Disk Theory Research Articles

The transitional states of a floating wind turbine during high levels of surge

With the deployment of floating offshore wind turbines, the effect of their wind- and wave-induced platform motion on the turbine’s performance and life is of concern and requires low-order models to develop appropriate control strategies. This work focusses on platform surge as one of the main additional degrees of freedom. The aerodynamic behaviour during the rotor’s transition into propeller state is explored by assessing the spanwise and rotor-integrated aerodynamic forces on the rotor of the NREL 5 MW turbine through a 3D URANS CFD simulation. The CFD findings are compared with a modified actuator disk (AD) theory and 2D Blade-Element Momentum (BEM) approach. In addition to propeller state, where both rotor torque and thrust are negative, two other states were observed with opposite signs of torque and thrust: a braking state, with negative torque but positive thrust, and the second a quasi-windmill state with thrust negative yet torque positive. The results demonstrate that BEM coupled with AD can reliably predict the transitions to these states during such high levels of platform surge and give hints to how the underpinning force balances are affected by temporal perturbations.

The high-energy probability distribution of accretion disc luminosity fluctuations

ABSTRACT The probability density function of accretion disc luminosity fluctuations at high observed energies (i.e. energies larger than the peak temperature scale of the disc) is derived, under the assumption that the temperature fluctuations are lognormally distributed. Thin disc theory is used throughout. While lognormal temperature fluctuations would imply that the disc’s bolometric luminosity is also lognormal, the observed Wien-like luminosity behaves very differently. For example, in contrast to a lognormal distribution, the standard deviation of the derived distribution is not linearly proportional to its mean. This means that these systems do not follow a linear rms-flux relationship. Instead they exhibit very high intrinsic variance, and undergo what amounts to a phase transition, in which the mode of the distribution (in the statistical sense) ceases to exist, even for physically reasonable values of the underlying temperature variance. The moments of this distribution are derived using asymptotic expansion techniques. A result that is important for interpreting observations is that the theory predicts that the fractional variability of these disc systems should increase as the observed frequency is increased. The derived distribution will be of practical utility in quantitatively understanding the variability of disc systems observed at energies above their peak temperature scale, including X-ray observations of tidal disruption events.

Numerical model of a tidal current acceleration structure

Advancements in technology have led to a rapid rise in the use of renewable energy sources in the past 25 years. The current work focuses on the potential of a novel hydraulic technology to contribute toward sustainable energy production. The tidal current acceleration structure is a simple structure that uses the basic principle of the Venturi effect in low-speed tides and rivers to accelerate the flow and, in turn, extract energy using turbines. The primary aim of the present study is to understand to what extent this newly proposed tidal flow structure is suitable for real applications. The shear stress transport k–ω model was utilized, and the parametric analysis based on angle variation, Froude number, and bed roughness was undertaken to optimize the performance of the structure. The potential power that could be extracted by an in-stream turbine was then estimated using actuator disk theory. The performance of the structure significantly increased for the configuration having the ratio of the opening and contraction width of 2.66.

Free Surface Double Actuator Disc Theory and Double Multiple Streamtube model for in-stream Darrieus hydrokinetic turbines

According to the literature, the last improvement to the Actuator Disc Theory for hydrokinetic turbines in channel flows included blockage ratio and Froude number as independent variables in a single-disc formulation. The authors attempted to solve these equations in a Blade Element Momentum model for axial-flow turbines, ensuring satisfactory results.To our best knowledge, no Blade Element Momentum code was developed for Darrieus rotors including an extended double-disc theory so far. In this perspective, a Free Surface double-disc LMADT is implemented to solve the governing equations in a Double Multiple Streamtube model, facilitating the assessment of In-Stream Darrieus hydrokinetic turbines. Rigid Lid equations, derived for higher-depth applications, are included for completeness. A further numerical procedure allows to set up the channel flow from downstream and update the inflow data due to the resistance induced by the turbine operation in a subcritical flow.In addition to corrections for flow curvature, shading of the downstream flow due to the shaft, and losses due to the mechanical struts, this work encloses an effective sub-model accounting for the turbine installation depth. The Double Multiple Streamtube model is finally validated with previous experimental data of a Darrieus turbine deployed in a narrow channel.

