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Wikipedia / Large eddy simulation

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 James Deardorff (1970). LES is currently applied in a wide variety of engineering applications, including combustion, acoustics, and simulations of the atmospheric boundary layer.

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 modelled, 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 and perform a spatial filtering operation, a temporal filtering operation, or both. The filtered field, denoted with a bar, is defined as:

is the filter convolution kernel. This can also be written as:

has an associated cutoff length scale may be split up into a filtered and sub-filtered (denoted with a prime) portion, as

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. 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 and Navier–Stokes equations are filtered, yielding the filtered incompressible continuity equation,

and the filtered Navier–Stokes equations,

is the rate-of-strain tensor evaluated using the filtered velocity. The nonlinear filtered advection 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), as:

is the residual stress tensor, so that the filtered Navier-Stokes equations become

, the Clark tensor, represents cross-scale interactions between large and small scales. must account for interactions among all scales, including filtered scales with unfiltered scales.

, such as mixture fraction or temperature, can be written as

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

Filtering the momentum equation results in

If we assume that filtering and differentiation commute, then

is known. A substitution is made:

. The resulting set of equations are the LES equations:

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 as:

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:

is the shear stress tensor, given for a Newtonian fluid by:

. The subgrid stress tensor for the Favre-filtered momentum field is given by

, which represents interactions among unresolved scales.

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:

to yield:

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.

can be positive or, 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).

Numerical methods

Large eddy simulation involves the solution to the discrete filtered governing equations using computational fluid dynamics, and as such a substantial portion of high wave number turbulent fluctuations must be resolved. This requires either high-order numerical schemes, or fine grid resolution if low-order numerical schemes are used. Chapter 13 of Pope. Ghosal. While even-order schemes have truncation error, they are non-dissipative, and because subfilter scale models are dissipative, even-order schemes will not affect the subfilter scale model contributions as strongly as dissipative schemes.

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. Chapter 8 of Sagaut (2006) covers LES numerics in greater detail.

Boundary conditions of large eddy simulations

Inlet boundary conditions affect the accuracy of LES significantly, and the treatment of inlet conditions for LES is a complicated problem. Theoretically, a good boundary condition for LES should contain the following features:

providing accurate information of flow characteristics, i.e. velocity and turbulence;
satisfying the Navier-Stokes equations and other physics;
being easy to implement and adjust to different cases.

Currently, methods of generating inlet conditions for LES are broadly divided into two categories classified by Tabor et al.:

The first method for generating turbulent inlets is to synthesize them according to particular cases, such as Fourier techniques, principle orthogonal decomposition (POD) and vortex methods. The synthesis techniques attempt to construct turbulent field at inlets that have suitable turbulence-like properties and make it easy to specify parameters of the turbulence, such as turbulent kinetic energy and turbulent dissipation rate. In addition, inlet conditions generated by using random numbers are computationally inexpensive. However, one serious drawback exists in the method. The synthesized turbulence does not satisfy the physical structure of fluid flow governed by Navier-Stokes equations.

The second method involves a separate and precursor calculation to generate a turbulent database which can be introduced into the main computation at the inlets. The database (sometimes named as ‘library’) can be generated in a number of ways, such as cyclic domains, pre-prepared library, and internal mapping. However, the method of generating turbulent inflow by precursor simulations requires large calculation capacity.

Researchers examining the application of various types of synthetic and precursor calculations have found that the more realistic the inlet turbulence, the more accurate LES predicts results.

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).

, but whose effects are dampened by the filter. Resolved sub-filter scales only exist when filters non-local in wave-space are used (such as a box or Gaussian filter). These resolved sub-filter scales must be modeled using filter reconstruction.

. The form of the SGS model depends on the filter implementation. As mentioned in the Numerical methods for LES section, if implicit LES is considered, no SGS model is implemented and the numerical effects of the discretization are assumed to mimic the physics of the unresolved turbulent motions.

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

is modeled as:

is the rate-of-strain tensor.

Based on dimensional analysis. Most eddy viscosity SGS models model the eddy viscosity as the product of a characteristic length scale and a characteristic velocity scale.

Smagorinsky–Lilly model

The first SGS model developed was the Smagorinsky–Lilly SGS model, which was developed by Joseph Smagorinsky and used in the first LES simulation by Deardorff. It models the eddy viscosity as:

is a constant.

Dynamic model (Germano et al. and beyond)

Germano et al.. Germano et is invariant of scale (see review could be determined. Lilly be determined using a least square fit by minimizing the residuals. This results in

Here

in a turbulent field is almost as likely to be negative as it is positive even though the integral over the fluid domain is always positive representing a net dissipation of energy in the large scales. A slight preponderance of positive values as opposed to strict positivity of the eddy-viscosity results in the observed net dissipation. This so-called "backscatter" of energy from small to large scales indeed corresponds to negative C values in the Smagorinsky model. Nevertheless, the Germano-Lilly formulation was found not to result in stable calculations. An ad hoc measure was adopted by averaging the numerator and denominator over homogeneous directions (where such directions exist in the flow)

was positive (or at least only rarely negative) stable calculations were possible. Simply setting the negative values to zero (a procedure called "clipping") with or without the averaging also resulted in stable calculations. Meneveau proposed an averaging over Lagrangian fluid trajectories with an exponentially decaying "memory". This can be applied to problems lacking homogeneous directions and can be stable if the effective time over which the averaging is done is long enough and yet not so long as to smooth out spatial inhomogenieties of interest.

Lilly's modification of the Germano method followed by a statistical averaging or synthetic removal of negative viscosity regions seems ad hoc, even if it could be made to "work". An alternate formulation of the least square minimization procedure known as the "Dynamic Localization Model" (DLM) was suggested by Ghosal et al. In this approach one first defines a quantity

is then pulled out of the hat operator

cannot be justified a posteriori. In the DLM approach one avoids this inconsistency by not invoking the step of removing C from the test filtering operation. Instead, one defines a global error over the entire flow domain by the quantity

must satisfy a Fredholm integral equation of the second kind

that is nonlinear

. The variational approach then immediately yields Lilly's result with averaging over homogeneous directions without any need for ad hoc modifications of a prior result.

One shortcoming of the DLM(+) model was that it did not describe backscatter which is known to be a real "thing" from analyzing DNS data. Two approaches were developed to address this. In one approach due to Carati et al. (follows by taking the trace of the Germano identity). To determine k we now use a transport equation

are positive coefficients representing kinetic energy dissipation and diffusion respectively. These can be determined following the dynamic procedure with constrained minimization as in DLM(+). This approach, though more expensive to implement than the DLM(+) was found to be stable and resulted in good agreement with experimental data for a variety of flows tested. Furthermore, it is mathematically impossible for the DLM(k) to result in an unstable computation as the sum of the large scale and SGS energies is non-increasing by construction. Both of these approaches incorporating backscatter works well. They yield models that are slightly less dissipative with somewhat improved performance over the DLM(+). The DLM(k) model additionally yields the subgrid kinetic energy, which may be a physical quantity of interest. These improvements are achieved at a somewhat increased cost in model implementation.

The Dynamic Model originated at the 1990 Summer Program of the Center for Turbulence Research (CTR) at Stanford University. A series of "CTR-Tea" seminars celebrated the 30th Anniversary of this important milestone in turbulence modeling.