https://www.cfd-online.com/W/index.php?title=Special:Contributions/Media777&feed=atom&limit=50&target=Media777&year=&month=CFD-Wiki - User contributions [en]2024-03-29T09:58:47ZFrom CFD-WikiMediaWiki 1.16.5https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-27T20:10:01Z<p>Media777: </p>
<hr />
<div>1. Those that use the physical hypothesis of scale similarity (Bardina et. al., 1980)<br />
<br />
:<math><br />
\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math><br />
<br />
<br />
2. Those derived by formal series expansions (Clark et. al., 1979)<br />
<br />
:<math><br />
\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}} <br />
</math><br />
<br />
3. Mixed models, which are based on linear combinations of the eddy-viscosity and structural types<br />
<br />
:<math><br />
\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
<br />
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)<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right) <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{G_{ij}}{G_{kk}}\right)<br />
</math><br />
The transport equation for the SGS kinetic energy can be used to close the model (Pomraning and Rutland, 2002; Lu et. al., 2008)<br />
:<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)<br />
</math><br />
or use a zero-equation procedure (Lu and Porte-Agel, 2010) to estimate the SGS kinetic energy.<br />
<br />
== References ==<br />
*{{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}}<br />
<br />
*{{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.}}<br />
<br />
*{{reference-paper|author=E. Pomraning and C. J. Rutland|year=2002|title=Dynamic one-equation nonviscosity large-eddy simulation model|rest=AIAA J.}}<br />
<br />
*{{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.}}<br />
<br />
*{{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}}<br />
<br />
*{{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}}</div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-25T22:28:34Z<p>Media777: </p>
<hr />
<div>1. Those that use the physical hypothesis of scale similarity (Bardina et. al., 1980)<br />
<br />
:<math><br />
\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math><br />
<br />
<br />
2. Those derived by formal series expansions (Clark et. al., 1979)<br />
<br />
:<math><br />
\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}} <br />
</math><br />
<br />
3. Mixed models, which are based on linear combinations of the eddy-viscosity and structural types<br />
<br />
:<math><br />
\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
<br />
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)<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right) <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{G_{ij}}{G_{kk}}\right)<br />
</math><br />
The transport equation for the SGS kinetic energy can be used to close the model (Pomraning and Rutland, 2002; Lu et. al., 2008)<br />
:<math> \frac{\partial \overline k_{\rm sgs}}{\partial t} + \frac{\partial \overline u_{j} \overline k_{sgs}} {\partial x_{j}} = - \tau_{ij} \frac{\partial \overline u_{i}}{\partial x_{j}} - C_{\varepsilon} \frac{k_{\rm sgs}^{3/2}}{\Delta} + \frac{\partial}{\partial x_{j}} \left( \nu_t \frac{\partial k_{\rm sgs}}{\partial x_{j}} \right)<br />
</math><br />
or use a zero-equation procedure (Lu and Porte-Agel, 2010) to predict the SGS kinetic energy.<br />
<br />
== References ==<br />
*{{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}}<br />
<br />
*{{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.}}<br />
<br />
*{{reference-paper|author=E. Pomraning and C. J. Rutland|year=2002|title=Dynamic one-equation nonviscosity large-eddy simulation model|rest=AIAA J.}}<br />
<br />
*{{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.}}<br />
<br />
*{{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}}<br />
<br />
*{{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}}</div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-25T22:27:46Z<p>Media777: </p>
<hr />
<div>1. Those that use the physical hypothesis of scale similarity (Bardina et. al., 1980)<br />
<br />
:<math><br />
\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math><br />
<br />
<br />
2. Those derived by formal series expansions (Clark et. al., 1979)<br />
<br />
:<math><br />
\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}} <br />
</math><br />
<br />
3. Mixed models, which are based on linear combinations of the eddy-viscosity and structural types<br />
<br />
:<math><br />
\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
<br />
