1. Introduction
Fast-running predictive models for the performance of turbomachines, including hydrokinetic/wind turbines and propellers, largely rely on momentum theory (Rankine Reference Rankine1865; Froude Reference Froude1878; Froude Reference Froude1889). Turbomachine rotors generate thrust forces that are either opposing (turbines) or in the direction of the flow (propellers). Given this thrust force, the velocity at the rotor cross-section differs from the free-stream velocity by the velocity induced by the thrust (induction) which is a consequence of the pressure difference (Sørensen Reference Sørensen2016). To predict the induced velocity, momentum theory represents the rotor as a porous actuator disk that imparts a uniform thrust force on the surrounding flow. Originally derived for propellers, momentum theory is now widely used in the design and analysis of hydrokinetic and wind turbines (Sørensen Reference Sørensen2011; Adcock et al. Reference Adcock, Draper, Willden and Vogel2021). Simple optimisation of the operating conditions (thrust coefficient) using the momentum theory model results in the well-known Lanchester–Betz–Joukowsky limit (Van Kuik Reference Van Kuik2007), which predicts the theoretical maximum efficiency (coefficient of power) for a turbine operating in unconfined flow. To yield a fully predictive model of rotor thrust, power and wake dynamics, the induction predictions from momentum theory can be coupled with a blade-element model (Glauert Reference Glauert1935) that predicts the thrust coefficient based on lift and drag forces on rotating hydro/aerofoils. The lift and drag forces depend on the velocity at the rotor (induction), yielding the blade-element momentum (BEM) approach (e.g. Vogel et al. Reference Vogel, Willden and Houlsby2018; Madsen et al. Reference Madsen, Larsen, Pirrung, Li and Zahle2020). Therefore, accurate BEM predictions rely on both a blade-element model that predicts blade-level forces given a known velocity field at the rotor, as well as a momentum closure that predicts the velocity field at the rotor considering the geometry and dynamics of the flow. However, in many engineering applications, turbomachines operate in flow configurations that are not appropriately described by classical momentum theory due to its simplifying assumptions, geometric arguments and boundary conditions.
Hydrokinetic turbines used in tidal and river energy harvesting often operate in confined settings where blockage, the ratio between the swept area of the turbine and the channel cross-section, is not negligible (Garrett & Cummins Reference Garrett and Cummins2005). Similarly, in experimental wind tunnels (Barlow et al. Reference Barlow, Rae and Pope1999) and water flumes (Ross & Polagye Reference Ross and Polagye2020), blockage can significantly affect the dynamics and performance of turbomachines (Chen & Liou Reference Chen and Liou2011). Although the swept area of wind turbines is often much smaller than the cross-section of the atmospheric boundary layer, shallow boundary layer heights and large, dense wind farms can also trigger significant blockage effects and associated pressure gradients (Ndindayino et al. Reference Ndindayino, Puel and Meyers2025).
Meanwhile, turbomachines often operate in non-ideal settings where the mean inflow is not aligned with the rotor. Such circumstances are common for hydrokinetic/wind turbines because of the chaotic, turbulent environmental flow in which they operate, along with slowly reacting yaw controllers (Fleming et al. Reference Fleming, Scholbrock, Jehu, Davoust, Osler, Wright and Clifton2014; Frost et al. Reference Frost, Morris, Mason-Jones, O’Doherty and O’Doherty2015, Reference Frost, Evans, Harrold, Mason-Jones, O’Doherty and O’Doherty2017; Piano et al. Reference Piano, Neill, Lewis, Robins, Hashemi, Davies, Ward and Roberts2017). Recent attention has also focused on wake steering, where the wake is laterally deflected by intentional yaw misalignment, to increase the performance of turbine arrays (Fleming et al. Reference Fleming2019; Howland et al. Reference Howland, Quesada, Martínez, Larrañaga, Yadav, Chawla, Sivaram and Dabiri2022; Meyers et al. Reference Meyers, Bottasso, Dykes, Fleming, Gebraad, Giebel, Göçmen and van Wingerden2022). Similarly, propellers frequently operate in misalignment, often called drift (Di Mascio et al. Reference Di Mascio, Muscari and Dubbioso2014).
Classical momentum theory does not account for the confined and misaligned settings described above, which can yield significant error in predictive models. The first-order effects of blockage and inflow-rotor misalignment are to generate pressure gradients and lateral velocities, respectively, which are not modelled in classical momentum theory. Substantial previous effort has been made to model the effect of blockage on rotor dynamics, starting with Glauert (Reference Glauert1935), which is based on a linearisation in the low-blockage regime. More recent studies (Mikkelsen & Sørensen Reference Mikkelsen and Sørensen2002; Bahaj et al. Reference Bahaj, Molland, Chaplin and Batten2007; Garrett & Cummins Reference Garrett and Cummins2007; Werle Reference Werle2010; Segalini & Inghels Reference Segalini and Inghels2014; Vogel et al. Reference Vogel, Willden and Houlsby2018; Ndindayino et al. Reference Ndindayino, Puel and Meyers2025) have developed improved models for the effects of flow confinement on rotor dynamics. A further limitation of existing blockage models and blockage corrections that are extensions of classical momentum theory is that they cannot predict the dynamics of high-thrust rotors (Ross & Polagye Reference Ross and Polagye2020), because momentum theory is not applicable for high-thrust rotors due to the assumption that the wake pressure recovers back to the free-stream pressure (Roshko Reference Roshko1955). This limitation has been recently addressed in unconfined flows (Steiros & Hultmark Reference Steiros and Hultmark2018; Liew et al. Reference Liew, Heck and Howland2024). Building on the model proposed in Steiros & Hultmark (Reference Steiros and Hultmark2018), Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) proposed a closed-form, analytical extension to simultaneously account for high-thrust rotors and confined flow, but it assumes perfect alignment between the rotor and the inflow. Separately, recent efforts have focused on modelling rotors operating in misalignment with the inflow (Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016; Shapiro et al. Reference Shapiro, Gayme and Meneveau2018; Heck et al. Reference Heck, Johlas and Howland2023; Tamaro et al. Reference Tamaro, Campagnolo and Bottasso2024). In parallel, Liew et al. (Reference Liew, Heck and Howland2024) developed an extension of classical momentum theory for unconfined flow that yields low error across the thrust coefficients and rotor-inflow misalignment angles relevant to wind turbines, but it does not capture confinement effects. To summarise, there is a need for a generalised momentum model which can address blockage, rotor-inflow misalignment and arbitrary thrust coefficients of realistic rotors in a unified framework for applications including BEM engineering modelling and blockage corrections. Such a model should both accurately predict key quantities at the rotor, such as the induction, thrust and power, and should also accurately predict the wake and bypass velocities and pressure that are consistent with conservation of mass, momentum and energy.
In this study, we extend the work of Liew et al. (Reference Liew, Heck and Howland2024) to develop a generalised momentum theory that accounts for blockage, rotor-inflow misalignment and arbitrary thrust coefficients. The analytical model, hereafter referred to as the Unified Blockage Model (UBM), is validated against large eddy simulations (LES) of an actuator disk across 80 independent combinations of blockage ratios, thrust coefficients and misalignment angles. We further incorporate the UBM within a BEM framework to predict the performance of bladed rotors under arbitrary blockage and misalignment, and validate the results against blade-resolved simulations. Finally, we present a new blockage correction method, which is a widely used approach for mapping rotor measurements between blockage ratios, based on the UBM. The blockage correction method is evaluated in comparison with computational fluid dynamics (CFD) simulations as well as experiments of an axial flow hydrokinetic turbine.
The remainder of this paper is organised as follows. The UBM, a generalised momentum model for confined rotors, is derived and integrated into BEM and blockage correction methods in § 2. The LES methodology used to validate the UBM is described in § 3. We present the results of the model predictions in § 4. Finally, the key findings of the work are summarised in § 5.
2. A generalised framework for modelling confined rotors
In this section, we derive the UBM in § 2.1, which is a generalisation of the Unified Momentum Model (UMM) (Liew et al. Reference Liew, Heck and Howland2024) to confined flow. Next, in § 2.2, the UBM is coupled with a blade-element closure yielding a fully predictive model for bladed rotors, termed the Unified BEM model. Finally, a method for applying the UBM to account for blockage in measurements of rotor thrust and power (a blockage correction) is developed in § 2.3.
2.1. A blockage model for misaligned actuator disks across thrust coefficients
Control volume for analysis of a confined, misaligned turbine. The streamwise and spanwise directions are
$x$
and
$y$
, respectively. The turbine is yaw/tilt misaligned at angle
$\gamma$
, where positive misalignment is a counter-clockwise rotation. The sides of the control volume are rigid so there is no mass flux through them. As in classical momentum theory, we consider four stations for the analysis, and the flow variables are labelled with the corresponding station as subscript numbers. The cross-sectional areas, streamwise velocities, spanwise velocities and pressures are denoted as
$A$
,
$u$
,
$v$
and
$p$
, respectively. The pressure at the outlet is
$p_{4,w}$
within the streamtube and
$p_{4}$
outside of the streamtube. The unit vector normal to the misaligned turbine is
$\hat {n}$
.
We aim to develop an analytical model for the induction, thrust and power of an actuator disk accounting for flow confinement (blockage), misalignment between the inflow and rotor, and arbitrary thrust coefficients, including high-thrust operation. The non-dimensional parameter describing the rotor-normal, rotor-averaged induced velocity is (Heck et al. Reference Heck, Johlas and Howland2023)
\begin{equation} a_n = 1 - \frac { \boldsymbol{u}_d \boldsymbol{\cdot }\hat {n} }{ u_{\infty } \cos (\gamma )}, \end{equation}
which is the generalisation of the standard (axial) induction factor in classical one-dimensional momentum modelling (Van Kuik Reference Van Kuik2007). The rotor-averaged velocity at the rotor disk is
$\boldsymbol{u}_d$
, the free-stream velocity is
$u_\infty$
, and the misalignment angle between the inflow and the rotor-normal vector
$\hat {n}$
is
$\gamma$
. As a result of misalignment, the rotor-induced velocity will vary azimuthally (Glauert et al. Reference Glauert1926). Following Heck et al. (Reference Heck, Johlas and Howland2023), the present analytical model is derived in terms of the rotor-averaged induction factor. Therefore, the model predictions will not depend on the azimuthal variations. To derive a model for the induction factor
$a_n$
, it is convenient to describe the rotor aerodynamics (i.e. blade aerofoil properties, number of blades, etc.) and control (tip-speed ratio, blade pitch angle, etc.) with a single parameter, the local thrust coefficient
$C_T^\prime$
. The thrust force for the actuator disk is thus defined as (Calaf et al. Reference Calaf, Meneveau and Meyers2010)
\begin{equation} \boldsymbol{F}_{\!T} = -\frac {1}{2} \rho C_T^{\prime } A_d (\boldsymbol{u}_{d} \boldsymbol{\cdot }\hat {n})^2 \hat {n}, \end{equation}
where
$\rho$
is the fluid density and
$A_d = \pi D^2/4$
is the rotor area, with
$D$
the rotor diameter. We will explicitly connect the value of
$C_T^\prime$
to the rotor design and control properties in § 2.2.
