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 6h x 6h exponentially with time. It is therefore important to be able to and h 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 ¼rðÞu 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. For Lagrangian metrics, the motion of a passive particle at position (Xp, Yp) in a flow field u ¼ (u,v) is determined Substituting Eq. (5) into Eq. (3) gives the ordinary differential by solving the advection equations equation dX dY p ; p ¼ ðÞuðx; y; tÞ; vðx; y; tÞ (2) f 0000ðyÞ2ðn2p=LÞ2f 00ðyÞþðn2p=LÞ4f ðyÞ¼0 (6) dt dt n n and the boundary conditions from Eq. (4) become That is, we assume that a particle instantaneously adjusts its ve- locity to the ambient flow (i.e., it is massless) and that it does not fnð6hÞ¼0 (7a) alter the flow (i.e., it is infinitesimally small) [1]. Since flows char- 0 acterized by chaotic advection have particle trajectories that are Cnfnð6hÞ¼61 (7b)

Fig. 1 General flow domain, and representative streamlines in the Stokes’ flow limit with U2=U1 ’ 0:8413 for (a) the first half of the flow period and (b) the second half of the flow pe- riod. Filled circles show the stagnation points used to define the flow protocol, and open circles show the points that exchange positions along the dotted streamlines when taking the parameter values h 5 1, Umax ’ 1:0, and s ’ 15:261

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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 The solution for this (steady) flow is given by is applied, which gives good near-wall resolution and fast trans- forms [13]. The use of the Fourier-Chebyshev formulation for the fnðyÞ¼y coshðn2ph=LÞ sinhðn2py=LÞ vorticity-streamfunction form of the incompressible Navier-Stokes (8a) h sinhðn2ph=LÞ coshðn2py=LÞ equations is well established [14,15]. The solution sought for any field variable in a domain which is periodic in x with a period of L 1 Cn ¼ 2½ sinhðÞþn4ph=L n4ph=L (8b) is of the form

XM KX=21 Thus, for a finite value of N, the velocity field is given by a ^ ik2px=L closed-form analytical solution. wðx; y; tÞ¼ wmkðtÞTmðyÞe (10a) From this steady flow we generate a time-dependent system m¼0 k¼K=2 consisting of two counter-rotating flow cells contained within XM KX=21 ik2px=L 0 x 6h by taking N ¼ 2 with the time-periodic boundary xðx; y; tÞ¼ x^ mkðtÞTmðyÞe (10b) conditions m¼0 k¼K=2 @w XM KX=21 ik2px=L ðÞ¼x; 6h; t 6½U1 sinðÞþ 2px=L þ uðÞt U2 sinðÞ 4px=L þ 2uðÞt u x; y; t u^ t T y e (10c) @y ð Þ¼ mkð Þ mð Þ m¼0 k¼K=2 (9a) XM KX=21 0 for t 2½ns; ðÞn þ 1=2 sÞ ik2px=L uðÞ¼t (9b) vðx; y; tÞ¼ v^mkðtÞTmðyÞe (10d) p for t 2½ðÞn þ 1=2 s; ðÞn þ 1 sÞ m¼0 k¼K=2

