## 2 Governing Equations and Subgrid-Scale Stress Modeling of LES
### Governing Equations
\[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]0 is an arbitrary constant, which has to be provided as an input. The Smagorinsky model needs specification of a value of the constant \[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]1 and this limits its applicability to various complex problems. When \[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]2 is set to the default value of 0.13 in the standard Smagorinsky model, it found that the subgrid dissipation is excessively high, generally resulting in a substantial underprediction of the jet width [17]. However, when \[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]3 is specified to 0.12, it is too dissipative and predicts incorrect asymptotic behavior near a wall or in laminar zones and overpredicts the length of the potential core [18]. Furthermore, it does not allow possible subgrid scale energy backscatter to the resolved scales [19]. To overcome the limitations, other SGS models have been developed. [PERSON] et al. [20] and [PERSON] [21] proposed a dynamic Smagorinsky-Lilly model (DSLM); \[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]4 can be calculated at each time step by double filtering of the flow variables. The DSLM can show the correct asymptotic behavior near the walls and in laminar regions and does not formally prohibit possible energy backscatter. A coefficient \[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]5, which is dynamically computed, replaces the square of \[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}\!=-\frac{\partial \overline{\rho}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left\{\mu \left[\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline {u}_{j}}{\partial x_{i}}\right]\right\}-\frac{\partial\tau_{ij}^{SGS}}{ \partial x_{j}} \tag{4}\]6 and depends on the local flow structure.