Forcing a low Reynolds number channel flow to generate synthetic turbulent-like structures

Computers & Fluids 55 (2012) 101–108

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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d

Forcing a low Reynolds number channel flow to generate synthetic turbulent-like structures Sedat Tardu Laboratoire des Ecoulements Géophysiques et Industriels-LEGI, B.P. 53-X, 38041 Grenoble Cédex, France

a r t i c l e

i n f o

Article history: Received 12 February 2011 Received in revised form 24 August 2011 Accepted 14 November 2011 Available online 25 November 2011 Keywords: Regeneration of vorticity Self-maintaining vortical structures Mixing Low Reynolds number flows

a b s t r a c t The effect of interacting vortices on the turbulent-like synthetic structures regeneration in a low Reynolds number flow Re = 10 is investigated through direct numerical simulations. The configuration simulates two forced synthetic jets staggered in the lateral spanwise direction. The initial vortices create wall normal vorticity layers near the wall. The stagnation flow induced by the triggering eddy compresses the wall normal vorticity associated with the mother structure that consequently disappears rapidly. That breaks up the spanwise symmetry resulting in a shear layer transforming into a compact streamwise vorticity zone. The regeneration of coherent vortical structures is subsequently triggered resulting rapidly in a local turbulent-like spot. Main characteristics of the perturbed flow field are in close qualitative similarity with a localized turbulent spot that is in principle inconceivable at such a small Reynolds number. The analysis of the passive scalar transport shows that the method leads to efficient mixing at a Prandtl number Pr = 10. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The main preponderant structures present in the inner layer of a turbulent wall flow are elongated vortices called commonly quasi-streamwises vortices. These eddies are responsible for the setup of Reynolds stress and turbulent mixing at low and moderate Reynolds numbers [1]. They self-maintain themselves through a complex regeneration mechanism. The wall turbulence community has been intensively working on this topic for the last two decades (see for example [2–4]). The different routes to the generation of self-maintaining quasi-streamwise vortices in the buffer layer are nicely reviewed in [2] but several questions are still unanswered. The regeneration process in the wall turbulence has to be related in some way to the pre-existing inner structures themselves, since their birth cadence, called commonly the ‘‘bursting frequency’’ scales with inner variables, at least at moderately large Reynolds numbers. The key element is the generation of streamwise vorticity zones near the wall that can roll up into distinguished quasi-streamwise structures, once sufficient conditions are fulfilled such as concentration and discontinuity in the vorticity distribution. Without a streamwise x dependent flow field, the stretching, tilting and twisting terms appearing in the production of the streamwise vorticity become zero and the self sustaining mechanism is broken. The induction of the x dependence near the elongated streamwise structures should be somewhat related

E-mail address: [email protected] 0045-7930/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2011.11.005

to their interaction with the wall flow, i.e. to the structures themselves in conformity with the ‘‘regeneration philosophy’’. Taking into account these observations, we recently proposed a mechanism based on a specific spatial combination of two longitudinal vortices shifted in the spanwise and wall normal directions [5]. We have shown that such a specific distribution of perturbations in a laminar Poiseuille base flow leads rapidly to the genesis of new active structures and to the development of a local turbulent spot. The mechanism we suggested has furthermore strong similarities with the regeneration of turbulent wall structures [6]. A primary vortex that is initially injected in the flow interacts with the wall normal vorticity layers near the wall, by compressing or stretching locally part of them through the straining motion it induces. The breakdown of spanwise symmetry, which is the key element of the process, leads to the rapid development of a new wall normal vorticity patch. The latter is subsequently tilted by the shear and rolls up into a new streamwise vortex. The process results in a localized turbulent spot at later stages of development. The Reynolds number does not directly intervene in this process. Consequently, we aim to see whether a similar strategy can be applied to create synthetic wall turbulent like structures in a low Reynolds number flow, by for example distributing in a specific way the wall actuators such as synthetic wall jets, both in time and space. We show through numerical experiments here that this is indeed plausible if the flow is forced in a specific way. Creating synthetic wall turbulence is far from being a new idea. Several experimental and theoretical attempts have been done in the past to mimic turbulent structures, their interaction with the

