Actuator disk¶

With mode = adm the loads OpenFAST computes on its blade nodes go into the flow as a disk, not as rotating lines. OpenFAST still sees the true blade positions; only the flow does not.

Rings¶

Each blade node, at radius \(r\) from the hub in the plane normal to the shaft, becomes a ring of num_points_t points at that radius, equally spaced in azimuth. Each point carries \(1/N_t\) of the node’s force. The axial component is kept; the radial and tangential components turn with the ring azimuth.

Kept exactly: the force along the shaft and the torque about it.
Averaged away: the net force in the rotor plane. A three-bladed rotor carries it only through differences between its blades at an instant.
The integrated source equals minus the thrust’s shaft component. The hub node keeps its own point.
At the first step ERF checks that the blade nodes lie within about 20 degrees of the plane normal to the shaft axis, so a mismatched orientation convention aborts.

With mode = none the turbine is driven by the flow but puts no force into it, as for a loads analysis in a precomputed flow.

Merged rings¶

With adm_spreading = merged_rings the blades’ nodes at the same radial station share one ring. Blade \(b\)’s node at station \(k\) has distance \(a_{bk}\) along the shaft, radius \(r_{bk}\), and force components \(f^{c}_{bk}\), \(c \in \{x, r, \theta\}\) (axial, radial, tangential, on the fluid). The shared ring of the \(N_b\) blades stands at the means and carries the sums:

\[\bar a_k = \frac{1}{N_b} \sum_b a_{bk}, \qquad \bar r_k = \frac{1}{N_b} \sum_b r_{bk}, \qquad F^{c}_k = \sum_b f^{c}_{bk}.\]

Rigid rotor: the same source as \(N_b\) rings, to rounding, with \(1/N_b\) of the points and of the spreading cost.
Shaft force: kept for any rotor.
Torque: differs at each station by \(\sum_b (r_{bk} - \bar r_k)(f^{\theta}_{bk} - F^{\theta}_k / N_b)\), which is \(N_b\) times the covariance of the radii and the tangential loads. It is exact when the radii (or the tangential loads) agree.
Flexible rotor: where the blades’ deflections differ by \(\Delta a\), the source near the tip changes by about \((\Delta a / \epsilon)^2\) of itself. This is small when \(\epsilon\) is several times the spread in deflection.

Sampling modes¶

The velocity at the rotor contains the rotor’s own induction. The sampling mode decides who accounts for it: the resolved flow or AeroDyn. ERF reads the AeroDyn file and checks Wake_Mod against the mode.

`sampling`	Velocity handed to OpenFAST	`Wake_Mod`	Behaviour
`disk`	At the blade nodes, inside the induction zone.	0	Loads too high when the grid does not resolve the disk.
`upstream`	Shifted `sample_diameters_upstream` ahead of the hub, along the horizontal projection of the shaft.	1 (BEMT)	BEM loads for that free stream, whatever the grid.
`disk_corrected` (default)	At the blade nodes, scaled to the recovered free stream.	1 (BEMT)	BEM loads from the disk velocity alone.

Plain disk. On a grid that does not resolve the smeared disk the sampled velocity is well above the momentum-theory disk velocity. The IEA 15 MW disk on 20 m cells measured 33 % high in power with a two-cell kernel, 29 % with 1.5 cells.

Upstream. In vortex-cylinder theory the rotor slows the flow one diameter ahead by 9 % of its disk-averaged induction, two diameters ahead by 3 %. Each node keeps its own height, so a tilted shaft does not move the points up or down. Limits are in Limitations.

Corrected disk¶

The velocity a Gaussian-smeared disk samples is not the thin-disk velocity \(U_\infty(1-a)\) but that velocity divided by the factor of Shapiro, Gayme and Meneveau (2019, eq. 25):

\[M = \left( 1 + \frac{C_T'}{4} \frac{\Delta / R}{\sqrt{3 \pi}} \right)^{-1}, \qquad C_T' = \frac{4 a}{1 - a} = \frac{C_T}{(1 - a)^2}.\]

\(R\) is the rotor radius and \(a\) the axial induction.
\(\Delta\) is the width of their filter \(G = (6 / \pi \Delta^2)^{3/2} \exp(-6 r^2 / \Delta^2)\). ERF’s kernel \(\exp(-r^2/\epsilon^2)\) is that filter with \(\Delta = \sqrt{6}\,\epsilon\).
The factor was derived for \(\Delta / R\) up to about 1.25. Keep \(\epsilon \le 0.5 R\); wider kernels give a warning.