Conceptual Errors in Actuator Disc Theory and Betz’s Law for Wind Turbines

This paper started with the explanation of the conditions for using the momentum equation and with the presentation of the actuator disc theory. Focusing on the flow model used in actuator disk theory, both the Froude-Rankine theorem and Betz’s law have been examined. It has been found that the Froude-Rankine theorem is not justified because a stream-tube that is used as the control volume does not really exist (pseudo stream-tube). The theorem is also not justified because an unfounded velocity (v2) is used to connect the thrust of the actuator disc with the total power loss. Two flaws have been identified in Betz’s law. First, the use of both the unjustified Froude-Rankine theorem and the incorrect flow model totally violates the condition of determining the thrust of the actuator disc using the momentum equation. Second, the unfounded velocity (v2) from the Froude-Rankine theorem is misinterpreted and used for the volumetric flow rate through the actuator disc. These two main flaws lead to diverse computational contradictions and paradoxes, particularly when considering the case of an impermeable circular disc. The flaws in Betz’s law become evident when the law is applied to a rectangular actuator plate of infinite length. The possible solution for the actuator disc flow has been presented. This includes the additional consideration of energy dissipation in the flow downstream of the actuator disc, similar to the method used to calculate the Borda-Carnot shock loss.

Experimental analysis of the effect of dynamic induction control on a wind turbine wake

Abstract. Dynamic induction control (DIC) has proven to be an effective method of increasing the power output for a wind farm in both simulation studies and wind tunnel experiments. By pitching the blades of a wind turbine periodically, the recovery of the low-velocity wake is accelerated, thereby increasing the energy available to downstream turbines. The wake itself of a turbine operating with DIC has not yet been studied experimentally. This paper presents a wind tunnel experiment where the wake of a wind turbine under periodic excitation is investigated. Using three-dimensional particle image velocimetry, the velocity field behind the turbine was reconstructed. Analysis of the velocity fields indicated that the available power in the wake increases when using DIC. This increase was partially due to a lower average thrust force experienced by the turbine with DIC. However, a large difference was seen between measurement results and actuator disk theory, indicating enhanced recovery of the wake is contributing to the increased energy. Instantaneous measurements visualizing the development of blade tip vortices also showed how the location of vortex breakdown, which is directly related to re-energizing the wake, shifts over time with DIC. We believe this shifting location is contributing to the enhanced wake recovery of DIC, providing more energy to downstream wind turbines.

Analytical Model for Ring-Wing Propulsors at Angle of Attack

A simple, accurate, analytical model is developed for predicting the forces and moments on subsonic ring-wing propulsors (aka fan ducts and shrouded propellers) at angle of attack. The method is applicable to drones, aircraft, ships, and more from hover to cruise. The new model is made up from elements of three previously published, closely related models, herein combined and modified for the current purpose: an empty shroud ring-wing force and moment linear aerodynamics model; a ring-wing propulsor axial force model employing linear aerodynamic, axial momentum, and propeller actuator disk theories; and a jet engine normal force model as semi-empirically adapted to ring-wing propulsors. The model’s efficacy is assessed using three independent datasets: two experimental and one Reynolds-averaged Navier–Stokes computational fluid dynamics. It is shown that the model’s algebraic solutions provide good engineering estimates for the lift, drag, normal, and axial forces as well as pitching moments at all operating conditions from hover to cruise and angles of attack up to 85°.