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wall, and their development in time and space. Acarlar and Smith [7,8] studied the hairpin vortices in a subcritical boundary layer generated either by the shedding from a hemispherical protuberance at the wall, or destabilization of the fluid injected through a streamwise slot. A series of experimental and theoretical investigations emerging from this group revealed strong similarities of the flow patterns generated by convecting hairpin vortices and fully developed turbulent boundary layer (see [9] for a synthetic resume). The development of a hairpin vortex into a turbulent spot was modeled through direct numerical simulations (DNSs) by Singer and Joslin [10]. Zhou et al. [11] showed also by DNS how a single hairpin vortex-like structure superposed to the mean turbulent field of a low Reynolds number channel flow generates packets of structures. The reader is refereed to Adrian [12] who nicely reviews these aspects. The main difference between these works and the study presented here is first the particularly low Reynolds number we investigate. The Reynolds number of the laminar base channel flow based on the Poiseuille centerline velocity and half channel width is only 10 here. The second difference lies on the direct application of the strategy we reported in Tardu and Nacereddine [5] to the low Reynolds number channel flow, showing the consistency of the methodology we developed. Finally, we investigate the effect of the local forcing on the passive scalar transport with the potential application on mixing in microdevices. The Reynolds number in micro systems is of the order of 1–10. The flow is therefore predominantly viscous, and the relating mixing process is diffusive and unacceptably slow. It is well known that mixing can ideally be achieved by quasi-streamwise vortices and one of the ways to enhance the mixing, in particular locally, lies doubtlessly in the capacity of either active or passive strategies to mimic the wall turbulence through genesis of turbulent like

synthetic vortical structures. The present investigation constitutes an attempt in this particular direction. The paper is divided into five parts. We briefly discuss the selfregenerating process in the next section detailed in [5,6]. The direct numerical simulations are described in Section 3 and the results in Section 4. 2. Interactive self-regenerating mixing structures The first condition to mimic the wall turbulence in low Reynolds laminar flows, if possible, is to create localized zones of the streamwise vorticity xx. The second target is to set-up a mechanism that is capable of self-maintaining the structures, and, furthermore to allow their regeneration (self reproduction) as in wall turbulence. The suggested strategy to achieve these goals is based on a specific interaction of the longitudinal vortices as shown in Fig. 1. We only give a brief outline of the mechanism we investigated in detail in [5,6] to make the paper self-contained. The longitudinal vortices have streamwise vorticity xx whose transport equation is

@ xx @ xx @ xx @ xx þ ðU þ uÞ þv þw @t @x @y @z   @u @w @U @u @ v @u  þ þ þ mr2 xx ¼ xx @x @x @y @y @x @z

ð1Þ

in a Poiseuille flow submitted to three dimensional perturbations. Hereafter x, y and z are respectively the streamwise, wall normal and spanwise coordinates with corresponding velocity components u, v and w. The fluctuating vorticity components in the x, y and z directions are denoted by xx, xy and xz. The mean velocity distri-

Fig. 1. (a) Cross sectional view of two pairs of counter rotating quasi-streamwise vortices in the plane y–z. Here x, y and z are respectively the longitudinal, wall normal and spanwise directions. The pair A regenerates wall normal vorticity layers denoted by xyA positive and negative respectively at the left and the right shown by gray bars. Similar vorticity layers xyB are generated by the pair B. The left vortex of the pair B compresses the xy+A breaking-up the spanwise symmetry; (b) schematic resume of the suggested regeneration mechanism.

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bution of the Poiseuille flow is U(y), and u, v and w depend on x, y, z and the time t, while m stands for viscosity. The first three terms at the right hand side of (1) are the production terms. The main production term of the streamwise vorticity comes from the tilting of the wall normal vorticity xy by the shear in the fully developed turbulent wall layer and plays also an important role in the linear instability of the streaks. A simple order of magnitude analysis shows that it is the leading production term during the   transient @U @U period of by-pass process and reduces to  @w þ @u .   @w @x @y @y @x @y The creation of compact zones with high concentration of streamwise gradient of the spanwise vorticity is therefore a key element in not only the by-pass transition mechanism but also the regeneration of the near wall coherent structures. Consider again the conceptual model given in Fig. 1 that shows two pairs of counter rotating vortices labeled respectively by A and B. Wall normal vorticity sheets xy are generated behind the vortices resulting from the kinematics induced by the near wall velocity distribution. This is a physical fact observed in the direct numerical simulations Jiménez [13] and can be easily understood through Fig. 1. A single quasi-streamwise vortex generates respectively a high-speed u > 0 streak at the sweep part of the induced near wall flow and a low speed streak u < 0 at the ejection side. The wall normal vorticity is xy ¼ @u  @w near the wall and in the @z @x neighborhood of elongated structures reduces to xy  @u . It appears @z essentially as ‘‘the sidewalls of the high and low speed streaks’’ at a certain time and location. This is shown schematically in Fig. 1 where layers of positive and negative vorticity are denoted respectively by xy+ and xy. Assume now that, there is a streamwise variation in the vorticity associated by the wall normal xy+ and xy layers induced by the mother eddy A and concentrated respectively at z < 0 and z > 0 of the domain   shown in Fig. 1. It can rigorously be shown that the average @w generated between xy+ and xy layers over any span@x wise symmetrical surface S at a given wall normal distance y is:

