EXPERIMENTAL STUDY OF FINE TURBULENT STRUCTURESIN A CONFINED JET FLOW

N. Kornev, V. Zhdanov and E. Hassel

University of Rostock, Germanynikolai.kornev@uni-rostock.de

1 IntroductionThe aim of this work is to analyse experimental PIVdata obtained by Valery Zhdanov which could sup-port or refutie some new concepts of modelling of finestructures in turbulent flows. In Kornev et al (2013) weproposed to simulate large scale motions on the gridwhereas the small scale ones using the grid free com-putational vortex method. The idea is based on theassumption that the turbulent flow at large Re num-bers consists of strong fine axisymmetric vortex struc-tures which are distributed intermittently while largevortices appear as clusters of fine ones. The method isefficient if the number of strong vortices is restricted.Within this method the fine structures are resolved di-rectly. The next concept to be proved is based on theuse of some universal functions like multiplier distri-butions which can be utilized for artificial synthesisof subgrid motions without their direct resolution (s.Burton et al (2002)).

Figure 1: Sketch of the flow. 1- knee bend of nozzle, 2- platefor damping of vortices shed from knee bend 1,3- outer tube, 4- support plates, 5- nozzle, 6- testsection, 7- water box.

2 Experimental setupThe flow is the turbulent axisymmetric jet developingin a coflow confined by a pipe of diameterD = 50mmand length 5000mm. A schematic of this flow systemis given in Fig. 1. Medium in both flows is water. Theinner tube had diameter d = 10mm and the length600mm. The test section was installed in a Perspexrectangular box filled with water to reduce refractioneffects. More detailed information about the hydrody-namic channel can be found in Zhdanov et al(2008).The ratio of the co-flow flow rate to that of the jet is

Figure 2: Mouthpieces of nozzles used in measurements: a)without tabs, b) with two plate tabs, c) with fourplate tabs and d) with four triangular tabs.

equal to 5.0 which corresponds to the jet -mode (Ko-rnev et al (2008)). Since the Reynolds number basedon the jet exit velocity Ud is Red = dUd/ν = 104

the jet can be considered as a fully- developed turbu-lent jet. PIV measurements were performed within thewindow 3.232mm×2.407mm with the pixel distanceof ∆ = 68.8µm. The laser thickness estimated as∼ 40µm is very thin.The measurement window was located on the center-line of the jet mixer at the distance within the rangefrom x/D = 1 to 9 measured from the nozzle. Timeresolution was limited by the 10 Hz frequency of laserused as an external trigger of the camera.

The issued jet conditions were changed by installa-tion of the mouthpieces with vortex generators (tabs)at the nozzle exit (Fig. 2). Two kinds of tabs weremanufactured: the square plate tabs with size of h =1.5 · 10−3m and the triangular tabs with the angle atthe apex of 900 and height of h = 1.3 · 10−3m. Inmeasurements four mouthpieces were used: the refer-ence one (without tabs) (Fig. 2a), with two (Fig. 2b)and four (Fig. 2c) plate tabs, and with four triangulartabs (Fig. 2d). Due to tabs, the jet exit cross sectionwas reduced by 5%, 11% and 8% in comparison withthe cross section of the reference case.

3 Data processingData filtering. The primary velocity field was ob-tained using Dantec processing programs. The vortic-ity component ωz =

∂uy

∂x −∂ux

∂y and two dimensional

dissipation rate ε = ν( ∂ui

∂xj− ∂uj

∂xi) ∂ui

∂xjwere calculated

using own processing code with the central differen-tial scheme (CDS). Due to limited concentration and

polydispersity of particles, the PIV processing proce-dure is not able to resolve the velocity field at all mea-surement points and delivers the data with three status: 0-valid, 1-replaced, 16-rejected. Relibale derivativecalculation requires neighbouring points with the sta-tus 0. The computational molecule of the CDS schemecontains only four points, whereas the Sobel operator,which contains a certain smoothing and therefore isoften used for calculation of derivatives based on ex-perimental data, involves nine points. Therefore, theprobability of the proper derivative calculations usingonly valid velocities is much higher for the CDS oper-ator than for the Sobel one.

