feat(solvers): BET part

contents/tech-solvers.tex

@@ -13,27 +13,114 @@
 \chapter{Solvers}
 \label{chap:tech:solvers}
 
+\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.}
 
 \section{Blade Element theory}
 \label{chap:tech:solvers:BET}
 
-Thrust, torque and power for the whole rotor:
+The Blade Element Theory postulates that a rotor can be represented as a
+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.
+
+\begin{figure}[ht]
+  \begin{center}
+    \incfig[0.8]{bladeElement}
+    \caption{Velocity diagram for a blade element $dy$ at location $y$.}
+    \label{fig:tech:blade-element}
+  \end{center}
+\end{figure}
+
+The representation in Figure~\ref{fig:tech:blade-element} showcases a single
+blade element with its associated velocity triangles and forces. The element
+lies at a total pitch angle $\aPitch$ with respect to the rotor disk. This pitch
+can be decomposed in different contributions:
+
+\begin{equation}
+  \aPitch = \aTwist + \aColl
+  \label{eq:tech:solv:pitch}
+\end{equation}
+
+Where $\aTwist$ is the twist angle of the blade (\ie a radial variation of
+pitch along the blade) and $\aColl$ is the collective pitch of the blade (\ie a
+modification of the pitch constant over the entire blade). Not that, by
+convention, the pitch angle is usually given with respect to the
+\emph{zero-lift line} and not the \emph{chord line} of the airfoil.
+
+The resultant velocity is the sum of the velocity components acting in the
+$\xtpp$ and $\ztpp$ directions, respectively $\vTg$ and $\vAx$.\footnote{There
+  is no contribution in $\ytpp$ as the BEMT represents the blade through 2D
+elements}.
+
+In oblique flows (\ie when the flow is not perfectly perpendicular to the rotor
+disk), the tangential velocity is dependent on blade instantaneous position,
+which can be represented though the azimuthal angle $\aAzim$.
+\begin{align}
+  \label{eq:tech:solvers:BET:vAx}
+  \vAx &= \airspeed \sin\aoaR + \iAx \\
+  \label{eq:tech:solvers:BET:vTg}
+  \vTg &= \airspeed \cos\aoaR \sin\aAzim + \angVel \radA - \iTg \\
+  \label{eq:tech:solvers:BET:vRel}
+  \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:
+\begin{equation}
+  \label{eq:tech:solvers:BET:aInd}
+  \tan^{-1}\aInd = \dfrac{\vAx}{\vTg}
+\end{equation}
+By convention, $\aInd$ 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
+difference of the element's pitch and the inflow angle. Note that, as the pitch
+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 = (\aPitch - \aoa_0) - \aInd
+\end{equation}
+
+The angle of attack, along with the Reynolds number can be used to determine the
+value of the airfoil lift and drag coefficient. The airfoil polar coefficients
+are usually given in the form of tabulated data, either directly from \XFOIL or
+\XFLR or via other means.
+\begin{equation}
+  \label{eq:tech:solvers:BET:clcd}
+  \cLift(\aoa,\reynolds) \qquad \text{and} \qquad \cDrag(\aoa,\reynolds)
+\end{equation}
+
+The elemental lift and drag can now be obtained though the known aerodynamic
+coefficients, alongside the dynamic pressure and the element's chord.
+\begin{equation}
+  \label{eq:tech:solvers:BET:dLdD}
+  d\lift = \dfrac{1}{2}\dens \vRel^2 \chord \cLift \dy
+  \qquad
+  \qquad
+  d\drag = \dfrac{1}{2}\dens \vRel^2 \chord \cDrag \dy
+\end{equation}
+
+Finally, the more useful notation consist in replacing the forces in the axes of
+the rotor and derive the thrust (\thrust), torque (\torque) and power (\power)
+contribution of the element
 \begin{align}
   \label{eq:tech:solvers:BET:thrust}
-  \thrust &= \nBlades (d\lift \sin\indAngle - d\drag \cos\indAngle)\\
+  d\thrust &= \nBlades (d\lift \sin\aInd - d\drag \cos\aInd)\\
   \label{eq:tech:solvers:BET:torque}
-  \torque &= \nBlades (d\lift \cos\indAngle + d\drag \sin\indAngle) \aRad\\
+  d\torque &= \nBlades (d\lift \cos\aInd + d\drag \sin\aInd) \radA\\
   \label{eq:tech:solvers:BET:power}
-  \power &= \nBlades (d\lift \cos\indAngle + d\drag \sin\indAngle) \aRad \angVel
+  d\power &= \nBlades (d\lift \cos\aInd + d\drag \sin\aInd) \radA \angVel
 \end{align}
-, where $\indAngle$ is the induced angle:
-\begin{equation}
-  \label{eq:tech:solvers:BET:indAngle}
-  \tan^{-1}\indAngle = \dfrac{\aVel_A}{\aVel_T}
-\end{equation}
 