We focus here on a domain with half-height 1 and boundary h ¼ where ^mkðtÞ are the unknown coefficients and Tm(y) ¼ cos(m conditions given by U1 ’ 0:62123 and U2 ’ 0:52264 (corre- cos1(y)) are the Chebyshev basis functions. Recall that x and y sponding to Umax ’ 1:0), and s ’ 15:261. These parameter values are normalized by h, so that 1 y 1 as required by this formu- are chosen so that the points highlighted in Fig. 1 move on exactly lation. The total number of terms, M K, representing each field periodic ‘figure-eight’ orbits in the Stokes’ flow case. During the variable is referred to as the resolution of the approximation. first half of a flow period, the (steady) streamlines are as shown in Substituting the spectral approximations for the field variables Fig. 1(a). The points marked by the open circles are advected into Eq. (1) results in along the streamline shown with the dashed line, and they exactly exchange their positions during the elapsed time Dt ¼ s=2. The K=21 XM X @x^ 1 flow is then switched instantaneously to that shown in Fig. 1(b). mk þQ^ x^ð2Þ ð2pk=LÞ2x^ @t mk Re mk mk At the instant the flow protocol is switched, the stagnation point m¼0 k¼K=2 marked in Fig. 1(a) coincides exactly with the right-most open ik2px=L circle in Fig. 1(b), the left-most open circle in Fig. 1(a) coincides TmðyÞe ¼ 0 (11a) exactly with the stagnation point marked in Fig. 1(b), and the third XM KX=21 hi (middle) points marked in each panel coincide. At the end of each ^ðÞ2 2 ^ ik2px=L wmk ðÞ2pk=L wmk þ x^ mk TmðÞy e ¼ 0 full period, the protocol is instantaneously switched back to the m¼0 k¼K=2 flow in Fig. 1(a). The trajectories traced out by the points marked (11b) in Fig. 1 are thus periodic in time with a period Dt ¼ 3s. where the superscript (2) refers to the matrix corresponding to the 2.2 Finite Reynolds Number Solution. The analytical second derivative operator of the Chebyshev polynomial, and Stokes’ solution in Sec. 2.1 is only correct when Re ¼ 0. To where include the effects of inertia in this system, Eqs. (1) are solved computationally for finite Reynolds number values using a spec- XM KX=21 ^ ik2px=L tral method algorithm. Spectral methods differ from finite element QmkTmðyÞe ¼ u rx (12) or finite difference methods in the choice of trial or basis functions m¼0 k¼K=2 [13]. In classical spectral methods, like the one used in this study, a single set of basis functions, obtained through a tensor product is the spectral projection of the nonlinear terms in Eq. (1a).By of 1D basis functions, is used for the entire domain. This use of a applying the orthogonality of the Fourier basis and the Chebyshev single set of basis functions affords solutions that are infinitely basis to Eq. (11a), M K sets of 1-D transient equations for vor- differentiable [13], unlike spectral element method solutions ticity and M K algebraic equations for the streamfunction are composed of multiple sets of basis functions, which are globally obtained. As in Ref. [15], the vorticity equations are discretized in 0 C continuous. Due to these properties, the global error for the time, with time step dt, using the semi-implicit Adams-Bash- spectral method used here decreases exponentially with the num- forth=Backward-Differentiation Scheme (AB=BDI2), where the ber of degrees of freedom. Lagrangian particle tracking is an im- linear (diffusive) terms are solved implicitly using a 2nd order portant tool for the analysis of chaotic advection, but since the Backward-Differentiation scheme, and the nonlinear (convective) flows are characterized by high sensitivity to initial conditions, terms are treated using an explicit Adams-Bashforth formulation errors grow at an exponential rate. Having velocity interpolants accurate to within a few orders of magnitude of machine precision nþ1 3Re x^ ð2Þ ð2pk=LÞ2 þ x^ nþ1 enables high accuracy in tracking particles. Thus, as compared to mk 2dt mk numerical studies of chaotic advection at finite Reynolds number h i hi Re that use traditional finite volume methods, which often have only n n1 ^n ^n1 (13a) ¼ 4x^ mk þ x^ mk þ Re 2Qmk Qmk 2nd order accuracy, the current method is able to achieve the 2dt needed accuracy with significantly fewer degrees of freedom. As in the analytical solution, the two-dimensional flow field is ^ðÞ2 2 ^ described by a stream function w(x,y,t), which enforces a diver- wmk ðÞ2pk=L wmk ¼x^ mk (13b) gence free velocity field. Since the flow domain under considera- tion is periodic in the x direction, the spectral basis of choice is To solve this system, four boundary conditions are required. the Fourier series. In the y direction the Chebyshev approximation Using a Lanczos-Tau formulation [16], the two boundary

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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 conditions given in Eq. (9a) are applied for the streamfunction, and the remaining two boundary conditions given in Eq. (4a) are integrated into the vorticity equation through the influence matrix technique [15]. The boundary velocities are changed instantane- ously at the end of each half of a flow cycle, just as in the Stokes’ flow case. In each case the flow is taken to be initially at rest. Unlike the Stokes’ flow case, the presence of inertial effects prevents the flow from immediately becoming periodic in time. The number of cycles needed for the flow to become periodic, denoted here by n*, increases with the Reynolds number of the flow, as shown in Fig. 2(a). In Fig. 3 we show instantaneous streamlines at time t ¼ n*s, a pattern that then repeats every flow period. Two counter-rotating flow cells similar to those in the Stokes’ flow case are evident for Re 30. For Re 50, inertial effects begin to develop a more complicated flow pattern, with the larger flow cell splitting into two smaller co-rotating cells. To ensure that the solutions are independent of the resolution used, the solutions obtained at different resolutions (M K ¼ 9 16, 17 32, 33 64, 65 128, 129 256, and 257 512) were interpolated onto an evenly spaced 100 600 spatial grid and compared at these points with a reference solution obtained at a resolution of M K ¼ 513 1024. The plot of the Frobenius norm, jj jjF, of the difference in these stream function values is shown in Fig. 2(b) for Re ¼ 0.01,1,10,30 and 100. Each case attains a geometric convergence rate, which shows that the flow is well resolved. The final resolutions M K used to obtain the velocity fields for our analysis were 33 64 for Re 1, 65 128 for Re ¼ 10, and 129 256 for Re > 10.

