Stability of helical tip vortices in a rotor far wake

2007 ◽  
Vol 576 ◽  
pp. 1-25 ◽  
Author(s):  
V. L. OKULOV ◽  
J. N. SØRENSEN
Keyword(s):  
Rotor Blades ◽  
Tip Vortex ◽  
Tip Vortices ◽  
Radial Extent ◽  
Helical Vortices ◽  
Constant Pitch ◽  
Bladed Rotor ◽  
The Stability ◽  
Further Development ◽  
Far Wake

As a means of analysing the stability of the wake behind a multi-bladed rotor the stability of a multiplicity of helical vortices embedded in an assigned flow field is addressed. In the model the tip vortices in the far wake are approximated by infinitely long helical vortices with constant pitch and radius. The work is a further development of a model developed in Okulov (J. Fluid Mech., vol. 521, p. 319) in which the linear stability of N equally azimuthally spaced helical vortices was considered. In the present work the analysis is extended to include an assigned vorticity field due to root vortices and the hub of the rotor. Thus the tip vortices are assumed to be embedded in an axisymmetric helical vortex field formed from the circulation of the inner part of the rotor blades and the hub. As examples of inner vortex fields we consider three generic axial columnar helical vortices, corresponding to Rankine, Gaussian and Scully vortices, at radial extents ranging from the core radius of a tip vortex to several rotor radii.The analysis shows that the stability of tip vortices largely depends on the radial extent of the hub vorticity as well as on the type of vorticity distribution. As part of the analysis it is shown that a model in which the vortex system is replaced by N tip vortices of strength Γ and a root vortex of strength − N/Γ is unconditionally unstable.

scholarly journals Some effects of flow expansion on the aerodynamics of horizontal-axis wind turbines

2021 ◽  
Author(s):  
David Wood ◽  
Eric Limacher
Keyword(s):  
Axial Velocity ◽  
Momentum Equation ◽  
Horizontal Axis ◽  
Speed Ratio ◽  
Axial Component ◽  
Horizontal Axis Wind Turbine ◽  
Tip Vortices ◽  
Tip Speed Ratio ◽  
Constant Pitch ◽  
Far Wake

Abstract. Upwind of an energy-extracting horizontal-axis wind turbine, the flow expands as it approaches the rotor, and the expansion continues in the vorticity-bearing wake behind the rotor. The upwind expansion has long been known to influence the axial momentum equation through the axial component of the pressure, although the extent of the influence has not been quantified. Starting with the impulse analysis of Limacher & Wood (2020), but making no further use of impulse techniques, we demonstrate that the expansion redistributes momentum from the external flow to the wake and derive its exact expression when the rotor is circumferentially uniform. This expression, which depends on the radial velocity and the axial induction factor, is added to the thrust equation containing the pressure on the back of the disk. Removing the pressure to obtain a practically useful equation shows the axial induction in the far-wake is twice the value at the rotor only at high tip speed ratio and only if the relationship between vortex pitch and axial induction in non-expanding flow carries over to the expanding case. At high tip speed ratio, we assume that the expanding wake approaches the "Joukowsky'' model of a hub vortex on the axis of rotation and tip vortices originating from each blade. The additional assumption that the helical tip vortices have constant pitch, allows a semi-analytic treatment of their effect on the rotor flow. Expansion modifies the relation between the pitch and induced axial velocity so that the far-wake area and induction are significantly less than twice the values at the rotor. There is a moderate decrease – about 6 % – in the power production and a similar size error occurs in the familiar axial momentum equation involving the axial velocity.

scholarly journals Some effects of flow expansion on the aerodynamics of horizontal-axis wind turbines

2021 ◽  
Vol 6 (6) ◽  
pp. 1413-1425
Author(s):  
David H. Wood ◽  
Eric J. Limacher
Keyword(s):  
Axial Velocity ◽  
Momentum Equation ◽  
Horizontal Axis ◽  
Speed Ratio ◽  
Axial Component ◽  
Horizontal Axis Wind Turbine ◽  
Tip Vortices ◽  
Tip Speed Ratio ◽  
Constant Pitch ◽  
Far Wake

