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Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and first explored by Deardorff (1970). LES is currently applied in a wide variety of engineering applications, including combustion, acoustics, and simulations of the atmospheric boundary layer.
Contents
- Filter definition and properties
- Filtered governing equations
- Incompressible flow
- Derivation
- Compressible governing equations
- Filtered kinetic energy equation
- Numerical methods for LES
- Filter implementation
- Modeling unresolved scales
- Sub grid scale models
- Functional eddyviscosity models
- References
The simulation of turbulent flows by numerically solving the Navier–Stokes equations requires resolving a very wide range of time and length scales, all of which affect the flow field. Such a resolution can be achieved with direct numerical simulation (DNS), but DNS is computationally expensive, and its cost prohibits simulation of practical engineering systems with complex geometry or flow configurations, such as turbulent jets, pumps, vehicles, and landing gear.
The principal idea behind LES is to reduce the computational cost by ignoring the smallest length scales, which are the most computationally expensive to resolve, via low-pass filtering of the Navier–Stokes equations. Such a low-pass filtering, which can be viewed as a time- and spatial-averaging, effectively removes small-scale information from the numerical solution. This information is not irrelevant, however, and its effect on the flow field must be modeled, a task which is an active area of research for problems in which small-scales can play an important role, such as near-wall flows , reacting flows, and multiphase flows.
Filter definition and properties
An LES filter can be applied to a spatial and temporal field
where
The filter kernel
It is important to note that the large eddy simulation filtering operation does not satisfy the properties of a Reynolds operator.
Filtered governing equations
The governing equations of LES are obtained by filtering the partial differential equations governing the flow field
Incompressible flow
For incompressible flow, the continuity equation and Navier–Stokes equations are filtered, yielding the filtered incompressible continuity equation,
and the filtered Navier–Stokes equations,
where
The filtered advection term can be split up, following Leonard (1974), as:
where
with the residual stress tensor
The filtered governing equation for a passive scalar
where
and can similarly be split up into contributions from interactions between various scales. This sub-filter tensor also requires a sub-filter model.
Derivation
Using Einstein notation, the Navier–Stokes equations for an incompressible fluid in Cartesian coordinates are
Filtering the momentum equation results in
If we assume that filtering and differentiation commute, then
This equation models the changes in time of the filtered variables
Let
Compressible governing equations
For the governing equations of compressible flow, each equation, starting with the conservation of mass, is filtered. This gives:
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 proposed a density-weighted filtering operation, called Favre filtering, defined for an arbitrary quantity
which, in the limit of incompressibility, becomes the normal filtering operation. This makes the conservation of mass equation:
This concept can then be extended to write the Favre-filtered momentum equation for compressible flow. Following Vreman:
where
and the term
By analogy, the Leonard decomposition may also be written for the residual stress tensor for a filtered triple product
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:
and the total filtered kinetic energy can be decomposed into two terms: the kinetic energy of the filtered velocity field
and the residual kinetic energy
such that
The conservation equation for
where
The terms on the left-hand side represent transport, and the terms on the right-hand side are sink terms that dissipate kinetic energy.
The
Numerical methods for LES
Large eddy simulation involves the solution to the discrete filtered governing equations using computational fluid dynamics. LES resolves scales from the domain size
Filter implementation
The filtering operation in large eddy simulation can be implicit or explicit. Implicit filtering recognizes that the subfilter scale model will dissipate in the same manner as many numerical schemes. In this way, the grid, or the numerical discretization scheme, can be assumed to be the LES low-pass filter. While this takes full advantage of the grid resolution, and eliminates the computational cost of calculating a subfilter scale model term, it is difficult to determine the shape of the LES filter that is associated with some numerical issues. Additionally, truncation error can also become an issue.
In explicit filtering, an LES filter is applied to the discretized Navier–Stokes equations, providing a well-defined filter shape and reducing the truncation error. However, explicit filtering requires a finer grid than implicit filtering, and the computational cost increases with
Modeling unresolved scales
To discuss the modeling of unresolved scales, first the unresolved scales must be classified. They fall into two groups: resolved sub-filter scales (SFS), and sub-grid scales(SGS).
The resolved sub-filter scales represent the scales with wave numbers larger than the cutoff wave number
Sub-grid scales are any scales that are smaller than the cutoff filter width
Sub-grid scale models
Without a universally valid description of turbulence, empirical information must be utilized when constructing and applying SGS models, supplemented with fundamental physical constraints such as Galilean invariance . Two classes of SGS models exist; the first class is functional models and the second class is structural models. Some models may be categorized as both.
Functional (eddy–viscosity) models
Functional models are simpler than structural models, focusing only on dissipating energy at a rate that is physically correct. These are based on an artificial eddy viscosity approach, where the effects of turbulence are lumped into a turbulent viscosity. The approach treats dissipation of kinetic energy at sub-grid scales as analogous to molecular diffusion. In this case, the deviatoric part of
where
Based on dimensional analysis, the eddy viscosity must have units of
Smagorinsky–Lilly model
The first SGS model developed was the Smagorinsky–Lilly SGS model, which was developed by Smagorinsky and used in the first LES simulation by Deardorff. It models the eddy viscosity as:
where
This method assumes that the energy production and dissipation of the small scales are in equilibrium - that is,
Germano dynamic model
Germano et al. identified a number of studies using the Smagorinsky model that each found different values for the Smagorinsky constant
which is also called the Germano identity. The quantity
where
However, this procedure was numerically unstable since the numerator could become negative and large fluctuations in
This has made the dynamic model more stable and making the method more widely applicable. Inherent in the procedure is the assumption that the coefficient