13.1.1 t he full potential equationcantwell/aa200_course_material/...bjc 13.1 5/31/13 c hapter 13 c...

bjc 13.1 5/31/13

C

HAPTER

13 C

OMPRESSIBLE

THIN

AIRFOIL

THEORY

13.1 C

OMPRESSIBLE

POTENTIAL

FLOW

13.1.1 T

HE

FULL

POTENTIAL

EQUATION

In compressible flow, both the lift and drag of a thin airfoil can be determined toa reasonable level of accuracy from an inviscid, irrotational model of the flow.Recall the equations developed in Chapter 6 governing steady, irrotational,homentropic ( ) flow in the absence of body forces.

. (13.1)

The gradient of the isentropic relation is

. (13.2)

Recall from the development in Chapter 6 that

(13.3)

Using (13.3) the momentum equation becomes

(13.4)

Substitute (13.2) into the continuity equation and use (13.3). The continuity equa-tion becomes

. (13.5)

Equate the Bernoulli integral to free stream conditions.

s 0=

U( ) 0=

U U�•2

--------------- P--------+ 0=

PP0------

0------=

P a2=

P--- 1–------------ P--------=

1–------------ P--- U U�•

2---------------+ 0=

U a2 1–( )a2 U�•+�• 0=

Compressible potential flow

5/31/13 13.2 bjc

(13.6)

Note that the momentum equation is essentially equivalent to the statement thatthe stagnation temperature is constant throughout the flow. Using (13.6) we

can write

(13.7)

The continuity equation finally becomes

(13.8)

The equations governing compressible, steady, inviscid, irrotational motionreduce to a single equation for the velocity vector . The irrotationality condition

permits the introduction of a velocity potential.

(13.9)

and (13.8) becomes

. (13.10)

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 athick airfoil or a bluff body for instance then the equation only applies to the sub-sonic Mach number regime at Mach numbers below the range where shocks beginto appear on the body. At high subsonic and supersonic Mach numbers wherethere are shocks then the homentropic assumption (13.2) breaks down. Equation(13.8) also applies to internal flows without shocks such as fully expanded nozzleflow.

13.1.2 T

HE

NONLINEAR

SMALL

DISTURBANCE

APPROXIMATION

In the case of a thin airfoil that only slightly disturbs the flow, equation (13.8) canbe simplified using small disturbance theory.

a2

1–------------ U U�•

2---------------+

a 2

1–------------

U2

2--------+

a 2

1–------------ 1 1–

2------------M2+ C pT t= = =

T t

a2

1–------------ ht

U U�•2

---------------–=

1–( ) htU U�•

2---------------– U U U U�•

2---------------�•–�• 0=

U

U× 0=

U =

1–( ) ht�•2

------------------------– 2 �•2

------------------------�•– 0=


bjc 13.3 5/31/13

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

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

(13.11)

where

. (13.12)

Similarly the state variables deviate only slightly from freestream values.

(13.13)

and

. (13.14)

This decomposition of variables is substituted into equation (13.8). Various termsare

(13.15)

and

U u+v

U

yx

y=f(x,z)

z

U U u+=

V v=W w=

u U 1 , v U 1 , w U 1«««

P P P'+=

T T T '+=

'+=

a a a'+=

U U�•2

---------------U2

2-------- uU u2

2----- v2

2----- w2

2------+ + + +=

U�• ux vy wz+ +=


5/31/13 13.4 bjc

(13.16)

as well as

(13.17)

and, finally

. (13.18)

Neglect terms in (13.17) and (13.18) that are of third order in the disturbancevelocities. Now

U U�•2

--------------- uxU uux vvx wwx ,+ + +(=

uyU uuy vvy wwy ,+ + +

uzU uuz vvz wwz+ + + )

1–( ) htU U�•

2---------------– U�• =

1–( ) htU2

2-------- uU u2

2----- v2

2----- w2

2------+ + + +– ux +

1–( ) htU2

2-------- uU u2

2----- v2

2----- w2

2------+ + + +– vy +

1–( ) htU2

2-------- uU u2

2----- v2

2----- w2

2------+ + + +– wz

U U U�•2

---------------�• uxU2 uuxU vvxU wwxU + + + +=

uuxU u2ux uvvx uwwx + + + +

vuyU vuuy v2vy vwwy + + + +

wuzU wuuz wvvz w2wz+ + +


bjc 13.5 5/31/13

(13.19)

or

(13.20)

Recall that . Equation (13.20) can be rearranged to read

(13.21)

Divide through by .

