M´ ethodes Exp´ erimentales en M´ ecanique des Fluides Optically-based visualisation Luc Pastur Master 1 PAM (P-PAM-305A) MEMF 2010-2011 1 / 43
Methodes Experimentales en Mecanique des FluidesOptically-based visualisation
Luc Pastur
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 1 / 43
Outline
1 Basics on Fourier optics
2 Shadowgraphy (ombroscopie)
3 Sclieren (strioscopie)
4 Interferometry and holography
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 2 / 43
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).
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 3 / 43
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′
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 4 / 43
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
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 5 / 43
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 ′
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 6 / 43
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
),
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 7 / 43
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
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 8 / 43
Examples of diffraction figures
circular pupil vertical slit vertical edge
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 9 / 43
Example of optical filtering
Figure: Optical Fourier transform (after Yves Usson).
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 10 / 43
Example of optical filtering
Figure: Low-pass optical filter.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 11 / 43
Example of optical filtering
Figure: High-pass optical filter.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 12 / 43
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
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 13 / 43
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
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 14 / 43
Dilute gaz
Gladstone-Dale law
n − 1 = Kρ
for air, K ' 0.22× 10−3m3 · kg−1
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 15 / 43
Non intrusive optical techniques
Shadowgraphy
Figure: Shock wave produced by a supersonic bullet. Source : Rochester Institute ofTechnology.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 16 / 43
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 .
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 17 / 43
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.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 18 / 43
Thermal boundary layer
T
y
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 19 / 43
Non intrusive optical techniques
Schlieren
Figure: Shock wave visualization by schlieren technique. Source : Rochester Institute ofTechnology.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 20 / 43
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 :
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 21 / 43
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))
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 22 / 43
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
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 23 / 43
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
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 24 / 43
Analytical signal
Consider ψ(x , y) = cos(
2πy
λ
), determine ψE and IE
Idem with ψ(x , y) = A(y) cos(ky + φ)
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 25 / 43
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)
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 26 / 43
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.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 27 / 43
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.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 28 / 43
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).
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 29 / 43
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.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 30 / 43
Shadowgraphy vs Schlieren visualisation
Figure: Left : shadowgrapohy, right : schlieren (vertical knife). Photographs by AndrewDavidhazy, Kennedy Space Center.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 31 / 43
Non intrusive optical techniques
Interferometry
Figure: Michelson (left) and Zehnder (right) interferometers.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 32 / 43
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.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 33 / 43
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.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 34 / 43
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
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 35 / 43
Non intrusive optical techniques
Holography
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 36 / 43
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
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 37 / 43
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
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 38 / 43
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.
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 39 / 43
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)
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 40 / 43
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
Master 1 PAM (P-PAM-305A) MEMF 2010-2011 41 / 43