Survival distribution of the stretching and tilting of vortical structures in isotropic turbulence. Anisotropic filtering analysis
UUnder revision for submission to PHYSICA D
Survival distribution of the stretching and tilting of vortical structures in isotropicturbulence. Anisotropic filtering analysis.
Daniela Tordella (cid:92) , ∗ Luca Sitzia (cid:93) , and Silvio Di Savino (cid:92) (cid:92)
Dipartimento di Ingegneria Meccanica e Aerospaziale,Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (cid:93)
Dottorato in Statistica e Matematica Applicata,Scuola di Dottorato ”V.Pareto” - Universita’ di Torino,via Real Collegio 30, 10024 Moncalieri (To), Italy (Dated: April 1, 2019)—————————————————————————————————————————–Using a Navier-Stokes isotropic turbulent field numerically simulated in a box with a discretization of1024 (Biferale L. et al. Physics of Fluids , (2), 021701/1-4 (2005)), we show that the probability ofhaving a stretching-tilting larger than twice the local enstrophy is negligible. By using an anisotropickind of filter in the Fourier space, where wavenumbers that have at least one component below athreshold or inside a range are removed, we analyze these survival statistics when the large, thesmall inertial or the small inertial and dissipation scales are filtered out. It can be observed that, inthe unfiltered isotropic field, the probability of the ratio ( | ω · ∇ U | / | ω | ) being higher than a giventhreshold is higher than in the fields where the large scales were filtered out. At the same time, itis lower than in the fields were the small inertial and dissipation range of scales is filtered out. Thisis basically due to the suppression of compact structures in the ranges that have been filtered indifferent ways. The partial removal of the background of filaments and sheets does not have a firstorder effect on these statistics. These results are discussed in the light of a hypothesized relationbetween vortical filaments, sheets and blobs in physical space and in Fourier space. The study infact can be viewed as a kind of test for this idea and tries to highlight its limits. We conclude thata qualitative relation in physical space and in Fourier space can be supposed to exist for blobs only.That is for the near isotropic structures which are sufficiently described by a single spatial scale anddo not suffer from the disambiguation problem as filaments and sheets do.Information is also given on the filtering effect on statistics concerning the inclination of the strainrate tensor eigenvectors with respect to vorticity. In all filtered ranges, eigenvector 2 reduces itsalignment, while eigenvector 3 reduces its misalignment. All filters increase the gap between themost extensional eigenvalue < λ > and the intermediate one < λ > and the gap between thislast < λ > and the contractile eigenvalue < λ > . When the large scales are missing, eigenvaluemodulus 1 and 3 become nearly equal, similar to the modulus of the related components of theenstrophy production.—————————————————————————————————————————– PACS numbers: ∗ Electronic address: [email protected]
I. INTRODUCTION
The formation of spatial and temporal internal scalescan in part be associated to the stretching and tilting of a r X i v : . [ phy s i c s . f l u - dyn ] J un vortical structures. Many aspects of the behavior of tur-bulent fields have been associated to this phenomenon:the onset of instability, vorticity intensification or damp-ing, or the three-dimensionalization of the flow field [1–3].In the standard picture of turbulence, the energy cascadeto smaller scales is interpreted in terms of the stretchingof vortices due to the interaction with similar eddy size(see for example [4]). A number of statistical details onthe stretching phenomenon and the closely related en-strophy production can be found in the monography byTsinober (2001, see in particular Chapter 6, [5]).Although the important physical role of these inertialphenomena is recognized, the literature does not ofteninclude statistical information on quantities such as themagnitude or the components of ω · ∇ U . For instance,in a letter to Nature (2003) dedicated to the measure-ments of intense rotation and dissipation in turbulentflows, Zeff et al. [6] observe that the understanding of thetemporal interactions between stretching and vorticity iscrucial to the science of extreme events in turbulence.However, the statistics presented there concern dissipa-tion and enstrophy and not directly stretching. The lit-erature more often includes statistical information con-cerning other gradient quantities such as the strain rateor the rate-of-rotation tensors, and, in particular, theirfundamental constituents: the longitudinal or transversevelocity derivatives. Over the last 20 years, statistics onthe skewness and flatness factors of the velocity deriva-tive have been considered in a number of laboratory andnumerical studies that show how these quantities increasemonotonically with the Reynolds number, see e.g. [7] andthe review by Sreenivasan and Antonia (1997)[8].In the case of turbulent wall flows, laboratory measure-ments of both the mean and the r.m.s. of fluctuationsof the stretching components across the two-dimensionalboundary layer have been reported by Andreapoulos andHonkan (2001) [9]. In this study, the normalized r.m.svalues of the