 Z Z @w 1 0 0 0 ¼ G D dx dy dz dS @x 4p S S R

ð2Þ

and:

D¼



@ xyþ

@ xy



@x0

@x0

and G is a weighting function. The concentration of the new vorticity @w and its roll-up into a new quasi-streamwise structure depend @x directly on the intensity of the asymmetry D. Presumably, when D appears in the distribution of the xy layers surrounding the flow induced by the mother eddy A, a local streamwise variation in w may be introduced and the set-up of the x dependence starts in the plane (x, z). The compression of one of the vorticity layers xyþA or xyA by the large positive straining induced by the left counter rotating vortex B in Fig. 1 may break up the symmetry and enhance the by-pass transition. It can be shown that for sufficiently large times the local dimensionless vorticity disappears exponentially in time according to xyþA / expðc t Þ under the stagnation flow with parameter c⁄ induced by the sweep motion induced by the structure B. The strain pffiffiffi v B where ⁄ stands for parameter associated with B is cB ¼  3 Re 4d2 B

the quantities non-dimensionalized by the viscosity m and a reference velocity such as the centerline velocity Uc. The distance of B to the wall is denoted by dB. The vortex Reynolds number is defined as Rev B ¼ 2KpBm, KB being the associated circulation. The negative xyA sidewall in return is located far away from the stagnation flow. It diffuses under the dominant effect of viscosity. The maximum vorticity in this layer decreases therefore as xyA / p1ffiffiffi . t For typically t   c2 the positive vorticity disappears almost instantaneously giving rise to a large @w > 0 wall layer that can sub@x

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sequently be tilted by the shear and roll up to a new streamwise structure (see [5,6]). A resume of the strategy is schematically shown in Fig. 1b. Beginning with existing vortices and their interaction with the wall, new structures of high mixing capability are regenerated continuously by mutual interactions. When the Reynolds number is large, say Re P 1000, an initial perturbation may lead to the generation of a turbulent spot and to the fully turbulence as we investigated in [6]. When Re 6 1000 all initial perturbation dies due to the viscosity and the limit 1000 is the upper bound that delimits the set-up of secondary instabilities in wall bounded flows (Orszag and Patera [14]). The mechanism we suggest can however been reproduced in low Reynolds number wall flows, when the system is suitably forced as we will show later in Section 4. 3. Direct numerical simulations and methodology A direct numerical simulation (DNS) code inferred from [15] is used for the present purpose. A second order finite difference scheme for spatial discretization is combined with the fractional step time advancement procedure [16] to maintain continuity. The viscous terms are discretized in time with the Crank–Nicholson scheme and the pressure is inserted at the non-solenoïdal step. The non-linear terms are explicitly treated by a third order Runge– Kutta scheme during the fractional step of the computational process. The number of modes is 256  128  128 in respectively streamwise x, wall normal y and spanwise z directions. The sizes of the computational domain extend from 16pa in x, 2a in y and to 8pa in z where a stands for the half height of the channel. The grid is clustered near the wall using stretched coordinates through a tangent hyperbolic function in the wall normal direction. Thus, the mesh size is as small as Dy = 103a in the wall normal direction near the wall, and Dx ¼ Dz ¼ 0:2a in the streamwise and spanwise directions. The resolution of the system is subsequently achieved in the transformed coordinate system wherein the mesh sizes are constant in all directions. Periodic boundary conditions are used in the homogeneous x and z directions. The Orlandi code [15] has been used by different groups and also by our team for a while in different configurations [6,17,18]. The time step is Dt ¼ 0:01a=U c with Uc standing for the centerline velocity. The Reynolds number based on the centerline velocity and half channel width is fixed to 10 through the whole study. In fully developed turbulent wall flows, it is usual to scale the time step by innerqvariables defined by the viscosity and the shear ffiffiffiffi  s ¼ sqw , where s w is the wall shear stress. The base being velocity u a laminar Poiseuille flow here, the time step in wall units is  2s which is an order of magnitude smaller than Dt ¼ 0:02m=u  2s that we typically use in fully developed turbulent Dt ¼ 0:1m=u  s =m ¼ 180. We further dechannel flows at for example Res ¼ au creased Dt by a factor of five in a run, and noticed only a few per cent modification of quantitative results. We therefore concluded that the dissipative numerical errors coming eventually from the fractional step process are negligible in the present configuration. Two pairs of counter rotating vortices were injected in the channel flow with stream functions of the form:

h i 2 2 w ¼ ef ðyÞðx0 =lx Þz0 exp ðx0 =lx Þ  ðz0 =lz Þ

ð3Þ

where x0 = x cos h  z sin h, z0 = x sin h  z cos h and h is the angle of the perturbation which has been set to h = 0 here. The perturbation flow field is given by:

ðu; v ; wÞ ¼ ðwy sin h; wz0 ; wy cos hÞ

ð4Þ

and f(y) = (1 + y)p(1  y)q. Hereafter the quantities are normalized with respect to a and the centerline velocity of the Poiseuille base flow. The velocity components in the longitudinal, wall normal

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and spanwise directions are denoted respectively by u, v and w. The perturbation expressed in (4) is of the same type as used in [19]. The difference however is in the modeling of the configuration given in Fig. 1. We combined two different w respectively for the structures A and B and shifted B in the z plane. More clearly we have chosen pA = qA = 2, and lxA = lzA = 4 for the structure A. The stream function corresponding to the structure B is of the same form but shifted in the spanwise direction by dz i.e.

wB ¼ eB fB ðyÞðx0 =lx Þðz0 þ dz Þ exp½ðx0 =lx Þ2  ðz0 þ d=lz Þ2 

ð5Þ

The quantities pB and qB have been monitored to allow B to sufficiently interact with the wall vorticity layers generated by A. Changing the parameter qB allows to control the distance of the structure B from the wall [6]. The intensity of the stagnation flow induced by B depends upon both pB and qB as it can clearly be seen from the Eq. (4) and (5). Thus, the parameters p and q are chosen in such a way that B is closer to the wall and interacts sufficiently strongly with the wall vorticity layers generated by A. A parametric study gave pB ¼ 1:5; qB ¼ 6, and lxB = lzB = 2 to fulfill this goal. The structure B is consequently centered at y = 0.6 closer to the wall, while A is at the centerline (y = 0 is the centerline, and y = 1 corresponds to the upper wall). The spanwise shift between the structures A and B is dz = 4 (scaled by a) through the whole study. The choice of the perturbation parameters eA and eB will be discussed in the next section. The simulation of passive scalar transport, is performed through the resolution of the equation:

@T @ðuj TÞ 1 @2T þ ¼ @t @xj RePr @x2j

turbations would be probably interesting to study in future investigations. Note how the amplitudes are small. Thus, the strategy consists of imposing a perturbation rather than a strong forcing. The specific interaction of the structure A with B (Fig. 1) leads to the rapid regeneration of vortical structures. Fig. 2a shows the quasi-streamwise vortices detected by the lambda-2 technique [20] and generated at t = 4 in the A + B case. Note the concentration of the structures in the zone z 6 0 wherein the interaction between A and B takes place. The amplitudes of the individual perturbations are so weak that the induction of strong enough vortices is not allowed to show the efficiency of the mechanism we suggest (Fig. 3). Here we intentionally chose small individual perturbations to show the feasibility and efficiency of the interactive mixing process. This is of course not an obligation in practical situations for which stronger singular excitations may be injected in the flow, and their interactions according to model given here will presumably result in a more rapid set-up of vortical structures. The regenerated vortices are l  4a long and their diameter is approximately g = 0.5a, as it can be seen from Fig. 2. These values compare surprisingly well with the characteristics of the quasi-streamwise Reynolds shear stress producing eddies in a fully developed low Reynolds number turbulent channel flow. It is indeed observed that at the low Reynolds number limit of fully developed wall turbulence Re = 2000, at which the equilibrium log-layer becomes to be appreciably thick, the streamwise extend of the active vortical structures is l  4a and their diameters are about 0.2a [6]. Thus, there is a reasonable correspondence between the characteristic

ð6Þ

where Pr is the Prandtl number. The resolution of this equation requires the calculation of the velocity field at the same time. So, the direct numerical and passive scalar transfer simulations have to be performed simultaneously. The numerical complexity is quite similar to the DNS resolution and similar tools can be employed. A Runge–Kutta (3rd order) scheme is applied for the time advancement while the diffusive term is resolved by a direct LU decomposition. The numerical cost of the passive scalar transfer resolution is quite low and corresponds approximately to 1/5 of the DNS one. 4. Results 4.1. Regeneration of turbulent-like synthetic structures As we indicated before, local forcing is necessary at low Reynolds numbers since the regeneration of the structures cannot be maintained when they result from perturbations constituting initial conditions and the Poiseuille flow is absolutely stable for Re < 1000. We first investigated local forcing by time modulating the strength of the perturbations eA and eB appearing in the equation (3) through sinusoidal intensities:

 t þ uA   eB ðtÞ ¼ aB sin 2CpB t þ uB

eA ðtÞ ¼ aA sin



2p

CA

ð7Þ

A parametric study has been conducted to optimize the efficiency of the regeneration mechanism discussed in the previous section, by modifying the amplitudes aA, aB together with the periods CA and CB. Best results have been obtained with aA = 8  103 and aB = 6  103 and CA = CB = C = 8. Recall that all the quantities are non-dimensionalized with respect to the centreline velocity Uc of the Poiseuille flow, and the channel half width a and that Re = Uca/m = 10. The system is forced only during one time period C in this investigation. The phases appearing in Eq. (7) were chosen as /A = /B = 0, although the effect of a phase shift between two per-

Fig. 2. Regeneration of the coherent structures in the A + B case at t = 2 (a) and t = 4. It is recalled that the imposed period is C = 8.

