In turbulent particulate flows, the occurrence of particle trajectory crossings (PTC) is the main constraint on classical monokinetic Eulerian methods. To handle such PTC, accounting for high-order moments of the particle velocity distribution is mandatory. In the simplest case, second-order moments are needed. To retrieve these moments, two solutions are proposed in the literature: the Algebraic-Closure-Based Moment Method (ACBMM) and the Kinetic-Based Moment Method (KBMM). The ACBMM provides constitutive relations for the random-uncorrelated-motion (RUM) particle kinetic stress tensor as algebraic closures based on physical arguments (Simonin et al. 2002; Kaufmann et al. 2008; Masi 2010; Masi & Simonin 2012). These closures rely on the internal energy, namely the RUM particle kinetic energy, which is obtained using an additional transport equation. Alternatively, it is possible to directly solve for the second-order moment by providing a closure for the third-order correlation. The KBMM proposes a kinetic description, that is, the number density function (NDF) is reconstructed based on the resolved moments and on a presumed shape. In the present work, an isotropic Gaussian and the anisotropic Gaussian closure of Vié et al. (2012) are used. The goal of the present study is to provide a first comparison between ACBMM and KBMM, using the same robust numerical methods, in order to highlight differences and common points. The test case is a 2D turbulent flow with a mean shear,