Méthodes Expérimentales en Mécanique des Fluides Optically ...
Transcript of Méthodes Expérimentales en Mécanique des Fluides Optically ...
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Methodes Experimentales en Mecanique des FluidesOptically-based visualisation
Luc Pastur
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Outline
1 Basics on Fourier optics
2 Shadowgraphy (ombroscopie)
3 Sclieren (strioscopie)
4 Interferometry and holography
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Basics on Fourier Optics
Domain of relevanceBased on the Huygens-Fresnel principle
Deals with solutions of the wave equation with boundary conditions
Diffraction ≡ object convolution by impulse response h(x , y) of the optical system
Approximations : Fresnel for the near field or Fraunhofer for the far field
Lenses ≡ optical processors (Fourier transform).
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Huygens-Fresnel principle
HuygensEach point P′ of a wave surface behaves as a (fictive) ponctual source, at the same frequencyas the parent source, with a phase which is the incident wave phase in P′
FresnelThe spherical wavelets emitted by such fictive sources propagate toward any point P where theyinterfer.
source
O
S ′ S
r0
r
θ
z
x ′
y ′
x
y
P
P′
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Amplitude of the light vibration at point P
ψ(P) =
∫∫S′
f (P′)e ikP′P
P′PK(a) dS(P′)
ψ(P) amplitude at point P,
f (P′)dS(P′) = ψSe ikr0
r0dS(P′) amplitude of sources over dS(P′) centered on P′,
e ikP′P
P′P for spherical propagation from P′ to P,
K(a) is the inclinaison factor introduced by Fresnel to take into account :
the distribution anisotropy of the diffracted energy
lack of “back” diffraction
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Basics of Fourier Optics
Helmholtz equation for wave propagationGreen function (diverging monochromatic spherical wave)
ψ(P) = −1
2π
∫∫S′ψ(P′)
(ik −
1
P′P
)e ikP′P
P′Pcos(~n,
−−→P′P) dS ′
⇒ Rayleigh-Sommerfeld integralWhen P′P � λ
2π(|ik| � 1
P′P ) :
ψ(P) = −i
λ
∫∫S′ψ(P′)
e ikP′P
P′Pcos(~n,
−−→P′P) dS ′
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Fresnel near-field approximation
Validity conditions
1 Paraxial optics :−−→P′P '
−→OP = ~r and cos(~n,
−−→P′P) ' cos θ
2 Far from the object : e ikP′P
P′P ' e ikP′P
z
3 About phase : P′P = z
(1 + 1
2
(x−x′
z
)2+ 1
2
(y−y′
z
)2)
+O((
x−x′
z
)4,(
y−y′
z
)4)
Resulting near field
ψ(P) =e ikz
iλzcos θ
∫∫S′ψ(P′)e ik
(r−r′)2
2z dS ′
Impulse response
h(x , y , z) =e ikz
iλzexp
(ik
x2 + y2
2z
),
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Fraunhofer far-field approximation
→ Further simplification when z � 12
k(x2 + y2)max :
ψ(P) = −e ikz
iλzcos θ e ik x2+y2
2z
∫∫S′ψ(P′)e−2iπ xx′+yy′
λz dS ′
ψ(P) = −e ikz
iλzcos θ e ik x2+y2
2z F{ψ(P′)}.
Spatial frequency : px = xλz, py = y
λz.
→ Severity of the approximation : (x2 + y2)max = 1 mm2, λ = 0.500 µm⇒ z � 6.3 m
→ Diffraction field in the focal plane of a converging lens
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Examples of diffraction figures
circular pupil vertical slit vertical edge
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Example of optical filtering
Figure: Optical Fourier transform (after Yves Usson).
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Example of optical filtering
Figure: Low-pass optical filter.
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Example of optical filtering
Figure: High-pass optical filter.
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Deflexion in a medium of index n = n(y)
Initial wave front Σ0 ; during δt, M and M′ moved by
δz(y) = c(y) δt, δz(y + δy) = c(y + δy) δt
Σδt tilted by δα′ :
tan δα′ =δz(y)− δz(y + δy)
δy= −
∂ δz
∂y
∣∣∣∣y
= −∂
∂y
(c0
n(y)δt
)
= − n(y)∂(1/n)
∂y
c0
n(y)δt =
1
n
∂n
∂yδz(y)
Weak deflexion : δα′ '1
n
∂n
∂yδz(y)
Total deflexion
α′ =
∫Cδα′ =
∫C
1
n
∂n
∂ydz
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Taking into account the vein-glasses thickness
Fluid enclosed in between windows of index n. Let na be the outside optical index, different fromn. Then (Snell-Descartes) :
nα′ ' na α,
Deflexion angle α at the vein outlet :
α =n
na
∫C
1
n
∂n
∂ydz.
1/n slowly changing within the fluid can be considered constant with respect to fluctuations ∂n∂y
.
