Turbulence

Turbulence is an irregular state of fluid motion in which the flow is highly disordered, rotational and three-dimensional. Its velocity changes randomly from moment to moment, and the flow contains eddies of widely different sizes. Turbulence is the counterpart of laminar flow, in which a fluid moves in smooth layers. It appears when the inertial forces in a flow become large compared with the viscous forces.

Most flows in fires are turbulent. The fire plume above a burning object is generally turbulent unless the fire is very small, and smoke flow is often turbulent near the fire, at openings and at obstacles. Turbulence mixes surrounding air into plumes and flames, increases the transfer of heat to walls and ceilings, and can accelerate a premixed flame until it becomes a detonation. Because of its random nature, a turbulent flow cannot be calculated exactly. Computer fire models therefore approximate it, for example with large eddy simulation.

Laminar and turbulent flow

Two kinds of force decide how a real fluid moves. Inertial forces arise from the velocity of the fluid, and viscous forces from its viscosity, its internal resistance to shear. The ratio of the two is the Reynolds number:

$$\mathrm{R}\mathrm{e}=\frac{Vd}{\nu },\nu =\frac{\mu }{\rho }$$

Symbol	Description	Unit
$\mathrm{R}\mathrm{e}$	Reynolds number	none
$V$	flow velocity	m/s
$d$	characteristic length, for example a pipe or duct diameter	m
$\nu$	kinematic viscosity	m²/s
$\mu$	dynamic viscosity	N·s/m²
$\rho$	density of the fluid	kg/m³

When the Reynolds number is small, the fluid moves in smooth layers and the flow is laminar. As the velocity increases, the inertial forces grow, the layers break up and the flow becomes turbulent. At a critical Reynolds number the flow becomes fully turbulent; for a flow entering a pipe through a sharp edge, this value is 2,700. For flow in a tube, Simmons gives a similar picture. Below a Reynolds number of 2,300 the flow is always laminar, above 3,200 it is usually turbulent, and in between it alternates between the two. The kinematic viscosity is the dynamic viscosity divided by the density.

Laminar flow (top) moves in smooth layers; turbulent flow (bottom) contains eddies of many sizes, and its velocity fluctuates around a mean value. Based on Kandola (2002), pp. 1-5 and 1-13.

Introduction to Fire Safety Engineering describes the difference in practical terms. In laminar flow the gas moves in thin layers along each other, with little mixing across the flow direction. In turbulent flow there are large and small eddies, also across the flow direction, and heat is mixed much more strongly. The change from laminar to turbulent flow happens as the velocity increases and where the flow meets obstacles.

Characteristics of turbulent flow

Randomness is the essential feature of turbulence. Kandola notes that turbulence cannot be defined in precise terms and that its flow cannot be predicted precisely. In practice, what matters is its effect on the flow. For this reason a turbulent flow is described as the sum of a mean component, which can be understood, and a fluctuating component, which is random:

$$q=\overline{q}+q^{\prime },\overline{q}=\frac{1}{\mathrm{\Delta }t}\underset{t_0}{\overset{t_0+\mathrm{\Delta }t}{\int }}q(t)\mathrm{d}t$$

Symbol	Description	Unit
$q$	instantaneous value of a flow quantity, such as a velocity component	unit of the quantity
$\overline{q}$	time-averaged (mean) value	unit of the quantity
$q^{\prime }$	fluctuating part, with a time average of zero	unit of the quantity
$\mathrm{\Delta }t$	averaging time	s

One of the main effects of turbulence is diffusion. A turbulent flow transports the fluid itself and anything carried with it, such as smoke particles or airborne pollution. Turbulence is therefore a feature of the flow, not of the fluid. The fluctuations come from a highly disordered set of eddies of widely different sizes, which move and rotate as the mean flow carries them along. During this process the larger eddies are distorted and stretched, and they break up into smaller ones.

The smallest turbulent eddies have a size known as the Kolmogorov scale. It depends on the kinematic viscosity and on the rate of viscous dissipation, the rate at which the kinetic energy of the motion is turned into heat by viscosity. The technical reference guide of the Fire Dynamics Simulator gives it as follows:

$$\eta ={\left(\frac{{\nu }^3}{\epsilon }\right)}^{1/4}$$

Here $\eta$ is the Kolmogorov scale, $\nu$ the kinematic viscosity and $\epsilon$ the rate of viscous dissipation. For a direct numerical simulation, which resolves all eddies, the grid must be finer than this scale. When the mean and fluctuating parts are inserted into the equations of motion, extra terms appear, known as Reynolds stresses. They act like an additional viscosity, often called the eddy viscosity. In almost all turbulent flows of practical interest they are much larger than the ordinary viscous stresses.

