\section{Title}
\newrefsegment

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\section*{Quality Assurance}
\begin{tabular}{@{}p{27mm}@{}p{0mm}p{\textwidth-52mm-48pt}p{15mm}p{10mm}}
\textbf{Date} & \textbf{:} & 16-09-2024  \\ 
\textbf{Author} & \textbf{:} & Victor Chavarrias  & \textbf{Initials:} & VC \\ 
\textbf{Review} & \textbf{:} & Bert Jagers  & \textbf{Initials:} & \ldots 
\end{tabular}

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\section*{Version information}
\begin{tabular}{@{}p{23mm}@{}p{0mm}p{\textwidth-23mm-30pt}}
\textbf{Date of study}& \textbf{:} & 16-09-2024 \\ 
\textbf{Executable}   & \textbf{:} & d1be1be456f8cdba4439163360d25a00ae56a7a5, 24/09/13 16:02:05 \\
\textbf{Location}     & \textbf{:} & \url{https://s3-console.deltares.nl/browser/dsc-testbench/Y2FzZXMvZTAyX2RmbG93Zm0vZjUwMF9pbXBsaWNpdDFkL2MwMV9oeWRyby8=} \\ 
\end{tabular}

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\section*{Purpose}
Testing the ability of the implicit 1D solver to capture steady hydrodynamics correctly. 

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\section*{Linked claims}
The implicit 1D solver correctly computes, on a network of channels involving confluences and bifurcations:
\begin{itemize}
\item normal flow,
\item an M1 backwater curve,
\item an M2 backwater curve.
\end{itemize}

%---------------------------------------------------------------------------------
\section*{Approach}
A multibranch domain is created in which two branches merge and then bifurcate. The domain is not symmetric in 2D, but it is in 1D. Steady state is reached and the result along different tracks is compared. The flow depth is compared to the approximate analytical solution.

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\section*{Model description}
The grid is shown in Figure \ref{fig:grid_01_e02_f500_c01_hydro_grid_xlim_01}. The space step is uniform and equal to \SI{10}{m}. A discharge boundary is imposed at \texttt{Node006} and \texttt{Node005}. A water-level boundary is imposed at \texttt{Node003} and \texttt{Node004}. The width of branches \texttt{Channel\_1D\_5}, \texttt{Channel\_1D\_4}, \texttt{Channel\_1D\_2}, \texttt{Channel\_1D\_3} is equal to \SI{50}{m} while the width of branches \texttt{Channel\_1D\_1} and \texttt{Channel\_1D\_6} is \SI{100}{m}. All branches have the same length. The non-dimensional friction coefficient is constant and equal to 0.007 (Ch\'{e}zy friction coefficient equal to \SI{37.435659}{m^{1/2}/s}). The slope is constant and equal to \num{7.1356e-4} (\num{0.007}/\num{9.81}, i.e., the slope for an infinitely wide channel with flow depth equal to \SI{1}{m} and a flow velocity equal to \SI{1}{m/s}). If the same boundary condition is imposed at \texttt{Node005} than at \texttt{Node006} and at the same time the same boundary condition is imposed at \texttt{Node004} than at \texttt{Node003}, the solution is symmetric in a purely one-dimensional sense. The naming is ``not in natural order'' to avoid errors in indexing while coding.

\begin{figure}[H]
    \centering
    \includegraphics*[width=0.85\textwidth]{figures/grid_01_e02_f500_c01_hydro_grid_xlim_01.png}
    \caption{Domain.}
		\label{fig:grid_01_e02_f500_c01_hydro_grid_xlim_01}
\end{figure}

Constant boundary conditions equal to \SI{50}{m^3/s} at each of the upstream ends and \SI{1.013333}{m} (the normal flow depth considering the hydraulic radius \footnote{
Computed iteratively. In Matlab$^{\text{\textregistered}}$ OpenEarthTools see \url{https://svn.oss.deltares.nl/repos/openearthtools/trunk/matlab/applications/vtools/general/normal_flow.m}. Use: \\
\texttt{s=normal\_flow('Q',50,'B',50,'Cf',0.007,'h',1,'hydraulic\_radius',2);} \\
\texttt{h=normal\_flow('Q',50,'B',50,'Cf',0.007,'s',s);}
}
) at the downstream end (the datum is the bed level at the downstream end). 