stretching components are very significantthroughout the boundary layer and reach values that are one order of magnitude larger than the mean span-wisecomponent (the only significant mean component, how-ever and only in the near wall region). The values ob-served for the r.m.s. of the stretching range from 0.04,close to the wall, to about 0.004 in the outer part.In a study concerning the structure and dynamics ofvorticity and rate of strain in incompressible homoge-neous turbulence, Nomura and Post (1998),[10] demon-strate the significance of both local dynamics (influenceof local vorticity) and spatial structure (influence throughnon-local pressure Hessian) in the interaction of the vor-ticity and strain rate tensor. The behaviour of high-amplitude rotation-dominated events cannot be solelyrepresented by local dynamics due to the formation ofdistinct spatial structure. Instead, high-amplitude straindominated regions are generated predominantly by lo-cal dynamics. The associated structure is less organizedand more discontinous than the one associated with ro-tation dominated events. They conclude that non-localeffects are significant in the dynamics of small scale mo-tion. This should be considered in the interpretation ofsingle-point statistics. Characterizations of small-scaleturbulence should consider not only the typical structuresthere present but also typical structure interactions. Inthis context these authors offer the radial distribution ofthe magnitude of the strain rate tensor normalized onthe enstrophy. In this paper the maximum value of thismagnitude is found close to 0.2.Laboratory statistical information on the stretching offield lines can be found in [11]. Here, probability densityfunctions of the logarithm of the local stretching in Ncycles were obtained for several two-dimensional time-periodic confined flows exhibiting chaotic advection. Thestretching fields were observed to be highly correlated inspace when N is large, and the probability distributionswere observed to be similar for different flows.However, a few examples in literature can also becited regarding direct results for stretching-tilting statis-tics. For instance, recently experimental and numeri-cal confirmation has been found of the predominanceof three dimensional turbulent vortex stretching in thepositive net enstrophy production. These aspects havebeen extensively considered in Tsinober (2000) [12] andin the 2001 monography [5], where a number of statisti-cal geometrical details concerning the vortex alignement,compression, tilting, and folding are outlined. Throughtwo papers, Constantin, Procaccia and Segel (1995) [13],Galanti, Procaccia and Segel (1996) [14] consider thestretching and its relationships with the amplification ofvorticity and the straightening of the vortex lines. Theyshow that the same stretching that amplifies the vorticityalso tends to straighten out the vortex lines. They alsoshow that in well-aligned vortex tubes, the self-stretchingrate of the vorticity is proportional to the ratio of thevorticity and the radius of curvature. In this context,[14]gives statistics on the stretching and vortex line curva-ture. Numerically this is seen as the appearance of highcorrelations between the stretching and the straightnessof the vortex lines. Regarding to this issue, an impor-tant universal feature of fully developed turbulent flowsis the preferential alignment of vorticity along the eigendirection of the intermediate eigenvalue of the strain-ratetensor. A number of works both experimental and nu-merical studies on this result are available (Tsinober, Kitand Dracos JFM (1992), [7], Kholmyansky, Tsinober andS. Yorish PoF (2001), [15], Gulitski et al. JFM (2007a,b,c), [16–18] and Chevillard et al. (2008), [19]). Itshould be noticed, however, that in the case of nonlo-cal strain rate, Hamlington, Schumacher and Dahm [20],have observed a direct assessment of vorticity alignmentwith the most extensional eigenvector by using data fromhighly resolved direct numerical simulations.In the present study, for the case of isotropic turbu-lence ( Re λ = 280, [21]), we consider statistics relatedto the intensity of the stretching term in the equationfor vorticity. If we consider the general instantaneouslocal intrinsic anisotropy of turbulent fields, looking atstretched structures as filaments and sheets, we would like to be able to disentangle them to follow and un-derstand better their evolution and detailed dynamics.Isotropic filtering is unable to carry out this job.We have conceived a probe function, the ratio betweenthe magnitude of the vortex stretching and the enstro-phy, to empirically and statistically measure the localactivity of the stretching phenomenon (see section II).In addition, we propose an alternative to the commonlyused isotropic filter: the cross filter. This is a new em-pirical, and at the moment limited, attempt to intro-duce an anisotropic filtering. In section III, we analyzethe survival function of the normalized stretching by us-ing the cross filter acting directly on the velocity Fourierspace. We do this in the hope of qualitatively highlight-ing aspects related to the role of the three-dimensionalstructures known as blobs, sheets and filaments and theirhypothetical Fourier counterparts. This study can beviewed as a kind of test for this idea and tries to high-light its limits. Concluding remarks are made in sectionIV. II. THE NORMALIZEDSTRETCHING-TILTING FUNCTION
