Editing Large eddy simulation (section)

=== Incompressible flow ===

For incompressible flow, the [[Continuity equation#Fluid dynamics|continuity equation]] and Navier–Stokes equations are filtered, yielding the filtered incompressible continuity equation,

:<math>
\frac{ \partial \bar{u_i} }{ \partial x_i } = 0
</math>

and the filtered Navier–Stokes equations,

:<math>
\frac{ \partial \bar{u_i} }{ \partial t }
+ \frac{ \partial }{ \partial x_j } \left( \overline{ u_i u_j } \right) 
= - \frac{1}{\rho} \frac{ \partial \overline{p} }{ \partial x_i } 
+ \nu \frac{\partial}{\partial x_j} \left( \frac{ \partial \bar{u_i} }{ \partial x_j } + \frac{ \partial \bar{u_j} }{ \partial x_i } \right)
= - \frac{1}{\rho} \frac{ \partial \overline{p} }{ \partial x_i } 
+ 2 \nu \frac{\partial}{\partial x_j} \bar{S}_{ij},
</math>

where <math>\bar{p}</math> is the filtered pressure field and <math>\bar{S}_{ij}</math> is the rate-of-strain tensor evaluated using the filtered velocity.  The [[nonlinear]] filtered advection term <math>\overline{u_i u_j}</math> is the chief cause of difficulty in LES modeling.  It requires knowledge of the unfiltered velocity field, which is unknown, so it must be modeled.  The analysis that follows illustrates the difficulty caused by the nonlinearity, namely, that it causes interaction between large and small scales, preventing separation of scales.

The filtered advection term can be split up, following Leonard (1975),<ref name="Leonard_1974">{{cite book |last=Leonard |first=A. |title=Turbulent Diffusion in Environmental Pollution, Proceedings of a Symposium held at Charlottesville |chapter=Energy cascade in large-eddy simulations of turbulent fluid flows |year=1975 |isbn=9780120188185 |series=Advances in Geophysics A |volume=18 |pages=237–248 |bibcode=1975AdGeo..18..237L |doi=10.1016/S0065-2687(08)60464-1 }}</ref> as:

:<math>
\overline{u_i u_j} = \tau_{ij} + \overline{u}_i \overline{u}_j
</math>

where <math>\tau_{ij}</math> is the residual stress tensor, so that the filtered Navier-Stokes equations become

:<math>
\frac{ \partial \bar{u_i} }{ \partial t }
+ \frac{ \partial }{ \partial x_j } \left( \overline{u}_i \overline{u}_j \right) 
= - \frac{1}{\rho} \frac{ \partial \overline{p} }{ \partial x_i } 
+ 2 \nu \frac{\partial}{\partial x_j} \bar{S}_{ij}
- \frac{ \partial \tau_{ij} }{ \partial x_j }
</math>

with the residual stress tensor <math>\tau_{ij}</math> grouping all unclosed terms.  Leonard decomposed this stress tensor as <math>\tau_{ij} = L_{ij} + C_{ij} + R_{ij}</math> and provided physical interpretations for each term.  <math>L_{ij} = \overline{ \bar{u}_{i} \bar{u}_{j} } - \bar{u}_{i} \bar{u}_{j}</math>, the Leonard tensor, represents interactions among large scales, <math>R_{ij} = \overline{u^{\prime}_{i} u^{\prime}_{j}}</math>, the Reynolds stress-like term, represents interactions among the sub-filter scales (SFS), and <math>C_{ij} = \overline{\bar{u}_{i} u^{\prime}_{j}} + \overline{\bar{u}_{j} u^{\prime}_{i}}
</math>, the Clark tensor,<ref name="Clark">{{cite journal
|last1=Clark
|first1=R.
|last2=Ferziger
|first2=J.
|last3=Reynolds
|author3-link=William Craig Reynolds
|first3=W.
|year=1979
|title=Evaluation of subgrid-scale models using an accurately simulated turbulent flow
|journal=[[Journal of Fluid Mechanics]]
|volume=91
|pages=1–16|bibcode = 1979JFM....91....1C |doi = 10.1017/S002211207900001X |s2cid=120228458
}}</ref> represents cross-scale interactions between large and small scales.<ref name="Leonard_1974"/>  Modeling the unclosed term <math>\tau_{ij}</math> is the task of sub-grid scale (SGS) models.  This is made challenging by the fact that the subgrid  stress tensor <math>\tau_{ij}</math> must account for interactions among all scales, including filtered scales with unfiltered scales.

