On the limitations of Froude's actuator disc concept

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On the limitations of Froude's actuator disc concept
Citation for published version (APA):
Kuik, van, G. A. M. (1991). On the limitations of Froude's actuator disc concept. [Phd Thesis 1 (Research TU/e /
Graduation TU/e), Applied Physics and Science Education]. Technische Universiteit Eindhoven.
https://doi.org/10.6100/IR345682
DOI:
10.6100/IR345682
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Published: 01/01/1991
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ON THE LIMITATIONS OF FROUDE'S
ACTUATOR DISC CONCEPT
G.A.M. VAN KUIK
ON THE LIMITATIONS OF
FROUDE,S ACTUATOR DISC CONCEPT
ON THE LIMITATIONS OF
FROUDE'S ACTUATOR DISC CONCEPT
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op
gezag van de Rector Magnificus, prof. ir. M. Tels, voor een commissie aangewezen
door het College van Dekanen, in het openbaar te verdedigen op
vrijdag 8 februari om 16.00 uur
door
Gijsbertus Arnoldus Maria van Knik
geboren te Mierlo
Dit proefschrift is goedgekeurd door de promotoren:
prof. dr. ir. G. Vossers
en
prof. dr. ir. P.P.J.M. Schram
Copromotor: dr. ir. A. Hirschberg
Het onderzoek, beschreven in dit proefschrift, is uitgevoerd aan de Technische Universiteit
Eindhoven, faculteit der Technische Natuurkunde, vakgroep Transportfysica, gebouw W & S,
postbus 513, 5600 MB Eindhoven.
Druk: Anraad, Utrecht
Illustratie omslag: Gerda Vink, Rotterdam. Copyright CEA, Rotterdam
SUMMARY
The actuator disc is the first mathematica! model of a propeller, helicopter- or wind
turbine rotor, introduced by R.E Fronde one century ago. His concept of the actuator
disc concerns a permeable surface covered with a uniform normal load, whlch
represents the action of the rotor. The performance predietien by momenturn theory
based on this concept is very simple, reason why it is still used to predict the
performance of rotors. However, the accuracy of this predietien leaves much to be
desired: a survey of experimental data shows that in all rotor flow states the average
velocity of the flow through the rotor is 10 to 15% higher than predicted by
momenturn theory. This suggests a 'natura!' concentration effect to be present:
artificial concentrator systems such as rotors with shrouds or tipvanes show this
increased mass flow through the rotor. The momenturn theory for these concentrator
systems shows that the appropriate way to account for this effect is to add edge forces
to Froude's concept. The characteristic of these edge forces is that they are
perpendicular to the local flow, so cannot perform work. Consequently, only a part of
the total load on the rotor perfarms work. In Froude's actuator disc concept this
distinction is not made: the entire load converts power.
The question arises whether the actuator disc including edge forces is a better
representation of the action of conventional rotors than Froude's concept is. A
confirmative answer is obtained in two ways. It is shown that the load on the
chordwise bound vorticity of the rotor blade does not perform work, which confiicts
with Froude's assumption of the load converting power being equal to the total load.
Furthermore an experiment on a model rotor in hover bas been conducted which
indeed shows the load converting power to besmaller than the totalload on the rotor.
The wake contraction bas been compared with the prediction by the momenturn
theory including edge forces. The agreement is very reasonable, in contrast with the
classica! prediction. The prediction of the average velocity through the rotor also
improves compared with the classica! prediction, but remains too low. The conclusion
of this part is that the addition of edge forces to Froude's concept improves the
performance prediction for rotors significantly, although discrepancies between
prediction and measurements remain.
i i
From the literature numerical as well as experimental data are known regarding flows
induced by actuator discs, which seem to conflict wîth Froude's results. Therefore the
second part of the work concerns the actuator disc itself, and focuses on the question
whether the edge farces are an inherent part of the load, instead of merely an addition
to Froude's concept in order to imprave the representation of real rotors. A numerical
as well as analytica! approach has been followed. A non-linear, viscous finite element
calculation has been set up for the flow of a two dimensional actuator strip with a
smooth load distribution. By increasing the Reynolds number and the steepness of the
distribution as far as the numerical metbod allowed, and by extrapolation of the data
the inviscid flow induced by a uniform load is approached. The results of this
extrapolation comply with the classica! performance prediction. Using the results of
our numerical method, most of the mentioned experimental and numerical data from
the literature can now be explained. For the remaining data ( one numerical and one
experimental) this still is impossible.