With reference to the phenomena described by the in-ertial nonlinear nonconvective part of the vorticity trans-port equation, let us introduce a local measure of theprocess of three-dimensional inner scales formation f ( x , t ) = | ω · ∇ U || ω | ( x , t ) = | ω · S i,j || ω | ( x , t ) . (1)where U is the velocity field, S i,j is the strain rate ten-sor, and ω is the vorticity vector. The numerator, the socalled stretching-tilting term of the vorticity equation, iszero in two-dimensional flows. In 3 D fields, it is com-monly believed to be responsible for the transfer of thekinetic energy from larger to smaller scales (positive orextensional stretching) and viceversa (negative or com-pressional stretching). According to definition (1), f de-pends on the local instantaneous velocity and vorticityfields. In this study, we leave aside the peculiarity asso-ciated to the convective forcing and focus on the actionof the fluctuation field only. For simplicity, we considerhere the fluctuation of an homogeneous isotropic turbu-lent field ([21]). Since the stretching term plays an im-portant role in the enstrophy production, in the previ-ous definition the normalization by | ω | was adopted. Itshould be recalled that the square of the vorticity mag-nitude is the only invariant of the rate of rotation tensorwhich is non zero and is also the square of the Frobeniusnorm, an invariant norm of the rate of strain tensor. Forthis reason, we considered the enstrophy a good candi-date as reference quantity for the product ω · ∇ U . Infact, as it can be seen below, the survival probability dis-tribution function of f is very small for values larger than O (1).Function f was evaluated over a fully resolved homo-geneous isotropic incompressible steady in the mean tur-bulence in order to look for the typical range of valuesof f ( x ) and to relate them to the behavior of the variousturbulence scales present in an isotropic field.The dataset consists of 1024 resolution grid point Di-rect Numerical Simulation (DNS) of an isotropic Navier-Stokes forced field at Reynolds Re λ = 280 [21]. Eachinstant in the simulation is statistically equivalent, andprovides a statistical set of a little more than 10 ele-ments. We considered the statistics that were obtainedaveraging over the full domain in one instant. The fieldhas been slightly modified in order to filter out instanta-neous effects of the forcing, in other words, a turbulentkinetic energy inhomogeneity of about 20% (in the spa-tial coordinate system). As this bias was generated bythe energy supply at the large scale range, the two largestscales have been filtered out. The resolved part of the en-ergy spectrum extends up to k ∼ k ∼
10 to k ∼
70, see the compensatedversion of the 3D spectrum in figure 1. The higher wave-
FIG. 1: Compensated 3D energy spectrum of one time instantof the turbulent isotropic field here considered. Open accessdatabase http://mp0806.cineca.it/icfd.php. Navier-Stokes di-rect numerical simulation in a box with a discretization of10243, Re λ = 280. See e.g. Biferale L. et al. Physics ofFluids, 17(2), 021701/1-4 (2005). numbers, which are affected by the aliasing error, are notshown.We focus now on a few statistical properties of ω ·∇ U | ω | .The pdf of the components of this vector (which are sta-tistically equivalent, since the field is isotropic) is shownin figure 2. Symmetry with the vertical axis is expectedbecause of isotropy; the skewness is in fact approximately10 − , which is not meaningfully far from zero. However,the distribution cannot be approximated with a Gaussian function. In fact, the actual kurtosis is approximately 55,which is very far from the Gaussian value of 3.The range of values attained by f ( x ) is wide. Val-ues as high as a few hundreds were observed at a sparsespatial points. In order to read the typical values of f ( x ), we study its survival function . By denoting F ( s ) = P ( f ( x ) ≤ s ) the cumulative distribution function (cdf) of f ( x ), the survival function is defined as the complementto 1 of the cdf, S ( s ) = P ( f ( x ) > s ) = 1 − F ( s ) . (2)For each threshold s , S ( s ) describes the probability that FIG. 2: Probability density function of one component ofthe vector ω ·∇ u | ω | in an isotropic velocity field with Re λ =280. Comparison with the Gaussian model. The Skewnessis negligible, about 10 − . The Kurtosis however is very highand reaches a value of about 55.FIG. 3: Survival probability of the normalized stretching-tilting function in a fully resolved isotropic 3D turbulent field( P ( f ( x ) ≥ s ), Re λ = 280. Unfiltered velocity field. Thedashed vertical line indicates the value of f where the proba-bility density function is maximum. f ( x ) takes greater values than s .It has been found that, when f ( x ) is evaluated on a wellresolved isotropic turbulent field, the probability that f ( x ) > f ( x ) = 2 canbe considered the maximum statistical value that f ( x )can reach when the turbulence is fully developed. FIG. 4: Schema of the anisotropic filter here named as CROSSfilter. Blue region: high-pass filter, the wave-numbers undera certain threshold are partially removed, see eq. (4), Redregion: band-stop filter, the wave-numbers inside a range arecut, see eqs. (5-7).
