Peclet Examples

GPU-accelerated CFD, granular dynamics, and gas–solid coupling — from Python, validated in the open.

peclet is an open-source simulation suite for incompressible CFD (cut-cell immersed boundaries over signed-distance geometry), discrete element granular dynamics (spheres and arbitrary SDF shapes), and coupled gas–solid CFD-DEM — driven entirely from Python, running on one code base from a laptop CPU to multi-GPU clusters (Kokkos: CUDA, HIP, OpenMP; MPI for multi-rank).

pip install peclet          # CPU wheels — every page here runs from this
pip install peclet-flow-cu13 peclet-dem-cu13   # CUDA wheels for the GPU sections

Why peclet?

• One stack, three physics. A validated Navier–Stokes solver, an XPBD discrete-element solver, and the volume-averaged coupling between them — the same SDF geometry and the same Python arrays end to end.
• Python-first, zero-copy. Fields and particle states are NumPy/CuPy views into solver memory — no file shuffling, no lock-in, matplotlib/PyVista straight from the GPU.
• Runs where you are. The identical script runs on a free Colab CPU, a workstation GPU, or an MPI cluster. Every example carries an Open in Colab badge.
• Evidence, not claims. Every figure on this site is produced by executing the page against the released package — analytical solutions, published benchmarks (Ghia, Schäfer–Turek, Zick–Homsy, MFIX-Exa, X-ray tomography beds), and textbook checks throughout.
• Honest scale. A million-particle fluidized bed — deposition, drag, projection, collisions — advances in seconds per step on a single GPU.

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Single-phase flow

The flow solver: exact and canonical benchmarks for the cut-cell immersed-boundary Navier–Stokes core — from machine-precision channel flow to buoyant convection and a turbulent channel DNS.

Plane Poiseuille — cut-cell IBM is exact

Channel flow past immersed no-slip walls, reproduced to machine precision — and the validation trap that makes it look inexact. flow · IBM · verification

Pipe flow — second-order convergence

The same parabolic flow through a curved wall, where the cut cells show their genuine O(h²) accuracy. flow · IBM · convergence

Taylor–Green vortex

An exact unsteady Navier–Stokes solution: divergence projected to ~10⁻¹⁵, kinetic energy decaying at the analytic rate. flow · projection · verification

Lid-driven cavity

The classic recirculating benchmark against the Ghia–Ghia–Shin tables, up to Re 1000. flow · benchmark

Developing channel flow

A uniform stream relaxes into the parabolic profile — the entrance region with inflow/outflow boundaries. flow · boundary conditions

Backward-facing step

Separation and reattachment: the canonical separated-flow benchmark, reattachment length tracked vs Reynolds number. flow · separation · benchmark

Cylinder vortex street

Kármán shedding past an immersed cylinder, drag/lift and Strouhal number vs the Schäfer–Turek benchmark. flow · IBM · benchmark

Rayleigh–Bénard convection

3-D Boussinesq convection on the GPU: critical Rayleigh number to 0.4%, Nusselt numbers matching published DNS. flow · heat transfer · GPU

Wall-bounded turbulence

A channel DNS against the Moser–Kim–Mansour data, with the HPC submit scripts to reproduce it. flow · DNS · MPI/HPC

Flow through packings & porous media

Where dem meets flow: build a packing with the particle solver, hand the geometry to the fluid solver as a signed-distance field, and measure drag and permeability against the literature.

Zick–Homsy sphere arrays

Stokes drag through ordered sphere lattices to <0.5% of the exact series solutions, across the full solid-fraction range. flow · IBM · verification

Random packed bed

Pour a packing with dem, solve the flow through it, recover Kozeny–Carman/Ergun permeability. dem → flow · permeability

Ring packed-bed permeability

A parametric ΔP study over beds of Raschig rings — non-spherical packing to pressure drop in one pipeline. dem → flow · non-spherical

Pall-ring bed flow

The frozen dem pack becomes flow geometry analytically; the pressure drop lands within 5% of Ergun fed the pack’s own measured porosity and Sauter diameter. flow · dem · porous

Pore-network extraction

Watershed the SDF into pores and read exact throat conductances straight off a resolved Stokes solution. pnm · flow