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)<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right) <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{G_{ij}}{G_{kk}}\right)<br />
</math><br />
The transport equation for the SGS kinetic energy can be used to close the model (Pomraning and Rutland, 2002; Lu et. al., 2008)<br />
:<math> \frac{\partial \overline k_{\rm sgs}}{\partial t} + \frac{\partial \overline u_{j} \overline k_{sgs}} {\partial x_{j}} = - \tau_{ij} \frac{\partial \overline u_{i}}{\partial x_{j}} - C_{\varepsilon} \frac{k_{\rm sgs}^{3/2}}{\Delta} + \frac{\partial}{\partial x_{j}} \left( \frac{\mu_t}{\sigma_k} \frac{\partial k_{\rm sgs}}{\partial x_{j}} \right)<br />
</math><br />
or use a zero-equation procedure (Lu and Porte-Agel, 2010) to predict the SGS kinetic energy.<br />
<br />
== References ==<br />
*{{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}}<br />
<br />
*{{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.}}<br />
<br />
*{{reference-paper|author=E. Pomraning and C. J. Rutland|year=2002|title=Dynamic one-equation nonviscosity large-eddy simulation model|rest=AIAA J.}}<br />
<br />
*{{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.}}<br />
<br />
*{{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}}<br />
<br />
*{{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}}</div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-25T20:19:34Z<p>Media777: </p>
<hr />
<div>1. Those that use the physical hypothesis of scale similarity (Bardina et. al., 1980)<br />
<br />
:<math><br />
\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math><br />
<br />
<br />
2. Those derived by formal series expansions (Clark et. al., 1979)<br />
<br />
:<math><br />
\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}} <br />
</math><br />
<br />
3. Mixed models, which are based on linear combinations of the eddy-viscosity and structural types<br />
<br />
:<math><br />
\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
<br />
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)<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right) <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{G_{ij}}{G_{kk}}\right)<br />
</math><br />
<br />
== References ==<br />
*{{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}}<br />
<br />
*{{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.}}<br />
<br />
*{{reference-paper|author=E. Pomraning and C. J. Rutland|year=2002|title=Dynamic one-equation nonviscosity large-eddy simulation model|rest=AIAA J.}}<br />
<br />
*{{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.}}<br />
<br />
*{{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}}<br />
<br />
*{{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}}</div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-25T20:06:26Z<p>Media777: </p>
<hr />
<div>1. Those that use the physical hypothesis of scale similarity (Bardina et al., 1980)<br />
<br />
:<math><br />
\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math><br />
<br />
<br />
2. Those derived by formal series expansions (Clark et al., 1979)<br />
<br />
:<math><br />
\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}} <br />
</math><br />
<br />
3. Mixed models, which are based on linear combinations of the eddy-viscosity and structural types<br />
<br />
:<math><br />
\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
<br />
4. Dynamic structure models (non-viscosity version)<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right) <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{G_{ij}}{G_{kk}}\right)<br />
</math><br />
<br />
== References ==<br />
*{{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}}<br />
<br />
*{{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.}}</div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-25T20:03:34Z<p>Media777: </p>
<hr />
<div>1. Those that use the physical hypothesis of scale similarity<br />
<br />
:<math><br />
\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math><br />
<br />
<br />
2. Those derived by formal series expansions<br />
<br />
:<math><br />
\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}} <br />
</math><br />
<br />
3. Mixed models, which are based on linear combinations of the eddy-viscosity and structural types<br />
<br />
:<math><br />
\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
<br />
4. Dynamic structure models (non-viscosity version)<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right) <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{G_{ij}}{G_{kk}}\right)<br />
</math><br />
<br />
== References ==<br />
<br />
