| 30 Dec 2025
Relating convergence of the actuator line method to its velocity sample using a novel description of the bound vortex in time-dependent fluid dynamics simulations
Abstract. The actuator line method (ALM) is a popular method for representing wind turbine blades in computational fluid simulations. It utilizes a model to compute aerodynamic forces that requires an undisturbed flow velocity relative to the blade. However, the input velocity and output forces are sampled and applied locally; within the bound vortex that forms around the blade, and is thus disturbed. This inherent conflict translates to a strong dependency of the outcome of the simulation, and its convergence, on the sampling approach. This work presents a quantitative description of the bound vortex, and thereby the disturbance on the sampled velocity, in time dependent simulations (e.g. large eddy simulations, LES). The properties of the bound vortex were found to be a function of the single parameter Λ = Δs/ε; the relative distance travelled by an ALM node within one time step, divided by the ALM kernel size. We use simulation data to validate this description and demonstrate that convergence of simulations can largely be attributed to the disturbance from sampling within the bound vortex. Consequently, this attribution to an erroneous process implies that convergence of the ALM is only partly related to the degree of representation of the relevant physical phenomena, which convergence is often assumed to imply. We propose that one must either ensure the disturbance is adequately accounted for, or that an alternative convergence criterion is to be used. We further use the description of the bound vortex to explain that a certain class of velocity sampling methods can indeed successfully account for the disturbance; removing the error and bringing simulations closer to convergence, yet that a correction based on the present description of the bound vortex would be more computationally efficient.
Relating convergence of the actuator line method to its velocity sample using a novel description of the bound vortex in time-dependent fluid dynamics simulations
This article presents an analysis of the actuator line model and its sensitivity to time-stepping. The article presents one key idea regarding the numerical parameter delta which represents the distance that an actuator point travels per time-step in relation to epsilon. While the main idea explored in the manuscript is relevant and worth investigating, the analysis as presented leaves too many unanswered questions and unresolved uncertainties to support the conclusions drawn.
While the authors have identified an issue to better understand and potentially improve the ALM, the analysis is unclear. The main issues discussed (sampling and applying forces at different locations and times) appear to arise from inconsistencies in the ALM implementation and numerical coupling, rather than from a fundamental limitation of the ALM formulation. The authors run cases with values of epsilon/dx< 2 that are not recommended in the literature which adds more uncertainty. The authors also mix numerical and physical parameters without clearly isolating the effects of each (epsilon vs dx).
The authors have not performed a sufficiently thorough literature review and their work could benefit from a better understanding of the known challenges and best practices in the ALM implementations.
The authors go from a simple 2D explanation of the bound vortex, to comparing full 3D simulations of a wind turbine. This is a good goal but adds more complexity and unknowns to the problem. The wind turbine problem is more complex than the simple 2D airfoil, and is not a good first case to understand the phenomena present in the problem.
I recommend that the authors first perform simulations that directly isolate the hypotheses being tested, like flow over a 2D airfoil using the ALM. This way, the authors can compare the exact solutions from their model to the numerical solution of the ALM.
The idea of a new parameter ds/epsilon is good. However, there are too many uncertainties on this manuscript to make clear conclusions. There are many parameters at play in the problem described in the manuscript, for example, grid resolution (dx), epsilon, time resolution (dt), and numerical implementation of the ALM. Then, there are combinations of these parameters (epsilon/dx, and ds/epsilon). There has been a lot of work in many of these areas and the authors should review the literature carefully and follow the guidelines. For example, there are many studies regarding epsilon/dx and there is a well-established limit of epsilon/dx>2.
This work presents an interesting hypothesis and could help improve the ALM and fits well within the scope of WES. However, I do not recommend that this article be published in its current form. The authors should consider performing 2D simulations to clearly isolate the issues being investigated. The modifications required would be significant and would likely warrant a substantially revised manuscript or a new publication.
Below are some detailed comments that can be used to improve the manuscript.
Manuscript:
“Despite its wide and frequent use, there is no thorough consensus on how the ALM is best operated, nor is it well understood how it affects the outcome of a simulation. In this work we aim to address the topics of convergence of simulations and velocity sampling using a quantitative description of the bound vortex, and discuss best practices regarding computational efficiency and accuracy. We first introduce how the ALM operates and how it is treated in literature before further introducing its problems.”
Reviewer Comment:
This statement is overstated and does not reflect the breadth of existing literature on ALM usage and parameter sensitivity. There is a wealth of literature on how to use the ALM and the influence of parameters. The authors need to do a deeper dive into the literature to better understand the challenges that are there and the ones that have been resolved.
