Editing Large eddy simulation (section)

== Filtered governing equations ==

The governing equations of LES are obtained by filtering the [[partial differential equations]] governing the flow field <math>\rho \boldsymbol{u}(\boldsymbol{x},t)</math>.  There are differences between the incompressible and compressible LES governing equations, which lead to the definition of a new filtering operation.

=== 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>

=== Compressible governing equations ===

For the governing equations of compressible flow, each equation, starting with the conservation of mass, is filtered.  This gives:

:<math>
\frac{\partial \overline{\rho}}{\partial t} + \frac{ \partial \overline{u_i \rho} }{\partial x_i} = 0
</math>

which results in an additional sub-filter term.  However, it is desirable to avoid having to model the sub-filter scales of the mass conservation equation.  For this reason, Favre<ref name="Favre_1983">{{cite journal
|author=Favre, Alexandre
|title=Turbulence: space-time statistical properties and behavior in supersonic flows
|year=1983
|journal=Physics of Fluids A
|volume=23
|issue=10
|pages=2851–2863
|doi=10.1063/1.864049|bibcode = 1983PhFl...26.2851F }}</ref> proposed a density-weighted filtering operation, called Favre filtering, defined for an arbitrary quantity <math>\phi</math> as:

:<math>
\tilde{\phi} = \frac{ \overline{\rho \phi} }{ \overline{\rho} }
</math>

which, in the limit of incompressibility, becomes the normal filtering operation.  This makes the conservation of mass equation:

:<math>
\frac{\partial \overline{\rho}}{\partial t} + \frac{ \partial \overline{\rho} \tilde{u_i} }{ \partial x_i } = 0.
</math>

This concept can then be extended to write the Favre-filtered momentum equation for compressible flow.  Following Vreman:<ref name="Vreman_1995">{{cite journal
|author1=Vreman, Bert
|author2=Geurts, Bernard
|author3=Kuerten, Hans
|journal=[[Applied Scientific Research]]
|year=1995
|volume=45
|issue=3
|doi=10.1007/BF00849116
|title=Subgrid-modelling in LES of compressible flow
|pages=191–203|bibcode=1995FTC....54..191V
|url=https://research.utwente.nl/en/publications/subgridmodelling-in-les-of-compressible-flow(7c54958d-ebdd-4422-bf9d-c17052984a68).html
}}</ref>

:<math>
\frac{ \partial \overline{\rho} \tilde{u_i} }{ \partial t }
+ \frac{ \partial \overline{\rho} \tilde{u_i} \tilde{u_j} }{ \partial x_j }
+ \frac{ \partial \overline{p} }{ \partial x_i }
- \frac{ \partial \tilde{\sigma}_{ij} }{ \partial x_j }
= - \frac{ \partial \overline{\rho} \tau_{ij}^{r} }{ \partial x_j }
+ \frac{ \partial }{ \partial x_j } \left( \overline{\sigma}_{ij} - \tilde{\sigma}_{ij} \right)
</math>

where <math>\sigma_{ij}</math> is the [[shear stress]] tensor, given for a Newtonian fluid by:

:<math>
\sigma_{ij} = 2 \mu(T) S_{ij} - \frac{2}{3} \mu(T) \delta_{ij} S_{kk}
</math>

and the term <math>\frac{ \partial }{\partial x_j} \left( \overline{\sigma}_{ij} - \tilde{\sigma}_{ij} \right)</math> represents a sub-filter viscous contribution from evaluating the viscosity <math>\mu(T)</math> using the Favre-filtered temperature <math>\tilde{T}</math>.  The subgrid stress tensor for the Favre-filtered momentum field is given by

:<math>
\tau_{ij}^{r} = \widetilde{ u_i \cdot u_j } - \tilde{u_i} \tilde{u_j}
</math>

