Changes between Version 19 and Version 20 of doc/tec/noncyclic
- Timestamp:
- Nov 12, 2015 1:02:42 PM (9 years ago)
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doc/tec/noncyclic
v19 v20 76 76 {{{ 77 77 #!Latex 78 $c_{\psi} = - \ frac{\partial_t \psi}{\partial_n \psi} \; . \quad (6)$78 $c_{\psi} = - \dfrac{\partial_t \psi}{\partial_n \psi} \; . \quad (6)$ 79 79 }}} 80 80 … … 84 84 {{{ 85 85 #!Latex 86 $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)$86 $c_{\psi} = - c_{max} \dfrac{\psi^t_{nx} - \psi^{t - \Delta t}_{nx} }{ \psi^{t-\Delta t}_{nx} - \psi^{t - \Delta t}_{nx-1} } \; , \quad (7)$ 87 87 }}} 88 88 with the maximum phase velocity 89 89 {{{ 90 90 #!Latex 91 $c_{max} = \ frac{\Delta x}{\Delta t} \; . \quad (8)$91 $c_{max} = \dfrac{\Delta x}{\Delta t} \; . \quad (8)$ 92 92 }}} 93 93 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). … … 96 96 {{{ 97 97 #!Latex 98 $\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)$98 $\psi^{t+\Delta t}_{nx+1} = \psi^{t}_{nx+1} - \dfrac{\overline{c}_{\psi}}{c_{max}} (\psi^{t}_{nx+1} - \psi^{t}_{nx}) \; , \quad (9)$ 99 99 }}} 100 100 with the phase velocity averaged parallel to the outflow: 101 101 {{{ 102 102 #!Latex 103 $\overline{c}_{\psi} = \ frac{1}{ny+1} \sum_{j=0}^{ny} c_{\psi, j} \; . \quad (10)$103 $\overline{c}_{\psi} = \dfrac{1}{ny+1} \sum_{j=0}^{ny} c_{\psi, j} \; . \quad (10)$ 104 104 }}} 105 105 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). … … 113 113 {{{ 114 114 #!Latex 115 $\begin{tabular}{|c |c |c |c| c|}115 \begin{tabular}{|c |c |c |c| c|} 116 116 \hline 117 117 & Right-left flow &\multicolumn{2}{c|}{ South-north flow} & North-south flow \\ 118 118 \hline 119 & $(nx + 1) \rightarrow 0$ & $(nx + 1) \rightarrow -1$ \multirow{ }{ }& \multirow{ }{ }& \\119 & $(nx + 1) \rightarrow 0$ & $(nx + 1) \rightarrow -1$ & & \\ 120 120 $\psi = u$ & $nx \rightarrow 1$ & $nx \rightarrow 0$ & &\\ 121 121 & $(nx - 1) \rightarrow 2$ & $(nx - 1) \rightarrow 1$ & & \\ … … 129 129 & $(nx - 1) \rightarrow 1$ & $(nx - 1) \rightarrow 1$ & & \\ 130 130 \hline 131 \end{tabular} $131 \end{tabular} 132 132 }}} 133 133 … … 155 155 {{{ 156 156 #!Latex 157 $\psi_{corr} = \ frac{\dot{m}_{inflow} - \dot{m}_{outflow}}{A} \; ,$157 $\psi_{corr} = \dfrac{\dot{m}_{inflow} - \dot{m}_{outflow}}{A} \; ,$ 158 158 }}} 159 159 where A is the area of the boundary