The filtered governing equation for a passive scalar <math>\phi</math>, such as mixture fraction or temperature, can be written as

:<math>
\frac{ \partial \overline{\phi} }{ \partial t }
+ \frac{\partial}{\partial x_j} \left( \overline{u}_j \overline{\phi} \right)
= \frac{\partial \overline{J_{\phi}} }{\partial x_j} 
+ \frac{ \partial q_j }{ \partial x_j }
</math>

where <math>J_{\phi}</math> is the diffusive flux of <math>\phi</math>, and <math>q_j</math> is the sub-filter flux for the scalar <math>\phi</math>.  The filtered diffusive flux <math>\overline{J_{\phi}}</math> is unclosed, unless a particular form is assumed for it, such as a gradient diffusion model <math>J_{\phi} = D_{\phi} \frac{ \partial \phi }{ \partial x_i }</math>.  <math>q_j</math> is defined analogously to <math>\tau_{ij}</math>,

:<math>
q_j = \bar{\phi} \overline{u}_j - \overline{\phi u_j}
</math>

and can similarly be split up into contributions from interactions between various scales.  This sub-filter flux also requires a sub-filter model.

====Derivation====
Using [[Einstein notation]], the Navier–Stokes equations for an incompressible fluid in Cartesian coordinates are
: <math> \frac{\partial u_i}{\partial x_i} = 0 </math>
: <math> \frac{\partial u_i}{\partial t} + \frac{\partial u_iu_j}{\partial x_j}
= - \frac{1}{\rho} \frac{\partial p}{\partial x_i}
+ \nu \frac{\partial^2 u_i}{\partial x_j \partial x_j}.
</math>

Filtering the momentum equation results in
: <math> \overline{\frac{\partial u_i}{\partial t}} + \overline{\frac{\partial u_iu_j}{\partial x_j}}
= - \overline{\frac{1}{\rho} \frac{\partial p}{\partial x_i}}
+ \overline{\nu \frac{\partial^2 u_i}{\partial x_j \partial x_j}}.
</math>

If we assume that filtering and differentiation commute, then
: <math> \frac{\partial \bar{u_i}}{\partial t} + \overline{\frac{\partial u_iu_j}{\partial x_j}}
= - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i}
+ \nu \frac{\partial^2 \bar{u_i}}{\partial x_j \partial x_j}.
</math>

This equation models the changes in time of the filtered variables <math>\bar{u_i}</math>.  Since the unfiltered variables <math>u_i</math> are not known, it is impossible to directly calculate <math>\overline{\frac{\partial u_iu_j}{\partial x_j}}</math>.  However, the quantity <math> \frac{\partial \bar{u_i}\bar{u_j}}{\partial x_j}</math> is known.  A substitution is made:
: <math> \frac{\partial \bar{u_i}}{\partial t} +  \frac{\partial \bar{u_i}\bar{u_j}}{\partial x_j}
= - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i}
+ \nu \frac{\partial^2 \bar{u_i}}{\partial x_j \partial x_j}
- \left(\overline{ \frac{\partial u_iu_j}{\partial x_j}} -  \frac{\partial \bar{u_i}\bar{u_j}}{\partial x_j}\right).
</math>

Let <math>\tau_{ij} = \overline{u_i u_j} - \bar{u}_{i} \bar{u}_{j}</math>. The resulting set of equations are the LES equations:
:<math> \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}
+ \nu \frac{\partial^2 \bar{u_i}}{\partial x_j \partial x_j}
- \frac{\partial\tau_{ij}}{\partial x_j}.
</math>