S. Tardu / Computers & Fluids 55 (2012) 101–108

Fig. 3. Structures induced by the perturbation A alone at t = 2.

scales of the vortical structures in low forced Reynolds number investigated here, and the structures in fully developed wall turbulence, when these lengths are related to the outer scale a. In fully developed turbulence, however, it is more likely that these characteristics scale with inner variables that are the viscosity m and the pffiffiffiffiffiffiffiffiffiffiffi shear velocity us ¼ sw =q where q is the density and sw is the wall shear stress. Thus, one has l+ = 300 and g+ = 20 at low Reynolds number wall turbulence at typically Re 6 104 . Hereafter ()+ designates quantities scaled by m and us. The streamwise extend l  4a of thepffiffiffiffiffiffiffiffi structures in forced laminar flow corresponds to þ l  4aþ ¼ 4 2Re ¼ 18, when one uses the shear velocity of the pffiffiffiffiffiffiffiffiffiffiffi Poiseuille base flow, i.e. us =U c ¼ 2=Re. The inner scaled streamwise length and diameters of the structures are one order of magnitude smaller than in fully developed wall turbulence. Inner scaling has a particular meaning in wall turbulence, and supposes the existence of an internal and external sublayer. Strictly speaking, the inner scaling is valid in the former and only at the limit of Re ? 1 (see for example Wosnik et al. [21]). For these reasons at least, the comparison between the forced low Reynolds number laminar flow and developed turbulence can of course be only qualitative. The regeneration process is somewhat weakened when the perturbation field reaches its maximum at the end of the acceleration phase at t = 4 (Fig. 2b). The structures progressively disappear during the deceleration phase. This behavior is in concordance with the fact that time and/or space acceleration destabilizes the flow in a way opposite to deceleration. To avoid the stabilizing effect of rapid deceleration we tested forcing the perturbation e in a dissymmetric way in time, by imposing a slow acceleration whose duration is longer than the subsequent rapid deceleration. This is schematically shown in Fig. 4. The maximum reached in eA(t) and eB(t) and the time period C were kept the same as in sinusoidal perturbations. The vortical activity developed continuously during the acceleration phase under the forcing dissymmetric in time. It persisted furthermore during the deceleration phase whose duration is smaller than the characteristic relaxation time scale of the induced structures. Fig. 5 shows the structures generated at the end of the acceleration phase at t = 7. The regeneration process could clearly be maintained during a time lapse longer than in sinusoidal perturbations. These results have two interesting features. First, they show how suitably choosing the temporal forcing waveform can efficiently control the vortical structures regeneration mechanism. Second, the management of forcing may lead to efficient mixing at prescribed target locations and times. We summarize in the rest of the paper the results we obtained through

105

Fig. 4. Dissymmetric forcing in time. The figure shows the temporal evolution of the intensity eA(t).

Fig. 5. Regeneration of the coherent structures in the A + B case with dissymmetric forcing at t = 7.

forcing the system by dissymmetric temporal waveform shown in Fig. 4. The synthetic vortical structures have a rich hierarchy of shapes and scales. Besides the quasi-streamwise elongated structures, arctype and spanwise vortices are clearly distinguishable in both Fig. 2a and Fig. 5. The former are commonly observed in the outer (logarithmic) layer of a fully developed wall bounded turbulent flow [1]. They are the detached heads of the hairpin structures generated in the low buffer layer, while the quasi-streamwise vortices are their remaining legs. The arc-vortices observed in Fig. 3a and Fig. 5 result presumably from a similar mechanism. Flow visualizations we performed indeed confirmed that some (but not all) of these structures are the heads of former hairpin vortices generated near the wall. The mean shear o U/ o y is small at this low Reynolds number. Thus, the forcing may give place to local fluctuating shear layers that may occasionally be larger than the mean shear. Consequently the stretching and twisting production terms appearing in Eq. (1) may play a certain role. The production terms in the spanwise vorticity transport equation that reads as

@ xz @ xz @ xz @ xz þ ðU þ uÞ þv þw @t @x @y @z   @ v @w @u @w @w @ v @U @u @w þ þ þ þ mr2 xz  ¼ @z @x @y @y @z @z @x @z @y

ð8Þ

may locally be enhanced, creating compact spanwise vorticity zones that result in spanwise vortices. This proposal may explain the emergence of near wall spanwise vortices in Fig. 3a and Fig. 5.

