Changes between Version 18 and Version 19 of doc/tec/noncyclic
- Timestamp:
- Nov 12, 2015 12:48:24 PM (9 years ago)
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doc/tec/noncyclic
v18 v19 27 27 {{{ 28 28 #!Latex 29 s^{t + \Delta t}(k,j,-1) = s_{init}(k) \; . \quad(1) 29 $s^{t + \Delta t}(k,j,-1) = s_{init}(k) \; . \quad(1)$ 30 30 }}} 31 31 t denotes the time, Δt the time step and s,,init,, the initialization profile of the scalar quantities which is constant in time. … … 34 34 {{{ 35 35 #!Latex 36 e^{t + \Delta t}(k,j,-1) = e^{t + \Delta t}(k,j,0) \; . \quad(2) 36 $e^{t + \Delta t}(k,j,-1) = e^{t + \Delta t}(k,j,0) \; . \quad(2)$ 37 37 }}} 38 38 To prevent gravity waves from being reflected at the inflow, a relaxation term can be added to the prognostic equations for the potential temperature θ (Davies, 1976): 39 39 {{{ 40 40 #!Latex 41 \theta^{t+1}(d) = ... - \Delta t \cdot K(d) \cdot \left( \theta^{t}(d) - \theta_{init} \right) \; . \quad(3) 41 $\theta^{t+1}(d) = ... - \Delta t \cdot K(d) \cdot \left( \theta^{t}(d) - \theta_{init} \right) \; . \quad(3)$ 42 42 }}} 43 43 Here, d is the distance normal to the wall and θ,,init,, the initial value of the potential temperature, which corresponds to the value at the inflow boundary. … … 46 46 {{{ 47 47 #!Latex 48 K(d) =48 $K(d) = 49 49 \begin{cases} 50 50 d_f \sin^2\left( \frac{\pi}{2} \frac{d_w - d}{d_w} \right) , \text{for } d < d_w \\ \qquad\quad 0 \qquad \quad \;\;\; , \text{for } d \ge d_w 51 \end{cases} . \quad (4) 51 \end{cases} . \quad (4)$ 52 52 }}} 53 53 d,,f,, is a damping factor to control the damping intensity, and d,,w,, is the width of the relaxation region extending from the inflow. Quantities d,,f,, and d,,w,, can be set with parameters [../../app/inipar/#pt_damping_factor pt_damping_factor] and [../../app/inipar/#pt_damping_width pt_damping_width], respectively. … … 65 65 {{{ 66 66 #!Latex 67 \partial_t \psi + c_{\psi} \partial_n \psi = 0 \; , \quad (5) 67 $\partial_t \psi + c_{\psi} \partial_n \psi = 0 \; , \quad (5)$ 68 68 }}} 69 69 which considers flow disturbances propagating with the mean flow and by waves. … … 76 76 {{{ 77 77 #!Latex 78 c_{\psi} = - \frac{\partial_t \psi}{\partial_n \psi} \; . \quad (6) 78 $c_{\psi} = - \frac{\partial_t \psi}{\partial_n \psi} \; . \quad (6)$ 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 $c_{\psi} = - c_{max} \frac{\psi^t_{nx} - \psi^{t - \Delta t}_{nx} }{ \psi^{t-\Delta t}_{nx} - \psi^{t - \Delta t}_{nx-1} } \; , \quad (7)$ 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 $c_{max} = \frac{\Delta x}{\Delta t} \; . \quad (8)$ 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 $\psi^{t+\Delta t}_{nx+1} = \psi^{t}_{nx+1} - \frac{\overline{c}_{\psi}}{c_{max}} (\psi^{t}_{nx+1} - \psi^{t}_{nx}) \; , \quad (9)$ 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 $\overline{c}_{\psi} = \frac{1}{ny+1} \sum_{j=0}^{ny} c_{\psi, j} \; . \quad (10)$ 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 \\ … … 129 129 & $(nx - 1) \rightarrow 1$ & $(nx - 1) \rightarrow 1$ & & \\ 130 130 \hline 131 \end{tabular} 131 \end{tabular} $ 132 132 }}} 133 133 … … 137 137 {{{ 138 138 #!Latex 139 \psi^{t + \Delta t}(k,j,nx+1) = \psi^{t}(k,j,nx) \; , \quad (11) 139 $\psi^{t + \Delta t}(k,j,nx+1) = \psi^{t}(k,j,nx) \; , \quad (11)$ 140 140 }}} 141 141 with ψ = {u,v,w}. … … 149 149 {{{ 150 150 #!Latex 151 \dot{m} = \sum_{k=1}^{nz-1} \Delta z(k) \sum_{l=0}^{nx_i} \psi(l,k) \Delta x_i \; . \quad (12) 151 $\dot{m} = \sum_{k=1}^{nz-1} \Delta z(k) \sum_{l=0}^{nx_i} \psi(l,k) \Delta x_i \; . \quad (12)$ 152 152 }}} 153 153 where Δx,,i,, and ψ is equal to Δy (Δx) and u (v) in case of [../../app/inipar/#bc_lr bc_lr] ([../../app/inipar/#bc_ns bc_ns]). … … 155 155 {{{ 156 156 #!Latex 157 \psi_{corr} = \frac{\dot{m}_{inflow} - \dot{m}_{outflow}}{A} \; , 157 $\psi_{corr} = \frac{\dot{m}_{inflow} - \dot{m}_{outflow}}{A} \; ,$ 158 158 }}} 159 159 where A is the area of the boundary 160 160 {{{ 161 161 #!Latex 162 A = \sum_1^{nz-1} \Delta z \sum_0^{nx_i} \Delta x_i \; . 162 $A = \sum_1^{nz-1} \Delta z \sum_0^{nx_i} \Delta x_i \; .$ 163 163 }}} 164 164 The streamwise velocity at the outflow is corrected by adding ψ,,corr,, at each grid point of the outflow between bottom and top (k=1:nz), in order to guarantee that the mass leaving the domain exactly balances the one entering it.