We now consider analysis of the control volume shown in figure 1, which has rigid sides that are slip walls. This is appropriate for wind (Mikkelsen & Sørensen Reference Mikkelsen and Sørensen2002) and water (Ross & Polagye Reference Ross and Polagye2020) tunnels and also for hydrokinetic turbine settings in low Froude number regimes (Garrett & Cummins Reference Garrett and Cummins2007). Following standard momentum theory, we consider turbulence-free, uniform inflow. We include the effects of the confining channel walls through a no-penetration condition only (i.e. slip walls); wall boundary layer effects such as shear and turbulence are not included. The flow is assumed to be inviscid (frictionless) within the control volume. Therefore, the model is applicable to the rotor and near wake in which turbulent mixing does not yet contribute to the dynamics (Burton et al. Reference Burton, Jenkins, Sharpe and Bossanyi2011; Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016). The exit of the control volume at station 4 is taken to be the location at which the shear layers around the potential core meet causing the wake to become fully turbulent, known as the near-wake length (Lee & Chu Reference Lee and Chu2003; Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016). The present model assumes uniform blockage and does not capture potential impacts of channel asymmetry on the rotor dynamics, relevant for applications such as tidal turbines in close proximity to walls/surfaces. For example, wall proximity could influence lateral induced velocities resulting from rotor-inflow misalignment. The free-stream velocity and pressure are denoted
$u_\infty$
and
$p_1$
, respectively. The velocity and pressure at the outlet outside of the streamtube are
$u_s$
and
$p_4$
, respectively. To allow for the additional pressure deficit within the wake associated with base-suction effects at high thrust (Liew et al. Reference Liew, Heck and Howland2024), the outlet pressure inside the streamtube is defined as
$p_{4,w}$
. The streamwise and lateral wake velocities at the streamtube outlet are
$u_4$
and
$v_4$
, respectively. Conservation of streamwise momentum in the control volume yields
\begin{equation} \boldsymbol{F}_{\!T} \boldsymbol{\cdot }\hat {i} = \rho u_4^2 A_4 + \rho u_s^2(A_{\textit{CV}}-A_4) - \rho u_\infty ^2 A_{\textit{CV}} + A_{\textit{CV}} (p_4 - p_1) + A_4(p_{4,w} - p_4), \end{equation}
where
$A_4$
is the area of the streamtube at station 4, and
$A_{\textit{CV}}$
is the cross-sectional area of the control volume in the
$yz$
-plane. Conservation of energy inside the streamtube implied by the Bernoulli equation yields (Liew et al. Reference Liew, Heck and Howland2024)
\begin{equation} a_n = 1 - \sqrt {\frac {u_\infty ^2 - u_4^2 - v_4^2}{C_T^\prime \cos ^2(\gamma ) u_\infty ^2}-\frac {(p_{4,w} - p_1)}{\frac {1}{2} \rho C_T^\prime \cos ^2(\gamma ) u_\infty ^2}}. \end{equation}
The Bernoulli equation applied along a streamline outside the streamtube that is inside that control volume is
\begin{equation} \frac {1}{2} \rho u_\infty ^2 + p_1= \frac {1}{2} \rho u_s^2 + p_4, \end{equation}
where
$u_s$
is the bypass velocity that can be larger than
$u_1$
due to the flow confinement. This application of the Bernoulli equation neglects any potential turbulent mixing between the wake and bypass flow (Nishino & Willden Reference Nishino and Willden2012, Reference Nishino and Willden2013), which is found to be negligible in the near wake within the present LES cases with zero free-stream turbulence inflow. Mass conservation in the streamtube between the actuator disk and outlet gives
\begin{equation} u_\infty (1-a_n)\cos (\gamma )A_d =u_4 A_4, \end{equation}
and mass conservation in the control volume gives
\begin{equation} A_{\textit{CV}}\ u_\infty = u_4 A_4 + (A_{\textit{CV}} - A_4) u_s. \end{equation}
The mass flux out of the control volume at the rigid sides is explicitly zero in this equation. Finally, the lateral velocity generated by misalignment is modelled based on Prandtl lifting line theory which yields an analytical solution for the constant transverse velocity induced by an elliptical lift distribution (Shapiro et al. Reference Shapiro, Gayme and Meneveau2018; Heck et al. Reference Heck, Johlas and Howland2023)
\begin{equation} v_4 = -\frac {1}{4} C_T^\prime (1-a_n)^2 \sin (\gamma ) \cos ^2(\gamma ) u_\infty . \end{equation}
This formulation was shown to accurately predict the lateral induced velocity of an unconfined actuator disk across a range of misalignment and thrust coefficients. However, the expression slightly overpredicted the lateral induced velocity for higher thrust values (
$C_T'\sim 10$
) (Liew et al. Reference Liew, Heck and Howland2024).
Combining (2.3)–(2.8) yields a system of six coupled nonlinear equations which is the final form of the UBM
\begin{align} a_n &= 1 - \sqrt {\frac {u_\infty ^2 - u_4^2 - v_4^2}{C_T^\prime \cos ^2(\gamma ) u_\infty ^2}+\frac {(p_1- p_{4,w})}{\frac {1}{2} \rho C_T^\prime \cos ^2(\gamma ) u_\infty ^2}}, \\[-12pt]\nonumber \end{align}
\begin{align} \frac {u_4}{u_\infty } &= \frac {A_d}{A_4}(1 - a_n) \cos (\gamma ), \\[-12pt]\nonumber \end{align}
\begin{align} \frac {v_4}{u_\infty } &= -\frac {1}{4} C_T^\prime (1-a_n)^2 \sin (\gamma ) \cos ^2(\gamma ), \\[-12pt]\nonumber \end{align}
\begin{align} \frac {u_s}{u_\infty } &= 1 + \frac {\beta A_4 (u_\infty - u_4)}{(A_d - \beta A_4)u_\infty }, \\[-12pt]\nonumber \end{align}
\begin{align} \frac {A_4}{A_d} &= \frac {\frac {1}{2}C_T'(1-a_n)^2 \cos ^3(\gamma ) u_\infty ^2 + (1/\beta )(u_s^2 - u_\infty ^2 - (p_1 - p_4)/\rho )}{(p_4 - p_{4,w})/\rho + u_s^2 - u_4^2}, \\[-12pt]\nonumber \end{align}
\begin{align} \frac {p_1 - p_4}{\rho u_\infty ^2} &= \frac {1}{2} (u_s^2 / u_\infty ^2 - 1), \end{align}
where
$\beta = A_d/A_{\textit{CV}}$
is the geometric blockage ratio. The UBM comprises six equations, but contains seven unknowns. The wake pressure
$p_{4, w}$
is the unclosed term. Liew et al. (Reference Liew, Heck and Howland2024) developed a model for the base suction of a turbine operating in unconfined flow. This is a region of persistent pressure deficit directly downstream of a highly loaded rotor, resulting from its high induction factor. The UMM predicts this base-suction pressure deficit using a rapid solution to the pressure Poisson equation associated with the two-dimensional inviscid Euler equations. In the following discussion, we describe the extension of the UMM pressure prediction to account for blockage.
Based on our suite of LES in this study, we observe that the wake pressure becomes homogeneous across the entire outlet (
$p_{4,w}=p_4$
) in the limit of high blockage. Specifically, the pressure difference between the wake and bypass flows becomes negligible compared with the much larger pressure gradient resulting from blockage. On the other hand, the near-wake wake pressure
$p_{4,w}$
significantly departs from the outer pressure
$p_4$
for low blockage and high thrust, as captured by the UMM (Liew et al. Reference Liew, Heck and Howland2024). When the blockage ratio is very small and the free-stream inflow has no turbulence, the turbulent mixing does not significantly contribute to the dynamics within the wake potential core until the near-wake length (Liew et al. Reference Liew, Heck and Howland2024; Heck et al. Reference Heck, Mata and Howland2026). In our zero free-stream turbulence LES of an actuator disk in confined flow, the role of turbulence entrainment in the near wake also has a small effect on the rotor dynamics, both within the potential core and in the bypass flow, compared with the large impact of the bypass flow pressure. Other studies have proposed turbulence entrainment from the bypass flow into the near wake as a mechanism for increasing the rotor power extraction (Nishino & Willden Reference Nishino and Willden2012, Reference Nishino and Willden2013; Dehtyriov Reference Dehtyriov2023). For example, using Reynolds-averaged Navier Stokes simulations of an actuator disk in confined flow, Nishino & Willden (Reference Nishino and Willden2012) found that near-wake entrainment increased rotor power and thrust when blade-induced or free-stream turbulence was added to the simulation, relative to a turbulence-free inflow baseline. Studying the near-wake turbulent transport across inflow conditions and rotor representations should be explored in future work. The UMM (Liew et al. Reference Liew, Heck and Howland2024) wake pressure solution is an inviscid solution based on unconfined flow that does not account for any potential effects from confinement, including possible turbulent shear stress at higherblockage ratios (Nishino & Willden Reference Nishino and Willden2012, Reference Nishino and Willden2013), free-stream turbulence (Nishino & Willden Reference Nishino and Willden2012; Dehtyriov Reference Dehtyriov2023) or possible inviscid effects associated with confinement on the solution of the pressure Poisson equation. As such, we model the pressure difference between wake and bypass flows as
$p_{4,w} - p_4 = (1 - \beta )p_{\textit{UMM}}(C_T', \gamma )$
, where
$p_{\textit{UMM}}$
is the pressure deficit given by the UMM (Liew et al. Reference Liew, Heck and Howland2024) for unconfined flows. This hybrid pressure model captures the departure of the wake pressure from the outer pressure in the low-blockage regimes (
$\beta \rightarrow 0$
) and captures the convergence of the wake and outer pressure for higher-blockage values. For completeness, a description of the UMM pressure closure is included in Appendix A and a sensitivity analysis to the hybrid pressure closure is presented in Appendix B. This formulation yields high accuracy across
$\beta$
and
$C_T'$
by yielding the correct limiting behaviour for high thrust and low blockage, and returning to a constant wake pressure for high blockage. Section 4.1 provides validation of the pressure closure against LES. Future work may consider extending the UBM to account for potential impacts of turbulent mixing between the wake and the bypass flow (Nishino & Willden Reference Nishino and Willden2013) on the wake pressure difference (
$p_{4,w} - p_4$
) measured at station 4 (where the shear layers around the potential core meet) in confined conditions. In the present LES data with zero free-stream turbulence inflow, the pressure difference between the wake and bypass flow at station 4 (
$p_{4,w} - p_4$
) is negligible relative to the pressure difference induced by confinement (
$p_{1} - p_4$
) for all blockage ratios except for approximately unblocked conditions (
$\beta =0.005$
).
As described previously, the local thrust coefficient
$C_T^\prime$
is a useful input quantity since it compactly represents the rotor design and control settings. Further,
$C_T^\prime$
is independent of the misalignment angle
$\gamma$
and the blockage ratio
$\beta$
. Often, rotors are characterised by a thrust coefficient
$C_T$
, which is defined as
$C_T = 2 {\|\boldsymbol{F}_{\!T}\|} / (\rho A_d u_\infty ^2),$
where
$\|\boldsymbol{F}_{\!T}\|$
is the magnitude of the thrust force. Equations (2.9)–(2.14) describing the UBM can be closed when
$C_T$
is used as the input thrust coefficient variable instead of
$C_T^\prime$
by adding an additional equation to the set above
\begin{equation} C_T' = \frac {C_T}{(1-a_n)^2\cos ^2(\gamma )}. \end{equation}
Note that the thrust coefficient
$C_T$
depends on the local thrust coefficient
$C_T^\prime$
, the misalignment angle
$\gamma$
and the blockage ratio
$\beta$
(the effect of the blockage ratio comes through the induction factor
$a_n$
).