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Fig. 6. Snapshot of alternate positive (red) and negative (blue) wall normal vorticity layers for A + B at t = 6 and y = 0.5. The maxima and minima are ±15 (scaled by Uc/ a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Close visual inspection of the general vorticity field in the A + B case revealed strong structural similarity with a local turbulent spot. That was curious enough given that the Reynolds number is only Re = 10 here. Fig. 6 shows a snapshot of the wall normal vorticity xy ¼ @u  @w in the x  z plane at t = 6 and y = 0.5. Keep in @y @x mind that the imposed forcing is dissymmetric in time hereafter. It is seen that negative and positive concentrated high vorticity zones alternate in a quasi-periodical fashion. The same behavior is observed in a transitional spot, and in fully developed wall turbulence wherein alternate thin xy layers separate the low and high-speed streaks the coherent structures induce near the wall. Fig. 7 shows the contours of streamwise vorticity xx ¼ @w  @@zv at @y the same epoch t = 6 and at x = 2 downstream the initial perturbation in the y  z plane. Positive and negative streamwise vorticity layers alternate in the spanwise direction as in fully developed turbulent wall flows wherein they correspond to counterrotating streamwise vortices. Despite these qualitative similarities, strong differences exist ineluctably between fully developed wall turbulence and the forced low Reynolds number flow investigated here. The xx layers induced by the forcing are for instance flattened near the wall and take the form of pancakes rather than circular vortical layers. The nondimensionalized xx vorticity peak intensity is as large as ±40 in Fig. 7. We computed the root mean square (rms) of the fluctuating streamwise vorticity in a box containing the vortical structures and found rxx ¼ 10. This value is significantly larger than what we have in low Reynolds number fully developed wall turbulence. At Re = 4000, the maximum rms streamwise vorticity near the wall is rþ xx ¼ 0:2 in inner variables [16]. Translating this value in outer scaling results in rxx ¼ 2 which is five times smaller than in forced laminar flow. The induced near wall activity in the forced flow depends upon the initial conditions. Furthermore, the flow is in a transitory stage since the forcing is only applied during one time period. 4.2. Effect on mixing The passive scalar transport and mixing under the interactive strategy will now be discussed. Fig. 8 shows the two-layer configuration used as initial conditions in the numerical experiments.

Fig. 7. Positive (red) and negative (blue) streamwise vorticity contours for A + B at t = 6 and x = 2. The maxima and minima are ±40 (scaled by Uc/a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Configuration of two fluid layers with different initial passive scalar values. The interface is slightly shifted from the channel centerline. The structure A is located at the centerline while B is at y = 0.6. The centerline is at y = 0, the lower and upper walls are respectively at y = 1 and y = 1. The quantities are scaled with the channel half width a and the centerline velocity Uc.

The bi-layer fluids have similar physical properties and are maintained at the scalar values T1 and T2 at t = 0. A relatively severe scalar difference is maintained between the two co-flows with T1 = 150 and T2 = 10. The interface is shifted from the centreline so that the mixing scalar reads for Tm = [d(T2  T1) + 2aT1]/2a and the dimensionless mixing parameter is #m = Tm/(T1 + T2) = 0.7 under the present conditions (Fig. 8). The Prandtl number is Pr = 10. The Reynolds number being Re = 10, the Péclet number in the present investigation is therefore Pe = 100. The Prandtl number could not be increased beyond Pr = 10 with the given resolution. The reason for this will be discussed later in this section. We investigate exclusively the impact of the forcing induced by interacting vertical structures on passive scalar transport and neglect voluntarily the buoyancy effects that are out of scope of the present study. Note that the initial buoyancy length scale H is zero in the configuration we analyze in Fig. 8, with a scalar singularity at 3 the interface. Thus, the Grashof number Gr ¼ gbDmTH is initially zero 2 at t = 0, where b is the coefficient of thermal expansion at constant pressure, g is the gravity, DT = T1  T2 and H is the efficient buoyancy length scale. pffiffiffiffiffiIn the absence of forcing and at short times H increases as H / at by pure diffusion and reaches H = 2a later when the conductive fully develops. Thus, the Richardson number ffi pffiffiffilayer gbDTð at Þ3 Gr Ri ¼ Re is presumably negligibly small at mixing times 2 / m2 Re2 investigated here. Fig. 9a shows the dimensionless scalar # = T/(T1 + T2) field at, x = 2 resulting from interactive mixing A + B, at t = 1, immediately after the asymmetric forcing has started. The interface begins to be distorted at this epoch, but the mixing is poor. The regeneration of new vortices progressively takes place when the forcing advances in time. At t = 7 that is approximately the end of the acceleration period, multiple structures appear increasing significantly the mixing as it can be seen in Fig. 9b. The vortical structures induce a truly expanded fluctuating velocity and passive scalar fields. These fields and the corresponding local mixing zone are intimately linked to the convection of compact vortices as is seen in Fig. 9c that shows the distribution of # at a given plane, and the direct effect of vortical structures on mixing. Thin filaments of scalar seen in Fig. 9b result from the straining induced by smallest scale vorticity. The thickness of these filaments deduced from the visualizations such as those given in Fig. 6 is gT0  0.2a, that is close to the Batchelor scale gT 0 ¼ Pr1=2 g, where g  0.5a is the smallest typical length scale of coherent vorticity. This particular point will be reconsidered in more detail at the end of this section. The mixing through the individual structures A (Fig. 10) or B is poor at this low Reynolds number. A typical mushroom structure is depicted from Fig. 10. Such structures are inherent to synthetic jets. Stirring of scalar is also present in this configuration, yet the spatial mixing spread is significantly limited compared to Fig. 9. We use the quantity Pm to analyze the mixing. To do that, we chose first a region S delimited by 10 < z < 10 and 0.9 < y < 0.9 at a given x. The portion of this section Sm into which the local instantaneous passive scalar #(x, y, z, t) verifies #m  D 6 # 6