*{{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.}}</div>Media777https://www.cfd-online.com/Wiki/Turbulence_modelingTurbulence modeling2013-06-25T20:00:50Z<p>Media777: /* Content of turbulence modeling section */</p>
<hr />
<div>Turbulence modeling is a key issue in most CFD simulations. Virtually all engineering applications are turbulent and hence require a turbulence model.<br />
<br />
==Classes of turbulence models==<br />
<br />
*RANS-based models<br />
**Linear eddy-viscosity models<br />
***Algebraic models<br />
***One and two equation models<br />
**Non-linear eddy viscosity models and algebraic stress models<br />
**Reynolds stress transport models<br />
<br />
*Large eddy simulations<br />
<br />
*Detached eddy simulations and other hybrid models<br />
<br />
*Direct numerical simulations<br />
<br />
==Content of turbulence modeling section==<br />
<br />
# '''[[Turbulence]]'''<br />
# '''[[RANS-based turbulence models]]'''<br />
## '''''[[Linear eddy viscosity models]]'''''<br />
### ''[[Algebraic turbulence models|Algebraic models]]''<br />
#### [[Cebeci-Smith model]]<br />
#### [[Baldwin-Lomax model]]<br />
#### [[Johnson-King model]]<br />
#### [[A roughness-dependent model]]<br />
### ''[[One equation turbulence models|One equation models]]''<br />
#### [[Prandtl's one-equation model]]<br />
#### [[Baldwin-Barth model]]<br />
#### [[Spalart-Allmaras model]]<br />
#### [[Rahman-Siikonen-Agarwal Model]]<br />
### ''[[Two equation models]]''<br />
#### [[k-epsilon models]]<br />
##### [[Standard k-epsilon model]]<br />
##### [[Realisable k-epsilon model]]<br />
##### [[RNG k-epsilon model]]<br />
##### [[Near-wall treatment for k-epsilon models|Near-wall treatment]]<br />
#### [[k-omega models]]<br />
##### [[Wilcox's k-omega model]]<br />
##### [[Wilcox's modified k-omega model]]<br />
##### [[SST k-omega model]]<br />
##### [[Near-wall treatment for k-omega models|Near-wall treatment]]<br />
#### [[Two equation turbulence model constraints and limiters|Realisability issues]]<br />
##### [[Kato-Launder modification]]<br />
##### [[Durbin's realizability constraint]]<br />
##### [[Yap correction]]<br />
##### [[Realisability and Schwarz' inequality]]<br />
## '''''[[Nonlinear eddy viscosity models]]'''''<br />
### [[Explicit nonlinear constitutive relation]]<br />
#### [[Cubic k-epsilon]]<br />
#### [[Explicit algebraic Reynolds stress models (EARSM)]]<br />
### [[v2-f models]]<br />
#### <math>\overline{\upsilon^2}-f</math> model<br />
#### <math>\zeta-f</math> model<br />
## '''''[[Reynolds stress model (RSM) ]]'''''<br />
# '''[[Large eddy simulation (LES) ]]'''<br />
## [[Smagorinsky-Lilly model]]<br />
## [[Dynamic subgrid-scale model]]<br />
## [[RNG-LES model]]<br />
## [[Wall-adapting local eddy-viscosity (WALE) model]]<br />
## [[Kinetic energy subgrid-scale model]]<br />
## [[Near-wall treatment for LES models]]<br />
## [[Structural modeling]]<br />
# '''[[Detached eddy simulation (DES) ]]'''<br />
# '''[[Direct numerical simulation (DNS) ]]'''<br />
# '''[[Turbulence near-wall modeling]]'''<br />
# '''[[Turbulence free-stream boundary conditions]]'''<br />
## [[Turbulence intensity]]<br />
## [[Turbulence length scale]]<br />
<br />
[[Category:Turbulence models]]</div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-24T22:09:32Z<p>Media777: </p>
<hr />
<div>Those that use the physical hypothesis of scale similarity<br />
<br />
:<math><br />
\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math><br />
<br />
<br />
Those derived by formal series expansions<br />
<br />
:<math><br />
\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}} <br />
</math><br />
<br />
Mixed models, which are based on linear combinations of the eddy-viscosity and structural types<br />
<br />
:<math><br />
\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
<br />
Dynamic structure models (non-viscosity version)<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right) <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \left(\frac{G_{ij}}{G_{kk}}\right)<br />
</math></div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-24T22:08:38Z<p>Media777: </p>
<hr />
<div>Those that use the physical hypothesis of scale similarity<br />
<br />
:<math><br />
\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math><br />
<br />
<br />
Those derived by formal series expansions<br />
<br />
:<math><br />
\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}} <br />
</math><br />
<br />
Mixed models, which are based on linear combinations of the eddy-viscosity and structural types<br />
<br />
:<math><br />
\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
<br />