1–( ) htU U�•

2---------------– U U U U�•

2---------------�•–�•

1–( ) htU2

2-------- uU+– ux +

1–( ) htU2

2-------- uU+– vy +

1–( ) htU2

2-------- uU+– wz uxU2 uuxU vvxU wwxU+ + +( ) ––

uuxU vuyU wuzU––

1–( ) htU U�•

2---------------– U U U U�•

2---------------�•–�•

1–( ) h uU–( ) ux vy wz+ +( ) uxU2 uuxU vvxU wwxU+ + +( )–

uuxU vuyU wuzU–––

1–( )h a2=

1–( ) htU U�•

2---------------– U U U U�•

2---------------�•–�•

a2 ux vy wz+ +( ) U2 ux– 1+( )uuxU vvxU wwxU+( )––

1–( ) uvyU uwzU+( ) vuyU wuzU––

a2


5/31/13 13.6 bjc

(13.22)

This equation contains both linear and quadratic terms in the velocity disturbancesand one might expect to be able to neglect the quadratic terms. But note that thefirst term becomes very small near . Thus in order to maintain the small

disturbance approximation at transonic Mach numbers the term must be

retained. The remaining quadratic terms are small at all Mach numbers and canbe dropped. Finally the small disturbance equation is

. (13.23)

The velocity potential is written in terms of a freestream potential and a distur-bance potential

. (13.24)

The small disturbance equation in terms of the disturbance potential becomes

(13.25)

Equation (13.25) is valid over the whole range of subsonic, transonic and super-sonic Mach numbers.

13.1.3 LINEARIZED POTENTIAL FLOW

If we restrict our attention to subsonic and supersonic flow, staying away fromMach numbers close to one, the nonlinear term on the right side of (13.25) can bedropped and the small disturbance potential equation reduces to the linear waveequation.

1–( ) htU U�•

2---------------– U U U U�•

2---------------�•–�•

1 M2–( )ux vy wz+ +1+( )M

a---------------------------uux ––

Ma--------- 1–( ) uvy uwz+( ) vuy wuz vvx wwx+ + + +( )

M 1=

uux

1 M2–( )ux vy wz+ +1+( )M

a---------------------------uux– 0=

U x x y z, ,( )+=

1 M2–( ) xx yy zz+ + 1+( )Ma--------- x xx=


bjc 13.7 5/31/13

. (13.26)

where . In two dimensions

. (13.27)

The general solution of (13.27) can be expressed as a sum of two arbitraryfunctions

. (13.28)

Note that if the 2-D linearized potential equation (13.27) is an elliptic

equation that can be rescaled to form Laplace’s equation and (13.28) expressesthe solution in terms of conjugate complex variables. In this case the subsonicflow can be analyzed using the methods of complex analysis. Presently we willrestrict our attention to the supersonic case. The subsonic case is treated later inthe chapter.

If 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 areconstant along the characteristic lines . The figure belowillustrates supersonic flow past a thin airfoil with several characteristics shown.Notice that in the linear approximation the characteristics are all parallel to oneanother and lie at the Mach angle of the free stream. Information about the

flow is carried in the value of the potential assigned to a given characteristic andin the spacing between characteristics for a given flow change. Right-leaningcharacteristics carry the information about the flow on the upper surface of thewing and left-leaning characteristics carry information about the flow on the lowersurface.

