__Optimal____ WindMILL Flow Theory____ __** **by Dr. S P. Farthing

After
completing the technical development of the Wing’d Mill, my research from 2004-2011 was into windmill
theory generally which produced 6 papers
with many original ideas. In 2011 I built a windmill model to test one
outstanding problem in Hawt theory and tested it in the wind tunnel of the
Danish Technical University at considerable personal and carbon expense. The
model validated the BEM and provided the first–ever visualization and then analysis of the process by which the
kinetic energy of the torque reaction flow is lost.

I have now extensively edited my working book
of windmill theory into a textbook, both
an introduction for university mechanics students with at least one fluid
mechanics course and a theory revisit for wind turbine engineers and
researchers. A publisher is sought, and sample electronic drafts (also paper
reprints) are available on request . __ Download the Table of Contents &
Introduction__ and click on sections for free download as a FREE
e-book. Email me at spfd@cantab.net with constructive comments)

Highlights of __ Optimal__ __Windmill Theory __ by Dr. Simon P. Farthing

Complete 2D
momentum analysis showing windpower lost by mixing, Betz limit approximate

The nature
of the singularity at the edge of the actuator disc solved

First
visualisation and analysis of flow
process in Hawt loss of torque reaction kinetic energy

Simple
analytic solution of Glauert’s Optimal Hawt including drag correction

First
solution and thorough explanation of robust optimality for windmills, including
near stall

Simple Trisection solution for Robust optimal Hawt
blade elements

Best
(unyawed) annual power through optimum blade twist and chord for typical winds
& tides

Clear
derivation of Prandtl tip loss and connection to isolated wing tip loss

First
correct (analytic) solution for Contra-rotating Hawts

First
sensible perturbation treatment of Hawt
boundary layers, with higher Coriolis stall delay

Inclusion of Tidal Stream Power in analysis

Clear and
throrough explanation of Hawt yawed power and moment and momentum balance

Constant
circulation vortex analysis of Vawt’s **&** tangent blade oscillating &
Schneider windmill

Analysis of
Vawts in common with Hawt for fair comparisons

First clear
proof of why for fixed blades Vawt has a narrower, lower peak C_{p} vs *X* than a Hawt

Optimal
variable and Robust cyclic pitch found for Vawt’s

**Frontspiece/Abstract/introduction/Summary**

This
short text concentrates on clearly presenting and extending simple analytic
understanding of windmills to give the reader as solid and extensive a
foundation as possible for coping with current windturbine design and research
with its daunting proprietary computer
codes, evolved company methodology, widespread empiricism and complex CFD. Prerequisites are courses in fluid mechanics
that includes integral methods and basic
aerodynamics and in univariate calculus
(especially optima) and vectors.

Whilst
immersed in the development of propellers and autogiros, Glauert gave the
basics of a distinct windmill theory consistent with the broad range of
windpumps to early wind turbines in his day. The author has visualised and
analysed the flow centrifuging phenomenon which actually achieves the key assumption
of Glauert’s method. The observed absence of centripetal pressure
also has implications for the wind blade boundary layer, explainning the
stronger Coriolis stall delay than
previously predicted with propeller boundary layer equations.

When
Glauert’s standard method is generalised
in modern vector terms, a simple
analytic solution of Glauert’s optimum rotor appears. The text also applies the generalised method and its solution to
Vertical Axis Windmills for more realistic and equitable performance
comparisons with Hawts. Waterpumping, contra-rotating and oscillating Hawts are
analysed as well

Glauert’s blade element design concentrated on a single
design point of min drag/lift ratio as in propellers whereas it is more
important for windmills to work well in widely varying windspeeds..Robust
optimality design for windmills is developed in the book as very simple soluble
and powerful method for all types and a major reason of the technical
superiority of Hawts.

Vortex
ring theory is clearly presented to explain the good Hawt performance in yaw
and the momentum balance reconciled due to lift on the rotortube. The text has
original material in explanation and extension of practically every standard
topic. The major original parts have been published in the refereed
literature but the book collects many
original extras too small to publish individually as papers. An emphasis is put
on optimisation which is one of the most revealing and underused tests of the
soundness of a model. So the text has much to offer someone practising in
windpower as well as students beginning in the subject. It has been extensively
edited and re-edited to develop the basic theories with the shortest clearest
most direct logic. A reviewer has predicted
it will become a classic of lifelong use to its readers.