3 Results One approach to evaluating mixing efficiency using a computa- tional analysis is to quantify the distribution of particles by the flow. The objective is to achieve a homogeneous distribution of particles. The ‘best’ mixed system can be defined as that system producing the most homogeneous distribution either in the limit of infinite time or by a given finite time; these two systems are not necessarily the same. Long-time homogenization can be charac- terized by examining chaotic advection using, for example, the Poincare´ section, while short-time homogenization can be charac- terized by observing the distribution of an initially concentrated passive scalar. Here we consider both points of view.

3.1 Chaotic Advection. The Poincare´ section is a standard tool for analyzing periodic dynamical systems. In two- dimensional, time-periodic flows, a Poincare´ section is generated by recording the location of an advected particle after each period of the system [1]. In order to apply this tool to the finite Reynolds number cases, we record particle positions at times t ¼ ps for p n* (with p an integer), when the flow has become periodic (see Fig. 2(a)). Thus, the long-time homogenization given by the Poin- care´ section neglects the effects of the initial transients in these cases. In general, the domain in a Poincare´ section will consist of Fig. 2 (a) Flow period n* at which the bulk flow becomes peri- ‘elliptic islands’, containing well-organized periodic and quasi- odic in time, (b) spectral convergence of the stream function, periodic trajectories, that are surrounded by the ‘chaotic sea’, con- and (c) order of convergence. Since the convergence is geomet- taining chaotic trajectories that eventually visit all of the space ric for a spectral method instead of algebraic, the order of con- occupied by the chaotic sea. The area contained within elliptic vergence (rate) varies with degrees of freedom, and often islands is isolated from the remainder of the flow (in the absence machine precision is reached. of diffusion), and thus these regions are mixed poorly. There may also be regions in the chaotic sea in which mixing is relatively slow; these regions are often in the vicinity of an elliptic island. in their immediate neighborhood. For Re ¼ 10 the entire flow do- Poincare´ sections for different Reynolds numbers are shown in main is filled by the chaotic sea. At Re ¼ 30 there are two obvious Fig. 4. Elliptic islands exist in three isolated regions of the domain elliptic islands in the flow domain, and although the area of these for Re 1. In the limit as Re ! 0 these small regions correspond nonchaotic regions is still reasonably small, we see in Sec. 3.2 to the presence of parabolic periodic points in the flow [9,11]. that they appear to have a detrimental impact on mixing. For the These periodic island regions are the ghost rods that generate a flows we considered with Re 50, the entire flow domain is cov- figure-eight stirring motion. The dynamics of passive particles ered by the chaotic sea. From these results, we might expect to get slow down in the vicinity of these regions, leading to poor mixing better mixing for Re ¼ 10, 50 and 100, as compared to Re 1 and

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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. 3 Instantaneous streamlines at time t 5 n*s, i.e. at the end of an advection cycle, once the flow has become periodic for Re 5 (a) 10, (b) 30, (c) 50 and (d) 100

Fig. 4 Poincare´ sections for Re 5 (a) 0.01, (b)1,(c) 10, (d) 30, (e) 50, and (f) 100

Re ¼ 30. However, Poincare` sections are not always useful for the bins) having the same aspect ratio as the half domain quantifying mixing efficacy at early times or for determining mix- (0 x 6), based on the scaling for the bin size [17] ing rates. For this, we consider the dispersion of a passive scalar 1=2 in the flow domain. s 2Np (14)