Turbulence in fire plumes and flames

A fire plume is the rising stream of hot gases above a fire. Heskestad describes it as generally turbulent, except when the fire source is very small. Because the flow fluctuates, plume properties are described by time-averaged values, such as the mean excess temperature and the mean velocity on the centreline of the plume. Air is drawn into the plume across its outer boundary, which at any instant is very sharp and highly convoluted. This entrainment makes the mass flow of the plume grow with height, so that the plume becomes wider and cooler and carries more and more diluted smoke. As a result, a smoke layer consists mostly of entrained air, and even a relatively small fire can produce a large flow of smoke and heat.

A turbulent fire plume: steady lower flame, intermittent upper flame, vortices shed from the base, and air entrained across a convoluted boundary. Based on Heskestad (2002), pp. 2-1–2-2, and Van Rossum (2027), §5.3.

The flame itself is also shaped by turbulence. In a laminar diffusion flame, such as a candle flame, oxygen reaches the fuel only by molecular diffusion. Because diffusion is slow, oxygen penetrates the flame only to a limited extent, and the outer part of a candle flame is the hottest.

In a turbulent diffusion flame, turbulence is largely responsible for the supply of oxygen. The flow pattern changes continuously, and the flame in effect pushes and pulls at the surrounding air. In fire flames, the lower part usually looks fairly steady, while the upper part is intermittent. Vortex structures sometimes form near the base of the flame and are shed upwards.

Fire flames also pulsate. The pulsation frequency depends mainly on the size of the fire. Cetegen and Ahmed collected measured pulsation frequencies for burners and pools from 0.03 to 20 m across, and fitted a simple relation:

Pulsation frequency of fire flames against burner or pool diameter, from the curve fit of Cetegen and Ahmed. Based on Heskestad (2002), pp. 2-14–2-15.

$$f=1.5D^{-1/2}$$

Here $f$ is the pulsation frequency in hertz and $D$ the diameter of the burner or pool in metres. Measured frequencies at a given diameter differ by up to a factor of two from each other. Large fires therefore pulsate slowly and small fires quickly.

Heat transfer and friction

Turbulence strongly increases heat transfer between a gas and a surface. In laminar flow, little mixing occurs across the flow and the transfer of heat to a wall or ceiling is limited. In turbulent flow, the eddies mix heat towards the surface and the transfer is larger. The rule of thumb is that a higher velocity and a more turbulent flow give a greater heat transfer. This effect is summarised in the convective heat transfer coefficient, which grows with turbulence and velocity. Typical values range from a few W/(m²·K) indoors to 50 W/(m²·K) in a very hot fire:

Situation	Convective heat transfer coefficient (W/(m²·K))
Normal indoor conditions	1 to 5
Smoke reservoir under a roof	3 to 10
Fully developed fire	20 to 25
Very hot fire	50
Outdoors in a storm	up to 200

For a standard fire, a value of 25 W/(m²·K) is usually assumed. Turbulence also increases friction at walls. In a turbulent boundary layer, the thin layer of flow next to a surface, the velocity changes much more steeply near the wall than in a laminar boundary layer. As a result, the frictional stress on the wall is much greater.

Turbulence in premixed flames and explosions

In a premixed flame, fuel and air are already mixed before they burn. Turbulence has a strong effect on the burning velocity of such flames. In a laminar flame the flow lines are parallel and the flame front is sharp. As the Reynolds number increases, the front becomes more and more blurred and takes on the appearance of a brush.

Local velocity fluctuations push parts of the front forward and back, which increases its effective area and therefore the burning velocity. For propane, ethene and acetylene burning in air, the burning velocity roughly doubles as the Reynolds number increases to 40,000. Theory suggests that it can increase by a factor of five when the turbulence is very strong.

In a duct or pipe, this effect can feed on itself. If the hot combustion products cannot vent, they push the flame and the unburnt gas ahead with increasing velocity. This creates more turbulence ahead of the flame front and a further increase in burning velocity.

In a long enough duct with rapid enough acceleration, the flame front acts as an accelerating piston and drives a shock wave ahead of it. The flame then becomes a detonation, typically travelling at 1,500 to 3,000 m/s. For gases initially at atmospheric pressure, the pressure behind the detonation can reach 20 bar, and up to 100 bar where it is reflected at the end of the duct. This process is the subject of the article on deflagration-to-detonation transition. The NUREG-1805 report likewise defines flame speed as dependent on the turbulence, the geometry of the equipment and the fundamental burning velocity.

