**3. Numerical Methodology**

The main limiting aspect of traditional methodologies for the validation of CI engines operation, which are based on reproducing the in-cylinder pressure profile averaged from a given number of recorded cycles using simple turbulence schemes (URANS), is the incapacity to assess the cyclical deviation and its subsequent effects on combustion.

The use of more complex turbulence approaches (LES) and the simulation of several consecutive cycles is currently the standard approach to reproduce the CCV in spark-ignited (SI) engines [31,32] in which the combustion variability due to the larger turbulent scales is traditionally considered as the main cause of the cycle-to-cycle variation [33]. However, the huge increase of computational burden hardly compromises the application of LES modelling in most of the industry cases. Furthermore, it is not clear that pure stochastic fluctuations could explain the whole variation of flow conditions among consecutive cycles in gasoline PPC mode.

Available literature on CCV [34] claims that CCV observed in CI engines is mostly originated from the casuistic variability in fuel mass or trapped-gas conditions, rather than from arising in random variations in the turbulent flow and combustion process.

Therefore, other authors [35,36] employed URANS schemes to analyse CCV sources in several LTC strategies, where CCV is larger than in conventional Diesel engines but still far from SI engines. They investigated the cyclical dispersion from the point of view of the operation uncertainties. The basis of their approach is to perform many parallel URANS simulations in which one or more key operating conditions are perturbed about the reference conditions.

Here, a methodology based on this latter approach was used in an attempt of reproducing the experimental CCV and its effects on the combustion. Therefore, conditions at the intake valves closing (IVC) of a baseline cycle simulated were considered as the reference. These conditions, after being validated to ensure realistic in-cylinder conditions, were artificially modified in order to imitate the variation of the injected fuel, trapped mass, injection timing, etc.

The selection of those parameters that significantly affect the combustion was made by taking into account the conclusions obtained by Klos and Kokjohn [35] to reduce the number of parallel simulations as much as possible. Klos and Kokjohn found clear relationships among the dispersion of three operating parameters (EGR, mean gas temperature at IVC and fuel mass) and the combustion behaviour. Thus, these parameters and their recommended variations were utilized for modifying IVC baseline conditions.

Nonetheless, an additional parameter must be considered in this study since the 2-stroke engine has a particular in-cylinder flow motion. While in traditional CI engines the swirl flow motion is clearly dominant, the tumble motion—characteristic of SI engines—prevails in this particular engine design. As reported by Vermorel et al. [31], the tumble intensity significantly changes among consecutive cycles. They observed from LES simulations data that the magnitude of this rotational velocity could vary up to 25% at IVC. Therefore, this parameter was also included together with the other three determined by Klos. The parameters and their maximum variations used to perturb baseline conditions are displayed in Table 2. Note that EGR value was replaced by the percentage of combustion products within the cylinder at IVC, this is an equivalent parameter that define the amount of inert gases available during the closed-cycle.

**Table 2.** Baseline operation parameters and ranges of variation for the alteration of the IVC conditions.


In addition to the baseline simulated case, ten parallel closed-cycle simulations were performed whereas the initial conditions at IVC were accordingly perturbed by randomly distributed variations of these four parameters to sample the four-dimensional domain.

This procedure was applied to the operation condition detailed in Table 3. This is defined by a medium speed (1500 rpm) and medium-high engine load (1.04 MPa of indicated mean effective pressure) that shows moderate levels of knock.