III. PROPERTIES OF THE SURVIVALFUNCTION OF THE NORMALIZEDSTRETCHING-TILTING TERM: ANALYSIS ONTHE ANISOTROPICALLY FILTERED FIELD
By means of suitable convolutions, the application offilters to the velocity field allows the behavior of the func-tion f ( x ) to be studied in relation to the different turbu-lence scale ranges. This analysis is carried out using twospectral filters, a high pass and a band-stop filter. Tofocus in an empirical way on the three principal kind ofgeometrical structures observed in turbulence, filaments,sheets and blobs, we use here a highly anisotropic kindof filter, which is less traditional than the axisymmetric-type filter. Of course, given the inadequacy of the spec-tral representation to account for the complex three-dimensional geometry of the turbulent structures com-monly seen through visualization tools, the approach weuse here is not rigorous and should be considered no morethan propaedeutical.The first filter is a sort of high-pass filter, which werefer to as cross filter and which allows the contributionof the structures that are characterized by at least one large dimension to be removed. From the Fourier pointof view, this means that the structures whose wave-vectorhas at least one small component are filtered out. Onecan here think about elongated structures as filamentsand sheets or very large globular structures. In figure 4,a graphical scheme of the filtering in the wave numberplane k , k can be seen. The first filter we consider ishere represented in blue and is a kind of high-pass filterwhich affects all wave-numbers that, along any possibledirection, have at least one component under a certainthreshold. Given the threshold k MIN , the filter reducesthe contribution of the modes with wave number compo-nents k < k MIN or k < k MIN or k < k MIN . The representation of this high-pass filter, g hp , can begiven by a function of the kind [22] g hp ( k ) = (cid:89) i φ ( k i ; k MIN ) , φ ( k i , k MIN ) =11 + e − ( k i − k MIN ) (3)Since function g hp filters any wavenumber that has atleast one component lower than the threshold k MIN , itreduces the kinetic energy of the filamentous (one compo-nent lower than k MIN ), layered (two components lowerthan k MIN ) and blobby (three components lower than k MIN ) structures. This filter is efficient in reducing theintegral scale of the turbulence [22].By varying the value of the threshold, k MIN , it ispossible to consider different scale ranges. The ranges0 − , − , −
40 are compared in figure 5. The firstfiltering affects the energy-containing range, while theother two also include a part of the inertial range, which
FIG. 5: Survival probability of the normalized stretching-tilting function in a high pass filtered isotropic turbulent field.CROSS filter, see in fig.4 the blue region. is visible in figure 1.The plots in figure 5 have coherent behavior. The sur-vival function S for the 0 −
10 filtering is slightly be-low the values of the distribution of the unfiltered turbu-lence. This trend is confirmed by the other two filterings,and the reduction grows as the threshold k MIN increases.The high-pass filter has the effect of decreasing the sta-tistical values taken by f ( x ) in the domain. The widerthe filtered range, the higher the effect on f .It is possible to say that when we reduce the weight ofthe large-scale structures (layers, filaments or blobs), thelocal stretching-tilting intensity decreases with respect tothe vorticity magnitude. On average, the values of f ( x )go down. The wider the range affected, the lower theprobability value becomes. This suggests that the largescales contribute more to the stretching-tilting (the nu-merator of f ) than to the magnitude of the vorticity fluc-tuation (the denominator of f ). It should be noted thatthis trend is consistent with the results in [20]. This con-sistency also includes results relevant to the behaviour ofthe stretching fluctuation and of the vorticity magnitudein boundary layer turbulence, see figures 6 and 9 in [9].For the wider range 0 −
40, a decrease of 30% in the cu-mulative probability is observed for a stretching-tilting ofabout one half of the local vorticity. The decrease goesup to 80% when statistically the stretching-tilting hasthe same magnitude of the vorticity, that is f is close to1, see figure 5.Let us now consider the behavior of f ( x ) when theinertial and dissipative ranges are affected by the filter-ing, namely a band-stop filtering. In this case, the bandwidth can be extended to obtain a low pass filtering.This filter can be obtained by reducing the contribu-tion of a variable band (see figure 4, part in red) k MIN < k < k MAX or k MIN < k < k MAX or k MIN < k < k MAX . This yields the filter function g bs g bs ( k ) = (cid:89) i φ ( k i ; k MIN , k