 \section{Momentum theory}
 \label{chap:tech:solvers:mom}
+
+
+
+
 Mass flow rate through the disk:
 
 \begin{equation}
@@ -54,7 +141,7 @@ Likewise, the torque is given by
 \begin{equation}
   \label{eq:tech:solvers:mom:torque}
   \begin{split}
-    d\torque & = \mFlow \aRad \indSwirl
+    d\torque & = \mFlow \radA \indSwirl
   \end{split}
 \end{equation}
 
@@ -91,7 +178,7 @@ We start by defining the axial and tangential (swirl) induction factors:
   \label{eq:tech:solvers:indfact:a}
   \aVel_A &= (1+\axFact)\aVel_\infty \\
   \label{eq:tech:solvers:indfact:b}
-  \aVel_T &= (1-b)\angVel \aRad
+  \aVel_T &= (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}).
@@ -106,16 +193,17 @@ Using the inflow factors, the thrust equations becomes:
 \begin{equation}
   \label{eq:tech:solvers:indfact:thurst}
   \begin{split}
-    d\thrust & = 4\pi \dens (\aVel_A + \indAxVel) \indAxVel \aRad dy \\
-             & = 4\pi \dens (1 + \axFact) a \aVel_\infty^2 \aRad dy
+    d\thrust & = 4\pi \dens (\aVel_A + \indAxVel) \indAxVel \radA dy \\
+             & = 4\pi \dens (1 + \axFact) a \aVel_\infty^2 \radA dy
   \end{split}
 \end{equation}
 , and the torque equation becomes
 \begin{equation}
   \label{eq:tech:solvers:indfact:thurst}
   \begin{split}
-    d\torque & = 2\pi \aRad \dens (\aVel_A + \indAxVel) (2\tgFact\angVel\aRad) \aRad dy \\
-             & = 4\pi \aRad^3 \dens (1 + \axFact) \aVel_\infty \tgFact \angVel dy
+    d\torque & = 2\pi \radA \dens (\aVel_A + \indAxVel) (2\tgFact\angVel\radA)
+    \radA dy \\
+             & = 4\pi \radA^3 \dens (1 + \axFact) \aVel_\infty \tgFact \angVel dy
   \end{split}
 \end{equation}
 

preamble.tex

@@ -96,5 +96,5 @@
 \usepackage{transparent}
 
 % Fixme for todo messages in text
-\usepackage[inline, marginclue]{fixme}
+\usepackage[draft, inline, marginclue]{fixme}
 \fxsetup{theme=color}

style/definitions.sty

@@ -28,6 +28,8 @@
 \newcommand\QBlade{\textsc{QBlade}\xspace}
 \newcommand\JBlade{\textsc{JBlade}\xspace}
 \newcommand\pyBEMT{\textsc{pyBEMT}\xspace}
+\newcommand\XFOIL{\textsc{XFOIL}\xspace}
+\newcommand\XFLR{\textsc{XFLR5}\xspace}
 \newcommand\AeroDyn{\textsc{AeroDyn}\xspace}
 \newcommand\git{\textsf{git}\xspace}
 

style/symbols.sty

@@ -21,112 +21,138 @@
 % --- Velocities ---------------------------------------------------------------
 
