**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 Benson [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.

Regarding the origin of the coefficients, while there have been some attempts to obtain them from computational fluid dynamics (CFD) simulations [11–13], 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 agreemen<sup>t</sup> 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 sugges<sup>t</sup> 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 Benson 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]. Typically, when these methods are applied to simple duct junctions, a single volume is used for the junction with appropriate effective areas and characteristic lengths at each connection with the adjacent ducts. As these connections contain information on vector quantities (including the orientation of the branch duct) the momentum equation can be solved, even in an approximate way, so that all the effects of the junction of the flow need not be included through the pressure loss coefficients. Additionally, it would be possible to use a refined mesh locally at the junction, so that a first-order estimate of any three-dimensional features could be obtained.

The objective of this paper is precisely to establish the potential of these ideas as a way to improve the description of the effect of simple duct junctions on an otherwise one-dimensional flow system, as the intake or exhaust of an internal combustion engine. Specific experiments have been performed in order to obtain precise and reliable results on the propagation of pressure pulses across junctions. The results obtained have been compared to simulations performed with different versions of a staggered mesh finite volume method and different meshes and, as a reference, also with the results of a more conventional pressure loss-based model.