Each step:

Sample \(u_d\), the axial velocity over the blade nodes, each weighted by its radius times its share of the span.
Evaluate \(C_T\) from the previous step’s shaft thrust and free stream, clamped to 0.96.
Set \(a = (1 - \sqrt{1 - C_T})/2\) and \(U_\infty = M u_d / (1 - a)\).
Scale the hub and blade node velocities by \(U_\infty / u_d\). Tower nodes keep the resolved flow. AeroDyn then applies its own induction.

The recovered free stream is checkpointed and logged in <output_root>_correction.csv.

Calibration, not theory alone

The factor above is for the disk velocity filtered twice: by the force kernel and by a sampling kernel of the same width. GALE samples point values, filtered once; the same derivation then gives \(1/\sqrt{6\pi}\) in place of \(1/\sqrt{3\pi}\). GALE keeps the twice-filtered form. With it the corrected disk matches the standalone BEM loads within 1 %; the once-filtered form recovers a free stream 3 to 4 % high on 20 m cells.

The 0.96 clamp is reached when the inferred thrust coefficient is that high: at high tip-speed ratio, below rated at fixed speed, in a wake, or on a coarse grid (a rotor spanning only 6 cells sat at the clamp). The induction then stays at 0.4, the recovered free stream reads a few per cent low, and the first clamped step warns.

The actuator line keeps disk sampling. It resolves its own induction, and the filtered lifting-line correction accounts for the kernel.

Under-relaxation¶

Why. The free stream is recovered from the last one, through the thrust coefficient it implies. Once the flow has answered the thrust, a small error in the previous free stream returns multiplied by the loop gain

\[G = -\frac{2 a (1 - a)(1 - \delta)}{(1 - 2 a)(1 - a + \delta a)}, \qquad \delta = \frac{\Delta / R}{\sqrt{3 \pi}}.\]

\(G\) is negative (the update overshoots) and reaches \(|G| = 1\) at \(C_T \approx 0.75 + 0.11\,\Delta / R\). Above that the plain update alternates and grows until the clamp stops it. A second mode comes from the flow: the disk velocity answers a thrust change only after the induced flow builds up, and a faster update can ring at that delay. An 8 m LES of the IEA 15 MW rang with a period of about 5 s.

The update is therefore relaxed:

\[U_\infty^{new} = (1 - \omega)\,U_\infty^{old} + \omega\,U_\infty^{update},\]

which turns the one-step gain into \(1 - \omega (1 - G)\).

What the options do.

correction_relax = -1 (default) takes \(\omega = 1/(1 - G)\), clamped to [0.2, 1], which cancels the step-to-step overshoot to first order. It also holds \(\omega \le \Delta t / (\tau (1 - G))\), so an error decays as \(e^{-t/\tau}\) whatever the gain.
\(\tau\) is correction_time: by default the rotor radius over the free stream, 11 s for the IEA 15 MW at 10.6 m/s. correction_time = 0 keeps the gain factor alone.
A fixed correction_relax in (0, 1] is used as given; 1 is the unrelaxed update.

The relaxation changes how the free stream is reached, not where it settles. The gain and \(\omega\) are logged in <output_root>_correction.csv.

Accuracy¶

IEA 15 MW in a uniform 10.59 m/s inflow (open boundaries, no sponge, no turbulence closure), as a ratio to the standalone OpenFAST BEM solution (thrust 2.482 MN, power 15.79 MW, momentum-theory disk velocity 7.67 m/s):

Disk set-up	Thrust	Power	Sampled velocity (m/s)
disk sampling, 20 m cells, kernel 1.5 dx	1.084	1.288	8.40
disk sampling, 20 m cells, kernel 2 dx	1.093	1.328	8.47
disk sampling, 20 m cells, kernel 3 dx	1.112	1.403	8.63
disk sampling, 10 m cells, kernel 2 dx	1.072	1.243	8.36
upstream sampling 1 D, BEM on, 20 m and 10 m	0.977	0.954	10.43
upstream sampling 2 D, BEM on, 20 m	0.993	0.999	10.58
disk_corrected, 20 m cells, kernel 2 dx	0.997	1.000	8.74 → 10.77
disk_corrected, 10 m cells, kernel 2 dx	1.001	0.993	8.44 → 10.82
disk_corrected, 20 m cells, kernel 3 dx	0.991	0.994	8.99 → 10.67

For the corrected disk the second velocity is the recovered free stream. It sits 1 to 2 % high, and the \(C_T\) it implies (0.76 against BEM’s 0.80) correspondingly low; the two biases cancel in the loads.
The one-diameter upstream sample sits 1.5 % below the free stream because of the rotor’s own induction. At two diameters, the turbine default, the bias is below 1 %.
The ratios do not depend on the advection scheme (within 0.1 %). A periodic box changes them by 1 to 2 %.
These runs used the unrelaxed update. The correction settled within about 50 s with no oscillation.

