### Updates from Maria concerning compressible flow docs.

Added description of non-smooth artificial viscosity and dealiasing.

(cherry picked from commit 2945d41d)

Conflicts:
docs/user-guide/solvers/compressible-flow.tex
parent 2e9b2475
 ... ... @@ -449,6 +449,24 @@ For the non-smooth artificial viscosity model the added artificial viscosity is \end{array} \right. \end{equation} To enable the non-smooth viscosity model, the following line has to be added to the \inltt{SOLVERINFO} section: \begin{lstlisting}[style=XmlStyle] \end{lstlisting} The diffusivity is controlled by the following parameters: \begin{lstlisting}[style=XmlStyle]

Skappa = -1.3

Kappa = 0.2

mu0 = 1.0

\end{lstlisting} where mu0 is the maximum values for the viscosity coefficient, Kappa is half of the width of the transition interval and Skappa is the value of the centre of the interval. \begin{figure}[!htbp] \begin{center} \includegraphics[width = 0.47 \textwidth]{img/Mach_P4.pdf} ... ... @@ -457,6 +475,7 @@ For the non-smooth artificial viscosity model the added artificial viscosity is \label{fig:} \end{center} \end{figure} \subsubsection{Smooth artificial viscosity model} For the smooth artificial viscosity model an extra PDE for the artificial viscosity is appended to the Euler system \begin{equation}\label{eq:eulerplusvis}\begin{split} ... ... @@ -511,5 +530,52 @@ The polynomial order in each element can be adjusted based on the sensor value t \right. \end{equation} For now, the threshold values $s_e$, $s_{ds}$, $s_{sm}$ and $s_{fl}$ are determined empirically by looking at the sensor distribution in the domain. Once these values are set, two .txt files are outputted, one that has the composites called VariablePComposites.txt and one with the expansions called VariablePExpansions.txt. These values have to copied into a new .xml file to create the adapted mesh. \subsection{De-Aliasing Techniques} Aliasing effects, arising as a consequence of the nonlinearity of the underlying problem, need to be address to stabilise the simulations. Aliasing appears when nonlinear quantities are calculated at an insufficient number of quadrature points. We can identify two types of nonlinearities: \begin{itemize} \item PDE nonlinearities, related to the nonlinear and quasi-linear fluxes. \item Geometrical nonlinearities, related to the deformed/curves meshes. \end{itemize} We consider two de-aliasing strategies based on the concept of consistent integration: \begin{itemize} \item Local dealiasing: It only targets the PDE-aliasing sources, applying a consistent integration of them locally. \item Global dealiasing: It targets both the PDE and the geometrical-aliasing sources. It requires a richer quadrature order to consistently integrate the nonlinear fluxes, the geometric factors, the mass matrix and the boundary term. \end{itemize} Since Nektar++ tackles separately the PDE and geometric aliasing during the projection and solution of the equations, to consistently integrate all the nonlinearities in the compressible NavierStokes equations, the quadrature points should be selected based on the maximum order of the nonlinearities: \begin{equation} Q_{min}= P_{exp}+\frac{max(2P_{exp},P_{geom})}{2} + \frac{3}{2} \end{equation} where $Q_{min}$ is the minimum required number of quadrature points to exactly integrate the highest-degree of nonlinearity, $P_{exp}$ being the order of the polynomial expansion and $P_{geom}$ being the geometric order of the mesh. Bear in mind thatwe are using a discontinuous discretisation, meaning that aliasing effect are not fully controlled, since the boundary terms introduce non-polynomial functions into the problem. To enable the global de-aliasing technique, modify the number of quadrature points by: \begin{lstlisting}[style=XmlStyle] \end{lstlisting} where \inltt{NUMMODES} corresponds to $P$+1, where $P$ is the order of the polynomial used to approximate the solution. \inltt{NUMPOINTS} specifies the number of quadrature points.
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