Chapter II: Theoretical Framework and Computational Methodology
2.1 Direct Numerical Simulations
The flows considered in this work are governed by the Navier-Stokes and scalar transport equations. The low-Mach number approximation is considered [80, 81].
This is a valid assumption for a wide range of combustion systems, and allows large density variations while removing acoustic effects. For example, Mach num- bers found in laminar and turbulent non-premixed flames are typically well below 0.1 [82].
Conservation of mass and momentum read
∂ ρ
∂t +∇ ·(ρu) =0, (2.1)
∂
∂t(ρu)+∇ ·(ρu⊗ u) =−∇p+∇ ·σ+ f, (2.2) respectively, wherepis the hydrodynamic pressure,uis the velocity,σis the stress tensor
σ= µ(∇u+∇uT)− 2
3µ(∇ ·u)I, (2.3)
where µis the dynamic viscosity of the mixture, and f represents all volumetric forces, including a gravitational term. The viscosity µis computed as [43]
µ= 1 2
* . ,
X
i=1
Xiµi+
X
i=1
Xi
µi
−1
+ / -
, (2.4)
where theith species viscosity is computed using its kinetic theory definition [83].
The transport equation for theith species, which was introduced in Ch. 1, and is repeated here for clarity, reads
∂
∂t(ρYi)+∇ ·(ρuYi) =−∇ · ji+ω˙i, (2.5)
2.1. Direct Numerical Simulations 15 The mixture-averaged diffusion coefficients of speciesi, Di,m, are calculated either by supplying a set of Lewis numbers, Eq. (1.1), or by using a mixture-averaged for- mulation,i.e., using Eq. (1.5). The binary diffusion coefficients Dki are computed using classical molecular gas theory [40]. In this work, Soret and Dufour effects are not considered. The energy equation reads
∂
∂t(ρT)+∇ ·(ρuT) =∇ ·(ρα∇T)+ω˙T − 1 cp
X
i
cp,iji· ∇T+ ρα cp
∇cp· ∇T, (2.6) whereT is the temperature,cp,iisith species specific heat at constant pressure, and ω˙T= −c−1p P
ihi(T)ω˙iis the temperature source term;hi(T) are the species specific enthalpies at temperature T. The heat diffusivityαis computed as
α= λ ρcp
, (2.7)
whereλis computed using [42]
λ= 1 2
* . ,
X
i=1
Xiλi+
X
i=1
Xi
λi
−1
+ / -
. (2.8)
In Eq. (2.8), the species thermal conductivitiesλi are computed using the modified Eucken formulation [83]. The equation of state is given by the perfect gas law
ρ= W P0
RT , (2.9)
whereW is the mixture molecular weight, P0 is the thermodynamic pressure, and Ris the universal gas constant.
2.1.2 Radiation model
For some simulations, radiation heat transfer is considered. This is done by means of a radiation source term added to the energy equation, i.e.,q˙rad/cp, whereq˙rad is modeled according to the RADCAL model of Grosshandler [84]
q˙rad =−4σX
i
piap,i(T4−T∞4), (2.10) where σ=5.669e-08 W/m2K4 is the Stefan-Boltzmann constant, pi and ap,i are the partial pressure and the Planck mean absorption coefficient of theith species, respectively, andT∞ is the background temperature. The ap,i coefficients are fitted as functions of the temperature for CO2, H2O, CH4, and CO, which account for most of the radiation heat losses. This model assumes that the gas is optically thin, and all radiation heat transfer is dispersed to the background.
2.1. Direct Numerical Simulations 16 In this work, radiation heat losses are included for two flames. The first, which is discussed in Ch. 3, is a laminar N2-diluted C2H4/air-coflow pressurized diffu- sion flame, which is one of the target flames of the International Sooting Flame (ISF) Workshops [27, 85, 86]. The purpose of Ch. 3, is to assess the validity of the constant non-unity Lewis number assumption over a wide range of combustion configurations. While the optically-thin RADCAL model may not be the ultimate choice for the laminar C2H4/air flame, it is used consistently across all four cases, and does not impact the outcome of that analysis.