Abstract. The flow upwind of an energy-extracting horizontal-axis wind turbine expands as it approaches the rotor, and the expansion continues in the vorticity-bearing wake behind the rotor. The upwind expansion has long been known to influence the axial momentum equation through the axial component of the pressure, although the extent of the influence has not been quantified. Starting with the impulse analysis of Limacher and Wood (2020), but making no further use of impulse techniques, we derive its exact expression when the rotor is a circumferentially uniform disc. This expression, which depends on the radial velocity and the axial induction factor, is added to the thrust equation containing the pressure on the back of the disc. Removing the pressure to obtain a practically useful equation shows the axial induction in the far wake is twice the value at the rotor only at high tip speed ratio and only if the relationship between vortex pitch and axial induction in non-expanding flow carries over to the expanding case. At high tip speed ratio, we assume that the expanding wake approaches the Joukowsky model of a hub vortex on the axis of rotation and tip vortices originating from each blade. The additional assumption that the helical tip vortices have constant pitch allows a semi-analytic treatment of their effect on the rotor flow. Expansion modifies the relation between the pitch and induced axial velocity so that the far-wake area and induction are significantly less than twice the values at the rotor. There is a moderate decrease – about 6 % – in the power production, and a similar size error occurs in the familiar axial momentum equation involving the axial velocity.

Investigating the Coupled Effects Between Rotor-Blade Aeroelasticity and Tip Vortex Stability

2020 ◽  
Author(s):  
Steven N. Rodriguez ◽  
Athanasios P. Iliopoulos ◽  
John G. Michopoulos ◽  
Justin W. Jaworski
Keyword(s):  
Rotor Blade ◽  
Turbine Rotor ◽  
Rotor Blades ◽  
Tip Vortex ◽  
Tip Vortices ◽  
Operational Conditions ◽  
Vortex Stability ◽  
Blade Vortex Interaction ◽  
Wake Interactions ◽  
The Relationship

Abstract The relationship between rotor-blade aeroelasticity and tip-vortex stability is investigated numerically. An aeroelastic framework based on the free-vortex wake and finite element methods is employed to model a subscaled helicopter rotor in hover and forward-tilted conditions. A linear eigenvalue stability analysis is performed on tip vortices to associate the coupled impact of aeroelastic effects and vortex evolution. Prior numerical investigations have shown that highly flexible wind turbine rotor-blades have the potential to decrease levels of the instability of tip vortices. The present work focuses on testing these findings against a subscaled rotor within the range of helicopter operational rotation frequencies. The presented work aims to develop further insight into rotor-wake interactions and blade-vortex interaction to explore the mitigation of adverse rotorcraft operational conditions, such as their effect on aerodynamic-induced airframe vibrations and the associated life-cycle fatigue performance.

On the motion of multiple helical vortices

2001 ◽  
Vol 447 ◽  
pp. 149-171 ◽  
Author(s):  
D. H. WOOD ◽  
J. BOERSMA
Keyword(s):  
Wind Turbines ◽  
The Self ◽  
Tip Vortices ◽  
Power Extraction ◽  
Small Pitch ◽  
Helical Vortices ◽  
The One ◽  
Induced Velocity ◽  
Far Wake

The analysis of the self-induced velocity of a single helical vortex (Boersma & Wood 1999) is extended to include equally spaced multiple vortices. This arrangement approximates the tip vortices in the far wake of multi-bladed wind turbines, propellers, or rotors in ascending, descending, or hovering flight. The problem is reduced to finding, from the Biot–Savart law, the additional velocity of a helix due to an identical helix displaced azimuthally. The resulting Biot–Savart integral is further reduced to a Mellin–Barnes integral representation which allows the asymptotic expansions to be determined for small and for large pitch. The Biot–Savart integral is also evaluated numerically for a total of two, three and four vortices over a range of pitch values. The previous finding that the self-induced velocity at small pitch is dominated by a term inversely proportional to the pitch carries over to multiple vortices. It is shown that a far wake dominated by helical tip vortices is consistent with the one-dimensional representation that leads to the Betz limit on the power output of wind turbines. The small-pitch approximation then allows the determination of the blade&s bound vorticity for optimum power extraction. The present analysis is shown to give reasonable estimates for the vortex circulation in experiments using a single hovering rotor and a four-bladed propeller.

