diff --git a/contents/tech-extensions.tex b/contents/tech-extensions.tex
index 8c3270469a592376be60ea60cdb3b75b07541b0b..2f9e75fad1a4ba7a3ef5d75f2db346153e7ba8ba 100644
--- a/contents/tech-extensions.tex
+++ b/contents/tech-extensions.tex
@@ -14,4 +14,6 @@
\label{chap:tech:ext}
+\section{Tip/hub loss}
+\label{sec:tech:ext:loss}
diff --git a/contents/tech-intro.tex b/contents/tech-intro.tex
index e15b563f83341585f05ac928482cab611954aa15..bdcab36ec66d3c67e010b386a9954335c88789cf 100644
--- a/contents/tech-intro.tex
+++ b/contents/tech-intro.tex
@@ -18,9 +18,10 @@ the code and its implementation. The goal is not to give a full lecture on the
Blade Element Momentum Theory, but rather to write out the main equations and
detail the way they are implemented and solved within \rotare.
-The first chapter details the architecture of the code itself, which can be
-useful if the developers that want to extend the possibilities offered by the
+The first chapter details the architecture of the code itself and is more
+aimed at developers that want to extend the possibilities offered by the
software.
-The following ones are more directed to users that want an in-depth knowledge of
-the equations and solving process employed by the program. Finally, some basic
-validation cases are provided against well-known literature examples.
+
+The following the chapters are directed towards users that want an in-depth
+knowledge of the equations and solving process implemented in \rotare. Finally,
+some basic validation cases are provided against well-known literature examples.
diff --git a/contents/tech-solvers.tex b/contents/tech-solvers.tex
index 29001ac24f8325b4c7bcd7d1cd239f04d86e689e..86dfa149fce20977ded40b86ce242dbd7077e34a 100644
--- a/contents/tech-solvers.tex
+++ b/contents/tech-solvers.tex
@@ -16,6 +16,78 @@
\fxfatal{While \rotare does not fully support coaxial rotors or oblique
flows yet, the current documentation is written as if it was already the case.}
+% ==============================================================================
+% COORDINATES
+% ==============================================================================
+
+\section{Coordinates systems}
+The rotors may not always be aligned with the freestream. It is therefore
+important to properly define the reference frames before developing the
+equations.
+In specific, the first reference frame will simply be the internal one. The
+second frame is defined as the tip plane path of the rotor and will be denoted
+TPP. This frame is defined solely be the rotor disk. In oblique flows, it is
+angled with respect to the inertial frame, while in axial flows it lies
+perpendicular to the freestream.
+\fxwarning{Add schematics}
+
+
+% ==============================================================================
+% COEFFICIENTS
+% ==============================================================================
+
+\section{Rotor coefficients}
+\label{chap:tech:solvers:coeff}
+In order to better compare different rotors or their performance under different
+conditions, it is often best to express the forces by means of non-dimensional
+coefficients. While the principle is the same for all applications, different
+factors are used to non-dimensionalize these forces in rotors, propellers or
+turbine applications.
+Moreover, the general convention applied in the United States for rotor differs
+from the one used in the rest of the world~\cite{leishman2006a}, where an
+additional one-half factor is used in the denominator.
+
+The Table~\ref{tab:tech:solv:coeff:def} lists the main coefficients in the
+US conventional notation. The other forces (longitudinal, lateral) or moments
+(rolling, pitching) are defined similarly to the thrust and torque coefficients
+respectively.