where s is the height of a bin. This scaling is chosen to ensure that 3.2 Finite Time Homogenization of a Passive Scalar. We for a perfectly random distribution of particles, 98% of the bins consider the effect of finite Reynolds number on finite time mix- will contain at least one particle. ing by directly observing the distribution of an initially concen- We consider three different measures to quantify how well the trated collection of passive scalars that move exactly with the flow distributes the particles among the bins. Each of these meas- ures is based on evaluating gi, the number of particles contained flow. A total of Np ¼ 40,000 particles were initially placed uni- th formly along the line y ¼ 0. In contrast to our calculation of the in the i bin at the instant the mixing measure is being calculated. Poincare´ sections, the advection of these particles starts with the The first two measures are defined by fluid at rest and includes the initial flow transient. In each of the XNb cases we consider, this line of particles is exponentially stretched 1 eg ¼ wi (15a) and folded, and the particles are quickly spread out over the do- Nb main, as shown in Fig. 5 for several example cases that we discuss i¼1 below. We quantify the rate at which the particles are distributed with across the domain using several mixing measures that rely on g =g if g < g dividing the domain into small bins. In the current study, the flow w ¼ i i (15b) i 1ifg g domain was divided into Nb ¼ 10,000 equally-sized boxes (i.e., i

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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. 5 Comparison of dispersion of passively advected particles for Re 5 1, Re 5 10, Re 5 30, Re 5 50 and Re 5 100 (top to bottom) at 1, 4, and 10 advection cycles (left to right)

If we take g* ¼ 1, then the stirring index, e1, gives the fraction of 5(d) for p ¼ 1 and Figs. 5(b) and 5(e) for p ¼ 4. The braiding bins that contain at least one particle. In a well-stirred system, motion of the ghost rods leads to exponential stretching and fold- eventually all bins will contain at least one particle, and e1 ! 1as ing of material lines for Re 10. the number of flow periods p !1. The evolution of 1 e1 is The presence of inertia in the Re ¼ 10 case does cause it to dif- shown in Fig. 6(a). fer slightly from the Re 1 cases. During the first few periods of If we instead take g* ¼ Np=Nb ¼ 4 in Eq. (15), then the homoge- motion, e1 indicates that particles have reached slightly fewer bins nization index, e4, gives a measure of the fraction of bins that con- in the Re ¼ 10 case as compared to Re 1, although e4 and r tain the homogenized limit of particles. Some credit is given for show that homogenization is essentially identical for each of good stirring that distributes at least one particle per bin, but the Re 10. For Re 10 there exist regions near the ‘ghost rods’ in weighting factor wi favors a homogeneous system, in which case which mixing is quite slow, as shown in Figs. 5(c) and 5(f), even e4 ! 1. The evolution of 1 e4 is shown in Fig. 6(b). though the Poincare´ sections in Figs. 4(b) and 4(c) show that there Our third mixing measure is similar to the homogenization are not large islands in these areas. These regions persist for large index and is essentially the same as the standard variance of con- p when Re ¼ 1, but they continue to shrink for Re ¼ 10, allowing centration measure [18], which we calculate here as mixing to progress more rapidly for this case when p & 5. The ghost rods for Re ¼ 10 can be seen as being more ‘leaky’ than 1 XNb those for Re ¼ 1. r ¼ ðg gÞ2 (16) For Re 10, at early times (p . 3 for r, p . 7 for e ) the N i g* b i¼1 decrease in the mixing rate is essentially monotonic with increas- ing Re. At intermediate times, the dynamics of the flow compli- where we take g* ¼ Np=Nb ¼ 4. Unlike the eg* measures in Eq. cates the relationship between Reynolds number and mixing. For (15), the variance of particle density r is penalized by bins for Re ¼ 30 and Re ¼ 50, the pattern of stretching and folding at early which gi > g*. As the system becomes homogenized, r ! 0. The times in the flow appears similar to that for Re 10, as seen by evolution of r is shown in Fig. 6(c). comparing Figs. 5(g) and 5(j) with Figs. 5(a) and 5(d). However, All three mixing measures support the same trends in the sys- Figs. 5(h) and 5(k) appear to suggest the absence of well-defined tem. Mixing progresses rapidly from the initial condition for ‘ghost rods’ that move on ‘figure-eight’ trajectories in the Re ¼ 30 Re 10. The efficiency of this mixing protocol can be explained and 50 cases. The Re ¼ 100 case exhibits the poorest mixing at by the presence of figure-eight ‘ghost rod’ trajectories [9,11] that early and intermediate times. The initial structure of the flow in are generated by the boundary motions of the lid-driven cavity. this case shows much less stretching and folding than all other Evidence of this underlying structure can be seen in Figs. 5(a) and cases considered, as illustrated in Fig. 5(m).Atp ¼ 4 the flow at

