz)

(14.16)

as well as

(� � 1)

✓ht �

U · U2

◆r · U =

(� � 1)

✓ht �

✓U1

2

2+ uU1 +

u2

2+

v2

2+

w2

2

◆◆ux+

(� � 1)

✓ht �

✓U1

2

2+ uU1 +

u2

2+

v2

2+

w2

2

◆◆uy+

(� � 1)

✓ht �

✓U1

2

2+ uU1 +

u2

2+

v2

2+

w2

2

◆◆uz

(14.17)

and, finally


(� � 1)

✓ht �

U · U2

◆r · U � U ·r

✓U · U2

◆⇠=

(� � 1)

✓ht �

✓U1

2

2+ uU1

◆◆ux+

(� � 1)

✓ht �

✓U1

2

2+ uU1

◆◆vy+

(� � 1)

✓ht �

✓U1

2

2+ uU1

◆◆wz�

�uxU1

2 + uuxU1 + vvxU1 + wwxU1��

uuxU1 � vuyU1 � wuzU1.

(14.18)

Neglect terms in (14.17) and (14.18) that are of third order in the disturbance velocities.Now

(� � 1)

✓ht �

U · U2

◆r · U � U ·r

✓U · U2

◆⇠=

(� � 1)

✓ht �

✓U1

2

2+ uU1

◆◆ux+

(� � 1)

✓ht �

✓U1

2

2+ uU1

◆◆vy+

(� � 1)

✓ht �

✓U1

2

2+ uU1

◆◆wz�

�uxU1


uuxU1 � vuyU1 � wuzU1

(14.19)

or


(� � 1)

✓ht �

U · U2

◆r · U � U ·r

✓U · U2

◆⇠=

(� � 1) (h1 � uU1) (ux + vy + wz)��uxU1


uuxU1 � vuyU1 � wuzU1.

(14.20)

Recall that (� � 1)h1 = a12. Equation (14.20) can be rearranged to read

(� � 1)

✓ht �

U · U2

◆r · U � U ·r

✓U · U2

◆⇠=

a12 (ux + vy + wz)� uxU1

2 � (� + 1)uuxU1 � (vvxU1 + wwxU1)�

(� � 1) (uvyU1 + uwzU1)� vuyU1 � wuzU1.

(14.21)

Divide through by a12.

✓� � 1

a12

◆✓ht �

U · U2

◆r · U �

✓1

a12

◆U ·r

✓U · U2

◆⇠=

�1�M1

2

�ux + vy + wz �

(� + 1)M1a1

uux�

M1a1

((� � 1) (uvy + uwz) + vuy + wuz + vvx + wwx)

(14.22)

Equation (14.22) contains both linear and quadratic terms in the velocity disturbancesand one might expect to be able to neglect the quadratic terms. But note that the firstterm becomes very small near M1 = 1. Thus in order to maintain the small disturbanceapproximation at transonic Mach numbers the uux term must be retained. The remainingquadratic terms are small at all Mach numbers and can be dropped. Finally the smalldisturbance equation is

�1�M1

2

�ux + vy + wz �

(� + 1)M1a1

uux = 0. (14.23)


The velocity potential is written in terms of a free-stream potential and a disturbancepotential.

� = U1x+ � (x, y, z) (14.24)

The small disturbance equation in terms of the disturbance potential is

�1�M1

2

��xx + �yy + �zz =

(� + 1)M1a1

�x�xx. (14.25)

Equation (14.25) is valid over the whole range of subsonic, transonic and supersonic Machnumbers.

14.1.3 Linearized potential flow

If we restrict our attention to subsonic and supersonic flow, staying away from Machnumbers close to one, the nonlinear term on the right side of (14.25) can be dropped andthe small disturbance potential equation reduces to the linear wave equation

�2�xx � (�yy + �zz) = 0 (14.26)

where � =qM1

2 � 1. In two dimensions

�2�xx � �yy = 0. (14.27)

The general solution of (14.27) can be expressed as a sum of two arbitrary functions

� (x, y) = F (x� �y) +G (x+ �y) . (14.28)

Note that if M12 < 1 the 2-D linearized potential equation (14.27) is an elliptic equation

that can be rescaled to form Laplace’s equation and (14.28) expresses the solution in termsof conjugate complex variables. In this case the subsonic flow can be analyzed using themethods of complex analysis. Presently we will restrict our attention to the supersoniccase. The subsonic case is treated later in the chapter.

