unsteady aerodynamics

8/15/2019 unsteady aerodynamics

1/29

Introduction to Unsteady Aerodynamics

AA200B

Lecture 13

November 27, 2007

AA200B - Applied Aerodynamics II 1


2/29

AA200B - Applied Aerodynamics II Lecture 13

Basic Concepts

Although many applications of interest in aerodynamics involve flowsthat may be considered steady – at least in some reference frame – manyothers are fundamentally unsteady phenomena. Here we consider some of the basic concepts and analysis methods appropriate for dealing with these

cases.

Unsteady aerodynamics may be important in analyzing aircraft stabilityand control, in assessing the tendency of a wing or other structure to flutter,or to understand how birds, insects, or ornithopters propel themselves byflapping. Although we could analyze these situations by simply specifying

boundary conditions and analyzing the time-dependent flow field with anunsteady CFD code, it is helpful for the understanding of such flows toconsider the many different aspects of unsteady flow that lead to differencesfrom steady flow theory.

2


3/29


4/29


In many unsteady flows of interest, the important unsteady aspects

involve not only the kinematic changes in boundary conditions caused bythe motion of a body, but the influence of an unsteady wake, and thechanges in the pressure-velocity relationship associated with the unsteadyform of Bernoulli’s equation.

As an introduction to some of these effects we consider unsteady wing

theory, starting from linear two-dimensional airfoil analysis and exploringsome nonlinear and three-dimensional phenomena.

4


5/29


Unsteady Airfoil Theory (2D)

Perhaps the simplest unsteady aerodynamic analysis involves a two-dimensional airfoil free to pitch and plunge as shown in the figure below.

!

h

Figure 1. Definition of terms and geometry for a pitching and plunging airfoil

Some of the initial theory for unsteady airfoils was developed by

Theodorsen, von Karman, and Sears in the 1930’s. Theodorsen [1] wasparticularly interested in this problem because of its importance to wingflutter, a coupling of aerodynamics and structural dynamics that can leadto wing instability and failure.

5


6/29


Theodorsen started by assuming that the motion could be described as

simple harmonic and involved only small perturbations.

We start with the expression for unsteady pressure on the airfoil inirrotational, inviscid flow (See [2]):

ρdφ

dt +

ρ

2V 2 + p − p∞ = const (1)

or:

p = const −ρ

2V 2 − ρ

dφ

dt (2)

The pressure difference between the upper and lower surfaces of a thinairfoil is then:

∆ p = ρ

2(V 2u − V

2l ) + ρ

d

dt(φu − φl) (3)

6


7/29


For a thin airfoil the upper and lower surface velocities can be written

as: V u = V ∞ + ∆V and V l = V ∞−

∆V where ∆V = γ (x)/2. And thepotential difference at some point on the airfoil x is related to the circulationby:

∆φ =

x0

(V u − V l)dx =

x0

γ (x)dx (4)

We can thus write the pressure difference in terms of the vorticity:

∆ p = ρV ∞γ (x) + ρ d

dt(

x0

γ (x)dx) (5)

This is expressed in terms of the dimensionless C p as:

∆C p = 2γ (x)

V ∞+

2

V 2∞

d

dt(

x0

γ (x)dx) (6)

7


8/29


In this expression the first term is the part associated with the steady

Bernoulli equation, while the second term comprises the unsteady part.(We note among other things that the first term can include a singularityat the leading edge, while the second term generally remains finite in thelimit as x → 0. This means that there is no leading edge suction due to theunsteady Bernoulli term: this part of the lift acts normal to the surface.)

Now we need to solve for the actual distribution of vorticity, γ (x), onthe airfoil. This can be done in a manner very similar to thin airfoil theory.One major difference, however, is the influence of the unsteady wake andhow the Kutta condition must be modified in unsteady flows.

y

x∞

gw

g

Figure 2. Vorticity on airfoil and in wake.

8


9/29


We assume that the vorticity on the airfoil is composed of two parts:

one that would be obtained without consideration of the unsteady wake(the quasi-steady part, γ 0), and a part that is added in order to compensatefor the induced velocities from the shed wake (γ 1). We can computethe first part using steady thin airfoil theory. The second part can alsobe computed this way by treating the velocities induced by the unsteadywake as changes in the boundary conditions. After some manipulation the

relationship between the bound vorticity component, γ 1 and the vorticityshed into the wake, γ w is:

γ 1(x, t) = 1

π

c − x

x

x∗c

x

x − c

γ w(x, t)

(x − x)dx (7)

The result is that the pressure can be expressed in terms of the quasi-steady circulation, its time rate of change, and the vorticity in the wake.