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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 4 Conclusions The characteristics of chaotic advection and mixing have been considered for a lid-driven cavity flow at finite Reynolds number. The flow protocol, as specified through imposed time-periodic boundary velocity conditions, are those that are known to generate a ‘figure-eight’ motion of ‘ghost rods’ in the Stokes’ flow limit [11]. Based on our calculations of Poincare´ sections and mixing measures, we observe that the mixing effectiveness in this flow does depend on Reynolds number, and there does exist a nonzero value of Reynolds number that produces optimal mixing in short times in the low Reynolds number regime. For Re 1 the mixing results are essentially independent of Reynolds number. Thus, as might be expected [6], previous analy- ses of transport and mixing in similar lid-driven cavity flows with Re ¼ 0[9,11] are applicable to the small Re regime. This result also suggests that three-dimensional channel flow variations on this system [10,12] can be extended to finite Reynolds number, although further work is needed to support this suggestion. After the first few advection cycles, the value of 1 e1 stays the lowest (over the time frame considered) for Re ¼ 10. Thus, overall the best mixing occurs in this system for the Re ¼ 10 case among the cases we have considered. For the parameters considered, this case provides just the right balance between viscous effects and inertial effects. The viscous effects generate stretching and folding through the motions imposed by the boundary velocities, while the inertial effects enhance mixing in the interior of the domain where poorly mixed regions exist for smaller Re. This result sup- ports the use of a Stokes’ flow analysis to establish a base-line design for a mixing system based on a figure-eight motion of peri- odic orbits, as long as the Reynolds number is kept small. For Re > 10, increasing the inertia of the fluid increases the complexity of the flow, as evidenced by the patterns in Fig. 5. However, this increase in complexity of the flow structure actually reduces the ability of the system to mix in short times for the stirring protocol used here. As can be seen in Figs. 5(h), 5(k) and 5(n),the desired figure-eight motion appears to be absent in the higher Reyn- olds number cases, which may explain the poor mixing at early times reflected in all three mixing measures. We speculate that this is related to the destruction of almost invariant sets (AIS) in the flow [11] that move on figure-eight trajectories, a line of inquiry that remains to be explored. For Re ¼ 30, elliptic islands appear in the do- main despite the presence of inertia, and these invariant regions lead to poor long-time mixing, a phenomena that has been observed in other systems [4]. For the other cases with Re > 10 (except for Re ¼ 30), inertial effects eventuallyleadtolong-timemixingthat approaches the homogenization exhibited by the Re ¼ 10 case. Thus it is seen that, with all other parameters held constant, gradually reducing the viscosity at first leads to an increase in mixing (at Re ¼ 10), and then the subsequent increase in inertial effects reduces the ability of the system to mix during the initial Fig. 6 Quantifying the time-dependent distribution of pas- times. However, the structure of the flow shown in Fig. 5(e) sug- gests that it may be possible to also produce a ‘figure-eight’ stir- sively advected particles using (a) the spreading index e1,(b) the homogeneity index e4, and (c) the variance of particle den- ring protocol for the higher Re cases by adjusting the parameters sity r, each as a function of the flow period p 5 t=s defining the flow protocol. It remains to be seen if this tuning can lead to better mixing at moderate Re.

Acknowledgment Re ¼ 100 appears more complex (see Fig. 5(n)), but clearly this All computations were performed using time provided by the pattern does not mix the fluid as well as the more organized Texas A&M University Supercomputing Center. P.R. acknowl- stretching and folding achieved in the other cases. edges the financial support of the Virginia Tech Department of At late times, the Re ¼ 30 case exhibits the poorest mixing Engineering Science and Mechanics and the Institute for Critical among the Re 10 cases, as indicated by all mixing measures in Technology and Applied Sciences (ICTAS). Fig. 6. This poor mixing is consistent with the appearance of two elliptic islands in the flow (see Fig. 4(d) and Fig. 5(i)) that act as barriers to transport. Thus, for late times the mixing achieved by Nomenclature ^ the Re ¼ 30 case is similar to that achieved in the Re 1 cases. Qmk ¼ nonlinear term, spectral transform The absence of large elliptic islands in the flow for the Re ¼ 50 Umax ¼ maximum boundary velocity and Re ¼ 100 cases enable them to eventually achieve a level of Cn ¼ analytical solution coefficient for nth flow field homogenization that is similar to the Re ¼ 10 case. N ¼ number of flow fields (Stokes’ solution)

