From 4b8ddf315815084e6644576293f02858579078c2 Mon Sep 17 00:00:00 2001
From: Thomas Lambert <t.lambert@uliege.be>
Date: Mon, 31 Oct 2022 11:01:29 +0100
Subject: [PATCH] style(nomencl): major symbols adjustments
---
contents/tech-solvers.tex | 75 +++++++++++++---------
preamble.tex | 4 +-
style/symbols.sty | 129 ++++++++++++++++++++++----------------
3 files changed, 122 insertions(+), 86 deletions(-)
diff --git a/contents/tech-solvers.tex b/contents/tech-solvers.tex
index 2c7ec4b..29001ac 100644
--- a/contents/tech-solvers.tex
+++ b/contents/tech-solvers.tex
@@ -6,7 +6,7 @@
% https://gitlab.uliege.be/thlamb/Rotare-doc
%
% License: CC-BY-SA-4.0
-% This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
+% This work is licensed under a Creative Commons Attribution-Share Alike 4.0
% International License (https://creativecommons.org/licenses/by-sa/4.0/)
% ------------------------------------------------------------------------------
@@ -24,6 +24,7 @@ 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}
@@ -35,16 +36,16 @@ be represented as a simple 2D airfoil section.
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
+lies at a total pitch angle $\pitch$ with respect to the rotor disk. This pitch
can be decomposed in different contributions:
\begin{equation}
- \aPitch = \aTwist + \aColl
+ \pitch = \twist + \coll
\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
+Where $\twist$ is the twist angle of the blade (\ie a radial variation of
+pitch along the blade) and $\coll$ 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.
@@ -56,12 +57,12 @@ 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 $\aAzim$.
+which can be represented though the azimuthal angle $\azim$.
\begin{align}
\label{eq:tech:solvers:BET:vAx}
- \vAx &= \airspeed \sin\aoaR + \iAx \\
+ \vAx &= \vAir \sin\aoaR + \viAx \\
\label{eq:tech:solvers:BET:vTg}
- \vTg &= \airspeed \cos\aoaR \sin\aAzim + \angVel \radA - \iTg \\
+ \vTg &= \vAir \cos\aoaR \sin\azim + \angVel \radA - \viTg \\
\label{eq:tech:solvers:BET:vRel}
\vRel &= \sqrt{\vAx^2+\vTg^2}
\end{align}
@@ -81,7 +82,7 @@ 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 = (\aPitch - \aoa_0) - \aInd
+ \aoa = (\pitch - \aoa_0) - \aInd
\end{equation}
The angle of attack, along with the Reynolds number can be used to determine the
@@ -118,30 +119,42 @@ contribution of the element
\section{Momentum theory}
\label{chap:tech:solvers:mom}
-
-
-
-Mass flow rate through the disk:
-
+The momentum theory considers the rotor as an infinitesimally thin actuator
+disk. Following this theory, we consider that
+\begin{itemize}
+ \item the flow velocity and pressure are constant on every section normal to
+ the throughflow in the rotor stream tube;
+ \item the rotor induces an abrupt change of the flow conditions by causing a
+ pressure jump across the disk.
+\end{itemize}
+Following the Blade Element Theory, the overall control volume can be further
+refined in infinitesimal concentrically arranged annuli whose area is expressed
+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 (\aVel_A + \indAxVel) d\area = \dens (\aVel_A + \indAxVel) 2 \pi y dy
+ \mFlow = \dens (\vAx + \viAx) d\area = \dens (\vAx + \viAx) 2 \pi y dy
\end{equation}
-Thrust:
+The axial and tangential momentum balance through the disk give:
\begin{equation}
\label{eq:tech:solvers:mom:thrust}
\begin{split}
- d\thrust & = \mFlow \indAxWake \\
- & = \mFlow (2\indAxVel) \\
- & = 4\pi \dens (\aVel_A + \indAxVel) \indAxVel y dy
+ d\thrust & = \mFlow (\vAx[d] - \vAx[u]) \\
+ & = \mFlow \vSlip \\
+ & = \mFlow (2\viAx) \\
+ & = 4\pi \dens (\vAx + \viAx) \viAx y dy
\end{split}
\end{equation}
+
+
+
+
Likewise, the torque is given by
\begin{equation}
\label{eq:tech:solvers:mom:torque}
\begin{split}
- d\torque & = \mFlow \radA \indSwirl
+ d\torque & = \mFlow \radA \viTg
\end{split}
\end{equation}
@@ -152,7 +165,7 @@ Likewise, the torque is given by
\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}).
+\todo{Ref for momentum EQ}).
These solver differs by introducing some additional assumptions or by modifying
the nonlinear system of equations (mostly the momentum equations) in order to
@@ -176,15 +189,15 @@ propellers in idle).
We start by defining the axial and tangential (swirl) induction factors:
\begin{align}
\label{eq:tech:solvers:indfact:a}
- \aVel_A &= (1+\axFact)\aVel_\infty \\
+ \vAx &= (1+\axFact)\vAir \\
\label{eq:tech:solvers:indfact:b}
- \aVel_T &= (1-b)\angVel \radA
+ \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}).
