In order to integrate the SparkCIMM and the level-set equations, the flow solver has to provide all mixture quantities required for the calculation of the propagation velocity of the mean turbulent flame front. Additional equations, conditioned on the unburnt, are solved for the mean mixture fraction eZ, its turbulent fluctuation fZ002, the mixture dilu- tion mass fraction eYDil, and the unburnt gas temperatures eT|Z=eZ and eT|Z=0, respectively.
SparkCIMM and the level-set method provide the 3D distribution of the flamelet pa- rameters eG(t) and gG002(t). Based on these parameters, the probability of finding a flame front in each computational grid cell can be obtained by Eq. (5.62). This probability ePf represents the coupling parameter of the G-equation flamelet model on which feedbacks of flame front propagation are exerted back onto the fluid flow. As already discussed in Sec. 5.2.5, these feedbacks are calculated as follows, exemplified on an arbitrary scalar e
S:
e
S(~x) = eSu· (1 −Pef) + eSbPef , (6.7)
with eSu and eSb as corresponding unburnt and burnt values, respectively. The applied temporal sub-cycling method within the combustion model, presented in the last sec- tions, allows for possibly large CFD time steps without causing instabilities in the G- equation integration scheme. However, if the flame front then actually moves very far within a single computational time step, the intense temporal change of ePf eventually
causes instabilities in the turbulent flow equations. This, in turn, increases the flame front propagation velocity leading to instabilities in the numerical solution. This unde- sired relation still tightly constraints the applicable CFD time step and the benefit from the sub-cycled G-equation approach is slim. Therefore, a second sub-cycling technique is also introduced for the coupling parameter ePf. This method increases its old time-step value ePnf to the new level ePn+1f throughout the outer PISO iterations i as follows:
e
Pif = ePnf +Pe
n+1 f −Penf
mPISO · i (6.8)
with ePn+1f defined by Eq. (5.62) in advance of the flow solution, and with mPISO as
the total number of PISO iterations applied during the time step. Then, the resulting distributions for the species mass fractions and temperatures are obtained by Eq. (6.7) in advance of each PISO iteration. These profiles are subsequently posed onto the flow solver. This approach greatly relaxes the numerical time step constraint since it ensures a continuous and smooth temporal change of species mass fractions and temperatures within each grid cell throughout the simulation.
Combustion Model Validation
In this chapter, the physical models for turbulent flame propagation as presented in the previous chapters are validated against fundamental test cases of premixed turbulent combustion, where detailed experimental data is available. At first, turbulent combus- tion is investigated in a non-moving cylindrical vessel. Sensitivities of the numerical simulation results to applied computational time steps and grid sizes are also investi- gated. In the second part, the G-equation is validated against a test case of increased physical complexity. A series of different operating points of a homogeneous charge natural gas engine, provided by Cummins Inc., is investigated. Pressure traces and heat release rates are compared to experimental data provided by Cummins Inc.
7.1
Turbulent Premixed Combustion in a Constant
Volume Chamber
In this section, the ignition and 3D combustion models are validated against experi- mental data from turbulent combustion measured in a constant volume cylindrical ves- sel. The vessel is homogeneously charged with an undiluted lean propane/air mixture (φ = 0.84), which is subjected to an axi-symmetric swirling flow. The experimental setup is presented in Fig. 7.1.
The cylindrical vessel has a diameter of d = 125 mm and an axial length of ` = 35 mm. It is charged through a swirl valve by means of a pressurized mixture tank, creating the swirling flow. This intake process has not been calculated using a three-dimensional numerical simulation. Instead, the computational domain is initialized at the time of intake valve closure, using the experimentally obtained initial conditions as given in Ta- ble 7.1, and approximations for the flow and turbulence field as presented in Fig. 7.2. The turbulent integral length scale `t is attenuated towards the walls to zero. The gener-
ated computational grid for the baseline validation test case comprises ∼ 50, 000 cells. The closed valve and the spark electrodes are not included in the combustion chamber
Figure 7.1: Experimental setup of the constant vol- ume cylindrical vessel test case [52].
mesh. The mixture is ignited at the center of the vessel (~xSpk= (0, 0, 0.0175)T). The
ignition duration is assumed to be 2 ms. Since a detailed chemistry premixed flamelet table is not available for propane kinetics, the correlation by M¨uller et al. [107] is em- ployed to approximate the laminar flame thickness and laminar burning velocity during the combustion simulation.
p 274 kPa T 345 K u0 ||e~u|| 0.3 `t min(0.1·d,κ ywall), κ=0.419 ε according to Eq. (2.17) φ 0.84 e G -10 m g G002 `2f m2 rK0 0.25 mm ˙ Qspk 20 J/s
Table 7.1: Initial conditions of the constant-volume cylindrical test case.
The experimental setup is taken from [52]. Other setups using stoichiometric mix- tures and mixture stratifications have also been investigated as described in [20, 28], [29], and in [30, 31, 32]. In this study, however, these cases are not considered for the validation of the combustion model.
Figure 7.2: Left: Exper- imentally obtained (sym- bols) and numerically ap- proximated (lines) radial distribution of the mean flow velocity immediately before ignition.
Right: Radial distribution of the turbulent viscosity νtand the turbulent kinetic
energy k [52].
using the G-equation flamelet model is compared to the experimentally obtained data. The comparison shows a satisfying agreement between the simulation results and the measurements. At three different timings after spark advance, the propagating mean turbulent flame front is visualized and also presented in Fig. 7.3.
To study the sensitivity of the results of the numerical simulation to grid size and time step variations, the combustion run is performed using three different time steps ∆t, denoted by ∆tI= 1.0e−4sec, ∆tII= 5.0e−5sec, and ∆tIII= 1.0e−5sec, and two different
computational grids, denoted by A ∼ 50, 000 cells and B ∼ 100, 000 cells, respectively. This sensitivity analysis is presented in Fig. 7.4, showing that the calculated pressure traces of the G-equation calculations are almost independent to variations of the applied time-steps and mesh sizes. This is attributed to the temporal sub-cycling and in-cell spatial interpolation scheme of the ignition and level-set flame front tracking method as discussed in Sec. 6.3.
Figure 7.3: Comparison of experimental (symbols) and numerically simulated pressure trace (line) for the constant-volume validation test case. The mean turbulent flame front is visualized at three different times after the start of energizing.
Figure 7.4: Sensitivity analysis of the numerical simulations to grid size (A ∼ 50, 000 cells; B ∼ 100, 000 cells) and time-step (I = 1.0e−4 s; II = 5.0e−5 s; III = 1.0e−5 s) variations.