Wave 53: Laminar-Turbulent Transition, Islamic Geometry & Blasius Boundary Layer

Wave 53 at a glance

〜 Laminar to Turbulent Transition

In 1883 Osborne Reynolds injected a thin thread of dye into a pipe and varied the flow speed. Below a critical velocity the dye formed a perfectly straight filament — laminar flow. Above a threshold the filament suddenly broke up and mixed across the entire pipe cross-section — turbulent flow. The dimensionless parameter governing the transition is the Reynolds number:

Laminar velocity profile

For steady, fully developed laminar flow in a circular pipe (Hagen-Poiseuille flow), the velocity profile is exactly parabolic:

u(r) = u_max (1 − r²/R²)

where r is the radial distance from the pipe centreline and R is the pipe radius. The maximum velocity at the centreline is twice the cross-sectional mean: u_max = 2 Ū. All fluid elements travel in straight, parallel paths — no mixing occurs.

Turbulent velocity profile

In fully turbulent pipe flow (Re ≫ 10 000) the time-averaged velocity profile is much flatter. The empirical 1/n power law gives a good approximation:

u(r)/u_c ≈ (1 − r/R)^1/n

where n ≈ 7 for Re ~ 10⁵. Superimposed on this mean profile are turbulent velocity fluctuations u′(t) and v′(t). Their root-mean-square magnitude relative to the mean is the turbulent intensity I = u_rms/Ū, typically 5–15% in well-developed turbulence.

The simulation uses particle tracers: coloured streams enter from the left. In laminar mode their paths remain straight and colour bands never mix. As Re crosses the transition threshold, v-component fluctuations scatter particles across the channel, reproducing the dye mixing seen in Reynolds' original experiment.

✦ Islamic Geometric Patterns

Islamic geometric art — found in mosques, palaces and manuscripts from Andalusia to Central Asia — is built from a small vocabulary of interlocking star polygons, rosettes and connector tiles. The patterns exhibit perfect translational and (locally) rotational symmetry while leaving no gaps or overlaps.

{n/k} star polygons

A regular star polygon {n/k} is drawn by placing n equally spaced points on a circle and connecting every k-th point. The notation was introduced by Schläfli. Key examples:

• {6/2} — Star of David (hexagram); gcd(6,2) = 2 produces two separate equilateral triangles.
• {8/3} — 8-pointed star (octagram); gcd(8,3) = 1 produces a single closed 8-point star — ubiquitous in Ottoman tile work and Girih patterns.
• {10/4} — 10-pointed Girih star; the canonical element of the Girih tile system found in the Darb-i Imam shrine, Isfahan (1453 CE). The remaining gap shapes (bowtie, pentagon, elongated hexagon, rhombus) fill the plane without overlaps.
• {12/5} — 12-pointed Ottoman star; common in Turkish İznik tilework and Moroccan zellige mosaics.

Tiling logic

The generator places star centres on either a square lattice (spacing d) or a hexagonal lattice (dx = d, dy = d√3⁄2 with alternating row offset). For each centre the {n/k} polygon paths are drawn and optionally filled. A smaller regular n-gon at the centre (radius ≈ 0.45 R) provides the inner fill region at reduced opacity, mimicking the contrast between the star body and the surrounding connector tiles.

Six historical colour palettes are available: Islamic Gold (amber and ochre), Alhambra (forest green), Ottoman Blue (cobalt and sky), Moroccan Terracotta (burnt sienna), Mughal White (off-white on black), and Neon Night (violet on near-black). The finished pattern can be exported as a lossless PNG.

〰 Blasius Boundary Layer

When a viscous fluid flows over a solid surface, the no-slip boundary condition forces the fluid velocity to zero at the wall. A thin layer — the boundary layer — develops in which velocity rises from zero to the freestream value U∞. Outside this layer the flow is essentially inviscid.

Blasius equation

In 1907–1908 Heinrich Blasius (a student of Prandtl) showed that for steady laminar flow over a semi-infinite flat plate the Navier-Stokes equations reduce, under boundary layer approximations, to a single third-order nonlinear ODE via the similarity variable η = y √(U∞ / νx):

f‴ + ½ f f″ = 0   with   f(0) = f′(0) = 0, f′(∞) = 1

Here f′(η) = u/U∞ is the dimensionless velocity profile. The solution is universal — all flat-plate laminar boundary layers collapse onto this single curve when plotted against η.

Shooting method and RK4

The boundary condition f′(∞) = 1 is not given at η = 0, so it cannot be integrated directly. Instead the shooting method is used: guess an initial value α = f″(0) ≈ 0.4696 (the well-known Blasius wall-shear constant), integrate forward with RK4 at step Δη = 0.02, and verify that f′ → 1 as η → 12. The resulting solution table is used at runtime via linear interpolation to evaluate u/U∞ at any (x, y) point on the canvas.

Key results

• Boundary layer thickness: δ₉₉(x) ≈ 5.0 √(νx/U∞)
• Local skin friction: C_f(x) = 0.664 / √Re_x
• Average skin friction over plate length L: C̄_f = 1.328 / √Re_L
• Wall shear stress: τ_w(x) = 0.332 ρ U∞² / √Re_x

The canvas shows the flat plate at the bottom with the δ₉₉(x) boundary layer envelope (orange fill), velocity profiles u/U∞ at four x-stations (white curves), and freestream arrows above the layer. Switching between water (ν = 1 × 10⁻⁶ m²/s) and air (ν = 1.5 × 10⁻⁵ m²/s) immediately updates the boundary layer thickness and skin friction values.

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