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Radar sensing of Wake Vortices in clear aira feasibility study

V. Brion* , N. Jeannin**

Wakenet workshop, 15-16 may 2013, DGAC STAC, Bonneuil-Sur-Marne

*Onera Paris**Onera Toulouse

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Introduction

In-house collaborative project at ONERA : DoCToR

Detection and Characterization of Wake Vortices by Radar and Lidar in clear air

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

� Fluid mechanics, radar and lidar teams involved

� Knowledge on vortex dynamics but little on clear air radar echo

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Bibliography

Gilson experiments ("Aircraft wake RCS measurement", NASA, 1990)

- VHF, UHF, L, S and C bands- R = 100m – 1km

- Power = 0.025 to 7 MW

Nespor et al. ("Doppler radar detection of vortex hazard indicators", NASA, 1994)

- C band

- Power ~ 1MW peak- RCS ~ - 80dBm2

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Bibliography

Shephard (GEC Marconi trial 1990)

- S and X band

- RCS ~ - 80dBm2

Thales , more recently, in rainy weather

- X band

- Orly & Roissy airports

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Bibliography

Main conclusions

– Proofs of radar detection in the 90'

– RCS ~ -80dBm2

– However no duplication of the tests

– Reports importance of the met. conditions (stratification, humidity)

– Independence upon jet exhaust is surprising

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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

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Illustration of the problem

ground radar

PtPr

R

wake vorticesin the far wake

> 10 wing spans

echo

radar - target

Pt: transmitted power

Pr: received power

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Illustration of the problem

ground radar

PtPr

Simplified radar equation

22 44 R

A

R

GPP et

r πσ

π=

RCS of the radar cell

wake vorticesin the far wake

> 10 wing spans

transmitted power

Ae

Ae: antenna surfaceG: antenna gain

echoradar cell

radar - target

R

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Origins of the radar echo in clear air

Sample of clear air

H2O (less than 1% in mass)

O2 (20%)

N2 (80%)

Main components

Humidity

� in the form of water vapor� characterized by Relative Humidity RH

satv

v

p

pRH 100=

pv partial pressure of water vapor

pvsat partial pressure of water vapor at saturation

� also specific humidity q

p

pq vv 622.0≈=

ρρ

−=

TTR

L

p

p

vsatv

satv 11

ln00,

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Origins of the radar echo in clear air

Consider a molecule of O 2

+q-q

By symmetry , the centers of positive and negative charges are at the same location

q = electric charge

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The capacitor model

Apply an electric field E 0

+q

-q

E0

++++++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

� the positive and negative charges shift in opposite directions, a distance d appart

d

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The induced dipole moment

Apply an electric field E 0

+q

-q

E0

++++++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

� 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

d p

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The atomic polarizability αααα

Apply an electric field E 0

+q

-q

E0

++++++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

� 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

d p

� In the case of linear dielectric, p is proportional to the electric field E0 : p=αE0

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Case of an ensemble of molecules

E0

++++++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

For more than one molecule, the resulting density of dipole moment is P=SUM(pi)/Volume

P

p1

p2 p3

p4

pi

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Case of an ensemble of molecules

E0

++++++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

'4 EP π−=

For more than one molecule, the resulting density of dipole moment is P=SUM(pi)/Volume

An opposite macroscopic electric field E' results, such that

P

p1

p2 p3

p4

pi

E'

opposed to the applied field E0

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Case of an ensemble of molecules

E0

++++++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

E'

'0 EEE +=

EP eχ=

( )EE eπχ410 +=

εεεεis the dielectric constant

E

Total macroscopic field

The resulting field yields

applied inducedtotal

Introducing the electric susceptibility χe

EE ε=0

p1

p2 p3

p4

pi

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Connecting εεεεto the flow variables temperature and density

Connecting εεεεand αααα

1- for gases at normal density

αχ Ne =2- therefore

απε N41+=

3 - which can be recast into

ραπεM

N A41+=

General formula found in the litterature

TKKK v

vd

ρρρε 3211 +++=K1, K2, K3 are constants

ρd is the density of dry air (80% N2 + 20% O2)

ρv is the density of water vapor (gas)

* M is the molar mass and NA the Avogadro number

E'E0

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Important remarks

Refraction index "n"

