C. Meola& - WIT Press · A. Cavaliere", G. de Felice\ F.M. Denaro\ C. Meola& Napoli, Italy ABSTRACT The experimental flow visualization of mixing phenomena in an array of planar jets

$Page 1: C. Meola& - WIT Press · A. Cavaliere", G. de Felice\ F.M. Denaro\ C. Meola& Napoli, Italy ABSTRACT The experimental flow visualization of mixing phenomena in an array of planar jets$
Eulerian and Lagrangian simulation of

transport phenomena in multiple or

periodical interacting planar jets and wakes

versus experimental results

A. Cavaliere", G. de Felice\ F.M. Denaro\ C. Meola&

Napoli, Italy

ABSTRACT

The experimental flow visualization of mixing phenomena in an array of planarjets has been carried out and compared with the corresponding numerical patternsobtained through flow simulation. To allow a correct evaluation of the results atheoretical analysis relative to Lagrangian as well as Eulerian approaches has beenoutlined. Qualitative comparisons indicate good agreement between experimentaland numerical patterns.

INTRODUCTION

The modeling of mixing phenomena inside unsteady shear flows and wakesis a fundamental prerequisite in order to perform an effective analysis and controlof the mixing itself and of many correlated processes like flames, water or airpollution etc.

The determination of a simple convective mixing even when it is producedby means of a laminar and known unsteady field can result a rather difficult task.Experimental investigations of mixing are mostly based on flow visualizationtechniques. For instance, local marking techniques, like those based on the elasticlight scattering from small particle clouds, smoke etc., as well as those based onthe light emission of suitable contaminants introduced in the main flow are used toproduce streak-lines or time-lines.It is well known how the interpretation of such laboratory flow visualizations isambiguous when oriented to the determination of fluid dynamic quantities of theflow field [1,2]. On the other side such visualization techniques besides beingusefully exploited in a direct way to evaluate some fundamental characteristics ofthe mixing itself, can also be employed to assess the quality of numerical flowcomputations by comparing experimental and numerically simulated patterns.

In this paper the viscous interaction of multiple jets obtained from the outletof an array of contiguous ducts has been experimentally investigated through the

Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

136 Computational Methods and Experimental Measurements

visualization of some flow patterns. At the chosen Reynolds numbers the flowresulted oscillatory but still laminar and in suitable regions could be well describedas a two-dimensional flow.A concurrent numerical simulation of the Navier-Stokes equations (in \|/,£ form)has been carried out by means of a Control Volume (CV) approach previouslydeveloped by the present authors [3].Comparisons between experimental and simulated results are reported anddiscussed also for flow configurations where square section obstacles are introducednear the outlet of the ducts.The numerical simulation of the streak-lines associated with the computed flowhas been performed mainly by means of the Lagrangian tracking of a suitablenumber of discrete particles. Moreover a traking has been performed also by meansof the above mentioned CV Eulerian approach. In this way it has been possible tocompare the discrete (Lagrangian) streak-lines with the continuous (Eulerian) onesobtained by the isolines of a suitable continuous, passive and not diffusive scalarmarker. With reference to such comparison, a theoretical analysis, showing thenumerical correlation between fixed mesh Eulerian methods and the Lagrangianaspects of the tracking problem of purely convected quantities, is presented.

THE STUDIED PROBLEM

A test case of mixing patterns has been chosen in order to evaluate the potentialsoffered by numerical simulation and optical diagnostics in evaluating mixingcharacteristics under the perspective of the theoretical analysis of the problem. Thephoenomenological choice has been guided by feasibility criteria and by thedeterministic behaviour of the involved flows. Therefore the design of the mixingtest was such that both the building of the experimental apparatus and the numericalsimulation were possible and meaningful. In this respect the most important featureswere considered to be the laminar and two-dimensional character of the flow, aswell as the choice of well defined, stable and easy to implement boundaryconditions. A further characteristic, i.e. the spatial periodicity of the flow, whichis quite desirable in a direct numerical simulation of the fluid-dynamic pattern, hasbeen disregarded because of the experimental difficulties in generating a peculiarperiodical mode for confined flows, according to what has been discussed in aprevious paper [4].A schematic representation of the chosen configuration is reported in Fig. 1. Severalparallel adjacent similar plane air jets (32 in number) are injected in a test section"a" through an array of ducts "b" each having the span-wise dimension much smallerof the other two (5 mm). Therefore, the cross section of the ducts has a very largeaspect ratio (20), which is a prerequisite of two-dimensionality. The length of theducts (0.8 m) is much larger than their width so that self-similar velocity profilescan develop. Furthermore the duct widths are chosen so small that, for the employedaverage velocities, relatively low Reynolds numbers are obtained and laminarPoiseuille-type flows can be reasonably supposed to develop. Finally, in order tocontrol the streak-lines roll-up process, one of the ducts (one adjacent to themid-plane) is fed with different average velocity with respect to the other ones.Special care is devoted to assure an even distribution through the remaining ducts.