9


10/29


The integrated dimensionless force coefficient, C l, can finally be written

(nontrivial mathematics omitted here) as:

C l(t) =

2Γ0

V ∞c−

2

V 2∞c

d

dt c

0 γ 0

(x

, t)(x

−

c

2)dx

+

1

V ∞ x∗=V ∞t+c

c

γ w(x, t)

x(x − c)dx(8)

In the derivation of equation 8, we see that the first term is just thequasi-steady lift coefficient. The second term is related to the unsteady

part of Bernoulli’s equation applied to the quasi-steady vorticity, and thethird term is a combination of both wake effects and the unsteady part inthe Bernoulli equation. The second term may also be interpreted as theunsteady part of the load that would exist if there were no wake, which

10


11/29


would happen if there were no net circulation on the airfoil. We often write:

C l = C l0 + C l1 + C l2

where the first term represents the quasi-steady lift; the second is associatedwith “apparent mass” (sometimes termed noncirculatory lift1); and the thirdterm represents the effect of the unsteady wake.

If we specify the motion and take Laplace transforms of this, we canrelate the wake strength to the quasi-steady circulation directly (using theidea of vorticity conservation). This leads to the idea that the unsteadywake term can be expressed as a factor multiplying the quasi-steady lift.

1The terminology is actually a bit confusing. It is not correct to associate the apparent mass term

with the noncirculatory lift. It is possible to have no net circulation, Γ = 0 and no quasi-steady vorticity,γ 0 = Γ0 = 0 and still have lift arising from the wake term. (If there were never any net circulation, then

the wake term would indeed be 0 when Γ0 was 0, but in general one might have no quasi-steady circulation,Γ0, on the section at a moment in time, and yet have wake effects from previous times.) In this case, all of the (noncirculatory) lift arises from the wake term, not the apparent mass term.

11


12/29


We also note that we could determine the wake strength by keeping track

of the vorticity shed throughout time. If, in a panel code, we included itseffect on the boundary conditions, we would have only part of C l2. Theother part would appear in the computation of pressures using the unsteadyBernoulli equation. The apparent mass term (C l1) can be computed moresimply because it is not affected by the wake (by construction). It could,therefore be evaluated using thin airfoil theory. We can write:

C l1 = −2

cV 2∞

d

dt

c0

γ 0(x −c

2)dx

or explicitly in terms of the local normal velocities on the airfoil:

C l1(t) = −4

cV 2∞

d

dt

c0

x(c − x)wa(x

, t)dx

12


13/29


And we note that

Γ0 = −2

c0

x

c − xwa(x

, t)

V ∞dx

If we first consider the case of pure plunging motion, wa is constant with x

so:

C l0 = 2Γ0V ∞c

= −4

c0

x

c − xwa(x

, t)

V ∞dx

=−

4wa(t)

V ∞ 1

0

x1 − xdx

= 2π

wa(t)

V ∞

This is just as expected: the quasi-steady lift coefficient is just 2π timesthe instantaneous angle of attack.

13


14/29


The apparent mass term in this case is:

C l1(t) = −4 ẇa(t)

cV 2∞

c0

x(c − x)dx

= −4 ẇa(t)c

V 2

∞

1

0 x

(1 − x)dx = −c ẇa(t)

2V 2

∞

π

This describes a force given by F = ma where m is the mass of air containedin a circle of diameter equal to the chord. (Of course the airfoil effectsa region of air much larger than a circle around the chord and does notproduce uniform acceleration of this air mass, but the integral does have avalue with this simple interpretation.)

14


15/29


At this point the lift (and similarly for moment, C p, and drag) must

either be computed numerically, or some simple type of motion prescribed.Theodorsen [1] assumed simple harmonic motion with:θ = θ0e

iωt and h = h0eiωt. where complex amplitudes θ0 and h0 can be

used to represent phase shifts.

It is convenient to define a dimensionless frequency, sometimes called

the reduced frequency, k = ωc2U ∞.

The number of chord lengths traveled in one cycle is:U ∞τ c

= U ∞2πωc

= πk

.

The value of k indicates the importance of unsteadiness in the flow,

with values of k less than 0.1 or 0.2 often indicating that unsteady effectsare not very important.

15


16/29


When simple harmonic motion is assumed, the previous integrals may

be evaluated in a simple form. In particular, an expression for the wakevorticity is derived by differentiating the condition that the total circulationin the field is 0:

Γ(t) +

U ∞tTE

γ w(x, t)dx = 0

The process of computing γ w and integrating the previous expressions for

lift is messy, but was done by Theodorsen without Mathematica. Detailsare posted on the course website.

16


17/29


The result can be written in terms of Bessel functions, easily evaluated

in MatLab or Excel, and lumped together in Theodorsen’s lift deficiencyfactor, C (k) = F (k) + iG(k), with:

F (k) = J 1(J 1 + Y 0) + Y 1(Y 1 − J 0)

(J 1 + Y 0)2 + (Y 1 − J 0)2

G(k) = −Y 1Y 0 + J 1J 0

(J 1 + Y 0)2 + (Y 1 − J 0)2

17


18/29


The final expression for the lift coefficient of an airfoil that is pitching

and plunging is given below. The incidence of the airfoil is θ and therotation axis is ac/2 behind the half-chord. The vertical position of therotation axis is h.

