Unsteady Lift and Pitching Moment of Airfoils

The potential equation is used to derive the unsteady lift and pitching moment of airfoils. We considered the case of incompressible flow in detail. Now we will consider the other cases, in increasing order of complexity:

Supersonic

Transonic

Subsonic

These problems, of small-amplitude, high-frequency oscillations of airfoils, were originally attacked due to the urgency of understanding flutter and other aeroelastic phenomena. In such problems, the precise amplitudes of fluctuations are of less interest than the frequencies, phase relationships and mode shapes. Hence we will find ourselves ignoring details of the boundary layer, skin-friction etc., and using potential theory with simple geometric shapes. The theoretical approaches of classical unsteady aerodynamics are seen here.

Our present interest stems from several applications, of which aeroelasticity is only one. We are also interested in the unsteady aerodynamics of:

a) insects, birds and micro-air-vehicles

b) helicopter rotor blades

c) combat aircraft executing maneuvers

d) missiles

e) parachutes

f) parafoils

There is no single textbook or other reference source which deals with all these issues. We therefore have to depend on multiple sources. The derivations from different texts bear strong resemblance to each other, but we have to rely on the nomenclature in one or the other to minimize confusion. The original expositions appear to be those in [Bisplinghoff], followed by [AGARD Manual on Aeroelasticity]. In what follows, the supersonic unsteady airfoil material is described succinctly in [Ashley and Landahl], but takes a long time for a new student to go from one step to the next. We follow the nomenclature from [Dowell et al.], because the derivations there are detailed. The subsonic unsteady airfoil material follows [Fong]. The transonic material follows the notes from [Sankar], with references back to [Dowell], [Fong] and [Ashley].

References:

1. Bisplinghoff.

2. AGARD Manual on Aeroelasticity.

3. Ashley, H., Landahl, M., "Aerodynamics of Wings and Bodies". Addison-Wesley, 1965.

4. Fong

5. Dowell, E.H., Curtiss, H.C., Scanlan, R.H., Sisto, F., "A Modern Course in Aeroelasticity". Sijthoff and Noordhoff, 1978.

6. Sankar

Oscillating Airfoils in Supersonic Flow

The airfoil considered has zero thickness and camber, with a stationary mean position. There is a uniform freestream U. The speed of sound is a. The Mach number U/a is greater than 1 everywhere in the flow. The Laplace transformation method used here is due to Stewartson. In the following derivation, we do not use the "infinity" subscript on velocity, density etc., because the only values of freestream velocity U, and density used are those in the freestream. This comes from the linearization of the potential equation.

When the flow is supersonic, conditions at any point are affected only by conditions upstream. Thus the chordwise distance has the same property as time: it is unidirectional [Ashley]. This fact is used in the approach: Laplace transforms are taken with respect to both streamwise distance and time.

The perturbation potential is

The potential equation can be written as:

The body surface bounday condition is:

for .

................(1)

The Kutta condition is not of much use here, because conditions at the trailing edge or in the wake do not influence the flow properties over the airfoil. This is also true for steady 2-d airfoils in supersonic flow, and for finite wings with supersonic trailing edges.

The potential equation can be rewritten as:

............(2)

Simple harmonic motion of the airfoil can be described by:

.................................(3)

Equations (1) and (2) become:

...........(4)

................(5)

Unlike in the incompressible or subsonic cases, here we can use the fact that disturbance is zero upstream of the airfoil. In this sense, the space variable (x) behaves much like the time variable (t). So we can use the Laplace transform in space.

For x < 0, etc. are zero. Taking the Laplace transform with respect to x,

.......................(6)

....................(7)

Transforming (4) and (5):

..................................................(8)

............................................(9)

where

Note:

Solutions to Eqns. (8)&(9) are:

......................................................(10)

Select A = 0 to keep the solution finite as z tends towards infinity.

. Determine B from (8):

. Thus,

Hence, .............................................(11)

Inverting Eq. 11, using the convolution theorem,

, .............................(12)

and in particular,

From a table of integral transforms [Bateman, H., "Table of Integral Transforms", McGraw-Hill, 1954];

where . Thus,

...................................(13)

may be computed by similar methods. In nondimensional terms,

......................(14)

where ; , and ;

Use Bernoulii's equation to compute p:

, or,

Introducing the Reduced Frequency k,

, where

Using Leibnitz's rule,

................(15)

Implications:

As frequency (steady flow), we get:

, and,

, where

Also,

High-frequency limit (highly unsteady flow)

The pressure result may be written as:

, which is the pressure on a piston in a long, narrow (1-d) tube where w is the velocity of the piston.

Note: In the limit of low and high frequency, the pressure at point x depends only upon the normalwash at that same point.

Gusts

Body motionless: boundary condition is that

where w_G is the specified vertical "gust" velocity and is the perturbation due to the body passing through the gust.

Thus, replace w by -w_a.

We will assume a frozen gust field with respect to the fluid-fixed coordinates x', y', z', t'.

Hence,

A sharp-edged gust can be expressed as:

for x'<0, and for x' >0. It can also be expressed as: for , and for

Transient Motion:

Take Laplace transform with respect to time and a Fourier transform with respect to streamwise coordinate x. The analogue to Eq. (11) is:

..................................(22)

is the Laplace transform variable, and is the Fourier transform variable ( was the Laplace transform variable in the previous simple harmonic motion result).

Inverting the Laplace transform and using * to denote a Fourier transform,

, or,

From the unsteady Bernoulli equation ........................(26)

we get, using the above,

. Thus,

................(24)

Inverting the Fourier transform,

...............................................................................(25)

The lift is obtained by using (24) and (25) in its definition:

............................(26)

In the 2nd term the integration over x has been carried out explicitly.

Lift due to airfoil motion

Consider a translating airfoil,

;

...................................................(27)

where

K may be simplified to:

Lift due to atmospheric gusts:

For a "frozen gust", .

x and t are coordinates fixed to the airfoil. x' and t' are fixed with respect to the atmosphere.

At t = t'=0, the airfoil enters the gust.

Boundary condition is: wa + wG = 0; wa=-wG on the airfoil.

The short and long-time limits correspond, respectively, to the high and low-frequency limits.

(i) Piston Theory (short-time / high frequency limit).

On the upper and lower airfoil surfaces,

Note: In acoustics, the relation between acoustic pressure and velocity fluctuations is: where c is the speed of sound. Thus,

A sharp-edged gust can be expressed as:

for x'<0, and for x' >0. It can also be expressed as: for , and for

for , and

, for ......................(28)

(ii) Static theory (large-time limit)

.............................(29)

The low and high frequency limits may be interpreted in the time domain for transient motion as the long and short time limits, respectively. This follows from the initial and final value Laplace Transform theorems. For example,

Step Change in angle of attack:

for t>0, and

for t<0.

for t>0, and for t<0.

Hence for short time (large frequency), .

For long time (small frequency), . The result for short time may also be deduced by applying Laplace transform with respect to time and taking the limit as t tends to zero of the formal inversion.

General Comments:

1) Solution for arbitrary time-dependent motion may be obtained by Fourier superposition of the simple harmonic motion result.

2) It is more efficient to use the Laplace transform with respect to time prior to inverting the spatial variable x.

3) Thickness and Lifting effects can be included as follows:

Thickness:

z=0+; w =wa; p=p+

z=0-; w = -wa; p=p+

Lifting:

z=0+; w=wa; p=p+

z=0-; w=wa; p=-p+

where p+ is the solution obtained.