By analogy, the Leonard decomposition may also be written for the residual stress tensor for a filtered triple product <math>\overline{\rho \phi \psi}</math>.  The triple product can be rewritten using the Favre filtering operator as <math>\overline{\rho} \widetilde{\phi \psi}</math>, which is an unclosed term (it requires knowledge of the fields <math>\phi</math> and <math>\psi</math>, when only the fields <math>\tilde{\phi}</math> and <math>\tilde{\psi}</math> are known).  It can be broken up in a manner analogous to <math>\overline{u_i u_j}</math> above, which results in a sub-filter stress tensor <math>\overline{\rho} \left( \widetilde{\phi \psi} - \tilde{\phi} \tilde{\psi} \right)</math>.  This sub-filter term can be split up into contributions from three types of interactions: the Leondard tensor <math>L_{ij}</math>, representing interactions among resolved scales; the Clark tensor <math>C_{ij}</math>, representing interactions between resolved and unresolved scales; and the Reynolds tensor <math>R_{ij}</math>, which represents interactions among unresolved scales.<ref name="Sagaut_2009">{{cite book
|author1=Garnier, E.
|author2=Adams, N.
|author3=Sagaut, P.
|title=Large eddy simulation for compressible flows
|year=2009
|publisher=Springer
|isbn=978-90-481-2818-1
|doi=10.1007/978-90-481-2819-8|url=https://cds.cern.ch/record/1339029
}}</ref>

=== Filtered kinetic energy equation ===

In addition to the filtered mass and momentum equations, filtering the kinetic energy equation can provide additional insight.  The kinetic energy field can be filtered to yield the total filtered kinetic energy:

:<math>
\overline{E} = \frac{1}{2} \overline{ u_i u_i }
</math>

and the total filtered kinetic energy can be decomposed into two terms: the kinetic energy of the filtered velocity field <math>E_f</math>,

:<math>
E_f = \frac{1}{2} \overline{u_i}  \, \overline{u_i}
</math>

and the residual kinetic energy <math>k_r</math>,

:<math>
k_r = \frac{1}{2} \overline{ u_i u_i } - \frac{1}{2} \overline{u_i} \, \overline{u_i} = \frac{1}{2} \tau_{ii}^{r}
</math>

such that <math>\overline{E} = E_f + k_r</math>.

The conservation equation for <math>E_f</math> can be obtained by multiplying the filtered momentum transport equation by <math>\overline{u_i}</math> to yield:

:<math>
\frac{\partial E_f}{\partial t} 
+ \overline{u_j} \frac{\partial E_f}{\partial x_j} 
+ \frac{1}{\rho} \frac{\partial \overline{u_i} \bar{p} }{ \partial x_i }
+ \frac{\partial \overline{u_i} \tau_{ij}^{r}}{\partial x_j} 
- 2 \nu \frac{ \partial \overline{u_i} \bar{S_{ij}} }{ \partial x_j }
= 
- \epsilon_{f} 
- \Pi
</math>

where <math>\epsilon_{f} = 2 \nu \bar{S_{ij}} \bar{S_{ij}}</math> is the dissipation of kinetic energy of the filtered velocity field by viscous stress, and <math>\Pi = -\tau_{ij}^{r} \bar{S_{ij}}</math> represents the sub-filter scale (SFS) dissipation of kinetic energy.

The terms on the left-hand side represent transport, and the terms on the right-hand side are sink terms that dissipate kinetic energy.<ref name="Pope_2000" />

The <math>\Pi</math> SFS dissipation term is of particular interest, since it represents the transfer of energy from large resolved scales to small unresolved scales.  On average, <math>\Pi</math> transfers energy from large to small scales.  However, instantaneously <math>\Pi</math> can be positive ''or'' negative, meaning it can also act as a source term for <math>E_f</math>, the kinetic energy of the filtered velocity field.  The transfer of energy from unresolved to resolved scales is called '''backscatter''' (and likewise the transfer of energy from resolved to unresolved scales is called '''forward-scatter''').<ref name="Piomelli_1991">{{cite journal
|author1=Piomelli, U.
|author2=Cabot, W.
|author3=Moin, P.
|author4=Lee, S.
|title=Subgrid-scale backscatter in turbulent and transitional flows
|journal=Physics of Fluids A
|year=1991
|volume=3
|issue=7
|pages=1766–1771
|doi=10.1063/1.857956|bibcode = 1991PhFlA...3.1766P
|s2cid=54904570
|authorlink3=Parviz Moin }}</ref>