Chapter2reviews the basic concepts of the key parameters determining the mixing.
The characteristics of turbulent flows are described at first. A brief introduction of droplet evaporation is followed. Finally, the characteristic mixing quantities are discussed and the scaling laws for mixing in inter-droplet space are listed.
Chapter 3 presents the detailed derivations of the scaling laws for spatial distri-butions of mixture fraction, the scalar dissipation and PDF in the Kolmogorov scale zone. When droplets move relative to the surrounding turbulent flow, a wake-like structure develops. The Kolmogorov scale zone (KSZ) is located beyond the quasi-laminar region (near droplet zone) where the diffusion effect is weak and turbulence determines the mixing. The determining parameters are the droplet evaporation rate, turbulent energy dissipation and the relative velocity between the droplet and its surrounding phase. The mixture fraction conditional scalar dissipation is derived by applying non-dimensional analysis (Π-theorem). The mixture fraction PDF is obtained by a quasi-steady-state relationship with the scalar dissipation. This re-lationship is derived by the integral of continuity and mixture fraction transport equations for the droplet evaporation in a spherically symmetric coordinate system using Robin boundary conditions at the droplet surface.
Chapter 4 introduces the computational frameworks of fully resolved DNS and CP-DNS. Fully resolved DNS will form the basis for the detailed analysis of all essential local interactions within the gas phase including the boundary layer at
1.4. THESIS OUTLINE 8 the droplet surface and the smallest turbulence scales. The standard conservation equations for mass, momentum, energy, species and an additional transport equation for the conserved scalar mixture fraction are solved simultaneously for the gas and liquid phases. They are coupled by boundary conditions at the droplet surface including the mass (flux) conservation, heat conduction and species diffusion to ensure the correct coupling. However, the random character of the droplet position is neglected in the fully resolved DNS and the CP-DNS can be used to overcome this limitation. The standard governing equations in the fully resolved DNS are extended by the Lagrangian formulation where the droplets are treated as point sources of mass, momentum and energy in the CP-DNS. The inflow turbulence generation method and the discretization schemes for solving the governing equations are also presented.
Chapter5presents the results of fully resolved DNS of droplet array evaporation and combustion in turbulent convective flows and modelling for mixing fields in inter-droplet space. A validation of the necessary mesh resolution and domain size is performed to ensure a good balance between computational cost and solution accuracy. The newly derived scaling laws are tested for the non-reacting cases under different turbulent intensities. The analysis is then extended to the reacting flows with a wider range of turbulent scales and investigated parameters. The transition between the near droplet zone and the Kolmogorov scale zone can be determined by the evaluation of variation of mixture fraction in axial and radial directions in inter-droplet space. After the Kolmogorov scale zone is located, the scaling laws for mixture fraction conditional dissipation and its PDF are assessed. Suitable modelling constants in the scaling laws are extracted from the DNS and functional dependencies of the parameters are suggested.
In Chapter6, spray evaporation in spatially decaying turbulence is simulated by a series of CP-DNS. The CP-DNS overcomes a limitation of fully resolved DNS of regular droplet arrays where the random character of droplet position is neglected.
The CP-DNS also covers a much wider parameter range than that was used to calibrate scaling laws in fully resolved DNS. The error introduced by the CP-DNS method is estimated by a comparison of the CP-DNS with results from fully resolved DNS as conducted in Chapter 5. It is concluded that CP-DNS data needs to be selected from beyond the insufficiently resolved quasi-laminar wake. The scaling laws are assessed by comparison with the CP-DNS statistics of the characteristic mixing quantities in the region where the small scale interactions between turbulence and evaporation determine the mixing.
In Chapter 7, carrier-phase direct numerical simulations (CP-DNS) are per-formed for turbulent spray combustion. The turbulent spray flame is statistically stabilized in the domain by dynamically controlling the mean velocity of the turbu-lent inflow and the fuel injection rate. Following the conclusions from Chapter 6for purely evaporating sprays, the DNS statistics are still taken beyond the unresolved quasi-laminar wake of the droplet for the assessment of the scaling laws for turbulent micro-mixing in spray combustion.
