Modelling turbulent flows

Many practical engineering flows are turbulent, particularly in combustion where the increased mixing introduced through the breakup of turbulent structures is used to improve the combustion performance. Turbulent flows are characterised as being chaotic, three-dimensional, unsteady, swirled and dissipative, where turbulent struc- tures known as eddies break down into increasingly smaller structures until viscous forces within the fluid transform the contained kinetic energy into heat. The Navier- Stokes equations for turbulent flows describe the evolution of the fluid dynamics by assuming the flow is a continuum.

The length- and time-scales of turbulent structures vary significantly in practical flows. The largest eddies are dependent upon the characteristic lengths of the geo-

metry being studied, where these structures will then transfer energy into increasingly smaller eddies until the dissipative force arising from the molecular viscosity of the fluid is large compared to the inertial force of the eddy, and the kinetic energy is dis- sipated into heat. These small dissipative scales can be several orders of magnitude smaller than the largest scales, both in terms of length- and time-scales. Calculations that directly solve the Navier-Stokes equations and resolve all of these scales are re- ferred to as direct numerical simulations (DNSs), however these computations require enormous resources, and practical cases require models to describe the behaviour of turbulent flows.

The methods of modelling turbulent fluid flows can be grouped into two approaches; Reynolds averaged Navier Stokes (RANS) methods and large eddy simulations (LESs). The most popular and time-effective approach is associated with applying RANS mod- els, whereby the solution is solved for temporally averaged quantities, and models are used to describe the full range of turbulent scales. Alternatively, LES approaches spatially filter the instantaneous flow, whereby geometry-dependent large-scale eddies are resolved, with smaller eddies, ideally self-similar structures dominated by viscous dissipation, are modelled. These two approaches are now discussed.

3.3.1.1 Reynolds averaged Navier-Stokes equations

In contrast to the instantaneous flow field, the distribution of time-averaged quantities vary smoothly in space and are relatively well behaved. By taking the Favre average, which is the density weighted average across time, of the Navier-Stokes equations it is possible to derive a new set of governing equations that are more readily solved. The first step is the assumption that the Reynolds decomposition can be applied to each scalar, such that its instantaneous value, φ, can be described as the composition of both the Favre averaged, ˜φ, and the fluctuating contribution, φ′, such as

φ(t) = ˜φ + φ′(t) (3.90)

Applying this operation to the Navier-Stokes equations for continuity and momentum, and applying the relation that ˜φ′ = 0, results in additional unclosed terms in the un- derlying equations, known as the Reynolds stresses.

One of the widely used approaches to modelling the Reynolds stresses involves in- troducing the Boussinesq hypothesis to relate the unclosed terms to an effective eddy viscosity, and there are a range of models that have been applied to represent the tur- bulent viscosity, with the most popular methods being the k-ǫ and k-ω models, which

introduce a further two transport equations. One of the main drawbacks to the eddy viscosity methods is that all turbulence is treated as isotropic, which could be an inac- curate assumption in cases with dominant anisotropic turbulence, such as in cases with highly swirling flows, which is common in combustion. Alternatively, Reynolds stress models have been developed which introduce additional transport equations for each of the Reynolds stress terms, however the additional equations also require further constants to close specific terms.

RANS calculations are often sensitive to the choice of turbulence model, although the Reynolds stress model is often shown to be more accurate than eddy viscosity models for swirled combusting flows [350–352]. However, the range of available models alludes to one of the weaknesses of the RANS approach; as all the turbulent structures are being modelled, including the largest geometry-dependent scales, the accuracy of the model is somewhat dependent upon the flow conditions and geometry being studied.

3.3.1.2 Large eddy simulations

In contrast to RANS calculations, the LES method solves the instantaneous Navier- Stokes equation, which has been spatially filtered. The filtering operation allows for the more coarser energy containing eddies to be directly simulated, while the smaller eddies, which are assumed to be self-similar isotropic structures, are modelled. While a range of filters have been suggested in the literature, it is most common to imply the Schumann filter [353], whereby the filter length is identical to the cell size. As simulations are often computationally restricted to the number of cells in the domain, it is necessary to pursue the smallest possible filter size permissible, which is realised by this choice of implicit filter.

