Radar sensing of Wake Vortices in clear air

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2 Radar sensing of Wake Vortices in clear air a feasibility study V. Brion*, N. Jeannin** Wakenet workshop, may 2013, DGAC STAC, Bonneuil-Sur-Marne *Onera Paris **Onera Toulouse 2

3 Introduction In-house collaborative project at ONERA : DoCToR Detection and Characterization of Wake Vortices by Radar and Lidar in clear air Fluid mechanics, radar and lidar teams involved Knowledge on vortex dynamics but little on clear air radar echo Part of the objectives : Understand radar echo in clear air Review the litterature for WV Model the physics Calculate the Radar Cross Section (RCS) Conclude on the feasability 3

4 Bibliography Gilson experiments ("Aircraft wake RCS measurement", NASA, 1990) - VHF, UHF, L, S and C bands - R = 100m 1km - Power = to 7 MW Nespor et al. ("Doppler radar detection of vortex hazard indicators", NASA, 1994) - C band - Power ~ 1MW peak - RCS ~ - 80dBm 2 4

5 Bibliography Shephard (GEC Marconi trial 1990) - S and X band - RCS ~ - 80dBm 2 Thales, more recently, in rainy weather - X band - Orly & Roissy airports 5

6 Bibliography Main conclusions Proofs of radar detection in the 90' RCS ~ -80dBm 2 However no duplication of the tests Reports importance of the met. conditions (stratification, humidity) Independence upon jet exhaust is surprising 6

7 Outline 1 How does radar echo in clear air occurs? theory of dielectrics 2 How can we model the physics? The Navier Stokes and Maxwell equations 3 How much is the Radar Cross Section of WV? Evaluation of the RCS with the fluid & ElectroMag numerical models 7

8 Illustration of the problem > 10 wing spans radar - target R echo wake vortices in the far wake Pr Pt Pt: transmitted power Pr: received power ground radar 8

9 Illustration of the problem > 10 wing spans radar - target wake vortices in the far wake R echo radar cell Simplified radar equation A e Pr Pt transmitted power P r = PG π σ A π t e R 4 R RCS of the radar cell ground radar A e : antenna surface G: antenna gain 9

10 Origins of the radar echo in clear air Sample of clear air Humidity in the form of water vapor characterized by Relative Humidity RH RH = 100 p p v sat v p v partial pressure of water vapor p v sat partial pressure of water vapor at saturation Main components p ln p L = Rv 1 T0 T sat v 1 sat v,0 N 2 (80%) O 2 (20%) H 2 O (less than 1% in mass) also specific humidity q q ρv = ρ pv p 10

11 Origins of the radar echo in clear air Consider a molecule of O 2 q = electric charge -q +q By symmetry, the centers of positive and negative charges are at the same location 11

12 The capacitor model Apply an electric field E E 0 +q -q d the positive and negative charges shift in opposite directions, a distance d appart 12

13 The induced dipole moment Apply an electric field E E 0 +q -q d p the positive and negative charges shift in opposite directions, a distance d appart A dipolar moment p results such that p=qd in magnitude and oriented upward 13

14 The atomic polarizability α Apply an electric field E E 0 +q -q d p the positive and negative charges shift in opposite directions, a distance d appart A dipolar moment p results such that p=qd in magnitude and oriented upward In the case of linear dielectric, p is proportional to the electric field E 0 : p=αe 0 14

15 Case of an ensemble of molecules p 2 p 3 p i E 0 p 1 p 4 P For more than one molecule, the resulting density of dipole moment is P=SUM(p i )/Volume 15

16 Case of an ensemble of molecules p 2 p 3 p i E 0 p 1 p 4 P E' For more than one molecule, the resulting density of dipole moment is P=SUM(p i )/Volume An opposite macroscopic electric field E' results, such that P = 4πE' opposed to the applied field E 0 16

17 p 2 p 3 Case of an ensemble of molecules Total macroscopic field p i E E 0 p 1 p 4 E' The resulting field yields E = E 0 + E' total applied Introducing the electric susceptibility χ e P = χee induced E ( 1 4πχ )E 0 = + E0 = εe e εis the dielectric constant 17

18 Connecting εto the flow variables temperature and density Connecting εand α 1- for gases at normal density χ Nα e = 2- therefore ε =1+ 4πNα E 0 E' 3 - which can be recast into ε =1+ 4π M N A α General formula found in the litterature = 1 + K ρ + K ρ + ρ K ε 1 d 2 v 3 * M is the molar mass and N A the Avogadro number ρv T K 1, K 2, K 3 are constants ρ d is the density of dry air (80% N % O 2 ) ρ v is the density of water vapor (gas) 18

