Actuator Disc Theory Research Articles

Solution of the flow over a non-uniform heavily loaded ducted actuator disk

AbstractThe paper presents an extension to ducted rotors of the nonlinear actuator disk theory of Conway (J. Fluid Mech., vol. 365, 1998, pp. 235–267) and it is exact for incompressible, axisymmetric and inviscid flows. The solution for the velocities and the Stokes stream function results from the superposition of ring vortices properly arranged along the duct surface and the wake region. Using a general analytical procedure the flow fields are given as a combination of one-dimensional integrals of expressions involving complete as well as incomplete elliptic integrals. The solution being exact, the proper shape of the slipstream whether converging or diverging is naturally accounted for, even for heavy loads. A semi-analytical method has been developed that enables the flow induced by an actuator disk housed in a contoured duct to be solved duly accounting for the nonlinear mutual interaction between the duct and the rotor. Non-uniform load distributions, rotor wake rotation and ducts of general shapes and thickness distribution can be dealt with. Thanks to its reduced numerical cost, the method is well suited for the design and/or analysis of ducted rotors for marine, wind and aeronautical applications.

Assessment of arrays of in-stream tidal turbines in the Bay of Fundy

Theories of in-stream turbines are adapted to analyse the potential electricity generation and impact of turbine arrays deployed in Minas Passage, Bay of Fundy. Linear momentum actuator disc theory (LMADT) is combined with a theory that calculates the flux through the passage to determine both the turbine power and the impact of rows of turbine fences. For realistically small blockage ratios, the theory predicts that extracting 2000-2500 MW of turbine power will result in a reduction in the flow of less than 5 per cent. The theory also suggests that there is little reason to tune the turbines if the blockage ratio remains small. A turbine array model is derived that extends LMADT by using the velocity field from a numerical simulation of the flow through Minas Passage and modelling the turbine wakes. The model calculates the resulting speed of the flow through and around a turbine array, allowing for the sequential positioning of turbines in regions of strongest flow. The model estimates that over 2000 MW of power is possible with only a 2.5 per cent reduction in the flow. If turbines are restricted to depths less than 50 m, the potential power generation is reduced substantially, down to 300 MW. For large turbine arrays, the blockage ratios remain small and the turbines can produce maximum power with a drag coefficient equal to the Betz-limit value.

Wind Turbines in Series; A Parametric Analysis

An analysis is made of wind turbines in a row by means of an extension to actuator disk theory and a representation of the turbulent diffusion in the wake by a velocity deficit scale and a single free parameter. Beyond this, no wake model is used. It is shown that when the thrust coefficient is ‘high’ a maximisation of overall power output leads to a large drop in power after the first turbine, followed by a fairly constant level and a rise at the end of the row; this behaviour is a natural consequence of optimisation, and on this basis a ‘deep array effect’ is to be expected. A variation of turbine size and the effect of impaired turbine performance are examined. The approach can also be used to calculate the turbine upstream velocity (with respect to a reference) from a distribution of measured power output and to make inferences about wake development. The approach could be useful in the assessment of wake models as well as turbine operation.

Evaluation of actuator disk theory for predicting indirect combustion noise

Indirect combustion noise is believed to be a key component of turbofan engine core noise, but existing experimental data have not been able to definitively determine its importance. Instead, actuator disk theory (ADT) as developed by Cumpsty and Marble [The interaction of entropy fluctuations with turbine blade rows; a mechanism of turbojet noise, Proceedings of the Royal Society of London A 357 (1977) 323–344] is commonly used to estimate its contribution based on combustor exit conditions and changes in the mean flow across blade rows. The theory, which assumes planar propagation of acoustic, entropic, and vortical waves in the long wavelength limit, is assessed by comparing its predictions to those from two-dimensional compressible Euler calculations of idealized entropy disturbances interacting with a 1980s era NASA turbine stator. Both low-frequency planar waves of constant frequency and higher-frequency, localized entropy disturbances are considered, with the former being within ADT's range of applicability and the latter outside of it. It is found that ADT performs well for the cut-on acoustic modes generated by the entropy-blade interaction but its accuracy suffers for the cut-off acoustic modes, which could impact indirect combustion noise predictions for turbines with closely spaced blade rows.

