Large eddy simulation

Large eddy simulation (LES) is a technique for calculating turbulent flows in which the larger eddies are computed directly on a numerical grid, while the effect of the smaller eddies is modelled or ignored. It is used in computational fluid dynamics to describe the turbulent mixing of fuel, combustion products and surrounding air in a fire. This mixing determines the burning rate in most fires and controls the spread of smoke and hot gases.

LES lies between two other approaches. Reynolds-averaged models solve only for time-averaged properties and represent all turbulent fluctuations with a turbulence model. Direct numerical simulation follows the fluctuations at every scale, but it is not yet possible at a practical scale. LES is the method used by the Fire Dynamics Simulator of the US National Institute of Standards and Technology. Introduction to Fire Safety Engineering names this program, with its viewer Smokeview, as by far the most used fire CFD software. The quality of an LES depends strongly on the size of the grid cells compared with the fire, which makes mesh resolution a central concern of every simulation.

Principle

A turbulent flow contains eddies over a wide range of sizes, each carrying part of the turbulent energy. In fire scenarios the smallest eddies are on the order of a millimetre. The basic idea of LES is that the eddies responsible for most of the mixing are large enough to be calculated from the equations of fluid dynamics. The hope, which must be justified by comparison with experiments, is that the small-scale motion can be roughly accounted for or ignored. Applied to fire, LES aims to extract more detail in time and space from simulations on the finer grids that ever faster computers allow.

Mathematically, the LES equations are obtained by applying a low-pass filter of a certain width to the equations for the transport of mass, momentum and energy. In practice, the filtered quantities can be thought of as mean values over a grid cell. In one dimension, the filtered density of a cell of width Δ is the average of the density over that width:

$$\overline{\rho }(x,t)=\frac{1}{\mathrm{\Delta }}\underset{x-\mathrm{\Delta }/2}{\overset{x+\mathrm{\Delta }/2}{\int }}\rho (r,t)\mathrm{d}r,\mathrm{\Delta }={\left(\delta x\delta y\delta z\right)}^{1/3}$$

Symbol	Description	Unit
$\overline{\rho }$	filtered (cell mean) density	kg/m³
$\rho$	local density	kg/m³
$x,r$	position	m
$t$	time	s
$\mathrm{\Delta }$	filter width	m
$\delta x,\delta y,\delta z$	cell dimensions in the three directions	m

In the Fire Dynamics Simulator the filter width equals the local cell size, a practice called implicit filtering. For cells that are not cubes, the filter width is taken as the cube root of the cell volume. The aim of the simulation is to calculate how the cell mean values of mass, momentum and energy develop in time. At the same time, it must account for the effect of transport and chemistry below the cell size on those means.

Cox and Kumar compare the approaches with thermocouples of different response times. A Reynolds-averaged model behaves like a thermocouple with a long time constant, which follows only an average. LES avoids time averaging of the fluctuations larger than the mesh size, so the influence of the larger eddies is represented rigorously. It is like a thermocouple with a much shorter time constant, which can follow the larger fluctuations but not the small-scale detail.

Velocity at one point against time: DNS follows every fluctuation, LES follows the larger swings, RANS gives only the mean. Based on Cox and Kumar (2002), p. 3-196.

Comparison with RANS and DNS

Early CFD fire models were almost all based on the Reynolds-averaged Navier–Stokes (RANS) equations, in particular the two-equation k–ε turbulence model. In that model, k is the kinetic energy of the turbulence and ε its rate of dissipation. Cox and Kumar note that full rigour is lost at this step. According to the FDS technical reference guide, the time averaging of RANS needs large eddy transport coefficients. These smooth the results, so that the evolution of the large eddy structures that characterise most fire plumes is lost, together with local transient events.

Turbulent energy against inverse eddy size: LES computes the large eddies and models the small ones, RANS averages over all scales, DNS resolves all. Based on Cox and Kumar (2002), p. 3-196.

Direct numerical simulation (DNS) resolves every eddy and needs no turbulence model, but its grid must be finer than the Kolmogorov scale, the size of the smallest turbulent eddies. In fire scenarios this scale is usually about one millimetre, so DNS is impractical for all but special research calculations of flames. For domains from metres to kilometres, the grid resolution that can be afforded in an LES fire calculation ranges from centimetres to metres.

Introduction to Fire Safety Engineering summarises the difference in one sentence. RANS models the average effect of turbulence, while LES simulates the larger eddies in the flow in more detail. Neither approach resolves the small length and time scales of the combustion itself. These can only be described by submodels.

