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 Cp vs X than a Hawt

Optimal variable and Robust cyclic pitch found for Vawt’s




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.