Sink fast and swim harder! Round-trip cost-of-transport for buoyant divers

Efficient locomotion between prey resources at depth and oxygen at the surface is crucial for breath-hold divers to maximize time spent in the foraging layer, and thereby net energy intake rates. The body density of divers, which changes with body condition, determines the apparent weight (buoyancy) of divers, which may affect round-trip cost-of-transport (COT) between the surface and depth. We evaluated alternative predictions from external-work and actuator-disc theory of how non-neutral buoyancy affects round-trip COT to depth, and the minimum COT speed for steady-state vertical transit. Not surprisingly, the models predict that one-way COT decreases (increases) when buoyancy aids (hinders) one-way transit. At extreme deviations from neutral buoyancy, gliding at terminal velocity is the minimum COT strategy in the direction aided by buoyancy. In the transit direction hindered by buoyancy, the external-work model predicted that minimum COT speeds would not change at greater deviations from neutral buoyancy, but minimum COT speeds were predicted to increase under the actuator disc model. As previously documented for grey seals, we found that vertical transit rates of 36 elephant seals increased in both directions as body density deviated from neutral buoyancy, indicating that actuator disc theory may more closely predict the power requirements of divers affected by gravity than an external work model. For both models, minor deviations from neutral buoyancy did not affect minimum COT speed or round-trip COT itself. However, at body-density extremes, both models predict that savings in the aided direction do not fully offset the increased COT imposed by the greater thrusting required in the hindered direction.

Effects of aerodynamic excitations on the dynamic behavior of helical gear system

PurposeThe purpose of this paper is to study the effects of aerodynamic excitations on the dynamic behavior of a helical two‐stage gear system.Design/methodology/approachThe methodology consists of developing a wind turbine model including a helical two‐stage gear train having 21 DOFs.FindingsThe results of this paper show that the dynamic behavior of the speed‐up gearbox inside the wind turbine is affected by various degrees of flexibility at different frequencies.Originality/valueThe aerodynamic forces are calculated by two main methods, the first is the actuator disc theory and the second is the Blade‐Element Method. Some correction functions are applied, such as the tip‐root loss functions and the Glauert correction factor. The convergence of the induction factors permits to increase the precision of predictions. Finally, the generator side is modeled by a simplified electric schematic based on steady state model.

An Evaluation for Predicting the Far Wake of Tidal Turbines Positioned in Array at Different Longitudinal Spaces

A study on tidal turbine using CFD simulation has been an economical and reliable method. However, large flow fields with multi-turbine arrays require high computer performance. Actuator disc theory therefore is widely applied. Actuator disc is the concept that imitates actual turbine by means of an energy absorption disc which has the same dimension and characteristics. Turbines installed in array may have disturbance effects on one another. Thus, the subject of this study is to analyze the far wake of these tidal turbines and compare to single turbine case. The main objects are to analyze two turbines positioned longitudinally at different spaces.

Acoustic–convective mode conversion in an aerofoil cascade

When an acoustic wave impinges on an aerofoil cascade, a convective vorticity mode is generated giving rise to transverse velocity perturbations. This mode conversion process is investigated to explain the flow dynamics observed when swirlers are submitted to incident acoustic disturbances. The phenomenon is first studied in the case of a two-dimensional aerofoil cascade using a model derived from an actuator disk theory. The model is simplified to deal with low-Mach-number flows. The velocity field on the downstream side of the cascade features two components, an axial perturbation associated with the transmitted acoustic wave and a transverse disturbance corresponding to the vorticity wave generated at the cascade trailing edge. The model provides the amplitude of both components and defines their phase shift. Numerical simulations are carried out in a second stage to validate this model in the case of a cascade operating at a low Reynolds number Rec = 2700 based on the chord length. Space–time diagrams of velocity perturbations deduced from these simulations are used to retrieve the two types of modes. Experiments are then carried out in the case of an axial swirler placed in a cylindrical duct and submitted to plane acoustic waves emitted on the upstream side of the swirler. The amplitude and phase of the two velocity components measured in the axial and azimuthal directions are found to be in good agreement with theoretical estimates and with numerical calculations. This analysis is motivated by combustion dynamics observed in flames stabilized by aerodynamic swirlers in continuous combustors.

