Mixing Analysis in a Lid-Driven Pradeep Rao1 Cavity Flow at Finite Reynolds e-mail: [email protected] Andrew Duggleby Numbers Department of Mechanical Engineering, The influence of inertial effects on chaotic advection and mixing is investigated for a two- Texas A&M University, dimensional, time-dependent lid-driven cavity flow. Previous work shows that this flow College Station, TX 77845 exhibits exponential stretching and folding of material lines due to the presence of figure- eight stirring patterns in the creeping flow regime. The high sensitivity to initial condi- tions and the exponential growth of errors in chaotic flows necessitate an accurate solu- Mark A. Stremler tion of the flow in order to calculate metrics based on Lagrangian particle tracking. The Department of Engineering streamfunction-vorticity formulation of the Navier-Stokes equations is solved using a Science and Mechanics, Fourier-Chebyshev spectral method, providing the necessary exponential convergence and Virginia Tech, machine-precision accuracy. Poincare´ sections and mixing measures are used to analyze cha- Blacksburg, VA 24061 otic advection and quantify the mixing efficiency. The calculated mixing characteristics are almost identical for Re 1. For the time range investigated, the best mixing in this system is observed for Re ¼ 10. Interestingly, increasing the Reynolds number to the range 10 < Re 100 results in an observed decrease in mixing efficacy. [DOI: 10.1115/1.4006361] 1 Introduction flow for Re > 10 leads to a loss in mixing efficiency. Clifford et al. [5] studied inertial effects in a simple planetary mixer and Fluid stretching and folding through stirring is a well- observed shrinking of nonchaotic islands with increasing Reyn- established means of enhancing mixing. This stretching and fold- olds number. They concluded that the best mixing protocol ing increases the interface(s) across which diffusion occurs, depends strongly on the Reynolds number of operation. Wang thereby increasing the mixing rate. In the cases where diffusion is et al. [6] found good agreement with Stokes’ flow predictions for slow relative to the time scale of the fluid motion, the stretching 2D cavity flows with Re 10. From these studies one can con- and folding of passively advected material surfaces can be viewed clude that, although Stokes’ flow solutions are good at establish- as an underlying template for the mixing process. An exponential ing basic flow configurations that can achieve chaotic advection, rate of stretching is achieved when particles on a material surface it is necessary to study these flows at finite Reynolds number for have chaotic trajectories; that is, when they undergo chaotic the simulation and optimization of real world mixing systems. advection [1]. The investigation of chaotic advection is particu- Recently, it has become understood that the relative motion of larly important in low Reynolds number flows, when the rapid stirring rods can have a fundamental impact on the rate of stretch- stretching and folding that naturally appears in turbulent flows ing and folding in a viscous fluid, and ‘figure-eight’ motions lead does not occur. Chaotic advection can be achieved in a wide vari- to excellent mixing [7]. Such stirring-like motions can be gener- ety of unsteady two-dimensional flows and in both steady and ated by outer boundary motions, creating what appear to be ‘ghost unsteady three-dimensional flows [1,2]. rods’ in the flow that move on ‘figure-eight’ orbits [8–11]. This Many studies of chaotic advection have focused on creeping perspective makes such a stirring approach possible in a variety of flows. One motivation for doing so is that this extreme of viscous- systems for which there are no physical rods. In these systems, dominated fluid motion can be particularly difficult to mix. These rapid mixing through stirring by ghost rods depends on the dy- flows are also attractive for the study of chaotic advection because namics of the fluid. However, the existing literature has focused they can often be described with exact analytical solutions or primarily on Stokes’ flow models. Understanding the effect of fi- highly accurate numerical approximations. Such representations nite Reynolds numbers on mixing in these figure-eight stirring enable accurate tracking of particle trajectories in the flow for the flows is important to understanding the extension of this mixing purposes of calculating chaos diagnostics and quantifying mixing. approach to more practical systems. The Stokes’ flow assumption, however, does not take into account In the current numerical study, the effect of inertia on chaotic the inertial effects present in most real flows. The presence of cha- advection and mixing is investigated for the two- dimensional lid- otic transport in a flow can depend sensitively on the details of the driven cavity