2xx yy zz+( )– 0=

M2 1–=

2xx yy– 0=

x y,( ) F x y–( ) G x y+( )+=

M2 1<

M2 1>

x y± cons ttan=

µ


5/31/13 13.8 bjc

All properties of the flow, velocity, pressure, temperature, etc. are constant alongthe characteristics. Since disturbances only propagate along downstream runningcharacteristics we can write the velocity potential for the upper and lower surfacesas

. (13.29)

Let define the coordinates of the upper surface of the wing and define the lower surface. The full nonlinear boundary condition on the

upper surface is

. (13.30)

In the spirit of the thin airfoil approximation this boundary condition can beapproximated by the linearized form

(13.31)

which we can write as

x

y

M 1>

x y– constant=

x y+ cons ttan=

µ

µ = Sin-1(1/M )

x y,( ) F x y–( ) y>0=x y,( ) G x y+( ) y<0=

y f x( )=y g x( )=

vU----

y f=

dfdx------=

vU--------

y 0=

dfdx------=


bjc 13.9 5/31/13

(13.32)

or

. (13.33)

On the lower surface the boundary condition is

. (13.34)

In the thin airfoil approximation the airfoil itself is, in effect, collapsed to a linealong the x-axis, the velocity potential is extended to the line and the sur-face boundary condition is applied at instead of at the physical airfoilsurface. The entire effect of the airfoil on the flow is accounted for by the verticalvelocity perturbation generated by the local slope of the wing. The linearizedboundary 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-

sonic flow where .

13.1.4 THE PRESSURE COEFFICIENT

Let’s work out the linearized pressure coefficient. The pressure coefficient is

. (13.35)

The stagnation enthalpy is constant throughout the flow thus

. (13.36)

Similarly the entropy is constant and thus the pressure and temperature are relatedby

x y,( )y

--------------------y 0=

U dfdx------=

F' x( )U-------- df

dx------–=

G' x( )U-------- dg

dx------=

y 0=y 0=

1 M2–( ) 1«

CPP P–12--- U2------------------- 2

M2------------ P

P------- 1–= =

TT------- 1 1

2C pT------------------ U2 U2 v2 w2+ +( )–( )+=


5/31/13 13.10 bjc

(13.37)

and the pressure coefficient is

. (13.38)

The velocity term in (13.37) is small

. (13.39)

Now use the binomial expansion to expandthe term in parentheses in (13.39). Note that the expansion has to be carried outto second order. The pressure coefficient is approximately

. (13.40)

Equation (13.40) is a valid approximation for small perturbations in subsonic orsupersonic 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

. (13.41)

For 3-D flows over slender approximately axisymmetric bodies we must retainthe last term and so

. (13.42)

PP------- 1 1

2CPT------------------- U2 U2 v2 w2+ +( )–( )+

1–------------

=

CP2

M2------------ 1 1

2C pT------------------ U2 U2 v2 w2+ +( )–( )+

1–------------

1–=

U2 U2 v2 w2+ +( )– 2uU u2 v2 w2+ + +( )–=

1 –( )n 1 n– n n 1–( )

2 2+

CP2uU-------- 1 M2–( )

u2

U2-------- v2 w2+

U2-------------------+ +–

CP 2 uU--------–

CP2uU-------- v2 w2+

U2-------------------+–


bjc 13.11 5/31/13

As was discussed above, if the airfoil is a 2-D shape defined by the function the boundary condition at the surface is

(13.43)

where is the angle between the airfoil surface and the horizontal. For a thin air-foil this is accurately approximated by

. (13.44)

For a thin airfoil in supersonic flow the wall pressure coefficient is

. (13.45)

where , which is valid for , has

been used. In the thin airfoil approximation in supersonic flow the local pressurecoefficient is determined by the local slope of the wing. The figure below showsthe wall pressure coefficient on a thin, symmetric biconvex wing.

y f x( )=

dfdx------ v

U u+------------------ tan= =

dfdx------ y

U--------

CPwall2

M2 1–( )1 2

-------------------------------- dfdx------=

dU2 U2 2 M 2 1–( )

1 2 d–= M 1

x

Cp

(+)

(-)


5/31/13 13.12 bjc

This airfoil will have shock waves at the leading and trailing edges and at firstsight this would seem to violate the isentropic assumption. But for small distur-bances 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 num-ber and zero angle of attack. The y-coordinate of the upper surface of theairfoil is given by the function

(13.46)

where is the airfoil chord. The airfoil thickness to chord ratio is small,. Determine the drag coefficient of the airfoil.

