Mesh resolution

Mesh resolution, also called grid resolution, is the fineness of the computational grid in a computational fluid dynamics (CFD) model. The model divides the space into many small cells, which together form the grid, and solves the flow equations in every cell. Mesh resolution is usually expressed as the cell size, written δx. In the Fire Dynamics Simulator (FDS), the cell size also sets the size of the smallest flow structures that the model calculates directly.

Smaller cells show more detail, but they do not automatically give a better or more reliable result. They also cost much more computing time. Halving the cell size in FDS reduces the numerical error by a factor of four, but it increases the computing time by a factor of sixteen. Engineers therefore choose a cell size with a rule of thumb based on the size of the fire, and then test that choice with a grid sensitivity study. The report of a CFD study should explain why the chosen grid is fine enough, which sensitivity check was carried out and which uncertainty remains.

Cells, grids and filter width

A CFD model breaks the three-dimensional space of a fire scenario into cells, and the collection of cells is called the computational grid or mesh. General-purpose CFD software can often use complex cell shapes, such as tetrahedra, to follow complex geometry. Fire-specific software such as FDS uses only rectangular cells, so every object is reduced to rectangular blocks. Rectangular cells are simpler and faster, but they represent fire and flow effects around complex shapes less accurately.

FDS prefers uniform meshes, in which all cells have the same size. The only numerical parameters that the user chooses are the three dimensions of the grid. Walls and other obstructions "snap" to the nearest cell faces, so the geometry is only as detailed as the grid allows. Several meshes can be combined and run in parallel on different processors. This allows a fine mesh near the fire and coarser meshes in areas where little happens, for example in a large car park.

The cell size has a second meaning in FDS, which is a large eddy simulation (LES) model. LES computes the average values of mass, momentum and energy in each cell. The effect of the smaller, unresolved motions inside a cell is described by submodels. The width of the averaging "filter" is linked directly to the cell size. For cells that are not perfect cubes, FDS uses the cube root of the cell volume:

$$\mathrm{\Delta }={\left(\delta x\delta y\delta z\right)}^{1/3}$$

A coarser grid therefore means that more of the turbulence and combustion has to be modelled rather than calculated. The cell shape matters as well. Good practice is to keep the cell proportions as close as possible to 1:1:1, so that cells are not strongly stretched in one direction.

Why cell size matters

CFD models solve an approximate form of the conservation equations of mass, momentum and energy. Replacing derivatives by differences between neighbouring cells introduces an error that depends on the size of the cells. In some numerical schemes, part of this error acts like extra diffusion, so sharp features in the flow are smoothed out. This is called numerical diffusion.

A grid can only describe features that are larger than its cells. The smallest turbulent eddies in a fire are about one millimetre in size. Calculating them directly, which is called direct numerical simulation (DNS), is impractical except for special research flames. Resolving the chemical reaction zone of a fire would likewise require cells smaller than 1 mm. For domains between metres and kilometres in size, the affordable cell size in most LES fire calculations ranges from centimetres to metres.

In large fire simulations, cells are typically tens of centimetres wide. The thin flame sheet then lies inside a single cell, and the computed cell-average temperature can be much lower than the temperature in the flame itself. Measured time-averaged gas temperatures in fires seldom exceed about 1300 °C. Because CFD averages temperatures over each cell, it is unlikely to predict such peak values.

The grid also limits the geometry. In the fire tests of the National Institute of Standards and Technology (NIST) for the World Trade Center investigation, a 10 cm grid described the test compartment to within about 5 cm. On a 50 cm grid, a 20 cm shift of the top of a window changed the depth of the hot upper layer by the same amount. The upper layer temperature then changed by roughly 100 °C. Small objects, such as cable trays and floor trusses, may not fit on a coarse grid at all.

Discretisation error and computing cost

FDS uses second-order accurate approximations in both time and space. The discretisation error is therefore proportional to the square of the cell size and of the time step. In theory, halving the cell size reduces this error by a factor of four. The price is high, because the number of cells grows by a factor of two in each of the three directions.

