## 1 Introduction  
Duct junctions are essential elements of numerous piping systems, including the intake and exhaust systems of reciprocating internal combustion engines. The use of one-dimensional time domain gas-dynamic codes has become commonplace in the numerical study of unsteady flows in such systems, both in terms of their effect on engine performance and on intake and exhaust orifice noise [1]. While assuming one-dimensional wave action may be acceptable when duct diameters are relatively small, as is the case in the majority of the ducts present in engine intake and exhaust systems of passenger car engines, in certain elements, and most notably in duct junctions, complex three-dimensional flow structures may occur [2]. Consideration of the effects of such structures on the one-dimensional flow in the adjacent ducts requires the definition of suitable boundary conditions at the junction, usually involving empirical information. The effects of a junction on the flow in the neighboring ducts arise in different ways. From the point of view of the passive propagation of small amplitude pressure waves (i.e., in the acoustic range) the effect can be characterized in terms of length corrections, which have been reported to depend on the type of side-branch and the branch width and length [3], and with a rapid increase in the duct length corrections being associated with the excitation of non-planar higher order modes, which also results in lower sound transmission. This sort of representation has been quite successfully applied to the prediction of the effect on intake noise of a multi-pipe junction in the intake manifold [4]. It has also been reported [5] that for low Strouhal numbers based on the duct diameter, the acoustic transmission properties of T-junctions can be acceptably described by using an incompressible quasi-steady model,the upper limit of the Strouhal number being defined by flow-acoustic interaction effects, which differ significantly between different flow configurations: waves incident on the junction at the downstream side are attenuated, whereas waves incident at the other branches may be either amplified or attenuated, depending on the Strouhal number [6]. Such flow-acoustic interactions due to the coupling of the flow and the geometry are common to all intake and exhaust system elements [7]. When the focus is on the effect of the junction on the propagation of finite amplitude pressure waves and the resulting influence on engine performance, different approaches are found in the literature, most of them inspired by the seminal work of [PERSON] [8]. The simplest approach is given by constant pressure models, in which is it assumed that the pressure at the end of all branches of the junction is the same at any time, so that the pressure is assumed to be uniform across the junction. The most comprehensive description of these models is given in [9], where it was shown that, besides the assumption of uniform pressure, additional closing equations must be added. While the choice of those equations is arbitrary, it was also shown in [9] that assuming that the total enthalpy for all the outgoing flows is the same provides suitable results. More elaborated approaches are based on the consideration of the pressure differences existing between the different branches, which are incorporated in a quasi-steady manner, i.e., steady pressure loss coefficients (or more properly, as discussed in detail in [10], energy change coefficients accounting partly for losses and partly to a mutual energy transfer between the partial flows) are applied at each time step. The solutions proposed differ mainly in the origin of the pressure loss coefficients, in the hypotheses underlying their determination, and in the precise implementation of the solution method.