Unsteady Aerodynamics: 2D Unsteady Airfoil Theory Flashcards
What are the hypothesis of 2D unsteady airfoil theory?
1) 2D irrotational (vorticity on boundary), incompressible flow undergoing unsteady motion in uniform flow
2) Airflow represented by a thin flap-plate (no thickness or camber effects), because we are considering the effects of the potential linear problem.
3) The shed wake is geometrically represented by a straight line from the TE to infinity in the direction of the flow field.
4) Both airfoil and wake are planar sheets of vorticity
5) The airfoil chord is equal to 2b
6) The position of the elastic axis is at ab, where a is the non dimensional position of the EA.
What are the boundary conditions?
The normal velocity of the body should be equal to the velocity of the flow in the z wa. The velocity wa is equal to the sum of velocity induced by wake vortexes λ and the one induced by airfoil bound vortex wb, so that wa(x) = λ(x) + wb(x), where the Biot-Savart expression is used to calculate λ and b.H
How can wake vorticity be expressed?
The bound circulation over the airfoil responsible of lift is the integral of the bound vorticity over the airfoil. Since Kelvin’s theorem states that for the total circulation Γt: dΓt/dt = 0 -> dΓw/dt = γwUoo = -dΓ/dt.
This means that at every instant at the TE a vortex γw is released in the wake with an equal intensity opposite to the bound vorticity variation. This vortex is convected in the flow at speed Uoo. The vorticity in the wake (x,t) is a function of the variation on the airfoil at time t-(x-b)/Uoo, when the vortex that travelled at Uoo speed was at the TE. This is a direct consequence of Kelvin’s theorem. Therefore, the way speed is induced over the airfoil is a function of the history of what happened over the airfoil.
What if the wake vorticity is transformed to frequency domain?
Transforming to frequency domain, it can be seen that the jump of potential along the wake is equal to the jump of potential at the TE computed at the time when the element of the wake was at the TE.
How can the wake vorticity eq be solved in oder order to calculate the uknowns?
We can use the solution for Fedholm integral equation of I kind and a transformation x = bcosθ and expressing the velocity in Fourier series.
λ(θ,t) =Σλn cosnθ dθ
wa(θ,τ) = Σ wan cosnθ dθ
and solve for ωan, λn. The integral of cosnθ is zero for n>=2.
Ultimately, we can calculate the bound vorticity and the circulation with the components λ0,λ1, wa0, wa1.
How can the bound vorticity be decomposed?
The bound vorticity can be decomposed into:
γbc: Circulatory bound vorticity
γbNC: Non circulatory bound vorticity.
γ = γbc + γbNC. γbc is responsible for circulation Γ, and so lift, but with zero induced speed by bound vortex.
How are the pressure distribution computed?
The unsteady Bernoulli theorem in unsteady and incomppresible flow reads
p(x,t) = -ρ(Uinf dφ/dx + dφ/dt). We are interested in the jump of pressure between the upper and lower surfaces, so we need Δφ = φ(x,0+,t) - φ(x,0-,t).
The bound vorticity per unit length over the airfoil is related to the jump in speed. We can also calculate the pressure difference with a Fourier series with the coefficients wan,λn. This dynamic system connects the distributed pressure to the dynamic of the velocity on the airfoil (the sum of the velocity of the body and the velocity induced by the wake)
How are the loads computed?
The lift is the integral over the chord of the pressure difference, where only f0 and f1 have integrals over the chord different than zero.
In the end, L = Lw + LQ + LNC, where:
*Lw = wake-lift due to shed wake-induced speed
*LQ = quasi-static lift. It is the only term present for the steady case where 2pi = CLa and (wa0 + 1/2wa1) = AoA.
*LNC = Non-circulatory lift. The non-circulatory lift is related to the fact that in an incompressible flow, the air has to follow exactly the movement of the body, so when the body accelerates, also the air accelerates and this generates inertia foces (added mass).
