| 49 | | Sect. [wiki:/doc/tec/particle particle code structure]). |
| | 49 | Sect. [wiki:/doc/tec/particle particle code structure]). Since Stoke's drag is only valid for radii ''≤ 30 ''μm (e.g., [#rogers1989 Rogers and Yau, 1989]), a nonlinear correction is applied to the Stokes's drag relaxation time scale: |
| | 50 | {{{ |
| | 51 | #!Latex |
| | 52 | \begin{align*} |
| | 53 | & \tau_\mathrm{p}^{-1} = |
| | 54 | \frac{9\,\nu\,\rho_0}{2\,r^2\,\rho_{\mathrm{p},0}}\,\cdot\,\left(1 + |
| | 55 | 0.15 \cdot {Re}_\mathrm{p}^{0.687} \right). |
| | 56 | \label{eq:lpm2} |
| | 57 | \end{align*} |
| | 58 | }}} |
| | 59 | Here, ''r'' is the radius of the particle, ''ν = 1.461 x 10^-5^'' m^2^ s the molecular viscosity of air, and ''ρ'',,p,0,, the density of the particle. The particle Reynolds number is given by |
| | 60 | {{{ |
| | 61 | #!Latex |
| | 62 | \begin{align*} |
| | 63 | & {Re}_\mathrm{p}=\frac{2\,r\,\left|\,u_i(x_{\mathrm{p},i}) - |
| | 64 | u_{\mathrm{p},i}\,\right|}{\nu}. |
| | 65 | \end{align*} |
| | 66 | }}} |
| | 67 | Following [#lamb1978 Lamb (1978)] and the concept of LES modeling, the Lagrangian velocity of a weightless particle can be split into |
| | 68 | a resolved-scale contribution ''u'',,p,,^res^ and an SGS contribution ''u'',,p,,^sgs^: |
| | 69 | {{{ |
| | 70 | #!Latex |
| | 71 | \begin{align*} |
| | 72 | u_{\mathrm{p},i} = u_{\mathrm{p},i}^{\text{res}} + |
| | 73 | u_{\mathrm{p},i}^{\text{sgs}}\,. |
| | 74 | \end{align*} |
| | 75 | }}} |
| | 76 | ''u'',,p,i,,^res^ is determined by interpolation of the respective LES velocity components ''u'',,i,, to the position of the particle. The SGS part of the particle velocity at time ''t'' is given by |
| | 77 | {{{ |
| | 78 | #!Latex |
| | 79 | \begin{align*} |
| | 80 | u_{\mathrm{p},i}^{\text{sgs}}(t) = u_{\mathrm{p},i}^{\text{sgs}}(t - |
| | 81 | \Delta t_\mathrm{L}) + \mathrm{d} u_{\mathrm{p},i}^{\text{sgs}}\,, |
| | 82 | \end{align*} |
| | 83 | }}} |
| | 84 | where d''u'',,p,i,,^sgs^ describes the temporal change of the SGS particle velocity during a time step of the LPM based on [#thomson1987 Thomson (1987)]. Note that the SGS part of ''u'',,p,i,, in the second equation of this section is always computed using the (1st-order) Euler |
| | 85 | time-stepping scheme. [#weil2004 Weil et al. (2004)] developed a formulation of the Langevin equation under assumption of isotropic Gaussian turbulence in order to treat the SGS particle dispersion in terms of a stochastic differential equation. This equation reads as |
| | 86 | {{{ |
| | 87 | #!Latex |
| | 88 | \begin{align*} |
| | 89 | \mathrm{d}u_{\mathrm{p},i}^{\text{sgs}} = &-\frac{3 c_{\text{sgs}} |
| | 90 | C_\mathrm{L}\epsilon}{4}\frac{u_{\mathrm{p},i}^{\text{sgs}}}{e} |
| | 91 | \Delta t_\mathrm{L} + \frac{1}{2} \left(\frac{1}{e} \frac{\mathrm{d} |
| | 92 | e}{\Delta t_\mathrm{L}} u_{\mathrm{p},i}^{\text{sgs}} + |
| | 93 | \frac{2}{3}\frac{\partial e}{\partial x_i} \right) \Delta |
| | 94 | t_\mathrm{L} + \left(c_{\text{sgs}} C_\mathrm{L} \epsilon |
| | 95 | \right)^{\frac{1}{2}} \mathrm{d}\zeta_i\, |
| | 96 | \end{align*} |
| | 97 | }}} |
| | 98 | and is used in PALM for the determination of the change in SGS particle velocities. Here, ''C'',,L,,'' = 3'' is a universal constant (''C'',,L,,'' = 4 ± 2'', see [#thomson1987 Thomson (1987)]). ''ζ'' is a vector composed of Gaussian-shaped random numbers, with each |
| | 99 | component neither spatially nor temporally correlated. The factor |
| | 100 | {{{ |
| | 101 | #!Latex |
| | 102 | \begin{align*} |
| | 103 | c_\text{sgs} = |
| | 104 | \frac{\langle\,e\,\rangle}{\langle\,e_\text{res}\,\rangle + |