Since na ' 1, one gets :
α =
∫C
∂n
∂ydz.
Light rays are deviated in the direction of increasing refraction index
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Dilute gaz
Gladstone-Dale law
n − 1 = Kρ
for air, K ' 0.22× 10−3m3 · kg−1
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Non intrusive optical techniques
Shadowgraphy
Figure: Shock wave produced by a supersonic bullet. Source : Rochester Institute ofTechnology.
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Observation of deflected rays
Shadowgraphy is developed by Dvorak in 1880 (Ernst Mach’s collaborator)
Very common phenomenon when air is hot ⇒ light ray distorsion
Two distinct incident rays, distant by ∆y at inlet and ∆yE on the screen. Bottom raydeflected by α ; top ray deflected by α+ dα.
Displacement on the screen : ∆yE = ∆y + zE dα
Optical contrast : I0−ISIS' −zE
∂α∂y
= −zE
na
∫∂2n
∂y2dz :
→ Connected to variations of ∂2n∂y2 .
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Shadowgraphy images
Widely used for visualizing supersonic and transonic flows ; reveals shocks, boundary layers, etc
Figure: Left : simulation of a space ship atmospheric entrance. Source : NASA. Right :turbulent jet, visualized by steam condensation on the left, by shadowgraphy on the right.Source : UC Irvine.
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Thermal boundary layer
T
y
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Non intrusive optical techniques
Schlieren
Figure: Shock wave visualization by schlieren technique. Source : Rochester Institute ofTechnology.
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Sclieren principle
Schlieren (german) developed by Foucault (1859), used for flow visualization byToepler (1864).
Principle : Foucault knife in the plane of L2 : suppression of the object positive spectralcomponents :
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Schlieren and optical filtering
→ In L2 focal plane :
ψ(px , py ) · H(px , py )
Filter H :
H(px , py ) =
{0 for py > 01 for py ≤ 0
=1
2(1− sgn(py ))
→ In the image plane E :
ψE (x , y) = 12F{ψ(px , py )− ψ(px , py ) · sgn(py )
}=
1
2
(ψ(−x ,−y)− ψ(−x ,−y) ∗
1
iπy
)
ψE =1
2(ψ(−x ,−y) + iH{ψ}(−x ,−y))
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Hilbert Transform
→ Convolution product of input signal ψ with 1/(πy) :
ψH (x , y) = H{ψ(x , y)} =1
π
∫ ∞−∞
ψ(x , y ′)
y − y ′dy ′
→ In the reciprocal space :
F {H{ψ}} = F {ψ ∗ (1/πy)}= F {ψ} · F {1/(πy)}= ψ(ν) · (−i sgn(py ))
→ ±π/2 rotation of the signal negative/positive spectral components
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Hilbert transform of some imput signal
Signal s(t) Hilbert transform H{s}(t)
1 0sin t − cos tcos t sin t
sin t
t
1− cos t
t
u(t)1
πln
∣∣∣∣∣ t + 12
t − 12
∣∣∣∣∣δ(t)
1
πt
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Analytical signal
Consider ψ(x , y) = cos(
2πy
λ
), determine ψE and IE
Idem with ψ(x , y) = A(y) cos(ky + φ)
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Complex demodulation
ψ(y) = A(y) · cos(ky + φ)
By Hilbert transform :
ψH (y) = H{ψ(y)} = A(y) · sin(ky + φ)
it follows the analytical signal :
ψ+(y) = ψ(y) + i ψH (y) = A(y) · e i(ky+φ)
from which can be extracted amplitude and phase :A2(y) = |ψ+(y)|2 = |ψ(y)|2 + |ψH (y)|2
tan(ky + φ) =ψH (y)
ψ(y)
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Schlieren of a phase modulation
→ Object ≡ weak phase modulation (a� 1) :
ψ(x , y) = e ia cos 2πyλ ' 1 + ia cos
2πy
λ
→ Hilbert transform :
ψH (x , y) = −ia sin2πy
λ
→ Intensity at screen is :
IE (x , y) =1
4
(1 + 2a sin
2πy
λ
).
→ Modulation factor :
M =Imax − Imin
Imax + Imin= 2a.
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Deflected rays with Schlieren
Let the couple D1 − D2 be a square hole - knife edge ; intensity at screen :
Id = I1ak +∆a
ak= I1
(1 + ∆a
ak
),
Contrast : ∆II1
= ∆aak
= αf2ak
= f2ak na
∫∂n∂y
dz.
⇒ Schlieren reveals variations of optical index gradient ∂n∂y
.
Brightest regions on screen ⇔ n increasing in the knife direction.
Axisymmetric flow ⇒ anti-symmetry of the intensity distribution apart of the symmetryplane defined by the knife.
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Couples of diaphragms
Large variety of couples of diaphragms D1 − D2
Balck and white images or coulor ;
Directional couples (square-knife), or non directional (circular mask).
