A symmetry-breaking inertial bifurcation in a cross-slot flow
Introduction
Bifurcations in fluid flows occur frequently [9]. Typically, beyond some critical value of the Reynolds number the solution branch of the “simple” base state, e.g. steady and/or symmetric, becomes unstable and is replaced with a more complex state or states. In the current paper we are interested in a particular class of bifurcation for internal duct flows where symmetry-breaking bifurcations occur but the bifurcated flow remains steady. In particular we are concerned with a “cross-slot”, or “cross-channel”, geometry formed by an “horizontal” planar channel along which two incoming fluid streams are made to impinge on each other, and an intersecting “vertical” channel which carries the outlet flow, with the other two streams now moving away from the central section and leaving through the vertical channel exits.
Perhaps the most famous, and certainly most studied, symmetry-breaking bifurcation occurring in internal duct flows is the case of flow through planar sudden expansions. For Newtonian fluids, as was first documented in Abbott and Kline [1], above a critical Reynolds number the flow field downstream of the expansion exhibits a stable asymmetric flow state (or, more precisely, two stable anti-symmetric flow states each corresponding to a shorter recirculation region attached to one of the two downstream walls). The critical Reynolds number at which the flow becomes asymmetric is dependent on the expansion ratio of the expansion (i.e. the ratio of the downstream to upstream channel heights) and, for three dimensional flows, the aspect ratio (i.e. the ratio of channel width to inlet channel or step height). Indeed the asymmetry is completely absent for expansion ratios less than 1.5. This asymmetry has been observed in both experimental ([7], [12] for example) and numerical [6], [11], [19], [21], [26] investigations. Drikakis [11] conducted an extensive study on the effect of expansion ratio and was able to demonstrate that the critical Reynolds number for asymmetric flow to occur decreases with increasing expansion ratio. For example for a 1:2 expansion the critical Re, based on the upstream channel maximum velocity and upstream channel height, is 216 which reduces to 52 for a 1:4 expansion and then to 28 for a 1:8 expansion. These critical Reynolds numbers are affected by the non-Newtonian character of the fluids, as shown by Neofytou and Drikakis [19] for Generalised Newtonian Fluids (viscous, shear-thinning), and Oliveira [21] and Rocha et al. [26] for viscoelastic fluids (constant-viscosity with elasticity). More recently evidence has emerged that even axisymmetric sudden expansions can also display stable asymmetric states [18], although in this case it appears a finite-amplitude perturbation is required to “kick” the solution onto the asymmetric branch in comparison to the linear instability observed in the planar case [29]. Planar sudden contractions have been significantly less studied than their sudden expansion counterparts. Chiang and Sheu [8] showed using a numerical technique that a supercritical pitchfork bifurcation occurs even for planar contraction flows if the Reynolds number is driven to high enough values. This bifurcation is related to the formation of lip vortices of differing size at the entrance to the smaller channel. Although for small contraction ratios the Reynolds numbers required appear unreasonably high for laminar flow conditions to be maintained, for example at a contraction ratio (CR) of two the critical Reynolds number for bifurcation is approximately 3080, at higher contraction ratios the Reynolds numbers at which asymmetric flow appears are more modest; for CR = 8, Recr = 1100 for example.
The emerging field of microfluidics has led to the discovery of a novel bifurcation in an inherently three-dimensional flow. One of the most basic microfluidic mixing geometries, which can be easily fabricated and integrated into more complex mixing geometries, first presented by Kockmann et al. [17], is a planar T-channel with two square inlets and an outlet of equal combined area (i.e. maintaining a constant bulk velocity in each channel arm). In this geometry, two-opposing planar channel streams join and turn through 90°. In their numerical study Kockmann et al. [17] found that the resulting flow in the outlet channel can be characterised by three flow regimes during steady flow: so-called “stratified” flow, “vortex” flow and “engulfment” flow. At low flowrates, so-called stratified flow was observed and this flow regime was defined as being where the flow streamlines remain primarily unidirectional and essentially follow the curvature of the geometry. At higher flowrates, where inertial effects become important and Dean-like rolls appear in the outlet channel [10], the flow was called “vortex” flow. Finally, at a critical Reynolds number ∼140 the flow breaks symmetry, although it remains steady, and this regime was dubbed “engulfment flow”. Since the original paper of Kockmann et al. [17], other studies have shown this effect experimentally [14], [16], [31] and probed the effect of different channel aspect ratios on the critical Reynolds number [25], [30]. Recently, Fani et al. [15] have conducted linear stability analysis of the engulfment regime in this T-mixer geometry with 3 arms, which is the closest in terms of geometrical configuration to the 4-arms cross-slot arrangement considered in the present work. For the cross-slot flow, no linear stability analysis could be found in the literature which is not surprising since the bifurcation instability we report here has not, as far as we are aware, been reported previously.
