Prandtl-Glauert-Goethert Transformation Potential · PDF fileThus the wing lift coefficient from slender wing theory is: The wing induced drag coefficient is: ... usual wings is dependent

Wings and Bodies in Compressible Flows

Prandtl-Glauert-Goethert Transformation

Potential equation:Potential equation:

Copyright 2009Narayanan Komerath

1

If we choose and

Laplace eqn.

The transformation has stretched the x co-ordinate by


2

Values of at corresponding points are identical.

Z co-ordinates remain the same, therefore at corresponding points are also the same.

Thus the equations (1) and (2) may be solved by solving equation (4) for a wing of greater Thus the equations (1) and (2) may be solved by solving equation (4) for a wing of greater sweep, smaller aspect ratio and same section shape.

Leading edge sweep angles are related by;Leading edge sweep angles are related by;

Similarly,

Therefore,


3

Also,

SinceSince,


4

Section Lift

Notes:

When , the equation reduces to the Laplace eqn.


5

“Supersonic Source Flow”

Source:

For , is real inside the Mach cone and imaginary outside.


6

Inside the Mach cone, velocity components are .

Thus, the supersonic flow about a slender, non-lifting body can be analyzed by superposing on the main flow the perturbation velocities of a line of supersonic sources of strengths c = on the main flow the perturbation velocities of a line of supersonic sources of strengths c c(x), whose Mach cones intersect the body surface upstream of any given surface point.

For 2-D flows, and derivatives are constant at every point on a given Mach line; but the perturbation velocities decrease with distance from the x-axis within the Mach cone.


7

Velocity potential for source distribution and uniform flow at a point P

where f(x)dx = 2 time source strength along dx.


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Problem:

Determine source density distributions f(x) such that the body surface is a streamline.

boundary condition

neglecting quadratic terms.


9

Von Karman showed that where , the rate of change of area dS ith f th b ddS with x of the body.

Assuming that can be represented by a Fourier seriesAssuming that can be represented by a Fourier series,


10

Then wave drag is and


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SLENDER WING THEORYWe saw that a wing at high subsonic M can be analyzed by transforming to a low-Aspect Ratio wing at Mach 0. How do you analyze a wing of low AR?

Assume: Small angle of attack: ; Small angle of attack: ;

Thin wing: .


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Notes:Notes:1. Pressure distribution has an infinite peak along the sloping sides of the wing.

2. Distribution along rays (lines of constant y/y1) is uniform.3. The center of pressure coincides with the centroid of the area.4. Maintenance of lift up to the trailing edge is associated solely with the case of zero width: does not exist for finite Aspect Ratio.5. Sections downstream of maximum width will not generate lift. A wake exists: no infinite suction peak downstream of the location of maximum width: Kutta condition.


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Results from Slender Wing Theory

The spanwise load distribution is:

This is an elliptical distribution, independent of planform shape.

The induced drag per unit chord is:The induced drag per unit chord is:


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Total Wing Lift is:

L =L =

Thus the wing lift coefficient from slender wing theory is:Thus the wing lift coefficient from slender wing theory is:

The wing induced drag coefficient is:

For small angle of attack.


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Lift, Drag and Pitching Moment Coefficientsg g

C b d? t b k t th th d i t t l f tt k i ti Cambered? must go back to the theory and integrate angle of attack variation with chordwise distance.


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SELECTED RESULTS FROM SLENDER BODY THEORY

References: Ashley.H., Landahl,M., "Aerodynamics of Wings and Bodies". Addison-Wesley, 1965, Chapters 6 and 9.

D ll t l Ch t 3 (D ’t h t t d th b k t d th t !)Dowell et al. , Chapter 3 (Don’t rush out to read these books: study the notes!)


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Crucial parameters are:

the cross-section area of the configuration at station x,

the "reduced cross-section area” where is the slenderness parameter,

For example, is the aspect ratio for a wing, or thickness ratio for a body of revolution.

and, and

E l f li d f t t i l tiExample: for a cylinder of constant, circular cross-secction,

;


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"Equivalence Rule", developed by Oswatitsch and Keune (1955) for transonic flow, and by Ward(1949) for supersonic flow.

a) Far away from a general slender body, the flow becomes axisymmetric and equal to the flow faround the equivalent body of revolution.

b) Near the slender body, the flow differs from that around the equivalent body of revolution by a 2-D constant-density cross-flow part which satisfies the tangency condition at the body surface.


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Transverse Forces and Moments on a Slender Body

(see Ashley and Landahl, Eq. 6-106)

Here the subscript B refers to the base section. Here the subscript B refers to the base section. Note: Lift depends on BASE dimensions. Consider if there is no wing (s=R) and the base is pointed (RB = 0). Lift is zero.