The time step must shrink as well, because the model advances in small steps of time. The size of each step is limited by the cell size and by the flow speed in the cell. The Courant-Friedrichs-Lewy (CFL) condition states that a fluid element should not travel through more than one cell in a single time step:

$$\mathrm{C}\mathrm{F}\mathrm{L}=\frac{\delta t\Vert \mathbf{𝐮}\Vert }{\delta x}\approx 1$$

Halving the cell size therefore also halves the time step. Together, these effects increase the computing time by a factor of 16. For LES, keeping the CFL number close to 1 also keeps the time step consistent with the cell size. If a faster run is needed, it is better to use a coarser grid than to take much larger time steps on a fine grid.

A validation study of FDS for nuclear power plants shows this trade-off in practice. A plume temperature in one test was calculated on grids of 10 cm, 7.5 cm and 5 cm. The 10 cm run took a few hours on a single 2.4 GHz processor, while the 5 cm run took a few days.

The 5 cm prediction was noticeably better, because the entrainment of air into the plume was described more faithfully. Further refinement made no noticeable difference to the result. The point of diminishing returns depends on the application and on the quantity of interest. A hot gas layer temperature usually needs a less fine grid than the heat flux to a target close to the fire.

The cost explains why very large problems use coarse cells. The NIST simulations of the fires in the World Trade Center towers used cells of 0.5 m by 0.5 m by 0.4 m, with about 125,000 cells per floor. One simulation of 105 minutes of fire in one tower took about a week on eight computers. By the factor of 16 described above, halving the cell size in all three directions would have raised this to about one million cells per floor and roughly sixteen weeks of computing time.

Characteristic fire diameter and the D*/δx ratio

Whether a cell is "small" depends on the size of the fire. The physical diameter of a fire is not always well defined: a burning room has no clear diameter, whereas a round pan of burning liquid has an obvious one. The dimensionless heat release rate $Q^{*}$ compares the heat release rate of a fire with its physical diameter $D$:

$$Q^{*}=\frac{\dot{Q}}{{\rho }_{\mathrm{\infty }}{c}_p{T}_{\mathrm{\infty }}\sqrt{gD}{D}^2}$$

A large value of $Q^{*}$ describes a fire with a high energy output for its size, such as an oil well blowout. A low value describes a fire such as a brush fire, and most common accidental fires have values of about 1. Because the physical diameter is often unclear, CFD practice uses the characteristic fire diameter $D^{*}$ instead:

$$D^{*}={\left(\frac{\dot{Q}}{{\rho }_{\mathrm{\infty }}{c}_p{T}_{\mathrm{\infty }}\sqrt{g}}\right)}^{2/5}$$

When $Q^{*}$ is about 1, $D^{*}$ is comparable to the physical diameter of the fire. The ratio $D^{*}/\delta x$ gives the number of cells that span this characteristic diameter. The greater the ratio, the more of the fire dynamics are resolved directly and the more accurate the simulation becomes.

Symbol	Description	Unit
$\mathrm{\Delta }$	LES filter width	m
$\delta x,\delta y,\delta z$	cell size in the three coordinate directions	m
$\delta t$	time step	s
$\Vert \mathbf{𝐮}\Vert$	local flow speed	m/s
$\dot{Q}$	heat release rate of the fire	kW
$Q^{*}$	dimensionless heat release rate	–
$D$	physical diameter of the fire	m
$D^{*}$	characteristic fire diameter	m
${\rho }_{\mathrm{\infty }}$	density of the ambient air	kg/m³
$c_p$	specific heat of air	kJ/(kg·K)
$T_{\mathrm{\infty }}$	ambient temperature	K
$g$	acceleration due to gravity	m/s²

The expressions describe buoyant fires. A value of $Q^{*}$ above about 2.5 indicates a high-momentum jet fire rather than a buoyant fire. The ratio is a guide for choosing a grid, not a measure of accuracy in itself.