Pore-space Voronoi mesh

Tessellate the void space of a packing into a graded polyhedral mesh with the voro engine. dem → voro · meshing

Navier–Stokes on a Voronoi mesh

Taylor–Green on an unstructured Voronoi mesh with voro’s covolume and collocated solvers: exact energy conservation vs second order. voro · flow · verification

Body-fitted walls: Poiseuille and sphere drag

SDF-clipped Voronoi cells as walls: the exact parabola with a quadratic wall gradient, and Zick–Homsy sphere-array drag with the semi-implicit Stokes march. voro · sdf · porous media

Redistributing pore-space seeds

Split, merge and relax interstitial seeds toward a uniform or wall-graded cell volume — the moves a position-only optimiser cannot make. voro · meshing

Two-phase flow (VoF)

Sharp interfaces on a fixed grid: geometric volume-of-fluid transport, a balanced-force continuum surface force, and momentum-consistent advection at large density ratios. Four pages are gated against something outside the solver — an exact algebraic identity, the exact viscous normal modes, a published benchmark — one composes all of it with a dem packing, and the last puts the whole stack to work on the question porous media actually ask: which pore fills first.

Bubble through a packed bed

A dem packing, cut-cell VoF, a contact angle and momentum-consistent transport in one run — the bubble squeezes through the throats, the gas volume holds to 1e-9. dem → flow · VoF · wetting

Pore-scale displacement

A pore doublet, a sphere packing and a micromodel: the contact angle alone decides whether the narrow pore fills or is bypassed — and a slip control shows where the wetting dynamics stop. flow · VoF · wetting · porous

VoF advection benchmarks

Zalesak’s slotted disc and LeVeque’s reversing deformation: shape error at order 2.16, and a volume drift that turns out to be the pressure solve’s divergence residual. flow · VoF · verification

Parasitic currents

The droplet that should not move: machine-zero spurious currents with an exact curvature, and a factor 3e15 riding on one operator pairing. flow · VoF · surface tension

Capillary oscillations

A ringing wave and a ringing drop against the exact viscous normal modes: the wave closes to 0.6%, the drop keeps a 4% deficit nobody has yet explained. flow · VoF · verification

Rising bubble (Hysing et al.)

Both benchmark cases at density ratios 10 and 1000 — centroid to 0.02%, and the momentum-consistent transport that buys a factor five. flow · VoF · benchmark

Droplet wetting

Set a contact angle on an immersed solid and get it back within a degree — plus the wall placement that silently freezes the contact line. flow · VoF · wetting

Trickle flow over a packing

Liquid dripped onto 34 wetting spheres with an open outlet below: holdup, wetted area and pressure drop, with the colour budget closing at the projection floor. flow · dem · VoF · porous media

Resolved particles & moving geometry

Bodies the fluid actually resolves: analytic CSG shapes as cut-cell geometry, walls and particles that move through the grid, and the exactly conservative reaction force and torque that couple them to dem. Every page here is gated against an exact solution — and two of them document the solver defects they uncovered on the way.

Oscillating sphere

Unsteady Stokes drag, in-phase and out-of-phase, within 0.23% of Stokes (1851) — and a periodic box that screens its own images. flow · moving geometry · exact

Moving sphere: does the drag spike?

The classic immersed-boundary pathology, measured — and the fresh-cell fix that removes a resolution-independent 2.6% drag bias. Finite Re within 0.4% of Blackburn (2002). flow · moving geometry · IBM

Rotating sphere torque

A negative result, then a fix: the missing wall term is μ(n×Ω) in closed form. −31% → +2.2% converging, and the first freely rotating resolved particle decays at 1.04× its box’s own drag. flow · torque · exact

Jeffery orbit

A spheroid tumbling in shear through a full resolved rotational loop — the half-period within 3–5% of Jeffery’s exact 1922 solution, and the bug hunt that made it possible. flow · dem · torque · exact

Dumbbell drag

Two fused spheres against the exact bispherical solutions — the axial drag to 0.1%, and the anisotropy that turns a settling doublet broadside-on. flow · CSG · exact

Sedimenting sphere (ten Cate)

The classic settling benchmark, resolved: E1 within 3% and E4 within 2% of the PIV at the coarsest rung — after two solver fixes and one geometry trap that hid for two campaigns. flow · dem · coupling · experiment