-As it can be seen directly in Eq.\ref{eq:tech:solvers:indfact:a}, this
-formulation falls down when the free stream velocity, $\aVel_\infty$, is zero.
+As it can be seen directly in EQ.\ref{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.
@@ -193,23 +206,23 @@ 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 \radA dy \\
- & = 4\pi \dens (1 + \axFact) a \aVel_\infty^2 \radA dy
+ d\thrust & = 4\pi \dens (\vAx + \viAx) \viAx \radA dy \\
+ & = 4\pi \dens (1 + \axFact) a \vAir^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 \radA \dens (\aVel_A + \indAxVel) (2\tgFact\angVel\radA)
+ d\torque & = 2\pi \radA \dens (\vAx + \viAx) (2\tgFact\angVel\radA)
\radA dy \\
- & = 4\pi \radA^3 \dens (1 + \axFact) \aVel_\infty \tgFact \angVel dy
+ & = 4\pi \radA^3 \dens (1 + \axFact) \vAir \tgFact \angVel dy
\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{fsolve} function in order to
+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.
@@ -223,7 +236,7 @@ rewrite or additional assumptions. It is a bit more formal, but has the benefit
of being directly usable in any flow condition.
\subsubsection{Resolution}
-The system is solved using \matlab's \lst{fsolve} function in order to determine
+The system is solved using \matlab's \lst{solve} function in order to determine
directly the induced velocities at the rotor disk. The solution is initialized
by considering that the axial induced velocity is 0.01 m/s and the tangential
component is 0.
diff --git a/preamble.tex b/preamble.tex
index 754cf6b..a9c6168 100644
--- a/preamble.tex
+++ b/preamble.tex
@@ -31,7 +31,7 @@
% Encoding and language
% ***********************************************
\PassOptionsToPackage{utf8}{inputenc}
-\usepackage{inputenc}
+%\usepackage{inputenc} Not needed for lualatex
\usepackage[english]{babel}
% Debugging
@@ -96,5 +96,5 @@
\usepackage{transparent}
% Fixme for todo messages in text
-\usepackage[draft, inline, marginclue]{fixme}
+\usepackage[draft, nomargin, inline, marginclue]{fixme}
\fxsetup{theme=color}
diff --git a/style/symbols.sty b/style/symbols.sty
index e9817b7..a1c9dbe 100644
--- a/style/symbols.sty
+++ b/style/symbols.sty
@@ -11,100 +11,118 @@
% https://gitlab.uliege.be/thlamb/rotare-doc
% ------------------------------------------------------------------------------
-% Package for the nomenclature and layout tweaks
-\RequirePackage[stdsubgroups]{nomencl}
+
+% ==============================================================================
+% PACKAGES AND OPTIONS
+% ==============================================================================
+
+\RequirePackage[stdsubgroups]{nomencl} % Nomenclature management
\makenomenclature
+\RequirePackage {siunitx} % SI units
+
+% Tweak layout and display of units in nomenclature
\newcommand{\nomunit}[1]{%
\renewcommand{\nomentryend}{\hspace*{\fill}\si{#1}}%
}
+
+% ==============================================================================
+% GENERAL SYMBOLS
+% ==============================================================================
+
% --- Velocities ---------------------------------------------------------------
% Main velocities components
-\newcommand{\vAx}{\ensuremath{V_\perp}}
-\newcommand{\vTg}{\ensuremath{V_\theta}}
-\newcommand{\vRel}{\ensuremath{V}}
-\newcommand{\airspeed}{\ensuremath{V_\infty}}
+\newcommand{\vAx}[1][]{\ensuremath{V_{#1}}}
+\newcommand{\vTg}[1][]{\ensuremath{U_{#1}}}
+\newcommand{\vRel}[1][]{\ensuremath{W}}
+\newcommand{\vAir}{\ensuremath{V_\infty}}
\nomenclature[A]{\vRel}{Relative velocity\nomunit{\m\per\s}}
\nomenclature[A]{\vAx}{Axial velocity\nomunit{\m\per\s}}
\nomenclature[A]{\vTg}{Tangential velocity\nomunit{\m\per\s}}
-\nomenclature[A]{\airspeed}{Freestream velocity\nomunit{\m\per\s}}
+\nomenclature[A]{\vAir}{Freestream velocity\nomunit{\m\per\s}}
% Induced velocities
-\newcommand\iAx[1][]{\ensuremath{v_{#1i}}}
-\newcommand{\iTg}[1][]{\ensuremath{u_{#1i}}}
+\newcommand{\viAx}{\ensuremath{v_{i}}}
+\newcommand{\viTg}{\ensuremath{u_{i}}}
+\newcommand{\vSlip}{\ensuremath{w}}
-\nomenclature[A]{\iAx}{Induced axial velocity\nomunit{\m\per\s}}
-\nomenclature[A]{\iTg}{Induced tangential velocity\nomunit{\m\per\s}}
-
-
-