The analytica! approach first proceeds from actuator discs with a smooth load
distribution. Illustrated by an exact salution of Wu's actuator disc equation, arelation
between the externally applied normal load and a tangential (in-plane) force density
has been established. If the thickness of the disc is zero, the order of this force density
is too low to yield a contribution to the resultant load. For a non-zero thickness the
in-plane force density yields a resultant load. In order to explain the physical origin of
this force density, we set up the hypothesis that it is the shear stress required to
generate vorticity on streamlines passing the disc, even if the Reynolds number based
on the disc radius is infinitely large. For one specific case this has been confirmed; in
general the interpretation of the origin is incomplete.
For the disc and strip covered with a uniform Ioad, the flow is singular at the edge.
The singularity is a vertex which, in case of steady flow, carries a non-zero edge force.
The order of this force is the order of the velocity at the edge. This order is unknown,
so the force can be infinitely smalt. The answer to the question of the existence of edge
forces as an inherent part of the load awaits an analytica! or numerical study towards
the entire flow field, induced by discs or strip with a constant uniform load.
iii
TABLE OF CONTENTS page
Summary
Table of contents iii
List of symbols vi
L Introduetion 1
2. The actuator disc problem 3
2.1 The role of Froude's actuator disc in performance prediction
by moment um theory 3
2.2 The classica! actuator disc performance compared with theory and
experiments 7
2.3 The force field of a real rotor and wing compared with the actuator
disc force field 11
2.4 The actuator disc concept extended with edge forces 15
Part I : The representation of a rotor by an actuator disc
19
3. The use of the modified actuator surface concept in the momentum theory 21
3.1 Momenturn theory including edge forces 21
3.2 Typical results for the actuator disc representing the ideal wind
turbine or propeller 25
3.3 The conservation laws for general non-uniform Ioad distributions 27
4. Experimental verification of the momentum theory including edge farces
on a model rotor in hover 29
4.1 The momenturn theory for real rofurs 29
4.2 The determination of the power 32
4.3 The measuring devices 34
4.4 The etaboration of the measurements 36
4.5 Comparison with momenturn theory results 40
4.6 Accuracy of the experiment 41
4.7 Discussion and conclusions 43
iv
Part II: The force and flow field of the actuator surface 47
5. The equation of motion 49
5.1 The equation of motion including the force field term 49
5.2 Definition of the actuator surface force field 52
5.3 Dimension analysis of actuator surface flow 53
6. Nnmerical analysis of the actuator strip flow 57
6.1. Inviscid calculations 57
6.1.1 The actuator cylinder of Madsen 57
6.1.2 The actuator strip of Lee & Greenberg 59
6.1.3 The actuator strip of Schouten 61
6.2. Viscous calculations 62
6.2.1 The force distribution and the definition of the Reynolds number 62
6.2.2 The computational metbod 65
6.2.3 Results for m=17, Re-+ oo 67
6.2.4 Extrapolation towards infinite m 67
6.2.5 The interpretation of the strip experiment of Lee & Greenberg 70
7. The generation of vorticity by force fields 73
7.1 The force density fin Wu•s equation illustrated by an exact solution 73
7.2 The smooth actuator surface force field 80
8. The edge singularity of an actuator surface with a constant normalload 84
8.1 A discrete vortex as edge singularity 84
8.2 The possible existence of edge forces 89
8.3 The comparison with other solutions 91
8.3.1 The spiral solution of Schmidt & Sparenberg 91
8.3.2 Greenherg's square root singularity 93
I
9. Discussion on the results; conclusions 97
9.1 Survey of the results 97
9.1.1 Part 1: the rotor problem 97
9.1.2 Part 11: the actuator surface problem 99
9.2 Discussion on the results 103
9.3 Conclusions 106
R.eferences 108
Appendices
A: Sparenherg's translation theorem 113
B: The exponential spiral with infinite length as the limit of a finite
length spiral 115
C: The viscous solution of the flow induced by a half-infini te, infinitely
thin actuator strip 123
D: The force on a 2-D discrete vortex obtained by an actuator strip limit 125
E: Justification of the experiment 127
F: Thè boundary layer flow for infinite Reynolds number 131
Samenvatting 135
Nawoord 137
Curriculum vitae 138