RMC-based severity metrics: possibilities and scalings Grégoire Winckelmans and Ivan De Visscher Université catholique de Louvain (UCL), Institute of Mechanics, Materials and Civil Engineering (immc) WaPT - Wake Prediction Technologies WakeNet-Europe Workshop 2014 13th 14th May 2014, Bretigny, France
Rolling Moment Coefficient (RMC) Γtot zv yv bf The RMC is the vortex induced rolling moment on the follower normalized by the follower parameters = 1 2 where is the induced rolling moment, wing surface and is the wing span is the flight speed, is the
Rolling Moment with and =, 2, / / the effective lift slope coefficient, = ( ) the chord distribution, the relative increase of the angle of attack due to the vortex induced vertical velocity Follower parameters - wing span bf - flight speed Vf - chord distribution c(y) - fuselage size dfus Vortex parameters - total circulation Γtot - circulation distribution Γ(r)/ Γtot - position w.r.t. aircraft (yv, zv)
Circulation distribution models with a core parameter ( ) Burnham-Hallock (B-H, Low-Order Algebraic) model: High-Order Algebraic (HOA) model: ( ) Lamb-Oseen (L-O, Gaussian) model: ( ) =1.256 = ( ( ) ) = ( ), with =1.781 = 1 exp is defined as the radius of maximum induced velocity, with
Energy of near wake and rolled-up wake For elliptical loading, the energy of the near wake is induced drag) = Γ (= For the rolled-up wake as a two-vortex system (2VS), the energy is = log, with = 0.5000 for the B-H model = 0.0562 for the L-O model = 0.2052 for the HOA model At early times, Γ = Γ and. For the case =, and for eliptical loading, the core size of the vortex is thus obtained as = 4.040% for the B-H model = 7.045% for the L-O model = 5.425% for the HOA model
Circulation and velocity distributions Distributions such that the rolled-up wake has the same energy as the near wake (case of elliptical loading)
Circulation distribution models without core 1) Betz model (case of rollup with elliptical loading) with 0 =, ( ) = sin, Source: AGARDograph No. 204, Vortex Wakes of Conventional Aircraft, C. du P. Donaldson and A. Bilanin, 1975
Betz model against experimental data Source: AGARDograph No. 204, Vortex Wakes of Conventional Aircraft, C. du P. Donaldson and A. Bilanin, 1975
Betz model against experimental data Source: AGARDograph No. 204, Vortex Wakes of Conventional Aircraft, C. du P. Donaldson and A. Bilanin, 1975
Note on experimental circulation distribution From experimental data, it appears that about 54% of the circulation is found in r/b 0.05 the rest being spread in a long tail beyond r/b = 0.05 Source: AGARDograph No. 204, Vortex Wakes of Conventional Aircraft, C. du P. Donaldson and A. Bilanin, 1975
Circulation distribution models without core 2) Kaden-Winckelmans model (case of rollup with elliptical loading) Γ( ) /2 = Γ 1 + ( 1) / /2 Considering the wake formed after roll-up, the model is well fitted on the obtained circulation profile when using 1.8 Moreover, imposing that the rolled-up wake has the same energy as the near wake ( = ) leads to = 1.7743 (indeed close 1.8!)
Results from roll-up with elliptical loading Gaussian regularization of the vortex sheet, using Spatially discretized using 16 layers of particles Animations
Fit of the model on the roll-up results 1 0.9 fitted model 0.8 numerical roll-up results 0.7 / 0 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 r/(b 0/2) 0.6 0.7 0.8 0.9 1
Fit of the model on the roll-up results 30 25 u (2 b 0/ 0) 20 15 10 fitted model 5 numerical roll-up results 0 0 0.1 0.2 0.3 0.4 0.5 r/(b 0/2) 0.6 0.7 0.8 0.9 1
Circulation and velocity distribution models Caution: the Betz model does not have the same energy as the near-wake
Modified Betz model so as to have the same energy as the near wake 1 2 = 8 sin 2, sin = 0.9257 Γ( ) = sin Γ
Assessment of the models
Assessment of the models Circulation distribution model ( = 0.05 ) Burnham-Hallock 0.605 Betz 0.523 KadenWinckelmans 0.576 Modified Betz 0.560 High-Order Algebraic 0.542 Lamb-Oseen 0.469
Computation of the corresponding RMC = The term ( ) Γ ( + ) accounts for the effective lift slope" correction in a wake encounter. C=4.0 has been obtained when considering an elliptical wing (AWIATOR results, see AW-UCL-114-001-D3, 2004). This value was also proposed by NLR in another report. The correction function depends on The vortex circulation distribution Γ(r)/Γtot The size of the leader compared to that of the follower bl/bf The position of the vortex relative to the follower (yv, zv) The chord distribution of the follower c(y)
Considered encounter cases 1) Vortex centered on the fuselage center = centered encounter 2) Vortex centered on the wing and fuselage edge (we assume dfus=10% bf) 3) Vortex positioned near the wing and fuselage edge (we assume dfus=10% bf), at a radius containing 50% of Γtot radius containing 50% of Γtot All cases consider an elliptical chord distribution
Correction function for a centered encounter bl/bf
Correction function for a centered encounter Circulation distribution model bl=bf bl=2 bf bl=0.5 bf Burnham-Hallock 0.851 0.725 0.922 Betz 0.812 0.659 0.904 Kaden-Winckelmans 0.829 0.694 0.913 Modified Betz 0.837 0.696 0.917 High-Order Algebraic 0.856 0.722 0.928 Lamb-Oseen 0.859 0.721 0.929
Correction function for encounter with a vortex centered on the wing and fuselage edge bl/bf
Correction function for encounter with a vortex centered on the wing and fuselage edge Circulation distribution model bl=bf bl=2 bf bl=0.5 bf Burnham-Hallock 0.833 0.710 0.904 Betz 0.794 0.646 0.884 Kaden-Winckelmans 0.811 0.679 0.893 Modified Betz 0.819 0.681 0.897 High-Order Algebraic 0.839 0.706 0.909 Lamb-Oseen 0.841 0.705 0.910
Correction function for encounter with a vortex positioned near the wing and fuselage edge, at a radius containing 50% of Γtot bl/bf
Correction function for encounter with a vortex positioned near the wing and fuselage edge, at a radius containing 50% of Γtot Circulation distribution model bl=bf bl=2 bf bl=0.5 bf Burnham-Hallock 0.753 0.579 0.859 Betz 0.697 0.491 0.831 Kaden-Winckelmans 0.732 0.547 0.851 Modified Betz 0.733 0.540 0.851 High-Order Algebraic 0.747 0.563 0.858 Lamb-Oseen 0.740 0.551 0.853
Conclusions The correction function that appears in the RMC metric takes into account the fact that the circulation of a wake vortex is distributed over a certain distance. Typically 50-60% of the circulation lies within a radius of 5% of the generator wing span; the rest of the circulation lies beyond that radius, and with a long tail in the distribution. All circulation distribution models (those with core and those witout core) have a characteristic lengthscale that is proportional to the generator wing span. Regardless of the chosen circulation distribution model, and provided that the considered rolled up wakes have the same energy, the correction function is found fairly universal. It is thus usable as such.