Lift of a body pointed at both ends is zero for small Lift of a body pointed at both ends is zero, for small angle of attack.


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For a Wing alone , so thatFor a Wing alone, , so that

So, lift coefficient referred to base area is simply, p y

If the body is pointed at the rear, this says that lift should be zero. There is only a pitching moment, and it is destabilizing. In reality, viscous forces will cause a small positive lift.

From the wing-alone result, , with ;

, where S is the wing planform area, we see that

, so that the lift coefficient referred to the planform area is:


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These equations hold only for wings with monotonically increasing span from pointed apex to the base. If the span decreases anywhere along the chord, a wake is formed. Thus, the lift on p y g , ,usual wings is dependent on the forward sections.

F l Eff t T t l LiftFuselage Effects on Total Lift


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Wave Drag of a Slender Body in Steady Supersonic FlowSource: Ashley, H., Landahl, M., "Aerodynamics of Wings and Bodies"Addison-Wesley, 1965.

For a slender body, f(x) = S'(x). If the trailing end of the body tapers to a point (no flat base), then S'(l) = 0. Under this condition,

The wave drag coefficient of a slender body in supersonic flow is independent of Mach b ifnumber, if

(a) the body has a pointed nose and(b) the trailing end is either pointed or cylindrical.( ) g p y

Of course DRAG still depends on density and square of velocity!!!


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Minimum Wave Drag, and Body Shape For Minimum Wave Drag

To find body shape for minimum wave drag, we proceed as follows:

Use the Glauert transformation

where goes from 0 at the base, where x = l, to at the nose where x = 0. The unknown source strength function f can be expressed as a Fourier sine series:

Giving wave drag

Mi i d i h th f ti f i h th t A 0 f ll >1Copyright 2009

Narayanan Komerath24

Minimum wave drag is when the function f is such that An = 0 for all n >1

Given that f is S', area distribution corresponding to this drag is found by integrating f over x.Given that f is S , area distribution corresponding to this drag is found by integrating f over x.

S() l2A1 Sin(2)

An

Sin(n1) Sin(n1)

S()

4A1

2

An n1 n1

n2

Integrating this, we get the total volume as


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Case 1: Given Base Area(this is the typical case of a missile forebody: note that the constant-diameter portion is not supposed to produce any wave drag, as seen above). At the base, , so only the coefficient A1 contributes to base area.At the base, , so only the coefficient A1 contributes to base area.

So, .........................................................(D-8)

All components contribute to drag, so minimum drag occurs when all coefficients

are zero, for . (See the argument for minimum induced drag of a wing in , ( g g gincompressible flow). The minimum drag has the value:

.................................(D-9)


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Drag coefficient referred to the base area isDrag coefficient referred to the base area is

The area distribution for this minimum drag is:g

This is called the von Karman ogive.


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Case 2: Minimum-Drag Body of Given VolumeThis is more similar to the case of a projectile: Body pointed at both ends, and with given volume. Now A1 =0, and

Minimum drag is when all other coefficients are zero This gives:Minimum drag is when all other coefficients are zero. This gives:

The drag coefficient is:

The area distribution is:


28This is called the Sears-Haack body.

Note:Both the von Karman ogive and the Sears-Haack body are slightly blunted (??).Linear theory gives decent results away from the blunted regions provided the bluntness is not excessive.

Generally, the drag is not very sensitive to small departures from the optimum shapes.Generally, the drag is not very sensitive to small departures from the optimum shapes.

Tangent Ogive ForebodyNote some features of the usage of theories like the above one. Esch (1979) points out that an g ( ) pabsolute limit of usefulness of the singularity-distribution method, or even the Karman-Tsien method, is reached when the leading Mach cone intersects the body surface. For example, consider a "tangent ogive" forebody of lN/D =3.5. This would be called a "3.5:1 tangent ogive consider a tangent ogive forebody of lN/D 3.5. This would be called a 3.5:1 tangent ogive forebody". Here, at Mach 3.5,


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So, beyond this Mach number, even the Karman-Tsien (source singularity) method will not give useful results. According to Esch(1979), the reliable regime of validity of linear singularity distributions, slender-body theory etc. is where the above ratio is between 0 and 0.05. The regime of validity of the Karman-Tsien method is where the ratio is between 0 and 0.4. So it should be noted that the above criterion, of the Mach wave running into the surface, is really extreme and constitutes the borderline of nonsense, rather than the borderline of accuracy.


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Prandtl-Glauert-Goethert Transformation Potential · PDF fileThus the wing lift coefficient from slender wing theory is: The wing induced drag coefficient is: ... usual wings is dependent