C l = 2πC (k)(ḣ

V ∞+ θ − (

1

4−

a

2)

cθ̇

V ∞) +

πc

2V 2∞

(ḧ + V ∞θ̇ −ac

2θ̈)

This can be rewritten, with the rotation axis xc behind the leading edge(a = 2x − 1):

C l = 2πC (k)(˙hV ∞

+ θ − (0.75−

x)˙θcV ∞

) + πc2V 2∞

(ḧ + V ∞θ̇ − (x − 0.5)cθ̈)

18


19/29


The apparent mass term is just one of the unsteady effects in this

equation. Although this term is directly related to the unsteady term inBernoulli’s equation, some of the difference between C (k) and 1.0 is also related to the unsteady term in the Bernoulli equation.

19


20/29


The following figure shows the value of C(k).

C(k)

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1k

C =

F + i G

F

G

Figure 3. Variation of C(k) with reduced frequency.

Many different approximations to the expression for C l are often made.

20


21/29


These include:

1. Steady (just use the instantaneous incidence as determined from asnapshot): C l = 2πθ

2. Quasi-steady (ignore unsteady wake and dφ/dt term in unsteady

Bernoulli equation): C l = 2π

ḣV ∞

+ θ − (0.75 − x) θ̇cV ∞

3. Unsteady, but ignoring wake effects: C l = C l0 + C l1 As noted below,this may be worse than quasi-steady.

4. Fully unsteady.

The fully unsteady computation requires either an assumption about

the motion, keeping track of the shed wake, or including it through aconvolution integral.

An analysis of the effect of frequency on C l1 and C l2 shows that we

21


22/29


should never ignore C l2 in comparison with the rest of the equation. At low

frequencies, C l1 is higher order in k (or ω) than C l2. At high frequenciesC (k) goes to 1/2, so C l2 again cannot be ignored in comparison with thequasi-steady term, C l0.

22


23/29


Substitution of the expression for C (k) into the equation for C l shows

how unsteady aerodynamics affects the time history of C l. The resultsdepend on the location of the rotation center, the amplitude and phaseof pitching relative to plunging, and the frequency of the motion. In thesimplest case of pure plunging motion (θ = 0), the expression for C l reducesto:

C l = 2πC (k)ḣ

V ∞+

πc

2V 2∞ḧ

In this case, the unsteady effects appear as an apparent mass effect, areduction in magnitude of the first term (more important as the frequencyincreases), and an apparent lag in the force due to the negative imaginarypart of C (k).

23


24/29


3D Unsteady Aerodynamics

The combination of the usual trailing vorticity from a 3D wing andthe transverse shed vorticity from the unsteady motion leads to vorticityin various directions in the wake of an oscillating wing. The figure belowshows how this might look for a flapping bird wing, but even a simple rigid

plunging wing has an interesting pattern of vorticity due to these basiceffects. Minimum induced power requirements fo r flapping fligh t 289

FIGURE

. Top view of bird in flight showing coordinate system and vortex filaments trailing and

shed vorticity) in unsteady wake. Harris’ hawk planform after Tucker

1992).

spanwise circulation varies with time. Nevertheless, as we will show, the Betz criterion

may be applied to flapping motion of wings (as well as helicopters in forward flight).

Furthermore, because the resulting theory deals with the far wake and not the details

of the flow about the wing itself, no simplifying assumptions regarding the reduced

frequency o r amplitud e of flapping motion are required. Therefore, the present theory

is applicable to high-frequency and/or large-amplitude flapping motions.

In 2, we describe the extension of the Betz criterion for

M.I.L.

propellers to

the case of the flapping motion of wings. In

3 ,

we apply the results of

2

to

the case of small-amplitude harmonic flapping motion. We show that the problem

of finding the optimum spanwise circulation distribution can be reduced to a one-

dimensional integral equation for the unknown circulation. This integral equation is

solved efficiently using numerical quadratu re. In we describe three-dimensional

Figure 4. Unsteady 3D wake – Figure from Hall and Hall [6].

24


25/29


Of course, these vortices are not straight, but roll-up under their own

self-induced velocities to form complex patterns.

86

K C.

all and S R all

FIGURE

. a) Vortex-ring wake. b ) Concertina or continuous-vortex wake. c) Ladder wake.