If M12 > 1 then (14.27) 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 alongthe characteristic lines x ± �y = constant. Figure 14.2 illustrates supersonic flow past a


thin airfoil with several characteristics shown. Notice that in the linear approximation thecharacteristics are all parallel to one another and lie at the Mach angle µ1 of the freestream. Information about the flow is carried in the value of the potential assigned toa 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 surfaceof the wing and left-leaning characteristics carry information about the flow on the lowersurface.

Figure 14.2: Right and left leaning characteristics on a thin 2-D airfoil.

All properties of the flow, velocity, pressure, temperature, etc. are constant along the char-acteristics. Since disturbances only propagate along downstream running characteristicswe can write the velocity potential for the upper and lower surfaces as

� (x, y) = F (x� �y) y > 0

� (x, y) = G (x+ �y) y < 0.(14.29)

Let y = f (x) define the coordinates of the upper surface of the wing and y = g (x) definethe lower surface. The full nonlinear boundary condition on the upper surface is

v

U

��y=f

=df

dx. (14.30)

In the spirit of the thin airfoil approximation this boundary condition can be approximatedby the linearized form

v

U

��y=0

=df

dx(14.31)

which we can write as


@� (x, y)

@y

��y=0

= U1df

dx(14.32)

or

F 0 (x) = �U1�

df

dx. (14.33)

On the lower surface the boundary condition is

G0 (x) =U1�

dg

dx. (14.34)

In the thin airfoil approximation the airfoil itself is, in e↵ect, collapsed to a line alongthe x -axis, the velocity potential is extended to the line y = 0 and the surface boundarycondition is applied at y = 0 instead of at the physical airfoil surface. The entire e↵ect ofthe airfoil on the flow is accounted for by the vertical velocity perturbation generated bythe local slope of the wing. The linearized boundary condition is valid on 2-D thin wingsand on 3-D wings of that are of ”thin planar form”.

Recall that (14.26) is only valid for subsonic and supersonic flow and not for transonic flowwhere 1�M1

2 << 1.

14.1.4 The pressure coe�cient

Let’s work out the linearized pressure coe�cient. The pressure coe�cient is

CP =P � P11

2

⇢1U12

=2

�M12

✓P

P1� 1

◆. (14.35)

The stagnation enthalpy is constant throughout the flow. Thus

T

T1= 1 +

1

2CpT1

�U1

2 ��U2 + v2 + w2

��. (14.36)

Similarly the entropy is constant and so the pressure and temperature are related by

P

P1=

✓1 +

1

2CpT1

�U1

2 ��U2 + v2 + w2

��◆�

��1

. (14.37)


The pressure coe�cient is

CP =2

�M12

✓1 +

1

2CpT1

�U1

2 ��U2 + v2 + w2

��◆�

��1

� 1

!. (14.38)

The velocity term in (14.37) is small.

U12 �

�U2 + v2 + w2

�= �

�2uU1 + u2 + v2 + w2

�(14.39)

Now use the binomial expansion (1� ")n ⇠= 1� n"+ n (n� 1) "2/2 to expand the term inparentheses in (14.39). Note that the expansion has to be carried out to second order. Thepressure coe�cient is approximately

CP⇠= �

✓2u

U1+�1�M1

2

� u2

U12

+v2 + w2

U12

◆. (14.40)

Equation (14.40) is a valid approximation for small perturbations in subsonic or supersonicflow.

