Miscible displacements in Hele-Shaw cells: three-dimensional Navier–Stokes simulations

Abstract

Three-dimensional Navier–Stokes simulations of viscously unstable, miscible Hele-Shaw displacements are discussed. Quasisteady fingers are observed whose tip velocity increases with the Péclet number and the unfavourable viscosity ratio. These fingers are widest near the tip, and become progressively narrower towards the root. The film of resident fluid left behind on the wall decreases in thickness towards the finger tip. The simulations reveal the detailed mechanism by which the initial spanwise vorticity of the base flow, when perturbed, gives rise to the cross-gap vorticity that drives the fingering instability in the classical Darcy sense. Cross-sections at constant streamwise locations reveal the existence of a streamwise vorticity quadrupole along the length of the finger. This streamwise vorticity convects resident fluid from the wall towards the centre of the gap in the cross-gap symmetry plane of the finger, while it transports injected fluid laterally away from the finger centre within the mid-gap plane. In this way, it results in the emergence of a longitudinal, inner splitting phenomenon some distance behind the tip that has not been reported previously. This inner splitting mechanism, which leaves the tip largely intact, is fundamentally different from the familiar tip-splitting mechanism. Since the inner splitting owes its existence to the presence of streamwise vorticity and cross-gap velocity, it cannot be captured by gap-averaged equations. It is furthermore observed that the role of the Péclet number in miscible displacements differs in some ways from that of the capillary number in immiscible flows. Specifically, larger Péclet numbers result in wider fingers, while immiscible flows display narrower fingers for larger capillary numbers. Furthermore, while higher capillary numbers are known to promote tip-splitting, inner splitting is delayed for larger Péclet numbers.

Three-dimensional Navier–Stokes simulations of buoyant, vertical miscible Hele-Shaw displacements

“…Some of these findings are consistent with previous observations for capillary tube flows by Chen & Meiburg (1996), Petitjeans & Maxworthy (1996) and Kuang, Petitjeans & Maxworthy (2004); see also the results of Soares & Thompson (2009) for immiscible displacements. The three-dimensional Navier-Stokes simulations show that the presence of strong streamwise vorticity quadrupoles leads to the formation of an inner splitting instability in the upward-propagating viscous finger, similar to the recent observations for neutrally buoyant flows by Oliveira & Meiburg (2011). For large unstable density differences, the Rayleigh-Taylor instability is seen to result in a reverse flow that gives rise to an anchor-like structure.…”

“…We will also make use of the alternative Reynolds number Re * = ρ 1 Ub/µ 1 = Re e M . Here Re will be more convenient for comparing our results with the linear stability results of Goyal et al (2007) for validation purposes, but we will employ Re * when making comparisons with the neutrally buoyant nonlinear simulation results of Oliveira & Meiburg (2011). The gravity number F = ρgb 2 /(µ 2 U) defines the ratio of gravitational to viscous forces and can be considered a measure of the density contrast between the two fluids.…”

“…Nevertheless, at least for immiscible displacements, it was established long ago that three-dimensional effects become important in the vicinity of the moving interface (Park & Homsy 1984). Recently, Oliveira & Meiburg (2011) presented the first fully three-dimensional Navier-Stokes simulations of neutrally buoyant, miscible Hele-Shaw displacements. These simulations revealed some surprising flow features, such as the presence of strong streamwise vorticity quadrupoles along the length of the finger.…”

Experimental study of a buoyancy-driven instability of a miscible horizontal displacement in a Hele-Shaw cell

“…Another dual role of Pe has already been evidenced in viscous instabilities. 22,23 Here the explanation can be understood as follows: recalling that we compare in Fig. 6 situations where the miscible interface has reached approximately the same position, this implies that for the smaller flow rate, it takes more time for the injected fluid to reach the same radius.…”

Estimation of mixing volumes in buoyant miscible displacement flows along near‐horizontal pipes

“…We also scale timet withL a /V 0 . This leads to: 17) where the Péclet number is simply Pe m =V 0R /D m 1. In the turbulent core the expression for W is:…”