scholarly journals Helical self-similarity of tip vortex cores

2018 ◽  
Vol 859 ◽  
pp. 1084-1097 ◽  
Author(s):  
Valery L. Okulov ◽  
Ivan K. Kabardin ◽  
Robert F. Mikkelsen ◽  
Igor V. Naumov ◽  
Jens N. Sørensen
Keyword(s):  
Experimental Studies ◽  
Simple Relation ◽  
Tip Vortex ◽  
Self Similarity ◽  
Local Flow ◽  
Tip Vortices ◽  
Longitudinal Vortices ◽  
The Core ◽  
Helical Vortices ◽  
Similarity Scaling

The present work investigates local flow structures and the downstream evolution of the core of helical tip vortices generated by a three-bladed rotor. Earlier experimental studies have shown that the core of a helical tip vortex exhibits a local helical symmetry with a simple relation between the axial and azimuthal velocities. In the present study, a self-similarity scaling argument further describes the downstream development of the vortex core. Self-similarity has up to now only been investigated for longitudinal vortices and it is the first time that helical vortices have become the subject of such an analysis. Combining symmetry arguments from previous studies on helical vortices with novel experiments and knowledge regarding the self-similarity evolution of the core of longitudinal vortices, a new model describing what is referred to as ‘helical self-similarity’ is proposed. The generality of the model is verified and supported by experimental data. The proposed model is important for fundamental understanding of the behaviour of helical vortices, with a range of applications in both industry and nature. Examples of this are tip vortices behind aerodynamic devices, such as vortex generators, and fixed and rotary aircraft, and in combustion chambers and cyclone separators.

The Tip Vortex System of a Four-Bladed Rotor in Dynamic Stall Conditions

2019 ◽  
Vol 64 (2) ◽  
pp. 1-14 ◽  
Author(s):  
C. Christian Wolf ◽  
Johannes N. Braukmann ◽  
Wolfgang Stauber ◽  
Till Schwermer ◽  
Markus Raffel
Keyword(s):  
Vortex Structure ◽  
Shear Layers ◽  
Stereoscopic Particle Image Velocimetry ◽  
Tip Vortex ◽  
Dynamic Stall ◽  
Test Case ◽  
Turbulent Structures ◽  
Vortex System ◽  
Tip Vortices ◽  
Constant Pitch

The tip vortex system downstream of a four-bladed instrumented rotor was investigated experimentally through the application of stereoscopic particle image velocimetry (PIV). A dynamic stall test case was facilitated by a high cyclic pitch setting of the swashplate, with additional attached-flow and constant-pitch test cases for comparison reasons. The phase-locked PIV system and a rotation of the swashplate assembly allowed for an acquisition of the tip vortex system over the entire dynamic stall cycle and vortex ages up to at least 235°. The vortex structure and its relation to the blade shear layers were studied by means of both phase-averaged flow fields and the identification of vortex properties such as circulation and swirl velocity distributions. When approaching dynamic stall, a breakdown of the vortex structure started at high vortex ages, accompanied by the entrainment of turbulent structures from the passing blade shear layers into the tip vortices. After the flow over the blade is fully separated and during large parts of the downstroke, the wake of the rotor tips appears as a highly turbulent area with no individual tip vortices traceable, before reestablishing an ordered tip vortex structure shortly before the minimum blade pitch angle.

The stability of dislocation networks under various oxygen-doping conditions in superconducting Bi2Sr2CaCu2Oy single crystals and their influence on the flux pinning properties

1994 ◽  
Vol 52 ◽  
pp. 802-803
Author(s):  
Y. Feng ◽  
X. Y. Cai ◽  
R. J. Kelley ◽  
D. C. Larbalestier
Keyword(s):  
Single Crystals ◽  
Oxygen Content ◽  
Basal Plane ◽  
Flux Pinning ◽  
Pinning Centers ◽  
Before And After ◽  
Anomalous Peak ◽  
Basal Plane Dislocation ◽  
The Stability ◽  
Further Development

The issue of strong flux pinning is crucial to the further development of high critical current density Bi-Sr-Ca-Cu-O (BSCCO) superconductors in conductor-like applications, yet the pinning mechanisms are still much debated. Anomalous peaks in the M-H (magnetization vs. magnetic field) loops are commonly observed in Bi2Sr2CaCu2Oy (Bi-2212) single crystals. Oxygen vacancies may be effective flux pinning centers in BSCCO, as has been found in YBCO. However, it has also been proposed that basal-plane dislocation networks also act as effective pinning centers. Yang et al. proposed that the characteristic scale of the basal-plane dislocation networksmay strongly depend on oxygen content and the anomalous peak in the M-H loop at ˜20-30K may be due tothe flux pinning of decoupled two-dimensional pancake vortices by the dislocation networks. In light of this, we have performed an insitu observation on the dislocation networks precisely at the same region before and after annealing in air, vacuumand oxygen, in order to verify whether the dislocation networks change with varying oxygen content Inall cases, we have not found any noticeable changes in dislocation structure, regardless of the drastic changes in Tc and the anomalous magnetization. Therefore, it does not appear that the anomalous peak in the M-H loops is controlled by the basal-plane dislocation networks.

