| | 96 | |
| | 97 | == Dynamic subgrid-scale model == |
| | 98 | |
| | 99 | The dynamic SGS model, based on an idea of [#germano Germano et al. (1991)], can be used as an alternative to the Moeng-Wyngaard version of the Deardorff model. In this case, ''K'',,m,, is calculated as |
| | 100 | {{{ |
| | 101 | #!Latex |
| | 102 | \begin{align*} |
| | 103 | K_\mathrm{m} = c_*\;\Delta_\mathrm{max}\;\sqrt{e}, |
| | 104 | \end{align*} |
| | 105 | }}} |
| | 106 | where ''Δ'',,max,, being the maximum of ''Δx'', ''Δy'', ''Δz''. |
| | 107 | The calculation of ''c'',,*,, is based on an idea of [#germano Germano et al. (1991)] to use a test filter, which is |
| | 108 | {{{ |
| | 109 | #!Latex |
| | 110 | \begin{align*} |
| | 111 | \Delta_T = 2\Delta_\mathrm{max} |
| | 112 | \end{align*} |
| | 113 | }}} |
| | 114 | in our case. The subgrid stress on the test filter scale is then |
| | 115 | {{{ |
| | 116 | #!Latex |
| | 117 | \begin{align*} |
| | 118 | T_{ij} = \widehat{\overline{u_iu_j}} - \widehat{\overline{u}}_i\widehat{\overline{u}}_j |
| | 119 | \end{align*} |
| | 120 | }}} |
| | 121 | (the hat denotes a filter operation on the test filter scale) which is also an unknown. The difference between subgrid stress on the test filter level and test filtered subgrid stress is described by the Germano identity |
| | 122 | {{{ |
| | 123 | #!Latex |
| | 124 | \begin{align*} |
| | 125 | L_{ij} = T_{ij} - \widehat{\tau}_{ij} = \widehat{\overline{u}_i\overline{u}_j} - \widehat{\overline{u}}_i\widehat{\overline{u}}_j |
| | 126 | \end{align*} |
| | 127 | }}} |
| | 128 | and can be calculated directly by application of the test filter on resolved quantities. ''c'',,*,, is then calculated via |
| | 129 | {{{ |
| | 130 | #!Latex |
| | 131 | \begin{equation*} |
| | 132 | c_*=-\frac{L_{ij}^d\widehat{\overline{S}}^d_{ij}}{2\nu_t^T\widehat{\overline{S}}_{lk}^d\widehat{\overline{S}}_{kl}^d}, |
| | 133 | \end{equation*} |
| | 134 | }}} |
| | 135 | where |
| | 136 | {{{ |
| | 137 | #!Latex |
| | 138 | \begin{align*} |
| | 139 | \overline{S}_{ij} = \frac{1}{2}\left(\frac{\partial\overline{u_i}}{\partial x_j} + \frac{\partial\overline{u_j}}{\partial x_i} \right) |
| | 140 | \end{align*} |
| | 141 | }}} |
| | 142 | the strain tensor and ''ν,,t,,'' the SGS viscosity. Unlike other dynamic models this formulation of ''c'',,*,, is not derived using model assumptions for the subgrid stress and the stress on the test filter level, but is based on proven turbulence properties ([#heinz2008 Heinz, 2008]; [#heinz2012 Heinz and Gopalan, 2012]). Furthermore, the stability of the simulation is ensured by using dynamic bounds that keep the values of ''c'',,*,, in the range |
| | 143 | {{{ |
| | 144 | #!Latex |
| | 145 | \begin{equation*} |
| | 146 | |c_*| \leq \frac{23}{24\sqrt{3}}\frac{e^{1/2}}{\Delta|\overline{S}|}, |
| | 147 | \end{equation*} |
| | 148 | }}} |
| | 149 | as was derived by [#mokhtarpoor2017 Mokhtarpoor and Heinz (2017)]. This model does not need artificial clipping for stable runs and allows the occurence of backscatter (negative values of ''ν,,t,,''). |
| | 150 | |
| | 151 | |