Climbing flight performance and load carrying in lesser dog-faced fruit bats (Cynopterus brachyotis)

The metabolic cost of flight increases with mass, so animals that fly tend to exhibit morphological traits that reduce body weight. However, all flying animals must sometimes fly while carrying loads. Load carrying is especially relevant for bats, which experience nightly and seasonal fluctuations in body mass of 40% or more. In this study, we examined how the climbing flight performance of fruit bats (Cynopterus brachyotis; N=4) was affected by added loads. The body weights of animals were experimentally increased by 0, 7, 14 or 21% by means of intra-peritoneal injections of saline solution, and flights were recorded as animals flew upwards in a small enclosure. Using a model based on actuator disk theory, we estimated the mechanical power expended by the bats as they flew and separated that cost into different components, including the estimated costs of hovering, climbing and increasing kinetic energy. We found that even our most heavily loaded bats were capable of upward flight, but as the magnitude of the load increased, flight performance diminished. Although the cost of flight increased with loading, bats did not vary total induced power across loading treatment. This resulted in a diminished vertical velocity and thus shallower climbing angle with increased loads. Among trials there was considerable variation in power production, and those with greater power production tended to exhibit higher wingbeat frequencies and lower wing stroke amplitudes than trials with lower power production. Changes in stroke plane angle, downstroke wingtip velocity and wing extension did not correlate significantly with changes in power output. We thus observed the manner in which bats modulated power output through changes in kinematics and conclude that the bats in our study did not respond to increases in loading with increased power output because their typical kinematics already resulted in sufficient aerodynamic power to accommodate even a 21% increase in body weight.

Modelling tidal energy extraction in a depth-averaged coastal domain

An extension of actuator disc theory is used to describe the properties of a tidal energy device, or row of tidal energy devices, within a depth-averaged numerical model. This approach allows a direct link to be made between an actual tidal device and its equivalent momentum sink in a depth-averaged domain. Extended actuator disc theory also leads to a measure of efficiency for an energy device in a tidal stream of finite Froude number, where efficiency is defined as the ratio of power extracted by one or more tidal devices to the total power removed from the tidal stream. To demonstrate the use of actuator disc theory in a depth-averaged model, tidal flow in a simple channel is approximated using the shallow water equations and the results are compared with the published analytical solutions.

Applications of Actuator Disk Theory to Membrane Flapping Wings

This study addresses the aerodynamics of elastic membrane flapping wings. Several applications of the actuator disk theory to the flapping wings of insects and birds are reviewed. In previous studies, to account for spatial and temporal variance in the wake behind the flapping wings, empirical corrections were proposed for the induced velocity and power. In the present paper, a new procedure for determination of the correction factor is proposed, using membrane-type flapping-wing devices. Wind-tunnel experiments were conducted and the stroke-averaged propulsive thrust was measured on 25-cm-wingspan (flat and 9% camber) and 74-cm-wingspan flapping-wing models. Either flapping frequency or input power was held constant during the tests. Obtained thrust forces were compared to theoretical values predicted by the actuator disk theory. Empirical correction factors to the actuator disk theory were determined, providing a best fit to the experimental data when the flapping axis aligned with freestream velocity. It is noteworthy that the numerical value for the correction factor for the 25 cm cambered wing agrees with the results obtained on large insects. The theoretical corrections for angle of attack of the flapping wing give satisfactory agreements with the experimental data only for relatively low forward speeds.

Robustly Optimal Wind Turbine for Swirl Expansion and Decay

The wind turbine optima of propeller actuator disc theories are problematic. The large optimal windmill interference doubles the wake area from the vortex cylinder used to justify the propeller blade element momentum (BEM) theory. The general momentum (GM) theory incorporates expansion, but its optimum windrotor power blows up below a tip speed ratio of about unity where the BEM and experimental optima are low. Even without expansion, the GM and vortex theory optimum can still affect infinitely more gross power flux than in undisturbed wind. The inner massive swirl of their optimal wake is unstable or unrealizable by many indicators. Rather than a cylindrical wake, the BEM is better founded on negligible suction from low swirl in a predominately axial wake expanded after instability and bursting of the hub vorticity outwards to cancel the axial component of the tip vorticity. If moderate swirl at outer high speed ratios can expand first with pressure regain, the optimum local power refines up to 5 per cent higher. Then only as this expanded outer axial vorticity is finally cancelled to relieve the suction from the swirl does the inner BEM flow slow and expand. The inner and outer axial velocities are equal with no radial velocity at local speed ratio 0.57 in the optimal rotor. Direct experimental verification of this dissipated expanded swirl head (DESH) refinement of the BEM is proposed further to support from measurements and computations in the literature. The design blade chord and pitch that make the DESH optimum robust to small changes in windspeed are little changed from the time-proven BEM design, whereas the GM robust chord has a singularity.