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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 Nb ¼ number of bins References Np ¼ total number of particles [1] Aref, H., 2002, “The Development of Chaotic Advection,” Phys. Fluids, 14, pp. th Tm(y) ¼ m Chebyshev polynomial 1315–1325. [2] Wiggins, S., and Ottino, J. M., 2004, “Foundations of Chaotic Mixing,” Philos. Un ¼ coefficient of nth term of boundary velocity X , Y particle coordinates Trans. R. Soc. London, Ser. A, 362, pp. 937–970. p p ¼ [3] Dutta, P., and Chevray, R., 1995, “Inertial Effects in Chaotic Mixing With Dif- Re ¼ Reynolds number, ¼ Umaxh=v fusion,” J. Fluid Mech., 285, pp. 1–16. ez ¼ unit vector in z direction [4] Hobbs, D. M., and Muzzio, F. J., 1998, “Reynolds Number Effects on Laminar u ¼ velocity vector Mixing in the Kenics Static Mixer,” Chem. Eng. J., 70, pp. 93–104. f y dependence of w (separation of variables) [5] Clifford, M. J., Cox, S. M., and Finn, M. D., 2004, “Reynolds Number Effects ¼ in a Simple Planetary Mixer,” Chem. Eng. Sci., 59, pp. 3371–3379. h ¼ domain half-height [6] Wang, J., Feng, L., Ottino, J. M., and Lueptow, R., 2009, “Inertial Effects on L ¼ spatial period in x Chaotic Advection and Mixing in a 2D Cavity Flow,” Ind. Eng. Chem. Res., n* ¼ flow period for onset of periodicity 48, pp. 2436–2442. p pth flow period [7] Boyland, P. L., Aref, H., and Stremler, M. A., 2000, “Topological Fluid ¼ Mechanics of Stirring,” J. Fluid Mech., 403, pp. 277–304. t ¼ time [8] Gouillart, E., Thiffeault, J.-L., and Finn, M. D., 2006, “Topological Mixing u,v ¼ velocity components with Ghost Rods,” Phys. Rev. E, 73, p. 036311. [9] Stremler, M. A., and Chen, J., 2007, “Generating Topological Chaos in Lid- wi ¼ weighting factor for mixing indices x,y coordinates positions driven Cavity Flow,” Phys. Fluids, 19(10), p. 103602. ¼ [10] Chen, J., and Stremler, M. A., 2009, “Topological Chaos and Mixing in a dt ¼ time step Three-Dimensional Channel Flow,” Phys. Fluids, 21(2), p. 021701. e1 ¼ stirring index [11] Stremler, M. A., Ross, S. D., Grover, P., and Kumar, P., 2011, “Topological Chaos and Periodic Braiding of Almost-Cyclic Sets,” Phys. Rev. Lett., 106,p.114101. e4 ¼ homogenization index g* critical particle number [12] Stroock, A. D., and McGraw, G. J., 2004, “Investigation of the Staggered Her- ¼ ringbone Mixer With a Simple Analytical Model,” Proc. R. Soc. A, 362(1818), gi ¼ number of particles in bin i pp. 971–986. s ¼ time period of flow [13] Canuto, C., Hussaini, M. Y., Quarteroni, A., Thomas, A., and Zang, J., 2006, ¼ kinematic viscosity Spectral Methods, Fundamentals in Single Domains, Springer-Verlag, Berlin. x z component of vorticity [14] Kim, J., Moin, P., and Moser, R., 1987, “Turbulent Statistics in Fully Developed ¼ Channel Flow at Low Reynolds Number,” J. Fluid Mech., 177, pp. 133–166. u ¼ phase for tangential boundary condition [15] Peyret, R., 2002, Spectral Methods for Incompressible Viscous Flow, Springer- w ¼ stream function Verlag, New York. r ¼ particle density variance [16] Lanczos, C., 1956, Applied Analysis, Prentice-Hall, Englewood Cliffs. k Fourier mode index [17] Liu, M., Muzzio, F. J., and Peskin, R. L., 1994, “Quantification of Mixing in ¼ Aperiodic Chaotic Flows,” Chaos, Solitons Fractals, 4, pp. 869–893. m ¼ Chebyshev mode index [18] Stremler, M. A., 2008, Encyclopedia of Microfluidics and Nanofluidics, n ¼ nth flow field (Stokes’ solution) Springer, New York, pp. 1376–1382.

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