Modelling turbulence

The Navier–Stokes equations describe fluid motion exactly, and they are valid for both laminar and turbulent flows. Analytical solutions exist only for very simple laminar cases, so practical turbulent flows are solved numerically. In fire engineering this is done with computational fluid dynamics. Introduction to Fire Safety Engineering stresses that turbulent flows, which occur in most fires, cannot be calculated exactly and can only be approximated. CFD can therefore give good qualitative insight, but its exact results must always be interpreted carefully and critically.

Two families of turbulence model are common. Reynolds-averaged Navier–Stokes (RANS) models represent the average effect of turbulence, while large eddy simulation (LES) computes the larger eddies in more detail. The technical reference guide of the Fire Dynamics Simulator explains why fire models moved towards LES. RANS models average over time and need large eddy transport coefficients, which smooth the results and lose the evolution of the large eddy structures that are characteristic of most fire plumes. In LES, the eddies that account for most of the mixing are large enough to be calculated, while the small-scale motion is approximated or ignored.

The same guide notes that turbulent mixing of fuel and combustion products with the surrounding air determines the burning rate in most fires and controls the spread of smoke and hot gases. This process is extremely difficult to predict accurately. Chemical reactions are even harder to capture, because their length and time scales may be orders of magnitude smaller than the grid can resolve. The rate of mixing in a computational cell is generally dominated by turbulence.

FDS estimates the mixing time in each cell from the fastest of three processes: molecular diffusion, advection by unresolved eddies and buoyant acceleration. Molecular diffusion controls mixing at the smallest scales. Above the Kolmogorov scale, in flows with a high enough Reynolds number, turbulent advection takes over, and at coarse scales buoyant acceleration is expected to control the mixing in fires. The guide also warns that fires differ from turbulent combustion in general, because the assumption of a locally high Reynolds number is frequently invalid in fires. The book adds that combustion models, turbulence models and models for extinguishing water are all approximations of sciences that are still developing.

Simpler two-zone models avoid turbulence altogether by assuming well-mixed layers. Their preconditions include a simple geometry without obstacles that make the smoke flow more turbulent, and a stable smoke layer without mixing at its interface. A common misconception is that turbulence is a property of hot smoke as such. Kandola stresses the opposite: turbulence is a feature of the flow, not of the fluid, so the same smoke can flow in a laminar or a turbulent way.

Worked example

The airflow in Klote's corridor example is used here to show how far flows at the scale of a building lie above the critical Reynolds number. The air moves at 1.45 m/s in a corridor 1.22 m wide. The kinematic viscosity of air at 300 K is 15.89 × 10⁻⁶ m²/s. All values below were calculated for this article, with the corridor width as the characteristic length.

$$\mathrm{R}\mathrm{e}=\frac{1.45\times 1.22}{15.89\times 10^{-6}}=111,000$$

This is about 40 times the critical value of 2,700, so the flow is fully turbulent. Hot gases have a much higher kinematic viscosity: for air at 600 K it is 52.69 × 10⁻⁶ m²/s, which lowers the Reynolds number of the same flow to about 33,600. That is still far into the turbulent range. For the flow at 300 K to stay below 2,700, the air would have to move at less than 0.035 m/s, which is barely a draught.

The second part of the example concerns a premixed flame. The maximum laminar burning velocity of a propane and air mixture is 41 cm/s. At a Reynolds number of 40,000 the burning velocity roughly doubles to about 0.82 m/s, and under very strong turbulence it could reach about 2.1 m/s. The pulsation relation gives a frequency of 1.5 Hz for a pool fire 1 m across and 0.93 Hz for a spill 2.6 m across. A large pool fire thus pulses roughly once per second.

See also

• Ceiling jet
• Heskestad plume
• Flame height
• Dimensionless heat release rate

References

• Atreya, A. (2002). Convection heat transfer. 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. 1-44–1-72). National Fire Protection Association; Society of Fire Protection Engineers.
• Heskestad, G. (2002). Fire plumes, flame height, and air entrainment. 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. 2-1–2-17). National Fire Protection Association; Society of Fire Protection Engineers.
• Iqbal, N., & Salley, M. H. (2004). Fire Dynamics Tools (FDTs): Quantitative fire hazard analysis methods for the U.S. Nuclear Regulatory Commission Fire Protection Inspection Program (NUREG-1805). U.S. Nuclear Regulatory Commission, Office of Nuclear Reactor Regulation.
• Kandola, B. S. (2002). Introduction to mechanics of fluids. 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. 1-1–1-26). National Fire Protection Association; Society of Fire Protection Engineers.
• Klote, J. H. (2002). Smoke control. 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. 4-274–4-291). National Fire Protection Association; Society of Fire Protection Engineers.
• 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
• Simmons, R. F. (2002). Premixed burning. 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. 1-144–1-154). National Fire Protection Association; Society of Fire Protection Engineers.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.