To decrease the number of invalid points (replacedand these with the status 16) we reduced the resolutionto 49 × 36 points. The simple arithmetic smoothingamong neighbours with the valid velocities is imple-mented to exclude remaining invalid points with thestatus 16:

uij =

i+1∑k=i−1

j+1∑n=j−1

δknukn/

i+1∑k=i−1

j+1∑n=j−1

δkn (1)

where uij is a velocity component at point ij andδkn = 1 if the status is zero and δkn = 0 otherwise.The analysis performed for the case x/D = 1 andRe = 104 shows that this replacement has no influ-ence on the vorticity field. Additionally we introducedthe filtering according to formula

uij =1

(2N + 1)2

i+N∑k=i−N

j+N∑n=j−N

ukn (2)

where 2N∆ is the filter width.Numerical determination of derivatives ∂u

∂x =u(x+∆x)−u(x)

∆x is a challenge on small scales (s. Wanget al(2007)). If ∆x → 0 the velocity u(x + ∆x)doesn’t tend to u(x) because of measurement errorswhich are commonly uncorrelated at adjacent points.As a result the derivative can become infinite in thelimit ∆x → 0. Therefore the results obtained furtherfor ωz and ε should be interpreted with a care. Theproblem is illustrated in Fig. 3 where the full threedimensional dissipation rate

ε = 6ν[(∂ux∂x

)2 + (∂ux∂y

)2 +∂ux∂y

∂uy∂x

]

calculated without and with 4 plate tabs are repre-sented depending on filter semi-width N∆. Since thelast fromula is valid only for the isotropic turbulence(Hinze (1975)), it was used for large distance from thenozzle at x/D = 9 where the isotropy assumption ismore plausible than at small x/D. Strong increase ofε at N → 0 can be due to reasons of either physical(resolution of small scales) or mathematical natures(experimental noise), which unfortunately can not beseparated. To estimate the plausibility of derivative

Figure 3: Influence of smoothing parameter N on 3D dissi-pation rate calculated using isotropy hypothesis atx/D = 9.0 without and with 4 plate tabes.

calculations, the Taylor microscale λϕ of the isotropicturbulence was determined from the formula

λ2ϕ = 30ν

u′2

ε

and compared with rough estimations λϕ ∼ 6...10η,where the Kolmogorov scale η is around ∼ 225µmaccording to Kornev et al(2008) , i.e. λϕ ∼1350µm...2250µm. The Taylor microscale calculatedfrom isotropic estimations is then varied between ≈500µm at N = 0 and ≈ 1900µm at N = 4. Obvi-ously the dissipation rate is strongly overestimated ifthe experimental data are not filtered (N = 0). Tak-ing these results into account, the dissipation rate pre-sented further was filtered with N = 4. In this casethe effect of scales less than ∼ 500µm is smoothed.Identification of structures. Identification ofstructures is a big challenge because the resolution∆ = 68.8µm we attained is not sufficient and noiseof experimental data is rather high in comparisonwith the real signal The vortices are identified fromthe analysis of the ω2

z field which is performedonly for the values larger than a certain thresholdω2z > ω2

z,tr. Due to inclination of vortex structuresto the measurement plane, reliable conclusions ofqualitative character can be drawn only in the limitwhen ω2

z,tr increases. It is supposed that in thislimit the vortices which are perpendicular to themeasurement plane are distinguished from others.

After cutting ω2z =

ω2z if ω2

z > ω2z,tr,

0 if ω2z < ω2

z,tr.the field

ω2z > ω2

z,tr is approximated. Then the identificationalgorithm is organized as follows:i) The point with the maximum ω2

z is found andassigned as the vortex center. The vortex structureis defined as the vicinity of the center where thecondition ω2

z > 0 is fullfilled.ii)K diametral lines are drawn through the center

at different angles αk = (k − 1)∆α with the con-stant angle increment ∆α = 2π/K. Axis lengthslk, k = 1,K are calculated along each diametral line.iii)Area of the structure A, its averaged radiusR =

