Rotor velocity averaging

Methods used by each software

A ladder of test cases

Rather than jumping straight to a waked rotor, the comparison walks through four velocity fields of increasing difficulty. Each step adds exactly one new way for a coarse stencil to go wrong, so a deviation can be attributed to a specific cause instead of to the case as a whole.

1. Uniform inflow — U/U∞ = 1
2. Linear vertical shear — U/U∞ = 1 + 0.2 (z/R)
3. Quadratic radial deficit — U/U∞ = 1 - 0.2 (r/R)²
4. Offset Gaussian velocity deficit — U/U∞ = 1 - 0.35 exp(-((y/R - 0.35)² + (z/R)²) / (2 · 0.28²))

Reference: 128-point Gauss-Legendre radial integration with 720 azimuthal points. Each software result is compared with a dense-disk reference using the same aggregation semantics.

1. Uniform inflow

U/U∞ = 1

A constant field. Any quadrature whose weights sum to one is exact here, so this verifies the harness and the normalization of each stencil rather than discriminating between methods.

Exact area average: 1

2. Linear vertical shear

U/U∞ = 1 + 0.2 (z/R)

A linear profile. Because all three stencils are symmetric about the hub height, the positive and negative contributions cancel and the area average is still recovered exactly. Point placement starts to matter only once the aggregation is nonlinear.

Exact area average: 1

3. Quadratic radial deficit

U/U∞ = 1 - 0.2 (r/R)²

The first case with real curvature. The exact area average is 0.9. Stencils clustered near the hub under-resolve the slower flow near the blade tips and overpredict the rotor average, which exposes how far each sampling pattern reaches toward the rotor edge.

Exact area average: 0.9

4. Offset Gaussian velocity deficit

U/U∞ = 1 - 0.35 exp(-((y/R - 0.35)² + (z/R)²) / (2 · 0.28²))

A laterally offset deficit that approximates a partially waked rotor. It is neither symmetric nor smooth on the scale of the coarse stencils, so it is the most demanding case and the one most representative of wake-model use.

Errors across the ladder

Because FLORIS uses a cubic mean while FOXES and PyWake use arithmetic averaging in these configurations, each error is measured against a dense-disk reference with matching aggregation semantics rather than against a single shared number.

How to interpret this comparison

• Every case uses one identical velocity field; differences come only from sampling and averaging.
• Uniform inflow is exact for all three methods, which confirms that each set of weights is normalized.
• Linear shear is exact for FOXES and PyWake because their stencils are symmetric about hub height. FLORIS reads 1.006623 against an exact area average of 1, because its cubic mean of a varying field is above the arithmetic mean no matter where it samples. Its dense-disk cubic reference is already 1.009902. The remaining 0.32% error against that reference is a sampling effect: FLORIS only samples out to ±R/2 on each axis, so it sees less of the shear than the full disk.
• The quadratic case separates the methods by radial reach. PyWake samples at(±R/3, ±R/3), no farther than r/R = 0.47 from the hub, and FLORIS reachesr/R = 0.71 at the corners of its ±R/2 square. Both miss the slower tip flow and read high. FOXES spans the full disk and lands slightly low (0.36%) because its four corner points sit at r/R = 1.06, beyond the rim, where the velocity is lowest, yet carry partial-cell weight.
• The offset Gaussian is the only case that is neither symmetric nor resolvable by a handful of points, which is why it is the most representative of partially waked operation. Both PyWake (3.86%) and FLORIS (1.76%) read too much deficit, while FOXES is within 0.11%. PyWake places two of its four points near the deficit core at y/R = 0.35 and none beyond R/3, so the core is over-weighted.
• These results isolate quadrature error and are not a ranking of full wake-model accuracy.