SolutionThe drag integral is

(13.47)

where the factor of 2 accounts for the drag of both the upper and lower sur-faces and is the local angle formed by the upper surface tangent to the airfoiland the x-axis. Since the airfoil is thin the angle is small and we can write thedrag coefficient as

(13.48)

The local tangent is determined by the local slope of the airfoil therefore and for small angles . Now the drag coeffi-

cient is

(13.49)

The pressure coefficient on the airfoil is given by thin airfoil theory (13.45) as

M

y x( ) A Sin xC------=

C2A C 1«

D 2 P P–( )Sin( ) xd0

C=

CDD

12--- U2 C------------------------ 2

P P–12--- U2------------------- ( )

xC----d

0

1= =

Tan( ) dy dx= dy dx=

CDD

12--- U2 C------------------------ 2

P P–12--- U2------------------- dy

dx------ x

C----d

0

1= =


bjc 13.13 5/31/13

(13.50)

and so the drag coefficient becomes

(13.51)

The drag coefficient is proportional to the square of the wing thickness-to-chordratio.

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 num-ber and a small angle of attack as shown below.

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

(13.52)

and the y-coordinate of the lower surface is

(13.53)

where and . The airfoil surface is defined by a dimension-less thickness function

, (13.54)

a dimensionless camber function

CPP P–12--- U2------------------- 2

M2 1–---------------------- dy

dx------= =

CD4

M2 1–----------------------- dy

dx------

2 xC----d

0

1 4A2

C2 M2 1–------------------------------ Cos2 x

C------ x

C------d

02A2 2

C2 M2 1–------------------------------= = =

M

x

y

Cy =

f xC---- A x

C---- B x

C---- x

C------–+=

g xC---- A– x

C---- B x

C---- x

C------–+=

2A C 1« C 1«

xC---- ; 0( ) 1( ) 0= =


5/31/13 13.14 bjc

(13.55)

and the angle of attack

(13.56)

Determine the lift and drag coefficients of the airfoil. Let

(13.57)

Solution

The lift integral is

(13.58)

where is the local angle formed by the tangent to the surface of the airfoiland the x-axis. Since the angle is everywhere small we can use .The lift coefficient is

(13.59)

The pressure coefficient on the upper surface is given by thin airfoil theory as

(13.60)

and on the lower surface the pressure coefficient is

. (13.61)

Substitute into the lift integral

xC---- ; 0( ) 1( ) 0= =

Tan( )C----–=

xC----=

L Plower P–( )Cos lower( ) xd0

CPupper P–( )Cos upper( ) xd

0

C–=

Cos( ) 1

CLD

12--- U2 C------------------------ CPlower

d0

1CPupper

d0

1–= =

CPupper

2

C M2 1–--------------------------- df

d------ 2

C M2 1–--------------------------- Ad

d------ Bd

d------- –+= =

CPlower

2

C M2 1–---------------------------– dg

d------ 2

C M2 1–---------------------------– A– d

d------ Bd

d------- –+= =


bjc 13.15 5/31/13

(13.62)

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

(13.63)

The drag integral is

(13.64)

Since the airfoil is thin the angle is small and we can write the drag coefficientas

(13.65)

The local tangent is determined by the local slope of the airfoil therefore

and for small angles .

Now the drag coefficient is

(13.66)

The drag coefficient becomes

(13.67)

Note that most of the cross terms cancel. The drag coefficient breaks into sev-eral terms.