√A/π, the minimum axis lj,min = minlk

and the maximum axis lj,max = maxlk, where j isthe number of the structure, are determined.iv) For vortex structures with lj,min > 3∆ the prob-ability density function of ratios lj,min/lj,maxand lj,min/∆ is computed. The choicelj,min > 3∆ = 204µm is dictated by the wishto exclude small vortices with lj,min ∼ ∆ which formand size can not be determined reliably.v) In the vicinity of r ≤ R all ω2

z are set to zero.vi) Go to the step i)

4 ResultsAnalysis of structures. Probability density func-

Figure 4: Probability density function of the ratio of mini-mum axis to maximum one. x/D = 3.0, Re =104, N = 1.

tions of ratios lj,min/lj,max and lj,min/∆ calculatedwith K = 10 are shown in Fig. 4 and 5. The ra-tio lj,min/lj,max shows the deviation of the structurefrom the circular form lj,min/lj,max = 1. It is ex-tremely difficult to confirm the circularity when sizeof the vortex is of a view cell sizes. Indeed, if a smallideal circular vortex is projected onto the grid obtainedfrom PIV processing program, the resulting figure israther ellipse than the circle because the vortex cen-tre doesn’t coincide with the grid nodes and the in-terpolation among the grid points disfigures the realstructure form. To get any proper conclusion, we anal-yse the tendency of p.d.f. for three increasing thresh-olds ω2

z,tr: 2.2ω2z,mean, 10ω2

z,mean and 15ω2z,mean,

where ω2z,mean is the mean enstrophy averaged both

over the window and in time. Sawtooth p.d.f. curvesin Figs. 4 and 5 are due to identification of small struc-

tures on the discrete PIV grid. When ω2z,tr grows,

the p.d.f of the ratio lj,min/lj,max is clearly shiftedto the right. This illustrates the tendency to circular-ity lj,min/lj,max → 1 for strong vortex structures,which are supposedly perpendicular to the measure-ment plane. Therefore, fine strong vortices of the tur-bulent flow at large Reynolds numbers can approxi-mately be modelled as axisymmetric ones. Fig. 5reveals that the most probable size of the structuresis around 4∆ = 272µm what is about of five Kol-mogorov scales, i.e. lj,min ∼ 5η. The p.d.f. distribu-tion for high Reynolds number 10000 correspondingto the full developed turbulence is similar to that forRed = 3000 which is a little higher than the transitionRe number. However, the structure of fine vortices inweak turbulence is similar to that in the full developedturbulence.

Figure 5: Probability density function of the ratio of mini-mum axis to cell size at two Re numbers. x/D =3.0. Structures were determined using threshold10ω2

z,mean.

Probability of lj,min/∆ ≥ 5 is almost zero, i.e.the presence of big strong structures is almost im-probable. The same conclusion can be drawn fromthe analysis of vorticity or ω2

z snapshots (Fig. 6).We revealed only structures which are much smallerthan the integral length. The vorticity field consists offine strong structures with sizes of a few Kolmogorovscales (η ∼ 0.05mm at x/D = 3) close to the Taylorscale (λϕ ∼ 0.3mm at x/D = 3) as well as of clustersof these structures. The distribution of strong vorticityis very uneven. These conclusions are valid both fororiginal (N=0) and smoothed (N=1, 3) fields.Multifractal properties of the turbulent velocityflow. Novikov (1971) introduced the multiplier distri-bution which implies the multifractal nature of the tur-bulence. Let us consider a positive stochastic functionof the turbulent flow y(x) and the fractional coefficientqr,l defined by the condition

qr,l(h, x) = yr(x1)/yl(x), (3)

Figure 6: Influence of smoothing parameter N on the vorticity (upper line) and vorticity squared ω2z (lower line) fields at

x/D = 1.0 and Re = 104.

where

yl(x) =1

l

∫ x+l/2

x−l/2y(x1)dx1,

−0.5 ≤ h =x1 − xl − r

≤ 0.5, r < l.

It is assumed that the function y(x) is locally homo-geneous and isotropic at scales smaller than a certainintegral length scale Lu and there exists the secondcharacteristic scale l∗ determined either by the viscos-ity or by the diffusion. If in the range between l∗ andLu there are no other characteristic scales influencingthe stochastic function y(x), then the probability den-sity function of the fractional coefficient qr,l dependsonly on the ratio l/r and |h| provided l and r satisfy thecondition L l > r l∗. A very strong assumptionof the Novikov’s theory is the statistical independencyof the consecutive fractional coefficients

qr,ρ(h, x+ h(l − ρ))qρ,l(h, x) = 0, (4)

where r < ρ < l at scales between l∗ and Lu.The scale similarity postulated by Novikov implies themultifractal nature of the turbulence closely related tothe cascade concept (Jimenez (2000)).