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\section*{Results}

Figure \ref{fig:map_1D_01_e02_f500_c01_hydro_200001020000_h_c1_val} shows the flow depth along branches [4,1,6,3]. The solution appears to be perfect. Normal flow depth imposed at the downstream end is maintained along the downstream branch ($s$=\SI{1500}{m}-\SI{2000}{m}, being $s$ [\si{m}] the streamwise coordinate measured from the upstream node under consideration). Along the center branch, the solution shows an M1 backwater curve. The analytical (approximated) flow depth at the upstream part of this branch ($s$=\SI{500}{m}) is \SI{1.007543}, and the actual value \SI{1.007459}{m}. The analytical value is computed with the linear approximation to the backwater equation:
\begin{equation}
\label{eq:bwc}
h=h_{\mathrm{n}}+\left(h_{0}-h_{\mathrm{n}}\right)\left(\frac{1}{2}\right)^{-\frac{s-s_0}{L_{1/2}}} \; ,
\end{equation}
where $h$ [\si{m}] is the flow depth at $s$ [\si{m}], $h_{\mathrm{n}}$ [\si{m}] is the normal flow depth, $h_{0}$ [\si{m}] is the flow depth at $s_0$ [\si{m}] and $L_{1/2}$ [\si{m}] is the adaptation half-length:
\begin{equation}
L_{1/2}=0.24\frac{h_{\mathrm{n}}}{S}\left(\frac{h_0}{h_{\mathrm{n}}}\right)^{4/3} \; ,
\end{equation}
where $S$ [$-$] is the slope.

For the central branch:
\begin{itemize}
\item $h_{\mathrm{n}}$=\SI{1.006667}{m} (see previous footnote on how to compute).
\item $h_0$=\SI{1.013333}{m} (i.e., the normal flow depth of the downstream branch).
\item $s$=\SI{500}{m}.
\item $s_0$=\SI{1500}{m}.
\item $S$=\num{7.1356e-4}.
\end{itemize}
Along the upstream branch, the analytical flow depth at the upstream end ($s$=0) is \SI{1.011254}{m} and the numerical one is \SI{1.011456}. The analytical solution is computed using Equation \ref{eq:bwc} using:
\begin{itemize}
\item $h_{\mathrm{n}}$=\SI{1.013333}{m}.
\item $h_0$=\SI{1.006667}{m} (i.e., the flow depth upstream of the central branch).
\item $s$=\SI{500}{m}.
\item $s_0$=\SI{0}{m}.
\item $S$=\num{7.1356e-4}.
\end{itemize}

The error is considered acceptable, given that the analytical solution is an approximation. 

\begin{figure}[ht]
    \centering
    \includegraphics[width=\textwidth]{figures/map_1D_01_e02_f500_c01_hydro_200001020000_h_c1_val.png}
		\caption{Flow depth along branches [4,1,6,3].}
    \label{fig:map_1D_01_e02_f500_c01_hydro_200001020000_h_c1_val}
\end{figure}

Figure \ref{fig:map_1D_01_e02_f500_c01_hydro_200001020000_h_c2_val} shows the flow depth along branches [5,1,6,2]. The solution ought to be the same as for branches [4,1,6,3] given the symmetry of the problem. We see that the solution is not exactly the same.

At the downstream branch ($s$=\SI{1500}{m}-\SI{2000}{m}) there is a small increase in downstream direction (M1 backwater curve) while the solution should be a constant flow depth. This could be due to a non-exact location of \texttt{Node003}. The coordinates are such that branch 2 is \SI{3}{cm} longer than branch 3. This could explain the extremely small backwater curve observed, although some other inaccuracy could be playing a role. 

\begin{figure}[ht]
    \centering
    \includegraphics[width=\textwidth]{figures/map_1D_01_e02_f500_c01_hydro_200001020000_h_c2_val.png}
		\caption{Flow depth along branches [5,1,6,2].}
    \label{fig:map_1D_01_e02_f500_c01_hydro_200001020000_h_c2_val}
\end{figure}

%---------------------------------------------------------------------------------
\section*{Conclusion} 

The implicit 1D solver correctly computes, on a network of channels involving confluences and bifurcations:
\begin{itemize}
\item normal flow,
\item an M1 backwater curve,
\item an M2 backwater curve.
\end{itemize}

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\printrefsegment