For 2-D flows over planar bodies it is su�cient to retain only the first term in (14.40) andwe use the expression

CP⇠= �2

u

U1. (14.41)

For 3-D flows over slender approximately axisymmetric, bodies we must retain the lastterm so

CP⇠= �

✓2u

U1+

v2 + w2

U12

.

◆. (14.42)

As was discussed above, if the airfoil is a 2-D shape defined by the function y = f (x) theboundary condition at the surface is

df

dx=

v

U1 + u= tan (✓) (14.43)

where ✓ is the angle between the airfoil surface and the horizontal. For a thin airfoil thisis accurately approximated by


df

dx⇠=

�y

U1⇠= ✓. (14.44)

For a thin airfoil in supersonic flow the wall pressure coe�cient is

CPwall=

2�M1

2 � 1�1/2

✓df

dx

◆(14.45)

where dU2/U2 = �✓2/qM1

2 � 1

◆d✓, which is valid for M1 � 1 has been used. In the

thin airfoil approximation in supersonic flow the local pressure coe�cient is determined bythe local slope of the wing. Figure 14.3 schematically shows the wall pressure coe�cienton a thin, symmetric biconvex wing.

Figure 14.3: Pressure distribution on a thin symmetric airfoil.

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

14.1.5 Drag coe�cient of a thin symmetric airfoil

A thin, 2-D, symmetric airfoil is situated in a supersonic stream at Mach number M1 andzero angle of attack. The y-coordinate of the upper surface of the airfoil is given by thefunction

y (x) = ASin⇣⇡xC

⌘(14.46)

where C is the airfoil chord. The airfoil thickness to chord ratio is small, 2A/C << 1.Determine the drag coe�cient of the airfoil.

Solution


The drag integral is

D = 2

Z C

0

(P � P1)Sin (↵) dx. (14.47)

where the factor of 2 accounts for the drag of both the upper and lower surfaces and ↵ isthe local angle formed by the upper surface tangent to the airfoil and the x-axis. Since theairfoil is thin the angle is small and we can write the drag coe�cient as

CD =D

1

2

⇢1U12C

= 2

Z1

0

P � P11

2

⇢1U12

!(↵) d

⇣ xC

⌘. (14.48)

The local tangent is determined by the local slope of the airfoil therefore Tan (↵) = dy/dxand for small angles ↵ = dy/dx. Now the drag coe�cient is

CD =D

1

2

⇢1U12C

= 2

Z1

0

P � P11

2

⇢1U12

!✓dy

dx

◆d⇣ xC

⌘. (14.49)

The pressure coe�cient on the airfoil is given by thin airfoil theory (14.45) as

CP =P � P11

2

⇢1U12

=2p

M12 � 1

✓dy

dx

◆(14.50)

and so the drag coe�cient becomes

CD =4p

M12 � 1

Z1

0

✓dy

dx

◆2

d⇣ xC

⌘=

4A2⇡

C2

pM1

2 � 1

Z1

0

Cos2⇣⇡xC

⌘d⇣⇡xC

⌘=

2A2⇡2

C2

pM1

2 � 1.

(14.51)

The drag coe�cient is proportional to the square of the wing thickness-to-chord ratio.

14.1.6 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 M1 anda small angle of attack as shown in Figure 14.4.

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


Figure 14.4: Trajectory of a piston and an expansion wave in the x-t plane.

f⇣ xC

⌘= A⌧

⇣ xC

⌘+B�

⇣ xC

⌘� �x

C(14.52)

and the y-coordinate of the lower surface is

g⇣ xC

⌘= �A⌧

⇣ xC

⌘+B�

⇣ xC

⌘� �x

C(14.53)

where 2A/C << 1 and �/C << 1. The airfoil surface is defined by a dimensionlessthickness function

⌧⇣ xC

⌘

⌧ (0) = ⌧ (1) = 0

(14.54)

a dimensionless camber function

�⇣ xC

⌘

� (0) = � (1) = 0

(14.55)

and the angle of attack

Tan (↵) = � �

C. (14.56)

Determine the lift and drag coe�cients of the airfoil. Let

⇠ =x

C. (14.57)

Solution


The lift integral is

L =

Z C

0

(Plower � P1)Cos (↵lower) dx�Z C

0

(Pupper � P1)Cos (↵upper) dx. (14.58)

