update theory doc

docs/src/DuctAPE/theory_latex/customcommands.tex

@@ -602,3 +602,21 @@ \section*{Chapter~\thechapter~Appendices}
 
 %%%%% ------ allow newlines inside table cells with the \makecell{} environment
 \usepackage{makecell}
+
+
+% fancy table alignment
+\usepackage{array}
+\newcommand{\PreserveBackslash}[1]{\let\temp=\\#1\let\\=\temp}
+\newcolumntype{C}[1]{>{\PreserveBackslash\centering}m{#1}}
+\newcolumntype{R}[1]{>{\PreserveBackslash\raggedleft}m{#1}}
+
+\usepackage{multirow}
+\usepackage{array, makecell}
+ \renewcommand\theadfont{}
+\renewcommand\theadalign{tc}
+\renewcommand\cellalign{tr}
+\setcellgapes{5pt}
+\newcolumntype{/}{!{\color{plotsgray!50}\vrule width 0.001pt}}
+
+% decimal aligned columns in table
+\usepackage{siunitx}  % for 'S' column type

docs/src/DuctAPE/theory_latex/ductape.tex

@@ -12,7 +12,7 @@ \chapter{DuctAPE Theory}
 
 \input{ductpostprocessing}
 
-\input{ductsolveralgorithm}
+\input{ductverification}
 
 % %%% --- Chapter Appendices --- %%%
 \clearpage

docs/src/DuctAPE/theory_latex/ductbodymethods.tex

@@ -225,7 +225,7 @@ \subsubsection{Discretizing bodies into panels}
 %
 \Cref{fig:flatbandgeometry} shows what is intended by a flat, axisymmetric band, and \cref{fig:flatpanelgeometry} shows the panel representation of said band.
 
-\begin{figure}[htb]
+\begin{figure}[h!]
      \centering
      \begin{subfigure}[t]{0.45\textwidth}
          \centering
@@ -329,28 +329,13 @@ \subsubsection{Calculating Panel Induced Velocities}
 
 Because the surface integrals of velocities induced by axisymmetric vortex rings are exceptionally difficult to solve analytically, we will take a numerical approach.
 %
-Specifically, we will utilize Gauss-Kronrod quadrature via the QuadGK.jl\sidenote{\url{https://juliamath.github.io/QuadGK.jl/stable/gauss-kronrod/}}
-package in the Julia\scite{Julia_2017} language.
-%
 In general, quadrature is the process of approximating an integral of a function using a sum of weighted samples of the function:
 
 \begin{equation}
     \int_a^b f(x) \d x \approx \sum_k^N w_k f(x_k),
 \end{equation}
 
 \where the main task of the setup is to decide where along the integration interval to place the sample points, \(x_k\), and what weights, \(w_k\), to apply to those samples.
-%
-Gauss-Kronrod quadrature is based on Gauss-Legendre quadrature, which uses orthogonal polynomial theory to select sample points and weights that allow for exact integration of polynomials up to degree \(2N-1\) (where \(N\) is the number of sample points), and other sufficiently smooth functions remarkably well.
-%
-Despite being based on Gauss-Legendre quadrature, however, Gauss-Kronrod quadrature is not quite as accurate as pure Gauss-Legendre quadrature in a one-to-one comparison.
-%
-Gauss-Kronrod quadrature can exactly integrate polynomials up to \(3N+1\) for \(2N+1\) sample points.
-%
-The decrease in accuracy for the same conditions is the trade off required to be able to both calculate the integral approximation and the error in the integral approximation.
-%
-The ability to quickly estimate error directly leads to capabilities for \(h\)-adaptive quadrature, which is an adaptive method that refines the integration range along portions that require further refinement for accuracy (such as sharp peaks, or perhaps discontinuities).
-%
-So although not as accurate out of the box, obtaining more accurate integrals over domains that would be difficult for pure Gauss-Legendre quadrature becomes relatively simple, and a worthwhile trade.
 
 In the nominal case when a panel induces velocity on the surface, but not on itself, we set things up as follows for a given panel and surface point, \(t\):
 %
@@ -1393,7 +1378,7 @@ \subsubsection{Multi-body System Verification}
 %
 We now place both in a single system in order to verify that multi-body systems work properly.
 
