Méthodes Expérimentales en Mécanique des Fluides Optically ...

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


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).


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′


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


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 ′


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

),


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


Examples of diffraction figures

circular pupil vertical slit vertical edge


Example of optical filtering

Figure: Optical Fourier transform (after Yves Usson).



Figure: Low-pass optical filter.



Figure: High-pass optical filter.


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


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


Dilute gaz

Gladstone-Dale law

n − 1 = Kρ

for air, K ' 0.22× 10−3m3 · kg−1


Non intrusive optical techniques

Shadowgraphy

Figure: Shock wave produced by a supersonic bullet. Source : Rochester Institute ofTechnology.


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 .


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.


Thermal boundary layer

T

y



Schlieren

Figure: Shock wave visualization by schlieren technique. Source : Rochester Institute ofTechnology.


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 :


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))


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


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


Analytical signal

Consider ψ(x , y) = cos(

2πy

λ

), determine ψE and IE

Idem with ψ(x , y) = A(y) cos(ky + φ)


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)


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.


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.


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).


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.


Shadowgraphy vs Schlieren visualisation

Figure: Left : shadowgrapohy, right : schlieren (vertical knife). Photographs by AndrewDavidhazy, Kennedy Space Center.



Interferometry

Figure: Michelson (left) and Zehnder (right) interferometers.


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.


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.


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



Holography


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


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


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.


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)


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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