Continuum reverberation mapping of the quasar PG 2130+099

Aims. We present the results of an intensive six-month optical continuum reverberation mapping campaign of the Seyfert 1 galaxy PG 2130+099 at redshift z = 0.063. The ground-based photometric monitoring was conducted on a daily basis with the robotic 46 cm telescope of the WISE observatory located in Israel. Specially designed narrowband filters were used to observe the central engine of the active galactic nucleus (AGN), avoiding line contamination from the broad-line region (BLR). We aim to measure inter-band continuum time lags across the optical range and determine the size-wavelength relation for this system. Methods. We used two methods, the traditional point-spread function photometry and the recently developed proper image subtraction technique, to independently perform the extraction of the continuum light curves. The inter-band time lags are measured with several methods, including the interpolated cross-correlation function, the z-transformed discrete correlation function, a von Neumann estimator, JAVELIN (in spectroscopic mode), and MICA. Results. PG 2130+099 displays correlated variability across the optical range, and we successfully detect significant time lags of up to ∼3 days between the multiband light curves. We find that the wavelength-dependent lags, τ(λ), generally follow the relation τ(λ)∝λ4/3, as expected for the temperature radial profile T ∝ R−3/4 of an optically thick, geometrically thin accretion disk. Despite that, the derived time lags can also be fitted by τ(λ)∝λ2, implying the possibility of a slim, rather than thin, accretion disk. Using the flux variation gradient method, we determined the AGN’s host-galaxy-subtracted rest frame 5100 Å luminosity at the time of our monitoring campaign with an uncertainty of ∼18% (λL5100 = (2.40 ± 0.42)×1044 erg s−1). While a continuum reprocessing model can fit the data reasonably well, our derived disk sizes are a factor of ∼2 − 6 larger than the theoretical disk sizes predicted from the AGN luminosity estimate of PG 2130+099. This result is in agreement with previous studies of AGN/quasars and suggests that the standard Shakura-Sunyaev disk theory has limitations in describing AGN accretion disks.

AGN light echoes and the accretion disc self-gravity limit

ABSTRACT Accretion disc theory predicts that an AGN disc becomes self-gravitating and breaks up into stars at an outer radius Rsg ≃ 12 light-days, with effectively no free parameter. We present evidence that the longest observed AGN light echoes are all close to 12 d in the AGN rest frames. These observations give a stringent test of the AGN disc theory. Further monitoring should offer insight into the formation angular momentum of the gas forming the disc. For distant AGN, observed lags significantly longer than 12 d give lower limits on their redshifts.

Fifty Years of Eclipsing Binary Analysis with the Wilson–Devinney Model

The Wilson–Devinney model has—over the last 50 years—become the standard in analyzing eclipsing binary observations. To provide orientation for both active binary and non-binary researchers, it is presented here in historical and on-going as well as astrophysical perspectives. Among the important advances that originated with the model are: the representation of star surfaces as equipotentials for circular and eccentric orbits, leading to four morphological types; simultaneous least-squares light and velocity curve analyses; efficient reflection computation, including multiple reflection; disk theory and disk modeling. Solutions in physical units allowed for the accurate estimation of parameters such as stellar masses and photometric distances; inclusion of types of observables, properly weighted.

The comparison of power capability between towed kite and horizontal axis fan

Airborne Wind Energy (AWE) mainly collects wind energy by tethered aircraft at a certain altitude. This paper discusses the recent development of AWE. The actuator disc theory is adopted to consider the influence of kites on wind flow obstruction. The difference of working mode between a horizontal-axis wind turbine (HAWT) and a crosswind kite power systems (CKPS) is compared, the power limit of HAWT and CKPS is calculated, and the reason of the limit power is analyzed. It is pointed out that CKPS has a wider range of flight and should be further analyzed and calculated rather than simply generalized by the system disk theory.

Bridging the Communication Gap Using DISC Theory of Leadership in the Hybrid Work Environment

The recent changes in work environment owing to the COVID 19 pandemic has made us rethink the utility of office space at a time when an exponential growth in the use of online platforms has been observed. With the flexibility of working space and timing, employees and the organizational heads have redefined work culture and the old adage of strict 9 to 5 job seems to have been completed changed. The current paper attempts to understand the four behavioral patterns of DISC exhibited by leaders in any work environment. It explains what are the four behavioral traits and how they influence communication. And in a hybrid work environment, with lot of flexibility in time and space, what steps are to be taken in order to make communication effective and avert miscommunication as half of the employees work offline while the other half works online.