Galilean invariance at finite Re

A sphere towed through fluid at rest against the same sphere held in a stream — one problem, two frames, and the moving-geometry machinery reproduces the fixed answer to 0.03% at Re 1.5. flow · moving geometry · exact

Confinement at finite Re

The 39% Faxén wall correction that inertia screens down to a tenth by Re 1.5, measured with the static twin of a settling experiment — and the periodic-box geometry trap that hid it. flow · CSG · verification

Drafting, kissing, tumbling

Two spheres released in line in a closed box draft, touch, and tumble apart side by side — the classic two-body resolved-coupling test, with nothing prescribed and contact handled by the DEM. flow · dem · coupling · two bodies

Non-spherical Stokes drag

Equal-volume drag ratios, validated against the exact Oberbeck spheroid — then a cube whose drag is orientation-independent to 0.14%, as its symmetry demands. flow · CSG · exact

Granular dynamics

The dem solver on its own: dense packings of spheres and arbitrary SDF shapes, moving container geometry, and hundred-thousand-particle scale on one GPU.

Custom-shaped particles

Pack arbitrary shapes — capsules, stars, imported meshes — via signed-distance collision. dem · SDF shapes

Rotating drum

A moving SDF wall drives continuous avalanching at a steady dynamic angle of repose. dem · moving walls

Stirred column

A pitched-blade impeller whose geometry actually sweeps, mixing 22 000 grains — Lacey index 0.06 → 0.93 in five revolutions. dem · moving geometry · mixing

Tennis-racket theorem

A racket authored as a CSG tree flips about its intermediate axis — within 0.02% of the torque-free Euler equations. dem · CSG · exact

Pall-ring packing

An industrial packing element as a certified CSG tree, poured into a periodic box — and an honest reading of its voidage. dem · CSG · packing

Tumbling cubes

100,000 sharp-edged cubes cascading in a drum at ~10 ms per step on one GPU — with the movie. dem · non-spherical · GPU

Gas–solid CFD-DEM

The coupling module runs both solvers together — volume-averaged gas, per-particle drag, momentum-conserving feedback — and reproduces published gas–solid benchmarks one-to-one.

Fluidized bed at engineering scale

One million 1 mm glass beads in air: the bed unpacks, bubbles, and carries its weight in the pressure drop — live CFD-DEM on one GPU. coupling · 10⁶ particles · GPU

Single-bubble injection

An injected bubble rises through an incipient bed — shape and rise velocity against MRI measurements and MFIX-Exa. coupling · validation

Bidisperse segregation

Jetsam sinks, flotsam floats: species segregation in a bubbling bed, following the MFIX-Exa configuration. coupling · segregation

Homogeneous cooling & clustering

Haff’s law, the clustering instability, and a one-to-one energy-decay overlay with the MFIX-Exa benchmark. coupling · benchmark

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Trust, but verify

The examples above show what the suite does; two growing collections prove it stays correct:

Sanity checks →

Small problems with closed-form answers — restitution, wall bounce, sliding-to-rolling, static stacks, spin–friction coupling, and a deep layered bed at rest. Each check records a number against theory in a pass/fail scorecard; together they are the regression tripwire run against every solver change. New checks are added as new physics lands. dem · regression · scorecard

Benchmarks →

Quantitative characterization where there is a trade-off to measure, not just a pass to record — discretization variants, convergence orders, accuracy-versus-cost. Start with staggered vs collocated Stokes-drag convergence. This collection grows as design decisions get measured. flow · convergence · characterization

Run any of this yourself

Every page is a self-contained notebook: it builds its geometry and sets every parameter inline. Click the Open in Colab badge on any example to run it on a free CPU runtime, or:

pip install peclet && git clone https://github.com/computational-chemical-engineering/peclet-examples
quarto render peclet-examples/examples/taylor-green/index.qmd --execute

The GPU sections (million-particle beds, cube drums, DNS) need the CUDA wheels or a local build — each page’s Reproduce this section gives the exact commands.

NoteHow these pages build

Solver-backed outputs are executed by the authors and frozen into the repository, so the site always builds without a GPU — and what you read is exactly what the released code produced.