 % Main velocities components
-\newcommand{\vAx}{\ensuremath{V}} % Absolute axial velocity
+\newcommand{\vAx}{\ensuremath{V_\perp}}
+\newcommand{\vTg}{\ensuremath{V_\theta}}
+\newcommand{\vRel}{\ensuremath{V}}
+\newcommand{\airspeed}{\ensuremath{V_\infty}}
+
+\nomenclature[A]{\vRel}{Relative velocity\nomunit{\m\per\s}}
 \nomenclature[A]{\vAx}{Axial velocity\nomunit{\m\per\s}}
-\newcommand{\vTg}{\ensuremath{U}} % Absolute tangential velocity
 \nomenclature[A]{\vTg}{Tangential velocity\nomunit{\m\per\s}}
-\newcommand{\rVel}{\ensuremath{W}} % Relative velocity
-\nomenclature[A]{\rVel}{Relative velocity\nomunit{\m\per\s}}
+\nomenclature[A]{\airspeed}{Freestream velocity\nomunit{\m\per\s}}
+
 % Induced velocities
-\newcommand\iAx[1][]{\ensuremath{v_{#1i}}} % Induced axial velocity
+\newcommand\iAx[1][]{\ensuremath{v_{#1i}}}
+\newcommand{\iTg}[1][]{\ensuremath{u_{#1i}}}
+
 \nomenclature[A]{\iAx}{Induced axial velocity\nomunit{\m\per\s}}
-\newcommand{\iTg}[1][]{\ensuremath{u_{#1i}}} % Induced tangential velocity
 \nomenclature[A]{\iTg}{Induced tangential velocity\nomunit{\m\per\s}}
 
-\newcommand{\indSwirl}{\ensuremath{u_i}} % Swirl velocity, in-plane toward rot blade
-\newcommand{\indAxVel}{\ensuremath{v_i}} % Induced velocity, normal to rotor dis
-\newcommand{\airspeed}{\ensuremath{V_\infty}} % Freestream velocity
-\nomenclature[A]{\airspeed}{Freestream velocity\nomunit{\m\per\s}}
-\newcommand{\aVel}{\ensuremath{V}} % Axial velocity component
-\newcommand{\indAxWake}{\ensuremath{v_w}} % Slipstream velocity downstream
-\newcommand{\indSwirlWake}{\ensuremath{u_w}} % Slipstream velocity downstream
-\newcommand{\axFact}{\ensuremath{a}} % Axial induction factor
-\nomenclature[A]{\axFact}{Axial induction factor\nomunit{-}}
-\newcommand{\tgFact}{\ensuremath{b}} % Tangential induciton factor
-\nomenclature[A]{\tgFact}{Tangential induction factor\nomunit{-}}
 
 
+\newcommand{\indSwirl}{\ensuremath{u_i}}
+\newcommand{\indAxVel}{\ensuremath{v_i}}
+\newcommand{\aVel}{\ensuremath{V}}
+\newcommand{\indAxWake}{\ensuremath{v_w}}
+\newcommand{\indSwirlWake}{\ensuremath{u_w}}
+
+\newcommand{\axFact}{\ensuremath{a}}
+\newcommand{\tgFact}{\ensuremath{b}}
+
+\nomenclature[A]{\axFact}{Axial induction factor\nomunit{-}}
+\nomenclature[A]{\tgFact}{Tangential induction factor\nomunit{-}}
 
 \newcommand{\angVel}{\ensuremath{\Omega}} % Angular velocity
 \nomenclature[G]{\angVel}{Angular velocity\nomunit{\per\s}}
 
