Axial Momentum Theory Research Articles

Circuit representation of ducted hydrokinetic turbines based on blade element momentum method

Hydrokinetic turbines are designed and deployed worldwide due to the vast scale of hydrokinetic energy existing in tidal streams, ocean currents, and riverine flow. Duct augmentation is a design feature potentially to improve power extraction, reduce maximum stresses, and reduce the cost of electricity. However, there is a lack of computationally efficient and accurate predictive models to evaluate and compare ducted and open turbines. This research proposes a lumped circuit representation model for ducted hydrokinetic turbines. The equivalent duct, rotor, and wake resistances are derived from the control volume analysis based on the axial momentum theory and blade element momentum method. The wake resistance, which represents the viscous mixing loss in the very far wake, is critical for turbine farm design. The duct resistances, which depend on the duct thrust, efficiency, and duct-rotor interactions are identified by computational fluid dynamics. Based on the proposed model, parametric investigations of distributed rotor thrust, torque, power, and wake loss for different duct and blade geometries are conducted. The model can facilitate preliminary parametric design, co-design, real-time control to enhance power extraction and dynamic stabilities, and turbine farm design. A design optimization case is demonstrated to illustrate the utility of the proposed model.

Induction study of a horizontal axis tidal turbine: Analytical models compared with experimental results

In a water channel, a scale horizontal axis tidal turbine is positioned in a low-disturbance uniform flow, and Particle Image Velocimetry (PIV) measurements are used to investigate turbulent-flow modifications in front of the turbine. The results confirm that even if the axial velocity deficit is mainly governed by the turbine rotational speed, a similar velocity profile is observed regardless of the rotational speed: the uniform flow evolves to a shear flow with a peak velocity deficit in front of the hub. The mean radial velocity component is not sensitive to the turbine rotational speed in front of the hub, whereas its amplitude increases near the tip of the blade as the Tip Speed Ratio increases. PIV database is also used to verify the validity of consolidated analytical induction models. Models based on the axial momentum theory, vortex method, and self-similar models exhibit some discrepancies with experimental measurements, especially in front of the hub. This paper proposes a turbine induction model that separately considers the influence of the hub and the rotating blades. The mean flow deficit in front of the operating turbine calculated using this new model is consistent with the experimental measurements.

Analysis and Validation of Glauert Rotor Design

The design of industrial wind turbine rotors is generally based on the blade-element/momentum (BEM) approach introduced by Glauert (1935). Essentially, the theory consists of combining a blade-element approach with axial momentum theory, and then introducing a tip correction to account for the finite number of rotor blades. This is required, as the momentum theory is based on representing the rotor by a disk, corresponding to a rotor with infinitely many blades. The optimum design properties are obtained by optimizing the local power coefficient at each blade element. This is typically accomplished by solving a simple analytical system of equations for the axial and tangential inductions factors, ignoring at first the tip correction. The tip correction is subsequently introduced to correct the design variables. In the present work we analyse the implications of including the tip correction directly in the optimization. As a result of the analysis, we find the somewhat surprising result that the maximum optimal interference factor is not 1/3, as normally encountered for optimum rotor design, but 2/5. In the paper we prove this analytically and use the theory to design optimum rotors, which subsequently are benchmarked by comparison to results from a numerical lifting line model.

Optimal Distribution of the Disk Load: Validity of the Betz–Joukowsky Limit

The popular axial momentum theory relies on the steady, incompressible, axisymmetric, and inviscid flow through a so-called actuator disk. The most important result of this theory is the famous Betz–Joukowsky limit, stating that the maximum power coefficient achievable by an open disk is limited to 16/27. Generally, this value is obtained assuming a priori that the disk is radially uniformly-loaded and the flow is axially one-dimensional. This, however, does not prove that the uniform type is the optimal load, or else that it returns the maximum value of the extracted power. For this reason, this paper preliminary shows that 16/27 is the exact value of the maximum power coefficient of a uniformly loaded disk, even if the flow is not assumed as one-dimensional. Then, it proves, using a calculus of variation approach, that the radially uniform load is optimal. The proof refers to an approximate classical local form of the axial momentum equation. Finally, the paper points out that, since the proof of the Betz–Joukowsky limit relies on this simplifying assumption, the exact evaluation of the optimal radial distribution of the disk load, leading to the maximum value of the power coefficient, is still an open question.