Fractal-like actuator disc theory for optimal energy extraction

The limit of power extraction by a device which makes use of constructive interference, i.e. local blockage, is investigated theoretically. The device is modelled using actuator disc theory in which we allow the device to be split into arrays and these then into sub-arrays an arbitrary number of times so as to construct an $n$ -level multi-scale device in which the original device undergoes $n-1$ sub-divisions. The alternative physical interpretation of the problem is a planar system of arrayed turbines in which groups of turbines are homogeneously arrayed at the smallest $n\mathrm {th}$ scale, and then these groups are homogeneously spaced relative to each other at the next smallest $n-1\mathrm {th}$ scale, with this pattern repeating at all subsequent larger scales. The scale-separation idea of Nishino & Willden (J. Fluid. Mech., vol. 708, 2012b, pp. 596–606) is employed, which assumes mixing within a sub-array occurs faster than mixing of the by-pass flow around that sub-array, so that in the $n$ -scale device mixing occurs from the inner scale to the outermost scale in that order. We investigate the behaviour of an arbitrary level multi-scale device, and determine the arrangement of actuator discs ( $n\mathrm {th}$ level devices) which maximises the power coefficient (ratio of power extracted to undisturbed kinetic energy flux through the net disc frontal area). We find that this optimal arrangement is close to fractal, and fractal arrangements give similar results. With the device placed in an infinitely wide channel, i.e. zero global blockage, we find that the optimum power coefficient tends to unity as the number of device scales tends to infinity, a 27/16 increase over the Lanchester–Betz limit of $0.593$ . For devices in finite width channels, i.e. non-zero global blockage, similar observations can be made with further uplift in the maximum power coefficient. We discuss the fluid mechanics of this energy extraction process and examine the scale distribution of thrust and wake velocity coefficients. Numerical demonstration of performance uplift due to multi-scale dynamics is also provided. We demonstrate that bypass flow remixing and ensuing energy losses increase the device power coefficient above the limits for single devices, so that although the power coefficient can be made to increase, this is at the expense of the overall efficiency of energy extraction which decreases as wake-scale remixing losses necessarily rise. For multi-scale devices in finite overall blockage two effects act to increase extractable power; an overall streamwise pressure gradient associated with finite blockage, and wake pressure recoveries associated with bypass-scale remixing.

Analytical and Computational Fluid Dynamics Models of Wells Turbines for Oscillating Water Column Systems

AbstractThe sea is an important renewable energy resource for its extension and the power conveyed by waves, currents, tides, and thermal gradients. Amongst these physical phenomena, sea waves are the source with the highest energy density and may contribute to fulfilling the global increase of power demand. Despite the potential of sea waves, their harnessing is still a technological challenge. Oscillating water column systems operating with Wells turbines represent one of the most straightforward and reliable solutions for the optimal exploitation of this resource. An analytical model and computational fluid dynamics models were developed to evaluate the functioning of monoplane isolated Wells turbines. For the former modeling typology, a blade element momentum code relying on the actuator disk theory was applied, considering the rotor as a set of airfoils. For the latter modeling typology, a three-dimensional multi-block technique was implemented to create the computational domain with a fully mapped mesh composed of hexahedral elements. The employment of circumferential periodic boundary conditions allowed for the reduction of computational power and time. The models use Reynolds-averaged Navier-Stokes (RANS) or u-RANS schemes with a multiple reference frame approach or the u-RANS formulation with a sliding mesh approach. The achieved results were compared with analytical and experimental literature data for validation. All the developed models showed good agreement. The analytical model is suitable for a fast prediction of the turbine operation on a wide set of configurations during the first design stages, while the computational fluid dynamics (CFD) models are indicated for the further investigation of the selected configurations.