 
 % --- Angles -------------------------------------------------------------------
-\newcommand{\sweep}{\ensuremath{\Lambda}} % Sweep
-\nomenclature[G]{\sweep}{Blade sweep angle\nomunit{}}
-\newcommand{\indAngle}{\ensuremath{\phi}} % Induced angle
-\nomenclature[G]{\phi}{Induced angle\nomunit{}}
-\newcommand{\aoa}{\ensuremath{\alpha}} % Angle of attack
+\newcommand{\aSweep}{\ensuremath{\Lambda}}
+\newcommand{\aInd}{\ensuremath{\phi}}
+\newcommand{\aoa}{\ensuremath{\alpha}}
+\newcommand{\aPitch}{\ensuremath{\beta}}
+\newcommand{\aColl}{\ensuremath{\beta_0}}
+\newcommand{\aTwist}{\ensuremath{\chi}}
+\newcommand{\aAzim}{\ensuremath{\psi}}
+\newcommand{\aoaR}{\ensuremath{\zeta}}
+
+\nomenclature[G]{\aSweep}{Blade sweep angle\nomunit{}}
+\nomenclature[G]{\aInd}{Induced angle\nomunit{}}
 \nomenclature[G]{\aoa}{Angle of attack\nomunit{}}
+\nomenclature[G]{\aPitch}{Pitch angle\nomunit{}}
+\nomenclature[G]{\aColl}{Collective pitch angle\nomunit{}}
+\nomenclature[G]{\aTwist}{Twist angle (stagger for propellers)\nomunit{}}
+\nomenclature[G]{\aAzim}{Azimuthal angle\nomunit{}}
+\nomenclature[G]{\aoaR}{Rotor angle of attack\nomunit{}}
 
 % --- Others -------------------------------------------------------------------
 \newcommand{\area}{\ensuremath{A}} % (Rotor) Area
 \nomenclature[A]{\area}{Area\nomunit{\m^2}}
 
 
+\newcommand{\chord}{\ensuremath{c}} % Rotor radius
 \newcommand{\radius}{\ensuremath{R}} % Rotor radius
-\nomenclature[A]{\radius}{Rotor radius\nomunit{\m}}
-\newcommand{\rRad}{\ensuremath{r}} % Nondimensional radial position along the blade
-\nomenclature[A]{\rRad}{Nondimensional radial position\nomunit{-}}
-\newcommand{\aRad}{\ensuremath{y}} % Radial distance along the blade
-\nomenclature[A]{\aRad}{Absolute radial position\nomunit{\m}}
+\newcommand{\radR}{\ensuremath{r}} % Nondimensional radial position along the blade
+\newcommand{\radA}{\ensuremath{y}} % Radial distance along the blade
 \newcommand{\advRat}{\ensuremath{\mathcal{J}}} % Advance ratio
-\nomenclature[A]{\advRat}{Advance ratio\nomunit{-}}
 \newcommand{\mFlow}{\ensuremath{\dot{m}}} % Mass flow rate
+\newcommand{\dy}{\ensuremath{dy}} % Mass flow rate
+
+\nomenclature[A]{\chord}{Rotor chord\nomunit{\m}}
+\nomenclature[A]{\radius}{Rotor radius\nomunit{\m}}
+\nomenclature[A]{\radR}{Nondimensional radial position\nomunit{-}}
+\nomenclature[A]{\radA}{Absolute radial position\nomunit{\m}}
+\nomenclature[A]{\advRat}{Advance ratio\nomunit{-}}
 \nomenclature[A]{\mFlow}{Mass flow rate\nomunit{\kg\per\s}}
 
 \newcommand{\mach}{\ensuremath{\mathcal{M}}} % Mach number
-\nomenclature[A]{\mach}{Mach number\nomunit{-}}
 \newcommand{\reynolds}{\ensuremath{Re}} % Reynolds number
-\nomenclature[A]{\reynolds}{Reynolds number\nomunit{-}}
 \newcommand{\dens}{\ensuremath{\rho}} % Density
-\nomenclature[A]{\dens}{Density\nomunit{kg\per\m^3}}
 \newcommand{\nBlades}{\ensuremath{N_b}} % Number of blades
+
+\nomenclature[A]{\mach}{Mach number\nomunit{-}}
+\nomenclature[A]{\reynolds}{Reynolds number\nomunit{-}}
+\nomenclature[A]{\dens}{Density\nomunit{kg\per\m^3}}
 \nomenclature[A]{\nBlades}{Number of blades\nomunit{-}}
 