Although a detailed literature search shows that an inertial bifurcation has not previously been studied in a cross-slot geometry, such instabilities have been observed in a related class of stagnation-point flows formed by two counterflowing jets. For example, Rolon et al. [28] observed multiple steady states in experiments of opposed or counterflowing jets of air. For the case when the jet mass flow rates are equal, the stagnation point is expected to be located half-way between the two inlets. Rolon et al. [28] illustrated the existence of two stable steady states, each having a stagnation point on the axis of symmetry, but equally displaced from the centre towards one or the other jet inlet. The effect was termed “bi-stability” and the two stable flow regimes observed were mirror images of each other indicating the possible existence of a pitchfork bifurcation. Motivated by these experimental results, Pawlowski et al. [23] used a numerical technique to investigate such counterflowing jets both for planar and axisymmetric jets. The stagnation-point flows formed in the planar case exhibit both steady-state multiplicity and time-dependent behaviour, depending on the jet separation, while axisymmetric jets exhibited only a steady-state multiplicity. Their stability analysis revealed transitions between a single (symmetric) steady state and multiple steady states or periodic steady states. Of particular relevance to the cross-slot geometry is the planar case when the jet separation distance is equal to one channel width (α = 1.0 in the nomenclature of Pawlowski et al. [23]) where a critical Reynolds number of 872 is observed. Finally, it is worth mentioning that the paper of Ait Mouheb et al. [2] show asymmetry in cross-slot experiments but at a very low Reynolds number (50). Their comparator numerical simulations exhibit symmetry and the authors attribute the experimentally-observed asymmetry to experimental artefacts.
Aside from the basic interest in bifurcations in fluid flows as discussed above, an additional motivation for the current study was our desire to support previous work on purely-elastic instabilities in the flow of viscoelastic flow through a cross slot [4], [24], [27]. Motivated by the steady asymmetry observed for visceolastic fluids in inertialess cross-slot flow, a natural question one might ask is whether, even for a Newtonian fluid, there will be a similar bifurcation and asymmetries at high Reynolds number, when inertial effects start dominating the dynamics of the flow. One of the purposes of the current paper is to investigate this very possibility.
The paper is structured as follows. In the following section the governing equations and the numerical method involved in their solution are discussed. This section is followed by information regarding the geometry, computational meshes and boundary conditions. A bifurcation parameter is then defined prior to the presentation of the results and accompanying discussion. Finally we draw conclusions.
Section snippets
Governing equations and numerical method
We are concerned with the isothermal, incompressible two-dimensional flow of a Newtonian fluid and hence the equations we need to solve are those of conservation of massand momentumwhere u = (u, v) is the local velocity vector, ρ the fluid density (assumed constant), p the pressure and τ the stress tensor which, for a Newtonian fluid is given bywhere μ is the dynamic viscosity. The numerical method applied in this work is the finite volume method. The
Geometry, computational meshes and boundary conditions
The cross-slot geometry is shown schematically in Fig. 1. Flow is provided in each inlet arm with bulk velocity U and channel width d, which served as velocity and length scales to define the Reynolds number Re = ρUd/μ. At the plane x = 0 the two streams meet and are turned through 90° into two outlet arms of identical width d. At the inlets we apply a velocity distribution corresponding to the analytical fully-developed channel-flow solution (see e.g. [32]) and at the outlets we apply a boundary
Bifurcation parameter
To estimate the critical Reynolds number for the bifurcation to asymmetric flow we define an asymmetry parameterwhere, as highlighted in Fig. 1, Yr1 is the length (normalised with d) of the recirculation region attached to the west face of the south outlet channel, Yr2 is the length of the recirculation region attached to the east face of the south outlet channel, Yr3 is the length of the recirculation region attached to the west face of the north
Symmetric flow prior to bifurcation
With increasing flowrate, four identical standing recirculating bubbles attached at the four corners of the cross-slot geometry grow significantly. In Fig. 5 the effect of increasing inertia on the size of these recirculation regions is highlighted for 200 ⩽ Re ⩽ 1000. Only one quarter of the cross-slot geometry is shown due to the symmetric nature of the flow field under these conditions. In this regime the length of the recirculation regions is found to increase linearly with Reynolds number
Conclusions
Motivated by the steady asymmetry observed for visceolastic fluids in inertialess cross-slot flow, the purpose of the current study was to investigate the possibility of a similar instability occurring in Newtonian fluids but driven by inertia. Indeed a transition is observed at a critical Reynolds number of Recr = 1490 ± 10 (based on the average inlet velocity and the channel width), above which the flow becomes asymmetric but the asymmetry is significantly different from the inertialess
Acknowledgements
GNR and PJO would like to thank the financial support by Fundação para a Ciência e a Tecnologia (FCT), Portugal, through Grant SFRH/BD/22644/2005 and Project PTDC/EME-MFE/98558/2008.
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