Experimental investigation of the effect of chordwise flexibility on the aerodynamics of flapping wings in hovering flight

Ornithopters or mechanical birds produce aerodynamic lift and thrust through the flapping motion of their wings. Here, we use an experimental apparatus to investigate the effects of a wing's twisting stiffness on the generated thrust force and the power required at different flapping frequencies. A flapping wing system and an experimental set-up were designed to measure the unsteady aerodynamic and inertial forces, power usage and angular speed of the flapping wing motion. A data acquisition system was set-up to record important data with the appropriate sampling frequency. The aerodynamic performance of the vehicle under hovering (i.e., no wind) conditions was investigated. The lift and thrust that were produced were measured for different flapping frequencies and for various wings with different chordwise flexibilities. The results show the manner in which the elastic deformation and inertial flapping forces affect the dynamical behavior of the wing. It is shown that the generalization of the actuator disk theory is, at most, only valid for rigid wings, and for flexible wings, the power P varies by a power of about 1.0 of the thrust T. This aerodynamic information can also be used as benchmark data for unsteady flow solvers.

A free-surface and blockage correction for tidal turbines

The effects of free-surface proximity on the flow field around tidal stream turbines are modelled using actuator disc theory. Theoretical results are presented for a blocked configuration of tidal stream turbines such as a linear array that account for the proximity of the free surface and the seabed. The theoretical results are compared to open channel flow experimental results in which the flow field has been simulated using a porous disc and strip. These results are complemented by more detailed measurements of the performance of a model horizontal-axis turbine carried out in a water flume and a wind tunnel. The two sets of experiments represent highly blocked and effectively unblocked cases, respectively. The theoretical model of the effects of free-surface proximity provides a blockage correction for axial induction that can be incorporated in blade element momentum codes. The performance predictions obtained with such a code are in good agreement with the experimental results for CP and CT at low tip-speed ratios. The agreement weakens with increasing tip-speed ratio, as the wake of turbine enters a reversed flow state. A correction following the philosophy of Maskell is applied to CT in this region, which provides a better agreement.

NPSHr Optimization of Axial-Flow Pumps

The two-step method for optimizing net positive suction head required (NPSHr) of axial-flow pumps is proposed in this paper. First, the NPSHr at the impeller tip is optimized with impeller diameter based on experimental data of 2D cascades in available wind tunnels. Then, it is optimized again with the velocity moment at the impeller outlet, which is expressed in terms of two parameters. The blade geometry is generated and flow details are clarified by using the radial equilibrium equation, actuator disk theory, and 2D vortex element method in the optimizing process. The NPSHr response surface has been established in terms of these two parameters. The results illustrate that the second optimization allows NPSHr to be reduced by 37.5% compared to the first optimization. Therefore, this two-step method is effective and expects to be applied in the axial-flow pump impeller blade design. The simulations of 3D turbulent flow with various cavitation models and related confirming experiments are going to be done in the axial-flow impellers designed with this method.