To summarise, the model solved using
$C_T'$
as the input thrust variable and (2.9)–(2.14) can be written compactly as
\begin{equation} [a_n,\; u_4/u_\infty ,\; v_4/u_\infty ,\; u_s/u_\infty ,\; A_4/A_d,\; (p_1-p_4)/\rho u_\infty ^2] = f_{\textit{Unified}}(C_T', \gamma , \beta ) \end{equation}
and will be referred to as the
$C_T'$
-formulation. When using
$C_T$
as the input thrust variable together with (2.9)–(2.15), the model can be written as
\begin{equation} [a_n,\; u_4/u_\infty ,\; v_4/u_\infty ,\; u_s/u_\infty ,\; A_4/A_d,\; (p_1-p_4)/\rho u_\infty ^2,\; C_T'] = f_{\textit{Unified}}(C_T, \gamma , \beta ) \end{equation}
and will be referred to as the
$C_T$
-formulation. Here,
$f_{\textit{Unified}}$
denotes the converged solution to the appropriate set of equations.
The nonlinear coupled system is solved using a standard root-finding algorithm. As with most such methods, fast and reliable convergence depends on providing a suitable initial guess. When solved with the
$C_T'$
-formulation, initial guesses for induction
$a_n$
and wake velocities
$u_4/u_\infty$
and
$v_4/u_\infty$
are obtained directly from the UMM (Liew et al. Reference Liew, Heck and Howland2024) which is the limit of
$\beta \rightarrow 0$
. These are combined with
$ A_4/A_d = (1 - a_n)\cos (\gamma )(u_\infty /u_4)$
, together with Eqs. 2.12 and 2.14 to determine
$u_s/u_\infty$
and
$(p_1 - p_4)/\rho u_\infty ^2$
. Since the initial guess is given by the
$\beta \rightarrow 0$
limit, this initial guess can be too far from the final solution for large values of
$\beta$
. So for larger values of
$\beta$
(e.g.
$\beta =0.3$
), we first solve the model for
$\beta =0.1$
based on initialisation from the
$\beta =0$
limiting case as described above, then we solve the model at
$\beta =0.2$
with initialisation from the
$\beta =0.1$
results and so on until the solution at the desired
$\beta$
is reached. When solved using the
$C_T$
-formulation, initial guesses are generated via pre-tabulation. First, the
$C_T'$
-formulation is solved across a range of
$C_T'$
,
$\gamma$
and
$\beta$
which gives the corresponding values of
$C_T$
using (2.15). Then, the desired initial guesses for the model outputs
$ (a_n,\; u_4/u_\infty ,\; v_4/u_\infty ,\; u_s/u_\infty ,\; A_4/A_d,\; (p_1 - p_4)/\rho u_\infty ^2,\; C_T')$
are tabulated as a function of the input variables
$(C_T, \gamma , \beta )$
. Initial guesses for any combination of input variables can then be obtained using linear interpolation. A reference implementation of the UBM is available as an open-source Python library (https://doi.org/10.5281/zenodo.20644355) (Liew et al. Reference Liew, Upfal, Gering, Heck and Howland2026b
).
In the limit of zero misalignment and low thrust coefficients, the UBM returns to the actuator disk model developed by Garrett & Cummins (Reference Garrett and Cummins2007) for confined flow, while in the limit of zero blockage (
$\beta \rightarrow 0$
), the model yields the UMM for unconfined flows developed by Liew et al. (Reference Liew, Heck and Howland2024). As will be shown in § 4.1, nonlinear interactions arise between blockage, thrust coefficient and misalignment angle, such that the previous models cannot be superposed and instead must be modelled jointly, as the present approach provides.
2.2. A generalised blade-element momentum model for rotors in confinement
The UBM presented in the previous section predicts the induced velocities, and therefore the velocity at the rotor, based on an input blockage ratio, misalignment angle and thrust coefficient. However, the thrust coefficient for an arbitrary rotor design and control strategy is not known a priori. Here, a blade-element model is used to calculate the thrust coefficient based on the velocity at the rotor, the aerofoil properties, misalignment angle, blade pitch angle and tip-speed ratio. The blade-element model is solved on a polar grid depending on radial and azimuthal position
$(\mu , \psi )$
. These positions are formulated non-dimensionally, with non-dimensional radius
$\mu =r/R$
, where
$R=D/2$
is the rotor radius and
$r$
is the radial coordinate, and
$\psi$
in radians. The tip-speed ratio is
$\lambda = R\varOmega /u_\infty$
, where
$\varOmega$
is the angular velocity of the rotor. The axial and tangential velocity components relative to the blades are
\begin{align} v_n(\mu ,\psi ) & = (1-a_n)\cos (\gamma )u_\infty, \\[-12pt]\nonumber \end{align}
\begin{align} v_t(\mu ,\psi ) & = (1 + a')\lambda \mu u_\infty - (1-a_n)\cos (\psi )\sin (\gamma )u_\infty , \end{align}
assuming uniform inflow, where
$a'$
is the tangential induction factor. The speed
$w$
and angle
$\phi$
of the flow relative to the blades are
\begin{align} w^2(\mu ,\psi ) & = v_n^2(\mu ,\psi ) + v_t^2(\mu ,\psi ), \\[-12pt]\nonumber \end{align}
\begin{align} \phi (\mu ,\psi ) & = \tan ^{-1}\left (\frac {v_n(\mu ,\psi )}{v_t(\mu ,\psi )}\right )\!. \end{align}
The lift and drag coefficients,
$C_l$
and
$C_d$
, are evaluated as a function of the local angle of attack
$\alpha (\mu ,\psi ) = \phi (\mu ,\psi ) - \theta _t(\mu ) - \theta _p$
, where
$\theta _t$
and
$\theta _p$
are the blade twist and pitch angles respectively. Transforming into the rotor frame, the rotor-normal and tangential force coefficients are
\begin{align} C_{n}(\mu ,\psi ) & = \cos (\phi )C_l(\mu , \alpha (\mu ,\psi )) + \sin (\phi )C_d(\mu , \alpha (\mu ,\psi )), \\[-12pt]\nonumber \end{align}
\begin{align} C_{\textit{tan}} (\mu ,\psi ) & = \sin (\phi )C_l(\mu , \alpha (\mu ,\psi )) - \cos (\phi )C_d(\mu , \alpha (\mu ,\psi )). \end{align}
Finally, the thrust coefficient is
\begin{equation} C_T(\mu , \psi ) = \sigma (\mu )C_{n}(\mu ,\psi ) \frac {w^2(\mu , \psi )}{u_\infty ^2}, \end{equation}
with blade solidity
$\sigma (\mu ) = \textit{Bc}/2\pi r$
, where
$B$
is the number of blades and
$c(\mu )$
is the blade chord. To account for reductions in the blade force coefficients due to the effects of tip vortices, we apply a standard tip loss correction factor
\begin{equation} F(\mu ) = \frac {2}{\pi }\arccos \left [\exp \left (-\frac {B(1-\mu )}{2\mu \sin {\phi }}\right )\right ]\!, \end{equation}
and use a modified axial force as input to the induction model (Madsen et al. Reference Madsen, Larsen, Pirrung, Li and Zahle2020)
\begin{equation} C_T^{\textit{corr}}(\mu , \psi ) = \frac {C_T(\mu , \psi )}{F(\mu )}. \end{equation}
The rotor-normal and tangential induction factors are initialised with
$a_n(\mu , \psi ) = 1/3$
and
$a'(\mu , \psi )=0$
, respectively. At each iteration, equations (2.18)–(2.26) are solved to update the rotor normal and tangential forces based on the current values of
$a_n(\mu , \psi )$
and
$a'(\mu , \psi )$
. The blade-element model is connected to the UBM derived in § 2.1 using the
$C_T$
-formulation
$f_{\textit{Unified}}(C_T^{\textit{corr}}(\mu , \psi ), \gamma , \beta )$
(equation (2.17)) to update
$a_n(\mu , \psi )$
. Here, the momentum model is solved separately at each point over the rotor to account for the azimuthal variation in thrust induced by misalignment (Madsen et al. Reference Madsen, Larsen, Pirrung, Li and Zahle2020). The tangential induction factor is updated using (Liew et al. Reference Liew, Heck and Howland2024)
\begin{equation} a'(\mu , \psi ) = \frac {\sigma (\mu )C_{\textit{tan}}(\mu ,\psi ) w^2(\mu , \psi )}{4 \lambda \mu F(\mu ) (1-a_n)\cos (\gamma ) u_\infty ^2}. \end{equation}
After the calculation of the new rotor-normal and tangential induction factors,
$a_n(\mu , \psi )$
and
$a'(\mu , \psi )$
, respectively, we return back to (2.18) to restart the loop. The coupled system is solved using fixed point iteration with relaxation until converged solutions for
$a_n(\mu , \phi )$
and
$a'(\mu , \psi )$
are reached. Finally, the output rotor-averaged thrust is calculated by integrating over the rotor area
\begin{equation} \bar {C}_T = \frac {1}{\pi }\int _0^1\int _0^{2\pi } \mu \sigma (\mu )C_{n}(\mu ,\psi ) \frac {w^2(\mu , \psi )}{u_\infty ^2} {\rm d}\psi {\rm d}\mu , \end{equation}
and similarly the rotor-averaged power coefficient is
\begin{equation} \bar {C}_P =\frac {1}{\pi } \int _0^1 \int _0^{2\pi } \lambda \mu ^2 \sigma (\mu )C_{\textit{tan}}(\mu ,\psi ) \frac {w^2(\mu , \psi )}{u_\infty ^2} {\rm d}\psi {\rm d}\mu . \end{equation}
2.3. A simple blockage correction based on the UBM
The Unified BEM method presented in the previous section predicts rotor thrust and power for arbitrary rotor design, tip-speed ratio, pitch angles, rotor-inflow misalignment and blockage ratio. This method solves for the rotor-normal induction using the aerofoil lift and drag coefficients as a thrust closure for the UBM. To compute the rotor power, the aerofoil data are then used to determine the torque contributions along the blades. However, if measurements of rotor thrust and power are available at a single blockage ratio, these can be used to predict
$C_T$
and
$C_{\!P}$
for arbitrary
$\beta$
without the need for a blade-element model, which can be beneficial because the blade-element model requires detailed information on the aerofoils. This methodology is widely referred to as a blockage correction (e.g. Ross & Polagye Reference Ross and Polagye2020).
