Large eddy simulation

A large eddy simulation computes the time-dependent large-scale motion of a turbulent flow while representing the influence of unresolved motion through a subgrid-scale model. A spatial averaging filter specifies which information is retained. The distinction concerns spatial resolution, rather than simply replacing a fluctuating flow by its time average.^[ref]

For an elementary illustration, take one component of a periodic velocity field,

u_{x} (y) = U sin (2 π y / L) + a sin (20 π y / L) .

(1)

An ideal filter retaining Fourier modes with integer index at most four keeps the first term and removes the second, whose index is ten. The removed term still contributes to products such as

u_{x}^{2}

. Filtering the nonlinear Navier-Stokes equations consequently introduces a subgrid-scale stress that cannot generally be reconstructed from the retained velocity field alone.

This gives a computational compromise. Direct numerical simulation must resolve the dynamically relevant dissipative motion, whereas a large eddy simulation resolves a chosen coarser field and models its missing influence. Doubling grid counts in all three directions multiplies the cell count by eight, before any additional time-step cost. The filter, numerical discretization and closure must work together; a coarse grid by itself does not certify a useful approximation.

An early concrete demonstration was James W. Deardorff's 1970 calculation of turbulent channel flow using 6720 uniform grid intervals and strain-dependent coefficients for subgrid effects. The paper compared computed statistics with experimental measurements and reported agreement ranging from good to marginal. That comparison tests a particular simulation, rather than establishing that every resolved structure or model is exact.^[ref]

Implementations can supply subgrid dissipation in different ways. The Cardinal documentation, for example, describes a spectral filter used to drain energy from the smallest retained wavelengths; that is a particular numerical treatment, not the definition of all large eddy simulation.^[ref]

Table of contents
- Spatial averaging filter Large eddy simulation
  - Grid filter Spatial averaging filter
  - Box averaging filter Spatial averaging filter

Spatial averaging filter ( $\overset{ˉ}{f} = G_{Δ} * f$ )

 0  0

A spatial averaging filter maps a spatially varying quantity to a weighted local average. For a translation-invariant kernel

G_{Δ}

of characteristic width

Δ

\overset{ˉ}{f} (x) = \int_{R^{d}} G_{Δ} (r) f (x - r) d r, \int_{R^{d}} G_{Δ} (r) d r = 1.

(1)

This is a convolution. The normalization preserves a constant value; the kernel determines how each spatial scale is changed. In large eddy simulation, the filter defines the resolved quantities whose evolution is calculated.^[ref]

For a one-dimensional box averaging filter,

\overset{ˉ}{f} (x) = \frac{1}{Δ} \int_{x - Δ/2}^{x + Δ/2} f (y) d y .

(2)

Integrating

f (x) = cos (k x)

explicitly gives

\overset{ˉ}{f} (x) = \frac{s i n ( k Δ/2 )}{k Δ/2} cos (k x) .

(3)

The multiplier is a sinc function. A wave with

k Δ = π

retains amplitude

2/ π

, while one with

k Δ = 2 π

is removed. This filter attenuates different Fourier modes differently; it is not an ideal sharp cutoff of all modes above one threshold.

A linear filter does not preserve products: generally

\overline{f g} \neq = \overset{ˉ}{f} \overset{g}{ˉ}

. That lost product information produces the subgrid-scale stress. Nor does every filter satisfy

\overline{\overset{ˉ}{f}} = \overset{ˉ}{f}

; applying the box average twice produces a different kernel.

With constant width on an unbounded or periodic region, differentiating the convolution gives

\partial_{j} \overset{ˉ}{f} = \overline{\partial_{j} f}

under suitable regularity. A width varying with position or a special near-wall construction can introduce additional terms. Those assumptions must be stated before commuting a filter through a partial derivative.

Grid filter ( $\overset{u}{ˉ} = G_{Δ} * u$ )

 0  0

A grid filter defines the resolved field associated with a large eddy simulation, commonly written

\overset{u}{ˉ} = G_{Δ} * u

. Its name reflects the connection between resolved scales and a computational grid, but the filtering operator and the mesh are different specifications. An explicit spatial averaging filter states its kernel; in other methods the retained representation and numerical operations supply an implicit scale selection.^[ref]

A characteristic width is often related to cell dimensions. One possible three-dimensional convention is

Δ = (h_{x} h_{y} h_{z})^{1/3} .

