Comparing large eddy simulations and experimental data

Introduction

Validation of large eddy simulations against experimental data is not straightforward:

• LES provide unsteady and spatially-filtered quantities. These instantaneous quantities cannot be directly compared to experimental flow fields. Only statistical quantities extracted from LES and experiments are expected to match as subgrid scale models are devised using statistical arguments (Pope⁴³¹, Pope⁴³²).

• To extract LES filtered quantities from experiments would require to measure three-dimensional instantaneous flow fields. Even though such experiments have already been reported (for example van der Bos et al.⁵¹⁷ or Tao et al.⁵⁰¹ who performed 3D PIV), three-dimensional instantaneous data are generally not available. Note also that such processing would require to specify the LES filter used in the simulation which is gener-ally not explicitly determined in practical computations.

• LES simulations provide Favre (mass weighted) filtered quantities whereas most diag-nostic techniques are expected to provide unweighted filtered quantities.

The following discussion compares statistical quantities such as means and variances in ex-periments and large eddy simulations. For clarity, the discussion is first conducted for constant density flows. Spatially filtered quantities are noted f (respectively ef when mass-weighted) and hf i denote usual time or ensemble averages (respectively {f } for Favre averages).

Constant density flows

The time-averaged spatially filtered quantity hf i of any variable f is given by:

hf i (x) = 1

The time (or ensemble) average of a filtered quantity f is equal to the filtered local averaged quantity because spatial filter and time-average operators may be exchanged. Assuming that the filter size ∆ remains small compared to the spatial evolution of hf i also allows to estimate hf i as hf i ≈ hf i.

The variance of the quantity f may be expressed as:

hf²i − hf i²=h Assuming that the filter size ∆ remains small compared to the spatial evolution of hf i and hf²i gives hf i ≈ hf i and hf²i ≈ hf²i. Eq. (4.93) becomes:

The variance of the quantity f is the sum of the variance of the filtered quantity f , provided by LES, and the time (or ensemble) average of the subgrid scale variance f²− f². To compare RMS quantities obtained in experiments and numerical simulations, this last contribution must be expressed:

• When f = ui is a velocity component, Eq. (4.94) becomes:

hu_i²i − hu_ii²≈h

hu_i²i − hu_ii²i +h

hu_iu_i− u_iu_iii

(4.95) where the subgrid scale variance, uiui−uiui, corresponding to the unresolved momentum transport, is modeled by the subgrid scale model and may be averaged over time. This relation is easily extended to subgrid scale Reynolds stresses:

huiuji − huiihuji ≈ [huiuji − huiihuji] + huiu_j− uiuji (4.96) where the first RHS term corresponds to the resolved momentum transport known in the simulation. The second term is the time-average of the subgrid scale Reynolds stresses, explicitly modeled in LES.

• When f is a scalar such as temperature, mixture or mass fraction, the subgrid scale variance in the last term of Eq. (4.94) is generally not explicitly modeled in simulation, even though some models are proposed in the framework of subgrid scale probability density functions. For example, Cook and Riley¹¹⁴propose a scale similarity assumption to model the subgrid scale variance of the mixture fraction Z (see § 6.5.2):

Z²− Z²= CZ

 Zc²− bZ

2

(4.97)

where CZ is a model parameter and bQ denotes a test filter larger than the LES filter.

Variable density flows

For variable density flows, Eq. (4.92) is rewritten for ρf , leading to:

hρ ef i = hρf i = hρi {f } (4.98)

where ef and {f } denote Favre (mass-weighted) spatial filter and averaging operators respec-tively. Comparison requires to extract from experiment the time (or ensemble) average hρf i of the quantity ρf , which is generally not available. Nevertheless, under the assumption that the filter size remains small against the spatial evolution of mean quantities, hρi{f } ≈ hρi{f } and hρi ≈ hρi lead to:

{f } =hρf i

hρi ≈ hρ ef i

hρi (4.99)

Accordingly, the Favre average {f } may be estimated by averaging the LES field ef weighted by the resolved density ρ. On the other hand, to simply average the LES field versus time will

not provide the actual Favre average {f }. Indeed, the time average value of ef is:

which is not equal to hρ ef i/hρi because filtering and averaging operator cannot be exchanged.

The variance of the quantity f may be expressed as:

hρi

Assuming that the spatial length scale of averaged quantities are small compared to the LES filter:

The RHS terms of this equation are the variance of the resolved field, which can be measured using the LES field, and the time-average of the subgrid scale variance, which must be modeled.

Previous comments in section 4.7.7 still hold:

• When f = uiis a velocity component, the subgrid scale varianceugiui−euiueifound in Eq.

(4.103) is described by the subgrid scale model. The previous result may be extended to Reynolds stresses: where the subgrid scale variance in the last RHS term is modeled in simulations.

• When f denotes a scalar field such as temperature, mixture fraction or species mass fractions, the subgrid variance ρ(ff²− ef²) is generally not explicitly modeled in simula-tions.

These findings show that the comparison of LES statistics with experimental data requires cautions, especially when Favre (mass-weighted) operators are involved. To estimate local vari-ances from LES results requires consideration of subgrid scale varivari-ances (Eq. 4.94 and 4.103).

An other technique would be to extract from experiments the filtered quantities computed in LES, but this approach is a challenge for experimentalists as it requires three-dimensional measurements (see van der Bos et al.⁵¹⁷ or Tao et al.⁵⁰¹) and the precise definition of the effective filter used in the simulations. Comparisons between experiments and LES should be clarified in the near future. First preliminary investigations, still to be confirmed, show that Eqs. (4.99) and (4.100) give similar results in practical simulations (private communications by S. Roux and H. Pitsch).