Inverse design-momentum, a method for the preliminary design of horizontal axis wind turbines

Wind turbine rotor prediction methods based on generalized momentum theory BEM routinely used in industry and vortex wake methods demand the use of airfoil tabulated data and geometrical specifications such as the blade spanwise chord distribution. They belong to the category of "direct design" methods. When, on the other hand, the geometry is deduced from some design objective, we refer to "inverse design" methods. This paper presents a method for the preliminary design of wind turbine rotors based on an inverse design approach. For this purpose, a generalized theory was developed without using classical tools such as BEM. Instead, it uses a simplified meridional flow analysis of axial turbomachines and is based on the assumption that knowing the vortex distribution and appropriate boundary conditions is tantamount to knowing the velocity distribution. The simple conservation properties of the vortex components consistently cope with the forces and specific work exchange expressions through the rotor. The method allows for rotor arbitrarily radial load distribution and includes the wake rotation and expansion. Radial pressure gradient is considered in the wake.The capability of the model is demonstrated first by a comparison with the classical actuator disk theory in investigating the consistency of the flow field, then the model is used to predict the blade planform of a commercial wind turbine. Based on these validations, the authors postulate the use of a different vortex distribution (i.e. not-uniform loading) for blade design and discuss the effect of such choices on blade chord and twist, force distribution and power coefficient. In addition to the method's straightforward application to the pre-design phase, the model clearly shows the link between blade geometry and performance allowing quick preliminary evaluation of non uniform loading on blade structural characteristics.

Recent advances in modeling of wind turbine wake vortical structure using a differential actuator disk theory

This paper presents the recent developments of a new CFD-based method aimed at predicting wind turbine aerodynamics, where velocity and pressure discontinuities are used to model the vortical system that creates lift on the turbine blades. To illustrate the ability of the present model to predict induced wake effect, the case of the taper wing is thoroughly analyzed and effects of both domain discretization and convection scheme are presented. Results are mitigated regarding predicted performance of induced drag, but accurate induced and upstream flow angles values are obtained. The method is even shown to be a useful calculator for the relationship between inflow angle measured upstream and effective angle of attack of a wing section. Interesting results for the NREL phase VI rotor have been obtained showing improvement of the method upon actuator-disk approach in handling tip vortices effect on the blade aerodynamics.

Time-Area-Averaged Momentum Stream Tube Model for Flapping Flight

This paper formulates a time-area-averaged momentum stream tube model to provide the useful functional relations for the cruise velocity, power, and propulsive efficiency in forward flapping flight. This model is a direct generalization of the classical actuator disk theory by taking into account the effects of the unsteadiness and spatial nonuniformity of velocity in a momentum stream tube. It is found that the functional relation for the time-areaaveraged power required in flapping flight is almost the same as that given by the classical lifting-line theory for a fixed wing. The flapping span efficiency is introduced, and its physical meaning and significant role in the power relationareinterpreted.The flappingpropulsiveefficiencyisrelatedtothe flappingspanefficiency,normalizedtotal fluctuating kinetic energy, and wing aspect ratio. The use of this model to fit the collected data for birds allows estimating the flapping span efficiency, parasite, and induced drag coefficients and propulsive efficiency. Nomenclature A = effective area of actuator disk or cross-section area of momentum stream tube Ab = circular area of a diameter of wingspan AR = wing aspect ratio b = wingspan CDPara = parasite drag coefficient

Wind Energy Conversion Efficiency Limit

The momentum analysis of Sharpe [3] is considered further. It is shown that the maximum wind energy conversion efficiency limit, or the Betz limit of 59.3%, deduced from conventional actuator disc theory is the limiting stationary maximum value as wind turbine rotor speed approaches infinity and rotor outlet swirl approaches zero. Theoretically, solution of the governing equations indicates 100% conversion efficiency could be possible with suitable wind turbine design and without wake limitations for all tip-speed ratios. Analogous to the thermodynamics laws for heat engines and similar to a Pelton wheel with hydro power, the wind turbine efficiency is not limited by the laws of fluid mechanics but by physical constraints only. Low tip-speed ratio would have greater prospects in achieving higher conversion efficiency in practice. Though the set of equations derived in the present technical note has assumed uniform bound circulation along the blade span, the same approach can be used for determination of optimum conditions for other blade designs or using other assumptions.

1809 アクチュエータディスク理論と数値計算を組み合わせた縦軸風車の性能予測(OS18-3 自然エネルギー資源の有効利用技術(3),OS18 自然エネルギー資源の有効利用技術,オーガナイズドセッション)

1809 アクチュエータディスク理論と数値計算を組み合わせた縦軸風車の性能予測(OS18-3 自然エネルギー資源の有効利用技術(3),OS18 自然エネルギー資源の有効利用技術,オーガナイズドセッション)