The experimental characterization of the flow is performed by means of avisualization technique based on the record of 2-D elastic scattering intensity fromvery small particles dispersed in the air of the separately fed jet. The scattered light


Computational Methods and Experimental Measurements 137

is proportional to the particle number concentration inside a very thin pulsed lasersheet. The details of this technique and some comments relative to its specific useare reported in the Appendix A.

a D

Fig. 1 Sketch of the test section

The plan of the experimentation was to yield mixing patterns which could becompared to those obtained by the numerical simulation in a wide range of inletconditions and configuration. Therefore, apart from the variation of the averagevelocity of the seeded and not-seeded jets, an other operative condition, more proneto generate vortical structures, was identified. It consists in placing rods (withcircular or square cross section) along the wider side of each rectangular jet in itsmedian plane inside or outside each of the ducts.

THE NUMERICAL SIMULATION OF MIXING PROCESSES

The numerical simulation of passive mixing phoenomena in fluids is basicallycorrelated with the simulation of the convective and diffusive transport. This canbe done following several approaches that, only schematically, can be classified asLagrangian or Eulerian. Indeed, some methods include features of both Eulerianand Lagrangian nature. A short survey of the approaches and the correspondingnumerical methods is presented in Appendix B.

For what it concerns the algorithms adopted here for the numerical simulation,it was taken into account that also the flow fields of interest must be computed.Indeed, quite in general, when one simulates a real mixing process, theNavier-Stokes (N.S.) equations must be solved as well. Therefore one must alsochoose an approach and the related algorithms for the treatment of the convectivetransport of vorticity or of the momentum components. Such quantities are diffusiveand not passive ones; then, also because of the need of solving some associatedelliptical equations (for the Stream-Function and/or for the pressure), Eulerian ormixed Eulerian-Lagrangian approaches are generally preferred when solving theN.S. for confined flows.For highly convective flows, the required computational stability and accuracy inthe treatment of the convective terms can be hardly obtained by means of Eulerianschemes when they are based on an unphysical direction splitting. In other words,



also for the Eulerian schemes, the treatment of the convection should comply withthe Lagrangian aspects of the convection itself i.e. with properties likeconservativity, directional transportivity and monotony (orpositivity of transportedquantity). Now, for viscous incompressible flows the property of monotony isallowed to be satisfied to a reduced extent so that the use of non-linear schemescan be avoided and linear, high order Eulerian schemes can be employed. One ofsuch schemes developped by the present authors [3], has been adopted in the presentwork for the transport of the vorticity. Indeed the \j/,£ formulation of the N.S.equations was employed to perform the two-dimensional simulation of the flowoccurring inside the experimental test section. As indicated also in [4], suitableboundary conditions were adopted i.e. Poiseuille inflow condition at the endingpart of each duct, no-slip conditions on solid walls and v, = 0as outflow condition,while when simulating flow situations in presence of obstacles special boundaryconditions where assumed for y on the obstacles (see [5]).

However, to produce the flow visualization of a computed flow field and tostudy the mixing effects on a single transported quantity, the plot of discretestreak-lines or time-lines is convenient and largely adopted. Here the discretestreak-lines have been numerically simulated for the various flow situationscomputed; such patterns have been satisfactorily obtained by means of theLagrangian approach also because numerically accurate solutions of the systemsof ordinary differential equations are easily obtained. Some streak-lines werenumerically obtained also by the use of the same CV Eulerian approach employedfor the treatment of vorticity convection allowing a comparison between Eulerianand Lagrangian traking methods.

