Structural modeling

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(Difference between revisions)

Latest revision as of 20:10, 27 June 2013

1. Those that use the physical hypothesis of scale similarity (Bardina et. al., 1980)

$\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}$

2. Those derived by formal series expansions (Clark et. al., 1979)

$\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}}$

3. Mixed models, which are based on linear combinations of the eddy-viscosity and structural types

$\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij}$

or

$\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij}$

4. Dynamic structure models, which divide the modeled SGS stress into a model for the SGS kinetic energy and a model for the structure of the SGS stress tensor (relative weights of each of the components) (Pomraning and Rutland, 2002; Lu et. al. 2007, 2008; Lu and Porte-Agel, 2010)

$\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right)$

or

$\tau_{ij} = 2k_{sgs} \left(\frac{G_{ij}}{G_{kk}}\right)$

The transport equation for the SGS kinetic energy can be used to close the model (Pomraning and Rutland, 2002; Lu et. al., 2008)

$\frac{\partial k_{sgs}}{\partial t} + \frac{\partial \overline u_{j} k_{sgs}} {\partial x_{j}} = - \tau_{ij} \frac{\partial \overline u_{i}}{\partial x_{j}} - C_{\varepsilon} \frac{k_{sgs}^{3/2}}{\Delta} + \frac{\partial}{\partial x_{j}} \left( \nu_t \frac{\partial k_{sgs}}{\partial x_{j}} \right)$

or use a zero-equation procedure (Lu and Porte-Agel, 2010) to estimate the SGS kinetic energy.

References

J. Bardina and J. H. Ferziger and W. C. Reynolds (1980), "Improved subgrid scale models for large eddy simulation", AIAA Paper No. 80-1357.

R. A. Clark and J. H. Ferziger and W. C. Reynolds (1979), "Evaluation of subgrid-scale models using an accurately simulated turbulent flow", J. Fluid Mech..

E. Pomraning and C. J. Rutland (2002), "Dynamic one-equation nonviscosity large-eddy simulation model", AIAA J..

H. Lu and C. J. Rutland and L. M. Smith (2007), "A priori tests of one-equation LES modeling of rotating turbulence", J. Turbul..

H. Lu and C. J. Rutland and L. M. Smith (2008), "A posteriori tests of one-equation LES modeling of rotating turbulence", Int. J. Mod. Phys. C.

H. Lu and F. Porte-Agel (2010), "A modulated gradient model for large-eddy simulation: application to a neutral atmospheric boundary layer", Phys. Fluids.

Structural modeling

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Latest revision as of 20:10, 27 June 2013

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@@ Line 1: / Line 1: @@
-Those that use the physical hypothesis of scale similarity
+. Those that use the physical hypothesis of scale similarity (Bardina et. al., 1980)
 :<math>
-\tau_{ij} = \bar{\bar{u}_i} \bar{\bar{u}_j} - \overline{\bar{u}_i \bar{u}_j}
+\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}
 </math>
+. Those derived by formal series expansions (Clark et. al., 1979)
+:<math>
+\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}}
+</math>
+. Mixed models, which are based on linear combinations of the eddy-viscosity and structural types
+:<math>
+\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij}
+</math>
+or
+:<math>
+\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij}
+</math>
+. Dynamic structure models, which divide the modeled SGS stress into a model for the SGS kinetic energy and a model for the structure of the SGS stress tensor (relative weights of each of the components) (Pomraning and Rutland, 2002; Lu et. al. 2007, 2008; Lu and Porte-Agel, 2010)
+:<math>
+\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right)
+</math>
+or
+:<math>
+\tau_{ij} = 2k_{sgs} \left(\frac{G_{ij}}{G_{kk}}\right)
+</math>
+The transport equation for the SGS kinetic energy can be used to close the model (Pomraning and Rutland, 2002; Lu et. al., 2008)
+:<math> \frac{\partial k_{sgs}}{\partial t} + \frac{\partial \overline u_{j} k_{sgs}} {\partial x_{j}} = - \tau_{ij} \frac{\partial \overline u_{i}}{\partial x_{j}}     - C_{\varepsilon} \frac{k_{sgs}^{3/2}}{\Delta} + \frac{\partial}{\partial x_{j}} \left( \nu_t \frac{\partial k_{sgs}}{\partial x_{j}}  \right)
+</math>
+or use a zero-equation procedure (Lu and Porte-Agel, 2010) to estimate the SGS kinetic energy.
+== References ==
+*{{reference-paper|author=J. Bardina and J. H. Ferziger and W. C. Reynolds|year=1980|title=Improved subgrid scale models for large eddy simulation|rest=AIAA Paper No. 80-1357}}
+*{{reference-paper|author=R. A. Clark and J. H. Ferziger and W. C. Reynolds|year=1979|title=Evaluation of subgrid-scale models using an accurately simulated turbulent flow|rest=J. Fluid Mech.}}
+*{{reference-paper|author=E. Pomraning and C. J. Rutland|year=2002|title=Dynamic one-equation nonviscosity large-eddy simulation model|rest=AIAA J.}}
+*{{reference-paper|author=H. Lu and C. J. Rutland and L. M. Smith|year=2007|title=A priori tests of one-equation LES modeling of rotating turbulence|rest=J. Turbul.}}
+*{{reference-paper|author=H. Lu and C. J. Rutland and L. M. Smith|year=2008|title=A posteriori tests of one-equation LES modeling of rotating turbulence|rest=Int. J. Mod. Phys. C}}
+*{{reference-paper|author=H. Lu and F. Porte-Agel|year=2010|title=A modulated gradient model for large-eddy simulation: application to a neutral atmospheric boundary layer|rest=Phys. Fluids}}