**About the author: **Simon Farthing is a lifelong sailor who has
spent many years researching and developing an oscillating wing pump that
involves large amplitude flutter. One of the foundations of this effort is to
thoroughly understand conventional windmills yet from a detached scientific
unbiased point of view. His Ph.D. in Fluid Mechanics at The Department of
Applied Mathematics (and Theoretical Physics) University of Cambridge was in
boundary layer secondary flow (in the aorta). He has published 8 original
refereed papers on windmill theory and recently built, flow-visualised, and flow-analysed a
wind-stator model to clarify a key point
in completing wind power theory and this text.

**Preface**: ** **Glauert’s only publication on wind power was an aside
to a somewhat disjointed collection of
early propeller theories in Durand’s voluminous compendium __Aerodynamic
Theory __[1935]. Before his part was finalised, he was tragically killed in a freak accident
in 1933. For 75 years the few windmill
pages and various selections from the propeller parts have been copied and now
encoded into windmill numerical simulations which have been patched and
patched.

** **In fact Glauert’s windmill aside to his
propeller preoccupation had missed a
simple indeed elegant “trisection” optimum solution of the nominal design apparent wind angle, real apparent wind angle *j*
and blade (¾ chord)angle of attack. It
is easy to show that designing the Hawt angle of attack to be at least .58
this ‘robustly optimal’ value will avoid axial interferences more than ½ at any
tipspeed ratio, so completely sidestepping
the whole intractable ‘turbulent’
windmill zone and all its encoded empiricism.

** **In many ways this text represents an
attempt to take up where Glauert was cut short and develop stream actuator disc
theories systematically and logically using modern vector representation. This
clarifies and generalises the basic ideas and then the trisection is trivially
proved. Glauert’s presentation of the
Prandtl tip correction is vastly simplified by use of the complex potential.
Consideration of the tip correction for a Vawt provides a second way of
understanding its functional dependence . Chapter 4 is a self-contained rational development of vortex ring yaw theory reconciling with
momentum theory without big multiple integrals.

Glauert was clearly aware of the design
of windpumps which experimentally refute the windmill predictions of Joukowski’s theory. The
retrogression to this theory in recent years is a fruit of the dismissal
by ‘modern’ wind electric specialists of
windpumps So the author makes no
apologies for including many alternative configurations such as out-of-fashion windpumps and now Vawts to
enrich and ground this text. He also did
his own wind tunnel visualisation to
find the simple flow phenomenon which refutes Joukowski’s assumptions and
justifies Glauert’s. The text stresses universal windmill principles and
parameters that should be measured and calculated for all windmills to keep
comparisons as rational as possible.

New progress is presented to the extent
that many of the features of conventional BEM numerical simulations can in fact be optimised analytically. This
reduces the necessity, complexity and possible errors of encoding and gives the
engineer a better starting point for design iteration and simulation with a
much more solid understanding and background. Naturally it is of much more
benefit to students than contemporary handbooks to the often proprietary
computer codes as they gain analytic skills which can serve them whatever their
ultimate work**. **Physical insight is lacking from many
‘advanced’ texts and “state of the art “ review articles which tackle major
complexities without thoroughly exploring and understanding simpler situations
first. The end result is a totally bewildered reader adrift in uncertainty and
complexity. Thus this text seeks to explain a few basic effects thoroughly with
as much certainty as possible rather than cover uncertain research topics
superficially, empirically or heuristically,

Every attempt is made to explain the
meaning and physical significance
of analytical and only failing
those, computed results An emphasis is put on optimisation which is
incidentally one of the most revealing tests of the soundness of a model, yet
often hardly mentioned in the computer code-oriented contemporary misnamed wind
‘energy’ texts. (Oil is finite **energy,**
wind is sustainable **power**).

The present monograph should be
suitable for students who have at least one course in fluid mechanics that
includes integral methods and basic
aerodynamics and who are familiar with univariate calculus (especially optima)
and vectors. Exercises are integral to
the text to cover side results with less interrupting of the main flow than
deriving them.