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 coe�cient is

CL =D

1

2

⇢1U12C

=

Z1

0

CPlowerd⇠ �

Z1

0

CPupperd⇠. (14.59)

The pressure coe�cient on the upper surface is given by thin airfoil theory as

CPupper =2

CpM1

2 � 1

✓df

d⇠

◆=

2

CpM1

2 � 1

✓Ad⌧

d⇠+B

d�

d⇠� �

◆. (14.60)

On the lower surface the pressure coe�cient is

CPlower= � 2

Cp

M12 � 1

✓dg

d⇠

◆= � 2

CpM1

2 � 1

✓�A

d⌧

d⇠+B

d�

d⇠� �

◆. (14.61)

Substitute into the lift integral

CL =�2

Cp

M12 � 1

✓Z1

0

df

d⇠d⇠ +

Z1

0

dg

d⇠d⇠

◆=

�2

CpM1

2 � 1

✓Z ��

0

df +

Z ��

0

dg

◆.

(14.62)

In the thin airfoil approximation, the lift is independent of the airfoil thickness.

CL =4p

M12 � 1

✓�

C

◆(14.63)

The drag integral is

D =

Z C

0

(Pupper � P1)Sin (↵upper) dx+

Z C

0

(Plower � P1)Sin (�↵lower) dx. (14.64)


Since the airfoil is thin the angle is small, and we can write the drag coe�cient as

CD =D

1

2

⇢1U12C

⇠=Z

1

0

Pupper � P1

1

2

⇢1U12

!(↵upper) d⇠ +

Z1

0

Plower � P1

1

2

⇢1U12

!(�↵lower) d⇠.

(14.65)

The local tangent is determined by the local slope of the airfoil therefore Tan (↵) = dy/dxand for small angles ↵ = dy/dx. Now the drag coe�cient is

CD =D

1

2

⇢1U12C

⇠=1

C

Z1

0

CPupper

✓dyupper

d⇠

◆d⇠ +

1

C

Z1

0

CPlower

✓�dylower

d⇠

◆d⇠. (14.66)

The drag coe�cient becomes

CD⇠=

2

C2

pM1

2 � 1

Z1

0

✓Ad⌧

d⇠+B

d�

d⇠� �

◆2

d⇠ +

Z1

0

✓�A

d⌧

d⇠+B

d�

d⇠� �

◆2

d⇠

!.

(14.67)

Note that most of the cross terms cancel. The drag coe�cient breaks into several terms.

CD⇠=

4pM1

2 � 1⇥

✓A

C

◆2

Z1

0

✓d⌧

d⇠

◆2

d⇠ +

✓B

C

◆2

Z1

0

✓d�

d⇠

◆2

d⇠ �✓B

C

◆✓�

C

◆Z1

0

✓d�

d⇠

◆d⇠ +

✓�

C

◆2

Z1

0

d⇠

!

(14.68)

The third term in (14.68) is

✓B

C

◆✓�

C

◆Z1

0

✓d�

d⇠

◆d⇠ =

✓B

C

◆✓�

C

◆Z1

0

d� =

✓B

C

◆✓�

C

◆(� (1)� � (0)) = 0. (14.69)

Finally

CD⇠=

4pM1

2 � 1

✓A

C

◆2

Z1

0

✓d⌧

d⇠

◆2

d⇠ +

✓B

C

◆2

Z1

0

✓d�

d⇠

◆2

d⇠ +

✓�

C

◆2

Z1

0

d⇠

!.

(14.70)


In the thin airfoil approximation, the drag is a sum of the drag due to thickness, drag dueto camber, and drag due to lift.