The Novikov’s concept has then been used in aseries of works on multifractals in turbulence (seeChabra and Sreenivasan (1992)) for dissipation ratesof velocity and scalar fields. Recently Burton (Burtonet al (2002)) developed LES SGS models utilizing themultifractal properties of the turbulence assuming thatthe enstrophy possesses multifractal properties. Weexamine the multiplier distributions for the two dimen-sional dissipation rate ε. The usual procedure which isalso applied in the present paper is the consideration ofa hierarchy of parents and children boxes of uniform

size. Each parent box with the size p is subdivided intoa number of children boxes, κ with sizes p1 = p/κ,and the ratios of the dissipation rates in the originalbox to those in the children sub-boxes are computed.A histogram of these ratios is then P (M). Accordingto the Novikov’s concept the shape of the distributionP (M) should remain invariant for the inertial rangeof scales. This fact is used for the development of theLES SGS models (Burton et al (2002)).

To obtain P (M) from measured data the experi-mental window is successively subdivided into squareboxes of equal size each of which is then bisected intotwo children boxes (a simple binomial cascade). Theratio

M =

∫child

ε(s)ds/

∫parent

ε(s)ds, (5)

computed for every couple of equal size p, contributesto the total probability density function P (M).

Like in our previous study for the scalar dissipa-tion (Kornev et al (2008)) the multiplier distributionsP (M) of the velocity dissipation rate computed forboxes of different sizes 227 < p < 2200 µm areproved to be dependent on the box size p (see Fig.7). There takes place a flattening of curves when pbecomes less, i.e. a irregular distribution of ε becomesmore and more probable when the size of the parentbox gets less. Although the results presented in Fig.7 confirm the intermittent character of the dissipationrate distribution, the difference in multiplier distribu-tions P (M) computed for different p calls the ideaof multifractal theory of turbulence into question. InKornev et al (2008) we mentioned the blocking effect(Jimenez (2000)) as a possible reason why the mul-tiplier distributions P (M) are not invariant and the

Figure 7: Multiplier distribution P (M) for different boxsizes at x/D = 9.

Novikov’s idea does not work. Due to the blocking ef-fect the main assumption of the Novikov’s theory (4)is not valid.

Alternative universal distribution functions forvorticity squared or dissipation. As an alternative

Figure 8: Parameter of the distribution intermittency εk ver-sust k/M for different smoothing parameter N .The diagonal line corresponds to the homogeneousdistribution of ω2

z .

to multiplier distribution we introduce the followingmeasure of the intermittency for ω2

z distribution. First,the measurement cells are sorted in order of descendof ω2

z , i.e. the first cell has the maximum value of ω2z

and the last one with the number M has the minimumvalue close to zero. The ratio

εk =

k∑i=1

ω2zi/

M∑i=1

ω2zi (6)

shows the contribution of k cells to the total amount

Figure 9: Parameter of the distribution intermittency εk ver-sust k/M at different cross sections along the jetmixer (upper, Re = 104) and Reynolds numbers(lower).

of enstrophy within the measurement window Ω2 =M∑i=1

ω2zi. Despite of strong smoothing seen in Fig. 6

the ratio (6) is almost the same for N ranging fromzero to three (s. Fig. 8). Obviously, the enstrophy isdistributed unevenly regardless of smoothing. Twentypercent of the cells contribute almost eighty percent ofthe total enstrophy. With the other words, only a smallfraction of vortex structures contributes to the total en-strophy. Surprisingly, the curve εk(k/M) is the samefor all x/D and Re numbers investigated in this paper(s. Fig.9). The ratio εk is also proved to be relativelynon sensible to the size of the measurement window (s.Fig.10) what is not observed for the multiplier distribu-tion. The curve εk(k/M) seems to be if not universalthen the most universal parameter characterizing theintermittent distribution of the enstrophy.Influence of initial condition on turbulence in theconfined jet. Initial conditions are manipulatedby mouthpieces (s. Fig. 2). Distributions ofReynolds stresses Ruxux

and Ruyuyand their ratio

Ruxux/Ruyuy

is shown in Fig. 11. The anisotropyRuxux

6= Ruyuyis very strong within the initial range

and remains up to x/D = 9.0. Downstream, the stressratio tends to the value of a round self similar jet (Hus-

Figure 10: Influence of the measurement window size onthe parameter of the distribution intermittency εkversust k/M .