-\begin{figure}[hb!]
+\begin{figure}[h!]
     \centering
         \input{figures/duct-and-hub-validation-geom.tikz}%
         \caption{Isolated duct and center body geometry together.}
@@ -1402,7 +1387,7 @@ \subsubsection{Multi-body System Verification}
 %
 As can be seen in \cref{fig:dfdclewiscomp}, the surface velocity on the hub and pressure on the duct match very well to DFDC, lending confidence in DuctAPE's ability to model both a duct and hub together.
 
-\begin{figure}[htb]
+\begin{figure}[h!]
      \centering
      \begin{subfigure}[b]{0.45\textwidth}
          \raggedright
@@ -1435,7 +1420,7 @@ \subsubsection{Multi-body System Verification}
 %
 We see in \cref{fig:dfdcexamplecomp} that DuctAPE also matches well with DFDC in this case.
 
-\begin{figure}[htb]
+\begin{figure}[h!]
      \centering
      \begin{subfigure}[b]{0.45\textwidth}
          \raggedright

docs/src/DuctAPE/theory_latex/ductintro.tex

@@ -5,7 +5,7 @@ \section{Introduction}
 %
 It may also be desirable to sacrifice some flexibility to further decrease computation time costs in the beginning stages of design.
 %
-To this end, a low-fidelity tool for the evaluation of ducted propulsors, catered specifically to electric ducted fans, is described in this chapter.
+To this end, a low-fidelity tool for the evaluation of ducted propulsors, catered specifically to electric ducted fans, is described in this document.
 %
 One of the major limitations in flexibility is that we require that:
 
@@ -24,11 +24,13 @@ \section{Introduction}
 
 \end{assumption}
 
-\noindent Therefore, we call our tool Ducted Axisymmetric Propulsor Evaluation, or DuctAPE for short.
+\noindent Therefore, we call our tool \textbf{Duct}ed \textbf{A}xisymmetric \textbf{P}ropulsor \textbf{E}valuation, or DuctAPE for short.
+%
+We also required a steady-state assumption, meaning the solution is stationary in time.
 %
 As alluded to, the overall goal of DuctAPE is to provide a computationally inexpensive tool to be used in a multidisciplinary design and optimization setting for preliminary and conceptual design.
 
-In this chapter we cover the methodology derivation required for DuctAPE as well as some implementation details and various verification and validation along the way.
+In this document we cover the methodology derivation required for DuctAPE as well as some implementation details and various verification and validation along the way.
 %
 The solver is comprised of two major components, which will be first presented in isolation before the full coupling is considered.
 %

docs/src/DuctAPE/theory_latex/ductrotorwakemethods.tex

@@ -298,7 +298,7 @@ \subsubsection{Rotor Profile Drag}
 %
 \Cref{fig:rvf_eif} shows visually this concept.
 %
-\begin{figure}[htb]
+\begin{figure}[h!]
      \centering
      \begin{subfigure}[t]{0.45\textwidth}
         \centering
@@ -488,7 +488,7 @@ \subsection{Wake Model}
     h_B \approx \frac{2\pi}{B} \left(\frac{W_m}{-W_\theta}\right).
 \end{equation}
 
-\begin{figure}[htb]
+\begin{figure}[h!]
      \centering
      \begin{subfigure}[t]{0.45\textwidth}
         \centering
@@ -1674,9 +1674,6 @@ \subsubsection{Elliptic Wake Grid}
 %
 In contrast to the centerbody trailing edge panel, we only apply the vorticity portion of the wake panel strength based on the last wake node along the streamline.
 
-\subsubsection{Solution Approach:}
-re-write
-
 
 \subsection{Verification and Validation of Isolated Rotor+Wake Aerodynamics}
 \label{ssec:rwvv}
@@ -1696,7 +1693,7 @@ \subsubsection{Verification of Induced Velocities}
 Note that the swirl velocity as modeled in DuctAPE is zero upstream of the rotor, and the far-field value at any point aft of the rotor as described by \cref{eqn:vtheta,eqn:vthetaself}.
 
 
-\begin{figure}[htb]
+\begin{figure}[h!]
      \centering
      \begin{subfigure}[t]{0.45\textwidth}
         \centering
@@ -1724,7 +1721,7 @@ \subsubsection{Validation of Thrust and Power Coefficients}
 %
 As can be seen in \cref{fig:rotval1}, DuctAPE matches well with BEMT, and both are within expectations when compared to experimental data.
 
-\begin{figure}[htb]
+\begin{figure}[h!]
      \centering
      \begin{subfigure}[t]{0.45\textwidth}
         \centering