+
+\begin{table}[ht]
+ \caption{Coefficients definitions for rotors, propellers and turbines -- US
+ notation.}
+ \label{tab:tech:solv:coeff:def}
+ \begin{center}
+ \begin{tabular}[c]{@{}lccc@{}}
+ \toprule
+ \textbf{Measure} & \textbf{Rotors} & \textbf{Propellers} &
+ \textbf{Turbines} \\
+ \midrule
+ Thrust coeff, \cThrust &
+ $\dfrac{\thrust}{\dens\area(\angVel\radius)^2}$ &
+ $\dfrac{\thrust}{\dens\rpsVel^2\diam^4}$ &
+ \fxnote{todo} \\\addlinespace
+ Thrust torque, \cTorque &
+ $\dfrac{\torque}{\dens\area\angVel^2\radius^3}$ &
+ $\dfrac{\torque}{\dens\rpsVel^2\diam^5}$ &
+ \fxnote{todo} \\\addlinespace
+ Thrust power, \cPower &
+ $\dfrac{\power}{\dens\area(\angVel\radius)^3}$ &
+ $\dfrac{\power}{\dens\rpsVel^3\diam^5}$ &
+ \fxnote{todo} \\
+ \bottomrule
+ \end{tabular}
+ \end{center}
+\end{table}
+
+Note that the angular velocity in the rotor notation is $\angVel$, expressed in
+[\si{1\per\sec}] while for the propellers it is $\rpsVel$, expressed in [RPS].
+
+
+
+% ==============================================================================
+% BLADE ELEMENT THEORY
+% ==============================================================================
+
\section{Blade Element theory}
\label{chap:tech:solvers:BET}
@@ -24,7 +96,6 @@ collection of 2D elements which are radially distributed over the blade. It
further assumes that the performance of an individual element is completely
independent of the influence of the other elements. Each element can therefore
be represented as a simple 2D airfoil section.
-\fxerror{Fix nomenclature on figure, once symbols name defined}
\begin{figure}[ht]
\begin{center}
@@ -67,13 +138,13 @@ which can be represented though the azimuthal angle $\azim$.
\vRel &= \sqrt{\vAx^2+\vTg^2}
\end{align}
-The angle induced (or inflow) angle $\aInd$ expresses the angle between the
-resultant velocity and the rotor disk:
+The inflow angle (sometimes called \emph{induced} angle) $\aInfl$ expresses
+the angle between the resultant velocity and the rotor disk:
\begin{equation}
- \label{eq:tech:solvers:BET:aInd}
- \tan^{-1}\aInd = \dfrac{\vAx}{\vTg}
+ \label{eq:tech:solvers:BET:aInfl}
+ \tan^{-1}\aInfl = \dfrac{\vAx}{\vTg}
\end{equation}
-By convention, $\aInd$ is positive when $\vRel$ is directed towards the disk and
+By convention, $\aInfl$ is positive when $\vRel$ is directed towards the disk and
negative when $\vRel$ points outwards.
The effective angle of attack at the element can therefore be obtained by the
@@ -82,7 +153,7 @@ angle is often given with respect to the airfoil \emph{zero-lift line}, the
zero-lift angle must be subtracted as well.
\begin{equation}
\label{eq:tech:solvers:BET:aoa}
- \aoa = (\pitch - \aoa_0) - \aInd
+ \aoa = (\pitch - \aoa_0) - \aInfl
\end{equation}
The angle of attack, along with the Reynolds number can be used to determine the
@@ -109,13 +180,18 @@ the rotor and derive the thrust (\thrust), torque (\torque) and power (\power)
contribution of the element
\begin{align}
\label{eq:tech:solvers:BET:thrust}
- d\thrust &= \nBlades (d\lift \sin\aInd - d\drag \cos\aInd)\\
+ d\thrust &= \nBlades (d\lift \sin\aInfl - d\drag \cos\aInfl)\\
\label{eq:tech:solvers:BET:torque}
- d\torque &= \nBlades (d\lift \cos\aInd + d\drag \sin\aInd) \radA\\
+ d\torque &= \nBlades (d\lift \cos\aInfl + d\drag \sin\aInfl) \radA\\
\label{eq:tech:solvers:BET:power}
- d\power &= \nBlades (d\lift \cos\aInd + d\drag \sin\aInd) \radA \angVel
+ d\power &= \nBlades (d\lift \cos\aInfl + d\drag \sin\aInfl) \radA \angVel
\end{align}
+
+% ==============================================================================
+% MOMENTUM THEORY
+% ==============================================================================
+
\section{Momentum theory}
\label{chap:tech:solvers:mom}
@@ -133,76 +209,213 @@ by $dA = 2\pi \radA \dy$. The mass flow rate through any individual annulus is
then given by:
\begin{equation}
\label{eq:tech:solvers:mom:mass}
- \mFlow = \dens (\vAx + \viAx) d\area = \dens (\vAx + \viAx) 2 \pi y dy
+ \mFlow = \dens (\vAx + \viAx) d\area = \dens (\vAx + \viAx) 2 \pi \radA \dy
\end{equation}
-The axial and tangential momentum balance through the disk give:
+Furthermore, it can be showed that axial induced velocity at the disk is equal
+to half the slipstream velocity in the far wake: $\viAx = \frac{\vSlip}{2}$.
