Changes between Version 4 and Version 5 of doc/tec/noncyclic

Timestamp:

Mar 12, 2013 6:10:35 PM (12 years ago)

Author:

fricke

Comment:

--

doc/tec/noncyclic

@@ -68 +68 @@
 }}}
 which considers flow disturbances propagating with the mean flow and by waves.
-Here ψ is the transported quantity and  $\partial_n$ is the derivative normal to the outflow boundary.
-In PALM, based on 5, the radiation boundary condition is realized in two ways as follows.
+Here ψ is the transported quantity and ∂,,n,, is the derivative normal to the outflow boundary.
+In PALM, based on (5), the radiation boundary condition is realized in two ways as follows.
 
+==== Radiation boundary condition I ====
+
+The phase velocity c,,ψ,, (also called convective velocity) is calculated after Orlanski (1976):
+{{{
+#!Latex
+   c_{\psi} = - \frac{\partial_t \psi}{\partial_n \psi}  \; . \quad (6)
+}}}
+
+''Left-right flow''
+
+If the outflow is defined at the right boundary (i = nx + 1, see Fig. 1), the phase velocity for each velocity components is calculated by
+{{{
+#!Latex
+   c_{\psi} = - c_{max} \frac{\psi^t_{nx} - \psi^{t - \Delta t}_{nx} }{ \psi^{t-\Delta t}_{nx} - \psi^{t - \Delta t}_{nx-1} }   \; , \quad (7)
+}}}
+with the maximum phase velocity
+{{{
+#!Latex
+  c_{max} = \frac{\Delta x}{\Delta t} \; . \quad (8)
+}}}
+The phase velocity has to be in the range of 0 ≤ c,,ψ,, < c,,max,, because negative values propagate waves back to the inner domain. c,,max,, represents the maximum phase velocity that ensures numerical stability (Courant-Friedrichs-Lewy condition).
+Both conditions are not ensured by (7), hence c,,ψ,, is set to zero for negative values and to c,,max,, for larger values than c,,max,,.
+The velocity components ψ at the outflow boundary are then calculated by
+{{{
+#!Latex
+  \psi^{t+\Delta t}_{nx+1} = \psi^{t}_{nx+1} - \frac{\overline{c}_{\psi}}{c_{max}} (\psi^{t}_{nx+1} - \psi^{t}_{nx}) \; , \quad (9)
+}}}
+with the phase velocity averaged parallel to the outflow:
+{{{
+#!Latex
+  \overline{c}_{\psi} = \frac{1}{ny+1} \sum_{j=0}^{ny}  c_{\psi, j} \; . \quad (10)
+}}}
+In Orlanskis work, the phase velocity c,,ψ,, was not averaged along the outflow, which is sufficient for simplified flows as shown by Yoshida and Watanabe (2010).
+However, we found that in simulations of shear driven convective boundary layer with a strong wind component along the outflow, the approach of Orlanski (1976) was unstable.
+Raymond and Kuo (1984) argued that locally calculated phase velocities leads to a large variation of phase speeds which can lead to numerical instabilities.
+Other works used a constant phase velocity or an average over the whole outflow area (see e.g. Fröhlich 2006, p.216) instead of the approach of Orlanski (1976).  
+With this in mind, we tested the average phase velocity calculated from (10), and the solution at the outflow becomes stable.
+We did not additionally average the phase velocity along the vertical direction, because the background wind usually varies with height. 
+
+For the other flow directions, the indices of equations (7)-(10) have to be replaced as follows: 
+{{{
+#!Latex
+\begin{tabular}{|c |c |c |c| c|}
+\hline
+  & Right-left flow  &\multicolumn{2}{c|}{ South-north flow} & North-south flow \\
+\hline
+  & $(nx + 1) \rightarrow 0$ & $(nx + 1) \rightarrow -1$   \multirow{ }{ }& \multirow{ }{ } &  \\
+ $\psi = u$ &   $nx \rightarrow 1$   &  $nx \rightarrow 0$ & &\\
+  & $(nx - 1) \rightarrow 2$ &  $(nx - 1) \rightarrow 1$   & & \\
+\cline{1-3}
+   & $(nx + 1) \rightarrow -1$ & $(nx + 1) \rightarrow 0$ &  &  $i \rightarrow j$  \\
+ $\psi = v$ &   $nx \rightarrow 0$   &  $nx \rightarrow 1$ & $i \rightarrow j$ &   \\
+  & $(nx - 1) \rightarrow 1$ &  $(nx - 1) \rightarrow 2$  &  &  $nx \rightarrow ny$ \\
+\cline{1-3}
+   & $(nx + 1) \rightarrow -1$ & $(nx + 1) \rightarrow -1$ &  &    \\
+ $\psi = w$ &   $nx \rightarrow 0$   &  $nx \rightarrow 0$ & & \\
+  & $(nx - 1) \rightarrow 1$ &  $(nx - 1) \rightarrow 1$  &  &  \\
+\hline
+\end{tabular} 
+}}}
+
+==== Radiation boundary condition II ====
+
+This is a simplified version of radiation boundary condition I, where just c,,ψ,, = c,,max,, is used in (5). This leads to a more simple radiation boundary condition (here e.g. for a left-right flow along positive x-direction):
+{{{
+#!Latex
+ \psi^{t + \Delta t}(k,j,nx+1) = \psi^{t}(k,j,nx) \; , \quad (11)
+}}}
+with ψ = {u,v,w}.
+This formulation of the radiation boundary condtions saves computational time compared to the formulation of equations (7) to (10).  
+Although, Orlanski (1976) suggested that this approach of radiation boundary condition leads to reflection for waves smaller than c,,max,, which may occur in complex geophysical flows, our simulations of a convective boundary layer with background wind have been stable so far.
+
+=== Mass flux correction ===