Influence of inhomogeneous stochasticity on the falsifiability of mean-field theories and examples from accretion disc modelling

ABSTRACT Despite spatial and temporal fluctuations in turbulent astrophysical systems, mean-field theories can be used to describe their secular evolution. However, observations taken over time scales much shorter than dynamical time scales capture a system in a single state of its turbulence ensemble. Comparing with mean-field theory can falsify the latter only if the theory is additionally supplied with a quantified precision. The central limit theorem provides appropriate estimates to the precision only when fluctuations contribute linearly to an observable and with constant coherent scales. Here, we introduce an error propagation formula that relaxes both limitations, allowing for non-linear functional forms of observables and inhomogeneous coherent scales and amplitudes of fluctuations. The method is exemplified in the context of accretion disc theories, where inhomogeneous fluctuations in the surface temperature are propagated to the disc emission spectrum – the latter being a non-linear and non-local function of the former. The derived precision depends non-monotonically on emission frequency. Using the same method, we investigate how binned spectral fluctuations in telescope data change with the spectral resolving power. We discuss the broader implications for falsifiability of a mean-field theory.

Performance and wake characteristics of tidal turbines in an infinitely large array

The efficiency of tidal stream turbines in a large array depends on the balance between negative effects of turbine-wake interactions and positive effects of bypass-flow acceleration due to local blockage, both of which are functions of the layout of turbines. In this study we investigate the hydrodynamics of turbines in an infinitely large array with aligned or staggered layouts for a range of streamwise and lateral turbine spacing. First, we present a theoretical analysis based on an extension of the linear momentum actuator disc theory for perfectly aligned and staggered layouts, employing a hybrid inviscid-viscous approach to account for the local blockage effect within each row of turbines and the viscous (turbulent) wake mixing behind each row in a coupled manner. We then perform large-eddy simulation (LES) of open-channel flow for 28 layouts of tidal turbines using an actuator line method with doubly periodic boundary conditions. Both theoretical and LES results show that the efficiency of turbines (or the power of turbines for a given bulk velocity) in an aligned array decreases as we reduce the streamwise turbine spacing, whereas that in a staggered array remains high and may even increase due to the positive local blockage effect (causing the local flow velocity upstream of each turbine to exceed the bulk velocity) if the lateral turbine spacing is sufficiently small. The LES results further reveal that the amplitude of wake meandering tends to decrease as we reduce the lateral turbine spacing, which leads to a lower wake recovery rate in the near-wake region. These results will help to understand and improve the efficiency of tidal turbines in future large arrays, even though the performance of real tidal arrays may depend not only on turbine-to-turbine interactions within the array but also on macro-scale interactions between the array and natural tidal currents, the latter of which are outside the scope of this study.

Null geodesics and QNMs in the field of regular black holes

In this paper, we analyze the null geodesics of regular black holes (BHs). A detailed analysis of geodesic structure, both null geodesics and timelike geodesics, has been investigated for the said BH. As an application of null geodesics, we calculate the radius of photon sphere and gravitational bending of light. We also study the shadow of the BH spacetime. Moreover, we determine the relation between radius of photon sphere [Formula: see text] and the shadow observed by a distance observer. Furthermore, we discuss the effect of various parameters on the radius of shadow [Formula: see text]. Also, we compute the angle of deflection for the photons as a physical application of null-circular geodesics. We find the relation between null geodesics and quasinormal mode (QNM) frequency in the eikonal approximation by computing the Lyapunov exponent. It is also shown that (in the eikonal limit) the QNMs of BHs are governed by the parameter of null-circular geodesics. The real part of QNMs frequency determines the angular frequency, whereas the imaginary part determines the instability timescale of the circular orbit. Next, we study the massless scalar perturbations and analyze the effective potential graphically. Massive scalar perturbations are also discussed. As an application of timelike geodesics, we compute the innermost stable circular orbit (ISCO) and marginally bound circular orbit (MBCO) of the regular BHs which are closely related to the BH accretion disk theory. In the appendix, we calculate the relation between angular frequency and Lyapunov exponent for null-circular geodesics.