+The thrust can then be expressed using the momentum balance in the axial
+direction as:
\begin{equation}
\label{eq:tech:solvers:mom:thrust}
\begin{split}
d\thrust & = \mFlow (\vAx[d] - \vAx[u]) \\
& = \mFlow \vSlip \\
& = \mFlow (2\viAx) \\
- & = 4\pi \dens (\vAx + \viAx) \viAx y dy
+ & = 4\pi \dens (\vAx + \viAx) \viAx \radA \dy
\end{split}
\end{equation}
-
-
-
-
-Likewise, the torque is given by
+The ideal power can be calculated similarly as it is expressed by the product of
+the thrust and the induced velocity:
\begin{equation}
- \label{eq:tech:solvers:mom:torque}
+ \label{eq:tech:solvers:mom:power}
\begin{split}
- d\torque & = \mFlow \radA \viTg
+ d\power & = d\thrust \viAx \\
+ & = 4\pi \dens (\vAx + \viAx) \viAx^2 \radA \dy
\end{split}
\end{equation}
+\fxfatal{torque equation}
+% ==============================================================================
+% SOLVERS
+% ==============================================================================
+
\section{Solvers}
\label{chap:tech:solvers:solvers}
-\rotare implements four different solvers, all of them based on the same set of
-initial equations
-(see~\ref{eq:tech:solvers:BET:thrust}-\ref{eq:tech:solvers:BET:power} and
-\todo{Ref for momentum EQ}).
-These solver differs by introducing some additional assumptions or by modifying
+The essence of the Blade Element Momentum Theory, is now to combine the Blade
+Element equations for the thrust and power (\ref{eq:tech:solvers:BET:thrust},
+\ref{eq:tech:solvers:BET:power}) and the corresponding momentum equations
+(\ref{eq:tech:solvers:mom:thrust}, \ref{eq:tech:solvers:mom:power}). This newly
+formed system can then be solved for the inflow angle and the induced velocity
+at the disk.
+
+Unfortunately, solving such a non-linear system is not trivial. \rotare does
+that by implementing four different solvers, all based on the same set of
+initial equations.
+These solvers differ by introducing some additional assumptions or by modifying
the nonlinear system of equations (mostly the momentum equations) in order to
-simplify its formulation or the convergence of its solution.
+simplify its formulation or ease the convergence of its solution. A quick
+comparison of the solvers is presented in
+Table~\ref{tab:tech:solvers:solvers:compa}.
+
+\begin{table}[ht]
+ \caption{Comparison \rotare's solvers}
+ \label{tab:tech:solvers:solvers:compa}
+ \begin{center}
+ \begin{tabularx}{\linewidth}{@{}XXXXX@{}}
+ \toprule
+ & \textbf{Leishman} & \textbf{indFact} & \textbf{indVel} &
+ \textbf{Stahlhut} \\
+ \midrule
+ Assumptions & Small angles & \multirow{3}*{-} & \multirow{3}*{-} & \multirow{3}*{-} \\
+ & $\vAx \ll \vTg$ & & & \\
+ & Drag $\ll$ Lift & & & \\
+ Applications & Hover/idle and slow axial flow & Any\footnotemark & Any & Any
+ \\
+ Convergence & Guaranteed & Medium & Medium & Easy \\
+ CPU time & Fastest & High & High & Medium \\
+ \bottomrule
+ \end{tabularx}
+ \end{center}
+\end{table}
+% Can't do normal footnotes in tables. Must do footnotemark and then
+%footnotetext outside of the table environment.