(1)

Cells with widths

0.01

0.02

and

0.04 m

then have

Δ = 0.02 m

. This single number preserves the volume scale but does not describe the strong directional differences of the cell. The filter shape and directional resolution still matter.

Sampling alone also cannot determine which unresolved oscillations were present. At eight equally spaced points on a periodic interval of length

L

, the modes with indices ten and two have identical sine samples:

sin (2 π 10 j /8) = sin (2 π 2 j /8), j \in Z .

(2)

This is aliasing. Removing the high mode by a specified filter before sampling and merely sampling the unfiltered field are different operations.

The subgrid-scale stress depends on the filter actually adopted. A test filter then applies an additional operation to the already resolved data. Reporting only a grid spacing, without the filtering and discretization choices, can leave the mathematical meaning of the resolved field and its closure ambiguous.

Box averaging filter ( $G_{Δ} (r) = Δ^{- 1} 1_{{∣ r ∣ \leq Δ/2}}$ )

 0  0

A box averaging filter gives equal weight to all positions within a specified window and zero weight outside it. In one spatial dimension, its normalized kernel and action are

G_{Δ} (r) = \frac{1}{Δ} 1_{{∣ r ∣ \leq Δ/2}}, \overset{ˉ}{f} (x) = \frac{1}{Δ} \int_{x - Δ/2}^{x + Δ/2} f (y) d y .

(1)

This is a spatial averaging filter expressed as a convolution. In several dimensions, averaging over an axis-aligned rectangular box uses the product of the one-dimensional kernels.

The discrete counterpart averages a fixed number of samples. On an infinite sequence that is zero except for value ten at index zero, a centred five-sample box average produces value two at indices

- 2, - 1, 0, 1, 2

and zero elsewhere. The total remains ten, while the isolated peak spreads. On a finite array, boundary extension must be specified; reflection, zero padding and wrapping can give different edge values.^[ref]

Applying the same box average twice is not the same operation as applying it once. The new kernel is the convolution of the two kernels:

(G_{Δ} * G_{Δ}) (r) = \frac{m a x ( Δ - ∣ r ∣ , 0 )}{Δ ^{2}} .

(2)

Indeed, the integral counts the length of overlap of two intervals, divided by

Δ^{2}

. The resulting support has width

2Δ

, and the weight decreases away from its centre.

For a Fourier mode

cos (k x)

, one application multiplies the amplitude by

sin (k Δ/2) / (k Δ/2)

; two applications square this multiplier. The sinc function response explains why a box average smooths a signal without implementing an ideal sharp frequency cutoff. In large eddy simulation, that distinction matters when relating a grid filter to a coarser test filter.

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Large Eddy Simulation (LES) is a computational fluid dynamics technique used to simulate turbulent fluid flows. It is particularly effective for resolving the large-scale motions of turbulence while modeling the smaller-scale motions. ### Key Components of LES: 1. **Spatial Filtering**: In LES, the governing equations of fluid dynamics (like the Navier-Stokes equations) are filtered to separate the large eddies (large-scale turbulent structures) from the small eddies (small-scale turbulent structures).

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Large eddy simulation

Spatial averaging filter ( $\overset{ˉ}{f} = G_{Δ} * f$ )

Grid filter ( $\overset{u}{ˉ} = G_{Δ} * u$ )

Box averaging filter ( $G_{Δ} (r) = Δ^{- 1} 1_{{∣ r ∣ \leq Δ/2}}$ )

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Large eddy simulation

Spatial averaging filter (fˉ​=GΔ​∗f)

Grid filter (uˉ=GΔ​∗u)

Box averaging filter (GΔ​(r)=Δ−11{∣r∣≤Δ/2}​)

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 Articles by others on the same topic (1)

Spatial averaging filter ( $\overset{ˉ}{f} = G_{Δ} * f$ )

Grid filter ( $\overset{u}{ˉ} = G_{Δ} * u$ )

Box averaging filter ( $G_{Δ} (r) = Δ^{- 1} 1_{{∣ r ∣ \leq Δ/2}}$ )

 Discussion (0) New discussion