MAX ) , (4) φ ( k i ; k ) = 11 + e − ( k i − k ) ,φ ( k i ; k MIN , k
MAX ) = [1 − φ ( k i ; k MIN )] + φ ( k i ; k MAX )The effects of the application of this band-stop filteron the probability P ( f ( x ) ≥ s ) are shown in figure 6.Let us now consider the inertial range in an extendedway , which includes the − range plus all the scaleswhich are not yet highly dissipative. The different bandsare 10 −
40, large scale inertial filtering, 40 −
70 interme-diate scale inertial filtering, 70 −
100 small scale inertialfiltering, 100 −
130 near dissipative, 30 −
150 intermediate-inertial/small scale filtering, 150 −
330 dissipative scalefiltering. Once again all the filtered ranges induce thesame effects: for s < /
2, a slight increase in the survivalprobability. For small values of s , the most effective fil-tering (i.e. the ones which produce the highest increase)is the 10 −
40, while, for higher statistically relevant val-ues, 0 . < s <
2, the most effective result is obtainedfiltering over the whole inertial range, 30 < k <
FIG. 6: Probability of the normalized stretching-tilting func-tion in a band pass filtered isotropic turbulent field of beinghigher than a threshold s . Control function: survival function1 − F ( x ), band-stop filtering in various portion of the inertialrange and in the dissipative range. this case, an increase of about 60% is observed for s = 1and of about 80% for s = 1 . < k < FIG. 7: Three dimensional visualization of the surfaces whereone vorticity component has the value 17 . sec − . The rootmean square value of the vorticity magnitude in the field is | ω | rms = 30 . sec − . As reference, we consider a water fieldwith Re λ = 284, u rms = 0 . m/sec , ν = 0 . ∗ − m /sec (water viscosity at 23 o C ), Taylor micro-scale λ = 3 mm andintegral scale l = 56 mm . Panel (a): unfiltered field; panel(b):the wave number range 0-20 is filtered out by using the high-pass cross filter, panel (c): the wave number range 30-150is filtered out by using the band-stop cross filter; panel (d):the wave number range 30-infinity is filtered out by using thelow-pass cross filter, i.e. by letting k MIN → ∞ , see figure 4.The visualization shows a 256 portion of the numerical fieldsimulated on a 1024 point grid. The field is visualized bymeans of VisIt (https://wci.llnl.gov/codes/visit/). are considering is homogeneous and isotropic, so all thecomponents are statistically alike and it suffices to ob-serve one component only. In panel (a) the unfilteredfield is visualized and a complex picture made of an elon-gated, thick, sleeve-like structures which are enfolded andtwisted can be seen. In panel (b) the structures with atleast one wavenumber component below 20 are smoothedout, see equation (3). One here sees a more sparse dis-tribution of mostly elongated and nearly flat structureswith a much shorter length with respect to panel (a). Themutual folding and twisting seems reduced. This imageis related to the survival distribution in figure 5, wherea depression of stretching-tilting over the vorticity mag-nitude is reported for ratio values above 0.3. Panel (c)shows the band-stop filtered field, where wavenumbers inbetween 30 and 150 are smoothed out. Basically, mostof the inertial and larger dissipative structures are re-moved. Here, it is interesting to observe that the surfaceis much more corrugated that in panel (a), which leaves adefinitive view of the small scale above wavenumber 150.Some of these structures are elongated, others are glob-ular. The image does not discourage the idea that thelarge and the very small scales directly interact. Finally,in panel (d) one sees the structures that have at leastone wavenumber component in the range 0 - 30. Here,large unruffled structures which are mainly globular canbe seen. The images in panel (c) and (d) represent in-stances of the statistical situation described in figure 6,where the survival ratio of the stretching and vorticityintensities is enhanced with respect to the natural sit-uation. Thus, it seems that the partial absence of theinertial range amplifies the stretching-tilting process. Itshould also be noted that the structure spatial densityis distributed in almost the same way in all the panels,though the density levels are different.We have tried to visualize the filtering effect also bymeans of contour plots and pseudo color imaging of thevorticity magnitude in a flat section of the 1024 f ield , seefigure 8. Here, the lines in the images in the left columnare the contours of iso-surfaces of the vorticity magni-tude. Starting from the top, one sees the unfiltered field,the high pass filtered field (the wavenumbers above 20 arekept), the 30 −