+\footnotetext{Technically it does not work with idle/hovering rotors directly.
+ This limitation has been circumvented in \rotare.}
+
+
+% Leishman
+% ***********************************************
\subsection{Leishman solver}
\label{chap:tech:solvers:leishman}
-This solver is based on the methodology described in \fxcite
+This solver is based on the methodology described by Leishman
+in~\cite{leishman2006a}. This solver makes some strong assumptions on the flow
+and the operation of the rotor in order to linearize the system as much as
+possible. These assumptions correspond to a rotor lightly loaded, which
+is linked to small angles approximations. This solver is therefore only suitable
+for hovering rotors and slow axial flows.
+
+\subsubsection{Assumptions}
+The following three assumptions are made in order to linearize the system:
+\begin{enumerate}
+ \item In-plane induced velocities are negligibles:
+ \begin{equation}
+ \label{eq:tech:solvers:leishman:assum:neglVel}
+ \vRel \approx \vTg
+ \qquad
+ \qquad
+ \viTg \approx 0
+ \end{equation}
+ \item Induced angle $\aInfl$ is small:
+ \begin{equation}
+ \label{eq:tech:solvers:leishman:assum:inflow}
+ \aInfl \approx \dfrac{\vAx}{\vTg}
+ \qquad
+ \qquad
+ \sin\aInfl \approx \aInfl
+ \qquad
+ \qquad
+ \cos\aInfl \approx 1
+ \end{equation}
+ \item Drag is much smaller than lift and does not contribute much to the
+ thrust so that:
+ \begin{equation}
+ \label{eq:tech:solvers:leishman:assum:forces}
+ d\drag \sin\aInfl \approx 0
+ \qquad
+ \qquad
+ d\drag \cos\aInfl \approx d\drag
+ \end{equation}
+ \item The lift coefficient is linear:
+ \begin{equation}
+ \label{eq:tech:solvers:leishman:assum:cl}
+ \cLift \simeq \cLiftSlope \aoa
+ \end{equation}
+\end{enumerate}
+Note that the fourth assumption can be pushed further by considering that
+$\cLiftSlope = 2\pi$ (thin airfoil theory). If the airfoil polars are provided,
+it is also possible to retrieve the true lift curve slope for more precision.
+Finally, \rotare allows to remove this hypothesis by calculating the proper lift
+coefficient using the polars provided instead of assuming the linear law.
+However, this requires an additional iterative scheme and extends a bit the
+calculation time.
+
+\subsubsection{Equations}
+\textit{The complete derivation of the equations is left out of this manual.
+Please refer to~\cite{leishman2006a} for more details.}
+
+The BEMT system formed by (\ref{eq:tech:solvers:BET:thrust},
+\ref{eq:tech:solvers:BET:power}, \ref{eq:tech:solvers:mom:thrust} and
+\ref{eq:tech:solvers:mom:power}) can be rewritten by taking into account the
+assumptions just described.
