LOS TECNÓCRATAS EXIGEN LA BOLSA O LA VIDA CON MÁS EFICACIA QUE LOS «MARINES»

Advection is the general motion of fluid and is studied predominantly using fluid dy- namics. Fluid dynamics is a large and active area of research and only the basic outlines of the principles are given here. A considerable number of texts on the general applica- tion of fluid dynamics (e.g. Batchelor (1967); Turner (1973)) and combustion theory (e.g. Williams (1985)) provide a more in-depth discussion.

A fundamental aspect of fluid dynamics and its application to the understanding of the motion of gases is the concept of continuity. The molecules of a gas are considered to be continuous and thus to behave as a fluid rather than a collection of independent particles. In conjunction with other key foundations of physics in general such as ther- modynamics and the conservation of quantities, a set of governing equations to describe the motion of fluids can be constructed. The fluidisedequations of motionform the basis of all fluid dynamics modelling.

General governing equations

A description of the rate of change of the density of particles in relation to the velocity of the particles and distribution of mass of particles provides a method of describing the continuity of the particles (Elliot 1993). By taking velocity moments of the density distribution (i.e. multiplying by powers of the velocity and integrating with respect to the velocity (" uk_{du, wherekis the moment index: 0, 1, 2. . . ), the fluidised equations of} motion are obtained which can be used to describe the continuity, conservation of mass, momentum and energy, etc. (Elliot 1993).

Whenk=0, the integration of density distribution results in the equation of continuity. If the particles are considered to have mass, then the continuity equation also describes the conservation of mass (Batchelor (1967, p. 74); Williams (1985, p. 625)):

∂ρ

∂t +∇.(ρu) = 0, (2.10)

whereρis density,tis time, anduis the fluid velocity (with vector componentsu,v, and

w) and∇.is the Laplacian or gradient operator (i.e. in three dimensionsi_∂x∂ +j_∂y∂ +k_∂z∂).

This is called thefluidised form of the continuity equation and is presented as a partial differential equation. Concisely, it says that the rate of accumulation of mass in a volume element is the same as the rate at which mass flows out of the volume element.

However, in order to solve this equation, the evolution ofu(i.e. ∂u

∂t) is needed. This incompleteness is known as the closure problem and is a characteristic of all the fluid equations of motion. The next order velocity moment (k = 1) can be taken and the evo- lution of the velocity field determined. This results in an equation for the force balance of the fluid or theconservation of momentumequation (Batchelor (1967, p. 136); Williams (1985, p. 625)):

∂ρu

∂t +∇.(ρu)u+∇p= 0, (2.11)

wherepis pressure. This equation balances the rate of increase of momentum with the inertial, pressure and viscous forces, and is, at its most basic, the application of Newton’s Second Law (F =ma) to small volumes of fluid; it is more formally known as the Navier-

Stokes equation (here presented in its simplified inviscid form for illustration). However, the evolution ofpis then needed to solve this equation. This can be determined by taking the second velocity moment (k=2) which provides an equation for the conservation of energy in the same manner as the First Law of Thermodynamics (Williams (1985, p. 626)):

ρ∂E

∂t +∇p−k∇T −φ=constant, (2.12)

whereEis energy,k is the Boltzmann constant (coefficient of heat conduction),T is ab- solute temperature andφ is the energy dissipation function (which takes into account

irreversible production of heat through dissipation of mechanical energy). This equation describes the fact that the sum of the thermal, chemical and kinetic energy in the system is equal to the sum of the energy lost from, and the work done by, the system.

This equation, too, needs a further, equally incomplete, equation to provide a solution for the evolution of energy. One can either continue determining higher order moments ad nauseumin order to provide a suitably approximate solution (as the series of equations can never be truly closed but will asymptote to an exact answer) or, as is more frequently done, utilise an equation of state to provide the closure mechanism. In fluid dynamics, the equation of state is generally that of the ideal gas law (i.e.pV =nRT).