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Examples of Schlieren visualization
Figure: Left : Shock wave visualization around a nose cone. Source : Georgia Tech. Right :Shock wave around a projectile. Source : Aerospace Sciences Corporation Pty. Ltd.
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Shadowgraphy vs Schlieren visualisation
Figure: Left : shadowgrapohy, right : schlieren (vertical knife). Photographs by AndrewDavidhazy, Kennedy Space Center.
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Non intrusive optical techniques
Interferometry
Figure: Michelson (left) and Zehnder (right) interferometers.
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Light coherence
Technique due to Ludwig Mach (son of) & Ludwig Zehnder.
Beam coherence featuresCoherence time∆t ∼ 1/∆ν : time duration over which the wave train keeps its mean frequency. For apure monochromatic wave (∆ν → 0), ∆t →∞. Over ∆t, wave behaves roughly as amonochromatic wave, its phase at a given point in space, in the direction of propagation,can be reasonnably predicted.
Coherence lengthL = c∆t : distance covered during ∆t. A few µm for mercury lamps ; several metres forsome laser.
Spatial coherencerelated to the finite spatial extension of the source. Two source points, distant by λ,usually emit at slightly different frequencies, with uncorrelated phases. If the two points,laterally displaced, remain on the same wave front, at a given time, then the wave trainsemitted by these two points are spatially coherent.
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Interference figures
Beams interfer under the same incidence ⇒ diffraction grating made of parallel fringeswith infinite inter-fringe.
If path difference ∆` = 0, uniform intensity on screen.Non zero path difference ⇒ ∆φ = 2π∆`
λ0= 2π
λ0
∫( 1
nprobe− 1
nref) dz
Amplitude superposition :
E = Eref + Eprobe = E0 cos
(2πcτ
λ
)+ E0 cos
(2πcτ
λ+ ∆φ
).
⇒ Intensity : I = |E |2 ' E 20 /2
(1 + 2 cos2
(∆φ
2
)).
⇒ Fringe grating.
Reference & probe beams cross with non zero angle θ ⇒ interference fringes, spaced out
by δ =λ/2
sin(θ/2)'λ
θ.
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Benefits and drawbacks of the technique
AssetsDirect access to fluctuations of n, and therefore ρ : most quantitative information aboutthe flow
More details in visualization than with shadowgraphy or schlieren
WeaknessesHigh setup stability required & highest quality of the optical elements (surface planeity< λ/10 !)
Inadapted to 3D flows, due to integration over the optical path
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Non intrusive optical techniques
Holography
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Holography principle
Interferometry : two beams, probe & reference, simultaneously propagate along differentpathes before being mixed
Holography : probe beam crossses the medium and interfers with reference beam. Figureof interference written on an holographic plate
Probe and reference beams travel again through the medium ⇒ different figure ofinterferenceGrating resulting from superposition ⇔ path difference in the fluid at rest and inmotion
⇒ Probe & reference beams, when writing and reading, follow the same path, at differenttimes
One propagation path ⇒ self-compensation of optical defects or misalignements⇒ Quality of optical components less rigorous
Reference beam diametre reduced ti a few mm
⇒ Lightened setup, less sensitive to surrounding perturbations ambiantes.
⇒ Simpler and cheaper technique than interferometry
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Differentes configurations
Figure: Interference grating writing on a holographic plate
Parallel superposition of images with and without flows ⇒ distorted parallel fringe grating
Superposition with a tilt θ 6= 0 of images with and without flow ⇒ distorted tilted fringegrating
Superposition of two images with flow ⇒ visualize relative differences in the flow
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Relecture
Information recorded on the plate :
I =∣∣Eref + Eprobe
∣∣2 = |Eref |2 +∣∣Eprobe
∣∣2 + E ref · Eprobe + Eref · E probe
When reading with the reference beam, the transmitted amplitude is
Electure = Eref
(|Eref |2 +
∣∣Eprobe
∣∣2) + Eprobe |Eref |2 + E probe E 2ref
(1) (2) (3)
(1) non diffracted transmitted beam (order 0)(2) reconstructed image of the probed object(3) conjugated image of the object.
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Images of interferometry/holography
Figure: Left : Compressing propeller put in vibration. Visualization of resonances usingholography. Source : Warwick University Eng. Dpt. Right : supersonic flow upon a wing ;visualization of lines of constant density using holography. Source : Lab. Thermique Appliqueeet Turbomachines (Lausanne)
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Techniques comparaison
Shadowgraphy
→ give access to variations of ∂2n∂y2 .
→ Implementation of high simplicity
Schlieren
→ give access to variations of ∂n∂y
→ Implementation relatively simple
→ High variety of filtering couples
Interferometry→ give access to variations of n(y)
→ Meticulous implementation & optical elements of high quality ⇒ expensive in time andmoney !
→ interferometric holography much simpler and less demanding concerning the opticalelements
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