Technology / Technical guide
Rotor Spatial Averaging: Point Wind, Area Mean & Equivalent Speed
A rotor encounters a field distributed across an area, while one wind reading belongs to one location. Spatial averaging condenses that field for a defined purpose. An average velocity, an average speed magnitude and an energy-equivalent scalar need not be interchangeable.
Choose what the average is intended to preserve.
An average is an operation applied to a specified quantity over a specified region. Making the area weights and direction explicit explains what is retained, and what information about a rotor-plane field is discarded.
01 / Point and area
One position does not represent every location.
A point reading can differ from neighbouring values because of shear, turbulence or a nonuniform disturbance.
A point reading can differ from neighbouring values because of shear, turbulence or a nonuniform disturbance. An area average represents values across a chosen plane. The geometric plane, coordinate directions and instant or time-averaging basis must be stated.
A hub-position value is not automatically the rotor-plane mean. Similarly, a rotor-plane mean is not the velocity seen by every moving blade section. The relative-flow guide treats that separate local moving-frame quantity.
02 / Area weights
Use the region represented by each value.
For scalar values Uj representing subareas Aj, an area-weighted mean is Σ(Aj Uj)/ΣAj.
For scalar values Uj representing subareas Aj, an area-weighted mean is Σ(Aj Uj)/ΣAj. Equal numerical weighting is equivalent only when the represented subareas are equal. A finer collection in one region should not accidentally make that region occupy more physical area.
A hypothetical disk divided into equal-area regions at 6 and 10 m/s has a mean of 8 m/s. If instead the 6 m/s region occupies three quarters of the area, the same two speeds give 7 m/s. The spatial weights change the result without changing either local reading.
03 / Vectors
Average components before interpreting direction.
For a vector average, each velocity component receives the same consistent area weights.
For a vector average, each velocity component receives the same consistent area weights. Taking the magnitude afterwards is generally different from averaging the local magnitudes. Directional differences can partly cancel even when local speed magnitudes remain large.
For equal-area vectors (6,2) and (6,−2) m/s, the vector average is (6,0), with magnitude 6 m/s. Each local magnitude is about 6.32 m/s. This invented geometry illustrates a definition difference, not a MARSTON inflow record.
04 / Equivalent scalar
A cubic mean preserves a different quantity.
For parallel flow normal to the area, uniform density and nonnegative local speeds, define Ueq = [Σ(Aj Uj³)/ΣAj]^(1/3).
For parallel flow normal to the area, uniform density and nonnegative local speeds, define Ueq = [Σ(Aj Uj³)/ΣAj]^(1/3). This preserves the area-integrated kinetic-energy flux of that particular incoming-flow representation; it is not the arithmetic mean.
The equal-area 6/10 m/s example has a cubic mean about 8.47 m/s, compared with its 8 m/s arithmetic mean. Oblique flow requires the appropriate normal flux factor as well as speed magnitude. No equivalent scalar here predicts electrical output or changes an actual power curve.
05 / Model quantity
A simulation average has a named physical role.
FAST.Farm documentation names a rotor-disk-averaged relative velocity and distinguishes its component normal to the disk.
FAST.Farm documentation names a rotor-disk-averaged relative velocity and distinguishes its component normal to the disk. That is a defined model input or aggregate, rather than permission to treat every possible average as the same number.
A normal component can support one momentum or loading normalisation while a local section calculation retains other components. Averages also discard spatial patterns: two fields with identical means can distribute higher and lower velocities differently across the blade path.
06 / Information loss
Keep spatial averaging separate from time averaging.
Time averaging follows one quantity through time; area averaging combines simultaneous spatial locations.
Time averaging follows one quantity through time; area averaging combines simultaneous spatial locations. Applying both requires stating their order and any nonlinear operation between them. The nonlinear-average guide explains why taking a cube changes this question.
This page does not supply a rotor-plane sampling procedure, a product-equivalent wind algorithm or a MARSTON yield estimate. It explains representation choices; nominal rotor diameter alone cannot reconstruct an actual spatial velocity field or its interaction with the machine.
Related technical guides: Wind Air Density: Temperature, Pressure & Humidity · Wind Turbulence Spectra, Time Scales & Spatial Scales · Nonlinear Averages: Why Cubing the Mean Changes the Result.
General background: OpenFAST FAST.Farm: defined disk-averaged relative-flow quantities. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Point value | One location and time basis | Not automatically the disk mean. |
| Area mean | ΣAjUj / ΣAj | Weights represent physical subareas. |
| Vector mean | Weighted component average | Magnitude taken after averaging differs. |
| Cubic equivalent | [ΣAjUj³ / ΣAj]^(1/3) | Parallel normal flow and uniform density. |
| Normal velocity | Component perpendicular to disk | Distinct from complete speed magnitude. |
Engineering questions.
Does a hub reading equal the rotor-plane mean?
Only under conditions making that representation valid; it is not an identity.
Can unequal subareas be equally weighted?
That would generally represent a different physical average.
Is mean speed the magnitude of mean velocity?
Not generally. Directional variations make them differ.
Is the cubic-equivalent speed a product power prediction?
No. It preserves only the specified incoming-flow energy-flux representation.
Can two fields share a mean but load a blade differently?
Yes. The mean removes information about spatial distribution and timing.
Are spatial and time averages the same operation?
No. They combine values over different domains and need explicit definitions.
Technical enquiry
Discuss the technical question.
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