14.2 Similarity rules for high speed flight

The Figure below shows the flow past a thin symmetric airfoil at zero angle of attack. Thefluid is assumed to be inviscid and the flow Mach number, U11

/a1 where a12 = �P1/⇢1

is the speed of sound, is assumed to be much less than one. The airfoil chord is c and themaximum thickness is t

1

. The subscript one is applied in anticipation of the fact that wewill shortly scale the airfoil to a new shape with subscript 2 and the same chord.

Figure 14.5: Pressure variation over a thin symmetric airfoil in low speed flow.

The surface pressure distribution is shown below the wing, expressed in terms of the pres-sure coe�cient.

CP1

=Ps � P11

2

⇢1U11

2

(14.71)

The pressure and flow speed throughout the flow satisfy the Bernoulli relation. Near theairfoil surface

P1 +1

2⇢1U1

1

2 = Ps1

+1

2⇢1Us

1

2. (14.72)

The pressure is high at the leading edge where the flow stagnates, then as the flow ac-celerates about the body, the pressure falls rapidly at first, then more slowly reaching aminimum 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.

r⇥ u1

= 0 (14.73)

This permits the velocity to be described by a potential function.

u1

= r�1

(14.74)

When this is combined with the condition of incompressibility, r · u1

= 0, the result isLaplace’s equation.

@2�1

@x1

2

+@2�

1

@y1

2

= 0 (14.75)

Let the shape of the airfoil surface in (x1

, y1

) be given by

y1

c= ⌧

1

g⇣x

1

c

⌘. (14.76)

where ⌧1

= t1

/c is the thickness to chord ratio of the airfoil. The boundary conditions thatthe velocity potential must satisfy are

✓@�

1

@y1

◆

y1

=c⌧1

g(x1

c )= U1

1

✓dy

1

dx1

◆

body

= U1⌧1

dg (x1

/c)

d (x1

/c)

✓@�

1

@x1

◆

1! U1

1

.

(14.77)

Any number of methods of solving for the velocity potential are available including theuse of complex variables. In the following we are going to restrict the airfoil to be thin,⌧1

<< 1. In this context we will take the velocity potential to be a perturbation potentialso that

u1

= U11

+ u1

0

v1

= v1

0(14.78)

or


u1

= U11

+@�

1

@x1

v1

=@�

1

@y1

.

(14.79)

The boundary conditions on the perturbation potential in the thin airfoil approximationare

✓@�

1

@y1

◆

y1

=0

= U11

✓dy

1

dx1

◆

body

= U11

⌧1

dg (x1

/c)

d (x1

/c)

✓@�

1

@x1

◆

1! 0.

(14.80)

The surface pressure coe�cient in the thin airfoil approximation is

CP1

= � 2

U11

✓@�

1

@x1

◆

y1

=0

. (14.81)

Note that the boundary condition on the vertical velocity is now applied on the line y1

= 0.In e↵ect the airfoil has been replaced with a line of volume sources whose strength isproportional to the local slope of the actual airfoil. This sort of approximation is reallyunnecessary in the low Mach number limit but it is essential when the Mach number isincreased and compressibility e↵ects come in to play. Equally, it is essential in this examplewhere we will map a compressible flow to the incompressible case.

14.2.1 Subsonic flow M1 < 1

Now imagine a second flow at a free-stream velocity, U12

in a new space (x2

, y2

) over a newairfoil of the same shape (defined by the function g (x/c) but with a new thickness ratio⌧2

= t2

/c << 1. Part of what we need to do is to determine how ⌧1

and ⌧1

are related toone another. The boundary conditions that the new perturbation velocity potential mustsatisfy are

✓@�

2

@y2

◆

y2

=0

= U12

✓dy

2

dx2

◆

body

= U12

⌧2

dg (x2

/c)

d (x2

/c)

✓@�

2

@x2

◆

1! 0.