Prescribed-Ct disk¶

type = ct_disk is the classical test rotor: a uniformly loaded disk of radius \(R\) and normal \(\mathbf{n}\), covered by a polar grid of points. It runs no OpenFAST.

\[T = \tfrac{1}{2}\, \rho\, C_T\, U\,|U|\, \pi R^2 .\]

\(U\) is the area-weighted normal velocity sampled on the same grid sample_diameters_upstream diameters upstream; \(\rho\) is the body’s air_density.
Each disk point carries the share of \(-T\mathbf{n}\) proportional to its area.
\(U\,|U|\) keeps the sign of the flow. A flow against the normal (\(U < 0\)) gives \(T < 0\), so the disk slows it too. The sample plane then lies downstream of the disk, in its own wake, so this thrust is a drag that damps the reversed flow, not a momentum-theory thrust.
The log records the upstream and disk-averaged speeds, the thrust, the power \(T U_d\) and the integrated source along the normal. The power column is \(T U_d\), the rate of work the disk extracts from the flow, not the turbine’s electrical power. It is negative only when the upstream and disk speeds have opposite signs (a turning flow), when the force speeds up the flow at the disk.
The upstream sample still feels the disk. In vortex-cylinder theory at \(C_T = 0.75\) it reads 0.977 of the free stream one diameter ahead, so the thrust is 4.6 % below nominal; at two diameters it is 1.4 % below.
air_density must match ERF’s density at the disk centre within density_tolerance.

With turbine = <name> the disk represents a published turbine. Its table in GALE’s turbine library (Turbines/Data/PowerCurves/<name>.txt, or erf.moving_bodies.turbine_data_dir) gives the thrust coefficient \(C_T(U)\) and the electrical power \(P(U)\) at a list of wind speeds, and the rotor diameter. At every update

\[T = \tfrac{1}{2}\,\rho\,C_T(U)\,U\,|U|\,\pi R^2, \qquad R = D/2,\]

where \(U\) is the sampled free-stream speed, \(\rho\) is air_density and \(D\) is the table’s rotor_diameter. Each table \(C_T\) must lie in [0, 2]; a constant ct lies in (0, 1). Between rows \(C_T\) and \(P\) are interpolated linearly; below the first and above the last speed (cut-in and cut-out) both are zero, so a parked rotor puts no force in the flow. Table speeds are not negative, so a reversed flow (\(U < 0\)) gives a table disk no thrust. The tables were made at sea-level density, 1.225 kg/m³. The disk log then adds the columns ct and power_curve (the table’s \(P(U)\) in W; power stays \(T U_d\)) for every disk of the run; a constant-ct disk in such a run logs its ct and T U_d. Runs where no disk names a turbine keep the five columns. The tables hold published \(C_T\) curves only. Turbines with only a power curve are left out: \(C_T\) from ideal momentum theory, \(C_p = 4a(1-a)^2\), \(C_T = 4a(1-a)\), came out 29 to 34 % below the published \(C_T\) at peak \(C_p\) for the eight NREL, IEA, DTU and LEANWIND reference tables (25 to 37 % over all 44). The sources and licences of the tables are in Turbines/Data/PowerCurves/README.md.

The thrust jumps at the first and last table speeds: from zero to the cut-in thrust, and from the cut-out thrust to zero. \(U\) is the instantaneous sampled speed, with no averaging and no hysteresis, so a disk whose free-stream speed fluctuates about a cut-in or cut-out speed switches its thrust on and off from one update to the next. Keep the inflow inside the table’s range when this matters.

The restart record of a disk holds its turbine name and its ct (zero with a table), so a restart that switches a disk between a table and a constant ct, or between tables, stops with a message. A table edited in place under the same name is not detected. The default table folder is the GALE source tree the code was built from; a relocated executable needs erf.moving_bodies.turbine_data_dir.