19 Important remarks Refraction index "n" ε n = ε n ' ' = 1 + K ρ + K ρ + 1 d 2 v K ' 3 ρv T Values given by Thayer* ' 6 3 K = m / kg 1 ' 6 3 K = m / kg 2 Limitations ' 3 K 1.74m. K / kg 3 = No dependence upon frequency, ~ true up to 22GHz (absorption ray of water) linear relationship between p and E : true if E no too strong * Thayer, "An improved equation for the radio refractive index of air", Radio Science,

20 Flow phenomena causing εvariations Atmospheric stratification temperature, pressure humidity mixing and transport by the wake vortices Additional effects compressibility in the vortex cores turbulence (increased mixing) jet exhaust 20

21 Radar echo in clear the full problem FLOW ELECTRO MAGNETIC RESPONSE ' ' n = 1 + K ρ + K ρ + K One way coupling : NS Maxwell through the formula for "n" 1 d 2 v ' 3 ρv T 21

22 Simplifications NS + Maxwell = too complicated Simplifications possible since flow is low speed incompressible mostly 2D harmonic electric field radar far away from the target scattered field = weak 22

23 Flow problem U Atmosphere stratification in ρ, p, T humidity sectional plane x -Γ +Γ - 2D z y b - equal strength Γ opposite vortices - separation b 23

24 Model for the flow Boussinesq model ( x t) = u ( x) u ( x t) u,, ( x t) = p ( x) p ( x t) p,, Atmospheric state + dynamics associated to the vortices ( x t) = ρ 0 ( x) ρ ( x, t), 1 ρ + NS incompressible Momentum Energy div( u 1 ) = 0 u t 1 ρ t + d1 1 Fr ( u1 ) u1 = p1 ρd1ez + u1 + u ρ 1 d1 = u 1 Pr. 1 ez + d 1 2 Re ρ 1 Re Parameters Fr = Froude number Re = Reynolds number Pr = Prandtl number Γ Fr = Re = 2 2πb N Γ 2πν k Pr = µ variable constant = 400 constant = 0.7 variable with Fr 24 c p N = Brunt-Vaisala freq. g dρ 0 N = ρ0 dz 1/ 2

25 Numerics Methodology - Finite element solver FreeFem++ - 2D computational domain - Only half of the domain calculated - 2 nd order time & space Computational domain and mesh structure fine mesh State of the Atmosphere - Standard atmosphere - Vapour pressure : p v = Pa/m vortex Flow initialization 2 Lamb-Oseen vortices separated by a distance b ω = 1 Γ πa 2 i i = 1 2 e ri a 2 2 Configuration = Airbus A Γ and b, a/b=0.2 25

26 Flow configuration b initial vortices initial altitude z 0 Γ refractive index (n-1) x 10 6 z (m) stratification ground y (m) 26

27 Illustration of the transport phenomenon, Fr=10 refractive index (n-1) x 10 6 time increases The WV become more visible with time 27

28 Illustration of the mixing phenomenon, Fr=10 refractive index (n-1) x 10 6 The background stratification is mixed by the vortices 28

29 Effect of the stratification on the vortex dynamics Fr= pure descent no stratification Fr=5 weak stratification formation of a secondary wake by the baroclinc torque dω 1 = ρ 2 dt ρ ( p) Fr=1 strong stratification damping of the descending motion and emission of gravity waves 29

30 Radar Cross Section evaluation (RCS) Setup ε vaccum ε radar cell ( θ, r) y ~ 1000m x ~ 2000m θ = 1 r = 5m Scattered electric field E s r E s r r 2 k 4π ( r) = o o E( r' ) ε ( r' ) V r r exp ( jk r r' ) r r' dv ' Hypothesis Far field approximation Born approximation E=E i ε r = ε r ε background Spherical incident wave E i 30

31 RCS evaluation Fourier transform of the ε r radial distribution ε r (a) HR=0% at ground level, Fr=1, t=47s (b) HR=70% at ground level 31

32 RCS results in X band (10GHz) RCS = 4πR 2 E E s i 2 2 (a) RH=0%, Fr=1, t=47s (b) RH=70% RCS -130dBm 2 < RCS < -95dBm 2 depending on humidity less than experimental values found in the litterature 32

33 Conclusion Summary Litterature review Origins of the radar echo in clear air Model and numerical evaluation of the RCS Feasability Strong dependence upon meteorological conditions (stratification, humidity), yet to be better investigated weak signal giving RCS ~ -95 / -130 dbm 2, less than experimental tests RCS(wake vortices) in clear air ~ RCS(1 droplet of water)! Such reflectivities are too weak to be measured by standard radars Need for high power transmitter and very sensitive receiver cost and size 33