Subgrid-scale models

The effect of the unresolved eddies appears in the filtered momentum equation as a subgrid-scale stress, which must be modelled. In LES, the "turbulence model" refers to this closure. FDS treats the subgrid fluxes of momentum and of other quantities as a diffusion process driven by the resolved gradients. The key quantity is then the turbulent or eddy viscosity, an extra viscosity that represents the mixing by the small eddies.

The classic model, by Smagorinsky, relates the eddy viscosity to the filter width and the local rate of strain of the resolved flow. FDS used this model, with a constant of 0.2, in its versions 1 to 5. The 2007 validation study for nuclear power plant applications and the World Trade Center investigation both used this Smagorinsky form of LES. The current default in FDS is a variation of the model of Deardorff. It was chosen as the default after comparisons with a wide variety of full-scale experiments. The two models are:

$${\mu }_t=\rho (C_s\mathrm{\Delta }{)}^2|S|\text{(Smagorinsky)},{\mu }_t=\rho C_{\nu }\mathrm{\Delta }\sqrt{k_{\mathrm{s}\mathrm{g}\mathrm{s}}}\text{(Deardorff)}$$

Symbol	Description	Unit
${\mu }_t$	turbulent (eddy) viscosity	kg/(m·s)
$\rho$	density	kg/m³
$C_s$	Smagorinsky constant, 0.2 in FDS versions 1 to 5	none
$C_{\nu }$	Deardorff model constant, 0.1	none
$\mathrm{\Delta }$	filter width	m
$|S|$	magnitude of the resolved rate of strain	1/s
$k_{\mathrm{s}\mathrm{g}\mathrm{s}}$	subgrid kinetic energy per unit mass	m²/s²

In the Deardorff model, the subgrid kinetic energy is estimated from the difference between the velocity in a cell and a weighted average over its neighbours. FDS also offers a dynamic Smagorinsky model, in which the constant is computed from the local flow, and the model of Vreman. Close to walls it uses the WALE model for the first cell next to the wall, because that model scales the eddy viscosity correctly near a surface.

LES in the Fire Dynamics Simulator

FDS was designed to simulate thermally driven flows in buildings and uses a rectilinear grid. Because it is an LES model, uniform meshing is preferred, and the only numerical parameters chosen by the user are the three dimensions of the grid cells. Obstructions such as walls are snapped to the underlying grid. Introduction to Fire Safety Engineering notes that FDS can only use rectangular cells, which are simpler and faster but represent complex geometry less accurately.

The World Trade Center investigation shows the scale of a practical LES. NIST modelled each tower floor with a grid of 128 by 128 by 9 cells, covering 64 m by 64 m by 3.6 m, and ran the floors in parallel on separate processors. The cells measured 0.5 m by 0.5 m by 0.4 m, while the simulations of the compartment experiments used cubes of 10 cm.

The investigation also tested what happens when the coarse grid is used for the compartment experiments. Many obstructions could not be resolved on a half-metre grid, so the main features were kept, such as the total window area. The predicted temperatures were encouraging, but the report stresses that such coarsening will, as a rule, not give such good results. The good agreement came from a very stable, uniform hot layer, for which energy conservation largely fixed the temperatures. Temperatures and heat fluxes near the fire plume were more uncertain, because the flow dynamics were greatly simplified.

Validation

Because the small eddies are only modelled, the results of an LES fire model must be checked against experiments. The 2007 NUREG-1824 study, carried out for nuclear power plant applications in the United States, assessed how well FDS predicted 13 quantities, such as hot gas layer temperature and smoke layer height. All 13 received either a green rating, meaning that the differences from the measurements were smaller than the combined experimental uncertainty, or a yellow rating, meaning that the model should be used with caution.

The same report notes an important property of CFD. Unlike a zone model or an empirical correlation, it can give ever more accurate results as the grid is refined. The price is computing time: FDS calculations take hours or days, depending on the size of the grid and the accuracy wanted. Introduction to Fire Safety Engineering mentions durations of hours, days or even weeks. Supplement 1 of 2016 increased the number of experiments from 26 to almost 500. This greatly widened the range of validity, while the measures of model bias and scatter changed very little in most cases.

Grid resolution

Because LES resolves only eddies larger than the grid cells, its accuracy depends on the cell size. The NUREG-1824 validation report states that the effectiveness of the technique depends largely on the ratio of the characteristic fire diameter to the cell size, $D^{*}/\delta x$. The characteristic diameter follows from the heat release rate and is explained in the article on the dimensionless heat release rate. A ratio of 5 to 10 usually gives favourable results at a moderate computational cost. The World Trade Center investigation used a ratio of about 10 for its compartment experiments.

