Sistemas de compensación e incentivos en instituciones de educación

-0.02 -0.01 0.01 0.02 0.03

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Θ=60°

Θ=45°

Θ=30°

Figure 2.5: Far-field scattered pressure directivity for a NACA 1112 aerofoil with k= 8, α_i = 0^◦, M∞= 0.7, k₃ = 0. The gust angle is varied from 30^◦ to 60^◦.

are more responsive to variations in the upstream gust angle than vertical momentum blocking.

Figure 2.6 shows the effects of varying the spanwise wavenumber, k₃. This has the result of altering w, hence we can only choose values of k₃ so that w =O(1), ensuring our asymptotic series remains valid. As w decreases, the frequency of both the leading-and trailing-edge fields decreases leading-and so too does the phase shift between them (since all are proportional to kw). We therefore expect, and indeed observe, that the effect of varyingk3 is to alter the modulation of the far-field pressure directivity. Sincewremains O(1) for all choices relating to this figure, we see little variation in the magnitude of the far-field pressure.

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k₃=0.5 k₃=0.3 k₃=0

Figure 2.6: Far-field scattered pressure directivity for a NACA 1112 aerofoil withk = 8, α_i = 0^◦, M_∞ = 0.7,θ_g = 45^◦. k₃ is varied from 0 to0.5.

gust perturbation is very useful.

Figure 2.7a shows the effect of altering the angle of attack on the unsteady pressure jump across the aerofoil. As we decrease the angle of attack, the pressure jump increases across the entire length of the aerofoil. This does not contradict the well-known result that the steady lift on an aerofoil increases as we increase its angle of attack (up to the stall angle, but given we assume α_i =O() we are not concerned with stall angles), because we must remember that the plot shows only the unsteady pressure experienced by the aerofoil due to the gust, not the total pressure. We therefore see that an incident gust affects the lift on aerofoils at lower (or negative) angles of attack more than aerofoils at positive angles of attack.

Figure 2.7b illustrates the effect of changing the frequency of the incident gust. As seen in Figure 2.6, where we varied k₃ which is equivalent to a variation in frequency, increasing the frequency increases the modulation of the pressure. For the unsteady surface pressure jump this is still caused by the interaction between the leading- and trailing-edge fields. We observe a slight variation in magnitude of the pressure jump as we alter k, because the leading-edge field scales as k⁻¹ and the trailing-edge field scales ask^−3/2.

0.5 1.0 1.5 2.0 0.1

0.2 0.3 0.4 0.5

Α_i=6^o Α_i=0^o Α_i=-6^o

(a) NACA 1112,k= 8, varying αi.

0.5 1.0 1.5 2.0

0.05 0.10 0.15 0.20 0.25 0.30 0.35

k=12 k=10 k=8

(b) NACA 1112, αi = 0^◦, varyingk.

0.5 1.0 1.5 2.0

0.1 0.2 0.3 0.4 0.5

0000 0006 0012

(c)k= 8,α_i = 0^◦, varying thickness.

Figure 2.7: Absolute value of the unsteady surface pressure jump across a NACA aerofoil, with θ_g = 45^◦, k₃ = 0, and M = 0.6. The horizontal axis denotes the chord position, and the vertical axis measures the non-dimensionalised pressure jump.

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-0.1 0.1 0.2

1112 0012

Figure 2.8: Unsteady lift on a NACA 1112 aerofoil, with θg = 45^◦ and αi = 0^◦. k varies from8 at the outermost point on the spiral, to 20at the innermost point on the spiral.

Axes show the real and imaginary parts of the non-dimensionalised lift.

Figure 2.7c shows the effect of altering the thickness of the aerofoil; there is little difference between a 12%and a6% thick aerofoil. However, there is a notable difference between aerofoils with non-zero thickness and flat plates; the flat plate generates a larger unsteady pressure jump than the thick aerofoils. For a thick aerofoil the gust is blocked horizontally by the bluff nose and the steady flow is slowed by the bluff aerofoil. A flat plate can only block the gust vertically and no flow slowing occurs. We see from Figure 2.4 that increasing thickness predominantly decreases the far-field pressure magnitude upstream of the aerofoil, but has little effect elsewhere, and away from the upstream re-gion the difference between the10%- and5%-thick aerofoils is smaller than the difference between the zero-thickness and5%-thick aerofoils.

As the frequency of the gust increases, we expect the unsteady lift on the aerofoil (i.e. the lift caused by the perturbation gust rather than the background steady flow) to decrease because the unsteady response by the aerofoil scales with inverse powers of k. Figure 2.8 illustrates this, but we also see that the complex value of the unsteady lift forms a spiral as we vary k. This is in agreement with the trend exhibited by the Sears function (Sears, 1941) which is proportional to the unsteady lift generated by a flat plate interacting with a sinusoidal gust in two-dimensional incompressible flow. We do not see lift values with large positive real parts for the smaller values of k in Figure 2.8 (something that is observed with the Sears function) because our solution is restricted to the high-frequency regime therefore we do not expect to recover any of the low-frequency behaviour that is exhibited by the Sears function.

We note here that there is an integrable singularity in the unsteady surface pressure at the leading edge predicted by the asymptotic solution as seen in Figure 2.7. This singularity violates the asymptotic assumption that the unsteady flow is much smaller than the steady flow, hence in a small region close to the nose the solution is not valid (but it does not invalidate any of our far-field solutions. It only invalidates our surface pressure solutions in a small region close to x = 0). This singularity is caused by the stagnation point of the steady flow, therefore we should consider a region around the stagnation point more carefully. To determine the size of this region we consider the true placing of the stagnation point for a lifting aerofoil. It lies at an O(²) distance from the leading edge, however due to the approximations enforced during the asymptotic work, we treat this point as if it were the same as the point directly at the leading edge.

This results in a square root type singularity in the surface pressure, x^−1/2, which should actually be (x+²)^−1/2 which is non-singular (see Van Dyke (1975) for further details).

This scaling implies that there should be a further asymptotic region close to the leading edge and stagnation point of the aerofoil that we need to treat differently in order to rectify the surface pressure singularity. If we were to introduce this new region to the solution in this chapter, we would have to retain terms of O(²) and therefore have to create a solution with many more terms than we have done currently. This is clearly not the optimal solution to the problem, therefore later, in Chapter 4, we choose a slightly different approach to solve for the flow around the stagnation point. The solution we obtain, however, can still be thought of as an “inner-inner” region to the current problem because it is shown to be consistent with the solutions in this chapter.