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Gresho Vortex Benchmark
Accurate Vortex Dynamics for Industrial Flow Design
Some systems don't explode or fail — they simply need to stay perfectly balanced over time. The Gresho Vortex is a classic test of that balance: a rotating flow that should keep its shape as it spins without any external forcing or dissipation.
With NumericalAI, engineers can instantly run this type of benchmark in the cloud — proving solver accuracy, stability, and reliability without any setup, installation, or hardware concerns. If a CFD solver can maintain a perfect vortex structure over hundreds of rotations, it demonstrates the numerical precision needed for real industrial applications.
Initial Vortex Structure (t = 0)
The initial condition shows a clean, symmetric density profile with a rotating core. The vortex is in exact equilibrium between centrifugal force and pressure gradient — any numerical error will cause degradation over time.
Later in Time (t = 700s)
After 700 seconds (approximately 100 full rotations), the structure remains intact. This demonstrates exceptional solver stability with negligible numerical dissipation — the vortex profile is virtually unchanged from its initial state.
Pressure Field Distribution
The pressure field remains smooth throughout the simulation with no spurious oscillations. The radial pressure gradient perfectly balances the centrifugal acceleration, maintaining the vortex in steady rotation.
Velocity Field — Stable Rotating Core
Velocity contours reveal the azimuthal (tangential) velocity profile that characterizes the Gresho vortex. The peak velocity occurs at mid-radius, smoothly transitioning to zero at the center and outer boundaries.
What is the Gresho Vortex?
The Gresho Vortex is a standard benchmark problem in computational fluid dynamics designed to test a solver's ability to maintain vortical structures without spurious numerical dissipation or dispersion. Originally proposed by Gresho and Chan (1990), it consists of:
Steady-state rotating flow with no external forcing
Exact balance between pressure gradient and centrifugal force
Sensitive to numerical errors — any artificial dissipation or pressure-velocity coupling issues will degrade the solution
Long-time integration test — the vortex should persist indefinitely without decay
Why This Test Matters
Many CFD solvers introduce artificial dissipation to maintain stability, but too much dissipation destroys important flow features like vortices, boundary layers, and shear layers. The Gresho Vortex benchmark proves that NumericalAI's solvers preserve these structures accurately — essential for industrial applications where flow features directly impact performance.
Simulation Configuration
Domain & Grid
Formulation:
2D Cartesian
Domain Size:
1 m × 1 m
Grid Resolution:
400 × 400
Total Cells:
160,000
Cell Size:
2.5 mm uniform
Initial Conditions
Vortex Radius:
R₀ = 0.2 m
Peak Velocity:
~0.5 m/s
Mach Number:
M ≈ 0.0015
Density:
ρ = 1.0 kg/m³
Ambient Pressure:
101.325 kPa
Temporal Settings
Simulation Time:
700 seconds
Rotations:
~100 full rotations
Time Step:
Adaptive CFL
CFL Number:
0.4
Output Frequency:
Every 10s
Solver Configuration
Spatial Scheme:
WENO5
Time Integration:
RK3-TVD
Riemann Solver:
HLLC
Equation of State:
Ideal Gas
Boundary:
Periodic
Computational Performance
~6 hrs
Wall-Clock Time
A100
NVIDIA GPU
~750 MB
Memory Usage
~7×
Speedup vs CPU
Validation Metrics & Accuracy
Quantitative Performance Indicators
To assess solver quality, we track several conservation and accuracy metrics:
Total Energy Conservation
Energy drift over 700s: < 0.02%
Indicates excellent time integration accuracy with minimal numerical dissipation.
Angular Momentum
Conservation error: < 0.01%
Confirms that rotational symmetry is respected by the numerical scheme.
L² Density Error
Relative error at t=700s: ~0.15%
Density profile remains nearly identical to initial condition — hallmark of low-dissipation schemes.
Peak Velocity Retention
Velocity decay: < 0.3%
The vortex maintains its strength without artificial damping effects.
Comparison with Literature
NumericalAI's results match or exceed published benchmarks from leading CFD research groups. The combination of WENO5 spatial reconstruction and RK3-TVD time integration delivers the accuracy expected from modern high-order methods — but now accessible through a simple cloud interface instead of months of code development.
Why Engineers Choose NumericalAI
Cloud-native simulation — run advanced CFD workflows without managing servers, installing software, or configuring HPC clusters
Proven accuracy — validated on industry-standard benchmarks like the Gresho Vortex, ensuring your results are trustworthy
Faster insights — launch, monitor, and visualize your results directly in your browser with GPU acceleration
Consistent results — reliable for long-duration, steady-state, and transient analysis across diverse applications
Industrial Relevance
Energy & Process Engineering
Predict circulation patterns, flow mixing, and residence time distributions in reactors, tanks, and separators. Accurate vortex modeling ensures optimal mixing and prevents dead zones.
Aerospace & Turbomachinery
Study tip vortices, secondary flows, and vortex breakdown in compressors, turbines, and propellers. Solver stability under rotation is critical for reliable performance predictions.
Manufacturing & Mixing
Simulate mixers, impellers, stirred tanks, and cooling channels. Vortex preservation ensures accurate prediction of blend time, homogeneity, and energy consumption.
R&D Validation
Demonstrate solver accuracy and numerical stability effortlessly. Use the Gresho Vortex as a quick sanity check before committing to expensive production runs.
Why It Matters
NumericalAI transforms advanced flow simulation into a fast, collaborative, and cloud-powered experience. No installation barriers, no hardware procurement delays — just instant access to validated, production-grade CFD.
A test like the Gresho Vortex demonstrates that if NumericalAI can maintain a perfect vortex structure over 100 rotations, it can handle your real-world designs — reliably, accurately, and efficiently, every time.
Ready to Validate Your Solver?
Run the Gresho Vortex benchmark yourself on NumericalAI's cloud platform. See the accuracy and stability in action.
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