CFD Prediction of Performance of Wind Turbines Integrated in the Existing Civil Infrastructure

Power generation from wind energy is almost entirely performed in rural locations or at sea, and very little attention has been given to the use of wind turbines in urban locations. Since the re-emergence of wind turbines, the majority of their applications are in large commercial wind farms in rural areas or out at sea, and there is an increasing focus on the use of wind turbines within an urban environment possibly using existing structures, such as bridges and viaducts. There are very few existing buildings which have been designed from the ground-up to include wind turbines in the structure. In order to estimate the wind resources and the performance of a turbine at a particular site, a CFD model is designed and CFD calculations are performed. In order to simplify the modelling of a wind turbine actuator, disc theory is applied. Actuator disc theory is used, as it allows the aerodynamic behaviour of a wind turbine to be analyzed by just considering the energy extraction process without a specific wind turbine design. The power output of wind turbines installed beneath an already existing civil infrastructure is determined and analyzed.

Vortex Ring Theory—An Alternative to the Existing Actuator Disk and Rotating Annular Stream Tube Theories

Currently, the actuator disk theory (ADT) and the rotating annular stream-tube theory (RAST), both of which predicate on the axial momentum and generalized momentum theories, among others, are commonly used in investigating the aerodynamic characteristics of horizontal axis wind turbines (HAWTs). These theories, which are based on a rotor with an infinite number of blades, typically do not properly capture the flow physics of wind blowing past the rotors of HAWTs. A vortex ring theory (VRT) that analyzes HAWTs based solely on the characteristics of fluids flowing past obstructions is proposed. The VRT is not predicated on the assertion that the induced velocity in the wake is twice the induced velocity at the rotor. On the contrary, it splits the axial induction factor in the wake into two components, namely, the induction or interference factor due to the solidity of the rotor and the induction factor due to the wake of the rotor aw; aw and its azimuthal counterpart are determined using the Biot–Savart law. The pressure differences across the rotor segments of a HAWT are derived from the Bernoulli equation for all the three theories. Blade segment/local areas based on the blade sectional geometry of the rotor are used in the case of the VRT to estimate the local forces. All the calculations in this study are based on the design parameters of the 5 MW National Renewable Energy Laboratory’s reference offshore wind turbine. Pressure differences are plotted as functions of local radii using the calculated axial and azimuthal induction factors for each theory. The local power coefficient is plotted as a function of the local tip-speed ratio, while the local thrust coefficient is plotted as a function of the local radii for all the three theories. There is piece-wise agreement between the VRT, the ADT, the RAST and numerical and experimental data available in the literature.

Spiral structures in gravito-turbulent gaseous disks

Context.Gravitational instabilities can drive small-scale turbulence and large-scale spiral arms in massive gaseous disks under conditions of slow radiative cooling. These motions affect the observed disk morphology, its mass accretion rate and variability, and could control the process of planet formation via dust grain concentration, processing, and collisional fragmentation.Aims.We study gravito-turbulence and its associated spiral structure in thin gaseous disks subject to a prescribed cooling law. We characterize the morphology, coherence, and propagation of the spirals and examine when the flow deviates from viscous disk models.Methods.We used the finite-volume code PLUTOto integrate the equations of self-gravitating hydrodynamics in three-dimensional spherical geometry. The gas was cooled over longer-than-orbital timescales to trigger the gravitational instability and sustain turbulence. We ran models for various disk masses and cooling rates.Results.In all cases considered, the turbulent gravitational stress transports angular momentum outward at a rate compatible with viscous disk theory. The dissipation of orbital energy happens via shocks in spiral density wakes, heating the disk back to a marginally stable thermal equilibrium. These wakes drive vertical motions and contribute to mix material from the disk with its corona. They are formed and destroyed intermittently, and they nearly corotate with the gas at every radius. As a consequence, large-scale spiral arms exhibit no long-term global coherence, and energy thermalization is an essentially local process.Conclusions.In the absence of radial substructures or tidal forcing, and provided a local cooling law, gravito-turbulence reduces to a local phenomenon in thin gaseous disks.