1.4. THESIS OUTLINE 10
Scalar Mixing in Turbulent Sprays
The basic concepts on the mixing modelling in inter-droplet space are reviewed in this chapter. The characteristics of turbulent flows such as turbulent length scales, turbulent fluctuation, turbulent kinetic energy and its dissipation are described at first. A brief introduction of droplet evaporation is followed in the second section and the derivation of a single droplet evaporation model is also reviewed there. Then, the definitions of mixture fraction, scalar dissipation and PDF in inter-droplet space are discussed, and the key parameters involving the mixing are illustrated. In the last section, the existing scaling laws for the near droplet zone and the newly derived scaling laws for the Kolmogorov scale zone are listed.
2.1 Turbulent Flows
The computational frameworks with different degrees of turbulence modelling are used to simulate turbulent flows. In the context of large eddy simulation (LES), large turbulent scales containing the most energetic eddies are resolved but small scale fluctuations are filtered and treated with models. Direct numerical simula-tion is required to resolve all the flow scales ranging from the Kolmogorov length scale and even smaller scales, e.g. the boundary layer at the droplet surface in the present study, to turbulent integral length scales. This leads to extremely high computational cost and DNS is usually limited to low Reynolds number flows (Re∼100). However, the ability of current DNS has been demonstrated to present the LES(RANS) sub-grid scale velocity and scalar fields in various turbulent non-reacting or non-reacting flows [134,138]. Considerable simplifications can be formulated if a turbulent flow reaches a statistically stationary state after a sufficiently long transient period [101]. The turbulent velocity can be decomposed into a constant
11
2.1. TURBULENT FLOWS 12 temporally independent mean velocity and statistically isotropic velocity fluctua-tions. The average magnitude of turbulent fluctuation, U0, is characterized by the turbulent kinetic energy which is given by [22,101, 106, 126]
U0 = r2k
3 . (2.1)
The turbulent kinetic energy,k, is supplied at the large turbulent scales by gradients of the mean velocity, followed by energy decay due to the break-up of large eddies into smaller ones, until the energy is finally dissipated into heat at the smallest (Kolmogorov) scales. This process is known as the principle of the energy cascade [106]. The large eddy length scale, lt, is of the order of the physical width of the flow and its more rigorous definition is the integral length scale. For an isotropic turbulent flow, the integral length scale can be calculated by [22, 101]
lt= Z ∞
0
Rii(rii)drii, (2.2)
where Rii is the two-point spatial correlation tensor and rii denotes the distance of two different locations in the flow field. The two-point correlation is defined as [22, 101]
Rii= u0i(x)u0i(x+rii) u0i2
, (2.3)
where u0i(x) represents the fluctuating velocity at a location, x and rii denotes a vector connecting two points. The large eddy (turn-over) time scale, which is also known as the time scale for the turbulent energy cascading from the large to the small eddy, is defined as [22, 101, 106,126]
τt= lt
U0. (2.4)
The rate of energy cascading is equal to the rate of energy that is dissipated into heat at the smallest scales. This rate is known as the turbulent kinetic energy dissipation rate, ε, which can be estimated as being of the order of k/τt and expressed as [22, 101, 106, 126]
ε∼ U03 lt
. (2.5)
Note that the energy dissipation rate can be directly computed in DNS of reacting flows as [23]
ε≡ 1^ ρτij∂u0i
∂xj. (2.6)
Here,τij denotes the viscous stress of a Newtonian fluid that is given by [22,101,126]
τij =µ ∂ui
∂xj +∂uj
∂xi
− 2 3µ∂uk
∂xkδij, (2.7)
where µ denotes the dynamic viscosity, δij represents the Kronecker delta that is written as [101]
δij =
1, if i=j, 0, if i6=j.
(2.8) According to Kolmogorov’s first similarity hypothesis [69], the smallest (Kol-mogorov) length (η) and time (τη) scales can be uniquely determined by the viscosity, ν, and energy dissipation, ε, viz.
η ≡ ν3/ε1/4
, (2.9)
τη ≡(ν/ε)1/2. (2.10)