Component Velocities and Turbulence Intensities within Ship Twin-Propeller Jet Using CFD and ADV

This study presents the decays of three components of velocity for a ship twin-propeller jet associated with turbulence intensities using the Acoustic Doppler Velocimetry (ADV) measurement and computational fluid dynamics (CFD) methods. Previous research has shown that a single-propeller jet consists of a zone of flow establishment and a zone of established flow. Twin-propeller jets are more complex than single-propeller jets, and can be divided into zones with four peaks, two peaks, and one peak. The axial velocity distribution is the main contributor and can be predicted using the Gaussian normal distribution. The axial velocity decay is described by linear equations using the maximum axial velocity in the efflux plane. The tangential and radial velocity decays show linear and nonlinear distributions in different zones. The turbulence intensity increases locally in the critical position of the noninterference zone and the interference zone. The current research converts the axial momentum theory of a single propeller into twin-propeller jet theory with a series of equations used to predict the overall twin-propeller jet structure.

Predictions of Wake and Central Mixing Region of Double Horizontal Axis Tidal Turbine

Predicting the velocity distribution of double horizontal axis tidal turbines (DHATTs) is significant for the effective development of tidal streams. This current research gives an account on double turbine wake theory and flow structure of DHATT connected to single support by using the joint axial momentum theory and computational fluid dynamics (CFD) method. Characteristics of single turbine wake were previously studied with two theoretical equations predicting the initial upstream velocity closer to the turbine, and it's lateral distributions along the downstream of the turbine. This current works agreed with the previous wake equations, which was used for predicting the velocity region along the downstream of the turbines. Flow field separating the two turbines is complicated in nature due to the indirect disturbance of turbines and no report was found on this central region. The Central region in the downstream flow is initially suppressed due to the blockage effects with a high velocity close to the free stream. Lateral expansion of two turbine wakes penetrated the central region with velocity reduction and followed by the flow recovery further downstream. This work provides more understandings of the wake and its central mixing region for double turbines with a proposed theoretical model.

Optimal relationship between power and design-driving loads for wind turbine rotors using 1-D models

Abstract. We investigate the optimal relationship between the aerodynamic power, thrust loading and size of a wind turbine rotor when its design is constrained by a static aerodynamic load. Based on 1-D axial momentum theory, the captured power P̃ for a uniformly loaded rotor can be expressed in terms of the rotor radius R and the rotor thrust coefficient CT. Common types of static design-driving load constraints (DDLCs), e.g., limits on the permissible root-bending moment or tip deflection, may be generalized into a form that also depends on CT and R. The developed model is based on simple relations and makes explorations of overall parameters possible in the early stage of the rotor design process. Using these relationships to maximize P̃ subject to a DDLC shows that operating the rotor at the Betz limit (maximum CP) does not lead to the highest power capture. Rather, it is possible to improve performance with a larger rotor radius and lower CT without violating the DDLC. As an example, a rotor design driven by a tip-deflection constraint may achieve 1.9 % extra power capture P̃ compared to the baseline (Betz limit) rotor. This method is extended to the optimization of rotors with respect to annual energy production (AEP), in which the thrust characteristics CT(V) need to be determined together with R. This results in a much higher relative potential for improvement since the constraint limit can be met over a larger range of wind speeds. For example, a relative gain in AEP of +5.7 % is possible for a rotor design constrained by tip deflections, compared to a rotor designed for optimal CP. The optimal solution for AEP leads to a thrust curve with three distinct operational regimes and so-called thrust clipping.

Semi-empirical wake structure model of rotors using joint axial momentum theory and DES-SA method

CFD simulation with DES-SA turbulence model are conducted to investigate the wake velocity characteristics. The turbine wake velocity distribution of axial, tangential and radial component is analysed by comparing numerical results with previous the experimental and theoretical results. The axial velocity component is continuously recovered along the axial direction. Based on the position of minimum axial velocity, turbine wake can be divided into two zones: zone of flow establishment and zone of established flow. In the zone of flow establishment, two-dipped velocity valleys of the axial velocity distribution appear along the radial direction. These two valleys are combined to one valley at the rotation axis in the zone of established flow. The tangential velocity component is the second largest and the radial velocity component accounts for only a small proportion. Both of tangential and radial velocity decreases along the axial direction. Two velocity peaks of both tangential and radial velocity appear along the radial direction. Several empirical equations of turbine wake are proposed to describe the velocity distribution.