We now use the UBM derived in § 2.1 to develop a simple blockage correction method. While previous work has sought to correct for blockage by identifying a so-called ‘equivalent unblocked free-stream velocity’ (Ross & Polagye Reference Ross and Polagye2020), we instead propose an alternative method that leverages the UBM to directly capture the change in rotor disk velocity induced by blockage. Though in practice these two approaches are similar, the current method provides a new interpretation of the blockage correction in terms of the local rotor performance. This enables an improved understanding of when blockage corrections may be expected to produce reliable predictions. Specifically, it is useful to normalise
$\lambda$
,
$C_T$
, and
$C_{\!P}$
by the rotor-normal disk velocity,
$\boldsymbol{u}_d \boldsymbol{\cdot }\hat {n}$
, instead of the free-stream velocity
$u_\infty$
. This yields the local tip-speed ratio
$\lambda '$
, local thrust coefficient
$C_T'$
and local power coefficient
$C_{\!P}'$
. Using
$\boldsymbol{u}_d \boldsymbol{\cdot }\hat {n} = (1 - a_n) \cos (\gamma ) u_\infty$
from (2.1) gives
\begin{equation} \lambda '=\frac {\lambda }{(1-a_n)\cos (\gamma )}, \quad C_T'=\frac {C_T}{(1-a_n)^2\cos ^2(\gamma )}, \quad C_{\!P}'=\frac {C_{\!P}}{(1-a_n)^3\cos ^3(\gamma )}. \end{equation}
This local description of rotor thrust and power in terms of the disk velocity,
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
, decouples the rotor performance, which is specific to every turbine, from the induction, in contrast to
$C_T$
and
$C_{\!P}$
where rotor performance and blockage effects are intermingled. As a result, we expect that
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
are independent of
$\beta$
if the aerofoil properties,
$C_l(\mu , \alpha )$
and
$C_d(\mu , \alpha )$
, yaw angle, and pitch angles do not change. A short analysis using the BEM equations is provided in Appendix C to show that
$C_T'$
and
$C_{\!P}'$
depend only on
$\lambda '$
,
$\theta _p$
and
$\gamma$
when
$C_l(\mu , \alpha )$
and
$C_d(\mu , \alpha )$
are fixed. Blade-resolved simulations are also used to confirm that
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
are constant across a wide range of
$\beta$
in § 4.3. The invariance of
$C_l$
and
$C_d$
to confinement is a critical assumption for all blockage corrections, and implications of non-constant
$C_l$
and
$C_d$
, for example due to Reynolds-number effects, are further discussed in § 4.3.
Given the local rotor performance scaling,
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
, we now apply it to develop a new blockage correction method. From measurements of
$C_T^{\beta _1}$
and
$C_{\!P}^{\beta _1}$
at an input blockage ratio
$\beta _1$
and input tip-speed ratio
$\lambda ^{\beta _1}$
, we determine the induction factor
$a_n^{\beta _1}$
using the
$C_T$
-formulation (equation (2.17)) of the UBM. Using the induction factor at
$\beta _1$
, we compute
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
with (2.30). Next, the
$C_T'$
-formulation (2.16) is used to predict the induction factor at an arbitrary output blockage ratio
$\beta _2$
. Finally, the induction at the output blockage ratio is used to determine
$\lambda ^{\beta _2}$
,
$C_T^{\beta _2}$
and
$C_{\!P}^{\beta _2}$
from
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
with (2.30). Therefore, we have a concise and physically justified blockage correction method to map measurements of thrust and power between different blockage ratios for a given rotor of interest. This new blockage correction method based on the UBM is summarised visually in figure 2 and is performed as follows:
-
(i) Solve the
$C_T$
-formulation
$f_{\textit{Unified}}(C_T, \gamma , \beta )$
(equation (2.17)) using the thrust coefficient
$C_T^{\beta _1}$
measured at blockage ratio
$\beta _1$
to determine the induction factor
$a_n^{\beta _1}$
(2.31)
\begin{equation} [a_n^{\beta _1}, \ldots ] = f_{\textit{Unified}}(C_T^{\beta _1}, \gamma , \beta _1). \end{equation}
-
(ii) Use
$a_n^{\beta _1}$
to compute the local tip-speed ratio
$\lambda '$
, local thrust coefficient
$C_T'$
, and local power coefficient
$C_{\!P}'$
with (2.30) from the data measured at
$\beta _1$
(2.32)
\begin{equation} \lambda ' = \frac {\lambda ^{\beta _1}}{(1 - a_n^{\beta _1})\cos (\gamma )}, \ C_T' = \frac {C_T^{\beta _1}}{(1 - a_n^{\beta _1})^2\cos ^2(\gamma )}, \ C_{\!P}' = \frac {C_{\!P}^{\beta _1}}{(1 - a_n^{\beta _1})^3\cos ^3(\gamma )}. \end{equation}
-
(iii) Solve the
$C_T'$
-formulation
$f_{\textit{Unified}}(C_T',\gamma , \beta )$
(2.16) to determine the induction factor
$a_n^{\beta _2}$
at the blockage ratio
$\beta _2$
(2.33)
\begin{equation} [a_n^{\beta _2}, \ldots ] = f_{\textit{Unified}}(C_T', \gamma , \beta _2). \end{equation}
-
(iv) Use
$a_n^{\beta _2}$
to determine the output control and performance parameters
$\lambda ^{\beta _2}$
,
$C_T^{\beta _2}$
and
$C_{\!P}^{\beta _2}$
with (2.30) (2.34)
\begin{align} \lambda ^{\beta _2} &= \lambda '(1 - a_n^{\beta _2})\cos (\gamma ), \\[-12pt]\nonumber \end{align}
(2.35)
\begin{align} C_T^{\beta _2} &= C_T'(1 - a_n^{\beta _2})^2\cos ^2(\gamma ), \\[-12pt]\nonumber \end{align}
(2.36)
\begin{align} C_{\!P}^{\beta _2} &= C_{\!P}'(1 - a_n^{\beta _2})^3\cos ^3(\gamma ). \end{align}
To summarise, this new blockage correction procedure is a mapping from one combination of blockage ratio, tip-speed ratio, thrust coefficient and power coefficient to another
\begin{equation} (\beta _1, \lambda ^{\beta _1}, C_T^{\beta _1}, C_{\!P}^{\beta _1}) \rightarrow (\beta _2, \lambda ^{\beta _2}, C_T^{\beta _2}, C_{\!P}^{\beta _2}), \end{equation}
where the mapping is performed by enforcing that
$C_T'$
and
$C_{\!P}'$
are constant for a given
$\lambda '$
, pitch angle and rotor-inflow misalignment.
Flow diagram of the blockage correction method presented in § 2.3.
These local parameters
$C_T'$
and
$C_{\!P}'$
can also be interpreted as the unknown rotor performance parameters predicted by the blade-element model. Therefore, measurements at a single
$\beta$
in conjunction with the UBM can be used in place of the blade-element model to make predictions at arbitrary
$\beta$
. This eliminates the need for aerofoil data, simplifying the modelling effort and reducing uncertainty introduced by the aerofoil polars.
3. Large eddy simulation numerical set-up
We perform LES with PadéOps (Ghate & Lele Reference Ghate and Lele2017; Howland et al. Reference Howland, Ghate and Lele2020; Heck & Howland Reference Heck and Howland2025), an open-source, incompressible, pseudo-spectral solver for the filtered Navier–Stokes equations (Subramaniam et al. Reference Subramaniam, Ghate, Ghaisas and Howland2021). Relevant details are provided in this section, and the complete numerical description and validation are available in published studies (e.g. Heck & Howland Reference Heck and Howland2025). The sigma subgrid-scale model is used for subgrid-scale stresses (Nicoud et al. Reference Nicoud, Toda, Cabrit, Bose and Lee2011). The simulations use inflow conditions consistent with the momentum modelling: uniform inflow with zero free-stream turbulence. This inflow is set by a fringe region (Nordström et al. Reference Nordström, Nordin and Henningson1999) in the
$x$
-direction. The turbine is placed
$x=5D$
from the inlet and centred laterally and vertically in the domain. No-flux boundary conditions are used at the rigid sides (
$y$
-direction), and periodic boundary conditions are used in the
$x$
- and
$z$
-directions. The simulation domains used for each blockage ratio are described in table 1. The grid sizes were selected to ensure that the resolution is approximately constant between blockage ratios.
Description of LES domains used for each blockage ratio. The domain lengths and number of grid points in the
$x$
,
$y$
and
$z$
directions are denoted
$L_x$
,
$L_y$
and
$L_z$
, and
$N_x$
,
$N_y$
and
$N_z$
, respectively.
Table 1. Long description
A table with four rows and five columns. The columns are labeled Blockage ratio, Lx/D, Ly/D, Lz/D, and Nx x Ny x Nz. The row labels are beta values: 0.005, 0.1, 0.2, and 0.3. Each row provides the corresponding values for Lx/D, Ly/D, Lz/D, and the grid points Nx x Ny x Nz. For beta = 0.005, Lx/D is 25, Ly/D and Lz/D are 12.5, and Nx x Ny x Nz is 640 x 800 x 800. For beta = 0.1, Lx/D is 25, Ly/D and Lz/D are 2.80, and Nx x Ny x Nz is 640 x 180 x 180. For beta = 0.2, Lx/D is 25, Ly/D and Lz/D are 1.98, and Nx x Ny x Nz is 640 x 128 x 128. For beta = 0.3, Lx/D is 25, Ly/D and Lz/D are 1.62, and Nx x Ny x Nz is 640 x 128 x 128.
The filtered actuator disk model (ADM) methodology developed by Shapiro et al. (Reference Shapiro, Gayme and Meneveau2019) is used to model the rotor. The thrust force depends on the local thrust coefficient
$C_T^\prime$
and the velocity at the disk
$\boldsymbol{u}_d \boldsymbol{\cdot }\hat {n}$
(Calaf et al. Reference Calaf, Meneveau and Meyers2010; Shapiro et al. Reference Shapiro, Gayme and Meneveau2019; Heck et al. Reference Heck, Johlas and Howland2023). The actuator disk power is defined as
$P = -\boldsymbol{F}_{\!T} \boldsymbol{\cdot }\boldsymbol{u}_d$
. For the inflow-aligned (
$\gamma =0$
)
$C_T'$
sweep, a filter width of
$\varDelta /D = 3.2\times 10^{-2}$
and a disk thickness of
$s = 3\Delta x/2$
are used, following Heck et al. (Reference Heck, Johlas and Howland2023). This narrow filter width minimises distortion of the induction, which occurs as a result of smoothing the ADM forcing through filtering. The filtered actuator disk correction factor from Shapiro et al. (Reference Shapiro, Gayme and Meneveau2019) is turned off since it is not needed because of the narrow filter width used. Additionally, since the correction factor is based on classical momentum theory, which is not valid for the confined case, it would introduce error to the induction of the confined rotor. To simulate misaligned turbines in unconfined flow, Heck et al. (Reference Heck, Johlas and Howland2023) rotated the angle of the inflow while keeping the actuator disk aligned with the computational grid. In the present study, simulations of misaligned rotors in a confined channel require the rotor itself to be rotated in order to maintain no-flux boundary conditions at the channel walls. This introduces a trade-off between maintaining a sufficiently narrow filter and ensuring a smooth spatial distribution of the turbine forcing for arbitrary misalignment angles. To balance these requirements, we employ a grid resolution approximately twice as fine in each spatial direction as Heck et al. (Reference Heck, Johlas and Howland2023), a filter width of
$\varDelta /D = 6.5\times 10^{-2}$
and a disk thickness of
$s = 3\Delta x$
for the misaligned cases. Although there is a small sensitivity to the numerical set-up, we find this to be at least an order of magnitude smaller than the effect of blockage. This configuration maintains a relatively narrow filter with minimal influence on induction while ensuring a smoothly distributed forcing for the misaligned ADM. The sensitivity of the simulation results to the numerical set-up of the ADM is presented in Appendix D.