 % --- Forces, torques, powers, etc ---------------------------------------------
 
 % Foces
 \newcommand{\lift}{\ensuremath{\mathcal{L}}} % Lift
-\nomenclature[A]{\lift}{Lift\nomunit{\N}}
 \newcommand{\drag}{\ensuremath{\mathcal{D}}} % Drag
-\nomenclature[A]{\drag}{Drag\nomunit{\N}}
 \newcommand{\thrust}{\ensuremath{\mathcal{T}}} % Thrust
-\nomenclature[A]{\thrust}{Thrust\nomunit{\N}}
 \newcommand{\torque}{\ensuremath{\mathcal{Q}}} % Torque
-\nomenclature[A]{\torque}{Torque\nomunit{\N.\m}}
 \newcommand{\power}{\ensuremath{\mathcal{P}}} % Power
-\nomenclature[A]{\power}{Power\nomunit{\W}}
 \newcommand{\ForceX}{\ensuremath{F_x}} % Rotor force along X_TPP-axis
 \newcommand{\ForceY}{\ensuremath{F_y}} % Rotor force along Y_TPP-axis
 \newcommand{\MomX}{\ensuremath{M_x}} % Rotor moment along X_TPP-axis
 \newcommand{\MomY}{\ensuremath{M_y}} % Rotor moment along Y_TPP-axis
 
+\nomenclature[A]{\lift}{Lift\nomunit{\N}}
+\nomenclature[A]{\drag}{Drag\nomunit{\N}}
+\nomenclature[A]{\thrust}{Thrust\nomunit{\N}}
+\nomenclature[A]{\torque}{Torque\nomunit{\N.\m}}
+\nomenclature[A]{\power}{Power\nomunit{\W}}
+
 % Coefficients
 \newcommand{\cLift}{\ensuremath{c_l}} % Sectional lift coefficient
-\nomenclature[A]{\cLift}{Lift coefficient\nomunit{-}}
 \newcommand{\cLiftSlope}{\ensuremath{c_{l,\alpha}}} % Lift curve slope
-\nomenclature[A]{\cLiftSlope}{Lift curve slope\nomunit{-}}
 \newcommand{\cDrag}{\ensuremath{c_d}} % Sectional drag coefficient
-\nomenclature[A]{\cDrag}{Drag coefficient\nomunit{-}}
 \newcommand{\cMom}{\ensuremath{c_m}} % Sectional moment coefficient coefficient
-\nomenclature[A]{\cMom}{Moment coefficient\nomunit{-}}
 \newcommand{\cThrust}{\ensuremath{C_{\thrust}}} % Thrust coefficient
-\nomenclature[A]{\cThrust}{Thrust coefficient\nomunit{-}}
 \newcommand{\cTorque}{\ensuremath{C_{\torque}}} % Torque coefficient
-\nomenclature[A]{\cTorque}{Torque coefficient\nomunit{-}}
 \newcommand{\cPower}{\ensuremath{C_{\power}}} % Power coefficient
-\nomenclature[A]{\cPower}{Power coefficient\nomunit{-}}
 \newcommand{\cForceX}{\ensuremath{C_{F_x}}} % Longitudinal force coefficient
 \newcommand{\cForceY}{\ensuremath{C_{F_y}}} % Lateral side force coefficient
 \newcommand{\cMomX}{\ensuremath{C_{M_x}}} % Rolling moment coefficient
 \newcommand{\cMomY}{\ensuremath{C_{M_y}}} % Pitching moment coefficient
 
-
-
-
+\nomenclature[A]{\cLift}{Lift coefficient\nomunit{-}}
+\nomenclature[A]{\cLiftSlope}{Lift curve slope\nomunit{-}}
+\nomenclature[A]{\cDrag}{Drag coefficient\nomunit{-}}
+\nomenclature[A]{\cMom}{Moment coefficient\nomunit{-}}
+\nomenclature[A]{\cThrust}{Thrust coefficient\nomunit{-}}
+\nomenclature[A]{\cTorque}{Torque coefficient\nomunit{-}}
+\nomenclature[A]{\cPower}{Power coefficient\nomunit{-}}
 
 % Subscripts
 \nomenclature[Z]{$\left(\ \right)_1$}{Upstream of the rotor}
 \nomenclature[Z]{$\left(\ \right)_2$}{At the rotor disk}
 \nomenclature[Z]{$\left(\ \right)_3$}{Far downstream of the rotor}
+
+
+% Coordinate system
+\newcommand{\xtpp}{\ensuremath{x_{TPP}}}
+\newcommand{\ytpp}{\ensuremath{y_{TPP}}}
+\newcommand{\ztpp}{\ensuremath{z_{TPP}}}
+