RESULTS

For the configuration without obstacles an ensemble of laser light scattering patternsrelative to different average velocities of the seeded and not seeded jets are reportedin Fig.2. The experimental conditions of the patterns have been selected with theaim to show a wide typology of mixing structures. Each picture of the figure hasbeen randomly chosen in a video-recorder sequence of frames and it has beenreported just as it appears in the original frame digitized with 16 gray levels.The five pattern on the left are relative to the seeded jet average velocity V^ = 0.4ms"\ while the average velocity of all the other jets V^ are fixed so that, from topto bottom, V^- V^ = AV = 0.25, 0.2, 0.0, -0.1, -0.3 ms ' respectively. The centralof these pictures, relative to the case AV = 0 presents a quite regular pattern withparallel interface in the initial part and mild oscillations developed downstream. Atthe increasing of absolute value of AV spiral structures are observed; the greaterIAVI the nearer to the jet outlet is the inception of the spirals. The only differenceconsists in the sense of the spiral convolution which depends on the sign of AV.The pictures on the right side are relative to V^ = 0.7 ms~* and, for these cases,from top to bottom, AV = 0.55,0.5, 0.3, 0.2 and 0.0 ms"\ Therefore, here only spiralstructures with upstream oriented braids are observed. The higher is AV, the higheris the convolution level which can be achieved. The inception of a vortex pairingcan be detected for the condition at the highest AV. Moreover it can be noticed thatthe enlargement of the seeded jet decreases quite significantly when IAVI decreases.Some of the pictures of Fig.2 have been processed through image enhancementtechniques, so that easier comparisons with numerical results can be done. Namelytreshold clipping, intensity scale expansion, background subtraction and intensityreversing have been used. The details of these techniques are not of interest here,in any case the results of this enhancement technique can be assessed by comparing



V .= 0.4m/sseed

ECNJo

(0

E

EIOCD

OII

X<D

10 cm

Fig. 2 -D Laser light scattering pattern of the median seeded jet (withoutobstacles)



10cm

= 0.7 m/s

= 0.2 m/s

• e

= °5 m/s

Fig. 3 Experimental (left side) versus numerical patterns.



the pictures reported on the left side of Fig.3 with the corresponding ones of Fig.2.The experimental patterns are, from top to bottom, in decreasing order of AV.

On the right side of the same figure the streak-line patterns obtained by thenumerical simulation are reported in the same order. The comparison betweenexperimental and numerical patterns is quite straight. The sense of convolution ofthe spiral structures is well predicted and only the couple of patterns relative to jetswith the same velocities show braids not exactly oriented in the same direction.This can be due to some inaccuracy of the experimental measurement of the airflow rates, since this type of patterns are expected to be very sensitive to velocitydifference variations. The enlargement of the seeded jet is, also quantitatively, wellpredicted, respecting the above mentioned trend as well. Finally the goodcorrespondence, on the seeded jet, of the inception of the experimental andnumerical spiral structures can be observed too. It is also interesting to notice that,in the condition for the largest AV, the aforementioned vortex pairing process seemsto be documented also in the numerical pattern. The spatial frequency distributionand the number of convolutions of the spiral structures seem, at a preliminaryanalysis, in good agreement. However, the assessment of such features requiresfurther morphological analyses which go beyond the scope of the present work.

In order to show the influence of the obstacles on the mixing patterns, cylmdicalobstacles were placed inside (-3 mm) or outside (+7 mm) each duct. In Fig.4 thepictures of some typical experimental patterns obtained with a magnification ratiohigher than that relative to Fig.2 are reported. More specifically they are relativeonly to the first 6 cm after the jet outlets, for V^ = 0.7 ms .In the top row of this figure the four pictures are relative to flow conditions withoutobstacles and are reported for sake of completeness. They can be considered ablow-up of the corresponding ones of Fig.2. Also in this case the formation of thefirst spiral structure is detectable in a clearer way when AV is higher.The pictures of the central row show the 2-D scattering patterns when the obstaclesare placed inside the ducts. Here more convoluted patterns are observed and, atleast for the conditions AV = 0.3 and 0.2 ms"\ the spatial periodicity of the braidsis quite regular. For the case with the same average velocities the seeding particlesare distributed more randomly and tiny filaments originate from the central regionoutwards suggesting a loss of two-dimensionality.The pictures in the bottom row are relative to rods placed outside the ducts. Thespatial periodicity of compact structures is more evident for AV = 0.5 and 0.3 ms" ,whereas, for AV = 0.2 and 0.0 ms"\ filament structures extending toward peripheralregions of the jets appear.The great variety of spiral structures presented in this last figure could be of interestin generating efficient mixing patterns, but this preliminary investigation showsthat the length scales of the structures are too small to be predicted by the numericalsimulation. This is because the computational effort required concurrently todetermine flow fields in non-linearly connected regions with a high degree ofcyclicity [australia] and, at the same time, to resolve the tiny filaments shown inthe most convoluted patterns is out of the possibilities of our workstation. Thereforea further experimental test configuration has been derived from the previous one.The ducts were widened up to 10 mm and obstacles with square cross section and3 mm thicknesss were adopted to have a better fitting with the numerical grid.The visualizations reported in Figs.5 are relative to this type of configuration whenV,^ = 0.2 ms"' and AV = 0.0.For the first one, i.e. that with the obstacles placed outside the ducts, the patternobtained by the numerical simulation show that the streak-lines separate behind therod. They are not equally divided from the obstacle both in respect of their numberand of their extension. Furthermore they become not-identifiable at least in two