There is a feedback structure between Lq and Lq. A variation of quasi-static lift causes a variation in vorticity that goes in the wake, affecting Lw.
What is the solution in the frequency domain?
We can express the circulatory lift as:
LC = LQ + LW = ρUinfG(k)Γ(k), where G(k) the Kutta-Joukowsky frequency response function that includes the Hankel function of the second type. For G(k) = 1, we have the steady case.
What is the Theodorsen model?
It is a model based on the unsteady 2D airfoil theory, which simplifies the type of movement in order to find an analytical solution.
It considered plunge movements (h) in the z direction and a rotation α around the EA.
Therefore the position of each point on the airfoil can be described with h and α and all the velocity of the flow in the z direction wa can be calculated, but also the Fourier components wa0, wa1 (for n>1 they are zero).
The movement considered in Theodorsen case is harmonic, so h(t) = h’e^jωt and α(t) = α’e^jωt.
The integrals can be computed to find the lift (L = Lw + LQ + LNC).
Then L(k) = LNC + LC, so the lift depends on the non-circulatory lift and the circulatory lift (Lw + LQ). LNC contains the Theodorsen function C(k) which is complex, since we are considering an harmoning oscillation in pitch and plunge. The theodorsen function is also called lift deficiency function. The unsteady circulatory lift of the airfoil is computed as the steady one if the angle of attack is measured at the three-quarter chord.The dynamic effect is due to C9K) that multiplied the steady circulatory.
C(k) = F(k) + jG(k).
Discuss about the Theodorsen function.
First of all, the Theodorsen lift deficiency function is equal to F(k) + iG(k). Since |C(k)| goes from 1 at low frequencies to 0.5 at high frequencies, the effect of the shed wake is to reduce the circulatory lift below the quasi-static value,
The circulatory term looks like a static airfoil coefficient which is multiplied by the lift deficiency.
Increasing k, the real part of C(k) becomes lower, meaning that given the same AoA, the lift generated is leess. Also, the imaginary part increases, so the change in Lift happens with a certain delay wrt to the AoA.
Discuss the lift variation due to a harmonic oscillation in pitch.
Plotting Lift vs AoA, we can see the effects on the CL for the circulatory part. The graph is an ellipse, with shows that when k increases, delay enters so when we reach the maximum AoA, the lift hasn’t reached it’s maximum. When the AoA increases, the lift is lower than the static value and higher when the AoA decreases. The circulatory lift lags behind the instantaneous AoA.
At k=0 and k = inf, the ellipse becomes a line (k=0 the slope is 2pi).
Also, plotting the normalized lift amplitude and phase angle vs the reduced frequency, it can be seen that at low frequency the circulatory component is dominant, while at high frequency the non-circulatory is dominant.
What is the Cicala, Kussner-Scwarz approach used for?
This approach can be used for any abritrary change of shape of the flat-plate (such as movable surphases or morhping airfoils).
Now the Theodorsen function is present in both the lift and moment.
What is the Wagner approach?
It computes the response of the aircraft to a step change in the AoA using the indicial function known as Wagner function.
What is the frozen gust front?
The gust speed vg is defined as a frozen distribution of speed fixed in space: vg = vg(ξ).
We compute ξA, which is the position of every point of the aircraft at a generic instant time in the fixed reference so that ξA = Uinft -(x+x0). Consequently the gust speed in the local reference frame of the aircraft is vg(ξΑ).
Transforming the gust speed in frequency domain, we can see two components: the spatial delay and the gust speed spectrum. The normal velocity of the body minus the gust speed should be equal to the velocity of the flow in the z direction, so vbz - vg = wa. Since the problem is linear we can only solve wa = -vg and superimpose.
If we assume the case where the gust spectrum is represented by W(k), we can find an expression for vg(θ,k) using the bessel functions of first kind.
Then we can calculate L(k), Mo(k), which depends on the Sears function (which includes the Theodorsen function). At low k, gust is similar to plunge, while at higher k the plunge delay enters.