+
+By introducing some interesting non-dimensional factors and ratios such as the
+local solidity $\solL$, the relative position of the element $\radR$ or the
+inflow ratio, $\infRat$; the thrust equation can be rewritten as
+\fxwarning{Verify system and solution}
+
+\begin{equation}
+ \label{eq:tech:solvers:leishman:eq:syst}
+ \infRat^2 + \left( \dfrac{\solL\cLiftSlope}{4\prandtl}\radR - \infRat_\infty \right)
+ \infRat - \dfrac{\solL\cLiftSlope}{4\prandtl}(\pitch)\radR^2 = 0
+\end{equation}
+where $\prandtl$ is the Prandtl tip-loss factor (see
+Section~\ref{sec:tech:ext:loss}). This equation is a quadratic expression for
+the inflow ratio $\infRat$ whose solution is:
+\begin{equation}
+ \label{eq:tech:solvers:leishman:eq:infRat}
+ \infRat(\radR,\infRat_\infty) = \sqrt{\left( \dfrac{\solL\cLiftSlope}{8\prandtl}
+ \radR - \dfrac{\infRat_\infty}{2} \right)^2 +
+ \dfrac{\solL\cLiftSlope}{4\prandtl}(\pitch)\radR^2} -
+ \left( \dfrac{\solL\cLiftSlope}{8\prandtl}\radR -
+ \dfrac{\infRat_\infty}{2} \right)^2
+\end{equation}
+
+If the linear lift coefficient assumption is removed, then the inflow equations
+becomes: \fxwarning{Verify system and solution}
+\begin{equation}
+ \label{eq:tech:solvers:leishman:eq:infRat2}
+ \infRat(\radR,\infRat_\infty) = \dfrac{\infRat_\infty}{2} +
+ \sqrt{\left( \dfrac{\solL\cLift \radR^2}{4\prandtl} -
+ \dfrac{\infRat_\infty^2}{4} \right)}
+\end{equation}
+% IndFact
+% ***********************************************
+
\subsection{Induction factor}
\label{chap:tech:solvers:indfact}
This solver is commonly used for the study of propellers or wind turbines. It
does not rely on additional assumptions, but rather on a reformulation of the
-momentum equations in terms of induction factors. This induction factors lighten
-a bit the formulation. The main drawback is that they are not compatible for the
-analysis of rotors with zero external velocity (such as helicopter in hover or
-propellers in idle).
+momentum equations in terms of induction factors. This formulation lighten a bit
+the expressions and has the benefit to be quite intuitive. Its main drawback is
+that the new equations are not directly compatible with the analysis of rotors
+at zero external velocity (such as helicopter in hover or propellers in idle).
-We start by defining the axial and tangential (swirl) induction factors:
+The equations are derived by first defining the axial and tangential (swirl)
+induction factors:
\begin{align}
\label{eq:tech:solvers:indfact:a}
\vAx &= (1+\axFact)\vAir \\
\label{eq:tech:solvers:indfact:b}
\vTg &= (1-b)\angVel \radA
\end{align}
-and then inject these in the momentum equations \ref{eq:tech:solvers:mom:thrust}
-and (\ref{eq:tech:solvers:mom:torque}).
+and then inject them in the momentum equations
+(\ref{eq:tech:solvers:mom:thrust}, \ref{eq:tech:solvers:mom:torque}).
-As it can be seen directly in EQ.\ref{eq:tech:solvers:indfact:a}, this
+As it can be seen directly in \eqref{eq:tech:solvers:indfact:a}, this
formulation falls down when the free stream velocity, $\vAir$, is zero.
-In that situation, it is more interesting to keep the original formulation for
-the thrust equation (in terms of velocities) and only use the tangential inflow
-ratio in the torque equation.
+To circumvent that limitation, \rotare keeps the original form of
+\eqref{eq:tech:solvers:mom:thrust} when the external velocity is zero but still
+uses \eqref{eq:tech:solvers:indfact:b} in the torque equation.
