## 1 Introduction
Regarding the origin of the coefficients, while there have been some attempts to obtain them from computational fluid dynamics (CFD) simulations [11; 12; 13], it appears that the results are strongly dependent on the numerical method used, both in the details of the flow and in the overall values of the coefficients obtained [14]. Therefore, usually the coefficients are either obtained from simple and robust models, or specific measurements are performed in order to characterize the junction under consideration. The most successful example of the first option is probably that presented in [15], where a remarkable agreement with experimental results was obtained from a model that extended the previous work performed in [16] and neglected any effects of mixing losses, compressibility and wall friction. Regarding the experimental characterization, it is usual to consider steady incompressible flow, as in [17], but more recently, specific studies accounting for the flow compressibility have been reported [18; 19] that suggest that the total pressure loss coefficient is mainly dependent on the Mach number, mass flow rate ratio, and area ratio, and is almost independent of the Reynolds number. Numerous implementations of the pressure loss model for multi-pipe junctions have been proposed in the literature, comprising implicit time formulations [20] and different explicit solutions, such as the supplier-collector strategy [21], the branch superposition method [22] and the generalization of the classical approach of [PERSON] presented in [23]. The limitations of these approaches lie mainly in the fact that, even if steady flow coefficients contain information on three-dimensional separation effects around the junction, the results will be significant only if quasi-steady flow can be assumed, which requires that mass and energy storage at the junction are very small, which may not be the case in real manifold flows. Additionally, any information regarding the wave refraction characteristics of the junction is lost in the quasi-steady approximation. Overcoming these limitations requires accounting for the unsteady and multi-dimensional character of the flow at the junction, but without incurring in an excessive computational cost. A suitable solution is thus to include a local multi-dimensional region within an otherwise one-dimensional wave-action engine simulation, as first suggested in [24]. In this first approach, an inviscid two-dimensional model was applied to the simulation of shock-wave propagation through different junctions, and the observed evolution of the wave fronts through the junctions and the measured high frequency pressure oscillations induced by the transverse reflections were successfully predicted. However, even if the increase in the computational cost was reasonable, it did not appear to be justified when compared with a conventional quasi-steady pressure loss model [25]. It appears, thus, that a full three-dimensional description of the junction should be used in order to describe its unsteady behavior. Such a description was presented in [26; 27], successfully reproducing the flow field and the associated non-plane-wave motion. However, even if coarse 3D grids were used in the first simulation cycles that were switched to more refined grids during the last simulation cycles, the computational cost and time may still be regarded as excessive for the practical design and evaluation of full intake and exhaust systems. A possible alternative to 1D-3D coupling, which could provide some accountancy for the three-dimensional effects at the junction and to the authors' knowledge has not been explored in some detail, would be the use in this context of a staggered mesh finite volume method [28]. Such methods have become standard in commercial codes, either as the core solver [29; 30], or used locally for elements exhibiting significant three-dimensional features, such as plenums and mufflers [31].