(14.82)


In this second flow the Mach number has been increased to the point where compressibilitye↵ects begin to occur: the density begins to vary significantly and the pressure distributionbegins to deviate from the incompressible case. As long as the Mach number is not toolarge and shock waves do not form, the flow will be nearly isentropic. In this instance the2-D steady compressible flow equations are

u2

@u2

@x2

+ v2

@u2

@y2

+1

⇢

@P

@x2

= 0

u2

@v2

@x2

+ v2

@v2

@y2

+1

⇢

@P

@y2

= 0

u2

@⇢

@x2

+ v2

@⇢

@y2

+ ⇢@u

2

@x2

+ ⇢@v

2

@y2

= 0

P

P1=

✓⇢

⇢1

◆�

.

(14.83)

Let

u2

= U12

+ u2

0

v2

= v2

⇢ = ⇢1 + ⇢0

P = P1 + P 0

(14.84)

where the primed quantities are assumed to be small compared to the free stream condi-tions. When quadratic terms in the equations of motion are neglected the equations (14.83)reduce to

✓1� ⇢1U1

2

2

�P1

◆@u

2

0

@x2

+@v

2

0

@y2

= 0. (14.85)

Introduce the perturbation velocity potential.

u2

0 =@�

2

@x2

v2

=@�

2

@y2

(14.86)


The equation governing the disturbance flow becomes

�1�M1

2

2

� @2�2

@x2

2

+@2�

2

@y2

2

= 0. (14.87)

Notice that (14.87) is valid for both sub and supersonic flow in the thin airfoil approxi-mation. Since the flow is isentropic, the pressure and velocity disturbances are related tolowest order by

P 0 + ⇢1U12

u2

0 = 0 (14.88)

and the surface pressure coe�cient retains the same basic form as in the incompressiblecase.

CP2

= � 2

U12

✓@�

2

@x2

◆

y2

=0

(14.89)

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 theleading edge of the airfoil where the velocity change is of the order of the free streamvelocity. For example for the ”thin” airfoil depicted in Figure 14.5 which is actually notall that thin, the pressure coe�cient is within �0.2 < Cp < 0.2 except over a very narrowportion of the chord near the leading edge.

Equation (14.87) can be transformed to Laplace’s equation, (14.75) using the followingchange of variables

x2

= x1

y2

=1q

1�M12

2

y1

�2

=1

A

✓U1

2

U11

◆�1

(14.90)

where, at the moment, A is an arbitrary constant. The velocity potentials are relatedby

�2

(x2

, y2

) =1

A

✓U1

2

U11

◆�1

(x1

, y1

) (14.91)


or

�1

(x1

, y1

) = A

✓U1

1

U12

◆�2

0

@x1

,1q

1�M12

2

y1

1

A (14.92)

and the boundary conditions transform as

✓@�

1

@y1

◆

y1

=0

= U11

0

@ A⌧2q

1�M12

2

1

A dg (x1

/c)

d (x1

/c)

✓@�

1

@x1

◆

1! 0

✓@�

2

@x2

◆

1! 0.

(14.93)

The transformation between flows one and two is completed by the correspondence

⌧1

=Aq

1�M12

2

⌧2

(14.94)

or

t1

c=

Aq1�M1

2

2

✓t2

c

◆. (14.95)

Finally the transformed pressure coe�cient is

CP1

= ACP2

. (14.96)

These results may be stated as follows. The solution for incompressible flow over a thinairfoil with shape g (x

1

/c) and thickness ratio, t1

/c at velocity U1

is identical to the subsoniccompressible flow at velocity, U

2

and Mach number M2

over an airfoil with a similar shapebut with the thickness ratio

t2

c=

q1�M1

2

2

A

✓t1

c

◆. (14.97)


The pressure coe�cient for the compressible case is derived by adjusting the incompressiblecoe�cient using CP

2

= CP1

/A. This result comprises several di↵erent similarity rules thatcan be found in the aeronautical literature depending on the choice of the free constant A.Perhaps the one of greatest interest is the Prandtl-Glauert rule that describes the variationof pressure coe�cient with Mach number for a body of a given shape and thickness ratio.In this case we select

A =q1�M1

2

2 (14.98)

so that the two bodies being compared in (14.97) have the same shape and thickness ratio.The pressure coe�cient for the compressible flow is

CP2

=CP

1q1�M1

2

2

. (14.99)

Several scaled profiles are shown in Figure 14.6. Keep in mind the lack of validity of (14.99)near the leading edge where the pressure coe�cient is scaled to inaccurate values.