Simulations are run for four flow-through times
$L_x / u_\infty$
, with the velocity and pressure fields averaged over the final two flow throughs, which is sufficient for convergence of first-order statistics in these zero free-stream turbulence inflow cases (Heck et al. Reference Heck, Johlas and Howland2023). To delineate between the wake and free-stream flow, we compute a streamtube seeded at the rotor disk (
$R_s = 0.5D$
) in the rotor plane, which may be rotated in the case of yaw misalignment. Wake flow quantities
$u_4$
and
$p_{4,w}$
are then extracted by averaging across the
$yz$
-plane in a region enclosed by this streamtube at the streamwise location
$x = x_0$
(Heck et al. Reference Heck, Johlas and Howland2023; Liew et al. Reference Liew, Heck and Howland2024). As described in Heck et al. (Reference Heck, Johlas and Howland2023), there are some numerical sensitivities in
$u_4$
and
$p_{4,w}$
to the rotor representation and to the streamtube seeding but they do not affect the qualitative outcomes of the study. The outlet wake area
$A_4$
is defined as the cross-sectional area of the streamtube at this location. The near-wake length
$x_0$
is defined as the streamwise location at which the minimum
$yz$
-averaged streamwise velocity occurred within the streamtube (i.e. the location where turbulent wake recovery begins).
Following Shapiro et al. (Reference Shapiro, Gayme and Meneveau2018), the spanwise initial wake velocity
$v_4$
is defined as the maximum
$yz$
-averaged spanwise velocity within the wake streamtube over
$0 \leqslant x/D \leqslant 5$
. To estimate the bypass flow quantities,
$u_s$
and
$p_4$
, a separate streamtube of radius
$R_b = 0.55D$
is seeded at the turbine location. The streamwise velocity and pressure are then
$yz$
-averaged over the channel cross-section outside this streamtube at
$x = x_0$
to obtain
$u_s$
and
$p_4$
, respectively.
4. Results
This section presents a comparison between the UBM and LES, followed by validation of the Unified BEM and blockage correction methods. The results of the UBM derived in § 2.1, and LES of an actuator disk are presented in § 4.1. In § 4.2, the Unified BEM model developed in § 2.2 is validated against blade-resolved simulations. In § 4.3, the new blockage correction method presented in § 2.3 is validated by mapping rotor thrust and power measurements from blade-resolved simulations, LES of an actuator line model and experimental measurements across a wide range of tip-speed and blockage ratios.
4.1. Analytical model of an actuator disk in misalignment and blockage
We begin by comparing the analytical model predictions with LES results for an inflow-aligned actuator disk (
$\gamma = 0$
), before considering misaligned cases. The induction factor
$a_n$
, thrust coefficient
$C_T$
and power coefficient
$C_{\!P}$
from LES and the UBM are shown in figure 3. The analytical model exhibits low error, compared with the LES results, across a wide range of blockage ratios and local thrust coefficients. In particular, the present model accurately represents the rotor dynamics in the limit of low blockage and high thrust, where classical momentum theory is not valid. Predictions from Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) are included for comparison. The UBM appears to slightly lower error at high local thrust coefficients and blockage ratios compared with the model of Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) in figure 3, but these relatively minor differences between the predictions for thrust and power coefficients from the models in yaw-aligned conditions are comparable to the numerical uncertainty in the LES (see Appendix D). Figure 3(
$a$
) shows that the induction factor
$a_n$
decreases with increasing blockage ratio
$\beta$
. Consequently, the thrust and power coefficients, which scale as
$C_T \sim (1-a_n)^2$
and
$C_{\!P} \sim (1-a_n)^3$
, respectively, both increase with blockage ratio (figure 3
b,c).
Comparison between the UBM (equations (2.9)–(2.14)) and LES for (a) rotor-normal induction, (b) thrust coefficient and (c) power coefficient across local thrust coefficients
$C_T'$
and blockage ratios
$\beta$
with rotor-aligned inflow (
$\gamma =0$
). The predictions of classical one-dimensional momentum theory and the blockage model of Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) are plotted for reference.
Comparison of (a) bypass velocity, (b) initial streamwise wake velocity, (c) spanwise initial wake velocity, (d) outlet wake area, (e) bypass pressure drop and ( f) streamtube pressure drop between the UBM developed in § 2.1 and LES of an actuator disk for various local thrust coefficients
$C_T'$
and rotor-aligned inflow (
$\gamma = 0$
).
In figure 4, the bypass velocity, streamwise and spanwise wake velocities, outlet wake area and outlet pressures are compared between LES and the UBM. Overall, the model predictions show strong agreement with the LES measurements. Figure 4(
$a$
) shows that the bypass velocity increases with blockage ratio to satisfy global mass conservation within the channel. Confinement also increases the streamwise wake velocity (figure 4
$b$
) as the induction factor is reduced. Consequently, the wake expands less, leading to a smaller outlet wake area at higher-blockage ratios (figure 4
$d$
). Figure 4(
$e$
) shows that rotor confinement induces a favourable pressure gradient,
$p_1 - p_4$
, due the acceleration of the bypass flow around the disk. In the absence of blockage, the pressure gradient outside of the streamtube,
$p_1 - p_4$
, is zero while base suction causes the change in pressure within the streamtube,
$p_1 - p_{4,w}$
, to be non-zero (figure 4
$f$
). The model captures the rotor behaviour in the limit of high thrust and low blockage, where classical momentum theory breaks down, by incorporating the base-suction pressure deficit predicted by the UMM for unconfined flows (Liew et al. Reference Liew, Heck and Howland2024). As
$\beta$
increases, the pressure gradient increases across the entire channel and dominates the contribution of base suction (
$p_1 - p_4 \gg p_4 - p_{4,w}$
). Figure 4(
$f$
) demonstrates that the model captures the combined blockage and base-suction pressure contributions across a wide range of
$\beta$
and
$C_T'$
.
The blockage-induced favourable pressure gradient,
$p_1 - p_4$
, allows additional momentum to be extracted from the pressure field. As a result, the disk velocity increases and the induction factor decreases. The blockage-induced pressure gradient increases with rotor thrust, such that blockage effects become more pronounced at higher
$C_T'$
. This is consistent with the limit of zero thrust coefficient, where there is no favourable pressure gradient from geometric confinement and therefore no blockage effects. Thus, the blockage effect depends not only on the geometric blockage ratio
$\beta$
but also on its coupling with the thrust coefficient (
$C_T$
or
$C_T^\prime$
).
Having established the rotor-aligned performance, the UBM is next assessed under rotor-misaligned inflow. In figure 5, model predictions of the rotor-normal induction factor
$a_n$
, thrust coefficient
$C_T$
and power coefficient
$C_{\!P}$
across misalignment angles
$\gamma$
and blockage ratios
$\beta$
are compared with LES for
$C_T' = 2$
. For an aligned rotor in unconfined flow,
$C_T^\prime =2$
results in
$C_T=8/9$
. Excellent agreement between the LES and the UBM is observed across all misalignment angles and blockage ratios. We note that predictions from the model of Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) are not shown since that study and model only consider perfect alignment between the turbine and inflow. In figure 5(
$a$
), induction decreases with increasing magnitude of misalignment and increasing blockage. As a result of the decreasing induction, the rotor thrust and power coefficients increase with increasing blockage (figure 5
b,c). As the misalignment angle increases, both thrust and power decrease, despite the reduction in induction. The quantitative outcome of the reduction in the thrust and power with the misalignment angle will depend on the competition between the geometric reduction in the rotor-normal component of the free-stream flow velocity against the reduction in induction associated with the misalignment. The competition between the geometry and the response of the induction factor is why the thrust and power for a misaligned actuator disk are not accurately modelled by geometry alone (i.e. modelling thrust and power reductions using
$\cos ^2(\gamma )$
and
$\cos ^3(\gamma )$
, respectively, will lead to errors).
Comparison between the UBM (equations (2.9)–(2.14)) and LES for (a) induction factor, (b) thrust coefficient and (c) power coefficient across misalignment angles
$\gamma$
and blockage ratios
$\beta$
with
$C_T'=2$
. Classical momentum theory and the blockage model of Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) are not shown as they assume perfect alignment between the rotor and inflow.
Comparison of (a) bypass velocity, (b) initial streamwise wake velocity, (c) spanwise initial wake velocity, (d) outlet wake area, (e) bypass pressure drop and ( f) streamtube pressure drop, between the UBM developed in § 2.1 and LES for various misalignment angles.
In figure 6, the bypass velocity, streamwise and spanwise wake velocities, outlet wake area and outlet pressures from LES and the UBM are shown for the misaligned actuator disk. Overall, the analytical model shows strong agreement with LES. The actuator disk induces a lower bypass velocity as it becomes misaligned with the inflow because both the induction and projected area decrease (figure 6
$a$
). The reduced induction also causes the streamwise wake velocity to increase with increasing blockage and magnitude of misalignment (figure 6
$b$
). Figure 6(
$c$
) shows that the analytical model captures the overall magnitude of the spanwise wake velocity. However, the variations with blockage are reversed relative to the LES predictions. Misalignment reduces induction, causing the wake to expand less, as well as incurring a
$\cos (\gamma )$
decrease in the projected area of the rotor, both of which reduce the wake area (figure 6
$d$
). Figure 6(
$e$
) demonstrates that the blockage-induced pressure gradient becomes weaker with increasing magnitude of misalignment as a result of the reduced bypass velocity. This reveals that misalignment weakens the blockage effect, as will be discussed further. Finally, figure 6(
$f$
) shows that the wake pressure measured in LES is also well captured by the model.
(a) Induction factor, (b) thrust coefficient and (c) power coefficient normalised by their inflow-aligned (
$\gamma =0$
) values for varying blockage ratios. UBM predictions ((2.9)–(2.14)) and LES results are shown for
$C_T' = 2$
. Increasing blockage leads to a more gradual reduction in induction with misalignment, while thrust and power decrease more rapidly with misalignment.
In figure 7, we replot the induction, thrust and power of the misaligned rotor normalised by their inflow-aligned (
$\gamma =0$
) values for each blockage ratio
$\beta$
. This reveals a central result of the present study: as the blockage ratio increases, the induction factor decreases more gradually with increasing misalignment angle. Correspondingly, as blockage increases, rotor thrust and power decrease more rapidly with increasing misalignment angle. This behaviour arises because rotor misalignment reduces the magnitude of the blockage effect by decreasing both the projected rotor area and the rotor thrust. By lowering the thrust force, misalignment reduces induction. However, the reduced thrust weakens the blockage effect, which would otherwise act to reduce induction as well. As a result, the decrease in induction with misalignment is more gradual when the rotor operates under confinement. The slower reduction in induction implies a lower disk velocity, which leads to a more rapid loss of thrust and power with misalignment when operating in a confined environment. Therefore, the effects of blockage and misalignment are coupled through the turbine thrust force. Due to this coupling, existing models for misalignment and blockage cannot be superimposed. Instead, these effects must be modelled jointly as provided by the UBM. In addition to implications of these results on measuring thrust and power for misaligned rotors in wind/water tunnels, these results may affect the future design of wake steering flow control for hydrokinetic turbines (Modali et al. Reference Modali, Vinod and Banerjee2021).