Extending Blade-Element Model to Contra-Rotating Configuration

Blade element model is one of the most popular models for rotating-wing analysis and design. Many legacy tools are based on blade-element and capable of analysing only a single-rotating configuration. The current scope suggests a simple and effective method for extending such applications to contra-rotating configurations. The model based on the classic axial-momentum theory and assumes rigid wake. The method is validated versus results from propeller, hovering rotor, and prop-fan experiments. The experiments present both single and dual-rotating wing configuration, which share the same blade geometry and test facilities. This enables the isolation of the single-versus dual-rotating configuration influences, thus validating the suggested model with high level-of-confidence. The validation shows that the same trends of the single-rotating model appears in the dual-rotating analyses. Moreover, the difference between single and dual rotating configuration is replicated accurately compared to the test results. This proves the validity of the suggested model extension. The model validation also exhibits the differences between the accuracy of the forward and rear rotating disk in the contra-rotating configuration. The rear disk contributes by the no-swirled wake and exhibits accurate results compared to the forward disk which operate similar to a single rotating wing, hence suffers from the same inaccuracies.

Filtered actuator disks: Theory and application to wind turbine models in large eddy simulation

SummaryThe actuator disk model (ADM) continues to be a popular wind turbine representation in large eddy simulations (LES) of large wind farms. Computational restrictions typically limit the number of grid points across the rotor of each actuator disk and require spatial filtering to smoothly distribute the applied force distribution on discrete grid points. At typical grid resolutions, simulations cannot capture all of the vorticity shed behind the disk and subsequently overpredict power by upwards of 10%. To correct these modeling errors, we propose a vortex cylinder model to quantify the shed vorticity when a filtered force distribution is applied at the actuator disk. This model is then used to derive a correction factor for numerical simulations that collapses the power curve for simulations at various filter widths and grid resolutions onto the curve obtained using axial momentum theory. The proposed correction, which is analytically derived from first principles, facilitates accurate power measurements in LES without resorting to highly refined numerical grids or empirical correction factors.

Tip-Bed Velocity and Scour Depth of Horizontal-Axis Tidal Turbine with Consideration of Tip Clearance

The scouring by a tidal turbine is investigated by using a joint theoretical and experimental approach in this work. The existence of a turbine obstructs a tidal flow to divert the flow passing through the narrow channel in between the blades and seabed. Flow suppression is the main cause behind inducing tidal turbine scouring, and its accelerated velocity is being termed as tip-bed velocity (Vtb). A theoretical equation is currently proposed to predict the tip-bed velocity based on the axial momentum theory and the conservation of mass. The proposed tip-bed velocity equation is a function of four variables of rotor radius (r), tip-bed clearance (C), efflux velocity (V0) and free flow velocity (V∞), and a constant of mass flow coefficient (Cm) of 0.25. An experimental apparatus was built to conduct the scour experiments. The results provide a better understanding of the scour mechanism of the horizontal axis tidal turbine-induced scour. The experimental results show that the scour depth is inversely proportional to tip-bed clearance. Turbine coefficient (Kt) is proposed based on the relationship between the tip-bed velocity and the experimental tidal turbine scour depth. Inclusion of turbine coefficient (Kt) into the existing pier scour equations can predict the maximum scour depth of a tidal turbine with an error range of 5–24%.

A ring-vortex actuator disk method for wind turbines including hub effects

Due to its robustness and simplicity, the simple actuator disk is still the most used and popular method for performance analysis of horizontal axis wind turbines. However, the hub blockage effect is generally disregarded in this approach, thus worsening the accuracy of its results. This is particularly true for small-sized wind turbines whose ratio between the hub and rotor radii can be as large as 25–30%. In order to obtain some insights onto the impact of the hub blockage-effect on the performance prediction, this paper presents a free-wake ring-vortex actuator disk method accounting for an axisymmetric hub of general shape. The fundamental concept of a uniformly loaded disk without wake rotation is adopted. In the proposed method, the flows induced by the disk and the hub are both modelled by ring sheet-vortices which are discretised through a classical panel method. An iterative solution procedure is developed to evaluate the density strength distribution of the sheets and the wake shape. To this aim, the homogeneous Dirichlet boundary condition is used for the velocity just beneath the hub sheet, while the free force condition is imposed all along the wake boundary. The latter is also required to be aligned with the overall flow field. The method is verified comparing its results with those of a Computational-Fluid-Dynamics-based approach, and of the classical axial momentum theory showing a very good agreement in both cases. Finally, some insights on the hub blockage effect are presented showing that, for a small-sized wind turbine, this effect significantly affect the flow distribution all along the blade span. In the analysed cases, the increment in the axial velocity at the disk, due to the hub blockage, goes from 20% in the hub proximity to 2% at the tip. The above values suggest that the hub blockage effect should never be disregarded in the analysis of small-sized wind turbines.

