Optics 066 · Diffraction, Fourier optics, and computational imaging

Schlieren Flow Viewer

How does a knife edge make invisible air flow visible, and why does turning it make some features vanish? Uneven air density bends light very slightly. Focus the light and put a knife edge at the focus: light bent toward the blade is cut off more, light bent away less, so gradients turn into brightness. Only bending across the blade changes how much it blocks. Rotate the blade, and gradients running along its edge drop out of the picture.

A Toepler lens-type Schlieren bench turns a known transparent index field into a grayscale image that depends on direction. Trace a point, rotate the opaque blade, change the source size and see where the image saturates.

Selected object-point deflection

\theta_x,\theta_y

$0.50$

Directional focal shift / raw point intensity

q,I_p

$50\%$

Empty-field background / saturated sample fraction

I_{\rm bg},S_{\rm clip}

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Schlieren Flow Viewer

Synthetic index field (names on the buttons below)

o

0

Signed refractive-index anomaly amplitude

\Delta n_0

-1\times10^{-5}

Uniform line-of-sight field thickness

L

40\,\mathrm{mm}

Transverse field-width scale

w

4\,\mathrm{mm}

Shared focal length (the conjugate optics move with it)

f

1000\,\mathrm{mm}

Source-image radius at knife plane

a

60\,\mu\mathrm m

Knife normal offset (positive darkens background)

b

0\,\mu\mathrm m

Knife sensitive normal (clockwise from object right)

\alpha

0^{\circ}

Knife: 0 removed, 1 inserted

k

1

Probe horizontal position in the original object

x_p

4\,\mathrm{mm}

Probe vertical position in the original object (positive down)

y_p

0\,\mathrm{mm}

Remove the knife edge and the transparent test field disappears. Insert it and invisible index gradients show up as bright and dark edges. Rotate the blade to see gradients in another direction; with the annular field, the apparent shading turns while the field itself stays put. Click the receiving image to follow one object point to the enlarged view of the source image at the cutoff. The blade is opaque, and light it blocks stops there. Source and cutoff positions, raw images and readouts all come from one geometric model, and images are not normalized automatically.

The index fields are prescribed and static with uniform thickness, and light follows straight paths in a geometric thin-screen approximation. The source is a uniform, incoherent disk, and the relay is ideal with unit magnification and an inverted image. The Lab does not run CFD, show real shocks, model shadowgraphy or BOS, or recover density. Diffraction and noise are not included, nothing is certified by experts, and apparatus parts are drawn at different scales.

Physics tutorial

Audit the physics of Schlieren Flow Viewer

BackgroundTransparent air can show light and dark patches once a knife edge turns its tiny refraction into partial blocking of the source. The deflection is $\boldsymbol\theta\approx\frac{L}{n_0}\nabla_{\perp}n$ . The Lab uses known, static index fields that are uniform along the line of sight; they are not measured flames or a fluid simulation.

Why it mattersHow does a knife edge make invisible air flow visible, and why does turning it make some features vanish?

Start with the essentials

Focus question: How does a knife edge make invisible air flow visible, and why does turning it make some features vanish?
One-sentence intuition: Only the part of the deflection along the blade normal changes the transmission: $q=f\boldsymbol\theta\cdot\hat{\mathbf e}_{\alpha}$ . Rotating the knife changes the shading although the density stays the same. A uniform index offset produces no contrast.

Core mathematical model

Integrating the index along a straight path

\begin{aligned}n(x,y,z)&=n_0+\Delta n_0F(x,y)\\\boldsymbol\theta&\approx\frac{L\Delta n_0}{n_0}\nabla F\end{aligned}

Coordinates are in millimetres: in the object, right and down, with propagation through the thickness. The small perturbation is collapsed into a thin screen; rays are not bent step by step through the full volume.

How much of a finite source disk passes the knife

\begin{aligned}u&=(q-b)/a\\I/I_0&=\frac12+\frac{\arcsin u+u\sqrt{1-u^2}}{\pi}\\&\qquad(-1<u<1)\end{aligned}

The transmission is the unblocked area of a uniform, incoherent source image of finite radius. Outside the interval shown, it is exactly zero or one. The blade blocks the half-plane on the low side of the normal coordinate, so a positive knife offset darkens the empty field.

Sensitivity and saturation

\begin{aligned}b=0:\quad I/I_0&\approx\frac12+\frac{2q}{\pi a}\\|q|&\ll a\\\left|q-b\right|&\ge a\Rightarrow I/I_0\in\{0,1\}\end{aligned}

The linear expression holds only near the centre; it is not a ramp that applies everywhere and is clipped at the ends. The image at every setting comes from the exact disk-area integral. In fully black or fully white regions, the intensity no longer corresponds to a unique gradient.

Two kinds of conjugate planes in one relay

\begin{aligned}\mathbf r_K&=f(\boldsymbol\eta+\boldsymbol\theta)\\\mathbf r_i&=-\mathbf r_o,\qquad m=-1\\z_K&=2f,\quad z_i=4f\end{aligned}

The test plane sits at zero, and the collector and relay lenses sit at one and three focal lengths. Each position on the source becomes an illumination angle. Changing the shared focal length moves the hardware but keeps the imaging inverted at unit magnification. The lens shapes are schematic and do not model aberrations.

Common difficulties

A bright edge is not a hot spot or a velocity arrow

Typical misconceptionBrightness directly measures temperature or flow speed, and a darker image always means a larger gradient.

Better mental modelThe image depends on the index gradient in one direction, integrated along the line of sight, and on the source size and knife position. Once the source disk is entirely passed or entirely blocked, the intensity saturates and no longer tells you how large the gradient is. A single view cannot recover a three-dimensional density field or a velocity.

Run the experiment

01
Scene 1: Invisible plume made visible
Remove the knife, then put it back. Click a bright or a dark pixel and compare its source disk with the absorbing blade.
What to observe: With the knife removed, this ideal in-focus thin screen gives unit intensity. The Lab does not simulate a shadowgraph.
02
Scene 2: Annular field: rotate the shading
Choose the annular field and rotate the sensitivity direction by a quarter turn. Compare the same raw profiles; the images are not rescaled automatically.
What to observe: The knife plane, where source position stands for angle, and the image plane are conjugate to different things. The camera inverts both transverse coordinates of the object.
03
Scene 3: Smooth front: saturation is lost information
Select the smooth front, then make the source small and the focal length long. Look for areas clipped to black or white, then turn the knife normal perpendicular to the gradient of the front.
What to observe: A smaller source gives higher sensitivity to small gradients but a narrower range of gradients before saturation.