(13.68)

CL2–

C M2 1–--------------------------- df

d------ dg

d------d

0

1+d

0

1 2–

C M2 1–--------------------------- df dg

0

–+

0

–= =

CL4

M2 1–----------------------

C----=

D Pupper P–( )Sin upper( ) xd0

CPlower P–( )Sin lower–( ) xd

0

C+=

CDD

12--- U2 C------------------------

Pupper P–12--- U2

------------------------------- upper( )d0

1 Plower P–12--- U2

------------------------------ lower–( )d0

1+= =

Tan( ) dy dx= dy dx=

CDD

12--- U2 C------------------------ 1

C---- CPupper

dyupperd

------------------- d0

1 1C---- CPlower

dylowerd

-------------------– d0

1+= =

CD2

C2M2 1–

-------------------------------- A dd------ Bd

d------- –+

2d

0

1A– d

d------ Bd

d------- –+

2d

0

1+=

CD4

M2 1–----------------------- A

C----

2 dd------

2 BC----

2 dd-------

2d

0

1 BC----

C----– d

d------- d

0

1C----

2d

0

1+ +d

0

1=

Similarity rules for high speed flight

5/31/13 13.16 bjc

The third term in (13.68) is

(13.69)

Finally

. (13.70)

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 ofattack. The fluid is assumed to be inviscid and the flow Mach number ,

where is the speed of sound, is assumed to be much less than

one. The airfoil chord is and the maximum thickness is . The subscript one is

applied in anticipation of the fact that we will shortly scale the airfoil to a newshape with subscript two and the same chord.

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

BC----

C---- d

d------- d

0

1 BC----

C---- d

0

1 BC----

C---- 1( ) 0( )–[ ] 0= = =

CD4

M2 1–----------------------- A

C----

2 dd------

2 BC----

2 dd-------

2d

01

C----

2+ +d

01

=

U 1 a

a2 P =

c t1

U 1

4x1c

---------

c

-2 -1 1 2

-1

-0.8

-0.6

-0.4

-0.2

0.2

C p–

t1


bjc 13.17 5/31/13

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

. (13.71)

The pressure and flow speed throughout the flow satisfy the Bernoulli relation, inparticular, near the airfoil surface,

. (13.72)

The pressure is high at the leading edge where the flow stagnates, then as the flowaccelerates about the body, the pressure falls rapidly at first, then more slowlyreaching a minimum at the point of maximum thickness. From there the surfacevelocity decreases and the pressure increases continuously to the trailing edge. Inthe absence of viscosity, the flow is irrotational.

(13.73)

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

(13.74)

when this is combined with the condition of incompressibility, , the

result is Laplace’s equation.

. (13.75)

Let the shape of the airfoil surface in be given by

(13.76)

where is the thickness to chord ratio of the airfoil. The boundary con-

ditions that the velocity potential must satisfy are,

CP1Ps P–12--- U 1

2----------------------=

P 12--- U 1

2+ Ps112--- Us1

2+=

u1× 0=

u1 1=

u1�• 0=

21

2x1

------------2

12y1

------------+ 0=

x1 y1,( )

y1c----- 1g x1 c[ ]=

1 t1 c=


5/31/13 13.18 bjc

. (13.77)

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 theairfoil to be thin, . In this context we will take the velocity potential to be

a perturbation potential so that.

(13.78)

or

. (13.79)

The boundary conditions on the perturbation potential in the thin airfoil approxi-mation are,

. (13.80)

The surface pressure coefficient in the thin airfoil approximation is,

. (13.81)

Note that the boundary condition on the vertical velocity is now applied on theline . 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 ofapproximation is really unnecessary in the low Mach number limit but it is essen-

1y1

---------y1 c 1g

x1c-----=

U 1dy1dx1---------

bodyU 1

dg x1 c[ ]

d x1 c( )------------------------= =

1x1U 1=

1 1«

u1 U 1 u1' ; v1+ v1'= =

u1 U 11

x1--------- ; v1+ 1

y1---------= =

1y1

---------y1 0=

U 1dy1dx1---------

bodyU 1

dg x1 c[ ]

d x1 c( )------------------------= =

1x10=

CP12

U 1----------- 1

x1---------

y1 0=–=

y1 0=


bjc 13.19 5/31/13

tial when the Mach number is increased and compressibility effects come in toplay. Equally, it is essential in this example where we will map a compressibleflow to the incompressible case.