Figure 11: Reynolds stressesRuxux andRuyuy (upper) andtheir ratio (lower) along the jet axis at differenttabs number.

Figure 12: Two dimensional dissipation rate along the jetaxis at different tabs number.

sein et al (1994)) (dashed line). The Reynolds stressescalculated in spirit of Large Eddy Simulation (LES)RLESii = (ui − ui)2 (referred to as the LES Reynoldsstress), where ui is the velocity averaged over the mea-surement window, shows clear tendency to isotropyRuxux

= Ruyuyvery quickly behind the nozzle at

x/D > 3. Locally the confined jet flow becomesisotrop. Two dimensional dissipation rate is shownin Fig. 12. The stresses and the dissipation rate de-pend strongly on the initial conditions only within therange 0 < x/D < 3. Downstream the differencebetween curves is comparable with the accuracy ofmeasurements and their processing procedures. Obvi-ously, the fine structure of the turbulent field becomesvery quickly (after x/D = 3 or x/d = 15) universalregardless of initial conditions. The Reynolds stressRuxux

behind the nozzle with two tabs is sufficientlyhigher than this with four tabs whereas the dissipa-tion rate shows the inverse trend. Apart from this,the difference for the dissipation rates with two andfour tabes is sufficiently smaller than that for Reynoldsstresses. This phenomenon can be explained if we re-member that the Reynolds stresses reflects the largescale velocity field inhomogeneity wheres the dissipa-tion rate shows the instantaneous spatial inhomogene-ity. Two tabs nozzle produces four large scale longi-tudinal vortices whereas four tabs one eight vorticies.The configuration with larger number of vortices isgetting faster unstable producing small scale vorticeswhich presence is illustrated in Fig. 13. This results inthe dissipation rate increase which is generally slightlyhigher than in case of two tabs.

Probability density function of the dissipation rate(Fig. 14) reveals three interesting facts:

Figure 14: Probability density function of the two dimen-sional dissipation. Black- x/D = 1, red- x/D =9.

• P.d.f. at small x/D depends strongly on the initialconditions. The p.d.f for the case without tabs differsufficiently from these with tabs at x/D = 1. Thedifference between two and four tabs is negligible.• At small x/D the maximum of p.d.f with tabs is at-tained not at ε = 0 at x/D = 1. This might be becausethe velocity field is tightly filled with vortices withoutgaps in contrast to the case without vortex generatingtabs. Due to this fact the prabability of zero dissipationrate is reduced.• Far from the nozzle at x/D = 9 the p.d.f distri-butions for all three cases are close each to other. Astrong deviation is observed only at large ε. Largedissipation is more frequent in case of four tabs. Thisphenomenon needs further investigations.

5 ConclusionsRegardless of initial conditions which were suffi-ciently different the following features of flow struc-tures can be documented:• Fine strong vortices of the turbulent flow at largeReynolds numbers are approximately axisymmetricand distributed very unevenly in space.• Only a small fraction of fine vortex structures con-tributes to the total enstrophy.• Existence of large strong vortices was not observed.

These facts confirm plausibility of the basic conceptof the method proposed in Kornev et al (2013).

• The most universal parameter characterizing the in-termittent distribution of the enstrophy is the ratioεk(k/M) (6).• The concept of the universal distribution of the mul-tiplier distribution for enstrophy has not been sup-ported in our measurements.

AcknowledgmentsThe study was supported by the German Research

Society (Deutsche Forschungsgemeinschaft) withinthe project HA 2226/14-1 .

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Figure 13: Vorticity fields ωz at x/D = 1 and different initial conditions.