Helicopters / Vortex theory. Filipe Szolnoky Cunha

Transcription

1 Vortex Theory Slide 1

2 Vortex Theory Slide 2

3 Vortex Theory µ=0.2 Slide 3

4 Vortex Theory µ=0.4 Slide 4

5 Vortex Theory Slide 5

6 Tip Vortex Trajectories Top view Slide 6

7 Definition Wake Age Slide 7

8 Assumptions: Tip Vortex Trajectories Wake undistorted in in the x-y plane Trajectories closely follow epicycloidal forms Then the trajectories can be described by the parametric equations: Slide 8

9 Blade Vortex Interactions Then locus of all potential BVI can be determined if both following equations are satisfied for r (on the blade) and ψ b : Slide 9

10 Blade Vortex Interactions With the solution : Only the real part is of interest: With the corresponding value of r: Slide 10

11 Blade Vortex Interactions Finally we can obtain the xand yvalues : With By solving for ψ b and r for numerical values of ψ w >0 we can determine all the locations of potential BVI intersection points Slide 11

12 Blade Vortex Interactions Slide 12

13 Blade Vortex Interactions Slide 13

14 Vortex Theory Extension of Prandtl s Lifting Line Theory Uses a combinationof Kutta-Joukowski Theorem Biot-Savart Law Empirical Prescribed Wake or Free Wake Representation of Tip Vortices and Inner Wake Robin Gray proposed the prescribed wake model in Landgrebe generalized Gray s model with extensive experimental data. was the extensively used in the 1970s and 1980s for rotor performance calculations, and is slowly giving way to CFD methods. Slide 14

15 Vortex Theory addresses some of the drawbacks of combined blade element-momentum theory methods,athighthrustsettings(highc T /σ). At these settings, the inflow velocity is affected by the contraction of the wake. Near the tip, there can be an upward directed inflow (rather than downward directed) due to this contraction, which increases the tip loading, and alters the tip power consumption. Slide 15

16 Kutta-Joukowsky Theorem The link between lift per unit length of span an the local circulation is: Since: Slide 16

17 Kutta-Joukowsky Theorem We had already seen that: So we can write: We a uniform circulation along the blade span, Helmholtz s theorem requires a single vortex of thesamestrengthtotrailfromthebladetips Slide 17

18 Representation of Bound and Trailing Vortices Since vorticity can not abruptly increase in space, trailing vortices develop.. Slide 18

19 Biot-Savart Law Fundamental to all vortex models is the requirement to compute the induced velocity at a point contributed by a vortex filament: Slide 19

20 Biot-Savart Law A different expression can be found: Control Point A Γ n B Vortex Segment Slide 20

21 Vortex Velocity model To avoid having a infinite velocity when r 0, the vortex is modelled having a outer potential flow region and inner solid body rotation region Slide 21

22 Vortex Velocity model The core radius, r c, is defined as the radial locationwherev θ ismaximum ThereforeV θ ismaximumat This boundary demarcates the inner (pure rotation) flow field from the outer(potential) flow Slide 22

23 Vortex Velocity model The simplest model is the Rankine vortex model where; The coreis modelled as asolidbodyrotation Velocity outside decreases hyperbolically with the distance Slide 23

24 Vortex Velocity model An alternative is the Oseen-Lamb vortex model, obtained through a simplified form of the Navier- Stokes equations: Whereα= Slide 24

25 Vortex Velocity model Newman has also derived exponential solutions for the three components of velocity in the vortex core based on a simplified Navier-Stokes formulation. The result for the swirl velocity is the same as that for the Ossen-Lamb model but Newman shows that the axial velocityin thevortexcoreis A is a constant that can be related to the drag on the generating lifting surface Slide 25

26 Vortex Velocity model A more general series of desingularized velocity profiles for columnar vortices with continuous distributions of flow quantities is given by Vastitas: Slide 26

27 Vortex Velocity model Slide 27

28 Vortex Velocity model Slide 28

29 Vortex Velocity model Slide 29

30 Vortex Core Growth The vortex core dimension is an important parameter that can be used to help define the structure and evolution of the tip vortices. The average viscous core radius can be considered as half the distance between the two velocity peaks. Slide 30

31 Vortex Core Growth Slide 31

32 Vortex Core Growth Slide 32

33 Vortex Core Growth A simple quantitative model of the growth in the vortex core radius with time can be based on Lamb s results for laminar flows Starting from the Lamb-Oseen swirls velocity profile: And with the change of variable: Slide 33