| | 105 | \langle\,e\,\rangle}\,,\quad 0 \leq c_\text{sgs} \leq 1\,, |
| | 106 | \end{align*} |
| | 107 | }}} |
| | 108 | where ''e'',,res,, is the resolved-scale TKE as resolved by the numerical grid, assures that the temporal change of the modeled SGS |
| | 109 | particle velocities is, on average (horizontal mean), smaller than the change of the resolved-scale particle velocities ([#weil2004 Weil et al., 2004]). Values of ''e'' and ''ε'' are provided by the SGS model described in Sect. [wiki:/doc/tec/sgs turbulence closure] (see Eqs. for ''∂e/∂t'' and for the dissipation rate ''ε'',respectively). The first term on the right-hand side of the Eq. for d''u'',,p,i,,^sgs^ represents the influence of the SGS particle velocity from the previous time step (i.e., inertial "memory"). This effect is considered by the Lagrangian time scale after [#weil2004 Weil et al. (2004)]: |
| | 110 | {{{ |
| | 111 | #!Latex |
| | 112 | \begin{align*} |
| | 113 | & |
| | 114 | \label{eq:LS5a} |
| | 115 | \tau_\mathrm{L} = \frac{4}{3}\frac{e}{c_{\text{sgs}} |
| | 116 | C_\mathrm{L}\epsilon}\,, |
| | 117 | \end{align*} |
| | 118 | }}} |
| | 119 | which describes the time span during which ''u'',,p,,^sgs^''(t - Δt'',,L,,'')'' is correlated to ''u'',,p,,^sgs^''(t)''. The applied time step of the particle model hence must not be larger than τ,,L,,. In PALM, the particle time step is set to be smaller than ''τ'',,L,,'' / 40''. The second term on the right-hand side of the Eq. for d''u'',,p,i,,^sgs^ ensures that the assumption of well-mixed conditions by [#thomson1987 Thomson (1987)] is fulfilled on the subgrid scales. This term can be considered as drift correction, which shall prevent an over-proportional accumulation of particles in regions of weak turbulence (#rodean1996 Rodean, 1996). The third term on the right-hand side is of stochastic nature and describes the SGS diffusion of particles by a Gaussian random process. For a detailed derivation and discussion of |
| | 120 | the Eq. for d''u'',,p,i,,^sgs^ see [#thomson1987 Thomson (1987)], [#rodean1996 Rodean (1996)] and [#weil2004 Weil et al. (2004)}. |
| | 121 | |
| | 122 | The required values of the resolved-scale particle velocity components, ''e'', and ''ε'' are obtained from the respective LES fields using the eight adjacent grid points of the LES and tri-linear interpolation on the current particle location (see Sect. [wiki:/doc/tec/particle particle code structure]). An exception is made in case of no-slip boundary conditions set for the resolved-scale horizontal wind components below the first vertical grid level above the surface. Here, the resolved-scale particle velocities are determined from MOST (see |
| | 123 | Sect. [wiki:/doc/tec/bc boundary conditions]) in order to capture the logarithmic wind profile within the height interval of ''z'',,0,, to ''z'',,MO,,. The available values of ''u'',,∗,,, |
| | 124 | {{{ |
| | 125 | #!Latex |
| | 126 | $\overline{w^{\prime\prime}u^{\prime\prime}}_0$, |
| | 127 | }}} |
| | 128 | and |
| | 129 | {{{ |
| | 130 | #!Latex |
| | 131 | $\overline{w^{\prime\prime}v^{\prime\prime}}_0$ |
| | 132 | }}} |
| | 133 | are first bi-linearly interpolated to the horizontal location of the particle. In a second step the velocities are determined using the Eqs. for ''u'',,∗,,, ''∂u/∂z'' and ''∂v/∂z'' (\ref{eq:most:begin})--(\ref{eq:most:end}). Resolved-scale horizontal |
| | 134 | velocities of particles residing at height levels below $z_0$ are set |
| | 135 | to zero. The LPM allows to switch off the transport by the SGS |
| | 136 | velocities. |