Dynamic structure models<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \frac{L_{ij}}{L_{kk}} <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = 2k_{sgs} \frac{G_{ij}}{G_{kk}} <br />
</math></div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-24T22:06:12Z<p>Media777: </p>
<hr />
<div>Those that use the physical hypothesis of scale similarity<br />
<br />
:<math><br />
\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math><br />
<br />
<br />
Those derived by formal series expansions<br />
<br />
:<math><br />
\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}} <br />
</math><br />
<br />
Mixed models, which are based on linear combinations of the eddy-viscosity and structural types<br />
<br />
:<math><br />
\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij} <br />
</math><br />
or<br />
:<math><br />
\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij} <br />
</math></div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-24T22:00:07Z<p>Media777: </p>
<hr />
<div>Those that use the physical hypothesis of scale similarity<br />
<br />
:<math><br />
\tau_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math><br />
<br />
<br />
Those derived by formal series expansions<br />
<br />
:<math><br />
\tau_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}} <br />
</math></div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-24T21:55:43Z<p>Media777: </p>
<hr />
<div>Those that use the physical hypothesis of scale similarity<br />
<br />
:<math><br />
\tau_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}<br />
</math></div>Media777https://www.cfd-online.com/Wiki/Structural_modelingStructural modeling2013-06-24T21:52:45Z<p>Media777: Created page with "Those that use the physical hypothesis of scale similarity :<math> \tau_{ij} = \bar{\bar{u}_i} \bar{\bar{u}_j} - \overline{\bar{u}_i \bar{u}_j} </math>"</p>
<hr />
<div>Those that use the physical hypothesis of scale similarity<br />
<br />
:<math><br />
\tau_{ij} = \bar{\bar{u}_i} \bar{\bar{u}_j} - \overline{\bar{u}_i \bar{u}_j}<br />
</math></div>Media777https://www.cfd-online.com/Wiki/Large_eddy_simulation_(LES)Large eddy simulation (LES)2013-06-24T21:50:08Z<p>Media777: /* Subgrid-scale models */</p>
<hr />
<div>==Introduction==<br />
Large eddy simulation (LES) is a popular technique for simulating turbulent flows. An implication of [[Kolmogorov]]'s (1941) theory of self similarity is that the large eddies of the flow are dependant on the geometry while the smaller scales more [[universal]]. This feature allows one to explicitly solve for the large eddies in a calculation and implicitly account for the small eddies by using a [[#Subgrid-scale models|subgrid-scale model]] (SGS model).<br />
<br />
Mathematically, one may think of separating the velocity field into a resolved and sub-grid part. The resolved part of the field represent the "large" eddies, while the subgrid part of the velocity represent the "small scales" whose effect on the resolved field is included through the subgrid-scale model. Formally, one may think of filtering as the convolution of a function with a [[LES filters|filtering kernel]] <math>G</math>:<br />
<br />
:<math><br />
\bar{u}_i(\vec{x}) = \int G(\vec{x}-\vec{\xi}) u(\vec{\xi})d\vec{\xi},</math><br />
<br />
resulting in<br />
<br />
:<math>u_i = \bar{u}_i + u'_i,</math><br />
<br />
where <math>\bar{u}_i</math> is the resolvable scale part and <math>u'_i</math> is the subgrid-scale part. However, most practical (and commercial) implementations of LES use the grid itself as the filter (the [[LES filters#Box filter|box filter]]) and perform no explicit filtering. More information about the theory and application of filters is found in the [[LES filters|LES filters article]].<br />
<br />
This page is mainly focused on LES of incompressible flows. For compressible flows, see [[Favre averaged Navier-Stokes equations]]. <br />
<br />
The filtered equations are developed from the incompressible [[Navier-Stokes equations]] of motion:<br />
<br />
:<math><br />
\frac{\partial{u_i}}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j}\left(\nu \frac{\partial u_i}{\partial x_j}\right). <br />
</math><br />
<br />
Substituting in the decomposition <math>u_i = \bar{u}_i + u'_i</math> and <math>p = \bar{p} + p'</math> and then filtering the resulting equation gives the equations of motion for the resolved field:<br />
<br />
:<math><br />
\frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i}<br />
+ \frac{\partial}{\partial x_j}\left(\nu \frac{\partial \bar{u}_i}{\partial x_j}\right)<br />
+ \frac {1}{\rho}\frac{\partial \tau_{ij}}{\partial x_j}.<br />