without obstacles

obstacles inside

obstacles outside

V = 0.2 m/s 0.4 m/s 0.5 m/s 0.7 m/s

1 cm

Fig.4 2-D Laser light scattering patterns of the median seeded jet.



10 mmi 1

= 0.2 m/s

= 0.2 m/s

Fig. 5 Experimental (left side) versus numerical patterns (with obstacles).

Fig. 6 Numerical patterns.Upper part: Lagrangian approach (streaklines);Lower part: Eulerian approach (isolines).



wide regions, where they clearly separate in dots distributed at low concentration.The comparisons with the flow visualization obtained by the light scatteringmeasurements are more difficult than those relative to the previous set of patterns.In fact in this case the seeded pattern appears less structured than the numerical oneand only the peripheral silhouettes keep some information content. For instance, ifthe sparse dots in the lower part of the numerical pattern are filled with a higherconcentration of dots, the two patterns become more similar, even though in anycase the experimental one seems more symmetric.The final example is relative to a configuration similar to the previous one, but withthe rods placed inside the ducts. Also in this case the streak-lines are separated bythe obstacle but they meet downstream, so that, differently from the previous case,the two peripheral streak-lines appear to be, in the whole pattern, the interfacebetween the marked and not-marked jets.The corresponding experimental pattern develops along the test section withwinding oscillation for which the contiguous interfaces nearly merge and are nolonger distinguishable. Therefore the whole jet silhouette appears as an expandingjet with sinuous interfaces. This type of characterization is also present in theexperimental patterns but, in the last cases the spatial frequencies of the peripheraloscillations is higher and the expansion is rather mild. Therefore, even though aqualitative similarity is still present, it is debatable whether the numerical andexperimental visualization can predict some common behaviours and which theycan be.

In fact, in general and particularly for flows after bluff bodies, great sensitivity(or even instability) of the streak-line patterns to the position where the markersare injected may be well expected [6,7]. It must be said, however, that the seedingtechnique of our experiment differs from other more usual as well as from thenumerical one inasmuch as, to perform a not-intrusive marking for the experiment,the particles have been nearly uniformly dispersed well upstream and at aconcentration well higher than that allowed by the numerical capabilities. Indeed,for this last example the streak-line representation is limited only to the first zonebecause, for the subsequent zone, the dots are so sparse that they can not berecognized in a clear pattern.

Finally, as anticipated above, by means of Fig.6 it is possible to compare twostreak-line patterns generated by two different techniques and relative to one of thefluid-dynamic situation previously reported. The first pattern is obtained following,as usual, the Lagrangian approach, while for the second an Eulerian approach hasbeen adopted. In fact the transport of a suitable scalar, passive and not-diffusivefunction convected by the computed flow field has been numerically simulatedusing the above mentioned CV scheme by adopting a linear distribution as inflowboundary condition. Subsequently a number of equally spaced isolines relative onlyto the interval of values which, in the inflow linear distribution, correspond to theseeded channel have been plotted. The comparison shows that, in order to tracenumerical mixing patterns, Eulerian techniques, though have reached a good degreeof overall accuracy and appear more economical, are a less immediate tool ofanalysis than Lagrangian techniques. In the next section limits for both approachesare outlined.

BASIC LIMITS OF LAGRANGIAN AND EULERIANSIMULATIONS



In order to evaluate in a correct way the numerical and experimental resultspresented, it is important to develop a simple theoretical analysis regarding somegeneral properties of the pure convective mixing (stirring) and the correlated limitsof the particle Lagrangian approximation. Moreover, for sake of completeness, theresults of an already mentioned and dual analyis regarding the limits of the fixedmesh single step time marching Eulerian approach for the simulation of a rigidmotion of a passive scalar will be briefly reported.