S. Tardu / Computers & Fluids 55 (2012) 101–108

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Fig. 11. Mixing efficiency at t = 7. See the text for related definitions.

Fig. 9. Non dimensional # = T/(T1 + T2) passive scalar field resulting from the interactions A + B at x = 2 and at t = 1 (a) and t = 7 (b). The forcing is asymmetric in time as in Fig. 4. The perfect mixing corresponds to #m = 0.7 (yellow–orange zones). Figure (c) shows the connection of the dimensionless scalar field at the plane x = 2 to the presence of the vortical structures. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Non dimensional passive scalar field resulting from A at t = 7 and x = 2. See Fig. 9 for legend. The forcing is asymmetric in time.

#m þ D is subsequently computed. The mixing parameter is simply Pm = Sm/S and characterizes the percentage of space wherein the mixing is realized to within D. The latter is D = 0.02 which is particularly small. It is used only to slightly smooth the data. Thus, the results presented in Fig. 11 and corresponding to t = 7 represent spatial mixing to within 3%. The configuration A + B rises in perfect mixing in 25% of the space at x > 0. The efficiency Pm hardly exceeds 0.1 for the individual structures, showing that the interactive mechanism is approximately three times more effective. The second interesting feature deduced from Fig. 11 is the localized character of mixing: Pm is indeed smoothly constant at x > 0 in the case A + B. We emphasize once more that the forcing is activated only during one time period T here, the aim of the present investigation being to show the feasibility of the present strategy through numerical experiments. Using multiple interacting structures and continuing the forcing for several periods can further improve mixing.

An estimate of the relative importance of the diffusion can now be given thanks to the periodical character of the imposed excitation. The characteristic diffusion length ‘ over the imposed period C can easily be estimated through the order of magnitude analysis of the time inertial andqdiffusion terms, through o T/ o t / (1/ ffiffiffiffi C Pe) o 2T/ o ‘2. One has ‘ ¼ Pe where it is recalled that the quantities are scaled by Uc and a. That gives roughly ‘  0.3 under the present conditions. The spanwise extend of the mixing zone wherein Pm reaches its maximum is about 4 (Fig. 8). The typical mixing length depicted from Fig. 6 corresponding to yellow–orange zones is also L = 4, that is one order of magnitude larger than ‘. Thus, the Péclet number is large enough to warrant that the stirring induced by forced vortical structures is the main cause of the mixing rather than the molecular diffusion. The reader should note that the strategy presented in this paper differs fundamentally from the shear superposition methodology [22]. Firstly, the perturbations used here are localized in a finite spatial domain. Secondly and more importantly, the aim is to regenerate vortical structures with high mixing capability by a somewhat deterministic scenario. Once the necessary conditions are fulfilled, and the process is triggered, the mixing is enhanced through the genesis of identifiable, coherent-structures with large vorticity. In that sense this method differs entirely from chaotic mixing [23,24]. It is now legitimate to ask why the analysis of higher Pr > 10 fluids could not be achieved in this study. That is because the local strain induced by short vortices stretches the passive scalar field into thin filaments when the diffusivity is small and the DNS resolution becomes inadequate. The process is exactly the same as the vorticity/scalar transport mechanism. Indeed, decomposing the velocity field into a phase averaged mean huii and fluctuating part u0i and the Reynolds averaging procedure results irremediably in the same transport equations as in turbulent flows (see for example [25]). Shortly, the dominant terms in the vorticity transport equation hx0i x0i i are the production by the fluctuating shear rate and the dissipation with hx0i x0i s0ij i  mhx0i;j x0i;j i in dimensional form where s0ij ¼ 1=2ðu0i;j þ u0j;i Þ is the fluctuating shear rate. We use short hand notations here with qi,j = o qi/ o xj, i = 1, 2, 3 referring to the longitudinal, wall normal and spanwise components. The order of magnitude of the major terms are s0i;j / x0i / t=k and x0i;j ¼ t=gk where t is the velocity scale, g is the dissipative smallest scale governing the vorticity gradients and k is an intermediate length scale (corresponding respectively to the Kolmogorov and Taylor scales in fully developed turbulence), between the large scale (channel half width) and g. The equation for the passive sca0 0 0 0 0 lar intensity is similar, namely, hhi hj i  ahhi;j hi;j i, where hi ¼ 0 @T =@xi . Yet the dissipation scales are different when the diffusivity a is different from the viscosity m. By making use of the orders of magnitude that have just been given one obtains gT 0 ¼ Pr1=2 g which is entirely similar to the Batchelor passive scalar microscale