The influence of flow confinement on measured thrust (drag) and power is an important consideration in experimental fluid dynamics. A common approach is to characterise blockage effects using only the geometric blockage (Schottler et al. Reference Schottler, Bartl, Mühle, Sætran, Peinke and Hölling2018; Upfal et al. Reference Upfal, Zhu, Handy-Cardenas and Breuer2025). However, our results reveal that this is insufficient to characterise the magnitude of blockage effects, owing to its dependence on the thrust coefficient. Consequently, there is no universal blockage ratio below which blockage effects may be considered negligible, as this threshold depends on
$C_T$
. We instead propose a more appropriate parameter based on the ratio of the streamwise component of the thrust force,
$\boldsymbol{F}_{\!T} \boldsymbol{\cdot }\hat {i}$
, to the free-stream momentum flux scale,
$({1}/{2})\rho A_{\textit{CV}} u_\infty ^2$
, yielding
$\beta C_T \cos (\gamma )$
. Note that in a confined wind/water tunnel setting, this
$C_T$
is what would be measured in the experiment, so all information in
$\beta C_T \cos (\gamma )$
would be known. To demonstrate this scaling, we quantify the magnitude of blockage effects as the change in thrust and power relative to an unblocked case with the same
$C_T'$
and
$\gamma$
, through the thrust ratio
$T_r = C_T/C_T(\beta =0)$
and power ratio
$P_r = C_{\!P}/C_{\!P}(\beta =0)$
. Figure 8 demonstrates that, unlike
$\beta$
, the proposed parameter captures the leading-order scaling of the blockage-induced increases in thrust and power, quantified by
$T_r$
and
$P_r$
, respectively. This scaling parameter cannot predict the influence of blockage with the precision of the model developed in § 2.1 but instead provides a simple method for estimating, to first order, the expected augmentation in thrust and power.
The blockage-induced change in thrust, quantified by
$T_r$
, is shown over
$0 \leqslant \gamma \leqslant 40^\circ$
and
$2 \leqslant C_T' \leqslant 10$
as a function of (a)
$\beta$
and (b)
$\beta C_T \cos (\gamma )$
. Panels (c, d) show the corresponding results for power quantified by
$P_r$
.
We find that this parameter captures the leading-order trend in
$T_r$
and
$P_r$
across a wide range of practical values of
$\beta$
,
$C_T$
and
$\gamma$
(see figure 8). The observed collapse persists beyond the range shown with some sensitivity of the linear fit to the range of
$\beta$
,
$C_T$
and
$\gamma$
used. This scaling can be readily evaluated and provides a practical metric for assessing blockage effects in experimental configurations to yield a simple, first-order, quantitative estimate of the approximate change in thrust and power due to blockage effects.
4.2. Confined blade-element momentum model
The UBM has been validated against LES of an actuator disk across a wide range of misalignment angles
$\gamma$
, local thrust coefficients
$C_T'$
and blockage ratios
$\beta$
in the previous section. However, the UBM is not fully predictive for a rotating turbine, since
$C_T'$
depends on the aerofoil characteristics as well as the blade pitch angle
$\theta _p$
, tip-speed ratio
$\lambda$
and yaw angle
$\gamma$
. Moreover, once
$a_n$
is determined, the power of a rotating turbine still explicitly depends on the aerofoil properties and the control setpoints. To address this, we develop a fully predictive Unified BEM model for a rotating turbine under arbitrary blockage and misalignment by coupling the UBM with a blade-element model in § 2.2. Here, we run the Unified BEM model using lift and drag polars of Wimshurst & Willden (Reference Wimshurst and Willden2016) for the RISO-A1-24 aerofoil with
$Re = 12 \times 10^6$
. A three-dimensional stall-delay correction was applied to the lift and drag coefficients of the aerofoil (Du & Selig Reference Du and Selig1998). MITRotor (Liew et al. Reference Liew, Upfal, Gering, Heck and Howland2026a
), an open source tool for physics-based rotor modelling and optimisation, is used to solve the BEM model described in § 2.2. The model is solved using 40 radial grid points.
In figure 9, the Unified BEM predictions are evaluated against the blade-resolved simulations of Wimshurst & Willden (Reference Wimshurst and Willden2016). The model reproduces the dependence of thrust and power on tip-speed ratio, and captures the systematic increase in both quantities with blockage. Importantly, the Unified BEM predictions capture the location of peak power across all blockage ratios considered. In general, BEM models rely on blade-element modelling to capture lift and drag forces, using two-dimensional aerofoil polars and empirical submodels for tip losses and three-dimensional stall-delay effects, both of which introduce uncertainty into the model predictions. These uncertainties are distinct from the momentum modelling described in the UBM (the momentum portion of BEM used here), and instead come from the blade-element portion of BEM. These uncertainties are illustrated by the shaded bands plotted in figure 9, encompassing the full spread of results obtained when the radial grid resolution is reduced to 20 points, and the tip loss, three-dimensional stall delay or tangential induction submodels are toggled on or off.
Quantifying the uncertainty associated with this reasonable range of modelling choices reveals substantial variability in the predictions and highlights the opportunity to improve existing engineering models for the aero/hydrodynamics of rotor blades. Despite these limitations, the overall agreement between the BEM predictions and the blade-resolved simulations is good and comparable to that reported in previous BEM validation studies. Figure 9 further compares the Unified BEM results with the model of Vogel et al. (Reference Vogel, Willden and Houlsby2018), which uses the momentum model developed by Garrett & Cummins (Reference Garrett and Cummins2007). The two BEM models demonstrate similar levels of agreement with the blade-resolved simulations of Wimshurst & Willden (Reference Wimshurst and Willden2016). Notably, the Unified BEM method generalises to rotor-inflow misalignment and high thrust coefficient by leveraging the UBM. In the following section, we demonstrate that the errors and uncertainties inherent to blade-element modelling can be eliminated by mapping rotor measurements between blockage ratios using a blockage correction method.
Validation of the Unified BEM method developed in § 2.2 with blade-resolved CFD simulations from Wimshurst & Willden (Reference Wimshurst and Willden2016). The earlier BEM model results from Vogel et al. (Reference Vogel, Willden and Houlsby2018) are also shown for comparison. The shaded bands indicate the full spread of results obtained when the radial grid resolution is varied, and the tip loss, three-dimensional stall delay or tangential induction submodels are toggled.
4.3. Blockage correction method based on the UBM
In § 4.2, we validated a fully predictive rotor model for arbitrary misalignment angles, thrust coefficients and blockage conditions against blade-resolved simulations. However, the reliance on two-dimensional aerofoil lift and drag polars, together with sub-models for tip losses and three-dimensional aerofoil corrections, introduces errors and uncertainties into BEM-based predictions. Alternatively, if measurements of rotor thrust and power are available at a single blockage ratio, these can be used instead of a blade-element model to make predictions at arbitrary blockage ratios. In § 2.3, we developed a new method that extracts the local thrust coefficient
$C_T'$
and local power coefficient
$C_{\!P}'$
as functions of the local tip-speed ratio
$\lambda '$
using the UBM. This scaling, resulting in blockage-invariant quantities
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
, describes the tip speed, thrust and power of the rotor in relation to the rotor-normal disk velocity instead of the free-stream velocity. In Appendix C, we show that when the aerofoil lift and drag curves are independent of blockage,
$C_T'$
and
$C_{\!P}'$
then only depend on
$\lambda '$
,
$\theta _p$
and
$\gamma$
. In this case, we expect such information to be sufficient for predicting the performance of a given rotor at other blockage ratios. In figure 10, we demonstrate the proposed scaling,
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
, using the blade-resolved simulations of Wimshurst & Willden (Reference Wimshurst and Willden2016). This scaling requires knowledge of the turbine induction, which is computed using the UBM. In the lowest blockage case (
$\beta = 0.01$
),
$C_T'$
and
$C_{\!P}'$
are slightly higher than in the blocked cases. Despite this deviation in the unblocked case, the data collapse well overall. The collapse of the data reveals that these local parameters are constant with blockage ratio, supporting the argument presented in Appendix C.
A scaling collapse of the (a) thrust and (b) power coefficients is demonstrated using the local parameters
$\lambda '$
,
$C_T'$
and
$C_{\!P}'$
for blade-resolved simulations from Wimshurst & Willden (Reference Wimshurst and Willden2016) over a range of blockage ratios
$\beta$
. The observed collapse supports the validity of the blockage correction method presented in § 2.3.
Blockage correction method developed in § 2.3 using the UBM is applied to (a) thrust and (c) power coefficients from the ADM LES results in (b) and (d), respectively. The (c) thrust and (d) power coefficients for low blockage
$\beta =0.005$
are predicted by correcting the thrust and power coefficients from the confined flows (
$\beta =0.1,$
$0.2$
and
$0.3$
).
Having verified the proposed scaling, we now apply the blockage correction method developed in § 2.3 to the ADM LES cases shown in § 4.1 and to blade-resolved simulations of Wimshurst & Willden (Reference Wimshurst and Willden2016). In figure 11, we apply the blockage correction method outlined in § 2.3 to ADM LES data. In this case, the measurements of the coefficient of thrust (figure 11
a) and the coefficient of power (figure 11
c) for the confined flows (
$\beta =0.1,$
$0.2$
and
$0.3$
) are corrected to the unconfined blockage value of
$\beta =0.005$
in figures 11(b) and 11(d), respectively. Similar to the model comparison with LES presented in § 4.1, the UBM-based blockage correction yields very low error. This very low error is expected, because the core assumption of the blockage correction method proposed in § 2.3 is that the local thrust coefficient
$C_T^\prime$
is constant, a feature which is enforced in these LES cases. Nonetheless, these results provide verification that the blockage correction method proposed in § 2.3 yields accuracy consistent with the UBM itself.
We next apply the blockage correction method to a more realistic setting using the blade-resolved simulations of Wimshurst & Willden (Reference Wimshurst and Willden2016). In figures 12(
$a$
) and 12(
$b$
), the thrust and power coefficients,
$C_T$
and
$C_{\!P}$
, from
$\beta = 0.20$
are mapped to
$\beta = 0.10$
and
$\beta = 0.01$
. The blockage-corrected thrust and power coefficients show low error relative to the
$\beta = 0.10$
case, reflecting the strong collapse of the blocked cases (
$\beta = 0.07, 0.10, 0.20$
) in figure 10. Larger errors occur when mapping to the
$\beta = 0.01$
case, consistent with the slight divergence of
$C_T'$
and
$C_{\!P}'$
between blocked and unblocked conditions. This indicates that the remaining error may originate from blockage-induced changes in local rotor performance not accounted for by the correction. In figures 12(
$c$
) and 12(
$d$
), we compare the present method with the closed-form corrections of Barnsley & Wellicome (Reference Barnsley and Wellicome1990) and Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) to map the
$\beta = 0.20$
measurements to
$\beta =0.01$
. The present method performs similarly to Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) while both methods yield notable improvements compared with Barnsley & Wellicome (Reference Barnsley and Wellicome1990) for higher
$C_T$
, achieved at larger values of the tip-speed ratio. Figures 12(
$e$
) and 12(
$f$
) shows that the present method and the correction of Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) both exhibit excellent agreement when mapping from
$\beta = 0.20$
to
$\beta = 0.10$
. The method of Barnsley & Wellicome (Reference Barnsley and Wellicome1990) is not shown as it only allows for mapping to
$\beta = 0$
.
Blockage correction method developed in § 2.3 using the UBM is applied to (
$a$
) thrust and (
$b)$
power coefficients from blade-resolved simulations by Wimshurst & Willden (Reference Wimshurst and Willden2016). The
$\beta =0.2$
results are mapped to
$\beta =0.1$
and
$\beta =0.01$
. (
$c, d$
) The present method is compared with the corrections of Barnsley & Wellicome (Reference Barnsley and Wellicome1990) and Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) to map the
$\beta = 0.2$
case to
$\beta =0.01$
. (
$e, f$
) The present method is compared with the correction of Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) to map the
$\beta = 0.2$
case to
$\beta =0.1$
. The correction of Barnsley & Wellicome (Reference Barnsley and Wellicome1990) is not shown as it only allows for mapping to
$\beta = 0$
.
