Categories
fluids

Joukowski Airfoil Performance

The Joukowski transformation is a great way to study basic airfoils using potential flow. It gives a nice way to derive the Kutta condition, as well as the classic lift slope . But in all those demonstrations, I never saw the lift and drag curves of the Joukowski airfoil! Luckily, I have XFOIL to remedy […]

Categories
image processing

Color Mapping: Swapping Image Palettes

Sometimes you want one picture to have the color scheme of another. This is called color mapping or transfer. When I was making the rearranged images, I was concerned that I would get bad matches if I didn’t make sure the images were similar enough. It was helpful to ensure that their brightness histograms matched, […]

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halftoning image processing penplotter

Spiral Halftoning

Let’s halftone an image with some overlapping wiggly spirals! I want to draw N lines, spiraling out from the center of the image, and have them each slightly offset. With the proper offset, this will create the (approximately) proper darkness. With image darkness k, line width w, and line spacing L, we can get a […]

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etcetera

Geometrically Approximating Pi

Archimedes found good estimates of pi by computing the perimeter of regular polygons that fit snugly inside and outside a circle. I’d like to take this basic approach and add some more modern twists. I say more modern, but I don’t mean cutting-edge. Taylor series are from the 1700s CE, compared to Archimedes living in […]

Categories
fluids

The Sleekest Airfoil

What does a minimum force airfoil look like? I basically want to make a fairing that minimizes the total force over the range of 0-5 degrees. This mostly means reducing the lift generated at non-zero angles of attack, while keeping the drag manageable. Minimizing lift on an airfoil feels oddly perverse. I suppose that’s why […]

Categories
fluids

Improving the NACA 0015 Airfoil

The four-digit NACA airfoil is perfectly decent, which is a bit surprising considering it was first published way back in 1933. Since it was designed before computers, I’m sure that it could be improved. The general form of the thickness equation is: This does not provide the proper thickness, but ensures a rounded leading edge […]

Categories
fluids

Airfoils with Circular Leading Edges

Airfoils are generally thickest somewhere around the middle. The classic NACA four-digit airfoils are thickest at 30% of the chord, and laminar flow airfoils are thicker around the 50% mark. Let’s throw out the received wisdom and make some airfoils that are thickest close to the front. Rather than working with a lifting body, let’s […]

Categories
fluids

Conformal Mapping: Point Vortex Near a Corner

I wanted to know what a point vortex would do near a corner with an arbitrary angle. Time to dust off the complex analysis! A really useful summary of using complex analysis to model potential flow is here. I will use the method of images to ensure that there is no flow through the wall, […]

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etcetera fluids halftoning image processing

Advection-Diffusion of Ink

I want to have the ink on a photo clump together, leading to a black and white image. This is a sort of flow, so my first thought is to take a look at the advection-diffusion equation. Let’s say that the ink distribution across an image is , the diffusion factor is , and the […]

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etcetera image processing

Rearrange a Picture into Itself

Previously, I took one photo and rearranged its squares to make a version of another photo. How about a sillier version, where you use one image to re-make itself? The simple answer is to just keep every square in its original place, but that’s no fun at all. Instead, we can add the requirement that […]