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in turbulent flows. The scalar dissipative length scale is limited by the resolution in the streamwise and spanwise directions with gT 0 ¼ 0:2a. The smallest scale associated with the vortical structures shown in Figs. 2 and 5 is their diameter that is roughly g  0.5a. Combining gives a critical Prandtl number Pr = 6, which is close to Pr = 10 used here. The estimation is satisfactory considering the roughness of the order of magnitude analysis. The way to resolve this problem is to increase the spatial resolution that is computationally heavy, and/or to use multigrid techniques in the neighborhood of imposed perturbations.

can of course be used in practical applications to increase the efficiency of the mixing. The local character of the mixing observed here points to the possibility of it’s active control that can be achieved by monitoring perturbations conveniently distributed in space. Optimization of several parameters to improve the mixing efficiency, such as the intensity and the location of the perturbations, is also necessary. We are actually conducting lattice Boltzmann simulations by simulating real wall jets through streamwise slots in a microchannel to show the physical feasibility of the mixing strategy proposed here. The first results are encouraging, and they will be published elsewhere.

4.3. Technical feasibility of the proposed strategy References Numerical experiments conducted here give encouraging results. The breakup of spanwise symmetry is the key element and the realization of the active process proposed here proves to be technically feasible. Primary perturbations can be generated either by synthetic jets [26], by wall actuators [27] or by the patterned surface charge technique in microsystems [28]. These actuators can subsequently be monitored to control mixing in time and space. Although that constitutes an extreme case, let us consider a microchannel and the use of net zero mass flux jets [26]. The dimensions of the synthetic jet used here are typically d = 4a and d = 2a to engender respectively the structure A and B (Fig. 1). Taking a microchannel flow with a ¼ 200 lm results in d ¼ 800  400 lm and the synthetic jets could be classified as mini rather than micro jets. The centerline jet velocity Uc for microchannel water flow at Re = 10 is U j ¼ 0:05 m=s under these circumstances. The maximum wall normal velocity corresponding to the perturbation parameters eA(t) and eB(t) reaches hv imax ¼ aU c ¼ 6U c during the oscillation cycle where Uc is the microchannel centerline velocity giving hvimax = 0.30 m/s. Thus, the maximum Reynolds number of the synthetic jet corresponding to the case d investigated here is Rej ¼ hv imax ¼ 200: The Strouhal number m

describing how fast a fluid element leaves the orifice region is S ¼ Th2vpi d ¼ 0:6 where T is the imposed period (T ¼ 0:032 s and max

the imposed frequency is f ¼ 31 Hz). It is seen that the Reynolds and Strouhal numbers are respectively large and low enough to ensure high momentum injection into the wall flow [26]. Note that the present investigation deals only with the interaction of a pair of disturbances. It is expected that the use of multiple actuators distributed in a spanwise-staggered manner can result in lesser energy and size requirement of individual perturbations. 5. Conclusion It is shown that staggered perturbations forced periodically in time of the form of streamwise longitudinal vortices interact to regenerate new structures even in low Reynolds number flows. The flow field has some strikingly similar characteristics with fully developed low Re number turbulent wall turbulence, although this is obviously not a fully developed turbulent flow wherein large and small scales can clearly be separated. The structures resulting from the interactive mechanism have a high capacity of mixing. Although the Prandtl number has to be limited to Pr = 10 because of the limited spatial resolution, the dynamic mixing length is an order of magnitude larger than the diffusive one. Thin filaments caused by the straining of vortical structures are observed in the passive scalar field. It has to be emphasized here that only two pairs of longitudinal vortices have been used as initial conditions, to show, on one hand the efficiency of the strategy, and to facilitate the data analysis, on the other. Several staggered protuberances

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