In figure 13, we apply the three corrections to the LES of an actuator line model performed by Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) for a high solidity (high
$C_T$
) rotor. The corrections are used to map the confined (
$\beta = 0.20$
) thrust and power coefficients to the unconfined (
$\beta = 0.03$
) results. The three corrections perform similarly at lower tip-speed ratios, but the method of Barnsley & Wellicome (Reference Barnsley and Wellicome1990) shows significant error at higher tip-speed ratios, corresponding to high thrust coefficients where classical momentum theory is no longer valid. The present model, which accurately captures the dynamics of high-thrust rotors, exhibits lower error across all tip-speed ratios.
Blockage correction method developed in § 2.3 using the UBM is applied to (
$a$
) thrust and (
$b$
) power coefficients from the actuator line simulations of Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) and compared with the methods of Barnsley & Wellicome (Reference Barnsley and Wellicome1990) and Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022). The confined (
$\beta =0.20$
) case is mapped to the near-unconfined (
$\beta =0.03$
) data.
Finally, we evaluate the blockage correction methods against experimental data. The experimental dataset of Ross & Polagye (Reference Ross and Polagye2020) compares the power and thrust of identical axial flow hydrokinetic turbines at two blockage ratios,
$\beta =0.02$
and
$\beta = 0.35$
. Figure 14 shows the blockage correction developed in § 2.3 as well as the Barnsley & Wellicome (Reference Barnsley and Wellicome1990) and Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022) methods applied to these measurements. All the corrections show similar discrepancies compared with the low-blockage reference data, particularly for
$C_{\!P}$
. The lower agreement for
$C_{\!P}$
is also reported in Ross & Polagye (Reference Ross and Polagye2020). The relatively large discrepancy in corrected
$C_{\!P}$
differs notably from the more accurate corrected
$C_{\!P}(\lambda )$
curves using simulation data shown previously (figures 12 and 13).
To elucidate challenges with blockage corrections in the experimental dataset, we show the scaling to
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
based on the local rotor velocity that underpins the blockage correction model from § 2.3. Figure 15 demonstrates that, unlike the results of Wimshurst & Willden (Reference Wimshurst and Willden2016), the Ross & Polagye (Reference Ross and Polagye2020) dataset does not collapse under the
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
scaling, suggesting that the local rotor performance may be changing with
$\beta$
.
Blockage correction methods applied to the (a) thrust and (b) power coefficients from the Ross & Polagye (Reference Ross and Polagye2020) dataset. The confined case (
$\beta = 0.35$
) case is mapped to the near-unconfined case (
$\beta = 0.02$
).
Scaling of the Ross & Polagye (Reference Ross and Polagye2020) measurements taken at two blockage ratios using the disk velocity computed with the UBM. The results show that (
$a$
) the local thrust coefficient and (
$b$
) the local power coefficient do not collapse with
$\lambda '$
.
This is acknowledged by Ross & Polagye (Reference Ross and Polagye2020), who note that blade-level Reynolds-number independence was not maintained across their tests. Specifically, Ross & Polagye (Reference Ross and Polagye2020) noted that rotor performance was sensitive to changes in chord-based Reynolds number. Using their
$C_T(\lambda )$
and
$C_{\!P}(\lambda )$
curves at two chord-based Reynolds numbers, we perform the scaling to
$C_T'(\lambda ')$
and
$C_{\!P}'(\lambda ')$
. As shown in figure 16, these measurements also fail to collapse under the proposed scaling, highlighting that the measurements were conducted under conditions where the local rotor performance – and consequently the lift and drag characteristics of the blades – changes with Reynolds number. Since the induction changes with blockage, blockage affects the chord-based Reynolds number, leading to different rotor performances when the blockage is changed.
This Reynolds-number sensitivity violates a key assumption of the blockage correction method, namely that the aerofoil lift and drag curves remain invariant with
$\beta$
. More generally, we expect that any turbine blockage correction approach will fail when trying to correct data that span a significant aero/hydrofoil Reynolds-number dependency unless those dependencies are explicitly and accurately captured in the blockage correction model. This analysis further reveals the need for experimental measurements of identical turbines at multiple blockage ratios and sufficiently high Reynolds number to achieve Reynolds-number independence. Such a dataset would be highly valuable to validate engineering models for rotors operating under blockage.
Scaling of the (
$a$
) thrust and (
$b$
) power measurements from Ross & Polagye (Reference Ross and Polagye2020) taken at two Reynolds numbers using the disk velocity computed with the UBM, revealing changes in local rotor performance with Reynolds number.
5. Conclusions
In confined flows, mass conservation requires acceleration of fluid around the rotor, which induces a favourable pressure gradient. This pressure gradient alters the balance of streamwise momentum, increasing the rotor disk velocity relative to unblocked conditions. Consequently, blockage reduces rotor induction while increasing thrust and power. Because the turbine thrust itself generates the blockage-induced pressure gradient, there is a coupled feedback between the thrust force and blockage effects.
Rotor misalignment reduces the rotor-normal disk velocity, lowering the turbine thrust and consequently reducing induction. When a rotor is misaligned in confined flow, misalignment and blockage effects are coupled through the thrust force: misalignment reduces the effective blockage by lowering both the thrust force and the projected rotor area. This mechanism causes the induction factor to decrease more gradually with misalignment in confined flows. As a result, the thrust and power reductions associated with misalignment are exacerbated in confined conditions. Since the misalignment and blockage effects are coupled, they must be modelled jointly as the present approach provides.
This work develops a UBM to predict the induced velocities of an actuator disk across arbitrary blockage ratios, misalignment angles and thrust coefficients. The analytical model is validated against LES, showing excellent agreement with the high-fidelity simulations and lower error than existing models. The model also accurately captures the high-thrust, low-blockage limit by including the base-suction pressure deficit from the UMM for unconfined rotors (Liew et al. Reference Liew, Heck and Howland2024).
Since the thrust coefficient for an arbitrary rotor design is not known a priori, the UBM is not fully predictive for a rotating turbine. Therefore, a Unified BEM model is developed by coupling the validated momentum model with a blade-element closure for the thrust coefficient. This yields a fully predictive rotor model for arbitrary blade design, blockage ratio and control settings. The resulting framework is validated against blade-resolved CFD simulations.
When measurements of rotor thrust and power are available, these can be used to extract the local thrust and power coefficients as functions of the local tip-speed ratio. We identify that if the blade lift and drag properties are independent of blockage, then the local rotor performance is also independent of blockage. This novel scaling is verified with blade-resolved simulation results. We then use this identified scaling to develop a blockage correction method for mapping measurements between arbitrary blockage ratios. The current method provides a new interpretation of the blockage correction in terms of the local rotor performance. This enables an improved understanding of when blockage corrections in general may be expected to produce reliable predictions. The method is validated against blade-resolved simulations, actuator line simulations and experimental measurements of an axial flow turbine. While the blockage correction method yields low errors when correcting simulation data, the blockage correction performs less well for existing experimental data due to the lack of Reynolds-number independence across the different experiments at different blockage ratios. Future work may consider experimental measurements that vary the blockage ratio while ensuring Reynolds-number independence.
The UBM captures the coupled effects of blockage, misalignment and arbitrary thrust coefficient on the induced velocities of an actuator disk. Based on this model, the new blockage correction provides a practical tool to account for blockage effects in turbine and propeller measurements. Finally, the Unified BEM method can be used to optimise the design and control of rotors operating in confined flows, such as hydrokinetic turbines in shallow water or wind turbines in shallow atmospheric boundary layers with stable capping inversions.
Acknowledgements
The authors acknowledge the use of the Stampede3 supercomputer through the NSF ACCESS project, allocation ATM170028, for performing the simulations.
Funding
This work was supported by the National Science Foundation under Grant No. 2542240. K.S.H. was supported by a National Science Foundation Graduate Research Fellowship under Grant No. DGE-2141064.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Base-suction pressure from UMM
The UMM describes the induction of an actuator disk operating in unconfined flow (
$\beta \rightarrow 0$
) for arbitrary thrust coefficient and misalignment angle (Liew et al. Reference Liew, Heck and Howland2024). The UMM captures the high-thrust regime of rotors by modelling the base-suction effect, which is a persistent pressure drop in the near-wake region. In confined flow, a uniform pressure gradient develops which spans the full channel cross-section and dominates over the contribution of the base-suction pressure deficit. In the present work, we include the base-suction pressure deficit predicted by the UMM to ensure high accuracy across arbitrary blockage levels, misalignment angles and thrust coefficients.
The UMM equations solve for the rotor-normal induction
$a_n$
, streamwise wake velocity
$u_4$
, lateral wake velocity
$v_4$
, near-wake length
$x_0$
and the pressure difference
$(p_{4,w}-p_1)$
. Note that in the context of UMM,
$p_{4,w}$
refers to the pressure within the streamtube at
$x=x_0$
. The final form of the UMM equations is
\begin{align} a_n &= 1 - \sqrt {\frac {u_\infty ^2 - u_4^2 - v_4^2}{C_T^\prime \cos ^2(\gamma ) u_\infty ^2}-\frac {(p_{4,w} - p_1)}{\frac {1}{2} \rho C_T^\prime \cos ^2(\gamma ) u_\infty ^2}}, \\[-12pt]\nonumber \end{align}
\begin{align} u_4 &= -\frac {1}{4} C_T^\prime (1-a_n) \cos ^2(\gamma )u_\infty + \frac {u_\infty }{2} + \notag \\ &\frac {1}{2}\sqrt {\left (\frac {1}{2} C_T^\prime (1-a_n) \cos ^2(\gamma )u_\infty - u_\infty \right )^2 - \frac {4(p_{4,w} - p_1)}{\rho }}, \\[-12pt]\nonumber \end{align}
\begin{align} v_4 &= -\frac {1}{4} C_T^\prime (1-a_n)^2 \sin (\gamma ) \cos ^2(\gamma ) u_\infty, \\[-12pt]\nonumber \end{align}
\begin{align} \frac {x_0}{D} &=\frac {\cos (\gamma )}{2\beta _s}\cfrac {u_\infty + u_4}{|u_\infty - u_4|}\sqrt {\frac {(1-a_n)\cos (\gamma )u_\infty }{u_\infty + u_4}}, \\[-12pt]\nonumber \end{align}
\begin{align} p_{4,w} - p_1 &= -\frac {1}{2\pi } \rho C_T^\prime (1-a_n)^2 \cos ^2(\gamma ) u_\infty ^2 \arctan \left [\frac {1}{2}\frac {D}{x_0}\right ] + p^{\textit{NL}}(C_T', \gamma , a_n, x_0), \end{align}
where the unknown shear layer growth rate parameter is
$\beta _s=0.1403$
, based on the analysis of fundamental turbulent jet flow. The near-wake length is predicted by modelling the growth of the shear layers surrounding the wake potential core (Lee & Chu Reference Lee and Chu2003; Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016; Liew et al. Reference Liew, Heck and Howland2024). Despite the presence of these shear layers, turbulent mixing does not significantly influence the dynamics before the near-wake length (Liew et al. Reference Liew, Heck and Howland2024). Therefore, the inviscid model is sufficient to predict this base-suction pressure term.
