Three-dimensional flow structures in X-shaped junctions: Effect of the Reynolds number and crossing angle

Abstract

We study numerically the three-dimensional (3D) dynamics of two facing flows in an X-shaped junction of two circular channels crossing at an angle α. The distribution of the fluids in the junction and in the outlet channels is determined as a function of α and the Reynolds number Re. Our goal is to describe the different flow regimes in the junction and their dependence on α and Re. We also explore to which extent two-dimensional (2D) simulations are able to describe the flow within a 3D geometry. In the 3D case, at large Re's (≳50) and α's (≳60 ○ ), axial vorticity (i.e., parallel to the outlet axis) of magnitude increasing both with α and Re develops in the outlet channels and cannot be reproduced by 2D numerical simulations. At lower angles (α ≲ 60 ○ ), instead, a mean vorticity component perpendicular to the junction plane is present: both its magnitude and the number of the corresponding vortices (i.e., recirculation zones) increase as α decreases. These vortices appear in both 2D and 3D simulations but at different threshold values of α and Re. At very low Re's (≲5) and α's (∼15 ○ ), the flow structure in 3D simulations is nearly 2D but its quantitative characteristics differ from 2D simulations. As Re increases, this two-dimensionality disappears, while vortices due to flow separation appear in the outlet channels.

Influence of aspect ratio on vortex formation in X-junctions: Direct numerical simulations and eigenmode decomposition

“…These former studies demonstrate therefore that the geometry of the channel section influences the critical Reynolds number for the transition between the different regimes. Similar observations were made in Y or T junctions 33 and in X-junctions with varying crossing angles 24 .…”

“…In our case, it is a symmetrical solution of Eq. (1) in which the liquid coming from each inlet splits equally between both outlets, with the streamlines completely segregated by the plane y = 0, as shown in Figures 1(b) and 1(c) 23,24,41 .…”

“…The Reynolds number is Re = U in W /ν, where ν is the viscosity of the liquid. We impose the same parabolic inflow conditions at both inlets (Poiseuille solution for rectangular pipes), a stress-free outflow for the outlets, and u = 0 at the walls 24 .…”

“…A convergence criterion of 10 −3 is used for the relative error defined by a weighted euclidean norm for two successive iteration steps (see Correa et al 24 ). In the stability problem given in equations ( 2), the eigenvalues were computed employing a variant of the implicitly restarted Arnoldi method in the ARPACK routine 46 .…”

“…At higher Re's, an axial vortex appears at the intersection of the channels and extends towards each outlet. In channels with a square cross-section 8,20,23 , the transition between the two regimes occurs for Re c ∼ 40 , while for circular channels 24 it is slightly higher with Re c ∼ 48.…”