By W.H.J.J. Van Staveren

This can be a Ph.D. dissertation. The reaction of plane to stochastic atmospheric turbulence performs a huge function in, for instance, airplane layout (load calculations) and flight simulation (handling characteristics examine and pilot training). with a view to simulate those plane responses, a correct mathematical version is needed. classical versions can be mentioned during this thesis that's the Delft collage of expertise (DUT) version and the 4 element plane (FPA) version. even if they're good validated, their constancy is still vague. The reason lies in a single of the necessities for approach id; it has constantly been essential to relate inputs to outputs to figure out, or establish, process dynamic features. From experiments, utilizing either the measured enter and the measured output, a mathematical version of any approach could be obtained.When contemplating an input-output process resembling an plane subjected to stochastic atmospheric turbulence, a big challenge emerges. in the course of flight checks, no useful hassle arises measuring the aircraft-system's outputs, similar to the angle-of-attack, the pitch-angle, the roll-angle, and so forth. in spite of the fact that, an immense challenge arises while the enter to the aircraft-system is taken into account; this enter is stochastic atmospheric turbulence during this thesis. presently, it nonetheless is still tremendous tricky to spot the total flowfield round an aircraft's geometry subjected to a turbulent box of movement; an enormous quantity of sensors will be required to spot the atmospheric turbulence pace component's distribution (the enter) over it. accordingly, it's tough, if no longer most unlikely, to spot atmospheric turbulence versions from flight checks.

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**Example text**

11: 2D Covariance function Cu3 u3 ( Lξ1g , ξ2 Lg , 0). 24) 26 The atmospheric turbulence model with i = 1, 2, 3, j = 1, 2, 3, δij the Kronecker delta, σ 2 the variance of atmospheric turbulence and Lg the turbulence scale length. 30) Reduced spatial frequency dimension examples Reduced spatial frequency dimension PSD examples are derived by integrating over nondimensional spatial frequencies Ωi Lg with i = 1, 2, 3. 4 The atmospheric turbulence PSD function matrix Ω22 L2g + Ω23 L2g 16πσ −Ω2 Ω1 L2g = (1 + Ω21 L2g + Ω22 L2g + Ω23 L2g )3 −Ω3 Ω1 L2g 2 −Ω1 Ω2 L2g 2 2 Ω1 Lg + Ω23 L2g −Ω3 Ω2 L2g −Ω1 Ω3 L2g −Ω2 Ω3 L2g Ω21 L2g + Ω22 L2g In the following, two examples of reduced spatial frequency dimension atmospheric turbulence will be given.

5. For example, consider the connection line between the Earth-Fixed Frame of Reference’s origin OE and the arbitrary point P (ξ1 , ξ2 , ξ3 ), spatially separated in three dimensions. If the atmospheric turbulence velocity components in origin OE and point P are decomposed in the direction of this connection line and perpendicular to it, the components of the correlation function matrix Rui uj (ξ) can be derived. 2 a simple 2D spatial separation example is given. Components of the covariance function matrix Cui uj (ξ), with i = 1, 2, 3 j = 1, 2, 3 and the spatial separation ξ = [ξ1 , ξ2 , 0]T , will be derived.

Similar to u1long and u1lat , u2long and u2lat are the decompositions of u2 on the line OE P and a line perpendicular to it (through point P ), respectively. The elements of the atmospheric turbulence covariance function matrix can be derived by calculating each Cui uj (ξ) = Cui uj (ξ1 , ξ2 , 0) = E {ui (0, 0, 0)uj (ξ1 , ξ2 , 0)} = E ui (0)uj (ξ) . 6. 17), E u1long (0)u1long (ξ) = σ 2 f (ξ) E u1lat (0)u1lat (ξ) = σ 2 g(ξ) E u1long (0)u1lat (ξ) = 0 E u1lat (0)u1long (ξ) = 0 with ξ = ξ12 + ξ22 . 11.