</math><br />
<br />
We have assumed that the filtering operation and the differentiation operation commute, which is not generally the case. It is thought that the errors associated with this assumption are usually small, though filters that commute with differentiation have been developed ("ref?"). The extra term <math> \frac{\partial \tau_{ij}}{\partial x_j} </math> arises from the non-linear advection terms, due to the fact that<br />
<br />
:<math><br />
\overline{ u_j \frac{\partial u_i}{\partial x_j} } \ne<br />
\bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j}<br />
</math><br />
<br />
and hence <br />
:<math><br />
\tau_{ij} = \bar{u}_i \bar{u}_j - \overline{u_i u_j}<br />
</math><br />
<br />
Similar equations can be derived for the subgrid-scale field (i.e. the residual field).<br />
<br />
Subgrid-scale turbulence models usually employ the [[Boussinesq eddy viscosity assumption|Boussinesq hypothesis]], and seek to calculate (the deviatoric part of) the SGS stress using: <br><br />
:<math><br />
\tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\mu_t \bar S_{ij} <br />
</math><br />
<br />
where <math><br />
\bar S_{ij} <br />
</math> is the rate-of-strain tensor for the resolved scale defined by <br><br />
<br />
:<math><br />
\bar S_{ij} = \frac{1}{2}\left( {\frac{{\partial \bar u_i }}{{\partial x_j }} + \frac{{\partial \bar u_j }}{{\partial x_i }}} \right)<br />
</math><br />
<br><br />
and <math> \nu_t </math> is the subgrid-scale turbulent viscosity. Substituting into the filtered Navier-Stokes equations, we then have<br />
<br />
:<math><br />
\frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \frac{\partial}{\partial x_j}\left([\nu+\nu_t]\frac{\partial\bar{u}_i}{\partial x_j}\right),<br />
</math><br />
<br />
where we have used the incompressibility constraint to simplify the equation and the pressure is now modified to include the trace term <math>\tau _{kk} \delta _{ij}/3</math>.<br />
<br />
== Subgrid-scale models ==<br />
<br />
*[[Smagorinsky-Lilly model|Smagorinsky model]] (Smagorinsky, 1963)<br />
*[[Dynamic subgrid-scale model|Algebraic Dynamic model]] (Germano, et. al., 1991)<br />
*[[Dynamic global-coefficient subgrid-scale model|Dynamic Global-Coefficient model]] (You & Moin, 2007)<br />
*[[Kinetic energy subgrid-scale model|Localized Dynamic model]] (Kim & Menon, 1993)<br />
*[[Wall-adapting local eddy-viscosity (WALE) model|WALE (Wall-Adapting Local Eddy-viscosity) model]] (Nicoud and Ducros, 1999)<br />
*[[RNG-LES model]]<br />
*[[Structural modeling]]<br />
<br />
== References ==<br />
<br />
*{{reference-paper|author=Germano, M., Piomelli, U., Moin, P. and Cabot, W. H.|year=1991|title=A dynamic sub-grid scale eddy viscosity model|rest=Physics of Fluids, A(3): pp 1760-1765, 1991}}<br />
*{{reference-paper|author=You, D. and Moin, P.|year=2007|title=A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries|rest=Physics of Fluids, 19(6): 065110, 2007}}<br />
*{{reference-paper|author=Kim, W and Menon, S.|year=1995|title=A new dynamic one-equation subgrid-scale model for large eddy simulation|rest=In 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995}}<br />
*{{reference-paper|author=Nicoud, F. and Ducros, F.|year=1999|title=Subgrid-scale modelling based on the square of the velocity gradient tensor|rest=Flow, Turbulence and Combustion, 62: pp- 183-200, 1999}}<br />
*{{reference-paper|author=Smagorinsky, J|year=1963|title=General circulation experiments with the primitive equations, i. the basic experiment. Monthly Weather Review|rest=91: pp 99-164, 1963}}<br />
<br />
[[Category:Turbulence models]]</div>Media777https://www.cfd-online.com/Wiki/Large_eddy_simulation_(LES)Large eddy simulation (LES)2007-02-12T02:14:53Z<p>Media777: /* Introduction */</p>
<hr />
<div>==Introduction==<br />
Large eddy simulation (LES) is a popular technique for simulating turbulent flows. An implication of [[Kolmogorov]]'s (1941) theory of self similarity is that the large eddies of the flow are dependant on the geometry while the smaller scales more [[universal]]. This feature allows one to explicitly solve for the large eddies in a calculation and implicitly account for the small eddies by using a [[#Subgrid-scale models|subgrid-scale model]] (SGS model).<br />
<br />
Mathematically, one may think of separating the velocity field into a resolved and sub-grid part. The resolved part of the field represent the "large" eddies, while the subgrid part of the velocity represent the "small scales" whose effect on the resolved field is included through the subgrid-scale model. Formally, one may think of filtering as the convolution of a function with a [[LES filters|filtering kernel]] <math>G</math>:<br />