13.2.1 SUBSONIC FLOW

Now imagine a second flow at a free stream velocity, in a new space

over a new airfoil of the same shape (defined by the function ) but with anew thickness ratio . Part of what we need to do is to determine

how and are related to one another. The boundary conditions that the new

perturbation velocity potential must satisfy are,

. (13.82)

In this second flow the Mach number has been increased to the point where com-pressibility effects begin to occur: the density begins to vary significantly and thepressure distribution begins to deviate from the incompressible case. As long asthe Mach number is not too large and shock waves do not form, the flow will benearly isentropic. In this instance the 2-D steady compressible flow equations are,

. (13.83)

Let

M 1<

U2

x2 y2,( )

g x c[ ]

2 t2 c 1«=

1 2

2y2

---------y2 0=

U 2dy2dx2---------

bodyU 2 2

dg x1 c[ ]

d x1 c( )------------------------= =

2x20=

u2u2x2

--------- v2u2y2

--------- 1--- Px2

---------+ + 0=

u2v2x2

--------- v2v2y2

-------- 1--- Py2

--------+ + 0=

u2 x2--------- v2 y2

--------u2x2

---------v2y2

--------+ + + 0=

PP------- -------=


5/31/13 13.20 bjc

(13.84)

where the primed quantities are assumed to be small compared to the free streamconditions. When quadratic terms in the equations of motion are neglected theequations (13.83) reduce to,

. (13.85)

Introduce the perturbation velocity potential

. (13.86)

The equation governing the disturbance flow becomes,

. (13.87)

Notice that (13.87) is valid for both sub and supersonic flow in the thin airfoilapproximation.

Since the flow is isentropic, the pressure and velocity disturbances are related tolowest order by,

(13.88)

and the surface pressure coefficient retains the same basic form as in the incom-pressible case,

. (13.89)

Since we are at a finite Mach number, this last relation is valid only within the thinairfoil, small disturbance approximation and therefore may be expected to beinvalid near the leading edge of the airfoil where the velocity change is of the orderof the free stream velocity. For example for the “thin” airfoil depicted in Figure

u2 U 2 u'2 ; v2+ v'2 ; ' ; P+ P P'+= = = =

1U 2

2

P-------------------–

u'2x2

----------v'2y2

---------+ 0=

u'22

x2--------- ; v'2

2y2

---------= =

1 M 22–( )

22

x22

------------2

2

y22

------------+ 0=

P' U 2u2'+ 0=

CP22

U 2----------- 2

x2---------

y2 0=–=


bjc 13.21 5/31/13

13.1 which is actually not all that thin, the pressure coefficient is within 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 fol-lowing change of variables.

(13.90)

where, at the moment, is an arbitrary constant. The velocity potentials arerelated by,

(13.91)

or,

(13.92)

and the boundary conditions transform as,

. (13.93)

The transformation between flows one and two is completed by thecorrespondence,

(13.94)

or

0.2 CP<– 0.2<

x2 x1 ; y21

1 M 22–

-------------------------y1 ; 21A---

U 2U 1----------- 1= = =

A

2 x2 y2,[ ]1A---

U 2U 1----------- 1 x1 y1,[ ]=

1 x1 y1,[ ] AU 1U 2----------- 2 x1

1

1 M 22–

-------------------------y1,=

1y1

---------y1 0=

U 1A 2

1 M 22–

-------------------------dg x1 c[ ]

d x1 c( )------------------------=

1x12x2

0= =

1A 2

1 M 22–

-------------------------=


5/31/13 13.22 bjc

. (13.95)

Finally the transformed pressure coefficient is

. (13.96)

These results may be stated as follows. The solution for incompressible flow overa thin airfoil with shape and thickness ratio, at velocity is iden-

tical to the subsonic compressible flow at velocity, and Mach number

over an airfoil with a similar shape but with the thickness ratio,

. (13.97)

The pressure coefficient for the compressible case is derived by adjusting theincompressible coefficient using . This result comprises several

different similarity rules that can be found in the aeronautical literature dependingon the choice of the free constant, .