Let us consider a general incompressible flow field defined, for sake ofsimplicity, on the whole geometrical space R^ (or R ), and let us suppose that at thetime t=0 a not diffusive quantity is distributed on it according to:

4)(P,0) = 4>o(P)where §<£P) is an integrable (in the sense of Lebesgue) density function defined inR^ (or R ). Let e' and e" be lower and upper bounds of (^ in R^ (or R ). Let usconsider firstly the most common physical case in which e' and e" result to belimited values; then one can always change the reference and the scale of the quantity(j> so that e' = 0 and e" = 1 . In other words, without any loss of generality, we willassume that the range YQ of <J>o is such that:

YoC[0, 1]The Lagrangian formulation of the transport in absence of diffusion easily

furnishes some fundamental properties for the function <p(P,t). Indeed from theequation:

^ = 0Dt

for any t, one obtains the integral:

where P(t) is the vector of the co-ordinates of a material point.Well known consequences of the above integral are the time invariance of thevolume and of the transported mass of any material subregion ofR (orR ). A furthersimple but fundamental consequence is that the range Y(t) of the function <|>(P,t) isalways identical to YQ.

To evaluate the mixing of (j)(P,t) it can be useful to consider the space-averagefunction (or, more in general, the filter):

where |i(V) is the measure of a suitably shaped region (e.g. a sphere) V of diameter8.

It can be shown, by means of simple integral unequalities, that if a' and a"are the lower and upper bounds of ((P,;) for P varying in R^ (or R\ then:

[a'(t,S),a"(t,S)]c[0,l] Vt (1)Of course a'(t, 6) and a"(t, 6) are referred to the same shape of V. During the timeevolution it may happen that a'(t, 8) and a"(t, 5) increase or decrease.If the flow is not so chaotic that measurable material regions of R^ (or R*) aredistorted in not measurable ones, then one can expect that the limits for 5 -» 0 ofa'(t, 5) and a"(t, 5) exist and result time invariant.For a not vanishing 5, the time dependent quantity:

m(t, 8) = log [a"(0, 8) - a'(0, 8)] - log [a"(t, 8) - a'(t, 8)1



can be assumed as a measure of the global mixing produced by the flow in the8-scale average, from the initial condition up to the current time t. Indeed, whenra(t, 8) —> oo, one has that (|)(P,t) is almost constant in R (or in R ). Moreover themixing that is produced by the flow during any time interval (ti, t%) can be simplymeasured as m( 8) - w(tj, 8).If a mixing has to be simulated, it seems fundamental to require that the Eq.(l) bealways verified.For instance the so-called monotony (or positivity) property of the schemes isequivalent to the fact that a'(t) must always be > 0.Now, let us consider which are the consequences of approximating the initialcondition §Q by a combination of Dirac measures or by any operation ofre-concentration. Even though an exact simulation of the dynamics of the particlescan be performed, the Eq.(l) can result no longer verified, at least with respect toits correct original meaning. Indeed, if a combination of Dirac measures have beenused, then the interval [e'o, e' J containing the initial range, becomes no longerlimited; e.g. for positive initial distributions it is ]0, +00[.The guarantee that:

[a'(t),a''(t)]c[e'o,e'y Vt (2)have no (or strongly reduced) meaning. If blobs or large particles are employed,then e'o and e"p are limited but still to verify simply the Eq.(2) will not guaranteethat (j) is less than one. Therefore an unphysical density increase could result forthe averaged quantities: an effect similar to that produced by the presence of anegative diffusion coefficient. This last point is not surprising because the blob orDirac approximation of the initial (^ is intrinsically antidiffusive.

Differently from particle methods Eulerian stable schemes, generally sufferfrom positive artificial diffusion effects. Indeed, first of all when, because of theflow stretching, the distribution of (J), modifies and acquires, for the directionsorthogonal to that of the stretching itself, smaller and smaller wavelengths, theselast ones will be out of the intrinsic limits of representation associated to fixed grids(as well as any spectral) description of modified <(>; this produces the disappearanceof such smaller wavelengths with an effect similar to that of the diffusion. MoreoverEulerian methods are diffusive also in absence of flow stretching. Therefore theyare checked by means of classical test cases, as the one early proposed by Crowley[8], concerning situations in which it is known that no mixing is produced, becausesuch test cases evidentiate any spurious diffusion. The difficulties of simulating,with a fixed rectangular Eulerian mesh, the convective transport in a rigid rotatingflow field appear somehow astonishing, but they can be fully understood if analysedin Lagrangian terms.For sake of simplicity we limit ourselves to consider a circular region concentricwith the center of rotation. Let us suppose that the scalar, not diffusive quantity (j)is distributed on it at the time t=0 can be described by the following Ritzrepresentation:

<KP,0)= Zy,(0) <b(P)i = 1

where the (P) are suitable shape functions, and y,(0) are the values of 0 on nodalpositions; in general, no re-concentration effects are present as it happens in therepresentation of the particle approaches. Let us also assume that one wants todescribe the time evolution by the approximation:

4)(P,t)= ly,(t) 4),(P)J = 1



such approximation is typical of the Eulerian approach with fixed mesh or by finiteelement methods.The time evolution of y depends only on the initial conditions and anytime-marching, linear, steady algorithm, either implicit or explicit, modelling theproblem can be expressed by a homogeneous linear relationship:

y"+'= Ry"the matrix R, for a fixed numerical scheme, depends upon physical and meshparameters. Let us assume that the time step At is such that T/At is equal to a positiveinteger p, T being the period of revolution; furthermore, for sake of simplicity, weconsider in the following only the cases when the rotation centre is not situated atany nodal position of the grid.It was demonstrated [3,9] that if and only if the algorithm is such that the associatedmatrix R, by means of a suitable reordering, can be put in a (k x k) block diagonalform, each block being a (p x p) p-cyclic submatrix, then it satisfies the followingproperties that are generally checked in this test case:a) Any constant distribution must be time-invariantb) Any positive distribution must remain positive. It can be shown that this isequivalent to the non negativity of coefficientsc) All the non axisymmetrical distribution are invariant after (and only after) atime equal to an integer number m of periods.

This entails also that p must be an integer divisor of the number N of nodalpoints of the mesh (i.e. N = k p). Then in the mentioned work the direct connectionbetween the Lagrangian character of the problem, the structure of the matrix R andthe chosen grid was shown. In fact every component of y", associated with a specificnodal point, must change with n so that it simulates the motion along a circulartrajectory; some of the trajectories may be identical, but each must be constitutedby p arcs. If we denote by s the maximum number of nodal point along a radius,then, since each of these nodal values should trace a p-arcs circular trajectory, thenumber k p should be greater or equal to s p. This shows that only a polar meshallows to determine a matrix R satisfying the properties a-c, while it is clear thatrectangular meshes are completely inadequate to the modeling of the rigid rotationtest case and that no benefit can be obtained by adopting finer but similarly structuredmeshes. Then on a rectangular grid the general property c) cannot be verified andan artificial diffusion will modify the distribution of <j> during its rotation. As aconclusion, in Eulerian approaches, when the physical diffusivity does not vanishand is sufficiently high, it is possible to adopt, at a fairly reduced computationalcost, a mesh size small enough to obtain a cell ScRe number such that the positiveartificial diffusivity does no*t affect the accuracy of the computation. Viceversa,when the transport is highly or purely convective and the flow field is not a constantone, since it is impossible to obtain everywhere a unit Courant number, positiveartificial diffusion overcomes the physical one and poor accuracy solutions areproduced unless special care is devoted to a vectorially correct convective transportof the shapes as it was shown in [3].

CONCLUSIONS

After Galileo, to give contributions to natural science means simply to fix thepossible correlations between the predictions of a theory or of a model and theresults of suitable experiments. From the obtained results structural similarities anddiscrepancies between experimental and simulated mixing patterns areevidentiated, however, even for simple laminar 2-D mixing processes, we are farfrom the possibility of performing also the quantitative comparisons we would like.



The reasons why the comparisons must be confined only to the morphological andqualitative aspects are the inherent limits relative to the following items:

Experimental set-up and flow visualizationCorrect boundary conditions flow modelling and numerical simulationTime sincronization between the experiment and the numerical simulation of

unsteady flow fields in oscillatory conditionsImplementation of space-time locally comparable numerical and experimental

mixing pattern tracersThe first two items were and are object of broad interest and continuous

research everywhere and some contributions have been presented also here;vice versa less care, at least here, has been devoted to the subsequent items.The authors wish to be able to develop further researches in these directions.Finally for taxonomical reasons it is also necessary to start from simple basic mixingtheories and fix the main global parameters to be studied in the processes of practicalinterest. In this way not only some of the insurmountable difficulties associated tothe above items can be, if not overcome, at least avoided but also a rationale canbe given to the global experimental and numerical effort.