Verification of the Axial Momentum Theory for Propellers with a Uniform Load Distribution

The paper provides an evaluation of the errors embodied in the Axial Momentum Theory (AMT) as applied to a uniformly loaded actuator disk model without wake rotation. Although this model exhibits some unphysical features, such as the tip singularity and the violation of the angular momentum equation, it is still considered a touchstone in the theoretical aerodynamics of propellers. To simplify the model, a purely mathematical assumption is commonly used in the differential form of the axial momentum equation, i.e., the contribution of the pressure forces on the lateral surface of the infinitesimal streamtubes swallowed by the disk is neglected. In this paper, the errors introduced by this simplifying assumption are evaluated by comparing the results of the AMT with those of a nonlinear method modelling the free wake as the superposition of ring vortices distributed along the wake boundary. Firstly, the validity of this method is verified in terms of global performance coefficients. Then, using a CFD approach, it is also verified in terms of local flow quantities. The comparison between the ring-vortices method and the AMT shows that, for a highly loaded propeller, significant errors exist in the axial velocity at the disk, especially near the tip. Moreover, despite the uniform load, the axial velocity at the disk varies in the radial direction. Instead, the velocity magnitude remains almost uniform only for values of the thrust coefficient lower than 1.

Ship Twin-propeller Jet Model used to Predict the Initial Velocity and Velocity Distribution within Diffusing Jet

The current research proposed the theoretical model for ship twin-propeller jet based on the axial momentum theory and Gaussian normal distribution. The twin-propeller jet model is compared to the more matured single propeller jet model with good agreement. Computational Fluid Dynamics (CFD) method is used to acquire the velocity distribution within the twin-propeller jet for understanding of flow characteristics and validation purposes. Efflux velocity is the maximum velocity within the entire jet with strong influences by the geometrical profiles of the blades. Twin-propeller jet model showed four-peaked profile at the initial plane downstream to the propeller compared to the two-peaked profile from a single-propeller. The four-peaked profile merges to be twopeaked velocity profile and then one-peaked profile due to the fluid mixing. Entrainment occurs between the ambient still water outside and the rotating flow within jet due to the high velocity gradient. The research proposes a twin-propeller jet theory with a serial of equations enabling the predictions of velocity magnitude within the jet.

Theoretical vertical-axis tidal-current-turbine wake model using axial momentum theory with CFD corrections

The wake from a tidal current turbine has a significant impact on a tidal farm. A single turbine wake would affect the turbine located adjacent or downstream. Two equations are proposed to predict the mean velocity within the wake of a vertical-axis turbine. The first equation used to predict the efflux velocity is derived based on the axial momentum theory and dimensional analysis. Efflux velocity is the minimum velocity closest to the turbine downstream. The second equation used to predict the lateral velocity distribution is derived based on Gaussian probability distribution. The predictions are compared with the existing experimental and numerical results. Validation of the equations gives a variation in the range of 0–1.13% for the efflux velocity by comparing the proposed theoretical works and Dai and Lam’s experimental measurements. These equations are the foundation of the analytical method for wake prediction of a vertical-axis turbine.

Novel energy coefficient used to predict efflux velocity of tidal current turbine

The efflux velocity is the basis for the prediction of turbine wake. A novel energy coefficient is defined to propose a new theoretical equation to predict the efflux velocity of tidal current turbine in this paper. Several CFD cases with different tip speed ratio and solidity is conducted using the DES-SA model. In order to overcome the limitations of the axial momentum theory, the effects of tip speed ratio and solidity on the efflux velocity are studied and the energy coefficients with different tip speed ratio and solidity are determined using the proposed equation based on the CFD results. Several semi-empirical efflux velocity equations are finally proposed by fitting the equation of the energy coefficient with tip speed ratio. The application of these equations in the prediction of wake flow and the power calculation of tidal turbine are also introduced in this paper.