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 thatdescribes the variation of pressure coefficient with Mach number for a body of agiven shape and thickness ratio. In this case we select,

t1c---- A

1 M 22–

-------------------------t2c----=

CP1 ACP2=

g x1 c[ ] t1 c U1

U2 M2

t2c----

1 M 22–

A-------------------------

t1c----=

CP2 CP1 A=

A

-2 -1 1 2

-1.5

-1

-0.5

CP–

4xc

------

M 0=

M 0.5= M 0.75=


bjc 13.23 5/31/13

(13.98)

so that the two bodies being compared in (13.97) have the same shape and thick-ness ratio. The pressure coefficient for the compressible flow is,

. (13.99)

Several scaled profiles are shown in Figure 13.2. Keep in mind the lack of validityof (13.99) near the leading edge where the pressure coefficient is scaled to inac-curate values.

13.2.2 SUPERSONIC SIMILARITY

All the theory developed in the previous section can be extended to the supersonic

case by simply replacing with . In this instance the mapping is

between the equation,

(13.100)

and the simple wave equation

(13.101)

A generalized form of the pressure coefficient valid for subsonic and supersonicflow is,

(13.102)

where is taken to be a function of .

A 1 M 22–=

CP2CP1

1 M 22–

-------------------------=

M 1>

1 M2– M2 1–

M 22 1–( )

22

x22

------------2

2

y22

------------– 0=

21

x12

------------2

1

y12

------------– 0=

C pA

------- FA 1 M2–------------------------------=

A 1 M2–


5/31/13 13.24 bjc

13.2.3 TRANSONIC SIMILARITY,

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 tothe nonlinear equation

. (13.103)

In terms of the perturbation potential,

. (13.104)

This equation is invariant under the change of variables,

(13.105)

where

. (13.106)

Notice that, due to the nonlinearity of the transonic equation (13.104), the con-stant is no longer arbitrary. The pressure coefficient becomes,

(13.107)

and the thickness ratios are related by,

M 1

1 M 12–( )

u'1x1

----------v'1y1

--------- 1 1+( )M 12

U 1---------------------------------u'1

u'1x1

----------–+ 0=

1 M 12–( )

21

x12

------------2

1

y12

------------ 1 1+( )M 12

U 1--------------------------------- 1

x1---------

21

x12

------------–+ 0=

x2 x1 ; y21 M 1

2–

1 M 22–

-------------------------y1 ; 21A---

U 2U 1----------- 1== =

A1 2+1 1+----------------

1 M 12–

1 M 22–

---------------------M 2

2

M 12------------=

A

CP11 2+1 1+----------------

1 M 12–

1 M 22–

---------------------M 2

2

M 12------------ C p2=

The effect of sweep

bjc 13.25 5/31/13

. (13.108)

In the transonic case, it is not possible to compare the same body at different Machnumbers or bodies with different thickness ratios at the same Mach number exceptby selecting gases with different . For a given gas it is only possible to map thepressure distribution for one airfoil to an airfoil with a different thickness ratio ata different Mach number. A generalized form of (13.102) valid from subsonic tosonic to supersonic Mach numbers is,

. (13.109)

Prior to the advent of supercomputers capable of solving the equations of highspeed flow, similarity methods and wind tunnel correlations were the only toolsavailable to the aircraft designer and these methods played a key role in the earlydevelopment of transonic and supersonic flight.