150 band-stop, and the low pass filteredfield (the range 30 − inf inity is removed). The contourplot technique is very popular, but, apart from clearlyshowing the reduction of the turbulence scale size whenthe large scales are missing, it does not give much infor-mation. Essentially, structure contours appear to havethe peanut shape which typically hosts vortex dipoles.In the enlarged views in the central and right columns,the pseudocolor plots are richer in information. Wherethe larger scales are removed, panels (e) and (f), the sur-vival probability of intense stretching is reduced. Therange of variation for the vorticity magnitude is verylarge, the root mean square value is about seven timessmaller than the maximum value. When only the largescales are left, panels (m) and (n), the rms value is aboutone third of the maximum value and the stretching is en-hanced. When the inertial scales are removed, the largescale appears to be wrapped by the small scales. The bigdipoles are surrounded by thin wavy-like sheets and thesmall scales are attached to the large ones. In this situa-tion it is not possible to neglect their direct interaction.In regions where a large scale is missing, the small scalesare also missing. Viceversa, a low level of direct inter-action between the largest and the smallest scales wouldhave been confirmed, if small scales had been sparsely dis-tributed in regions where the large scales are not presentand would not have surrounded the large structures inregions where the latter are present.To complement the understanding of the visualizationin relation to the anisotropic filtering here used, we haveconsidered the alignment between the eigenvectors of thestrain rate tensor and the direction of the vorticity. Func-tion f in fact, see equation (1), can also be written as f ( x ) = | S || ω | [ s i ( e i ∗ e ω ) ] / where | S | is the magnitudeof the strain rate tensor, s i are the eigenvalues of thestrain rate tensor S ij normalized by | S | , s i = λ i / | S | , and ( e i ∗ e ω ) describe the alignments between the eigenvec-tors of S ij , denoted e i , and the direction of the vortic-ity e ω . Figure 9 shows the probability density functionof these alignments in the reference unfiltered field andin two filtered cases described in this work: the highpass filter where the smallest 20 wavenumbers are re-moved, see equation (3) and the band-stop filter wherewavenumbers 30 −
150 are removed, see equation (4).We first observe that the standard trend of alignments isnot fully spoiled by the filtering. In both filtering cases,eigenvector 2 reduces its alignment, while eigenvector 3reduces its misalignment. Conversely, eigenvector 1 in-stead shows a different behavior. In the band-stop fil-tering case (large scale dominate) eigenvector 1 slightlyincreases the alignment. In the high-pass filtering, eigen-vector 1 reduces the alignment that becomes statisticallyequal to that of the eigenvector 3. This is confirmed,see Table 1, by considering the ratio among the field av-eraged strain rate tensor eigenvalues and related com-ponent of the enstrophy production, < σ i > / < σ tot > ,where < · > is the average over all the computational do-main (1024 grid point box), σ i = ω λ i cos ( e ω , e i ) and σ tot = < (cid:80) i ω λ i cos ( e ω , e i ) > . All filters increase thegap between the eigenvalue < λ > and < λ > and thegap between < λ > and < λ > . However, when thelarge scales are missing, eigenvalues 1 and 3 are very closein modulus. The same happens to the modulus of theirrelated enstrophy production components. When the in-ertial scales and part, or the entire, dissipative range areremoved the mutual relation among the eigenvalues andthe modulus of the production components changes lesswith respect to the natural turbulence.In all the three cases the filtering reduces the averagevalues of the largest and intermediate eigenvalue, < λ > and < λ > .At this point, let us consider the dual nature of the fil-aments and