APPENDIX A: THE TEST SET-UP AND DIAGNOSTICS

The test section of the set-up is a rectangular duct built with plexiglas to have opticalaccess to all the four sides. It is fed at its inflow section by several adjacent air jetseach coming from a smaller separate rectangular duct. The two outermost ductswere not fed for the test cases reported in this paper. In fact it was observed that,when fed, a more active and instable roll-up of the flow occurred.The separately fed duct is also seeded by submicronic particles of TiC . They arepreviously mixed with isopropilic alcohol by means of a magnetic stirring device,so that a slurry with uniform concentration is formed. This mixture is atomized byan air assisted nebulizer and the alcohol evaporates along the guiding vane; thendry particles are uniformly dispersed in the air before reaching the test section. Theparticles are so small (0.1 - 0.3 jum) that the Stokes number (the ratio of the particleaerodynamic response time to the characteristic time of the flow field) is also verysmall (following the formula of Chung et al. f 10] within the order of 10 ); therefore,the inertia force being negligible, they practically move with the fluid convectedby the aeroynamic drag force. This means that the behaviour of the particle cloudsis similar to that of a contaminant with a Schmidt number (Sc = v/D) very large.In other words the particle diffusivity D can be considered much smaller than themomentum diffusivity. This is an unique feature for the characterization of pureconvection of a contaminant in gaseous flow, because the molecular mixing of anygaseous species cannot be avoided unlike what occurs in liquid systems for whichimmiscible species with negligible surface tension can be selected in order to followthe pure convection.The particle cloud is illuminated by a pulsed (10 nsec) laser sheet which is shapedby a set of cylindrical lenses. The height of the laser sheet can be varied accordingto the control volume required in the different test cases, whereas the depth is fixedwithin the order of some hundreds of microns (~ 200 Jim). For all the measurementsthe second harmonic of Nd:Yag pulse laser (k = 532 nm) is used and the beamcrosses the central part of the measure section. The scattered light is collected ona CCD camera by means of an objective lens system equipped with an interferencefilter which rejects the light outside a narrow spectral range (~ 10 nm) centered atthe laser wavelength. The scattered light intensity is proportional to the particlenumber concentration because the particles have the same sizes and therefore



approximately scatter the same light intensity. The particle intensity is chosen sohigh that the region of seeded flow appears on the image as a continuous area sincethe CCD resolution (as it will be discussed later) is lower than inter-particle distance.The images are recordered on a videotape and eventually printed on a hard copysystem. 2The CCD camera is an imaging array with total sensitive area of 8x6 mm andpixel dimension of 16 x 11 Jim . According to different magnification ratios, thesingle pixel collects light from a parallelepipedic (square) volume (objective pixels)of variable height and width (according to the required magnification ratio) whereasthe depth is fixed by the laser depth. For instance the image from the widest objectivepixel reported in this paper is relative to a test section with height of 22 cm,corresponding to a single discretized volume with height of 340 |im. The finitenessof this volume entails that, even assuming uniform particle concentration and theiruniform illumination, the recordered scattered light can be not-uniform. In fact theobjective pixel, in which one or more interfaces of the seeded-not-seeded flowregions are present, is only partially filled with the seeded fuid flow and thereforethe particle number is smaller than that relative to pixels which are placed far fromthe interfaces. This smearing effect must be taken into account when comparingnumerical and experimental results.

APPENDIX B :LAGRANGIAN AND EULERIAN APPROACHES

In the short survey of the methods we are going to present, we renounce to giveany literature reference because it would be in any case oversized but not exaustiveand perhaps not correct for what it concerns the priority of the ideas.

Typical Lagrangian methods are the particles methods where the timedependent quantity is approximated at each time by a linear combination of Diracmeasures in the space. Purely convective mixing processes produced by unsteadyflows can be simulated by tracking the positions of a discrete number of Diracmeasures of invariant intensity. In this way the simulated mixing results to bemass-conservative. The time dependent co-ordinates of a discrete number ofpoint-like particles associated with the Dirac measures, can be computed by solvingthe corresponding set of systems of first order ordinary differential equations. Inmost cases one is interested in the transport of quantities described by quite generalfunctions, then the problem of approximating any initial distribution by a linearcombination of Dirac measures arises. For this and other reasons (some of whichwill be illustrated later) in other methods the Dirac particles have been substitutedby blobs or suitable shape functions which assume not vanishing, limited valueson particles of small but not infinitesimal area.In order to discretize the initial condition both the above methods require a sort ofre-concentration of the transported quantities.