-Using the inflow factors, the thrust equations becomes:
+Using the inflow factors, the thrust equations becomes
\begin{equation}
\label{eq:tech:solvers:indfact:thurst}
\begin{split}
@@ -210,9 +423,9 @@ Using the inflow factors, the thrust equations becomes:
& = 4\pi \dens (1 + \axFact) a \vAir^2 \radA dy
\end{split}
\end{equation}
-, and the torque equation becomes
+and the torque equation becomes
\begin{equation}
- \label{eq:tech:solvers:indfact:thurst}
+ \label{eq:tech:solvers:indfact:torque}
\begin{split}
d\torque & = 2\pi \radA \dens (\vAx + \viAx) (2\tgFact\angVel\radA)
\radA dy \\
@@ -220,13 +433,17 @@ Using the inflow factors, the thrust equations becomes:
\end{split}
\end{equation}
+
\subsubsection{Resolution}
The system made of the Blade Element equations and the new form of the momentum
equations is then solved using \matlab's \lst{solve} function in order to
determine the value of both induction factors. The solution is initialized with
-an axial inflow factor of 0.01 and assuming no swirl is induced.
+an axial inflow factor of 0.01 and a tangential induction factor of 0 (no
+swirl).
+% IndVel
+% ***********************************************
\subsection{Induced velocities}
\label{chap:tech:solvers:indvel}
@@ -242,6 +459,9 @@ by considering that the axial induced velocity is 0.01 m/s and the tangential
component is 0.
+% Stahlhut
+% ***********************************************
+
\subsection{Stahlhut solver}
\label{chap:tech:solvers:stahlhut}
This solver relies on a complete rewriting of the system in a single nonlinear
@@ -251,6 +471,10 @@ the equation formed is much more complex and less intuitive than the ones at the
base of the system.
+% ==============================================================================
+% COAXIAL
+% ==============================================================================
+
\section{Coaxial rotors}
\label{chap:tech:solver:coax}
@@ -266,6 +490,10 @@ The second rotor is then evaluated as if it was in isolation as well, but with
the new inlet velocity profile.
+% ==============================================================================
+% OBLIQUE
+% ==============================================================================
+
\section{Oblique flows}
\label{chap:tech:solver:oblique}
diff --git a/preamble.tex b/preamble.tex
index a9c61683e10c485b8cda45f6a62e00d800eb89af..2efd8b7b29b5612e869b72e4d1efd87adfe92d87 100644
--- a/preamble.tex
+++ b/preamble.tex
@@ -80,7 +80,8 @@
% Style
\usepackage{ULiege-colors} % ULiege color theme
\usepackage{scrhack} % So other packages play nice with KOMA-Scripts
-\usepackage{booktabs} % Better looking tables
+\usepackage{booktabs, tabularx, multirow} % Better looking tables
+
% Functionality
\usepackage{amsmath} % Math
diff --git a/rotare-bib.bib b/rotare-bib.bib
index f6dbe1b2821fc86a705c6c21e614bd0d5b7e0cdb..e72ca7e53eb08202e7d8f76e7e54a0f4fe8d882b 100644
--- a/rotare-bib.bib
+++ b/rotare-bib.bib
@@ -1,14 +1,26 @@
+@book{leishman2006a,
+ title = {Principles of Helicopter Aerodynamics},
+ author = {Leishman, J. Gordon},
+ date = {2006},
+ series = {Cambridge Aerospace Series},
+ edition = {Second edition},
+ publisher = {{Cambridge University Press}},
+ location = {{New York, NY, USA}},
+ isbn = {978-0-521-85860-1},
+ langid = {english},
+ pagetotal = {826},
+ keywords = {helicopters},
+ annotation = {OCLC: ocm61463625}
+}
-@inproceedings{stahlhut2012,
+@inproceedings{stahlhut2012a,
title = {Aerodynamic Design Optimization of Proprotors for Convertible-Rotor Concepts},
booktitle = {American {{Helicopter Society}} 68th {{Annual Forum}}},
author = {Stahlhut, Conor W. and Leishman, J Gordon},
- year = {2012},
- month = may,
+ date = {2012-05-01/2012-05-03},
pages = {1--24},
publisher = {{American Helicopter Society}},
- address = {{Fort Worth, Texas, USA}},
+ location = {{Fort Worth, Texas, USA}},
keywords = {\#nosource,helicopters,IMPORTANT,methods/BEMT,rotors}
}
-