34 Vortex Core Growth The core radius r c corresponds to the value of r when V θ reaches a maximum: The solution is x=1.1209so the core radius grows with Slide 34

35 Vortex Core Growth Where α= see Lamb s vortex model. Is practice, because of turbulence generation the actual diffusion of vorticity contained in the vortexis knowntobemuchquickerthatthis. This effect, albeit very complicated on a fundamental level, can be incorporated into a model core growth equation using an average turbulent viscosity coefficient: Slide 35

36 Howto use thistheory Wethis theorywecan: Simulate vortex sheets/ tip vortex Calculatede vortexgrowth withwake age Calculate the vortex induced velocity anywhere Letusapplythis theorytoahelicopterrotor Slide 36

37 Blade Representation Slide 37

38 Solve for Vortex Strength At the blade control points: Set-up matrix equation to solve for unknown Γ s Slide 38

39 Modelling the Vortex wake Slide 39

40 Landgrebe s Model Inner wake descendsfaster near thetipthan at theroot. Tip Vortex has a Contraction that can be fitted with an exponential curve fit. Slide 40

41 Radial Contraction Radial position of the tip vortex With the empirical values Slide 41

42 Landgrebe s Curve Fit for the Tip Vortex Contraction R w v 2v Ψ Slide 42

43 Landgrebe s Curve Fit for Tip Vortex Descent Rate Slide 43

44 Outer end Landgrebe s Curve Fit for the Vortex Sheet Inner end Slide 44

45 With Landgrebe s Curve Fit for the Vortex Sheet Slide 45

46 Wakecomparison Slide 46

47 Wakecomparison Slide 47

48 Vortex Wake Models for Forward Vortexring: Flight Stacked of vortex rings(vortex tube) Each ring is the vortex trailed by a blade during a rotation The position of the ring is defined by simple momentum theory An analytic solution for the induced velocity can be obtained Slide 48

49 Vortex Wake Models for Forward Flight Rigid or undistorted wake Trailed vortices are represented by skewed helical filaments The position of the vortex filaments is defined geometrically based on flight conditions and momentum theory conditions There are no self or mutual-interactions between vortex filaments Slide 49

50 Vortex Wake Models for Forward Flight If the tip vortex is the only one considered then its position is given by the parametric equations: Slide 50

51 Vortex Wake Models for Forward Flight Modifications for the rigid wake Other type of wake models were based on experimental results like Egolf & Landgrebe: Slide 51

52 Vortex Wake Models for Forward E is theamplitude Flight G is thegeometric function Slide 52

53 Vortex Wake Models for Forward Flight Modifications to Rigid Vortex Wake models The advantage is that with a small increase in computational effort, much better estimates of the rotor wake geometry can be obtained compared with the rigid wake Slide 53

54 Tip Vortex Representation in Computational Analyses The tip vortex is a continuous helical structure. This continuous structure is broken into piecewise straight line segments, each representing 15 degrees to 30 degrees of vortex age. The tip vortex strength is assumed to be the maximum bound circulation. Some calculations assume it to be 80% of the peak circulation. The vortex is assumed to have a small core of an empirically prescribed radius, to keep induced velocities finite. Slide 54

55 Overview of Vortex Theory Based Computations (Code supplied) Compute inflow using BEM first, using Biot-Savart law during subsequent iterations. Compute radial distribution of Loads. Convert these loads into circulation strengths. Compute the peak circulation strength. This is the strength of the tip vortex. Assume a prescribed vortex trajectory. Discard the induced velocities from BEM, use induced velocities from Biot-Savart law. Repeat until everything converges. During each iteration, adjust the blade pitch angle (trim it) if C T computed is too small or too large, compared to the supplied value. Slide 55

56 Free Wake Models These models remove the need for empirical prescription of the tip vortex structure. We march in time, starting with an initial guess for the wake. The end points of the segments are allowed to freely move in space, convected the self-induced velocity at these end points. Their positions are updated at the end of each time step. Slide 56

57 Free Wake Models Slide 57

58 Vortex Calculation (top View) Slide 58

59 Vortex Calculation (Side View) Slide 59

60 Prof. Leishman Calculations Helicopter Rotor in Low Speed Axial Descent (Incipient VRS), Followed by Transition into Low-Speed Forward Flight: Slide 60

61 Prof. Leishman Calculations Tandem Helicopter Rotor Operating in Low Speed Vertical Descent Slide 61