Manuscript:
“Various corrections can be found in literature to make these 2D polars more applicable to the 3D situation, most commonly addressing tip behaviour”
Reviewer Comment:
The corrections that are referenced in this are not to correct the 2D polars. They are meant to include the missing induction from an unresolved ALM. The authors should study the referenced papers to better understand the challenges that are being addressed in the literature.
Manuscript:
“Practically, convergence is often associated with how well the relevant physical phenomena are represented, since adding more detail (thereby changing an input) to a simulation that already appropriately represents all relevant physical phenomena will not lead to a better representation (it was already appropriate).”Reviewer Comment: This sentence is not clear. Convergence has to do with the solution approaching the limit as the resolution is increased. It is not about how well the relevant phenomena are represented, that should be accuracy.
The authors are talking about different things in this section. There is the convergence of grid spacing (dx, dy, dz). And there is also convergence in time (dt). Apart from that, there is the sensitivity to ALM parameters such as epsilon. The authors are mixing the many elements of the problem, but they should expand this section and clarify what convergence is. The literature has a good grasp of “convergence” related to epsilon/dx. However, the authors should make the point here that the convergence of epsilon / dt * U has not been studied thoroughly.
Manuscript:
“These three steps (velocity sampling, aerodynamic routine, force projection) are executed at each node at each time step before advancing to the next, such that the velocity is sampled a distance ∆s away from its previous location relative to the flow, where the force projection was centred.”
Reviewer Comment:
This is not necessarily true and a properly implemented ALM algorithm should use a numerical method where there is consistency between the sampling and application of the force. For example, I recommend that the authors look at the work of Kuhn et al (WE 2025, Sec 2.5.2, eq 36).
In the case of sampling and applying the force at different locations, depending on the numerical algorithm used in the solver, the implementation of the ALM might not be consistent with the numerical method of the code used.
Manuscript:
“We can therefore deduce the vortex properties of the flow induced by the AL by considering this graph of cumulative velocity increments. The cumulative velocity increment at any location xand time step nis as follows;”
Reviewer Comment:
This is making an approximation regarding the solution of the moving problem as a collection of lamboseen vortices that appear and disappear in a discrete way. However, this is not the right formulation. This is assuming that at every time, there is a collection of discrete vortices in the flow. However, that is not the way that the solution will look like.
In continuous forcing, the induced vorticity/velocity field evolves continuously and the discrete superposition interpretation must be justified against the governing equations/time integration.
Manuscript:
“The velocity sample V s that we use for validation is taken at the location of where the future force projection will be.”
Reviewer Comment:
This raises concerns regarding the consistency of the force-velocity coupling and requires further clarification. The authors should explicitly document how velocity sampling and force application are synchronized in time and space.
Manuscript:
“2.4 Software”
Reviewer Comment:
This section should be revisited to talk about the numerical details of the ALM implementation in regard to the time-stepping scheme and the location of the actuator point.
Manuscript:
“We observe that most points lie close to the y= xline, thereby confirming that flow near AL nodes generally behaves as described by Eq. (13) and by extension, Eq. (5).”
Reviewer Comment:
The results are quite off from the line, I would not consider this to be good agreement, it is at best a first-order approximation.
Manuscript:
“Lastly, it is observed that simulations using the smallest kernel sizes (ϵ= ∆x, and ϵ= 1.5∆x to a lesser extent) deviate downwards for larger values on the x-axis. …”
Reviewer Comment:
These values of epsilon (epsilon/dx<2) are not recommended in the literature. The authors later justify including these; however, including those doesn’t seem to add any value to the discussion, other than adding more error and uncertainty. I recommend that the authors remove this discussion and revisit their choice for range of epsilon to be epsilon/dx > 2.
Manuscript: “Figure 5 shows the thrust coefficient Ct of the wind turbine for various time steps (∆t) and kernel sizes (ϵ).”
Reviewer Comment:
This figure result is unexpected. This strong dependency on time-step suggests that there could be an inconsistency in the ALM implementation. I also recommend removing results of epsilon/dx<2. Also, it is not clear, what is the value of epsilon in these simulations? Is it changing with the grid?
Manuscript:
“there is an erroneous process (i.e. sampling within the bound vortex yielding V d ̸= 0) affecting the outcome of the simulation.”
Reviewer Comment:
This is the key element of this manuscript. As written, it appears more consistent with an inconsistency in the numerical force-velocity coupling than with a fundamental limitation of the ALM formulation.