<br />
:<math><br />
\bar{u}_i(\vec{x}) = \int G(\vec{x}-\vec{\xi}) u(\vec{x})d\vec{\xi},</math><br />
<br />
resulting in<br />
<br />
:<math>u_i = \bar{u}_i + u'_i,</math><br />
<br />
where <math>\bar{u}_i</math> is the resolvable scale part and <math>u'_i</math> is the subgrid-scale part. However, most practical (and commercial) implementations of LES use the grid itself as the filter (the [[LES filters#Box filter|box filter]]) and perform no explicit filtering. More information about the theory and application of filters is found in the [[LES filters|LES filters article]].<br />
<br />
This page is mainly focused on LES of incompressible flows. For compressible flows, see [[Favre averaged Navier-Stokes equations]]. <br />
<br />
The filtered equations are developed from the incompressible [[Navier-Stokes equations]] of motion:<br />
<br />
:<math><br />
\frac{\partial{u_i}}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j}\left(\nu \frac{\partial u_i}{\partial x_j}\right). <br />
</math><br />
<br />
Substituting in the decomposition <math>u_i = \bar{u}_i + u'_i</math> and <math>p = \bar{p} + p'</math> and then filtering the resulting equation gives the equations of motion for the resolved field:<br />
<br />
:<math><br />
\frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i}<br />
+ \frac{\partial}{\partial x_j}\left(\nu \frac{\partial \bar{u}_i}{\partial x_j}\right)<br />
- \frac {1}{\rho}\frac{\partial \tau_{ij}}{\partial x_j}.<br />
</math><br />
<br />
We have assumed that the filtering operation and the differentiation operation commute, which is not generally the case. It is thought that the errors associated with this assumption are usually small, though filters that commute with differentiation have been developed ("ref?"). The extra term <math> \frac{\partial \tau_{ij}}{\partial x_j} </math> arises from the non-linear advection terms, due to the fact that<br />
<br />
:<math><br />
\overline{ u_j \frac{\partial u_i}{\partial x_j} } \ne<br />
\bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j}<br />
</math><br />
<br />
and hence <br />
:<math><br />
\tau_{ij} = \overline{u_i u_j}- \bar{u}_i \bar{u}_j <br />
</math><br />
<br />
Similar equations can be derived for the subgrid-scale field (i.e. the residual field).<br />
<br />
Subgrid-scale turbulence models usually employ the [[Boussinesq eddy viscosity assumption|Boussinesq hypothesis]], and seek to calculate (the deviatoric part of) the SGS stress using: <br><br />
:<math><br />
\tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\mu_t \bar S_{ij} <br />
</math><br />
<br />
where <math><br />
\bar S_{ij} <br />
</math> is the rate-of-strain tensor for the resolved scale defined by <br><br />
<br />
:<math><br />
\bar S_{ij} = \frac{1}{2}\left( {\frac{{\partial \bar u_i }}{{\partial x_j }} + \frac{{\partial \bar u_j }}{{\partial x_i }}} \right)<br />
</math><br />
<br><br />
and <math> \nu_t </math> is the subgrid-scale turbulent viscosity. Substituting into the filtered Navier-Stokes equations, we then have<br />
<br />
:<math><br />
\frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \frac{\partial}{\partial x_j}\left([\nu+\nu_t]\frac{\partial\bar{u}_i}{\partial x_j}\right),<br />
</math><br />
<br />
where we have used the incompressibility constraint to simplify the equation and the pressure is now modified to include the trace term <math>\tau _{kk} \delta _{ij}/3</math>.<br />
<br />
== Subgrid-scale models ==<br />
<br />
*[[Smagorinsky-Lilly model|Smagorinsky model]] (Smagorinsky, 1963)<br />
*[[Dynamic subgrid-scale model|Algebraic Dynamic model]] (Germano, et. al., 1991)<br />
*[[Kinetic energy subgrid-scale model|Localized Dynamic model]] (Kim & Menon, 1993)<br />
*[[Wall-adapting local eddy-viscosity (WALE) model|WALE (Wall-Adapting Local Eddy-viscosity) model]] (Nicoud and Ducros, 1999)<br />
*[[RNG-LES model]]<br />
<br />
== References ==<br />
<br />
*<b>J. Smagorinsky.</b> General circulation experiments with the primitive equations, i. the basic experiment. Monthly Weather Review, 91: 99-164, 1963.<br />
*<b>M. Germano, U. Piomelli, P. Moin, and W. H. Cabot.</b> A dynamic sub-grid scale eddy viscosity model. Physics of Fluids, A(3): 1760-1765, 1991.<br />
*<b>W. Kim and S. Menon.</b> A new dynamic one-equation subgrid-scale model for large eddy simulation. In 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995.<br />
*<b>F. Nicoud and F. Ducros.</b> Subgrid-scale modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62: 183-200, 1999.<br />
<br />
[[Category:Turbulence models]]</div>Media777