Modelling Fuel Injector Spray Characteristics in Jet Engines by Using Vine Copulas

Abstract

Summary The emission requirements for jet engines are becoming more stringent and the combustion process determines pollutant emissions. Therefore, we model the distribution of fuel drops generated by a fuel injector in a jet engine, which can be assumed to be a five-dimensional problem in terms of drop size, x-position, y-position, x-velocity and y-velocity. The data are generated by numerical simulations of the fuel atomization process for several jet engine operating conditions. In combustion simulations, the variables are usually assumed to be independent at the start of the simulation, which is clearly not so as our data show. The dependence between some of the variables is non-monotone and asymmetric, which makes the modelling task difficult. Our aim is to provide a realistic parametric model for the dependence structure. For this, we employ vine copulas which provide a flexible way to construct a multivariate distribution function. However, we need to use non-standard bivariate copulas as building blocks. Using this copula representation enables us to create realistic samples of fuel spray droplets which improve the prediction of the combustion process and the pollutant emissions. Moreover, this approach is significantly faster than solving the set of differential equations describing fuel disintegration.

A D-vine copula mixed model for joint meta-analysis and comparison of diagnostic tests

“…This truncation or simplification, as per the terminology in Brechmann et al (2012), offers a substantial reduction of the dependence parameters. For example, this enabled Coblenz et al (2020) to provide a parsimonious parametric model for the dependence structure of fuel drops generated by a fuel injector in a jet engine. In our case there are 10 dependence parameters less, which is extremely useful for estimation purposes given the typically small number of primary studies involved in the meta-analysis.…”

MATVines: A vine copula package for MATLAB

“…In the last two decades, vine copulas have become an important tool for modeling multivariate dependence structures with applications in finance [1], biology [2], genetics [3], seismology [4], engineering [5], and hydrology [6] among others. Vine copulas provide a way to construct a d-dimensional copula, which is a special kind of multivariate distribution function.…”

“…Also, it implements the more general class of R-vine copulas, which comprises C-and D-vine copulas. R-vine copulas are often needed in applications because they provide more modeling flexibility than C-or D-vine copulas, e.g., see [5,[12][13][14].…”

“…For example, [26] uses the package in a simulation study for a non-simplified vine copula model that relies on tessellations of conditioning spaces. Furthermore, the MATVines package has been used to model the dependence structure of fuel droplets in the combustion process of a jet engine [5].…”

Modeling Multivariate Spray Characteristics with Gaussian Mixture Models

“…The spray is usually injected at discrete points distributed evenly around the rotation axis at one or more specified radial distances [7,16]. Some authors employ more sophisticated approaches that incorporate some multivariate stochastic dependencies of the droplet features [19,20]. One approach that should be highlighted is the multivariate injection model by Coblenz et al [20], which employs vine copulas to accurately reproduce the multivariate dependencies of the individual droplet diameters, velocities, and positions determined through 2D numerical simulation of the primary atomization process [21] and allows for the sampling from these determined statistical distributions.…”