sheets, as regards their inclusion in the cat-egories of the small and large scales. A filament whichis filtered out by the filter g hp because it has a small0 FIG. 8: Visualization of the vorticity magnitude in a sec-tion parallel to one face of the computational box. Refer-ence isotropic turbulence: Re λ = 280, u rms = 0 . m/sec (rms of the velocity), ν = 0 . ∗ − m /sec (water viscosityat 23 o C ), Taylor micro-scale λ = 3 mm and integral scale l = 56 mm , | ω | rms = 30 . sec − (root-mean-square of thevorticity magnitude), 1024 grid domain points. First row(a,b,c): unfiltered field. Second row (e,f,g): the wave numberrange 0-20 is filtered out by using the high-pass cross filter.Third row (g,h,i): the wave number range 30-150 is filteredout by using the band-stop cross filter. Fourth row (l,m,n)the wave number range 30-infinity is filtered out by using thelow-pass cross filter. It is possible to see: in the first column(a,d,g,l) the vorticity magnitude countourplots of the entire1024 grid domain. In the second and third column (b,e,h,mand c,f,i,n) Pseudocolor plots of a 256 portion of the grid(black box in the previous column) are shown. In the thirdcolumn the range of magnitude values in between 10 and 30is visualized to show the details of the part of the field wherethe vorticity is below its rms value. The part above the rmsis in fact visualized in the central column, where all the rangeof values is included. The Pseudocolor method maps the datavalues of a scalar variable to color. The plot then draws thecolors onto the computational mesh. The field is visualizedby means of VisIt (https://wci.llnl.gov/codes/visit/) .wave number (the axial wave number component), willhave two large wave number components (the ones nor-mal to the filament axis). Due to these wave components,it will also be filtered out by the filter g bs . A similar FIG. 9: PDFs of the cosine of the angle between vorticity( ω ), and the eigenvector, e i , of the rate of strain tensor ( S ij ).The red lines refer to unfiltered field; the green lines refer tothe wave number range 0-20 filtered out by using the high-pass cross filter; the blue lines refer to the wave number range30-150 filtered out by using the band-stop cross filter. situation also holds for the sheets. Thus filaments andsheets are always partially removed when a filtering, ei-ther anisotropic or isotropic, is applied. The situationchanges with compact structures (the blob), which nonambiguously belong either to the large scale range or tothe intermediate-small scale range. The different behav-ior shown in figures 5 and 6 is therefore mainly due tothe blob contributions, and, since the variation in the cu-mulative distributions is opposite and almost of the samemagnitude, it is possible to deduce that the partial re-1moval of the background filaments and sheets, which isalways done regardless of the filter typology (high pass,band pass, low pass,..), is not statistically relevant. Inother words, it appears reasonable to conclude that untila better way to select and remove anisotropic structuressuch as filaments and sheets is found, first order statisti-cal modification associated to their presence/absence willnot be clearly seen.Lastly, it is interesting to observe that box filteringsmall scales modifies the stretching statistics to a greatextent. A field filtered in such a way shows a finiteprobability of having more than twice the amount ofstretching/tilting compared to the enstrophy [23, 24]. Inthe context of the Large Eddy Simulation methodology,where this kind of filtering is commonly used, it is pos-sible to deduce that, when a fluctuating field shows sucha feature, the field is unresolved. As a consequence, itis possible to build a criterion that locates the regionsof the field where the inclusion of a subgrid term in thegoverning equations is advisable (see also [25]). IV. CONCLUSIONS
To summerize, we have collected a set of statisti-cal information about the stretching and tilting inten-sity of vortical structures normalized by the enstrophy, f ( x ) = | ω ·∇ U || ω | ( x ), in isotropic turbulence. A first re-sult is that there is a very small probability of having alarger stretching/tilting of intensity than the double ofthe square of the vorticity magnitude. Then, if compactstructures (blobs) in the inertial range are filtered out, itcan be seen that the probability of having higher f thana given threshold s increases by 20% at s = 0 .
5, and by60-70% at s = 1 .