The nonlinear pressure field
$p^{\textit{NL}}(x,y)$
as a function of streamwise
$x$
and spanwise
$y$
directions is computed as
\begin{equation} p^{\textit{NL}}(x, y) = \frac {\rho }{2\pi }\left (g_x \circledast \frac {x}{x^2 + y^2} + g_y \circledast \frac {y}{x^2 + y^2}\right )\!, \end{equation}
where
$\circledast$
is the two-dimensional convolution operator with advection terms
\begin{align} g_x &= -\left (w_x\frac {\partial w_x}{\partial x} + w_y\frac {\partial w_x}{\partial y}\right )\!, \\[-12pt]\nonumber \end{align}
\begin{align} g_y &= -\left (w_x\frac {\partial w_y}{\partial x} + w_y\frac {\partial w_y}{\partial y}\right )\!. \end{align}
The velocity field is decomposed into background flow and actuator disk-induced component
$u=u_\infty + w_x$
and
$v=w_y$
. There is no free-stream flow
$v_\infty$
in the lateral direction. The velocity components are updated using the inviscid Euler equations for the nonlinear induced velocities. This system is solved by iteration and evaluated at
$P^{\textit{NL}}(x=x_0, y=0)$
which is written as
$p^{\textit{NL}}(C_T', \gamma , a_n, x_0)$
. The converged solution to the base-suction pressure in (A5) gives the
$p_{\textit{UMM}}(C_T', \gamma )$
term used to close the UBM.
(a) Blockage-induced pressure gradient and (b) pressure difference between bypass and streamtube flows.
Appendix B. Model sensitivity to hybrid pressure closure
The hybrid pressure closure presented in § 2.1 combines the blockage-induced pressure gradient with the additional pressure deficit resulting from the base-suction effect of highly loaded rotors. The blockage-induced pressure gradient in the bypass flow
$p_1 - p_4$
is plotted in figure 17(
$a$
) and the difference in pressure between the wake and bypass flows
$p_4 - p_{4,w}$
resulting from base-suction effects is plotted in figure 17(
$b$
). To eliminate sensitivity to the numerical set-up as described in Appendix D, the results are plotted with respect to
$C_T$
. The wake pressure deficit
$p_4 - p_{4,w}$
is found to decrease as the blockage ratio increases. This is consistent with the limiting case of
$\beta \to 1$
where the pressure difference must go to zero. Figure 17 shows that the linear pressure closure
$(1 - \beta )p_{\textit{UMM}}$
of the UBM provides a simple upper bound to the pressure difference and returns to the UMM closure in the unblocked case.
Next, we compare the proposed linear hybrid pressure closure used in § 2.1,
$p_{4,w} - p_4 = (1 - \beta )p_{\textit{UMM}}$
, with the two limiting cases in which the full base-suction term from the UMM is included,
$p_{4,w} - p_4 = p_{\textit{UMM}}$
, or the base-suction pressure is set to zero,
$p_{4,w} - p_4 = 0$
. The sensitivity of the model predictions for induction, thrust coefficient and power coefficient to the choice of hybrid pressure closure is shown in figure 18. This test generally demonstrates minimal sensitivity between the different closures. However, omitting the base-suction term entirely introduces significant predictive errors for the very low-blockage cases, as shown in figure 19. This is because in the low-blockage cases, the base-suction pressure term contributes significantly to the dynamics in the absence of a blockage-induced pressure gradient. Conversely, for confined flows, blockage-induced pressure gradients tend to dominate base-suction pressure gradients as shown in figure 17, rendering the interpolation method as a secondary impact for confined flows. Thus, including the hybrid closure is critical because it allows predictions in the low-blockage case (where base suction is a primary pressure effect) and in the higher-blockage cases (where the pressure effect from confinement tends to be more important than the base-suction pressure), though the specific form of the interpolation has minimal impact.
Sensitivity of (
$a$
) induction factor, (
$b$
) thrust coefficient and (
$c$
) power coefficient to pressure closure. The linear interpolation pressure closure,
$p_4 - p_{4,w} = (1- \beta )p_{ {UMM}}$
, is compared with the two limiting cases:
$p_4 - p_{4,w} = p_{ {UMM}}$
and
$p_4 - p_{4,w} = 0$
.
Percent error in (
$a$
) induction factor, (
$b$
) thrust coefficient and (
$c$
) power coefficient relative to LES for each hybrid pressure closure.
Appendix C. Local normalisation of blade-element equations
Here, we show that, under a specific set of assumptions, we expect the local thrust and power coefficients
$C_T'$
and
$C_{\!P}'$
to depend only on
$\lambda '$
,
$\theta _p$
and
$\gamma$
, and they are therefore independent of the blockage ratio
$\beta$
. For this analysis, we neglect tangential induction, assume
$a_n$
to be constant over the rotor, and assume the aerofoil lift
$C_l(\mu , \alpha )$
and drag
$C_d(\mu , \alpha )$
are independent of
$\beta$
. These assumptions constrain the inflow velocity at the rotor and the aerofoil performance.
We begin by normalising the axial and tangential relative velocities (equations (2.18) and (2.19)) by the rotor-normal disk velocity
$\boldsymbol{u}_d \boldsymbol{\cdot }\hat {n} = (1 - a_n) \cos (\gamma ) u_\infty$
\begin{align} \frac {v_n'(\mu ,\psi )}{u_\infty } & = \frac {v_n(\mu ,\psi )}{(1-a_n)\cos (\gamma )u_\infty } = 1, \\[-12pt]\nonumber \end{align}
\begin{align} \frac {v_t'(\mu ,\psi )}{u_\infty } &= \frac {v_t(\mu ,\psi )}{(1-a_n)\cos (\gamma )u_\infty }= \lambda ' \mu - \cos (\psi )\tan (\gamma ), \end{align}
which now depend only on
$\lambda '$
and
$\gamma$
, removing dependence on
$a_n$
and therefore also removing dependence on
$\beta$
. Using (2.20) and (2.21) the relative flow speed and angle become
\begin{align} \frac {w'(\mu , \psi )^2}{u_\infty ^2} &= \frac {w(\mu , \psi )^2}{(1 - a_n)^2 \cos ^2(\gamma ) u_\infty ^2} = 1 + (\lambda ' \mu - \cos (\psi )\tan (\gamma ))^2, \\[-12pt]\nonumber \end{align}
\begin{align} \phi &= \tan ^{-1} \left (\frac {1}{\lambda ' \mu - \cos (\psi )\tan (\gamma )}\right )\!. \end{align}
Finally, we can show that
$C_{n}$
and
$C_{\textit{tan}}$
no longer depend on
$a_n$
or
$\beta$
. Since
$\alpha (\mu ,\psi ) = \phi (\mu ,\psi ) - \theta _t(\mu ) - \theta _p$
and
$\phi$
is a function of
$\lambda '$
and
$\gamma$
only, then
$C_{n}(\mu , \alpha )$
and
$C_{\textit{tan}}(\mu , \alpha )$
depend only on
$\lambda '$
,
$\theta _p$
and
$\gamma$
.
Sensitivity of the LES results to the numerical set-up for (
$a$
) induction factor and (
$b,c$
) wake pressure deficit. The wake pressure deficit is plotted with respect to the local thrust coefficient in (
$b$
), and the resolved thrust coefficient in (
$c$
).
Using the above expressions, we write the rotor-averaged local thrust coefficient as
\begin{equation} \overline {C}'_T = \frac {1}{\pi } \int _0^1 \int _0^{2\pi } \mu \sigma C_{n} w'(\mu , \psi )^2 {\rm d}\mu {\rm d}\psi, \end{equation}
and rotor-averaged local power coefficient as
\begin{equation} \overline {C}'_P = \frac {1}{\pi } \int _0^1 \int _0^{2\pi } \mu ^2 \lambda ' \sigma C_{ tan} w'(\mu , \psi )^2 {\rm d}\mu {\rm d}\psi, \end{equation}
which no longer depend explicitly on
$a_n$
and therefore are also independent of
$\beta$
.
This analysis demonstrates that
$C_T'$
and
$C_{\!P}'$
depend only on
$\lambda '$
,
$\theta _p$
and
$\gamma$
, and are independent of both
$a_n$
and the blockage ratio
$\beta$
. Consequently,
$C_T'$
and
$C_{\!P}'$
are identical between flows with different blockage ratios when
$\lambda '$
,
$\theta _p$
and
$\gamma$
are matched, enabling the scaling proposed in § 2.3.
Appendix D. Sensitivity of large eddy simulations to numerical set-up
We assess the sensitivity of the LES results to the numerical set-up by conducting simulations while varying the ADM filter width, toggling the Shapiro correction factor on or off (Shapiro et al. Reference Shapiro, Gayme and Meneveau2019), and changing the resolution of the simulations. Additional simulations, which are coarsened to the resolution used in Heck et al. (Reference Heck, Johlas and Howland2023), are added to compare with the high resolution simulations presented in § 3.
Figure 20(
$a$
) shows the spread in rotor-normal induction resulting from varying the numerical set-up of the simulations. All combinations of wide/narrow filter width, ADM correction on/off and high/low resolution are shown with the exception of the high-resolution
$\beta = 0.005$
case which was not varied due to computational cost. Similarly, figure 20(
$b$
) indicates the variation in wake pressure deficit resulting from changing the numerical set-up of the simulations. This sensitivity is due to the direct effect of the numerical set-up on the wake development, as well as the indirect effect of the variation in induction that is triggered by different numerical parameters. In figure 20(
$c$
), we replot the wake pressure deficit with respect to
$C_T$
which eliminates the influence of the change in induction. By plotting the results with respect to the realised thrust force, the results collapse well. This collapse when plotted with respect to the resolved thrust coefficient indicates that the primary sensitivity comes from the change in induction due to the numerical set-up and that the relationship between the thrust coefficient and the induction factor, which is what is modelled in this study, is not sensitive to the numerical set-up. The influence of the ADM set-up on the induction factor has been studied extensively and is well understood (Shapiro et al. Reference Shapiro, Gayme and Meneveau2019; Heck et al. Reference Heck, Johlas and Howland2023). In this work, we use a narrow filter width which minimises the influence of the filtering on the induction of the ADM (Heck et al. Reference Heck, Johlas and Howland2023). While the LES results show some quantitative sensitivity to the numerical set-up, this does not change the qualitative conclusions of the study. This numerical sensitivity further motivates future experimental work to measure rotors in confinement and rotor-inflow misalignment across thrust coefficients in the Reynolds-number independent flow regime.
x
y
γ
A
u
v
p
p4,w
p4
n^
x
y
z
Lx
Ly
Lz
Nx
Ny
Nz
CT′
β
γ=0
CT′
γ=0
γ
β
CT′=2
γ=0
CT′=2
Tr
0⩽γ⩽40∘
2⩽CT′⩽10
β
βCTcos(γ)
Pr
λ′
CT′
CP′
β
β=0.005
β=0.1,
0.2
0.3
a
b)
β=0.2
β=0.1
β=0.01
c,d
β=0.2
β=0.01
e,f
β=0.2
β=0.1
β=0
a
b
β=0.20
β=0.03
β=0.35
β=0.02
a
b
λ′
a
b
a
b
c
p4−p4,w=(1−β)pUMM
p4−p4,w=pUMM
p4−p4,w=0
a
b
c
a
b,c
b
c
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