The validation report for nuclear power plants states that past experience has shown a ratio of 5 to 10 to produce favourable results at a moderate computational cost. The NIST World Trade Center investigation used a ratio of 10 when it chose a 10 cm grid for its laboratory fire tests. A later study compared FDS flame heights with an empirical correlation for $Q^{*}$ values from 0.1 to 10,000. It used ratios of 5, 10 and 20. Except at the lowest values of $Q^{*}$, the predictions fell within the uncertainty of the correlation itself. The table lists the ratios used in the earlier validation study.

Test series	Heat release rate (kW)	$D^{*}$ (m)	$\delta x$ (m)	$D^{*}/\delta x$
ICFMP BE #2	1800–3600	1.2–1.6	0.13	9–12
ICFMP BE #3	400–2300	0.7–1.3	0.10	7–13
ICFMP BE #4	3500	1.6	0.10	16
ICFMP BE #5	400	0.7	0.10	7
FM/SNL	500	0.7	0.05	14
NBS Multi-Room	100	0.4	0.10	4

The NBS Multi-Room tests show that the ratio alone does not decide the outcome. Only four cells spanned the burner, yet the predicted temperatures in all three rooms were within the experimental uncertainty. The ceiling was low, so the plume was less important for the average room conditions.

Worked example

A worked example in the New Zealand commentary on Verification Method C/VM2 (2013) models a hotel corridor with FDS. The fire grows to 340 kW, when sprinklers activate, and the guest room, corridor and lift lobby use cells of 0.1 m. The stairway, further from the fire, uses cells of 0.2 m. The ambient values are taken as $T_{\mathrm{\infty }}$ = 293 K, ${\rho }_{\mathrm{\infty }}$ = 1.2 kg/m³, $c_p$ = 1.0 kJ/(kg·K) and $g$ = 9.81 m/s².

The denominator of the expression for $D^{*}$ is 1.2 × 1.0 × 293 × √9.81, which is about 1101 kW/m^5/2. The characteristic diameter is then (340 / 1101)^2/5, or about 0.62 m. With 0.1 m cells, the ratio $D^{*}/\delta x$ is about 6.2, inside the range of 5 to 10.

A ratio between 5 and 10 would need cells between about 0.062 m and 0.12 m. Halving the cells to 0.05 m would raise the ratio to about 12, at roughly 16 times the computing cost. The stairway cells of 0.2 m give a ratio of only about 3.1. The commentary uses these coarser cells because the stairway is further from the fire.

The same ambient values reproduce the table above. A 500 kW fire gives $D^{*}$ = 0.73 m, which the validation study lists as 0.7 m. A 3500 kW fire gives 1.59 m, listed as 1.6 m.

Grid sensitivity studies

Determining which cell size to use for a given calculation is known as a grid sensitivity study. The idea is to run the same scenario on successively finer grids and to compare the results. The aim is the coarsest grid that still gives reliable results. In the validation study for nuclear power plants, each grid was refined until the results no longer changed from run to run.

A chapter on CFD in the SFPE Handbook of Fire Protection Engineering gives a checklist for such studies. The chapter deals mainly with Reynolds-averaged models rather than with LES. The grid study should preferably use at least three grid resolutions, ideally by doubling the number of cells in all directions. The vertical spacing needs special attention in buoyant flows.

The mesh should be fine at the fire source and at openings. The aspect ratio of the cells should typically not exceed 50. The time step must suit the grid, and global mass and heat balances should be better than 95 %.

A converged solution is not the same as a grid-independent solution. An iterative solver can reach very small residuals on a grid that is still too coarse. The solution must also be shown to be independent of the grid. In the terms of verification and validation, a grid study is a verification task. It checks whether the model equations are solved with adequate accuracy, not whether the physics in the model is correct.