-\newcommand{\indSwirl}{\ensuremath{u_i}}
-\newcommand{\indAxVel}{\ensuremath{v_i}}
-\newcommand{\aVel}{\ensuremath{V}}
-\newcommand{\indAxWake}{\ensuremath{v_w}}
-\newcommand{\indSwirlWake}{\ensuremath{u_w}}
+\nomenclature[A]{\viAx}{Induced axial velocity\nomunit{\m\per\s}}
+\nomenclature[A]{\viTg}{Induced tangential velocity\nomunit{\m\per\s}}
+\nomenclature[A]{\vSlip}{Slipsteam velocity in the far wake\nomunit{\m\per\s}}
+% Induction factors
\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
+% Other velocities
+\newcommand{\angVel}{\ensuremath{\Omega}}
\nomenclature[G]{\angVel}{Angular velocity\nomunit{\per\s}}
% --- Angles -------------------------------------------------------------------
-\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{\twist}{\ensuremath{\chi}}
+\newcommand{\sweep}{\ensuremath{\Lambda}}
+\newcommand{\coll}{\ensuremath{\beta_0}}
+\newcommand{\pitch}{\ensuremath{\beta}}
\newcommand{\aoaR}{\ensuremath{\zeta}}
+\newcommand{\azim}{\ensuremath{\psi}}
+\newcommand{\aInd}{\ensuremath{\phi}}
-\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]{\twist}{Twist angle (stagger for propellers)\nomunit{}}
+\nomenclature[G]{\sweep}{Blade sweep angle\nomunit{}}
+\nomenclature[G]{\coll}{Collective pitch angle\nomunit{}}
+\nomenclature[G]{\pitch}{Pitch angle\nomunit{}}
\nomenclature[G]{\aoaR}{Rotor angle of attack\nomunit{}}
+\nomenclature[G]{\azim}{Azimuthal angle\nomunit{}}
+\nomenclature[G]{\aInd}{Induced angle\nomunit{}}
-% --- Others -------------------------------------------------------------------
-\newcommand{\area}{\ensuremath{A}} % (Rotor) Area
-\nomenclature[A]{\area}{Area\nomunit{\m^2}}
-
+% --- Lengths ------------------------------------------------------------------
-\newcommand{\chord}{\ensuremath{c}} % Rotor radius
-\newcommand{\radius}{\ensuremath{R}} % Rotor radius
-\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
-\newcommand{\mFlow}{\ensuremath{\dot{m}}} % Mass flow rate
-\newcommand{\dy}{\ensuremath{dy}} % Mass flow rate
+% Lengths
+\newcommand{\chord}{\ensuremath{c}}
+\newcommand{\radius}{\ensuremath{R}}
+\newcommand{\radR}{\ensuremath{r}}
+\newcommand{\radA}{\ensuremath{y}}
+\newcommand{\dy}{\ensuremath{dy}}
\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}}
+% Surfaces
+\newcommand{\area}{\ensuremath{A}} % (Rotor) Area
+\nomenclature[A]{\area}{Area\nomunit{\m^2}}
+
+% --- Non-dimensional ----------------------------------------------------------
+\newcommand{\advRat}{\ensuremath{\mathcal{J}}} % Advance ratio
\newcommand{\mach}{\ensuremath{\mathcal{M}}} % Mach number
\newcommand{\reynolds}{\ensuremath{Re}} % Reynolds number
-\newcommand{\dens}{\ensuremath{\rho}} % Density
-\newcommand{\nBlades}{\ensuremath{N_b}} % Number of blades
+% solidity
+\nomenclature[A]{\advRat}{Advance ratio\nomunit{-}}
\nomenclature[A]{\mach}{Mach number\nomunit{-}}
\nomenclature[A]{\reynolds}{Reynolds number\nomunit{-}}
+
+
+% --- Others -------------------------------------------------------------------
+\newcommand{\dens}{\ensuremath{\rho}}
+\newcommand{\mFlow}{\ensuremath{\dot{m}}}
+
\nomenclature[A]{\dens}{Density\nomunit{kg\per\m^3}}
+\nomenclature[A]{\mFlow}{Mass flow rate\nomunit{\kg\per\s}}
+
+\newcommand{\nBlades}{\ensuremath{N_b}}
\nomenclature[A]{\nBlades}{Number of blades\nomunit{-}}
+
% --- Forces, torques, powers, etc ---------------------------------------------
% Foces
@@ -145,14 +163,19 @@
\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}
+% ==============================================================================
+% Subscripts
+% ==============================================================================
+
+\nomenclature[Z]{$\left(\ \right)_u$}{Upstream of the rotor}
+\nomenclature[Z]{$\left(\ \right)_d$}{Far downstream of the rotor (slipstream)}
+
+
+% ==============================================================================
+% COORDINATES
+% ==============================================================================
-% Coordinate system
\newcommand{\xtpp}{\ensuremath{x_{TPP}}}
\newcommand{\ytpp}{\ensuremath{y_{TPP}}}
\newcommand{\ztpp}{\ensuremath{z_{TPP}}}
-
--
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