The rule of thumb does not replace a grid resolution study, in which the grid is refined until the results no longer change. The book describes the same practice as a grid independence study, which looks for the coarsest grid that still gives reliable results. It also advises cell proportions close to 1:1:1, finer grids near the fire and coarser grids elsewhere. The report should state why the chosen grid is suitable, which sensitivity check was carried out and what uncertainty remains.

Worked example

The example estimates suitable cell sizes for an average living-room sofa fire of about 2.5 MW. With the air properties of the NUREG-1824 example, its characteristic diameter is 1.39 m. All numbers below were calculated for this article.

For a ratio of 5 to 10, the cell size should lie between 1.39/5 = 0.28 m and 1.39/10 = 0.14 m. A grid of 10 cm cubes gives a ratio of 13.9, finer than this range and therefore better resolved but more costly. The cells used for the World Trade Center floors, 0.5 m by 0.5 m by 0.4 m, have a filter width of:

A 2.5 MW fire (characteristic diameter 1.39 m) on a 10 cm grid (left, D*/δx about 14) and on 0.46 m cells (right, D*/δx about 3). Based on McGrattan (2007), p. 6-5, and McGrattan et al. (2005), p. 48.

$$\mathrm{\Delta }=(0.5\times 0.5\times 0.4{)}^{1/3}=0.464\mathrm{m}$$

The ratio for the sofa fire would then be only 3.0, below the usual range. Refining the grid has a high price. One tower floor of 64 m by 64 m by 3.6 m needed 147,456 cells of the coarse size, but would need about 14.7 million cells as 10 cm cubes, 100 times more. Resolving the Kolmogorov scale of about one millimetre would require about 1.5 × 10¹³ cells for the same floor. That is about 100 million times the LES grid, which shows why DNS is impractical for fire engineering.

Limitations

LES does not resolve the combustion itself. The length and time scales of the chemical reactions may be orders of magnitude smaller than the grid can resolve, so the reactions depend on submodels. Most LES submodels are valid where turbulent advection controls the mixing, which needs a high enough Reynolds number. The FDS guide warns that in fires the assumption of a locally high Reynolds number is frequently invalid. The treatment of the small eddies remains a working hypothesis that must be justified by comparison with experiments.

The book lists further limits of CFD that apply to LES fire models. The model does not predict the fire on its own, because the modeller must prescribe the fire scenario and its heat release rate. Modelling ventilation-controlled fires is limited and model-dependent, and coarse cells give no or only limited information at a fine scale. Even with the best input and expert use, CFD models are mainly reliable for qualitative comparisons of situations. For quantitative analyses, comparison with laboratory experiments remains the best measure. The book therefore does not regard CFD as a tool for everyone: it advises a university background in physics or a related field, together with much practical experience.

A common misconception is that a finer grid always gives a better answer. The book points out that finer cells give more specific results, but not automatically better or more reliable ones. Another is that a good match with measured temperatures proves the whole flow field correct. The World Trade Center tests showed that a coarse grid can reproduce layer temperatures largely through energy conservation, while details near the plume remain uncertain.

See also

• Verification and validation
• Fire modelling
• Sensitivity analysis
• CFAST

References

• Cox, G., & Kumar, S. (2002). Modeling enclosure fires using CFD. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 3-194–3-218). National Fire Protection Association; Society of Fire Protection Engineers.
• McGrattan, K. (2007). Verification and validation of selected fire models for nuclear power plant applications: Vol. 7. Fire Dynamics Simulator (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
• McGrattan, K. B., Bouldin, C., & Forney, G. P. (2005). Computer simulation of the fires in the World Trade Center towers (NIST NCSTAR 1-5F). National Institute of Standards and Technology.
• McGrattan, K., Hostikka, S., Floyd, J., McDermott, R., Vanella, M., Mueller, E., & Paul, C. (2026). Fire Dynamics Simulator technical reference guide: Vol. 1. Mathematical model (6th ed., NIST Special Publication 1018-1). National Institute of Standards and Technology. https://doi.org/10.6028/NIST.SP.1018
• Najafi, B., Joglar, F., & Dreisbach, J. (2007). Verification and validation of selected fire models for nuclear power plant applications: Vol. 1. Main report (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
• Stroup, D., Lindeman, A., McGrattan, K., Peacock, R., Overholt, K., Joglar, F., LeStrange, S., & Montanez, S. (2016). Verification and validation of selected fire models for nuclear power plant applications: Supplement 1 (NUREG-1824, Supplement 1; EPRI 3002002182). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.