The result of a grid study depends on the quantity examined. In one validation test, a 20 cm grid still predicted the average hot gas layer temperature reasonably well. However, the cable trays were too small to include, and the door and vent were described less accurately. The compartment pressure therefore had a greater error. Coarse grids can give reasonable values for quantities that follow directly from the conservation of mass and energy, such as average temperatures and pressures. Other results become less reliable or cannot be obtained at all.

NIST tested its World Trade Center grid with office workstation fires on cells of 25 cm and 50 cm. Both grids gave similar results, because the burning rate depended mainly on heat from the hot upper layer. The report warns that this is not guaranteed for other applications. A fire that is small compared with its compartment needs a much finer grid, because coarse grids smear out steep gradients.

Mesh resolution in design practice

Design models often combine meshes of different sizes, with a fine mesh near the fire and coarser meshes elsewhere. In the validation study, the two test series with a small fire in a large space used a fine grid in the plume, where the entrainment of fresh air matters most. Coarse grids were used elsewhere.

Building regulations sometimes have to allow for the grid. In New Zealand, C/VM2 (second edition, 2025) lets the leakage through walls be modelled in CFD as two vents, one at floor level and one at ceiling level. If the required leakage area is smaller than one cell, it may be increased to fit the grid. The combined modelled area must not exceed 5 times the calculated leakage area. The 2013 commentary example does this for a 10 mm gap around a door. It models the gap as two vents of 0.1 m by 0.1 m, which fit the 0.1 m mesh.

The same edition of C/VM2 asks for the design fire in a CFD model to be a burning area sized from a heat release rate per square metre. For most buildings, it allows 500 to 1000 kW/m², a range that accommodates different heat release rates and mesh sizes. A zone model uses a single value of 250 kW/m² instead.

The quantity of interest also guides the grid near particular features. Predicting the activation of a heat detector or sprinkler requires a realistic ceiling jet, and therefore a reasonably fine mesh just below the ceiling. Openings need enough cells as well. In one NIST test, a window was spanned by seven cells of 10 cm. The model predicted the inflow of air well, but it could not resolve the details of the outflow velocity, which varied by at least 3 m/s over the window height.

Limits and common misconceptions

A frequent misconception is that a finer grid always gives a better answer. Finer cells give a more specific result, but this does not by itself make the result more reliable. The combustion, turbulence and water spray models in CFD remain approximations, and the modeller still has to prescribe the fire scenario and its heat release rate.

A second misconception is that the rule of thumb for $D^{*}/\delta x$ replaces a grid study. The validation study states explicitly that it does not. The ratio describes the fire plume, while other features, such as openings, targets or a ceiling jet, may need their own resolution.

A third misconception is that good agreement on a coarse grid proves that the grid is adequate. NIST obtained encouraging results when it simulated a laboratory test on its coarse World Trade Center grid. The agreement occurred because a stable, uniform hot layer formed, so the conservation of energy controlled the temperatures. NIST stressed that coarsening a grid to that extent does not, as a rule, give such good results.

Even a careful grid study leaves some uncertainty. The NUREG-1824 validation reports treat the grid size of a CFD model as part of the intrinsic uncertainty of the model, which they did not try to quantify. Coarse cells give little or no reliable information at scales smaller than the cells. It is also easy to produce impressive images of a flow, so the sensitivity of the solution to further mesh refinement has to be shown, not assumed.

See also

• Fire modelling
• Zone model
• Sensitivity analysis
• Finite element analysis
• Hand calculations

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.
• Hamins, A., & McGrattan, K. (2007). Verification and validation of selected fire models for nuclear power plant applications: Vol. 2. Experimental uncertainty (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
• 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
• Ministry of Business, Innovation and Employment. (2013). Commentary for Building Code clauses C1–C6 and Verification Method C/VM2.
• Ministry of Business, Innovation and Employment. (2025b). Verification Method C/VM2: Framework for fire safety design (2nd ed.).
• National Institute of Standards and Technology. (2005). Final report on the collapse of the World Trade Center towers (NIST NCSTAR 1).
• 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.