13.3 THE EFFECT OF SWEEP

Return to the continuity equation (13.8) written as

(13.110)

Using the irrotationality condition , (13.110) can expressed as

(13.111)

t2c----

1 1+1 2+----------------

1 M 22–

1 M 12–

---------------------

32---

M 12

M 22------------

t1c----=

CP 1+( )M2( )

1 3

2 3------------------------------------------------- F

1 M2–

1+( )M2( )

2 3--------------------------------------------=

U 1

a2-----U U U�•2

---------------�•–�• 0=

U× 0=

1 U2

a2-------– U

x------- 1 V 2

a2------– V

y------- 1 W 2

a2--------– W

z-------- 2 UV

a2-------- U

y------- WV

a2--------- V

z------- UW

a2---------- U

z-------+ +–+ + 0=

The effect of sweep

5/31/13 13.26 bjc

The figure below shows a plan view of of a long slender wing with the free streamflow approaching at a sweep angle . The component of the free stream flow par-allel to the long axis of the wing is assumed to play little role in determining thepressure 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 plane normal to the long axis of the wingwill be nearly two-dimensional so that and can be assumed tobe very small. Now

(13.112)

Assume small disturbances.

x y,( )

W z U z

1 U2

a2-------– Ux

------- 1 V 2

a2------– Vy

------- 2Va---- U

a---- U

y------- W

a----- V

z-------+–+ 0=

The effect of sweep

bjc 13.27 5/31/13

(13.113)

Neglect terms that are cubic in the small disturbances. Equation (13.112) becomes

(13.114)

The quantity 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

(13.115)

For the swept wing the disturbance potential is governed by the component of thefree stream velocity normal to the leading edge of the wing . If

then the normal flow is supersonic and we would use supersonic

theory developed above to relate the pressure coefficient to the slope of the airfoilsurface in the plane.

If 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 thinairfoil theory can be used to determine the disturbance potential . The

pressure coefficient of the solution is used with the Prandtl-Glauert rule(13.99) to determine the pressure coefficient on the swept wing.

(13.116)

U U Cos( ) u+=

V v=W U Sin( ) w+=

a a a'+=

1U2 Cos2

( )

a2------------------------------– Ux

------- Vy

------- 2 Va------

U Cos( )

a--------------------------- U

y-------

U Sin( )

a------------------------- V

z-------+–+ 0=

V a

1 M2 Cos2( )–( )

2

x2---------2

y2---------+ 0=

U Cos( )

M2 Cos2( ) 1>

x y,( )

M2 Cos2( ) 1<

M 0= 1

M 0=

CPM Cos( )

CP1

1 M2 Cos2( )–

--------------------------------------------=

The effect of sweep

5/31/13 13.28 bjc

The best discussion of this topic that I know of appears in NACA Technical Report863 by R. T. Jones published in 1945. The rest of the discussion of this topic com-prises images taken directly from this report. Where Jones refers to equations (4)and (6) he is referring to Laplaces equation and the wave equation.

The effect of sweep

bjc 13.29 5/31/13

The effect of sweep

5/31/13 13.30 bjc

The effect of sweep

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The effect of sweep

5/31/13 13.32 bjc

The effect of sweep

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The effect of sweep

5/31/13 13.34 bjc

The effect of sweep

bjc 13.35 5/31/13

The effect of sweep

5/31/13 13.36 bjc

Problems

bjc 13.37 5/31/13

13.4 PROBLEMS

Problem 1 - A thin, 2-D, airfoil is situated in a supersonic stream at Machnumber and a small angle of attack as shown below.

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

and the y-coordinate of the lower surface is

where and . Determine the lift and drag coefficients of theairfoil.

M

x

y

Cy =

f x( ) A xC---- 1 x

C----– C

----x–=

g x( ) A– xC---- 1 x

C----– C

----x–=

2A C 1« C 1«

Problems

5/31/13 13.38 bjc

13.1.1 t he full potential equationcantwell/aa200_course_material/...bjc 13.1 5/31/13 c hapter 13 c...

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