Alternatively one can introduce a Lagrangian mesh (or a Lagrangian finiteelement distribution) moving with the fluid. This corresponds to consider a set oflarge particles each constituted by a cell (or a finite element) of the mesh. Suchdiscretization does not require any approximation producing re-concentrationeffects as it happens with particles methods. Though, for incompressible flows andnot diffusive quantities the mass and the volume of each material cell remainstime-invariant, the dynamic of the cell itself is much more complicated of that ofa particle because of the stretching and distortion. Then such a dynamic cannot belimited to the description of the position of the baricentral point of the cell. A timeevolution of the material boundaries of each cell must be computed but, becauseof the too high level of distortion and of the computational cost, it is necessary after



some time to resort to a re-definition or re-zoning of the mesh. Such operation is acritical one and generally is source of errors of diffusive character.

The diffusive character of the error results to be also the main problem of fixedmesh or fixed finite element methods. These are typical in the approaches basedon the Eulerian formulation of the balance, so that fixed mesh methods are generallyclassified as Eulerian. Eulerian approaches have many advantages and, for highlydiffusive transports, can produce very accurate results with a reduced computationaleffort. On the contrary their use in the simulation of purely or highly convectivetransport can be critical as it is evident from the ever increasing number of proposedalgorithms. An indicator of such difficulties is the great deal of reported results fora simple test case like the popular rotating spot. The reasons of such difficulties,have been briefly reported in a previous section.

Finally further approaches, that can be considered of intermediate character,have been proposed and developed. Such are those where the mesh moves but notin Lagrangian ways and without strong distortions. The change of the mesh can bealso due to the needs of adapting the mesh to the best representation of the solutionor to the minimization of the residual of the equations. Of course the choice ofadaptive grids can be also correlated to the stretching produced by the flow field;in this case such an approach could be also considered as a Lagrangian one.

ACKNOWLEDGMENTS

This work has been partly sponsored by MURST funds

REFERENCES

1. I. Gursul, D. Lusseyran and D. Rockwell 'On interpretation of flowvisualization of unsteady shear flows' Experiments in Fluids, Vol.9, pp. 257-266,19902. I. Gursul and D. Rockwell 'Effect of concentration of vortices on streaklinepatterns' Experiments in Fluids, Vol.10, pp. 294-296, 1991, Technical note3. G. de Felice, F.M. Denaro, C. Meola 'Multidimensional Single Step VectorUp winded Schemes for Highly Convective Transport Problems' To appear onNumerical Heat Transfer, 19934. A. Cavaliere, G. de Felice, F.M. Denaro, C. Meola, R. Ragucci, G. Vanacore,C. Venitozzi 'Mixing Characteristics of Quasi Two-Dimensional Periodic Seriesof Jets' Int. Symp. on Spatio-Temp. Struc. and Chaos in Heat and Mass Transf.Pmc&Mg,s, Athens 1992.5. G. de Felice, F. M. Denaro, C. Meola, Stream-Function based multiple BluffBodies 2D flow analysis, Second Int. Coll. on Bluff Body Aerodynamics andApplications, Melbourne, Australia, Dec., 19926. J.M. Cimbala, H.M. Nagib, A. Roshko, 'Wake instability leading to new largescale structures downstream of bluff bodies', Bull. Am. Phys. Soc. 26, 1256, 1981.7. J.M. Cimbala, H.M. Nagib, A. Roshko, 'Large structures in the far wakes oftwo-dimensional bluff bodies',./. F/wWMfdz. 190, 265-298, 1988.8. W. P. Crowley, 'Numerical Advection Experiments', Mon. Weath. Rev., vol.96, pp. 1-11, 1968.9. G. de Felice, F.M. Denaro, C. Meola 'A general approach to upwind schemesfor regular solutions' D.E.TE.C. Internal Report, Nov, 199210. Chung J.N., Troutt T.R., Crowe C.T. 'Modelling of Droplet-Gas Interactionin a Spray Combustion Engine' Int. Symp. COMODIA 90JSME, pp. 601-605,1990


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C. Meola& - WIT Press · A. Cavaliere", G. de Felice\ F.M. Denaro\ C. Meola& Napoli, Italy ABSTRACT The experimental flow visualization of mixing phenomena in an array of planar jets