The second is Sandia flame B, a CH4/air, partially premixed, turbulent jet flame, discussed in Chs. 5, 6, and 7, which is one of the target flames of the Interna- tional Workshop on Measurement and Computation of Turbulent Nonpremixed Flames (TNF) [72]. While RADCAL is the suggested radiation model for both flames [72,86], enabling a computationally-inexpensive treatment of radiation, the optically-thin assumption is known to overpredict the radiant fraction for hydro- carbon flames [87]. Frank et al. [87] performed radiant fraction and multiscalar measurements for Sandia flames C-F. In their work, radiation calculations using RADCAL were coupled with temperature and species mean profiles, and it was found that, for Sandia flame D, the total radiant power for the emission-only case was 39% greater than emission/absorption computations. It is suggested that this over-prediction of radiative heat losses is due to the strong absorption by CO2 at the 4.3 µm wavelength [87–89]. However, the emphasis of the work of Frank et al. [87] is on the prediction of nitric oxide (NO) formation, which is known to be highly sensitive to the radiation model [90–93]. For the simulation of Sandia flame B discussed in Chs.5,6, and7, the NO chemistry is not considered in the chemistry model. While not the primary goal of those chapters, radiation heat loss is consid- ered to assess if a good agreement with experiments can be achieved. Further, in Ch. 7it is shown that including radiation heat loss does not affect the analysis carried out in that chapter. For all these reasons, the RADCAL model is a valid, practical choice for the present work.
Scalar transport is controlled by a balance of three main contributions, including convection, diffusion, and the scalar source term. For the flames considered in this work, the radiation source term, q˙rad/cp, is expected to be small compared to the dominant terms in Eq. (2.6). For instance, the radiant fraction, frad, values measured for the Sandia flames C-F by Frank et al.[87], were found to vary from 6.4% for flame C, to 3.0% for flame F. In their work, the radiant fraction was defined
2.1. Direct Numerical Simulations 17 as the ratio of the total radiated power,S˙rad, to the power released in the combustion reaction
frad≡ S˙rad
˙
mfuel∆Hcomb, (2.11)
wherem˙fuel and∆H are the mass flow rate of the fuel and the heat of combustion, respectively. The reduction of frad with increasing jet Reynolds number was pri- marily ascribed to the reduced residence time [87]. For Sandia flame B, fradshould be higher, yet close to the measured value for flame C, which is relatively small.
For reference, the reported energy release of the pilot flame in flames C-F is only approximately 6% of the main jet [72]. It will be shown in Ch.6that the maximum temperature for the radiating simulation is about 70K less than the adiabatic case (which is approximately a 3% reduction).
2.1.3 Numerical approach
The governing equations are solved using the finite differences, discretely energy- conserving code NGA, designed for the simulation of variable density laminar and turbulent flows [78]. The low-Mach Navier-Stokes equations are solved using a fractional step approach [94], whereby the momentum, scalar, and density fields are staggered in time. Time marching is carried out using a semi-implicit Crank- Nicolson scheme, where the integration of the chemical source terms is carried out using the preconditioned iterative method of Savard et al. [95] (four subit- erations are used). With this semi-implicit time-stepping, the time step is con- strained by the stiffness of the chemical source terms in laminar flames, while for the turbulent flames the convective CFL becomes more restrictive. Conser- vation of mass is then enforced through a pressure Poisson equation. For most simulations, the Lawrence Livermore National Laboratories HYPRE libraries are used for the pressure solver [96]. In the present work, the scheme used for the Navier-Stokes equations is second-order accurate both in space and in time. The scalars (Yi, T) are transported using the third-order accurate Bounded QUICK (BQUICK) scheme [97], which ensures boundedness of the transported scalars.
The one-dimensional flames are solved using the FlameMaster code [79]. NGA and FlameMaster give the same simulation results in one-dimensional calculations.
The NGA code has been extensively tested to verify the order of accuracy of the numerical methods. Desjardinet al.[78] simulated a range of canonical, constant density flows, to assess the influence of the order of the spatial numerical schemes on the solution. Savardet al.[95] considered one-dimensional premixed flames to
2.2. The flamelet model 18