flow studied in Ref. [11], which is closely related to dynamics, so it is not immediately clear to what extent the results that considered in Refs. [9,12]. The fluid motion is fully resolved from a Stokes’ flow analysis can be used to predict behavior of a by solving the Navier-Stokes equations (conservation of mass and system when operated at finite Reynolds number. It is therefore of momentum) using a spectral method algorithm, for low to moder- value to study how inertial effects influence mixing. ate Reynolds numbers (0.01 Re 100). The Reynolds number Dutta and Chevray [3] observed that transient effects due to is increased by decreasing the value of viscosity, and mixing is an- inertia significantly enhanced mixing in a time-periodic annular alyzed by considering the dispersion of a passive scalar in the flow between two eccentric cylinders, even at a low Reynolds flow domain. We find that the mixing depends nontrivially on the number (Re ¼ 0.45). Hobbs and Muzzio [4] found that, for a Reynolds number. Kenics static mixer, the formation of nonchaotic islands in the 1Corresponding author. Present address: Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061. 2 Problem Setup Contributed by the Fluids Engineering Division of ASME for publication in the The basic mixing protocol is a ‘figure-eight’ type motion of a JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 8, 2011; final manu- script received March 8, 2012; published online April 20, 2012. Assoc. Editor: Z. C. viscous fluid driven by time-periodic tangential boundary veloc- Zheng. ities in a two-dimensional rectangular domain, as illustrated in Journal of Fluids Engineering Copyright VC 2012 by ASME APRIL 2012, Vol. 134 / 041203-1 Downloaded 08 May 2012 to 128.173.163.249. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Fig. 1. The basic flow structure consists of counter-rotating cells extremely sensitive to initial position, small errors get magnified of different size in a rectangular domain with À6h x 6h exponentially with time. It is therefore important to be able to and Àh y h. Velocity boundary conditions are specified on determine the velocity vector u accurately at any point in the flow y ¼ 6h as described in Sec. 2.1, and the domain is spatially peri- field. For the analysis presented here, all velocity fields are deter- odic in x with period L ¼ 12h. By the symmetry of the system, mined either analytically or with spectral accuracy in space. A 4th straight vertical streamlines always exist at x ¼ 6qh (with q an in- order Runge Kutta method is used to integrate Eq. (2) with respect teger), giving side wall boundaries with no penetration but with to time. slip at x ¼ 0, 66h. Time-dependence is introduced by periodically ‘blinking’ the system between two different (constant) boundary velocity distributions. In the Stokes’ flow limit, the mixing proto- 2.1 Stokes’ Flow Solution. When the system is dominated col amounts to blinking between the two steady state flows shown by viscous effects (in the limit Re ! 0), Eqs. (1) reduce to the in Fig. 1. Due to the symmetry of the system and the fact that the biharmonic equation for the streamfunction, lines x ¼ 0,6h are vertical streamlines, subsequent discussion and results focus on 0 x 6h. This Stokes’ flow is identical to that r4w ¼ 0 (3) considered in Ref. [11], and the periodic counter-rotating flow structure is similar to that examined in Refs. [9,12]. in which there is no explicit dependence on time, and temporal In our analysis, the velocity field u is determined using the changes in the applied boundary velocities are reflected instantly mass and momentum conservation equations for an incompressi- in the flow domain. Thus, in the Stokes’ flow case, the flow is im- ble, Newtonian fluid in the streamfunction-vorticity form, which mediately periodic in time when subjected to time periodic bound- for a two-dimensional flow are given by ary conditions. Consider a two-dimensional flow with boundary conditions @x Re þ u rx ¼r2x (1a) @t wðx; 6hÞ¼0 (4a) 2 XN x ¼rÂðÞu Á ez ¼rw (1b) @w ðÞ¼x; 6h 6 U sinðÞn2px=L (4b) @ n y n¼1 where w is the streamfunction satisfying u ¼r(wez). The Reynolds number is defined as Re ¼ Umax h=, where Umax is the maximum value of the boundary velocity and is the kinematic Due to the linearity of Eq. (3), the streamfunction for this flow can viscosity. In this formulation, the divergence-free condition (mass be written in the form conservation) is automatically imposed, which simplifies the nu- XN merical and analytical solutions significantly. From here onwards, wðx; yÞ¼ U w ðx; yÞ velocity u, vorticity x, streamfunction w, and time t are nondi- n n n¼1 mensionalized by U and h. (5) max XN To characterize the effects of Reynolds number on mixing, we ¼ U C f ðyÞ sinðn2px=LÞ examine the Lagrangian transport of point tracers in each calcu- n n n n¼1 lated flow field.