Effect of Upstream Rotor Vortical Disturbances on the Time-Averaged Performance of Axial Compressor Stators: Part 1—Framework of Technical Approach and Wake–Stator Blade Interactions

1999 ◽  
Vol 121 (3) ◽  
pp. 377-386 ◽  
Author(s):  
T. V. Valkov ◽  
C. S. Tan
Keyword(s):  
Total Pressure ◽  
Axial Compressor ◽  
Tip Vortex ◽  
Technical Approach ◽  
Tip Vortices ◽  
Tip Leakage ◽  
Unsteady Interaction ◽  
Flow Approximation ◽  
Stator Blade ◽  
The Impact

In a two-part paper, key computed results from a set of first-of-a-kind numerical simulations on the unsteady interaction of axial compressor stators with upstream rotor wakes and tip leakage vortices are employed to elucidate their impact on the time-averaged performance of the stator. Detailed interrogation of the computed flow field showed that for both wakes and tip leakage vortices, the impact of these mechanisms can be described on the same physical basis. Specifically, there are two generic mechanisms with significant influence on performance: reversible recovery of the energy in the wakes/tip vortices (beneficial) and the associated nontransitional boundary layer response (detrimental). In the presence of flow unsteadiness associated with rotor wakes and tip vortices, the efficiency of the stator under consideration is higher than that obtained using a mixed-out steady flow approximation. The effects of tip vortices and wakes are of comparable importance. The impact of stator interaction with upstream wakes and vortices depends on the following parameters: axial spacing, loading, and the frequency of wake fluctuations in the rotor frame. At reduced spacing, this impact becomes significant. The most important aspect of the tip vortex is the relative velocity defect and the associated relative total pressure defect, which is perceived by the stator in the same manner as a wake. In Part 1, the focus will be on the framework of technical approach, and the interaction of stator with the moving upstream rotor wakes.

Tip vortex effects on oscillating rotor blades in hovering flight

AIAA Journal ◽  
1971 ◽  
Vol 9 (1) ◽  
pp. 106-113 ◽  
Author(s):  
W. P. JONES ◽  
B. M. RAO
Keyword(s):  
Rotor Blades ◽  
Tip Vortex ◽  
Hovering Flight

A Robust Implementation of a Reynolds Stress Model for Turbomachinery Applications in a Coupled Solver Environment

2020 ◽  
Author(s):  
David Roos Launchbury ◽  
Luca Mangani ◽  
Ernesto Casartelli ◽  
Francesco Del Citto
Keyword(s):  
Reynolds Stress ◽  
Turbulence Modeling ◽  
Computational Cost ◽  
Reynolds Stress Model ◽  
Tip Vortex ◽  
Stability And Convergence ◽  
Stress Model ◽  
Robust Implementation ◽  
Flow Phenomena ◽  
The Stability

Abstract In the industrial simulation of flow phenomena, turbulence modeling is of prime importance. Due to their low computational cost, Reynolds-averaged methods (RANS) are predominantly used for this purpose. However, eddy viscosity RANS models are often unable to adequately capture important flow physics, specifically when strongly anisotropic turbulence and vortex structures are present. In such cases the more costly 7-equation Reynolds stress models often lead to significantly better results. Unfortunately, these models are not widely used in the industry. The reason for this is not mainly the increased computational cost, but the stability and convergence issues such models usually exhibit. In this paper we present a robust implementation of a Reynolds stress model that is solved in a coupled manner, increasing stability and convergence speed significantly compared to segregated implementations. In addition, the decoupling of the velocity and Reynolds stress fields is addressed for the coupled equation formulation. A special wall function is presented that conserves the anisotropic properties of the model near the walls on coarser meshes. The presented Reynolds stress model is validated on a series of semi-academic test cases and then applied to two industrially relevant situations, namely the tip vortex of a NACA0012 profile and the Aachen Radiver radial compressor case.

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