The above equations are in the form of the Euler equations for illustrative purposes. In the case of bushfire, where chemical reactions provide additional sources and sinks of mass, momentum and energy, the right hand side of the fluidised equations of motion may be non-zero and will depend on additional models of gas phase and solid phase species formation and consumption as well as chemical and enthalpy source closure (Cox 1998). Similar non-fluidised equations for the conservation of mass and energy for the solid phase can be constructed (assuming that solid phase fuel does not move and thus

§2.3 Physics of combustion and heat transfer 37

not have momentum (Porterieet al. 2000)). A typical conservation equation for a species places the accumulation rate of a given species in a given volume equal to the sum of convection of the species out of the volume, diffusion of the species into the volume and production of the species via chemical reaction.

Due to the complexity of the governing equations, they often cannot be solved an- alytically and must be solved numerically. A branch of fluid dynamics called compu- tational fluid dynamics (CFD) has arisen that utilises computers to numerically solve the equations of motion. Many methods exist to numerically solve the set of equations of motion (Shampine 1980) and much effort is exerted in developing faster and more efficient numerical solver methods (Ferziger and Peri´c 1996). The method of choice de- pends on many factors, including the form of the equations (e.g. Lagrangian, Eulerian), the spatial dimension of the equations (i.e. 1D, 2D or 3D), the method of discretisation of the equations (finite difference, finite element/volume), the spatial and temporal resolu- tions used, the treatment of boundary conditions, the computational capability, and the co-ordinate system involved.

Due to the considerable computational resources required to solve them precisely at the necessary scales (for example, using the method of direct numerical simulation (DNS) (Ferziger and Peri´c 1996, p. 249)), trade-offs are made in order to improve the speed of computation at the cost of precision. Buoyancy and turbulence are two aspects of fluid flow that that are particularly difficult to solve exactly via DNS and are instead modelled via separate mechanisms to improve computational feasibility.

Buoyancy, convection and turbulence

The action of heat release from the chemical reactions within a combustion zone results in heated gases, both in the form of combustion products as well ambient air heated by, or entrained into, the combustion products. The reduction in density caused by the heating of the gas increases the buoyancy of the gas and results in the gas rising as a plume via convection. The interaction of the rising gas with an air flow can then lead to turbulence in the flow (Turner 1973).

Because modelling of a bushfire necessarily requires modelling of atmospheric pro- cesses, the regime of the flow is of high-order Reynolds number (in the order of 60,000- 90,000) which is well outside the bounds of laminar flow (Jim´enez 2006). Thus turbulence is a key component of air flow in the open. Turbulence acts over the entire range of scales in the atmosphere, from the fine scale of flame to the scale of the atmospheric boundary layer. Non-linearity produces eddy motions of smaller and smaller scales until viscous dissipation causes the cascading of energy to smaller scales to stop (Richardson 1922; Kolmogorov 1941). Interaction of the flow with elements on the earth’s surface, such as terrain, vegetation or structures, through the effect of drag and mechanical disturbances, can increase the rate at which energy is cascaded down the scales (Finnigan and Brunet 1995; Finnigan 2000).

Turbulence mixes heated gases from the fire with ambient air and acts to increase the entrainment of cooler air into the convection column. Turbulence also mixes the heated gases of the fire with unburnt solid phase fuels around the fire and to immerse the fuel in flame, resulting in greater transfer of heat and increased fuel ignition rates (Finney et al. 2006). This aspect of turbulent buoyant flow in and around the fuel is critical to the understanding of the behaviour of bushfires, particularly in complicated

fuel structures. Turbulence also affects the transport of solid phase fuel, such as that of firebrands, resulting in spotfires downwind of the main burning front (see Section 2.3.3). It can also act to increase the rate of solid phase combustion by improving gas-solid interface exchange and removing insulating ash.

The study and modelling of turbulence is a very active research field. Suitably for- mulated Navier-Stokes equations can be used to incorporate the effects of buoyancy and turbulence as separately modelled components (such as turbulent kinetic energy, energy dissipated via the energy cascade or Boussinesq’s eddy viscosity for the modelling of turbulence or a renormalised perturbation variable for modelling the effects of buoy- ancy (McComb 2006)) in such a way that the small-scale turbulent fluctuations do not have to be directly simulated.