TKKKn v

vd

ρρρ '3

'2

'11 +++=

kgmK /10223 36'1

−×=

kgKmK /.74.1 3'3 =

kgmK /10299 36'2

−×=

21

0

=

εε

n

Limitations

� No dependence upon frequency, ~ true up to 22GHz (absorption ray of water)

� linear relationship between p and E : true if E no too strong

Values given by Thayer*

* Thayer, "An improved equation for the radio refractive index of air", Radio Science, 1974

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Flow phenomena causing εεεεvariations

Additional effects– compressibility in the vortex cores

– turbulence (increased mixing)– jet exhaust

Atmospheric stratification– temperature, pressure

– humidity– mixing and transport by the wake vortices

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Radar echo in clear – the full problem

One way coupling : NS � Maxwell through the formula for "n"

FLOW ELECTRO MAGNETIC RESPONSE

TKKKn v

vd

ρρρ '3

'2

'11 +++=

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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

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Flow problem

sectional plane

y

z

x

Atmosphere� stratification in ρ, p, T� humidity

U∞

- 2D- equal strength |Γ| opposite vortices

- separation bb

-Γ +Γ

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Model for the flow

Boussinesq model

( ) 01 =udiv

( ) 1121111

Re

11ue

Frpuu

t

uzd ∆+−−∇=∇+

∂∂ ρ

11111

RePr

1. dzd

d euut

ρρρ ∆+=∇+∂

∂

( ) ( ) ( )txuxutxu ,, 10 +=

( ) ( ) ( )txpxptxp ,, 10 +=

( ) ( ) ( )txxtx ,, 10 ρρρ +=

Atmospheric state + dynamics associated to the vortices

incompressible

Momentum

Energy

Fr = Froude number Re = Reynolds number Pr = Prandtl number

NbFr

22πΓ= πν2

ReΓ=

pc

k

µ=Pr

N = Brunt-Vaisala freq.2/1

0

0

−=

dz

dgN

ρρ

Parameters

variable constant = 400 constant = 0.7 variable with Fr

NS

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Numerics

Methodology- Finite element solver FreeFem++

- 2D computational domain

- Only half of the domain calculated- 2nd order time & space

Computational domain and mesh structure

vortex

∑ =

−Γ= 2

1 21

2

2

ia

r

ii

eaπ

ω

State of the Atmosphere- Standard atmosphere

- Vapour pressure : ∆pv = -0.35 Pa/m

Flow initialization

� 2 Lamb-Oseen vortices separated by a distance b

fine mesh

� Configuration = Airbus A340-600 �Γ and b, a/b=0.2

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Flow configuration

initial altitude z0

z (m)

y (m)

(n-1) x 106

initial vortices

stratification

refractive index

b

Γ

ground

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Illustration of the transport phenomenon, Fr=10

(n-1) x 106

refractive index

time increases

The WV become more visible with time

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Illustration of the mixing phenomenon, Fr=10

(n-1) x 106

refractive index

The background stratification is mixed by the vortices

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Effect of the stratification on the vortex dynamics

Fr=∞

Fr=5

Fr=1

no stratification

weak stratification

strong stratification

formation of a secondary wakeby the baroclinc torque

pure descent

damping of the descending motionand emission of gravity waves

( )pdt

d ∇×∇= ρρ

ω2

1

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Radar Cross Section evaluation (RCS)

Setup

Scattered electric field E s

Hypothesis– Far field approximation– Born approximation E=Ei

– Spherical incident wave Ei

radar cell (∆θ,∆r)

∆y ~ 1000m

∆x ~ 2000m

∆θ = 1°

∆r = 5m

( ) ( ) ( ) ( )'

'

'exp''

4

2

dVrr

rrjkrrE

koorE

V rs −−

∆××−= ∫ επ

rrrr

backgroundrr εεε −=∆

vaccumε

ε

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RCS evaluation

���� Fourier transform of the ∆∆∆∆εεεεr radial distribution

(a) HR=0% at ground level, Fr=1, t=47s (b) HR=70% at ground level

∆∆∆∆εεεεr

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RCS results in X band (10GHz)

RCS

(a) RH=0%, Fr=1, t=47s (b) RH=70%

� -130dBm2 < RCS < -95dBm2 depending on humidity

� less than experimental values found in the litterature

2

2

24i

s

E

ERRCS π=

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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…