| | 2 | |
| | 3 | = Lagrangian particle model (LPM) = |
| | 4 | |
| | 5 | The embedded LPM allows for studying transport and dispersion processes within turbulent flows. In the following we will describe |
| | 6 | the general modeling of particles, including passive particles that do not show any feedback on the turbulent flow. In Sect. [wiki:/doc/tec/lcm Lagrangian cloud model] we will describe the use of Lagrangian particles as explicit cloud droplets. |
| | 7 | |
| | 8 | == Formulation of the LPM == |
| | 9 | |
| | 10 | Lagrangian particles can be released in prescribed source volumes at |
| | 11 | different points in time. The particles then obey |
| | 12 | {{{ |
| | 13 | #!Latex |
| | 14 | \begin{align*} |
| | 15 | & \frac{\mathrm{d} x_{\mathrm{p}, i}}{\mathrm{d}t} = u_{\mathrm{p},i}(t) |
| | 16 | \end{align*} |
| | 17 | }}} |
| | 18 | where ''x'',,p,i,, describes the particle location in ''x'',,i,, direction (''i ∈ {1, 2, 3}'') and ''u'',,p,i,, is the |
| | 19 | respective velocity component of the particle. Particle trajectories are calculated by means of the turbulent flow fields provided by PALM |
| | 20 | for each time step. The location of a certain particle at time ''t + Δ t,,L,,'' is calculated by |
| | 21 | {{{ |
| | 22 | #!Latex |
| | 23 | \begin{align*} |
| | 24 | & |
| | 25 | x_{\mathrm{p},i}(x_{\text{ps},i},t + \Delta t_\mathrm{L}) = |
| | 26 | x_{\mathrm{p},i}(x_{\text{ps},i},t) + \int \limits_t^{t + \Delta |
| | 27 | t_\mathrm{L}} u_{\mathrm{p},i}(\hat{t}) \mathrm{d}\hat{t}\,, |
| | 28 | \end{align*} |
| | 29 | }}} |
| | 30 | where ''x'',,ps,i,, is the spatial coordinate of the particle source point and ''Δ t'',,L,, is the applied time step in the Lagrangian particle model. Note that the latter is not necessarily equal to the time step of the LES model. The integral in the Eq. above is evaluated using either a Runge-Kutta (2nd- or 3rd-order) or the (1st-order) Euler time-stepping scheme. |
| | 31 | |
| | 32 | The velocity of a weightless particle that is transported passively by the flow is determined by |
| | 33 | {{{ |
| | 34 | #!Latex |
| | 35 | \begin{align*} |
| | 36 | & u_{\mathrm{p},i} = u_i(x_{\mathrm{p},i})\;, |
| | 37 | \end{align*} |
| | 38 | }}} |
| | 39 | and for non-passive particles (e.g., cloud droplets) by |
| | 40 | {{{ |
| | 41 | #!Latex |
| | 42 | \begin{align*} |
| | 43 | & \frac{\mathrm{d}u_{\mathrm{p},i}}{\mathrm{d}t} = |
| | 44 | \frac{1}{\tau_\mathrm{p}} \left(u_i(x_{\mathrm{p},i}) - u_{\mathrm{p},i}\right) |
| | 45 | -\delta_{i3} \left(1 - \frac{\rho_0}{\rho_{\mathrm{l},0}} \right) g, |
| | 46 | \end{align*} |
| | 47 | }}} |
| | 48 | considering Stoke's drag, gravity and buoyancy (on the right-hand side, from left to right). Note that this Eq. is solved analytically assuming all variables but ''u'',,p,i,, as constants for one time step. Here, ''u'',,i,,''(x'',,p,i,,'')'' is the velocity of air at the particles location gathered from the eight adjacent grid points of the LES by tri-linear interpolation (see |
| | 49 | Sect. [wiki:/doc/tec/particle particle code structure]). |
| | 50 | |
| | 51 | == Boundary conditions and release of particles == |
| | 52 | |
| | 53 | == Recent applications == |