0. If, on the other hand, larger blobsare filtered, an opposite situation occurs. The unfilteredfield is thus a separatrix for the cumulative probabilityfunction. This behavior - high fluctuation vorticity mag-nitude → low stretching, and viceversa - agrees with gen-eral aspects highlighted by a number of laboratory and numerical analyses, [11, 20] also in near wall turbulentflow configurations[7]. The present observations need tobe associated to the non discriminating effect of filteringon filaments and sheets, which is due to their specific na-ture that cannot be reconciled inside either a categoryof small or large scales. It has also been shown that ahigh intermittency is associated to f , whose kurtosis isas high as 55.The probability density function of the alignments be-tween the strain rate eigenvectors and the vorticity isin part modified by the anisotropic filtering here investi-gated. In particular, we observe that, though the stan-dard trend of alignments is not fully spoiled, eigenvec-tor 2 reduces its alignment, while eigenvector 3 reducesits misalignment. Conversely eigenvector 1 shows a dif-ferent behaviour. In the band stop filtering case (largescale dominate) eigenvector 1 slightly increases the align-ment. In the high pass filtering case (inertial scales dom-inate), eigenvector 1 reduces the alignment that becomesstatistically equal to that of the eigenvector 3. This isconfirmed by considering the mutual ratio among the av-eraged strain rate eigenvalues and related components ofthe enstrophy production. Both filters increase the gapbetween the most extensional eigenvalue < λ > and theintermediate one < λ > and the gap between this last < λ > and the contractile eigenvalue < λ > . How-ever, when the large scales are missing, the modulus ofeigenvalues 1 and 3 become nearly equal, similar to themodulus of the related components of the enstrophy pro-duction. V. ACKNOWLEDGMENTS
We are grateful to Professor James J. Riley for fruitfuldiscussions. We thank the referees for their constructivereviews and the many suggestions. This work was carriedout in cooperation with the International Collaborationfor Turbulence Research.2 [1] A.S. Monin and A.M. Yaglom
Statistical Fluid Me-chanics.
Vol.s 1-2, The MIT Press, Cambridge, Mas-sachusetts, and London, England., (1971, 1975).[2] H. Tennekes and J.L. Lumley
A first course in turbulence ,The MIT Press, Cambridge, Massachusetts, and London,England., (1972).[3] S.B. Pope,
Turbulent flows , Cambridge University Press,(2000).[4] U. Frisch,
Turbulence - The legacy of A.N. Kolmogorov ,Cambridge University Press, (1995).[5] A. Tsinober,
An Informal Introduction to Turbulence, 1thedition , Springer, (2001).[6] B.W. Zeff, D. Lanterman, R. McAllister, R. Roy, J.Kostelich and D.P. Lathrop , Nature , 146-149 (2003).[7] A. Tsinober, E. Kit and T. Dracos, ”Experimental in-vestigation of the field of velocity gradients in turbulentflows“, J. Fluid Mech. , 169-192 (1992).[8] K. R. Sreenivasan and R. A. Antonia, Annu. Rev. FluidMech. , 435 (1997).[9] Y. Andreapoulos and A. Honkan, J. Fluid Mech., ,131 (2001).[10] K.K. Nomura and G.K. Post, J. Fluid Mech. , 65-97(1998).[11] P.E. Arratia and J.P. Gollub, J. Stat. Phys., , Nos.516, 805 (2005). J. Fluid Mech., , 131 (2001).[12] A. Tsinober Turbulence structure and Vortex Dynamics ,J.C.R. Hunt nad J.C. Vassilicos editors, Cambridge Uni-versity Press, 164 (2000).[13] P. Constantin, I. Procaccia and D. Segel, Phys. Rev. E. , 3207 (1994). [14] B. Galanti, I. Procaccia and D. Segel, Phys. Rev. E. ,5122 (1996).[15] M. Kholmyansky, A. Tsinober and S. Yorish, Physics offluids, , 311 (2000).[16] G. Gulitski, M. Kholmyansky, W. Kinzelbach, B. Lthi,A. Tsinober and S. Yorish, J. Fluid Mech. , 57-81(2007).[17] G. Gulitski , M. Kholmyansky, W. Kinzelbach, B. Lthi,A. Tsinober and S. Yorish, J. Fluid Mech. , 83-102(2007).[18] G. Gulitski , M. Kholmyansky, W. Kinzelbach, B. Lthi,A. Tsinober and S. Yorish, J. Fluid Mech. , 103-123(2007).[19] L. Chevillard, C. Meneveau, L. Biferale, F. Toschi, Phys.Fluids , 101504 (2008).[20] P.E. Hamlington, J. Schumacher and W.J.A. DahmPhys. of Fluids, , 111703 (2008).[21] L. Biferale, G. Boffetta, A. Celani, A. Lanotte, F. Toschi,Phys. of Fluids. (2), 021701/1-4 (2005).[22] D. Tordella and M. Iovieno, J. Fluid Mech. , 441(2006).[23] D. Tordella, M. Iovieno, S. Massaglia, Comput. Phys.Commun., , 539 (2007).[24] D. Tordella, M. Iovieno, S. Massaglia, Comput. PhysicsCommun.178