Figure 14.6: Pressure coe�cient over the airfoil in Figure 14.5 at several Mach numbersas estimated using the Prandtl-Glauert rule (14.99).

14.2.2 Supersonic similarity M1 > 1

All the theory developed in the previous section can be extended to the supersonic caseby simply replacing 1�M1

2 with M12 � 1. In this instance the mapping is between the

equation

�M1

2

2 � 1� @2�

2

@x2

2

� @2�2

@y2

2

= 0 (14.100)


and the simple wave equation

@2�1

@x1

2

� @2�1

@y1

2

= 0. (14.101)

A generalized form of the pressure coe�cient valid for subsonic and supersonic flow is

CP

A= F

0

@ ⌧

Aq��1�M1

2

��

1

A (14.102)

where A is taken to be a function of��1�M1

2

��.

14.2.3 Transonic similarity M1 ⇠= 1

When the Mach number is close to one, the simple linearization used to obtain (14.87)from (14.83) loses accuracy. In this case the equations (14.83) reduce to the nonlinearequation

�1�M1

1

2

� @u1

0

@x1

+@v

1

0

@y1

� (�1

+ 1)M11

2

U11

u1

0@u10

@x1

= 0. (14.103)

In terms of the perturbation potential

�1�M1

1

2

� @2�1

@x1

2

+@2�

1

@y1

2

� (�1

+ 1)M11

2

U11

@�1

@x1

@2�1

@x1

2

= 0. (14.104)

This equation is invariant under the change of variables

x2

= x1

y2

=

q1�M1

1

2

q1�M1

2

2

y1

�2

=1

A

✓U1

2

U11

◆�1

(14.105)

where


A =

✓1 + �

2

1 + �1

◆✓1�M1

1

2

1�M12

2

◆✓M1

2

2

M11

2

◆. (14.106)

Notice that, due to the nonlinearity of the transonic equation (14.104) the constant A isno longer arbitrary. The pressure coe�cient becomes

CP1

=

✓1 + �

2

1 + �1

◆✓1�M1

1

2

1�M12

2

◆✓M1

2

2

M11

2

◆CP

2

(14.107)

and the thickness ratios are related by

t2

c=

✓1 + �

1

1 + �2

◆✓1�M1

2

2

1�M11

2

◆ 3

2

✓M1

1

2

M12

2

◆t1

c. (14.108)

In the transonic case, it is not possible to compare the same body at di↵erent Mach numbersor bodies with di↵erent thickness ratios at the same Mach number except by selecting gaseswith di↵erent �. For a given gas it is only possible to map the pressure distribution forone airfoil to an airfoil with a di↵erent thickness ratio at a di↵erent Mach number. Ageneralized form of (14.102) valid from subsonic to sonic to supersonic Mach numbersis

CP�(� + 1)M1

2

� 1

3

⌧2

3

= F

0

@ 1�M12

�⌧ (� + 1)M1

2

� 2

3

1

A . (14.109)

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 aircraftdesigner and these methods played a key role in the early development of transonic andsupersonic flight.

14.3 Problems

Problem 1 - A thin, 2-D, airfoil is situated in a supersonic stream at Mach number M1and a small angle of attack as shown in Figure 14.7.

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

f (x) = Ax

C

⇣1� x

C

⌘� �

x

C(14.110)

CHAPTER 14. THIN AIRFOIL THEORY 25

Figure 14.7: Bi-convex airfoil at angle of attack

and the y-coordinate of the lower surface is

g (x) = �Ax

C

⇣1� x

C

⌘� �

x

C(14.111)

where 2A/C << 1 and �/C << 1. Determine the lift and drag coe�cients of the air-foil.

thin airfoil theory - stanford universitycantwell/aa210a_course_material/aa210a... · chapter 14...

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