### Updates from Maria concerning compressible flow docs.

Added description of non-smooth artificial viscosity and dealiasing.
parent 623a8b06
 ... ... @@ -221,7 +221,7 @@ to screen; \item \inltt{TInf} farfield temperature (i.e. $T_{\infty}$). Default value = 288.15 $K$; \item \inltt{Twall} temperature at the wall when isothermal boundary conditions are employed (i.e. $T_{w}$). Default value = 300.15$K$; \item \inltt{uInf} farfield $X$-component of the velocity (i.e. $u_{\infty}$). Default value = 0.1 $m/s$; \item \inltt{uint} farfield $X$-component of the velocity (i.e. $u_{\infty}$). Default value = 0.1 $m/s$; \item \inltt{vInf} farfield $Y$-component of the velocity (i.e. $v_{\infty}$). Default value = 0.0 $m/s$; \item \inltt{wInf} farfield $Z$-component of the velocity (i.e. $w_{\infty}$). Default value = 0.0 $m/s$; \item \inltt{mu} dynamic viscosity (i.e. $\mu_{\infty}$). Default value = 1.78e-05 $Pa s$; ... ... @@ -437,7 +437,6 @@ Compressible flow is characterised by abrupt changes in density within the flow \begin{equation}\label{eq:sensor} S_e=\frac{||\rho^p_e-\rho^{p-1}_e||_{L_2}}{||\rho_e^p||_{L_2}} \end{equation} by default the comparison is made with the $p-1$ solution, but this can be changed by setting the parameter \inltt{SensorOffset}. An artificial diffusion term is introduced locally to the Euler equations to deal with flow discontinuity and the consequential numerical oscillations. Two models are implemented, a non-smooth and a smooth artificial viscosity model. \subsubsection{Non-smooth artificial viscosity model} For the non-smooth artificial viscosity model the added artificial viscosity is constant in each element and discontinuous between the elements. The Euler system is augmented by an added laplacian term on right hand side of equation \ref{eq:euler}. The diffusivity of the system is controlled by a variable viscosity coefficient $\epsilon$. The value of $\epsilon$ is dependent on $\epsilon_0$, which is the maximum viscosity that is dependent on the polynomial order ($p$), the mesh size ($h$) and the maximum wave speed and the local sensor value. Based on pre-defined sensor threshold values, the variable viscosity is set accordingly ... ... @@ -450,6 +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} ... ... @@ -458,6 +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} ... ... @@ -512,54 +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: \subsection{Quasi-1D nozzle flow} A quasi-1D inviscid flow (flow with area variation) can be obtained using the \inltt{Quasi1D} forcing in a 1D solution of the Euler equations: \begin{lstlisting}[style=XMLStyle] Area \end{lstlisting} in this case a function named \inltt{Area} must be specified in the \inltt{CONDITIONS} section of the session file. \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} In this case, it is possible to prescribe the inflow conditions in terms of stagnation properties (density and pressure) by using the following boundary condition \begin{lstlisting}[style=XmlStyle] \end{lstlisting} 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} \subsection{Axi-symmetric flow} An axi-symmetric inviscid flow (with symmetry axis on x=0) can be obtained using the \inltt{AxiSymmetric} forcing in a 2D solution of the Euler equations: \begin{lstlisting}[style=XMLStyle] \end{lstlisting} The \inltt{StagnationInflow} boundary condition can also be used in this case. 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: Also, by defining the geometry with \inltt{} (i.e. a two-dimensional mesh in three-dimensional space) and adding the \inltt{rhow} variable, we obtain an axi-symmetric flow with swirl, in which case the \inltt{StagnationInflow} boundary condition allows prescribing \inltt{rhow}: \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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