Sketches after Pennycuick

1988).

was seeded with neutrally buoyant soap bubbles filled with helium. Trained birds and

bats then flew through the seeded air, and the resulting flow structure in the wake

of the animals was captured on film using stroboscopic photography. Rayner 1991)

surveyed the available experimental data, and found that the structure of the wakes

behind birds and bats falls into one of two distinct patterns. In slow flight, the wake

appears to be composed of a series of vortex rings, one ring for each downstroke

of the wings see figure 1). In fast flight, the wake is composed of two undulating

vortices which trail behind the animal. Furthermore, no transverse vorticity is

observed. Quoting Rayner, “The absence of transverse vortices is not surprising, since

the interaction of transverse vortices with the vortex on the wing can dramatically

increase induced drag.”

Based on these experimental observations, Rayner 1991, 1993) has proposed that

birds use two distinct gaits: the vortex-ring gait and the continuous-wake gait. The

vortex-ring gait is used in slow flight. According to Rayner, during the upstroke,

the wing is flexed so that the span of the wing is reduced. Furthermore, the wing is

aerodynamically inactive little or no circulation is generated along the span of the

wing). During the downstroke, the wing is fully extended and nearly flat, and the wing

circulation along the span of the wing generates thrust and lift. The continuous-wake

gait, on the other hand, is used in fast flight. The wing is aerodynamically active

during both the downstroke and the upstroke with constant total circulation. During

the downstroke, the wing is fully extended. During the upstroke, however, the wing is

flexed or swept slightly to shorten the span of the wing, reducing the instantaneous

lift while maintaining constant circulation.

Rayner 1991) has asserted that all vertebrates use one of these two gaits in forward

fli ht. Althou h not et ex erimentall observed in vertebrates Penn cuick 1988 has

Figure 5. Rolled-up wake geometries from birds.

These wakes and those shed from the tails of swimming animals havebeen studied extensively, but often more significance is attributed to thecomplex geometry than may be necessary to understand some of the basic

flow physics. In particular, just as one does not need to compute the detailsof the 3D wake roll-up process to very accurately compute induced drag,much of the important unsteady aerodynamics may be understood withmuch simpler models of the wake motion. This is suggested by Hall and

25


26/29


Hall [6] in their analysis of flapping flight, and in the case of small amplitude

unsteady motion, the analysis is even simpler.Just as we argued in the case of induced drag, the velocity induced by

the wake becomes less important as it is convected farther downstream.And since the near wake geometry is simpler than the far wake, it is oftena very good approximation to ignore the self-induced wake motion and

compute the pressures on the lifting surface itself. Just as in the case of induced drag we must be careful if we compute forces based on far-fieldvelocities and simplified wakes, while near-field calculations are less sensitiveto the assumed wake geometry.

In fact, many interesting cases can be studied with a simple vortex panel

code in which the wake propagates downstream in a simple lattice. Trailingvorticity trails aft in the freestream direction while transverse vorticity isshed in proportion to the change in circulation on the wing at the specifiedspanwise station.

26


27/29


This is the standard approach used by codes such as NASTRAN

for aeroelastic flutter analysis. Of course this is a very simpleapproximation, ignoring, not just the effects of wake deformation, butnonlinear compressibility effects, separation and other viscous effects. Adiscussion of certain of these phenomena, of particular interest in helicopteraerodynamics, is given in Ref. [7].

Even simpler approximations are also common, including the use of unsteady 2D results in a kind of modified strip theory in which 3D steadyresults are used in place of the 2D steady lift curve slopes in the previously-described derivation. This is less commonly used these days, as true 3Dcodes are more computationally tractable.

27


28/29


References

[1] Theodorsen T., “General Theory of Aerodynamic Instability and theMechanism of Flutter,”NACA Report No. 496, 1935.

[2] von Karman, T., and Sears, W. R., Airfoil Theory for Non-UniformMotion, Journal of the Aeronautical Sciences, Vol. 5, No. 10, August,

1938, pp. 378-390.[3] Sears, W. R., A Systematic Presentation of the Theory of Thin Airfoils

in Non-Uniform Motion, PhD. Thesis, California Institute of Technology,1938.

[4] Garrick, I. E., A Review of Unsteady Aerodynamics of Potential

Flows, Applied Mechanics Review, Vol. 5, No. 3, March 1952, pp. 89-91.

[5] Bisplinghoff, R., Ashley, H., Halfman, R., Aeroelasticity, AddisonWesley, 1955, pp. 288-293.

28


29/29


[6] Hall, K.C., and Hall, S.R., “Minimum Induced Power Requirements

for Flapping Flight,” J. Fluid Mech. (1996), vol. 323, pp. 285-315,Cambridge University Press.

[7] W. J. McCroskey, W.J., “Unsteady Airfoils,” Ann. Rev. Fluid Mech.1982. 14:285-311

29

Date post:	04-Jul-2018
Category:	Documents
Upload:	samik4u
View:	258 times
Download:	0 times

unsteady aerodynamics

Documents