| 208 | | In case |
| | 208 | In case of laminar inflow, Dirichlet boundary conditions are used for all quantities, except for the SGS-TKE ''e'' and perturbation pressure |
| | 209 | ''π^∗^'' for which Neumann boundary conditions are used. Vertical profiles, as taken for the initialization of the simulation, are used for the Dirichlet boundary conditions. In order to allow for a fast turbulence development, random perturbations can be imposed on the velocity fields within a certain area behind the inflow boundary (inlet). These perturbations may persist for the entire simulation. For the purpose of preventing gravity waves from being reflected at the inlet, a relaxation area can be defined after [#davies1976 Davies (1976)]. So far, it was found to be sufficient to implement this method for temperature only. This is hence realized by an additional term in the prognostic equation for ''θ'' (see third equation in Sect. [wiki:/doc/tec/gov governing equation]): |
| | 210 | {{{ |
| | 211 | #!Latex |
| | 212 | \begin{align*} |
| | 213 | \frac{\partial \theta}{\partial t} = \ldots - C_{\text{relax}} |
| | 214 | \left(\theta - \theta_{\text{inlet}}\,\right). |
| | 215 | \end{align*} |
| | 216 | }}} |
| | 217 | Here, ''θ'',,inlet,, is the stationary inflow profile of ''θ'', and ''C'',,relax,, is a relaxation coefficient, depending on the distance ''d'' from the inlet, viz. |
| | 218 | {{{ |
| | 219 | #!Latex |
| | 220 | \begin{align*} |
| | 221 | &C_{\text{relax}}(d) = |
| | 222 | \begin{cases} |
| | 223 | F_{\text{inlet}} \cdot \sin^2 \left(\frac{\pi}{2} \frac{D - d}{D} \right) & \text{for~} d < D,\\ |
| | 224 | 0 & \text{for~} d \ge D,\\ |
| | 225 | \end{cases} |
| | 226 | \end{align*} |
| | 227 | }}} |
| | 228 | with ''D'' being the length of the relaxation region and ''F'',,inlet,, being a damping factor. |
| | 231 | If non-cyclic horizontal boundary conditions are used, PALM offers the possibility of generating time-dependent turbulent inflow data by using a turbulence recycling method. The method follows the one described by [#lund1998 Lund et al. (1998)], with the modifications introduced by [#kataoka2002 Kataoka and Mizuno (2002)]. Figure 3 gives an overview of the recycling method used in PALM. |
| | 232 | |
| | 233 | [[Image(03.png,600px,border=1)]] |
| | 234 | |
| | 235 | |
| | 236 | The turbulent signal ''φ^'^(y, z, t)'' is taken from a recycling plane which is located at a fixed distance ''x'',,recycle,, from the inlet: |
| | 237 | {{{ |
| | 238 | #!Latex |
| | 239 | \begin{align*} |
| | 240 | & \varphi^{\prime}(y, z, t) = \varphi(x_{\text{recycle}},y, z, t) - |
| | 241 | \langle \varphi\rangle_y(z, t), |
| | 242 | \end{align*} |
| | 243 | }}} |
| | 244 | where ''<φ>,,y,,(z, t)'' is the line average of a prognostic variable ''φ ∈ {u, v, w, θ, e}'' along ''y'' at ''x = x,,recycle,,''. ''φ^'^(y, z, t)'' is then added to the mean inflow profile ''<φ,,inflow,,>,,y,,(z)'' at ''x,,inlet,,'' after each time step: |
| | 245 | {{{ |
| | 246 | #!Latex |
| | 247 | \begin{align*} |
| | 248 | & \varphi_{\text{inlet}}(y, z, t) = \langle |
| | 249 | \varphi_{\text{inlet}}\rangle_y(z) + \phi(z) \varphi^{\prime}(y, z, t), |
| | 250 | \end{align*} |
| | 251 | }}} |
| | 252 | with the inflow damping function ''Φ(z)'', which has a value of ''1'' below the initial boundary layer height, and which is linearly damped to ''0'' above, in order to inhibit growth of the boundary layer depth. ''<φ,,inlet,,>,,y,,(z)'' is constant in time and either calculated from the results of the precursor run or prescribed by the user. The distance ''x'',,recycle,, has to be chosen much larger than the integral length scale of the respective turbulent flow. Otherwise, the same turbulent structures could be recycled repeatedly, so that the turbulence spectrum is illegally modified. It is thus recommended to use a precursor run for generating the initial turbulence field of the main run. The precursor run can have a comparatively small domain along the horizontal directions. In that case the domain of the main run is filled by cyclic repetition of the precursor run data. Note that the turbulence recycling has not been adapted for humidity and passive scalars so far. |
| | 253 | |
| | 254 | Turbulence recycling is frequently used for simulations with urban topography. In such a case, topography elements should be placed |
| | 255 | sufficiently downstream of ''x'',,recycle to prevent effects on the turbulence at the inlet. |