Applying the filter to the Navier-Stokes equations results in unclosed terms ana- logous to the Reynolds stress terms in RANS calculations. However, unlike RANS calculations, only the sub-filter scales need to be modelled, with the large geometry dependent scales being simulated without any further approximations. Very fine spa- tial scales are required in order to ensure that only the geometry independent structures are modelled, which often imposes a far higher spatial dependency than is required in RANS calculations. Furthermore, LES calculations require the full resolution of specific turbulent structures, which means that the modelled domain must be fully three-dimensional and unsteady.

LES has been successfully applied to model coal combustion in air-fired and oxy- fuel conditions [203, 210, 213, 214, 216, 354–356]. Studies comparing RANS predic-

tions to LES results often show improvement in agreement with measured data using the LES approach [73, 216]. However qualitative trends are often similar to RANS approaches, suggesting that the results are almost equally sensitive to the other sub- models that are used to predict combustion.

3.3.1.3 Particle motion and heat transfer

Simulating multiple phase flows introduces additional complexity, as the physical properties of different phases vary significantly. For coal combustion, the density of solid-phase coal particles is significantly higher than the density of the combustion environment, and it is necessary to separate the phases. As coal particles occupy a small volume fraction in pulverised fuel combustion systems, modelling approaches will often track the particles in a Lagrangian frame without accounting for the volume of the particles within the fluid, and treat the mass and energy transfer as source terms in the relevant equations. When particle-particle interactions and particle feedback on the fluid is significant, it is necessary to consider alternative Eulerian multiphase mod- els, however this approach introduces further challenges associated with accurately modelling the inter-phase exchange rates [357], and the Lagrangian method is often preferred in coal combustion systems.

In Lagrangian tracking, the motion of the discrete particle phase is modelled by treatment of the forces acting on the particle [357]. The change in a particle’s velocity is calculated using the drag and gravitational forces for particles as

d ˆvp

dt = FD ˆv− ˆvp
+

ˆg ρp− ρ
ρp

(3.91)

where ˆvp is the velocity of the particle, ˆv is the velocity of the fluid, FDis the drag force, ˆg is the gravity force vector, ρpis the density of the particle and ρ is the density of the fluid. The drag force is calculated in turbulent flows from as[207]

FD= 3 4 µCDRe ρpdp2 (3.92)

where µ is the dynamic viscosity of the fluid, dp is the particle diameter, CD is the drag coefficient for the particle and Re is the Reynolds number of the particle, which is calculated as Re = ρdp    ˆvp− ˆv    µ (3.93)

particles, however correlations based on non-spherical particles are also used for solid fuel combustion [218]. Additional forces, such as the thermophoretic and Brownian forces, can be included as additional terms on the right hand side of Equation (3.91), however the two forces that are shown are often dominant for pulverised fuel particles, which are usually greater than 1 μm in diameter.

The heat transfer towards a particle, assuming that the particle is at a constant in- ternal temperature, is described by the following equation [218]

mpCp dTp dt = hAp(T∞− Tp) + fh dmp dt H + QabsσsbAp(T 4 rad− T 4 p) (3.94)

where mp, Cp, Tp and Ap is the mass, specific heat, temperature and external surface area of the particle respectively, h is the heat transfer coefficient, which describes con- vective and conductive heat transfer, Qabsis the absorption efficiency of the particle, σsbis the Stefan-Boltzmann constant and Trad is the radiation temperature, which is defined by the local incident radiation intensity,

Trad=          R 4πIdΩ 4σsb          1 4 (3.95)

The heat transfer coefficient h is often calculated from correlations based on a spherical particle shape, such as the Ranz and Marshall [358] method.

The above description of particle motion and heat transfer requires correlations and assumptions for the derivation of the drag and heat transfer coefficients, as well as as- sumptions on the particle being spherical and thermally thin. Furthermore, the equa- tions require terms for the instantaneous velocity, density, temperature and incident radiation for the fluid phase, which are values that are not necessarily available for either RANS or LES calculations. Models are often used to describe the instantan- eous or subgrid velocity fluctuations by incorporating a stochastic perturbation of the particle’s velocity, such as the discrete random walk model for RANS [359] or the Bini and Jones [360] method for LES. However, the unresolved temperature, incid- ent radiation or species composition is often neglected. These fields are crucial in calculating the heat transfer to the particle and the progression of combustion, and neglecting the instantaneous values may induce significant errors. This is particularly the case in RANS calculations, as the time-averaged quantities for fields such as O₂ concentration could potentially vary considerably from instantaneous values.