The methods for solving the Navier-Stokes equations represent a spectrum of ap- proaches: from DNS, which explicitly computes everything up to and including the en- ergy dissipation scales; Reynolds-averaged Navier-Stokes (RANS) simulations (Ferziger and Peri´c 1996, p. 265), in which the solution variables in the instantaneous (exact) Navier-Stokes equations are decomposed into the mean (ensemble-averaged or time- averaged) and fluctuating components; and large-eddy simulations (LES) (Mason 1994) which explicitly computes large scale eddies directly but treats the dissipation and in- ertial cascade at smaller scales using sub-grid-scale approximations. The formulation of equations used in RANS are are not closed due to use of unknown Reynolds stress terms and must be closed through estimated eddy viscosities (such as the κ ₋ε method in-

volving the evolution of fluctuating kinetic energy (κ) and eddy dissipation (ε) (Jim´enez

2006)) or Reynolds stress evolution methods (known as the Reynolds stress transport model (RSM) (Launderet al. 1975)), in which the RANS equations are closed by equa- tions for the Reynolds stresses.

All these methods provide a direct estimation of the kinetic energy associated with turbulent motion into the conservation of energy equation. The application of rigorous statistical methods from quantum field theory (i.e. renormalisation group theory (RNG) (Yakot and Orszag 1986; McComb 2004)) have led to improved formulations of some of these turbulence models (McComb 2006).

Atmospheric interactions

In addition to the exchange of heat released from the fire to the flow of the air immedi- ately around the fire, the interaction of the transport of the gas phase products from the combustion processes with the wider-scale atmosphere flow also plays a significant role in determining the behaviour of the fire (Clark et al. 1996a). If only a subsection of the atmosphere in which the fire is present is modelled, suitable boundary conditions and the interactions across those boundaries are needed to adequately model the larger scale effects of the fire. For example, the condition of the atmosphere, particularly the lapse rate (the negative rate of change of air temperature with increasing height which dictates the ease with which heated parcels of air rise within the atmosphere), controls the impact the buoyancy of the heated air from the combustion zone will have on the atmosphere and the fire (Byram 1959b).

Changes in the ambient meteorological conditions, such as changes in wind speed and direction, moisture, temperature, lapse rate, etc, both at the surface and higher in the atmosphere, can have a significant impact on the state of the fuel (moisture content),

§2.3 Physics of combustion and heat transfer 39

the behaviour of a fire, its growth, and, in turn, the impact the fire can have on the atmo- sphere itself. Structures in the atmospheric flow such as fire whirls or, on a larger scale, pyro-cumulus cloud (Mills 2005b), can be generated. These structures might themselves in turn affect the spread and behaviour of the fire.

Topographic interactions

The topography in which a fire is burning also plays a part in the way in which energy is transferred to unburnt fuel and the ambient atmosphere. The ground acts as an imper- vious boundary that defines the bottom of the atmosphere (the atmospheric boundary layer) and acts as a source of friction to the flow over it (Gibson and Launder 1978). Veg- etation on that boundary increases the amount of friction through the roughness length and provides the layer (and the fuel) through which the fire moves (Finnigan 2000).

It has long been recognised that fires burn faster upslope than they do down (McArthur 1967; Van Wagner 1977). This is thought to be due to increased transfer of radiant heat due to the change in the geometry between the fuel on the slope and the flame, however recent work by Wuet al. (2000) suggests that there is also increased ad- vection in these cases.

The complicated interaction of the topography with the atmosphere results in changes to the flow over the land surface. The surface, particularly complex geometry surfaces as found in hilly or mountainous terrain, induces turbulence in the air flow over it by causing flow separation and the formation of eddies (Belcher and Hunt 1998) and can lead to wind directions at odds with the bulk (synoptic) motion of the air (e.g. gully or valley winds). Differential solar heating of the surface can lead to the generation of local vortices (commonly called whirlwinds or willy-willies) and also differential in fuel moisture contents. Diurnal heating and cooling of the ground can result in generation of upslope (anabatic) or downslope (katabatic) winds.