Performance analysis of ducted marine propellers. Part II – Accelerating duct

This paper completes the work presented in the companion paper [Bontempo et al., Appl. Ocean Res., 58 (2016) 322–330] by presenting the investigation of the flow around a propeller ducted with a so-called accelerating duct. To this aim, both the axial momentum theory and a nonlinear actuator disk method are used. The straightforward application of the first approach reveals that if the duct and rotor thrusts are concordant, then a beneficial effect on the propulsive efficiency can be readily obtained by enclosing a propeller in an accelerating duct. When the more advanced nonliner actuator disk method is applied to verify the outcomes of the axial momentum theory additional information on the performance of the device are obtained. Moreover, the nonlinear actuator disk method is also employed to investigate, through experimental design techniques, the effect of the key geometrical parameters of the duct onto the efficiency and robustness of this kind of propulsive system. In particular, it has been found that a propulsive efficiency gain can be achieved through a duct thickness, camber and chord increase, and through an incidence decrease.

The axial momentum theory as applied to wind turbines: some exact solutions of the flow through a rotor with radially variable load

The paper gives the exact solutions of the equations involved in the axial momentum theory for several kinds of radially variable load distributions. Specifically, a relation between the power and thrust coefficients is obtained for a load distribution of polynomial type well representing physically relevant loads. The paper also analyses simpler loads of the trapezoidal, triangular and linear type. The mass flow swallowed by the rotor is also evaluated for all cases. By providing a set of exact solutions of the axial momentum theory for rather complex load shapes, the study offers the unique opportunity to verify the correctness of existing or new numerical methods based on the actuator disk approach for other than the trivial, though widely used, uniform load solution.

Experimental study of flow field of an aerofoil shaped diffuser with a porous screen simulating the rotor

This study presents an experimental investigation on a diffuser augmented wind turbine (DAWT). A screen mesh is used to simulate the energy extraction mechanisms of a wind turbine in experiment. Different screen porosities corresponding to different turbine loading coefficients are tested. Measurements of the axial force and of the velocity distribution in radial direction are reported. The general purpose is to highlight the dependency between the diffuser and the screen, and to compare the radial velocity distributions in the diffuser between unloaded and loaded conditions. It is shown that the thrust on an unshrouded screen is lower than on a shrouded screen, under the same inflow condition. Moreover, the thrust on the diffuser largely depends on the screen loading. For the present configuration, the thrust on the screen with high loading coefficient contributes for more than 70% of the total thrust on the DAWT. Smoke visualizations and radial velocity profiles reveal that the high loading screen induces flow separation on the outer surface of the diffuser, justifying the results of the thrust measurements. It is also inferred that the flow separation leads to loss of thrust and has a great effect on the total pressure drag. It should be emphasized that the experimental results indicate that the flow field around the diffuser is strongly affected by the choice of screen porosity, that is, turbine loading. And that, the thrust coefficient of the diffuser does not show a linear dependence on the thrust coefficient of the screen. The axial momentum theory, therefore, is not a solid predictor for DAWT performance with high loaded screens.

Effects of the duct thrust on the performance of ducted wind turbines

This work investigates the performance of ducted wind turbines (DWTs) through the axial momentum theory (AMT) as well as through a semi-analytical approach. Although the AMT points out that the duct thrust plays a key role in the enhancement of the power extraction, it does not allow for the evaluation of the flow field around the duct. For this reason, a semi-analytical model is also used to investigate the local and global features of the flow through a DWT. In comparison to the AMT, the proposed semi-analytical method can properly evaluate the performance of the device for each prescribed rotor load distribution and duct geometry. Moreover, in comparison to other linearised methods, this approach fully takes into account the wake rotation and divergence, and the mutual interaction between the turbine and the shroud. The analysis shows the opportunity to significantly increase the power output by enclosing the turbine in a duct and that the growth in the duct thrust has a beneficial effect onto the device performance. Finally, some insights on the changes occurring to the performance coefficients with the rotor thrust and the duct camber are obtained through a close inspection of the local features of the flow field.